Deck 2: An Introduction to Linear Programming

Question 1

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 1

A

Question 2

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 2

C

Question 3

The constraint 5x₁ -2x₂ $\le$ 0 passes through the point (20, 50).

Answer 3

True

Question 4

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 5

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 6

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 7

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 8

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 9

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

Question 10

A redundant constraint is a binding constraint.

Question 11

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

Question 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.

Question 13

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 $\le$ constraint is smaller than the right side.
C)is the amount by which the left side of a $\ge$ constraint is larger than the right side.
D)exists for each variable in a linear programming problem.

Question 14

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 15

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

Question 16

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 17

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 18

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

Question 19

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 20

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 21

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

Question 22

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 23

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

Question 24

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

Question 25

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

Question 26

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

Question 27

Explain the concepts of proportionality, additivity, and divisibility.

Question 28

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

Question 29

Explain how to graph the line x₁ -2x₂ $\ge$ 0.

Question 30

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 31

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

Question 32

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

Question 33

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.

Question 34

For the following linear programming problem, determine the optimal solution by the graphical solution method
Max
-X + 2Y
s.t.
6X - 2Y $\le$ 3
-2X + 3Y $\le$ 6
X + Y $\le$ 3
X , Y $\ge$ 0

Question 35

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

Question 36

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

Question 37

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.

Question 38

Consider the following linear programming problem
Max
8X + 7Y
s.t.
15X + 5Y $\le$ 75
10X + 6Y $\le$ 60
X + Y $\le$ 8
X , Y $\ge$ 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 39

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

Question 40

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

Question 41

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 $\le$ 3
X- 2Y $\ge$ 0
Y $\le$ 1
X , Y $\ge$ 0

Question 42

Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below. $$\begin{array} { l | c c | c } & \text { Fliptop Model } & \text { Tiptop Model } & \text { Available } \\\hline \text { Plastic } & 3 & 4 & 36 \\\text { Ink Assembly } & 5 & 4 & 40 \\\text { Molding Time } & 5 & 2 & 30\end{array}$$ 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?

Question 43

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 44

Find the complete optimal solution to this linear programming problem.
Max
2X + 3Y
s.t.
4X + 9Y $\le$ 72
10X + 11Y $\le$ 110
17X + 9Y $\le$ 153
X , Y $\ge$ 0

Question 45

Use this graph to answer the questions. Max
20X + 10Y
s.t.
12X + 15Y $\le$ 180
15X + 10Y $\le$ 150
3X - 8Y $\le$ 0
X , Y $\ge$ 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 46

And the complete optimal solution to this linear programming problem.
Max
5X + 3Y
s.t.
2X + 3Y $\le$ 30
2X + 5Y $\le$ 40
6X - 5Y $\le$ 0
X , Y $\ge$ 0

Question 47

Find the complete optimal solution to this linear programming problem.
Min
5X + 6Y
s.t.
3X + Y $\ge$ 15
X + 2Y $\ge$ 12
3X + 2Y $\ge$ 24
X , Y $\ge$ 0

Question 48

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. 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? Which targets will be exceeded? How much will the blend cost? $$\begin{array} { l | c c } & \text { Type } A & \text { Type B } \\\hline \text { Shade Tolerance } & 1 & 1 \\\text { Traffic Resistance } & 2 & 1 \\\text { Drought Resistance } & 2 & 5\end{array}$$

Question 49

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 50

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

Question 51

Find the complete optimal solution to this linear programming problem.
Min
3X + 3Y
s.t.
12X + 4Y $\ge$ 48
10X + 5Y $\ge$ 50
4X + 8Y $\ge$ 32
X , Y $\ge$ 0

Question 52

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