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1. 6 1 85 T 103 retr trybip ind look all MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2X1 X2 2 X3 X4 2 X5 X6 2 X7 3 XB lt 13 1 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 4 X1 X3 X55 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END GIN 8 ters go LP OPTIMUM FOUND AT STEP 5 OBJECTIVE VALUE 19 3000000 NEW INTEGER SOLUTION OF 1 00000000 AT BRANCH 14 PIVOT 43 BOUND ON OPTIMUM 16 82500 NEW INTEGER SOLUTION OF 3 00000000 AT BRANCH 20 PIVOT 59 BOUND ON OPTIMUM 16 82500 NEW INTEGER SOLUTION OF 9 00000000 AT BRANCH 25 PIVOT 69 BOUND ON OPTIMUM 16 30000 NEW INTEGER SOLUTION OF 15 0000000 AT BRANCH 29 PIVOT 81 BOUND ON OPTIMUM 16 30000 ENUMERATION COMPLETE BRANCHES 30 PIVOTS 88 LAST INTEGER SOLUTION IS THE BEST FOUND RE INSTALLING BEST SOLUTION retr trybip ind bip 10 go LP OPTIMUM FOUND AT STEP 5 OBJECTIVE VALUE 19 3000000 NEW INTEGER SOLUTION OF 14 0000000 AT BRANCH 35 PIVOT BOUND ON OPTIMUM 18 00000 NEW INTEGER SOLUTION OF 15 0000000 AT BRANCH 41 PIVOT BOUND ON OPTIMUM 17 07500 ENUMERATION COMPLETE BRANCHES 43 PIVOTS 110 LAST INTEGER SOLUTIO
2. LP OPTIMUM FOUND AT STEP 5 OBJECTIVE VALUE 15 0000000 solu Print a solution report OBJECTIVE FUNCTION VALUE 1 15 0000000 VARIABLE VALUE REDUCED COST X1 16 000000 000000 X2 13 000000 000000 X3 3 000000 000000 X4 000000 3 000000 X5 4 000000 000000 X6 2 000000 000000 X7 000000 2 000000 X8 11 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 4 000000 3 000000 1 000000 4 000000 1 000000 5 000000 1 000000 6 000000 1 000000 7 000000 1 000000 8 000000 000000 NO ITERATIONS 5 COMMAND LINE COMMANDS DMPS creates a detailed MPS format solution report The following illustrates 22 X3 794 X3 Here is our model 77 X1 6 X2 3 X3 6 X4 33 X5 13 X6 110 X7 785 X9 lt 435 X9 lt 1 42 X4 21 X5 760 X6 1000 53 X4 234 X5 32 X6 200 Here is the standard LINDO solution report DMPS look all MAX 21 X8 47 X9 SUBJECT TO SPACE 774 X1 76 X2 818 X7 62 X8 WEIGHT 67 Xl 27 X2 792 X7 97 X8 END INTE 9 solu OBJECTIVE FUNCTION VALUE 1 170 0000 VARIABLE VALUE X1 0 0000 X2 0 0000 X3 0 0000 X4 1 0000 X5 1 0000 X6 0 0000 X7 1 0000 X8 1 0000 x9 0 0000 ROW SLACK OR SUR SPACE 57 0000 WEIGHT 24 0000 NO ITERATIONS 31 BRANCHES 5 DETERM dmps ty Or Or O Om tS QOU SOG O O O O O10 O E G
3. PAGE The PAGE command sets the length of the page or screen size in lines For instance PAGE 25 will cause the display to pause after 25 lines and await a carriage return before displaying the next 25 lines When 0 is entered as the argument to PAGE paging is turned off entirely LINDO will no longer stop output to wait for a carriage return PAGE 0 is very useful at the start of any LINDO batch command file 9 User Supplied Subroutines USER The USER Command calls the subroutine with the name USER A dummy version is supplied with LINDO that does nothing except print a brief message However the user may replace the dummy USER subroutine with one of his own making This subroutine may call any number of internal LINDO routines for building models solving them and requesting solution output For more information on this programming interface to the LINDO libraries please refer to the Chapter 8 LINDO Callable Libraries COMMAND LINE COMMANDS 183 10 Miscellaneous INVERT INVERT re solves the set of simultaneous linear equations implied by the current set of basic variables The inverse is represented by a product of elementary matrices 1 e identity matrix except for one column Inversion tries to permute the rows and columns of the basis so the matrix is as close to triangular as possible The rows that cannot be triangularized constitute the bump The columns in the b
4. LINDO COMMANDS BY CATEGORY FOR INFORMATION ON A SPECIFIC COMMAND TYPE HELP FOLLOWED BY THE COMMAND NAME 1 INFORMATION HELP COM LOCAL CAT 2 INPUT AX MIN RETR RMPS TAKE LEAV RDBC FINS FBR 3 DISPLAY PIC TABL LOOK NONZ SHOC SOLU RANGE BPIC CPRI RPRI DMPS 4 FILE OUTPUT SAVE DIVE RVRT SMPS SDBC FBS FPUN 5 SOLUTION GO PIV GLEX 6 PROBLE EDITING ALT EXT DEL SUB APPC SLB FREE 7 QUIT QUIT 8 INTEGER QUADRATIC AND PARAMETRIC PROGRAMS INT QCP PARA POSD TITAN BIP GIN IPTOL 9 CONVERSATIONAL PARAMETERS WIDTH TERS VERB BAT PAGE PAUS 10 USER SUPPLIED ROUTINES USER 11 MISCELLANEOUS INV STAT BUG DI GI w n a H TITL 120 CHAPTER 3 LOCAL The LOCAL command displays information regarding your specific version of LINDO On multi user systems and mainframes this command may give information supplied by the System Operator TIME The TIME command displays the cumulative processor time used since opening LINDO This command is useful for measuring your use of computer resources on a time shared computer system or in determining the time required to solve a model The TIME command takes no parameters and can be entered only at the colon prompt The TIME command does not affect the current model or the current solution in memory An example of the use of the Time command follows time CUMULATIVE HR MIN SEC 0 0 19 07 The user input is in bold here This co
5. Now let s look at the solution report LINDO delivered LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1 145 0000 VARIABLE VALUE REDUCED COST STD 10 000000 0 000000 DLX 3 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 2 500000 3 9 000000 0 000000 4 0 000000 7 500000 NO ITERATIONS 2 Taken in order this tells you first that LINDO took 2 iterations to solve the model second that the maximum profit attainable from the two variables as you ve constrained them is 145 and third the variables STD and DLX take the values 10 and 3 respectively What s interesting to note is that we make less of the relatively more profitable Deluxe model due to it s intensive use of our limited supply of labor We ll explain REDUCED COSTS SLACK OR SURPLUS and DUAL PRICES a little later in this chapter Printing the Model and Solution Command line versions of LINDO do not contain a facility for directly printing models and their solutions This may be accomplished quite easily however using the DIVERT command to send screen output to a file which can then be printed either from the operating system or a word processing package To do this on any platform enter the following command at the LINDO colon prompt DIVERT lt FileName gt where lt FileName gt is the name of the file to which you want output sent If you have a solved model in memory entering the command LOOK ALL f
6. change the right hand side add a variable to one row change the name of a row or delete a variable from one row 160 CHAPTER 3 The syntax of the command is as follows ALTER lt Row gt lt VarName DIR NAME RHS gt where lt Row gt is the name or index of the row where you wish to make the alteration and lt VarName gt is the name of the variable to be altered ALTER requires the row and the variable you wish to change to be identified If you enter ALTER without identifying the row or variable LINDO will prompt you for the row or variable You cannot use a variable index number to identify a variable Use the variable name with ALTER You can use RHS in place of the variable name to change the right hand side of any row except row number one the objective function If you wish to change the right hand sides of all the rows use the APPCOL command instead Use DIR in place of the variable name to change the relational operator lt or of any row except row number one the objective function For row 1 only use DIR in place of the variable name to change the sense of the objective function MAX or MIN You can use NAME in place of the variable name to change the name of any row but a row name for row number one the objective function is never displayed The current solution is no longer considered optimal after ALTER is used unless you only change the name of a row ALTER can be us
7. DRPVAR J PRIMAL DUAL Double PRIMAL DUAL Integer J Call this routine after optimization to get the value and reduced cost for variable J On entry J contains the index of a variable On output the double arguments PRIMAL and DUAL will contain the value and reduced cost for variable J If you don t know the index of a variable you can have LINDO look it up with the NDXOFYV routine Note that DRPVAR is a double precision equivalent of the REPVAR routine FPRIME N1 N2 Integer N1 N2 Returns in M2 the smallest prime number greater than or equal to NZ 234 CHAPTER 8 FREEIT J Integer J Makes variable J a free variable In other words variable J may take on any real value positive or negative If you don t know the index of a variable you can have LINDO look it up with the NDXOFV routine This routine corresponds to the FREE command line command GETCOL J KNAME 8 NONZ VAL NONZ IRO NONZ SUBJ TROUBLE Character KNAME Float VAL SUBJ Integer J NONZ IRO TROUBLE Retrieves the description of variable J If you don t know the index of a particular variable you can have LINDO look it up with the NDXOFV routine KNAME should be declared as a character array of at least 8 bytes and will return the variable s name NONZ returns the number of nonzero coefficients in the variable s column VAL is a single precision array which returns the values of the nonzero coefficients JRO is an array
8. You have input an invalid variable name Processing of the command is terminated Check the spelling of the name and try again 274 25 26 27 28 29 30 APPENDIX A lt Not In Use gt FILE TYPE NOT RECOGNIZED You tried to read a model file which is not in a format recognized by LINDO This error typically occurs when trying to read a LINDO text file into the command line version of LINDO using the RETR Command which only recognizes LINDO packed pk files In order to open a LINDO text file in the command line versions the TAKE command must be used MODEL SIZES OF lt m gt ROWS lt gt INTEGERS lt n gt VARIABLES lt z gt NONZEROS ARE TOO BIG FOR VERSION LIMITS OF lt M gt lt I gt lt N gt lt Z gt The model you are attempting to read is too big for the limits on your version of LINDO Either cut back on the model s size or use a larger version of LINDO that has the capacity for this model In Windows versions of LINDO the Help About LINDO Command shows the limits on the version of LINDO you are using This also shows how to contact LINDO Systems Inc to order an upgrade to a larger version Command line versions of LINDO can use the HELP Command alone to get the same information Please refer to your manual for information on special versions of LINDO NO OF VARIABLES MUST EQUAL NO OF CONSTRAINTS In a quadratic programming model there is one dual variable for each real constraint and one
9. You have specified an improper variable index while trying to set a simple upper bound on a variable The index must lie in the range of 1 to N where N is the total number of variables INVALID SUB lt Value gt FOR VAR lt VariableIndex gt You have specified an invalid value for a simple upper bound on a variable Be sure that the bound is numeric and that it is non negative 51 52 53 54 55 56 57 58 59 ERROR MESSAGES 277 lt Not In Use gt UNBOUNDED SOLUTION AT STEP lt N gt REDUCED COST lt Value gt USE THE DEBUG COMMAND FOR MORE INFORMATION The objective function can be increased without bound You have most likely misspelled a variable name or have omitted some constraints Run the DEBUG command to narrow down the problem ITERATION LIMIT EXCEEDED LINDO has hit the limit on iterations You can specify your own limit as part of the GO command by typing GO N where N is an integer value that will be used as an iteration limit Typically this error will prompt the user as to how many additional iterations should be allowed NO FEASIBLE SOLUTION AT STEP lt N There is no solution that simultaneously satisfies all the constraints Run the DEBUG command in command line versions and the Solve Debug command on Windows versions to try to narrow down the problem NO INTEGER SOLUTION WAS FOUND Your model is LP feasible but IP infeasible However no solution was found once the integer
10. AppWizard will create a new skeleton project with the following specifications Application type of LNDMSYC Dialog Based Application targeting Win32 Classes to be created Application CLNDMSYCApp in LNDMSYC h and LNDMS VC cpp Dialog CLNDMSYCDlg in LNDMSYCDIg h and LNDMSYCDlg cpp Features About box on system menu 3D Controls Uses shared DLL implementation MFC42 DLL Activex Controls support enabled Localizable text in English United States Project Directory C Program Files Microsoft Visual Studio MyProjects LNDMS C Cancel 7 Finally press the OK button and AppWizard will generate the base code set LINDO CALLABLE LIBRARIES 253 Use the IDE s resource editor to modify the application s main dialog box so it resembles the following fz LINDO MSVC Example Objective Value X Y Next you will need to use the Class Wizard in VCS to assign member variables to the three output fields in the dialog box Specifically the following member variables should be assigned to the three output fields all members are of type float Output Field Member Variable float Objective Value m_fObjective X m fX You need to add definitions for the LINDO library routines to the dialog class The LINDO definitions for VC5 can be found in the file LINDO DLL32 MSVC LINDMSVC H Open up the header file for the dialog class and add the include statement shown below in bold
11. As an example the following command will print the names and slacks on all non binding rows RPRI N P If the row attribute list is omitted only the number of rows satisfying the conditional expression is printed For instance RPR Ss P will return an integer value corresponding to the number of binding rows RPRI report output may be sent to an external file by using RPRI in conjunction with the DIVERT command The resulting output may then be loaded into other programs such as spreadsheets and databases If an attribute s numeric value is excessively large it will print as a series of asterisks in the RPRI report Next we will list a number of RPRI examples for the following model and solution P gt 0 0 MAX 9 X1 X2 4 X3 2 X4 8 X5 2 SUBJECT TO 2 X1 c Oe 4 X1 5 xl 6 7 8 X2 X4 X5 END OBJ 1 VARIABLE X1 X2 X3 X4 x5 X6 X7 X8 ROW 2 3 oNN opener rpri 3 X2 2 X3 3 X4 X5 2 X3 X5 9 X2 X4 3 X3 X6 X7 12 X6 X8 9 X7 X8 15 ECTIVE FUNCTION VALUE 15 0000000 VALUE REDUCE 16 000000 13 000000 3 000000 000000 3 4 000000 2 000000 000000 2 11 000000 SLACK OR SURPLUS DUAL 000000 4 000000 1 000000 1 000000 1 000000 1 000000 S 000000 NUMBER OF MATCH rpri n ES 8 List the row names X2 2 X3 X4 2 X5 X6
12. Each of these categories includes a number of commands which we will discuss in depth in this chapter Before proceeding however you may want to glance at the section below which gives a brief comprehensive list of all the commands available under LINDO for Windows LINDO for Windows Commands in Brief In this section we list each of the menu commands available to the LINDO user under Windows The commands are broken down into the six main menu categories File Edit Solve Reports Window and Help A brief description of each command is included In depth discussions of each of these commands may be found in the sections immediately following 26 CHAPTER 2 1 File Menu Command _ Description Opens a View Only model from disk Save As Saves a Model Window to disk and prompts for a new name Closes the active window Prints the contents of the active window Printer Setup Used to configure your printer Log Output Opens or closes a log file used for recording the results of your session Take Commands Runs a LINDO script file Mt Saves the active model s current solution to disk Date Displays the date the session of LINDO Exits LINDO 2 Edit Menu Command Description Undoes last edit operation Cut Removes selected text from window and places it in clipboard Copies selected text to clipboard Pastes clipboard s contents into active window Removes selected text from window Find Repl
13. IV Primal T Dual T Rim mF GraphT ype mG Graph Style I Upper Bound C Area Bar 2D I Lower Bound C Line Pie CD E Type C Point J Nonzeros Graphics Format H Graph Title optional C Condition optional Nonzero Column Values pod Now our graph includes only the days where the number of starting employees is greater than 0 Nonzero Column Values H Primal WINDOWS COMMANDS 89 Picture Alt 5 The Picture command displays a model in matrix form You may request either a text or graphical representation of the matrix of nonzero coefficients This command is a useful way to obtain a visual impression of your model Viewing the model in matrix form can be helpful in a couple of instances First and perhaps most importantly is the use of nonzero pictures in debugging formulations Most linear programming models have strong repetitive structure Incorrectly entered sections of the model will stand out in a nonzero picture Secondly a nonzero picture can be helpful when you are attempting to identify special structure in your model As an example if your model displays strong block angular structure then algorithms that decompose the model into smaller fragments might prove fruitful To view a nonzero picture of a model select the model s window by clicking on it with the mouse Next issue the Picture command You should now see the Non
14. This routine corresponds to the RDBC command line command 242 CHAPTER 8 RDMPS LUNIT NDIR Integer LUNIT NDIR Reads an MPS format model from the file associated with unit number LUNIT To associate a file with a unit number see routines LUNOPN and LUNGET On entry set NDIR as follows 1 for minimization 1 for maximization and 0 if the user is to be prompted for the direction of the objective LUNIT will return 1 if the attempt to read failed This routine corresponds to the RMPS command line command REPROW IROW PRIMAL DUAL Float PRIMAL DUAL Integer IROW Determines after optimization i e calling routine GO the slack value and dual price on row JROW On return the variables PRIMAL and DUAL will contain the slack value and dual price for row JROW REPVAR JCOL PRIMAL DUAL Float PRIMAL DUAL Integer JCOL Determines after optimization i e calling routine GO the value and reduced cost for variable JCOL On return the single precision variables PRIMAL and DUAL will contain the value and reduced cost for the variable To retrieve this same information in double precision see routine DRPVAR RETR LUNIT Integer LUNIT Retrieves a model from a file attached to logical unit LUNIT The file should have been created using the SAVE routine To attach a file to a logical unit see routines LUNOPN and LUNGET This routine corresponds to the RETRIEVE command line command RNGBGN
15. cccccccccesseseceeeeeeeeeeeennteeeeeeeeeeeeeees 25 The Commands in Depth crer vce feo o ces tel poh eon Oc tees ie a ieee eee ena 28 File MON UW a tavsasiveesdiaenvia e a a e E a aa E ie aa aio 29 Edit MORU r e a aa aar aE a Te e e a Ar naa dantet poderat 41 Solve MOM se atea a e E E A AA NOTE ARE 58 Reports Men stot eeter ss decd ata aia EE ant aaae eE se EART A ARE i i 71 Window Me ucrencesec e a E a a a A Ea e a aa 102 Help Men pmnan ra bosusloctaepaseveceacuencr ae iaka SeA AE Aa EEEN iaa 110 Ch 3 LINDO for Command line Environments cccccceeeeeeeeeeeeeeeeeee 115 Commands in Brief iseina ta n E A E RA 115 The Commands in DOP Mixgcscdeesscccesdeecactessaeacestassesigeea Sviedee de ter dr e Ea 118 Information ComMmmandS cise cassoses snes eetet can sedee ves tecguecpens eaeeetn env dive eneectnnest een deveanstel 118 Inp utGommandS sasct ita tsclat seared a da an teens a eve T atot iia 120 Display Commands iens ie hain 127 File Output Commands sissa ae ae dia raaa ko estos 144 Solution Commands 2 0 cece nice ere eee aaa Ea pE e TEE vant 151 Problem Editing Commands ic 2c 45 cele Sali ene eee ee 159 Integer Quadratic amp Parametric Programs CommandSs ceeeeeteeeeeees 167 Conversational Parameters COMMANAS ccceeeeeeeeeeeeeeeeeeeteeeeeeeeeeeneeeeeees 180 User Supplied Subroutines 20 0 2 eee cece eeeeceeeeeeeeeeeeeeeeeaaeeeeeeeaaeeeeeensaaeeeeneaas 182 Miscellaneous Command sees scacseete ti
16. terse go LP OPTIMUM FOUND AT STEP 18 OBJECTIVE VALUE 8942181 00 EW INTEGER SOLUTION OF 8929170 AT BRANCH 7 PIVOT 74 BOUND ON OPTIMUM 8936519 ENUMERATION COMPLETE BRANCHES 7 PIVOTS 74 LAST INTEGER SOLUTION IS THE BEST FOUND REINSTALLING BEST SOLUTION 8 Conversational Parameters TERSE VERBOSE TERSE turns off automatic display of the solution after solving the model with the GO command VERBOSE reverses a TERSE and turns on automatic display of the solution after a GO The default setting for LINDO is VERBOSE For a linear program the output from GO when TERSE has been given is just two lines showing the number of steps and the objective function value For an integer program the output from GO when a TERSE command has been given is first the number of steps and the objective value for the linear relaxation of the integer program Then the value of each integer solution is given as it is found with the branch and pivot numbers and the bound on the optimum Finally LINDO reports when it has found the best solution and is preparing it for output In the following example we use the TERSE command before the GO command Notice how the standard solution report is suppressed In its place we receive a brief message indicating the optimal solution was found the step at which it was found and the optimal objective value retr model lpk terse go LP OPTIMUM FOUND AT
17. 165 000000 see how many pivots it takes using FINS warm start retr tran2 fins tranl fpu ters go LP OPTIMUM FOUND AT STEP 1 OBJECTIVE VALUE 165 000000 quit FINS As illustrated in the previous example FINS performs the same operation as FBR except it retrieves the current basis in an industry standard format known as MPS These are bases saved using the FPUN Command in the current model This allows you to export the basis information to other programs or systems which may require it See below for details on how this command is used with FPUN to create a new model from an old basis in MPS format RMPS Syntax RMPS lt FileName gt The RMPS command reads an MPS format file from disk While MPS files tend not to be very compact they are an industry standard and as such readily allow for porting models from one environment to another MPS files are text files and may be read into any text editor 124 CHAPTER 3 After issuing the RMPS command the MPS file is loaded into memory and becomes the current model See also the SMPS command The form of the command is RMPS lt FileName gt where lt FileName gt is the name of the MPS format file you wish to retrieve If you attempt to retrieve a file which is not the correct format or does not exist LINDO will give an error message If you enter nothing after the RMPS command LINDO will prompt you for the name of the file After fin
18. 194 CHAPTER 4 Setting an Optimality Tolerance On difficult IP problems it may be useful to curtail the search with the Integer Programming Optimality Tolerance IPTOL Use of IPTOL can frequently reduce the solution time dramatically You can set the IPTOL value using either the Edit Options command in Windows versions of LINDO or the IPTOL command in command line versions The IPTOL value can be set to any fractional value say f from 0 to 1 Once a feasible IP solution is found a branch in the tree will be pursued only if it can improve on the current best solution by at least the fraction f The end result is that the objective value of the final solution returned by LINDO will be within 100 f percent of the true optimal objective value Suppose for example we have an IPTOL of 02 we are maximizing and the branch and bound procedure has just found a solution with value 100 A branch of the search tree will not be pursued if it cannot lead to a solution better than 102 The end result is the solution returned by LINDO will be no farther than 2 percent away from the true optimal solution Exploiting a Known Good Solution to an IP If you have a known good solution but not necessarily optimal to an IP then you may be able to exploit this knowledge In the early stages of the branch and bound search branches that are clearly not optimal in light of this prior information need not be examined The IP Objective Hurdle value allows yo
19. 3 Use the LINDO library definitions file LINDO DLL32 LNDDLL32 DEF to build an import library with the following DOS command LIB DEF LNDDLL32 DEF OUT LNDDLL32 LIB Note that the LIB application is included with Powerstation FORTRAN 4 Use the nsert File Into Project command to add the sample FORTRAN application file LINDO DLL32 MSFORT LINDOPS FOR and the import library from the previous step in the project 5 Use the Build Build command to create the executable The contents of the FORTRAN source file are reproduced below program lindops Gok ee Gah Ee ee ee A ee a Pn eo c c A Microsoft Powerstation FORTRAN program to define and c solve the following model using the 32 bit LINDO Dynamic c Link Library c G Max 20 X 30 Y e Sele c Xx lt 50 c Y lt 60 c X 2Y lt 120 Cc End c implicit none c Inspsint h contains the definitions for all the LINDO c callable routines include lndpsint h c define variables character 8 kname integer 4 i idrow istat irowx 3 irowy 3 logical trouble real obj x y dual real rhs 3 valx 3 valy 3 260 CHAPTER 8 c data initialization data rhs 50 60 120 data valx 20 1 1 data valy 30 1 2 data irowx 1 2 4 data irowy 1 3 4 c Initialize LINDO by calling ILINDO and INIT call ilindo call init c Put LINDO in TERSE mode call quiet 1 c Call DEFROW to define objective row call defrow
20. FRI XMON XWED XTHU XFRI SAT XWED XTHU XFRI XSAT END 180 160 150 160 190 140 120 gt gt gt gt gt gt gt WINDOWS COMMANDS 77 Next we issue the Parametrics command and are presented with this dialog box Row Parametrics Parametric Row New RHS Value OK Cancel Help gt Report Types T Text I Graphics Graph Type C 2D 3D First we must select a row and enter it into the box labeled Parametric Row Suppose we want to see how increasing our staffing requirement on Saturdays influences our staffing costs In this case the row labeled SAT will be our parametric row We can type SAT into the Parametrics Row box or simply double click on SAT in the list box below Our dialog box will then look like Row Parametrics Parametric Row New RHS Yalue 120 OK ok Cancel eT Help r Report Types T Text I Graphics Graph Type5 20 3 78 CHAPTER 2 Once we select our row LINDO displays the row s type lt or gt and current right hand side value in the lower left corner of the dialog box Let s suppose we want to trace the effects of increasing Saturday s staff to a total of forty employees We would change the current right hand side value of 120 to 400 in the box labeled New RHS Value Next we move to the Report Types section in the lower right corner of the dialog box where we have the opt
21. Note If required LINDO will erase output from the top of the Reports Window in order to make room for new output at the bottom of the window 8 CHAPTER 1 Getting back to our example the Reports Window now contains the solution to our model and should resemble the following J Reports Window LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1 145 0000 VARIABLE VALUE REDUCED COST STD 10 000000 0 000000 3 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 0 000000 2 500000 9 000000 0 000000 0 000000 7 500000 NO ITERATIONS 2 Taken in order this tells you first that LINDO took 2 iterations to solve the model second that the maximum profit attainable from the two variables as you ve constrained them is 145 and third the variables STD and DLX take the values 10 and 3 respectively What s interesting to note is that we make less of the relatively more profitable Deluxe computer due to it s intensive use of our limited supply of labor We ll explain REDUCED COSTS SLACK OR SURPLUS and DUAL PRICES a little later in this chapter The next section deals with versions of LINDO running on platforms other than Windows At this point the Windows user will want to skip ahead to the section titled Model Syntax Entering A Model from the Command Line If you are running LINDO on a platform other than a Windows based PC then you will interface with LINDO through the means of a command line prompt Specifically all in
22. Such a variable is effectively a slack If you did not explicitly add slack variables to your model and the SINGLE COLS count is greater than zero then it suggests a misspelled variable name The REDUNDANT COLS statistic is a count of the number of columns with the same nonzero structure You may want to examine the model and try to consolidate Perusing A Model for Errors For more detailed analysis the Reports Peruse command page 81 in Windows versions or the CPRI and RPRI commands page 138 in command line versions are useful They allow you to select a subset of either rows or columns that satisfy a specified condition and print various attributes of each column or row in the subset For example suppose we suspect a variable name is misspelled The misspelling will show up as a variable with only one nonzero To request a listing of variables with a single nonzero element in command line versions of LINDO give the following CPRI command CPRI N Z2 1 ANALYZING amp DEBUGGING A MODEL 207 In Windows versions of LINDO select the Reports Peruse command and fill in the dialog box as follows Peruse Model Report Parameters r Orientation Columns C Rows mB View Items lV Name IV Primal I Dual I Rim I Upper Bound I Lower Bound T Type I Nonzeros C Condition optional Bt Help Cancel r Report Format mD Report Types IV Text E Text Report Format IV In
23. You have attempted a command that requires that the current model be compiled Please issue the Solve Compile Model command and then retry 1039 The current window is not a Model Window Please select a Model Window first You have issued a command that requires you to first select a Model Window Click ona window that contains the desired model and then reissue the command 1040 The selected option field contains an invalid argument You have input an invalid value for one of the fields in the Edit Options dialog box LINDO will highlight the first invalid value that it finds Please correct the highlighted value or press the Cancel button 1041 Please select an Edit Window to paste into You have attempted to paste text into a window that doesn t accept pasted data e g a view or graphics window The only windows in LINDO that accept pasted data are Edit Windows Edit Windows are created using either the File Open or File New commands 1042 If you wish to choose the pivot variable you must specify a variable name in the My Variable Selection box You have elected to choose the pivot variable as part of the So ve Pivot command but have neglected to specify the variable Either choose a variable or let LINDO select the pivot variable 1043 1044 1045 1046 1047 1048 1049 ERROR MESSAGES 289 If you wish to choose the pivot row you must specify a row name in the My Row Selection box You
24. 0 The GLEX command may be applied to both linear and integer programs PIVOT Syntax PIVOT lt Variable gt lt Row gt The PIVOT command is almost identical to entering GO 1 LINDO performs one pivot of the simplex algorithm The name of the entering variable is given it s primal value the number of the row that bounded the variable i e the pivot row and the objective function value The syntax of the PIVOT command is as follows PIVOT lt Variable gt lt Row gt You may optionally specify a variable name or index to enter into the basis If you do not specify a variable LINDO will make the selection automatically If you do specify a variable then you also have the option of selecting the pivot row name or row index When a variable is specified but not a row PIVOT brings that variable into the basis in the ordinary fashion of the simplex algorithm When a variable and row are specified PIVOT brings that variable into the basis using the specified row as the pivot row You can start with a feasible or even optimal solution and PIVOT in variables which render the basis infeasible LINDO may also report the objective function is unbounded when you specify both the variable and the row after PIVOT if the row specified does not bound the variable specified The TABLEAU and BPICTURE commands may provide additional insight into each step of the algorithm COMMAND LINE COMMANDS 159 In the following ex
25. 115 120 Local optimum 201 Log Output command 26 33 35 214 Logical operators 87 139 142 LOOK 10 11 116 127 216 lpr command 215 LSEXIT 257 LUNOPN 249 255 Macros 35 102 115 Mainframes 120 Martin cuts 193 Match Case box 44 Mathematical programming 2 Matrix 127 coefficients 92 193 form 89 generators 245 63 notation 201 permuting 235 picture of 132 positive definite 101 175 quadratic programs 197 rotating 95 MAX MIN 5 9 12 116 120 21 160 Memory limit 155 Minus 12 Miscellaneous command line commands 118 183 89 Mitra G 176 Model windows 4 33 Models changing a line 11 compiling 6 entering in Windows 4 8 printing 11 solving 6 splitting lines of 13 syntax 12 21 Monitoring the Solver 7 MPS Punch files 30 32 37 38 146 150 51 Multiple optima 67 155 202 Multiplication 87 139 142 Multi user systems 120 N Natural logarithm 87 139 142 NECESSARY SET 61 183 210 Negative definite 201 Negative variables 18 Nesting files 124 219 New command 26 29 259 New Features v New Project command 222 246 NEWIP 264 Nonzeros in reports 71 limit 51 statistics 81 206 NONZEROS 116 137 Not equal to 87 139 142 Notepad 2 31 211 Numerical Aesthetics 269 Numerical considerations 267 70 O OBJ COEFFICIENT RANGES 74 130 Objective coefficient 24 Objective function 4 9 12 72 81 127 158 187 203 changing 159 q
26. 400 3 Yl Y2 Y3 Z3 lt 500 4 Zl Z2 lt 500 END go UNBOUNDED SOLUTION AT STEP 1 REDUCED COST 26 0000 UNBOUNDED VARIABLES ARE Z3 Y2 OBJECTIVE FUNCTION VALUE 1 0 9999990E 08 VARIABLE VALUE REDUCED COST X1 000000 312 000000 X2 400 000000 000000 Y1 000000 571 000000 Y2 99999903 000000 000000 Z1 000000 598 000000 Z2 500 000000 000000 X3 000000 26 000000 Y3 000000 25 000000 Z3 99999903 000000 27 000000 186 CHAPTER 3 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 13 000000 3 000000 25 000000 4 000000 28 000000 NO ITERATIONS 1 debug SUFFICIENT SET COLS TO CORRECT Z3 NECESSARY SET COLS TO CORRECT Y2 Here the NECESSARY SET contains the variables Y2 and Z3 and the crucial SUFFICIENT SET variable is Z3 Although at first glance all variables seem to be constrained a look at row 3 reveals that Z3 can be increased indefinitely As long as the sum of the Y variables minus Z3 is less than 500 the constraint will be satisfied Removing Z3 from the model or constraining it separately will cause the problem to become bounded In general the kind of correction required in a column is one or more of the following change the objective coefficient change a coefficient in some constraint change the direction of some constraint in which the variable appears or make either the upper or lower bound finite STATS The STATS command give
27. Debugging a QP and the POSD Command You will in general be interested in whether the optimum found by the quadratic solver is a global optimum or simply a local optimum If the submatrix corresponding to the quadratic part satisfies certain conditions then the solution found will be a global optimum One condition sufficient to guarantee global optimality is positive definiteness In matrix notation a quadratic program may be represented as Min x Qx cx Subject To Ax lt b The term x Qx is the quadratic form The matrix Q is defined to be positive definite if and only if x Qx gt 0 for all nonzero x If Q is positive definite then the objective function is convex and the solution found is guaranteed to be the unique optimal The list of all possible conditions for Q is positive definite if x Qx gt 0 for all nonzero x positive semidefinite if x Qx 0 for all nonzero x negative semidefinite if x Qx lt 0 for all nonzero x negative definite if x Qx lt 0 for all nonzero x indefinite if x Qx gt 0 for some x and x Qx lt 0 for some x Pv eho If your problem does not satisfy either condition 1 or condition 2 then the results from LINDO s solver will be unpredictable and should not be relied upon A physical interpretation of positive definiteness is that a matrix is positive definite if the quadratic expression corresponding to the matrix is strictly convex A matrix is said to be positive semidef
28. FRI 000000 000000 SAT 000000 000000 SUN 000000 000000 NO ITERATIONS 4 As the solution currently stands all of the overstaffing occurs on Monday and Tuesday as indicated by the nonzero values in the SLACK OR SURPLUS column for these two days Thus on Monday and Tuesday we have two extra staff members while during the remainder of the week we have none WINDOWS COMMANDS 69 We would like to have extra staff if possible but more than one extra employee per day is not beneficial So we ll let X denote the extra workers up to a maximum of one on each day Then using the Preemptive Goal command in place of the standard Solve command we can first minimize cost fix cost at its optimal value and then maximize the sum of the X variables The modified model is then IS C LINDO61 SAMPLES STAFFPG LTX ME A HIN COST EXTRA SUBJECT TO MON TUE WED THU FRI AAR SSeS be Ne Re Re Saag gn 2 Ae a al fig ea WAT F S N N F S S 1 1 1 1 1 I I COST 9M 9R OF 95 SN 9T 99 0 EXTRA EXTRA XM XT XWV XR XR XS XN ol END We want to minimize total cost first so we place the variable COST first in the objective The variable COST is defined in the row that is also named COST After minimizing COST and fixing it to its optimal level we want to maximize the excess staff for each day up to at most one person per day Note that
29. Forcing X to be binary you might guess that the optimal solution would be for X to be 0 because 4 is closer to 0 than it is to 1 If we round X to 0 and optimize for A and B we get an objective of 84 In reality a considerably better solution is obtained at X 1 A 10 and B 1 for an objective of 112 In general rounded continuous solutions may be non optimal and at worst infeasible Based on this one can imagine that it can be very time consuming to obtain the optimal solution to a model with many binary variables This is typically true and you are best off utilizing the INT feature only when absolutely necessary As a final note the INT command also accepts an integer value argument in place of a variable name The number corresponds to the number of variables you want to be general integers These variables must appear first in the formulation of the model For more details and examples on the use of the INT please jump to page 167 SUB and SLB STATEMENTs If you do not specify otherwise LINDO assumes that variables are continuous bounded below by zero and unbounded from above That is variables can be any non negative fractional number increasing indefinitely In many applications this assumption may not be realistic Suppose your facilities limit the quantity produced of an item In this case the variable that represents the quantity produced is bounded from above Or suppose you want to allow for backordering in a system An
30. LINDO did not set aside enough memory to complete the operation you specified Adding additional memory to your machine will not alleviate the problem Call technical support for assistance 312 988 9421 INVALID COMMAND lt Command gt TYPE COM TO SEE VALID COMMANDS You have input an unknown command to LINDO command line prompt Please check the spelling of the command or give the COMMANDS command to get a list of available commands AMBIGUOUS COMMAND lt Command gt LINDO allows you to abbreviate commands as long as there is no possibility for confusion When you abbreviate a command too much it may be confused with other commands and LINDO issues this error message lt Not In Use gt INADMISSABLE VALUE lt Value gt COMMAND IGNORED You have entered an invalid argument for a command Please check the documentation or help file for this command and try again lt Not In Use gt lt Not In Use gt INVALID ROW DISREGARDED You have specified an invalid row name or number Please check the spelling of names or be sure that row numbers are within the range of total rows in the model COMMAND DISREGARDED SOLVE MODEL FIRST You have attempted a command that requires a valid solution in memory Use the GO command to solve the model and then try again WARNING OVERLENGTH LINE CHARACTERS MAY HAVE BEEN LOST OFF END USE CARRIAGE RETURN TO BREAK UP OVER SEVERAL INPUT LINES You have attempted to input a line tha
31. Optimal 65 0 TS TE ig 13 Elapsed Time 00 00 00 Update Interval fi Now suppose we know a feasible integer solution exists with an objective value 14 If we set the IP hurdle to this value in the Options Dialog box as follows pore IP Objective Hurdle IP Var Fixing Tol 49 50 CHAPTER 2 and then re solve we get the following results LINDO Solver Status m Optimizer Status Status Optimal Iterations 22 Infeasibility 0 Objective 15 Best IP Ale IP Bound 1S Branches z Elapsed Time 00 00 00 Update Interval 1 Notice the total number of iterations fell by more than 50 percent from 68 to 32 Reductions of this magnitude can be a big help when tackling large IP models However if IP hurdles are set too aggressively LINDO may return an infeasible message In our previous example a hurdle value of 16 would be infeasible because the objective is only 15 The default IP Objective Hurdle value is None In other words the feature is disabled IP Variable Fixing Tolerance LINDO begins solving IP models by solving a linear relaxation of the original IP model This linear relaxation merely lifts the integrality requirements and solves the remaining model as a pure linear programming problem You may optionally have LINDO find integer variables that have an integer value and high reduced cost in the linear relaxation solution LINDO will initially fix
32. SENSITIVITY ANALYSIS y RANGES IN WHICH THE BASIS IS UNCHANGED OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE COMP1 20 000000 INFINITY 5 000000 COMP2 30 000000 10 000000 30 000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREAS 2 60 000000 60 000000 60 000000 3 70 000000 INFINITY 40 000000 4 120 000000 80 000000 60 000000 131 132 CHAPTER 3 Now we will change the objective coefficient of COMP1 to 14 decreasing it by 1 more than the allowable decrease and re solve alt 1 VAR comp1 NEW COEFFICIENT 14 go LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1 1800 000 VARIABLE VALUE REDUCED COST COMP1 000000 1 000000 COMP2 60 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 60 000000 000000 3 10 000000 000000 4 000000 15 000000 NO ITERATIONS 1 As expressed by the 000000 VALUE for COMP in the solution report the variable COMP 1 has now been forced out of the solution and COMP2 has doubled in size PICTURE The PICTURE command displays the model in matrix form using letters for many of the numbers For small to medium sized models the PICTURE command is a useful way to obtain a visual impression of the model COMMAND LINE COMMANDS 133 The following letter codes are used to represent numbers in the PICTURE output Letter Code Coefficient Range 0
33. X6 8 X7 12 X8 2 X7 3 X8 lt 12 X6 X7 X8 lt D COST 0000 0000 0000 0000 0000 0000 0000 0000 00 00 00 00 00 00 00 00 PRICES 0000 0000 0000 0000 0000 0000 0000 This command simply counts the rows 00 00 00 00 00 00 00 144 CHAPTER 3 NAME AANHADUBWNE List the name nonzero count and RHS Ifor all less than or equal to rows rpri nz r t lt NAME NONZEROES RIM 2 8 12 3 8 List the name and RHS values for all rows with RHS greater than 2 and less than 10 rpri nr r gt 2 and r lt 10 NAME RIM 5 3 7 9 List name and nonzero count on all rows with a nonzero count less than 4 rprinz z lt 4 NAME NONZEROE 4 5 7 8 Print the value of the objective row rpri np n I NAME PRIMAL 1 S WWWWN 4 File output SAVE Syntax SAVE lt FileName gt The SAVE command stores the model currently in memory using the LINDO packed format Ipk at the path and filename you specify For example SAVE lt FileName gt where lt FileName gt is the name of the file you wish to save the model under will save the file in the current working directory If you enter nothing after the SAVE command LINDO will prompt you for the name of the file Note Command line versions of LINDO do NOT warn you if you already have a file with that name Any existing file under the same name will be overwritten Use
34. less than 8 characters then the remaining elements in KNAME must be filled with blanks APPCOL is the quickest way to load a model into LINDO Internally LINDO stores a model in column major form so APPCOL does not need to do any shuffling of data to store the new column Given this APPCOL is much faster than routines such as INSROW and INSERT discussed below Note in order to append a column with APPCOL all the rows in the model should have been previously defined by calls to the DEFROW routine The APPCOL routine corresponds to the APPC command line command BINVT DDA Double DDA On entry DDA contains a double precision vector BINVT pre multiplies this vector by the inverse of the current basis matrix For example to get column J of the current basis inverse DDA should contain all zeros except for a 1 0 in the J th position in DDA CAPOUT LUNIT Integer LUNIT Captures all of LINDO s standard terminal output in a file assigned to unit number LUNIT This may be required in GUI environments such as Windows in order to keep your application from crashing To assign a unit number to a file see routines LUNGET and LUNOPN To restore LINDO s output to the standard output device call CAPOUT with LUNIT lt 0 CLRBAS Clears the current basis without altering the current model formulation LINDO CALLABLE LIBRARIES 233 D2DMY NS1800 NDAY MON NYR Integer NS1800 NDAY MON NYR On entry NS7800 con
35. refer to the documentation on the SET command line command On entry NPARAM contains the index of the parameter to fetch On return DPVAL will contain the parameter s current value DERR will be set to 0 if there was no problem or if the parameter index was invalid To set the values of parameters see the LSPUTP routine 238 CHAPTER 8 LSGTPR DZ KZ IRTCOD Character KZ IRTCOD Double DZ Integer IRTCOD Retrieves a formulation saved in memory using the LSAVPR routine On entry the double precision array DZ contains the numeric part of the model and the character array KZ contains the text parts of the model These two arrays are written by the LSAVPR routine and their formats are not of practical interest to the average user On return JRTCOD will be 0 if all went well Otherwise it returns a 1 to indicate failure LSGTRO IROW NNZMX NNZ COF NNZMX ISCOL NNZMX Float COF Integer IROW NNZMX NNZ ISCOL Retrieves information on a row On entry JROW contains the index of the row you want to retrieve and NNZMX gives the maximum number of nonzero values you want returned in the COF and ISCOL arrays On return the arrays COF and ISCOL will contain the row s nonzero coefficients and the corresponding column indices respectively NNZ returns the number of nonzeros in these two arrays LSINSR LUNIT Integer LUNIT Reads in a solution basis from an MPS PUNCH format file attached to logical unit number L
36. restrictions were restored Be sure your model is correctly specified and that there are no misspelled words WARNING SOLUTION MAY BE NONOPTIMAL NONFEASIBLE The current solution in memory is either non optimal or infeasible Use the GO command to solve the model and then try again NESTED TAKE FILES ARE NOT ALLOWED LINDO does not allow nested TAKE files In other words you cannot place a TAKE command in a file you intend to do a TAKE on MORE THAN lt N gt COLUMNS HAD NONZERO OBJECTIVE COEFFICIENTS THIS EXCEEDS THE LIMITS OF THIS VERSION You have too many goals in a goal programming model Try to eliminate some goals FORMULATION ANOMALIES PRESENT FIRST ANOMALY WAS IN COLUMN lt N gt OBJECTIVE COEFFICIENT NOT 1 The objective coefficients in a goal programming model should always be 1 278 60 61 62 63 64 65 66 67 APPENDIX A FORMULATION ANOMALIES PRESENT FIRST ANOMALY WAS IN COLUMN lt N gt MORE THAN 2 NONZEROS IN COLUMN WITH 1 IN OBJECTIVE ROW The goal variable in a goal programming model should only have 2 nonzeros one in the objective and the other in the row defining the goal lt Not In Use gt lt Not In Use gt INFEASIBLE MODEL ENCOUNTERED DURING GOAL PROGRAMMING You have attempted to solve a goal programming model that does not have any solution that simultaneously satisfies all the constraints Run the DEBUG command to try to narrow down the problem UNBOUNDED MO
37. since we are maximizing EXTRA it has a negative coefficient in the objective 70 CHAPTER 2 After applying the Preemptive Goal command we obtain the new solution 3S Reports Window aok OBJECTIVE FUNCTION VALUE 1 2 000000 VARIABLE VALUE REDUCED COST COST 9 000000 0 000000 7 000000 1 000000 9 000000 000000 0 000000 000000 0 000000 000000 4 000000 000000 0 000000 000000 1 000000 000000 0 000000 000000 1 000000 000000 0 000000 000000 1 000000 000000 1 000000 000000 0 000000 000000 1 000000 000000 1 000000 000000 ooqcoaconocoacoa ac ROW SLACK OR SURPLUS DUAL PRICES MON 9 000000 0 000000 TUE 000000 000000 WED 000000 000000 THU 000000 000000 FRI 000000 000000 SAT 000000 000000 SUN 000000 000000 XH 000000 000000 XT 000000 000000 XW 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 CODD 0G DGO OCA C eN cooroooncancoocnoaoS EXTRA NO ITERATIONS ol Note we now have one extra person on each of the four days Monday Tuesday Saturday and Sunday as indicated by the values for the variables XM XT XS and XN Thus we have spread our beneficial excess staffing out to two additional days without increasing our staffing costs The Preemptive Goal command may be applied to both linear and integer programs It may not be applied to quadratic
38. this means when the formulation is solved as an LP the solution should look a lot like the IP solution e g many of the IP variables are naturally integer and the LP objective value is approximately equal to the IP objective value INTEGER PROGRAMMING 195 LINDO has a command TITAN which will do some of this tightening It will do two things a deduce tighter upper and lower bounds for continuous variables and then using this information b reduce the coefficients of integer variables where justified The net effect of b is that when the problem is solved as an LP the nonzero IP variables will be closer to 1 than they would otherwise have been For example if you have the constraint 7 X 4 Y 2 Z 2 5 where all variables have a lower bound of zero and X is declared to be an integer variable then TITAN will transform this constraint to 5 X 4Y 2225 196 CHAPTER 4 197 5 Quadratic Programming LINDO has a quadratic capability which allows the user to consider models with a quadratic objective function That is objectives that contain the product of two variables In matrix notation a quadratic program may be written as Minimize x Qx cx Subject To Ax lt b In words a quadratic optimization problem is considered to be a quadratic when 1 all constraints are linear and 2 the objective contains at least one quadratic second degree term As an example Model 1 is a quadratic program but Model 2 is not due to
39. to 3 B2 6 C2 C1 C2 C3 ll O O o C4 A3 B3 4 63 24 2 B3 4 C3 9 A4 lt 12 B4 lt 12 C4 lt 12 6 B4 3 C4 Demand Demand Demand Demand Supply Supply Supply for for for for cust cust cust cust AUNE limit for A limit for B limit for C INTERFACING 213 Now we make individual substitutions to make the right hand sides correct This gives the following MIN ST A1 A2 A3 A4 A1 Bl Cl END 2 Al 4 Bl Di Cle els B 1 4 B2 B3 4 B 4 4 A2 B2 C2 4 A2 3 A3 5 A4 3 B2 2 BIF 6 Bay 2b 6 c2 4 C3 3 C4 Cl 9 C2 7 C3 6 C4 8 A3 A4 lt 12 B3 B4 lt 11 C3 C4 lt 13 Demand for Demand for Demand for Demand for l 1 l cust cust cust cust BwWNHE Supply limit for A Supply limit for B Supply limit for C Now we want to store this model in standard text format On some editors this requires some care The default file format for many editors is a special format unique to that editor We do not want to store the model in this special format because it will be unreadable by LINDO Most such editors have an optional file save command for saving in generic text form For example in Microsoft Word you choose the File Save As command and from the list box labeled Save as type choose the Text Only txt format and press the Save button as illustra
40. 000000 5 000000 40 000000 0 000000 0 000000 15 000000 NO ITERATIONS 2 74 CHAPTER 2 and the Range command yields the following range report J Reports Window RANGES IN WHICH THE BASIS IS UNCHANGED VARIABLE COMP1 COMP2 CURRENT COEF 20 000000 30 000000 CURRENT RHS 60 000000 70 000000 120 000000 OBJ COEFFICIENT RANGES ALLOWABLE INCREASE INFINITY 10 000000 RIGHTHAND SIDE RANGES ALLOWABLE INCREASE 60 000000 INFINITY 80 000000 ALLOWABLE DECREASE 5 000000 30 000000 ALLOWABLE DECREASE 60 000000 40 000000 60 000000 Interpretation of the range report is as follows the OBJ COEFFICIENT RANGES for each variable are the amount by which that objective coefficient can be increased or decreased without causing a change in the basis the values of the set of nonzero variables The RIGHTHAND SIDE RANGES for each row are the amount by which that right hand side can be increased or decreased without causing a change in the basis We ll now make some changes to demonstrate these concepts Note that the allowable decrease on COMP is 5 Thus changing COMP1 s objective coefficient to 15 1 20 4 9 should not cause a change in the variable values However reducing the objective coefficient by one tenth more to 15 would begin to effect the solution Making this modification to the objective we have IS C LINDO61 SAMPLES RANGE LTX DEAR MAX 15 1 COMP1 30 COMP2 SUBJECT TO 2 COMP1 lt 6
41. A L Brearley G Mitra and H P William s Analysis of Mathematical Programming Problems Prior to Applying the Simplex Algorithm Mathematical Programming 1975 Vol 8 pp 54 83 In the following sample session a general integer program is displayed and solved before using TITAN Two branches and five pivots were required The same model is displayed and solved after using TITAN Note after using the TITAN command no branches and only two pivots were required look all MAX X Y SUBJECT TO GIN 2 COMMAND LINE COMMANDS 177 go LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE 4 30000000 EW INTEGER SOLUTION OF 4 00000000 AT BRANCH 2 PIVOT 5 BOUND ON OPTIMUM 4 000000 ENUMERATION COMPLETE BRANCHES 2 PIVOTS 5 LAST INTEGER SOLUTION IS THE BEST FOUND RE INSTALLING BEST SOLUTION titan REDUCTIONS BNDS 8 IN 3 PASSES COEFFICIENTS 0 look all MAX X Y SUBJECT TO 2 X 2 5 3 lt Qe 4 X Y lt 4 3 END SUB X 2 00000 GI X SUB Y 2 00000 GI Y go LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE 4 00000000 ENUMERATION COMPLETE BRANCHES 0 PIVOTS 2 LAST INTEGER SOLUTION IS THE BEST FOUND RE INSTALLING BEST SOLUTION BIP Syntax BIP lt ObjectiveValue gt The BIP command is used
42. C3 6 C2 o 0o N wo A4 lt B4 lt C4 lt 4 B1 3 B2 ME Ge 3 C4 12 13 Sending LINDO Output to External Editors Now suppose we wish to direct the output from a LINDO session to a file that might be processed by an editor or sent to a printer If you are using a Windows version of LINDO you can log all Reports Window output to an external text file that will be readable by any text editor or may be printed from the operating system To open and close a log file use the File Log Output command This command is discussed in more detail on page 33 If you are using a command line version of LINDO you can redirect report output to a file with the DIVERT command See page 145 for the details of the DIVERT command INTERFACING 215 As an example of this feature suppose we are running a command line version of LINDO and we have the transportation model listed above in memory We will then solve the model and send a copy of the model to one file and a solution report to another as follows look all MIN 2 Al 4 A2 3 A3 5 A4 4 Bl 3 B2 2 B3 6 B4 iB ET ee SO C2P ae AL C3 3 4 SUBJECT TO 2 Al B1 Cl 9 3 A2 B2 C2 7 4 A3 B3 C3 6 5 A4 B4 C4 8 6 Al A2 A3 A4 lt 12 7 B1 B2 B3 B4 lt 11 8 Cl C2 C3 C4 lt 13 END terse Suppress the solution go Solve the model LP OPTIMUM FOUND AT
43. E2T1 6 B2T1 12 I1T2 E1T2 6 B1T2 13 I2T2 E2T2 6 B2T2 14 P1T1 N1T1 B1T1 15 P1T2 N1T1 N1T2 16 P2T1 N2T1 B2T1 Re P2T2 N2T1 N2T2 18 CITI R1T1 B1T1 19 c1iT2 C2T1 RiT 1 20 Cc2T1 R2T1 B2T1 21 C1 T1 C2T2 R2T1 o o 6 B1T2 T1T2 4 B2T2 T2T2 Tea Sl E aa N O e ee Aa ey Bk END The objective should be 8942180 WINDOWS COMMANDS 95 First we optimize the model by selecting the Solve command from the Solve menu Then we select the Basis Picture command from the Reports menu and LINDO sends the following report to the Reports Window I gt Reports Window SEK A COLUMN ROW 10110101112001012001 118029324671735590648 TITI BITI 1 Tpi B2T1 Cart a ia BiTA PITZ B2T2 Talt P2T2 P2T1 PITI NDT1 I2T2 TITZ E1T2 I1T1 L2T1 E2T1 eHeeeqrri In the report you will note that the basis matrix has been rotated by 90 degrees The names of the variables appearing in the basis are displayed to the left The names of the rows that the variables are basic in run from right to left along the top Positive integer coefficients from 1 to 9 appear unaltered in the report For the remaining nonzero coefficients plus symbols are displayed for positive values and minus symbols are displayed for negative values LINDO can triangularize the basis for some models i e all nonzero elements contained on or below the diagonal Models with a
44. END TERSE Suppresses solution report GO 100 Solves model DIVERT SOLUTION TXT Opens output file CPRI NP P gt 0O Print names and values of nonz vars RVRT Close output file In general the PAGE 0 command should be the first command in any LINDO script This disables LINDO s terminal paging feature where it pauses after each screen of output This allows your script file to run without interruption The TERSE command is added to suppress the standard solution report There is no reason to generate the entire solution report when we need only the nonzero variable values The GO 100 command solves the model using the value of 100 as a pivot limit When running scripts it is always a good idea to provide a pivot limit sufficient to solve the model If not LINDO computes a limit based on a model s dimensions which may not be adequate In which case you can end up with a non optimal solution The DIVERT command opens a file for the solution values The CPRI command prints the name N and primal values P for all variables such that their value is greater than 0 P gt 0 Since we have an open DIVERT file the CPRI output is redirected to the file Finally we close the DIVERT file using the RVRT command INTERFACING 219 To execute this script use the File Take Commands command in Windows or the TAKE command in command line versions of LINDO Opening the solution file SOLUTION TXT after running the script reveals the
45. FEASIBLE SOLUTION AT STEP 1 SUM OF INFEASIBILITIES 1 00000 VIOLATED ROWS HAVE EFGATIVE SLACK OR EQUALITY ROWS ONZERO SLACKS ROWS CONTRIBUTING TO INFEASIBILITY HAVE NONZERO DUAL PRICE OBJECTIVE FUNCTION VALUE 1 1 00000000 VARIABLE VALUE REDUCED COST X 1 000000 000000 Y 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 1 000000 3 1 000000 1 000000 NO ITERATIONS al The returned variable values have little meaning when a model is infeasible There is other information in the solution report however that may prove useful in helping to track down the source of the infeasibility First off the slack or surplus value for violated constraints will be negative In the example above constraint 3 is violated by 1 unit Second the dual prices have a useful meaning in this situation They are the amount by which the total sum of infeasibility the total amount that all constraints are violated is reduced if the right hand side is increased by one unit Referring back to our example once again we see that increasing the RHS of constraint 2 by one unit will reduce the infeasibility by one unit while increasing the RHS of row 3 by one unit will result in a corresponding increase in the infeasibilities In this small model it is easy to verify this fact by eyeball If you would like a more mechanical method for tracing down infeasibilities please refer to the DEBUG command The DEBUG command can narrow an infeasible model
46. General Optimizer Options Nonzero Limit for details on how to change the limit and why you would want to LINDO uses the nonzero allocation as a work area during optimization Specifically LINDO stores the basis inverse matrix along with the structural nonzeros in this shared memory location Given this you should try to allocate a good number of additional nonzeros over and above what is required by just the structural nonzeros If LINDO runs out of nonzero space it halts the optimization and returns with an error The additional boxes in the screen list the serial number type of user and any expiration if applicable These may be needed in updating or upgrading to larger versions 115 3 LINDO for Command line Environments This chapter discusses all of the command line commands available to the LINDO user On platforms other than Windows based PCs the user interfaces with LINDO entirely through text commands issued to LINDO s command line colon prompt LINDO commands may also be used to build command scripts or macros contained in an external file These scripts may be run automatically at startup and or whenever the user desires Command scripts can be run by both Windows users and command line users of LINDO and as such are of interest to both classes of users We will first list the commands according to their general function then explain them one by one with some examples of their application The Commands in Brief We
