For this optimal simplex tableau, the right-hand sides for the two original $\ge$ constraints were 300 and 250. The problem was a minimization. \[\begin{array} { c c | c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \\ \text { Basis } & \mathrm { C } _ { \mathrm { B } } & - 100 & - 110 & - 95 & 0 & 0 & \\ \hline \mathrm { x } _ { 3 } & - 95 & 0 & - 1.5 & 1 & 1 & - 1.5 & 75 \\ \mathrm { x } _ { 1 } & - 100 & 1 & 2 & 0 & - 1 & 1 & 50 \\ \hline & \mathrm { z } _ { \mathrm { j } } & - 100 & - 57.5 & - 95 & 5 & 42.5 & - 12125 \\ & \mathrm { C } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & - 52.5 & 0 & - 5 & - 42.5 & \end{array}\] a.What would the new solution be if the right-hand side value in the first constraint had been 325? b.What would the new solution be if the right-hand side value for the second constraint had been 220?

a.x 3 = 50, x 1 = 75, Z = 12250 b.x 3 = 30, x 1 = 80, Z = 10850

Given the following linear programming problem Max 10x1 + 12x2 s.t. 1x1 + 2x2 $\ge$ 40 5x1 + 8x2 $\le$160 1x1 + 1x2 $\le$ 40 x1, x2 $\ge$ 0 the final tableau is \[\begin{array} { c c | c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\ \text { Basis } & \mathrm { c } _ { \mathrm { B } } & 10 & 12 & 0 & 0 & 0 & \\ \hline \mathrm { x } _ { 2 } & 12 & 0 & 1 & - 2.5 & - .5 & 0 & 20 \\ \mathrm { x } _ { 1 } & 10 & 1 & 0 & 4 & 1 & 0 & 0 \\ \mathrm {~s} _ { 3 } & 0 & 0 & 0 & - 1.5 & - 5 & 1 & 20 \\ \hline & \mathrm { z } _ { \mathrm { j } } & 10 & 12 & 10 & 4 & 0 & 240 \\ & \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & 0 & - 10 & - 4 & 0 & \end{array}\] a.Find the range of optimality for c1 and c2. b.Find the range of feasibility for b1, b2, and b3. c.Find the dual prices.

a.7.5 @#LAT-DLM& c 1 2 @#LAT-DLM& 1

If the dual price for b1 is 2.7, the range of feasibility is 20 $\le$ b1 $\le$ 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. 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.

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 $\le$ c2 $\le$ 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. A) will decrease by 12. B) will decrease by 2. C) will not change. D) cannot be determined from this information.

For the following linear programming problem Max Z -2x1 + x2 - x3 s.t. 2x1 + x2 $\le$ 7 1x1 + x2 + x3 $\ge$ 4 the final tableau is \[\begin{array} { c c | c c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm { a } _ { 2 } & \\ \text { Basis } & \mathrm { C } _ { \mathrm { B } } & - 2 & 1 & - 1 & 0 & 0 & - \mathrm { M } & \\ \hline \mathrm { s } _ { 2 } & 0 & 1 & 0 & - 1 & 1 & 1 & - 1 & 3 \\ \mathrm { x } _ { 2 } & 1 & 2 & 1 & 0 & 1 & 0 & 0 & 7 \\ \hline & z _ { j } & 2 & 1 & 0 & 1 & 0 & 0 & 7 \\ & \mathrm { C } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & - 4 & 0 & 1 & - 1 & 0 & - \mathrm { M } & \end{array}\] a.Find the range of optimality for c1, c2 , c3.c4, c5 , and c6. b.Find the range of feasibility for b1, and b2.

a.c 1 @#LAT-DLM& 2 0 @#LAT-DLM&

Exam 18: Simplex-Based Sensitivity Analysis and Duality

For this optimal simplex tableau, the right-hand sides for the two original $\ge$ constraints were 300 and 250. The problem was a minimization. Basis -100 -110 -95 0 0 -95 0 -1.5 1 1 -1.5 75 -100 1 2 0 -1 1 50 -100 -57.5 -95 5 42.5 -12125 - 0 -52.5 0 -5 -42.5 a.What would the new solution be if the right-hand side value in the first constraint had been 325? b.What would the new solution be if the right-hand side value for the second constraint had been 220?

