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

  1. 3x1+ 5x2 + x4 ≤ 120
  2. x1 + x3≤ 25
  3. 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.

  1. A) can
  2. B) cannot
  3. C) may
  4. 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.

  1. A) integer
  2. B) non-linear
  3. C) linear
  4. 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.

  1. A) ≤
  2. B) <
  3. C) ≥
  4. D) >
  5. 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.

  1. A) first
  2. B) second
  3. C) last
  4. 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.

  1. A) converted
  2. B) restrained
  3. C) divided
  4. 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 __________.

  1. A) 0
  2. B) 1
  3. C) -1
  4. 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 __________.

  1. A) 0
  2. B) 1
  3. C) -1
  4. 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.

  1. A) linear
  2. B) slack
  3. C) non-linear
  4. 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.

  1. A) adding, to
  2. B) multiplying, by
  3. C) dividing, by
  4. 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 __________.

  1. A) maximum positive value
  2. B) minimum negative value
  3. C) minimum positive value
  4. 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.

  1. A) sometimes does
  2. B) does
  3. C) does not
  4. 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.

  1. A) surplus, subtracted
  2. B) slack, added
  3. C) artificial, added
  4. D) artificial, subtracted
  5. 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:

  1. A) +M
  2. B) -M
  3. C) 0
  4. D) 1
  5. 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:

  1. A) is always zero
  2. B) is always greater than zero
  3. C) is always less than zero
  4. 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

  1. A) the solution is infeasible
  2. B) the solution is unbounded
  3. C) there are multiple optimal solutions
  4. 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

  1. A) the solution is infeasible
  2. B) the solution is unbounded
  3. C) there are multiple optimal solutions
  4. 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

  1. A) the solution is infeasible
  2. B) the solution is unbounded
  3. C) there are multiple optimal solutions
  4. 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.

  1. A) standard
  2. B) primal
  3. C) feasible
  4. D) dual
  5. E) simplex

Answer:  D

Diff: 2

Main Heading:  The Simplex Method

Key words:  dual

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