QM10 Tif ch07

Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 141
CHAPTER 7
Linear Programming Models: Graphical and Computer Methods
TRUE/FALSE
7.1 Management resources that need control include machinery usage, labor volume, money spent, time
used, warehouse space used, and material usage.
ANSWER: TRUE {easy, INTRODUCTION}
7.2 In the term linear programming, the word programming comes from the phrase computer programming.
ANSWER: FALSE {moderate, INTRODUCTION}
7.3 One of the assumptions of LP is “simultaneity.”
ANSWER: FALSE {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.4 Any linear programming problem can be solved using the graphical solution procedure.
ANSWER: FALSE {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.5 An LP formulation typically requires finding the maximum value of an objective while simultaneously
maximizing usage of the resource constraints.
ANSWER: FALSE {moderate, FORMULATING LP PROBLEMS}
7.6 There are no limitations on the number of constraints or variables that can be graphed to solve an LP
problem.
ANSWER: FALSE {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.7 Resource restrictions are called constraints.
ANSWER: TRUE {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.8 One of the assumptions of LP is “proportionality.”
ANSWER: TRUE {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.9 The set of solution points that satisfies all of a linear programming problem's constraints simultaneously
is defined as the feasible region in graphical linear programming.
ANSWER: TRUE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.10 An objective function is necessary in a maximization problem but is not required in a minimization
problem.
ANSWER: FALSE {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall.

142 Linear Programming Models: Graphical and Computer Methods l CHAPTER 7
7.11 In some instances, an infeasible solution may be the optimum found by the corner point method.
ANSWER: FALSE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.12 The rationality assumption implies that solutions need not be in whole numbers (integers).
7.13 The solution to a linear programming problem must always lie on a constraint.
ANSWER: TRUE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.14 In a linear program, the constraints must be linear, but the objective function may be nonlinear.
7.15 Resource mix problems use LP to decide how much of each product to make, given a series of resource
restrictions.
ANSWER: FALSE {moderate, FORMULATING LP PROBLEMS}
7.16 The existence of nonnegativity constraints in a two-variable linear program implies that we are always
working in the northwest quadrant of a graph.
ANSWER: FALSE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.17 In linear programming terminology, dual price and sensitivity price are synonyms.
ANSWER: FALSE {moderate, SENSITIVITY ANALYSIS}
7.18 Any time that we have an isoprofit line that is parallel to a constraint, we have the possibility of multiple
solutions.
ANSWER: TRUE {moderate, FOUR SPECIAL CASES IN LP}
7.19 If the isoprofit line is not parallel to a constraint, then the solution must be unique.
ANSWER: TRUE {moderate, FOUR SPECIAL CASES IN LP, AACSB: Reflective Thinking}
7.20 When two or more constraints conflict with one another, we have a condition called unboundedness.
ANSWER: FALSE {moderate, FOUR SPECIAL CASES IN LP}
7.21 The addition of a redundant constraint lowers the isoprofit line.
ANSWER: FALSE {moderate, FOUR SPECIAL CASES IN LP}
7.22 Sensitivity analysis enables us to look at the effects of changing the coefficients in the objective
function, one at a time.
ANSWER: TRUE {moderate, SENSITIVITY ANALYSIS}

MULTIPLE CHOICE
7.23 Typical resources of an organization include __________________.
(a) machinery usage

(b) labor volume
(c) warehouse space utilization
(d) raw material usage
(e) all of the above
ANSWER: e {easy, INTRODUCTION}
7.24 Which of the following is not a property of all linear programming problems?
(a) the presence of restrictions

(b) optimization of some objective
(c) a computer program
(d) alternate courses of action to choose from
(e) usage of only linear equations and inequalities
ANSWER: c {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.25 A feasible solution to a linear programming problem
(a) must be a corner point of the feasible region.

(b) must satisfy all of the problem's constraints simultaneously.
(c) need not satisfy all of the constraints, only the non-negativity constraints.
(d) must give the maximum possible profit.
(e) must give the minimum possible cost.
ANSWER: b {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.26 Infeasibility in a linear programming problem occurs when
(a) there is an infinite solution.

