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The Midland Tool Company

The Midland Tool Company manufactures prefabricated metal housings and covers for consumer products under contracts from finished goods producers on four heavy presses of different sizes and different requirements. Currently, the company has a contract to manufacture three products that each require operations on any one of the four presses. The contract requires 400 units of product 1; 570 units of product 2; and 320 units of product 3. The time required, in minutes, for each product on each press follows.
Product 1 2 3 4
1 35 41 34 39
2 40 36 32 43
3 38 37 33 40
Machine 1 is available for 150 hours during the contract period, machine 2 is similarly available for 240 hours, machine 3 is available for 200 hours, and machine 4 is available for 250 hours. The manufacture of these products also results in different contributions to profit according to the press utilized due to time, waste and operating cost. These contributions are summarized as follows.
Product 1 2 3 4
1 $7.8 7.8 8.2 7.9
2 6.7 8.9 9.2 6.3
3 8.4 8.1 9.0 5.8

a. Formulate a linear programming model that can be used to determine the optimal production schedule for the presses to meet the contract requirements that yields the maximum total contribution to profit.
b. Determine the optimal production schedule using the Management Scientist software, including the number of units of each product produced on each press and the total profit contribution. Provide a narrative that explains the Management Scientist solution used.