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Introduction to Management Science, 11e (Taylor)
Module A: The Simplex Solution Method
1) The simplex method cannot be used to solve quadratic programming problems.
Answer: TRUE
Diff: 3
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
2) The simplex method is a general mathematical solution technique for solving linear programming
problems.
Answer: TRUE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
3) In the simplex method, the model is put into the form of a table, and then a number of mathematical
steps are performed on the table.
Answer: TRUE
Diff: 3
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
4) The simplex method can be used to solve quadratic programming problems.
Answer: FALSE
Diff: 3
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
5) The simplex method is a general mathematical solution technique for solving nonlinear programming
problems.
Answer: FALSE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
6) The simplex method moves from one better solution to another until the best one is found, and then it
stops.
Answer: TRUE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
7) The mathematical steps in the simplex method replicate the process in graphical analysis of moving
from one extreme point on the solution boundary to another.
Answer: TRUE
Diff: 1
1
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Section Heading: Converting the Model into Standard Form
Keywords: simplex method
8) The first step in solving a linear programming model manually with the simplex method is to convert
the model into standard form.
Answer: TRUE
Diff: 1
Section Heading: Converting the Model into Standard Form
Keywords: standard form, simplex method
9) The last step in solving a linear programming model manually with the simplex method is to convert
the model into standard form.
Answer: FALSE
Diff: 1
Section Heading: Converting the Model into Standard Form
Keywords: standard form, simplex method
10) Slack variables are added to constraints and represent unused resources.
Answer: TRUE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: slack variables, simplex method
11) Artificial variables are added to constraints and represent unused resources.
Answer: FALSE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: artificial variables, slack variables, simplex method
12) A basic feasible solution satisfies the model constraints and has the same number of variables with
non-negative values as there are constraints.
Answer: TRUE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: basic feasible solution
13) A basic feasible solution satisfies the model constraints and has the same number of variables with
negative values as there are constraints.
Answer: FALSE
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: basic feasible solution
14) Row operations are used to solve simultaneous equations where equations are multiplied by
constants and added or subtracted from each other.
Answer: TRUE
Diff: 2
Section Heading: Converting the Model into Standard Form
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Keywords:
simultaneous equations, row operations
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15) The basic feasible solution in the initial simplex tableau is the origin where all decision variables
equal zero.
Answer: TRUE
Diff: 1
Section Heading: The Simplex Method
Keywords: basic feasible solution, initial simplex tableau
16) At the initial basic feasible solution at the origin, only slack variables have a value greater than zero.
Answer: TRUE
Diff: 2
Section Heading: The Simplex Method
Keywords: basic/initial basic feasible solution, slack variables
17) At the initial basic feasible solution at the origin, only slack variables have a value greater than 1.
Answer: FALSE
Diff: 2
Section Heading: The Simplex Method
Keywords: basic/initial basic feasible solution, slack variables
18) In using the simplex method, the number of basic variables is equal to the number of constraints.
Answer: TRUE
Diff: 2
Section Heading: The Simplex Method
Keywords: basic feasible solution, constraints
19) The simplex method does not guarantee an integer solution.
Answer: TRUE
Diff: 1
Section Heading: The Simplex Method
Keywords: simplex method
20) In solving a linear programming problem with simplex method, the number of basic variables is the
same as the number of constraints in the original problem
Answer: TRUE
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex method
21) A change in the objective function coefficient of a basic variable cannot change the value of zj for a
non-basic variable in the final simplex tableau.
Answer: FALSE
Diff: 3
Section Heading: The Simplex Method
Keywords: simplex method
4
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22) When solving a linear programming problem, a decision variable that leaves the basis in one
iteration of the simplex method can return to the basis on a later iteration.
Answer: TRUE
Diff: 3
Section Heading: The Simplex Method
Keywords: simplex method
23) Final tableaus cannot be used to conduct sensitivity analysis.
Answer: FALSE
Diff: 1
Section Heading: The Simplex Method
Keywords: simplex method
24) The dual form of a linear program is used to determine how much one should pay for additional
resources.
Answer: TRUE
Diff: 1
Section Heading: The Simplex Method
Keywords: dual
25) Multiple optimal solutions cannot be determined from the simplex method.
Answer: FALSE
Diff: 1
Section Heading: The Simplex Method
Keywords: simplex method
26) The theoretical limit on the number of decision variables that can be handled by the simplex method
is 50.
Answer: FALSE
Diff: 1
Section Heading: The Simplex Method
Keywords: simplex method
27) The ________ column is the column corresponding to the entering variable.
Answer: pivot
Diff: 2
Keywords: pivot column
28) The variable with the largest positive cj - zj is the ________ variable.