(b) a constraint is redundant.
(c) more than one solution is optimal.
(d) the feasible region is unbounded.
(e) there is no solution that satisfies all the constraints given.
ANSWER: e {moderate, FOUR SPECIAL CASES IN LP}
7.27 In a maximization problem, when one or more of the solution variables and the profit can be made
infinitely large without violating any constraints, then the linear program has
(a) an infeasible solution.

(b) an unbounded solution.
(c) a redundant constraint.
(d) alternate optimal solutions.
(e) none of the above

ANSWER: b {moderate, FOUR SPECIAL CASES IN LP}

7.28 Which of the following is not a part of every linear programming problem formulation?
(a) an objective function

(b) a set of constraints
(c) non-negativity constraints
(d) a redundant constraint
(e) maximization or minimization of a linear function
ANSWER: d {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.29 When appropriate, the optimal solution to a maximization linear programming problem can be found by
graphing the feasible region and
(a) finding the profit at every corner point of the feasible region to see which one gives the highest
value.
(b) moving the isoprofit lines towards the origin in a parallel fashion until the last point in the
feasible region is encountered.
(c) locating the point that is highest on the graph.
(d) none of the above
(e) all of the above
ANSWER: a {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.30 Which of the following is not a property of linear programs?
(a) one objective function

(b) at least two separate feasible regions
(c) alternative courses of action
(d) one or more constraints
(e) objective function and constraints are linear
ANSWER: b {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.31 The corner point solution method
(a) will always provide one, and only one, optimum.

(b) will yield different results from the isoprofit line solution method.
(c) requires that the profit from all corners of the feasible region be compared.
(d) requires that all corners created by all constraints be compared.
(e) will not provide a solution at an intersection or corner where a non-negativity constraint is
involved.
ANSWER: c {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.32 When a constraint line bounding a feasible region has the same slope as an isoprofit line,

(a) there may be more than one optimum solution.

(b) the problem involves redundancy.
(c) an error has been made in the problem formulation.
(d) a condition of infeasibility exists.
ANSWER: a {moderate, FOUR SPECIAL CASES IN LP}
7.33 The simultaneous equation method is
(a) an alternative to the corner point method.

(b) useful only in minimization methods.
(c) an algebraic means for solving the intersection of two or more constraint equations.
(d) useful only when more than two product variables exist in a product mix problem.
ANSWER: c {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.34 Consider the following linear programming problem:
Maximize 12X + 10Y

Subject to: 4X + 3Y  480
2X + 3Y  360
all variables  0
The maximum possible value for the objective function is
(a) 360.
(b) 480.
(c) 1520.
(d) 1560.
ANSWER: c {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}

Maximize 4X + 10Y
4X + 2Y  360
all variables  0
The feasible corner points are (48,84), (0,120), (0,0), (90,0). What is the maximum possible value for
the objective function?
(a) 1032
(b) 1200
(c) 360
(d) 1600
ANSWER: b {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
Maximize 5X + 6Y
1X + 2Y  120
all variables  0
Which of the following points (X,Y) is not a feasible corner point?
(a) (0,60)
(b) (105,0)
(c) (120,0)
(d) (100,10)
ANSWER: c {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
Maximize 5X + 6Y
1X + 2Y  120

all variables  0
Which of the following points (X,Y) is not feasible?

(a) (50,40)
(b) (20,50)
(c) (60,30)
(d) (90,10)
ANSWER: a {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
7.38 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. A linear
programming model is used to determine the production schedule. The formulation is as follows:
Maximize profit = 50X + 60 Y

Subject to: 8X + 10Y  800 (labor hours)
X + Y  120 (total units demanded)
4X + 5Y  500 (raw materials)
all variables  0
The optimal solution is X = 100 Y = 0.
How many units of the regular model would be produced based on this solution?
(a) 0
(b) 100
(c) 50
(d) 120
ANSWER: b {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
7.39 Two models of a product – Regular (X) and Deluxe (Y) – are produced by a company. A linear
programming model is used to determine the production schedule. The formulation is as follows:
Maximize profit = 50X + 60Y

Subject to: 8X + 10Y  800 (labor hours)
X + Y  120 (total units demanded)
4X +10Y  500 (raw materials)
X, Y  0

The optimal solution is X=100, Y=0.
Which of these constraints is redundant?
(a) the first constraint

