Use the following Management Scientist output to answer the questions. MIN 4X1+5X2+6X3 S.T. 1) X1+X2+X3280 3) 2X1+4X2+4X3>320 Objective Function Value = 400.000 \[\begin{array} { c r c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\ \text { X1 } & 0.000 & 1.500 \\ \text { X2 } & 80.000 & 0.000 \\ \text { X3 } & 0.000 & 1.000 \end{array}\] $\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Frice } \\ 1 & 5.000 & 0.000 \\ 2 & 40.000 & 0.000 \\ 3 & 0.000 & - 1.250 \end{array}$ OBJECTIVE COEFFICIENT RANGES $\begin{array}{lllc} \text { Variable }&\text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline \mathrm{X} 1 & 2.500 & 4.000 & \text { No Upper Limit } \\ \mathrm{X} 2 & 0.000 & 5.000 & 6.000 \\ \mathrm{X} 3 & 5.000 & 6.000 & \text { No Upper Limit } \end{array}$ $\text { RIGHT HAND SIDE RANGES }$ $\begin{array}{cccc} \text { Constraint }&\text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline 1 & 80.000 & 85.000 & \text { No Upper Limit } \\ 2 & \text { No Lower Limit } & 280.000 & 320.000 \\ 3 & 280.000 & 320.000 & 340.000 \end{array}$ a.What is the optimal solution, and what is the value of the profit contribution? b.Which constraints are binding? c.What are the dual prices for each resource? Interpret. d.Compute and interpret the ranges of optimality. e.Compute and interpret the ranges of feasibility.

a.x 1 = 0, x 2 = 80, x 3 = 0, s 1 = 5, s 2 = 40, s

Portions of a Management Scientist output are shown below. Use what you know about the solution of linear programs to fill in the ten blanks. LINEAR PROGRAMMING PROBLEM MAX 12X1+9X2+7X3 S.T. 1) 3X1+5X2+4X3116 OPTIMAL SOLUTION Objective Function Value = 336.000 \[\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\ \text { X1 } &- & 0.000 \\ \text { X2 } &24.000&- \\ \text { X3 } &- &3.500 \\ \hline \end{array}\] \[\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\ \hline 1 & 0.000 & 15.000 \\ 2 & --& 0.000 \\ 3 & --& 0.000 \\ 4 & 0.000 &-- \\ \hline \end{array}\] $\text { OBJECTIVE COEFFICIENT RANGES }$ $\begin{array}{cccc} \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit }\\\hline \mathrm{X} 1 & 5.400 & 12.000 & \text { No Upper Limit } \\ \mathrm{X} 2 & 2.000 & 9.000 & 20.000 \\ \mathrm{X} 3 & \text { No Lower Limit } & 7.000 & 10.500 \end{array}$ $\text { RIGHT HAND SIDE RANGES }$ $\begin{array}{lllr} \text { Constraint } & \text { Lower Limit }& \text { Current Value } & \text { Upper Limit }\\\hline 1 & 145.000 & 150.000 & 156.667 \\ 2 &- &- & 64.000 \\ 3 & -&-& 80.000 \\ 4 &110.286 &116.000& 120.000 \end{array}$

x 3 = 0 because the reduced cost is positi

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

Use the following Management Scientist output to answer the questions. MIN 4X1+5X2+6X3 S.T. 1) X1+X2+X3<85 2) 3X1+4X2+2X3>280 3) 2X1+4X2+4X3>320 Objective Function Value = 400.000 Variable Value Reduced Cost X1 0.000 1.500 X2 80.000 0.000 X3 0.000 1.000 Constraint Slack/Surplus Dual Frice 1 5.000 0.000 2 40.000 0.000 3 0.000 -1.250 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit 1 2.500 4.000 No Upper Limit 2 0.000 5.000 6.000 3 5.000 6.000 No Upper Limit $\text { RIGHT HAND SIDE RANGES }$ Constraint Lower Limit Current Value Upper Limit 1 80.000 85.000 No Upper Limit 2 No Lower Limit 280.000 320.000 3 280.000 320.000 340.000 a.What is the optimal solution, and what is the value of the profit contribution? b.Which constraints are binding? c.What are the dual prices for each resource? Interpret. d.Compute and interpret the ranges of optimality. e.Compute and interpret the ranges of feasibility.

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Question 41

Portions of a Management Scientist output are shown below. Use what you know about the solution of linear programs to fill in the ten blanks. LINEAR PROGRAMMING PROBLEM MAX 12X1+9X2+7X3 S.T. 1) 3X1+5X2+4X3<150 2) 2X1+1X2+1X3<64 3) 1X1+2X2+1X3<80 4) 2X1+4X2+3X3>116 OPTIMAL SOLUTION Objective Function Value = 336.000 Variable Value Reduced Cost X1 - 0.000 X2 24.000 - X3 - 3.500 Constraint Slack/Surplus Dual Price 1 0.000 15.000 2 -- 0.000 3 -- 0.000 4 0.000 -- $\text { OBJECTIVE COEFFICIENT RANGES }$ Variable Lower Limit Current Value Upper Limit 1 5.400 12.000 No Upper Limit 2 2.000 9.000 20.000 3 No Lower Limit 7.000 10.500 $\text { RIGHT HAND SIDE RANGES }$ Constraint Lower Limit Current Value Upper Limit 1 145.000 150.000 156.667 2 - - 64.000 3 - - 80.000 4 110.286 116.000 120.000

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Question 42

The dual price measures, per unit increase in the right hand side,

(Multiple Choice)

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Question 43

When the cost of a resource is sunk, then the dual price can be interpreted as the

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Question 44

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

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Question 45

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

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Question 46

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

(True/False)

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Question 47

The dual price measures, per unit increase in the right hand side,

When the cost of a resource is sunk, then the dual price can be interpreted as the

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

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

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

Introduction

An Introduction to Linear Programming

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Markov Processes

Linear Programming: Simplex Method

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

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