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Introduction to Management Science, 13e (Taylor)
Chapter 2 Linear Programming: Model Formulation and Graphical Solution
1) Linear programming is a model consisting of linear relationships representing a firm’s
decisions given an objective and resource constraints.
Answer: TRUE
Diff: 2 Page Ref: 34
Section Heading: Model Formulation
Keywords: model formulation
AACSB: Analytical thinking
2) The objective function always consists of either maximizing or minimizing some value.
Answer: TRUE
Diff: 2 Page Ref: 34
Section Heading: Model Formulation
Keywords: objective function
AACSB: Analytical thinking
3) The objective function is a linear relationship reflecting the objective of an operation.
Answer: TRUE
Diff: 1 Page Ref: 34
Section Heading: Model Formulation
Keywords: model formulation
AACSB: Analytical thinking
4) Both objective functions and constraints contain parameters.
Answer: TRUE
Diff: 2 Page Ref: 34
Section Heading: Model Formulation
Keywords: model formulation
AACSB: Analytical thinking
5) Proportionality means the slope of a constraint is proportional to the slope of the objective
function.
Answer: FALSE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear programming models, proportionality
AACSB: Analytical thinking
6) The terms in the objective function or constraints are additive.
Answer: TRUE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear programming models, additive
AACSB: Analytical thinking
1
Copyright ยฉ 2019 Pearson Education, Inc.
7) The terms in the objective function or constraints are multiplicative.
Answer: FALSE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear programming models, additive
AACSB: Analytical thinking
8) All linear programming models exhibit a set of constraints.
Answer: TRUE
Diff: 1 Page Ref: 34
Section Heading: Model Formulation
Keywords: properties of linear programming models, constraints
AACSB: Analytical thinking
9) When using the graphical method, only one of the four quadrants of an xy-axis needs to be
drawn.
Answer: TRUE
Diff: 1 Page Ref: 39
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical linear programming
AACSB: Analytical thinking
10) Linear programming models exhibit linearity among all constraint relationships and the
objective function.
Answer: TRUE
Diff: 1 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear prog models, linearity, proportionality
AACSB: Analytical thinking
11) The equation 8xy = 32 satisfies the proportionality property of linear programming.
Answer: FALSE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: graphical solution, proportionality
AACSB: Analytical thinking
12) Typically, finding a corner point for the feasible region involves solving a set of three
simultaneous equations.
Answer: FALSE
Diff: 2 Page Ref: 45
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, extreme points, feasible region
AACSB: Analytical thinking
2
Copyright ยฉ 2019 Pearson Education, Inc.
13) Objective functions in linear programs always minimize costs.
Answer: FALSE
Diff: 2 Page Ref: 34
Section Heading: Model Formulation
Keywords: properties of linear programming models, objective function
AACSB: Analytical thinking
14) The feasible solution area contains infinite solutions to the linear program.
Answer: TRUE
Diff: 1 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: properties of linear programming models, feasible solution area
AACSB: Analytical thinking
15) There is exactly one optimal solution point to a linear program.
Answer: FALSE
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: properties of linear programming models, optimal solution point
AACSB: Analytical thinking
16) The following equation represents a resource constraint for a maximization problem: X + Y
โฅ 20.
Answer: FALSE
Diff: 2 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: properties of linear programming models, constraints
AACSB: Analytical thinking
17) The optimal solution for a graphical linear programming problem is the corner point that is
the farthest from the origin.
Answer: FALSE
Diff: 2 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: feasibility, constraints
AACSB: Analytical thinking
18) A minimization model of a linear program contains only surplus variables.
Answer: FALSE
Diff: 1 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: properties of linear programming models, surplus variables
AACSB: Analytical thinking
3
Copyright ยฉ 2019 Pearson Education, Inc.
19) In the graphical approach, simultaneous equations may be used to solve for the optimal
solution point.
Answer: TRUE
Diff: 2 Page Ref: 45
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
20) Slack variables are only associated with maximization problems.
Answer: FALSE
Diff: 2 Page Ref: 47
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, slack variables
AACSB: Analytical thinking
21) Surplus variables are only associated with minimization problems.
Answer: FALSE
Diff: 2 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: graphical solution, surplus variable
AACSB: Analytical thinking
22) If the objective function is parallel to a constraint, the constraint is infeasible.
Answer: FALSE
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solution
AACSB: Analytical thinking
23) Multiple optimal solutions occur when constraints are parallel to each other.
Answer: FALSE
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solution
AACSB: Analytical thinking
24) Graphical solutions to linear programming problems have an infinite number of possible
objective function lines.
Answer: TRUE
Diff: 2 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, objective function line
AACSB: Analytical thinking
4
Copyright ยฉ 2019 Pearson Education, Inc.
25) The first step in formulating a linear programming model is to define the objective function.
Answer: FALSE
Diff: 2 Page Ref: 34
Section Heading: Introduction
Keywords: linear programming problems, formulation
AACSB: Analytical thinking
26) A linear programming problem requires a choice between alternative courses of action.
Answer: TRUE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming problems, formulation
AACSB: Application of knowledge
27) The term continuous is synonymous with divisible in the context of linear programming.
Answer: TRUE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming problems, formulation
AACSB: Application of knowledge
28) Linear programming problems can model decreasing marginal returns.
Answer: FALSE
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming problems, formulation
AACSB: Application of knowledge
29) One of the most frequent objectives of business forms is to gain the most profit possible.
Answer: TRUE
Diff: 1 Page Ref: 36
Section Heading: Introduction
Keywords: linear programming problems, maximization
AACSB: Analytical thinking
30) We have George Dantzig to thank for developing linear programming.
Answer: TRUE
Diff: 2 Page Ref: 37
Section Heading: A Maximization Model Example
Keywords: linear programming, Dantzig
AACSB: Analytical thinking
5
Copyright ยฉ 2019 Pearson Education, Inc.
31) In the absence of nonnegativity constraints, our solution cannot have zero values for decision
variables.