(b) the second constraint
(c) the third constraint
(d) all of the above
ANSWER: b {hard, FOUR SPECIAL CASES IN LP, AACSB: Analytic Skills}
Minimize 20X + 30Y

Subject to 2X + 4Y  800
6X + 3Y  300
X, Y  0
What is the optimum solution to this problem (X,Y)?
(a) (0,0).
(b) (50,0).
(c) (0,100).
(d) (400,0).
ANSWER: b {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
Maximize 20X + 30Y

Subject to: X + Y  80
6X + 12Y  600
X, Y  0
This is a special case of a linear programming problem in which
(a) there is no feasible solution.

(b) there is a redundant constraint.
(c) there are multiple optimal solutions.

(d) this cannot be solved graphically.

ANSWER: e {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
Maximize 20X + 30Y

Subject to X + Y  80
8X + 9Y  600
3X + 2Y  400
X, Y  0

ANSWER: a {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
7.43 Adding a constraint to a linear programming (maximization) problem may result in __________
(a) a decrease in the value of the objective function.

(b) an increase in the value of the objective function.
(c) no change to the objective function.
(d) either (c) or (a) depending upon the constraint.
(e) either (c) or (b) depending upon the constraint.
ANSWER: d {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}

7.44 Deleting a constraint from a linear programming (maximization) problem may result in _________
(a) a decrease in the value of the objective function.

(b) an increase in the value of the objective function.
(c) no change to the objective function.
(d) either (c) or (a) depending upon the constraint.
(e) either (c) or (b) depending upon the constraint.
ANSWER: e {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
7.45 Which of the following is not acceptable as a constraint in a linear programming problem
(maximization)?
Constraint 1 X + XY + Y  12
Constraint 2 X  2Y  20
Constraint 3 X + 3Y = 48
Constraint 4 X + Y + Z  150
(a) Constraint 1
(b) Constraint 2
(c) Constraint 3
(d) Constraint 4
ANSWER: a {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.46 If two corner points tie for the best value of the objective function, then
(a) the solution is infeasible.

(b) there are an infinite number of optimal solutions.
(c) the problem is unbounded.
(d) the problem is degenerate.
ANSWER: b {hard, FOUR SPECIAL CASES IN LP}
7.47 If one changes the contribution rates in the objective function of an LP,
(a) the feasible region will change.

(b) the slope of the isoprofit or iso-cost line will change.
(c) the optimal solution to the LP is sure to no longer be optimal.
ANSWER: b {moderate, SENSITIVITY ANALYSIS}

7.48 Sensitivity analysis may also be called
(a) postoptimality analysis.

(b) parametric programming.
(c) optimality analysis.
ANSWER: d {moderate, SENSITIVITY ANALYSIS}
7.49 Sensitivity analyses are used to examine the effects of changes in
(a) contribution rates for each variable.

(b) technological coefficients.
(c) available resources.
ANSWER: d {moderate, SENSITIVITY ANALYSIS}
7.50 Which of the following is a basic assumption of linear programming?
(a) The condition of uncertainty exists.

(b) Independence exists for the activities.
(c) Proportionality exists in the objective function and constraints.
(d) Divisibility does not exist, allowing only integer solutions.
(e) Solutions or variables may take values from   to + .
ANSWER: c {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.51 The condition when there is no solution that satisfies all the constraints is called
(a) boundedness.
(b) redundancy.
(c) optimality.
(d) dependency.
ANSWER: e {moderate, FOUR SPECIAL CASES IN LP}
7.52 In a minimization problem, when one or more of the solution variables and the cost can be made
infinitely large without violating any constraints, then the linear program has
(a) an infeasible solution.

(b) an unbounded solution.
(c) a redundant constraint.
(d) alternate optimal solutions.
ANSWER: e {hard, SOLVING MINIMIZATION PROBLEMS}
7.53 If the addition of a constraint to a linear programming problem does not change the solution, the
constraint is said to be

(a) unbounded.
(b) non-negative.
(c) infeasible.
(d) redundant.
(e) bounded.
ANSWER: d {moderate, FOUR SPECIAL CASES IN LP}
7.54 Which of the following is not an assumption of LP?
(a) simultaneity
(b) certainty
(c) proportionality
(d) divisibility
(e) additivity
ANSWER: a {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.55 In order for a linear programming problem to have a unique solution, the solution must exist
(a) at the intersection of the non-negativity constraints.

