Question

A manufacturer sells two types of products. Product 1 is sold at a price of $50 per unit and product 2 at a price of $60 per unit. Three units of raw material and 1.5 labor hours are needed to manufacture one unit of product 1. Six units of raw material and 2 labor hours are needed to manufacture one unit of product 2. The unit variable cost for product 1 is $30, and for product 2 is $20. A total of 15,000 units of raw material and 10,000 labor hours are available. If any product 1 is produced, a setup cost of $20,000 is incurred; if any product 2 is produced, a setup cost of $35,000 is incurred. Determine how to maximize the manufacturer’s profit.

a) What is the effective capacity for product 1 and product 2, respectively?

b) In the optimal solution, which product(s) will be manufactured? What is the optimal production quantity? What is the optimal profit?

Answer 1

(a)

		Per unit consumption		Max. possible production
Resources	Available	Product 1	Product 2	Product 1	Product 2
Raw material	15,000	3	6	5000	2500
Labor hours	10,000	1.5	2	6666.67	5000

Considering the minimums of the maximum possible production, we can say that 5000 units of Product-1 and 2500 units of Product-2 are the capacities.

(b)

Let P1 and P2 be the quantities of Product-1 and Product-2 to be produced. Also, let Yj be the binary integer such that Yj=1 when the Product-j is produced. j=1,2

Maximize Z = total profit = (50 - 30) P₁ + (60 - 20) P₂ - 20000 Y₁ - 35000 Y₂

Subject to,

3 P₁ + 6 P₂ <= 15000

1.5 P₁ + 2 P₂ <= 10000

P₁ - 5000 Y₁ <= 0

P₂ - 2500 Y₂ <= 0

P₁, P₂ >= 0

Y₁, Y₂ = {0,1}

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LINDO Code

Max 20 P1 + 40 P2 - 20000 Y1 - 35000 Y2
s.t.
3 P1 + 6 P2 < 15000
1.5 P1 + 2 P2 < 10000
P1 - 5000 Y1 < 0
P2 - 2500 Y2 < 0
end
INT Y1 INT Y2

Solution

P1 = 5000; P2 = 0; Y1 = 1; Y2 = 0 and the Profit = $80,000