The management at Hackney's Southwestern Regional (HSR) are interested in determining the optimal fleet purchase mix for the coming year. There are four types of vehicles that can be purchased: standard, intermediate, compact and subcompact. All cars purchased are depreciated and paid off for over a two-year period, after which they are sold in a secondary market. Table 1 presents the wholesale purchase price and revenues (the second year includes the resale value) over the two-year period for each type of car:

HSR has established a budget of $5 million for funding the purchases this year. It can either pay the entire amount for the cars or make a down payment and pay the remaining balance over a two-year period. The financing company requires at least a 20% down payment and that at least 50% of the purchase price plus interest must be paid by the end of the first year. The financing company is currently charging 8%. HSR uses a 10% discount rate for financial planning. Hackney management estimates that they can rent all of the cars that are purchased. However, in order to meet overall market demand they wish that each vehicle category represent at least 15 percent and no more than 50 percent of the total number of vehicles purchased. a. Formulate a linear programming model for this problem. Define the basic assumptions associated with this problem and determine the optimal fleet-mix to purchase including the payment schedule.

Table 1 Purchase Price and Net Revenue Summary Type Standard Intermediate Compact Subcompact Price (S) 20,000 15,000 12,000 9,000 1st yr. (S) 14,000 9,000 7,000 5,000 2nd yr. ($) 23,000 15,000 13,000 9,000