# An Introduction to Management Science: Quantitative Approach, 15th Edition Test Bank

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CH 02 – An Introduction to Linear Programming
True / False
1. In a linear programming problem, the objective function and the constraints must be linear functions of the decision
variables.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Remember
2. Only binding constraints form the shape (boundaries) of the feasible region.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Remember
3. It is not possible to have more than one optimal solution to a linear programming problem.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
4. A linear programming problem can be both unbounded and infeasible.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
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Page 1
CH 02 – An Introduction to Linear Programming
KEYWORDS:
Bloom’s: Understand
5. An infeasible problem is one in which the objective function can be increased to infinity.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Understand
6. An unbounded feasible region might not result in an unbounded solution for a minimization or maximization problem.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Understand
7. An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the
problem.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.03 – 2.3
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.3 Extreme Points and the Optimal Solution
KEYWORDS:
Bloom’s: Understand
8. The optimal solution to any linear programming problem is the same as the optimal solution to the standard form of the
problem.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
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Page 2
CH 02 – An Introduction to Linear Programming
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Understand
9. The constraint 2×1โ x2 = 0 passes through the point (200, 100).
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
10. The point (3, 2) is feasible for the constraint 2×1 + 6×2โค 30.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
11. No matter what value it has, each objective function line is parallel to every other objective function line in a problem.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
12. Constraints limit the degree to which the objective in a linear programming problem is satisfied.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
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Page 3
CH 02 – An Introduction to Linear Programming
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Remember
13. Alternative optimal solutions occur when there is no feasible solution to the problem.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Understand
14. Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given
positive coefficients in the objective function.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Understand
15. A redundant constraint cannot be removed from the problem without affecting the feasible region.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
16. The constraint 5×1โ 2×2โค 0 passes through the point (20, 50).
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Moderate
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Page 4
CH 02 – An Introduction to Linear Programming
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
17. At a problem’s optimal solution, a redundant constraint will have zero slack.
a. True
b. False
ANSWER:
False
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Blooms: Understand
18. If a constraint is redundant, it can be removed from the problem without affecting the feasible region.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
19. For a minimization problem, the solution is considered to be unbounded if the value may be made infinitely small.
a. True
b. False
ANSWER:
True
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Remember
Multiple Choice
20. The maximization or minimization of a desired quantity is the
a. goal of management science.
b. decision for decision analysis.
c. constraint of operations research.
d. objective of linear programming.
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Page 5
CH 02 – An Introduction to Linear Programming
ANSWER:
d
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Remember
21. Decision variables
a. are values that are used to determine how much or how many of something to produce, invest, etc.
b. represent the values of the constraints.
c. are values that measure the objective function.
d. must be unique for each constraint.
ANSWER:
a
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Understand
22. Which of the following is a valid objective function for a linear programming problem?
a. Min 8xy
b. Min 4x + 3y + (1/2)z
c. Min 5×2 + 6y2
d. Max (x1 + x2)/x3
ANSWER:
b
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Understand
23. Which of the following statements is NOT true?
a. A feasible solution satisfies all constraints.
b. An optimal solution satisfies all constraints.
c. An infeasible solution violates all constraints.
d. A feasible solution point does not have to lie on the boundary of the feasible region.
ANSWER:
c
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
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Page 6
CH 02 – An Introduction to Linear Programming
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
24. When no solution to the linear programming problem satisfies all the constraints, including the nonnegativity
conditions, it is considered
a. optimal.
b. feasible.
c. infeasible.
d. semifeasible.
ANSWER:
c
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Understand
25. The amount by which the left side of a less-than-or-equal-to constraint is smaller than the right side
a. is known as a surplus.
b. is known as slack.
c. is optimized for the linear programming problem.
d. exists for each variable in a linear programming problem.
ANSWER:
b
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
26. To find the optimal solution to a linear programming problem using the graphical method,
a. find the feasible point that is the farthest away from the origin.
b. find the feasible point that is at the highest location.
c. find the feasible point that is closest to the origin.
d. None of these are correct.
