FiltersDone

Question type

Essay

Multiple Choice

Short Answer

True False

Matching

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

Show Answer

Share

---

All types

Filters

Study Flashcards

Question 1

Essay

Review LaterAdd Note

Review Later

Use the following Management Scientist output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX 31X1+35X2+32X3 S.T. 1) 3X1+5X2+2X3>90 2) 6X1+7X2+8X3<150 3) 5X1+3X2+3X3<120 OPTIMAL SOLUTION Objective Function Value = 763.333 $\begin{array} { c r c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { X1 } & 13.333 & 0.000 \\\text { X2 } & 10.000 & 0.000 \\\text { X3 } & 0.000 & 10.889\end{array}$ $\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\ 1 & 0.000 & - 0.778 \\ 2 & 0.000 & 5.556 \\ 3 & 23.333 & 0.000 \end{array}$ OBJECTIVE COEFFICIENT RANGES $\begin{array}{cccc}\text { Variable }&\text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline\text { X1 } & 30.000 & 31.000 & \text { No Upper Limit } \\\text { X2 } & \text { No Lower Limit } & 35.000 & 36.167 \\\text { X3 } & \text { No Lower Limit } & 32.000 & 42.889\end{array}$ $\text { RIGHT HAND SIDE RANGES }$ $\begin{array}{cccc}\text { Constraint }&\text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline1 & 77.647 & 90.000 & 107.143 \\2 & 126.000 & 150.000 & 163.125 \\3 & 96.667 & 120.000 & \text { No Upper Limit }\end{array}$ a.Give the solution to the problem. b.Which constraints are binding? c.What would happen if the coefficient of x₁ increased by 3? d.What would happen if the right-hand side of constraint 1 increased by 10?

Correct Answer

verified

a.x₁ = 13.33, x₂ = 10, x₃ = 0, s₁ = 0, s₂ = 0...

View Answer

Show Answer

Question 2

True/False

Review LaterAdd Note

Review Later

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

A) True
B) False

Correct Answer

verified

Show Answer

Question 3

Essay

Review LaterAdd Note

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

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.

The amount that the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the

The range of feasibility measures

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

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.

Output from a computer package is precise and answers should never be rounded.

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

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

To solve a linear programming problem with thousands of variables and constraints

The amount by which an objective function coefficient would have to improve before it would be possible for the corresponding variable to assume a positive value in the optimal solution is called the

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

LINDO output is given for the following linear programming problem. MIN 12 X1 + 10 X2 + 9 X3 SUBJECT TO 2) 5 X1 + 8 X2 + 5 X3 > = 60 3) 8 X1 + 10 X2 + 5 X3 > = 80 END LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1) 80.000000 $\begin{array} { c c c } \text { VARIABLE } & \text { VALUE } & \text { REDUCED COST } \\\text { X1 } & .000000 & 4.000000 \\\text { X2 } & 8.000000 & .000000 \\\text { X3 } & .000000 & 4.000000\end{array}$ $\begin{array}{rrr}\text { ROW } & \text { SLACK OR SURPLUS } & \text { DUAL PRICE }\\\text { 2) } & 4.000000 & .000000 \\3) & .000000 & -1.000000\end{array}$ NO. ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: $\begin{array} { c c c c } && { \text { OBJ. COEFFICIENT RANGES } } \\ & \text { CURRENT } & \text { ALLOWABLE } & \text { ALLOWABLE } \\\text { VARIABLE }&\text { COEFFICIENT }& \text { INCREASE } & \text { DECREASE } \\\hline \text { X1 } & 12.000000 & \text { INFINITY } & 4.000000 \\\text { X2 } & 10.000000 & 5.000000 & 10.000000 \\\text { X3 } & 9.000000 &\text { INFINITY } & 4.000000\\\end{array}$ $\begin{array}{cccc}&&&\text { RIGHT HAND SIDE RANGES }\\&\text { CURRENT } & \text { ALLOWABLE }& \text { ALLOWABLE }\\\text { ROW } & \text { RHS } & \text { INCREASE } & \text { DECREASE } \\\hline2 & 60.000000 & 4.000000 & \text { INFINITY } \\3 & 80.000000 & \text { INFINITY } & 5.000000\end{array}$ a.What is the solution to the problem? b.Which constraints are binding? c.Interpret the reduced cost for x₁. d.Interpret the dual price for constraint 2. e.What would happen if the cost of x₁ dropped to 10 and the cost of x₂ increased to 12?

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.

The 100% Rule compares

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

If the range of feasibility indicates that the original amount of a resource, which was 20, can increase by 5, then the amount of the resource can increase to 25.

If a decision variable is not positive in the optimal solution, its reduced cost is