Question

A company needs to purchase larger aircraft. The options included 20 of type A​ and/or type B aircraft. To aid in their​ decision, executives at the company analyzed the following data. Type A Type B Direct Operating Cost ​$1600 per hour ​$500 per hour Payload 32 comma 000 pounds 4000 pounds The company was faced with the following constraints. ​1) Hourly operating cost was limited to ​$40 comma 000. ​2) Total payload had to be at least 512 comma 000 pounds. ​3) Only twenty type A aircraft were available. Given the​ constraints, how many of each kind of aircraft should the company purchase to maximize the number of​ aircraft? To maximize the number of​ aircraft, the company should purchase nothing type A aircraft and nothing type B aircraft.

Answer 1

Let the company purchase x type A and y type B aircrafts.

The hourly operating cost of these aircrafts is 1600x+500 y so that 1600x+500 y ? 40000 or, 16x+5y ? 400.

Total payload of these aircrafts is 32000x+4000y so that 32000x+4000y ? 512000 or, 8x+y ? 128.

Since only twenty type A aircraft were available, we have x ? 20.

A graph of the lines 16x+5y = 400 or, y = -(16/5)x +80…(1) , y = -8x+128 …(2)and x = 20…(3) is attached. The feasible region is in the 1^st quadrant as x ?0, y ?0) , to the left of the line x = 20 and including it (red line), on or above the line y = 128-8x (in purple) and on or below the line y = -(16/5)x +80 (in green). The 1^st and the 2^nd lines intersect at the point (10,48), the 2^nd and the 3^rd lines do not intersect in the 1^st quadrant and the 1^st and the 3^rd lines intersect at the point (20,16). At these points, the number of aircrafts is x+y =10+48 = 58 or, 20+16 = 36.

Thus, in order to maximize the number of? aircrafts, the company should purchase 10 type A aircrafts and 48 type B aircrafts. This will satisfy all the given constraints.