A truck company has allocated $800,000 for the purchase of new vehicles and is considering three types. Vehicle A has a 10-ton payload capacity and is expected to average 45 mph; it costs $26,000. Vehicle B has a 20-ton payload capacity and is expected to average 40 mph; it costs $36,000. Vehicle C is a modified form of B and carries sleeping quarters for one driver. This modification reduces the capacity to an 18-ton payload and raises the cost to $42,000, but its operating speed is still expected to average 40 mph.

Vehicle A requires a crew of one driver and, if driven on three shifts per day, could be operated for an average of 18 hr per day. Vehicles B and C must have crews of two drivers each to meet local legal requirements. Vehicle B could be driven an average of 18 hr per day with three shifts, and vehicle C could average 21 hr per day with three shifts.
The company has 150 drivers available each day to make up crews and will not be able to hire additional trained crews in the near future. The local labor union prohibits any driver from working more than one shift per day. Also, maintenance facilities are such that the total number of vehicles must not exceed 30. Formulate a mathematical model to help determine the number of each type of vehicle the company should purchase to maximize its shipping capacity in ton-miles per day.

a. Identify the decision variables: What decision is to be made?

b. Formulate the objective function: How do these decisions affect the objective?

c. Formulate the constraint set: What constraints must be satisfied? Be sure to consider whether negative values of the decision variables are allowed by the problem, and ensure they are so constrained if required.