47. LINDO will compile the model if it hasn t been compiled already then the following dialog box appears Basis Read Folders OK c lindo61 samples fc gt LINDOEB1 gt SAMPLES Cancel C Samptext List files of type Drives Punch pun x E Network This is a standard Windows dialog box for selecting the basis file to read You can use the list box in the lower left corner to select a file filter based on the three basis file formats used by LINDO e Punch MPS Punch format e FBS File Basis Save format e SDBC Save DataBase Column format Once you have selected the correct basis file LINDO will attempt to install the solution into the model in the active window Typically your next step will be to solve the model If the basis contained an optimal solution to the model the required solution time should be quite short In fact usually no iterations are required at all If you have modified the model slightly from the model used to save the solution then LINDO may require several iterations to find the new optimum Regardless this beats having to wait through conceivably thousands of iterations on a large model Reading the basis of models with integer restrictions or quadratic programming generally has little effect on the time required to re solve the model One final note As long as a model is open in LINDO LINDO automatically keeps a basis in memory It is not until you physically
48. Line command to jump to this line and correct the syntax error Keep in mind that LINDO does its best to point to the line where the syntax error actually occurred but it may not always nail the line down precisely So be sure to examine surrounding lines in addition to the one specified by LINDO 1027 Model is too large for an Edit Window Please use the File View command instead You have attempted to open a model with the File Open command that contains more characters than can be handled in a standard Edit Window Standard Edit Windows can only handle up to about 64 000 characters To open a large file use the File View command which is limited only by available memory 1028 There is no compiled model in memory You have issued a command that requires a compiled model in memory Select a Model Window and issue the So ve Compile Model command and then try again 1029 lt Not In Use gt 1030 lt Not In Use gt 1031 The model must be compiled first Please choose the SolvelCompile Model command first Or alternatively save the model using LINDO Text Itx format You have attempted to save a file that is to be stored in either LINDO packed format Ipk or MPS format mps that has not been successfully compiled Please select the Solve Compile Model command and then retry If you are unable to compile for some reason you may always save the file in LINDO text format Itx 1032 A graph may not have more than 8000 points Th
49. MAX 10 ST Det I 5 DLX SUBJECT TO LINDO would give a syntax error message because the variable STD is split between lines and the coefficient 15 is also 14 CHAPTER 1 LINDO is not case sensitive All input is converted to upper case internally by LINDO For example the following model is valid Max x SE X lt 1 eNd and contains a single variable X rather than the two variables x and X Only constant values not variables are permitted on the right hand side of a constraint equation Thus an entry such as X gt Y would be rejected by LINDO Such an entry could be made as MoS 0 Conversely only variables and their coefficients are permitted on the left hand side of constraints For instance the constraint 3X 4Y 10 0 is not permitted due to the constant term of 10 on the left hand side Of course the constraint may be recast as 3X 4Y 10 in order to comply with LINDO syntax GETTING STARTED 15 Optional Modeling Statements In addition to the three required model components of an objective function variables and constraints LINDO also has a number of other optional modeling statements which may appear after the END statement in a model In command line versions of LINDO you enter these statements as system commands to the command level prompt after the END statement terminating the model s constraints In Windows versions of LINDO the statements should be entered as part of the model text
50. NVALU Using the pointer vector INDX swaps the first largest item with the item in position N and then reheaps the first N 1 items User must reset N to N 1 M is the dimension of NVALU Thus HEAPFR removes items from a heap in decreasing order ILINDO Performs one time initialization for LINDO ILINDO must be called once before calling any other LINDO routines INIT Initializes storage in preparation for a new model INIT should be called whenever creating a new formulation INSERT I J AMT NOADD Float AMT Integer l J NOADD Adjusts the element in row J column J of the matrix If NOADD 1 then AMT replaces the old value at position else AMT is added to the old value The right hand side is indicated by setting J 0 Row and column J must have been previously defined by DEFROW and APPCOL calls INSROW J NONZ LSTCOL NONZ VALIST NONZ Float AMT Integer l J NOADD Inserts one or more elements into row J of the formulation Row J must have been defined previously e g with a call to DEFROW The scalar NONZ contains the number of nonzeros to be inserted LSTCOL a vector contains the list of column indices to be inserted The corresponding nonzero values are given in the vector VALIJST This routine is more efficient than repeated calls to the routine INSERT if more than one element is to be inserted However APPCOL is the most efficient routine for adding nonzeros to the model s
51. PAGE 0 should always be the first command lin a LINDO script It disables the terminal paging feature PAGE 0 l Here is the model MAX 20X 30Y anl lt 50 2Y lt 120 D Suppress the standard solution report ERSE He e A i KON A D fo Solve the model and be sure to allow plenty of pivots GO 1000 l Open a solution file DIVERT SOLUTION TXT j Send the primal values to Ithe solution file CPRI N P l Close the solution file RVRT Quit LINDO QUIT Assuming we named the batch file lindo bat your screen should appear as follows when you run this application gt chmod a x lindo bat gt lindo bat X and Y 50 35 gt Note that any operating system that allows for I O redirection and batch programming will support this form of integration with LINDO 222 CHAPTER 7 A Visual Basic Front End for Windows LINDO The following example uses Microsoft Visual Basic 3 0 and the Windows version of LINDO to build and solve this simple model Maximize 20 X 30 Y Sab 0 0 Y lt 120 x KX lt 5 lt 6 2 End If you would prefer to simply examine this sample application without going to the trouble of entering it it may be found in the file LINDO SHELL SHELL MAK The approach used in this example is straightforward We write a LINDO script file containing the model and various commands using Visual Basic VB use the Shell command in VB
52. WWORK SWORK SUWORK THU MWORK TWORK WWORK THWORK SUWORK FRI MWORK TWORK WWORK THWORK FWORK SAT TWORK WWORK THWORK FWORK SWORK SUN WWORK THWORK FWORK SWORK SUWORK END gt 20 VVVVVV 13 10 12 16 18 20 23 Now you re ready to solve the model In Windows versions of LINDO press the Solve button Sh on the toolbar at the top of the frame window In command line versions give the GO command at the colon prompt LINDO will solve the model and return the following solution OBJECTIVE FUNCTION VALUE 1 7750 000 VARIABLE VALUE REDUCED COST WORK 2 000000 0 000000 TWORK 0 000000 100 000000 WWORK 2 000000 0 000000 THWORK 7 000000 0 000000 FWORK 5 000000 0 000000 SWORK 4 000000 0 000000 SUWORK 2 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES MON 0 000000 100 000000 TUE 0 000000 0 000000 WED 0 000000 100 000000 THU 1 000000 0 000000 FRI 0 000000 100 000000 SAT 0 000000 25 000000 SUN 0 000000 135 000000 NO ITERATIONS 8 Your weekly salary cost is minimized to 7750 and all staffing needs have been met You need to start 2 people on Monday nobody on Tuesday 2 on Wednesday 7 on Thursday 5 on Friday 4 on Saturday and 2 on Sunday to meet your requirements However there s more in the solution report than just an optimal objective If you look in the row titled THU you see that there s a slack of 1 returned on the THU c
53. Width in the Options dialog box in the following edit box pore Terminal Width eens However the minimum and maximum Terminal Width values allowed are 40 and 132 When LINDO reads an input file any characters going beyond the Terminal Width limit set here will not be recognized by LINDO and may affect the way the model is solved Constraints and objective functions may be split over multiple lines or combined on single lines You may split a line anywhere except in the middle of a variable name or a coefficient Be sure to use the return key to any rows that exceed this width on the next line The default Terminal Width is 80 characters Go To Line Ctrl T The Go To Line command allows you to jump to a specified line in the current window You can also select to jump to either the top or the bottom of the window 56 CHAPTER 2 When issuing the Go To Line command you will be presented with the following dialog box Go To Line Line number to go to _ Goto Top Go to End Cancel Help If you want to go to a specific line enter the line number in the edit box labeled Line number to go to and click the OK button LINDO will move the cursor to the start of the specified line scrolling it into view if required To go to the top or the bottom of the window click either Go to Top or Go to End Paste Symbol Ctrl P The Paste Symbol command displays a dialog box which assi
54. X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 4 Xl X3 X5 9 9 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 Xe 15 9 X1 X2 X3 X4 lt 116 10 24 X8 X9 gt 0 END APPCOL Syntax APPCOL lt NewVarName RHS gt The APPCOL command is used to add variables or to change the right hand sides of the model currently in memory If you do not type anything after APPCOL LINDO will prompt you for the name of the variable The form of the command is APPCOL lt NewVarName RHS gt If you use RHS as an argument LINDO assumes you want to change the model s right hand side values All other argument values are assumed to be a name of a new variable you want to add to the model After entering the first line of the command LINDO gives a question mark prompt Enter the row numbers and coefficients in pairs one pair per line with a space in between For all row numbers not entered the coefficient will be assumed as zero in the model Complete the command by entering 0 for the row number You may use the word END instead of the 0 if you wish If RHS is entered as the command argument the command changes all right hand sides After entering APPCOL RHS enter the row numbers and new right hand sides in pairs one pair per line with
55. XMON XTUE XWED XSAT XSUN gt 16 THU XMON XTUE XWED XTHU XSUN gt 19 FRI XMON XTUE XWED XTHU XFRI gt 14 SAT XTUE XWED XTHU XFRI XSAT gt 12 END COMMAND LINE COMMANDS 149 Now here s the same model in MPS format NAME MIN ROWS N 1 G SU G MO G TUE G WED G THU G FRI G SAT COLUMNS XMON I 100 0000000 XMON MO 1 0000000 XMON TUE 1 0000000 XMON WED 1 0000000 XMON THU 1 0000000 XMON FRI 1 0000000 XTUE 100 0000000 XTUE 1 0000000 XTUE L 0000000 XTUE 1 0000000 XTUE L 0000000 XTUE 1 0000000 XWED 100 0000000 XWED 0000000 XWED L 0000000 XWED 1 0000000 XWED 1 0000000 XWED 1 0000000 XTHU i 100 0000000 XTHU SUN 1 0000000 XTHU MO 1 0000000 XTHU THU 1 0000000 XTHU FRI 1 0000000 XTHU SAT 1 0000000 XFRI 100 0000000 XFRI SU 1 0000000 XFRI MON 1 0000000 XFRI TUE 1 0000000 XFRI FRI 1 0000000 XFRI SAT 1 0000000 XSAT 1 100 0000000 XSAT SU 1 0000000 XSAT MO 1 0000000 XSAT TUE 1 0000000 XSAT WED 1 0000000 XSAT SAT 1 0000000 XSUN 1 100 0000000 XSUN SUN 1 0000000 XSUN MO 1 0000000 XSUN TUE 1 0000000 XSUN WED 1 0000000 XSUN THU 1 0000000 RHS 150 CHAPTER 3 RHS SU 18 0000000 RHS MO 16 0000000 RHS TUE 15 0000000 RHS WED 16 0000000 RHS THU 19 0000000 RHS FRI 14 0000000 RHS SAT 12 0000000 ENDATA One thing you will immediately notice is a relatively simple 11 line
56. XWHIC3 XWHIC4 XWH2C4 XWH3C2 XWH3C3 XWH3C4 Issuing the Tile command we receive the following dialog box Tile Windows Style OK Horizontal Help Vertical A e WINDOWS COMMANDS 109 If we press the Horizontal button and then press the OK button our windows are repositioned as below 35 Peruse Graph CALINDOG 1iSamples TRAN LTX J Reports Window LP OPTIMUM FOUND AT STEP 6 OBJECTIVE FUNCTION VALUE 1 161 0000 VARIABLE VALUE REDUCED COST XWH1C1 2 000000 0 000000 1 gt C LINDO61 Samples TRAN LTX E EE warehouse 4 customer transportation model EWH lt i gt C lt j gt amount shipped from warehouse lt i gt to customer lt j gt MIN 6 XWH1C1 2 XWH1C2 6 XWH1C3 7 XWH1C4 4 XWH2C1 9 XWH2C2 5 XWH2C3 3 XWH2C4 8 XWH3C1 8 XWH3C2 XWH3C3 5 XWH3C4 SUBJECT TO Close All Alt X The Close All command closes all open windows and dialog boxes If you have made a change to a Model Window without saving it you will be prompted to save the model before it is closed Note Any changes you have made since the last save will be lost if the model is not saved before closing It is recommended to always choose yes at the save prompt to be sure Arrange Icons Alt l If you have minimized any open windows so they appear as icons on the screen you can issue the Arrange Icons command to line all the icons up in the lower left hand corner of the frame window 110 CHAPTER 2 6
57. You will have to wait until the report is complete before closing the window If the report is excessively long you may want to use the File Log Output command to divert it to disk Diverting a file to disk is quicker than sending it to the Reports Window Windows is running low on space Try closing some windows LINDO has received a low memory warning from Windows This generally occurs when you are trying to place too much text in an Edit Window created with either the File New or File Open commands Edit Windows can only hold up to about 64 000 characters If you need more capacity than this use a View Window instead View Windows may be opened with the File View command Please select some text first You have issued a command that requires you to select a box of text Please select some text by clicking with the mouse and dragging and then try the command again lt Not In Use gt Nothing in clipboard to paste You have issued the Edit Paste command when there was no text in the Windows clipboard Use the Copy command in LINDO or some other application to select some text and try Paste again Unable to Solve no model currently in memory You have issued a Solve command without a model in memory Select a Model Window by clicking on it and reissue the Solve command LINDO is busy right now Try again later LINDO is busy performing critical background computations and is unable to handle your request at the moment Please try th
58. an old basis FPUN Syntax FPUN lt FileName gt The FPUN command saves the solution currently in memory to a file using the file name that you specify If you don t supply a path the file will be saved in the current working directory If you enter nothing after the FPUN command LINDO will prompt you for the name of the file The FPUN command is an abbreviation for File PUNch and was named in deference to the old PUNCH as in punching a card deck command in MPS FPUN saves the current basis in MPS PUNCH format for later insertion as the starting point for a model A basis file created with FPUN is reloaded into LINDO using the FINS File INSert command described above In the following example we read in a model solve it from scratch and save a basis in FPUN format We then reread the model load the FPUN basis file created in the previous step and re solve The interesting point to note is when starting with the benefit of a FPUN basis file the number of iterations required to optimize the model falls from 741 to 0 retr etamacro lpk Load a model from disk stats Examine the stats ROWS 401 VARS 688 INTEGER VARS 0 0 0 1 QCP 0 NONZEROS 2513 CONSTRAINT NONZ 2409 1026 1 DENSITY 0 009 SMALLEST AND LARGEST ELEMENTS IN ABSOLUTE VALUE 0 833574E 02 10000 0 OBJ MAX NO lt gt 48 272 80 GUBS lt 182 VUBS gt 267 SINGLE COLS 32 REDUNDANT COLS
59. any time If you do close it it may be restored at any time with the Open Status Window command For more information on the interpretation and use of the Status Window please refer to the table on page 7 Send to Back Ctrl B The Send To Back command sends the active window behind all other windows on the screen This command is particularly useful when switching between a Model Window and the Reports Window WINDOWS COMMANDS 105 As an example suppose our screen is configured with a Model Window appearing before the Reports Window SEE 3 warehouse 4 customer tran a EWH lt i gt C lt j gt amount shipped fri MIN 6 XWH1C1 2 XWH1C2 6 4 XWH2C1 9 XWH2C2 5 8 XWH3C1 8 XWH3C2 SUBJECT TO lt Lil EXWH1C2 17 000000 Issuing the Send to Back command causes the windows to reverse position The Reports Window moves to the front and becomes the active window C LINDO61 Samples TRAN 5 x Dor 6 OBJECTIVE FUNCTION VALUE T 161 0000 VARIABLE VALUE XWH1C1 2 000000 XWH1C2 17 000000 i This example is shown with the Report Window and Model Window smaller than the full screen to allow the reader to see the switch of the windows In actual use this command would be most beneficial when the windows are maximized and you want to toggle between them quickly The Send to Back button is provided to do this with the click of a mouse Cascade Ctrl A The Cascade command arrang
60. are about to overwrite an existing log file If Append output is checked LINDO will append output to the end of any preexisting log file The Log Output command works as a toggle When you first select the command you open a file for logging output As long as this file is open a checkmark will appear next to the command in the File menu indicating that you are in Log Output mode Select the command again and the log file is closed The checkmark is removed from the menu as well As an example suppose you wish to create a disk file containing a copy of the solution report to the active model First solve the model using the Solve command in the Solve menu Next select the Log Output command from the File menu to open the file for the solution report Once you have opened the file pull down the File menu again and note the checkmark adjacent to the Log Output command indicating that you are now in Log Output mode You have now opened the log file but as yet it contains no data To remedy this select the Solution command from the Reports menu The solution will then be sent to the log file Note The solution will not appear on your screen unless you also checked the Echo to screen box in the Log Output dialog box WINDOWS COMMANDS 35 Finally close the log file by once again selecting the Log Output command You should now have a disk copy of the solution report suitable for such things as routing to the printer p
61. can refer to the companion LINDO textbook Optimization Modeling with LINDO by Linus Schrage Preprocess An optional phase to the integer programming IP solver in LINDO is known as preprocessing During the preprocessing phase the solver does extensive evaluation of your model in order to add IP cuts JP cuts are simply constraints added by the LINDO solver These constraints cut away sections of the feasible region to the continuous model i e the model with integer restrictions dropped which are not contained in the feasible region to the integer model On most IP models this will accomplish two things First solutions to the continuous problem will tend to be more naturally integer Thus the branch and bound solver will have to branch on fewer variables Secondly the bounds that come out of solutions at intermediate nodes will tend to be tighter allowing the branch and bound solver to fathom i e drop from consideration branches sooner These improvements can really turbocharge the IP solver on many models 46 CHAPTER 2 However there is a downside to IP preprocessing which you should be aware of As you can imagine determining a valid and meaningful set of cuts can be time consuming If your model is naturally tight or if there are just a few integer variables then the time to generate cuts can exceed the savings in solution times resulting from those cuts In these cases you are better off skipping the prepro
62. carriage return and you will be returned to command level Sample output for a small model follows look all AX gt ae ae A SUBJECT TO 2 X Y lt a END go LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1 1 00000000 VARIABLE VALUE REDUCED COST X 1 000000 000000 Y 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 1 000000 NO ITERATIONS al DO RANGE SENSITIVITY ANALYSIS Y RANGES IN WHICH THE BASIS IS UNCHANGED OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X 1 000000 INFINITY 000000 Y 1 000000 000000 INFINITY RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 1 000000 INFINITY 1 000000 2 EXAMPLE OF INFEASIBLE SOLUTION OUTPUT If your model is overly constrained to the point that no solution can simultaneously satisfy all the constraints then you will get an infeasible solution message from LINDO A small infeasible example follows look all AX X Y SUBJECT TO 2 X Y lt 1 3 X Y gt 2 END COMMAND LINE COMMANDS 153 This is obviously infeasible since X Y cannot simultaneously be lt 1 AND gt 2 go NO
63. command 26 42 Cutting stock 67 D Databases 143 Date command 26 39 DEBUG 118 153 154 183 205 Debug command 27 60 61 63 205 209 10 Debugging 118 183 205 10 infeasible models 152 184 quadratic programs 201 2 script files 181 219 Decimal arithmetic 269 Decimal points 267 Decision arguments 2 DEFROW 249 256 261 DELETE 117 163 Density of a model 81 206 Disable AutoUpdate button 113 Display command line commands 116 127 44 Displaying the model 127 147 cat command 215 NONZ command 137 DIVERT 11 116 145 218 Diverting a model 145 216 Division 87 139 142 DLLs See Dynamic Link Libraries DMPS 116 129 Double indirection 264 Downloading 112 Dual prices 24 72 127 137 142 153 range analysis 24 Dual variables 198 Dynamic Link Libraries DLLs 231 66 samples 245 63 e 87 139 142 Echo to screen 34 Edit menu 26 41 58 Edit windows 29 41 Editing 41 58 42 211 17 Elapsed Time command 26 40 END 5 10 12 15 23 Entering a command line model 8 12 Entering a Model in Windows 4 8 Entering Variable Tolerance 51 53 Equal to 12 87 139 142 Error messages 271 89 Exit command 26 41 Exponentiation 87 139 142 EXTEND 117 161 62 External editors 2 211 17 218 output 214 F FBR 116 121 23 126 FBS 116 126 145 FBS files 37 38 Feasible region 268 Feasible solution 97 File menu 26 29 File output comman
64. command useful in tracking down the error The effect of the command is to cause all input whether from a file or from the terminal to be echoed to the terminal Thus you can see exactly what was read from the file up to the point where the error was encountered Place a BATCH command at the very start of the script file to turn on the feature each time the file is read Place a BATCH command at the very end of the file to toggle off the feature before exiting the script file Integrating LINDO Into Other Applications Developers of application specific turnkey systems which require optimization capability need to access LINDO in an automated or batch mode Perhaps the most seamless way to accomplish this is by using the LINDO callable libraries or the LINDO Windows DLL see Chapter 8 LINDO Callable Libraries but this may take intensive programming and development efforts The more casual system developer may find it expedient to have a front end application build an input file for LINDO run LINDO to process the input file and create output files and then upon returning to the front end application parse the output files from LINDO and present the results to the user in a custom report 220 CHAPTER 7 format In this section we will illustrate how to accomplish this using three different styles First off we will explore the use of input and output redirection as a tool for linking up your application with command line versions of LINDO N
65. continuing here See Save c ommand above 52 CHAPTER 2 Upon restarting the new nonzero limit will be established If the Nonzero Limit is set to a size too large for available memory on your computer LINDO will automatically scale back the limit to a size which will allow the program to run You can verify the true physical Nonzero Limit at any time by selecting the About LINDO command from the Help menu This command displays the LINDO release number copyright information and at the bottom you should find a box similar to the following Maximum Model Size Constraints 32000 Variables 200000 Integer Variables 20000 Nonzeros 2000000 This box lists the constraint and variable limits of your copy of LINDO along with the physical nonzero limit We should also mention that LINDO uses the nonzero allocation as a work area during optimization Specifically LINDO stores the basis inverse matrix along with the structural nonzeros in this shared memory location Given this you should try to allocate a good number of additional nonzeros over and above what is required by just the structural nonzeros If LINDO runs out of nonzero space it halts the optimization and returns with an error Iteration Limit The LINDO solver uses a form of the simplex method to perform optimization The simplex method performs a series of iterations also called pivots until an optimal solution is found At each iteration a favorable
66. copy of your model you can always create one by sending the output from the Reports Formulation command in Windows or the LOOK ALL command in command line versions to a file Running Command Scripts with the Take Command LINDO allows the user to construct an external text file containing a series of commands The commands in these files are run by using the File Take Commands command in Windows or the TAKE command in command line versions of LINDO 218 CHAPTER 7 In the section above one of the things we did was to build a model in an external editor and import it using the TAKE command This was actually your first view of a command script Normally however we will want to place additional commands in our script along with the model text As an example we will take the transportation model from the previous section and expand on it by adding commands to solve the model and send the nonzero variable values to a separate file The following is a script file that should do what we want PAGE 0 Turns off terminal paging MIN 2 Al 4 A2 3 A3 5 A4 4 Bl 3 B2 2 B3 6 B4 5 C1 6 C2 4 C3 3 C4 ST Al B1 Cl A2 B2 C2 A3 B3 4 C3 A4 B4 C4 Al A2 A3 A4 lt 12 Bl B2 B3 B4 lt 11 Cl C2 C38 C4 413 Demand for cust Demand for cust Demand for cust Demand for cust l l l ooN BwWNE Supply limit for A Supply limit for B Supply limit for C
67. cost of the variable type of variable free continuous integer and simple upper bound Sis a o The SDBC command may also be used when you intend later retrieval of the solution A solution saved using SDBC may be reloaded using the RDBC command For saving solutions to be restored at a later date the FBS and FPUN commands are generally more robust than SDBC SDBC does not store complete basis information so RDBC is not always able to precisely restore a solution SMPS Syntax SMPS lt FileName gt SMPS saves the model currently in memory to the path and filename that you give If you don t supply a path the file will be saved in the current working directory If you enter nothing after the SMPS command LINDO will prompt you for the name of the file The model is stored in MPS format which is a common format in industry originally developed for IBM s MPSX system MPS files are text files and may be ported to other platforms Being text files the curious user may wish to examine a sample MPS file We reproduce a small model below and its corresponding MPS file First here s the model in standard equation format MIN 100 XMON 100 XTUE 100 XWED 100 XTHU 100 XFRI 100 XSAT 100 XSUN SUBJECT TO SUN XWED XTHU XFRI XSAT XSUN gt 18 MON XMON XTHU XFRI XSAT XSUN gt 16 TUE XMON XTUE XFRI XSAT XSUN gt ae ED
68. cutting and pasting it into the password dialog box Cut the password from the e mail that contains it and paste it directly into the LINDO password dialog box with the Ctrl V shortcut 4 CHAPTER 1 LINDO users on platforms other than Windows should refer to the installation instructions that came with your copy of LINDO for platform specific instructions Now let s illustrate how to get started using LINDO by entering a small model Entering A Model in Windows This next section illustrates how to input and solve a small model in the Windows version of LINDO If you are running LINDO on a platform other than Windows please jump ahead to the next section Entering a Model from the Command Line When you start LINDO your screen should resemble the following 2 LINDO File Edit Solve Reports Window Help 3 lt untitled gt DER The outer window labeled LINDO is the main frame window All other windows will be contained within this window The main frame window also contains all the command menus and the command toolbar The smaller child window labeled lt untitled gt is a new blank Model Window We will type our sample model directly into this window For our sample model let us assume XYZ Corporation produces two models of computer Standard and Deluxe At the XYZ facilities the Standard can be produced at 10 computers per day while the Deluxe exceeds that at 12 computers per day Furthermore XYZ has a limited supply of
69. easy way to model this is to allow an inventory variable to go negative In which case you would like to circumvent the default lower bound of zero The SUB and SLB statements are used to alter the GETTING STARTED 19 bounds on a variable SLB stands for Simple Lower Bound and is used to set lower bounds Similarly SUB stands for Simple Upper Bound and is used to set upper bounds The following small examples illustrate the use of the SUB and SLB statements for Windows versions and command line versions of LINDO MAX 20X 30Y ST X 2Y lt 120 END SLB X 20 SUB X 50 SLB Y 40 SUB Y 70 LOOK ALL MAX 20K 30Y A MAX 20 X 30 Y ST E SUBJECT TO X 27 lt 120 2 X 2Y lt 120 END SLB X 20 00000 SUB X 50 00000 SLB Y 40 00000 SUB Y 70 00000 Windows Command Line Note that in the Windows version the SUB SLB statements appear as part of the model s text in the Model Window In command line versions of LINDO SUB and SLB are entered as actual system commands to the command level colon prompt after the model s objective and constraints have been entered In this example we could have just as easily used constraints to represent the bounds Specifically we could have entered our small model as follows max 20x 30y St 2y lt 120 20 50 40 70 lt x eX NVANV end Of course this formulation would yield the same results but there are two points to keep in mind First off SUBs and SL
70. first order condition constraint for each real variable Thus in an expanded quadratic model with the first order conditions present the total number of variables must equal the total number of constraints LINDO has detected that this is not the case Please check your model for mistakes and typographical errors OUT OF SPACE WHILE ADDING SLACK VARIABLES When LINDO solves a model it adds one slack or surplus variable to each inequality row to convert it internally to an equality row These additional variables count against the variable limit in your version of LINDO which you have exceeded Try running a larger version of LINDO If this is not possible you may be able to convert some inequality constraints to equalities without loss of generality In Windows versions of LINDO the Help About LINDO Command shows the limits on the version of LINDO you are using This also shows how to contact LINDO Systems Inc to order an upgrade to a larger version Command line versions of LINDO can use the HELP Command alone to get the same information Please refer to your manual for information on special versions of LINDO FILE COULD NOT BE OPENED LINDO was unable to open a file that you specified as part of a command Make sure you have spelled the filename correctly the path specified is correct and you have access to the file 31 32 33 34 35 36 37 38 39 40 ERROR MESSAGES 275 NOT ENOUGH WORKING MEMORY
71. full basis information For installing solutions into current models the FBS and FBR commands are more robust than SDBC and RDBC Because SDBC does not store full basis information RDBC is not always able to restore the exact solution as left by SDBC FBS stores complete basis information Note that retrieving solutions contained in basis files is of most benefit for linear programming models only In general retrieving a basis file cannot restore an optimal solution to an integer programming IP model You can use the SOLUTION command immediately after RDBC to observe the current solution and determine whether it has been properly installed COMMAND LINE COMMANDS 127 3 Display LOOK Syntax LOOK lt Row StartRow EndRow ALL gt The LOOK command displays part or all of the current model It may well be the most commonly used command in LINDO If LOOK is typed alone LINDO asks for a row specification Some responses might be 3 or 1 2 or ALL causing rows 3 through 2 or all the rows to be printed to the screen respectively With a three line model entering LOOK ALL or LOOK 1 3 displays the entire model I X1 4 X2 3 X5 SUBJECT TO SECOND X1 4 X2 3 X5 lt 11 3 X3 gt 2 END Entering LOOK 1 displays IN X1 4 X2 3 X5 Entering LOOK SECOND or LOOK 2 displays SECOND x1 4 X2 3 X5 lt 1 1 SOLUTION The SOLUTION command displays the current solution of the model in memory This includes A w
72. have broken LINDO s commands down into 11 categories These categories are presented below along with a brief description of the commands in the category and their functions Detailed examples of each of the commands follow 1 Information Command Description List categories of commands List commands by category HELP Give help on individual commands LOCAL Give information specific to your installation TIME Return cumulative time of current session 116 CHAPTER 3 2 Input Command Description Retrieve a basis saved with FBS command FINS Retrieve a basis saved with FPUN command 3 Display Command Description BPICTURE Display a picture of the basis ordering the rows according to the last inversion Obtain selected information about a user specified subset of a model s columns RPRI Obtain selected information about a user specified subset of a model s rows SHOCOLUMN Display a column variable of the model SOLUTION Display standard solution report TABLEAU Display current tableau 4 File output Command Description LEAVE SDBC Save a solution in column database format SMPS Save current model in MPS format COMMAND LINE COMMANDS 5 Solution Command Description GLEX Perform Lexico optimization Go o Go solve the model PIVOT Do a simplex pivot iteration 6 Problem editing Command Description ALTER Alter some element of the model APPC Append a new colum
73. identifies a set of constraints as well as column bounds that constitute a NECESSARY SET Such a set has the feature that it is infeasible However if any member of this set is deleted then the set becomes feasible Thus it is necessary to make at least one correction in the NECESSARY SET ROWS if the model is to be feasible Running the Debug command on the above model generates the following report SUFFICIENT SET ROWS CORRECT ONE OF 4 0 55 X Y gt 4 NECESSARY SET ROWS CORRECT ONE OF 2 X 2 Y lt 3 Notice row 2 which happens to be correct does not appear in the SUFFICIENT SET list of possibly faulty rows while the faulty one row 4 does appear So the Debug command can be useful in reducing the number of rows over which you need search for a bug The Debug command may be used in a similar fashion on unbounded models to determine sets of potentially faulty variables that need bounding 211 7 Interfacing with the Outside World Although LINDO has an extensive set of commands for interactive model development and solution there are instances where it may be more convenient or necessary to build models outside of LINDO This chapter explores three instances where we hook LINDO to external sources The first instance deals with editing model and solution files outside of LINDO LINDO s editing capabilities are somewhat limited and many users may find themselves longing for the
74. in depth 29 115 Word 2 31 211 X XYZ Corporation 4 9 Z Zoom tool 93
75. in its place 2 CHAPTER 1 A Note About This Manual The commands in this manual are separated into Windows and Command line sections Although they are presented separately the Windows version of LINDO does have a Command Window with all the command line commands available This manual is intended to be a comprehensive manual for both Windows and Command line users Please refer to the Table of Contents or Index for reference to sections on the two versions There are a few conventions used in this manual that although they may be the standard for many user s manuals it would be good to take note of First in all interactive examples of LINDO commands the characters to be entered by the user are in boldface lettering and the output by LINDO is interspersed throughout in regular type Second the Command line commands and all Windows commands not available in the menus are displayed in boldface capital letters The arguments to be entered by the user with these commands are in italics and surrounded by the lt and gt signs Finally when there is a choice between using one of two arguments for a command the arguments are shown as follows lt argument 1 argument2 gt with the character signifying the decision This manual also uses certain terms interchangeably to help the user understand how the LINDO internal solver transforms a linear programming model into a matrix the computer can comprehend Variables in a linear program
76. in memory The form of the command is EXTEND lt Rowl gt lt Row2 gt END where lt Row gt and lt Row2 gt are the constraints to be added After you enter the EXTEND command LINDO gives a question mark prompt Enter the new constraint rows as you normally would in a model Refer to those commands for additional syntax specifications However you cannot use MAX MIN ST SUCH THAT or SUBJECT TO in this context You may only enter new constraints Complete the command by entering the word END The EXTEND command changes the current model by adding to the end of the model the constraints which you enter The row numbers of the new rows follow the last row number already in the model The current solution is no longer considered optimal after EXTEND is used In this following sample session two new constraints are appended to the model look all there is our model MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 4 XE X 3 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END extend use EXTEND to add two more constraints x1 x2 x3 x4 lt 116 24x8 x9 gt 0 end 162 CHAPTER 3 look all look at model with new constraints appended to end MAX 9 X1 X2 4 X3 2
77. labor In particular there are a total of 16 units of labor Standard computers require one unit of labor while Deluxe computers are relatively more labor intense with a requirement of 2 units of labor A LINDO model has a minimum requirement of three things It needs an objective variables and constraints GETTING STARTED 5 The first requirement an objective is just what it sounds like a goal You have the choice of two goals MAX or MIN which stand for maximize and minimize In a typical business situation for instance you might want to maximize profit or minimize cost The first word in a LINDO model must be either MAX or MIN The formula you enter after the MAX or MIN is called the objective function In this example XYZ wants to maximize profit achievable with the limited labor and production facilities available We will let STD and DLX be our variables which are things you want LINDO to adjust to reach your maximum In this sample model STD has a profit contribution of 10 and DLX has a profit contribution of 15 The important thing is that they are variable In LINDO the minute you use a variable in your model it exists You don t have to do anything other than enter it in a formula You enter MAX 10 STD 15 DLX followed by a return Now let s look at constraints In some ways constraints are the most important part of your model and they require some real thinking In the little example we re considering if y
78. may not use the DELETE command on the objective row To edit the objective row use the ALTER command on row 1 END INVALID EXCEPT AFTER A COMPLETE ROW LINDO found an END statement before the completion of model input This indicates that a syntax error has occurred You should enter several carriage returns to retreat to command level examine the model with the LOOK command make sure END is on the last line by itself and enter the remainder of the model with the EXTEND command EXPECTING A RHS VALUE The LINDO model parser was expecting a right hand side value but encountered something else Chances are you tried to place a variable on the right hand side which is not allowed by LINDO All variables must be carried over to the left hand side of your constraints You should enter several carriage returns to retreat to command level examine the model with the LOOK command make sure all variables are on the left hand side of the constraints and enter the remainder of the model with the EXTEND command 15 16 17 18 19 20 21 22 23 24 ERROR MESSAGES 273 INVALID DATA SET NAME PLEASE REENTER You have input an invalid file name Please check the spelling and path specification and try again lt Not In Use gt lt Not In Use gt THERE IS NO CURRENT FORMULATION IN MEMORY You have issued a command that requires a formulation be in memory Please enter a new model with the MAX or MIN commands or
79. model with the constraint 6 13X 12Y 1 08Z gt 1 2 204 CHAPTER 5 we get the solution QP OPTIMUM FOUND AT STEP 6 OBJECTIVE FUNCTION VALUE 1 739300400 VARIABLE VALUE REDUCED COST X 420234 000000 Y 229572 000000 Z 350194 000000 UNITY 12 342430 000000 RETURN 11 517520 000000 XFRAC 000000 329766 Y FRAC 000000 520428 ZFRAC 000000 399806 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 420234 3 000000 229572 4 000000 350194 5 000000 12 342430 6 000000 11 517520 7 329766 000000 8 520428 000000 9 399806 000000 Notice that the dual price on row 6 is at a value intermediate between the prices at the two points that bracket 1 2 i e the dual price is not constant between breakpoints 6 Analyzing amp Debugging a Model Analyzing a model becomes difficult when the model is large For our purposes a model is large if neither it s formulation nor it s solution report fit on one printed page You may be interested in only parts of the model or parts of the solution report The difficulty should be apparent if these parts are scattered over several pages One of the major reasons for analyzing a model is the concern that it may be invalid A large model is like a large computer program it must be debugged Bugs in a model come in a variety of species A variable name may be spelled incorrectly the sign of a coefficient may b
80. model without problems However if LINDO seems to have a hard time solving the model you may want to attempt to scale your data INVALID VALUE DISREGARDED You have entered an invalid argument value as part of a command Please refer to the documentation or help file for this command and try again 68 69 70 71 72 73 74 75 76 ERROR MESSAGES 279 END ASSUMED TOO MANY ELEMENTS You have too many nonzero elements in an APPCOL command Please check your data and try again VARIABLE ALREADY EXISTS COMMAND DISREGARDED You have attempted to append a column with a name that already exists Please use a different name and try again BAD COLUMN INDEX lt N gt TO INSERT ROW You have attempted to insert a row with a bad column index Check your data and try again OUT OF SPACE IN INSERT ROW ADDING ROW lt l gt SPACE NEEDED lt J gt SPACE AVAILABLE lt K gt You have run out of nonzero space adding a constraint to LINDO Increasing the nonzero limit may help see message 72 However if you have already maximized the nonzero space you may need to reduce the size of the model or use a larger version of LINDO In Windows versions of LINDO the Help About LINDO Command shows the limits on the version of LINDO you are using This also shows how to contact LINDO Systems Inc to order an upgrade to a larger version Command line versions of LINDO can use the HELP Command alone to get the same informati
81. most large real world models less than one percent of the variables will appear in the average row or constraint Thus in any given row most variables have a coefficient of zero Storing all these zeros would be wasteful with respect to both performance and storage Therefore LINDO only stores the nonzero coefficient values When LINDO starts it sets aside a fixed amount of memory as space for the nonzero coefficients You can determine this limit by examining the Nonzero Limit field in the Options dialog box If you have a model with a particularly large amount of nonzeros you may want to increase the Nonzero Limit to allow LINDO to optimize the model On the other hand if you are trying to conserve memory you might want to decrease the Nonzero Limit Each nonzero you allocate requires eight bytes of storage This memory will be taken up regardless of how large or small your models are To change the nonzero limit enter a new value in the Nonzero Limit field then press the Save button At this point you will be presented with the message i You have changed the nonzero limit You will need to Save the current options and restart LINDO to establish this change LINDO cannot physically change the Nonzero Limit in midstream In order to accomplish this you will need to exit LINDO and restart Note When you restart LINDO you will lose any changes you have made since your last save It is recommended you save your work before
82. of the linear and integer solvers Windows DLL version of the callable library Model debugging capability for finding a minimal set of constraints variables leading to an infeasible unbounded model Additional callable routines in the programming library Lexico optimization feature for performing goal programming Improved basis save and retrieve functions Permuted nonzero picture for viewing models in as close as possible to lower triangular form MPSX compatible solution reports We hope you enjoy this new release of the LINDO software Please feel free to contact us at any time regarding questions or suggestions at LINDO Systems Inc 1415 N Dayton Chicago IL 60622 Tel 312 988 7422 Fax 312 988 9065 E mail info lindo com WWW http www lindo com vi PREFACE 1 Getting Started with LINDO What Is LINDO LINDO Linear INteractive and Discrete Optimizer is a convenient but powerful tool for solving linear integer and quadratic programming problems These problems occur in areas of business industry research and government Specific application areas where LINDO has proven to be of great use would include product distribution ingredient blending production and personnel scheduling inventory management The list could easily occupy the rest of this chapter The guiding design philosophy for LINDO has been that if a user wants to do something simple then there should not be a large setup cost
83. off the output ERS Retrieve the model from disk ETR MYFILE LPK Solve the model GO DBC MYFILE SDB Then save the solution to MYFILE SDB Beep G Last close the TAKE file and end the script staying in LINDO RAVE T R S I RDBC Syntax RDBC lt FileName gt The RDBC command reads a file from disk which was created with the SDBC command or is in SDBC format LINDO attempts to create a basis from the SDBC file There must already be a model in memory The form of the command is RDBC lt FileName gt where lt FileName gt is the name of the file you wish to retrieve If you try to retrieve a nonexistent file or one with an incorrect format LINDO gives an error message If you enter nothing after the RDBC command LINDO prompts you for the name of the file RDBC doesn t affect the current model in memory but it does change the current solution in memory If LINDO succeeds in creating the basis from the RDBC file the solution read from disk becomes the current solution However LINDO does not know whether the current solution is optimal Solution display commands should show the right solution but will also warn the solution may be infeasible or not optimal If LINDO fails to create the basis no message is given Usually a few pivots will be necessary either with GO or PIVOT to fully install the solution because the SDBC file does not contain
84. open the solution file printed by Rem LINDO Open C LINDO SHELL LNDOUT TXT J For Input As 1 Rem Break out of loop if successful If Err 0 Then Exit Do Rem Let other tasks run DoEvents Loop On Error GoTo 0 Keep in mind that the Shell command in VB runs applications asynchronously Therefore we must wait for LINDO to build the output file before we attempt to read it in the VB application The above loop iterates until we successfully open the LINDO output file At that point we drop out of the loop and retrieve the results from LINDO As a final note if you don t want LINDO to display the Status Window during the solution process you can disable it by running LINDO interactively and using the Edit Options command to disable the Status Window Be sure to save the options after making any changes LINDO will now be configured to suppress the Status Window during all subsequent runs For more details please see page 54 A C Front End for LINDO The following example uses Microsoft Visual C 4 0 and the command line version of LINDO to build and solve the following simple model Maximize 20 X 30 Y Su Es xX lt 50 Y lt 60 K 2 o lt 120 End INTERFACING 227 The approach used in this example is straightforward We have a small C program that builds a LINDO script file containing the model and various LINDO commands We then use the shell function in C to load the command line versi
85. powerful features of modern text editors for model development Secondly we will examine the use of command script files These are text files that contain a series of LINDO commands Finally developers of turnkey systems requiring optimization may need to access LINDO in an automated or batch mode We will examine these three areas in detail throughout the remainder of this chapter Using External Editors with LINDO If you are familiar with the general purpose text editors e g MS Notepad or word processors e g MS Word available on most computers you may find it efficient to use a text editor for preparing input or examining output The major commands in LINDO for facilitating this use of external editors are the File Open and File Log Output commands in Windows versions of LINDO and the TAKE and DIVERT commands in command line versions of LINDO Reading Models Created with External Editors As an example suppose we want to prepare the following transportation model in a text editor outside of LINDO MIN 2 Al 4 A2 3 A3 5 A4 4 Bl 3 B2 2 B3 6 B4 5 Cl 6 C2 4 C3 3 C4 ST Al BL C1 AQ A gt B2 4 C2 AS 4 BIs C3 A4 B4 C4 Demand for cust Demand for cust Demand for cust Demand for cust l iow ow DANIO PUNE Al A2 A3 A4 lt 12 Supply limit for A Bl B2 B3 B4 lt 11 Supply limit for B Cd G23 4 Caw lt 1 3 I Supply limit for C 212 CHAP
86. quality that the feasible region does not suffer a dramatic change if only a small change is made in a coefficient In mathematical terms a model should not contain a constraint that is almost linearly dependent upon other constraints For the above example constraint 4 or 3 or 2 should not have been in the formulation When Z 1 constraint 4 is either redundant or inconsistent with respect to 2 and 3 depending upon a very small change in a coefficient If you delete constraint 4 you will discover NUMERICAL CONSIDERATIONS 269 that the formulation is now robust and the solution is only slightly affected by small changes in the coefficients of Z in 2 and 3 The large dual prices 9998 34 in the first solution are a clue that the formulation contains linearly dependent but not quite consistent constraints Large dual prices indicate that a small change in a parameter may have a big effect on the results Numerical Aesthetics If numbers like 99999 instead of 1 in a solution report displease you then you may be interested in some of the causes of these un aesthetic round off errors One fundamental cause is that computers that use binary rather than decimal arithmetic are unable to represent most fractions with complete accuracy For example most computers cannot represent a perfectly innocent fraction like 1 with complete accuracy Thus if your input formulation had fractions then the formulation stored inside the compute
87. reading an MPS format basis file Perhaps this basis file does not correspond to the current model or the variable name is spelled incorrectly Please correct and retry THE FOLLOWING RECORD HAD AN INVALID STATUS lt CardText gt THE RECORD IS IGNORED LINES READ lt N gt LINDO did not recognize a record type in an MPS format basis file The types recognized by LINDO are XL XU LL UL or IV Please correct and retry NONEXISTENT ROW NAME ON THE FOLLOWING BASIS RECORD lt CardText gt LINES READ lt N gt LINDO encountered an invalid row name while reading an MPS format basis file Perhaps this basis file does not correspond to the current model or the row name is spelled incorrectly Please correct and retry 282 92 93 94 95 96 97 98 99 100 101 102 APPENDIX A THE MODEL MUST BE SOLVED FIRST You have attempted to print an MPS format solution report without first solving your model Please solve the model using the Solve command in Windows versions and the GO command in command line versions then try again THE ROW ON THE FOLLOWING LINE IS NOT BASIC lt CardText gt Tried to make a column basic in a row that already has a column assigned to it This is merely a warning If it is unexpected you should correct the problem and retry ROW lt RowName gt HAS WRONG BOUND OPPOSITE BOUND SET Tried to set a slack variable to an infinite bound THE COLUMN ON THE FOLLOWING LINE IS BA
88. representation of the model created from compiling the text in your Model Window Thus all comments and special formatting will be absent WINDOWS COMMANDS 99 To run the Formulation command you must select the Model Window you want to generate the report for Next select the Formulation command from the Reports menu and you will be presented with the following dialog box Formulation Report Options x gt Rows to view All C Selected Selected rows Beginning row fi Ending row m In the box labeled Rows to view you have the option of viewing all the rows in the model or selected rows only If you choose to view selected rows then the box labeled Selected rows becomes enabled and you can specify starting and ending rows to view The default LINDO opens up to is All the rows Let s illustrate the Formulation command using the following small model which we have entered into a Model Window lt untitled gt BEE This is a small LP Maximize 20 x 30 y T Here are the two simple bound constraints x lt 50 y lt 70 Here is the joint constraint x 2y lt 120 End Issuing the Formulation command for all rows of this model causes the following to appear in the Reports Window MAX 20 X 30 Y SUBJECT TO 100 CHAPTER 2 As mentioned this is LINDO s internal representation and not an exact replica of the model you typed into your Model Window As such
89. resembles LINDO Visual Basic Example MBR Now you will need to add handler code for the two buttons on the form The Quit button is relatively straightforward All the Quit button does when the user clicks on it is exit the application So double click on the Quit button and enter the following code Private Sub Quit _ Click End End Sub A little more work is required to handle the Solve button This button is programmed below When the user presses the Solve button the model passes to LINDO LINDO solves the model and the solution is then printed in the upper left corner of the form LINDO CALLABLE LIBRARIES 247 As a first step all the external LINDO functions must be declared The declarations for Visual Basic may be found in the file LINDO DLL32 MSVB LNDDECL TXT which is a standard text file readable into any text editor Read the declarations into the text editor of your choice copy them and then paste them into the General code section of the VB project When looking at the General code section of your project you should now see the LINDO routine declarations Project Microsoft Visual Basic design LINDO Code BEE Gl Fie Edit Yiew Project Format Debug Run Query Diagram Tools Add Ins Window Help la x B S SSE 2 O M om gt n YTA SRAA General Private Declare APPCOL Lib kB Private Declare BINVT Lib Private Declare CAPOUT Lib lndc A fabi Private Declare CHKPDF Lib lndc ea z Private Decla
90. returning the row indices of the nonzeros Be sure VAL and JRO have been declared to be of adequate length to hold the entire column SUBJ returns the simple upper bound on the variable TROUBLE returns 1 if J is not a valid variable index GO LIMGO ISTAT Integer LIMGO ISTAT Solves the current model On entry LIMGO contains a limit on the number of pivots i e solver iterations LIMGO 0 means let LINDO set its own sensible limit If LINDO reaches this limit it will interrupt the solution process and return to the calling application If you don t want LINDO to interrupt the solution process set LIMGO to a sufficiently large value STAT on return describes the solution status STAT 4 means optimal 5 means unbounded and 2 means infeasible APPCOL DEFROW etc may be called after GO to do on the fly column or row generation This routine corresponds to the GO command line command HEAPIT N M INDX N NVALU M Integer N M INDX NVALU Creates a heap of the items in the NVALU vector using the JVDX vector as a pointer Used in conjunction with the HEAPFR routine HEAPIT is useful for performing efficient sorts On entry V contains the number of items to be heaped and M contains the dimension of the NVALU vector On return JVDX has been adjusted to heap form i e NVALU INDX J lt NVALU INDX _J 2 for J 2 3 N LINDO CALLABLE LIBRARIES 235 HEAPFR N M INDX N NVALU M Integer N M INDX
91. searches for a 32 bit DLL called WATSUP DLL when it starts If this file is present LINDO will call the first routine in WATSUP DLL once per iteration of the simplex algorithm passing two arrays The first array which we will call JSTTUS contains two 32 bit integers The second array ENVIRN contains four 32 bit float values C and FORTRAN definitions of WATSUP are given below Cs void WATSUP long ISTTUS 4 float ENVIRN 4 FORTRAN SUBROUTINE WATSUP ISTTUS ENVIRN INTEGER 4 ISTTUS 4 REAL 4 ENVIRN 4 266 CHAPTER 8 The contents of the elements in these arrays are listed in the tables below ISTTUS Index Contents 1 1 if current solution is infeasible 2 if feasible but not LP optimal and 3 if solver is doing branch and bound Current iteration pivot number ENVIRN Index Contents Current sum of infeasibilities Current objective value thus far value 267 9 Numerical Considerations The size of model that can be conveniently solved depends mainly on the speed of the machine used memory limitations of the machine and how well the model is formulated On a standard personal computer with 8Mb of addressable memory the capability is for 8000 rows and 16000 columns On machines with less stringent memory limitations there are versions of LINDO that will accept problems with 64 000 rows and several 100 000 columns The user should scale rows and columns to avoid numerical di
92. solution in some sense Examples are cutting stock or staff scheduling applications where a number of demand requirements must be satisfied Traditional formulations of such problems frequently have multiple optima solutions some of which greatly over satisfy one or two requirements and just barely satisfy all others The user typically has a slight preference for solutions which spread out the over satisfaction more evenly A Goal Programming formulation would have a first objective of minimizing the cost of satisfying all requirements and perhaps a second objective of minimizing the maximum over satisfaction of any requirement The Lexico approach saves the user from having to worry about exact tradeoff rates between the cost of a solution and the value of over satisfaction 156 CHAPTER 3 In the following example we look at the GLEX command applied to a small staff scheduling model We have staffing requirements for each of the seven days of the week Employees work for 5 days with two days off each week M represents the number of employees starting on Monday T is the number starting on Tuesday and so on Typically you must accept some overstaffing in problems of this sort but you may want to spread the overstaffing across the week Note the clustered overstaffing in the solution look all MIN M T Wt R FH S 4N S