Answer: FALSE
Diff: 2 Page Ref: 37
Section Heading: A Maximization Model Example
Keywords: nonnegativity, linear programming
AACSB: Analytical thinking
32) If there are no feasible solutions to a linear programming model, then the best course of
action for a manager is to choose a solution that violates at least one constraint.
Answer: FALSE
Diff: 1 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: properties of linear programming models, feasible solution area
AACSB: Analytical thinking
33) ________ are mathematical symbols representing levels of activity.
Answer: Decision variables
Diff: 1 Page Ref: 34
Section Heading: Model Formulation
Keywords: decision variables, model formulation
AACSB: Analytical thinking
34) A(n) ________ is a linear relationship representing a restriction on decision making.
Answer: constraint
Diff: 1 Page Ref: 34
Section Heading: Model Formulation
Keywords: constraint, model formulation
AACSB: Analytical thinking
35) If at least one constraint in a linear programming model is violated, the solution is said to be
________.
Answer: infeasible
Diff: 1 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: constraint, infeasible solution
AACSB: Analytical thinking
36) A graphical solution is limited to solving linear programming problems with ________
decision variables.
Answer: two
Diff: 1 Page Ref: 38
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
6
Copyright ยฉ 2019 Pearson Education, Inc.
37) The ________ solution area is an area bounded by the constraint equations.
Answer: feasible
Diff: 1 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
38) Multiple optimal solutions can occur when the objective function line is ________ to a
constraint line.
Answer: parallel
Diff: 2 Page Ref: 47
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, multiple optimal solutions
AACSB: Analytical thinking
39) XY Corporation makes two products, X and Y, and uses graphical linear programming to
determine their monthly product mix. This November, their only production constraint is X โค 75.
November’s production problem is ________.
Answer: unbounded
Diff: 2 Page Ref: 58
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solution, unbounded problem
AACSB: Analytical thinking
40) The best feasible solution is ________.
Answer: optimal
Diff: 1 Page Ref: 43
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: optimal solutions
AACSB: Analytical thinking
41) In a constraint, the ________ variable represents unused resources.
Answer: slack
Diff: 1 Page Ref: 47
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, surplus variable
AACSB: Analytical thinking
42) ________ is the difference between the left- and right-hand sides of a greater than or equal to
constraint.
Answer: Surplus
Diff: 1 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: surplus
AACSB: Analytical thinking
7
Copyright ยฉ 2019 Pearson Education, Inc.
43) If the objective function is parallel to a constraint, the linear program could have ________.
Answer: multiple optimal solutions
Diff: 2 Page Ref: 47
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solutions, multiple optimal solutions
AACSB: Analytical thinking
44) Corner points on the boundary of the feasible solution area are called ________ points.
Answer: extreme
Diff: 1 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: feasibility, constraints
AACSB: Analytical thinking
45) ________ are at the endpoints of the constraint line segment that the objective function
parallels.
Answer: Alternate optimal solutions
Diff: 3 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: alternative optimal solutions, multiple optimal solutions
AACSB: Analytical thinking
46) The ________ step in formulating a linear programming model is to define the decision
variables.
Answer: first
Diff: 1 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: linear programming, formulation
AACSB: Analytical thinking
47) The management scientist constructed a linear program to help the alchemist maximize his
gold production process. The computer model chugged away for a few minutes and returned an
answer of infinite profit., which is what might be expected from a(n) ________ problem.
Answer: unbounded
Diff: 1 Page Ref: 58
Section Heading: Irregular Types of Linear Programming Problems
Keywords: unbounded
AACSB: Analytical thinking
48) The ________ property of linear programming models indicates that the rate of change, or
slope, of the objective function or a constraint is constant.
Answer: proportionality or linearity
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear programming models, certainty
AACSB: Analytical thinking
8
Copyright ยฉ 2019 Pearson Education, Inc.
49) The objective function 3x + 2y + 4xy violates the assumption of ________.
Answer: additivity
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming properties
AACSB: Application of knowledge
50) Mildred is attempting to prepare an optimal quantity of macaroni and cheese for the potluck
supper this Sunday. The instructions indicate that one cup of water is needed for each box she
needs to prepare. She sleeps well on Saturday night, secure in her knowledge that she knows the
precise amount of water she will need the next day. This knowledge illustrates the assumption of
________.
Answer: certainty
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming properties
AACSB: Application of knowledge
51) Tim! airlines procurement division works with their linear programming algorithm to secure
contracts for gasoline for the coming year. After twenty minutes of thinking, the computer
suggests that they secure 425.8125 contracts with their suppliers. This value illustrates the
assumption of ________ in linear programming models.
Answer: divisibility or continuous
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming properties
AACSB: Application of knowledge
52) In a linear programming problem, the binding constraints for the optimal solution are:
5×1 + 3×2 โค 30
2×1 + 5×2 โค 20
As long as the slope of the objective function stays between ________ and ________, the current
optimal solution point will remain optimal.
Answer: -5/3, -2/5
Diff: 3 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: optimal solution, solution interpretation, slope
AACSB: Analytical thinking
9
Copyright ยฉ 2019 Pearson Education, Inc.
53) In a graphical approach to a linear programming problem, the objective function is
represented by a(n) ________.
Answer: (straight) line
Diff: 2 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: optimal solution, solution interpretation, slope
AACSB: Analytical thinking
54) In a graphical approach to a linear programming problem, the objective function has a
maximum value when it is ________ the origin.
Answer: farthest away from
Diff: 2 Page Ref: 45
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: optimal solution, solution interpretation, slope
AACSB: Analytical thinking
55) The three types of linear programming constraints are ________, ________, and ________.
Answer: โฅ, โค, = (greater than or equal to, less than or equal to, equal to)
Diff: 1 Page Ref: 52
Section Heading: A Minimization Model Example
Keywords: linear programming, constraint
AACSB: Analytical thinking
56) In a linear programming problem, the binding constraints for the optimal solution are:
5×1 + 3×2 โค 30
2×1 + 5×2 โค 20
Which of these objective functions will lead to the same optimal solution?