(b) at the intersection of a non-negativity constraint and a resource constraint.
(c) at the intersection of the objective function and a constraint.
(d) at the intersection of two or more constraints.
ANSWER: d {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}
7.56 In order for a linear programming problem to have multiple solutions, the solution must exist
(a) at the intersection of the non-negativity constraints.

(b) on a non-redundant constraint parallel to the objective function.
(c) at the intersection of the objective function and a constraint.
(d) at the intersection of three or more constraints.
ANSWER: b {moderate, FOUR SPECIAL CASES IN LP}

Maximize 12X + 10Y

4X + 3Y  360
all variables  0
The maximum possible value for the objective function is
(a) 360.
(b) 480.
(c) 1520.
(d) 1560.
ANSWER: e {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
Maximize 12X + 10Y

2X + 3Y  360
all variables  0
Which of the following points (X,Y) is feasible?
(a) (10,120)
(b) (120,10)
(c) (30,100)
(d) (60,90)

Maximize 5X + 6Y
1X + 2Y  120
all variables  0
Which of the following points (X,Y) is in the feasible region?

(a) (30,60)
(b) (105,5)
(c) (0,210)
(d) (100,10)
ANSWER: d {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}

Maximize 5X + 6Y
1X + 2Y  120
all variables  0
Which of the following points (X,Y) is feasible?
(a) (50,40)
(b) (30,50)
(c) (60,30)
(d) (90,20)
7.61 Which of the following is not an assumption of LP?
(a) certainty
(b) proportionality
(c) divisibility
(d) multiplicativity
(e) additivity
ANSWER: d {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

Maximize 20X + 30Y
Subject to: X + Y  80
12X + 12Y  600
3X + 2Y 400
X, Y  0



7.63 Which of the following functions is not linear?
(a) 5X + 3Z
(b) 3X + 4Y + Z − 3
(c) 2X + 5YZ
(d) Z
(e) 2X – 5Y + 2Z
ANSWER: c {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.64 Which of the following is not one of the steps in formulating a linear program?
(a) Graph the constraints to determine the feasible region.

(b) Define the decision variables.
(c) Use the decision variables to write mathematical expressions for the objective function and the
constraints.
(d) Identify the objective and the constraints.
(e) Completely understand the managerial problem being faced.
ANSWER: a {moderate, FORMULATING LP PROBLEMS}
7.65 Which of the following is not acceptable as a constraint in a linear programming problem
(minimization)?
Constraint 1 X + Y  12
Constraint 2 X - 2Y  20
Constraint 3 X + 3Y = 48
Constraint 4 X + Y + Z  150
Constraint 5 2X - 3Y + Z > 75
(a) Constraint 1
(b) Constraint 2
(c) Constraint 3

(d) Constraint 4
(e) Constraint 5
ANSWER: e {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.66 What type of problems use LP to decide how much of each product to make, given a series of resource
restrictions?
(a) resource mix

(b) resource restriction
(c) product restriction
(d) resource allocation
(e) product mix
ANSWER: e {moderate, FORMULATING LP PROBLEMS}
Maximize 10X + 30Y

Subject to X + 2Y  80
8X + 16Y  640
4X + 2Y  100
X, Y  0

7.68 Consider the following constraints from a linear programming problem:
2X + Y  200
X + 2Y  200
X, Y  0
If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?
(a) (0, 0)
(b) (0, 200)
(c) (0,100)
(d) (100, 0)
(e) (66.67, 66.67)
ANSWER: b {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}

7.69 Consider the following constraints from a linear programming problem:
2X + Y  200
X + 2Y  200
X, Y  0
If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?
(a) (0, 0)
(b) (0, 100)
(c) (65, 65)
(d) (100, 0)
(e) (66.67, 66.67)
ANSWER: c {hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
PROBLEMS
7.70 As a supervisor of a production department, you must decide the daily production totals of a certain
product that has two models, the Deluxe and the Special. The profit on the Deluxe model is $12 per
unit and the Special's profit is $10. Each model goes through two phases in the production process, and
there are only 100 hours available daily at the construction stage and only 80 hours available at the
finishing and inspection stage. Each Deluxe model requires 20 minutes of construction time and 10
minutes of finishing and inspection time. Each Special model requires 15 minutes of construction time
and 15 minutes of finishing and inspection time. The company has also decided that the Special model
must comprise at least 40 percent of the production total.
(a) Formulate this as a linear programming problem.

