uantitative Analysis BA 452 Homework 3 Questions Homework 3 covers the theory and applications in Lessons I-6 and I-7. This document has four parts: Objectives of doing your homework. Assignment of homework questions, with suggestions about which other questions may help you understand the homework questions. Homework 3 Supplemented Questions listing 32 questions: 4 of them are your homework, others may help you understand the homework questions, and the rest may help you understand fine points. Some supplemental questions refer to a section of a chapter in the textbook (for example, Section 2.1 means Chapter 2 Section 1 of your textbook). Homework 3 Supplemented Answers listing answers to all 32 questions excluding your 4 homework questions. Quantitative Analysis BA 452 Homework 3 Questions Objectives By working through the homework questions and the supplemental questions, you will: 1. Understand what happens in graphical solutions when coefficients of the objective function change. 2. Be able to interpret the range for an objective function coefficient. 3. Understand what happens in graphical solutions when right-hand sides change. 4. Be able to interpret the dual price. 5. Be able to interpret the range for a right-hand side. 6. Learn how to formulate, solve and interpret the solution for linear programs with more than two decision variables. 7. Understand the following terms: sensitivity analysis dual price reduced cost 100 percent rule sunk cost relevant cost Quantitative Analysis BA 452 Homework 3 Questions On the definition of the dual price ... In this course, I defined 1) the dual price of a constraint is the rate of improvement in the value of the optimal solution per unit increase in the right-hand-side constant. For example, if a production line is already operating at its maximum 40 hour limit, the dual price would be the maximum price the manager trying to maximize profit or minimize cost would be willing to pay for operating the line for an additional hour, based on the benefits he would get from that change. That definition of \"dual price\" is used on all your homework and exams, and is consistent with the definition used in the Management Scientist program and in the Anderson text for editions 12 and earlier. However, the Anderson text for editions 13 and later and answers to some of the supplemental questions below use this altered definition 2) the dual price of a constraint is the rate of change in the value of the optimal solution per unit increase in the right-hand-side constant. In the case of minimization problems, definition 2) changes the sign of the dual price from definition 1). For example, reconsider a production line that is already operating at its maximum 40 hour limit. Suppose the maximum price the manager would be willing to pay for operating the line for an additional hour is $100. Suppose the objective of the manager is to minimize cost. In that case, operating the line for an additional hour lowers costs by $100. Under definition 1), the \"dual price\" is positive $100, since running the line improves (lowers) costs by $100. But under definition 2), the \"dual price\" is negative $100, since running the line for an additional hour changes costs by -$100. So watch out for this difference in definitions when considering the supplemental questions below. Quantitative Analysis BA 452 Homework 3 Questions Assignment Questions 15, 18 (skip 18d and 18f), 23, and 26 is your homework assignment. All questions can be answered with notes and computers. To supplement those homework questions, you should consider (but not turn in) the following questions. Questions answered without notes or computers: 3 and 4. Questions answered with notes and computers: 5-32, excluding the homework problems (15, 18, 23, and 26). Tip: Those homework questions and supplementary questions are grouped into sets of similar type. Once you have mastered the questions in a set, you can skip the rest of the questions in that set. Tip: Some of your Exam 1 questions will be variations of some of those homework questions. Quantitative Analysis BA 452 Homework 3 Questions Homework 3 Supplemented Questions 1. Consider the following linear program: 3 + 2 . . 1 + 1 10 3 + 1 24 1 + 2 16 , 0 a. Use the graphical solution procedure to find the optimal solution. b. Assume that the objective function coefficient for A changes from 3 to 5. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. c. Assume that the objective function coefficient for A remains 3, but the objective function coefficient for B changes from 2 to 4. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. d. The computer solution for the linear program in part (a) provides the following objective coefficient range information: Variable A B Objective Coefficient 3.00000 2.00000 Allowable Increase 3.00000 1.00000 Allowable Decrease 1.00000 1.00000 Use this objective coefficient range information to answer parts (b) and (c) Quantitative Analysis BA 452 Homework 3 Questions 2. Consider the linear program in Problem 1. The value of the optimal solution is 27. Suppose that the right-hand side for constraint 1 is increased from 10 to 11. a. Use the graphical solution procedure to find the new optimal solution. b. Use the solution to part (a) to determine the dual value or constraint 1. c. The computer solution for the linear program in Problem 1 provides the following righthand-side range information: Constraint 1 2 3 RHS Value 10.00000 24.00000 16.00000 Allowable Increase 1.20000 6.00000 Infinite Allowable Decrease 2.00000 6.00000 3.00000 d. The dual value for constraint 2 is 0.5. Using this dual value and the right-hand-side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint 2? Quantitative Analysis BA 452 Homework 3 Questions 3. Consider the following linear program: Min s.t. 8X + 12Y 1X + 3Y 9 2X + 2Y 10 6X + 2Y 18 A, B 0 a. Use the graphical solution procedure to find the optimal solution. b. Assume that the objective function coefficient for X changes from 8 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. c. Assume that the objective function coefficient for S remains 8, but the objective function coefficient for Y changes from 12 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. d. The computer solution for the linear program in part (a) provides the following objective coefficient range information: Variable X Y Objective Coefficient 8.00000 12.00000 Allowable Increase 4.00000 12.00000 Allowable Decrease 4.00000 4.00000 How would this objective coefficient range information help you answer parts (b) and (c) prior to re-solving the problem? Quantitative Analysis BA 452 Homework 3 Questions 4. Consider the linear program in Problem 3. The value of the optimal solution is 48. Suppose that the right-hand side for constraint 1 is increased from 9 to 10. a. Use the graphical solution procedure to find the new optimal solution. b. Use the solution to part (a) to determine the dual value for constraint 1. c. The computer solution for the linear program in Problem 3 provides the following right-hand-side range information: Constraint 1 2 3 RHS Value 9.00000 10.00000 18.00000 Allowable Increase 2.00000 8.00000 4.00000 Allowable Decrease 4.00000 1.00000 Infinite What does the right-hand-side range information for constraint 1 tell you about the dual value for constraint 1? d. The dual value for constraint 2 is 3. Using this dual value and the right-hand-side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint 2? Quantitative Analysis BA 452 Homework 3 Questions 5. Refer to the Kelson Sporting Equipment problem (Chapter 2, Problem 24). Letting R=number of regular gloves C=number of catcher's mitts Leads to the following formulation: . . 