C02 ed11 p1-34 - 1 Which of the following mathematical...

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Unformatted text preview: 1. Which of the following mathematical relationships could be found in a linear progrannningj model, and which could not? For the relationships that are unacceptable for linear pro- grams, state why. a. —lx1 + 2x2 — 1x3 5 70 b. 2x1 — 2x3 = 50 c. 1.171 - 235% + 4153 E 10 5: {1. 3x51 + 2x2 — 1x3 2 15 g i 9. 1x1 + 1x2 + 'lx3 = 6 f. 2x1 + 5362 + 1):le “£- 25 2. Find the feasible solution points for the following constraints: a. 4x1 + 2x2 5 16 b. 43c1 + 2x2 2 16 8. 4x1 -|- 2x2 = 16 3. Show a separate graph of the constraint lines and feasible solutions for each of the fol—- lowing constraints: a. 3x1 + 2x2 5 18 b. 12x1 + 81172 a 480 «3. 5x1 + 10562 = 200 Show a separate graph of the constraint lines and feasible solutions for each of the fol- lowing constraints: a. 3x1 — 4x2 3: 60 b. #6331 + 5x2 5 60 C. 53cI -— 2x2 5 O Show a separate graph of the constraint lines and feasible solutions for each of the fol- lowing constraints: a. x1 2 0.25 (x1 + x2) b. 352 S {110(Jr1 + x2) c. x1 E 0.50031 + x2) - Three objective functions for linear programming problems are 7x1 + 10:52, 61:1 + 4x2, and — 4x1 + 7x2. Determine the 510pe of each objective function. Show the graph of each for objective function values equal to 420. Identify the feasible region for the following set of constraints: 1f2):1 + 14x: 3 30 1x1 + 5x2 33: 250 143:1 + 1/2352 5 50 x1, x2 3 0 8. Identify the feasible region for the following set of constraints: 2x1 - hr2 i: 0 “-le + 1.5a:2 :3 200 x1, x2 3 0 9. Identify the feasible region for the following set of constraints: 3x1 -* 2x2 33 0 2x1 — 1.7462 1: 200 1.:rl E 150 x1, x2 3: 0 10. For the linear program 10. 11. For the linear program Max 2x1 + 3x2 SI. 1x1 + 2x2 5 6 5161 + 3x2 5 15 3'- xv):2 _ 0 find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution? Solve the following linear program using the graphical solution procedure. Max 5x1 + 53c2 S.t. 1x1 5 100 1x2 5 80 2x1 + 4x2 5 400 xhxz E 0 12. Consider the following linear programming model: PF.” 13. Consider the following linear program: i 14. Consider the following linear program: Max 3x1 + 3x2 S.t. 2x1 + 4x2 5 12 6x1 + 4x2 5 24 :1"- x1,x2 _ 0 Find the optimal solution using the graphical solution procedure. If the objective function is changed to 2x1 + 6x2, what will the optimal solution be? How many extreme points are there? What are the values of x1 and x2 at each ex- Ti '3‘ treme point? Max 3x] + 2x2 S.t. 2x1 + 2):2 E 8 3x1 + 2x2 5 12 1x1 + 0.5x2 E 3 xvxz E 0 Find the optimal solution using the graphical solution procedure. What is the value of the objective function? Does this linear program have a redundant constraint? If so, what is it? Does the so“ if lution change if the redundant constraint is removed from the model? Explain. - '3': 14. Consider the following linear program: Max 1.1:] + 2.122 SJ. 1x1 3-: 5 I 1.1::2 E 4 2x1 + 2172 = 12 x1,x2 E 0 a. Show the feasible region. I}. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical procedure. 16. Refer to the feasible region for the Par, Inc., problem'in Figure 2.13. a. Develop an objective function that will make extreme point @ the optimal extreme point. I). What is the optimal solution using the objective function you selected in part (a)? c. What are the values of the slack variables associated with this solution? 17. Write the following linear program in standard form: Max 5x1 + 2x2 + 8x3 S.t. 1x] + 2x2 + 1/5173 S 420 2x1 + 3x2 — 1x3 5 610 6x1 _ lx2 -+- 3x3 5 125 x1, x2,x3 2': 0 13. For the linear program Max 4x1 + 1x2 8.1:. 10x1 + 2352 E 30 3x1 + 2x2 *5 12 2361 + 2x2 10 x1, x2 2 0 [AI a. Write this linear program in standard form. b. Find the optimal solution using the graphical solution procedure. c. What are the values of the three slack variables at the optimal solution? - 19. Given the linear program / 19. 20. Given the linear program a. Write the linear program in standard form. b. Find the optimal solution using the graphical solution procedure. c. What are the values of the three slack variables at the optimal solution? Embassy Motorcycles (EM) manufactures two lightweight motorcycles designed for easy handling and safety. The EZ—Rider model has a new engine and a low profile that make it easy to balance. The Lady-Sport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders. Embassy produces the engines for both models at its Des Moines, Iowa, plant. Each EZ—Rider engine requires 6 hours of manufacturing time and each Lady-Sport engine requires 3 hours of manufacturing time. The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period. Embassy’s motorcycle frame supplier can supply as many EZ-Rider frames as needed. However, the Lady-Sport frame is more complex and the supplier can provide only up to 280 Lady-Sport frames for the next production period. Final assembly and testing requires 2 hours for each EZ—Rider model and 2.5 hours for each Lady-Sport model. A maximum of 1000 hours of assembly and testing time are available for the next production period. The company’s accounting department projects a profit contribution of $2400 for each EZ-Rider produced and $1800 for each Lady—Sport produced. a. Formulate a linear programming model that can be used to determine the number of units of each model that shOuld be produced in order to maximize the total contribu- tion to profit. b. Find the optimal solution using the graphical solution procedure. Which constraints are binding? 21. RMC, lnc., 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 2/2, ton of mate- rial 1 and 3/2, of material 3. A ton of solvent base is a mixture of 1/2 ton of material 1, 1/5 ton of material 2, and 9/10 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. Is there any unused material? If so, how much? (I. Are there any redundant constraints? If so, which ones? 22. Kclson Sporting Equipment, Inc, makes two different types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cut- ting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table. Production Time (hours) Cutting ' Packaging Model ' and Sewing Finishing and Shipping Profit/Glove Regular model 1 1/2; 1/3 $5 Catcher’s model 3/3 1/3 ‘ 1/4 $8 Assuming that the company is interested in maximizing the total profit contribution, an- swer 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 listed production quantities? 