Deck 18: Simplex-Based Sensitivity Analysis and Duality

Question 1

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

Answer 1

True

Question 2

There is a dual price associated with each decision variable.

Answer 2

False

Question 3

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 $\leq$ c₂ $\leq$ 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.

Answer 3

will decrease by 12.

Question 4

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)each of the above is true.

Question 5

The dual price is the improvement in value of the optimal solution per unit increase in the value of the right-hand-side associated with a linear programming problem.

Question 6

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 7

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.

Question 8

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.

Question 9

The dual variable represents

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

Question 10

The entries in the associated slack column of the final tableau indicate the changes in the values of the current basic variables corresponding to a one-unit increase in the right-hand side.

Question 11

A one-sided range of optimality

A)always occurs for non-basic 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 12

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

Question 13

If the dual price for b₁ is 2.7, the range of feasibility is 20 $\leq$ b₁ $\leq$ 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)Each of the above is true.

Question 14

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

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

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

Question 17

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 which are in the basis will not change.
D)more simplex iterations must be performed.

Question 18

The range of optimality is useful only for basic variables.

Question 19

For the basic feasible solution to remain optimal

A)all c_j - z_j values must remain $\leq$ 0.
B)no objective function coefficients are allowed to change.
C)the value of the objective function must not change.
D)each of the above is true.

Question 20

The number of constraints to the dual of the following problem is: Max Z = 3x₁ + 2x₂ + 6x₃
S.t.4x₁ + 2x₂ + 3x₃ $\geq$ 100
2x₁ + x₂ - 2x₃ $\leq$ 200
4x₂ + x₃ $\geq$ 200

A)1.
B)2.
C)3.
D)4.