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Six Chapter 6 - Linear Programming Applications (Six Problems) Due: 10:00 pm on 02/19 #1 The Heinlein and Krampf Brokerage firm has just been instructed
Six Chapter 6 - Linear Programming Applications (Six Problems) Due: 10:00 pm on 02/19 #1 The Heinlein and Krampf Brokerage firm has just been instructed by one of its clients to invest $300,000 of her money obtained recently through the sale of land holdings in Ohio. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm selects whatever stocks and bonds they believe are well rated, but within the following guidelines: (a) Municipal bonds should constitute at least 20% of the investment. (b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers. (c) No more than 50% of the amount invested in municipal bonds should be placed in a highrisk, high-yield nursing home stock. Subject to these restraints, the client's goal is to maximize projected return on investments. The analysts at Heinlein and Krampf, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return: Investment Projected Rate of Return (%) Los Angles municipal bonds 5.3 Thompson Electronics, Inc. 6.8 United Aerospace Corp. 8.9 Palmer Drugs 6.4 Happy Days Nursing Homes 11.8 (a) Formulate this portfolio selection problem using LP. (b) Solve this problem. #2 The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3 A.M., 7 A.M., 11 A.M., 3 P.M., 7 P.M., or 11 P.M., and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang's scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day's operation. (Hint: Let equal the number of waiters and busboys beginning work in time period i, where i = 1, 2, 3, 4, 5, 6.) 1 Number of Waiters Period Time And Busboys Required 1 3 A.M. - 7 A.M. 5 2 7 A.M. - 11 A.M. 12 3 11 A.M. - 3 p.M. 15 4 3 p.M. - 7 p.M. 10 5 7 p.M. - 11 p.M. 11 6 11 p.M. - 3 p.M. 4 #3 The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston's historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses' diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table on this page shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes. In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds. Feed Mix Diet Oat Enriched Mineral Minimum Daily Requirement Product Grain Product Requirement (Ingredients) (Units/lb) (Units/lb) (Units/lb) (Units) A 2 3 1 6 B 0.5 1 0.5 2 C 3 5 6 9 D 1 1.5 2 8 E 0.5 0.5 1.5 5 Cost/lb $0.09 $0.14 $0.17 #4 The I. Kruger Paint and Wallpaper Store is a large retail distributor of the Supertrex brand of vinyl wallcoverings. Kruger will enhance its citywide image in Miami if it can outsell other local stores in total number of rolls of Supertrex next year. It is able to estimate the demand function as follows: 2 Number of rolls of Supertrex sold = 20 Dollars spent on advertising + 7.2 Dollars spent on in-store displays + 16 Dollars invested in on-hand wallpaper inventory - 73,000 Percentage markup taken above wholesale cost of a roll The store budgets a total of $21,000 for advertising, in-store displays, and on-hand inventory of Supertrex for next year. It decides it must spend at least $3,000 on advertising; in addition, at least 5% of the amount invested in on-hand inventory should be devoted to displays. Markups on Supertrex have seen at other local stores range from 20% to 45%. Kruger decides that its markup had best be in this range as well. (a) Formulate as an LP problem. (b) Solve the problem. (c) What is the difficulty with the answer? (d) What constraint would you add? #5 Daniel Grady is the financial advisor for a number of professional athletes. An analysis of the long-term goals for many of these athletes has resulted in a recommendation to purchase stocks with some of their income that is set aside for investments. Five stocks have been identified as having very favorable expectations for future performance. Although the expected return is important in these investments, the risk, as measured by the beta of the stock, is also important. (A high value of beta indicates that the stock has a relatively high risk.) The expected return and the betas for five stocks are as follows: Stock 1 2 3 4 5 Expected Return (%) 11.0 9.0 6.5 15.0 13.0 Beta 1.20 0.75 0.65 1.60 1.35 Daniel would like to minimize the beta of the stock portfolio (calculated using a weighted average of the amounts put into the different stocks) while maintaining an expected return of at least 11%. Since future conditions may change, Daniel has decided that no more than 42% of the portfolio should be invested in any one stock. (a) Formulate this as a linear program. (Hint: Define the variables to be the proportion of the total investment that would be put in each stock. Include a constraint that restricts the sum of these variables to be 1.) (b) Solve this problem. What are the expected return and beta for this portfolio? 3 #6 Raptor Fuels produces three grades of gasoline - Regular, Premium, and Super. All of these are produced by blending two types of crude oil Crude A and Crude B. The two types of crude contain specific ingredients which help in determining the octane rating of gasoline. The important ingredients and the costs are contained in the following table: CRUDE A CRUDE B Cost per gallon $0.83 $0.98 Ingredient 1 40% 52% Other ingredients 60% 48% In order to achieve the desired octane ratings, at least 41% of Regular gasoline should be Ingredient 1; at least 44% of Premium gasoline must be Ingredient 1, and at least 48% of Super gasoline must be Ingredient 1. Due to current contract commitments, Raptor Fuels must produce as least 25,000 gallons of Regular, at least 18,000 gallons of Premium, and at least 15,000 gallons of Super. Formulate a linear program that could be used to determine how much of Crude A and Crude B should be used in each of the gasolines to meet the demands at the minimum cost. What is the minimum cost? How much of Crude A and Crude B are used in each gallon of the different types of gasoline? Formulate this problem as a Linear Program and use Solver to find the answers. 4
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