Question

Question 3

The Fly-High Airplane Company builds small jet airplanes to sell to corporations for use by their executives. To meet the needs of these executives, the company's customers sometimes order a custom design of the airplanes being purchased. When this occurs, a substantial start-up cost is incurred to initiate the production of these airplanes.

Fly-High has recently received purchase requests from three customers with short deadlines. However, because the company's production facilities already are almost completely tied up filling previous orders, it will not be able to accept all three orders. Therefore, a decision now needs to be made on the number of airplanes the company will agree to produce (if any) for each of the three customers.

The relevant data are given in the table below. The first row gives the start-up cost required to initiate the production of the airplanes for each customer. Once production is under way, the marginal net revenue from each airplane produced is shown in the second row. The marginal net revenue is the purchase price minus the marginal production cost. The third row gives the percentage of the available production capacity that would be used for each plane produced. The last row indicates the maximum number of airplane requested by each customer (but less will be accepted).

	Customer 1	Customer 2	Customer 3
Start-up cost	$3 million	$2 million	0
Marginal net revenue	$2 million	$3 million	$0.8 million
Capacity used per plane	20%	40%	20%
Maximum order	3 planes	2 planes	5 planes

Fly-High now wants to determine how many airplanes to produce for each customer (if any) to maximize the company's total profit (total net revenue minus start-up costs). Formulate the mixed integer programming model and solve it using Excel solver for this problem.

Answer 1

Let Xj = number of planes produced for the j-th customer; j=1,2,3
Let Yj be a binary integer such that Yj=1 when Xj >0 and Yj=0 otherwise; j=1,2,3

Objective Function: Maximize Z = Total profit
Z = 2X1 + 3X2 + 0.8X3 - (3Y1 + 2Y2 + 0Y3)

Subject to,

0.2X1 + 0.4X2 + 0.2X3 <= 1

X1 - 3*Y1 <= 0
X2 - 2*Y2 <= 0
X3 - 5*Y3 <= 0

Xj = Positive integers, Yj = [0,1}

Setting up in excel:

Initial formulation:

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