Deck 22: Linear Programming: The Simplex Method

Question 1

A surplus variable is added to a constraint in order to create an equality, and represents a quantity of unused resource.

Answer 1

False

Question 2

We can solve a minimization problem by maximizing the negative of the minimization problem's objective function.

Answer 2

True

Question 3

In a maximization problem, if a variable is to enter the solution, it must have a positive coefficient in the C_j - Z_j row.

Answer 3

True

Question 4

The constraint 5 X₁ + 6 X₂ ≤ 30, when converted to an = constraint for use in the simplex algorithm, will be 5 X₁ + 6 X₂ - S = 30.

Question 5

The simplex method begins with an initial feasible solution in which all real variables are set equal to 0.

Question 6

A basic feasible solution to a system of n equations is found by setting n variables equal to 0 and solving for the other variables.

Question 7

In a maximization problem, the C_j - Z_j row gives the net loss from introducing one unit of each variable into the solution.

Question 8

Artificial variables can be used in both maximization and minimization problems but are most often used in minimization problems.

Question 9

In the simplex process, the new pivot row is found by dividing each number in the row by the pivot number.

Question 10

Slack and surplus variables are used in simplex only for the solution of maximization problems.

Question 11

In linear programs with more than two decision variables, the area of feasible solutions is represented by an n-dimensional polyhedron.

Question 12

For a maximization problem, the Z_j values in the body of the simplex table represent the gross profit given up by adding one unit of this variable into the current solution.

Question 13

A surplus variable is added to an = constraint in order to utilize the simplex algorithm.

Question 14

A basic feasible solution is a solution to a linear programming problem that corresponds to a corner point of the feasible region.

Question 15

Because slack variables yield no profit, they are not included in the objective function when creating the initial simplex tableau.

Question 16

The constraint 5 X₁ + 6 X₂ = 30, when converted to an = constraint for use in the simplex algorithm, will be 5 X₁ + 6 X₂ - A = 30.

Question 17

The constraint 5X₁ + 6X₂ ≥ 30, when converted to an = constraint for use in the simplex algorithm, will be 5 X₁ + 6 X₂ - S = 30.

Question 18

In any linear programming problem, if a variable is to enter the solution, it must have a positive coefficient in the C_j - Z_j row.

Question 19

A correctly formulated linear program, when solved with the simplex algorithm, will always yield a single optimal solution.

Question 20

In a maximization problem, if a variable is to enter the solution, it must have a negative coefficient in the C_j - Z_j row.

Question 21

An infeasible solution is indicated when all the C_j - Z_j row entries are of the proper sign to imply optimality, but an artificial variable remains in the solution.

Question 22

How does Karmarkar's Algorithm differ from the simplex method?

A)It follows a path of integer solutions.
B)It follows a path inside the feasible region.
C)It follows a counterclockwise path around the outside edges of the feasible region.
D)It follows a clockwise path around the outside edges of the feasible region.

Question 23

The shadow price is the value of one additional unit of a scarce resource across the range (-∞,∞).

Question 24

Karmarkar's algorithm is especially useful in solving very large-scale LP problems.

Question 25

Sensitivity testing of basic variables involves reworking the initial simplex tableau.

Question 26

A minimization problem with two decision variables and three ≥ constraints can have two non-basic surplus variables.

Question 27

Shadow prices are the positives of the numbers in the C_j - Z_j row's slack variable columns.

Question 28

In 1984, who developed an alternative to the simplex algorithm, which often takes significantly less computer time to solve very large-scale LP problems?

A)Carmichael
B)Kruskev
C)Karmarkar
D)Krakov

Question 29

Karmarkar's algorithm reaches a solution quickly by compressing the edges of the feasible region.

Question 30

Basic variables may be in the solution mix or out of the solution mix, just not at the same time.

Question 31

A simplex formulation that suffers from degeneracy means that the problem has no optimal solution.

Question 32

As we are doing the ratio calculations for a simplex iteration, if there is a tie for the smallest ratio, the problem is degenerate.

Question 33

Which of the following is not true about slack variables in a simplex tableau?

A)They are used to convert ≤ constraint inequalities to equations.
B)They represent unused resources.
C)They require the addition of an artificial variable.
D)They may represent machine time, labor hours, or warehouse space.

Question 34

If, at an optimal solution, the C_j - Z_j value for a real variable that is not in the solution mix has a value of one, there are multiple optimal solutions.

