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Chapter 19 Linear Programming

True / False Questions 1. Linear programming techniques will always produce an optimal solution to an LP problem. True

False

2. LP problems must have a single goal or objective specified. True

False

3. Constraints limit the alternatives available to a decision maker; removing constraints adds viable alternative solutions. True

False

4. Profit maximization, like cost minimization, could be an objective of an LP problem, but neither would be an actual decision variable. True

False

5. The feasible solution space only contains points that satisfy all constraints. True

False

6. The equation 5x + 7y = 10 is linear. True

False

7. The equation 3xy = 9 is linear. True

False

8. Graphical linear programming can handle problems that involve any number of decision variables. True

False

9. An objective function represents a family of parallel lines. True

False

10. The term isoprofit line means that all points on the line will yield the same profit. True

False

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11. The feasible solution space is the set of all feasible combinations of decision variables as defined by only binding constraints. True

False

12. The value of an objective function decreases as it is moved away from the origin. True

False

13. A linear programming problem can have multiple optimal solutions. True

False

14. A maximization problem may be characterized by all greater than or equal to constraints. True

False

15. If a single optimal solution exists to a graphical LP problem, it will exist at a corner point. True

False

16. The simplex method is a general-purpose LP algorithm that can be used for solving only problems with more than six variables. True

False

17. A change in the value of an objective function coefficient does not change the optimal solution. True

False

18. The term range of optimality refers to a constraint's right-hand-side quantity. True

False

19. A shadow price indicates how much a one-unit decrease/increase in the right-handside value of a constraint will decrease/increase the optimal value of the objective function. True

False

20. The term range of feasibility refers to coefficients of the objective function. True

False

21. Nonzero slack or surplus is associated with a binding constraint. True

False

22. In the range of feasibility, the value of the shadow price remains constant. True

False

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23. Every change in the value of an objective function coefficient will lead to changes in the optimal solution. True

False

24. Nonbinding constraints are not associated with the feasible solution space; i.e., they are redundant and can be eliminated from the matrix. True

False

25. When a change in the value of an objective function coefficient remains within the range of optimality, the optimal solution also remains the same. True

False

26. Using the enumeration approach, optimality is obtained by evaluating every coordinate. True

False

Multiple Choice Questions 27. The linear optimization technique for allocating constrained resources among different products is:

A. linear regression analysis. B. linear disaggregation. C. linear decomposition. D. linear programming. E. linear tracking analysis.

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28. Which of the following is not a component of the structure of a linear programming model?

A. constrain ts B. decision variables C. paramete rs D. a goal or objective E. environmental uncertainty 29. Coordinates of all corner points are substituted into the objective function when we use the approach called:

A. least squares. B. regressio n. C. enumeratio n. D. graphical linear programming. E. constraint assignment. 30. Which of the following could not be a linear programming problem constraint?

A. 1A + 2B ≤ 3 B. 1A + 2B ≥ 3 C. 1A + 2B = 3 D. 1A + 2B + 3C + 4D ≤5 E. 1A + 2B

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31. For the products A, B, C, and D, which of the following could be a linear programming objective function?

A. Z = 1A + 2B + 3C + 4D B. Z = 1A + 2BC + 3D C. Z = 1A + 2AB + 3ABC + 4ABCD D. Z = 1A + 2B/C + 3D E. Z = 1A + 2B 1CD 32. The logical approach, from beginning to end, for assembling a linear programming model begins with:

A. identifying the decision variables. B. identifying the objective function. C. specifying the objective function parameters. D. identifying the constraints. E. specifying the constraint parameters. 33. The region which satisfies all of the constraints in graphical linear programming is called the:

A. optimum solution space. B. region of optimality. C. lower left hand quadrant. D. region of nonnegativity. E. feasible solution space.

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34. In graphical linear programming, the objective function is: (I) a family of parallel lines. (II) a family of isoprofit lines. (III) interpolated. (IV) linear.

A. I only B. II only C. III and IV only D. I, II, and IV only E. I, II, III, and IV 35. Which objective function has the same slope as this one: $4x + $2y = $20?

A. $4x + $2y = $10 B. $2x + $4y = $20 C. $2x - $4y = $20 D. $4x - $2y = $20 E. $8x + $8y = $20 36. For the following constraints, which point is in the feasible solution space of this maximization problem?

