Deck 18: Simplex-Based Sensitivity Analysis and Duality

Question 1

The improvement in the value of the optimal solution per unit increase in a constraint's right-hand side is

A)the slack value.
B)the dual price.
C)never negative.
D)the 100% rule.

Answer 1

B

Question 2

We can often avoid the process of formulating and solving a modified linear programming problem by using the range of optimality to determine whether a change in an objective function coefficient is large enough to cause a change in the optimal solution.

Answer 2

True

Question 3

Dual prices and ranges for objective function coefficients and right-hand-side values are found by considering

A)dual analysis.
B)optimality analysis.
C)ranging analysis.
D)sensitivity analysis.

Answer 3

D

Question 4

The range of optimality is calculated by considering changes in the c_j − z_j value of the variable in question.

Question 5

As long as the actual value of the objective function coefficient is within the range of optimality,the current basic feasible solution will remain optimal.

Question 6

A one-sided range of optimality

A)always occurs for nonbasic variables.
B)always occurs for basic variables.
C)indicates changes in more than one coefficient.
D)indicates changes in a slack variable's coefficient.

Question 7

The dual price is the improvement in value of the optimal solution per unit increase in a constraint's right-hand-side value.

Question 8

If the simplex tableau is from a maximization converted from a minimization,the signs and directions of the inequalities that give the objective function ranges will need to be adjusted to apply to the original coefficients.

Question 9

The range of optimality for a basic variable defines the objective function coefficient values for which that variable will remain part of the current optimal basic feasible solution.

Question 10

The ranges for which the right-hand-side values are valid are the same as the ranges over which the dual prices are valid.

Question 11

The range of feasibility indicates right-hand-side values for which

A)the value of the objective function will not change.
B)the values of the decision variables will not change.
C)those variables that are in the basis will not change.
D)more simplex iterations must be performed.

Question 12

The entries in the associated slack column of the final tableau can also be interpreted as the changes in the values of the current basic variables corresponding to a one-unit increase in the right-hand side.​

Question 13

The dual variable represents the

A)marginal value of the constraint.
B)right-hand-side value of the constraint.
C)artificial variable.
D)technical coefficient of the constraint.

Question 14

The range of optimality is useful only for basic variables.

Question 15

As long as the objective function coefficient remains within the range of optimality,the variable values will not change although the value of the objective function could.

Question 16

For the basic feasible solution to remain optimal,

A)all c_j − z_j values must remain ≤ 0.
B)no objective function coefficients are allowed to change.
C)the value of the objective function must not change.
D)All of these are correct.

Question 17

The dual price for an equality constraint is the z_j value for its artificial variable.

Question 18

A linear programming problem with the objective function 3x₁ + 8x₂ has the optimal solution x₁ = 5,x₂ = 6.If c₂ decreases by 2 and the range of optimality shows 5 ≤ c₂ ≤ 12,the value of Z

A)will decrease by 12.
B)will decrease by 2.
C)will not change.
D)cannot be determined from this information.

Question 19

Within the concept of duality is the original formulation of a linear programming problem known as the primal problem.

Question 20

A dual price is associated with each decision variable.

Question 21

For the following linear programming problem ​
the final tableau is ​
a.Find the range of optimality for c_1, c_2, c_3, c_4, c_5, and c₆.
b.Find the range of feasibility for b₁,b₂,and b₃.
​

Question 22

If the dual price for b₁ is 2.7,the range of feasibility is 20 ≤ b₁ ≤ 50,and the original value of b₁ was 30,which of the following is true?

A)There currently is no slack in the first constraint.
B)We would be willing to pay up to $2.70 per unit for up to 20 more units of resource 1.
C)If only 25 units of resource 1 were available,profit would drop by $13.50.
D)All of these are correct.

Question 23

The primal problem is as follows: ​
The final tableau for its dual problem is as follows: ​
Give the complete solution to the primal problem.

Question 24

Creative Kitchen Tools manufactures a wide line of gourmet cooking tools from stainless steel.For the coming production period,there is demand of 1200 for eight-quart stock pots and unlimited demand for three-quart mixing bowls and large slotted spoons.In the following model,the three variables measure the number of pots,bowls,and spoons to make.The objective function measures profit.Constraint 1 measures steel,constraint 2 measures manufacturing time,constraint 3 measures finishing time,and constraint 4 measures the stock pot demand. ​
The final tableau is as follows: ​
a.Calculate the range of optimality for c₁,c₂,and c₃.
b.Calculate the range of feasibility for b₁,b₂,b₃,and b₄.
c.​
Suppose that the inventory records were incorrect and the company really has only 14,000 units of steel.What effect will this have on your solution?
d.​
Suppose that a cost increase will change the profit on the pots to $4.62.What effect will this have on your solution?
e.​
​
Assume that the cost of time in production and finishing is relevant.Would you be willing to pay a $1.00 premium over the normal cost for 1000 more hours in the production department? What would this do to your solution?
​

Question 25

For the following linear programming problem ​
the final tableau is ​
a.Find the range of optimality for c₁,c₂,c₃,c₄,c₅,and c₆.
b.Find the range of feasibility for b₁ and b₂.
​

Question 26

For the following linear programming problem ​
the final tableau is ​
a.Find the range of optimality for c₁ and c₂.
b.Find the range of feasibility for b₁,b₂,and b₃.
c.Find the dual prices.

Question 27

Write the dual of the following problem: ​
​

Question 28

For this optimal simplex tableau,the original right-hand sides were 100 and 90.The problem was a maximization. ​
a.What would the new solution be if there had been 150 units available in the first constraint?
b.What would the new solution be if there had been 70 units available in the second constraint?
​
​

Question 29

An LP maximization problem with all less-than-or-equal-to constraints and nonnegativity requirements for the decision variables is known as

A)a canonical form for a minimization problem.
B)a canonical form for a maximization problem.
C)always unbounded.
D)None of these are correct.

Question 30

For this optimal simplex tableau,the right-hand sides for the two original ≥ constraints were 300 and 250.The problem was a minimization. ​
a.What would the new solution be if the right-hand-side value in the first constraint had been 325?
b.What would the new solution be if the right-hand-side value for the second constraint had been 220?

Question 31

Write the dual to the following problem.

Question 32

Given the simplex tableau for the optimal primal solution,

A)the values of the dual variables can be found from the c_j − z_j values of the slack/surplus variable columns.
B)the values of the dual surplus variables can be found from the c_j − z_j values of the primal decision variable columns.
C)the value of the dual objective function will be the same as the objective function value for the primal problem.
D)All of these are correct.