93. the branch and bound search Once the branch and bound enumeration process is completed the best solution found is reinstalled Thus report generating commands can be used to examine the best solution Note however that the reduced costs and dual prices resulting from an integer model are essentially meaningless to the casual user and therefore should be disregarded General integer variables are represented directly LINDO does not use the so called binary expansion representation of a general integer variable as a sum of 0 1 variables For most practical problems the binary expansion method increases solution times prohibitively 192 CHAPTER 4 The following is a small example of a valid IP model MIN X 3 Y 22 ST2 gt 2 20 2 Pe E ae e N Dir ofS Qe E END INTE INTE INTE ANA RX ER Y RZ Notice the INTEGER statements appear after the END statement These three INTEGER statements declare our three variables as being zero one If we solve this model LINDO displays the following report LP OPTIMUM FOUND AT STEP 3 OBJECTIVE VALUE 2 25714278 SET Y TO gt AT 1 BND 4 000 TWIN 3 000 12 EW INTEGER SOLUTION OF 4 00000000 AT BRANCH 1 PIVOT 12 BOUND ON OPTIMUM 3 000000 FLIP Y TO lt 0 AT 1 WITH BND 3 0000000 R Pp EW INTEGER SOLUTION OF 3 00000000 AT BRANCH 1 PIVOT 12 BOUND ON OPTIMUM 3 000000 DELETE Y AT LEVEL 1 ENUMERATION COMPLETE BRANCH
94. the View command When you read a model into LINDO using the Open command the model is placed into an Edit Window When you read a model with the View command the model is placed into a View Window You can determine when a model is in a View Window by checking the window s title bar If it is a View Window a v will appear after the file s name in the title bar 42 CHAPTER 2 The distinction between View Windows and Edit Windows is important when it comes to editing your files Edit Windows can only handle files with up to approximately 64 000 characters while View Windows can handle files as large as available memory allows Edit Windows offer an advantage over View Windows in that there are many more editing features available Thus many of the commands discussed in this section are only available in Edit Windows The table below lists the editing features supported by both classes of windows and should help in understanding the differences between them Editing Feature View Window Edit Window Capacity Limited only by 64K e eal Uno Te cw o S S i Coy S S i Base ooo o Clear o d S i Find Replace m GoToLine e BE Paste Symbol E Selecta S i O Ceara S Choose New Font m m If you need extensive editing capabilities for a large model we recommend that you use an external text editor to do your work The model must be saved as Text Only in the external editor Once this is done
95. the basis for some models i e all nonzero elements contained on or below the diagonal Models with a triangular basis tend to solve rapidly BPICTURE may be used with the TABLEAU and PIVOT commands to obtain additional insight into each step of the algorithm The following shows the optimal basis to the following model look all MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 XI X2 2 X3 X4 2 X5 X6 2 X7 3 X9 lt 12 3 XL 3X2 2 X3 3 X4 X52 XO XP XB lt 6 4 X1 X3 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END terse go LP OPTIMUM FOUND AT STEP 5 OBJECTIVE VALUE 15 0000000 bpic COLUMN ROW 18342567 X8 X2 4 X3 Sia XLS St XJi X6 zaps ART The term ART stands for artificial variable which are devices used to jump start the solver You may also see a column titled SLK n where n is an integer which stands for the slack variable in the row n a slack variable is added internally to each inequality constraint to convert it to an equality CPRI Syntax CPRI lt ColAfttList gt lt CondExp gt The CPRI command is a powerful command for obtaining selected information about a specific subset of a model s columns i e variables that satisfy a user specified condition You specify the conditions you want the subset of columns t
96. the cubic term in the objective and the quadratic terms in the first constraint Model 1 MAX X ST XEY oe BK LOE AY Y lt 10 7 6 KXXX NA END Model 2 3 AX X ST X Y 3 X 10 Y N y lt 10 KK XxX ae lt 6 END To use LINDO to solve quadratic programs you must convert models to an equivalent linear form This is accomplished by writing a LINDO model with the first order conditions also called the Karush Kuhn Tucker LaGrange conditions as the first set of rows and the Ax lt b constraints which we refer to as the real constraints as the second set of rows The first order conditions are the optimality conditions which must be satisfied by a solution to a quadratic model It turns out that the first order conditions to a quadratic model are all linear LINDO exploits this fact to use it s linear solver to solve what is effectively a nonlinear model If you are uncertain as to exactly what first order conditions are and how they are used you can refer to any introductory calculus text 198 CHAPTER 5 The QCP statement identifies the end of the first order condition constraints and the beginning of the real constraints In Windows versions of LINDO the QCP statement appears as part of the model text after the END statement In command line versions of LINDO the QCP statement is entered as a command to the command level colon prompt after the model s END statement In con
97. the model may be read into LINDO using the View command Undo Ctrl Z The Undo command is used to reverse the last action taken in an Edit Window The Undo command is not available in View Windows For a discussion on the distinction between Edit and View Windows please refer to the Open and View commands in the File menu section above Cut Ctrl X The Cut command is used to remove selected text from an Edit Window and place it into the Windows clipboard for future pasting at another point in the window other Edit windows in LINDO or other applications To select text for cutting simply press down on the left mouse button at the beginning of the text and then drag the mouse pointer to the end of the text and release the mouse button The selected text should now be displayed in reverse video white letters black background Selecting the Cut command at this point will cause the selected text to be removed from the window and placed into the Windows clipboard For pasting this text refer to the Paste Command below Copy Ctrl C The Copy command copies selected text from an Edit Window into the system clipboard The text is not deleted from the Edit Window when it is copied Once there it may be pasted into other applications or other Edit Windows in LINDO To select text for copying simply press down on the WINDOWS COMMANDS 43 left mouse button at the beginning of the text and then drag the mouse pointer to the end of the text and r
98. these two arguments ACTUAL and BOUND Just before calling NEWIP LINDO sets ACTUAL to the cost of the current solution as perceived by LINDO and BOUND to the bound on the best possible IP solution NEWIP lets you reset the argument ACTUAL but not BOUND LINDO CALLABLE LIBRARIES 265 You can use the NEWIP interface in a number of ways For example the C version of NEWIP shown below makes LINDO search for all integer solutions greater than 1000 and displays each such solution Note that this NEWIP will cause longer solution times than the usual void NEWIP float fActual float fBound Print this solution int nOne 1 OUTSPC amp nOne Now claim that this solution was not very good fActual float 1000 return The following version of NEWIP written in FORTRAN causes LINDO to consider future branches in the search tree only if they might lead to solutions at least 15 better than the incumbent solution This search process will tend to be a little shorter than the standard search procedure Maximization is presumed SUBROUTINE NEWIP ACTUAL BOUND C PRINT OUT THIS SOLUTION CALL OUTSPC 1 C NOW CLAIM THIS SOLUTION WAS 15 BETTER THAN IT REALLY IS ACTUAL ACTUAL 15 ABS ACTUAL RETURN END Monitoring the Solver At each iteration of the simplex algorithm LINDO can call a user supplied callback routine with information on the status of the current solution The LINDO DLL
99. these variables to the integer values obtained in the relaxation and optimize over a restricted branch and bound tree The underlying logic in doing this is that integer valued variables with a high reduced cost would incur a large expense if they were to move away from their bounds Given this they have a relatively high probability of remaining at their current values in the optimal integer solution By initially fixing these variables and solving the restricted tree the hope is to obtain good objective bounds early on allowing large portions of the full tree to be fathomed once we free up the fixed integer variables To exploit this feature input a numeric value in the IP Variable Fixing Tolerance box in the Options window When solving IPs LINDO will initially fix all integer variables with reduced costs exceeding this value The default IP Variable Fixing Tolerance value is None In other words the feature is disabled WINDOWS COMMANDS 51 GENERAL OPTIMIZER Options Options included here influence the optimizer s operation on all classes of models and include the following Nonzero Limit Iteration Limit Initial Constraint Tolerance Final Constraint Tolerance Entering Variable Tolerance Pivot Size Tolerance Nonzero Limit When we speak of nonzeros we are referring to the coefficients on the variables in the constraint equations It would be unusual for all variables in a model to appear in a single constraint In fact in
100. to learn the necessary features of LINDO If for example a user wishes to Maximize 2X 3Y Subject to 4X 3Y lt 10 Sx sho Wh 2 then that is exactly what the user types into LINDO immediately after starting the program At the other extreme LINDO has been used to solve real industrial linear quadratic and integer programs of respectable size For commercial applications LINDO is frequently used to solve problems with tens of thousands of constraints and hundreds of thousands of variables There are three basic styles of using the LINDO software For small to medium sized problems LINDO is simple to use interactively from the keyboard Entering a model is quite easy to do It s also possible to use LINDO with externally created files which contain scripts of commands and input data to produce files for reporting purposes Finally custom created subroutines may be linked directly with LINDO to form an integrated program containing both your code and the LINDO optimization libraries The LINDO software is designed to be simple to learn and use This is particularly true for small problems In the remainder of this chapter we ll look at the basic commands and syntax necessary to enter and solve a small problem Then take a quick look at one larger real world problem 1 The strict inequality lt is interpreted to mean the loose inequality lt Most keyboards do not have the latter so LINDO allows the former to be used
101. to specify a bound on the worst objective value of an integer program This bound is usually based on a known feasible solution The bound is used in the branch and bound algorithm to narrow the search for the optimum When LINDO is searching for an initial integer solution it can ignore branches with objective values worse than your BIP value Depending on the problem a good BIP value can greatly speed solution times If you enter BIP without lt ObjectiveValue gt LINDO will prompt you for the number For a maximization problem the default value for BIP is negative infinity For a minimization problem it is positive infinity If BIP is set too large for a maximization problem or too small for a minimization problem LINDO will report that no integer solution could be found The BIP specification is not saved with any files created by LINDO The BIP command should usually reduce the amount of time required to solve the model However this is not guaranteed as seen in the example below 178 CHAPTER 3 In the following example an integer program with optimal value 15 is first solved with the default setting of BIP LINDO finds four integer solutions requiring 30 branches and 88 pivots When BIP is set to 10 the optimal solution is found in 43 branches and 110 pivots When BIP is set to 14 5 the optimal solution is found in only 13 branches and 41 pivots Thus a bad bound may not be helpful while a good bound can be very helpful
102. triangular basis tend to solve faster than non triangular ones In this particular example the optimal basis matrix is mostly triangular with only seven columns protruding beyond the diagonal B2T1 BIT1 C2T1 T1T2 BIT2 T2T2 and P1T2 There may not always be a full complement of variables in a basis In which case you will see variables in the Basis Picture report titled ART This stands for an artificial variable which LINDO uses as a placeholder when it has not yet assigned a variable to a row The Basis Picture report complies with the Terminal Width parameter which may be reset using the Edit Options command Thus if the width of the picture exceeds the Terminal Width parameter the report will be continued on additional pages You may want to increase the Terminal Width to maximize the amount of output per page in the report 96 CHAPTER 2 Tableau Alt 7 The Tableau command shows the current simplex tableau It is a useful way to observe the simplex algorithm at each step especially when used in conjunction with the Pivot command The Tableau command displays the variables across the top the row numbers in the first column on the left the basic variable in each row in the second column the current coefficients for each variable and the current right hand side on the far right The Tableau command may be combined with the Pivot command to monitor the internal workings of LINDO s solver In the following example we alterna
103. values of the columns for which these values are not zero e dual prices of the rows for which these values are not zero e number of iterations pivots required to find a solution e number of branches in the branch and bound algorithm for integer programming and e the determinant of the basis matrix for an integer program integer programming determinants close to 1 are preferred because if the determinant is 1 or 1 and the right hand sides are integer the solution will be naturally integer The NONZEROS command does not solve the model It only displays the current solution Use GO to solve the model The NONZEROS command is often used when the TERSE command has been used since TERSE prevents automatic output of the solution with GO For large models you may wish to see only the nonzero primal values Using TERSE GO and NONZEROS eliminates a lot of uninteresting output in this case NONZEROS may be used with DIVERT to print the solution or save the solution to a file See the DIVERT command for more information on this 138 CHAPTER 3 BPICTURE The BPICTURE command displays a picture of the current basis ordering the rows according to the last inversion or triangularization If a model has just been retrieved the GO PIVOT or INVERT command must be used to obtain the first inversion Plus symbols are displayed for positive values and minus symbols are displayed for negative values LINDO can triangularize
104. variable lt VariableName gt as being 0 1 The second form identifies the first N variables in the current formulation as being 0 1 The order of the variables is determined by their order encountered in the model This order can be verified by observing the order of variables in the solution report The second form of this command is more powerful because it allows the user to identify several integer variables with one line but it requires the user to be aware of the exact order of the variables This may be confusing if not all variables appear in the objective If there are several variables but they are scattered throughout the internal representation of the model the variables would have to be identified as integers on separate lines General integer variables are identified with the GIN statement The GIN command is used exactly as the INT command For example GIN 4 makes the first 4 variables general integer GIN TONIC makes the variable TONIC a general integer variable The solution method used on an integer program IP is the branch and bound search process It will typically find a sequence of better and better solutions If LINDO is in the default verbose output mode then a log of the search process will be displayed If you have placed LINDO into terse output mode from either the Options command in Windows versions or the TERSE command in command line versions then only a short message is printed as each better solution is found in
105. windows If you choose a text based report then you also have the options in the Text Report Format box of suppressing column headers and comma delimiting the data If you select a graphics report you can specify one of any of the following in the Graph Type box area bar line pie and point You can also opt in the Graph Style box to display graphs in either two or three dimensions Finally in graphics reports you can optionally specify a graph title in the Graph Title Optional box Once you ve sorted through these various options press the OK button to generate the report As an example consider the following staff scheduling model XS C LINDO61 SAMPLES STAFF2 LTX MIN XM XT XW XR XF XS EN SUBJECT TO MON TUE WED THU FRI SAT SUN 84 CHAPTER 2 In this model X represents the number of employees that start on day i of the week Each employee works for five consecutive days with two days off When we solve this model we get the following solution report I Reports Window MER A LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE t3 7 000000 VARIABLE VALUE REDUCED COST XH 3 000000 000000 1 000000 000000 2 000000 000000 0 000000 000000 0 1 0 000000 000000 000000 000000 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES HON 1 000000 000000 TUE 000000 000000 WED 000000 000000 THU 000000 000000 FRI 000000 000000 SAT 0
106. you will notice how all comments have been removed and the text has been reformatted Of course even though LINDO s internal representation appears different from the model you input the two formulations are mathematically equivalent The Formulation report complies with the Terminal Width parameter Thus if the width of a constraint exceeds the Terminal Width parameter the constraint will be continued onto additional lines You may want to increase the Terminal Width using the Edit Options command to increase the amount of output per line in the Formulation report Show Column Alt 9 When you are dealing with a large model it can be useful to be able to display a selected row or column without sifting through the entire formulation The Formulation command allows you to view a single row while the Show Column command allows you to view the details of a single column To run the Show Column command you must first select the Model Window of interest by clicking it once with the mouse Next issue the Show Column command and select the variable you want to generate the report for Press the OK button and the Show Column report is sent to the Reports Window As an example suppose you have solved the following small transportation model 25 C LINDO61 Samples TRAN LTX 3 warehouse 4 customer transportation model XWH lt i gt C lt j gt amount shipped from warehouse lt i gt to customer lt j gt MIN 6 XWH1C1 2 XWH1C2 6
107. 0 70 3 COMP2 lt 4 COMP1 2 COMP2 lt 120 WINDOWS COMMANDS 75 and re solving we get J Reports Window LP OPTIMUM FOUND AT STEP 0 OBJECTIVE FUNCTION VALUE T 1806 000 VARIABLE VALUE REDUCED COST COMP1 60 000000 0 000000 COMP2 30 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 0 000000 0 100000 40 000000 0 000000 0 000000 15 000000 NO ITERATIONS 0 As predicted by the range report the optimal variable values are unchanged Now let s push COMP1 s objective coefficient just past the allowable decrease to 14 9 so we have 43 C LINDO61 SAMPLES RANGE LTX DEAR MAX 14 9 COMP1 30 COMP2 SUBJECT TO COMP1 lt 60 COMP2 lt 70 2 3 4 COMP1 2 COMP2 lt 120 Re solving we get the new solution J Reports Window LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1 1800 000 VARIABLE VALUE REDUCED COST COMP1 0 000000 0 100000 COMP2 60 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 60 000000 0 000000 10 000000 0 000000 0 000000 15 000000 NO ITERATIONS 1 76 CHAPTER 2 Note that the variable values have now changed with COMP falling out of the solution Before moving on we should mention that the allowable ranges in range reports are minimal values You can always alter a single right hand side or objective coefficient up to the allowable range without affecting the variable values However you may need to go significantly past the allowable range before you ac
108. 0 terse Suppress solution report go Solve it LP OPTIMUM FOUND AT STEP 741 OBJECTIVE VALUE YD00 TLSE fpun etamacro pun Save the basis retr etamacro lpk Load the problem again fins etamacro pun Load the basis we just saved terse Suppress solution report go Solve again LP OPTIMUM FOUND AT STEP 0 OBJECTIVE VALUE TOD flo 27 a Note how starting with an optimal basis cuts steps to 0 Basis files created with the FPUN command are in text format and the curious user may examine them with a text editor if desired SDBC Syntax SDBC lt FileName gt The SDBC command saves the solution currently in memory to a file using the path and filename that you give If you don t supply a path the file will be saved in the current working directory If you enter nothing after the SDBC command LINDO will prompt you for the name of the file COMMAND LINE COMMANDS 147 The file saved to disk is in a convenient form for perusing the solution entering into other software etc SDBC files are text files and may be read into any text editor for examination As an example a small model is presented below along with its solution and corresponding SDBC file Here s the model formulation look all MIN 100 XMON 100 XTUE 100 XWED 100 100 XFRI
109. 0 X2 13 000000 000000 X3 3 000000 000000 X4 000000 3 000000 x5 4 000000 000000 X6 2 000000 000000 X7 000000 2 000000 X8 11 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 4 000000 3 000000 1 000000 4 000000 1 000000 5 000000 1 000000 6 000000 1 000000 7 000000 1 000000 8 000000 000000 cpri This simply counts the number of columns NUMBER OF MATCHES 8 cpri NAME X1 X2 X3 X4 X5 X6 X7 X8 COMMAND LINE COMMANDS n Here we list the names of all columns List the names duals i e reduced costs and value of all columns cpri NAME DUA X1 X2 X3 X4 X9 X6 X7 X8 nd p PRIMAL 16 13 ODNODWOCOWCCAOE FON BOW 1 display name and value for all variables with a value greater than 5 cpri np p gt 5 NAME PRIMAL X1 16 X2 13 X8 TI display n and primal value for all columns that have a value greater than 5 and less than 15 cpri NAMI X2 X8 np p gt 5 and p lt 15 E PRIMAL T3 47 Display name primal and dual for all columns with a nonzero dual cpri NAMI X4 X7 ndp d 0 E PRIMAL DUAL 0 3 0 2 display the name primal and dual for all colums with a 4 as the second character in its name cpri NAMI x4 n d p n 4 E DUAL PRIMAL 3 0 141 142 CHAPTER 3 RPRI Syntax RPRI lt RowAttList gt lt CondExp gt The RPRI command is a powerful command for obtain
110. 0 T2T2 lt 70000 LET Oe Lets 1000 I1T2 I2T2 lt 1000 ITITI E1T1 6 BiTi T2TL E2T1 I1T2 E1T2 I2T2 E2T2 PITI N1T1 Pitz N1T1 P2T1 N2T1 P2T2 N2T1 C1T1 RETI C1T2 C2T1 C2T1 R2T1 C1T1 C2T2 130 190 nnn wt S 0 0 6 BIT2 T1T2 4 NNHEHHHHHHHH HH H O o o a D A ea W N H O D O a oD AN a o Se aa a r ar ar ee ee a r Elite arm iG wae al lc cal bag i e tibet tae t utun BeT2 T2T2 We will choose to not have LINDO permute the rows and columns into lower triangular form and we will generate both text and graphics reports So we issue the Picture command and fill in the dialog box as follows Nonzero Picture r Row Var Permutation None Lower Triangular m Picture Type C Graph Text Graph and Text 91 WINDOWS COMMANDS After we press the OK button our text and graphics reports are generated and will appear as follows as z a a 2 S a a Le dpm R asutveyevvvyv ooo won wo wy Wow Hm HANM TN DMN cM Text Nonzero Picture Report E te 3 3 te a 3 E E a Graphics Nonzero Picture Report 92 CHAPTER 2 In the text report variable names are printed vertically along the top row names appear on the left and right hand side values appear on the right along with the direction of the row Nonzero matrix coefficients appear in their appropriate positions in the matrix Note that coefficients which are
111. 00 iterations Then re solving the model using an IPTOL value of 1 finds the solution in only 74 iterations The true objective value of 8 929 390 is a mere 0 002 better than the objective of 8 929 170 found using an IPTOL value retrieve testipt read in our test model stats print the model s statistics ROWS 21 VARS 34 INTEGER VARS 34 0 0 1 QCP 0 NONZEROS 102 CONSTRAINT NONZ 64 52 1 DENSITY 0 139 SMALLEST AND LARGEST ELEMENTS IN ABSOLUTE VALUE 1 80000 gt 6 14 0 GUBS lt 9 VUBS gt 8 OBJ MAX NO lt SINGLE COLS 0 REDUNDANT COLS 2 terse suppress some output go 100000 solve the model give it lots of iterations LP OPTIMUM FOUND AT STEP 18 OBJECTIVE VALUE 8942181 00 EW INTEGER SOLUTION OF 8929170 AT BRANCH 7 PIVOT 74 BOUND ON OPTIMUM 8936519 NEW INTEGER SOLUTION OF 8929230 AT BRANCH 98 PIVOT 630 BOUND ON OPTIMUM 8936519 EW INTEGER SOLUTION OF 8929390 AT BRANCH 766 PIVOT 4484 BOUND ON OPTIMUM 8936519 ENUMERATION COMPLETE BRANCHES 1715 PIVOTS 10064 LAST INTEGER SOLUTION IS THE BEST FOUND REINSTALLING BEST SOLUTION 180 CHAPTER 3 We will now re solve but use an IPTOL this time retr testipt iptol 01
112. 000 4 210 000000 INFINITY 50 000000 Now illustrate the PARAMETRICS command on row 4 para ROW 4 NEW RHS VAL 1 VAR VAR PIVOT RHS DUAL PRICE OBJ OUT IN ROW VAL BEFORE PIVOT VAL 210 000 0 000000 2700 00 SLK 4 SIK 3 4 160 000 0 000000 2700 00 COSMO SLK 2 3 60 0000 15 0000 1200 00 ASTRO ART 2 0 000000 20 0000 0 000000 1 00000 INFINITY INFEASIBLE Ostensibly the PARAMETRICS capability allows you to investigate the impact of changes to the RHS of at most one constraint at a time Realize however that a simple device allows you to change several right hand sides simultaneously in a linear fashion Let the change column be called D It may have coefficients in any or all rows Add a constraint D 0 and then do parametric analysis on the RHS of this constraint COMMAND LINE COMMANDS 175 POSD The POSD command is used with quadratic programs to check whether you have a guarantee of global optimality POSD examines the submatrix of constraints corresponding to the quadratic form to determine whether this submatrix is POSitive Definite If the matrix is positive definite then the objective function of the quadratic program is convex and the solution found is guaranteed to be a global optimal solution By specifying which rows are the first order conditions of the quadratic program with the QCP command you also specify the submatrix corresponding to the quadratic form If you specify QCP k where k is an integer then the POSD comman
113. 00000 000000 SUN 000000 000000 NO ITERATIONS 4 lt iil WINDOWS COMMANDS 85 Let s create a text report and a graphics report to display the column values For the graphics format we ll use a 3D bar chart The Peruse command dialog box should be filled in so it resembles the following Peruse Model m Report Parameters mA Orientation Columns C Rows mB View Items MV Name V Primal Dual Rim Upper Bound Lower Bound I Type I Nonzeros C Condition optional _ Help Cancel r Report Format mD Report Types IV Text IV Graphics mE Text Report Format IV Include Headers J Comma Delimited r Graphics Format F GraphT ype Point C Area Bar 2D Line Pie G Graph Style 3D H Graph Title optional Column Values The text report is sent to the Reports Window and appears as follows I gt Reports Window Bec PRIMAL 86 CHAPTER 2 The graphics report is sent to an individual window and is shown below Column Values H Primal Value XM XT Xw XR XF XS XN Column As we mentioned above you can input a condition as part of the Peruse command to filter out unwanted columns and rows This feature is particularly useful when perusing large models where unconditional reports wou
114. 08 2S 1 12 Max fraction of X multiplier is XFRAC XS A015 Max fraction of Y multiplier is YFRAC YS E Max fraction of Z multiplier is ZFRAC ZS ED END QCP 5 Now suppose we wish to trace out the objective value as we increase the RHS of the growth constraint constraint number 6 1 3 X 1 2 Y 1 08 Z gt 1 12 QUADRATIC PROGRAMMING 203 The highest possible value we could hope to achieve is 1 3 which corresponds to putting all weight on variable X To perform the parametric analysis on this constraint run the Reports Parametrics command in Windows versions of LINDO or the PARAMETRICS command in command line versions You should receive the following report RIGHTHANDSIDE PARAMETRICS REPORT FOR ROW 6 VAR VAR PIVOT RHS DUAL PRICE OBJ OUT IN ROW VAL BEFORE PIVOT VAL 1 12000 0 Q000000E 00 0 417375 SLK 6 RETURN 6 1 14410 0 000000E 00 0 417375 SLK 7 FRAC 7 1 26947 25 8299 2 03651 Z ART 0 1 27500 28 7500 2 18750 1 30000 INFINITY INFEASIBLE One line is produced in the report for each value of the RHS at which some variable hits a bound The parametric analysis indicates that constraint 6 is not binding for any RHS value below 1 441 Above this value the return constraint becomes binding At a RHS value of 1 26947 the constraint X lt 75 becomes binding The objective value at this RHS value has increased to 2 03651 The dual price on row 6 at this va
115. 1 range analysis 7 27 sensitivity analysis 11 APPCOL 117 162 63 249 256 Append output option 34 Applications integrating with 219 28 AppWizard 251 Arguments 2 231 character 231 double 231 float 231 integers 231 passing by reference 232 264 Arrange Icons command 28 109 ART artificial variable 95 96 135 138 AUTOLD DAT 125 181 219 225 Automatic Take Command 125 Automating files 211 219 AutoUpdate 112 B Backordering 18 Basic 231 264 291 INDEX Basis 94 183 creating 126 determinant 127 erasing 61 in memory 38 inverting 52 118 matrix 95 127 optimal 94 range report 130 retrieving 38 116 121 123 124 126 241 saving 36 116 145 triangularizing 95 138 183 Basis Picture command 27 94 95 Basis Read command 26 37 38 Basis Save command 26 36 37 BATCH 117 124 181 219 Batch mode 211 Benders decomposition 194 Binary arithmetic 269 Binary expansion 191 Binary integers 15 17 117 167 191 206 Binding constraints 203 267 BIP 117 177 Boldface lettering 2 BPICTURE 116 135 138 Branch and bound 45 73 127 169 172 191 192 93 264 Breakpoints 79 203 Brearley A L 176 BUG 118 183 Building an application 246 251 259 C C C 231 264 Callable libraries 219 231 66 application requirements 244 routines 232 44 samples 245 Capitalization 14 264 CAPOUT 249 255 Cascade command 28 105 7 CAT 115 119 cd
116. 1 0 idrow trouble c Now call DEFROW to define constraint rows do i 1 3 call defrow 1 rhs i idrow trouble enddo c Call APPCOL to define columns Note that all character c data is passed using double indirection Define X kname X call appcol loc kname 3 valx irowx trouble c Define Y kname y call appcol loc kname 3 valy irowy trouble c Display the formulation call look 1 4 c Call GO to solve the model call go 1000 istat c Get the objective valu call reprow 1 obj dual c Get the value of x call repvar 1 x dual c Get the value of y call repvar 2 y dual c Report values write 6 10 obj x y 10 format Objective f8 1 X f8 1 Y 8 1 pause Press lt Enter gt to exit c Shut down LINDO by calling LSEXIT call lsexit c All Done end LINDO CALLABLE LIBRARIES 261 When calling LINDO from Powerstation you must include the definitions of the LINDO routines so the compiler formats the calls to the LINDO library correctly The following code accomplishes this c Inspsint h contains the definitions for all the LINDO c callable routines include lIndpsint h Note you can find the definitions file LNDPSINT H in the directory LINDO DLL32 MSFORT LINDO is initialized with calls to LINDO and INIT call ilindo call init ILINDO performs one time initialization for LINDO It should be called once at the start of
117. 100 XSAT 100 XSUN SUBJECT TO SUN XWED XTHU XFRI XSAT XSUN MON XMON XTHU XFRI XSAT XSUN TUE XMON XTUE XFRI XSAT XSUN WED XMON XTUE XWED XSAT XSUN THU XMON XTUE XWED XTHU XSUN FRI XMON XTUE XWED XTHU XFRI SAT XTUE XWED XTHU XFRI XSAT END Here s the standard solution report GO OBJECTIVE FUNCTION VALUE 1 2200 000 VARIABLE VALUE REDUCED COST XMO 2 000000 0 000000 XTUE 2 000000 0 000000 XWED 4 000000 0 000000 XTHU 3 000000 0 000000 XFRI 3 000000 0 000000 XSAT 0 000000 0 000000 XSUN 8 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES SUN 0 000000 20 000000 MON 0 000000 20 000000 TUE 0 000000 20 000000 WED 0 000000 20 000000 THU 0 000000 20 000000 FRI 0 000000 20 000000 SAT 0 000000 20 000000 NO ITERATIONS 0 SDBC EXAMPLE 1tx XTHU 18 16 15 16 14 12 148 CHAPTER 3 And finally the SDBC file RDBC EXAMPLE 1tx SDBC 2200 0000 1 0000000 F 0 10000000E 31 XMO 2 0000000 0 00000000E 00C 0 10000000E 31 XTUE 2 0000000 0 00000000E 00C 0 10000000E 31 XWED 4 0000000 0 00000000E 00C 0 10000000E 31 XTHU 3 0000000 0 00000000E 00C 0 10000000E 31 XFRI 3 0000000 0 00000000E 00C 0 10000000E 31 XSAT 0 00000000E 00 0 00000000E 00C 0 10000000E 31 XSU 8 0000000 0 00000000E 00C 0 10000000E 31 The columns in the SDBC file stores the following variable name primal value of the variable reduced
118. 10000 099999 Single digit integers are shown explicitly rather than being displayed as a code This is especially handy because many models have a large number of coefficients of positive or negative 1 which can affect the solution procedure Negative signs are also displayed The following example shows a small model and its PICTURE report look all MI X1 4 X2 3 X5 SUBJECT TO SECOND X1 100 X2 3 X5 lt 11 3 X3 gt 2 END picture xX X X X 12 5 3 1 143 MIN SECOND 1 B3 lt B Si Se Note the right hand side of the SECOND row is 11 and is represented as B from the table above The PIC command maintains row names Variables are listed across the top The sense of the objective function and the sense of each row are also shown Spaces stand in for zeroes and single quote marks are inserted to give a grid like background on large PICTUREs SHOCOLUMN Syntax SHOCOLUMN lt Variable gt The SHOCOLUMN command shows all information pertaining to a variable its internal index its nonzero coefficients including a nonzero coefficient in the objective function bounds on the variable if they exist and whether the variable is integer 134 CHAPTER 3 If an optimal solution to the model is currently in memory SHOCOLUMN displays the row numbers variable coefficients and dual prices for each row in which the variable has a nonzero The primal value and reduced cost of the variable are also shown on the bott
119. 1indmsvcDlg h header file if defined AFX_LNDMSVCDLG H_ __INCLUDED_ define AFX_LNDMSVCDLG_H_ __ INCLUDED if MSC_VER gt 1000 pragma once endif MSC VER gt 1000 include lindmsvc h LILLLLTTITITLTTTAA TTT AAT AAA AAA ATTA ATT CLndmsvcDlg dialog class CLndmsvcDlg public CDialog Construction public CLndmsvcDlg CWnd pParent NULL standard constructor 254 CHAPTER 8 The next task is to attach a handler routine to the Solve button to pass the model to LINDO and solve it Use the Class Wizard to map the BN_ CLICKED message from the Solve button to a handler routine called OnSolve Once you have the stub handler in place edit it and add the code listed below void CLndmsvcD1lg OnSolve fi Led ae When the user presses the Solve button we use the 32 bit LINDO DLL LNDDLL32 DLL to generate and solve the following small LP Max 20 X 30 Y Sok xX lt 50 Y lt 60 X 2Y lt 120 End Results are posted to the dialog box Initialize LINDO by calling ILINDO and INIT LINDO NIT T I Capture any standard output in a file F e irst open a file har cKfname LINDO OUT long lUnit 60 lFname 9 lInOrOut 0 1NotFmt 0 lLutrmi 0 lLutrmo 0 LUNOPN amp lUnit amp lFname amp cKfname amp lInOrOut amp lNo
120. 21 OF ih gt Demand constraints Oo EL XWH1C1 XWH2C1 XWH3Cl1 gt LS o G2 XWH1C2 XWH2C2 XWH3C2 gt 17 C3 XWH1C3 XWH2C3 XWH3C3 gt 22 C4 XWH1C4 XWH2C4 XWH3C4 gt 12 END terse go LP OPTIMUM FOUND AT STEP 6 OBJECTIVE VALUE 161 000000 save the model save tranl save the basis in FBS form fbs tranl fbs save the basis in FPUN form fpun tranl fpu retrieve the formulation retr tranl add a new customer ext BEGIN EXTEND WITH ROW 9 c5 xwhlc5 xwh2c5 xwh3c5 gt 2 end look c5 C5 XWH1C5 XWH2C5 XWH3C5 gt 2 END alt 1 xwhic5 VARIABLE NOT IN THIS ROW WANT IT INCLUDED Y NEW COEFFICIENT 2 alt 1 xwh2c5 VARIABLE NOT IN THIS ROW WANT IT INCLUDED Y NEW COEFFICIENT 6 COMMAND LINE COMMANDS 123 alt 1 xwh3c5 VARIABLE NOT IN THIS ROW WANT IT INCLUDED Yy NEW COEFFICIENT 5 look 1 IN 6 XWH1C1 2 XWH1C2 6 XWH1C3 7 XWH1C4 4 XWH2C1 9 XWH2C2 5 XWH2C3 3 XWH2C4 8 XWH3C1 8 XWH3C2 XWH3C3 5 XWH3C4 2 XWH1C5 6 XWH2C5 5 XWH3C5 save the modified model save tran2 see how many pivots it takes without warm start ters go LP OPTIMUM FOUND AT STEP 8 OBJECTIVE VALUE 165 000000 ot see how many pivots it takes using FBR warm start retr tran2 fbr tranl fbs ters go LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE
121. 3 a simple lower bound of 1 with the SLB command and simple upper bound of 20 with the SUB command and to be general integer with the GIN command The result is the model below MAX 4 X3 9 X1 X2 2 X4 8 X5 2 X6 58 Ks L2 X8 SUBJECT TO 2 wo 2 E DK RD ee KA 2 XSi KO 2 X7 3 X8 lt 12 3 2 X3 XE 3 X2 3 X4 X5 2 X6 X7 X8 lt 6 4 X3 X1l X5 9 5 x1 X2 X4 3 6 X3 X2 X6 X7 12 7 X4 X6 X8 9 8 XB te XT eS XB Sy 15 END SLB X3 1 00000 SUB X3 20 00000 GIN X3 This model was then solved with the GO command Following the GO SHOCOLUMN X3 was entered at the colon prompt with the following output x3 1 ROW COEF DUAL PRICE 1 4 00000 1 00000 2 2 00000 333333 3 2 00000 000000 4 1 00000 T3333 6 1 00000 666667 SLB 1 00000 SUB 20 0000 INTEGER VARIABLE VALUE 3 00000 REDUCED COST 666666 Note that SHOCOLUMN now displays the bounds and the integrality requirement as well TABLEAU The TABLEAU command shows the current simplex tableau It is a useful way to observe the simplex algorithm at each step especially when used in conjunction with the PIVOT command TABLEAU displays the variables across the top the row numbers in the first column on the left the basic variable in each row in the second column the coefficients on each variable and the current right hand side on the far right The term ART shown
122. 4 pivot size tolerance 53 variable fixing tolerance 50 tool palette 222 Toolbars 28 Solve button 23 Trademarks ii Transportation model 100 211 Triangularization 95 138 183 Turnkey systems 219 Tutorial See Getting Started with LINDO U Unbounded models 61 118 151 153 54 158 183 185 86 Undo command 26 42 58 Updating 112 USER 118 182 User Supplied Subroutines 118 Vv Variable fixing tolerance 45 50 Variables 2 4 9 12 133 adding 159 162 298 INDEX artificial 95 96 135 138 binary 17 117 191 bounds 187 branching 193 continuous 16 18 195 defining 11 deleting 159 favorable 53 integers 16 46 80 117 170 191 names 12 92 95 120 negative 18 117 pivoting 158 primal values 137 rounding 53 54 slack 96 statistics 80 186 VERBOSE 117 180 Verbose mode 54 193 View command 7 26 31 View windows 29 31 41 Visual Basic 220 222 26 sample application 245 51 Visual C C 220 226 28 sample application 251 58 W Warranties ii WATSUP 265 WIDTH 117 180 Wildcards 87 139 142 William H P 176 Window menu 28 102 9 Windows Command window See Command window Edit windows See Edit windows Entering a model 4 8 mainframe window 4 Model windows See Model windows Reports window See Reports window Solver Status See Solver Status Window View windows See View windows Windows commands 25 114 in brief 25 28
123. ALUE 48 4615402 EW INTEGER SOLUTION OF 40 0000000 AT BRANCH 0 PIVOT 4 BOUND ON OPTIMUM 40 00000 ENUMERATION COMPLETE BRANCHES 0 PIVOTS 4 LAST INTEGER SOLUTION IS THE BEST FOUND REINSTALLING BEST SOLUTION COMMAND LINE COMMANDS 169 nonz Note that Y is no longer used OBJECTIVE FUNCTION VALUE 1 40 00000 VARIABLE VALUE REDUCED COST X 1 000000 30 000000 Z 1 000000 10 000000 NO ITERATIONS 4 BRANCHES 0 DETERM 1 000E 0 The INTEGER lt NumberOfVars gt form of the command is most useful when you want to turn all the variables in the model to binary integers If you don t remember the total number of variables in the model then give the STATS command If you want to make a subset of variables binary using INTEGER lt NumberOfVars gt then those variables must appear first in LINDO s internal ordering of variables A variable s internal order is determined by the order in which it initially appears in the formulation If you aren t sure what the internal order of the variables is enter the command CPRI N to see the variables listed in their internal order The INTEGER lt Variable gt form of the command is most useful when you want to designate a small subset of the variables to be binary This is particularly true when the variables are scattered throughout the formulation and therefore not all f
124. AMPLES E Samptext File Format Drives Punch pun Y v Network This is the standard Windows dialog box for selection to save a file Once you have selected the file to save the basis in you must also select a format LINDO offers the following three different basis formats e Punch MPS Punch format e FBS File Basis Save format e SDBC Save DataBase Column format The details of these formats are of little interest Be aware however that your best results will be obtained with either the Punch or FBS formats SDBC does not save all the details required of a true basis but is supported for backward compatibility with earlier releases of LINDO Saving a basis generally has little effect on re solving the model for integer and quadratic programs For information on restoring a saved basis please refer to the following Basis Read command Basis Read F12 The Basis Read command is used to retrieve a solution to a model saved using the Basis Save command For more information on how and why you save a solution refer to the previous command Basis Save 38 CHAPTER 2 To restore a solution using the Basis Read command you must first select the Model Window you wish to associate the solution with by clicking on it with the mouse This model then becomes the active model Once a Model Window is activated the Basis Read command should be available in the File menu When you choose the Basis Read command
125. Bs are handled implicitly by the solver and therefore are more efficient from a performance point of view than constraints Secondly SUBs and SLBs do not count against the constraint limit in LINDO allowing you to solve larger models within the limits of your version of LINDO For more details and examples on the use of SUBs and SLBs refer to pages 163 166 20 CHAPTER 1 QCP STATEMENT The QCP statement is used in quadratic programming models to indicate the first real constraint as opposed to the first order condition constraints In the Windows version the QCP statement appears as part of the model s text in the Model Window In command line versions of LINDO QCP is entered as an actual system command to the command level colon prompt after the model has been entered The details of formulating a quadratic program are somewhat involved and the reader is asked to refer to page 196 for more information on formulating quadratic programs TITLE STATEMENT This statement is used to associate a title with a model The title may be any alphanumeric string of up to 74 characters in length The title of the current model may be displayed using the File Title command in Windows versions or the TITLE command in command line versions Unlike all the other statements that must appear after the END statement the Title statement may appear before the objective or after the END statement of a model Here is an example of a small model with a ti
126. DDLL32 LIB This creates the import library under the name LNDDLL32 LIB You must add this import library to your project with the Project Add to Project Files command At this point you should be able to build and run the application 258 CHAPTER 8 Running the Application In order to run the application you need to make the LINDO DLL available to the system To do this simply copy LINDO DLL32 LNDDLL32 DLL to either the Windows directory or to the directory containing your application Once you have done this you can issue the Build Execute command in VCS to begin execution You should see the dialog box displayed on the screen f LINDO MSVC Example X Objective Value 0 X 0 ie 0 sae Press the Solve button to build and solve the model At which point you should see the solution displayed in the dialog box f LINDO MSVC Example X Objective Value 2050 x 50 y 35 sae Finally press the OK button to exit the application LINDO CALLABLE LIBRARIES 259 A Sample FORTRAN Application This section illustrates Powerstation FORTRAN 4 0 code which generates and solves the simple two variable model discussed in the sections above Building the Application The following discussion walks you through the steps to build the sample executable 1 Start the Powerstation IDE 2 Create a new project workspace using File New command Select Console Application as the project type
127. DEL ENCOUNTERED DURING GOAL PROGRAMMING You have attempted to solve a goal programming model that has a goal that may be increased without bound Run the DEBUG command to try to narrow down the problem MAXIMUM ROW LIMIT OF lt M gt EXCEEDED LINDO has a limit on the total number of constraints in a model which you have exceeded You can move to a larger version of LINDO or eliminate some constraints Keep in mind that the default lower limit for variables in LINDO is 0 Thus constraints of the form X gt 0 are not required Also using the SUB and SLB statements to set simple bounds does not count against the constraint limit In Windows versions of LINDO the Help About LINDO Command shows the limits on the version of LINDO you are using This also shows how to contact LINDO Systems Inc to order an upgrade to a larger version Command line version of LINDO can use the HELP Command alone to get the same information Please refer to your manual for information on special versions of LINDO WARNING PROBLEM IS POORLY SCALED THE UNITS OF THE ROWS AND VARIABLES SHOULD BE CHANGED SO THE COEFFICIENTS COVER A MUCH SMALLER RANGE LINDO prints this message when some nonzero coefficients are abnormally large or small in value It also computes the ratio of the smallest coefficient to the largest coefficient and will print this message if it finds this ratio excessive This is really just a warning and you may be able to go ahead and solve the
128. DO output file and reports the results in the Visual Basic application window To save on typing you may also paste this code into your Visual Basic project from the one provided on your disk under the name LINDO SHELL SHELL MAK Next click on the Quit button and input the following code Sub Command2_Click End End Sub This code will terminate the application when the user presses the Quit button To run the application select the VB Run Start command and then press the Solve button in the sample application s dialog box If you ve entered everything correctly then you should see the following solution displayed in the VB application window LINDO Staff Example Begin optimization x and 50 35 1 This sample code assumes that you have installed LINDO in the default C LINDO subdirectory If this is not the case then you will need to modify the code to account for LINDO s actual directory INTERFACING 225 Finally press the Quit button to exit this sample application and return to the Visual Basic development environment The VB code in this example is all straightforward with some minor exceptions In particular consider the following line I Shell C LINDO LINDO J t C LINDO SHELL LNDIN TXT 2 Note that the J character should not actually be entered in the code We have simply placed that character there to indicate that we had to break the line in two to have it fit
129. Does the initial setup for range analysis It must be called once just before the first call to either RNGCOL or RNGROW See these two routines for details on performing range analysis RNGBGN in conjunction with RNGCOL and RNGROW correspond to the RANGE command line command RNGCOL JCOL COBJ TU TD Float COBJ TU TD Integer JCOL Does range analysis on the objective row coefficient of column JCOL RNGBGN must have been called just before the first call to RNGCOL On return from RNGCOL COB is the objective coefficient of column JCOL TU is the allowable change up and 7D is the allowable change down in COBJ before a basis change is necessary to retain optimality LINDO CALLABLE LIBRARIES 243 RNGCOL in conjunction with RNGBGN and RNGROW correspond to the RANGE command line command RNGROW IROW RHS TU TD Float RHS TU TD Integer IROW RNGROW does range analysis on the right hand side of row JROW RNGBGN must have been called just before the first call to RNGROW On return from RNGROW RHS is the right hand side coefficient of row I TU is the allowable change up and 7D is the allowable change down in the right hand side before a basis change is necessary to retain feasibility If 7 lt 0 then Z is interpreted as a column number and the analysis is done on the implicit constraint X J gt 0 RNGROW in conjunction with RNGBGN and RNGCOL corresponds to the RANGE command line command SAVE LUNIT Integ
130. ES 1 PIVOTS 12 LAST INTEGER SOLUTION IS THE BEST FOUND REINSTALLING BEST SOLUTION OBJECTIVE FUNCTION VALUE 1 3 000000 VARIABLE VALUE REDUCED COST X 1 000000 1 000000 Y 0 000000 3 000000 Z 1 000000 2 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 3 600000 0 000000 3 0 100000 0 000000 NO ITERATIONS 12 BRANCHES 1 DETERM 1 000E 0 Branch and Bound Solution Method The solution method used by LINDO is based around the enumerative method known as branch and bound Some understanding of this mechanism may help you produce formulations that are easier to solve The basic idea is to solve the problem as a linear program and pray the solution is integer If the prayer algorithm does not work i e some variable that should be integer is fractional then some fractional variable say X is selected as a branch variable Suppose X was declared as a general integer However in the solution of the problem as an LP variable X was equal to 3 7 Two new subproblems are created by alternately appending one of the two constraints X lt 3 or X 4 This branching is continued as long as there are fractional variables and various feasibility tests are satisfied INTEGER PROGRAMMING 193 If LINDO first pursues the branch X lt 3 and it is in verbose output mode then it will print a message resembling t
131. HERWISE Further LINDO Systems Inc reserves the right to revise this software and related documentation and make changes to the content hereof without obligation to notify any person of such revisions or changes ACKNOWLEDGMENTS We gratefully acknowledge Mr Fritz Raffensperger for his contributions to the LINDO Systems Inc user s manuals Copyright 2003 by LINDO Systems Inc All rights reserved Published by ct LINDO SYSTEMS INC 1415 North Dayton Street Chicago Illinois 60622 Technical Support 812 988 9421 E mail tech lindo com WWW http www lindo com Table Of Contents Table ot Contents icisscasesscvcscscecasacestvanacccacocasasactsauscnctancesauseasasscteaascodsesasenians iii PRO lA CG e E E E E ET v Ch 1 Getting Started with LINDO ccceeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeees 1 A Note About This Manual cecceeccceeeeeeeeneeeeeeeeeeeeeeeeaeeneneeeeeeeesessseeeenseeeeeees 2 lastalling eeii a ep eed edt te od eel A ea eh Maen ded dc 3 Entering a Model in Windows i lt 3 J 206t aot hece eee tener Sh iad oe a eerste ale wie 4 Entering a Model from the Command Line ccccceceeeeeeeeeeeeeeeeeeeeeeeeeneeeeeeeeneeees 8 DOGS SV A oa ee aces aceasta he is ats oat ea eae he a Ae A ETETE EERE 12 A Staff Scheduling example sic ntss eh ch ade ik A nner nennseeee 21 Ch 2 LINDO for Windows as icceeres aes cues ectes ecaees tierce caer 25 LINDO for Windows Commands in Brief
132. Help Menu The Help menu is shown at left and contains Conteris Fi several commands for accessing LINDO s help Search for Help On AIF eT These commands are discussed below in etail How to Use Help Ctrl F1 Register AutoUpdate About LINDO Contents F1 The Contents Command opens LINDO s help window to the contents section You may examine the contents section for subjects of interest and il to a desired topic by pressing on its text The Contents command can be invoked by the button The button is used to invoke context sensitive help Selecting this command turns the mouse cursor into a question mark Once you have the question mark prompt selecting any command or button will take you to the relevant section of the help file Search for Help On Alt F1 The Search for Help On Command allows you to search the LINDO help system for a key word or topic How to Use Help Ctrl F1 The How to Use Help command provides you with useful information on navigating the LINDO help system WINDOWS COMMANDS 111 Register The Register command is used to register your version of LINDO online You will need a connection to the internet open for this command to work When issuing the Register command the following dialog box will appear Lindo Systems Product Registration What is your company s main business Educational C Consulting C Manufacturing C Accounting C Governmental C Petrochemical C Agricultural C Medic
133. IENT SET constraint The user quickly noticed that the RHS of the SUFFICIENT SET constraint was incorrect Debugging Infeasible Models Suppose an LP model contains a single typographical mistake that makes the model infeasible The constraint containing the mistake will have a nonzero dual price in the solution report Unfortunately there may be a large number of other constraints that also have a dual price not equal to zero The nonzero dual price on a constraint means that relaxing the constraint may reduce the sum of the infeasibilities The following model illustrates The coefficient 55 in row 4 should have been 5 5 MAX DASE ae SUBJECT TO 2 X 2 Y lt 3 3 2X Y lt 2 4 0 55 X Y gt 4 END When we attempt to solve this formulation we get the following NO FEASIBLE SOLUTION AT STEP 1 SUM OF INFEASIBILITIES 2 483333 VIOLATED ROWS HAVE NEGATIVE SLACK I R OR EQUALITY ROWS NONZERO SLACKS ROWS CONTRIBUTING TO INFEASIBILITY HAVE NONZERO DUAL PRICE OBJECTIVE FUNCTION VALUE 1 10 33333 VARIABLE VALUE REDUCED COST X 0333333 0 000000 Y 1 333333 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 0 483333 3 0 000000 0 033333 4 2 483333 1 000000 All the constraints have nonzero dual prices so we do not have much of a clue in tracking down the culprit COMMAND LINE COMMANDS 185 T
134. IN nails manufactured For more information on the application of integer programming IP you may refer to the LINDO textbook Optimization Modeling with LINDO by Linus Schrage The form of the command is GIN lt Variable NumberOfVars gt where lt Variable gt is the name of an individual variable you want to designate as general integer or lt NumberOfVars gt is an integer value corresponding to the number of variables you wish to make binary Both forms of the GIN command are illustrated below look all 2 END go Solve as a continuous model LP OPTIMUM FOUN OBJ 1 VARIABLE X Y Z ROW 2 NO ITERATIONS AX 20 X 14 Y 132 SUBJECT TO 13 Y 112 lt DO RANGE SE N 93 PRICI Syntax GIN lt Variable NumberOfVars gt The GIN command is used to specify general integer variables A general integer variable can take on any non negative integer value 0 1 2 General integer variables are useful for modeling decision values which are difficult to round to the nearest integer value For instance if you have a planning model for building 2 5 factories whether you round to 2 or 3 will have a big impact on the company s future In this case you would probably want the number of factories represented in a LINDO model using a general integer variable On the other hand if the objective function of a model says to ma
135. INDO box listing the features of the upgraded license Verify that these features correspond to the license you intended to install Note Ifyou were e mailed your password then you have the option of cutting and pasting it into the password dialog box Cut the password from the e mail that contains it then paste it into the LINDO File License dialog box with the Ctrl V shortcut WINDOWS COMMANDS 41 Exit Shift F6 The Exit command causes LINDO to quit and returns you to the operating system You will be prompted to save any files which have modified since they were last saved Note You will lose any changes you made to the open models if you don t save them before quitting see the SAVE command 2 Edit Menu EG Solve Reports Window F The LINDO Edit menu is pictured at left This menu contains the Undo Ctrl z commands that in general support the full screen editing options in LINDO An in depth discussion of each of the commands in the Cut Ctrl x Edit menu is below Before jumping into detailed discussions of Copy Ctrl C the Edit menu commands it would be useful at this point to Paste Ctrl discuss some of the general properties of the LINDO editor Clear Del Find Replace Ctrl F Options Alt O Go To Line Ctrl T Paste Symbol Ctrl P Select All Ctrl A Clear All Choose New Font There are two commands contained in the File menu which open a model in LINDO These are the Open command and
136. If you wish to use PARAMETRICS twice you must use GO to re optimize the model between PARAMETRICS commands In the following example observe that the allowable decrease for the right hand side of row 4 is 50 as reported by the RANGE command We use the PARAMETRICS command to see how the objective value would change if the right hand side of row 4 were to change to 1 well beyond the allowable decrease Max 20 astro 30 cosmo subject to astro lt 60 cosmo lt 70 astro 2 cosmo lt 210 end 174 CHAPTER 3 terse go LP OPTIMUM FOUND AT STEP 2 OBJ VALUE 2700 00000 solution OBJECTIVE FUNCTION VALUE 1 2700 00000 VARIABLE VALUE REDUCED COST ASTRO 60 000000 0 000000 COSMO 50 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 20 000000 3 0 000000 30 000000 4 50 000000 0 000000 NO ITERATIONS 2 Now use the RANGE command to get the allowable decrease on row 4 range RANGES IN WHICH THE BASIS IS UNCHANGED OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE ASTRO 20 000000 INFINITY 20 000000 COSMO 30 000000 INFINITY 30 000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 60 000000 50 000000 60 000000 3 50 000000 25 000000 50 000
137. LINDO User s Manual LINDO Systems Inc 1415 North Dayton Street Chicago Illinois 60622 Phone 312 988 7422 Fax 312 988 9065 E mail info lindo com WWW Attp Avww lindo com COPYRIGHT LINDO software and its related documentation are copyrighted You may not copy the LINDO software or related documentation except in the manner authorized in the related documentation or with the written permission of LINDO systems Inc TRADEMARKS LINGO is a trademark and LINDO and What sBest are registered trademarks of LINDO Systems Inc Other product and company names mentioned herein are the property of their respective owners DISCLAIMER LINDO Systems Inc warrants that on the date of receipt of your payment the disk enclosed in the disk envelope contains an accurate reproduction of the LINDO software and that the copy of the related documentation is accurately reproduced Due to the inherent complexity of computer programs and computer models the LINDO software may not be completely free of errors You are advised to verify your answers before basing decisions on them NEITHER LINDO SYSTEMS INC NOR ANYONE ELSE ASSOCIATED IN THE CREATION PRODUCTION OR DISTRIBUTION OF THE LINDO SOFTWARE MAKES ANY OTHER EXPRESSED WARRANTIES REGARDING THE DISKS OR DOCUMENTATION AND MAKES NO WARRANTIES AT ALL EITHER EXPRESSED OR IMPLIED REGARDING THE LINDO SOFTWARE INCLUDING THE IMPLIED WARRANTIES OF MERCHANTABILITY FITNESS FOR A PARTICULAR PURPOSE OR OT
138. LINDO are ordered from left to right Comments may be placed anywhere in a model A comment is denoted by an exclamation mark Anything following the exclamation mark on the current line will be considered a comment For example the small example discussed earlier in this chapter is recast below using comments MAX 10 STD 15 DLX Max profit SUBJECT TO Here are our factory capacity constraints for Standard and Deluxe computers STD lt 10 DLX lt 12 Here is the constraint on labor availability STD 2 DLX lt 16 END Command line versions of LINDO allow you to input comments but they will not be stored with the model Comments are preserved in Windows versions of LINDO as long as the file is saved in text format Saving the file in either compressed LPK or MPS format MPS will cause comments and any special formatting to be stripped from the model Note Comments will not be preserved by command line versions of LINDO or when a file is saved in compressed formats See above Constraints and objective functions may be split over multiple lines or combined on single lines You may split a line anywhere except in the middle of a variable name or a coefficient The following would be mathematically equivalent to our example although not quite as easy to read MAX 10 STD 15 DLX SUBJECT TO STD 10 dix lt 12 STD 2 dlx lt 16 end However if the objective function appeared as follows