(b) Find the solution that gives the maximum profit.
ANSWER:
(a) Let X1 = number of Deluxe models produced

X2 = number of Special models produced

Maximize 12X1 + 10X2
Subject to: 1/3X1 + 1/4X2  100
1/6X1 + 1/4X2  80
0.4X1 + 0.6X2  0
X1, X2  0
(b) Optimal solution: X1 = 120, X2 = 240 Profit = $3,840

{hard, FORMULATING LP PROBLEMS and GRAPHICAL SOLUTION TO AN LP PROBLEM,
AACSB: Analytic Skills}
7.71 The Fido Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken
and liver-flavored biscuits) that meets certain nutritional requirements. The liver-flavored biscuits
contain 1 unit of nutrient A and 2 units of nutrient B, while the chicken-flavored ones contain 1 unit of
nutrient A and 4 units of nutrient B. According to federal requirements, there must be at least 40 units
of nutrient A and 60 units of nutrient B in a package of the new mix. In addition, the company has
decided that there can be no more than 15 liver-flavored biscuits in a package. If it costs 1 cent to make
a liver-flavored biscuit and 2 cents to make a chicken-flavored one, what is the optimal product mix for
a package of the biscuits in order to minimize the firm's cost?
(a) Formulate this as a linear programming problem.

(b) Find the optimal solution for this problem graphically
(c) Are any constraints redundant? If so, which one or ones?
(d) What is the total cost of a package of dog biscuits using the optimal mix?
ANSWER:
(a) Let X1 = number of liver-flavored biscuits in a package

X2 = number of chicken-flavored biscuits in a package
Minimize X1 + 2X2
Subject to: X1 + X2  40
2X1 + 4X2  60
X1  15
X1, X2  0
(b) Corner points (0,40) and (15,25)

Optimal solution is (15,25) with cost of 65.
(c) 2X1 + 4X2  60 is redundant
(d) minimum cost = 65 cents

7.72 Consider the following linear program:


Subject to: 3X1 + X2  300
X1 + X2  200
X1  100
X2  50
X1  X2  0
X1, X2  0
(a) Solve the problem graphically. Is there more than one optimal solution? Explain.
(b) Are there any redundant constraints?
ANSWER:
(a) Corner points (0,50), (0,200), (50,50), (75,75), (50,150)
Optimum solutions: (75,75) and (50,150). Both yield a profit of $3,000.
(b) The constraint X1  100 is redundant since 3X1 + X2  300 also means that X1 cannot exceed 100.
{hard, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}
7.73 Solve the following linear programming problem using the corner point method:
Maximize 10X + 1Y
2X + 4Y  40
Y 3
X, Y  0
ANSWER:
Feasible corner points (X,Y):

(0,3) (0,10) (2.4,8.8) (6.75,3)
Maximum profit 70.5 at (6.75,3).
7.74 Solve the following linear programming problem using the corner point method:
Maximize 3 X + 5Y
1X + 2Y  20
Y2
X, Y  0
ANSWER:

Feasible corner points (X,Y): (0,2) (0,10) (4,8) (10,2)
Maximum profit is 52 at (4,8).

7.75 Billy Penny is trying to determine how many units of two types of lawn mowers to produce each day.
One of these is the Standard model, while the other is the Deluxe model. The profit per unit on the
Standard model is $60, while the profit per unit on the Deluxe model is $40. The Standard model
requires 20 minutes of assembly time, while the Deluxe model requires 35 minutes of assembly time.
The Standard model requires 10 minutes of inspection time, while the Deluxe model requires 15 minutes
of inspection time. The company must fill an order for 6 Deluxe models. There are 450 minutes of
assembly time and 180 minutes of inspection time available each day. How many units of each product
should be manufactured to maximize profits?
ANSWER:
Let X = number of Standard models to produce
Y = number of Deluxe models to produce
Maximize 60X + 40Y

10X + 15Y  180
Y6
X, Y  0
Maximum profit is $780 by producing 9 Standard and 6 Deluxe models.