Question 35

The dual of a primal minimization problem with two decision variables and three constraints will be a maximization problem with three decision variables in the objective function.

Question 36

Dual variables represent the potential value of resources.

Question 37

The substitution rates give

A)the number of units of each basic variable that must be removed from the solution if a new variable is entered.
B)the gross profit or loss given up by adding one unit of a variable into the solution.
C)the net profit or loss that will result from introducing one unit of the variable indicated in that column into the solution.
D)the maximal value a variable can take on and still have all the constraints satisfied.

Question 38

If all the numbers in the pivot column are negative, this implies that the solution is unbounded.

Question 39

The dual problem formulation can be solved using the same simplex process used for the primal formulation.

Question 40

The dual of a maximization problem with three decision variables and two constraints will be a maximization problem with only two decision variables in the objective function.

Question 41

Table M7-1
According to Table M7-1, all of the resources are being used.If the amount of resource A were changed from 64 to 65, then the maximum possible total profit would be

A)416.
B)417.
C)416.5.
D)415.5.

Question 42

How should the constraint, 5X - 2Y = 6, be converted into simplex tableau form?

A)5X - 2Y + A = 6
B)5X - 2Y - A = 6
C)5X - 2Y + A₁ - A₂ = 6
D)5X - 2Y + S = 6

Question 43

In LP problems with more than two variables, the area of feasible solutions is known as an n-dimensional

A)pentagon.
B)cube.
C)octagon.
D)polyhedron.

Question 44

Consider the following linear programming problem: Maximize 40 X1 + 30 X2 + 60X3
Subject to: X1 + X2 + X3 ≥ 90
12 X1 + 8 X2 + 10 X3 ≤ 1500
X1 , X2 , X3 ≥ 0
How many slack, surplus, and artificial variables would be necessary if the simplex were used to solve this problem?

A)3 slack, 3 surplus, and 3 artificial
B)1 slack, 2 surplus, and 2 artificial
C)1 slack, 4 surplus, and 4 artificial
D)1 slack, 1 surplus, and 1 artificial

Question 45

The substitution rates

A)decrease with each iteration of the simplex method.
B)tell us the number of units of a basic variable that must be removed from the solution for another variable to enter the basis.
C)tell the amount of one resource that can be substituted for another.
D)tell us the amount of a resource that must be used to gain another unit of profit.

Question 46

The number -2 in the X₂ column and X₁ row of a simplex tableau implies that

A)if 1 unit of X₂ is added to the solution, X₁ will decrease by 2.
B)if 1 unit of X₁ is added to the solution, X₂ will decrease by 2.
C)if 1 unit of X₂ is added to the solution, X₁ will increase by 2.
D)if 1 unit of X₁ is added to the solution, X₂ will increase by 2.

Question 47

Table M7-1
According to Table M7-1, it is currently profitable to produce some units of X₁ and the current profit per unit of X₁ is $20.What is the lowest value that this could be to allow this variable to remain in the basis?

A)8
B)16
C)20
D)30

Question 48

If one changes the contribution rates in the objective function of an LP problem

A)the feasible region will change.
B)the slope of the isoprofit or isocost line will change.
C)the optimal solution to the LP will no longer be optimal.
D)the number of decision variables will change

Question 49

Consider the following general form of a linear programming problem: Maximize Profit
Subject to: Amount of resource A used ≤ 100 units
Amount of resource B used ≤ 240 units
Amount of resource C used ≤ 50 units
The shadow price for S1 is 25, for S2 is 0, and for S3 is 40.If the right-hand side of constraint 3 were changed from 150 to 151, what would happen to maximum possible profit?

A)It would not change.
B)It would increase by 25.
C)It would decrease by 25.
D)It would increase by 40.

Question 50

Shadow prices represent

A)the value of one additional unit of a basic variable.
B)the value of one less unit of a basic variable.
C)the value of one less unit of a specific resource.
D)the value of one additional unit of a specific resource.

Question 51

Table M7-2
According to Table M7-2, which is a summarized solution output from simplex analysis, the optimal solution to this problem is

A)X₁ = 0, X₂ = 16.667, S₁ = 300, S₂ = 0.
B)X₁ = 0, X₂ = 16.667, S₁ = 0, S₂ = 0.
C)X₁ = 12, X₂ = 30, S₁ = 0, S₂ = 0.625.
D)X₁ = 12, X₂ = 30, S₁ = 0.625, S₂ = 0.