5 + 8 3 2 + 900 1 2 1 3 + 300 1 8 1 4 + 100 , 0 The computer solution is shown I Figure 3.13. Quantitative Analysis BA 452 Homework 3 Questions a. b. c. d. 6. What is the optimal solution, and what is the value of the total profit contribution? Which constraints are binding? What are the dual values for the resources? Interpret each. If overtime can be scheduled in one of the departments, where would you recommend doing so? Refer to the computer solution of the Kelson Sporting Equipment problem in Figure 3.13 (see Problem 5). a. Determine the objective coefficient ranges. b. Interpret the ranges in part (a). c. Interpret the right-hand-sides ranges. d. How much will the value of the optimal solution improve if 20 extra hours of packaging and shipping time are made available? Quantitative Analysis BA 452 Homework 3 Questions 7. Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows: . . 3 + 5 25 + 50 80,000 0.50 + 0.25 700 1 1000 . . , 0 The computer solution of this problem is shown in Figure 3.14. Quantitative Analysis BA 452 Homework 3 Questions a. What is the optimal solution, and what is the value of the total annual return? b. Which constraints are binding? What is your interpretation of these constraints in terms of the problem? c. What are the dual values for the constraints? Interpret each. d. Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not? 8. Refer to Figure 3.14, which shows the computer solution of Problem 7. a. How much would the return for U.S. Oil have to increase before it would be beneficial to increase the investment in this stock? b. How much would the return for Huber steel have to decrease before it would be beneficial to reduce the investment in this tock? c. How much would the total annual return be reduced if the U.S. Oil maximum were reduced to 900 shares? Quantitative Analysis BA 452 Homework 3 Questions 9. Recall the Tom's, Inc., problem (Chapter 2, Problem 28). Letting W=jars of western Foods Salsa M=jars of Mexico City Salsa Leads to the formulation: 1 + 1.25 . . 5 + 7 4480 3 + 1 2080 2 + 2 1600 , 0 The computer solution is shown in Figure 3.15. a. What is the optimal solution, and what are the optimal production quantities? b. Specify the objective function ranges. c. What are the dual values for each constraint? Interpret each. d. Identify each of the right-hand-side ranges. Quantitative Analysis BA 452 Homework 3 Questions 10. Recall the Innis Investments problem (Chapter 2, Problem 39). Letting S=units purchased in the stock fund M=units purchased in the money market fund leads to the following formulation: . . 8 + 3 50 + 100 1,200,000 5 + 4 60,000 3,000 , 0 The computer solution is shown in Figure 3.16. a. What is the optimal solution, and what is the minimum total risk? b. Specify the objective coefficient ranges. c. How much annual income will be earned by the portfolio? d. What is the rate of return for the portfolio? e. What is the dual value for the funds available constraint? Quantitative Analysis BA 452 Homework 3 Questions f. What is the marginal rate of return on extra funds added to the portfolio? 11. Refer to Problem 10 and the computer solution shown in Figure 3.16. a. Suppose the risk index for the stock fund (the value of Cs) increases from its current value of 8 to 12. How does the optimal solution change, if at all? b. Suppose the risk index for the money market fund (the value of Cm) increases from its current value of 3 to 3.5. how does the optimal solution change, if at all? c. Suppose Cs increases to 12 and Cm increases to 3.5. How does the optimal solution change, if at all? 12. Quality Air conditioning manufactures three hoe air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows: Economy Standard Deluxe Number of Fans 1 1 1 Number of Cooling Coils 1 2 4 Manufacturing Time(hours) 8 12 14 For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows: . . 63 + 95 + 135 1 + 1 + 1 200 Fan motors 1 + 2 + 4 320 Cooling coils 8 + 12 + 14 2400 Manufacturing time , , 0 The computer solution is shown in Figure 3.17. a. b. c. d. What is the optimal solution, and what is the value of the objective function Which constraints are binding? Which constraint shows extra capacity? How much? If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change? Use the information in Figure 3.17 to answer this question. Quantitative Analysis BA 452 Homework 3 Questions 13. Refer to the computer solution of Problem 12 in Figure 3.17 a. Identify the range of optimality for each objective function coefficient. b. Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by 42 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be? c. Identify the range of feasibility for the right-hand-side values. d. If the number of fan motor available for production is increased by 100, will the dual value for that constraint change? Explain. Quantitative Analysis BA 452 Homework 3 Questions 14. Digital Controls, Inc. (DCI), manufactures two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B has an accuracy of plus or minus 3 miles per hour. For the next week, the company has orders for 100 units of model A and 150 units of model B. Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires 4 minutes of injection-molding time and 6 minutes of assembly time. Each model B case requires 3 minutes of injection-molding time and 8 minutes of assembly time. Each model B case requires 3 minutes of injection-molding time and 8 minutes of assembly time. For next week, the Newark plant has 600 minutes of injectionmolding time available and 1080 minutes of assembly time available. The manufacturing cost is $10 per case for model A and $6 per case for model B. Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise. The purchase cost is $14 for each model A case and $9 for each model B case. Management wants to develop a minimum cost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should be purchased. The following decision variables were used to formulate a linear programming model for this problem: AM=number of cases of model A manufactured BM=number of cases of model B manufactured AP=number of cases of model A purchased BP=number of cases of model B purchased The linear programming model that can be used to solve this problem is as follows: . . 