A) 2×1 + 1×2
B) 7×1 + 8×2
C) 80×1 + 60×2
D) 25×1 + 15×2
Answer: D
Diff: 3 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: optimal solution, solution interpretation, slope
AACSB: Analytical thinking
10
Copyright ยฉ 2019 Pearson Education, Inc.
57) Decision variables:
A) measure the objective function.
B) measure how much or how many items to produce, purchase, hire, etc.
C) always exist for each constraint.
D) measure the values of each constraint.
Answer: B
Diff: 2 Page Ref: 34
Section Heading: Model Formulation
Keywords: decision variables
AACSB: Analytical thinking
58) In a linear programming problem, a valid objective function can be represented as:
A) Max Z = 5xy
B) Max Z 5×2 + 2y2
C) Max 3x + 3y + 1/3 z
D) Min (x1 + x2) / x3
Answer: C
Diff: 3 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: objective function
AACSB: Analytical thinking
59) Which of the following could not be a linear programming problem constraint?
A) 1A + 2B โ 3
B) 1A + 2B = 3
C) 1A + 2B โค 3
D) 1A + 2B โฅ 3
Answer: A
Diff: 2 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: formulation, constraints
AACSB: Analytical thinking
60) Which of the following could be a linear programming objective function?
A) Z = 1A + 2BC + 3D
B) Z = 1A + 2B + 3C + 4D
C) Z = 1A + 2B / C + 3D
D) Z = 1A + 2B2 + 3D
Answer: B
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: objective function
AACSB: Analytical thinking
11
Copyright ยฉ 2019 Pearson Education, Inc.
The campaign manager for a doomed candidate is considering the which states to visit during the
last frenzied campaign week leading up to the nationwide election. Pennsylvania (P), Wisconsin
(W), Florida (F), New York (Y), and North Carolina (C) are all aching for one last visit, but the
candidate has only 80 hours and $250 million left in her campaign fund. A visit to Pennsylvania
takes 10 hours and costs $15 million but earns 1% of the electorate. A visit to Wisconsin takes
15 hours and costs $20 million and earns 1.5%; a visit to Florida is only $8 million but takes 16
hours and earns 2%, and a visit to New York costs $25 million, requires 2 hours and earns 2% of
the electorate. North Carolina requires 18 hours and $22 million per trip but earns 3% of the
electorate.
61) What is the objective function?
A) MIN 10P+15W+16F+2Y+18C
B) MAX 10P+15W+16F+2Y+18C
C) MIN 15P+20W+8F+25Y+22C
D) MAX P+1.5W+2F+2Y+3C
Answer: D
Diff: 2 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: formulation, objective function
AACSB: Analytical thinking
62) What is the time constraint?
A) P+1.5W+2F+2Y+3C โค 250
B) P+1.5W+2F+2Y+3C โค 80
C) 10P+15W+16F+2Y+18C โค 80
D) 15P+20W+8F+25Y+22C โค 80
Answer: C
Diff: 2 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: formulation, constraints
AACSB: Analytical thinking
63) What is the financial constraint?
A) P+1.5W+2F+2Y+3C โค 250
B) 15P+20W+8F+25Y+22C โค 250
C) 15P+20W+8F+25Y+22C โค 80
D) 10P+15W+16F+2Y+18C โค 250
Answer: B
Diff: 2 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: formulation, constraints
AACSB: Analytical thinking
12
Copyright ยฉ 2019 Pearson Education, Inc.
64) The ________ property of linear programming models indicates that the rate of change or
slope of the objective function or a constraint is constant.
A) additive
B) divisibility
C) certainty
D) proportionality
Answer: D
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear programming models
AACSB: Analytical thinking
65) The ________ property of linear programming models indicates that the values of all the
model parameters are known and are assumed to be constant.
A) additive
B) divisibility
C) certainty
D) proportionality
Answer: C
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: properties of linear programming models
AACSB: Analytical thinking
66) The region that satisfies all of the constraints in a graphical linear programming problem is
called the:
A) region of optimality.
B) feasible solution space.
C) region of non-negativity.
D) optimal solution space.
Answer: B
Diff: 1 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, feasibility
AACSB: Analytical thinking
67) In the formulation of a โฅ constraint:
A) a surplus variable is subtracted.
B) a surplus variable is added.
C) a slack variable is subtracted.
D) a slack variable is added.
Answer: A
Diff: 1 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: surplus
AACSB: Analytical thinking
13
Copyright ยฉ 2019 Pearson Education, Inc.
68) Which of the following statements is not true?
A) An infeasible solution violates all constraints.
B) A feasible solution point does not have to lie on the boundary of the feasible solution.
C) A feasible solution satisfies all constraints.
D) An optimal solution satisfies all constraints.
Answer: A
Diff: 2 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, feasibility
AACSB: Analytical thinking
69) A hot dog manufacturer wishes to minimize the cost in dollars of producing a low-cost
niched product while meeting the dietary guidelines for protein and sodium. Once the model has
been run, the surplus variable in the sodium constraint has a value of 1300 milligrams. The best
interpretation of this outcome is:
A) the value of the sodium in a hot dog is 1300.
B) the amount of sodium in a single hot dog should be 1300 milligrams.
C) the minimum cost hot dog has 1300 milligrams more sodium than required.
D) a hot dog should have at least 1300 milligrams of sodium.
Answer: C
Diff: 2 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: surplus
AACSB: Analytical thinking
70) Which of these statements is best?
A) An unbounded problem is also infeasible.
B) An infeasible problem is also unbounded.
C) An unbounded problem has feasible solutions.
D) An infeasible problem has unbounded solutions.
Answer: C
Diff: 2 Page Ref: 58
Section Heading: Irregular Types of Linear Programming Problems
Keywords: infeasible problem, infeasible solution
AACSB: Analytical thinking
71) The optimal solution to a linear programming model that has been solved using the graphical
approach:
A) is typically located at the origin.