139. M R N T W XR gt 8 FRI M R t F 4T W XF gt 8 SAT R F S T W XS gt 3 SUN R F S N W XN gt 3 XM XM lt 1 XT XT lt 1 XW XW lt 1 XR XR lt 1 XF XF lt 1 XS XS lt 1 XN XN lt 1 COST COST 9M 9R 9F 9S ON 9T OW 0 EXTRA EXTRA XM XT XW XR XF XS XN 0 END glex LP OPTIMUM FOUND AT STEP 4 OBJECTIVE VALUE 72 0000000 LP OPTIMUM FOUND AT STEP 7 OBJECTIVE VALUE 68 0000000 OBJECTIVE FUNCTION VALUE 1 68 00000 VARIABLE VALUE REDUCED COST COST 72 000000 0 000000 EXTRA 4 000000 1 000000 M 4 000000 0 000000 R 0 000000 0 000000 F 0 000000 0 000000 S 0 000000 9 000000 N 0 000000 9 000000 XM 1 000000 0 000000 lt Note T 0 000000 0 000000 that XT 1 000000 0 000000 lt we ve spread wW 4 000000 0 000000 our surplus XW 0 000000 0 000000 workers XR 0 000000 0 000000 across 4 XF 0 000000 9 000000 days M T Xs 1 000000 0 000000 lt S amp N XN 1 000000 0 000000 lt 158 CHAPTER 3 ROW SLACK OR SURPLUS DUAL PRICES MON 0 000000 0 000000 TUE 0 000000 0 000000 WED 0 000000 0 000000 THU 0 000000 0 000000 FRI 0 000000 9 000000 SAT 0 000000 0 000000 SUN 0 000000 0 000000 XM 0 000000 0 000000 XT 0 000000 0 000000 XW 1 000000 0 000000 XR 1 000000 0 000000 XF 1 000000 0 000000 XS 0 000000 0 000000 XN 0 000000 0 000000 COST 0 000000 1 000000 EXTRA 0 000000 0 000000 NO ITERATIONS
140. MDZ KZ 8 MEMKZ MEMKZ IRTCOD Character KZ Double DZ Integer MEMDZ MEMKZ IRTCOD Saves the current model to memory Once saved the model may be restored by calling the LSGTPR routine On entry MEMDZ contains the length of the numeric DZ array in double words 8 bytes and MEMKZ contains the length of the character KZ array in double words On output DZ contains the numeric part of the model KZ the character part and JRTCOD 0 if there was enough space in the DZ and KZ arrays to store the model The number of double words required in the KZ array for storing the model s character data is M N T 8 where M number of rows in model N number of variables T length of model s title if any LINDO CALLABLE LIBRARIES 237 The number of double words required in the NZ array depends on many things including the number of rows variables inequality rows and nonzero elements In general it would be difficult for a calling application to compute the exact number of double words required in NZ However if there is insufficient space to store the model JRTCOD returns the number of double words required in the NZ array Thus a useful tactic is to call LSAVPR once with little or no space allotted LSAVPR will then return with the required number of elements for the NZ array in JRTCOD Using this value in IRTCOD your application can then allocate an array of appropriate length for storing the numeric part of the model and then r
141. N IS THE BEST FOUND RE INSTALLING BEST SOLUTION COMMAND LINE COMMANDS 179 retr trybip ind bip 14 5 go LP OPTIMUM FOUND AT STEP 5 OBJECTIVE VALUE 19 3000000 NEW INTEGER SOLUTION OF 15 0000000 AT BRANCH 10 PIVOT 31 BOUND ON OPTIMUM 17 07500 ENUMERATION COMPLETE BRANCHES 13 PIVOTS 41 LAST INTEGER SOLUTION IS THE BEST FOUND RE INSTALLING BEST SOLUTION IPTOL Syntax IPTOL lt Fraction gt The IPTOL command specifies a fraction f which indicates to the branch and bound solver that it should only search for solutions with objective values 100 better than the best one considered so far The end results of using an IPTOL are twofold First on the positive side solution times will tend to be considerably faster On the negative side however the use of IPTOL can mean LINDO may not find the optimal solution and you will not receive any warning to this effect in the solution output You will know however that the solution found is within 100 of the true optimal On large integer models the potential of getting a solution within say 2 of optimal in a few minutes versus getting the true optimal in a few days makes the use of an IPTOL value quite attractive In the following example we solve a small IP without the benefit of an IPTOL value and we see that it solves to optimality in a little over 10 0
142. NAL WIDTH The text report line length implied by a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO exceeds the terminal width The terminal width may be increased using the Edit Options command in Windows versions or the WIDTH command in command line versions Alternatively you can cut down on the amount of requested output If you are running under windows and all you want is a graphical view of the data you can cancel the request for a text based report 114 INVALID QCP ROW COMMAND DISREGARDED You have specified an invalid row number as part of the QCP command Check your model to determine the valid range for rows Be sure to use the row s number and not its name Error codes 1000 and above pertain only to Windows versions of LINDO 1000 Toolbar installation failed LINDO was unable to install the command toolbar into memory on startup You are probably low on memory If other applications are running try exiting them before starting LINDO 1001 Please enter a valid line number to go to You have input an invalid line number in the Edit Go To Line dialog box Please reenter a non negative integer value corresponding to the line you wish to jump to 1002 No help available on this command The LINDO Help System does not currently have any help available on this command 1003 The row values are not valid Please reenter them You have input invalid row number
143. NDO Text equation equivalents before being displayed in a window For example readers familiar with the MPS file format will find if they were to read the following MPS file into LINDO NAME ROWS N 1 L2 L 3 L4 COLUMNS X 1 20 0000000 X 2 1 0000000 X 4 1 0000000 Y a 30 0000000 Y 3 1 0000000 Y 4 2 0000000 RHS RHS 2 50 0000000 RHS 3 60 0000000 RHS 4 120 0000000 WINDOWS COMMANDS 31 LINDO converts the model into the more readable LINDO Text format before displaying it as follows IS C LINDO61 Samples TEST MPS O X MIN 20 X 30 SUBJECT TO LINDO keeps track of a file s original format Thus when you save the model LINDO uses the original format unless you specify otherwise Please refer to the Save command for more information View F4 The View command reads a saved model file from disk and places it in a View Window Unlike the Open command which uses Edit Windows that are limited to files of 64 000 characters or less the View command can read a file of any size and is limited only by available memory A View Window is primarily useful in viewing models and submitting them to LINDO s solver Although you can read large models into a View Window you will not have all the editing capabilities of an Edit Window You can scroll through a View Window use the Go To command to jump to a particular line use the Find Replace command to search for a selected string and optionally replace the string If y
144. NDO is waiting for a command from you As each line is entered LINDO checks these entries for correct syntax and mathematical sense If there is a syntax or logic mistake an error message will appear and LINDO will prompt you to correct that line of the model Use the LOOK ALL command at the colon prompt to check the internal representation of the model currently in memory for errors The internal representation of this sample model is shown below LOOK ALL AX 10 STD 15 DLX UBJECT TO TD lt 10 LX lt 12 TD 2 DLX lt 16 D ANnNnUNN GETTING STARTED 11 The ALTER command can be used to change a line of the model which may have been typed incorrectly To do this on any platform enter the following at the colon prompt ALTER lt Row gt lt VarName DIR NAME RHS gt where lt Row gt is the name or index of the row where you wish to make the alteration and lt VarName gt is the name of the variable to be altered Please refer to the Problem Editing section of Chapter 3 Command line Commands for details on the syntax and use of the ALTER command Once the model is typed correctly you can solve the problem by typing GO followed by a return at the colon prompt The answer will come back and LINDO will ask you whether or not you want sensitivity analysis on the model Eventually you ll be able to make use of that information but for now just type N at the question mark prompt
145. O would select the variable and the pivot row In most cases as a minimum you will want to select the pivot variable If you select the pivot variable then you will also have the option of selecting the pivot row Be careful when selecting your pivot row LINDO will report that the objective function is unbounded if the row specified does not bound the pivot variable If you do not select a row LINDO will select the correct one for you WINDOWS COMMANDS 65 Getting back to our example let s pivot on Y first since it has the highest objective coefficient Since we want to select the pivot variable we must click on the button marked Use Mine in the box labeled Variable Selection Next we type the variable name Y into the edit box labeled My Variable Selection Because we are also specifying the pivot row we click on the button marked Use Mine in the box labeled Row Selection Finally we type the pivot row name into the box labeled My Row Selection Examining our model we find row 4 places the tightest restriction on Y so we make that our pivot row The Pivot command dialog box will now resemble the following m Pivot Variable r Pivot Row Variable Selection Row Selection C LINDO s C LINDO s 4 Cancel Use Mine Use Mine Help My Variable Selection My Row Selection At this point data input is finished for our first pivot so we press the OK button LINDO t
146. SIC Tried to make a column basic which is already basic THE COLUMN ON THE FOLLOWING CARD IS NOT AN INTEGER VARIABLE lt CardText gt LINES READ lt N gt Tried to fix a variable that is not an integer variable using the IV status SEARCH TERMINATED DUE TO USER INTERRUPT OR NUMERICALPROBLEMS The LINDO solver was interrupted prematurely This is generally the result of the user interrupting the solver In this case LINDO will return with the best solution found so far If you did not interrupt the solver then LINDO encountered numerical problems and you should not rely on the reported results lt Not In Use gt lt Not In Use gt lt Value gt IS NOT A VALID INTEGER VALUE You have input a non integral value to a command that was expecting an integer Please retry specifying an integer value BAD INPUT TO NDXOFV lt 1 gt lt 2 gt lt VariableName gt You have passed invalid values to the user interface routine NDXOFV Please see the documentation on this routine and correct your input values SOLUTION STATUS NOT OPTIMAL AT START OF PARAMETRICS LINDO must have an optimal solution in memory in order to perform right hand side parametrics Please use the Solve command in Windows or the GO command in command line versions to optimize the model and then retry 103 104 105 106 107 108 109 110 111 ERROR MESSAGES 283 INVALID OPTION FIELD IN COMMAND You have specified an invalid op
147. STEP 18 OBJECTIVE VALUE 8942181 00 WIDTH Syntax WIDTH lt NumberOfChars gt WIDTH tells LINDO the number of characters per line your screen or printer supports LINDO will wrap output lines in order not to overflow the WIDTH limit For a wide screen or printer you may wish to set WIDTH to a large value For a narrow screen or printer you may wish to set it to a small value You may use any integer between 40 and 132 If you enter nothing after WIDTH LINDO will prompt you for the value The default value for WIDTH is dependent on the platform you are running on but in general will lie somewhere between 68 and 80 COMMAND LINE COMMANDS 181 WIDTH also affects input When reading input LINDO will ignore any input that lies outside the terminal width For example if you have WIDTH 40 but you enter more than 40 characters on one line LINDO will ignore anything following the 40th column The WIDTH parameter is not saved when the model is saved with SAVE SMPS or the DIVERT LOOK ALL sequence You can use different values of WIDTH during one LINDO session Thus if your screen is 68 columns wide you probably will wish to use the default setting If your printer or workstation screen is 132 columns wide you may wish to enter the command WIDTH 132 If you want a terminal width different from the default each time you run LINDO you may want to place a WIDTH command in your AUTOLD DAT so it is executed each time LINDO starts up For more i
148. STEP 5 OBJECTIVE VALUE 77 0000000 divert model txt Put the model in a file look all rvrt divert solu txt Put the solution in a file solution OBJECTIVE FUNCTION VALUE 1 77 00000 rvrt We could read the solution or model files into a text editor display them on our terminal using the the cat command in Unix or send them to the printer using the the lpr command in Unix Doing so would reveal the following two files MIN 2 Al 4 A2 3 A3 5 A4 4 Bl 3 B2 2 B3 6 B4 5 Cl 6 C2 4 C3 3 C4 SUBJECT T O 2 Al Bl Cl 9 A2 B2 C2 7 AS BSN tt EIIES 6 A4 B4 C4 8 Al A2 A3 A4 lt 12 Bl B2 B3 B4 lt 11 C1 C2 C3 C4 lt 13 END 216 CHAPTER 7 MODEL TXT OBJECTIVE FUNCTION VALUE 1 77 00000 VARIABLE VALUE REDUCED COST Al 9 000000 0 000000 A2 2 000000 0 000000 A3 0 000000 0 000000 A4 0 000000 2 000000 B1 0 000000 3 000000 B2 5 000000 0 000000 B3 6 000000 0 000000 B4 0 000000 4 000000 c1 0 000000 3 000000 c2 0 000000 2 000000 c3 0 000000 1 000000 C4 8 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 2 000000 3 0 000000 4 000000 4 0 000000 3 000000 5 0 000000 3 000000 6 1 000000 0 000000 7 0 000000 1 000000 8 5 000000 0 000000 NO ITERATIONS 5 SOLU TXT Note that since command line versions of LINDO do not contain a print command you will need to re
149. T TO a dre eee by nee eee 2 2 X Y lt 12 3 X 3 Y gt 1 X END T GIN 2 Windows Command Line Note that in the Windows version the GIN statements appear as part of the model s text in the Model Window In command line versions of LINDO GIN is entered as a system command to the command level colon prompt after the model s objective and constraints have been entered Had we not specified X and Y to be general integers in this model LINDO would not have found the optimal solution of X 6 and Y 0 Instead LINDO would have treated X and Y as continuous and returned the solution of X 5 29 and Y 1 43 Note also that simply rounding the continuous solution to the nearest integer values does not yield the optimal solution in this example In general rounded continuous solutions may be non optimal and at worst infeasible Based on this one can imagine that it can be very time consuming to obtain the optimal solution to a model with many integer variables This is typically true and you are best off utilizing the GIN feature only when absolutely necessary As a final note the GIN command also accepts an integer value argument in place of a variable name The number corresponds to the number of variables you want to be general integers These variables must appear first in the formulation of the model Thus in this simple example we could have replaced our two GIN statements with the single statement GIN 2 For more deta
150. T1 C2T2 R2T1 130 190 80000 80000 70000 70000 0 0 6 BiT2 T1T2 4 B2T2 T2T2 47 The default in LINDO is to use no IP Optimality Tolerance When we stick with this default we find the following in the Solver Status Window once optimization is complete LINDO Solver Status m Optimizer Status Status Iterations Infeasibility Objective Best IP IP Bound Branches Elapsed Time Optimal 10958 0 8 92939e 006 8 92939e 006 8 93652e 006 1728 00 00 12 Update Interval fi 48 CHAPTER 2 Note that it has taken us 10 064 iterations 1 715 branches and 19 seconds to reach optimality Now we set the IP Optimality Tolerance to 01 in the Options Dialog Box as follows IP Optimality Tol D1 IP Objective Hurdle By setting this value to 01 we are telling the solver that we are happy with any integer solution that is no more than 1 worse than the true optimal solution Re solving the same model yields the following statistics in the Solver Status Window LINDO Solver Status Optimizer Status Status Optimal Iterations 171 Infeasibility 0 Objective 8 92917e 006 Best IP 8 92917e 006 IP Bound 8 93652e 006 Branches 14 Elapsed Time 00 00 01 Update Interval 1 This time the model solves in only 74 iterations 7 branches and two seconds The true objective value of 8 929 390 is a mere 0 002 better than the objective of 8 929 170 found usin
151. TER 7 The following session illustrates how to do this With the text editor we open a file and enter the following subset of the model MIN ST A1 A1 END 2 A1 4 A2 3 B2 6 C2 4 Bl Be GI 4 BL 4 A2 4 CL 3 A3 5 A4 2 B3 4 C3 9 A3 A4 lt 12 6 B4 3 C4 Demand for cust 1 Supply limit for A Most popular editors have some variation of a copy command Using it we make three additional copies of the demand constraint and two additional copies of the supply constraint so the model looks as follows MIN ST A1 A1 A1 A1 A1 A1 A1 END 2 A1 B1 SugL A A2 4 B1 4 4 B1 4 4 BL 4 4 B1 4 A 4 4 A2 4 4 A2 3 B2 6 C2 C1 Cl Cl Cl A3 A3 4 A3 4 3 A3 5 A4 dt AUS 4 C3 O O Co 9 A4 lt 12 A4 lt 12 A4 lt 12 6 B4 3 C4 Demand Demand Demand Demand Supply Supply Supply for for for for cust cust cust cust PRR limit for A limit for A limit for A Most editors also have some form of a Find Replace command Using it we replace every occurrence of 1 in the second demand constraint by 2 etc every occurrence of A in the second supply constraint by B etc When done the file should appear as follows MIN 2 Al 4 A2 3 A3 5 A4 ST Al A2 A3 A4 Al Bl Cl END 4 Bl SGE 4 B 1 4 oh B2 4 B 3 4 B 4 4 A2 4 B2 4 fe C2 4
152. TIVE FUNCTION VALUE 1 107 0000 VARIABLE VALUE REDUCED COST X 2 000000 20 000000 Y 2 000000 14 000000 Z 3 000000 13 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 0 000000 NO ITERATIONS 72 BRANCHES 22 DETERM 1 000E 0 Note that the integer solution is considerably different from the continuous solution Rounding in this case would not have been appropriate gin 0 Turn off all integers look all AX 20 X 14 Y 132 SUBJECT TO 2 E X te OLS a EZ 93 END This time use the GIN lt NumberOfVariables gt form to make the variables integer gin 3 172 CHAPTER 3 look all AX 205 Xe dey E38 ZF SUBJECT TO 2 DX A Ye Ln E 93 END GIN 3 The GIN lt NumberOfVars gt form of the command is most useful when you want to turn all the variables in the model to general integers If you don t remember the total number of variables in the model then give the STATS command If you want to make a subset of variables general integer using GIN lt NumberOfVars gt then those variables must appear first in LINDO s internal ordering of variables A variable s internal order is determined by the order in which it initially appears in the formulation If you aren t sure what the internal order of the variables is enter CPRI N to see the variables listed in internal order The GIN lt Variable gt form of the command is most useful when you want to designate a small subset of the varia
153. The Solve command is the primary command you will use from the Solve menu This command causes LINDO to begin solving your model If it is a relatively small model the Solve command will execute almost instantaneously If it is a large tough model the Solve command could potentially run for hours WINDOWS COMMANDS 59 When you invoke the Solve command LINDO will check to see if the model has been modified and needs to be compiled Assuming no compilation errors exist LINDO posts a window called LINDO Solver Status This window allows you to monitor the progress of the solver towards a solution A sample of the window appears below LINDO Solver Status Optimizer Status Status Optimal Iterations 2 Infeasibility 0 Objective 145 Best IP Nva IP Bound NYA Branches N A Elapsed Time 00 00 01 Update Interval f1 Interrupt Solver A description of the various fields and controls within the Status Window appears in the table below Field Control Description Status Gives status of the current solution Possible values e a Optimal ease neshleand aed Number of solver iterations Best IP Objective value of best integer solution found Only for ee ee ere IP Bound Theoretical bound on the objective for IPs Only for integer e in em Branches The number of integer variables branched on by LINDO s IP solver Only for integer programming models Update Interval The frequency in seconds that the Status Window is u
154. UBJECT TO MON M R F 4 S N gt 3 TUE M T FHt S N gt 3 WED M T W S N gt 8 THU M T W R N gt D 8 FRI M T W R tF gt _ 8 SAT T Wt tR F tS gt 3 SUN WtR FHt S N gt 3 END go LP OPTIMUM FOUND AT STEP 4 OBJECTIVE VALUE 8 00000000 OBJECTIVE FUNCTION VALUE 1 8 000000 VARIABLE VALUE REDUCED COST 5 000000 0 000000 T 0 000000 0 000000 W 3 000000 0 000000 R 0 000000 0 000000 F 0 000000 0 000000 S 0 000000 1 000000 N 0 000000 1 000000 ROW SLACK OR SURPLUS DUAL PRICES MON 2 000000 0 000000 lt Note that we TUE 2 000000 0 000000 lt have extra WED 0 000000 0 000000 workers only THU 0 000000 0 000000 on M E amp T FRI 0 000000 1 000000 SAT 0 000000 0 000000 SUN 0 000000 0 000000 NO ITERATIONS 4 COMMAND LINE COMMANDS 157 We would like to have extra staff if possible but more than one extra employee per day is not beneficial So we ll let X denote the extra workers up to a maximum of one on each day Then using GLEX we can first minimize cost fix cost at its optimal value and then maximize the sum of the X variables Note that since we are maximizing EXTRA it has a negative coefficient in the objective Here is the new formulation setup up for the GLEX command look all MIN COST EXTRA SUBJECT TO MON M R F S N XM gt 3 TUE M Ft S N 4 T XT gt 3 WED M S N T W XW gt 8 THU
155. UNIT This solution basis will be used as a starting point when you solve the current model The model must be in memory before calling LSINSR To attach a file to a unit see routines LUNOPN and LUNGET To create an MPS PUNCH format file see routine LSPUNC This routine corresponds to the FINS command line command LSPUNC LUNIT XDATA 8 Character XDATA Integer LUNIT Sends a solution basis in MPS PUNCH format to the file associated with unit number LUNIT see LUNOPN and LUNGET XDATA contains an 8 byte name to be written to the first line of the report This routine corresponds to the FPUN command line command LSPUTP DERR NPARAM DPVAL Double DERR DPVAL Integer NPARAM Sets various internal parameters used by LINDO For a list of these parameters refer to the documentation on the SET command line command On entry NPARAM contains the index of the parameter to set and DPVAL contains the new value for the parameter DERR will be set to 0 if there was no problem if the parameter index was invalid or 2 if the new parameter value was invalid To retrieve the current setting of a parameter see the LSFETP command LSPUTP corresponds to the SET command line command LINDO CALLABLE LIBRARIES 239 LUNGET LUNIT INROUT NOTFMT Integer LUNIT INROUT NOTFMT Prompts the user for a filename and opens it on logical unit LUNIT On entry set IVROUT to 1 if you are opening the file for input Otherwise set it to 0 Se
156. W 4 OBJ VALUE 1900 0 tabl THE TABLEAU ROW BASIS A Cc SLK 2 SLK 3 1 ART 000 000 000 10 000 1900 000 2 SIK 2 000 000 1 000 2 000 40 000 3 G 000 1 000 000 1 000 50 000 4 A 1 000 000 000 2 000 20 000 ART 4 ART 000 000 000 10 000 000 pivot SLK 3 ENTERS AT VALUE 20 000 IN ROW 2 OBJ VALUE 2100 0 tabl THE TABLEAU ROW BASIS A G SLK 2 SLK 3 1 ART 000 000 5 000 000 2100 000 2 SLK 3 000 000 2500 1 000 20 000 3 Cc 000 1 000 500 000 30 000 4 A 1 000 000 1 000 000 60 000 pivot LP OPTIMUM FOUND AT STEP 3 COMMAND LINE COMMANDS 137 OBJECTIVE FUNCTION VALUE 1 2100 00000 VARIABLE VALUE REDUCED COST A 60 000000 000000 Cc 30 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 5 000000 3 20 000000 000000 4 000000 15 000000 NO ITERATIONS 3 tabl THE TABLEAU ROW BASIS A Cc SLK 2 SIK 3 1 ART 000 000 5 000 000 2100 000 2 A 1 000 000 1 000 000 60 000 3 SLK 3 000 000 500 1 000 20 000 4 Cc 000 1 000 500 000 30 000 Refer to the Tableau Command in the Windows Commands in Depth section of Chapter 2 LINDO for Windows for a more detailed description of how the Tableau Command works in LINDO NONZEROS The NONZEROS command displays the nonzero values of the current solution This includes e a warning if the current solution is not guaranteed to be optimal or feasible e objective function value e primal
157. XWH1C3 7 XWH1C4 4 XWH2C1 9 XWH2C2 5 XWH2C3 3 XWH2C4 8 XWH3C1 8 XWH3C2 XWH3C3 5 XWH3C4 SUBJECT TO Demand constraints C1 XWH1C1 XWH2C1 EWH3C1 C2 XWHIC2 XWH2C2 EWH3C2 C3 XWHIC3 XWH2C3 EWH3C3 C4 XWHIC4 XWH2C4 EWH3C4 Supply constraints WH1 XWH1C1 XWH1C2 XWH1C3 XWH1C4 lt WH2 XWH2C1 XWH2C2 XWH2C3 XWH2C4 lt WH3 XWH3C1 XWH3C2 XWH3C3 XWH3C4 lt END WINDOWS COMMANDS 101 When we issue the Show Column command LINDO presents the following dialog box to assist you in selecting a column name Show Column T Column to Show Cancel DHe Help The list box labeled Column to Show contains all the variable names in the current model You may type in a name directly or select a column name from the list box In this example we have chosen the variable XWH1C2 Pressing the OK button causes the following report to be sent to the Reports Window I Reports Window XWH1C2 7 2 RoW COEF DUAL PRICE 2 00000 1 00000 1 00000 2 00000 1 00000 0 000000E 00 VALUE 17 0000 REDUCED COST 0 000000E 00 In the first line we have the variable name and its internal index In this case the internal index is 2 which means this was the second variable LINDO encountered when it compiled the model In the next section of the report there is one line for each row in which the column has a nonzero coefficient In each line the row number the nonzero co
158. Y ST X Y gt 5 X Y gt 7 END Te FREE Y lt untitled gt MER LOOK ALL AA MIN 5X Y SUBJECT TO 2 X Y gt 5 3 X Y gt 7 END FREE Y Windows Command Line Note that in the Windows version FREE appears as part of the model s text in the Model Window In command line versions of LINDO FREE is entered as a system command after the model has been entered Had we not set Y to be a free variable in this example LINDO would not have found the optimal solution of X 6 and Y 1 Instead given the default lower bound of 0 on Y LINDO would have returned the solution X 7 and Y 0 For more information and examples on FREE please refer to page 166 GIN STATEMENT By default LINDO assumes that all variables are continuous In other words unless told otherwise LINDO assumes variables can be any non negative fractional number In many applications fractional values may be of little use e g 2 5 employees In these instances you will want to make use of the general integer statement GIN GIN followed by a variable name restricts the value of the variable to the non negative integers 0 1 2 GETTING STARTED 17 The following small examples illustrate the use of the GIN statement for Windows versions and command line versions of LINDO MAX 11X 10Y ST 2X Y lt 12 X 3Y gt 1 END GIN X GIN Y i lt untitled gt AR LOOK ALL 11X 10Y w MAX 11 x 10 SUBJEC
159. Y 7 2XY XZ 0 8 YZ Subject To We start with 1 X Y 2Z 1 We want to end with at least 1 12 Tee sXe WEA Ye O A Se Le dD No asset may constitute more than 75 of the portfolio Xx lt 0 75 Yo x 0119 Z lt 0 75 The input procedure for LINDO requires this model be converted to true linear form by writing the first order conditions To do this we must introduce a dual variable or LaGrange multiplier for each constraint For the above 5 constraints we will use 5 dual variables denoted respectively as UNITY RETURN XFRAC YFRAC and ZFRAC QUADRATIC PROGRAMMING 199 The LaGrangean expression corresponding to this model is then Min X Y Z 3 X 2 Y 7 2XY XZ 8 YZ X Y 2 1 UNITY 1 12 1 3 X 1 2 Y 1 08 Z RETURN X 75 XFRAC t Y 75 YFRAC t Z 75 ZFRAC Basically we have moved all the constraints into the objective weighting them by their corresponding dual variables The next step is to compute the first order conditions The first order conditions are computed by taking the partial derivatives of f X Y Z with respect to each of the decision variables and setting them to be non negative For example for the variable X the first order condition is 6X 2 Y Z UNITY 1 3 RETURN XFRAC 2 0 Qualitatively speaking this constraint says that at the optimum the cost of increasing X must be non negative If it were negative then decreasing X could reduce the object
160. a plus sign in front of variable Z3 in constraint 3 A look at row 3 reveals that Z3 can be increased indefinitely leading to an unbounded objective 42 C LINDO61 SAMPLESWUNBOUND LTX DER HAX 12 X1 13 X2 22 Y1 25 Y2 23 Z1 28 72 X B Y3 Z3 SUBJECT TO 2 X1 X2 X3 lt 400 3 Yl Y2 Y3 Z3 lt 500 4 Zi Z2 lt 500 WINDOWS COMMANDS 63 The resulting model is unbounded and when issuing the Solve command we receive the unbounded error message LINDO Error Message pet 52 Error text UNBOUNDED SOLUTION AT STEP 3 REDUCED COST 26 000 USE THE DEBUG COMMAND FOR MORE INFORMATION Taking the cue from the error message and issuing the Debug command from the Solve menu we receive the following breakdown i3 Reports Window SUFFICIENT SET COLS CORRECT ONE OF Z3 NECESSARY SET COLS CORRECT ONE OF v2 The Debug command has successfully determined that bounding Z3 is sufficient to bound the entire model This illustrates that Debug can be very useful for narrowing down the search for problems in an unbounded model In general the kind of correction required in a column is one or more of the following change the objective coefficient change a coefficient in some constraint change the direction of the inequality sign of some constraint where the variable in question appears make either the upper or lower bound finite Pivot Ctrl N The fundamental operation in the sim
161. a space in between as mentioned above The APPCOL command modifies the current model The current solution is no longer considered optimal after APPCOL is used In the following sample session a variable name Y is appended to the model look all MAX 9 Xl X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 4 X1 X33 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 EN F E COMMAND LINE COMMANDS 163 appcol y 1 11 3 16 5 5 22 0 look all MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 11 Y SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 Xl 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 16 5 Y lt 6 4 X1 X3 xX5 9 5 Xl X2 X4 22 Y 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 DELETE Syntax DELETE lt Row BegRowEndRow gt The DELETE command deletes a row or range of rows from your model After DELETE enter the row name row number or range of row numbers you want to delete If you type nothing after the DELETE command LINDO will prompt you for the row s to delete Some examples of the DELETE command are DELETE 3 Deletes row 3 DELETE CAPACITY Deletes the row called CAPACITY DELETE 3 5 Deletes rows 3 through 5 Th
162. able and row name tables required by the Paste Symbol command Finally running the compile WINDOWS COMMANDS 61 command eliminates any stored basis for the model If you have solved the model during the current session LINDO maintains a basis i e solution in memory to use as a starting point should you modify and re solve it again Issuing the compile command causes LINDO to erase the basis for the model Thus all subsequent Solve commands will commence from scratch Debug Ctrl D In the ideal world all models solved by LINDO would return an optimal solution Unfortunately this is not the case Sooner or later you are bound to run across either an infeasible or unbounded model This is particularly true in the development phase of a project when the model will tend to suffer from typographical errors Tracking down an error in a large model can prove to be a daunting task The Debug command is useful in narrowing the search for problems in both infeasible and unbounded models When Debug encounters a model with no feasible solution it first tries to identify one or more crucial constraints A constraint is crucial if dropping just that constraint from the entire model is sufficient to make the model feasible These constraints are identified by Debug as the SUFFICIENT SET ROWS Not every infeasible model has a crucial constraint Regardless of whether any crucial constraints were found the Debug command also identifies a set
163. ables first in the formulation i e you must define them using APPCOL before defining any other variables and must set the simple upper and lower bounds appropriately see SETSUB and SETSLB OUTSPC J Integer J Sends a standard LINDO solution report to the standard output device On entry set J to 0 to display nonzero values only or to display the full report To capture this output in a file see the CAPOUT routine This routine corresponds to the SOLU and NONZ command line commands PARBGN IROW PAML Float PAML Integer IROW Sets up for doing right hand side parametrics On entry set ROW to the index of the row you want to do parametrics on and PAML to the target right hand side value The model should be optimized beforehand using the GO routine Once you have called PARBGN to setup for parametrics you will then need to call PARSTP iteratively to perform the actual parametric steps PARBGN in conjunction with PARSTP correspond to the PARA command line command LINDO CALLABLE LIBRARIES 241 PARSTP PV JI JO Float PV Integer JI JO Performs a single step in parametric analysis of a right hand side value You must setup for parametrics by calling PARBGN once before making any calls to PARSTP PARSTP returns the change in the value of the right hand side in the single precision variable PV the index of the entering variable in JI and the index of the departing variable in JO If no departing variable cou
164. ace Finds a given string of text in active window and optionally allows for replacement Used to configure LINDO options and reserved symbols into active window Clear All Clears all the text in the active window Choose New Font Resets font in active window WINDOWS COMMANDS 27 3 Solve Menu Command Description Solve Solves model in the active window Compile Model Compiles model in the active window unbounded the active window programming on active window 4 Reports Menu Command _ Description Model Window constraint Statistics Displays statistics for the model in the active window Peruse Used to generate text reports on and or graphics of selected items of the active model Creates a text and or graphics picture of the nonzero structure of the current model Creates a text based picture of the basis matrix for active model Displays the simplex tableau for the active model Displays the active model Show Column Displays a column Vvariable of the active model Positive Definite Determines if the constraint matrix of a quadratic programming model is positive definite 28 CHAPTER 2 5 Window Menu Command _ Description Open Command Window Opens a window that accepts traditional LINDO commands which are discussed in the following chapter Command Line Commands used to monitor the solver s progress on a model Send To Back Sends the active window to the back bringing windows to the front which
165. after the END statement in the Model Window These statements and their functions appear in the table below Model Statement Function FREE lt Variable gt Removes all bounds on lt Variable gt allowing lt Variable gt to take on any real value positive or negative GIN lt Variable gt Makes lt Variable gt a general integer i e restricts it to the set of non negative integers INT lt Variable gt Makes lt Variable gt binary i e restricts it to be either 0 or 1 SLB lt Variable gt lt Value gt Places a simple lower bound on lt Variable gt of lt Value gt Use in place of constraints of form X 2 r SUB lt Variable gt lt Value gt Places a simple upper bound on lt Variable gt of lt Value gt Use in place of constraints of form X lt r QCP lt Constraint gt Marks the beginning of the real constraints in a quadratic programming model TITLE lt Title gt Makes lt Title gt the title of the model Next we will briefly illustrate the use of each of these statements FREE STATEMENT The default lower bound for a variable is 0 In other words unless you specify otherwise LINDO will not let a variable be negative The FREE statement allows you to remove all bounds on a variable so it may take on any real value positive or negative 16 CHAPTER 1 The following small examples illustrate the use of the FREE statement for the Windows version and command line versions of LINDO MIN 5X
166. al Transportation C Financial C Market Research C Other C Telecommunications Insurance What other optimization packages have you used a What will be your primary application of our product meee ee Comments EE By entering your personal information and select the Register button your information will be sent directly to LINDO Systems via the Internet 112 CHAPTER 2 Once your registration is complete the following dialog box will appear on your screen Registration Complete Online Registration Submitted Selecting the OK button returns you to the main LINDO environment LINDO Systems is constantly working to make our products faster and easier to use Registering your software with LINDO ensures that you will be kept up to date on the latest enhancements and other product news You can also register through the mail or by fax using the registration card included with your software package AutoUpdate The AutoUpdate command is used to automatically check every time you start the LINDO software whether there is a more recent version of LINDO available for download on the LINDO Systems website You will need a connection to the internet open for this command to work When issuing the AutoUpdate command or starting a version of LINDO with AutoUpdate enabled LINDO will search the internet to see if an updated version of the LINDO software is available for download If you currently have the most recent versio
167. amount of penalty you would have to pay to introduce a variable into the solution Reduced costs are valid only over a finite range of a variable s value Next in the rows section of the report you will find one line for each constraint giving the constraint s name if any its slack or surplus value and its dual price The SLACK OR SURPLUS column tells you how close you are to the right hand side limit on each constraint This quantity on less than or equal to constraints is generally referred to as slack Similarly on greater than or equal to constraints it is called a surplus If a constraint is exactly satisfied as an equality the SLACK OR SURPLUS value will be zero If a constraint is violated as in an infeasible solution the SLACK OR SURPLUS value will be negative Knowing this can help you find the violated constraints in an infeasible model The LINDO solution report also gives a DUAL PRICE figure for each constraint You can interpret the dual price as the amount by which the objective would improve given a unit of increase in the right hand side of the constraint Dual prices are sometimes called shadow prices because they tell you how much you should be willing to pay for additional units of a resource Dual prices are meaningful in linear and quadratic models but in general should be ignored when solving integer programming models WINDOWS COMMANDS 73 As with reduced costs dual prices are valid only over a limited range T
168. ample we use successive PIVOT commands to optimize a small model look all MAX 9 XI X2 4 X 3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 XL X2 2 X3 KA 2X5 XO 2 KT 3 X8 lt 12 3 XL 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 4 Xl X3 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END piv x1 Here we pick the variable X1 ENTERS AT VALUE 3 0000 IN ROW 5 OBJ VALUE 27 0 piv x2 2 Now we pick both the variable and the row X2 ENTERS AT VALUE 2 0000 IN ROW 2 OBJ VALUE 43 0 piv Let LINDO make the selection X8 ENTERS AT VALUE 4 0000 IN ROW 4 OBJ VALUE 27 0 piv X3 ENTERS AT VALUE 15 000 IN ROW 7 OBJ VALUE 27 0 piv X5 ENTERS AT VALUE 3 0000 IN ROW 7 OBJ VALUE 15 0 piv X6 ENTERS AT VALUE 1 6667 IN ROW 8 OBJ VALUE 13 0 piv X3 ENTERS AT VALUE 3 0000 IN ROW 3 OBJ VALUE 15 0 piv LP OPTIMUM FOUND AT STEP 7 OBJECTIVE FUNCTION VALUE 1 15 00000 6 Problem editing ALTER Syntax ALTER lt Row gt lt VarName DIR NAME RHS gt The ALTER command can be used to change the sense of the objective function MAX or MIN change the coefficient of a variable including the sign change the direction of a row lt 5 or gt
169. andard input to the script file built earlier in the application and redirect all standard output to an additional file Thus the running of LINDO is entirely automated with no output appearing on the screen By doing this we make it appear as if LINDO is seamlessly integrated into the front end C application This example should be portable to other C compilers and hardware platforms with only minor modifications Nothing in the code is dependent on Microsoft Visual C Thus you should be able to make use of the above code with any standard C compiler using a command line version of LINDO on any hardware platform that allows for redirection of input and output INTERFACING 229 Conclusion We hope we have successfully demonstrated a number of ways to easily hookup your applications to LINDO without having to go to the effort of linking to the LINDO callable libraries Many users have been very successful in building impressive applications around LINDO utilizing the techniques demonstrated in this section However for the user that requires the maximum in performance and control the callable libraries are still the best bet We examine the use of these libraries in Chapter 8 LINDO Callable Libraries 230 CHAPTER 7 231 8 LINDO Callable Libraries Most versions of LINDO are available in a form that lets you link your own programs directly with LINDO The internal structure of LINDO is designed so you can combine LINDO with your code to
170. ariables 72 Solve command 6 27 58 60 Solve menu 27 58 70 Solver Status Window 6 7 59 104 226 fields 59 removing 54 Solvers 231 monitoring 7 Solving 6 11 23 54 58 151 218 iterations 8 11 maximum 8 spawnlp function 228 Splitting lines 13 Spreadsheets 143 145 Staff scheduling 67 example 21 24 Start command 224 250 Starting LINDO 4 Statistics command 27 80 81 205 6 Statistics reports 186 205 207 STATS 118 169 186 205 6 Strict inequality 12 SUB 15 18 19 117 163 64 SUBJECT TO 5 10 12 22 120 Submatrix 117 175 201 Subroutines Callable Libraries 182 user created 182 Subtraction 87 120 139 142 SUFFICIENT SET 61 183 Syntax 12 56 errors 6 10 13 60 81 271 names 12 optional modeling statements 15 21 System Operator 120 System parameters 44 INDEX 297 System requirements 3 T TABLEAU 116 135 37 Tableau command 27 63 96 98 116 TAKE 116 124 25 216 217 19 Take command 217 19 Take Commands command 26 35 36 102 124 Technical support v 2 183 Terminal width 55 117 181 TERSE 117 180 Terse mode 54 Text files 30 32 213 217 Text reports 78 89 91 207 Tile command 28 107 9 TIME 115 120 130 TITAN 117 175 195 TITLE 15 20 21 118 188 Title command 26 39 Tolerances 45 51 188 entering variable tolerance 53 final constraint tolerance 53 initial constraint tolerance 53 IP optimality tolerance 46 19
171. arning if the current solution is not guaranteed to be optimal or feasible Objective function value Primal values and reduced costs of all the columns Slacks or surpluses and dual prices of all the rows Iterations pivots required to find the solution Number of branches in the branch and bound algorithm for integer programming IP Determinant of the basis matrix for an integer program for integer programming IP determinants close to 1 are preferred because if the determinant is 1 or 1 and all right hand side values are integer then the solution will be naturally integer The SOLUTION command is often used with the TERSE command since TERSE prevents automatic output of the solution with GO SOLUTION may be used with DIVERT to print the solution or save it to a file See the DIVERT command for more information on this 128 CHAPTER 3 In the following example we use the TERSE command to suppress the automatic generation of a solution report solve the model and then use the SOLUTION command to request a report look all MAX 9 XI X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 Xl 3 X2 2 X3 3 X4 X5 2 X6 X7 XB lt 6 4 X1 X3 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END ters Suppress automatic solution report go Solve the model
172. asting into a report or loading into a text editor Capturing Lengthy Sessions As you are running LINDO output messages and reports are all directed to the Reports Window Each new line of output is appended to the end of all other output in the Reports Window The Reports Window can only maintain up to about 64 000 characters of data Once it approaches this limit LINDO deletes old output from the top of the Reports Window to make room for new output at the bottom This makes it impossible to maintain entire copies of lengthy sessions in the Reports Window If you would like to save a copy of an entire session that will exceed the 64 000 character limit simply open a log file before beginning your work and check the Echo to screen box in the Log Output dialog box All output to the Reports Window will also be directed in duplicate to the selected log file When you are finished running LINDO a copy of the output for the entire session will be contained in the log file Take Commands F11 LINDO for Windows supports a command language for building script or macro files The commands available are discussed in detail in Chapter 3 LINDO for Command line Environments If you build a script file you may run it using the Take Commands command As an example suppose we have a situation where we run a model using several values for a particular right hand side coefficient saving a solution after each modification to the right hand side The
173. bles and constraints The first requirement an objective is just what it sounds like a goal You have the choice of two goals MAX or MIN which stand for maximize and minimize In a typical business situation for instance you might want to maximize profit or minimize cost The first word in a LINDO model must be either MAX or MIN The formula you enter after the MAX or MIN is called the objective function In this example XYZ wants to maximize profit achievable with the limited labor and production facilities available We will let STD and DLX be our variables which are things you want LINDO to adjust to reach your maximum profit In this sample model STD has a profit contribution of 10 and DLX has a profit contribution of 15 These will be the coefficients Coefficients are any non variable numbers that appear before a variable The important thing is that because STD and DLX are not explicitly defined they are variable In LINDO the minute you use a variable in your model it exists You don t have to do anything other than enter it in a formula At the colon prompt for a new model you enter AX 10 STD 15 DLX followed by a return Your screen will look like the screen below LINDO COPYRIGHT C LINDO SYSTEMS ICENSED MATERIAL ALL RIGHTS RESERVED MAX 10 STD 15 DLX 2 LINDO is waiting for further input from you Now let s look at constraints In some ways constraints are the m
174. bles to be general integer This is particularly true when the variables are scattered throughout the formulation and therefore not all first in the formulation For efficiency reasons LINDO permutes all integer variables to the front of the internal ordering of variables Thus in solution reports the values of the integer variables will appear before the values of the continuous variables The GIN specification is saved when the model is saved with SAVE SMPS and the DIVERT LOOK ALL sequence If you wish to convert an integer program to a linear program with all continuous variables just enter GIN 0 This specifies that there are no integer variables When you specify integer variables with either INTEGER or the GIN command LINDO solves the problem using the branch and bound technique which is an algorithm to narrow the search for the best answer that satisfies the requirement for integrality The branch and bound technique however does slow the solution process considerably In the example above the continuous model solved in only one iteration while the integer version took 72 iterations This should illustrate that if time is of the essence in solving a large problem try to formulate it without using integer variables and round the fractional variables whenever possible Of course rounding may not always be viable In the continuous example above we would have had to round Z down to 8 Z rounded up to 9 would be infeasible whi
175. bsequent information output to the Reports Window will be sent This is useful for creating disk copies of solution and range reports These log files may then be read into other applications or sent to your printer 34 CHAPTER 2 When selecting the Log Output command you will be presented with the following dialog box Open Log File File Name Directories OK f x c lindo61 samples Cec LINDO jae Samples Cancel Help C Samptext Network List Files of Type Drives Echo to screen all files x ci J Append output This is a standard Windows file selection dialog box allowing you to select a particular file for logging output to There are two additional features to note however in the lower right corner of the dialog box These features are the two check boxes labeled Echo to screen and Append output The default is for both these boxes to be unchecked If you leave Echo to screen unchecked output will be directed only to the log file and not to the Reports Window Checking Echo to screen will cause output to be directed to both the log file and the Reports Window If you are directing a lengthy solution to disk using the Log Output command the process can be sped up substantially by suppressing simultaneous output to the Reports Window If Append output is not checked any preexisting log file under the selected name will be overwritten and lost LINDO will warn you if you
176. by pressing the Paste button One final important concept to keep in mind when using the Paste Symbol command is that LINDO must be able to compile see the Compile Model command the current model in order to build the variable and row name lists If LINDO can t compile the model the Paste Symbol command will not execute 58 CHAPTER 2 Select All Ctrl A The Select All command sets the current selection to be all the text in the current window That is all the text in the current window is highlighted This is useful when you want to copy the entire contents of a window and place it into a new window or into a different application Clear All El The Clear All command erases the entire contents of the current window This is useful when you want to clear old output from the Reports Window and start with a clean slate If you mistakenly run the Clear All command it may be reversed by the Undo command Choose New Font Use the Choose New Font command to select a new font in which to display or print the active window You may find it easier to read models and solution reports if you select a mono spaced font such as Courier 3 Solve Menu ei Reports Window Hel The Solve menu appears to the left and contains all the commands Solve Ctrl s which invoke the core LINDO solver These commands are discussed Compile Model Ctri below in detail Debug Ctrl D Pivot Ctrl Preemptive Goal Ctrl G Solve Ctrl S
177. can be sure the proper number is specified Also in the first row the QCP statistic indicates the row in which the first real constraint as opposed to first order conditions starts in a quadratic program The number of nonzeros gives another measure of model size In the second row the constraint nonzeros is the count when the objective function is not included If the constraint nonzeros are 1 or 1 the model tends to be easier to solve so this count is also given here The density is the fraction not the percent of the elements in the model that are nonzero If a decimal point in a coefficient is grievously misplaced it will manifest itself as either an extremely small number or an extremely large number Thus the largest and smallest coefficients in absolute value are given on the third line If either of these are out of line you are on the trail of a bug The fourth line of the statistics report gives the count of the number of rows of each type The GUBS Generalized Upper Bounds statistic is a measure of model simplicity It is an upper bound on the number of non intersecting constraints in the model If all the constraints were non intersecting the model could be solved by inspection by considering each constraint as a separate model A model for which the GUBS statistic is high relative to the row count tends to be easier to solve Finally the SINGLE COLS statistic is a count of the number of variables that appear in only one row
178. ce command you will be presented with the following dialog box Find Replace Find What OO Repl Replace With if eplace Replace All Close IT Match Case Help If you just want to find a string of text enter it in the box labeled Find What then press the Find Next button LINDO will begin the search from the current cursor position If you want to begin the search from the top of the file you may want to use the Go To Line command to reposition the cursor to the top of the file before issuing the Find Replace command 44 CHAPTER 2 If you want to replace the search string with an alternate string enter the replacement text in the box labeled Replace With then press the Replace button The first time you press replace LINDO will simply jump to the next occurrence of the search string Pressing replace again causes LINDO to replace the search string and then jump to the next occurrence Using this technique you can pass through a window clicking Replace when you want to replace an occurrence or Find Next when you want to skip to the next occurrence without replacing the current selection To replace all occurrences of the search string from the current insertion point to the end of the document press the Replace All button LINDO will inform you as to the total number of replacements performed The default action for LINDO when doing a Find Replace is to ignore the case of the text If y
179. cessing phase LINDO defaults to performing the preprocessing step Preferred Branch When LINDO is solving an IP one of the fundamental operations of the solver is branching Initially the solver lets the integer variables be continuous The solver searches the integer variables to determine if any have fractional values If they do then the solver will branch on these variables When the solver branches on a variable it fixes its value to one of the nearest integer values and continues Since there are always two close integer values to any fractional number one just greater than and one just less than the fractional number LINDO can initially branch either up or down With some models always branching up first or down first can make a large difference in performance The Preferred Branch option gives you control over this parameter The default is for LINDO to pick the branching direction In this case LINDO will make an educated decision whether to branch up or down by analyzing the direction that looks most favorable IP Optimality Tolerance The IP Optimality Tolerance command specifies a fraction f that indicates the solver should only search for solutions with objective values 100 f better than the best one considered so far For command line users of LINDO the IP Optimality Tolerance corresponds to the IPTOL value The end results of using an IP Optimality Tolerance are twofold First on the positive side solution times will
180. ch yields an objective of 104 Solving as a true integer problem we get a considerably different solution Y 2 Z 3 with an objective of 107 QCP Syntax QCP lt FirstRealConstraint gt The QCP command is used in conjunction with quadratic programs The QCP command identifies the end of the first order condition constraints and the beginning of the real constraints The QCP command is entered to the command level colon prompt after the model s END statement In conjunction with the QCP command you must enter an integer specifying the index of the first real constraint that immediately follows the first order condition constraints Acceptable argument values for the QCP command run from 2 to one more than the number of constraints COMMAND LINE COMMANDS 173 The QCP specification is saved with the model when you use the SAVE command or a DIVERT LOOK ALL sequence The QCP specification is NOT supported under the MPS file format and will not be saved in model files created with the SMPS command For additional information on the use of the QCP command please turn to the Chapter 5 Quadratic Programming on page 196 PARAMETRICS Syntax PARAMETRICS lt Row gt lt NewRHS gt The PARAMETRICS command does a parametric sensitivity analysis on the right hand side of one constraint for linear and quadratic programs It allows you to automatically trace out the effect of varying a right hand side over a wide range You specify a new right hand s
181. close a model or exit and reenter LINDO that you need to read in a basis If for some reason you want to eliminate the basis in memory to force LINDO to start from scratch on a model you can do this by issuing the Compile Model command in the Solve menu When you explicitly issue the compile command LINDO discards all internal basis information WINDOWS COMMANDS 39 Title Shift F3 You can use the Title command to give a title to a model In the following example we have dubbed the model Product Mix Example IS C LINDO61 SAMPLES WPRODMIX LTX a EX TITLE PRODUCT MIX EXAMPLE MAX 20X 30Y ST E lt 50 Y lt 260 Et 2 lt 120 END Running the Title command will cause the model s title to be sent to the Reports Window and you should see the following 13 Reports Window mf oe PRODUCT MIX EXAMPLE The Title command is useful for attaching headers to solution reports Otherwise it is sometimes difficult to look at a solution and determine its corresponding model Date Shift F4 The Date command sends the current date and time to the Reports Window This is useful for date stamping reports Sample output from the Date command appears below 3S Reports Window ES IMM DD Y Y 0871572002 HH MH 55S 09 43 29 40 CHAPTER 2 Elapsed Time Shift F5 The Elapsed Time command sends the elapsed runtime of your current session to the Reports Window This command is useful when you want to determ
182. clude Headers I Comma Delimited Graphics Format rf iraphT ype C Area f Bar Line Pie Paint H Graph Title optional m Note we have specified Columns for our orientation we ve placed the condition Z 1 in the Condition box and requested a text report The key in both cases is the use of the Z 1 condition This restricts the report to columns with one nonzero element As another example suppose we have a model that doesn t seem to be behaving correctly and we suspect a problem in the formulation So we generate a statistics report and find the following ROWS 85 VARS 84 NO INTEG NONZEROS 1848 CONSTRAINT NONZ 1680 1680 ARE SMALLEST AND LARG EST EIL EMEN 1 NO SING lt 84 NO E COLS 0 NO ER VARS 0 QCP 0 1 D ENSITY 256 TS IN ABSOLUTE VA UE 1 gt 0 OBJ MAX GUBS 0 5 0 lt 4 208 CHAPTER 6 The fact that we have a column with one nonzero SINGLE COLS 1 looks suspicious Fortunately we happen to know that every variable whose name is such that its first character is R and its third character is C should have 21 nonzero elements It is easy to check this feature with Reports Peruse Windows command or the CPRI command line command In Windows we fill out the Reports Peruse dialog box as follows Peruse Model r Report Parameters Help Cancel
183. command and find that LINDO concurs with our analysis J Reports Window DER Aa X ENTERS AT VALUE 50 000 IN ROW 2 OBJ VALUE 2050 0 v 0 g Running the Tableau command once again reveals iS Reports Window ox THE TABLEAU ROW BASIS SLE lt 2 SLE 3 1 ART 5 00 0 000 2050 000 2 X 1 00 0 000 50 000 3 SIK 0 50 1 000 35 000 4 Y 0 50 0 000 35 000 98 CHAPTER 2 At this point there are no further attractive variables to introduce into the solution Therefore we have arrived at the optimal solution Issuing one more Pivot command verifies this 3 Reports Window aor A LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1 2050 000 VARIABLE VALUE REDUCED COST X 50 000000 0 000000 35 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 0 000000 5 000000 35 000000 0 000000 0 000000 15 000000 NO ITERATIONS 2 The Tableau report complies with the Terminal Width parameter Thus if the width of the report exceeds the Terminal Width parameter it will be continued on additional pages You may want to increase the Terminal Width using the Edit Options command to maximize the amount of output per page in the report Formulation Alt 8 The Formulation command is used to display all or selected segments of your model in the Reports Window Longtime LINDO users will recognize the Formulation command is equivalent to the LOOK command in command line versions The Formulation command displays LINDO s internal
184. create an integrated customized system Furthermore you can design your system such that the user is unaware that a linear programming solver is being accessed The LINDO callable libraries come in two forms Windows versions of LINDO provide the callable libraries in Windows Dynamic Link Library DLL format The advantage of this format is that it is compiler independent Thus you may build your application in any language that can call a DLL effectively all current PC compilers On all other platforms the callable libraries are supplied in linkable object code library format In order to link successfully with an object code library you must use the same development environment that is used in creating the object code library For information on the required development environment for the version of LINDO that you are using please refer to the Installation Instructions provided with your copy of LINDO We proceed next by introducing the routines available through the LINDO library After this we will illustrate the use of the libraries through a number of examples LINDO Callable Library Routines Introduction The LINDO library routines reproduced below in alphabetical order can be called from your program to access LINDO s functionality There are many routines listed below but most are concerned with one of three areas 1 passing a model to LINDO 2 solving a model and 3 retrieving the solution to a model If a routine has an
