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Linear Programming : Profit Contribution and Maximizing Profit

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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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Wal-Mart, a discount store chain, is planning to build a new store in Rock Springs, Maryland.. The parcel of land the company owns is large enough to accommodate a store with 140,000 square feet of floor space. Based on marketing and demographic surveys of the area and historical data from its other stores, Wal-Mart estimates its annual profit contribution per square foot for each of the store's departments to be as shown in the following table.

(see chart in attached file)

Each department must have at least 15,000 ft2 of floor space and no department can have more than 20% of the total retail floor space. Men's women's and children's clothing plus housewares keep all their stock on the retail floor; however, toys, electronics, and auto supplies keep some items (bicycles, televisions, tires, etc.) in inventory. Thus, 10% of the total retail floor space devoted to these three departments must be set aside outside the retail area for stocking inventory.

a. Formulate a linear programming model that can be used to determine the floor space that should be devoted to each department in order to maximize profit contribution.

b. Determine the optimal floor space allocation and the resulting total contribution to profit.

c. Wal-Mart is considering the purchase of a parcel of land adjacent to this planned building site. The cost of the parcel is $190,000 and it would enable Wal-Mart to increase the size of the store to 160,000 ft2. Company policy requires that acquisitions of new land for expansion be offset by additional contribution to profit within five years. Historically, however, profit contributions decline in all departments by 10% if a store size increases past 150,000 ft2 (slower stock turnover, increased inventory costs, etc). Provide a justified recommendation with respect to this expansion option.

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