Using Excel to Formulate and Solve Transportation Problems

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Using Excel to Formulate and Solve Transportation Problems

As described in Sec. 3.5, the process of using a spreadsheet to formulate a linear programming model for a problem begins by developing answers to three questions. What are the decisions to be made? What are the constraints on these decisions? What is the overall mea- sure of performance for these decisions? Since a transportation problem is a special type of

Introduction to Operations Research-0110

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linear programming problem, addressing these questions also is a suitable starting point for formulating this kind of problem on a spreadsheet. The design of the spreadsheet then revolves around laying out this information and the associated data in a logical way.

To illustrate, consider the P & T Co. problem again. The decisions to be made are the number of truckloads of peas to ship from each cannery to each warehouse. The con- straints on these decisions are that the total amount shipped from each cannery must equal its output (the supply) and the total amount received at each warehouse must equal its al- location (the demand). The overall measure of performance is the total shipping cost, so the objective is to minimize this quantity.

This information leads to the spreadsheet model shown in Fig. 9.4. All the data pro- vided in Table 9.2 are displayed in the following data cells: UnitCost (D5:G7), Supply (J12:J14), and Demand (D17:G17). The decisions on shipping quantities are given by the changing cells, ShipmentQuantity (D12:G14). The output cells are TotalShipped (H12:H14) and TotalReceived (D15:G15), where the SUM functions entered into these cells are shown near the bottom of Fig. 9.4. The constraints, TotalShipped (H12:H14) = Supply (J12:J14) and TotalReceived (D15:G15) = Demand (D17:G17), have been specified on the spread- sheet and entered into Solver. The objective cell is TotalCost (J17), where its SUMPROD- UCT function is shown in the lower right-hand corner of Fig. 9.4. The Solver parameters box specifies that the objective is to minimize this objective cell. Choosing the Make Vari-bles Nonnegative option specifies that all shipment quantities must be nonnegative. The Simplex LP solving method is chosen because this is a linear programming problem.

To begin the process of solving the problem, any value (such as 0) can be entered in each of the changing cells. After clicking on the Solve button, Solver will use the simplex method to solve the transportation problem and determine the best value for each of

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Introduction to Operations Research-0113

Note that Solver simply uses the general simplex method to solve a transportation problem rather than a streamlined version that is specially designed for solving trans- portation problems very efficiently, such as the transportation simplex method presented in the next section. Therefore, a software package that includes such a streamlined version should solve a large transportation problem much faster than Solver.

We mentioned earlier that some problems do not quite fit the model for a transportation problem because they violate the requirements assumption, but that it is possible to re- formulate such a problem to fit this model by introducing a dummy destination or a dummy source. When using Solver, it is not necessary to do this reformulation since the simplex method can solve the original model where the supply constraints are in < form or the demand constraints are in > form. (The Excel files for the next two examples in your OR Courseware illustrate spreadsheet formulations that retain either the supply constraints or the demand constraints in their original inequality form.) However, the larger the problem, the more worthwhile it becomes to do the reformulation and use the transportation simplex method (or equivalent) instead with another software package.

The next two examples illustrate how to do this kind of reformulation.

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