3:9 1:2,- .2" .'?-“-‘ rig?” "i\ 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 op— tions: (1) a bond fund and (2) a stock fund. The projected returns over the life of the in— vestments 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 investrnent alternatives. b. Find the optimal solution using the graphical solution procedure. The Sea Wharf Restaurant would like to determine the best way to allocate a monthly adver— tising budget of $1000 between newspaper advertising and radio advertising. Management has decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on local newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant has 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 local newspaper 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 au— dience exposure. [1. Find the optimal Solution using the graphical solution procedure. {’25. 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 has decided to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue Chip fund. The Internet fund has a projected annual return of 12%, while the Blue Chip. fund has a projected annual return of 9%. The investment adviser 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 has developed 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 questionnaireresults have classified the current client as a moderate investor. B&R recommends that a client who is a moderate in- vestor 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 re- turn for the portfolio? _ . b. Suppose that a second client with $50,000 to invest has been classified as an aggres- sive investor. B&R recommends that the maximum portfolio risk rating for an aggres- sive investor is 320. What is the recommended investment portfolio for this aggressive investor? 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 ' investor. B&R recommends that the maximum portfolio risk rating for a conservative in— vestor is 1 60. Develop the recommended inve strnent portfolio for the conservative investor. Discuss the interpretation of the slack variable for the total investment fund constraint. 26. 27. 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. 3. Develop a linear programming model that will enable Tom’s to determine the mix of salsa products that will maximize the total profit contribution. I), Find the optimal solution. Autoignite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’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 com— ponent 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 compo- nent 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 Buf— falo and Dayton is sent to "Cleveland for assembly of the ignition systems on the follow— ing work day. 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 ig- nition systems at Cleveland. 0&1} b. Find the optimal solution. -'|_X"-+!. '31 A financial advisor at Diehl Investments identified two companies that are likely candi- "'" dates 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 Com- Switch is currently trading for $25 per share. If the takeovers occur, the financial adviser } t; estimates that the price of Eastern Cable. will go to $55 per share and ComSwitch will go I to $43 per share. At this point in time, the financial adviser identified CemSwitch as the f 53' higher risk alternative. Assume that a client who indicated a willingness to invest a maxi— ‘ mum of $50,000 in the two companies wants to invest at least $15,000 in Eastern Cable 1. - %:.-‘aiid at least 33 10,000 in ComSwitch. Because of the higher risk associated with ComSwitch, the financial adviser recommends 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 CemSwitch 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. 1 £33 . /.i :L_~" a 29. Consider the following linear program: Min 3x1 + 4x2 8.1:. 1x1 + 3x2 2 6 1x1 + 1x2 2 4 x1, x2 3 0 Identify the feasible region and find the optimal solution using the graphical solution pro— cedure. What is the value of the objective function? 30. Identify the three extreme-point solutions for the M&D Chemicals problem (see Section 2.5). Identify the value of the objective function and the values of the slack and surplus variables at each extreme point. 32. Consider the following linear program: Min 2x1 + 2.752 8.1:. 1x1 + 3x2 E 12 3x1 + 1x2_13 1x x1 — Li:2 = 3 xv):2 2 0 a. Shevv the feasible region. b. What are the extreme points of the feasible region? c. Find the Optimal solution using the graphical solution procedure. 33. For the linear program Min 6x] + 4952 31. 2x1 + 1x2 2 12 1x1 + 1x2 2 10 1x2 5 4 x1, x2 2 0 3. Write the linear program in standard form. b. Find the optimal solution using the graphical solution procedure. c. What are the values of the slack and surplus variables? As part of a quality improvement initiative, Consolidated Electronics employees complete a three—day training program on teaming and a two—day training program on problem solv- ing. The manager of quality improvement requested that at least 8 training programs on teaming and at least 10 training programs on problem solving be offered during the next six months. In addition, senior—level management specified that at least 25 training pro- grams must be offered during this period. Consolidated Electronics uses a consultant to teach the training programs. During the next six months, the consultant has 84 days of training time available. Each training program on teaming costs $10,000 and each training program on problem solving costs $8,000. a. Formulate a linear programming model that can be used to determine the number of training programs on teaming and the number of training programs on problem solv- ing that should be offered in order to minimize total cost. 1}. Graph the feasible region. c. Determine the coordinates of each extreme point. {1. Solve for the minimum- -cost solution ...
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