Answer: entering
Diff: 2
Keywords: entering variable
29) ________ variables are added to constraints and represent unused resources.
Answer: Slack
Diff: 2
Keywords: slack variables
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30) The first step in solving a linear programming model manually with the simplex method is to
convert the model into ________ form.
Answer: standard
Diff: 2
Keywords: standard form
31) The ________ values are contribution to profit for each variable.
Answer: cj
Diff: 2
Keywords: cj values, contribution to profit.
32) The ________ values are computed by multiplying the cj column values by the variable column
values and summing.
Answer: zj
Diff: 2
Keywords: zj values
33) The ________ variable allows for an initial basic feasible solution, but it has no meaning. Therefore,
after we get the simplex tableau started, they are discarded in later iterations.
Answer: artificial
Diff: 2
Keywords: artificial variables
34) In solving a minimization problem, artificial variables are assigned a ________ in the objective
function to eliminate them from the final solution.
Answer: large cost
Diff: 2
Keywords: artificial variables
35) A(n) ________ maximization linear programming problem has an artificial variable in the final
simplex tableau where all cj - zj values are less than or equal to zero.
Answer: infeasible
Diff: 2
Keywords: infeasible problem, infeasible solution
36) In using the simplex method, ________ optimal solutions are identified by cj - zj = 0 for a non-basic
variable.
Answer: multiple or alternative
Diff: 2
Keywords: alternative optimal solutions, multiple optimal solutions
37) A primal maximization model with ≤ constraints converts to a ________ minimization model with
constraints.
Answer: dual
Diff: 2
Keywords: dual model
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38) The quantity values on the right-hand side of the primal inequality constraints are the ________
coefficients in the dual.
Answer: objective function
Diff: 2
Keywords: dual model
39) If the primal problem has three constraints, then the corresponding dual problem will have three
________.
Answer: decision variables
Diff: 2
Keywords: dual model
40) Whereas the maximization primal model has ≤ constraints, the ________ dual model has ≥
constraints.
Answer: minimization
Diff: 1
Keywords: dual model
41) ________ in linear programming is when a basic variable takes on a value of zero (i.e., a zero in the
right-hand side of the constraints of the tableau).
Answer: Degeneracy
Diff: 2
Keywords: degeneracy
42) In a ________ problem, artificial variables are assigned a very high cost.
Answer: minimization
Diff: 1
Keywords: artificial variables
43) A(n) ________ problem can be identified in the simplex procedure when it is not possible to select a
pivot row.
Answer: unbounded
Diff: 2
Keywords: simplex irregularity, unbounded solution
44) The ________ form of a linear program is used to determine how much one should pay for
additional resources.
Answer: dual
Diff: 2
Keywords: dual
45) To determine the sensitivity range for the coefficient of a variable in the objective function,
calculations are performed such that all values in the cj - zj row are ________.
Answer: less than or equal to zero
Diff: 2
Keywords: sensitivity analysis
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46) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What are the basic variables in the initial tableau?
Answer: S1, S2
Diff: 1
Section Heading: The Simplex Method
Keywords: basic variables, initial tableau
47) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What are the Cj values for the basic variables?
Answer: 0, 0
Diff: 1
Section Heading: The Simplex Method
Keywords: basic variables, objective function coefficients
48) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the (Cj - Zj) value for S1 at the initial solution?
Answer: 0
Diff: 1
Section Heading: The Simplex Method
Keywords: Cj - Zj values
49) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the (Ci- Zi) value for S2 at the initial solution?
Answer: 0
Diff: 1
Section Heading: The Simplex Method
Keywords: Cj - Zj values
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50) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the value of X1 in the final tableau?
Answer: 0 or 4.25
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex method, simplex tableaus
51) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the value of x2 in the final tableau?
Answer: 2 or 7.667
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex method, simplex tableaus
52) Solve the following problem using the simplex method.
Minimize Z = 3x1 + 4x2 + 8x3
Subject to:
2x1 + x2 ≥ 6
x2 + 2x3 ≥ 4
x1, x2 ≥ 0
Answer: x1 = 1, x2 = 4, x3 = 0 and Z = 19
Diff: 3
Section Heading: The Simplex Method
Keywords: simplex method, simplex tableaus
53) Solve the following problem using the simplex method.
Minimize Z = 2x1 + 6x2
Subject to:
2x1 + 4x2 ≤ 12
3x1 + 2x2 ≥ 9
x1, x2 ≥ 0
Answer: x1 = 1.5, x2 = 2.25, and Z = 16.5
Diff: 3
Section Heading: The Simplex Method
Keywords: simplex method, simplex tableaus
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54) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
What is the optimal value of this objective function?
