Purchasing Various Trucks--A truck company has allocated $800,000 for the purcha

Question

Purchasing Various Trucks--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 45mph; 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, coube be operated for an average of 18 hr per day. Vehicle 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, maintainence 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.

Explanation / Answer

vehicle A costs 26000, vehicle B costs 36000, and vehicle C costs 42000

As 800,000 is available, 26000 A + 36000 B + 42000 C <= 800000

As there may be only 30 vehicles, A + B + C <= 30

All 3 vehicles may be driven 3 shifts, with A requiring 1 man and B and C 2 men

A and B may be driven 18 hours and C 21

150 drivers are available

Thus, 3A + 6B + 6C <= 150

vehicle A has a 10-ton capacity and may be driven 45 mph; B has a 20-ton capacity and may be driven at 40 mph; vehicle C has 18-ton capacity and may be driven at 40 mph

Thus, the shipping capacity is 45*10*18 A + 40*20*18 B + 40*18*21 C

We maximize 8100A + 14400B + 15120 C

Of course, A, B, and C >= 0 and A, B, and C are integers.

Then, the solution is

maximize 8100A + 14400B + 15120 C subject to

26000 A + 36000 B + 42000 C <= 800000

A + B + C <= 30

3A + 6B + 6C <= 150

A, B, C >= 0 and A, B, and C are integers.

I solved the simplex in Excel and it had integer solutions, so this is the optimal solution.

The solution is A = 10, B = 0, and C = 20

We generate 383,400 ton miles a day

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