Question

In: Operations Management

a) A chemical manufacturer processes two chemicals, Arkon and Zenon, in varying proportions to produce three...

a) A chemical manufacturer processes two chemicals, Arkon and Zenon, in varying proportions to produce three products, A, B, and C. He wishes to produce at least 150 units of A, 200 units of B, and 60 units of C. Each ton of Arkon yields 3 of A, 5 of B and 3 of C. Each ton of Zenon yields 5 of A, 5 of B and 1 of C.

i. If Arkon cost Ghc 40 per ton and Zenon Ghc 50 per ton, advise the manufacturer how to minimize cost.

ii. What is the dual of the minimizing LP problem?

Solutions

Expert Solution

i.

Let us assume the following:

Tons of Arkon produced = X1

Tons of Zenon produced = X2

The objective function is cost minimization:

Minimize Z = 40X1 + 50X2

Constraint 1: At least 150 units of A:

3X1 + 5X2 >= 150

Constraint 2: At least 200 units of B:

5X1 + 5X2 >= 200

Cosntraint 3: At least 60 units of C:

3X1 + 1X2 >= 60

Therefore, the linear programming model is formulated as follows:

Minimize Z = 40X1 + 50X2

subject to:

3X1 + 5X2 >= 150

5X1 + 5X2 >= 200

3X1 + 1X2 >= 60

X1, X2 >= 0

Solving the problem using MS Excel Solver:

Therefore, the optimal solution is as follows:

Tons of Arkon produced = X1 = 25

Tons of Zenon produced = X2= 15

Total Cost = ( 40 x 25) + ( 15 x 50) = 1000 + 750 =  Ghc 1750

ii.

Dual of the Minimization problem shall be a maximization problem.

Maximize Zd = 150Y1 + 200Y2 + 60Y3

subject to:

3Y1 + 5Y2 + 3Y3 <= 40

5Y1 + 5Y2 + 1Y3 <= 50

Y1, Y2, Y3 >= 0

keosha answered 1 year ago
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