ANSWER:
d
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.03 – 2.3
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.3 Extreme Points and the Optimal Solution
KEYWORDS:
Blooms: Understand
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Page 7
CH 02 – An Introduction to Linear Programming
27. Which of the following special cases does NOT require reformulation of the problem in order to obtain a solution?
a. alternative optimality
b. infeasibility
c. unboundedness
d. Each case requires a reformulation.
ANSWER:
a
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Understand
28. Infeasibility means that the number of solutions to the linear programming models that satisfies all constraints is
a. at least 1.
b. 0.
c. an infinite number.
d. at least 2.
ANSWER:
b
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom’s: Remember
29. A constraint that does NOT affect the feasible region of the solution is a
a. nonnegativity constraint.
b. redundant constraint.
c. standard constraint.
d. slack constraint.
ANSWER:
b
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Remember
30. Whenever all the constraints in a linear program are expressed as equalities, the linear program is said to be written in
a. standard form.
b. bounded form.
c. feasible form.
d. alternative form.
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Page 8
CH 02 – An Introduction to Linear Programming
ANSWER:
a
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Remember
31. All of the following statements about a redundant constraint are correct EXCEPT
a. a redundant constraint does not affect the optimal solution.
b. a redundant constraint does not affect the feasible region.
c. recognizing a redundant constraint is easy with the graphical solution method.
d. at the optimal solution, a redundant constraint will have zero slack.
ANSWER:
d
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom’s: Understand
32. All linear programming problems have all of the following properties EXCEPT
a. a linear objective function that is to be maximized or minimized.
b. a set of linear constraints.
c. alternative optimal solutions.
d. variables that are all restricted to nonnegative values.
ANSWER:
c
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom’s: Understand
33. If there is a maximum of 4,000 hours of labor available per month and 300 ping-pong balls (x1) or 125 wiffle balls (x2)
can be produced per hour of labor, which of the following constraints reflects this situation?
a. 300×1 + 125×2 > 4,000
b. 300×1 + 125×2 < 4,000
c. 425(x1 + x2) < 4,000
d. 300×1 + 125×2 = 4,000
ANSWER:
POINTS:
DIFFICULTY:
b
1
Moderate
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Page 9
CH 02 – An Introduction to Linear Programming
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom's: Apply
34. In which part(s) of a linear programming formulation would the decision variables be stated?
a. objective function and the left-hand side of each constraint
b. objective function and the right-hand side of each constraint
c. the left-hand side of each constraint only
d. the objective function only
ANSWER:
a
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom's: Understand
35. The three assumptions necessary for a linear programming model to be appropriate include all of the following
EXCEPT
a. proportionality.
b. additivity.
c. divisibility.
d. normality.
ANSWER:
d
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom's: Remember
36. A redundant constraint results in
a. no change in the optimal solution(s).
b. an unbounded solution.
c. no feasible solution.
d. alternative optimal solutions.
ANSWER:
a
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Remember
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Page 10
CH 02 – An Introduction to Linear Programming
37. A variable added to the left-hand side of a less-than-or-equal-to constraint to convert the constraint into an equality is
a
a. standard variable.
b. slack variable.
c. surplus variable.
d. nonnegative variable.
ANSWER:
b
POINTS:
1
DIFFICULTY:
Easy
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Reflective Thinking
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Remember
Subjective Short Answer
38. Solve the following system of simultaneous equations.
6X + 2Y = 50
2X + 4Y = 20
ANSWER:
X = 8, Y =1
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Apply
39. Solve the following system of simultaneous equations.
6X + 4Y = 40
2X + 3Y = 20
ANSWER:
X = 4, Y = 4
POINTS:
1
DIFFICULTY:
Moderate
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Apply
40. Consider the following linear programming problem:
Max
s.t.
