An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 400 first class, 900 tourist class, and 1500 economy class passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

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MRROYALGO MRROYALGO · Answer 1

The relationship between airplane 1 and airplane 2 is an illustration of objective function

14 of airplane 1 and 7 of airplane 2 should be used to minimize the operation cost

Represent P1 and P2 with x and y, respectively.

From the question, we ave have the following parameters:

x y Minimum

First class 20 18 400

Tourist 50 30 900

Economy 110 44 1500

Cost 10000 85000

So, the objective cost function to minimize would be:

C = 10000x + 8500y

And the constraints are:

20x + 18y ≥ 400

50x + 30y ≥ 900

110x + 44y ≥ 1500

x, y ≥ 0

Next, we plot the graphs of the constraints

From the graph (see attachment), we have the following feasible solutions

(x,y) = {(4.9,21.8), (8.5,12.7), (14,6.7)}

Substitute these values in the objective function.

C(5,22) = 10000 * 5 + 8500 * 22 = 237000

C(9,13) = 10000 * 9 + 8500 * 13 = 200500

C(14,7) = 10000 * 14 + 8500 * 7 = 199500

The minimum value is:

C(14,7) = 199500

Hence, 14 of airplane 1 and 7 of airplane 2 should be used to minimize the operation cost

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