The Simplex Method
Problem Set 01
1) The simplex method cannot be used to solve quadratic programming problems.
Answer: TRUE
Diff: 3
Main Heading: Converting the Model into Standard Form
Key words: simplex method
2) The simplex method is a general mathematical solution technique for solving linear programming problems.
Answer: TRUE
Diff: 2
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: simplex method
4) The simplex method can be used to solve quadratic programming problems.
Answer: FALSE
Diff: 3
Main Heading: Converting the Model into Standard Form
Key words: simplex method
5) The simplex method is a general mathematical solution technique for solving nonlinear programming problems.
Answer: FALSE
Diff: 2
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: standard form, simplex method
10) Slack variables are added to constraints and represent unused resources.
Answer: TRUE
Diff: 2
Main Heading: Converting the Model into Standard Form
Key words: slack variables, simplex method
11) Slack variables are added to constraints and represent unused resources.
Answer: FALSE
Diff: 2
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: simultaneous equations, row operations
15) The basic feasible solution in the initial simplex tableau is the origin where all decision variables equal zero.
Answer: TRUE
Diff: 1
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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 one.
Answer: FALSE
Diff: 2
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: basic feasible solution, constraints
19) The simplex method does not guarantee an integer solution.
Answer: TRUE
Diff: 1
Main Heading: The Simplex Method
Key words: 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 n the original problem.
Answer: TRUE
Diff: 2
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: simplex method
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
Main Heading: The Simplex Method
Key words: simplex method
23) Final tableaus cannot be used to conduct sensitivity analysis.
Answer: FALSE
Diff: 1
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: dual
25) Multiple optimal solutions cannot be determined from the simplex method.
Answer: FALSE
Diff: 1
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: simplex method
27) The __________ column is the column corresponding to the entering variable.
Answer: pivot
Diff: 2
Key words: pivot column
28) The variable with the largest positive cj - zj is the __________ variable.
Answer: entering
Diff: 2
Key words: entering variable
29) __________ variables are added to constraints and represent unused resources.
Answer: Slack
Diff: 2
Key words: slack variables
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
Key words: standard form
31) The __________ values are contribution to profit for each variable.
Answer: cj
Diff: 2
Key words: 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
Key words: 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
Key words: artificial variables
34) In solving a minimization problem, artificial variables are assigned a __________ in the objective function to eliminate them form the final solution.
Answer: large cost
Diff: 2
Key words: 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
Key words: 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
Key words: alternative optimal solutions, multiple optimal solutions
37) A primal maximization model with ≤ constraints converts to a __________ minimization model with constraints.
Answer: dual
Diff: 2
Key words: dual model
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
Key words: dual model
39) If the primal problem has three constraints, then the corresponding dual problem will have three __________.
Answer: decision variables
Diff: 2
Key words: dual model
40) Whereas the maximization primal model has ≥ constraints, the __________ dual model has constraints.
Answer: minimization
Diff: 1
Key words: 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
Key words: degeneracy
42) In a __________ problem, artificial variables are assigned a very high cost.
Answer: minimization
Diff: 1
Key words: 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
Key words: 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
Key words: 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
Key words: sensitivity analysis
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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 (Cj - Zj) value for S2 at the initial solution?
Answer: 0
Diff: 1
Main Heading: The Simplex Method
Key words: cj - zj values
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: simplex method, simplex tableaus
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: simplex tableaus, simplex iterations
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: slack variables, simplex iterations
The linear programming problem whose output follows determines how many fire red nail polishes, bright red nail polishes, basil green nail polishes, and basic 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
- 3x1+ 5x2 + x4 ≤ 120
- x1 + x3≤ 25
- 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
Main Heading: The Simplex Method
Key words: computer output of linear programming method, slack variables
62) How much time will be used?
Answer: 57
Diff: 2
Main Heading: The Simplex Method
Key words: computer output of linear programming problems, slack variables
63) By how much will the second marketing restriction be exceeded?
Answer: 0
Diff: 2
Main Heading: The Simplex Method
Key words: computer output of linear programming problems, slack variables
64) What is the profit?
Answer: 7475
Diff: 2
Main Heading: The Simplex Method
Key words: comp output of linear prog problems, objective function value
65) To what value can the profit on fire red nail polish drop before the solution would change?
Answer: 87.5
Diff: 2
Main Heading: The Simplex Method
Key words: computer output of linear prog problems, sensitivity analysis
66) By how much can the profit on basil green nail polish increase before the solution would change?
Answer: 12.5
Diff: 2
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: computer output of linear prog problems, sensitivity analysis
69) Consider the following linear programming problem:
MAX Z = 10 x1 + 30 x2
s.t. 4 x1 + 6 x2 ≤ 12
8 x1 + 4 x2 ≤ 16
Use the two tables below to create the initial tableau and perform 1 pivot.
Answer:
Diff: 2
Main Heading: The Simplex Method
Key words: simplex procedure
70) Consider the following linear programming problem and the corresponding final tableau.
MAX Z = 3 x1 + 5 x2
s.t. x1≤ 4
2 x2 ≤ 12
3 x1 + 2 x2 ≥ 18
What is the shadow price for each constraint?
Answer: Constraint 1, 3; constraint 2, 2.5; constraint 3, 0.
Diff: 2
Main Heading: The Simplex Method
Key words: sensitivity analysis, shadow price
71) Consider the following linear programming problem and the corresponding final tableau.
MAX Z = 3 x1 + 5 x2
s.t. x1≤ 4
2 x2 ≤ 12
3 x1 + 2 x2 ≥ 18
What is the sensitivity range for the first constraint?
Answer: maximum decrease of 2, and an infinite increase.
Diff: 2
Main Heading: The Simplex Method
Key words: sensitivity analysis, quantity ranges for constraints
72) Write the dual form of the following linear program.
MAX Z = 3 x1 + 5 x2
s.t. x1≤ 4
2 x2 ≤ 12
3 x1 + 2 x2 ≥ 18
Answer: MIN Z2 = 4 y1 + 12 y2 + 18 y3
s.t. y1 + 3 y3 ≥ 3
2 y2 + 2 y3 ≥ 5
Diff: 2
Main Heading: The Simplex Method
Key words: dual form
73) The simplex method __________ be used to solve quadratic programming problems.
- A) can
- B) cannot
- C) may
- D) should
Answer: B
Diff: 2
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: simplex method
75) Slack variables are added to __________ constraints and represent unused resources.
- A) ≤
- B) <
- C) ≥
- D) >
- E) =
Answer: A
Diff: 2
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: Converting the Model into Standard Form
Key words: standard form, simplex method
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
Main Heading: Converting the Model into Standard Form
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: basic feasible solution
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: simplex method
84) For a maximization linear programming problem, a __________ 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
Main Heading: The Simplex Method
Key words: slack variables
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: slack variable, shadow price
87) In the simplex procedure, if cj - zj = 0 for a nonbasic 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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: simplex irregularity, unbounded solution
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
Main Heading: The Simplex Method
Key words: 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
Main Heading: The Simplex Method
Key words: dual
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