8X + 7Y
15X + 5Y โค 75
10X + 6Y โค 60
X+ Yโค8
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Page 11
CH 02 – An Introduction to Linear Programming
X, Y โฅ 0
a.
Use a graph to show each constraint and the feasible region.
Identify the optimal solution point on your graph. What are the values of X and Y at the
b.
optimal solution?
c. What is the optimal value of the objective function?
ANSWER:
a.
b.
The optimal solution occurs at the intersection of constraints 2 and 3. The point is X = 3,
Y = 5.
The value of the objective function is 59.
c.
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Apply
41. For the following linear programming problem, determine the optimal solution using the graphical solution method.
โX + 2Y
6X โ 2Y โค 3
โ2X + 3Y โค 6
X+ Yโค3
X, Y โฅ 0
ANSWER:
Max
s.t.
X = 0.6 and Y = 2.4
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Page 12
CH 02 – An Introduction to Linear Programming
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Apply
42. Use this graph to answer the questions.
Max
s.t.
20X + 10Y
12X + 15Y โค 180
15X + 10Y โค 150
3X โ 8Y โค 0
X, Y โฅ 0
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Page 13
CH 02 – An Introduction to Linear Programming
a.
b.
c.
d.
Which area (I, II, III, IV, or V) forms the feasible region?
Which point (A, B, C, D, or E) is optimal?
Which constraints are binding?
Which slack variables equal zero?
ANSWER:
a.
b.
c.
d.
Area III is the feasible region.
Point D is optimal.
Constraints 2 and 3 are binding.
S2 and S3 are equal to 0.
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
43. Find the complete optimal solution to this linear programming problem.
Min
s.t.
5X + 6Y
3X + Y โฅ 15
X + 2Y โฅ 12
3X + 2Y โฅ 24
X, Y โฅ 0
ANSWER:
The complete optimal solution is X = 6, Y = 3, Z = 48, S1 = 6, S2 = 0, S3 = 0
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
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Page 14
CH 02 – An Introduction to Linear Programming
TOPICS:
KEYWORDS:
2.2 Graphical Solution Procedure
Bloom's: Apply
44. Find the complete optimal solution to this linear programming problem.
Max
s.t.
5X + 3Y
2X + 3Y โค 30
2X + 5Y โค 40
6X โ 5Y โค 0
X, Y โฅ 0
ANSWER:
The complete optimal solution is X = 15, Y = 0, Z = 75, S1 = 0, S2 = 10, S3 = 90
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
45. Find the complete optimal solution to this linear programming problem.
Max
s.t.
2X + 3Y
4X + 9Y โค 72
10X + 11Y โค 110
17X + 9Y โค 153
X, Y โฅ 0
ANSWER:
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Page 15
CH 02 – An Introduction to Linear Programming
The complete optimal solution is X = 4.304, Y = 6.087, Z = 26.87, S1 = 0, S2 = 0, S3 =
25.043
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
46. Find the complete optimal solution to this linear programming problem.
Min
s.t.
3X + 3Y
12X + 4Y โฅ 48
10X + 5Y โฅ 50
4X + 8Y โฅ 32
X, Y โฅ 0
ANSWER:
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Page 16
CH 02 – An Introduction to Linear Programming
The complete optimal solution is X = 4, Y = 2, Z = 18, S1 = 8, S2 = 0, S3 = 0
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
47. For the following linear programming problem, determine the optimal solution using the graphical solution method.
Are any of the constraints redundant? If yes, identify the constraint that is redundant.
Max
s.t.
X + 2Y
X+ Yโค3
X โ 2Y โฅ 0
Yโค1
X, Y โฅ 0
ANSWER:
X = 2 and Y = 1 Yes, there is a redundant constraint; Y โค 1
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CH 02 – An Introduction to Linear Programming
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
48. Maxwell Manufacturing makes two models of felt-tip marking pens. Requirements for each lot of pens are given
below.