Question 52

Shadow prices

A)can be derived from the coefficients of the slack variables in the C_j - Z_j row of an optimal simplex tableau.
B)represent the value of a one-unit increase of the contribution rate.
C)are found in the solution to the primal LP.
D)reflect the optimal solution in a parallel universe.

Question 53

Table M7-1
According to Table M7-1, all of the resources are being used.If the amount of resource B were changed from 96 to 97, then the maximum possible total profit would be

A)416.
B)417.
C)419.
D)420.

Question 54

If one changes a nonbasic objective function coefficient, the optimal solution of a maximization problem will remain optimal if

A)the increase in the coefficient does not exceed the value of the Z_j associated with that nonbasic variable.
B)the increase in the coefficient does not exceed the values of the Z_j's of every basic variable.
C)the decrease in the coefficient does not exceed the value of the Z_j associated with the nonbasic variable.
D)the new C_j - Z_j associated with the nonbasic variable remains positive.

Question 55

How should the constraint, 5X - 2Y ≥ 6, be converted into simplex tableau form?

A)5X - 2Y + S + A = 6
B)5X - 2Y - S + A = 6
C)5X - 2Y - S = 6
D)5X - 2Y + A = 6

Question 56

A slack variable

A)is added to each ≤ constraint to facilitate the simplex process.
B)is added to each ≥ constraint to facilitate the simplex process.
C)is added to each ≤ or = constraint to facilitate the simplex process.
D)is added to each = constraint to facilitate the simplex process.

Question 57

Which of the following is not true of artificial variables?

A)They have no meaning in a physical sense-nothing more than a computational tool.
B)In all linear programs, they appear in the objective function with a very low cost ($M).
C)They are usually used with ≥ constraints.
D)They are usually used with = constraints.

Question 58

Table M7-2
According to Table M7-2, which is a summarized solution output from simplex analysis, if the amount of resource A were decreased so that there were only 550 units available instead of 600, what would happen to total profits?

A)They would decrease.
B)They would increase.
C)They would not change.
D)Unable to determine from the given information.

Question 59

The C_j - Z_j row of a simplex tableau gives

A)the number of units of each basic variable that must be removed from the solution if a new variable is entered.
B)the gross profit or loss given up by adding one unit of a variable into the solution.
C)the net profit or loss that will result from introducing one unit of the variable indicated in that column into the solution.
D)the maximal value a variable can take on and still have all the constraints satisfied.

Question 60

Using the simplex method, we know we have an optimal solution when

A)all slack variables have a zero value.
B)all basic variables are negative.
C)all the real variables have a nonzero value.
D)the C_j - Z_j row has no positive values.

Question 61

Nonbasic variables in the simplex method of linear programming are

A)variables in the solution mix.
B)variables not in the solution mix.
C)the real variables in the initial solution.
D)the slack variables in the optimum solution.

Question 62

Table M7-3
According to Table M7-3, which is the final simplex tableau for a linear programming problem (maximization), what would happen to profits if the X₁ column were selected as the pivot column and another iteration of the simplex algorithm were performed?

A)Total profits would increase.
B)Total profits would decrease.
C)An infeasible solution would be found.
D)Another optimal solution would be found.

Question 63

If, in the optimal tableau of a linear programming problem, an artificial variable is present in the solution mix, this implies

A)infeasibility.
B)unboundedness.
C)degeneracy.
D)alternate optimal solutions.

Question 64

Sensitivity analyses are used to examine the effects of changes in

A)objective function switching from maximization to minimization or vice versa.
B)technological coefficients.
C)the types of resources available.
D)the number of constraints.

Question 65

Consider the following linear programming problem: Maximize 40 X1 + 30 X2 + 60X3
Subject to: X1 + X2 + X3 ≥ 90
12 X1 + 8 X2 + 10 X3 ≤ 1500
X1 = 20
X3 ≤ 100
X1 , X2 , X3 ≥ 0
How many slack, surplus, and artificial variables would be necessary if the simplex algorithm were used to solve this problem?