Answer: 23
Diff: 2
Section Heading: The Simplex Method
Keywords: objective function value, simplex tableaus
55) Given the following linear programming problem:
maximize 4x1 + 3x2
subject to 4x1 + 3x2 ≤ 23
5x1 - x2 ≤ 5
x1, x2 ≥ 0
How many iterations did we have to perform before reaching the final tableau?
Answer: 3
Diff: 3
Section Heading: The Simplex Method
Keywords: simplex tableaus, simplex iterations
56) Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the optimal value for X1?
Answer: 16
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex tableaus, simplex iterations
57) Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the optimal value for X2?
Answer: 12
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex tableaus, simplex iterations
10
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58) Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the value for S2 in the optimal tableau?
Answer: 0
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex tableaus, simplex iterations
59) Given the following linear programming problem:
maximize
Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the optimal value for the objective function?
Answer: $2560
Diff: 2
Section Heading: The Simplex Method
Keywords: objective function value
60) Given the following linear programming problem:
maximize Z = $100x1 + 80x2
subject to x1 + 2x2 ≤ 40
3x1 + x2 ≤ 60
x1, x2 ≥ 0
Using the simplex method, what is the value for S1 in the final basic feasible solution?
Answer: 0
Diff: 2
Section Heading: The Simplex Method
Keywords: slack variables, simplex iterations
11
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The linear programming problem whose output follows determines how many red nail polishes, blue
nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective
function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures
display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4
are marketing restrictions.
MAX
100x1 + 120x2 + 150x3 + 125x4
Subject to 1. x1 + 2x2 + 2x3 + 2x4 ≤ 108
2. 3x1 + 5x2 + x4 ≤ 120
3. x1 + x3 ≤ 25
4. x2 + x3 + x4 > 50
x1, x2, x3, x4 ≤ 0
Optimal Solution:
Objective Function Value = 7475.000
Objective Coefficient Ranges
Right Hand Side Ranges
61) How much space will be left unused?
Answer: 0
Diff: 1
Section Heading: The Simplex Method
12
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Keywords:
computer output of linear programming method, slack variables
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62) How much time will be used?
Answer: 57
Diff: 2
Section Heading: The Simplex Method
Keywords: computer output of linear programming problems, slack variables
63) By how much will the second marketing restriction be exceeded?
Answer: 0
Diff: 2
Section Heading: The Simplex Method
Keywords: computer output of linear programming problems, slack variables
64) What is the profit?
Answer: 7475
Diff: 2
Section Heading: The Simplex Method
Keywords: comp output of linear prog problems, objective function value
65) To what value can the profit on red nail polish drop before the solution would change?
Answer: 87.5
Diff: 2
Section Heading: The Simplex Method
Keywords: computer output of linear prog problems, sensitivity analysis
66) By how much can the profit on green nail polish increase before the solution would change?
Answer: 12.5
Diff: 2
Section Heading: The Simplex Method
Keywords: computer output of linear prog problems, sensitivity analysis
67) By how much can the amount of space decrease before there is a change in the profit?
Answer: 0
Diff: 2
Section Heading: The Simplex Method
Keywords: computer output of linear prog problems, sensitivity analysis
68) You are offered the chance to obtain more space. The offer is for 15 units and the total price is 1500.
What should you do?
Answer: Reject the offer.
Diff: 2
Section Heading: The Simplex Method
Keywords: computer output of linear prog problems, sensitivity analysis
14
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69) Consider the following linear programming problem:
MAX
s.t.
Z = 10x1 + 30x2
4x1 + 6x2 ≤ 12
8x1 + 4x2 ≤ 16
Use the two tables below to create the initial tableau and perform 1 pivot.
Answer:
15
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Diff: 2
Section Heading: The Simplex Method
Keywords: simplex procedure
70) Consider the following linear programming problem and the corresponding final tableau.
MAX
s.t.
Z = 3x1 + 5x2
x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≥ 18
What is the shadow price for each constraint?
Answer: constraint 1, 3; constraint 2, 2.5; constraint 3, 0
Diff: 2
Section Heading: The Simplex Method
Keywords: sensitivity analysis, shadow price
16
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71) Consider the following linear programming problem and the corresponding final tableau.
MAX
s.t.
Z = 3x1 + 5x2
x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≥ 18
What is the sensitivity range for the first constraint?
Answer: maximum decrease of 2, and an infinite increase
Diff: 2
Section Heading: The Simplex Method
Keywords: sensitivity analysis, quantity ranges for constraints
72) Write the dual form of the following linear program.
MAX
s.t.