7.76 Two advertising media are being considered for promotion of a product. Radio ads cost $400 each,
while newspaper ads cost $600 each. The total budget is $7,200 per week. The total number of ads
should be at least 15, with at least 2 of each type. Each newspaper ad reaches 6,000 people, while each
radio ad reaches 2,000 people. The company wishes to reach as many people as possible while meeting
all the constraints stated. How many ads of each type should be placed?
ANSWER:
Let R = number of radio ads placed

N = number of newspaper ads placed
Maximize 2000R + 6000N

Subject to: R+ N  15

400R + 600N  7200

R 2
N2
R, N  0
Feasible corner points (R,N): (9,6) (13,2) (15,2)

Maximum exposure 54,000 with 9 radio and 6 newspaper ads.
7.77 Suppose a linear programming (minimization) problem has been solved and the optimal value of the
objective function is $300. Suppose an additional constraint is added to this problem. Explain how this
might affect each of the following:
(a) the feasible region

(b) the optimal value of the objective function
ANSWER:
(a) Adding a new constraint will reduce the size of the feasible region unless it is a redundant
constraint. It can never make the feasible region any larger. However, it could make the problem
infeasible.
(b) A new constraint can only reduce the size of the feasible region; therefore, the value of the objective
function will either increase or remain the same. If the original solution is still feasible, it will
remain the optimal solution.
{moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM and FOUR SPECIAL CASES IN LP,
7.78 Upon retirement, Mr. Klaws started to make two types of children’s wooden toys in his shop, Wuns and
Toos. Wuns yield a variable profit of $9 each and Toos have a contribution margin of $8 each. Even
though his electric saw overheats, he can make 7 Wuns or 14 Toos each day. Since he doesn't have
equipment for drying the lacquer finish he puts on the toys, the drying operation limits him to 16 Wuns
or 8 Toos per day.
(a) Solve this problem using the corner point method.

(b) For what profit ratios would the optimum solution remain the optimum solution?
ANSWER:
Let X1 = numbers of wuns/day

X2 = number of toos/day
Maximize 9X1 + 8X2

Subject to: 2X1 + 1X2  14
1X1 + 2X2  16
X1, X2  0
Corner points (0,0), (7,0), (0,8), (4,6)
Optimum profit $84 at (4,6).


7.79 Susanna Nanna is the production manager for a furniture manufacturing company. The company
produces tables (X) and chairs (Y). Each table generates a profit of $80 and requires 3 hours of
assembly time and 4 hours of finishing time. Each chair generates $50 of profit and requires 3 hours of
assembly time and 2 hours of finishing time. There are 360 hours of assembly time and 240 hours of
finishing time available each month. The following linear programming problem represents this
situation.
Maximize 80X + 50Y

4X + 2Y  240
X, Y 0
The optimal solution is X = 0, and Y = 120.
(a) What would the maximum possible profit be?

(b) How many hours of assembly time would be used to maximize profit?
(c) If a new constraint, 2X + 2Y  400, were added, what would happen to the maximum possible
profit?
ANSWERS: (a) 6000, (b) 360, (c) It would not change.

7.80 As a supervisor of a production department, you must decide the daily production totals of a certain
product that has two models, the Deluxe and the special. The profit on the Deluxe model is $12 per
unit, and the special's profit is $10. Each model goes through two phases in the production process, and
there are only 100 hours available daily at the construction stage and only 80 hours available at the
finishing and inspection stage. Each Deluxe model requires 20 minutes of construction time and 10
minutes of finishing and inspection time. Each special model requires 15 minutes of construction time
and 15 minutes of finishing and inspection time. The company has also decided that the special model
must comprise at most 60 percent of the production total.
Formulate this as a linear programming problem.
ANSWER:
Let X1 = number of Deluxe models produced

X2 = number of special models produced
Subject to: 1/3X1 + 1/4X2  100

1/6X1 + 1/4X2  80
1.5X1 + X2  0
X1, X2  0
{moderate, FORMULATING LP PROBLEMS, AACSB: Analytic Skills}

7.81 Determine where the following two constraints intersect.

5X + 23Y ≤ 1000
10X + 26Y ≤ 1600
ANSWER: (108, 20) {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic

Skills}
7.82 Determine where the following two constraints intersect.

