Teach Integer Programming Add-in - Benders' Algorithm
 The examples for this section are in the demo workbook (teachip_benders.xls). To solve or change the model or construct a new model, you must have the Teach IP add-in loaded. The Jensen LP/IP Solver add-in must also be loaded. Use the Relink buttons command to establish links to the worksheet buttons. We do not provide a complete description of the algorithm here. Download the PDF Bender's Algorithm document for details. It is easy to run the algorithm. Create a Benders' model from the Teach IP menu. Click the Setup button on the worksheet. Click the button to Do Benders' Algorithm. Then click the Solve Dual button. Then sequentially click the Solve Master Button and Solve Dual buttons until the bounds become equal.

Problem Definition

Begin a Benders' problem by clicking on the Benders item on the Teach IP menu. The program presents the problem definition dialog to accept model data.

The program presents a possible name in the Name field such as TeachIP1. There can be multiple IP models in a workbook, and the integer number at the end of the name will advance as IP models are added. The name is used as prefix for a number of named ranges on the worksheet, so once the name is specified it should not be changed. Names must obey the Excel rules for naming ranges. The name should begin with a letter, and include no spaces or punctuation marks (a period is OK). The underline symbol may be used and is often handy for two-part names such as P_1. A name like P1 can't be used because it can be confused with a cell reference.

Fields are available for number of variables, number of constraints and number of integer variables. If the latter is fewer than the number of variables, the variables with the smaller indices are assumed integer. For a reasonable demonstration, keep the numbers of variables and constraints small. The model must be expressed as a maximization. For Benders' algorithm some variables must be real and some must be integer. If the Make Random Problem box is checked, the program will provide random coefficients for the objective function and constraints. Bender's problems must have only less than or equal to constraints. All coefficients are integer and positive. A random problem will always have a feasible solution. When the model is displayed the student can change any of the data.

One of four modes of operation is chosen with a button in the Solution Options frame.

• Demonstration Option: In this mode, the program takes the student through the steps of the procedure. Message boxes describe the steps as they occur. The student presses OK buttons to progress.
• Instruction Option: This is almost the same as the the next option, but the program places the active line at each the Dual or Primal problem as appropriate.
• Do it Yourself Option: The student must make all the decisions directing the algorithm.
• Run without Stopping Option: The program runs the procedure with no student interaction.

Operations Research Models and Methods
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by Paul A. Jensen