185. d line commands 116 144 51 File types 32 Compressed files 116 External files 115 FBS File Basis Save 37 38 LINDO text Itx 30 32 Macro files 35 115 MPS Punch files 30 32 33 37 38 116 146 150 51 Packed files lpk 30 32 33 121 144 Save DataBase Column SDB 37 38 116 Script files 35 115 116 124 Text files Itx 145 146 148 213 217 Final constraint tolerance 51 53 188 Find Next button 43 Find Replace command 26 43 44 212 FINS 116 123 First order conditions 197 199 Formatting 58 100 saving 33 Formulating a model 267 268 Formulation command 27 98 100 216 FORTRAN 231 264 INDEX 293 sample application 259 63 FPUN 116 146 Fractions 269 FREE 15 16 117 164 165 15 16 G General integers 15 16 117 170 191 206 General Optimizer Options 51 54 Generalized upper bounds 187 206 Getting Started with LINDO 1 24 GIN 15 16 17 117 135 170 72 191 GLEX 117 Global optimums 101 175 201 GO 11 23 117 151 158 218 Go To Line command 26 55 Goal programming 67 155 Graphic reports 78 81 83 89 91 printing 93 Greater than 12 87 139 142 H Handler code 246 HELP 115 118 Help menu 28 110 14 How to Use Help command 28 110 I ILINDO 255 261 Infeasible models 61 118 151 152 53 158 183 184 85 Information command line commands 115 118 20 INIT 255 261 Initial constraint tolerance 51 53 188 Input com
186. d tests the matrix defined by rows 2 through k excluding the dual variables required by the first order conditions to the quadratic program The POSD command also checks to see that the submatrix is symmetric and computes the rank of the submatrix For more details on the use of the POSD command please turn to page 201 TITAN The TITAN command adds valid constraints to a binary or general integer program to allow for faster solving These additional constraints commonly called cuts do not violate the integer feasible region of the model They do however intentionally violate the linear feasible region of the model with smaller or new upper bounds and smaller constraint coefficients The idea is that we should be able to get tighter objective bounds and or less fractional solutions during the branch and bound procedure used by the LINDO solver on integer programs The result being faster solution times The TITAN command directly modifies the model currently in memory The new constraints in the form of added bounds on the variables are displayed after TITAN is entered 176 CHAPTER 3 The following graph pertains to the example below and illustrates how these cuts can reduce time spent searching for a solution 0 1 2 3 4 a IP Feasible Linear Feasible Region Region The bounds created by the TITAN command remove from the search the areas outside the IP Feasible region A reference for the theory behind TITAN is
187. debugging your LINDO application it can be useful to view the contents of this LINDO output file for any helpful system messages In order to enhance performance the QUIET routine is called next Call QUIET 0 This puts LINDO into terse output mode minimizing the amount of output to the log file In general it is best to leave out this call to QUIET until you have completely debugged your application By suppressing LINDO s output you may miss important system messages As mentioned above you must make one call to DEFROW for each row in your model The following code does this once to define the objective row and three more times in a loop to define the three constraints Define objective row Call DEFROW 1 O Idrow Trouble Define constraint rows Rhs 1 50 Rhs 2 60 Rhs 3 120 For I 1 To 3 Call DEFROW 1 Rhs I Idrow Trouble Next I Once the rows are defined APPCOL can be used to define the columns Here APPCOL is called twice once for each column in our model Define column Xx Iro 1 1 Iro 2 2 Iro 3 4 Value 1 20 Value 2 1 Value 3 1 Kname X n Nonz 3 Call APPCOL Kname Nonz Value 1 Iro 1 Trouble Define column Yy Iro 1 1 Iro 2 3 Iro 3 4 Value 1 30 Value 2 1 Value 3 2 Kname Y Hi Nonz 3 Call APPCOL Kname Nonz Value l1 Iro 1 Trouble INSROW or INSERT could also have been used to add the nonzero elements
188. ding the file LINDO will prompt you for the row number of the objective function If the model has more than one objective then it suggests which row or rows are candidates to be the objective function This is usually row 1 Following this LINDO prompts you as to whether the model is a maximization or a minimization If the MPS file was created with LINDO s SMPS command this information appears on the screen in the title block in parentheses Finally when the model is created in memory the STATS command automatically displays various statistics about the model The RMPS command erases any current model in memory and any current solution The model that is read from the RMPS file becomes the current model in memory You create MPS files with LINDO s SMPS command The MPS format files generally take longer to read and write than the SAVE RETRIEVE format TAKE Syntax TAKE lt FileName gt The TAKE command is used to execute a file from disk which contains a sequence or script file of LINDO commands stored in a text file These files are automatically open and the commands in them executed A file used with the TAKE command can include any valid LINDO command except the command TAKE See the Take Commands command in Chapter 2 LINDO for Windows for why these files would be useful The syntax of the TAKE command is TAKE lt FileName gt where lt FileName gt is the name of the file you wish to retrieve If you enter nothing after
189. double pointer there is a single pointer cover function for each routine that passes character data To call the single pointer cover function simply place the letter X on the end of the routine name For example APPCOL expects a character name passed by placing a pointer to a pointer on the stack Calling APPCOLX accesses the single pointer cover function All LINDO routines conform to the cdecl standard where the calling routine is responsible for cleaning up the stack However many development environments use the standard calling convention where the called routine is responsible for cleaning the stack If you re compiler can t generate cdecl calls then LINDO provides standard call cover functions for each of its routines To access the standard cover function for each routine append the string STD to the end of the routine s name As an example the standard cover for APPCOL has the name APPCOL STD Integer Programming User Interface LINDO uses the branch and bound search technique to solve IP models This search technique finds a series of better and better integer solutions until optimality is reached You have the option of providing a callback routine to monitor and influence the branch and bound procedure The LINDO DLL searches for a 32 bit DLL called NEWIP DLL when it starts If this file is present LINDO will call NEWIP DLL whenever a new integer solution is found and pass two 32 bit float values by reference We call
190. down to the smallest subset of constraints leading to an infeasibility 3 EXAMPLE OF AN UNBOUNDED SOLUTION When LINDO can improve on the objective without bound then the model is said to be unbounded In most applications this would imply a problem in the formulation because in the real world we know it is impossible to maximize profit without limit A sample of an unbounded model follows look all AX X Y SUBJECT TO 32 Xot Y gt s 2 END 154 CHAPTER 3 go UNBOUNDED SOLUTION AT STEP 0 UNBOUNDED VARIABLES ARE SLK 2 X OBJECTIVE FUNCTION VALUE 1 2 00000000 VARIABLE VALUE REDUCED COST Xx 99999900 000000 000000 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 3 000000 1 000000 NO ITERATIONS 0 There are a couple things you can do to help trace down the problem in your model when it is unbounded First off when a model is unbounded LINDO helps by showing you the variables it can increase to drive the objective to infinity by assigning them a large value in the solution report In the above example we can see_X is one of the culprits due to its large value of 99999900 Secondly you can make use of the DEBUG command The DEBUG command can find a minimal subset of variables which may be used to force the objective to infinity See the DEBUG command for more information 4 EXAMPLE OF HITTING THE PIVOT LIMIT If an upper limit on the number o
191. e DELETE command changes the current model by deleting the constraints you choose The current solution is no longer considered optimal after DELETE is used If you to want to clear an entire model from memory and begin entering a new one simply use the MAX or MIN command LINDO will clear the old model from memory before parsing the new one SUB Syntax SUB lt Variable gt lt UpperBound gt The SUB command specifies a simple upper bound for one variable In other words the SUB command can be used to represent constraints of the form X lt b where b is some constant non negative bound LINDO uses these upper bounds directly in the simplex algorithm Alternatively you can specify simple upper bounds as an explicit constraint in the model However LINDO can solve the model more efficiently when the SUB command is used in place of an explicit constraint The form of the command is SUB lt Variable gt lt UpperBound gt where lt Variable gt is the name of the variable you want to bound and lt UpperBound gt is a non negative bound value The value for lt UpperBound gt must be non negative even if you use the FREE command with the variable However a negative upper bound can be specified in an explicit constraint in the model for variables designated as FREE If you type nothing after the SUB LINDO will prompt you for lt Variable gt and lt UpperBound gt 164 CHAPTER 3 The FREE command and SUB are contradictory If you use
192. e JCOL to the value SUB For efficiency use this feature in place of explicit constraints of type X lt SUB The SETSUB routine corresponds to the SUB command line command STATS IDTAIL Integer IDTAIL Sends a model statistics report to the standard output device If DTA L gt 0 then it prints the model statistics report as would be generated by the STATS command If JDTA L 0 then it just checks the model for poor scaling To capture the output from STATS in a file see the CAPOUT command General Application Requirements There is a wealth of different applications one could devise using the LINDO libraries Fundamentally all these different applications will as a minimum always perform the following functions building a model solving the model and retrieving the solution to the model The primary routines used to perform each of these functions are listed below Operation Routines Called Building a Model INIT DEFROW APPCOL Solving a Model GO Retrieving the Solution REPROW REPVAR Almost every application linking to LINDO will make extensive use of the routines listed above Some small examples are illustrated below LINDO CALLABLE LIBRARIES 245 Sample Matrix Generators As a simple example suppose we want to use the callable libraries to pass LINDO the following two variable model Maximize 20X 30Y Subject to xX lt 50 Y lt 60 X 2Y lt 120 End In addition we want to solve the mode
193. e SLB command specifies a simple lower bound for one variable In other words the SLB command can be used to represent constraints of the form X gt b where b is some constant bound and may be negative LINDO uses these lower bounds directly in the simplex algorithm Alternatively you can specify simple lower bounds as an explicit constraint in the model However LINDO can solve the model more efficiently when the SLB command is used in place of an explicit constraint The form of the command is SLB lt Variable gt lt LowerBound gt where lt Variable gt is the name of the variable you wish to bound and lt LowerBound gt is any real number If you type nothing after the SLB LINDO will prompt you for lt Variable gt and lt LowerBound gt The FREE command and SLB are contradictory If you use the FREE command with a variable after using the SLB command with that variable the lower bound is erased If you wish to remove a SLB on a variable and return to the default lower bound of 0 enter the command SLB lt Variable gt 0 Every variable except a variable specified as FREE can have a SLB The SLB specification does not count against LINDO s constraint limit Whereas an explicit constraint does The SLB specification is saved when the model is saved with SAVE SMPS and the DIVERT LOOK ALL sequence The current solution is no longer considered optimal after SLB is used The following illustrates the use of the SLB co
194. e accessed by pressing the following button f Along with accessing commands by mouse and buttons most commands may also be accessed by a single unique keystroke known as an accelerator The equivalent accelerator key is listed alongside each command in the menus For example the File New command for creating a new Model Window can be accessed by pressing the F2 function key 1 File Menu Edt Solve Reports Window Help At left is a picture of LINDO s File menu This menu contains N F2 eu as commands which generally pertain to the movement of files and data View F4 in and out of LINDO Each of the commands contained in the File ae a menu are discussed below in depth Save As F Close F7 Print Fe Printer Setup F9 Log Output F10 Take Commands Fil Basis Read Fi2 Basis Save Shift F2 Title Shift F3 Date Shift F4 Elapsed Time Shift F5 License Exit Shift F6 New F2 The New command opens a new blank Model Window You may enter a model directly into this window or paste in text from the clipboard Open F3 The Open command reads a saved model file from disk and places it in an Edit Window All of the standard editing features e g cut copy and paste are available in an Edit Window The one restriction is that Edit Windows are limited to handling files with up to 64 000 characters If you try to open a file that is too large for an Edit Window to handle you will be prompted to place
195. e graph you are trying to create has too many data points selected Try reducing the number of data points contained in the graph by restricting the selection more 1033 The selected row was not found in the current model You have specified a row name or row number that is not valid for the current model Please check the row number and retry the command 288 APPENDIX A 1034 The right hand side value is invalid You have specified a right hand side value for a constraint that is not valid Please be sure that you have entered a numeric value and retry the command 1035 Right hand side parametrics are not permitted on the objective row You have attempted to perform right hand side parametric analysis on the objective row which is not allowed Either cancel the command or specify a constraint 1036 The current solution is non optimal Please use the Solve command first You have attempted to issue a command that requires a model that has been solved to optimality Please issue the Solve command first and then retry 1037 Not enough memory to generate graph There was insufficient free memory to complete a request for graphics Try closing other applications and any windows not currently required Also if you have allocated an excessive amount of nonzero elements using Edit Options there may be inadequate memory available for other functions 1038 The current model is not compiled Please choose the Solvel Compile Model command first
196. e request later Error writing to LOG file Perhaps the disk is full LINDO has encountered an error writing to a log file open with the File Log Output command The most likely cause is a full disk If the disk your writing to is not full then you may want to run diagnostics on it to determine if there is a problem 286 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 APPENDIX A Unable to open the LOG file The log file specified as part of the File Log Output command could not be opened Perhaps you have specified an invalid drive or a drive to which you do not have write access Try reissuing the command using a different drive Error writing to basis file An error has occurred writing to a basis file opened with the File Basis Save command Perhaps the disk is full or you do not have write access to the drive Not enough memory for Find command LINDO did not have enough memory to execute an Edit Find command Try closing other applications and any windows not currently required Also if you have allocated an excessive amount of nonzero elements using Edit Options there may be inadequate memory available for other functions lt Not In Use gt Cannot save file lt FileName gt LINDO was unable to perform a successful save of the file lt F ileName gt Perhaps the disk is full or you do not have write access to the specified drive Error writing to file lt FileName gt An error occu
197. e wrong the coefficient may be in the wrong constraint or the decimal point may be misplaced The main commands for analyzing a model in Windows versions of LINDO are Reports Statistics Reports Peruse and Solve Debug In command line versions of LINDO the corresponding commands are STATS CPRI RPRI and DEBUG respectively Model Statistics The Reports Statistics command in Windows versions of LINDO and the STATS command in command line versions gives some simple summary statistics on the current model We illustrate this command by entering the following example into LINDO MAX 2x 3y 2z ST 4x 3z lt 8 2x 3y z 1 4x 3y 2z gt 2 END INT 2 The statistics report for this model is as follows ROWS 4 VARS 3 INTEGER VARS 2 2 0 1 QCP 0 NONZEROS 14 CONSTRAINT NONZ 8 1 1 DENSITY 0 875 SMALLEST AND LARGEST ELEMENTS IN ABSOLUTE VALUE 1 0 8 0 OBJ MAX NO lt gt 1 1 1 GUBS lt 1 VUBS gt 0 SINGLE COLS 0 REDUNDANT COLS 0 206 CHAPTER 6 Most statistics in the report are straightforward such as the row and column counts in the first row If a variable name is misspelled it should manifest itself via a column count that is greater than expected This is because the typical variable appears several places in the model but is misspelled only one place The number of general and binary integers is also shown here so you
198. ecall LSAVPR to perform the actual save LSCTTN Adds valid constraints to a binary or general integer program to allow for faster solving These additional constraints commonly called cuts do not violate the integer feasible region of the model They do however intentionally violate the linear feasible region of the model with smaller or new upper bounds and smaller constraint coefficients The idea is that we should be able to get tighter objective bounds and or less fractional solutions during the branch and bound procedure used by the LINDO solver on integer programs The result being faster solution times LSCTTN corresponds to the TITAN command line command LSDLVR J Integer J Deletes variable J from the current formulation Variables with indices greater than J are all shifted down by one If you don t know the index of a variable you can have LINDO look it up with the NDXOFV routine LSDMPS LUNIT Integer LUNIT Sends an MPS format solution report to the file associated with output unit LUNIT To associate a file with a unit number see routine LUNOPN or LUNGET LSDMPS corresponds to the DMPS command line command LSEXIT Call LSEXIT when your application no longer needs access to the LINDO routines LSEXIT performs LINDO s final clean up LSFETP DERR NPARAM DPVAL Double DERR DPVAL Integer NPARAM Fetches the values of internal parameters used by LINDO For a list of these parameters
199. ecking for certain types of errors in your model For example if there is a misspelled variable name you may find SINGLE COLS when you don t expect any If you misplace a decimal point in the model you may find a variable with an unexpectedly large or small absolute value The STATS command for the following model is illustrated below look all MAX 9 XL X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 XL X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 Xl 3 X2 2 X3 3 X4 X5 2 X6 X7 XB lt 6 4 X1 X33 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END GIN 8 stats ROWS 8 VARS 8 INTEGER VARS 8 0 0 1 QCP 0 NONZEROES 47 CONSTRAINT NONZ 32 23 1 DENSITY 653 SMALLEST AND LARGEST ELEMENTS IN ABSOLUTE VALUE 1 00000 15 0000 OBJ MAX NO lt gt 2 5 0 GUBS lt 2 VUBS gt 0 SINGLE COLS 0 REDUNDANT COLS 0 188 CHAPTER 3 SET Syntax SET lt ParameterlD gt lt NewValue gt The SET command can be used to set internal parameters used by LINDO These parameters are Param ID Description final constraint tolerance initial constraint tolerance 3 entering variable reduced cost tolerance 4 fixing threshold for integer variables pivot size threshold whether LINDO should do integer program preprocessing 0
200. ecl standard 264 Choose New Font command 26 58 292 INDEX Class Wizard 253 Clear command 26 43 58 Close command 26 28 33 109 Coefficients 9 changing 159 202 determining 120 matrix 193 nonzeros 81 186 objective 24 Column to Show list box 101 Columns lt ColAttList gt 139 limits 267 matrix 2 redundant 81 187 statistics 206 COM 115 119 Command language 102 Command line Entering a model 8 12 prompts See also Prompts Command menus 25 Command window 2 33 103 Command line commands 115 89 in brief 115 18 in depth 118 89 Commands Windows 25 114 in brief 25 28 in Depth 29 115 Comments 13 117 181 saving 32 Compile Model command 27 57 60 Compilers 228 Compiling the model 6 Conditional selection 139 142 Constraints 4 9 12 57 81 121 adding 117 161 additional cuts 175 binding 203 crucial 183 210 defining 5 first order conditions 20 172 197 199 generalized upper bound 81 infeasibilities 53 limits 19 164 naming 12 nonbinding 267 real 20 172 197 removing 117 submatrix 101 syntax 13 14 variable upper bounds 81 Contacting LINDO v 2 Contents command 28 110 Continuous variables 16 Conventions 2 Conversational Parameters command line commands 117 180 82 Convexity 101 201 Copy command 26 42 Copyright ii 8 Covariance matrix 201 CPRI 116 138 41 206 8 218 Crucial constraints 61 183 210 Cut
201. ed in a TAKE file to automatically change parts of the model This is an advanced use of LINDO and the TAKE command See the TAKE command for more information Suppose you want to change 2 WRNG in the objective function to RGHT in the following model MIN 130 X 120 Y 100 Z 2 WRNG SUBJECT TO 2 XY 42 gt g 3 2X 8 y ILZ 4 WRNG X Y gt 6 END At the LINDO command prompt type the commands listed in the session below alter This is the Alter command ROW LINDO asks what row we want to change 1 We want to change Row 1 VAR LINDO asks which variable we want to change wrng We want to change 2 WRNG to RGHT NEW COEFFICIENT LINDO asks for the new coefficient on Ithe variable 0 Enter a zero to get rid of it COMMAND LINE COMMANDS 161 alt Do another ALT to put RGHT in the obj ROW LINDO asks what row number we want to change 1 We want to change Row 1 again VAR LINDO asks the variable we want to change rght We want to add RGHT to the row VARIABLE NOT USED IN THIS PROBLEM BEFORE WANT IT INCLUDED y Yes we want to include it NEW COEFFICIENT LINDO prompts for the new coefficient 1 The new coefficient for the RGHT variable look all Run LOOK 1 to examine the altered objective MIN 130 X 120 Y 100 Z RGHT EXTEND The EXTEND command is used to add constraints to the model currently
202. eeeeeeeeeeeeeeee 211 Using External Editors with LINDO ic 20 si 0d dra iael nese nie earns 211 Running Command Scripts with the Take Command ccceeceeeeeeeeeneeeeeees 217 Integrating LINDO Into Other Applications cceceeeeeeeeeeeeeeeeeeeeeeeeeeeteneeeeeees 219 MCOMCIUSION sss coe fa ete a ote estate ne ce Nita tn E E meee rene ens 229 Ch 8 LINDO Callable Libraries cssesssseeeeneeeeeneeeeeeeeeeeneneeeeenenees 231 LINDO Gallable Library Routines s 0 4 000ie a ree eA eh 231 General Application Requirement cccccccccceeeeseeceeeeeeceeeeeeeeneeeseeneeeeeeeeneeees 244 Sample Matrix Generators s c c ce eee hen ee ee 245 Calling LINDO from Other Languages c ccceccceeeeeeeeeeceeeeeeeneeeteeseeeeeeeeneeeees 264 Integer Programming User Interface ccccceeeeeeeeeneeeeeeeeeeeeeeeeeeeeeeeeeeneeeneees 264 Monitoring the SOIVER ses oS cd ckitcns ie h suttel oscil lade eae eects ace chiara 265 Ch 9 Numerical Considerations ccccceseneneeeeeeeeeeeeeeneeeeeeeeeeeeeeeeeeeees 267 Appendix A Error Messages List eeee ttt i i innninninieeeeeeeeeeenees 271 Vv Preface We have been busy adding a number of features to LINDO since the manual was last published The most significant new features include Windows version with pull down menus full editing capabilities graphics and numerous additional user friendly features Improved performance and robustness
203. efficient and the row s dual price appear Finally at the bottom of the report the column s value and reduced cost are printed Positive Definite The Positive Definite command is used with quadratic programs to check whether you have a guarantee of global optimality It examines the submatrix of constraints corresponding to the quadratic form to determine whether this submatrix is positive definite If the matrix is positive definite then the objective function of the quadratic program is convex and the solution found is guaranteed to be a global optimal solution For more detailed information on the POSD command please see page 201 102 CHAPTER 2 5 Window Menu Open Command Window Alt C Open Status Window Send to Back Ctrl B Cascade Alt A Tile Alt T Close All Alt x Arrange Icons Alt I v 1 C ALINDO61 Samples Tests Itx Open Command Window LINDO has a macro or command language for accessing its features For more details on the command language refer to the next chapter LINDO for Command line Environments A script file containing LINDO commands may be run using the Take Commands command in the File menu Alternatively you can interactively enter commands into LINDO s Command Window To bring up the Command Window issue the Open Command Window command The following window should appear on your screen The Window menu is shown at left and contains all the commands which assist in managing the vari
204. elease the mouse button The selected text should now be displayed in reverse video white letters black background Selecting the Copy command at this point will cause the selected text to be inserted into the Windows clipboard For pasting this text refer to the Paste Command below Paste Ctrl V The Paste command pastes text from the Windows clipboard into an Edit Window at the current cursor position If text is currently selected when the Paste Command is given the selected text will be erased and replaced with the clipboard text In LINDO you may only paste text Graphics are not allowed Clear Del The Clear command is used to remove selected text from an Edit Window To select text for clearing simply press down on the left mouse button at the beginning of the text and then drag the mouse pointer to the end of the text and release the mouse button The selected text should now be displayed in reverse video white letters black background Selecting the Clear command at this point will cause the selected text to be removed from the window Note The text that is removed using the Clear command or Delete key is not placed into the Windows clipboard and will be lost once cleared The Undo command is not available here Find Replace Ctrl F The Find Replace command can be used in either an Edit or a View Window to find some specified string of text and optionally replace it with an alternate text string After invoking the Find Repla
205. element is very small then given the finite precision of floating point arithmetic there is a chance that the pivot element is merely a phantom consisting of noise due to round off error Assigning a variable to such a row can lead to erratic behavior of the solver The Pivot Size Tolerance is the dividing line between what is and what is not considered a valid pivot element The default value for the Pivot Size Tolerance is 1 Output Options These options allow you to control the amount and type of output you receive from LINDO Status Window Whenever LINDO begins solving a model it displays a Solver Status Window which allows you to monitor the progress of the solution process If you want to keep this window from popping up every time you solve a model remove the checkmark from the Status Window box in the Options dialog box as illustrated here Dutput I Status Window U I Tarna ite Once you have done this LINDO no longer automatically displays the Solver Status Window However you can still bring it up at any time during optimization by issuing the Open Status Window command from the Window menu See Window menu section below LINDO defaults to displaying the Solver Status Window whenever the Solver is invoked Terse Output Unless specified otherwise in the Options dialog box LINDO automatically routes a solution report to the Reports Window after solving a model You can suppress this by checking the Ter
206. ent to complete our model which should resemble the following MIN X Y Z UNITY RETURN XFRAC YFRAC ZFRAC ST First order condition for X 6X 2 Y Z UNITY 1 3 RETURN XFRAC gt 0 First order condition for Y 2X 4Y 0 8 Z UNITY 1 2 RETURN YFRAC gt 0 First order condition for Z X 0 8 Y 2 Z UNITY 1 08 RETURN ZFRAC gt 0 I SSS Sse Start of real constraints Budget constraint multiplier is UNITY X Y 2 1 Growth constraint multiplier is RETURN 1 82 Xo 1s2 Y 108 eS E2 Max fraction of X multiplier is XFRAC x lS Max fraction of Y multiplier is YFRAC YO SID Max fraction of Z multiplier is ZFRAC Z lt 75 END QCP 5 After solving our model we obtain the following solution OBJECTIVE FUNCTION VALUE 1 0 4173749 VARIABLE VALUE REDUCED COST X 0 154863 0 000000 Y 0 250236 0 000000 Z 0 594901 0 000000 UNITY 0 834750 0 000000 RETURN 0 000000 0 024098 XFRAC 0 000000 0 595137 Y FRAC 0 000000 0 499764 ZFRAC 0 000000 0 155099 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 0 154863 3 0 000000 0 250236 4 0 000000 0 594901 5 0 000000 0 834750 6 0 024098 0 000000 7 0 595137 0 000000 8 0 499764 0 000000 9 0 155099 0 000000 NO ITERATIONS 7 Thus our portfolio variance is minimized at 417 by investing 15 5 in X 25 in Y and 59 5 in Z QUADRATIC PROGRAMMING 201
207. equivalent command line command we make note of it so you can refer to the command for additional insights The arguments to each of the routines listed below are enclosed in parentheses following the name of the routine Arguments to the routines are classified as Character Integer Float or Double The equivalents of these various argument types in Basic C C and FORTRAN are listed below Argument Type Basic C C FORTRAN CHARACTER INTEGER 4 REAL 4 REAL 8 232 CHAPTER 8 A pair of square brackets will follow arguments that are arrays If the length of the array is known it will be contained in the brackets Arrays of unknown length will contain no length value in their brackets One important fact to keep in mind when calling LINDO routines particularly from C C is that all arguments are passed by reference Library Routine Definitions APPCOL KNAME 8 NONZ VAL NONZ IRO TROUBLE Character KNAME Float VAL Integer NONZ IRO TROUBLE Appends a new column i e a variable whose name is stored in character format in the 8 byte character vector KNAME The column has NONZ nonzero coefficients stored in the float vector VAL with the associated row numbers stored in the vector JRO On return TROUBLE will be set to 1 if LINDO ran out of space else it will be 0 The contents of KNAME should be in uppercase KNAME must be declared as an array with at least 8 elements of character type If the name of the column has
208. er LUNIT Saves a model in a proprietary compressed format to the file associated with logical unit LUNIT The model may be restored using the RETR routine To associate a file with a unit number see routines LUNGET and LUNOPN This routine corresponds to the SAVE command line command SDBC LUNIT Integer LUNIT Writes a column report to the file associated with logical unit LUNIT To associate a file with a unit number see routines LUNGET and LUNOPN This routine corresponds to the SDBC command line command SETIPT TOL Float TOL Sets the acceptance tolerance for solving integer programs to the value of TOL For example if TOL 1 then the branch and bound search is terminated as soon as the unknown optimum is guaranteed to be no more than 10 better than the known incumbent SETIPT must be called after each call to INIT SETIPT corresponds to the IPTOL command line command SETQCP IROW Integer IROW Indicates that row JROW is the first real constraint in a quadratic programming model SETQCP corresponds to the QCP command line command 244 CHAPTER 8 SETSLB JCOL SLB Float SLB Integer JCOL Sets the simple lower bound of variable JCOL to the value SLB For efficiency use this feature in place of explicit constraints of the type X SLB The SETSLB routine corresponds to the SLB command line command SETSUB JCOL SUB Float SUB Integer JCOL Sets the simple upper bound of variabl
209. er variable LI lower bounded integer variable Only one RANGES set is recognized in the RANGES section GUB constraint types are not recognized MODIFY sections are not recognized SCALE lines are accepted but have no effect COMMAND LINE COMMANDS 151 Although embedded blanks are permitted in names in an MPS file they aren t recommended For example even though MY NAME is an acceptable name for a row in an MPS file the ALTER command couldn t be used to change elements in this row If you typed ALTER MY NAME LINDO would try to alter the element in row MY called NAME Lowercase names are permitted but for consistency also for ease in distinguishing between 1 one and L you should consider using only uppercase 5 Solution GO Syntax GO lt PivotLimit gt GO attempts to solve the model GO can take at most one parameter an optional integer limit on the maximum number of pivots you wish LINDO to make For example GO 100000 will invoke the LINDO solver with a limit of 100 000 iterations If you do not specify a pivot limit then the default limit is 3 M N 6 I M where M is the number of rows N is the number of columns and I is the number of integer variables The GO command does not affect the current model However it destroys the current solution and creates a new one GO generally starts with the current solution If you have a solution or basis for the current model saved to a file you may retrieve
210. es all the open windows in a cascade from upper left to lower right with the currently active window on top 106 CHAPTER 2 For instance suppose our windows are positioned as follows 35 LINDO File Edit Solve Reports Window Help A 3 warehouse 4 customer EWH lt i gt C lt j gt amount shippe PERI r J Peruse Graph ER CALINDOGIGampksiTRAN LTX SUBJECT Tq lt 1 gt Reports M NO ITERA NAME XWH1C1 XWH1C2 XWH1C3 XWH1C4 WINDOWS COMMANDS 107 After we issue the Cascade command our windows will be repositioned in the following manner maok 15 C _LINDO61 Samples TRA 13 Reports Window 3 Peruse Graph DAR CALINDOGISampksTRAN LTX Tile Alt T EB The Tile command arranges all the open windows in a tiled pattern Each window is re sized so all windows appear on the screen and are of roughly equivalent size You may choose to tile either horizontally or vertically LINDO will try and maximize the horizontal or vertical dimension of each window based on your selection If there are more than three open windows LINDO will tile the windows but the choice of horizontal or vertical will no longer have an effect 108 CHAPTER 2 For instance suppose we have three open windows positioned as follows 23 LINDO File Edit Solve Reports Window Help Heelse INDO61 Samples TRAN LTX i Reports Window I Peruse Graph CALINDO61 Samples TRAN LTX 3 E prmal XWH1IC2
211. es ntact es ee ed Aa acid viceeiv ees 183 QUES coisas itieeicie hs a ent ee ae needa a eee 189 Ch 4 Integer Programming sicn25 secs aac occupa coca psccehoaticcteces idciecepiccisdaciectsit 191 Branch and Bound Solution Method 0cccceeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeees 192 Solving Difficult Integer Programs cccecccceeeeececeeeseeeeeeeeeesaeeeseenseeeseenseeeees 193 Setting an Optimality Tolerance cceececeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeaeeeeeeenaaeeees 194 iv TABLE OF CONTENTS Exploiting a Known Good Solution to an IP ccceeeceeeeeeeeeeeeeeeeeeeeeeeeeeeeneaas 194 Benders DecornpmSiti Oris ccce et cata sco tei eee ce eile Siete psec oaks ane 194 Tightening Loose IP Formulations cccccceeeeeeeeeeeeceeeeeeeeeseseeeeeneeeeeeeeeees 194 Ch 5 Quadratic Programming sssssssssssnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn 197 Debugging Quadratic Programs and the POSD Command ccccceeeeeeeee 201 Parametric Analysis of Quadratic Programs cccccecceceeeeeeeeeeeeeeeeeeeeeteeeeeeeeee 202 Ch 6 Analyzing amp Debugging a Model ssssssssssnnnnnnnnnnnnnnnnnnnnnnnnnnnnns 205 Model StatistiCs scx e ea cies Sete nte lavas a Neen eee ee teense eh 205 Perusing a Model for Errorsi s s gesce 2s cata tien eee eee eee 206 Deb g Command o are eee ee a ear aE Ee na aaae na e dsa 209 Ch 7 Interfacing with the Outside World c ceeeeseeeeeeeee
212. ext we will use Microsoft Visual Basic to develop a simple front end to the Windows version of LINDO Finally we will use Microsoft Visual C C to develop a simple front end to a command line version of LINDO Inout and Output Redirection One of the simplest ways to tie LINDO into your application is by using input and output redirection This is a technique which allows you to trick LINDO into reading input from a file rather than from the keyboard and sending output to a file instead of to the screen This feature is more a function of the operating system than of LINDO and therefore is not available on all versions of LINDO Operating systems that support I O redirection currently include all the various versions of Unix You set up for I O redirection when you initially start LINDO as follows lindo lt input txt gt output txt The less than sign lt is the input redirection operator while the greater than sign gt is the output redirection operator In this example we are telling LINDO to get input from the file input txt rather than the keyboard Similarly we are telling LINDO to send all output to the file output txt instead of the screen You need not specify both an input and output file Specifying either one by itself is also allowed By specifying an input file you no longer need to interactively enter commands Instead you may place all required commands in a file and entirely automate the input process By specif
213. f pivots is reached LINDO prompts you as to how many more iterations you wish to do You may specify an additional number of new pivots to perform or enter a 0 to halt optimization and return to command level with the best solution found so far In the next example we read in a model and issue a GO command with a limit of 10 pivots LINDO hits this limit before finding a solution and prompts us for an additional number of pivots We reply with an additional 30 pivots which allows LINDO to finish solving the model Read in a small model retr test2 1lpk terse Go into terse output mode Use GO to solve the model set pivot limit to 10 go 10 PIVOT LIMIT OF 10 EXCEEDED HOW MANY MORE ALLOWED 30 LP OPTIMUM FOUND AT STEP 18 OBJECTIVE VALUE 8942181 00 COMMAND LINE COMMANDS 155 5 INTERRUPTING THE SOLVER If you have a large model and don t want to wait for LINDO to find the optimal solution the Windows version of LINDO allows you to interrupt the optimizer and return to command level with the best solution found up to that point To interrupt the solver press the interrupt button in the Status Window If the Status Window is not on screen from the Windows menu select the Open Status Window command When you interrupt the optimizer LINDO will inform you how many pivots it has done and ask you whether you want to do more If you truly want to interrupt reply with a 0 LINDO wi
214. fficulties A rule of thumb is that there should be no nonzero coefficient whose absolute value is greater than 100 000 or less than 0001 If LINDO feels the matrix is poorly scaled it will print a warning On poorly scaled problems LINDO will do significance checking of crucial calculations e g to determine in which situations a number like 00000325 is really zero and in which situations it is really a perfectly valid nonzero number In very large problems with lots of non binding constraints there is a slight computational advantage to converting greater than or equal to constraints to less than or equal to constraints The expense in terms of both space and time of carrying along a non binding constraint is slightly less for the latter LINDO assumes your input data is accurate to about 6 decimal digits Generally the answers from LINDO will have almost the same accuracy as the input data However if you do not keep in mind the accuracy LINDO expects in the input data it is very easy to produce a formulation for which the accuracy of the answers is considerably less than the accuracy of the input data As an example of a careless formulation consider the following MAX Z SUBJECT TO 2 0 6666 Z X 0 3 0 3333 2 Y 0 4 Z X Ye 0 5 Z lt 1 END A possible interpretation of the user s intentions is that he wanted 2 X 2 3 2 3 Y 1 3 2 4 XY Z Di ee L 268 CHAPTER 9 In this case the solutio
215. fly in this manual it will not be our primary focus For the interested reader an in depth discussion of modeling in an array of application areas and the details on the variations of the Simplex method algorithm are contained in the companion LINDO textbook Optimization Modeling with LINDO by Linus Schrage Users who do not already have a copy of the textbook might wish to consider acquiring one as well Please refer to the cover page for information on how to contact LINDO for technical support and acquiring other products GETTING STARTED 3 Installing LINDO Installing the LINDO software is straightforward on most platforms To run LINDO for Windows we recommend a computer with at least a 386 processor and coprocessor capabilities running Windows 3 1 Windows 95 or Windows NT In general you will need at least 16Mb of RAM and 10Mb of free disk space A faster processor and additional memory may allow LINDO to solve tougher problems and or improve performance It should be noted that these are minimums and there may be higher requirements needed to solve models that approach the limits of the various versions To install a PC Windows version of LINDO under Windows 3 1 and Windows NT simply insert the LINDO CD double click on the LINDO folder to open it and then double click on the Setup icon to run LINDO s setup program Setup will do all the required work to install LINDO on your system and will prompt you for any required information O
216. followed by a space and a number such as min x 5 is ambiguous Here the 5 will be interpreted as a coefficient and rejected by LINDO The coefficients must be to the left of the variable names Do not use asterisks for multiplication and do not enter nonlinear formulas Any number of spaces may be used but leave at least one space after MAX or MIN If you run out of space at the end of a line for your objective function simply press enter to continue typing the model You will get a question mark for a prompt Again you end the objective function with ST SUBJECT TO S T or SUCH THAT You will get another question mark prompt at which point you can begin entering constraints or you may type the command END to indicate you have completed the model An example using the MIN command to input a small model follows min 20x 30y st x lt 50 y lt 60 x 2y lt 120 end ee DD ww For additional information on the required syntax of a LINDO model please refer to the section titled Model Syntax in Chapter 2 LINDO for Windows page 12 RETRIEVE Syntax RETRIEVE lt FileName gt The RETRIEVE command reads a file from disk that was created with the SAVE command That is only LINDO packed Ipk format files The file is loaded into memory and becomes the current model Any previous model in memory will be lost The form of the command is RETRIEVE lt FileName gt where lt FileName gt is the file you wis
217. following file containing all the nonzero variable values NAME PRIMAL Al 9 A2 2 B2 5 B3 6 c4 8 This discussion of script files is intended only as a brief introduction Many more options are available to the creative user One possibility that leaps to mind is the use of the ALTER command in a script in an iterative fashion to repeatedly modify a model and re solve saving solutions along the way All of LINDO s command line commands described in Chapter 3 Command line Commands are available for use in script files with the one exception being the TAKE command itself LINDO does not allow for nested or multiple TAKE files Thus a TAKE command may not be embedded in a script file that is to be read by the TAKE command Auto Loading Script File AUTOLD DAT When LINDO starts it looks for a file called AUTOLD DAT in the default directory If it does not find a file with that name it continues in the standard fashion If the file is found then LINDO starts reading commands from the file Effectively LINDO automatically executes the command File Take Commands or TAKE in command line versions on AUTOLD DAT as soon as it starts This feature is particularly useful if you always want to issue a standard set of commands at the start of a LINDO session Error Detection in Script Files If LINDO encounters an error while reading a script file and you have difficulty identifying the location of the error then you may find the BATCH
218. fprin Close tf fp RVRT n tf fp QUIT n LINDO command file fclose fp 2 This sample code assumes that the C executable file is in the same directory as the LINDO software If this is not the case then you will need to modify the code to account for LINDO s actual directory 228 CHAPTER 7 Redirect standard input to the script file nStdIn dup 0 fpIn fopen cFileIn r dup2 fileno fpIn 0 Redirect standard output to a file nStdOut dup 1 fpOut fopen cFileOut w dup2 fileno fpOut 1 Run LINDO and process script file spawnlp P WAIT LINDO NULL Flush i o streams fflush stdin fflush stdout Restore standard i o dup2 nStdIn 0 dup2 nStdOut 1 Close files fclose fpIn fclose fpOut Open solution file fpSolution fopen cFileSolu r Display the solution fscanf fpSolution Sf nSf amp fXVal amp fYVal printf X and Y Sf Sf n f XVal fYVal Close solution file fclose fpSolution Assuming you name the compiled executable shell then when you run this application your screen should appear as follows gt shell X and Y 50 000000 35 000000 gt The crucial point to observe in this application is that we are exploiting the fact that LINDO reads from the standard input device and writes to the standard output device Before invoking LINDO with the C spawnlp function we redirect st
219. g an IPTOL value Differences of this magnitude applied to a large IP model can make for dramatic improvements in runtimes The default IP Optimality Tolerance value is None In other words the feature is disabled IP Objective Hurdle When IP Objective Hurdle is used LINDO will only search for integer solutions in which the objective value is better than the specified hurdle This bound is usually based on a known feasible solution The bound is used in the branch and bound algorithm to narrow the search for the optimum When LINDO is searching for a initial integer solution it can ignore branches with objective values worse than your hurdle value because LINDO knows that a better solution the hurdle will be found on a subsequent branch In other words branches must have objective values more attractive than the hurdle value before they will be considered Depending on the problem a good hurdle value can reduce solution times As an example consider the following small IP gt C LINDO61 Samples TESTBIP LTX WINDOWS COMMANDS 9 X1 X2 4 X3 2 X4 8 XS 2 X6 8 E 12 X8 SUBJECT TO X2 2 X3 4 2 X5 X6 2 E 3 X8 lt 13 1 3 X2 2 X3 3 X4 X5 2 X6 E X8 lt 6 1 X3 X2 X3 Running without an IP hurdle we obtain the following results with the solver LINDO Solver Status m Optimizer Status Iterations Infeasibility Objective Best IP IP Bound Branches Status
220. graphics report you will notice three tools represented by the following icons ch P WINDOWS COMMANDS 93 The uppermost tool the magnifying glass with a plus sign is the zoom in tool To select this tool click the button immediately to its left The mouse cursor will then take on the appearance of the zoom in tool y You may then position the mouse over an area of the matrix you wish to examine more closely click the mouse and LINDO will zoom in around the area you selected Using the model presented above we used the zoom in tool to enlarge rows 4 and 5 and columns B1T1 through T1T2 35 LINDO Nonzero Picture z3 Fie Edit Solve Reports Window Help CALINDO61 SAMPLES TEST6 LTX BITI TATL B1T2 T1T2 T m Rows 21 Columns 34 Nonzeros 90 Note By zooming in there is enough space to begin displaying variable names along the top of the picture and larger numbers in their boxes In addition scroll bars have been added to the picture to allow you to scroll vertically and horizontally through the matrix You may zoom out in a identical manner by selecting the zoom out tool Q The arrow tool can be used to select a specific area of the matrix to enlarge Point to the upper left corner of the area that you want enlarged click down on the mouse button and drag to the lower right corner of the area of interest Finally release the mouse button and LINDO will enlarge the selected area so it fills the ent
221. h to retrieve If you attempt to retrieve a file which is not the correct format or does not exist LINDO will give an error message If you enter nothing after the RETRIEVE command LINDO will prompt you for the name of the file You may also specify a drive and path as part of the full file name FBR Syntax FBR lt FileName gt The FBR command retrieves the current basis 1 e solution of a linear programming model in LINDO s proprietary format Once retrieved in this way the file will then become the basis of the model currently in memory This is particularly useful when you have solved a large model and want to make a small change in the formulation and don t want to wait for a new solution In order to do this you would save the basis with FBS after solving See FBS command below alter the model install the old basis with FBR and solve again The following example illustrates 122 CHAPTER 3 take tran dat A 3 warehouse 4 customer transportation model 1 XWH lt i gt C lt j gt amt shipped from warehouse i to customer j 1 7 MIN 6 XWH1C1 2 XWH1C2 6 XWH1C3 7 XWH1C4 2 4 XWH2C1 9 XWH2C2 5 XWH2C3 3 XWH2C4 2 8 XWH3C1 8 XWH3C2 XWH3C3 5 XWH3C4 SUBJECT TO 1 Supply constraints WH1 XWH1C1 XWH1C2 XWH1C3 XWH1C4 lt 30 gt WH2 XWH2C1 XWH2C2 XWH2C3 XWH2C4 lt 25 WH3 XWH3C1 XWH3C2 XWH3C3 XWH3C4 lt
222. have a first objective of minimizing the cost of satisfying all requirements and perhaps a second objective of minimizing the maximum over satisfaction of any requirement The Lexico approach saves the user from having to worry about exact tradeoff rates between the cost of a solution and the value of over satisfaction In the following example we look at the Preemptive Goal command applied to a small staff scheduling model We have staffing requirements for each of the seven days of the week Employees work for 5 days with two days off each week M represents the number of employees starting on Monday T is the number starting on Tuesday and so on The formulation for this model is 42 C LINDO61 SAMPLES STAFF LTX DER HIN M T SUBJECT TO MON TUE WED THU FRI SAT SUN ZOOM mmm N 68 CHAPTER 2 Typically you must accept some overstaffing in problems of this sort but you may want to spread the overstaffing across the week After optimizing with the Solve command you will notice the clustered overstaffing in the solution I gt Reports Window fale a LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1 8 000000 VARIABLE VALUE REDUCED COST M 5 000000 0 000000 0 000000 0 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 ROW SLACK OR SURPLUS PRICES 2 MON 000000 000000 TUE 000000 000000 WED 000000 0 000000 THU 000000 000000
223. have elected to choose the pivot row as part of the So ve Pivot command but have neglected to specify the row Either choose a row or let LINDO select the pivot row The selected variable was not found in the current model You have specified a variable that does not exist in the current model Please be sure you have spelled the variable s name correctly and retry Objective row missing from beginning of model The objective row which is signified by either MAX or MIN must appear first in a model Either you have omitted the objective or have placed some constraints before it Please edit the model to correct the problem and retry You may not have both an AUTOLD DAT file and a command line input file Please use one or the other LINDO looks for a command script file called AUTOLD DAT when it starts up If this file is found LINDO will automatically execute the commands contained therein You may also specify a command script file as a command line argument to LINDO LINDO restricts you to doing one or the other however If you attempt to use both files you will receive this error message Try combining the commands in the files to either the AUTOLD DAT file or the command script file LINDO was unable to allocate sufficient space for nonzero elements LINDO will quit and return to Windows On startup LINDO needs to allocate a minimal amount of storage for nonzero elements You will get this error message when this alloca
224. he LOOK command title this is the title of this model look all TITLE THIS IS THE TITLE OF THIS MODEL MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 4 X1 X3 X55 9 5 X1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END 11 Quit QUIT The QUIT command exits the LINDO environment and returns you to the operating system QUIT has no parameters Note You will lose any changes you made to the current model if you don t save it before quitting see the SAVE command 190 CHAPTER 3 191 4 Integer Programming Two kinds of integer variables are recognized by LINDO zero one variables binary and general integer variables Zero one variables are restricted to the values 0 or 1 They are useful for representing go no go type decisions General integer variables are restricted to the non negative integer values 0 1 2 GIN and INTEGER statements are used respectively to identify general and binary integer variables The statements should appear after the END statement in your model Variables that are restricted to the values 0 or are identified using the INTEGER statement It is used in one of two forms INTEGER lt VariableName gt or INTEGER lt N gt The first and recommended form identifies
225. he script by issuing the command TAKE MYSCRIPT LTX For additional insight as to how the TAKE command is used please refer to Chapter 7 Interfacing with the Outside World section on Running Command Scripts with the TAKE Command Automatic Take Command AUTOLD DAT When you open LINDO it searches the current directory for a file called AUTOLD DAT If it doesn t find one it opens normally and awaits a command However if the AUTOLD DAT file is found LINDO reads and executes commands from the file automatically What LINDO does is to automatically execute the command TAKE AUTOLD DAT as soon as it opens If you always want to issue a certain set of commands when LINDO starts write the commands in the form of a TAKE file and save the file as text only with the name AUTOLD DAT to the directory where LINDO is located LEAVE The LEAVE command indicates to LINDO that a TAKE command script file has ended LINDO then closes the TAKE file This command should be the last line of any TAKE file On some computer systems especially older ones an error may occur if LEAVE is left out of the TAKE file The LEAVE command may not be necessary on your system However it s a good habit to include it 126 CHAPTER 3 In the example below we have a small command script which is terminated with a LEAVE command This file retrieves amp solves MYFILE LPK then saves the solution First turn on echoing of input BATCH Turn
226. he DEBUG command will identify a minimal set of constraints so if any one constraint in the set is dropped the formulation becomes feasible Let us illustrate use of the DEBUG command on the above model debug n UFFICIENT SET ROWS TO CORRECT 4 0 55 X Y gt 4 NECESSARY SET ROWS TO CORRECT 3 2X Y lt 2 Notice that row 2 which happens to be correct does not appear in the list of possibly faulty rows while the faulty one row 4 does appear So DEBUG helps reduce the number of rows over which you need search for a bug If the complete model would be feasible except for a bug in a single row that row will be listed in the SUFFICIENT SET of rows The NECESSARY SET of rows is a set such that as long as all of them are present the model remains infeasible In the above example as long as rows 3 and 4 are present the model remains infeasible In general the kind of correction required in a constraint is one or more of the following change the RHS change the direction change the coefficient of a variable in the constraint or change the upper or lower bound of a variable in the constraint Debugging Unbounded Models DEBUG operates in a similar manner for unbounded models Consider the following example MAX 12 X1 13 X2 22 Y1 25 Y2 23 Z1 28 Z2 X3 Y 3 23 SUBJECT TO 2 X1 X2 X3 lt
227. he Range command may also be used to determine the extent of these valid ranges The final line in the solution report gives the number of iterations or simplex pivots required to solve your model In integer programming models LINDO will also print a branch count which is the number of variables the branch and bound solver had to branch on to arrive at a solution Range Alt 1 The Range command generates a range report i e sensitivity analysis for the active Model Window Note you will need to solve the model before attempting to use the Range command This range report includes e the variable names with their current objective function coefficients and allowable increases and decreases on these coefficients and e the row numbers or names with their current right hand side values and allowable increases and decreases for these right hand side values The range report is relevant only for linear models A range report for integer and quadratic programs is of little practical use To illustrate the Range command consider the small linear program 3 C LINDO61 SAMPLES RANGE LTX MAX 20 COMP1 30 COMP2 SUBJECT TO COMP1 lt 60 COMP2 lt 70 COMP1 2 COMP2 lt 120 Solving this model we get the solution I gt Reports Window LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE i 2100 000 VARIABLE VALUE REDUCED COST COMP1 60 000000 0 000000 COMP2 30 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 0
228. he following SET xX lt 3 AT 1 WITH BND 3 20000 TWIN 2 8000 The 3 2 means there are no solutions better than 3 2 down the branch with X lt 3 There are no solutions better than 2 8 down the branch with X 4 When LINDO later returns to examine the branch X 4 it prints the message FLIP X TO gt 4 AT LEVEL 1 After having fully examined all offspring of both branches it prints the message DELETE X AT LEVEL 1 The selection of the variable upon which to branch is biased towards variables which 1 have a fractional part close to 5 2 appear early in the problem 3 cause a large change in the objective value if their values are changed Superimposed upon this variable based branching LINDO may use a set based branching In a hypothetical IP suppose variables X3 X7 and X8 have value zero when solved as an LP If the reduced costs of these variables are large suggesting variables should remain at zero LINDO may create two new subproblems by alternately adding the Martin due to Kipp Martin cuts X3 X7 X8 0 or X3 X7 X8 1 When the first cut is added a message of the flavor FIXING ALL VARIABLES WITH RC gt 9 2 is printed While when the second replaces the first a message like RELEASING FIXED VARIABLES is printed The subproblem with the first constraint is investigated first because we suspect we will quickly find a good rounded solution This set based branch
229. he model This means LINDO will determine whether the model makes mathematical sense and whether it conforms to syntactical requirements If the model doesn t pass these tests you ll be informed with the following error message An error occurred during compilation on line n LINDO will then jump to the line where the error occurred You should examine this line for any syntax errors and correct them If there are no formulation errors during the compilation phase LINDO will then begin to actually solve the model When LINDO s internal solver initiates it displays a Status Window on your screen that looks like the following LINDO Solver Status m Optimizer Status Status Optimal Iterations Infeasibility Objective Best IP IP Bound Branches Elapsed Time 00 00 00 Update Interval fi GETTING STARTED 7 This Status Window is useful for monitoring the progress of the solver A description of the various fields and controls within the Status Window appear in the table below Field Control Description Status Gives status of current solution Possible values include Optimal Feasible Infeasible and Unbounded Iterations Number of solver iterations Infeasibility Amount by which constraints are violated Objective Current value of the objective function Best IP Objective value of best integer solution found Only relevant in integer programming IP models IP Bound Theoretical bound on the ob
