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7
Introduction to Linear Programming
MULTIPLE CHOICE
1. The maximization or minimization of a quantity is the
a. goal of management science.
b. decision for decision analysis.
c. constraint of operations research.
d. objective of linear programming.
ANSWER: d
TOPIC: Introduction
2. Decision variables
a. tell how much or how many of something to produce, invest, purchase,
hire, etc.
b. represent the values of the constraints.
c. measure the objective function.
d. must exist for each constraint.
ANSWER: a
TOPIC: Objective function
3. Which of the following is a valid objective function for a linear programming
problem?
a. Max 5xy
b. Min 4x + 3y + (2/3)z
c. Max 5x + 6y 2
d. Min (x + x )/x
1 2 3
ASNWER: b
TOPIC: Objective function
4. 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
TOPIC: Graphical solution
1 2 Chapter Introduction to Linear Programming
5. A solution that satisfies all the constraints of a linear programming problem
except the nonnegativity constraints is called
a. optimal.
b. feasible.
c. infeasible.
d. semi-feasible.
ANSWER: c
TOPIC: Graphical solution
6. Slack
a. is the difference between the left and right sides of a constraint.
b. is the amount by which the left side of a < constraint is smaller than the
right side.
c. is the amount by which the left side of a > constraint is larger than the
right side.
d. exists for each variable in a linear programming problem.
ANSWER: b
TOPIC: Slack variables
7. 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 the alternatives is correct.
ANSWER: d
TOPIC: Extreme points
8. Which of the following special cases does not require reformulation of the
problem in order to obtain a solution?
a. alternate optimality
b. infeasibility
c. unboundedness
d. each case requires a reformulation.
ANSWER: a
TOPIC: Special cases
9. The improvement in the value of the objective function per unit increase in a
right-hand side is the
a. sensitivity value.
b. dual price.
c. constraint coefficient.
d. slack value.
ANSWER: b
TOPIC: Right-hand sides Chapter Introduction to Linear Programming 3
10. As long as the slope of the objective function stays between the slopes of the
binding constraints
a. the value of the objective function won’t change.
b. there will be alternative optimal solutions.
c. the values of the dual variables won’t change.
d. there will be no slack in the solution.
ANSWER: c
TOPIC: Objective function
11. 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
TOPIC: Alternate Optimal Solution
12. A constraint that does not affect the feasible region is a
a. non-negativity constraint.
b. redundant constraint.
c. standard constraint.
d. slack constraint.
ANSWER: b
TOPIC: Feasible regions
13. 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.
ANSWER: a
TOPIC: Slack variables
14. 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. 4 Chapter Introduction to Linear Programming
ANSWER: d
TOPIC: Slack variables
15. 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
TOPIC: Problem formulation
TRUE/FALSE
1. Increasing the right-hand side of a nonbinding constraint will not cause a change
in the optimal solution.
ANSWER: False
TOPIC: Introduction
2. In a linear programming problem, the objective function and the constraints must
be linear functions of the decision variables.
ANSWER: True
TOPIC: Mathematical statement of the RMC Problem
3. In a feasible problem, an equal-to constraint cannot be nonbinding.
ANSWER: True
TOPIC: Graphical solution
4. Only binding constraints form the shape (boundaries) of the feasible region.
ANSWER: False
TOPIC: Graphical solution
5. The constraint 5x1- 2x2< 0 passes through the point (20, 50).
ANSWER: True
TOPIC: Graphing lines
6. A redundant constraint is a binding constraint.
ANSWER: False
TOPIC: Slack variables
7. Because surplus variables represent the amount by which the solution exceeds
a minimum target, they are given positive coefficients in the objective function.
ANSWER: False
TOPIC: Slack variables Chapter Introduction to Linear Programming 5
8. Alternative optimal solutions occur when there is no feasible solution to the
problem.
ANSWER: False
TOPIC: Alternative optimal solutions
9. A range of optimality is applicable only if the other coefficient remains at its
original value.
ANSWER: True
TOPIC: Simultaneous changes
10. Because the dual price represents the improvement in the value of the optimal
solution per unit increase in right-hand-side, a dual price cannot be negative.
ANSWER: False
TOPIC: Right-hand sides
11. Decision variables limit the degree to which the objective in a linear
programming problem is satisfied.
ANSWER: False
TOPIC: Introduction
12. No matter what value it has, each objective function line is parallel to every other
objective function line in a problem.
ANSWER: True
TOPIC: Graphical solution
13. The point (3, 2) is feasible for the constrai1t 2x2+ 6x ≤ 30.
ANSWER: True
TOPIC: Graphical solution
14. The constraint 2x1- x2= 0 passes through the point (200,100).
ANSWER: False
TOPIC: A note on graphing lines
15. The standard form of a linear programming problem will have the same solution
as the original problem.