Plastic
Ink assembly
Molding time
Fliptop Model
3
5
5
Tiptop Model
4
4
2
Available
36
40
30
The profit for either model is $1000 per lot.
a. What is the linear programming model for this problem?
b. Find the optimal solution.
c. Will there be excess capacity in any resource?
ANSWER:
a.
Let F = number of lots of Fliptop pens to produce
T = number of lots of Tiptop pens to produce
Max
s.t.
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1000F + 1000T
3F + 4T โค 36
5F + 4T โค 40
5F + 2T โค 30
F, T โฅ 0
Page 18
CH 02 – An Introduction to Linear Programming
b.
The complete optimal solution is F = 2, T = 7.5, Z = 9500, S1 = 0, S2 = 0, S3 = 5
c. There is an excess of 5 units of molding time available.
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.1 A Simple Maximization Problem
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
49. The Sanders Garden Shop mixes two types of grass seed into a blend. Each type of grass has been rated (per pound)
according to its shade tolerance, ability to stand up to traffic, and drought resistance, as shown in the table. Type A seed
costs $1 and Type B seed costs $2.
Shade tolerance
Traffic resistance
Drought resistance
Type A
1
2
2
Type B
1
1
5
a. If the blend needs to score at least 300 points for shade tolerance, 400 points for traffic resistance, and 750 points for
drought resistance, how many pounds of each seed should be in the blend?
b. Which targets will be exceeded?
c. How much will the blend cost?
ANSWER:
a. Let A = pounds of Type A seed in the blend
B = pounds of Type B seed in the blend
Min
s.t.
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1A + 2B
1A + 1B โฅ 300
2A + 1B โฅ 400
2A + 5B โฅ 750
A, B โฅ 0
Page 19
CH 02 – An Introduction to Linear Programming
The optimal solution is at A = 250, B = 50.
b. Constraint 2 has a surplus value of 150.
c. The cost is 350.
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom's: Analyze
50. Muir Manufacturing produces two popular grades of commercial carpeting among its many other products. In the
coming production period, Muir needs to decide how many rolls of each grade should be produced in order to maximize
profit. Each roll of Grade X carpet uses 50 units of synthetic fiber, requires 25 hours of production time, and needs 20
units of foam backing. Each roll of Grade Y carpet uses 40 units of synthetic fiber, requires 28 hours of production time,
and needs 15 units of foam backing.
The profit per roll of Grade X carpet is $200, and the profit per roll of Grade Y carpet is $160. In the coming production
period, Muir has 3000 units of synthetic fiber available for use. Workers have been scheduled to provide at least 1800
hours of production time (overtime is a possibility). The company has 1500 units of foam backing available for use.
Develop and solve a linear programming model for this problem.
ANSWER:
Let X = number of rolls of Grade X carpet to make
Y = number of rolls of Grade Y carpet to make
Max
200X + 160Y
s.t.
50X + 40Y โค 3000
25X + 28Y โฅ 1800
20X + 15Y โค 1500
X, Y โฅ 0
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CH 02 – An Introduction to Linear Programming
The complete optimal solution is X = 30, Y = 37.5, Z = 12,000, S1 = 0, S2 = 0, S3 = 337.5
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.01 – 2.1
IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
2.1 A Simple Maximization Problem
KEYWORDS:
Bloom's: Analyze
51. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternative optimal
solutions? Explain.
Min
s.t.
3X + 3Y
1X + 2Y โค 16
1X + 1Y โค 10
5X + 3Y โค 45
X, Y โฅ 0
ANSWER:
The problem has alternative optimal solutions.
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Page 21
CH 02 – An Introduction to Linear Programming
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom's: Analyze
52. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternative optimal
solutions? Explain.
Min
s.t.
1X + 1Y
5X + 3Y โค 30
3X + 4Y โฅ 36
Yโค7
X, Y โฅ 0
ANSWER:
The problem is infeasible.