Z = 3x1 + 5x2
x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≥ 18
Answer:
MIN Zd = 4y1 + 12y2 + 18y3
s.t.
y1 + 3y3 ≥ 3
2y2 + 2y3 ≥ 5
Diff: 2
Section Heading: The Simplex Method
Keywords: dual form
17
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73) The simplex method ________ be used to solve quadratic programming problems.
A) can
B) cannot
C) may
D) should
Answer: B
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
74) The simplex method is a general mathematical solution technique for solving ________
programming problems.
A) integer
B) non-linear
C) linear
D) A, B, and C
Answer: C
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: simplex method
75) Slack variables are added to ________ constraints and represent unused resources.
A) ≤
B) <
C) ≥
D) >
E) =
Answer: A
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: slack variables
76) The ________ step in solving a linear programming model manually with the simplex method is to
convert the model into standard form.
A) first
B) second
C) last
D) only
Answer: A
Diff: 2
Section Heading: Converting the Model into Standard Form
Keywords: standard form, simplex method
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77) Row operations are used to solve simultaneous equations where equations are ________ by
constants and added to or subtracted from each other.
A) converted
B) restrained
C) divided
D) multiplied
Answer: D
Diff: 3
Section Heading: Converting the Model into Standard Form
Keywords: row operations, simultaneous equations
78) The basic feasible solution in the initial simplex tableau is the origin where all decision variables
equal:
A) 0
B) 1
C) -1
D) 1 or -1
Answer: A
Diff: 3
Section Heading: The Simplex Method
Keywords: basic feasible solution, initial simplex tableau
79) At the initial basic feasible solution at the origin, only slack variables have a value greater than:
A) 0
B) 1
C) -1
D) 1 or -1
Answer: A
Diff: 3
Section Heading: The Simplex Method
Keywords: basic feasible solution, initial simplex tableau
80) At the initial basic feasible solution at the origin, only ________ variables have a value greater than
zero.
A) linear
B) slack
C) non-linear
D) integer
Answer: B
Diff: 1
Section Heading: The Simplex Method
Keywords: basic feasible solution
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81) The leaving variable is determined by ________ the quantity values ________ the pivot column
values and selecting the minimum possible value or zero.
A) adding, to
B) multiplying, by
C) dividing, by
D) subtracting, from
Answer: C
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex tableau, leaving variable
82) The leaving variable is determined by dividing the quantity values by the pivot column values and
selecting the
A) maximum positive value.
B) minimum negative value.
C) minimum positive value.
D) maximum negative value.
Answer: C
Diff: 3
Section Heading: The Simplex Method
Keywords: simplex tableau, leaving variable
83) The simplex method ________ guarantee integer solutions.
A) sometimes does
B) does
C) does not
D) may
Answer: C
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex method
84) For a maximization linear programming problem, a(n) ________ is ________ for a
less-than-or-equal-to constraint.
A) surplus, subtracted
B) slack, added
C) artificial, added
D) artificial, subtracted
E) surplus, added
Answer: B
Diff: 2
Section Heading: The Simplex Method
Keywords: slack variables
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85) The objective function coefficient of an artificial variable for a minimization linear programming
problem is:
A) +M
B) -M
C) 0
D) 1
E) an arbitrary value between 0 and positive infinity
Answer: A
Diff: 2
Section Heading: The Simplex Method
Keywords: artificial variables
86) If a slack variable has a positive value (is basic) in the optimal solution to a linear programming
problem, then the shadow price of the associated constraint
A) is always zero.
B) is always greater than zero.
C) is always less than zero.
D) could be any value (i.e., zero greater than zero or less than zero).
Answer: A
Diff: 2
Section Heading: The Simplex Method
Keywords: slack variable, shadow price
87) In the simplex procedure, if cj - zj = 0 for a non-basic variable, this indicates that
A) the solution is infeasible.
B) the solution is unbounded.
C) there are multiple optimal solutions.
D) the formulation is incorrect.
Answer: C
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex irregularity, multiple optimal solutions
88) In the simplex procedure, if it is not possible to select a pivot row, this indicates that
A) the solution is infeasible.
B) the solution is unbounded.
C) there are multiple optimal solutions.
D) the formulation is incorrect.
Answer: B
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex irregularity, unbounded solution
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89) In the simplex procedure, if all cj - zj ≤ 0 and one or more of the basic variables are artificial, this
indicates that
A) the solution is infeasible.
B) the solution is unbounded.
C) there are multiple optimal solutions.
D) the formulation is incorrect.
Answer: A
Diff: 2
Section Heading: The Simplex Method
Keywords: simplex irregularity, infeasible solution
90) The ________ form of a linear program is used to determine how much one should pay for
additional resources.
A) standard
B) primal
C) feasible
D) dual
E) simplex
Answer: D
Diff: 2
Section Heading: The Simplex Method
Keywords: dual
22
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