Question

16. United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade, as follows:

                                                Mill

Aluminum Grade                    1          2         

High                                        6          2      12

Medium                                   2          2      8

Low                                         4          10     5

The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminum, 8 tons of medium-grade aluminum, and 5 tons of low-grade aluminum. It costs United $6,000 per day to operate mill 1 and $7,000 per day to operate mill 2. The company wants to know the number of days to operate each mill in order to meet the contract at the minimum cost.

Formulate a linear programming model for this problem.

17. Solve the linear programming model formulated in Problem 16 for United Aluminum Company graphically.

a. How much Extra (i.e., surplus) high- , medium- , and low-grade Aluminum does the company produce at the optimal solution?

b. What would be the effect on the optimal solution if the cost of operating mill 1 increased from $6000 to $7500 per day ?

c. What would be the effect on the optimal solution if the company could supply only 10 tons of high-grade Aluminum. ?

Answer 1

16.

Linear programming model is formulated as below:

Let X1 and X2 be the number of days Mill 1 and Mill 2 to be operated

Minimize 6000X1+7000X2

s.t.

6X1+2X2 >= 12

2X1+2X2 >= 8

4X1+10X2 >= 5

X1, X2 >= 0

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