Question

The Heat-Aire Company has two plants that produce identical heat pump units. Let X represent the #  of units produced at plant 1 and Y represent the # of units produced at plant 2.  The total cost of production at each plant is X^2 + 0.5X + 20 and Y^2 + 0.3 Y + 10 respectively.   Neither plant can make more than 450 heat pumps. Heat pumps can be shipped from either plant to satisfy demand from three different customers. The unit shipping costs and demands for each customer are summarized in the following table.

	Customer 1	Customer 2	Customer 3
Plant 1	$23	$20	$18
Plant 2	$29	$17	$25
Demand	300	250	150

What is the optimal production and shipping plan, if management wants to meet customer demand at the lowest total cost?

1. Formulate an NLP model for this problem.

b. What is the optimal solution?  Clearly show how many pumps are produced at each plant, how many pumps are shipped from each plant to each customer, the total cost of production, and the total cost of shipping.

Chen is president of Chen cabinets Inc., a firm that manufactures two types of metal file cabinets.  Chen has a weekly labor capacity of 1,300 hours, with each smaller cabinet taking 1 hour to produce and the larger cabinet requiring 2 hours each. One wooden plank is used for each smaller cabinet and 1.5 planks are used for each larger cabinet.  Chen can get a supply of a maximum of 1000 planks each week.  Each two-drawer model sold yields a $10 profit, and the profit for the larger model is $25.

Chen has the following goals (1). Maximize profit, (2). Maximize number of cabinets produced.

1. Formulate this Goal programming problem.
2. Solve the goal programming problem to minimize the maximum % deviation from each goal. Show the solution, target values and % deviation from each goal.
3. What will be new solution if Goal # 1 was twice as important as Goal # 2?

Answer 1

Q1.

a)

NLP model is formulated as under:

Let X and Y be the total number of heat pumps produced at plant 1 and 2 respectively.

X1, X2, X3 be the number of heat pumps to be shipped from plant 1 to customers 1,2,3 resp

Y1, Y2, Y3 be the number of heat pumps to be shipped from plant 2 to customers 1,2,3 resp

Min X^2 + 0.5X + 20 + Y^2 + 0.3Y + 10 + 23X1 + 20X2 + 18X3 + 29Y1 + 17Y2 + 25Y3

s.t.

X-X1-X2-X3 = 0

Y-Y1-Y2-Y3 = 0

X1+X2+X3 <= 450

Y1+Y2+Y3 <= 450

X1+Y1 = 300

X2+Y2 = 250

X3+Y3 = 150

X, X1, X2, X3, Y, Y1, Y2, Y3 >= 0

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b)

Create Excel model as follows:

Enter Solver Parameters as follows:

Click Solve to generate the solution. Values appear automatically in yellow cells, after pressing Solve button

Number of pumps to be produced at plant 1 = 351

Number of pumps to be produced at plant 2 = 349

Number of pumps to be shipped from each plant to each customer are shown in the following table

	Customer 1	Customer 2	Customer 3
Plant 1	201	0	150
Plant 2	99	250	0

Total cost of production = 123396.50 + 121915.70 = $ 245,312.20

Total cost of shipping = 201*23+150*18+99*29+250*17 = $ 14,444