Exam 9: Linear Programming

If Q_A > 0, then the marginal value of inputs employed:

If slack exists in the solution of the primal LP, the dual shadow price variable is:

Linear programming assumes:

For managerial decision problems analyzed using the LP approach:

Optimal Production. Ozark Telephone, Inc. (OTI) is a small telephone company offering local dial-tone service to its franchised areas in rural southeastern Missouri. A new office park development site is being planned within OTI's territory and John Sample, a network engineer, has to maximize the conversation capacity per line under cost and technology constraints using both traditional copper-wire lines and new fiber-optic lines. OTI wants to gradually move into the all-digital communication environment possible with fiber-optics, so a company policy has been adopted specifying that at least 3 fiber-optic lines be employed for every 2 copper lines on new installations. To minimize the need to quickly retrain its linemen, OTI wants at least 30% of new telephone lines installed to be copper. No existing telephone facilities run to the development site, and OTI must use its own facilities to carry the traffic (it cannot lease capacity from any other local telephone company). Finally, current costs and technologies dictate that 1 fiber line can carry the equivalent of 5 copper lines at the same cost to OTI. That is, if one copper line can carry one telephone conversation, fiber optic lines can carry five conversations at no cost penalty. Sample's objective is to maximize the capacity per line of the transmissions facilities being built to carry traffic to/from the office park. A. Using the inequality form of the constraint conditions, set up and interpret the linear programming problem Sample would use to determine the optimal percentage of copper and fiber-optic lines. Also formulate the problem using the equality form of the constraint conditions. B. With a graph, determine the optimal solution; check your solution algebraically. Fully interpret solution values. C. Holding all else equal, how much would the capacity of fiber optic lines have to fall to alter the optimal construction mix determined in part B? D. Calculate the opportunity cost, measured in terms of conversation capacity per line, of OTI's 30% copper line constraint.

Optimal Lending. Penny Lane is a senior loan officer with Citrus National Bank in Tampa, Florida. Lane has both corporate and personal lending customers. On average, the profit contribution margin or interest rate spread is 1.75% on corporate loans and 2.25% on personal loans. This return difference reflects the fact that personal loans tend to be riskier than corporate loans. Lane seeks to maximize the total dollar profit contribution earned, subject to a variety of restrictions on her lending practices. In order to limit default risk, Lane must restrict personal loans to no more than 40% of total loans outstanding. Similarly, to ensure adequate diversification against business cycle risk, corporate lending cannot exceed 80% of loaned funds. To maintain good customer relations by serving the basic needs of the local business community, Lane has decided to extend at least 40% of her total credit authorization to corporate customers on an ongoing basis. Finally, Lane cannot exceed her current total credit authorization of $50 million. A. Using the inequality form of the constraint conditions, set up and interpret the linear programming problem Lane would use to determine the optimal dollar amount of credit to extend to corporate (C) and personal (P) lending customers. Also formulate the LP problem using the equality form of the constraint conditions. B. Use a graph to determine the optimal solution, and check your solution algebraically. Fully interpret solution values.

Net profits equals profit contribution minus:

If the primal objective function is to minimize cost subject to output constraints, the dual objective function is to maximize:

Profit Maximization. Samantha Spade & Associates, Ltd. is a small architectural firm located in Portland, Oregon, specializing in the preparation of multi-family residential housing complex, R, and commercial retails, C, architectural designs. Prevailing prices in the market are $10,000 for residential housing designs and $25,000 for commercial retail designs. Six architects run the firm, and work a 50-hour workweek, 50 weeks per year. They are assisted by six drafting personnel and two secretaries, all of whom work a typical 40-hour workweek, 50 weeks per year. The firm must decide how to target its promotional efforts so as to best use its resources during the coming year. Based on previous experience, the firm expects that an average of 150 hours of architect and 100 hours of drafting time will be required for each residential housing complex design, whereas commercial retail design will require an average of 250 architect hours and 200 drafting hours. Fifty hours of secretarial time will also be required for each architectural design. In addition, variable computer and other processing costs are expected to average $1,000 per residential design and $1,500 per commercial retail design. A. Set up the linear programming problem the firm would use to determine the profit-maximizing output levels for residential and commercial designs. Show both the inequality and equality forms of the constraint conditions. B. Completely solve and interpret the solution values for the linear programming problem. C. Calculate maximum possible net profits per year for the firm assuming that architects draw a salary of $100,000 per year, drafting personnel earn $65,000 per year, secretaries are paid $20 per hour, and fixed overhead (including promotion and other expenses) averages $50,000. D. After considering the above data, one senior architect recommended reducing one drafting personnel to part-time status (adjusting salary accordingly) while retaining the rest of the current staff full-time. What are net profits per year under this suggestion?

Optimal Production. Canine Products, Inc., produces and markets a new moist and chewy nugget dog food called "Chow Hound" being test marketed in the San Diego market. This product is similar to several others offered by Canine Products, and can be produced with currently available equipment and personnel using any of three alternative product methods. Method A requires 6 hours of labor and 1 processing facility hour to produce 100 bags of dog food, Q_A. Method B requires 3 labor and 3 processing facility hours per Q_B. Method C requires 2 labor hours and 4 processing facility hours per unit of Q_C. Because of slack demand for other products CP currently has 15 labor hours and 5 processing facility hours available per week for producing Chow Hound. A. Using the equality form of the constraint conditions, set up and interpret the linear program Canine Products would use to maximize production of Chow Hound given currently available resources. B. Calculate and interpret all solution values.

When an LP objective function is to maximize profits:

Linear Programming Concepts. Indicate whether each of the following statements is true or false. Defend your answer. A. Linear programming can be used for solving any type of constrained optimization problem where the relations involved can be approximated by linear equations. B. Linear revenue, cost and profit relations will be observed when output prices, input prices, and average variable costs are constant. C. Equal distances along a given production process ray in a linear programming problem always represent an identical level of output D. At isoquant segment midpoints, each adjacent production process must be used to produce 50% of output efficiently. E. Maximizing a LP profit contribution objective function always results in also maximizing total net profits.