Question 1

Describe each of the sections of output that come from The Management Scientist and how you would use each.

Accepted Answer

The answer of Describe each of the sections of output...

Question 2

The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.

Accepted Answer

The dual price, or shadow price, represents the improvement in the objective function value per unit increase, not decrease, in the right-hand side of a constraint.

Question 3

The binding constraints for this problem are the first and second. Min x₁ + 2x₂ s.t. x₁ + x₂ $\ge$ 300 2x₁ + x₂ $\ge$ 400 2x₁ + 5x₂ $\le$ 750 x₁ , x₂ $\ge$ 0 a.Keeping c₂ fixed at 2, over what range can c₁ vary before there is a change in the optimal solution point? b.Keeping c₁ fixed at 1, over what range can c₂ vary before there is a change in the optimal solution point? c.If the objective function becomes Min 1.5x₁ + 2x₂, what will be the optimal values of x₁, x₂, and the objective function? d.If the objective function becomes Min 7x₁ + 6x₂, what constraints will be binding? e.Find the dual price for each constraint in the original problem.

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The answer of The binding constraints for this problem are...

Question 4

For a minimization problem, a positive dual price indicates the value of the objective function will increase.

Accepted Answer

In a minimization problem, a positive dual price indicates the value of the objective function will decrease if the corresponding constraint is relaxed.

Question 5

In a linear programming problem, the binding constraints for the optimal solution are
5X + 3Y $\le$ 30
2X + 5Y $\le$ 20
a.Fill in the blanks in the following sentence:
As long as the slope of the objective function stays between _______ and _______, the current optimal solution point will remain optimal.
b.Which of these objective functions will lead to the same optimal solution?
1) 2X + 1Y 2) 7X + 8Y 3) 80X + 60Y 4) 25X + 35Y

Accepted Answer

The answer of In a linear programming problem, the binding...

Question 6

Decision variables must be clearly defined before constraints can be written.

Accepted Answer

This is because constraints are based on the values that the decision variables can take, so it is necessary to define them first.

Question 7

Decreasing the objective function coefficient of a variable to its lower limit will create a revised problem that is unbounded.

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The answer of Decreasing the objective function coefficient of a...

Question 8

Explain the two interpretations of dual prices based on the accounting assumptions made in calculating the objective function coefficients.

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The answer of Explain the two interpretations of dual prices...

Question 9

How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for both the right-hand side values and objective function.

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The answer of How would sensitivity analysis of a linear...

Question 10

The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or decreases in that percentage.

Accepted Answer

The answer of The dual price for a percentage constraint...

Question 11

How is sensitivity analysis used in linear programming? Given an example of what type of questions that can be answered.

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The answer of How is sensitivity analysis used in linear...

Question 12

If the optimal value of a decision variable is zero and its reduced cost is zero, this indicates that alternative optimal solutions exist.

Accepted Answer

When the optimal value of a decision variable is zero and its reduced cost is zero, it means that increasing or decreasing its value will not affect the objective function. This suggests that there may be alternative solutions where this decision variable takes on non-zero values while still maintaining the optimality of the solution. Therefore, alternative optimal solutions are possible.

Question 13

There is a dual price for every decision variable in a model.

Accepted Answer

Not every decision variable in a model has a dual price; dual prices are associated with the constraints of a linear programming problem, not directly with decision variables.

Question 14

A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective.

Accepted Answer

A negative dual price for a constraint in linear programming indicates that the objective function would decrease if the right-hand side of the constraint were increased, meaning that it is detrimental to the objective.

Question 15

The amount of a sunk cost will vary depending on the values of the decision variables.

Accepted Answer

Sunk costs are costs that have already been incurred and cannot be recovered, regardless of the value of decision variables. They are independent of current or future decision-making processes.

Question 16

For any constraint, either its slack/surplus value must be zero or its dual price must be zero.

Accepted Answer

This statement is true according to the strong duality theorem in linear programming. If a constraint has a non-zero slack/surplus value, it means it is not binding and its corresponding dual variable (dual price) is zero. On the other hand, if a constraint has a non-zero dual price, it means it is binding and its corresponding slack/surplus value is zero. Therefore, either the slack/surplus value or dual price of a constraint must be zero.

Question 17

The 100% Rule does not imply that the optimal solution will necessarily change if the percentage exceeds 100%.

Accepted Answer

The 100% Rule states that the sum of the resource allocations should not exceed 100%. However, exceeding this limit does not necessarily mean that the solution needs to be changed. It simply means that the resource utilization is at its maximum capacity. It is possible that the current solution is still the most optimal one, even with resource utilization above 100%.

Question 18

The optimal solution of the linear programming problem is at the intersection of constraints 1 and 2. Max 2x₁ + x₂ s.t. 4x₁ + 1x₂ $\le$ 400 4x₁ + 3x₂ $\le$ 600 1x₁ + 2x₂ $\le$ 300 x₁ , x₂ $\ge$ 0 a.Over what range can the coefficient of x₁ vary before the current solution is no longer optimal? b.Over what range can the coefficient of x₂ vary before the current solution is no longer optimal? c.Compute the dual prices for the three constraints.

Accepted Answer

The answer of The optimal solution of the linear programming...

Question 19

Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of feasibility.

Accepted Answer

The answer of Explain the connection between reduced costs and...

Question 20

How can the interpretation of dual prices help provide an economic justification for new technology?

Accepted Answer

The answer of How can the interpretation of dual prices...

Quiz 3: Linear Programming: Sensitivity Analysis and Interpretation of Solution