Deck 2: An Introduction to Linear Programming

Question 1

A redundant constraint is a binding constraint.

Answer 1

False

Question 2

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 2

False

Question 3

The standard form of a linear programming problem will have the same solution as the original problem.

Answer 3

True

Question 4

The point (3, 2) is feasible for the constraint 2x₁ + 6x₂ ≤ 30.

Question 5

No matter what value it has, each objective function line is parallel to every other objective function line in a problem.

Question 6

A range of optimality is applicable only if the other coefficient remains at its original value.

Question 7

Alternative optimal solutions occur when there is no feasible solution to the problem.

Question 8

Only binding constraints form the shape (boundaries) of the feasible region.

Question 9

An infeasible problem is one in which the objective function can be increased to infinity.

Question 10

Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given positive coefficients in the objective function.

Question 11

In a feasible problem, an equal-to constraint cannot be nonbinding.

Question 12

Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution.

Question 13

A linear programming problem can be both unbounded and infeasible.

Question 14

Decision variables limit the degree to which the objective in a linear programming problem is satisfied.

Question 15

It is possible to have exactly two optimal solutions to a linear programming problem.

Question 16

The constraint 2x₁ − x₂ = 0 passes through the point (200,100).

Question 17

An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the problem.

Question 18

In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables.

Question 19

An unbounded feasible region might not result in an unbounded solution for a minimization or maximization problem.

Question 20

The constraint 5x₁ − 2x₂ ≤ 0 passes through the point (20, 50).

Question 21

​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

Question 22

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.

Question 23

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.

Question 24

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.

Question 25

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.

Question 26

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.

Question 27

​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

Question 28

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.

Question 29

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.

Question 30

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 5x² + 6y²
D) Min (x₁ + x₂)/x₃

Question 31

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.

Question 32

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.

Question 33

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.

Question 34

​In what 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

Question 35

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.

Question 36

​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) ​a slack variable
C) ​a surplus variable
D) ​a non-negative variable

Question 37

​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) ​300x₁ + 125x₂ > 4,000
B) ​300x₁ + 125x₂ < 4,000
C) ​425(x₁ + x₂) < 4,000
D) ​300x₁ + 125x₂ = 4,000

Question 38

A constraint that does not affect the feasible region is a

A) non-negativity constraint.
B) redundant constraint.
C) standard constraint.
D) slack constraint.

Question 39

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.

Question 40

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.

Question 41

Does the following linear programming problem exhibit infeasibility, unboundedness, or alternate optimal solutions? Explain.
Min
3X + 3Y
s.t.
1X + 2Y ≤ 16
1X + 1Y ≤ 10
5X + 3Y ≤ 45
X , Y ≥ 0

Question 42

Given the following linear program:
Min
150X + 210Y
s.t.
3.8X + 1.2Y ≥ 22.8
Y ≥ 6
Y ≤ 15
45X + 30Y = 630
X, Y ≥ 0
​
Solve the problem graphically. How many extreme points exist for this problem?

Question 43

Does the following linear programming problem exhibit infeasibility, unboundedness, or alternate optimal solutions? Explain.
Min
1X + 1Y
s.t.
5X + 3Y ≤ 30
3X + 4Y ≥ 36
Y ≤ 7
X , Y ≥ 0

Question 44

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.

Question 45

Solve the following linear program graphically.
Max
5X + 7Y
s.t.
X ≤ 6
2X + 3Y ≤ 19
X + Y ≤ 8
X, Y ≥ 0

Question 46

Solve the following system of simultaneous equations.
6X + 2Y = 50
2X + 4Y = 20

Question 47

Find the complete optimal solution to this linear programming problem.
Max
5X + 3Y
s.t.
2X + 3Y ≤ 30
2X + 5Y ≤ 40
6X − 5Y ≤ 0
X , Y ≥ 0

Question 48

​Explain how to graph the line x₁ − 2x₂ ≥ 0.

Question 49

​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?

Question 50

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

Question 51

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

Question 52

Solve the following system of simultaneous equations.
6X + 4Y = 40
2X + 3Y = 20

Question 53

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?

Question 54

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

Question 55

Consider the following linear program:
Max
60X + 43Y
s.t.
X + 3Y ≥ 9
6X − 2Y = 12
X + 2Y ≤ 10
X, Y ≥ 0
​
a.
Write the problem in standard form.
b.
What is the feasible region for the problem?
c.
Show that regardless of the values of the actual objective function coefficients, the optimal solution will occur at one of two points. Solve for these points and then determine which one maximizes the current objective function.

Question 56

A businessman is considering opening a small specialized trucking firm. To make the firm profitable, it is estimated that it must have a daily trucking capacity of at least 84,000 cu. ft. Two types of trucks are appropriate for the specialized operation. Their characteristics and costs are summarized in the table below. Note that truck 2 requires 3 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. ​
Solve the problem graphically and note there are alternate optimal solutions. Which optimal solution:
a.
uses only one type of truck?
b.
utilizes the minimum total number of trucks?
c.
uses the same number of small and large trucks?

Question 57

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

Question 58

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

Question 59

Use this graph to answer the questions. 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?

Question 60

Solve the following linear program by the graphical method.
Max
4X + 5Y
s.t.
X + 3Y ≤ 22
−X + Y ≤ 4
Y ≤ 6
2X − 5Y ≤ 0
X, Y ≥ 0

Question 61

​Explain the concepts of proportionality, additivity, and divisibility.

Question 62

​Explain the steps of the graphical solution procedure for a minimization problem.

Question 63

​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.

Question 64

​Explain the steps necessary to put a linear program in standard form.

Question 65

​Explain what to look for in problems that are infeasible or unbounded.

Question 66

​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.