A. x = 1, y =5 B. x = -1, y =1 C. x = 4, y =4 D. x = 2, y =1 E. x = 2, y =8

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37. Which of the following choices constitutes a simultaneous solution to these equations?

A. x = 2, y = .5 B. x = 4, y = -.5 C. x = 2, y =1 D. x = y E. y = 2x 38. Which of the following choices constitutes a simultaneous solution to these equations?

A. x = 1, y = 1.5 B. x = .5, y =2 C. x = 0, y =3 D. x = 2, y =0 E. x = 0, y =0

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39. What combination of x and y will yield the optimum for this problem?

A. x = 2, y =0 B. x = 0, y =0 C. x = 0, y =3 D. x = 1, y =5 E. x = 0, y =4 40. In graphical linear programming, when the objective function is parallel to one of the binding constraints, then:

A. the solution is suboptimal. B. multiple optimal solutions exist. C. a single corner point solution exists. D. no feasible solution exists. E. the constraint must be changed or eliminated. 41. For the constraints given below, which point is in the feasible solution space of this minimization problem?

A. x = .5, y =5 B. x = 0, y =4 C. x = 2, y =5 D. x = 1, y =2 E. x = 2, y =1 19-8 Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

42. What combination of x and y will provide a minimum for this problem?

A. x = 0, y =0 B. x = 0, y =3 C. x = 0, y =5 D. x = 1, y = 2.5 E. x = 6, y =0 43. The theoretical limit on the number of decision variables that can be handled by the simplex method in a single problem is:

A. 1 . B. 2 . C. 3 . D. 4 . E. unlimite d. 44. The theoretical limit on the number of constraints that can be handled by the simplex method in a single problem is:

A. 1 . B. 2 . C. 3 . D. 4 . E. unlimite d.

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45. A shadow price reflects which of the following in a maximization problem?

A. marginal cost of adding additional resources B. marginal gain in the objective that would be realized by adding one unit of a resource C. net gain in the objective that would be realized by adding one unit of a resource D. marginal gain in the objective that would be realized by subtracting one unit of a resource E. expected value of perfect information 46. In linear programming, a nonzero reduced cost is associated with a:

A. decision variable in the solution. B. decision variable not in the solution. C. constraint for which there is slack. D. constraint for which there is surplus. E. constraint for which there is no slack or surplus. 47. A constraint that does not form a unique boundary of the feasible solution space is a:

A. redundant constraint. B. binding constraint. C. nonbinding constraint. D. feasible solution constraint. E. constraint that equals zero.

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48. In linear programming, sensitivity analysis is associated with: (I) the objective function coefficient. (II) right-hand-side values of constraints. (III) the constraint coefficient.

A. I and II only B. II and III only C. I, II, and III D. I and III only E. I only 49. In a linear programming problem, the objective function was specified as follows: Z = 2A + 4B + 3C The optimal solution calls for A to equal 4, B to equal 6, and C to equal 3. It has also been determined that the coefficient associated with A can range from 1.75 to 2.25 without the optimal solution changing. This range is called A's:

A. range of optimality. B. range of feasibility. C. shadow price. D. slack . E. surplu s.

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50. An analyst, having solved a linear programming problem, determined that he had 10 more units of resource Q than previously believed. Upon modifying his program, he observed that the optimal solution did not change, but the value of the objective function increased by $30. This means that resource's Q's shadow price was:

A. $1.5 0. B. $3.0 0. C. $6.0 0. D. $15.0 0. E. $30.0 0. 51. In a linear programming problem involving minimization, at least one constraint must be of the __________ type.

A. less than or equal B. integ er C. greater than or equal D. binar y E. slac k 52. In a linear programming problem involving maximization, at least one constraint must be of the __________ type.

A. greater than or equal B. integ er C. binar y D. less than or equal E. surplu s

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53. In the graphical approach to linear programming, finding values for the decision variables at the intersection of corners requires the solving of:

A. linear constraints. B. surplus variables. C. slack variables. D. simultaneous equations. E. binding constraints. 54. A redundant constraint is one that:

A. is parallel to the objective function. B. has no coefficient for at least one decision variable. C. has a zero coefficient for at least one decision variable. D. has multiple coefficients for at least one decision variable. E. does not form a unique boundary of the feasible solution space. 55. The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B). Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. What is the objective function?

