BUSI2013 – Decision Analysis Individual Problem Set 5 Chapter 8: • Problem 28 (14 points) Chapter 9: • Problem 2 (8 points) • Problem 18 (10 points) • Problem 21 (6 points) Chapter 10: • Problem 3 (8 points) • Problem 7 (10 points)Chapter 8 28) The Pfeiffer Company manages approximately $15 million for clients. For each client, Pfeiffer chooses a mix of three investment vehicles: a growth stock fund, an income fund, and a money market fund. Each client has different investment objectives and different tolerances for risk. To accommodate these differences, Pfeiffer places limits on the percentage of each portfolio that may be invested in the three funds and assigns a portfolio risk index to each client. Here's how the system works for Dennis Hartmann, one of Pfeiffer's clients. Based on an evaluation of Hartmann's risk tolerance, Pfeiffer has assigned Hartmann's portfolio a risk index of 0.05. Furthermore, to maintain diversity, the fraction of Hartmann's portfolio invested in the growth and income funds must be at least 10% for each, and at least 20% must be in the money market fund. The risk ratings for the growth, income, and money market funds are 0.10, 0.05, and 0.01, respectively. A portfolio risk index is computed as a weighted average of the risk ratings for the three funds, where the weights are the fraction of the portfolio invested in each of the funds. Hartmann has given Pfeiffer $300,000 to manage. Pfeiffer is currently forecasting a yield of 20% on the growth fund, 10% on the income fund, and 6% on the money market fund. a. Develop a linear programming model to select the best mix of investments for Hartmann's portfolio. b. Solve the model you developed in part (a). c. How much may the yields on the three funds vary before it will be necessary for Pfeiffer to modify Hartmann's portfolio? d. If Hartmann were more risk tolerant, how much of a yield increase could he expect? For instance, what if his portfolio risk index is increased to 0.06? e. If Pfeiffer revised the yield estimate for the growth fund downward to 0.10, how would you recommend modifying Hartmann's portfolio? f. What information must Pfeiffer maintain on each client in order to use this system to manage client portfolios? g. On a weekly basis Pfeiffer revises the yield estimates for the three funds. Suppose Pfeiffer has 50 clients. Describe how you would envision Pfeiffer making weekly modifications in each client's portfolio and allocating the total funds managed among the three investment funds. Chapter 9 2) The management of Hartman Company is trying to determine the amount of each of two products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability: Product (hours/unit) Department 1 2 Labor-Hours Available A 1.00 0.35 100 B 0.30 0.20 36 C 0.20 0.50 50 profit contribution / unit $30.00 $15.00a. Develop a linear programming model of the Hartman Company problem. Solve the model to determine the optimal production quantities of products 1 and 2. b. In computing the profit contribution per unit, management doesn't deduct labor costs because they are considered fixed for the upcoming planning period. However, sup- pose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime? How much would you be willing to pay per hour of overtime in each department? c. Suppose that 10, 6, and 8 hours of overtime may be scheduled in departments A, B, and C, respectively. The cost per hour of overtime is $18 in department A, $22.50 in department B, and $12 in department C. Formulate a linear programming model that can be used to determine the optimal production quantities if overtime is made available. What are the optimal production quantities, and what is the revised total contribution to profit? How much overtime do you recommend using in each department? What is the increase in the total contribution to profit if overtime is used? 18) The Two-Rivers Oil Company near Pittsburgh transports gasoline to its distributors by truck. The company recently contracted to supply gasoline distributors in southern Ohio, and it has $600,000 available to spend on the necessary expansion of its fleet of gasoline tank trucks. Three models of gasoline tank trucks are available. Truck Model Capacity (gallons) Purchase Cost Monthly Operating Cost, Including Depreciation Super Tanker 5000 $67,000 $550 Regular Line 2500 $55,000 $425 Econo-Tanker 1000 $46,000 $350 The company estimates that the monthly demand for the region will be 550,000 gallons of gasoline. Because of the size and speed differences of the trucks, the number of de- liveries or round trips possible per month for each truck model will vary. Trip capacities are estimated at 15 trips per month for the Super Tanker, 20 trips per month for the Regular Line, and 25 trips per month for the EconoTanker. Based on maintenance and driver availability, the firm does not want to add more than 15 new vehicles to its fleet. In addition, the company has decided to purchase at least three of the new EconoTankers for use on short-run, low-demand routes. As a final constraint, the company does not want more than half the new models to be Super Tankers. a. If the company wishes to satisfy the gasoline demand with a minimum monthly operating expense, how many models of each truck should be purchased? b. If the company did not require at least three Econo-Tankers and did not limit the number of Super Tankers to at most half the new models, how many models of each truck should be purchased?21) Star Power Company is a power company in the Midwest region of the United States. Star buys and sells energy on the spot market. Star can store power in a high-capacity battery that can store up to 60 kWh (kilowatt hours). During a particular period, Star can buy or sell electricity at the market price known as LMP (Locational Marginal Price). The maximum rate that power can be injected or withdrawn from the battery is 20 kWh per period. Star has forecasted the following LMPs for the next 10 periods: Period LMP ($/kWh) 1 $5 2 $27 3 $2 4 $25 5 $22 6 $29 7 $24 8 $20 9 $61 10 $66 The battery is full at the beginning of period 1; that is, at the start of the planning horizon, the battery contains 60 kWh of electricity. a. Develop a linear programming model Star Power can use to determine when to buy and sell electricity in order to maximize profit over these 10 weeks. What is the maxi- mum achievable profit? b. Your solution to part (a) should result in a battery level of 0 at the end of period 10. Why does this make sense? Modify your model with the requirement that the battery should be full (60 kWh) at the end of period 10. How does this impact the optimal profit? c. To further investigate the impact of requirements on the battery level at the end of period 10, solve your model from part (b) with the constraint on the ending battery level varying from 0 kWh to 60 kWh in increments of 10 kWh. Develop a graph with profit on the vertical axis and required ending battery level on the horizontal axis. Given that Star has not forecasted LMPs for periods 11, 12, and so on, what ending battery level do you recommend that Star use in its optimization model? Chapter 10 3) Tri-County Utilities, Inc., supplies natural gas to customers in a three-county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quantities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows:From To Hamilton Butler Clermont Southern Gas 10 20 15 Northwest Gas 12 15 18 a. Develop a network representation of this problem. b. Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs. c. Describe the distribution plan and show the total distribution cost. d. Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as 100 units. Which supplier should Tri-County contract with to supply the additional capacity? 7) Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in Los Angeles, Tulsa, and Seattle. The following table shows Aggie Power Generation's major residential markets, the annual demand in each market (in megawatts or MW), and the cost to supply electricity to each market from each power generation plant (prices are in $/MW). Distribution Costs City Los Angeles Tulsa Seattle Demand (MW) Seattle $356.25 $593.75 $59.38 950.00 Portland $356.25 $593.75 $178.13 831.25 San Francisco $178.13 $475.00 $296.88 2375.00 Boise $356.25 $475.00 $296.88 593.75 Reno $237.50 $475.00 $356.25 950.00 Bozeman $415.63 $415.63 $296.88 593.75 Laramie $356.25 $415.63 $356.25 1187.50 Park City $356.25 $356.25 $475.00 712.50 Flagstaff $178.13 $475.00 $593.75 1187.50 Durango $356.25 $296.88 $593.75 1543.75 a. If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem? Which cities should be supplied by which power plants? What is the total annual power distribution cost for this solution? b. If at most 4000 MW of power can be supplied by any one of the power plants, what is the optimal solution? What is the annual increase in power distribution cost that results from adding these constraints to the original formulation?