2X − 4Y = 800
−X + 6Y ≥ −200
ANSWER: (500, 50) {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic

Skills}
7.83 Two advertising media are being considered for promotion of a product. Radio ads cost $400 each,
while newspaper ads cost $600 each. The total budget is $7,200 per week. The total number of ads
should be at least 15, with at least 2 of each type, and there should be no more than 19 ads in total. The
company does not want the number of newspaper ads to exceed the number of radio ads by more than
25 percent. Each newspaper ad reaches 6,000 people, 50 percent of whom will respond; while each
radio ad reaches 2,000 people, 20 percent of whom will respond. The company wishes to reach as
many respondents as possible while meeting all the constraints stated. Develop the appropriate LP
model for determining the number of ads of each type that should be placed.
ANSWER:
Let R = number of radio ads placed
N = number of newspaper ads placed
Maximize: 0.20*2000R + 0.50*6000N

or Maximize: 400R + 3000N
Subject to: R + N  15
R + N  19
400R + 600N  7200
1R - N 
R2
N2
R, N  0
{moderate, FORMULATING LP PROBLEMS, AACSB: Analytic Skills}

7.84 Suppose a linear programming (maximization) problem has been solved and the optimal value of the
objective function is $300. Suppose a constraint is removed from this problem. Explain how this might
affect each of the following:
(a) the feasible region

(b) the optimal value of the objective function
ANSWER:
(a) Removing a constraint may, if the constraint is not redundant, increase the size of the feasible
region. It can never make the feasible region any smaller. If the constraint was active in the
solution, removing it will also result in a new optimal solution. However, removing an essential
constraint could cause the problem to become unbounded.
(b) Removal of a constraint can only increase or leave the same the size of the feasible region;
therefore, the value of the objective function will either increase or remain the same, assuming
the problem has not become unbounded.
{moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM and FOUR SPECIAL CASES IN LP,
7.85 Consider the following constraints from a two-variable linear program.
(1) X ≥ 0
(2) Y ≥ 0
(3) X + Y ≤ 50
If the optimal corner point lies at the intersection of constraints (2) and (3), what is the optimal solution
(X, Y)?
ANSWER: Y = 0, so X + 0 = 50, or X = 50. Thus the solution is (50, 0).

{easy, GRAPHCIAL SOLUTION TO AN LP PROBLEM, AACSB: Analytic Skills}

7.86 Consider a product mix problem, where the decision involves determining the optimal production levels
for products X and Y. A unit of X requires 4 hours of labor in department 1 and 6 hours a labor in
department 2. A unit of Y requires 3 hours of labor in department 1 and 8 hours of labor in department
2. Currently, 1000 hours of labor time are available in department 1, and 1200 hours of labor time are
available in department 2. Furthermore, 400 additional hours of cross-trained workers are available to
assign to either department (or split between both). Each unit of X sold returns a $50 profit, while each
unit of Y sold returns a $60 profit. All units produced can be sold. Formulate this problem as a linear
program. (Hint: Consider introducing other decision variables in addition to the production amounts for
X and Y.)
ANSWER: Let X = the number of units of product X sold

Y = the number of units of product Y sold
C1 = the number of cross-trained labor hours allocated to department 1
C2 = the number of cross-trained labor hours allocated to department 2
Max 50X + 60Y

Subject to
4X + 3Y ≤ 1000 + C1
6X + 8Y ≤ 1200 + C2
C1 + C2 ≤ 400
X, Y ≥ 0
{hard, FORMULATING LP PROBLEMS: AACSB: Analytic Skills}
SHORT ANSWER/ESSAY
7.87 Define dual price.
ANSWER: The dual price for a constraint is the improvement in the objective function value that
results from a one-unit increase in the right-hand side of the constraint. {moderate, SENSITIVITY
ANALYSIS}
7.88 One basic assumption of linear programming is proportionality. Explain its need.
ANSWER: Rates of consumption exist and are constant. For example, if the production of 1 unit
requires 4 units of a resource, then if 10 units are produced, 40 units of the resource are required. A
change in the variable value results in a proportional change in the objective function value.
{moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}
7.89 One basic assumption of linear programming is divisibility. Explain its need.
ANSWER: Solutions can have fractional values and need not be whole numbers. If fractional values
would not make sense, then integer programming would be required. {moderate, REQUIREMENTS
OF A LINEAR PROGRAMMING PROBLEM}
7.90 Define infeasibility with respect to an LP solution.
ANSWER: This occurs when there is no solution that can satisfy all constraints. {moderate, FOUR
SPECIAL CASES IN LP}

