Textbook Problem

Consider again Grand Prix’s problem of shipping automobiles from three plants to four regions. However, the problem is now extended in two directions. First, we assume that Grand Prix not only ships the autos, but it manufactures them at the plants and sells them in the various regions. Second, we assume that this problem takes place in a global context. The effect is that the unit production costs vary by plant, the unit selling prices vary by region, and the tax rates on profits vary according to the plant at which the autos are produced (regardless of where they are sold). The capacities of the plants, the demands of the regions, and the unit shipping costs are the same as before, as shown earlier in Table 5.1. In addition, the unit production costs and tax rates are given in Table 5.2, and the unit selling prices in Table 5.3. For example, if plant 1 produces an auto and ships it to region 2, where it is sold, the profit before taxes is $20,520 − $14,350 − $218 = $5,952. This is taxed at plant 1’s rate of 30%, so the after-tax profit is $5,952(1 − 0.3) = $4,166.40. The company now needs to find a production and shipping plan that maximizes its after-tax profit.

Modify the original Grand Prix example as follows. Increase the demands at the regions by 200 each, so that total demand is well above total plant capacity. This means that some demands cannot be supplied. Suppose there is a unit penalty cost at each region for not supplying an automobile. Let these unit penalty costs be $600, $750, $625, and $550 for the four regions. Develop a model to minimize the sum of shipping costs and penalty costs for unsatisfied demands. (Hint: This requires a trick. Introduce a fourth plant with plenty of capacity, and set its “unit shipping costs” to the regions equal to the unit penalty costs. Then interpret an auto shipped from this fictitious plant to a region as a unit of demand not satisfied.)