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.

In the Grand Prix example with varying tax rates, the optimal solution uses all available plant capacity and more than satisfies customer demands. Will this always be the case? Experiment with the unit selling prices and/or tax rates to see whether the company ever uses less than its total capacity.