Problems Consider the following linear program: SELF test Max 3A + 28 S.t. 1A + 18 = 10 3A + 18 : 24 14 + 28 = 16 A, B 20 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 op- timal 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 func- tion 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 sensitivity report for the linear program in part (a) provides the following objec tive coefficient range information: Objective Allowable Allowable Variable Coefficient Increase Decrease A 3.000 3.000 1.000 B 2.000 1.000 1.000 Use this objective coefficient range information to answer parts (b) and (c). 2. Consider the linear program in Problem 1. The value of the optimal solution is 27. Sup- SELF test pose that the right-hand side for constraint I is increased from 10 to 1 1. a. Use the graphical solution procedure to find the new optimal solution. b. Use the solution to part (a) to determine the shadow price for constraint 1. C. The sensitivity report for the linear program in Problem I provides the following right- hand-side range information: Constraint Allowable Allowable Constraint R.H. Side Increase Decrease 10.000 1.200 2.000 24.000 6.000 6.000 16.000 Infinite 3.000 What does the right-hand-side range information for constraint I tell you about the shadow price for constraint 1?Problems 333 The shadow price for constraint 2 is 0.5. Using this shadow price and the right-hand-side range information in part (c), what conclusion can you draw about the effect of changes to the right-hand side of constraint 2? 3. Consider the following linear program: Min 8X + 12Y S.t. 1X + 31 = 9 2X + 2Y = 10 6X + 2Y = 18 X. Y20 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 opti- mal solution. C. Assume that the objective function coefficient for X remains 8, but the objective func- tion coefficient for Ychanges from 12 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution. d. The sensitivity report for the linear program in part (a) provides the following objec tive coefficient range information: Objective Allowable Allowable Variable Coefficient Increase Decrease X 8.000 4,000 4,000 12.000 12.000 4.000 How would this objective coefficient range information help you answer parts (b) and (c) prior to resolving the problem? 4. Consider the linear program in Problem 3. The value of the optimal solution is 48. Sup- pose 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 shadow price for constraint 1. c. The sensitivity report for the linear program in Problem 3 provides the following right- hand-side range information: Constraint Allowable Allowable Constraint R.H. Side Increase Decrease 9.000 2.000 4.000 W N - 10.000 8.000 1.000 18.000 4.000 Infinite4. Consider the linear program in Problem 3. The value of the optimal solution is 48. Sup- pose 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 shadow price for constraint 1. c. The sensitivity report for the linear program in Problem 3 provides the following right- hand-side range information: Constraint Allowable Allowable Constraint R.H. Side Increase Decrease 9.000 2.000 4.000 WN = 10.000 8.000 1.000 18.000 4.000 Infinite What does the right-hand-side range information for constraint 1 tell you about the shadow price for constraint 1? d. The shadow price for constraint 2 is 3. Using this shadow price 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?Problems 341 17. The Porsche Club of America sponsors driver education events that provide high- performance driving instruction on actual racetracks. Because safety is a primary conside eration 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 ex- isting 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 assem- bly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of man- ufacturing 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 DRW. The lin- ear programming model for this problem is as follows: Max 200DRB + 280DRW s.t. 20DRB + 25DRW = 40,000 Steel available 40DRB + 100DRW = 120,000 Manufacturing minutes 60DRB + 40DRW = 96.000 Assembly minutes DRB, DRW 2: 0 The sensitivity report is shown in Figure 8.21. 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 shadow price for the manufacturing time constraint change? Explain