Chapter 2 Problems (HW # 2)

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.'l SELFzest A" SELFzest Problems 63 13. 14. 15. 16. 17. Consider the following linear program: Max 1A + 2B s.t. 1A =5 1B=4 2A +2B =12 A, B=0 a. Show the feasible region. b. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical procedure. RMC, Inc., is a small firm that produces a variety of chemical products. In a particular pro- duction process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of % ton of material 1 and % of material 3. A ton of solvent base is a mixture of ¥2 ton of material 1, ¥ ton of material 2, and %o ton of material 3. After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive produced and $30 for every ton of solvent base produced. RMC’s production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material: Raw Material Amount Available for Production Material 1 20 tons Material 2 5 tons Material 3 21 tons Assuming that RMC is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many tons of each product should be produced, and what is the projected total profit contribution? c. Isthere any unused material? If so, how much? d. Are any of the constraints redundant? If so, which ones? Refer to the Par, Inc., problem described in Section 2.1. Suppose that Par, Inc., management encounters the following situations: a. The accounting department revises its estimate of the profit contribution fer the deluxe bag to $18 per bag. b. A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.) c. New sewing equipment is available that would increase the sewing operation capacity to 750 hours. (Assume that 104 + 98B is the appropriate objective function.) If each of these situations is encountered separately, what is the optimal solution and the total profit contribution? Refer to the feasible region for Par, Inc., problem in Figure 2.13. a. Develop an objective function that will make extreme point (3the optimal extreme point. b. What is the optimal solution for the objective function you selected in part (a)? ¢. What are the values of the slack variables associated with this solution? Write the following linear program in standard form: Max 5A + 2B S.t. 1A 2B =420 2A + 3B = 610 6A 1B = 125 A,B=0
Problems 67 25. 26. 27. Assuming that the company is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many gloves of each model should Kelson manufacture? c¢. What is the total profit contribution Kelson can earn with the given production quantities? d. How many hours of production time will be scheduled in each department? e. What is the slack time in each department? George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%. a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. b. Solve the problem using the graphical solution procedure. The Sea Wharf Restaurant would like to determine the best way to allocate a monthly ad- vertising budget of $1000 between digital advertising and radio advertising. Management decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on digital advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for digital advertising is 50 and the value of the index for spot radio advertising is 80, how should the restaurant allocate its advertising budget in order to maximize the value of total audience exposure? a. Formulate a linear programming model that can be used to determine how the restau- rant should allocate its advertising budget in order to maximize the value of total audi- ence exposure. b. Solve the problem using the graphical solution procedure. Blair & Rosen, Inc. (B&R), is a brokerage firm that specializes in investment portfolios de- signed to meet the specific risk tolerances of its clients. A client who contacted B&R this past week has a maximum of $50,000 to invest. B&R’s investment advisor decides to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue Chip fund. The In- ternet fund has a projected annual return of 12%, whereas the Blue Chip fund has a projected annual return of 9%. The investment advisor requires that at most $35,000 of the client’s funds should be invested in the Internet fund. B&R services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000 is invested in each of the two investment funds, B&R'’s risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally, B&R de- veloped a questionnaire to measure each client’s risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the ques- tionnaire results classified the current client as a moderate investor. B&R recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 240. a. What is the recommended investment portfolio for this client? What is the annual return for the portfolio? b. Suppose that a second client with $50,000 to invest has been classified as an aggressive investor. B&R recommends that the maximum portfolio risk rating for an aggressive investor is 320. What is the recommended investment portfolio for this aggressive in- vestor? Discuss what happens to the portfolio under the aggressive investor strategy. c. Suppose that a third client with $50,000 to invest has been classified as a conservative in- vestor. B&R recommends that the maximum portfolio risk rating for a conservative inves- tor is 160. Develop the recommended investment portfolio for the conservative investor. Discuss the interpretation of the slack variable for the total investment fund constraint.
68 L" SELFrest Chapter 2 An Introduction fo Linear Programming 28. 29. 30. 31. Tom’s, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom’s, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Tom’s, Inc., can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom’s, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom’s contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa. a. Develop a linear programming mode] that will enable Tom’s to determine the mix of salsa products that will maximize the total profit contribution. b. Find the optimal solution. Autolgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at Autolgnite’s plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components each day. For instance, 60% of Buffalo’s production time could be used to produce component 1 and 40% of Buffalo’s production time could be used to produce component 2; in this case, the Buffalo plant would be able to produce 0.6(2000) = 1200 units of component 1 each day and 0.4(1000) = 400 units of component 2 each day. The Dayton plant can produce 600 units of component 1, 1400 units of component 2, or any combination of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Cleveland for assembly of the ignition systems on the following workday. a. Formulate a linear programming model that can be used to develop a daily production schedule for the Buffalo and Dayton plants that will maximize daily production of ignition systems at Cleveland. b. Find the optimal solution. A financial advisor at Diehl Investments identified two companies that are likely candidates for a takeover in the near future. Eastern Cable is a leading manufacturer of flexible cable systems used in the construction industry, and ComSwitch is a new firm specializing in digital switching systems. Eastern Cable is currently trading for $40 per share, and ComSwitch is currently trading for $25 per share. If the takeovers occur, the financial advisor estimates that the price of Eastern Cable will go to $55 per share and ComSwitch will go to $43 per share. At this point in time, the financial advisor has identified ComSwitch as the higher risk alterna- tive. Assume that a client indicated a willingness to invest a maximum of $50,000 in the two companies. The client wants to invest at least $15,000 in Eastern Cable and at least $10,000 in ComSwitch. Because of the higher risk associated with ComSwitch, the financial advisor has recommended that at most $25,000 should be invested in ComSwitch. a. Formulate a linear programming model that can be used to determine the number of shares of Eastern Cable and the number of shares of ComSwitch that will meet the in- vestment constraints and maximize the total return for the investment, b. Graph the feasible region. c. Determine the coordinates of each extreme point. , d. Find the optimal solution. Consider the following linear program: Min 3A + 4B S.t. 1A+3B=6 IA+1B=4 A, B=0
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