7.91 Define unboundedness with respect to an LP solution.
ANSWER: This occurs when a linear program has no finite solution. The result implies that the
formulation is missing one or more crucial constraints. {moderate, FOUR SPECIAL CASES IN LP}
7.92 Define alternate optimal solutions with respect to an LP solution.
ANSWER: More than one optimal solution point exist because the objective function is parallel to a
binding constraint. {moderate, FOUR SPECIAL CASES IN LP}
7.93 How does the case of alternate optimal solutions, as a special case in linear programming, compare to
the two other special cases of infeasibility and unboundedness?
ANSWER: With multiple alternate solutions, any of those answers is correct. In the other two cases, no
single answer can be generated. Alternate solutions can occur when a problem is correctly formulated
whereas the other two cases most likely have an incorrect formulation. {moderate, FOUR SPECIAL
CASES IN LP, AACSB: Reflective Thinking}

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Linear Programming Models: Graphical and Computer Methods l CHAPTER 7 141

ANSWER: TRUE {easy, INTRODUCTION}

ANSWER: FALSE {moderate, INTRODUCTION}

7.3 One of the assumptions of LP is “simultaneity.”

ANSWER: FALSE {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

ANSWER: FALSE {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM}

ANSWER: FALSE {moderate, FORMULATING LP PROBLEMS}

ANSWER: FALSE {easy, GRAPHICAL SOLUTION TO AN LP PROBLEM}

7.7 Resource restrictions are called constraints.

ANSWER: TRUE {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

7.8 One of the assumptions of LP is “proportionality.”

ANSWER: TRUE {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

ANSWER: TRUE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

ANSWER: FALSE {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall.

ANSWER: FALSE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

ANSWER: FALSE {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

ANSWER: TRUE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

ANSWER: FALSE {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

ANSWER: FALSE {moderate, FORMULATING LP PROBLEMS}

ANSWER: FALSE {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

ANSWER: FALSE {moderate, SENSITIVITY ANALYSIS}

ANSWER: TRUE {moderate, FOUR SPECIAL CASES IN LP}

ANSWER: FALSE {moderate, FOUR SPECIAL CASES IN LP}

7.21 The addition of a redundant constraint lowers the isoprofit line.

ANSWER: FALSE {moderate, FOUR SPECIAL CASES IN LP}

ANSWER: TRUE {moderate, SENSITIVITY ANALYSIS}

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall.

(a) machinery usage

ANSWER: e {easy, INTRODUCTION}

(a) the presence of restrictions

ANSWER: c {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

7.25 A feasible solution to a linear programming problem

(a) must be a corner point of the feasible region.

ANSWER: b {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

7.26 Infeasibility in a linear programming problem occurs when

(a) there is an infinite solution.

ANSWER: e {moderate, FOUR SPECIAL CASES IN LP}

(a) an infeasible solution.

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall.

ANSWER: b {moderate, FOUR SPECIAL CASES IN LP}

(a) an objective function

ANSWER: d {moderate, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

ANSWER: a {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

7.30 Which of the following is not a property of linear programs?

(a) one objective function

ANSWER: b {easy, REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM}

7.31 The corner point solution method

(a) will always provide one, and only one, optimum.

ANSWER: c {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall.

(a) there may be more than one optimum solution.

ANSWER: a {moderate, FOUR SPECIAL CASES IN LP}

7.33 The simultaneous equation method is

(a) an alternative to the corner point method.

ANSWER: c {moderate, GRAPHICAL SOLUTION TO AN LP PROBLEM}

7.34 Consider the following linear programming problem:

Maximize 12X + 10Y

The maximum possible value for the objective function is