B) must be below and on the left side of all constraint lines.
C) must be above and the right of all constraint lines.
D) is typically at some corner of the feasible region.
Answer: D
Diff: 1 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: solution
AACSB: Analytical thinking
14
Copyright ยฉ 2019 Pearson Education, Inc.
72) Without satisfying the non-negativity constraint, a solution that satisfies all the other
constraints of a linear programming problem is called:
A) feasible.
B) infeasible.
C) semi-feasible.
D) optimal.
Answer: B
Diff: 3 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, feasibility
AACSB: Analytical thinking
73) An intern sets up a linear program to optimize the use of paper products in the men’s
washroom. The system of equations he develops is:
Max 2T + 3S + 4ST
s.t 3T + 6S โค 40
10T + 10S โค 66
10T + 15S โค 99
His mentor studies the model, frowns, and admonishes the intern for violating which of the
following properties of linear programming models?
A) divisibility
B) proportionality
C) certainty
D) additivity
Answer: D
Diff: 1 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: additivity
AACSB: Analytical thinking
74) Which of the following is not a typical characteristic of a linear programming problem?
A) Restrictions exist.
B) A choice among alternatives is required.
C) The problem can be solved graphically.
D) The problem has an objective.
Answer: C
Diff: 1 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: graphical solution
AACSB: Analytical thinking
15
Copyright ยฉ 2019 Pearson Education, Inc.
The campaign manager for a doomed candidate is considering the which states to visit during the
last frenzied campaign week leading up to the nationwide election. Pennsylvania (P), Wisconsin
(W), Florida (F), New York (Y), and North Carolina (C) are all aching for one last visit, but the
candidate has only 80 hours and $250 million left in her campaign fund. A visit to Pennsylvania
takes 10 hours and costs $15 million but earns 1% of the electorate. A visit to Wisconsin takes
15 hours and costs $20 million and earns 1.5%; a visit to Florida is only $8 million but takes 16
hours and earns 2%, and a visit to New York costs $25 million, requires 2 hours and earns 2% of
the electorate. North Carolina requires 18 hours and $22 million per trip but earns 3% of the
electorate.
75) Which of the following is not a feasible schedule?
A) two trips each to Pennsylvania and Wisconsin and one trip each to Florida, New York, and
North Carolina
B) four trips each to New York and North Carolina
C) two trips each to Pennsylvania and North Carolina and one trip to Florida
D) four trips to Wisconsin and five trips to New York
Answer: A
Diff: 3 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, feasibility
AACSB: Analytical thinking
76) What is the total percentage increase if the candidate makes the following schedule?
A) 10%
B) 11%
C) 12%
D) 13%
Answer: D
Diff: 2 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
77) ________ is used to analyze changes in model parameters.
A) Optimal solution
B) Feasible solution
C) Sensitivity analysis
D) A slack variable
Answer: C
Diff: 2 Page Ref: 47
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: sensitivity analysis
AACSB: Analytical thinking
16
Copyright ยฉ 2019 Pearson Education, Inc.
Lame Example Furniture Company makes two products for its adoring public: chairs (C)and
tables (T). Each chair requires 5 hours of labor (L) and 4 linear feet of rich mahogany (M), and
each table requires 3 hours of labor and 20 linear feet of rich mahogany. The company has 240
labor hours available this week, and the warehouse has 700 linear feet of rich mahogany
available. Profit for each chair is $300 and for each table is $1500.
78) Which of the following is not a feasible production plan?
A) 35 chairs and 20 tables
B) 20 chairs and 35 tables
C) 25 chairs and 30 tables
D) 30 chairs and 25 tables
Answer: B
Diff: 3 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: formulation, feasibility
AACSB: Analytical thinking
79) What is the maximum profit?
A) $52,500
B) $48,000
C) $55,000
D) $56,250
Answer: A
Diff: 3 Page Ref: 43
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
80) In order to maximize profit, how many tables and how many chairs should be produced?
A) T = 35, C = 0
B) T = 0, C = 48
C) T = 26.3, C = 32.8
D) T = 28.9, C = 30.7
Answer: A
Diff: 3 Page Ref: 43
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
17
Copyright ยฉ 2019 Pearson Education, Inc.
81) The theoretical limit on the number of constraints that can be handled by a linear
programming problem is:
A) 2.
B) 3.
C) 4.
D) unlimited.
Answer: D
Diff: 1 Page Ref: 34
Section Heading: Model Formulation
Keywords: constraints
AACSB: Analytical thinking
82) Consider the following maximization problem.
MAX z = x + 2y
s.t.
2x + 3y โค 6
5x + 6y โค 30
yโฅ1
The optimal solution:
A) occurs where x = 4.67 and y = 1.11.
B) occurs where x = 0 and y = 2.
C) occurs where x = 6 and y = 0.
D) results in an objective function value of 12.
Answer: B
Diff: 1 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, extreme points, feasible region
AACSB: Analytical thinking
18
Copyright ยฉ 2019 Pearson Education, Inc.
Figure 1: The following is a graph of a linear programming problem. The feasible solution space
is shaded, and the optimal solution is at the point labeled Z*.
83) This linear programming problem shown in Figure 1 is a(n):
A) maximization problem.
B) minimization problem.
C) irregular problem.
D) cannot tell from the information given
Answer: B
Diff: 1 Page Ref: 52
Section Heading: A Minimization Model Example
Keywords: graphical solution
AACSB: Analytical thinking
84) The equation for constraint DH as shown in Figure 1 is:
A) 4X + 8Y โฅ 32.
B) 8X + 4Y โฅ 32.
C) X + 2Y โฅ 8.
D) 2X + Y โฅ 8.
Answer: C
Diff: 3 Page Ref: 52
Section Heading: A Minimization Model Example
Keywords: graphical solution, constraints
AACSB: Analytical thinking
19
Copyright ยฉ 2019 Pearson Education, Inc.
85) Which of the following points is not feasible for the graph shown in Figure 1?