A)3 slack, 6 surplus, and 6 artificial
B)2 slack, 1 surplus, and 2 artificial
C)1 slack, 2 surplus, and 2 artificial
D)1 slack, 2 surplus, and 1 artificial

Question 66

In solving a linear programming minimization problem using the simplex method

A)every time an artificial variable is added, a surplus variable must be subtracted.
B)every time a surplus variable is added, an artificial variable must be added.
C)every time a surplus variable is added, an artificial variable must be subtracted.
D)every time a surplus variable is subtracted, an artificial variable must be added.

Question 67

Table M7-3
According to Table M7-3, which is the final simplex tableau for a problem with two variables and two constraints, what is the maximum possible profit (objective function value)for this problem?

A)20
B)40
C)800
D)26.667

Question 68

Which of the following is true about surplus variables in a simplex tableau?

A)They are used to convert ≤ constraint inequalities to equations.
B)They require the addition of an artificial variable.
C)They represent unused resources.
D)They should be added to the left-hand side of the equation.

Question 69

Variables in the solution mix are called

A)standard.
B)surplus.
C)real.
D)basic.

Question 70

In applying the simplex solution procedure to a minimization problem to determine which variable enters the solution mix

A)pick the one with the most positive C_j - Z_j.
B)pick the one with the largest C_j.
C)pick the one with the smallest Z_j.
D)pick the one with the most negative C_j - Z_j.

Question 71

Table M7-3
According to Table M7-3, which is the final simplex tableau for a problem with two variables and two constraints, the 0.667 in the X₁ column means that

A)to produce 1 unit of X₁, 0.667 units of X₂ must be given up.
B)to produce 1 unit of X₂, 0.667 units of X₁ must be given up.
C)if 1 unit of X₁ is produced, profits on X₂ will decrease by 0.667.
D)if 1 unit of X₁ is produced, profits on X₂ will increase by 0.667.

Question 72

If, in the final optimal simplex tableau, the C_j - Z_j value for a nonbasic variable is zero, this implies

A)infeasibility.
B)unboundedness.
C)degeneracy.
D)alternate optimal solutions.

Question 73

A solved LP problem indicated that the optimal solution was X₁ = 5, X₂ = 10, A₁ = 40.The first constraint was: 4X₁ + 2X₂ ≥ 80.This solution is

A)indeterminant.
B)degenerate.
C)infeasible.
D)unbounded.

Question 74

Sensitivity analysis cannot be used to examine the effects of

A)changes in the contribution rates for each variable.
B)changes in the technological coefficients.
C)changes in the available resources.
D)the addition or deletion of a constraint.

Question 75

An artificial variable has no physical interpretation but

A)is added to each ≤ constraint to facilitate the simplex process.
B)is added to each ≥ constraint to facilitate the simplex process.
C)is added to each ≤ or = constraint to facilitate the simplex process.
D)is merely another manner of introducing a negative slack.

Question 76

Table M7-3
According to Table M7-3, which is the final simplex tableau for a problem with two variables and two constraints, what can be said about the optimal solution and the constraints?

A)There is slack remaining in the first constraint.
B)There is slack remaining in the second constraint.
C)There is slack remaining in both constraints.
D)There is no slack remaining in either constraint.

Question 77

Right-hand side ranging

A)tells us the number of units of a constraint that may be added or subtracted without changing the profit (objective function value).
B)requires computation of the primal before being accomplished.
C)is a method to find the range over which shadow prices vary the greatest.
D)rely on left-hand side dominance.

Question 78

Sensitivity analysis may be used to

A)experiment with possible future changes in the firm that may affect profits.
B)determine whether a corner point of the feasible region actually yields optimal profit.
C)replace the simplex method.
D)reduce the number of variables in a complex LP problem.

Question 79

Table M7-3
According to Table M7-3, which is the final simplex tableau for a problem with two variables and two constraints, what are the values for all the variables in this solution?

A)X₁ = 20, X₂ = 40, S₁ = 0, S₂ = 0
B)X₁ = 0, X₂ = 40, S₁ = 0, S₂ = 40
C)X₁ = 0, X₂ = 20, S₁ = 0, S₂ = 40
D)X₁ = 0, X₂ = 0, S₁ = 0, S₂ = 0

Question 80

In applying the simplex solution procedure to a maximization problem to determine which variable enters the solution mix

A)pick the one with the largest positive C_j - Z_j.
B)pick the one with the smallest C_j - Z_j.
C)pick the one with the largest C_j.
D)pick the one with the smallest Z_j.