230. hen performs the pivot operation and returns with a summary of the results in the Reports Window i Reports Window X ENTERS AT VALUE 50 000 IN ROW 2 OBJ VALUE 2050 0 LINDO confirms that it has pivoted on Y in row 4 Y now has a value of 60 and the objective row has a value of 1800 It is instructional to view the simplex tableau before performing the next pivot To display the tableau select the Reports Tableau command LINDO should display the following 23 Reports Window Dor A THE TABLEAU ROW BASIS X Y SIK 2 SIK3 SIK 4 1 ART 0 000 0 000 0 000 15 00 1800 000 2 SIK 2 0 000 1 000 0 000 0 00 50 000 3 SLK 3 0 000 0 000 1 000 0 50 10 000 4 1 000 0 000 0 000 0 50 60 000 66 CHAPTER 2 The reader experienced with the theory of the simplex algorithm can see that the only attractive variable to pivot on is X due to its negative reduced cost in row 1 Furthermore the row binding X will be row 2 Thus this must be our pivot row Entering all this into the Pivot command dialog box we have the following Pivot Variable Pivot Row Variable Selection Row Selection C LINDO s LINDO s Cancel Use Mine Use Mine Help My Variable Selection My Row Selection Clicking OK to perform the pivot results in this summary from LINDO i Reports Window X ENTERS AT VALUE 50 000 IN ROW 2 OBJ VALUE After X enters the solution the objective value climbs to 2050 the opt
231. ialog box UpdateData FALSE Shut LINDO down LSEXIT All done return First off in the above code LINDO is initialized with the calls ILINDO INIT amp lTruble amp lTruble 255 ILINDO performs one time initialization for LINDO It should be called once at the start of your code Next the INIT command performs initialization for a new model INIT should be called each time your code begins input of a new model Next a file called LINDO OUT is open with the LUNOPN routine and LINDO s standard output is redirected to this file via a call to CAPOUT char cKfname LINDO OUT long 1Unit 60 1Fname 9 l1InOrOut 0 INotFmt 0 lLutrmi 0 lLutrmo 0 LUNOPN amp lUnit amp lFname amp cKfname amp lInOrOut amp lNotFmt amp lLutrmi amp lLutrmo Now tell LINDO to divert all standard output to this unit CAPOUT amp lUnit 256 CHAPTER 8 Had LINDO s standard output not been redirected the application would crash under Windows the first time LINDO wrote anything to the standard output device In addition if you are having difficulty debugging your LINDO application it can be useful to view the contents of this LINDO output file for any helpful system messages Note in the calls to LUNOPN and CAPOUT all arguments are passed by reference This is true for all LINDO routines all arguments must be passed by reference rather than by value Next in order
232. ide and LINDO then changes the current right hand side in steps to the new right hand side value displaying the objective function value at each step The form of the command is PARAMETRICS lt Row gt lt NewRHS gt where lt Row gt is the number or name of the row you wish to analyze and lt NewRHS gt is the right hand side value you wish to extrapolate to If you type nothing after PARAMETRICS LINDO will prompt you for the row and the new right hand side value You must have a solution LINDO considers optimal in memory to use this command properly If no optimal solution is in memory LINDO attempts to carry out the command but also warns you that the solution is not optimal PARAMETRICS shows the entering and leaving variables for every pivot the right hand side value as it changes from the current value to lt NewRHS gt the dual price and the objective value of the new solution The PARAMETRICS command does not apply to integer programs It is only intended for linear and quadratic programs Please refer to the RANGE command for related sensitivity analysis You may wish to use the RANGE command to choose an interesting value for lt NewRHS gt The PARAMETRICS command does not change the row you specify or the current model in any way However the current solution is no longer considered optimal after PARAMETRICS is used because the LINDO solver must change the internal representation of the solution to do the analysis
233. ient to make the model feasible These constraints are identified by DEBUG as the SUFFICIENT SET ROWS Not every infeasible model has a crucial constraint Regardless of whether any crucial constraints were found the DEBUG command also identifies a set of constraints as well as column bounds that constitute a NECESSARY SET ROWS Such a set has the feature that it is infeasible However if any member of this set is deleted then the set becomes feasible Thus it is necessary to make at least one correction in the NECESSARY SET ROWS if the model is to be feasible 184 CHAPTER 3 When DEBUG encounters an unbounded model it first tries to identify one or more crucial variables These variables are identified by DEBUG as SUFFICIENT SET COLS A variable is crucial if fixing it is sufficient to make the model bounded Regardless of whether any crucial variables were found DEBUG also identifies a set of variables that constitutes a NECESSARY SET COLS Such a set has the feature that it is unbounded but if any variable in this set is fixed the set becomes bounded Hence constraints ROWS are infeasible and variables COLS are unbounded Typically DEBUG helps substantially reduce the search effort The first version of DEBUG was implemented in response to a user who had an infeasible model The user had spent a day searching for a bug in a model with 400 rows DEBUG quickly found a NECESSARY SET with 55 constraints as well as one SUFFIC
234. if there is a misspelled variable name you may find columns with a single nonzero element when you don t expect any If you misplace a decimal point in the model you may find a value that is unexpectedly large or small in absolute value Peruse Alt 4 The Peruse command can be used to view or peruse selected portions of a model s solution and or structure For old time LINDO users the Peruse command is an enhanced version of the CPRI and RPRI commands Peruse reports are generated in text and or graphics format The ability to focus on specific parts of a model and its solution through the use of Peruse is particularly useful on large models where attempting to sift through an entire formulation or solution can prove to be an overwhelming task 82 CHAPTER 2 In order to use all the features of the Peruse command you should first run the Solve command on your model although this is not required Next issue the Peruse command and LINDO posts the following dialog box Peruse Model r Report Parameters A Orientation Help Cancel OK ok Columns Report Format C Rows mD Report Types mE Text Report Format B View Items IV Text IV Include Headers V Name IV Graphics IT Comma Delimited IV Primal I Dual Rim MF GraphT ype G Graph Style I Upper Bound C Area Bar C20 I Lower Bound C Line Pie CD E Type Point I Nonzeros r Graphics Format H Graph Title optional C C
235. ils and examples on the use of the GIN please refer to page 170 INT STATEMENT Using INT restricts a variable to being either 0 or 1 These variables are often referred to as binary variables In many applications binary variables can be very useful in modeling all or nothing situations Examples might include such things as taking on a fixed cost building a new plant or buying a minimum level of some resource to receive a quantity discount For more information on the many useful applications of binary variables you can refer to the companion LINDO textbook Optimization Modeling with LINDO by Linus Schrage 18 CHAPTER 1 The following small examples illustrate the use of the INT statement for Windows versions and command line versions of LINDO MAX 100X 20A 12B ST A 10X lt 0 A B lt 11 B lt 7 END i lt untitled gt mA INT X IMake X 0 1 MAX 100X 20A 12B a LOOR ADI ST MAX 100 X 20 A 12 B SUBJECT TO 2 10 X A lt 0 3 A B lt 11 4 B lt 7 END INTE 1 Windows Command Line Note that in the Windows version the INT statements appear as part of the model s text in the Model Window In command line versions of LINDO INT is entered as a system command to the command level colon prompt after the model has been entered Had we not specified X to be binary in this example LINDO would have returned a solution of X 4 A 4 and B 7 for an objective value of 124
236. imal solution If you would like a full solution report select the Reports Solution command You should receive the following I Reports Window DER OBJECTIVE FUNCTION VALUE 1 2050 000 VARIABLE VALUE REDUCED COST X 50 000000 0 000000 35 000000 0 000000 1 000000 35 000000 0 500000 0 000000 0 500000 SLACK OR SURPLUS DUAL PRICES 0 000000 NO ITERATIONS 2 Preemptive Goal Ctrl G The Preemptive Goal command performs Lexico optimization on a model This allows the user to specify an ordered list of objectives LINDO begins by optimizing the first objective Given the optimal value for the first objective it then optimizes the second objective subject to the first objective being equal to its optimal value Given the optimal values for the first and second objectives it then optimizes the third objective etc WINDOWS COMMANDS 67 A typical situation has a first objective of minimizing cost or maximizing profit and a second objective of smoothing the solution in some sense Examples are cutting stock or staff scheduling applications where a number of demand requirements must be satisfied Traditional formulations of such problems frequently have multiple optima solutions some of which greatly over satisfy one or two requirements and just barely satisfy all others The user typically has a slight preference for solutions which spread out the over satisfaction more evenly A Goal Programming formulation would
237. in the example below stands for artificial variable which are devices used to jump start the solver The term SLK n where n is an integer stands for the n th slack variable a slack variable is added internally to each inequality constraint to convert it to an equality TABLEAU may be combined with PIVOT to monitor the internal workings of LINDO s solvers Also the BPICTURE command may provide additional insight into each step of the algorithm 136 CHAPTER 3 In the following example we alternate between TABLEAU and PIVOT commands to keep tabs on LINDO as it optimizes the following small model Note LINDO always chooses the entering variable with the lowest reduced cost look all MAX 20 A 30 C SUBJECT TO 2 A lt 60 3 C lt 50 4 A 2Ce 120 END tabl THE TABLEAU ROW BASIS A SLK 2 SIK 3 1 ART 20 000 30 000 000 000 000 2 SLK 2 1 000 000 1 000 000 60 000 3 SLK 3 000 1 000 000 1 000 50 000 4 ART 1 000 2 000 000 000 120 000 ART 4 ART 1 000 2 000 000 000 120 000 pivot C ENTERS AT VALUE 50 000 IN ROW 3 OBJ VALUE 1500 0 tabl THE TABLEAU ROW BASIS A Cc SLK 2 SLK 3 1 ART 20 000 000 000 30 000 1500 000 2 SLK 2 1 000 000 1 000 000 60 000 3 c 000 1 000 000 1 000 50 000 4 ART 1 000 000 000 2 000 20 000 ART 4 ART 1 000 000 000 2 000 20 000 pivot A ENTERS AT VALUE 20 000 IN RO
238. ine consists of number of rows number of variables number of integer variables with the number that are 0 1 or binary in parentheses and the index of the first real constraint in a quadratic program 0 indicates this is not a quadratic program WINDOWS COMMANDS 81 The second line consists of e number of nonzero coefficients in the whole model e number of nonzero coefficients in the constraints with the number that are 1 or 1 in parentheses and e model density defined as number of nonzeros number of rows number of columns 1 The third line consists of e absolute values of the smallest and largest nonzeros respectively The fourth line consists of e sense of the objective function MIN or MAX e number of less than or equal to equality and greater than or equal to constraints e upper bound estimate of the number of generalized upper bound GUBS constraints constraints which have no variable in common and e lower bound estimate of the number of variable upper bounds VUBS For example the constraint X1 X2 X3 0 contains the implications X3 0 implies X1 0 X3 0 implies X2 0 The last line consists of e the number of columns with only one nonzero coefficient and e the number of redundant columns i e columns identical to some other column except possibly for the bounds The Statistics command can be used for checking for certain types of errors in your model For example
239. ine runtimes Sample output from the Elapsed Time command follows i Reports Window CUMULATIVE HR MIN SEC 0 15 12 96 In this example LINDO has been running for 17 minutes and 12 11 seconds License Some versions of LINDO require the user to input a password Think of the password as a key that unlocks the LINDO application If you upgrade your copy of LINDO then you will need to enter a new password The File License command prompts you for a new password When you run the File License command you will be presented with the following dialog box LINDO Password Please enter your LINDO password below OO If you don t have a password you can press Demo Version to automatically generate a temporary license for a demonstration version of LINDO Demo versions function the same as standard versions with the one exception that maximum problem dimensions are restricted If your password is available in the Windows clipboard you may paste it into this dialog box by pressing Ctrl Otherwise carefully enter your password as one long string without carriage returns being sure to include all hyphens You can access this dialog at any time using the FilelLicense command Help Cancel Demo Version Carefully enter the password into the edit field including hyphens making sure that each character is correct Click the OK button and assuming the password was entered correctly LINDO will display the Help About L
240. ing IP options that influence the solvers approach on IP models b General general solver options that pertain to all classes of models 2 Output Options options that affect the amount and style of output from LINDO The remainder of this section explores the options available in each of these categories Optimizer Options The group box labeled Optimizer contains all the options that influence the functioning of the optimizer This group box consists of the two subgroup boxes Integer Programming and General The options in the Integer Programming IP subgroup influence how LINDO s branch and bound solver handles integer programming models The options in the General subgroup pertain to the way the solver handles all classes of models linear integer and quadratic Integer Programming IP OPTIMIZER Options Given that integer programming problems can at times be extremely difficult to solve the following IP options will be of interest if you plan on solving anything other than the most trivial IP models The integer programming options included in LINDO and discussed below are Preprocessing Preferred branching direction IP optimality tolerance IP objective hurdle IP variable fixing tolerance Much of the discussion of these parameters assumes a certain amount of knowledge about the branch and bound solution algorithm for IP models If you are unfamiliar with this algorithm and would like to learn more you
241. ing if the current solution is not guaranteed to be optimal or feasible e variable names with their current objective function coefficients and allowable increases and decreases and e row numbers or names with their current right hand side values and allowable increases and decreases RANGE may be used with DIVERT to save the report to a file Refer to the DIVERT command for more information on this Interpretation of the range report is as follows the OBJ COEFFICIENT RANGES for each variable are the amount by which the objective coefficient can be increased or decreased without causing a change in the basis the set of nonzero variables The RIGHTHAND SIDE RANGES for each row are the amount by which the right hand side can be increased or decreased without causing a change in the basis In the following small model we ve made changes after solving to demonstrate COMMAND LINE COMMANDS look all MAX 20 COMP1 30 COMP2 SUBJECT TO 2 COMP1 lt 60 3 COMP2 lt 70 4 COMP1 2 COMP2 lt 120 END go LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1 2100 000 VARIABLE VALUE REDUCED COST COMP1 60 000000 000000 COMP2 30 000000 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 000000 5 000000 3 40 000000 000000 4 000000 15 000000 NO ITERATIONS 2 DO RANGE
242. ing selected information about a specific subset of model s rows i e objective and constraints that satisfy a user specified condition You specify the conditions you want the subset of rows to satisfy and the information you want listed for each row This feature is extremely useful for perusing large models and their solutions Refer to the CPRI command above for similar output regarding columns The row attribute list lt RowAttList gt may be any of the characters in the left column in the following table The interpretation of the attribute is in the right column RowAtt Interpretation DiDi e E P Slack surplus value o T Type of row lt for less than or equal to for Loe equality and gt for greater than or equal to Simple upper bound Number of nonzeros in the row The conditional expression lt CondExp gt may include any of the operators listed in the table below in conjunction with the row attributes in the table above to form conditional expressions Operator Function Wild card character placeholder for use in PU Rimingrownane l Subtraction S O Division Multiplication Compare equalto O COMMAND LINE COMMANDS 143 LINDO evaluates the conditional expression for each row If the conditional expression evaluates to True LINDO will print the values for all the row attributes listed in the row attribute list to the left of the semicolon in the command
243. ing strategy is particularly appropriate for large knapsack problems where a simple variable based branching strategy would perform an embarrassingly large number of branches just to find the first integer solution When set based branching is used you will notice in solution reports that there are additional rows in the problem corresponding to the added cuts These constraints will be deleted from the problem when the solution process is finished Solving Difficult Integer Programs A usual feature of a good IP formulation is that when it is solved as an LP most of the integer variables will naturally be integer valued You can sometimes predict this before hand If the coefficient matrix consists entirely of 0 s 1 s and 1 s then the following are good indicators each column has at most two nonzeros and they are of opposite sign each row has at most two nonzeros and they are of opposite sign or each column has more than two nonzeros but they are in consecutive rows and of the same sign For problems not satisfying the above you may nevertheless be able to predict the integrality of the LP solution by considering special cases of the problem For example in a production scheduling IP a useful case might be to suppose demands are the same for all products and demands amp costs are constant from period to period In this case you may be able to deduce the optimal LP solution and assess it s integrality without numerically solving it
244. inite if the corresponding quadratic expression is convex but not strictly convex i e it has some linear sections For many applications of quadratic programming the quadratic part of the model can be expected to be positive definite For example in portfolio models the quadratic part is really a covariance matrix and this matrix is naturally positive definite or at worst positive semidefinite LINDO has a command for ascertaining the definiteness of the submatrix corresponding to the quadratic part of the objective In Windows versions of LINDO the command is Reports Positive Definite In command line versions of LINDO the command is POSD This command first checks if the matrix is symmetric If not symmetric LINDO will identify the elements in violation Thus the Positive Definite command is useful for debugging a quadratic programming model After checking the symmetry of the submatrix the Positive Definite command will ascertain the positive definiteness of the submatrix and report on the results To reemphasize if you find the model is neither positive definite nor positive semidefinite then the results from LINDO s solver will be unpredictable and should not be relied upon Finally LINDO will tell you the rank of the submatrix If you think of the submatrix as the coefficients of a set of linear equations then the rank is the number of independent equations in the set 202 CHAPTER 5 It is a good idea to apply the POSD command
245. integer valued in the range 1 to 9 are printed without modification Alphabetic letters are substituted for all other coefficients using the following scheme Letter Code Coefficient Range Z 000000 000001 000001 000009 010000 099999 G So line 4 would read C B1T1 B T1T1 lt E where the variables are in parentheses and C B and E refer to the ranges specified above 000010 000099 C E F The text version of the nonzeros picture report complies with the Terminal Width parameter which may be reset using the Edit Options command Thus if the width of the picture exceeds the Terminal Width parameter the report will be continued on additional pages You may want to increase the Terminal Width to maximize the amount of output per page in the text report The graphics version of the nonzero picture report displays the model s name across the top the model dimensions across the bottom and space permitting the row names along the left and variable names across the top Negative coefficients are displayed using red rectangles while positive coefficients are displayed in blue If there is enough space in a rectangle the actual coefficient value will also be displayed The difference between the graphics and text version of a nonzero picture report are that the graphics version doesn t have the right hand side values or direction but it does include the exact numbers of the larger coefficients Along the right of the
246. ion of selecting a text report and or a graphics report If you select a graphics report you may also choose to display the graphics in either two or three dimensions For our purposes let s select both a text report and a 3D graphics report Input to the Parametrics dialog is complete and we have Parametric Row New RHS Value 40q Row Parametrics Report Types Ce Cancel Help Iv Text IV Graphics pices Type gt Next we click on the OK button to generate our parametrics report The text report is sent to the Reports Window and appears as follows I Reports Window RIGHTHANDSIDE PARAMETRICS REPORT FOR ROW SAT VAR OUT EMON XSUN XSAT XTUE VAR IN PIVOT ROW RHS VAL 120 170 260 320 340 400 ooo ooo ooo ooo ooo ooo DUAL PRICE BEFORE PIVOT 20 0000 20 0000 33 3333 100 000 100 000 100 000 Maok OBJ VAL 22000 23000 26000 32000 34000 40000 WINDOWS COMMANDS 79 Starting with our original requirement for 120 people on Saturday shown in the first RHS VAL row we have a dual price of 20 That is it costs 20 to increase the staffing requirement by one on Saturday The increase to 170 people shown in row two above increases the objective value by 1000 50 20 However when the staffing requirement is increased to 185 people the dual price goes up to 33 33 That is it now costs 33 33 13 33 more to increase the staffing requirement b
247. ire picture At present graphical nonzero picture reports cannot be printed directly from LINDO However you can issue the Copy command from the Edit menu This places the nonzero picture into the Windows clipboard From the clipboard the picture can be pasted into any graphics program e g Microsoft Paint Microsoft Powerpoint etc and printed from there 94 CHAPTER 2 Basis Picture Alt 6 The Basis Picture command displays a text format report containing a picture of the current basis ordering the rows and columns according to the last inversion or triangularization performed by the solver The Basis Picture report is sent to the Reports Window We will use the Basis Picture command to display a picture of the optimal basis for the following model SER The PROTRAC planning model form EPPEN amp GOULD s Quantitative Concepts for Management Prentice Hall I 29970 P1T1 29970 P1T2 29910 P2T1 29910 P2T2 1000 I1T1 1000 I1T2 1000 I2T1 1000 I2T2 1200 E1T1 1200 E1T2 1200 E2T1 1200 E2T2 20 C1T1 20 C1T2 20 C2T1 20 C2T2 3000 NDT1 3000 NDT2 20 R1T1 20 R1T2 20 R2T1 20 R2T2 300 N1T1 300 N1T2 300 N2T1 300 N2T2 SUBJECT TO 2 P1T1 P2T1 NDT1 130 3 P1T2 P2T2 NDT2 190 4 900 B1T1 90 TI1T1 lt 80000 5 900 B1T2 90 T1T2 lt 80000 6 600 B2T1 60 T2T1 lt 70000 T3 600 B2T2 60 T2T2 lt 70000 8 ELTS I2T1 lt 1000 9 I1T2 I2T2 lt 1000 10 LETI E1T1 6 B1T1 11 I2T1
248. irst in the formulation For efficiency reasons LINDO permutes all integer variables to the front of the internal ordering of variables Thus in solution reports the values of the integer variables will appear before the values of the continuous variables The INTEGER specification is saved when the model is saved with SAVE SMPS and the DIVERT LOOK ALL sequence The INTEGER command is equivalent to using the GIN command below with a simple upper bound of 1 SUB lt Variable gt 1 This is how LINDO interprets the INTEGER command internally If you wish to convert an integer program to a linear program with all continuous variables just enter INTEGER 0 This says there are no integer variables However this will not remove the simple upper bound of 1 on the binary variables You can use the FREE command to do this above When you specify integer variables with either INTEGER or the following GIN command LINDO solves the problem using the branch and bound technique a special algorithm to narrow the search for the best answer that satisfies the requirement for integrality The branch and bound technique however does slow the solution process considerably If time is of the essence in solving a large problem try to formulate it without using integer variables and round the fractional variables whenever possible However keep in mind that rounded continuous solutions may be non optimal and at worst infeasible 170 CHAPTER 3 G
249. ive The input procedure is LP based and requires a pseudo objective row even though there is no explicit objective listed in the first order conditions The pseudo objective row serves an important purpose however in that it is used to identify the order of variables which in turn determines the correspondence between variables and rows This is important because LINDO tries to find a feasible solution to the first order conditions which satisfies the complementary slackness conditions For the above example the complementary slackness conditions require that if X is strictly greater than zero then the above constraint 6 X 2 Y Z UNITY 1 3 RETURN XFRAC 2 0 must hold as an equality Similarly if Y is nonzero then it s first order condition 2 X 4 Y 0 8 Z UNITY 1 2 RETURN YFRAC 0 must hold as an equality and so forth Thus LINDO must know this correspondence between variables and constraints to enforce these conditions There is one final restriction on the input of quadratic models Namely the real constraints which are X tY ta m L T3 Te2y L082 2 gt L12 X lt 75 YS als Zs 75 in our example must appear last 200 CHAPTER 5 Finally the command QCP must be used to specify which constraint is the first of the real constraints Since we have an objective and one first order condition for each of the three variables the real constraints begin with row 5 Thus we will need a QCP 5 statem
250. jective for IP models Only relevant in integer programming IP models Branches The number of integer variables branched on by LINDO s IP solver Only relevant in integer programming IP models Elapsed Time Update Interval The frequency in seconds the Status Window is updated You can set this to any non negative value desired Setting the interval to zero will tend to increase solution times and have it return the current best solution found Press this button to close the Status Window Optimization will continue Selecting the Status Window command from the Window menu will reopen the Status Window When the solver is finished it will prompt you to determine if you wish to do sensitivity and range analysis Eventually you ll be able to make use of this information but for now just press the No button and then close the Status Window There will now be a new window on your screen titled Reports Window The Reports Window is where LINDO sends all text based reporting output The window can hold up to 64 000 characters of information If you have a lengthy solution report that you need to examine in its entirety you can log all information sent to the Report Window in a disk file using the File Log Output command This file can then be examined using an external text editor or using the File View command Refer to the File menu section in Chapter 2 LINDO for Windows for details on how to use these commands
251. jective to the file and retry ERROR IN MPS MODEL ON LINE lt N gt lt CardText gt An invalid record was encountered in an MPS file on line number lt N gt Please correct the MPS file and try again OUT OF PLACE INTEGER MARKER CARD IN LINE lt CardText gt LINDO requires that integer variables appear first in the variable set in an MPS file Please move integers to the front and try again WARNING UNTERMINATED MARKER SET The integer variable set was not terminated All variables will be made integer This is just a warning and it is not mandatory that you correct this in order to proceed BAD VARIABLE IN BOUNDS SECTION OF MPS MODEL IN LINE lt N gt An invalid variable name was specified in the BOUNDS section of an MPS file on line number lt N gt Please check the spelling of the variable name and retry BAD BOUND lt BoundValue gt IN BOUNDS SECTION OF MPS MODEL IN LINE lt N gt An invalid bound value was specified in the BOUNDS section of an MPS file Be sure that you have specified a numeric value and be sure that the bound value is lined up in the correct columns and retry UNRECOGNIZED BOUND TYPE IN MPS MODEL ON LINE lt N gt You have used an invalid bound type in an MPS model file on line lt N gt The bound types recognized by LINDO are FX UP LO FR BV UI and LI Please correct the bound type and try again BAD RANGE IN MPS MODEL ON LINE lt CardText gt You have specified a range on a row that does
252. junction with the QCP command you will need to enter an integer specifying the index of the first real constraint that immediately follows the first order condition constraints Acceptable argument values for the QCP command run from 2 to one more than the number of constraints By specifying that the first real row number is one more than the largest row number in the model you indicate there are no real constraints in the problem At this point a small example may help in understanding the function of the QCP command Suppose we have the following budget constrained investment portfolio optimization problem There are three candidate assets X Y and Z for our portfolio We want to determine what fraction should be devoted to each asset so an expected return of at least 12 equivalently a growth factor of 1 12 is obtained while minimizing the variance in return and not exceeding a budget constraint The expected returns for X Y and Z are 30 20 and 8 respectively We also have the restriction that any given asset can constitute at most 75 of the portfolio Finally the covariance matrix for the assets is Let the symbols X Y and Z represent the fraction of the portfolio devoted to each of the three assets The variance of the entire portfolio will then be SK eB OP DY RS 0S OZ Since variance is a measure of risk we want to minimize it The complete model is then Minimize portfolio risk i e variance Minimize 3 X 2
253. k all Display the model IN BO HY SUBJECT TO 2 X Y lt 3 3 X Y gt 1 END go Now solve it LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1 1 000000 VARIABLE VALUE REDUCED COST X 0 000000 4 000000 Y 1 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 2 000000 0 000000 3 0 000000 1 000000 NO ITERATIONS T DO RANGE SENSITIVITY ANALYSIS N COMMAND LINE COMMANDS 167 free x Let x go negative look all Notice that x is now free I 5X t Y SUBJECT TO 2 X Y lt 3 3 X Y gt 1 END FREE X go Re solve and note how x becomes negative LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1 3 000000 VARIABLE VALUE REDUCED COST X 1 000000 0 000000 Y 2 000000 0 000000 7 Integer quadratic amp parametric programs INTEGER Syntax INTEGER lt Variable NumberOfVars gt The INTEGER command is used to specify variables as binary integer A binary integer variable can take on only one of the two values 0 or 1 Binary variables are very useful for modeling go no go situations an example of which would be whether or not to incur a fixed cost For more information on the application of binary variables you may refer to the LINDO textbook Optimization Modeling with LINDO by Linus Schrage The form of the command is INTEGER lt Variable NumberOfVars gt where lt Variable gt is the name of an individual variable you wan
254. ke sure all variable 272 10 11 12 14 APPENDIX A names begin with an alphabetic character and enter the remainder of the model with the EXTEND command VARIABLE NAMES MUST BE lt 8 CHARACTERS The maximum number of characters allowed in LINDO variable and row names is 8 You should enter several carriage returns to retreat to command level examine the model with the LOOK command make sure all variable names are less than 9 characters and enter the remainder of the model with the EXTEND command INVALID VARIABLE NAME You have specified a variable name that LINDO does not recognize You should enter several carriage returns to retreat to command level examine the model with the LOOK command check the spelling of all variable names and enter the remainder of the model with the EXTEND command INVALID ROW NUMBER REENTER ROW NO 1 TO M You have entered an invalid row number or a row number that is out of bounds Please try again and enter a row number in the range of 1 to M where M is the number of rows in the current model ROW 1 HAS NO RHS REENTER ROW NUMBER You have attempted to ALTER the right hand side coefficient on the objective function The objective function in LINDO is not allowed to have a right hand side value It is always row 1 and must begin with a MAX or MIN CANNOT DELETE ROW 1 REENTER ROW NUMBER LINDO requires that an objective function always be present in a model Thus you
255. l get the solution from LINDO and report the solution to our user The following sections illustrate how to accomplish this using three development environments Visual Basic Visual C C and Powerstation FORTRAN The examples assume you the user have a base knowledge of the programming language All examples make use of the LINDO DLL for Windows For linking instructions on platforms other than Windows refer to the Installation Instructions provided with your specific version of LINDO In order to simplify the following examples all forms of error checking have been omitted in our sample code A well written application however would check all returned error codes to guard against potential problems A Sample Visual Basic Application This section illustrates a Microsoft Visual Basic application to interface with the LINDO DLL which builds and solves the simple two variable model presented above If you would prefer to skip the details of constructing this project and jump ahead to experimenting with the completed project you can find it under the name LINDO DLL32 MSVB LNDMSVB VBP You should be able to load this project directly into VB 5 0 246 CHAPTER 8 Building the Application The first step in building the project is as follows 1 Start Visual Basic and then issue the File New Project command 2 Select Standard Exe and then click the OK button 3 Use the button tool to add two buttons to the project s form so it
256. l text editor or using the View command from the LINDO File menu If the model is linear then after writing the solution report LINDO will prompt you to determine if you wish to do sensitivity and range analysis If you request to see this report it will also be sent to the Reports Window You can also request this report at any time by using the Reports Range command For more information on this type of analysis please refer to the Range command on page 130 Compile Model Ctrl E Before LINDO can solve your model it must compile i e translate the model into the arithmetic form required by the LINDO solver LINDO will automatically do this if required but you may request a compile at any time by running the Compile Model Command While compiling your model LINDO maintains the following window to keep you posted on the progress of the compilation Compiling Model Constraints Variables Nonzeros This window displays a progress bar showing the percentage of work completed along with a running count of constraints variables and nonzero coefficients encountered so far in the model If LINDO finds a syntax error during compilation it will inform you of the line number where the error occurred and move the cursor to that line Requesting a compile may be useful in several instances First off you can use the compile command to check the syntax of a model as you are developing it Secondly compiling a model builds the vari
257. ld be found e g the solution becomes unbounded then JO will return 0 If no entering variable could be found e g the solution becomes infeasible then JI returns 0 In general you will want to iteratively call PARSTP until one of three things occur 1 PV reaches your target value 2 JO returns 0 unbounded or 3 JI returns 0 infeasible PARSTP in conjunction with PARBGN correspond to the PARA command line command QUIET NOISE Integer NOISE Sets LINDO s output level On entry set NOJSE to the following desired output level from LINDO NOISE LINDO Message Level Essentially no output 0 Corresponds to TERSE mode 1 Default or VERBOSE mode Message at each pivot step Be sure you have thoroughly debugged your application before calling QUIET 1 Otherwise you may miss important error system messages from LINDO As an alternative to QUIET all messages may be routed to a file by use of the CAPOUT routine RDBC LUNIT Integer LUNIT Reads in a solution basis from the file associated with logical unit number LUNIT To associate a file with a unit number see routines LUNOPN and LUNGET The file must have been created with a call to the SDBC routine This solution basis will be used as a starting point when solving the current model The model must be in memory before calling RDBC Note that at this point routines LXBWRT and LXBRED are more powerful than SDBC and RDBC at saving and restoring a basis
258. ld be unwieldy The conditional expression is evaluated for each row and column If the expression evaluates to true then the row or column is added to the report If the expression is false it is skipped When constructing conditional expressions there is a set of eight symbols that you may use that correspond to the eight report items mentioned above These symbols are listed below Report Item Symbol Primal O P O Dul D Upper bound U WINDOWS COMMANDS 87 In addition to these symbols the following table lists a number of different operators which may be applied to the symbols when constructing conditional expressions Operator Function Wild card character placeholder for use in T Addition Subtraction SSS A Exponentiation re Compare equalto i y O Parentheses for specifying precedence 88 CHAPTER 2 As an illustration in our example above there were several days of the week when we do not start any employees Suppose we want to eliminate these days from our reports We can do this by only including the columns with primal values greater than 0 A conditional expression to enforce this is P gt 0 We enter this expression into the edit box labeled Condition as shown here Peruse Model Report Parameters Help Cancel rA Orientation Columns m Report Format c Hons rD Report Types pE Text Report Format a I Text I Include Headers I Name M Graphics J Comma Delimited
259. ll return to command level with the best solution found so far in memory On platforms that don t support the interrupt feature you can usually do a Ctrl C or a Ctrl Z to break out of LINDO Note When a solution is interrupted by pressing Ctrl C or Ctrl Z keep in mind you will lose everything in volatile memory and will be returned to the operating system 6 RUNNING OUT OF MEMORY If LINDO runs out of memory it is usually because you attempted to load a model very close to its limits on inputs and it ran out of working space in memory In general you will need to either move to a larger version of LINDO or make your model smaller If you do not know the limits of your version of LINDO run the HELP command If LINDO says you have hit a nonzero coefficient limit you may be able to increase the limit on some platforms In Windows versions you can specify a larger nonzero limit by using Edit Options command GLEX The GLEX command for Lexico optimization allows the user to specify an ordered list of objectives GLEX begins by optimizing the first objective Given the optimal value for the first objective it then optimizes the second objective subject to the first objective being equal to its optimal value Given the optimal values for the first and second objectives it then optimizes the third objective etc A typical situation has a first objective of minimizing cost or maximizing profit and a second objective of smoothing the
260. lowing names would be considered valid XYZ MY VAR A12 SHIP LA whereas the following would not THISONEISTOOLONG A HY PHEN LINFRONT The first example contains more than eight characters while the second contains a forbidden hyphen and the last example does not begin with an alphabetic character You may optionally name constraints in a model Constraint names make many of LINDO s output reports easier to interpret Constraint names must follow the same conventions as variable names To name a constraint you must start the constraint with its name terminated with a right parenthesis After the right parenthesis you enter the constraint as before As an example the following constraint is given the name XBOUND XBOUND X lt 10 LINDO recognizes only five operators plus minus greater than gt less than lt and equals When you enter the strict inequality operators greater than gt or less than lt LINDO will interpret them as the loose inequality operators greater than or equal to and less than or equal to S respectively This is because many keyboards do not have the loose inequality operators On systems that do have the loose operators LINDO will not recognize them However if you prefer you may enter gt and lt in place of gt and lt GETTING STARTED 13 LINDO will not accept parentheses as indicators of a preferred order of precedence All operations in
261. lt FileName gt where lt FileName gt is the name of the file you wish to create If you enter nothing after the DIVERT command LINDO will prompt you for the name of the file When DIVERTing output keep in mind that you will see little or no output on your screen This is because the output is being sent to the output file rather than the screen The RVRT command reverses a DIVERT command and returns normal output to the screen after closing the DIVERT file As an example the following series of commands creates a file called MYFILE TXT and then sends a copy of the model and its solution to the file DIVERT MYFILE TXT LOOK ALL SOLU RVRT A file created in this manner could be read into a word processor to become part of a larger report or routed to your site s printer RVRT This command undoes the preceding DIVERT restoring subsequent output to screen See above for details FBS Syntax FBS lt FileName gt The FBS command saves the current basis i e solution of a linear programming model in LINDO s proprietary format Once saved in this way after solution the FBS file can be retrieved with FBR and 146 CHAPTER 3 will then become the basis of the model currently in memory This is particularly useful when you have solved a large model and want to make a small change in the formulation without waiting for a new solution See above for details on how this command is used with FBR to create a new model from
262. lue is 25 8299 This means that if we make a very small decrease in the RHS of row 6 then the objective value will decrease at the rate of 25 8299 Note however that in quadratic programs the dual price does not remain constant between the points where variables enter and leave the solution Interpolation of the Objective Value in Parametric Analysis In linear programs the objective value changes linearly with the RHS between breakpoints in the parametric analysis In quadratic programs however the objective function changes quadratically with the RHS value Thus to get the objective value at some intermediate point requires a bit more analysis One could of course re solve the model at the intermediate point but there is a modest interpolation formula that will also give the objective value Suppose we wish to interpolate between two adjacent breakpoints which we label 1 and 2 We will use the following notation V objective value at point i for i 1 or 2 R RHS value at point i fori 1 or 2 D dual price just before point 2 r some value between R and R w R r R2 R The objective value with a RHS ofr is Vo Do Ro r Vi Vz Dy Rz R w For example consider an r 1 2 This falls between R 1 1441 and R 1 26947 Thus Dy 25 8299 V 417375 and V2 2 03651 Substituting these values into the interpolation formula implies an objective value of 7393 To check if we re solve the
263. ly on this ability to build external output files for printing Using the Take Command on Divert Files One useful feature of the output from the Reports Formulation command in Windows versions of LINDO and the LOOK ALL command in command line versions is that it is readable as an external model file As an example we will use the file MODEL TXT created above but we will use an external text editor to modify the demand constraint right hand sides yielding MIN 2 Al 4 A2 3 A3 5 A4 4 B1 3 B2 2 B3 6 B4 5 Cl 6 C2 4 C3 3 C4 SUBJECT TO 2 Al B1 Cl 10 3 A2 B2 C2 9 4 A3 B3 C3 4 5 A4 B4 C4 9 6 Al A2 A3 A4 lt 12 7 Bl B2 B3 B4 lt IAA 8 Cl C2 C3 C4 lt 13 END INTERFACING 217 Now we use the File Open command in Windows or the TAKE command in command line versions to read the model back into LINDO We will illustrate this with a sample session from a command line version of LINDO take model txt look all MIN 2 Al 4 A2 3 A3 5 A4 4 B1 3 B2 2 B3 6 B4 eo Clk 6 620 tb A683 bh BC SUBJECT TO 2 Al B1 Cl 10 3 A2 B2 C2 9 4 A3 B3 C3 4 5 A4 B4 C4 9 6 Al A2 A3 A4 lt 12 7 B1 B2 B3 B4 lt 11 8 Cl C2 C3 C4 lt 13 END terse go LP OPTIMUM FOUND AT STEP 5 OBJECTIVE VALUE 84 0000000 So even if you don t currently have a text
264. m Orientation Columns r Report Format pees mD Report Types pE Text Report Format B View Items I Text IV Include Headers V Name J Graphics I Comma Delimited I Primal Tl Dual Tl Rim I Upper Bound T Lower Bound l Type IV Nonzeros C Condition optional In 1 c and z 21 In command line versions we issue the following CPRI command cpri n z n r c and z 21 The key in both cases is that we use the condition n r c and z 21 This requests LINDO to display all the variables that have a name where the first character is r and the second is c n r c and the nonzero count does not equal 21 z 21 After issuing this command we received the following report NAME NONZEROS R1C1 41 R1C6 29 RSC1 T R5C6 13 RDC4 22 RDC6 20 Thus we easily see that there are a half dozen variables in error The Reports Show Column command in Windows and the SHOCOLUMN command line command may be used to look at each of the above columns ANALYZING amp DEBUGGING A MODEL 209 Debug Command Suppose an LP model contains a single typographical mistake that makes the model infeasible The constraint containing the mistake will have a nonzero dual price in the solution report Unfortunately there may be a large number of other constraints which also have a dual price not equal to zero The nonzero dual price on a constraint mea
265. mand line commands 116 120 26 Input Output redirection 220 21 228 INSERT 249 256 INSROW 249 256 Installing the software 3 4 Integer binary 15 167 general 15 INTEGER 15 17 18 117 167 69 191 Integer programming 139 172 191 95 264 basis matrix 137 Benders decomposition 194 bounding 117 177 branch and bound 137 294 INDEX cuts 175 dual prices 191 exploiting solutions 194 optimality tolerance 117 options 45 50 reduced costs 191 solving difficult 193 96 tightening the formulation 194 user interface 264 Integer quadratic and parametric programs command line commands 117 167 80 Integers 80 186 binary 17 191 206 general 16 117 170 191 206 Integers 46 Integrating 219 28 C Front End 226 28 Visual Basic Front End 222 26 Interfacing LINDO 211 29 Internal parameters 188 Internal representation 98 100 169 172 191 Internal solver 104 matrix 2 monitoring 265 Interrupting the solver 53 151 155 INVERT 118 183 IP cuts 45 IP objective hurdle 194 IPTOL 117 179 80 194 Iterations 117 limiting 51 52 151 154 reducing 179 K Karush Kuhn Tucker LaGrange conditions 197 L LaGrange multiplier 198 LEAVE 116 125 Left hand sides 14 Less than 12 87 139 142 Lexico optimization 66 67 117 155 Libraries Callable 219 231 66 application requirements 244 linkable object code 231 routines 232 44 samples 245 63 Linear programming 231 LOCAL
266. matically translated into ltx format before being displayed on the screen Using ltx files will avoid this additional translation step For these reasons we recommend you always try using the LINDO Text format to save your models When saving with the Save command LINDO automatically uses the file s previous format You must use the Save As command if you wish to override a file s format Close F7 The Close command closes the active window If you have made changes to the contents of this window without saving you will be prompted to perform a save before the window closes Note You will lose all changes made since the last save if you close the window without saving Print F8 The Print command sends the contents of the active window to the printer You may print any type of window Reports Command or Model If for some reason the output from the printer is obscured or incorrect you may need to configure your printer with the Printer Setup command see below Printer Setup F9 The Printer Setup Command presents you with the standard Windows printer setup dialog box This allows you to control many options influencing how your document is printed The print facility in LINDO is fairly simple so many of these options are not applicable In general you should not need this command if you have already been successfully printing from your machine Log Output F10 The Log Output command prompts you for a file name where all su
267. matrix Columns named in LSTCOL must have been previously defined e g with calls to APPCOL and must not already appear in row J INVPRM N IPERM N Integer N IPERM Performs an in place invert of a permutation matrix On entry N contains the number of items in the permutation and PERM contains an array where the K th element of JPERM is the index of the item in the K th position of the permutation On return JPERM is permuted so the K th element gives the position containing the K th item 236 CHAPTER 8 KLOSE LUNIT Integer LUNIT Closes a file assigned to logical unit LUNIT that was opened by a call to the LUNOPN or LUNGET routines LOOK J1 J2 Integer J1 J2 Prints rows J through J2 of the current formulation in natural format Output is routed to the standard output device You can capture LINDO s standard output in a file via a call to the CAPOUT routine This routine corresponds to the LOOK command line command LSALTN IRTCOD KTEXT J1 J2 INROW Character KTEXT Integer IRTCOD J1 J2 INROW Modifies a constraint s name On entry KTEXT a character array contains the new variable name J points to the start of the name in KTEXT J2 points to the end and JVROW contains the index of the row to receive the new name JRTCOD will be set to 0 if there was no problem 1 if the name already exists 2 if IVROW is out of range or 3 if no name was found in KTEXT LSAVPR DZ MEMDZ ME
268. means no 1 means yes not in use whether to display the pop up solution window automatically 0 means no 1 means yes The form of the command is SET lt ParameterID gt lt NewValue gt where lt ParameterID gt is one of the Parameter IDs listed in the table above and lt NewValue gt is the new value you wish to change the parameter to For a more detailed discussion about each of these parameters please see page 45 The SET command does not affect the current model or the current solution Values for SET are not saved with any model or solution file TITLE Syntax TITLE lt TitleString gt The TITLE command associates a text string with the model The string is displayed at the beginning of the model preceded by the word TITLE The title string can be no more than 73 characters long If you enter a longer string it will be truncated If you enter nothing after TITLE LINDO will display the current TITLE The default value is blank There does not need to be a blank between the command TITLE and the title string However you may wish to insert a blank between TITLE and title string to make the TITLE more readable There must be exactly one newline enter character in the command which ends the title The TITLE will be saved when the model is saved with SAVE SMPS or the DIVERT LOOK ALL sequence COMMAND LINE COMMANDS 189 The following example illustrates how to input a title and how the title is displayed when you run t
269. ming model appear to the internal solver as columns in a matrix The constraints appear as the rows of this matrix Therefore throughout this user s manual variables are referred to as columns constraints are referred to as rows and vice versa One last distinction to make about this manual is the meaning of the term external editor An editor includes any application that has the capability to read an ASCII text file and make changes to the characters in that file This includes applications that are commonly referred to as wordprocessors such as MS Word However wordprocessors tend to have many more capabilities than just text editing that are not required to edit a LINDO text file MS Notepad is a text editor that is generally included with new PCs Any of these applications would qualify as an external editor as defined in this manual The underlying algorithm used by LINDO s internal engine is the Revised Simplex Method with Product form Inverse However the intent of the LINDO User s Manual is to focus on the mechanics of using the LINDO software Perhaps the most challenging and rewarding aspect of mathematical programming is the ability to develop a concise and accurate model of a particular problem A well crafted model can be solved in a short amount of time whereas a model that is not as well thought out might take more than a reasonable amount of time on the fastest hardware available Although we touch on the topic of modeling brie
270. minimize to zero Obviously everything else being equal it s cheapest to hire nobody at So you have to constrain each day s labor force to meet the daily staffing requirement We indicate the beginning of our constraint set by typing SUBJECT TO Each day the number of workers on each shift that covers the day will have to be at least as great as the minimum required for that day On Monday for instance 20 workers are the minimum so MON MWORK THWORK FWORK SWORK SUWORK gt 20 Notice the variables TWORK and WWORK do not appear in this constraint This is because workers who start on Tuesday and Wednesday are not on duty on Monday Note also we have made use of the row naming feature and have assigned this row the name MON Moving down through the week the rest of the constraints are TUE MWORK TWORK FWORK SWORK SUWORK gt 13 WED MWORK TWORK WWORK SWORK SUWORK gt 10 THU MWORK TWORK WWORK THWORK SUWORK gt 12 FRI MWORK TWORK WWORK THWORK FWORK gt 16 SAT TWORK WWORK T HWORK FWORK SWORK gt 18 SUN WWORK THWORK FWORK SWORK SUWORK gt 20 GETTING STARTED Finally we denote the end of the model by typing END In its entirety the model is as follows MIN 300 MWORK 325 TWORK 360 WWORK 360 THWORK 360 FWORK 360 SWORK 335 SUWORK SUBJECT TO MON MWORK THWORK FWORK SWORK SUWORK TUE MWORK TWORK FWORK SWORK SUWORK WED MWORK TWORK
271. mmand on a small model look all MAX 2K a Y SUBJECT TO 2 X gt 2 3 Y gt 3 4 3 X Y lt 12 END terse go Solve the model LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE 10 0000000 delete 3 Remove the lower bound constraints delete 2 look all AX 2X yY SUBJECT TO 2 3X Y lt 12 END 166 CHAPTER 3 slb x 2 Reenter bounds with the SLB command slb y 3 look all AX 2xX yY SUBJECT TO 2 3 X Y lt 12 END SLB X 2 00000 SLB Y 3 00000 go Re solve we should get the same objective LP OPTIMUM FOUND AT STEP 0 OBJECTIVE VALUE 10 0000000 FREE Syntax FREE lt Variable gt The FREE command specifies that a variable has no upper or lower bound It allows variables to take on any real value After typing FREE enter the name or number of the variable you choose If you do not specify a variable LINDO will prompt you for one The FREE command and simple bounding commands i e SUB and SLB are contradictory If you use the FREE command with a variable after using the SUB and or SLB commands with that variable the bounds are erased The FREE specification is saved when the model is saved with SAVE SMPS and the DIVERT LOOK ALL sequence The current solution is no longer considered optimal after FREE is used The following session illustrates the use of the FREE command loo
272. mns Here APPCOL is called twice once for each column in the model c Define X kname X call appcol loc kname 3 valx irowx trouble c Define Y kname yY call appcol loc kname 3 valy irowy trouble INSROW or INSERT could also have been used to add the nonzero elements to the model However because LINDO stores models internally in column order using APPCOL is more efficient 262 CHAPTER 8 At this point the entire model is built A useful debugging device is to display the model s formulation This is done with LINDO s LOOK routine c Display the formulation call look 1 4 The next step is to solve the model using LINDO s GO routine call go 1000 istat Once the model is solved the solution is retrieved from LINDO and displayed c Get the objective value call reprow 1 obj dual c Get the value of x call repvar 1 x dual c Get the value of y call repvar 2 y dual c Report values write 6 10 obj x y 10 format Objective f8 1 x Spee ly ot Y 8 1 pause Press lt Enter gt to exit The LINDO routine REPROW is used to get the value of the objective row Likewise REPVAR is used twice to get the values for the two variables Finally LINDO is closed down with a call to LSEXIT call lsexit LINDO CALLABLE LIBRARIES 263 Running the Application In order to run the application you need to make the LINDO DLL available to the system To do thi
273. model balloons to more than 60 lines in MPS format As a general rule MPS files tend to be quite large Another point to notice is an MPS file basically consists of a long list of column row coordinate pairs followed by a nonzero value As such files in this format are not easily interpreted by merely looking at the text On the other hand if you are writing model generators to build formulations to pass to LINDO you may find it easier to build MPS files as opposed to equation format files Note Information regarding any solution is not saved in an MPS file and internal parameters such as BIP PAGE IPTOL SET QCP and WIDTH are not saved as well Use the RMPS command to read a file from disk created with SMPS MPS format files generally take longer to read and write than the files created with LINDO s SAVE format Some Notes on MPS Format Files Most of the commonly used features of the MPS format are understood by LINDO subject to the following restrictions e Leading blanks in variable and row names are ignored All other characters including embedded blanks are allowed e Only one free row type N row is kept from the ROWS section after input That is the one selected as the objective e Only a single BOUNDS set is recognized in the BOUNDS section The bound types recognized are UP upper bound LO lower bound FR free variable FX fixed variable BV bivalent variable i e 0 or 1 UI upper bounded integ
