Deck 7: Introduction to Linear Programming

Question 1

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

Answer 1

True

Question 2

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

Answer 2

True

Question 3

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 3

A

Question 4

A redundant constraint is a binding constraint.

Question 5

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

Question 6

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 7

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 8

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 9

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 10

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 11

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

Question 12

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

Question 13

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 14

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 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.

Question 16

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 17

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 18

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 19

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 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

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

Question 22

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 - 6
X + Y $\le$ 3
X,Y $\ge$ 0

Question 23

A linear programming problem can be both unbounded and infeasible.

Question 24

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

Question 25

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 26

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 27

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

Question 28

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

Question 29

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

Question 30

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

Question 31

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

Question 32

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

Question 33

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

Question 34

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

Question 35

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

Question 36

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

Question 37

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

Question 38

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

Question 39

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 40

Consider the following linear programming problem
Max 8X + 7Y
s.t.15X + 5Y < 75
10X + 6Y < 60
X + Y < 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 41

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 42

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 43

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 44

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 $\ge$ 0

Question 45

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 46

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 47

Consider the following linear program:
MAX 60X + 43Y
s.t.X + 3Y > 9
6X -0 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 48

Solve the following linear program graphically.
MAX 5X + 7Y
s.t.X < 6
2X + 3Y < 19
X + Y < 8
X,Y > 0

Question 49

Maxwell Manufacturing makes two models of felt tip marking pens.Requirements for each lot of pens are given below.
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 50

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 51

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 52

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?

Question 53

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 54

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