10 + 6 + 14 + 9 1 + + 1 + = 100 1 + 1 = 150 4 + 3 600 6 + 8 1080 , , , 0 Quantitative Analysis BA 452 Homework 3 Questions The computer solution is shown in Figure 3.18. a. b. c. d. What is the optimal solution and what is the optimal value of the objective function? Which constraints are binding? What are the dual values? Interpret each. If you could change the right-hand side of one constraint by one unit, which one would you choose? Why? Quantitative Analysis BA 452 Homework 3 Questions 15. Refer to the computer solution to Problem 14 in Figure 3.18. a. Interpret the ranges of optimality for the objective function coefficients. b. Suppose that the manufacturing cost increases to $11.20 per case for model A. What is the new optimal solution? c. Suppose that the manufacturing cost increases to $11.20 per case for model A and the manufacturing cost for model B decreases to $5 per unit. Would the optimal solution change? Quantitative Analysis BA 452 Homework 3 Questions 16. Tucker Inc. produces high-quality suits and sport coats for men. Each suit requires 1.2 hours of cutting time and 0.7 hours of sewing time, uses 6 yards of material, and provides a profit contribution of $190. Each sport coat requires 0.8 hours of cutting time and 0.6 hours of sewing time, uses 4 yards of material, and provides a profit contribution of $150. For the coming week, 200 hours of cutting time, 18- hours of sewing time, and 1200 yards of fabric are available. Additional cutting and sewing time can be obtained by scheduling overtime for these operations. Each hour of overtime for the cutting operation increase the hourly cost by $15, and each hour of overtime for the sewing operation increase the hourly cost by $10. A maximum of 100 hours of overtime can be scheduled. Marketing requirements specify a minimum production of 100 suits and 75 sport coats. Let S=number of suits produced SC=number of sport oats produced D1=hours of overtime for the cutting operation D2=hours of overtime for the sewing operation The computer solution is shown in Figure 3.19. a. What is the optimal solution, and what is the total profit? What is the plan for the use of overtime? b. A price increase fir suits is being considered that would result in a profit contribution of $210 per suit. If this price increase is undertaken, how will the optimal solution change? c. Discuss the need for additional material during the coming week. If a rush order for material can be placed at the usual price plus an extra $8 per yard for handling, would you recommend the company consider placing a rush order for material? What is the maximum price Tucker would be willing to pay for an additional yard of material? How many additional yards of material should Tucker consider ordering? d. Suppose the minimum production requirement for suits is lowered to 75. Would this change help or hurt profit? Explain. Quantitative Analysis BA 452 Homework 3 Questions Quantitative Analysis BA 452 Homework 3 Questions 17. The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the highalloy steel will be available next quarter. In addition, Deegan estimates that 20000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRB. The linear programming model for this problem is as follows: . . 200 + 20 + 25 40,000 40 + 100 120,000 60 + 40 96,000 , 0 The computer solution is shown in Figure 3.20. a. What are the optimal solution and the total profit contribution/ b. Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain. c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain. d. Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain. e. If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain. Quantitative Analysis BA 452 Homework 3 Questions Quantitative Analysis BA 452 Homework 3 Questions 18. Davison Electronics manufactures two LCD television monitors, identified as model A and model B. Each model has its lowest possible production cost when produced on Davison's new production line. However, the new production line does not have the capacity to handle the total production of both models. As a result, as least some of the production must be routed to a higher-cost, old production line. The following table shows the minimum production requirements for next month, the production line table shows the minimum production requirements for next month, the production line capacities in units per month, and the production cost per unit for each production line: Model New Line A B Production Line Capacity $30 $25 80,000 Let: Production Cost per Unit Old Line $50 $40 60,000 Minimum Production Requirements 50,000 70,000 AN= Units of model A produced on the new production line AO= Units of model A produced on the old production line BN = Units of model B produced on the new production line BO= Units of model B produced on the old production line Davison's objective is to determine the minimum cost production plan. The computer solution is shown below. a. Formulate the linear programming model for this problem using the following four constraints: i. Constraint 1: Minimum production for model A ii. Constraint 2: Minimum production for model B iii. Constraint 3: Capacity of the new production line iv. Constraint 4: Capacity of the old production line b. Using computer solution in Figure 3.21, what is the optimal solution, and what is the total production cost associated with this solution? c. Which constraints are binding? Explain. d. The production manager noted that the only constraint with a positive dual values is the constraint on the capacity of the new production line. The manager's interpretation of the dual value was that a one-unit increase in the right-hand side of this constraint would actually increase the total production cost by $15 per unit. Do you agree with this interpretation? Would an increase in capacity for the new production line be desirable? Explain. e. Would you recommend increasing the capacity of the old production line? Explain. Quantitative Analysis BA 452 Homework 3 Questions f. The production cost for model A on the old production line is $50 per unit. How much would this cost have to change to make it worthwhile to produce model A on the old production line? Explain. g. Suppose that the minimum production requirement for model B is reduced from 70,000 units to 60,000 units. What effect would this change have on the total production cost? Explain. Optimal Objective Value Variable AN AO BN BO = 3850000.00000 Value 50000.00000 0.0000 30000.00000 40000.00000 Reduced Cost 0.00000 5.00000 0.00000 0.00000 Constraint 1 2 3 4 Slack/Surplus 0.00000 0.00000 0.00000 20000.00000 Dual Value 45.00000 40.00000 -15.00000 0.00000 OBJECTIVE COEFFICIENT RANGES Variable Objective Coefficient AN 30.00000 AO 50.00000 BN 25.00000 BO 40.00000 Allowable Increase 5.00000 Infinite 15.00000 5.00000 Allowable Decrease Infinite 5.00000 5.00000 15.00000 RIGHT HAND SIDE RANGES Constraint RHS Value 1 50000.00000 2 70000.00000 3 80000.00000 4 60000.00000 Allowable Increase 20000.00000 20000.00000 40000.00000 Infinite Allowable Decrease 40000.00000 40000.00000 20000.00000 20000.00000 Quantitative Analysis BA 452 Homework 3 Questions 19. Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit are as follows: Category Profit/unit Machine 1 time/unit Machine 2 time/unit Product 1 $30 0.5 1.0 Product 2 $50 2.0 1.0 Product 3 $20 0.75 0.5 Two operators are required for machine 1; thus, 2 hours of labor must ne scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced. a. How many units of each product should be produced to maximize the total profit contribution? What is the projected weekly profit associated with your solution? b. How many hours of production time will be scheduled on each machine? c. What is the value of an additional hour of labor? d. Assume that labor capacity can be increased to 120 hours. Would you be interested in using the additional 20 hours available for this resource? Develop the optimal product mix assuming the extra hours are made available. 20. a. b. c. d. e. Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 7% for home loans, 12% for personal loans, and 9% for automobile loans. The bank's planning committee has decided that at least 40% of new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans. Formulate a linear programming model that can be used to determine the amount of finds ASB should allocate to each type of loan in order to maximize the total annual return for the new funds. How much should be allocated to each type of loan? What is the total annual return? What is the annual percentage return? If the interest rate on home loans increased to 9%, would the amount allocated to each type of loan change? Explain. Suppose the total amount of new funds available was increased by $10,000. What effect would this have on the total annual return? Explain. Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change? How much would the annual percentage return change? Quantitative Analysis BA 452 Homework 3 Questions 21. Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows: Room Type I Type II Super Saver $30 $20 Deluxe $35 $30 Business -$40 Type I rooms do not have Internet access and are not available for the Business rental class. Round Tree's management makes a forecast of the demand by rental class for each m=night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rental sin the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms. a. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. Is the demand by any rental class not satisfied? Explain. b. How many reservations can be accommodated in each rental class? c. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5. Should this incentive be offered? d. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? Why? e. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change? Quantitative Analysis BA 452 Homework 3 Questions 22. Industrial Design has been awarded a contract design label for a new wine produced by Lake View Winery. The company estimates that 150 hours will be required to complete the project. The firm's three graphics designers available for assignment to this project are Lisa, a senior designer and team leader; David, a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management specified that Lisa must be assigned at least 40% of the total number of hours assigned to the two senior designers. To provide label-designing experience for Sarah, Sarah must be assigned at least 15% of the total project time. However, the total number of hours assigned to Sarah must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Lisa has a maximum of 50 hours available to work on this project. Hourly wage rates are $30 for Lisa, $25 for David, and $18 for Sarah. a. Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project in order to minimize total cost. b. How many hours should each graphic designer be assigned to the project? What is the total cost? c. Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? Explain. d. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain. 23. Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a sharper and a grinder. The times (in minutes) required on each machine are as follows: Component 1 2 3 Machine Sharper 6 4 4 Grinder 4 5 2 The sharper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1,2, and 3 are $8, $6, and $9, respectively. a. Formulate and solve for the recommended production quantities. b. What are the objective coefficient ranges for the three components? Interpret these ranges for company management. c. What are the right-hand-side ranges? Interpret these ranges for company management. d. If more time could be made available on the grinder, how much would it be worth? e. If more units of component 3 can be sold by reducing the sales price by $4, should the company reduce the price? Quantitative Analysis BA 452 Homework 3 Questions 24. National Insurance Associates carries an investment portfolio of stocks, bonds, and other investment alternatives. Currently $200,000 of funds are available and must be considered fro new investment opportunities. The four stock options National is considering and the relevant financial data are as follows: Stock A B C D Price per share $100 $50 $80 $40 Annual rate of return 0.12 0.08 0.06 0.10 Risk measure per dollar invested 0.10 0.07 0.05 0.08 The risk measure indicates the relative uncertainty associated with the stock in terms of it realizing the projected annual return; higher values indicate greater risk. The risk measures are provided by the firm's top financial advisor. National's top management has stipulated the following investment guidelines: The annual rate of return for the portfolio must be at least 9% and o one stock can account of more than 50% of the total dollar investment. a. Use linear programming to develop an investment portfolio that minimizes risk. b. If the firm ignores risk and uses a maximum return-on-investment strategy, what is the investment portfolio? c. What is the dollar difference between the portfolios in parts (a) and (b)? Why might the company prefer the solution developed in part (a)? Quantitative Analysis BA 452 Homework 3 Questions 25. Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here. Cabinetmaker 1 Cabinetmaker 2 Cabinetmaker 3 Hours required to complete all the oak cabinets Hours required to complete all the cherry cabinets Hours available 50 42 30 60 48 35 40 30 35 Cost per hour $36 $42 $55 For example, Cabinetmaker 1 estimates it will take 50 hours to complete all the oak cabinets and 60 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 40 hours available for the final finishing operations. Thus, Cabinetmaker 1 can only complete 40/50 = 0.80, or 80%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/60 = 0.67, or 67%, of the cherry cabinets if it worked only on cherry cabinets. a. Formulate a linear programming model that can be used to determine the percentage of the oak cabinets and the percentage of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects. b. Solve the model formulated in part (a). What percentage of the oak cabinets and what percentage of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? Explain. d. If Cabinetmaker 2 has additional hours available, would the optimal solution change? Explain. e. Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain. Quantitative Analysis BA 452 Homework 3 Questions 26. Benson Electronics manufactures three components used to produce cell telephones and other communication devices. In a given production period, demand for the three components may exceed Benson's manufacturing capacity. Tin this case, the company meets demand by purchasing the components from another manufacturer at an increased cost per unit. Benson's manufacturing cost per unit and purchasing cost per unit for the three components are as follows: Source Component 1 Component 2 Component 3 Manufacture $4.50 $5.00 $2.75 Purchase $6.50 $8.80 $7.00 Manufacturing times in minutes per unit for Benson's three departments are as follows: Department Component 1 Component 2 Component 3 Production 2 3 4 Assembly 1 1.5 3 Testing & Packaging 1.5 2 5 For instance, each unit for component 1 that Benson manufactures requires 2 minutes of production time, 1 minute of assembly time, and 1.5 minutes of testing and packaging time. For the next production period, Benson has capacities of 360 hours in the production department, 250 hours in the assembly department, and 300 hours in the testing and packaging department. a. Formulate a linear programming model that can be used to determine how many units of each component to manufacture and how many units of each component to purchase. Assume that component demands that must be satisfied are 6000 units for component 1, 4000 units for component 2, and 3500 units for component 3. The objective is to minimize the total manufacturing and purchasing costs. b. What is the optimal solution? How many units of each component should be manufactured and how many units of each component should be purchased? c. Which departments are limiting Benson's manufacturing quantities? Use the dual value to determine the value of an extra hour in each of these departments. d. Suppose that Benson had to obtain one additional unit of component 2. Discuss what the dual value for the component 2 constraints tells us about the cost to obtain the additional unit. Quantitative Analysis BA 452 Homework 3 Questions 27. Golf Shafts, Inc. (GSI), produces graphite shafts for several manufacturers of golf clubs. Two GSI manufacturing facilities, one located in San Diego and the other in Tampa, have the capability to produce shafts in varying degrees of stiffness, ranging from regular models used primarily by average golfers to extra stiff models used primarily by low-handicap and professional golfers. GSI just received a contract for the production of 200,000 regular shafts for previous orders, neither plant has sufficient capacity by itself to fill the new order. The San Diego plant can produce up to a total of 180,000 shafts. Because the equipment differences at each of the plants and differing labor costs, the per-unit production costs vary as shown here: San Diego Cost Tampa Cost Regular shaft $5.25 $4.95 Stiff shaft $5.45 $5.70 a. Formulate a linear programming model to determine how GSI should schedule production for the new order in order to minimize the total production cost. b. Solve the model that you developed in part (a). c. Suppose that some of the previous orders at the Tampa plant could be rescheduled in order to free up additional capacity for the new order. Would this option be worthwhile? Explain. d. Suppose that the cost to produce a stiff shaft in Tampa had been incorrectly computed, and that the correct cost is $5.30 per shaft. What effect, if any, would the correct cost have on the optimal solution developed in part (b)? What effect would it have on the total production cost? Quantitative Analysis BA 452 Homework 3 Questions 28. The Pfeiffer Company manages approximately $15 million for clients. For each client, Pfeiffer choose 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 to each client. Here's how the system works for Dennis Hartman, 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% 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 be 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. Quantitative Analysis BA 452 Homework 3 Questions 29. La Jolla Beverage Products is considering producing a wine cooler that would be a blend of white wine, a rose wine, and fruit juice. To meet taste specifications, the wine cooler must consist of at least 50% white wine, at least 20% and no more than 30% rose wine, and exactly 20% fruit juice. La Jolla purchases wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period, 10,000 gallons of white wine and 8,000 gallons of rose wine can be purchased; an unlimited amount of fruit juice can be ordered. The costs for the wine are $1.00 per gallon for the white and $1.50 per gallon for the rose; the fruit juice can be purchased for $0.50 per gallon. La Jolla Beverage Products can sell all of the wine cooler they produce for $2.50 per gallon. a. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation? Explain. b. Formulate a linear program to determine the blend of the three ingredients that will maximize the total profit contribution. Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution they will realize from this blend. c. If La Jolla could obtain additional amounts of the white wine, should they do so? If so, how much should they be willing to pay for each additional gallon, and how many additional gallons would they want to purchase? d. If La Jolla Beverage Products could obtain additional amounts of the rose wine, should they do so? If so, how much should they be willing to pay for each additional gallon, and how many additional gallons would they want to purchase? e. Interpret the dual value for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this dual value? f. Interpret the dual value for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this dual value? Quantitative Analysis BA 452 Homework 3 Questions 30. The program manager for Channel 10 would like to determine the best way to allocate the time for the 11:00-11:30 evening news broadcast. Specifically, she would like to determine the number of minutes of broadcast time to devote to local news, national news, weather, and sports. Over the 30minute broadcast, 10 minutes are set aside for advertising. The station's broadcast policy states that at least 15% of the time available should be devoted to local news coverage; the time devoted to local news or national news must be at least 50% of the total broadcast time; the time devoted to the weather segment must be less than or equal to the time devoted to the sports segment; the time devoted to the sports segment should be no longer than the total time spent on the local and national news; and at least 20% of the time should be devoted to the weather segment. The production costs per minute are $300 for local news, $200 for national news, $100 for weather, and $100 for sports. a. Formulate and solve a linear program that can determine how the 20 available minutes should be used to minimize the total cost of producing the program. b. Interpret the dual value for the constraint corresponding to the available time. What advice would you give the station manager given this dual value? c. Interpret the dual value for the constraint corresponding to the requirement that at least 15% of the available time should be devoted to local coverage. What advice would you give the station manager given this dual value? d. Interpret the dual value for the constraint corresponding to the requirement that the time devoted to the local and the national news must be at least 50% of the total broadcast time. What advice would you give the station manager given this dual value? e. Interpret the dual value for the constraint corresponding to the requirement that the time devoted to the weather segment must be less than or equal to the time devoted to the sports segment. What advice would you give the station manager given this dual value? Quantitative Analysis BA 452 Homework 3 Questions 31. Gulf Coast Electronics is ready to award contracts for printing their annual report. For the past several years, the four-color annual report has been printed by Johnson Printing and Lakeside Litho. A new firm, Benson Printing, inquired into the possibility of doing a portion of the printing. The quality and service level provided by Lakeside Litho has been extremely high; in fact, only 0.5% of their reports have had to be discarded because of quality problems. Johnson Printing has also had a high quality level historically, producing an average of only 1% unacceptable reports. Because Gulf Coast Electronics has had no experience with Benson Printing, they estimated their defective rate to be 10%. Gulf Coast would like to determine how many reports should be printed by each firm to obtain 75,000 acceptable-quality reports. To ensure that Benson Printing will receive some of the contract, management specified that the number of reports awarded to Benson Printing must be at least 10% of the volume given to Johnson Printing. In addition, the total volume assigned to Benson Printing, Johnson Printing, and Lakeside Litho should not exceed 30,000, 50,000, and 50,000 copies, respectively. Because of the long-term relationship with lakeside Litho, management also specified that at least 30,000 reports should be awarded to Lakeside Litho. The cost per copy is $2.45 for Benson Printing, $2.50 for Johnson Printing, and $2.75 for Lakeside Litho. a. Formulate and solve a linear program for determining how many copies should be assigned to each printing firm to minimize the total cost of obtaining 75,000 acceptable quality reports. b. Suppose that the quality level for Benson Printing is much better than estimated. What effect, if any, would this quality level have? c. Suppose that management is willing to reconsider their requirement that the Lakeside Litho be awarded at least 30,000 reports. What effect, if any, would this consideration have? Quantitative Analysis BA 452 Homework 3 Questions 32. PhotoTech, Inc., a manufacturer of rechargeable batteries for digital cameras, signed a contract with a digital photography company to produce three different lithium-ion battery packs for a new line of digital cameras. The contract calls for the following: Battery Pack Production Quantity PT-100 200,000 PT-200 100,000 PT-300 150,000 PhotoTech can manufacture the battery packs at manufacturing plants located in the Philippines and Mexico. The unit cost of the battery packs differs at the two plants because of differences in production equipment and wage rates. The unit costs for each battery pack at each manufacturing plant are as follows: Plant Product Philippines Mexico PT-100 $0.95 $0.98 PT-200 $0.98 $1.06 PT-300 $1.34 $1.15 The PT-100 and PT-200 battery packs are produced using similar production equipment available at both plants. However, each plant has a limited capacity for the total number of PT-100 and PT-200 battery packs produced. The combined PT-100 and PT-200 production capacities are 175,000 units at the Philippines plant and 160,000 units at the Mexico plant. The PT-300 production capacities are 75,000 units at the Philippines plant and 100,000 units at the Mexico plant. The cost of shipping from the Philippines plant is $0.18 per unit, and the cost of shipping from the Mexico plant is $0.10 per unit, and the cost of shipping from the Mexico plant is $0.10 per unit. a. Develop a linear program that PhotoTech can use to determine how many units of each battery pack to produce at each plant in order to minimize the total production and shipping cost associated with the new contract. b. Solve the linear program developed in part (a) to determine the optimal production plan. c. Use sensitivity analysis to determine how much the production and/or shipping cost per unit would have to change in order to produce additional units of the PT-100 in the Philippines plant. d. Use sensitivity analysis to determine how much the production and/or shipping cost per unit would have to change in order to produce additional units of the PT-200 in the Mexico plant. Quantitative Analysis BA 452 Homework 3 Questions Homework 3 Supplemented Answers 1. a. b. The same extreme point, A = 7 and B = 3, remains optimal. The value of the objective function becomes 5(7) + 2(3) = 41 c. A new extreme point, A = 4 and B = 6, becomes optimal. The value of the objective function becomes 3(4) + 4(6) = 36. d. The objective coefficient range for variable A is 2 to 6. Since the change in part (b) is within this range, we know the optimal solution, A = 7 and B = 3, will not change. The objective coefficient range for variable B is 1 to 3. Since the change in part (c) is outside this range, we have to re-solve the problem to find the new optimal solution. Quantitative Analysis BA 452 Homework 3 Questions 2. a. b. The value of the optimal solution to the revised problem is 3(6.5) + 2(4.5) = 