POINTS:
DIFFICULTY:
1
Challenging
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Page 22
CH 02 – An Introduction to Linear Programming
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom's: Analyze
53. A businessman is considering opening a small specialized trucking firm. To make the firm profitable, it must have a
daily trucking capacity of at least 84,000 cubic feet. Two types of trucks are appropriate for the specialized operation.
Their characteristics and costs are summarized in the table below. Note that truck two requires three drivers for long haul
trips. There are 41 potential drivers available, and there are facilities for at most 40 trucks. The businessman's objective is
to minimize the total cost outlay for trucks.
Truck
Small
Large
Cost
$18,000
$45,000
Capacity
(cu. ft.)
2,400
6,000
Drivers
Needed
1
3
Solve the problem graphically and note that there are alternative optimal solutions.
a. Which optimal solution uses only one type of truck?
b. Which optimal solution utilizes the minimum total number of trucks?
c. Which optimal solution uses the same number of small and large trucks?
ANSWER:
a. 35 small, 0 large
b. 5 small, 12 large
c. 10 small, 10 large
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.06 – 2.6
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.6 Special Cases
KEYWORDS:
Bloom's: Analyze
54. Consider the following linear program:
Max
s.t.
60X + 43Y
X + 3Y โฅ 9
6X โ 2Y = 12
X + 2Y โค 10
X, Y โฅ 0
a.
b.
Write the problem in standard form.
What is the feasible region for the problem?
Show that regardless of the values of the actual objective function coefficients, the optimal
c. solution will occur at one of two points. Solve for these points and then determine which one
maximizes the current objective function.
ANSWER:
a. Max
60X + 43Y
s.t.
X + 3Y โ S1 = 9
6X โ 2Y = 12
X + 2Y + S3 = 10
X, Y, S1, S3โฅ 0
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CH 02 – An Introduction to Linear Programming
b. Line segment of 6X โ 2Y = 12 between (22/7, 24/7) and (27/10, 21/10).
c. Extreme points: (22/7, 24/7) and (27/10, 21/10). First one is optimal, giving Z = 336.
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.03 – 2.3
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.3 Extreme Points and the Optimal Solution
KEYWORDS:
Bloom's: Analyze
55. Solve the following linear program graphically.
Max
s.t.
5X + 7Y
X
โค6
2X + 3Y โค 19
X+ Yโค8
X, Y โฅ 0
ANSWER:
From the graph below, we see that the optimal solution occurs at X = 5, Y = 3, and Z = 46.
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
KEYWORDS:
Bloom's: Analyze
56. Solve the following linear program graphically. How many extreme points exist for this problem?
Min
s.t.
150X + 210Y
3.8X + 1.2Y โฅ 22.8
Yโฅ6
Y โค 15
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CH 02 – An Introduction to Linear Programming
45X + 30Y = 630
X, Y โฅ 0
ANSWER:
Two extreme points exist (points A and B below). The optimal solution is X = 10, Y = 6, and
Z = 2760 (point B).
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
IMS.ASWC.19.02.03 – 2.3
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
2.3 Extreme Points and the Optimal Solution
KEYWORDS:
Bloom's: Analyze
57. Solve the following linear program graphically.
Max
s.t.
4X + 5Y
X + 3Y โค 22
โX + Y โค 4
Yโค6
2X โ 5Y โค 0
X, Y โฅ 0
ANSWER:
Two extreme points exist (points A and B below). The optimal solution is X = 10, Y = 6, and
Z = 2760 (point B).
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Page 25
CH 02 – An Introduction to Linear Programming
POINTS:
1
DIFFICULTY:
Challenging
LEARNING OBJECTIVES: IMS.ASWC.19.02.02 – 2.2
IMS.ASWC.19.02.03 – 2.3
NATIONAL STANDARDS: United States – BUSPROG: Analytic
TOPICS:
2.2 Graphical Solution Procedure
2.3 Extreme Points and the Optimal Solution
KEYWORDS:
Bloom's: Analyze
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