Introduction to Management Science, 11e (Taylor) Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis

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1 Instant download and all chapters Test Bank Introduction to Management Science 11th Edition Bernard W. Taylor III Introduction to Management Science, 11e (Taylor) Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis 1) The shadow price for a positive decision variable is 0. Answer: TRUE Diff: 2 Page Ref: 90 Keywords: shadow price, sensitivity analysis 2) The simplex method is a graphical technique used to solve all management science problems. Diff: 1 Page Ref: 73 Section Heading: Computer Solution Keywords: simplex method 3) When the right-hand sides of two constraints are both increased by one unit, the value of the objective function will be adjusted by the sum of the constraints' prices. Diff: 3 Page Ref: 90 Keywords: sensitive analysis, right-hand-side 4) Most computer linear programming packages readily accept constraints entered in fractional form, such as X1/X3. Diff: 1 Page Ref: 77 Section Heading: Computer Solution Keywords: computer solution, formulation, constraint 5) Sensitivity ranges can be computed only for the right-hand sides of constraints. Diff: 1 Page Ref: 88 Keywords: computer solution

2 6) The marginal value of any scarce resource is the dollar amount one should be willing to pay for one additional unit of that scarce resource. Answer: TRUE Diff: 1 Page Ref: 78 Section Heading: Computer Solution Keywords: marginal value

3 7) Sensitivity analysis determines how a change in a parameter affects the optimal solution. Answer: TRUE Diff: 2 Page Ref: 80 8) Because the management science model requires that parameters are known with certainty, sensitivity analysis is not used in practical, real-world applications of linear programming. Diff: 1 Page Ref: 80 9) The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal. Answer: TRUE Diff: 2 Page Ref: 81, objective coefficient 10) The accepted sequence for sensitivity analysis is objective function, left-hand side, and righthand side. Diff: 1 Page Ref: 80 11) The sensitivity range for an objective function coefficient is the range of values over which the profit does not change. Diff: 2 Page Ref: 81, objective coefficient 12) The sensitivity range for a constraint quantity value is the range over which the shadow price is valid. Answer: TRUE, objective coefficient

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5 13) If we change the constraint quantity to a value outside the sensitivity range for that constraint quantity, the shadow price will change. Answer: TRUE, shadow price 14) The sensitivity range for a constraint quantity value is the range over which the optimal values of the decision variables do not change., objective coefficient 15) A change in the value of an objective function coefficient will always change the value of the optimal solution. Diff: 2 Page Ref: 81 16) The terms shadow price and dual price mean the same thing. Answer: TRUE Diff: 1 Page Ref: 90 17) Sensitivity analysis can be used to determine the effect on the solution for changing several parameters at once. Diff: 2 Page Ref: 90 18) For a profit maximization problem, if the allowable increase for a coefficient in the objective function is infinite, then profits are unbounded. Diff: 2 Page Ref: 85

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7 19) The reduced cost (shadow price) for a positive decision variable is. Answer: zero Diff: 2 Page Ref: 90, shadow price 20) The sensitivity range for a is the range of values over which the quantity values can change without changing the shadow price. Answer: constraint quantity Keywords: computer solution, right-hand-side value 21) is the analysis of the effect of parameter changes on the optimal solution. Answer: Sensitivity analysis Diff: 2 Page Ref: 80 22) The sensitivity range for a constraint quantity value is also the range over which the is valid. Answer: shadow price, shadow price 23) The sensitivity range for a(n) coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal. Answer: objective function Diff: 1 Page Ref: 81, objective coefficient

8 Consider the following linear program, which maximizes profit for two products--regular (R) and super (S): MAX 50R + 75S s.t. 1.2 R S 600 assembly (hours) 0.8 R S 300 paint (hours).16 R S 100 inspection (hours) Sensitivity Report: Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$7 Regular = $C$7 Super = Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$3 Assembly (hr/unit) E $E$4 Paint (hr/unit) $E$5 Inspect (hr/unit) ) The optimal number of regular products to produce is, and the optimal number of super products to produce is, for total profits of. Answer: , , $24,583 Diff: 1 Page Ref: 76 Section Heading: Computer Solution Keywords: computer solution 25) If the company wanted to increase the available hours for one of their constraints (assembly, painting, or inspection) by two hours, they should increase. Answer: inspection Diff: 2 Page Ref: 85 Keywords: computer solution, sensitivity analysis 26) The profit on the super product could increase by without affecting the product mix. Answer: $50 Diff: 1 Page Ref: 85 Keywords: computer solution