28.5. The one-unit increase in the right-hand side of constraint 1 has improved the value of the optimal solution by 28.5 - 27 = 1.5. Thus, the dual value for constraint 1 is 1.5. c. The right-hand-side range for constraint 1 is 8 to 11.2. As long as the right-hand side stays within this range, the dual value of 1.5 is applicable. d. The improvement in the value of the optimal solution will be 0.5 for every unit increase in the right-hand side of constraint 2 as long as the right-hand side is between 18 and 30. Quantitative Analysis BA 452 Homework 3 Questions 3. a. b. The same extreme point, X = 3 and Y = 2, remains optimal. The value of the objective function becomes 6(3) + 12(2) = 42. c. A new extreme point, X = 2 and Y = 3, becomes optimal. The value of the objective function becomes 8(2) + 6(3) = 34. d. The objective coefficient range for variable X is 4 to 12. Since the change in part (b) is within this range, we know that the optimal solution, X = 3 and Y = 2, will not change. The objective coefficient range for variable Y is 8 to 24. Since the change in part (c) is outside this range, we have to re-solve the problem to find the new optimal solution. Quantitative Analysis BA 452 Homework 3 Questions 4. a. b. The value of the optimal solution to the revised problem is 8(2.5) + 12(2.5) = 50. Compared to the original problem, the value of the optimal solution has increased by 50 - 48 = 2. Thus, the dual value is 2. c. The right-hand side range for constraint 1 is 5 to 11. As long as the right-hand side stays within this range, the dual value of 2 is applicable. Since increasing the right-hand side does not improve the value of the optimal solution, decreasing the right-hand side of constraint 1 would b desirable. d. As long as the right-hand side of constraint 2 is between 9 and 18, a unit increase in the righthand side will cause the value of the optimal solution to increase by 3. 5. a. Regular Glove = 500 Catcher's Mitt = 150 Value = 3700 b. The finishing and packaging and shipping constraints are binding. c. Cutting and Sewing = 0 Finishing = 3 Packaging and Shipping = 28 Additional finishing time is worth $3 per unit and additional packaging and shipping time is worth $28 per unit. d. In the packaging and shipping department. Each additional hour is worth $28. Quantitative Analysis BA 452 Homework 3 Questions 6. a. Variable Regular Glove Catcher's Mitt Objective Coefficient Range 4 to 12 3.33 to 10 b. As long as the profit contribution for the regular glove is between $4.00 and $12.00, the current solution is optimal. As long as the profit contribution for the catcher's mitt stays between $3.33 and $10.00, the current solution is optimal. The optimal solution is not sensitive to small changes in the profit contributions for the gloves. c. The dual values for the resources are applicable over the following ranges: Constraint Cutting and Sewing Finishing Packaging 7. d. Amount of increase = (28) (20) = $560 a. U = 800 H = 1200 Estimated Annual Return Right-Hand-Side Range 725 to No Upper Limit 133.33 to 400 75 to 135 = $8400 b. Constraints 1 and 2. All funds available are being utilized and the maximum permissible risk is being incurred. c. Constraint Funds Avail. Risk Max U.S. Oil Max 8. c. Dual Values 0.09 1.33 0 d. No, the optimal solution does not call for investing the maximum amount in U.S. Oil. a. By more than $7.00 per share. b. By more than $3.50 per share. None. This is only a reduction of 100 shares and the allowable decrease is 200. management may want to address. Quantitative Analysis BA 452 Homework 3 Questions 9. a. Optimal solution calls for the production of 560 jars of Western Foods Salsa and 240 jars of Mexico City Salsa; profit is $860. b. Variable Western Foods Salsa Mexico City Salsa Objective Coefficient Range 0.893 to 1.250 1.000 to 1.400 c. Constraint Dual Value Interpretation 1 0.125 One more ounce of whole tomatoes will increase profits by $0.125 2 0.000 Additional ounces of tomato sauce will not improve profits; slack of 160 ounces. 3 0.187 One more ounce of tomato paste will increase profits by $0.187 d. Constraint 1 2 3 10. a. Right-Hand-Side Range 4320 to 5600 1920 to No Upper Limit 1280 to 1640 S = 4000 M = 10,000 Total risk = 62,000 b. Variable S M Objective Coefficient Range 3.75 to No Upper Limit No Upper Limit to 6.4 c. 5(4000) + 4(10,000) = $60,000 d. 60,000/1,200,000 = 0.05 or 5% e. -0.057 risk units f. -0.057(100) = -5.7% Quantitative Analysis BA 452 Homework 3 Questions 11. 12. a. No change in optimal solution; there is no upper limit for the range of optimality for the objective coefficient for S. b. No change in the optimal solution; the objective coefficient for M can increase to 6.4. c. The optimal solution does not change. a. E = 80, S = 120, D = 0 Profit = $16,440 b. Fan motors and cooling coils c. Labor hours; 320 hours available. d. Objective function coefficient range of optimality No lower limit to 159. Since $150 is in this range, the optimal solution would not change. 13. a. Range of optimality E 47.5 to 75 S 87 to 126 D No lower limit to 159. b. The optimal solution does not change, but the change in total profit will be: E 80 unit @ + $6 $48 = 0 S 120 unit @ - $2 -240 = $24 0 Profit = $16,440 + 240 = 16,680. c. Range of feasibility Constraint 1 Constraint 2 Constraint 3 d. 160 to 180 200 to 400 2080 to No Upper Limit Yes, the dual value will change since 100 is greater than the allowable increase of 80. Quantitative Analysis BA 452 Homework 3 Questions 14. a. Manufacture 100 cases of model A Manufacture 60 cases of model B Purchase 90 cases of model B Total Cost = $2170 b. Demand for model A Demand for model B Assembly time c. Constraint 1 2 3 4 Dual Value 12.25 9.0 0 -.375 If demand for model A increases by 1 unit, total cost will increase by $12.25 If demand for model B increases by 1 unit, total cost will increase by $9.00 If an additional minute of assembly time is available, total cost will decrease by $.375 d. The assembly time constraint. Each additional minute of assembly time will decrease costs by $.375. Note that this will be true up to a value of 1133.33 hours. Some students may say that the demand constraint for model A should be selected because decreasing the demand by one unit will decrease cost by $12.25. But, carrying this argument to the extreme would argue for a demand of 0. Quantitative Analysis BA 452 Homework 3 Questions 15. a. Decision toRanges My answer thisof Optimality homework question will Variable No lower limit to 11.75 go here.AM To find your answer, you may want BM 3.667 to 9 12.25 to No Upper to studyAPthe answers to some of the similar Limit BP 6 to 11.333 questions. Provided a single change of an objective function coefficient is within its above range, the optimal solution AM = 100, BM = 60, AP = 0, and BP = 90 will not change. b. This change is within the