Software Options for Solving Such Models

Software Options for Solving Such Models

All the software packages featured in your OR Courseware (Excel, LINGO/LINDO, and MPL/Solvers) include an algorithm for solving (pure or mixed) BIP models, as well as an algorithm for solving general (pure or mixed) IP models where variables need to be integer but not binary. However, since binary variables are considerably easier to deal with than general integer variables, the former algorithm generally can solve substantially larger problems than the latter algorithm.

When using Solver (or ASPE’s Solver), the procedure is basically the same as for linear programming. The one difference arises when you click on the “Add” button on the Solver dialog box to add the constraints. In addition to the constraints that fit linear programming, you also need to add the integer constraints. In the case of integer variables that are not binary, this is accomplished in the Add Constraint dialog box by choosing the range of integer-restricted variables on the left-hand side and then choosing “int” from the pop-up menu. In the case of binary variables, choose “bin” from the pop-up menu instead.

One of the Excel files for this chapter shows the complete spreadsheet formulation and solution for the California Manufacturing Co. example. The Solved Examples section of the book’s website also includes a small minimization example with two integer- restricted variables. This example illustrates the formulation of the IP model and its graphical solution, along with a spreadsheet formulation and solution.

A LINGO model uses the function @BIN() to specify that the variable named inside the parentheses is a binary variable. For a general integer variable (one restricted to inte- ger values but not just binary values), the function @GIN() is used in the same way. In either case, the function can be embedded inside an @FOR statement to impose this bi- nary or integer constraint on an entire set of variables.

In a LINDO syntax model, the binary or integer constraints are inserted after the END statement. A variable X is specified to be a general integer variable by entering GIN X. Alternatively, for any positive integer value of n, the statement GIN n specifies that the first n variables are general integer variables. Binary variables are handled in the same way except for substituting the word INTEGER for GIN.

For an MPL model, the keyword INTEGER is used to designate general integer variables, whereas BINARY is used for binary variables. In the variables section of an MPL model, all you need to do is add the appropriate adjective (INTEGER or BINARY) in front of the label VARIABLES to specify that the set of variables listed below the label is of that type. Alternatively, you can ignore this specification in the variables section and instead place the integer or binary constraints in the model section anywhere after the other constraints. In this case, the label over the set of variables becomes just INTEGER or BINARY.

The student version of MPL includes four elite solvers for linear programming — CPLEX, GUROBI, CoinMP, and SULUM — and all four also include state-of-the-art algorithms for solving pure or mixed IP or BIP models. When using CPLEX, for example, by selecting the MIP Strategy tab from the CPLEX Parameters dialog box in the Options menu, an experienced practitioner can even choose from a wide variety of options for ex- actly how to execute the algorithm to best fit the particular problem.

These instructions for how to use the various software packages become clearer when you see them applied to examples. The Excel, LINGO/LINDO, and MPL/Solvers files for this chapter in your OR Courseware show how each of these software options would be applied to the prototype example introduced in this section, as well as to the subsequent IP examples.

The latter part of the chapter will focus on IP algorithms that are similar to those used in these software packages. Section 12.6 will use the prototype example to illustrate the application of the pure BIP algorithm presented there.