ANSWER: True
TOPIC: Surplus variables
16. An optimal solution to a linear programming problem can be found at an extreme
point of the feasible region for the problem.
ANSWER: True
TOPIC: Extreme Points
17. An unbounded feasible region might not result in an unbounded solution for a
minimization or maximization problem.
ANSWER: True 6 Chapter Introduction to Linear Programming
TOPIC: Special cases: unbounded
18. An infeasible problem is one in which the objective function can be increased to
infinity.
ANSWER: False
TOPIC: Special cases: infeasibility
19. A linear programming problem can be both unbounded and infeasible.
ANSWER: False
TOPIC: Special cases: infeasibility and unbounded
20. It is possible to have exactly two optimal solutions to a linear programming
problem.
ANSWER: False
TOPIC: Special cases: alternative optimal solutions
SHORT ANSWER
1. Explain the difference between profit and contribution in an objective function.
Why is it important for the decision maker to know which of these the objective
function coefficients represent?
TOPIC: Objective function
2. Explain how to graph the line 1 - 22 > 0.
TOPIC: Graphing lines
3. Create a linear programming problem with two decision variables and three
constraints that will include both a slack and a surplus variable in standard form.
Write your problem in standard form.
TOPIC: Standard form
4. Explain what to look for in problems that are infeasible or unbounded.
TOPIC: Special cases
5. Use a graph to illustrate why a change in an objective function coefficient does
not necessarily lead to a change in the optimal values of the decision variables,
but a change in the right-hand sides of a binding constraint does lead to new
values.
TOPIC: Graphical sensitivity analysis
6. Explain the concepts of proportionality, additivity, and divisibility.
TOPIC: Notes and comments Chapter Introduction to Linear Programming 7
PROBLEMS
1. Solve the following system of simultaneous equations.
6X + 2Y = 50
2X + 4Y = 20
TOPIC: Simultaneous equations
2. Solve the following system of simultaneous equations.
6X + 4Y = 40
2X + 3Y = 20
TOPIC: Simultaneous equations
3. Consider the following linear programming problem
Max 8X + 7Y
s.t. 15X + 5Y < 75
10X + 6Y < 60
X + Y < 8
X, Y ≥ 0
a. Use a graph to show each constraint and the feasible region.
b. Identify the optimal solution point on your graph. What are the values of
X and Y at the optimal solution?
c. What is the optimal value of the objective function?
TOPIC: Graphical solution
4. For the following linear programming problem, determine the optimal solution by
the graphical solution method
Max X + 2Y
s.t. 6X – 2Y ≤ 3
2X + 3Y ≤ 6
X + Y ≤ 3
X, Y ≥ 0
TOPIC: Graphical solution 8 Chapter 7Introduction to Linear Programming
5. Use this graph to answer the questions.
15
A I
10
B
5 III II
C
D
V
IV E
0
0 5 10 15
Max 20X + 10Y
s.t. 12X + 15Y < 180
15X + 10Y < 150
3X - 8Y < 0
X, Y > 0
a. Which area (I, II, III, IV, or V) forms the feasible region?
b. Which point (A, B, C, D, or E) is optimal?
c. Which constraints are binding?
d. Which slack variables are zero?
TOPIC: Graphical solution
6. Find the complete optimal solution to this linear programming problem.
Min 5X + 6Y
s.t. 3X + Y > 15
X + 2Y > 12
3X + 2Y > 24
X, Y > 0
TOPIC: Graphical solution
7. Find the complete optimal solution to this linear programming problem.
Max 5X + 3Y
s.t. 2X + 3Y < 30
2X + 5Y < 40 ChapterIntroduction to Linear Programming 9
6X - 5Y < 0
X, Y > 0
TOPIC: Graphical solution
8. Find the complete optimal solution to this linear programming problem.
Max 2X + 3Y
s.t. 4X + 9Y < 72
10X + 11Y < 110
17X + 9Y < 153
X, Y > 0
TOPIC: Graphical solution
9. Find the complete optimal solution to this linear programming problem.
Min 3X + 3Y
s.t. 12X + 4Y > 48
10X + 5Y > 50
4X + 8Y > 32
X, Y > 0
TOPIC: Graphical solution
10. For the following linear programming problem, determine the optimal solution by
the graphical solution method. Are any of the constraints redundant? If yes,
then identify the constraint that is redundant.
Max X + 2Y
s.t. X + Y < 3
X 2Y > 0
Y < 1
X, Y ≥ 0
TOPIC: Graphical solution
11. Maxwell Manufacturing makes two models of felt tip marking pens.
Requirements for each lot of pens are given below.
Fliptop Model Tiptop Model Available
Plastic 3 4 36
Ink 5 4 40
Assembly
Molding 5 2 30
Time
The profit for either model is $1000 per lot. 10 Chapter Introduction to Linear Programming
a. What is the linear programming model for

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