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    Linear Programming : Profit Contribution and Maximizing Profit



    Skillings Industrial Chemicals, Inc., operates a refinery in southwestern Ohio near the Ohio River. The company's primary product is manufactured from chemical process that requires that use of two raw materials-material A and material B. The production of 1 pound of the primary product requires the use of 1 pound of material A and 2 pounds of material B. The output of the chemical process is 1 pound of the primary product, 1 pound of of liquid waste material, and 1 pound of solid waste by product. The solid waste by product is given to a local fertilizer plant as payment for picking it up and disposing of it. The liquid waste material has no market value, so the refinery has been dumping it directly into the Ohio River.
    Government pollution guidelines established by the EPA will longer permit the disposal of the liquid waste directly into the river. The refinery's research group developed the following set of alternatives uses for the liquid waste material.
    1. Produce a secondary product K by adding 1 pound of raw material A to every pound of liquid waste.
    2. Produce a secondary product M by adding 1 pound of raw material B to every pound of liquid waste.
    3. Specially treat the liquid waste so that it meets pollution standards before dumping it into the river.
    The company's management knows that the secondary products will be low in quality and may not be profitable. However, management also recognizes that the special treatment alternative will be a relatively expensive operation. The company's problem is to determine how to satisfy the pollution regulations and still maintain the highest possible profit. How should the liquid waste material be handled? Should Skillings produce product K, produce product M, use the special treatment, or employ some combination of the three alternatives?
    Last month 10,000 pounds of the company's primary product was produced. The accounting department has prepared a cost report showing the breakdown of fixed and variable expenses that were incurred during the month.
    Cost Analysis for 10,000 Pounds
    Of Primary Product

    Fixed Cost
    Administrative expenses $12,000
    Refinery overhead $ 4,000
    Variable Costs
    Raw Material A $15,000
    Raw Material B $16,000
    Direct Labor $ 5,000
    Total $52,000
    In this cost analysis, the fixed cost portion of the expenses is the same every month regardless of the production level. Direct labor costs are expected to run $0.20 per pound for product K and $0.10 per pound for product M.
    The company's primary product sells for $5.70 per pound. Secondary product K sells for $0.85 and product M will sell for $0.65 per pound. The special treatment of the liquid waste will cost $0.25 per pound.
    For the upcoming production period, 5000 pounds of raw material A and 7000 pounds of raw material B will be available.

    Please show a cost analysis showing the profit contribution per pound for the primary product, product K, and product M.
    The optimal production quantities and waste disposal plan, including the projected profit.
    Show linear programming and the constraints.

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    Solution Summary

    An LP problem is solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.