A. $1A + $2B =Z B. $12A + $8B =Z C. $2A + $1B =Z D. $8A + $12B =Z E. $4A + $8B =Z

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56. The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B). Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. What is the Columbia bean constraint?

A. 1A + 2B ≤ 4,800 B. 12A + 8B ≤ 4,800 C. 2A + 1B ≤ 4,800 D. 8A + 12B ≤ 4,800 E. 4A + 8B ≤ 4,800 57. The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B). Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. What is the Dominican bean constraint?

A. 12A + 8B ≤ 4,800 B. 8A + 12B ≤ 4,800 C. 4A + 8B ≤ 3,200 D. 8A + 4B ≤ 3,200 E. 4A + 8B ≤ 4,800

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58. The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B). Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. Which of the following is not a feasible production combination?

A. 0 A and 0 B B. 0 A and 400 B C. 200 A and 300 B D. 400 A and 0 B E. 400 A and 400 B 59. The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B). Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. What are optimal weekly profits?

A. $ 0 B. $40 0 C. $70 0 D. $80 0 E. $90 0

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60. The production planner for Fine Coffees, Inc., produces two coffee blends: American (A) and British (B). Two of his resources are constrained: Columbia beans, of which he can get at most 300 pounds (4,800 ounces) per week; and Dominican beans, of which he can get at most 200 pounds (3,200 ounces) per week. Each pound of American blend coffee requires 12 ounces of Colombian beans and 4 ounces of Dominican beans, while a pound of British blend coffee uses 8 ounces of each type of bean. Profits for the American blend are $2.00 per pound, and profits for the British blend are $1.00 per pound. For the production combination of 0 American and 400 British, which resource is "slack" (not fully used)?

A. Colombian beans (only) B. Dominican beans (only) C. both Colombian beans and Dominican beans D. neither Colombian beans nor Dominican beans E. cannot be determined exactly 61. The operations manager for the Blue Moon Brewing Co. produces two beers: Lite (L) and Dark (D). Two of his resources are constrained: production time, which is limited to 8 hours (480 minutes) per day; and malt extract (one of his ingredients), of which he can get only 675 gallons each day. To produce a keg of Lite beer requires 2 minutes of time and 5 gallons of malt extract, while each keg of Dark beer needs 4 minutes of time and 3 gallons of malt extract. Profits for Lite beer are $3.00 per keg, and profits for Dark beer are $2.00 per keg. What is the objective function?

A. $2L + $3D =Z B. $2L + $4D =Z C. $3L + $2D =Z D. $4L + $2D =Z E. $5L + $3D =Z

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62. The operations manager for the Blue Moon Brewing Co. produces two beers: Lite (L) and Dark (D). Two of his resources are constrained: production time, which is limited to 8 hours (480 minutes) per day; and malt extract (one of his ingredients), of which he can get only 675 gallons each day. To produce a keg of Lite beer requires 2 minutes of time and 5 gallons of malt extract, while each keg of Dark beer needs 4 minutes of time and 3 gallons of malt extract. Profits for Lite beer are $3.00 per keg, and profits for Dark beer are $2.00 per keg. What is the time constraint?

A. 2L + 3D ≤ 480 B. 2L + 4D ≤ 480 C. 3L + 2D ≤ 480 D. 4L + 2D ≤ 480 E. 5L + 3D ≤ 480 63. The operations manager for the Blue Moon Brewing Co. produces two beers: Lite (L) and Dark (D). Two of his resources are constrained: production time, which is limited to 8 hours (480 minutes) per day; and malt extract (one of his ingredients), of which he can get only 675 gallons each day. To produce a keg of Lite beer requires 2 minutes of time and 5 gallons of malt extract, while each keg of Dark beer needs 4 minutes of time and 3 gallons of malt extract. Profits for Lite beer are $3.00 per keg, and profits for Dark beer are $2.00 per keg. Which of the following is not a feasible production combination?

A. 0 L and 0 D B. 0 L and 120 D C. 90 L and 75 D D. 135 L and 0 D E. 135 L and 120 D

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64. The operations manager for the Blue Moon Brewing Co. produces two b...