A) A
B) B
C) H
D) G
Answer: D
Diff: 1 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, feasible point
AACSB: Analytical thinking
86) Which line in Figure 1 is represented by the equation 2X + Y โฅ 8?
A) BF
B) CG
C) DH
D) AJ
Answer: A
Diff: 2 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, constraints
AACSB: Analytical thinking
87) Which of the following constraints shown in Figure 1 has a surplus greater than 0?
A) BF
B) CG
C) DH
D) AJ
Answer: C
Diff: 2 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: graphical solution, constraints
AACSB: Analytical thinking
88) In Figure 1, the constraint AJ:
A) is a binding constraint.
B) has no surplus.
C) does not contain feasible points.
D) contains the optimal solution.
Answer: B
Diff: 3 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: graphical solution, constraints
AACSB: Analytical thinking
20
Copyright ยฉ 2019 Pearson Education, Inc.
Figure 2
89) Consider the optimization problem represented by the graph in Figure 2. Which of the
following statements is best?
A) This is a maximization problem with a feasible solution.
B) This is a maximization problem with no feasible solution.
C) This is a minimization problem with a feasible solution.
D) This is a minimization problem with no feasible solution.
Answer: C
Diff: 1 Page Ref: 56
Section Heading: A Minimization Model Example
Keywords: graphical solution, feasibility
AACSB: Analytical thinking
90) Line segment GH in Figure 2 represents the objective function. Which constraint has
surplus?
A) AB
B) CD
C) EF
D) none of the constraints has surplus
Answer: A
Diff: 2 Page Ref: 55
Section Heading: A Minimization Model Example
Keywords: graphical solution, surplus variable
AACSB: Analytical thinking
21
Copyright ยฉ 2019 Pearson Education, Inc.
91) What is the equation for the constraint AB shown in Figure 2?
A) 3X + 12Y โฅ 15
B) X + 4Y โฅ 12
C) X + Y โฅ 15
D) 12X + 3Y โฅ 36
Answer: D
Diff: 3 Page Ref: 53
Section Heading: A Minimization Model Example
Keywords: graphical solution, constraints
AACSB: Analytical thinking
92) What is the equation for constraint EF shown in Figure 2?
A) 4X + 8Y โฅ 64
B) 4X + 8Y โฅ 12
C) 16X + 8Y โฅ 24
D) 16X + 8Y โฅ 32
Answer: A
Diff: 3 Page Ref: 53
Section Heading: A Minimization Model Example
Keywords: graphical solution, constraints
AACSB: Analytical thinking
93) Consider the optimization problem represented by the graph in Figure 2. The objective
function is represented by line GH. Where is the optimal solution?
A) the intersection of lines AB and EF
B) the intersection of lines AB and CD
C) the intersection of lines CD and EF
D) the upper right corner of the shaded region
Answer: C
Diff: 1 Page Ref: 53
Section Heading: A Minimization Model Example
Keywords: graphical solution, objective function line
AACSB: Analytical thinking
94) Consider the optimization problem represented by the graph in Figure 2. Line GH represents
the objective function. Which of the following statements is best?
A) This is a single optimal solution.
B) All points along GH are optimal.
C) All points on lines AB, CD and DE that touch the shaded region are optimal.
D) All points in the shaded region are optimal
Answer: A
Diff: 1 Page Ref: 53
Section Heading: A Minimization Model Example
Keywords: graphical solution, multiple optimal solutions
AACSB: Analytical thinking
22
Copyright ยฉ 2019 Pearson Education, Inc.
95) In order for an optimization problem to have multiple optimal solutions:
A) the objective function and one constraint must have the same y-intercept.
B) the objective function and one constraint must have the same slope.
C) two or more of the constraints must not have intersection points.
D) two or more of the constraints must have the same slope.
Answer: B
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solutions, multiple optimal solutions
AACSB: Analytical thinking
96) An optimization problem that has multiple optimal solutions:
A) means that there are actually no optimal solutions.
B) is reflected by the entire feasible region being optimal
C) means that the surplus for a third constraint cannot be calculated.
D) provides the decision-maker with increased flexibility.
Answer: D
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solutions, multiple optimal solutions
AACSB: Analytical thinking
97) How would multiple optimal solutions typically appear on a graphical solution?
A) a point
B) a line
C) a plane
D) a cube
Answer: B
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solutions, multiple optimal solutions
AACSB: Analytical thinking
98) Which of the following statements about infeasible problems is best?
A) All of the possible solutions violate at least one constraint.
B) All of the possible solutions violate all of the constraints.
C) At least one of the possible solutions violates all of the constraints.
D) At least one of the possible solutions violates at least one of the constraints.
Answer: A
Diff: 1 Page Ref: 58
Section Heading: Irregular Types of Linear Programming Problems
Keywords: infeasible problem, infeasible solution
AACSB: Analytical thinking
23
Copyright ยฉ 2019 Pearson Education, Inc.
99) Greg, a young entrepreneur, has developed an aggressive business plan and is presenting his
profit projections on the popular show Shark Tank in hopes of securing some venture capital. He
concludes his presentation with an LP model of his planned product mix, and is convinced he
will seal the deal by demonstrating that his profits are limitless since his LP model is unbounded.
What should the sharks tell him?
A) “Limitless profits sound fantastic, here’s a blank check.”
B) “Limitless profits are possible only in minimization models, and we want you to maximize
profits.”
C) “Unlimited profits aren’t possible. You must have made a mistake in your LP model.”
D) “Limitless profits are possible only in maximization models, and we want you to minimize
profits.”
Answer: C
Diff: 1 Page Ref: 58
Section Heading: Irregular Types of Linear Programming Problems
Keywords: unbounded
AACSB: Analytical thinking
100) Multiple optimal solutions can occur when the objective function is ________ a constraint
line.
A) unequal to
B) equal to
C) perpendicular to
D) parallel to
Answer: D
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: irregular types of linear programming problems
AACSB: Analytical thinking
101) A slack variable:
A) is the amount by which the left side of a โฅ= constraint is larger than the right side.
