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Linear Programming

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The Pittsburg Chemical Company of Akron, Ohio, produces a variety of products within its four divisions for agricultural and industrial customers. Division II will produce only two products this week for sale to wholesale distributors and all production will be sold. Product 1 yields a contribution margin (contribution to profit) of $25 per ton and product 2 yields a profit contribution of $10 per ton. Both products are manufactured by mixing raw materials from inventory. There are 12,000 tons of material 1, 4,000 tons of material 2, and 6,000 tons of material 3 in inventory in Division II this week. Product 1 is manufactured by mixing materials 1 and 2 in ratios of 60% and 40% respectively. Product 2 is manufactured by mixing 50% of material 1, 10% of material 2, and 40% of material 3. Any unused materials are maintained in inventory at insignificant cost for future production.
a. Formulate a linear programming model that can be used to determine the optimal production scheduling mix that yields the best contribution solution while meeting the inventory capacity restrictions of the division for the week.
b. Determine the optimal production scheduling mix for this division of Pittsburg Chemical Company for the week using the Management Scientist software, including the quantity of each product line manufactured, the quantity of unused materials (if any) remaining, and the total profit contribution for the week. Provide a narrative that explains the Management Scientist solution used. Include all information available relative to division resource utilization, and resources remaining, total production, and profit contribution.

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Linear Programming solutions are provided in details.