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10 27) If downtime reduced the available capacity for painting by 40 hours (from 300 to 260 hours), profits would be reduced by. Answer: $1,333 Keywords: computer solution, sensitivity analysis 28) A change in the market has increased the profit on the super product by $5. Total profit will increase by. Answer: $667 Diff: 2 Page Ref: 84 Keywords: computer solution, sensitivity analysis

11 Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management. Max R = 14Z + 13Y + 17X subject to: Beef 2Z + 3Y + 4X 28 Cheese 9Z + 8Y + 11X 80 Beans 4Z + 4Y + 2X 68 X,Y,Z 0 The sensitivity report from the computer model reads as follows: 29) Taco Loco should try to purchase additional, but should not buy more. Answer: beef and cheese, beans Diff: 2 Page Ref: 90 Keywords: shadow price, sensitivity analysis 30) Taco Loco should produce both but should not make any. Answer: Z and Y, X Diff: 1 Page Ref: 77 Section Heading: Computer Solution Keywords: computer solution, spreadsheets 31) Taco Loco will make the same quantity of X, Y, and Z if the amount of cheese at their disposal is between pounds and pounds.

12 Answer: 74.67, 126 Diff: 2 Page Ref: 88, right-hand-side value 32) For humanitarian reasons, Taco Loco decides they would rather make product X than product Y. The dollar amount that they can both increase the price of Y and reduce the price of X by to accomplish this reversal of demand is. Answer: 82 cents (81.8 cents) Diff: 2 Page Ref: 81, parameter changes 33) If Taco Loco reduces the price of the X product by about 82 cents, then their optimal product mix will contain X. Answer: 0 units (none) Diff: 1 Page Ref: 91 Keywords: reduced cost

13 Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor and $30 on each lawn mower, and they sell all they can produce. The time requirements in the machine shop, fabrication, and tractor assembly are given in the table. Formulation: Let x = number of tractors produced per period y = number of lawn mowers produced per period MAX 30x + 30y subject to 2x + y 60 2x + 3y 120 x 45 The graphical solution is shown below. 34) How many tractors and saws should be produced to maximize profit, and how much profit will they make? Answer: 15 tractors and 30 saws for $1350 in profit Diff: 2 Page Ref: 79 Section Heading: Computer Solution Keywords: graphical solution, simultaneous solution 35) Determine the sensitivity range for the profit for tractors. Answer: 20 x 60

14 Keywords: graphical solution, sensitivity analysis 36) What is the shadow price for assembly? Answer: 0 Diff: 1 Page Ref: Keywords: graphical solution, sensitivity analysis 37) What is the shadow price for fabrication? Answer: $7.50 Diff: 2 Page Ref: 84, right-hand-side value 38) What is the maximum amount a manager would be willing to pay for one additional hour of machining time? Answer: $ $1350 = $7.50 Diff: 3 Page Ref: 84 and Graphical Solution Keywords: graphical solution, sensitivity analysis 39) A breakdown in fabrication causes the available hours to drop from 120 to 90 hours. How will this impact the optimal number of tractors and mowers produced? Answer: x = 22.5, y = 15, Z = 1125, so profits will fall by $ $1125 = $225. Students may also answer the question by determining the sensitivity range, which is from 60 to 180 hours, resulting in a profit change of 30 $7.5 = $225. Diff: 3 Page Ref: 84 and Graphical Solution Keywords: graphical solution, sensitivity analysis 40) What is the range for the shadow price for assembly? Answer: allowable decrease = = 30, and allowable increase is. and Graphical Solution Keywords: graphical solution, sensitivity analysis

15 The production manager for the Whoppy soft drink company is considering the production of two kinds of soft drinks: regular (R) and diet (D). The company operates one 8-hour shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup, is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case.