B) is the amount by which the left side of a โค= constraint is smaller than the right side.
C) is the difference between the left and right side of a constraint.
D) exists for each variable in a linear programming problem.
Answer: B
Diff: 2 Page Ref: 46
Section Heading: Slack Variables
Keywords: slack variables
AACSB: Analytical thinking
24
Copyright ยฉ 2019 Pearson Education, Inc.
The campaign manager for a doomed candidate is considering the which states to visit during the
last frenzied campaign week leading up to the nationwide election. Pennsylvania (P), Wisconsin
(W), Florida (F), New York (Y), and North Carolina (C) are all aching for one last visit, but the
candidate has only 80 hours and $250 million left in her campaign fund. A visit to Pennsylvania
takes 10 hours and costs $15 million but earns 1% of the electorate. A visit to Wisconsin takes
15 hours and costs $20 million and earns 1.5%; a visit to Florida is only $8 million but takes 16
hours and earns 2%, and a visit to New York costs $25 million, requires 2 hours and earns 2% of
the electorate. North Carolina requires 18 hours and $22 million per trip but earns 3% of the
electorate.
102) The campaign manager elects to take one trip each of Pennsylvania, Florida and North
Carolina, two trips to Wisconsin, and three trips to New York. Which resources will be
completely used?
A) only money
B) only time
C) time and money
D) neither time nor money
Answer: B
Diff: 2 Page Ref: 48
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: slack variables
AACSB: Analytical thinking
Lame Example Furniture Company makes two products for its adoring public: chairs (C)and
tables (T). Each chair requires 5 hours of labor (L) and 4 linear feet of rich mahogany (M), and
each table requires 3 hours of labor and 20 linear feet of rich mahogany. The company has 240
labor hours available this week, and the warehouse has 700 linear feet of rich mahogany
available. Profit for each chair is $300 and for each table is $1500.
103) If the furniture company produces twenty tables and thirty-six chairs, which of the two
resources will be completely used?
A) labor only
B) rich mahogany only
C) both labor and rich mahogany
D) neither labor and rich mahogany
Answer: A
Diff: 2 Page Ref: 41
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: slack variables
AACSB: Analytical thinking
25
Copyright ยฉ 2019 Pearson Education, Inc.
104) Consider the following linear program:
MAX z = 5x + 3y
s.t. x – y โค 6
xโค1
The optimal solution:
A) is infeasible.
B) occurs where x = 1 and y = 0.
C) occurs where x = 0 and y = 1.
D) results in an objective function value of 5.
Answer: D
Diff: 2 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: slack variables
AACSB: Analytical thinking
105) The first step in solving a graphical linear programming model is to:
A) plot the model constraints as equations on the graph and indicate the feasible solution area.
B) plot the objective function and move this line out from the origin to locate the optimal
solution point.
C) solve simultaneous equations at each corner point to find the solution values at each point.
D) determine which constraints are binding.
Answer: A
Diff: 1 Page Ref: 39
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphic solution, steps for solving a graphical linear prog model
AACSB: Analytical thinking
106) The optimal solution of a minimization problem is at the extreme point ________ the
origin.
A) farthest from
B) closest to
C) exactly at
D) parallel to
Answer: B
Diff: 2 Page Ref: 53
Section Heading: A Minimization Model Example
Keywords: minimization problem
AACSB: Analytical thinking
26
Copyright ยฉ 2019 Pearson Education, Inc.
107) Multiple optimal solutions provide ________ flexibility to the decision maker.
A) greater
B) less
C) greater or equal
D) less or equal
Answer: A
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: irregular types of linear programming problems
AACSB: Analytical thinking
108) Which of the following special cases does not require reformulation of the problem in order
to obtain a solution?
A) unboundedness
B) infeasibility
C) alternate optimality
D) Each one of these cases requires reformulation.
Answer: C
Diff: 3 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: irregular types of linear programming problems
AACSB: Analytical thinking
109) If the feasible region for a linear programming problem is unbounded, then the solution to
the corresponding linear programming problem is ________ unbounded.
A) always
B) sometimes
C) never
D) There is not enough information to complete this statement.
Answer: B
Diff: 3 Page Ref: 58
Section Heading: Irregular Types of Linear Programming Problems
Keywords: irregular types of linear programming problems, unboundedness
AACSB: Analytical thinking
27
Copyright ยฉ 2019 Pearson Education, Inc.
110) Solve the following graphically:
Max z = 3×1 + 4×2
s.t. x1 + 2×2 โค 16
2×1 + 3×2 โค 18
x1 โฅ 2
x2 โค 10
x1, x2 โฅ 0
What are the optimal values of x1, x2, and z?
Answer: x1 = 9, x2 = 0, z = 27
Diff: 3 Page Ref: 39
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, simultaneous solution
AACSB: Analytical thinking
28
Copyright ยฉ 2019 Pearson Education, Inc.
111) A novice business analyst develops the following model to determine the optimal
combination of socks and underwear to take on his next business trip. The model is as follows:
Maximize 5S + 7U
subject to:
3S – 2U โค 45
7S + 3U โค 33
2S + 8U โค 70
Solve this problem graphically and determine how many of each item the analyst should pack.
Answer: The optimal solution lies at the point representing 1.08 socks and 8.48 underwear. I
suppose this is why I referred to the analyst as a novice.
Corner points and the objective function value in (Socks,Underwear) order are:
Z(0,0) = 0
Z(4.714,0) = 23.57
Z(0,8.75) = 61.25
Z(1.08. 8.48) = 64.76 optimal
Diff: 3 Page Ref: 39
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution
AACSB: Analytical thinking
29
Copyright ยฉ 2019 Pearson Education, Inc.