allowable increase. The optimal solution remains AM = 100, BM = 60, AP = 0, and BP = 90. The $11.20 - $10.00 = $1.20 per unit cost increase will increase the total cost to $2170 = $1.20(100) = $2290. c. Yes, it is necessary to solve the model again. Resolving the problem provides the new optimal solution: AM = 0, BM = 135, AP = 100, and BP = 15; the total cost is $22,100. Quantitative Analysis BA 452 Homework 3 Questions 16. a. The optimal solution calls for the production of 100 suits and 150 sport coats. Forty hours of cutting overtime should be scheduled, and no hours of sewing overtime should be scheduled. The total profit is $40,900. b. The objective coefficient range for suits shows and upper limit of $225. Thus, the optimal solution will not change. But, the value of the optimal solution will increase by ($210-$190)100 = $2000. Thus, the total profit becomes $42,990. c. The slack for the material coefficient is 0. Because this is a binding constraint, Tucker should consider ordering additional material. The dual value of $34.50 is the maximum extra cost per yard that should be paid. Because the additional handling cost is only $8 per yard, Tucker should order additional material. Note that the dual value of $34.50 is valid up to 1333.33 -1200 = 133.33 additional yards. d. The dual value of $35 for the minimum suit requirement constraint tells us that lowering the minimum requirement by 25 suits will increase profit by $35(25) = $875. 17. a. Produce 1000 units of model DRB and 800 units of model DRW Total profit contribution = $424,000 b. The dual value for constraint 1 (steel available) is 8.80. Thus, each additional pound of steel will increase profit by $8.80. At $2 per pound Deegan should purchase the additional 500 pounds. Note: the upper limit on the right hand side range for constraint 1 is approximately 40,909. Thus, the dual value of $8.80 is applicable for an increase of as much as 909 pounds. c. Constraint 3 (assembly time) has a slack of 4000 hours. Increasing the number of hours of assembly time is not worthwhile. d. The objective coefficient range for model DRB shows a lower limit of $112. Thus, the optimal solution will not change; the value of the optimal solution will be $175(1000) + $280(800) = $399,000. e. An increase of 500 hours or 60(500) = 30,000 minutes will result in 150,000 minutes of manufacturing time being available. Because the allowable increase is 40,000 minutes, the dual value of $0.60 per minute will not change. Quantitative Analysis BA 452 Homework 3 Questions 18. a. The linear programming model is as follows: b. Optimal solution: My answer to this homework question will Min 30AN + 50AO + 25BN + 40BO go here. To find your answer, you may want s.t. AN + AO 50,000 to study the answers to some the similar BN + BO of 70,000 AN + BN 80,000 questions. AO + BO 60,000 Model A Model B New Line 50,000 30,000 Old Line 0 40,000 Total Cost $3,850,000 c. The first three constraints are binding because the values in the Slack/Surplus column for these constraints are zero. The fourth constraint, with a slack of 0 is nonbinding. d. The dual value for the new production line capacity constraint is -15. Because the dual value is negative, increasing the right-hand side of constraint 3 will cause the objective function value to decrease. Thus, every one unit increase in the right hand side of this constraint will reduce the total production cost by $15. In other words, an increase in capacity for the new production line is desirable. e. Because constraint 4 is not a binding constraint, any increase in the production line capacity of the old production line will have no effect on the optimal solution. Thus, there is no benefit in increasing the capacity of the old production line. f. The reduced cost for Model A made on the old production line is 5. Thus, the cost would have to decrease by at least $5 before any units of model A would be produced on the old production line. g. The right hand side range for constraint 2 shows an allowable decrease of 20,000. Thus, if the minimum production requirement is reduced 10,000 units to 60,000, the dual value of 40 is applicable. Thus, total cost would decrease by 10,000(40) = $400,000. Quantitative Analysis BA 452 Homework 3 Questions 19. a. Let P1 = units of product 1 P2 = units of product 2 P3 = units of product 3 Max s.t. 30P1 0.5P1 P1 2P1 0.5P1 -0.2P1 P1, P2, P3 0 + + + + - 50P2 2P2 P2 5P2 0.5P2 0.2P2 + + + + + 20P3 0.75P3 0.5P3 2P3 0.5P3 0.8P3 40 40 100 0 0 Machine 1 Machine 2 Labor Max P1 Min P3 The optimal solution is Optimal Objective Value 1250.00000 Value 25.00000 0.00000 25.00000 Reduced Cost 0.00000 -7.50000 0.00000 Slack/Surplus 8.75000 2.50000 0.00000 0.00000 15.00000 Dual Value 0.00000 0.00000 12.50000 10.00000 0.00000 Objective Coefficient 30.00000 50.00000 20.00000 Allowable Increase Infinite 7.50000 10.00000 Allowable Decrease 10.00000 Infinite 4.28571 RHS Value 40.00000 40.00000 100.00000 0.00000 Allowable Increase Infinite Infinite 6.66667 5.00000 Allowable Decrease 8.75000 2.50000 100.00000 25.00000 Variable P1 P2 P3 Constraint 1 2 3 4 5 Quantitative Analysis BA 452 Homework 3 Questions 0.00000 15.00000 b. Machine Hours Schedule: Machine 1 31.25 Hours Machine 2 37.50 Hours c. $12.50 d. Increase labor hours to 120; the new optimal product mix is P1 = 24 P2 = 8 P3 = 16 Profit = $1440 Infinite Quantitative Analysis BA 452 Homework 3 Questions 20. a. Let H P A Max s.t. b. = amount allocated to home loans = amount allocated to personal loans = amount allocated to automobile loans 0.07H + 0.12P + 0.09A H 0.6H + - P 0.4P P + - A 0.4A 0.6A = 1,000,000 0 0 Amount of New Funds Minimum Home Loans Personal Loan Requirement H = $400,000 P = $225,000 A = $375,000 Total annual return = $88,750 Annual percentage return = 8.875% c. The objective coefficient range for H is No Lower Limit to 0.101. Since 0.09 is within the range, the solution obtained in part (b) will not change. d. The dual value for constraint 1 is 0.089. The right-hand-side range for constraint 1 is 0 to No Upper Limit. Therefore, increasing the amount of new funds available by $10,000 will increase the total annual return by 0.089 (10,000) = $890. e. The second constraint now becomes -0.61H - 0.39P - 0.39A 0 The new optimal solution is H = $390,000 P = $228,750 A = $381,250 Total annual return = $89,062.50, an increase of $312.50 Annual percentage return = 8.906%, an increase of approximately 0.031%. Quantitative Analysis BA 452 Homework 3 Questions 21. a. Let S1 = S2 D1 D2 B1 SuperSaver rentals allocated to room type I = SuperSaver rentals allocated to room type II = Deluxe rentals allocated to room type I = Deluxe rentals allocated to room type II = Business rentals allocated to room type II The linear programming formulation and solution is given. MAX 30S1+20S2+35D1+30D2+40B2 S.T. 1) 2) 3) 4) 5) 1S1+1S2<130 constrain