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.

a.1.33@#LAT-DLM& c 1 @#LAT-DLM& 4 b..5 @#LAT-DLM& c 2

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: \[\begin{array} { c c c c } && { \text { OBJ. COEFFICIENT RANGES } } \\ & \text { CURRENT } & \text { ALLOWABLE } & \text { ALLOWABLE } \\ \text { VARIABLE }&\text { COEFFICIENT }& \text { INCREASE } & \text { DECREASE } \\ \hline \text { X1 } & 12.000000 & \text { INFINITY } & 4.000000 \\ \text { X2 } & 10.000000 & 5.000000 & 10.000000 \\ \text { X3 } & 9.000000 &\text { INFINITY } & 4.000000\\ \end{array}\] $\begin{array}{cccc}&&&\text { RIGHT HAND SIDE RANGES }\\ &\text { CURRENT } & \text { ALLOWABLE }& \text { ALLOWABLE }\\ \text { ROW } & \text { RHS } & \text { INCREASE } & \text { DECREASE } \\\hline 2 & 60.000000 & 4.000000 & \text { INFINITY } \\ 3 & 80.000000 & \text { INFINITY } & 5.000000 \end{array}$ 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?

a.x 1 = 0, x 2 = 8, x 3 = 0, s 1 = 4, s 2 = 0, z =

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

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

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

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

The first interpretation is based on the assumption of constant opportunity costs. In this interpretation, the dual prices represent the shadow prices of the resources used in the production process. These shadow prices reflect the value of using one additional unit of a resource in the production process, taking into account the opportunity cost of using that resource in an alternative way. This interpretation is based on the assumption that the opportunity cost of resources remains constant regardless of the level of resource usage.

The second interpretation is based on the assumption of variable opportunity costs. In this interpretation, the dual prices represent the marginal values of the resources used in the production process. These marginal values reflect the change in the objective function value for a small change in the availability of a resource. This interpretation is based on the assumption that the opportunity cost of resources varies with the level of resource usage.

Both interpretations provide valuable insights into the economic implications of resource allocation in the production process. The choice of interpretation depends on the specific accounting assumptions and economic conditions relevant to the production process under consideration.

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

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Question 4

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

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Question 5

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

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Question 6

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.

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Question 7

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

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Question 8

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.

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Question 9

The 100% Rule compares

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Question 10

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 VARIABLE VALUE REDUCED COST X1 .000000 4.000000 X2 8.000000 .000000 X3 .000000 4.000000 ROW SLACK OR SURPLUS DUAL PRICE 2) 4.000000 .000000 3) .000000 -1.000000 NO. ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES CURRENT ALLOWABLE ALLOWABLE VARIABLE COEFFICIENT INCREASE DECREASE X1 12.000000 INFINITY 4.000000 X2 10.000000 5.000000 10.000000 X3 9.000000 INFINITY 4.000000 RIGHT HAND SIDE RANGES CURRENT ALLOWABLE ALLOWABLE ROW RHS INCREASE DECREASE 2 60.000000 4.000000 INFINITY 3 80.000000 INFINITY 5.000000 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?

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Question 11

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 Variable Value Reduced Cost X1 13.333 0.000 X2 10.000 0.000 X3 0.000 10.889 Constraint Slack/Surplus Dual Price 1 0.000 -0.778 2 0.000 5.556 3 23.333 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 30.000 31.000 No Upper Limit X2 No Lower Limit 35.000 36.167 X3 No Lower Limit 32.000 42.889 $\text { RIGHT HAND SIDE RANGES }$ Constraint Lower Limit Current Value Upper Limit 1 77.647 90.000 107.143 2 126.000 150.000 163.125 3 96.667 120.000 No Upper Limit 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?

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Question 12

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

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Question 13

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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Question 14

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

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Question 15

The range of feasibility measures

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Question 16

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

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Question 17

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

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Question 18

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

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Question 19

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

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Question 20

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

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

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.

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

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

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

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

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.

The 100% Rule compares

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

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

The range of feasibility measures

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

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

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.

Introduction

An Introduction to Linear Programming

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

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Integer Linear Programming

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Project Scheduling: Pertcpm

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Linear Programming: Simplex Method

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