Short Answer (4 points each)

1. Define the following:

1. Slack variable
2. Surplus variable

2. Explain the difference between an optimal solution and a feasible solution.

3. Describe briefly the management science approach to problem solving. Give an example.

4. What is the economic interpretation of shadow prices?

5. What is LP sensitivity analysis and why is it used?

Problems

1. The Leather Company produces baseball gloves and leather straps using raw leather that is processed on a machine. For the coming week, there are 1000 square meters of leather and 40 hours of machine time available. As Production Manager, you want to determine a production plan for this week (how many units of each product to make) that maximizes net corporate profits.

a. Identify and define the decision variables. (3 points)

b. Identify and assign a symbolic name to each additional data value you would have to obtain to be able to formulate an LP model. (3 points)

c. Formulate an LP model using the variable names from (a) and the symbolic names for the data from (b). (4 points)

2. The two liquid products that Case Chemical makes (CS-01 and CS-02)b yield excessive amounts of three different pollutants: A, B, and C. The state government has ordered the company to install and to employ antipollution devices. The following table provides the current daily emissions of each of the three pollutants, in kg/1000 liters of product (kilograms per 1000 liters), and the maximum of each pollutant allowed in kg.

The manager of the Production Department has approved the installation of two antipollution devices. The emissions from each product can be handled by either device in any proportion. (The emissions are sent through a device only once, that is, the output of one device cannot be the input to the other or back to itself.) The following table shows the percentage of each pollutant from each product that is removed by each device.

For example, if the emission from CS-01 is sent through Device 1, 40% of pollutant A, 60% of pollutant B, and 55% of pollutant C are removed. Manufacturing considerations dictate that CS-01 and CS-02 be produced in the ratio of 2 to 1. Formulate an LP model to determine a plan that maximizes the total daily production (amount of CS-01 plus amount of CS-02) while meeting governmental requirements. (10 points)

3. Incredible Indelible Ink Company mixes three additives, A1, A2, and A3, in different proportions, and a base to obtain different colors of ink. Red ink is obtained by mixing A1, A2, and A3 in the ratio of 3:1:2, blue ink in the ratio of 2:3:4, and green in the ratio of 1:2:3. After these additives are mixed to make a certain color, an equal amount of base is added to finish making this color. The company currently has 1000 gallons of A1, 1500 gallons of A2, 2000 gallons of A3, and 4000 gallons of base. Given that the selling price per gallon for each type of ink is the same, formulate an LP model to determine how these resources should be used to obtain the maximum revenue. (12 points)

4. The Portfolio Manager of Pension Planners, Inc. has been asked to invest $1,000,000 of a large pension fund. The Investment Research Department has identified six mutual funds with varying investment strategies, resulting in different expected returns and associated risks, as summarized in the following table.

One way to control the risk is to limit the amount of money invested in the various funds. To that end, the management of Pension Planners, Inc. has specified the following guidelines:

1. The total amount invested in high-risk funds must be between 50% and 75% of the portfolio.
2. The total amount invested in medium-risk funds must be between 20% and 30% of the portfolio.
3. The total amount invested in low-risk funds must be at least 5% of the portfolio.

A second way to control risk is to diversify—that is, to spread the risk by investing in many different alternatives. The management of Pension Planners, Inc, has specified that the amount invested in the high-risk Funds 1, 2 and 3 should be in the ratio 1:2:3, respectively. The amount invested in the medium risk Funds 4 and 5 should be in the ratio 1:2.

With these guidelines, formulate a single LP model to maximize the expected rate of return. The complete model formulation is available on Bb in the Assignments area.

To determine what portfolio you, the Portfolio Manager, should recommend so as to maximize the expected rate of return, obtain the formulation and prepare a managerial report along the following guidelines.

1. Use Excel’s Solver to solve the linear programming formulation. Prepare a table indicating the amount of money invested in each fund. (6 points)
2. Indicate the highest return that can be achieved if no money is invested in Fund 6. (4 points)
3. Discuss the effect that increasing the amount that must be invested in the low-risk fund would have on the optimal rate of return. (6 points)
4. Provide the optimal investment strategy if the amount invested in Fund 4 is to be the same as that invested in Fund 5. (4 points)

5. Mineral Mining Company sends one truckload of iron and copper ore daily from the mine to the processing plant. The truck has a weight capacity of 10 tons and a volume capacity of 1200 cubic feet. Each pound of iron ore takes up 0.04 cubic feet of space and yields a net profit of $0.30 when processed. Each pound of copper ore uses 0.08 cubic feet of space and provides $0.50 of net profit. The following LP model is proposed, where I is the number of pounds of iron ore to load on the truck and C is number of pounds of copper ore to load on the truck:

a. Solve the problem graphically. Explain the optimal solution and objective function value in the context of the problem. (6 points)

6. Graphically and algebraically determine the sensitivity range of each objective function coefficient at the optimal solution of the linear program in Problem 5. Explain the meaning of each range in the context of the problem. (6 points)

7. For each constraint that holds with equality at the optimal solution of the linear program in Problem 5, perform the following:

a. Graphically and algebraically determine the sensitivity range of the right-hand-side value. : (4 points)

b. Compute the shadow price associated with each sensitivity range found in part (a). Explain the meaning of the shadow price in the context of the problem. : (6 points)

8. For the problem of Mineral Mining Company in Problem 5, management is considering leasing one of the following two new trucks to replace the existing one:

Should the company replace the old truck? And, if so, with which one of the new trucks should the company replace? Explain. (Use results from Problem 7) (6 points)

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Solution: The downloadable solution consists of 19 pages, 2456 words and 12 charts.
Deliverable: Word Document