274. n i e variable to the formulation DEL sd Delete a specified constraint FREE Declare a variable unconstrained in sign SLB Enter a Simple Lower Bound for a variable SUB Enter a Simple Upper Bound for a variable 7 Integer quadratic and parametric programs Command Description Set an optimality tolerance on an IP model PARAMETRICS Perform RHS parametric analysis POSD Examine the submatrix of constraints to determine whether this submatrix is POSitive Definite Indicate first real constraint in a quadratic model TITAN Tighten up an IP 8 Conversational Parameters DEL Extend problem by adding constraints SLB SUB Command Description ft Insertacomment O BATCH Indicate this is a batch run PAGE Set page screen length PAUSE Pause for keyboard input TERSE Set conversational style to terse VERB Set conversational style to verbose WIDTH Set terminal display width 117 118 CHAPTER 3 9 User Supplied Subroutines Command Description 10 Miscellaneous Command Description INVERT STATS Print summary model statistics BUG Display information on how to report a bug in LINDO DEBUG Help in debugging infeasible and unbounded models TITLE Enter change or return model title 11 Quit Command Description QUIT Quit LINDO The Commands in Depth All LINDO commands are listed and explained below by category 1 Information HELP The HELP comma
275. n or AutoUpdate is in snooze mode see below then you will be returned to the main LINDO environment If you have an old version of the software you will be presented with the following dialog box AutoUpdate An updated version of your Lindo Systems Product was found on the website Would you like AutoUpdate to retrieve it for you Remind me in 7 Days Disable AutoUpdate By entering a number of days in the textbox and selecting the Remind me in button AutoUpdate is put in a type of snooze mode A checkmark next to the command in the Help menu indicates that the command is still enabled However LINDO will suppress the AutoUpdate dialog box on startup until the time limit has elapsed WINDOWS COMMANDS 113 Select the Yes button to download the new version of the software from the web LINDO will start downloading the new installation file and present the following dialog box to show the status of downloading Auto Update Downloading the new Install file Cancel Upon completion of downloading the InstallShield Wizard will open to guide you through the setup process to update LINDO For your convenience your license key will be automatically transferred to the new version during installation Select the Disable AutoUpdate button from the AutoUpdate dialog box to disable the AutoUpdate feature The AutoUpdate feature is disabled by default Keeping your software up to date helps ensure that you are using the mo
276. n problem such as our example interpreting the dual price requires some thought The dual price on the FRI constraint for example is 100 This means raising the right hand side of the FRI constraint by one unit would cause the objective to improve by negative 100 That is it would increase by 100 In other words the marginal cost of providing one additional staff member on Fridays would be 100 in additional salary expense Dual prices are sometimes called shadow prices because they tell you how much you should be willing to pay for additional units of a resource Going back to the Friday constraint if we use the shadow price interpretation of the dual price we can also say that we should be willing to pay up to 100 to have an additional staff member present on Friday If temporary employees are available for less than 100 per day then we might want to consider this option As with reduced costs dual prices are valid only over a limited range Range analysis will be discussed in more depth in Chapter 2 LINDO for Windows 25 2 LINDO for Windows This chapter explores all the commands available to the LINDO user running under Windows In the Windows environment LINDO partitions all commands into the following six categories 1 File 2 Edit 3 Solve 4 Reports 5 Window 6 Help These categories are simply the menus available to you when running LINDO as the following picture of the LINDO command menu illustrates
277. n should be Z 1 X 2 3 Y 1 3 LINDO however literally interprets the coefficients that were entered and proposes the following solution correctly as the optimum LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1 0 0000000E 00 VARIABLE VALUE REDUCED COST Z 0 000000 0 000000 X 0 000000 0 000000 Y 0 000000 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 9998 340820 3 0 000000 9998 340820 4 0 000000 9998 340820 5 1 000000 0 000000 The reason is that for the given input data constraint 4 is consistent with constraints 2 and 3 only if X Y Z 0 If you sum constraints 2 and 3 you get 9999 Z X Y 0 This is consistent with 4 to five decimal digits only if X Y Z 0 On the other hand if you specify the fractions to 6 decimal places as below MAX Z SUBJECT TO 2 0 666666 Z X 0 3 0 333333 2 Y 0 4 Z X Y 0 5 Z lt T END then LINDO gives the solution LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1 1 000000 VARIABLE VALUE REDUCED COST Z 1 000000 0 000000 X 0 666666 0 000000 Y 07333333 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 0 000000 3 0 000000 0 000000 4 0 000001 0 000000 5 0 000000 1 000000 This is because it considers 999999 and 1 indistinguishable The general moral to this little example is that a good formulation has the
278. nce you exit LINDO s Setup program you can run LINDO by double clicking on the LINDO icon in the LINDO folder which should open on your desktop Some versions of LINDO require you to enter a password the first time LINDO is run If you need to enter a password LINDO will display the following dialog box LINDO Password Please enter your LINDO password below If you don t have a password you can press Demo Version to automatically generate a temporary license for a demonstration version of LINDO Demo versions function the same as standard versions with the one exception that maximum problem dimensions are restricted If your password is available in the Windows clipboard you may paste it into this dialog box by pressing Ctrl Otherwise carefully enter your password as one long string without carriage returns being sure to include all hyphens You can access this dialog at any time using the FilelLicense command Help Cancel Demo Version If you don t know your password check for it on the back of the CD sleeve Carefully enter your password into the edit field including hyphens making sure that each character is correct Click the OK button and assuming the password was entered correctly LINDO will display the Help About box listing the features of your license Verify that these features correspond to the license you intended to install Note Ifyou were e mailed your password then you have the option of
279. nd by itself will give you general information about your version of LINDO along with the maximum number of constraints variables and nonzeros which your version of LINDO can handle The HELP command combined with another LINDO command gives specific information on a command This information is usually quite brief but is often all you need COMMAND LINE COMMANDS 119 For example typing HELP GO displays the following GO COMMAND SOLVES THE CURRENT MODEL THE MODEL REMAINS INTACT THROUGH THE SOLUTION PROCESS A POSITIVE INTEGER TYPED AFTER GO IS INTERPRETED AS AN UPPER LIMIT ON THE NUMBER OF PIVOTS CAT The CAT command lists the eleven categories displayed in COM but does not display the actual commands Then it asks you which category is of interest by number 1 through 11 If you enter a category number the list of commands in the category is displayed followed by another prompt asking for a category number To escape from the prompt simply press the enter key instead of a category number COM The COM command lists all commands arranged in eleven categories When you know LINDO can do a particular function and you can t remember which command to use the COM command is helpful COM displays the following
280. ne column Xx Iro 1 1 Iro 2 2 Iro 3 4 Value 1 20 Value 2 1 Value 3 1 Kname X Nonz 3 Call APPCOL Kname Nonz Value l1 Iro 1 Trouble Define column y Iro 1 1 Iro 2 3 Iro 3 4 Value 1 30 Value 2 1 Value 3 2 Kname Y y Nonz 3 Call APPCOL Kname Nonz Value 1 Iro 1 Trouble Solve the model Call GO 0 Istat Print objective value w qA K O H H gt t o EPROW I Primal Dual Objective Value Primal Print variable values Call REPVAR 1 Primal Dual Print Print X Primal Call REPVAR 2 Primal Dual Print Y Primal Shut down LINDO Call LSEXIT End Sub First off in the above code LINDO is initialized with the calls Call ILINDO Call INIT ILINDO performs one time initialization for LINDO It should be called once at the start of your code Then INIT performs initialization for a new model INIT should be called each time your code begins input of a new model LINDO CALLABLE LIBRARIES 249 Next the file called LINDO OUT is opened with the LUNOPN routine Then LINDO s standard output is redirected to this file via a call to CAPOUT Call LUNOPN 60 9 LINDO OUT _ 0 O O QO Call CAPOUT 60 Had LINDO s standard output not been redirected the application would crash under Windows the first time LINDO wrote anything to the standard output device As a note if you are having difficulty
281. nformation see the TAKE Command in the File Input section Comment An exclamation point may be added almost anywhere in a LINDO input line The remainder of any line following the exclamation point is ignored by LINDO The carriage return ends the comment These are stripped out by LINDO as soon as the model is entered into memory For example if you type the following at the colon prompt for a new model MAX X Y The Objective Function 2 As soon as you hit the carriage return the comment is lost As you can see entering comments in interactive mode is of little or no help However when an external editor is used to develop a LINDO TAKE file these comments can help make your model easier to read during development while having no effect on the final solution BATCH The BATCH command in LINDO causes the following two things to happen 1 all input is echoed to the screen and 2 if LINDO s input contains more than 5 consecutive errors LINDO will automatically exit to the operating system The BATCH command is a toggle Typing a second BATCH command undoes the first The BATCH command is useful when submitting command scripts to LINDO to run in batch i e non interactive mode In this case without the BATCH command all input to the program would not be visible and should there be errors in the file LINDO would not stop until it reached the end of the command script The BATCH command is also useful in an interacti
282. not exist Check the spelling of the row name and try again WARNING lt N gt ERRORS INPUTTING MPS FILE CORRECT EXECUTION UNLIKELY LINDO encountered lt N gt errors while processing your MPS file This is a warning and will not terminate processing of the file However if any of the errors were unexpected you should correct them before proceeding 86 87 88 89 90 91 ERROR MESSAGES 281 PROBLEM TOO BIG DATA INPUT IS FINISHED lt M gt ROWS lt N gt COLUMNS lt Z gt NONZERO ELEMENTS Your MPS model has hit an internal limit of your version of LINDO LINDO prints the total number of rows columns and nonzeros read before interrupting processing of the file To determine the limits of your version give the Help About LINDO command in Windows or the HELP command in command line versions Note in some versions of LINDO you can increase the nonzero limit see error message 72 RETRIEVAL ABORTED LINDO has interrupted processing of your MPS file due to errors in the file Please correct any errors and retry END OF FILE ENCOUNTERED IN MPS FILE LINES READ lt N gt LINDO has encountered a premature end of file in your MPS format file Perhaps part of the file is missing or you omitted the ENDATA card at the end of the file Please correct and retry NONEXISTENT COLUMN NAME ON THE FOLLOWING BASIS RECORD lt CardText gt LINES READ lt N gt LINDO encountered an invalid variable name while
283. nput the name of the file to the prompt For additional insight as to how the Take Commands Command is used please refer to Chapter 7 Interfacing with the Outside World section on Running Command Scripts with the TAKE Command TAKE is the corresponding Command line Command of the File Take Commands Command in Windows Basis Save Shift F2 In the parlance of operations research a solution to a linear programming model is referred to as a basis Thus the Basis Save command is simply a command to save the solution to your model Why would you want to save a model s solution Suppose you need to run the model at a later date perhaps with some minor modifications Starting from the old optimal solution could save you considerable time especially if the model is large These saved solutions are different from the solution reports that LINDO generates with the Solution command In general solutions saved with the Basis Save command are difficult if not impossible to interpret They are intended primarily for loading back into LINDO at some later date to use as a starting point for creating a similar model WINDOWS COMMANDS 37 Assuming you have solved the active model you can save the solution at any point by running the Basis Save command When you issue the Basis Save command you will be presented with a dialog box similar to the following Save Basis As Directories OK c lindo61 samples E os Cancel gt LINDOG1 E gt S
284. ns that relaxing that constraint may reduce the sum of the infeasibilities The following example illustrates The coefficient 55 in row 4 should have been 5 5 MAX 3 X OEY SUBJECT TO 2 X 2 Y lt 3 3 2X Y lt 2 4 0 55 X Y gt 4 END When we attempt to solve this formulation we get the following solution report NO FEASIBLE SOLUTION AT STEP 1 SUM OF INFEASIBILITIES 2 483333 VIOLATED ROWS HAVE EGATIVE SLACK OR EQUALITY ROWS NONZERO SLACKS ROWS CONTRIBUTING TO INFEASIBILITY HAVE NONZERO DUAL PRICE OBJECTIVE FUNCTION VALUE 1 10 33333 VARIABLE VALUE REDUCED COST X 04333333 0 000000 Y 1333333 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 0 483333 3 0 000000 0 033333 4 2 483333 1 000000 All the constraints have nonzero dual prices so we do not have much of a clue in tracking down the culprit 210 CHAPTER 6 The Solve Debug page 61 command in Windows and the DEBUG command line command page 138 will try to identify one or more crucial constraints A constraint is crucial if dropping just that constraint from the entire model is sufficient to make the model feasible These constraints are identified by Debug as the SUFFICIENT SET Not every infeasible model has a crucial constraint Regardless of whether any crucial constraints were found the Debug command also
285. nufacture 3 741 543 35 two inch nails whether you round up or down is of little consequence The cost in time to solve the model largely outweighs the benefits of getting an exact nail count In this case for solution efficiency reasons you should use a continuous variable to represent the number of ED COST 0909 363636 0000 09 00 ES 181818 Now use the GIN lt Variable gt command to make each variable general integer gin x gin y gin z COMMAND LINE COMMANDS 171 look all AX 20 Kote Da ok ob 137 SUBJECT TO 2 17 X 13 Y d112 lt 93 END GIN 3 The integer solver may print copious output if not in TERSE mode terse Solve the integer form of the model allow for 1000 iterations go 1000 LP OPTIMUM FOUND AT STEP 1 OBJECTIVE VALUE 109 909088 NEW INTEGER SOLUTION OF 104 AT BRANCH 1 PIVOT 8 BOUND ON OPTIMUM 109 8333 NEW INTEGER SOLUTION OF 105 AT BRANCH 4 PIVOT 17 BOUND ON OPTIMUM 109 8333 EW INTEGER SOLUTION OF 106 AT BRANCH 8 PIVOT 27 BOUND ON OPTIMUM 109 8333 EW INI SOLUTION OF 107 AT BRANCH 17 PIVOT 59 D Z Q Uw BOUND ON OPTIMUM 108 5455 ENUMERATION COMPLETE BRANCHES 22 PIVOTS 72 LAST INTEGER SOLUTION IS THE BEST FOUND I REINSTALLING BEST SOLUTION solution OBJEC
286. nvention will be used throughout the command line section of this user s manual 2 Input MAX MIN Every model in LINDO must start with either MAX or MIN Either of these two words when entered at the command line indicates to LINDO that a new model is being entered Complete the objective function with the command ST SUCH THAT S T or SUBJECT TO Note The MAX and MIN commands erase any current model in memory and any current solution The new model you enter becomes the current model Leave a space or a carriage return after MAX or MIN This ensures that the command is unambiguous Following these you may enter numbers for objective function coefficients or strings that begin with an alphabetic character for variable names LINDO looks at plus and minus symbols and letters to determine which characters are coefficients and which are variable names For example mAx 9xl x2 4x3 2 x4 8 XS 2x6 8x7 12 x8 is interpreted as MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 A number following a plus minus MIN or MAX is interpreted as a coefficient e g the 9 above A number immediately following a letter is interpreted as part of the variable name such as the 6 above If the variable name string is too long the input will be rejected by LINDO For example most versions of LINDO allow variable names of 8 characters in length so the variable name X12345678 would be rejected COMMAND LINE COMMANDS 121 A letter
287. o internal tolerances which dictate the maximum amount of violation allowed on a constraint in order to consider it still to be satisfied These tolerances are called the Initial Constraint Tolerance and the Final Constraint Tolerance The Initial Constraint Tolerance is used when the solver first begins and can be relatively loose in order to speed solution times Once LINDO has closed in on an optimal solution it switches to the Final Constraint Tolerance which should be relatively tight in order to guarantee an accurate final solution The default value for the Initial Constraint Tolerance is 8 while the default for the Final Constraint Tolerance is 1 One instance where these tolerances can be of use is when LINDO reports a model is infeasible by a small amount When a model is infeasible LINDO reports a sum of infeasibilities which is the amount of violation on all the constraints and bounds If this number is relatively small then you might want to consider loosening the constraint tolerances This is particularly true in a model where scaling is poor and the units of measurement on some constraints are such that minor violations are insignificant For instance suppose you have a budget constraint measured in millions of dollars In this case a violation of a few pennies would be of no consequence Short of the preferred method of re scaling your model loosening tolerances may be the most expedient way around a problem of this natu
288. o satisfy and the information you want listed for each column This feature is extremely useful for perusing large models and their solutions See the RPRI command for similar output regarding rows constraints COMMAND LINE COMMANDS 139 The column attributes lt Co AttList gt may be any of the characters in the left column in the following table The interpretation of the attribute is listed as well ColAtt U Interpretation Reduced cost Simple lower bound Column name Primal value i e the solution value Objective coefficient Rim Type of variable C for continuous T for integer and F for free Simple upper bound Number of nonzeros in the column The conditional expression lt CondExp gt may include any of the operators listed in the table below in conjunction with the column attributes in the table above to form conditional expressions Operator LOG x EXP x ABS x AND OR NOT Function Wild card character placeholder for use in forming variable name templates Addition Subtraction Division Multiplication Exponentiation Natural logarithm of x Absolute value of x Logical and Logical or Logical not Compare greater than Compare less than Compare equal to Compare not equal to Parentheses for specifying precedence LINDO evaluates the conditional expression for each column If the conditional expression evaluates to True LINDO will print the val
289. o the command file Print 1 MAX 20X 30Y Prine tip ST Print 1 X lt 50 Print 1 Y lt 60 Print 1 X 2Y lt 120 Print 1 END Rem Suppress the standard solution report Print 1 TERSE Rem Solve the model Print 1 GO Rem Open a file for Print 1 4 Rem Send the solutio Print 1 CPRI Rem Close output fil Print 1 RVRT Rem Quit LINDO Print QUIT storing the solution DIVERT C LINDO SHELL LNDOUT TXT n to the output file N p amp Rem Close LINDO command file Close Rem Shell to LINDO Rem command file in Rem means run LINDO I Shell C LIN t C LINDO SHE passing the name of the the command line The 2 minimized DO LINDO J LL LNDIN TXT 2 Rem We now need to w Rem our input file On Error Resume N Do Err 0 Rem Try to open the Rem LINDO Open C LINDO For Input As ait until LINDO creates ext solution file printed by SHELL LNDOUT TXT J 1 224 CHAPTER 7 Rem Break out of loop if successful If Err 0 Then Exit Do Rem Let other tasks run DoEvents Loop On Error GoTo 0 Rem Read in the variable values Input 1 X Y Rem Close solution file Close 1 Rem Print solution on screen Print X and Y X Y Page224cEnd Sub This is the code that writes a script file uses a shell command to invoke LINDO for solving it reads the results from a LIN
290. of constraints as well as column bounds that constitute a NECESSARY SET ROWS Such a set has the feature that it is infeasible However if any member of this set is deleted then the set becomes feasible Thus it is necessary to make at least one correction in the NECESSARY SET ROWS if the model is to be feasible When Debug encounters an unbounded model it first tries to identify one or more crucial variables These variables are identified by Debug as SUFFICIENT SET COLS A variable is crucial if fixing it is sufficient to make the model bounded Regardless of whether any crucial variables were found Debug also identifies a set of variables that constitutes a NECESSARY SET COLS Such a set has the feature that it is unbounded However if any variable in this set is fixed the set becomes bounded In summary if the complete model would be feasible except for a bug in a single row that row will be listed in the SUFFICIENT SET of rows The NECESSARY SET of rows is a set such that if as long as all of them are present the model remains infeasible Typically Debug helps substantially reduce the search effort The first version of Debug was implemented in response to a user who had an infeasible model The user had spent a day searching for a bug in a model with 400 rows Debug quickly found a NECESSARY SET with 55 constraints as well as one SUFFICIENT SET constraint The user quickly noticed that the RHS of the SUFFICIENT SET constraint
291. ollowed by the command SOLUTION will send the model formulation and its solution report to lt FileName gt You close the file and return to screen output with the RVRT command 12 CHAPTER 1 Note The SAVE command in LINDO saves the formulation only in a compressed format Use DIVERT as shown above to save the model or solution in a format your word processor can make use of For more detailed information about the syntax and use of the DIVERT command please refer to the File Output section of Chapter 3 Command line Commands Model Syntax This section details the syntax required in a LINDO model Fortunately the list of rules is rather short and easy to learn As shown in the example above a LINDO model has a minimum requirement of three things an objective variables and constraints The objective function must always be at the start of the model and is initiated with either MAX for maximize or MIN for minimize The end of the objective function and the beginning of the constraints is signified with any of the following SUBJECT TO SUCH THAT Se Ty ST The end of the constraints is signified with the word END LINDO has a limit of eight characters in a variable name Names must begin with an alphabetic character A to Z which may then be followed by up to seven additional characters These additional characters may include anything with the exception of the following lt gt So as an example the fol
292. olutions 17 18 269 Rows changing 11 direction of 159 limits 267 names 92 statistics 80 186 206 RPRI 116 142 44 206 8 Running an application 250 258 263 Running minimized 225 Runtimes 40 46 48 RVRT 11 116 145 218 S SAVE 116 121 144 45 189 Save As command 26 32 33 213 Save command 26 32 33 Saving 32 33 144 internal parameters 150 solutions 146 Scaling 267 Scripting 35 124 217 19 SDBC 116 126 146 48 SDBC files 37 38 Search for Help On command 28 110 Select All command 26 58 Send to Back command 28 104 5 Sensitivity analysis 11 73 130 151 173 SET 118 188 Shadow prices 24 72 Shell command 222 226 Shell function 227 SHOCOLUMN 116 133 35 208 Show Column command 27 100 101 208 Simple lower bounds 15 18 19 117 142 165 Simple upper bounds 15 18 19 117 142 163 Simplex Method See Revised Simplex Method Simplex tableau 135 SINGLE COLS 206 Size of a model 267 Slack or surplus 23 72 127 142 SLB 15 18 19 117 165 66 Smoothing 67 SMPS 116 148 SOLUTION 11 116 127 Solution command 27 71 73 Solution command line commands 117 151 59 Solution reports 8 11 21 60 71 116 153 191 269 constraints 72 DUAL PRICE 24 72 feasible 137 headers 39 iterations pivots 137 optimal 137 printing 11 137 range analysis 24 REDUCED COST 24 72 retrieving 37 saving 33 34 36 137 SLACK OR SURPLUS 23 72 suppressing 54 v
293. om You may specify either a variable index or name as illustrated below SHOCOLUMN lt Varlndex gt Displays the coefficients of variable index number lt Varlndex gt This is LINDO s internal index for the variable SHOCOLUMN lt VariableName gt Displays the coefficients of the variable named lt VariableName gt Consider the following model MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8B lt 6 4 Xl X3 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X 8 15 END Before we solve the model entering the command SHOCOLUMN 8 or SHOCOLUMN X8 generates the following SHOCOLUMN report x8 8 ROW COEF DUAL PRICE 1 12 0000 000000 2 3 00000 000000 3 1 00000 000000 7 1 00000 000000 8 1 00000 000000 The 8 in the top line of the output is LINDO s internal index for variable X8 After solving the model with the current solution in memory the same command displays X8 8 ROW COEF DUAL PRICE 1 12 0000 1 00000 2 3 00000 4 00000 3 1 00000 1 00000 7 1 00000 1 00000 8 1 00000 000000 VALUE 11 0000 REDUCED COST 000000 Note the primal value and reduced cost from the solution are now shown COMMAND LINE COMMANDS 135 The above formulation was modified to give variable X
294. on Please refer to your manual for information on special versions of LINDO FATAL OUT OF SPACE IN INVERT AT STEP lt N gt LINDO has run out of working nonzero space during a basis inversion In Windows versions you can use the Edit Options command to increase the nonzero limit In other versions you may be able to increase the maximum nonzero level by creating a file in LINDO s directory called LINDO CMNF Set the first line in the file to be MAXNZ z where z is the maximum number of nonzero elements you desire Restart LINDO and run the HELP command to see if you have obtained the desired number of nonzero elements NO BASIS ORDER EXISTS USE INVERT FIRST The Basis Picture command BPIC requires that an active basis be present in memory Run the INVERT command and try again MODEL TOO BIG TO PERMUTE LINDO has run out of space while attempting to permute a model into lower triangular form for a nonzero picture Increasing the nonzero limit may help see message 72 DUPLICATE ROW IN MPS MODEL lt Row gt A row has been declared twice in a MPS file Please correct the MPS file and try again BAD CARD IN MPS MODEL lt CardText gt An invalid record was encountered in an MPS file Please correct the MPS file and try again 280 77 78 79 80 81 82 83 84 85 APPENDIX A NO OBJECTIVE ROW WAS FOUND IN THE MPS MODEL LINDO did not find an objective row in an MPS model Please add an ob
295. on of LINDO as a subtask to process the script file Finally we return to the C front end to read the solution file created by LINDO and print the results on the screen Although a real life application would do much more than this the basic framework used here should help provide insights The C source code for this sample application is as follows include lt stdio h gt for file i o include lt io h gt for dup functions include lt process h gt for spawn function void main WEA Tf char char char float FILE int n Open a fp Always first fprin Send t fprintf Suppre fprin Solve fprin Open a fprin Send t fprin fprintf fp X lt 50 n cFileiIn LNDIN TXT cFileOut LNDOUT TXT cFileSolu SOLUTION TXT XVal f YVal fp fpIn fpOut fpSolution Stdin nStdOut LINDO script file fopen cFileIn w a good idea to do a PAGE 0 off in LINDO script files tf fp PAGE O n he formulation to the script file fp MAX 20X 30Y n fp ST n fp Y lt 60 n fp X 2Y lt 120 n fp END n ss the standard solution report tf fp TERSE n the model t fp GO n file for storing the solution tf fp DIVERT Ss n cFileSolu he solution to the output file tf fp CPRI N P n Close output file fprin Quit LINDO
296. on the vertical axis 80 CHAPTER 2 Ostensibly the Parametrics command allows you to investigate changes in the right hand side of at most one constraint at a time Realize however that a simple device allows you to do this for several right hand sides simultaneously in a linear fashion Let the change column be called D It may have coefficients in any or all rows Add a constraint D 0 and then do parametric analysis on the RHS of this constraint The Parametrics command may be applied to linear and quadratic programs but is not applicable to integer programs Statistics Alt 3 The Statistics command sends a small report to the Reports Window This report contains a number of summary statistics concerning the active Model Window As an illustration consider the following model IS C _LINDO61 SAMPLES SMALLIP LTX 9 X1 X2 4 X3 2 X4 8 XS 2 X6 8 X7 12 8 X X2 2 X3 X4 2 X5 X6 2 X7 3 XB lt 13 1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 1 9 3 X7 12 9 15 O A S A Issuing the Statistics command for this model yields the following report i Reports Window DER A ROWS 8 VARS 8 INTEGER VARS 8 0 071 OQCP 0 NONZEROS 4 CONSTRAINT NONZ 32 23 1 DENSITY 0 653 SMALLEST AND LARGEST ELEMENTS IN ABSOLUTE VALUE 1 00000 15 0000 OBJ MAX NO lt Sees 2 5 0 GUBS lt 2 VUBS gt 0 SINGLE COLS 0 REDUNDANT COLS The first l
297. ondition optional CALINDO61 SAMPLES SMALLIP LTX Ip gt 0 The Peruse command will report on either columns or rows So depending on the type of report desired you will need to highlight either the Columns or Rows button in the box labeled Orientation Next you need to check off the items that you want included in the report in the box labeled View Items A list of candidate items and their definitions are listed below Note the definition of a report item depends on whether the report is oriented towards columns or rows Report Item Column Definition Row Definition Row slack surplus Objective coefficient Right hand side Upper bound Upper bound WINDOWS COMMANDS 83 C for continuous F lt for for free and T for less than or equal to 69 integer for equality and gt for greater than or equal to Nonzeros Column nonzero count Row nonzero count Next you may optionally input a condition in the Condition Optional box which you can use to filter out unwanted columns or rows from the report We will discuss this feature in more detail below The remaining options deal with the format of your report The first option in this regard is to select the type of report in the Report Types box You may select a text based report a graphical report or both Text reports are routed to the Reports Window Graphics reports are placed in individual
298. onstraint This means in order to satisfy all the requirements of the model LINDO had to have one more person than required available on Thursday Overstaffing always seems wasteful but because of uneven staffing requirements sometimes it is unavoidable Slack or Surplus The SLACK OR SURPLUS column in a LINDO solution report tells you how close you are to the right hand side RHS limit on each constraint That is the limit that appears to the right of the 24 CHAPTER 1 inequality or equal sign in any given constraint This quantity on less than or equal lt constraints is generally referred to as slack while on greater than or equal to constraints it is called a surplus If a constraint is exactly satisfied as an equality the SLACK OR SURPLUS value will be zero Ifa constraint is violated as in an infeasible solution the SLACK OR SURPLUS value will be negative Knowing this can help you find the violated constraints in an infeasible model Reduced Costs In a LINDO solution report you Il find a REDUCED COST figure for each variable There are two valid equivalent interpretations of a reduced cost First you may interpret a variable s reduced cost as the amount by which the objective coefficient of the variable would have to improve before it would become profitable to bring that variable into the solution at a nonzero value In the staffing example we ve been looking at only the variable TWORK is not in the solu
299. orkers receives 300 per week for a regular schedule with 25 extra for Saturday work and 35 extra for Sunday work Each worker can work only five days a week and must have two consecutive days off Second you have minimum staffing needs which must be met In table form that requirement looks like this DAY M T W Th F S Su REQ 20 13 10 12 16 18 20 22 CHAPTER 1 Under the restrictions outlined above the following schedules are available with varying weekly pay rates where the symbol W denotes a day worked and the symbol Q denotes a day off Schedule Weekly Start Day Mon Tue Wed Thu Fri Sat Sun Pay Mon m m m mj m of of 30 Here s how this information translates into a LINDO model First let s determine what the objective will be If you want to minimize your cost that means minimizing the sum of all weekly salaries paid If you have a variable called MWORK that represents the number of workers who start on Monday and the salary for that shift is 300 then 300 MWORK is the amount paid to those Monday Start workers Similarly TWORK can represent Tuesday Start workers and 325 TWORK according to the table above is the amount paid to the those workers Proceeding through the week you can formulate your objective like this MIN 300 MWORK 325 TWORK 360 WWORK 360 THWORK 360 FWORK 360 SWORK 335 SUWORK However if you were to solve the model right now the solution would
300. ost important part of your model and they require some real thinking 10 CHAPTER 1 In the little example we re considering if you were to maximize 10 STD 15 DLX now there s no limit to how many Standard and Deluxe computers you could produce Of course there must be some limit In this example the limit is the factory output and the labor supply So let s constrain both STD and DLX to be less than the factory capacity available per day of 10 and 12 respectively You do this by entering the words SUBJECT TO at the question mark prompt or just the letters ST pressing the return key and entering STD lt 10 and DLX lt 12 on the following two lines Note that LINDO interprets the lt symbol as meaning less than or equal to rather than strictly less than If you prefer you may alternatively enter lt in place of the lt character Very well we ve constrained the variables by computer production available at the factories We now add the labor constraint of 16 units of labor by entering STD 2 DLX lt 16 on the next line Finally on the line after that we designate the end of the constraints by adding END Now your screen should look like this LINDO COPYRIGHT C LINDO SYSTEMS LICENSED MATERIAL ALL RIGHTS RESERVED MAX 20 STD 15 DLX SUBJECT TO STD lt 10 DLX lt 12 STD 2 DLX lt 16 END 8D DD dD oe The model is in memory and LI
301. ou want LINDO to match the case of the text exactly click the Match Case box in the lower left corner of the dialog box before proceeding with the command Options Alt O The Options command is used to modify default system parameters When issuing the Options command the following dialog box will appear Options m Optimizer m Output Integer Programming p General IV Status Window IV Preprocess r Prefered Branch Default Up C Down IP Optimality Tol None IP Objective Hurdle None IP Var Fixing Tol None Nonzero Limit 2000000 Iteration Limit None Initial Constraint Tol 8e 005 Final Constraint Tol fi e005 Entering Yar Tol 5e 007 Pivot Size Tol 1e 010 Terse Output Fage Length Limit None Terminal Width OK Cancel Save Default Help LINDO options remain in effect for the entire session If you modify any options and wish to preserve their setting from one LINDO session to the next press the Save button This will save the current options so they can be restored when LINDO is restarted If at anytime you wish to return to the default set of options press the Default button The defaults are identified in detail below WINDOWS COMMANDS 45 Options in LINDO are broken down into the following categories 1 Optimizer Options options that pertain to the functioning of the solver engine a Integer Programm
302. ou need to do extensive editing on a large model we suggest that you use a word processor e g MS Word to make your changes or any text editor e g MS Notepad and then load the file into LINDO with the View command If you use an external editor be sure to have the external editor save the model as Text Only Otherwise LINDO will not be able to read the file With the exception of using a View Window the View command functions exactly as the Open command does Please refer to the documentation directly above on the Open command for additional insight into the functioning of the View command 32 CHAPTER 2 Save F5 Save As F6 The Save and Save As commands save the contents of the active window to a disk file Save uses the model s existing file name and format while Save As prompts you for a new name and format to save under When issuing the Save As command you will see the following standard File Save As dialog box File Name Directories lindo Its c Mindo61 samples Cancel jo cs ea gt LINDOS1 Help E gt Samples 9 Samptext File Format Drives LINDO Text Itx x c Network This dialog box has all the standard features which allow you to navigate through your system to save a file The one non standard feature is the list box in the lower left corner titled File Format When saving a model you have a choice of three different formats The File Format list box is used
303. ou were to maximize 10 STD 15 DLX now there s no limit to how many Standard and Deluxe computers you could produce Of course there must be some limit In this example the limit is the factory output and the labor supply So let s constrain both STD and DLX to be less than the factory capacity available per day of 10 and 12 respectively You do this by entering the words SUBJECT TO on the next line of your model or just the letters ST pressing the return key and entering STD lt 10 and DLX lt 12 on the following two lines Note that LINDO interprets the lt symbol as meaning less than or equal to rather than strictly less than If you prefer you may alternatively enter lt in place of the lt character Very well we ve constrained the variables by computer production available at the factories We now add the labor constraint of 16 units of labor by entering STD 2 DLX lt 16 on the next line Finally on the line after that we designate the end of the constraints by adding END 6 CHAPTER 1 After entering the above your screen should look like this i gt lt untitled gt DER ST STD lt 10 DLX lt 12 STD 2 DLX lt 16 END Your model has now been entered and it s ready to be solved To begin solving the model select the Solve command from the Solve menu or press the Solve button Sy on the toolbar at the top of the window LINDO will begin by trying to compile t
304. ous windows created by LINDO These commands are discussed below in detail Alt C Command Window mf The colon in the upper left hand corner is LINDO s familiar command line prompt You may proceed by entering any of the valid LINDO commands WINDOWS COMMANDS 103 In this next example we enter a small model into the Command Window display the formulation and then solve it 3 Command Window max 20x 30y oy ut2y lt 120 end look all MAX 20 X 30 SUBJECT TO LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1 2050 000 VARIABLE VALUE REDUCED COST X 50 000000 0 000000 35 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 0 000000 5 000000 25 000000 0 000000 0 000000 15 000000 NO ITERATIONS 2 IDO RANGE SENSITIVITY ANALYSIS n In general you will probably prefer to use the menu and toolbar interface when running LINDO interactively The Command Window feature is primarily provided for users wishing to interactively test out ideas for LINDO scripts 104 CHAPTER 2 Open Status Window When LINDO s internal solver initiates it displays a Status Window on your screen resembling the following LINDO Solver Status Optimizer Status Status Optimal Iterations Infeasibility Objective Best IP IP Bound Branches Elapsed Time 00 00 00 Update Interval 1 Interrupt Solver This window allows you to monitor the progress of the solver You can close this Status Window at
305. pdated Setting this to zero tends to increase solution times Interrupt Solver Press this button to interrupt the solver and return the current best solution found Press this button to close the Status Window 60 CHAPTER 2 When the Solve command is finished the solution will be sent to the Reports Window where it can be reviewed edited and printed If your model is formulated correctly the Status Window will indicate that you have an optimal solution If the Status Window indicates the model is infeasible or unbounded then there is a problem in your formulation and you should not attempt to use the results from LINDO If the model contains no integer variables and is not a quadratic programming model the LINDO Debug command can assist you in finding the flaws in your model when it is either unbounded or infeasible If you want to suppress the automatic generation of a solution report at the end of the solve command click on Terse output in the Edit Options command dialog box The Reports Window can hold up to 64 000 characters of information If required LINDO will erase output from the top of the Reports Window in order to make room for new output at the bottom of the window If you have a lengthy solution report which you need to examine in its entirety you can log all information sent to the Report Window in a disk file using the Log Output command from the LINDO File menu This file can then be examined using an externa
306. plex algorithm used by the LINDO solver involves introducing a variable which formerly had a value of zero into the solution while simultaneously driving another variable to zero This operation is referred to as a pivot The Pivot command allows you to perform individual pivots optionally specifying the variable and constraint to perform the pivot on The Pivot command is not of much practical use but it is of great potential interest to students of the simplex algorithm particularly when coupled with the Tableau command The Tableau command can be used to view the current simplex tableau to assist in selecting an attractive pivot variable and its pivot row Users unfamiliar with the simplex algorithm can refer to any introductory operations research text to learn more 64 CHAPTER 2 As an example of the use of the Pivot command consider the following model I C ALINDO61 SAMPLES PIVOT LTX f2 EX MAX 20X 30Y ST ZH lt 50 2 IJ 56 270 4 X 2 lt 120 ENT We will use the Pivot command twice to introduce both X and Y into the solution and in so doing obtain the optimal solution When we first issue the Pivot command we are presented with the following dialog box Pivot Row Row Selection LINDO s r Pivot Variable m Variable Selection LINDO s C Use Mine C Use Mine My Variable Selection zi My Row Selection z If you were to click the OK button at this point LIND
307. programming models WINDOWS COMMANDS 71 4 Reports Menu Reports Window Help The Reports menu is shown at left and contains all the commands which Solution Alt 0 may be used to generate reports related to your model To generate a Range Alt 1 report for a window you must first make it the active window by Parametrics Alt 2 clicking on it with the mouse Then you can issue the desired command Statistics Alt 3 from the Reports menu The commands in this menu are discussed below Peruse Alt 4 in detail Picture Alt S Basis Picture Alt 6 Tableau Alt Formulation Alt 8 Show Column Alt 9 Positive Definite Solution Alt 0 The Solution command sends a standard solution report to the Reports Window for the currently active Model Window Normally LINDO automatically generates a solution report as part of the Solve command This automatic solution report is suppressed if you have used the Options command to place LINDO in Terse mode If LINDO is in Terse mode you will need to run the Solution command to get a solution report When you run the Solution command you will be presented with the following dialog box Solution Report Options X m Values OK All Values Cancel C Nonzeros Only Help Basically LINDO is prompting you to determine if you want all values included in the solution report or just the nonzero values If you request nonzeros only then LINDO will only include the variables wi
308. q wn oot 1 0005 Dy REDUCED COST 77 000000 6 000000 3 000000 6 000000 33 000000 13 000000 110 000000 21 000000 47 000000 DUAL PRICES 0 000000 0 000000 0 Now let s view the solution in MPS format 129 130 CHAPTER 3 SOLUTION OPTIMAL TIME 53350 88 SECS ITERATION NUMBER 3 NAME ACTIVITY DEFINED AS FUNCTIONAL 279 70786 SECTION 1 ROWS NUMBER ROW AT ACTIVITY SLACK ACTIVITY LOWER LIMIT UPPER LIMIT DUAL ACTIVITY sak BS 279 70786 279 70786 NONE NONE 1 00000 2 SPACE UL 1000 00000 NONE 1000 00000 0 854718E O1 3 WEIGHT UL 1200 00000 fi NONE 1200 00000 0 16186 SECTION 2 COLUMNS NUMBER COLUMN AT ACTIVITY INPUT COST LOWER LIMIT UPPER LIMIT REDUCED COST 4 x1 BS 0 31865 77 00000 f NONE A 5 2 IL 6 00000 NONE 4 86617 6 x3 LL 3 00000 z NONE 127 39992 7 xa IL 6 00000 NONE 6 16857 8 Z5 IL 33 00000 NONE 6 67094 9 X6 IL 13 00000 5 NONE 57 13818 10 x LL 110 00000 NONE 88 11172 11 x8 BS 12 15104 21 00000 7 NONE 12 x9 LL A 47 00000 A NONE 90 50592 The report is a standard MPS report including all range and dual price information The TIME is the total seconds elapsed since the last midnight FUNCTIONAL refers to the objective RANGE The RANGE command displays the sensitivity analysis of the currently installed solution of the model in memory This includes e a warn
309. r may be very slightly different from the formulation you entered As a result the solution may be very slightly different from what you think it should be To reduce un aesthetic round off you may wish to avoid fractions in your input data Fractions that are obviously truncated such as 66666 should be avoided A constraint like 66666 Z X 0 might be better rewritten as 2Z 3X 0 Also a constraint like X 1Y lt 0 might be better rewritten as 10 Y lt 0 270 CHAPTER 9 271 A Error Messages Listed below by code number are the error messages you may encounter when using LINDO Brief suggestions for overcoming these errors are also included 1 UNRECOGNIZED COMMAND The last command you entered was not recognized by the LINDO command processor Check the spelling of the command and try again 2 EXPECTING SUBJECT TO ENTER THAT OR END The LINDO model parser was expecting to find SUBJECT TO or ST to indicate the start of the constraints This indicates that a syntax error has occurred You should enter several carriage returns to retreat to command level examine the model with the LOOK command make sure there is a SUBJECT TO SUCH THAT S T or ST after the MAX MIN objective and at the start of the constraints and enter the remainder of the model with the EXTEND command 3 FIRST ROW SHOULD NOT HAVE A RELATIONAL OPERATOR The first row in a LINDO model must always be the objective row Given this
310. re Entering Variable Tolerance The fundamental operation in the algorithm used by the LINDO solver involves introducing a variable that formerly had a value of zero into the solution while simultaneously driving another variable to zero Variables are only introduced into the solution if they are favorable to the progress of the objective Due to round off error in floating point calculations a variable may compute as being favorable when in fact it may have no impact or worse a negative impact on the objective The Entering Variable Tolerance is the dividing line between what we consider favorable and unfavorable Variables with negative reduced costs whose absolute values exceed the Entering Variable Tolerance will be considered as potential candidates for entering into the solution The default value for the Entering Variable Tolerance is 57 Pivot Size Tolerance When LINDO introduces a variable into the solution it must assign the variable to a constraint or row Each row has one variable assigned to it and this assigned row is used in computing the value of the 54 CHAPTER 2 variable The row in which a variable is assigned is called its pivot row In order for LINDO to assign a variable to a row there must currently be a nonzero in the row which is called the pivot element The exact value of the pivot element will vary depending on all the matrix operations that were performed prior to assigning a variable to the row If the pivot
311. re CLRBAS Lib Indc geese a Private Declare D2DMY Lib lnddl Quit CommandButton V e Private Declare DEFROW Lib lndc Private Declare DELCON Lib lndec fene E8 Es Private Declare DMY2D Lib lnddl Returns the name used in code to identify an object Ay Form Layout j Private Declare DRPVAR Lib lndcw a rf The next step is to add handler code for the Solve button which references the LINDO routines To do this double click on the Solve button and enter the following code Private Sub Solve_Click A simple Visual Basic 5 0 program to define and solve the following model using the 32 bit LINDO DLL Sidhe X lt 50 Y lt 60 X 2Y lt 120 End Dim Nonz As Long Istat As Long Dim I As Long Idir As Long Dim Trouble As Long Idrow As Long Static Iro 3 As Long Dim Primal As Single Dual As Single Static Rhs 3 As Single Static Value 3 As Single Dim Kname As String Initialize LINDO Call ILINDO Call INIT Redirect LINDO s standard output to a file Call LUNOPN 60 9 LINDO OUT 0 0 Oy 0 Call CAPOUT 60 i Max 20 X 30 Y 248 CHAPTER 8 Put LINDO in TERSE model note not generally good practice until your app is fully debugged Call QUIET 0 Define objective row Call DEFROW 1 O Idrow Trouble Define constraint rows Rhs 1 50 Rhs 2 60 Rhs 3 120 For I 1 To 3 Call DEFROW 1 Rhs I Idrow Trouble Next I Defi
312. read in a model from disk using the RETRIEVE or TAKE commands lt Not In Use gt UGH RETREATING TO COMMAND LEVEL USE LOOK ALTER DELETE AND EXTEND TO VIEW DAMAGE MAKE MINOR REPAIRS DELETE A CONSTRAINT AND GET BACK IN INPUT MODE LINDO has encountered several contiguous errors in the text of your model Rather than attempt to go on LINDO halts model input and returns to command level At this point you can begin entering a new model again with the MAX or MIN commands or read in a model from disk using the RETRIEVE or TAKE commands COMMAND DISREGARDED You have input an invalid argument to a command or you have chosen to back out of a command by entering a blank line to a prompt In either case processing of the command is interrupted and you will be returned to command level lt Not In Use gt TOO MANY INTEGER VARS MUST BE lt lt n gt LINDO places a limit on the maximum number of integer variables in a model which you have exceeded In general this limit is the same as the limit on the total number of variables However some special versions of LINDO may have a lower limit on integer variables than there is on total variables In Windows versions of LINDO the Help About LINDO Command shows the limit on the total number of variables Command line versions of LINDO can use the HELP Command alone to get the same information Please refer to your manual for information on special versions of LINDO BAD VARIABLE NAME
313. reemptive goal command 66 70 Preferred branch 45 46 Preprocessing 45 188 Primal values 127 139 158 Print command 26 33 Printer Setup command 26 33 Printing a model 11 33 215 Command line 145 Problem editing command line commands 117 159 67 Product form Inverse 2 Prompts 8 backing out 9 296 INDEX colon 8 11 102 question mark 9 121 Pseudo objective row 199 Punch files 37 38 146 Q QCP 15 20 117 172 73 198 200 statistics 206 Quadratic programming 20 101 117 172 175 186 197 204 parametric analysis 202 4 real constraints 15 statistics 80 QUIET 249 256 261 QUIT 118 189 Quit command 41 118 189 R Raffensperger Fritz ii RANGE 116 130 32 173 Range analysis 27 116 130 rows 73 variables 73 Range command 27 73 76 RDBC 116 126 Real constraints 197 Reduced costs 24 127 188 negative 53 range analysis 24 REDUNDANT COLS 206 Register 111 Remind me in button 112 Replacing text See Find Replace command Reports menu 27 71 101 Reports window 7 33 34 39 60 capturing lengthy sessions 35 REPROW 250 262 REPVAR 257 262 Reserved symbols 56 Resuming optimization 53 RETRIEVE 116 121 144 Revised Simplex Method 2 52 63 66 73 96 117 135 158 Phase I 96 WATSUP subroutine 265 RIGHTHAND SIDE RANGES 74 130 Right hand sides 142 changing 76 79 159 162 173 202 saving with scripts 35 syntax 14 RMPS 116 123 24 Rounding s
314. rred writing to the file lt FileName gt Perhaps the disk is full If not you may need to run diagnostics on your drive to determine if there is a problem Could not open file lt FileName gt LINDO could not open the file lt FileName gt Perhaps you do not have the required privileges to access this file or you may need to run diagnostics on your drive to determine if there is a problem lt Not In Use gt Could not read file lt FileName gt LINDO experienced a read error on file lt FileName gt Perhaps you do not have the required privileges to access this file or you may need to run diagnostics on your drive to determine if there is a problem Not enough memory to complete command LINDO was unable to access enough memory to complete the command you just issued Try closing other applications and any windows not currently required Also if you have allocated an excessive amount of nonzero elements using Edit Options there may be inadequate memory available for other functions ERROR MESSAGES 287 1024 Error attempting to load the file into a window This error is most likely the result of either low memory see previous message 1023 a file too large for a Model Window see message 1027 or a disk error see message 1022 1025 lt Not In Use gt 1026 An error occurred during compilation on line lt N gt LINDO was unable to successfully compile your model due to a syntax error on line lt N gt Use the Edit Go To
315. s simply copy LINDO DLL32 LNDDLL32 DLL to either the Windows directory or to the directory containing your application Once you have done this execute the application and you should see the following displayed on your screen es C LINDO61 DII32 MsfortWebug lindofrt exe MEE ES MAR 26 36 SUBJECT TO Objective 2050 0 R 56 6 y 35 6 Press lt Enter gt to exit 264 CHAPTER 8 Calling LINDO from Other Languages Even if you aren t using Basic C or FORTRAN you should still be able to successfully link to the LINDO DLL Any current development environment for Windows should be able to call a DLL We suggest you thoroughly review your compiler s documentation on calling DLLs Some additional guidelines are LINDO conforms to the 32 bit or NT DLL standard Be sure you are making 32 bit calls and not the old style 16 bit calls All LINDO routine names are in uppercase Be aware of your compiler s name decoration strategies Some compilers may append underscores and other strange characters to symbol names This will lead to unresolved external symbols when linking with LINDO In general however there should be an option in the compiler to suppress name decoration All arguments are passed to LINDO by reference rather than by value All character arguments are passed to LINDO using double indirection i e by passing a pointer to a pointer If your development environment isn t capable of generating a
316. s it has a feasible solution in this model this row will be dropped from the tableau WINDOWS COMMANDS 97 Glancing at the tableau we see variable Y is the non basic variable with the lowest reduced cost 30 in row 1 Performing the simplex ratio test on Y shows variable Y is bounded tightest by row 4 which limits Y to 60 120 2 Thus a good pivot candidate would be variable Y in row 4 for a value of 60 To see if LINDO agrees issue the Pivot command in the Solve menu When we do this we find LINDO agrees with the above analysis I Reports Window DER a Y ENTERS AT VALUE 60 000 IN ROW 4 OBJ VALUE 1800 0 F we Now we run the Tableau command once more to see the effects of bringing Y into the basis J Reports Window THE TABLEAU ROW BASIS T SILK 2 SLK 3 1 ART 0 000 0 000 0 000 2 SLK 2 0 000 1 000 0 000 3 SLK 3 0 000 0 000 1 000 4 E 1 000 0 000 0 000 Y has replaced SLK 4 in the list of basic variables on the left The current objective value of 1800 appears in the upper right hand corner Y s value of 60 appears in the last row in the far right hand column which contains the updated right hand sides Notice the last row from the previous Tableau report is gone because the first pivot was performed and LINDO knows there is a feasible solution Variable X is the only non basic variable which now offers an attractive negative reduced cost The bounding row is row 2 and limits X to 50 Once again we run the Pivot
317. s or names to the Reports Formulation dialog box Please enter row numbers or names that are valid for the current model 1004 The dimensions of your model have exceeded the limits of this version The model is too large for the limits of this version of LINDO You can determine the limits of your version with the Help About LINDO command You will need to cut back on the size of your model or use a version of LINDO that has a larger capacity Keep in mind that simple bounds e g X gt 6 or X lt 10 entered with the SUB and SLB statements do not count against the row limit Thus converting simple bound constraints to use SUB and SLB statements can reduce the row count in a model Also the default lower bound for variables is zero Thus constraints of the form X gt 0 are not required 1005 1006 1007 1008 1009 1010 1011 1012 1013 ERROR MESSAGES 285 LINDO must finish the current task before an exit is allowed You have attempted to exit LINDO while it is performing background calculations Typically this is because the solver is currently optimizing a model Press the Interrupt Solver button on the Solver Status dialog box then try exiting again If the Solver Status dialog box is not visible use the Window Open Status Window command to display it Report Window may not be closed at this time You have attempted to close the Reports Window while LINDO is in the middle of generating a report
318. s various statistics about the model The first line of the report consists of number of rows number of variables number of integer variables with the number that are 0 1 or binary in parentheses and the index of the first real constraint in a quadratic program 0 indicates this is not a quadratic program The second line consists of e number of nonzero coefficients in the whole model e number of nonzero coefficients in the constraints with the number that are 1 or 1 in parentheses and e model density defined as number of nonzeros number of rows number of columns 1 The third line consists of e absolute values of the smallest and largest nonzeros respectively COMMAND LINE COMMANDS 187 The fourth line consists of sense of the objective function MIN or MAX number of less than or equal to equality and greater than or equal to constraints upper bound estimate of the number of generalized upper bound GUBS constraints constraints that have no variable in common and lower bound estimate of the number of variable upper bounds VUBS For example the constraint X X2 X3 0 contains the implications X3 0 implies X 0 X3 0 implies X2 0 The last line consists of e the number of columns with only one nonzero coefficient and e the number of redundant columns i e columns identical to some other column except possibly for the bounds The STATS command can be used for ch
319. se Output box in the Options dialog box as shown here Sets oo 7 V Terse Output D f Lr Note that you can always request a solution report at any time using the Reports Solution command See Reports menu below Switching to Terse Output will also suppress much of the trace output sent to the Reports Window when solving an integer programming model The default in LINDO is to have the Terse Output option unchecked That is to run in Verbose mode WINDOWS COMMANDS 55 Page Length Limit You can enter a Page Length Limit to request that LINDO interrupt output to the Reports Window every few lines and wait for a response from you This is useful if you want to examine a long solution before it has a chance to scroll out of the Reports Window For instance if you enter a page length limit of 25 as shown here M ov Teseoupu T Page Lenath Limit 25 then every time LINDO sends 25 lines of output to the Reports Window it will stop and display this dialog box LDO X E LINDO will not resume output until you press the More button in this box The default Page Length Limit is None In other words this feature is disabled Terminal Width When doing sequential I O operations LINDO respects a Terminal Width i e maximum characters allowed per line Thus output lines to the Reports Window are wrapped to fit within the Terminal Width and input records are truncated to fit You may change the Terminal