112) Nathan enters the final exam period needing to pull off a miracle to pass his three toughest
classes, Healthy Life Choices, Success Central, and Walking Fitness. Naturally he would also
prefer to expend as little effort as possible doing so and as luck would have it, he knows a guy
that can help optimize his time and GPA using the magic of management science. The model
they develop is built around the notion of time spent studying and doing all the assignments he
has neglected throughout the semester. The model is as follows, where S represents time spent
studying (in minutes) and A represents time spent making up assignments (also in minutes).
Maximize Z = 6S + 4A
subject to:
HLC 12S + 10A โฅ 100
SC 6S + 8A โฅ 64
W 7S – 3A โฅ 36
Graphing was never one of Nathan’s strengths, so it is up to you to develop a graphical solution
to his problem and advise him on how much time should be invested in studying and how much
time should be spent catching up on assignments.
Answer: The two corner points meriting investigation are (in (Studying, Assignments) order)
Z(10.67,0) = 64
Z(6.48,3.13) = 51.46 the optimal solution
So, 6 minutes of studying and 3 minutes of working on assignments was all that was required for
my first born to successfully complete his first semester with something other than a 0.0 GPA.
Sad, but true.
Diff: 2 Page Ref: 53
Section Heading: A Minimization Model Example
Keywords: graphical solution
AACSB: Analytical thinking
30
Copyright ยฉ 2019 Pearson Education, Inc.
113) Consider the following linear program:
MAX Z = 25A + 30B
s.t. 12A + 15B โค 300
8A + 7B โค 168
10A + 14B โค 280
Solve this linear program graphically and determine the optimal quantities of A, B, and the value
of Z.
Answer: Solution shown below. A = 11.67, B = 10.67
Diff: 2 Page Ref: 39
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical linear programming
AACSB: Analytical thinking
31
Copyright ยฉ 2019 Pearson Education, Inc.
114) Consider the following linear program:
MIN Z = 50A + 60B
s.t. 6A + 8B โค 300
14A + 7B โฅ 196
A โฅ 10
Bโฅ8
Solve this linear program graphically and determine the optimal quantities of A, B, and the value
of Z.
Answer: A = 10, B = 8, Z = 980
Diff: 2 Page Ref: 39
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical linear programming
AACSB: Analytical thinking
32
Copyright ยฉ 2019 Pearson Education, Inc.
115) A graphical representation of a linear program is shown below. The shaded area represents
the feasible region, and the dashed line in the middle is the slope of the objective function.
If this is a maximization, which extreme point is the optimal solution?
Answer: E
Diff: 1 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, extreme points, feasible region
AACSB: Analytical thinking
33
Copyright ยฉ 2019 Pearson Education, Inc.
116) A graphical representation of a linear program is shown below. The shaded area represents
the feasible region, and the dashed line in the middle is the slope of the objective function.
What are the equations for any two greater than or equal constraints for this problem?
Answer: The three greater than or equal constraints are Aโฅ10, Bโฅ8, 14A+7Bโฅ98
Diff: 3 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, extreme points, feasible region
AACSB: Analytical thinking
34
Copyright ยฉ 2019 Pearson Education, Inc.
117) A graphical representation of a linear program is shown below. The shaded area represents
the feasible region, and the dashed line in the middle is the slope of the objective function.
Provide a full description of the type of constraint is represented by line JK.
Answer: Line JK is a nonbinding, greater than or equal constraint. It cannot be a less than or
equal constraint because then the problem would be infeasible.
Diff: 2 Page Ref: 57
Section Heading: Irregular Types of Linear Programming Problems
Keywords: graphical solution, multiple optimal solutions
AACSB: Analytical thinking
118) Consider the following linear programming problem:
Max Z = $15x + $20y
Subject to: 8x + 5y โค 40
0.4x + y โฅ 4
x, y โฅ 0
Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is
the amount of slack associated with the first constraint?
Answer: x = 0, y = 8, revenue = $160, s1= 0
Diff: 2 Page Ref: 48
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: graphical solution, slack variables
AACSB: Analytical thinking
35
Copyright ยฉ 2019 Pearson Education, Inc.
119) Given this model
Maximize Z = 6S + 4A
subject to:
12S + 10A โฅ 100
6S + 8A โฅ 64
7S – 3A โฅ 36
What is the optimal solution and the surplus associated with the first constraint?
Answer: The optimal solution lies at S = 6.48 and A = 3.13.
The s1 variable is 9.1892
Diff: 2 Page Ref: 54
Section Heading: A Minimization Model Example
Keywords: surplus
AACSB: Analytical thinking
120) The poultry farmer decided to make his own chicken scratch by combining alfalfa and corn
in rail car quantities. A rail car of corn costs $400 and a rail car of alfalfa costs $200. The
farmer’s chickens have a minimum daily requirement of vitamin K (500 milligrams) and iron
(400 milligrams), but it doesn’t matter whether those elements come from corn, alfalfa, or some
other grain. A unit of corn contains 150 milligrams of vitamin K and 75 milligrams of iron. A
unit of alfalfa contains 250 milligrams of vitamin K and 50 milligrams of iron. Formulate the
linear programming model for this situation.
Answer: Min Z = $4005C + $200A
Subject to: 150C + 250A โฅ 500
75C + 50A โฅ 400
C, A โฅ 0
Diff: 3 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: constraint, model formulation
AACSB: Analytical thinking
121) Consider the following linear programming problem:
MIN Z = 3×1 + 2×2
Subject to: 2×1 + 3×2 โฅ 12
5×1 + 8×2 โฅ 37
x1, x2 โฅ 0
What is minimum cost and the value of x1 and x2 at the optimal solution?
Answer: 9.25 at x1 = 0 and x2 = 4.625
Diff: 3 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: minimization problem
AACSB: Analytical thinking
36
Copyright ยฉ 2019 Pearson Education, Inc.
122) Consider the following linear programming problem:
MIN Z = 3×1 + 2×2
Subject to: 2×1 + 3×2 โฅ 12
5×1 + 8×2 โฅ 37
x1, x2 โฅ 0
What is minimum cost and the value of x1 and x2 at the optimal solution?