320. se operations would require us to input a number of commands If we needed to do this on an ongoing basis inputting all the required commands would be tiresome and prone to error It would be better to build a script file which we can run by issuing a single Take Commands command Such a script file might look as follows BAT A 3 warehouse 4 customer transportation model l XWH lt i gt C lt j gt amount shipped from 15 warehouse lt i gt to customer lt j gt MIN 6 XWH1C1 2 XWH1C2 6 XWH1C3 7 XWH1C4 4 XWH2C1 9 XWH2C2 5 XWH2C3 3 XWH2C4 8 XWH3C1 8 XWH3C2 XWH3C3 5 XWH3C4 SUBJECT TO Demand constraints C1 XWH1C1 XWH2C1 XWH3C1 gt 10 C2 XWH1C2 XWH2C2 XWH3C2 gt 17 C3 XWH1C3 XWH2C3 XWH3C3 gt 22 C4 XWH1C4 XWH2C4 XWH3C4 gt 12 Supply constraints WH1 XWH1C1 XWH1C2 XWH1C3 XWH1C4 lt 30 WH2 XWH2C1 XWH2C2 XWH2C3 XWH2C4 lt 25 WH3 XWH3C1 XWH3C2 XWH3C3 XWH3C4 lt 21 END Solve the model TERSE GO 36 CHAPTER 2 Divert a solution DIVERT SOLU 10 SOLU RVRT Set RHS of Cl to 15 re solve and save report ALT Cl RHS t5 TERS GO DIVERT SOLU 15 SOLU RVRT Set RHS of C1 to 20 re solve and save report ALT C1 RHS 20 TERS GO DIVERT SOLU 20 SOLU RVRT BAT LEAVE To have LINDO run the commands contained in this file simply run the Take Commands command and i
321. sion in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO Try reducing the number of text operands in the conditional expression and try again NUMERIC STACK OVERFLOW IN CONDITIONAL EXPRESSION The conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO is too complex Try simplifying or reducing the size of the conditional expression and try again TEXT STACK OVERFLOW IN CONDITIONAL EXPRESSION The conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO is too complex Try simplifying or reducing the size of the conditional expression and try again UNDEFINED OPERATION IN CONDITIONAL EXPRESSION The conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO contains an undefined operation e g division by 0 Correct this problem and try again 284 APPENDIX A 112 CONDITIONAL EXPRESSION DOES NOT EVALUATE TO A BOOLEAN VALUE The conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO must evaluate to either TRUE or FALSE to be considered valid Correct this problem and try again 113 REQUESTED OUTPUT LINE LENGTH EXCEEDS TERMI
322. st recent version of the software that you are entitled to and that compatibility and operational problems are kept to a minimum 114 CHAPTER 2 About LINDO The About LINDO command opens a system window that provides you with information about the software you are using and LINDO Systems Inc The first box shows the version of LINDO being used the release number and date and the LINDO copyright respectively The following box shows the mailing address telephone e mail address and Internet address for LINDO Systems This may be useful for future reference about updated versions of the software and other LINDO Systems products Extended LINDO PC Release 6 1 15 Jan 02 E Copyright 2002 LINDO Systems Inc 312 988 7422 1415 North Dayton St info lindo com Chicago IL 60622 http www lindo com m Maximum Model Size Constraints 32000 Variables 200000 Integer Variables 20000 Nonzeros 2000000 Additional Information License LDPC3 99999 m License Scope License Expiration Single User Perpetual License Usage The Maximum Model Size box lists the constraint variable and integer variable limits of your copy of LINDO along with the physical nonzero limit The constraint variable and integer variable limits are determined by the version of LINDO you are using contact LINDO for details The Nonzeros limit can be set to whatever you would like by selecting Options from the Edit menu Refer to Edit menu
323. structions are issued to LINDO in the form of text command strings When you start LINDO you ll see the copyright notice and a colon prompt on the last line This means that LINDO is waiting for you to start entering commands Your screen should resemble the following LINDO COPYRIGHT C LINDO SYSTEMS ICENSED MATERIAL ALL RIGHTS RESERVED GETTING STARTED 9 First a word about LINDO s prompts When you see the colon prompt LINDO is expecting a command When you see the question mark prompt you have already initiated a command and LINDO is asking you to supply additional information related to this command such as a number or a name If you wish to back out of a command that you ve already started you may enter a blank line in response to the question mark prompt and LINDO should return you to the command level colon prompt For our sample model let us assume the XYZ Corporation produces two models of computer the Standard and the Deluxe At the XYZ facilities the Standard can be produced at 10 computers per day while the Deluxe exceeds that at 12 computers per day Furthermore XYZ has a limited supply of labor In particular there are a total of 16 units of labor and Standard computers require one unit of labor while Deluxe computers are relatively more labor intense and require 2 units of labor A LINDO model has a minimum requirement of three things It needs an objective varia
324. sts you in building a model This dialog box contains all the reserved symbols in the LINDO syntax plus all the variable and row names defined so far in the model The dialog box appears as follows Paste Symbol m Symbol Lists Reserved Variables m Paste Buffer aS Clear Help Close WINDOWS COMMANDS 57 When you double click on a symbol it is appended to the edit box titled Paste Buffer Once you have finished constructing the constraint you press the Paste button to send the contents of the paste buffer into the current Model Window at the cursor point You may also type directly into the paste buffer if desired Press the Clear button to erase the contents of the paste buffer As an example suppose we wanted to use the above dialog box to add the constraint XSAT XSUN gt 4 to our model We would double click on the items 1 XSAT in the Variables list box 2 The plus sign in the Reserved list box 3 XSUN in the Variables list box 4 The gt sign in the Reserved list box Finally we would enter the right hand side of 4 directly into the paste buffer At this point the dialog box should resemble the following Paste Symbol r Symbol Lists Reserved Variables r Paste Buffer KSAT xSUN gt 4 Clear Help Close Note that the paste buffer contains the desired constraint At this point we may insert it into the Model Window at the current cursor location
325. t NOTFMT to 1 if the file is not formatted i e binary Otherwise set NOTFMT to 0 On return LUNIT will be 1 if the open was unsuccessful Otherwise it will contain the logical unit number of the file When you are finished using the file it should be closed with a call to the KLOSE routine LUNOPN LUNIT LFNAME KFNAME LFNAME INROUT NOTFMT LUTRMI LUTRMO Character KFNAME Integer LUNIT LFNAME INROUT NOTFMT LUTRMI LUTRMO Opens a file on unit number LUNIT On entry the character array KFNAME contains the filename Set LFNAME to the number of characters in the filename JVROUT to for file input or 0 for file output NOTFMT to 1 for a binary or 0 otherwise LUTRMI to the standard input unit number normally 5 and LUTRMO to the standard output unit number normally 6 On return LUNIT will be 1 if the open was unsuccessful Otherwise it will contain the logical unit number assigned to the opened file LXBRED LUNIT INERR Integer LUNIT INERR Reads in a solution basis saved with the LXBWRT routine from the file associated with the logical unit LUNIT Use routine LUNOPN to assign a unit number to a file The solution basis will be used as a starting point when solving the current model The model must be in memory before calling LXBRED On return JVERR will be nonzero if an error occurred This routine corresponds to the FBR command line command LXBWRT LUNIT INERR Integer LUNIT INERR Sa
326. t has exceeded or is close to exceeding the terminal width limit Either shorten your input lines or use the WIDTH command line command or the Windows Edit Options command to increase the terminal width 276 41 42 43 44 45 46 47 48 49 50 APPENDIX A lt Not In Use gt NAME ALREADY IN USE You have attempted to use the ALTER command to change a row name to a name that is already in use Try using a different name INPUT WAS INVALID You have specified invalid input for a command Please refer to the documentation or help file for the command and try again lt Not In Use gt WARNING VARIABLE lt Variable gt HAS THE LOWER AND UPPER BOUNDS lt Lower gt lt Upper gt You have specified inconsistent bounds This usually means that you have specified a lower bound that is greater than the upper bound Please reenter the bounds on the variable VARIABLE NOT RECOGNIZED You have specified an invalid variable name as part of a command Please check the spelling of the name and try again lt VariableIndex gt IS NOT A VALID INTEGER VARIABLE You have specified an improper variable index while trying to make a variable integer The index must lie in the range of 1 to N where N is the total number of variables INVALID BOUND VALUE You have specified an invalid value for a variable bound Be sure that the bound value is numeric and retry SUB FOR INVALID VARIABLE lt Variablelndex gt
327. t to designate as binary or lt NumberOfVars gt is an integer value corresponding to the number of variables you wish to make binary Both forms of the INTEGER command are illustrated below look all AX 30 X 20 Y 10 SUBJECT TO 2 17 X 13 Y 4112 lt 29 END Make all the variables 0 1 using the INTEGER lt Variable gt form int x int y int z look all MAX 30 X 20 Y 10 Z 168 CHAPTER 3 SUBJECT TO 2 17 X 13 Y 112 lt 29 END INTE 3 Note the INTE 3 indicating that the first three variables are binary int 0 Now we remove the integrality condition look all AX 30 X 20 Y 10 2 SUBJECT TO 2 17 X 13 Y 11 Z lt 29 END SUB X 1 00000 SUB Y 1 00000 SUB Z 1 00000 Note that the SUBS remain but the binary integer restrictions are gone terse go Now we solve the continuous version LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE 48 4615402 nonz OBJECTIVE FUNCTION VALUE 1 48 46154 VARIABLE VALUE REDUCED COST X 1 000000 3 846154 Y 0 923077 0 000000 ROW SLACK OR SURPLUS DUAL PRICES 2 0 000000 1 538462 NO ITERATIONS 2 int 3 Restore integers using INTEGER n command look all AX 30 X 20 Y 10 Zz SUBJECT TO 2 17 X 13 Y 112 lt 29 END INTE 3 terse go Solve for an integer solution LP OPTIMUM FOUND AT STEP T OBJECTIVE V
328. tFmt amp lLutrmi amp lLutrmo Now tell LINDO to divert all standard output to this unit CAPOUT amp 1Unit Put LINDO in TERSE mode long lTerse 0 QUIET amp lTerse Define the rows long i long lIdir 1 lIdrow l1Truble float fRhs 0 f float fConstraintRhs 3 50 f 60 120 f Define objective row DEFROW amp lIdir amp fRhs amp lIdrow amp l1Truble Define constraint rows lidir 1 for i 2 i lt 4 i DEFROW amp lIdir amp fConstraintRhs i 2 amp lidrow amp lTruble LINDO CALLABLE LIBRARIES Define our two columns char cNameX X ma char cNameY Y ma long lNonz 3 float fValX 20 f 1 f 1 f float fValY 30 f 1 f 2 f long lIroX 1 2 4 long 1lIroY 1 3 4 Call APPCOL to send LINDO column X APPCOL amp cNameX amp lNonz fValX lIroxX Call APPCOL to send LINDO column Y APPCOL amp cNameY amp lNonz fValY lIroyY Call GO to optimize the model long 1LIMGO 0 LISTAT GO amp l1LIMGO amp lISTAT Get the solution from LINDO Get the objective valu float fDual Jos a REPROW amp i amp m fObjective amp fDual Get the value of X REPVAR amp i amp m_ fX amp fDual Get the value of Y A REPVAR amp i amp m fY amp fDual Post solution values in d
329. tains the number of days since 28 Feb 1800 On return NDAY will contain the corresponding day of the month MON will contain the corresponding month of the year ranging from 1 to 12 and NYR will contain the corresponding year as a 4 digit integer for the current date DMY2D NDAY MON NYR NS1800 NDOFWK Integer NDAY MON NYR NS1800 NDOFWK On entry NDAY contains the day of the month MON the month of the year 1 to 12 and NYR the year as a 4 digit integer On return NS7800 will contain the number of days since 28 Feb 1800 and NDOFWK will contain the day of the week for the current date where Mon 1 Tue 2 DEFROW IDIR RHS IDROW TROUBLE Float RHS Integer IDIR IDROW TROUBLE Appends a new row to the end of the current formulation All rows are defined via this routine A row must be defined before you can use APPCOL or INSROW to add nonzero elements to it On entry set IDIR to the direction of the row 1 if a gt row or MAX objective 0 if an row and 1 if a lt row or MIN objective and set RHS to the right hand side value of the row On return DROW will contain the row number assigned to this row and TROUBLE returns 1 if trouble occurred e g an out of space condition in DEFROW DELCON J Integer J Deletes constraint number J All higher numbered constraints are renumbered down by 1 The objective function row 1 cannot be deleted This routine corresponds to the DELETE command line command
330. te between the Tableau command and Pivot command to keep tabs on LINDO as it optimizes the following small model The following discussion assumes a modest knowledge of the workings of the simplex algorithm Interested readers can refer to any introductory operations research textbook to learn more or Optimization Modeling with LINDO by Linus Schrage Issuing the Tableau command we receive a report containing the initial tableau i gt Reports Window THE TABLEAU ROW BASIS 1 ART 2 SLK 3 SIK 4 SLK ART The term ART stands for artificial variable which are devices used to jump start the simplex algorithm An artificial variable is used whenever a variable has not yet been assigned to a row The term SLK n stands for the slack variable for the n th row Slack variables are added internally by the solver to each inequality constraint to convert it to an equality The current basic variables are listed in the second column on the left Note LINDO has already added all the slack variables into the basis The variable names appear along the top The updated column nonzero values are displayed beneath their column names The updated right hand side values are displayed to the right Finally note LINDO has added an additional row at the bottom of the tableau This additional row corresponds to the simplex algorithm s Phase I objective of minimizing the sum of infeasibilities Once LINDO performs the first pivot and verifie
331. ted here Save in LINDO61 x a amp ei E Tools History Zax My Documents g My Network Places Name ODelkeys nliz Samples Shell Type File Folder File Folder File Folder File Folder Date Modified 4 3 2002 9 13 AM 4 3 2002 9 13 AM 8 15 2002 1 36 PM 4 3 2002 9 13 AM File name fsample lex v i Save as type Plain Text Ext x Cancel A 214 CHAPTER 7 Suppose we save this model in the text file called SAMPLE LTX and we have restarted LINDO If you are using a Windows version of LINDO select the File Open command and direct LINDO to SAMPLE LTX After pressing the OK button the transportation model should appear on your screen in a Model Window as follows Ta a C LINDO61 SAMPLE LTX MIN 2 Al 4 A2 3 A3 5 Ad ST 4 Bl 3 B2 Cl 6 C2 C1 2 B3 4 C3 6 B4 3 C4 Demand Demand Denand Demand Supply l Supply Supply aok for cust for cust for cust for cust limit for A limit for B limit for C If you are running a command line version of LINDO you can use the TAKE command to read SAMPLE LTX from disk as illustrated in this next sample session take FILE NAME sample 1tx look all MIN 2 Al 2 BS 4 SUBJECT TO 2 Al 4 3 A2 4 4 A3 4 5 A4 4 6 Al 7 B1 8 Cl END 4 A2 3 A3 B4 5 Cl BI FCI B2 C2 BS G3 B4 C4 A2 A3 B2 B3 C2
332. tend to be considerably faster On the negative side use of this tolerance can mean LINDO may not find the optimal solution and you will not receive any warning to this effect in the solution report You are guaranteed however that the solution found is within 100 f of the true optimum On large integer models the alternatives of getting a solution within say 2 of optimal in a few minutes versus the true optimum in a few days makes an IP Optimality Tolerance value quite attractive As an illustration of the potential power of the IP Optimality Tolerance consider the following IP 23 C LINDO SAMPLES TEST2 LTX 29970 PITI 29970 P1iT2 WINDOWS COMMANDS SEE 29910 PZT1 29910 P2T2 1000 I1T1 iS 1000 IITA 1000 I2T1 1000 I2Tz 1200 EITI 1200 E1T2 1200 E2T1 1200 E2T2 20 C1T1 20 C1T2 20 C2Ti 20 C2T2 3000 NDT1 3000 NDT2 20 R1IT1 20 R1iT2 20 R2T1 20 R2T2 300 H1T1 300 HiT2 300 H2T1 300 HATZ SUBJECT TO 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 P1T1 P2T1 NDT1 PiT2 P2T2 NDT2 900 PITI 90 T1T1 900 B1iT2 90 TITS 600 B2T1 60 T2T1 600 B2T2 60 T2T2 lt EIT I1T2 IiTi I2T2 I1T2 I2T2 PiTi PiT2 P2T1 P2T2 citi c1itT2 c2T1 city I2Ti lt 1000 I2T2 lt 1000 E1TL 6 DITIS E2T1 6 B2T1 ETA 6 BiTA E2T2 6 B2T2 Niti y Ee ie NiTi N1T2 N2T1 B2T1 N2T1 N2T2 RiTi BiTi C2T1 R1IT1 R2T1 B2
333. th values other than 0 and only the rows that are binding Requesting just the nonzeros can cut down considerably on the size of the solution report 72 CHAPTER 2 Once you determine the scope of the report click the OK button and LINDO will begin generating the report An example of a small solution report follows i Reports Window elk A OBJECTIVE FUNCTION VALUE 1 2050 000 VARIABLE VALUE REDUCED COST X 50 000000 0 000000 35 000000 0 000000 SLACK OR SURPLUS DUAL PRICES 0 000000 1 000000 35 000000 0 500000 0 000000 0 500000 NO ITERATIONS 2 The first information at the top of the solution report is the value of the objective function After this the solution report is broken down into two sections The first section reports on the variables i e the columns and the second section reports on the rows i e the constraints In the variable section there is one line for each variable giving the variable s name its value and its reduced cost Reduced costs are meaningful in linear and quadratic models but in general should be ignored when solving integer programming IP models There are two valid equivalent interpretations of a reduced cost First you may interpret a variable s reduced cost as the amount by which the objective coefficient of the variable would have to improve before it would become profitable to bring that variable into the solution Second the reduced cost may be interpreted as the
334. that solution or basis into the current model before issuing the GO command GO will then try to use this as an initial starting point refer to the SDBC FBS and FPUN commands for information on saving a solution Six things can happen as a result of typing GO 1 an optimal solution is found 2 the model is infeasible i e there is no solution which simultaneously satisfies all the constraints 3 the model is unbounded 1 e the model is such that the objective may be increased without bound 4 an upper limit on the number of pivots i e solver iterations is reached 5 you get tired of waiting for the model to be solved or you are pleased with the best solution found so far so you interrupt the solver 6 LINDO runs out of memory Each of these potential scenarios is discussed in more depth below 1 EXAMPLE OF OPTIMAL SOLUTION OUTPUT Ideally your model is formulated such that LINDO returns an optimal solution in a short amount of processing time When LINDO returns with an optimal solution to a continuous model you are asked if you wish to see the range report which is the sensitivity analysis For more information on the range report refer to the RANGE command If you do wish to see the range report enter y or yes after the following prompt DO RANGE SENSITIVITY ANALYSIS 152 CHAPTER 3 6699 66 If you do not wish to see the range report you can enter n no or a
335. the FREE command with a variable after using the SUB command with that variable the upper bound is erased You can use this to remove the SUB Entering FREE lt Variable gt eliminates the SUB Then entering SLB lt Variable gt 0 see the SLB command restores the default simple lower bound of 0 Every variable except a variable specified as FREE can have a SUB The SUB specification does not count against LINDO s constraint limit Whereas an explicit constraint does The SUB specification is saved when the model is saved with SAVE SMPS and the DIVERT LOOK ALL sequence The current solution is no longer considered optimal after SUB is used The following session illustrates the use of the SUB command on a small model look all MAX 20 X 30 Y SUBJECT TO 2 X lt 50 3 Y lt 60 4 X 2 Y lt 120 END terse go Solve the model LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE 2050 00000 delete 3 Delete the simple bound constraints delete 2 look all AX 20 Xe os 30 Y SUBJECT TO 2 X 2 Y lt 120 END sub x 50 Bound the variables with SUB commands sub y 60 look all AX 20 X 30 Y SUBJECT TO 2 X 2 Y lt 120 END SUB X 50 00000 SUB Y 60 00000 go Re solve we should get the same objective LP OPTIMUM FOUND AT STEP T OBJECTIVE VALUE 2050 00000 COMMAND LINE COMMANDS 165 SLB Syntax SLB lt Variable gt lt LowerBound gt Th
336. the RETRIEVE command to retrieve a file from disk which was created with SAVE COMMAND LINE COMMANDS 145 Large models may be saved and retrieved quickly with this format in contrast with the TAKE command for LINDO text files Itx which can take longer to retrieve big models Information regarding any solution internal parameters such as BIP PAGE IPTOL SET and WIDTH and comments are not saved in the file GIN INTEGER SLB SUB QCP and TITLE specifications are saved The file is on disk in a very compact text format Because the file is text it may be ported to other LINDO platforms Unfortunately since the file has been compressed it is unusable for human interpretation If you need to save a readable text copy of your model to disk use the DIVERT command to open a file a LOOK ALL command to send the model to the file then close the file with the RVRT command If desired files created this way may be read back into LINDO with the TAKE command DIVERT Syntax DIVERT lt FileName gt The DIVERT command opens a file and diverts all subsequent reports e g SOLUTION RANGE LOOK from the screen to the specified file This command is used to save models as a LINDO text format ltx file in the command line versions of LINDO These exported DIVERT files may then be loaded into other programs e g word processors and spreadsheets or they may be routed to your system s printer The form of the DIVERT command is DIVERT
337. the TAKE command LINDO will prompt you for the name of the file TAKE commands may not be nested i e once a TAKE command is in effect you may not issue another If you attempt to TAKE a file that contains an invalid command or that does not exist LINDO will give an error message If an invalid command is encountered in the TAKE file the message INVALID COMMAND lt command gt is displayed To echo the contents of the TAKE file on your screen as it is read give the BATCH command before doing a TAKE A comment may be added almost anywhere in a TAKE file The remainder of any line following an exclamation point is treated as a comment by LINDO COMMAND LINE COMMANDS 125 Script files are created in any text format known by LINDO LINDO text Itx LINDO packed Ipk and MPS files mps As an example the following text put in a script file reads in a model solves it then writes a solution file This file retrieves amp solves MYFILE LPK then saves the solution First turn on echoing of input D TO T urn off the output 5 W n Retrieve the model from disk LPK Solve the model O DBC MYFILE SDB Then save the solution to MYFILE SDB Beep G Last close the TAKE file and end the script staying in LINDO EAVE Ho NO DHA Wee E cal zo Z K Hy H Suppose we named this command script file MYSCRIPT LTX Then we could have LINDO run t
338. the file in a View Window instead more on these when we get to the View command 30 CHAPTER 2 When you issue the Open command you will be presented with the Windows standard File Open dialog box as follows Model Open File name Folders 0K fa c lindo61 samples Cancel lindo Itx cs Test2 Itx LINDOS1 Test3 ltx E gt Sampl Test4 ltx amps Test5 ltx 5 Samptext List files of type Drives LINDO Text Itx Network This dialog box has all the standard features for navigating your system in search of a file The only non standard feature of the File Open dialog box is the list box in the lower left corner labeled List files of type This allows the user to select one of the following four filters used for selecting files LINDO Text ltx LINDO Packed Ipk MPS mps and All Files The first three filters correspond to the three different file formats supported by LINDO for storing models In most cases we recommend that you use the LINDO Text Itx format This saves your model in text form exactly as it appears on your screen For specifics on the other file formats refer to the Save command below After selecting a file to open LINDO examines the file to determine the format it was saved under If the model is in LINDO Text format or some unknown format it is read directly into LINDO without modification MPS format and LINDO Packed format files are converted to their LI
339. the first row should never contain a relational operator e g lt or gt You should enter several carriage returns to retreat to command level examine the model with the LOOK command make sure the objective is in the first row starting with a MAX MIN and enter the remainder of the model with the EXTEND command 4 EXPECTING SIGN OR RELATIONAL OPERATOR The LINDO model parser was expecting to find a coefficient sign or a relational operator but found something else This indicates that a syntax error has occurred You should enter several carriage returns to retreat to command level examine the model with the LOOK command check and edit if necessary the operators of all constraints and enter the remainder of the model with the EXTEND command 5 EXPECTING A VARIABLE COEFFICIENT The LINDO model parser was unable to find a variable coefficient This indicates that a syntax error has occurred You should enter several carriage returns to retreat to command level examine the model with the LOOK command check and edit if necessary all variable coefficients and enter the remainder of the model with the EXTEND command 6 FIRST CHARACTER OF A VARIABLE NAME MUST BE A LETTER The LINDO model parser did not find a variable name beginning with an alphabetic character This indicates that a syntax error has occurred You should enter several carriage returns to retreat to command level examine the model with the LOOK command ma
340. tion no workers start on Tuesday The reduced cost of 100 for TWORK the number of workers who start on Tuesday means that TWORK s objective coefficient of 325 the weekly pay of Tuesday start workers would have to improve by at least 100 before 7WORK would appear in the solution In other words the weekly pay of workers starting on Tuesday would have to reduce to 225 before it would be profitable for a worker to start on Tuesday Note that an improvement in a minimization problem is a decrease in the objective A reduced cost of zero as in the case of MWORK for instance indicates that the variable is already in the solution Second the reduced cost may be interpreted as the amount of penalty you would have to pay to introduce a variable into the solution In this case to start one worker on Tuesday with the weekly pay remaining 325 would increase the objective function value total labor cost by 100 to 7850 Reduced costs are valid only over a range of values LINDO provides a Range Analysis facility for determining the valid ranges of the reduced costs Range analysis will be discussed in more depth in Chapter 2 LINDO for Windows Dual Prices The LINDO solution report also gives a DUAL PRICE figure for each constraint You can interpret the dual price as the amount by which the objective would improve given a unit of increase in the right hand side of the constraint Notice that improve is a relative term In a minimizatio
341. tion fails Try closing other applications and any windows not currently required LINDO was unable to save the configuration values to a file Check for write access to your working directory LINDO stores non default options values in a file called LINDO CNF When you change the options in the Edit Options dialog box and press the Save button LINDO writes a new copy of the LINDO CMF file in the working directory If this process fails you will receive this error message Either you do not have write access to the working directory the disk is full or you are experiencing a hardware error You requested a graph with lt N gt points Graphs can have up to 500 points pie charts can have 100 Use a condition to narrow the selection You have used the Reports Peruse command to request a graph which has more points than LINDO can plot Use the condition field to specify a filter on data points to reduce their number For more information refer to the Reports Peruse command on page 81 290 APPENDIX A Symbol exclamation mark 13 117 181 32 bit DLL standard 264 A About LINDO command 28 52 114 Absolute values 81 87 139 142 186 206 Accelerator keystrokes 28 Accuracy decimals 267 269 Addition 87 120 139 142 Algorithm in LINDO See Revised Simplex Method All or nothing situations 17 Allowable increases decreases 73 130 ALTER 11 117 159 Analysis parametric analysis 27 76 80 117 173 74 202 4 24
342. tion in the print list of a CPRI or RPRI command Valid options are N P D R U T and Z Refer to the documentation or help files on these commands for help and try again INVALID SYMBOL IN COMMAND You have used an invalid symbol in the conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO Refer to the documentation or help file for these commands for help UNMATCHED PARENTHESES IN CONDITIONAL EXPRESSION You have unmatched parentheses in the conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO Add or remove parentheses as required and retry INSTRUCTION LIST OVERFLOW IN CONDITIONAL EXPRESSION You have too many instructions in the conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO Try reducing the size of the conditional expression and try again TEMPORARY OPERATOR STACK OVERFLOW IN CONDITIONAL EXPRESSION The conditional expression in a Reports Peruse command in a Windows version of LINDO or in a CPRI RPRI command in a command driven version of LINDO is too complex Try simplifying or reducing the size of the conditional expression and try again IMMEDIATE TEXT LIST OVERFLOW IN CONDITIONAL EXPRESSION You have too many text operands in the conditional expres
343. tle in a Windows version of LINDO il lt untitled gt DAR TITLE Your Title Here A MAX 20K 307 ST X lt 50 lt 60 E 2 lt 120 END When we issue the Title command from the File menu the model s title is sent to the Reports Window as follows I gt Reports Window DER TITLE YOUR TITLE HERE GETTING STARTED 21 In this following example we are using a command line version of LINDO to input a small model with a title display the model with the LOOK ALL command and view the title using the TITLE command TITLE Your Title Here MAX 20X 30Y ST X lt 50 Y lt 60 2 X 2Y lt 120 END LOOK ALL TITLE YOUR TITLE HERE MAX 20 X 30 Y SUBJECT TO 2 xX lt 50 3 Y lt 60 4 X 2 Y lt 120 END TITLE TITLE YOUR TITLE HERE A Staff Scheduling Example In this next section we develop a small staff scheduling example and delve a little more deeply into the details of the LINDO solution report Although this model is still quite small it is perhaps our first example in this user s manual of a potential real world application Let s say you operate a retail business You want to know how many clerks you need to hire and what days to assign them in order to minimize your labor cost while meeting your needs at the store There are two limiting factors that govern how many people you hire and what days they work First each of your w
344. to add the nonzero elements to the model However because LINDO stores models internally in column order using APPCOL is more efficient At this point the entire model is built The next step is to solve it using LINDO s GO routine Call GO to optimize the model long 1LIMGO 0 LISTAT GO amp 1LIMGO amp lISTAT LINDO CALLABLE LIBRARIES 257 Once the model is solved the solution is retrieved from LINDO and displayed Get the solution from LINDO Get the objective valu float fDual qo Se he REPROW amp i amp m_fObjective amp fDual Get the value of X REPVAR amp 1i amp m fX amp fDual Get the value of Y i SF REPVAR amp i amp m fY amp fDual Post solution values in dialog box UpdateData FALSE The LINDO routine REPROW is used to get the value of the objective row Likewise REPVAR is used twice to get the values for the two variables Finally LINDO is closed down with a call to LSEXIT Shut LINDO down LSEXIT C Now an import library needs to be created to add to our project which resolves the external references to the LINDO library routines The import library is built from the definitions or DEF file for the LINDO library The DEF file can be found in location LINDO DLL32 LNDDLL32 DEF The LIB utility supplied with MSVC is used to create the import library with the following DOS command LIB DEF LNDDLL32 DEF OUT LN
345. to any quadratic program when you first develop the model In most cases in particular with financial portfolio models one would expect the Positive Definite command to declare the matrix positive definite or at least positive semi definite If you do not get either of these two messages then it probably means there is an error in the data or the logic If the model is indefinite it may have multiple local optima which optimum you will get as a solution is arbitrary Below is the results for running the Positive Definite command on our 3 asset portfolio model presented above SUB MATRIX IS POSITIVE DEFINITE RANK 3 OUT OF 3 Parametric Analysis of Quadratic Programs LINDO allows you to perform parametric analysis on right hand side RHS coefficients of quadratic models as well as linear models There are some subtle differences however that the user should be aware of We illustrate with the following portfolio example from the previous sections MIN X Y Z UNITY RETURN XFRAC YFRAC ZFRAC ST First order condition for X 6 X 2 Y Z UNITY 1 3 RETUR XFRAC gt 0 First order condition for Y 2X 4yY 0 8 Z UNITY 1 2 RETURN YFRAC gt 0 First order condition for Z X 0 8 Y 2 Z UNITY 1 08 RETURN ZFRAC gt 0 b SeeseSs Start of real constraints Budget constraint multiplier is UNITY X Y FAS Growth constraint multiplier is RETURN TSS D E 6 1 2 Yor 1
346. to enhance performance the QUIET routine is called long 1Terse 0 QUIET amp l1Terse This puts LINDO into terse output mode minimizing the amount of output to the log file In general it s best to leave out this call to QUIET until you have completely debugged your application By suppressing LINDO s output you may miss important system messages As mentioned at the beginning of this chapter you must make one call to DEFROW for each row in your model The following code does this once to define the objective row and three more times in a loop to define our three constraints Define objective row DEFROW amp lIdir amp fRhs amp lIdrow amp 1Truble Define constraint rows lidir 1 for i 2 i lt 4 itt is EFROW amp lIdir amp fConstraintRhs i 2 amp lidrow amp lTruble Once the rows are defined APPCOL can be used to define the columns Here APPCOL is called twice once for each column in our model Define our two columns char cNameX X ue char cNameY Y Ws long 1Nonz 3 float fValX 20 f 1 f 1 7 oat Valy 3008 Tity 2 ong lIroX 1 2 4 ong lIroY 1 3 4 f I J Call APPCOL to send LINDO column X APPCOL amp cNameX amp lNonz fValX lIroX amp lTruble Cc all APPCOL to send LINDO column Y APPCOL amp cNameY amp lNonz fValY lIroY amp lTruble INSROW or INSERT could also have been used
347. to exit A Sample Visual C C Application The next section uses Visual C C 5 0 VC5 to build and solve the simple two variable model discussed above This section assumes the reader is well versed in the mechanics of using VCS If you would rather skip the details involved in constructing this project and would prefer to experiment with a completed version of the project you can load the project file LINDO DLL32 MSVC LNDMSVC DSW directly into the VC5 IDE Building the Application The following discussion walks you through the steps involved in building a VC5 project to interface with LINDO VC5 s AppWizard is used to provide the base code for a dialog based application Ina dialog based application the user interface consists of a single dialog box Additional code is added to the AppWizard code to perform the model generation solution and reporting 252 CHAPTER 8 The steps required to create the base code with App Wizard follow 1 Start up the VC5 IDE 2 Issue the File New command 3 In the New dialog box select the Projects tab click on the MFC AppWizard exe option and click OK 4 Assign the project a name For this example the name LNDMSVC was chosen 5 Click the OK button 6 You should now see the MFC AppWizard Step 1 dialog box Click on the Dialog based radio button and then press the Finish button You should now see the following summary of your choices New Project Information
348. to load the Windows version of LINDO for processing the script file and finally return to the VB front end to read a solution file created by LINDO The results are then displayed in the VB application window Although a real life application would do much more than this the basic framework used here should help to provide some insights into the process To begin entering this example start VB and select the File New Project command Visual Basic will present you with a blank form Use the tool palette to add two new buttons to the form Label the first button Solve and the second button Quit At this point your form should resemble the following LINDO Staff Example Next double click on the Solve button then input the following code Sub Command1_Click Rem Sample VB matrix generator for LINDO Uses Rem the LINDO executable for Windows to perform Rem the optimization Dim I X Y INTERFACING 223 Rem Let the user know we are about to start optimization Print Begin optimization Rem Eliminate any old work files On Error Resume Next Kill C LINDO SHI Kill C LINDO SHI On Error GoTo 0 NDIN TXT NNDOUT TXT ELL ELL Rem Open a LINDO Command file Open C LINDO SH For Output As 1 ELL LNDIN TXT J E Rem Always a good idea to do a PAGE 0 Rem first off in LIN DO command files Print 1 PAGE 0 Rem Send the formulation t
349. to select the desired format The available formats are summarized in the table below Format Description LINDO Text Te Window is saved in text format exactly as it appears on the screen Available for Model Reports and Command Windows Comments and special formatting are saved with the model LINDO Ipk Model is saved in a proprietary compressed format Packed The file is a text file and may be easily ported to other platforms but the contents are unreadable by other text editors or wordprocessors All comments and special formatting are stripped from the model before saving mps Model is saved in industry standard MPS format This is an inefficient format in terms of required disk space and is difficult to interpret The advantage of this format is that it is a widely accepted standard for optimization models and is portable to many other solvers All comments and special formatting are stripped from the model before saving WINDOWS COMMANDS 33 Note that the Reports Window and Command Window may only be saved in the Itx text format Model Windows may be saved in any of the three formats Note The LINDO Packed and MPS formats do not support comments or special formatting e g indenting and spacing Be forewarned that should you save a model in either of these formats all comments will be stripped and LINDO will reformat the model In addition when lpk and mps files are read back into LINDO they are auto
350. to the model However because LINDO stores models internally in column order using APPCOL is much more efficient At this point the entire model is built The next step is to solve it using LINDO s GO routine Call GO 0 Istat 250 CHAPTER 8 Once the model is solved the solution is retrieved from LINDO and displayed Print objective value IT 1 Call REPROW I Primal Dual Print Objective Value Primal Print variable values Call REPVAR 1 Primal Dual Print Print X Primal Call REPVAR 2 Primal Dual Print Y Primal The LINDO routine REPROW is used to get the value of the objective row Likewise REPVAR is used twice to get the values for the two variables Finally LINDO is closed down with a call to LSEXIT Call LSEXIT Running the Application In order to run the application you need to make the LINDO DLL available to the system To do this simply copy LINDO DLL32 LNDDLL32 DLL to either the Windows directory or to the directory containing your application Once you have done this you can issue the Run Start command in VB to begin execution You should see the dialog box displayed on the screen LINDO Visual Basic Example LINDO CALLABLE LIBRARIES 251 Press the Solve button to build and solve the model at which point you should see the solution displayed in the dialog box LINDO Visual Basic Example Objective Value x 50 Y 35 Finally press the Quit button
351. tually experience a change in the variable values Parametrics Alt 2 When reading about the Range command above you may have been curious as to what happens to your model when you vary an objective coefficient or a right hand side value over its entire range not just the allowable range listed in the range report Of course you could do this by hand as follows alter a coefficient just beyond its allowable range re solve and repeat However this could quickly become tedious LINDO s Parametrics command will automate this process for right hand side values at present LINDO does not support objective coefficient parametrics The Parametrics command will generate a report and or graph detailing the changes in the objective value as a function of changes in a specified right hand side value You specify a new right hand side and LINDO then changes the current right hand side in steps to the new right hand side value displaying the objective function value at each step To have LINDO parametrically vary a right hand side value solve your model then select the Parametrics command To illustrate suppose we have just solved the following staff scheduling model 2 C LINDO61 SAMPLES STAFFX LTX MIN 100 XMON 100 XTUE 100 XWED 100 XTHU 100 XFRI 100 XSAT 100 XSUN SUBJECT TO SUN XWVED XTHU XFRI XSAT XSUN MON XMON XTHU XFRI XSAT XSUN TUE XMON XFRI XSAT XSUN WED XMON XWED XSAT XSUN THU XMON XWED XTHU XSUN
352. u to exploit this prior knowledge You can set this value using either the Edit Options command in Windows versions or the BIP command in command line versions of LINDO As an example suppose you know the optimal solution to a certain IP is greater than 1492 Further the objective is MAX You should set the IP Objective Hurdle to 1492 LINDO will not waste time examining solutions with objective values less than or equal to 1492 Benders Decomposition For certain integer programs with special structure a two level iterative solution approach known as Benders Decomposition can be useful LINDO has a special subroutine interface to facilitate the use of Benders Decomposition This solution approach involves the repeated solution of a master integer program by LINDO and the solution of a very simple problem specific linear program subproblem by a user supplied subroutine Each time LINDO obtains a new integer solution it calls a subroutine NEWIP ACTUAL BOUND To use Benders Decomposition the user should rewrite this subroutine so it solves the subproblem Generally this will result in a new feasible solution to the entire problem In the subroutine NEWIP the user should set the argument ACTUAL to the value of this solution to the entire problem For further details on the NEWIP interface see page 264 Tightening Loose IP Formulations Because IP problems tend to be difficult to solve it is important they are tight Loosely speaking
353. uadratic programming 101 syntax 12 13 120 Objective hurdle 45 48 Online registration 111 Open command 26 29 31 217 Open Command Window command 28 102 3 Open Status Window command 28 104 Operators 12 Optimal solution 151 52 155 Optimality tolerance 45 46 117 194 Optimization Modeling with LINDO 2 17 45 96 167 170 Optimizer options 45 54 Optional modeling statements 15 21 Options optimizer options 45 54 output options 54 57 page length limit 55 resetting defaults 44 saving for later sessions 44 solver status window 54 terminal width 55 terse output 54 INDEX 295 Options command 26 44 55 Output 2 DIVERT RVRT 11 external editors 214 options 54 57 terse mode 117 180 verbose mode 117 180 Overstaffing 23 68 P Packed files 30 32 121 144 PAGE 117 182 218 Page length limit 55 Parameter defaults 44 Parametric analysis 202 4 241 PARAMETRICS 117 173 74 Parametrics command 27 76 80 Parentheses 13 87 139 142 Paste command 26 43 57 60 Paste Symbol command 26 56 PAUSE 117 182 Permuting a model 89 Peruse command 27 81 88 206 8 filtering 86 PICTURE 116 132 33 Picture command 27 89 93 PIVOT 117 135 Pivot command 27 63 66 96 97 Pivot Size Tolerance 51 53 Pivots 52 73 79 117 127 151 element 54 limits 154 218 row 54 Plus 12 POSD 117 175 201 2 201 Positive Definite command 27 101 117 175 201 Prayer algorithm 192 P
354. ues for all the column attributes listed in the column attribute list to the left of the semicolon in the command As an example if you have an integer programming IP model with binary integer variables the following CPRI N P P gt 0 AND P lt 1 AND T T will display the names and primal values of the fractional binary variables 140 CHAPTER 3 If the column attribute list is omitted only the number of columns satisfying the conditional expression is printed For instance CPRI P gt 0 will return an integer value corresponding to the number of variables with values greater than 0 CPRI report output may be sent to an external file by using CPRI in conjunction with a DIVERT command The resulting output may then be loaded into other programs such as spreadsheets and databases If an attribute s numeric value is excessively large it will print as a series of asterisks in the CPRI report Next we will list a number of CPRI examples for the following model and solution MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 X1 X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 Xl 3 X2 2 X3 3 X4 X5 2 X6 X7 X8 lt 6 4 X1 X3 X5 9 5 Xl X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END OBJECTIVE FUNCTION VALUE 1 15 0000000 VARIABLE VALUE REDUCED COST X1 16 000000 00000
355. ump which protrude above the diagonal are called spikes INVERT prints a short summary describing the basis the new inverse and the bump and spike structure You can use the BPICTURE command after INVERT to observe the structure of the basis The model below was partially solved then the INVERT command was given look all MAX 9 X1 X2 4 X3 2 X4 8 X5 2 X6 8 X7 12 X8 SUBJECT TO 2 2 XI X2 2 X3 X4 2 X5 X6 2 X7 3 X8 lt 12 3 X1 3 X2 2 X3 3 X4 X5 2 X6 X7 X8B lt 6 4 X1 X3 X5 9 5 x1 X2 X4 3 6 X2 X3 X6 X7 12 7 X4 X6 X8 9 8 X5 X7 X8 15 END invert INVERT AT STEP 3 CAUSE 1 STATUS 0 Z 6 0000 CYCLING 0 OLD INV NONZEROES I 7 IN U 23 BASIS COLS 6 ELEMENTS 30 SLACKS O ARTS 1 NEW INV IN L 10 IN U 22 BUMPSIZE 5 SPIKES 4 SINGLE COLS 2 TRIANGULAR COLS 2 DETERM 4 00E 0 BUG The BUG command displays the address and telephone number of the technical support department at LINDO Systems with the request that suggestions or errors in LINDO be reported DEBUG The DEBUG command is useful in debugging both infeasible and unbounded models When DEBUG encounters a model with no feasible solution it first tries to identify one or more crucial constraints A constraint is crucial if dropping just that constraint from the entire model is suffic
356. variable is introduced into the solution while another variable is simultaneously driven to zero and dropped from the current solution In some cases you may want to place an upper limit on the number of iterations LINDO performs You can do this by typing a limit value into the box labeled Iteration Limit in the Options dialog box as follows Chedeieetieded Iteration Limit 100 Initial Constraint Tol Iiae Should LINDO hit the Iteration Limit before optimization is complete you will be presented with the following dialog box PIVOT LIMIT OF 100 EXCEEDED HOW MANY MORE ALLOWED EEN Cancel WINDOWS COMMANDS 53 If you want to halt optimization press the button labeled Cancel LINDO will end optimization and return with the best answer found so far If you change your mind and you want to continue with optimization input a number corresponding to the additional number of pivots you want LINDO to perform and then press OK The optimizer will resume and should it hit the new limit before completion will prompt again for an additional increment to the Iteration Limit The default Iteration Limit is None In other words the number of iterations is not limited Initial Constraint Tolerance Final Constraint Tolerance Due to the finite precision available for floating point operations on digital computers LINDO can t always satisfy each constraint exactly Given this LINDO has tw
357. ve environment When you execute a TAKE command on a LINDO script file you may find it useful to have the input of the script file echoed to your screen as LINDO reads it This is particularly useful when you are attempting to debug a LINDO script file To do this simply make both the first and last commands in the script BATCH The first BATCH command enables echoing while the last command disables it 182 CHAPTER 3 PAUSE Syntax PAUSE lt Message gt The PAUSE command suspends LINDO until you press the enter key When you press the enter key LINDO continues processing command input This is useful when you need to view intermediate output from a BATCH file without having it scroll off the screen A message following the PAUSE command is optional If a message is present it must be on the same line as PAUSE LINDO will echo the message on the screen before pausing and waiting for you to press enter As an example of PAUSE suppose you have the following LINDO command script contained in the file MYSCRIPT TXT MAX 20 X 30 Y ST xX lt 50 Y lt 60 X 2Y lt 120 Zz iw ERSE AUSE The objective should be 2050 Press lt Enter gt to esume EAVE B Qa EH O When you run this command script you should see the following on your screen take myscript txt LP OPTIMUM FOUND AT STEP 2 OBJECTIVE VALUE 2050 00000 THE OBJECTIVE SHOULD BE 2050 PRESS lt ENTER gt TO RESUME
358. ves the current solution in a basis file The format of this file is proprietary and is of little use outside the LINDO software The basis is sent to the file associated with the logical unit number LUNIT see LUNGET and LUNOPN JNERR will return 0 if everything was successful otherwise it will be nonzero This routine corresponds to the FBS command line command 240 CHAPTER 8 MAKINT J Integer J Makes variable J a general integer variable The SLB and SUB of variable J remain unchanged LINDO permutes integer variables to appear first in the internal ordering Thus this routine may change the internal ordering of variables To make variable J binary make two calls first call SETSUB to set the upper bound to 1 then call MAKINT to make the variable integer If you don t know the index of a variable you can have LINDO look it up with the NDXOFYV routine NDXOFV KLINE I2 11 12 J Character KLINE Integer N 12 J Returns the internal index J of the variable whose name is stored one character per element in the character array KLINE starting at position and extending no further than position 2 On return 2 will point to the last byte in the name J will be 0 if the name is invalid or one more than the number of existing variables if the name was valid but did not match any existing variable NINTEQ NOINT Integer NOINT Sets the number of integer variables to NOJNT You must place the desired integer vari
359. was incorrect The following example illustrates The coefficient 55 in row 4 should have been 5 5 IS C ALINDO61 SAMPLES INFEAS LTX 2 A R 62 CHAPTER 2 When we attempt to solve this formulation we get the following error code LINDO Error Message m Error code aml Error text NO FEASIBLE SOLUTION AT STEP 2 SUM OF INFEASIBILITIES 2 483333349227906 VIOLATED ROWS HAVE NEGATIVE SLACK OR EQUALITY ROWS NONZERO SLACKS ROWS CONTRIBUTING TO INFEASIBILITY HAVE A NONZERO DUAL PRICE USE THE DEBUG COMMAND FOR MORE INFORMATION Now if we run the Debug command we are presented with the following report in the Reports Window i Reports Window maok A SUFFICIENT SET ROWS CORRECT ONE OF G 4 0 55 X Y gt 4 NECESSARY SET ROWS CORRECT ONE OF SUBJECT TO 2 X 2Y lt 3 The Debug command has correctly identified that the erroneous constraint number 4 when eliminated is sufficient to guarantee a feasible solution Thus Debug can be very useful in tracking down potential problems in a model In general the kind of correction required in a constraint is one or more of the following change the RHS change the direction change the coefficient of a variable in the constraint or change the upper or lower bound of a variable in the constraint Debug operates in a similar manner for unbounded models In the following example we introduced an error by placing a minus sign instead of
360. were previously in the back cascade style Repositions all open windows in a tiled style Close All Closes allopenwindows __ _ Arrange Icons If any windows are minimized the icons for the minimized windows will be arranged at the bottom of the screen 6 Help Menu Command Description system Performs a search of the Help system for a user specified topic How to Use Help Gives some assistance on using the Help system Registers your version of LINGO online AutoUpdate Prompts you to download updated software when available About LINDO Displays specific information about your version of LINDO including the version number and maximum problem capacity The Commands in Depth All LINDO menu commands available under Windows are listed and explained below by category The commands are broken down into the six main menu categories File Edit Solve Reports Window and Help Along with the description for each command we also list any equivalent button from the toolbar and any equivalent accelerator keystroke combination The toolbar runs along the top of the screen and is illustrated in the following picture Dake Eere WINDOWS COMMANDS 29 Each button on the toolbar corresponds to a menu command Not all menu commands have a toolbar button but in general the most frequently used commands will have an equivalent button For example the File New command for creating a new Model Window can b
361. within the margins of this document This line of code loads the executable version of LINDO and tells LINDO to run our script file by adding the following string to the command line t C LINDO SHELL VB LNDIN TXT When LINDO for Windows starts it checks the command line for the presence of the following three commands Command Action at Runtime t filename LINDO executes a File Zake Commands command on the file filename If an AUTOLD DAT file is present it will be queued before filename filename LINDO executes a File Log Output command causing all Reports Window output to be routed to filename If LINDO finds any of these commands in the command line it will attempt to perform the associated action immediately after starting up So by adding the t command above to LINDO s command line we are simply telling LINDO to run our script file on startup The second argument value of 2 tells LINDO to run minimized When LINDO runs minimized it appears as a small icon at the bottom of your screen Also when LINDO runs minimized the initial banner is not displayed The end result is that your application remains displayed on the screen while LINDO runs giving the appearance of tight integration with LINDO 226 CHAPTER 7 One other aspect of this example to consider is the following section of code Rem We now need to wait until LINDO creates Rem our input file On Error Resume Next Do Err 0 Rem Try to
362. y one more on Saturday The increase to 260 people shown in row 4 above increases the objective value by 2500 333 33 7 5 over the value at 185 people Yet again however the dual price rises to 100 when the staffing requirement on Saturday reaches 340 people in row 5 above This increases the objective value by 8000 80 100 over the value at 260 people From here for this model the staffing requirement for Saturday could increase indefinitely from here and each one more increased would add 100 to the objective value Some models however if increased large enough would become either infeasible or unbounded The Parametrics text report shows the entering and leaving variables for every pivot the right hand side value as it changes from the current value to the new right hand side value the dual price and the objective value of the new solution Each line in the report represents a breakpoint in the piecewise linear or piecewise quadratic if the model is quadratic relationship between the objective value and the right hand side value The graphics report clearly illustrates this relationship This report if requested will appear in an individual window The graphics generated for this small example appears as follows RHS Parametrics for Row SAT Objective MIN 30000 25000 100 150 200 250 300 350 400 Righthand Side gt The right hand side is represented on the horizontal axis and the objective value
363. ying an output file you can capture output for use later by your application and or suppress unwanted output so it does not appear on the screen Skillful use of input and output redirection coupled with applications of your own and operating system batch files can result in applications that appear to be tightly integrated with LINDO As a simple example of the use of I O redirection with LINDO we will have a batch file that invokes LINDO sending it the following model for solution Maximize 20 X 30 Y Del s End When LINDO has solved the model our batch program will display the results on the screen from a file created by LINDO Here is the listing of the batch file we will use lindo lt input txt gt output txt echo X and Y cat solution txt INTERFACING 221 In the first line we run LINDO passing it input from the script file named input txt and directing all of LINDO s screen output to the file output txt In this simple example we will have input txt made up beforehand but a more realistic application would probably build the input txt file based on user input or some new data set The second line merely sends the string X and Y to the users screen The final line sends the solution to the screen from a file created during the LINDO run The interesting part of this application is in the LINDO command script file input txt which is reproduced below with comments to document the purpose of each command
364. your code Next you should call the INIT command which performs initialization for a new model INIT should be called each time your code begins input of a new model When demonstrating Basic and C the next thing the sample code above does is redirect LINDO s output to a file This is required when building Windows GUI applications In this case it is a console application in which the standard output device is available Thus LINDO can default to writing to the standard output device Therefore redirection of output is unnecessary On the other hand if you were using Powerstation to build a Windows GUI application you would need to redirect LINDO s output In order to minimize the amount of LINDO s output to the console the QUIET routine is called call quiet 1 In general it s best to leave out this call to QUIET until you have completely debugged your application By suppressing LINDO s output you may miss important system messages As mentioned at the beginning of this chapter you must make one call to DEFROW for each row in your model The following code does this once to define the objective row and three more times in a loop to define the three constraints c Call DEFROW to define objective row call defrow 1 0 idrow trouble c Now call DEFROW to define constraint rows do i 1 3 call defrow 1 rhs i idrow trouble enddo Once the rows are defined APPCOL can be used to define the colu
365. zero Picture dialog box Nonzero Picture Row Var Permutation None C Lower Triangular Cancel OK m Picture Type Help Graph C Text Graph and Text Your first option is to decide if you want LINDO to reorder i e permute the rows and columns in an attempt to maximize the number of nonzeros appearing beneath the matrix diagonal A model that can be permuted with most of the nonzero elements appearing beneath the diagonal tends to be relatively easier to solve If you want LINDO to permute the rows and columns press the button labeled Lower Triangular Otherwise press the None button Your next option is to select the Picture Type You can select a graphics report a text based report or both Text reports are routed to the Reports Window Graphics reports will be sent to individual windows 90 CHAPTER 2 As an illustration we will create nonzero picture reports for the following model X C LINDO61 SAMPLES TEST6 LTX 29970 PITI 29970 PIT2 29910 P2Tl 29910 P2T2 1000 I1T1 1000 I1T2 1000 I2T1 1000 I2T2 1200 E1T1 1200 E1T2 1200 E2T1 1200 E2T2 20 C1T1 20 C1T2 20 C2T1 20 C2T2 3000 NDT1 3000 NDT2 20 R1T1 20 R1T2 20 R2T1 20 R2T2 300 NiT1 300 NiT2 300 N2T1 300 N2T2 SUBJECT TO P1T1 P2T1 NDT1 PIT 2 P2T2 NDT2 900 B1T1 90 T1T1 lt 80000 900 BiT2 90 T1T2 lt 80000 600 B2T1 60 T2T1 lt 70000 600 B2T2 6

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