Answer: 9.25 at x1 = 0 and x2 = 4.625
Diff: 3 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: minimization problem
AACSB: Analytical thinking
123) Ponder the following linear programming problem:
MIN Z = 3×1 + 8×2
Subject to: 3×1 + 4×2 โฅ 52
3×1 + 4×2 โฅ 38
x1, x2 โฅ 0
What is minimum cost and the value of x1 and x2 at the optimal solution?
Answer: 52 at x1 = 17.33 and x2 = 0.0
Diff: 3 Page Ref: 44
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: minimization problem
AACSB: Analytical thinking
124) The international man of mystery knew the finest haberdashers the world over and
constantly sought to expand his dazzling array of fine suits, ties, and cufflinks. Closet space was
at a premium however, so purchases were carefully weighed. Each suit provides 23 units of
dazzlement, each tie 14, and a set of cufflinks is worth an easy 8. A suit takes up 0.5 cubic feet of
closet space and $900 of budget. A tie costs $135 and cufflinks cost $100 per set. Cufflinks are
tiny โ even in the original box, they take up only .01 cubic feet while ties occupy a lusty .25
cubic feet. He has budgeted $12,000 for clothes on this trip and has 20 cubic feet of closet space
left to fill.
Formulate an objective function and constraints to model this situation.
Answer: Max Dazzlement = 23S + 14T + 8C
subject to:
900S + 135T + 100C โค 12,000
0.5S + 0.25T + 0.01C โค 20
Diff: 3 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: linear programming formulation
AACSB: Analytical thinking
37
Copyright ยฉ 2019 Pearson Education, Inc.
125) Ponder the following linear programming problem:
Max Z = 5×1 + 6×2
Subject to: 3×1 + 4×2 โค 76
8×1 + 9×2 โค 123
3×1 + 3×2 โค 56
x1, x2 โฅ 0
What is the optimal solution point?
Answer: 12.31 at x1 and 2.72 at x2 for an objective function value of 77.897
Diff: 3 Page Ref: 42
Section Heading: A Maximization Model Example
Keywords: optimal solutions
AACSB: Analytical thinking
126) List the four properties of linear programming models and provide an example of a
violation of each.
Answer: Properties and brief discussions are contained in the table. Counter examples will vary.
Proportionality
Additivity
Divisibility
Certainty
The slope of a constraint or objective function is constant. There
are no increasing or decreasing marginal returns on either.
Strictly linear functions – there are no interaction effects among
decision variables.
Non-integer values of decision variables are OK.
All model parameters are known exactly.
Diff: 2 Page Ref: 59
Section Heading: Characteristics of Linear Programming Problems
Keywords: linear programming properties
AACSB: Application of knowledge
127) Formulate all elements of linear program to model your university effort. Include a
narrative that explains each of the components.
Answer: Answers will vary, perhaps dramatically. A noble objective function would seek to
maximize a GPA or minimize total cost. Constraints would likely include budget, hours in a day,
financial capital, conflicts with social endeavors, and others.
Diff: 2 Page Ref: 34
Section Heading: Model Formulation
Keywords: linear programming properties
AACSB: Application of knowledge
38
Copyright ยฉ 2019 Pearson Education, Inc.
128) Consider the following linear programming problem:
MIN Z = 10×1 + 20×2
Subject to: x1 + x2 โฅ 12
2×1 + 5×2 โฅ 40
x2 โค 13
x1, x2 โฅ 0
At the optimal solution, what is the value of surplus associated with constraint 1 and constraint 3,
respectively?
Answer: constraint 1: (0 surplus), constraint 2: (7.667 surplus)
Diff: 2 Page Ref: 52
Section Heading: A Minimization Model Example
Keywords: graphical solution
AACSB: Analytical thinking
129) Given this set of constraints, for what objective function is the point x = 5, y = 3 in the
feasible region?
s.t 3x + 6y โค 30
10x + 10y โค 60
10x + 15y โค 90
Answer: No objective function can move that point into the feasible region.
Diff: 2 Page Ref: 42
Section Heading: Graphical Solutions of Linear Programming Models
Keywords: feasibility, constraints
AACSB: Analytical thinking
130) Consider the following linear programming problem:
MIN Z = 2×1 + 3×2
Subject to: x1 + 2×2 โค 20
5×1 + x2 โค 40
4×1 + 6×2 โค 60
x1, x2 โฅ 0
What is the optimal solution?
Answer: Multiple optimal solutions exist between the extreme point (0,10) and (6.92,5.38) along
the line with a slope of -2/3.
Diff: 2 Page Ref: 52
Section Heading: A Minimization Model Example
Keywords: graphical solution, multiple optimal solutions
AACSB: Analytical thinking
39
Copyright ยฉ 2019 Pearson Education, Inc.
131) A company producing a standard line and a deluxe line of fidget spinners has the following
time requirements (in minutes) in departments where either model can be processed.
Stamping
Extruding
Fidget Testing
Standard
0.3
0.25
1
Deluxe
0.4
0.5
1.5
The standard models contribute $2 each and the deluxe $3 each to profits. Because the company
produces other items that share resources used to make the fidget spinners, the stamping machine
is available only 15 minutes per hour, on average. The extruding unit has 20 minutes available
each hour. There are two ADHD certified inspectors for fidget testing, but their availability is
only 45 minutes per hour because they’re easily distracted.
Let S = number of standard fidget spinners produced per hour
D = number of deluxe fidget spinners produced per hour
Write the formulation for this linear program and solve it using the graphical method.
40
Copyright ยฉ 2019 Pearson Education, Inc.
Answer: Max $2S + $3D
s.t 0.3S + 0.4D โค 15
0.25S + 0.5D โค 20
1S + 1.5D โค 45
The optimal product mix is to make 30 Deluxe units and no Standard units.
Diff: 3 Page Ref: 36
Section Heading: A Maximization Model Example
Keywords: formulation, objective function, constraints
AACSB: Analytical thinking
41
Copyright ยฉ 2019 Pearson Education, Inc.

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