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Question 1

Correct Answer:

Verified

a.x₃ = 50, x₁ = 75, Z = 12250
b.x₃ = 30, x₁ = 80, Z = 10850

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

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Question 2

Correct Answer:

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True

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.

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Question 3

Correct Answer:

Verified

True

Given the following linear programming problem Max 10x₁ + 12x₂ s.t. 1x₁ + 2x₂ $\ge$ 40 5x₁ + 8x₂ $\le$ 160 1x₁ + 1x₂ $\le$ 40 x₁, x₂ $\ge$ 0 the final tableau is Basis 10 12 0 0 0 12 0 1 -2.5 -.5 0 20 10 1 0 4 1 0 0 0 0 0 -1.5 -5 1 20 10 12 10 4 0 240 - 0 0 -10 -4 0 a.Find the range of optimality for c₁ and c₂. b.Find the range of feasibility for b₁, b₂, and b₃. c.Find the dual prices.

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Question 4

The range of feasibility indicates right-hand side values for which

(Multiple Choice)

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Question 5

The linear programming problem Max 6x₁ + 2x₂ + 3x₃ + 4x₄ s.t. x₁ + x₂ + x₃ + x₄ $\le$ 100 4x₁ + x₂ + x₃ + x₄ $\le$ 160 3x₁ + x₂ + 2x₃ + 3x₄ $\le$ 240 x₁, x₂, x₃, x₄ $\ge$ 0 has the final tableau Basis 6 2 3 4 0 0 0 2 0 1 1/2 0 3/2 0 -1/2 30 6 1 0 0 0 -1/3 1/3 0 20 4 0 0 1/2 1 -1/6 -1/3 1/2 50 6 2 3 4 2.33 .67 1 380 - 0 0 0 0 -2.33 -.67 -1 Fill in the table below to show what you would have found if you had used The Management Scientist to solve this problem. LINEAR PROGRAMMING PROBLEM MAX 6X1+2X2+3X3+4X4 S.T. 1) 1X1 + 1X2 + 1X3 + 1X4 < 100 2) 4X1 + 1X2 + 1X3 + 1X4 < 160 3) 3X1 + 1X2 + 2X3 + 3X4 < 240 OPTIMAL SOLUTION Objective Function Value $=$ Variable Value Reduced Cost X1 - - X2 - - X3 - - X4 - - Constraint Slack/Surplus Dual Price 1 - - 2 - - 3 - - $\text { OBJECTIVE COEFFICIENT RANGES }$ Variable Lower Limit Current Value Upper Limit 1 - - - 2 - - - 3 - - - 4 - - - $\text { RIGHT HAND SIDE RANGES }$ Constraint Lower Limit Current Value Upper Limit 1 - - - 2 - - - 3 - - -

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Question 6

Write the dual of the following problem Min Z = 2x₁ -3x₂ + 5x₃ s.t. -3x₁ + 2x₂ + 5x₃ $\ge$ 7 2x₁ -x₃ $\ge$ 5 4x₂ + 3x₃ $\ge$ 8.

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Question 7

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

(True/False)

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Question 8

Explain the simplex tableau location of the dual constraint for each type of constraint.

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Question 9

If the dual price for b₁ is 2.7, the range of feasibility is 20 $\le$ b₁ $\le$ 50, and the original value of b₁ was 30, which of the following is true?

(Multiple Choice)

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Question 10

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.

(True/False)

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Question 11

When sensitivity calculations yield several potential upper bounds and several lower bounds, how is the range determined?

(Short Answer)

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Question 12

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 $\le$ c₂ $\le$ 12, the value of Z

(Multiple Choice)

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Question 13

For the following linear programming problem Max Z -2x₁ + x₂ - x₃ s.t. 2x₁ + x₂ $\le$ 7 1x₁ + x₂ + x₃ $\ge$ 4 the final tableau is Basis -2 1 -1 0 0 - 0 1 0 -1 1 1 -1 3 1 2 1 0 1 0 0 7 2 1 0 1 0 0 7 - -4 0 1 -1 0 - a.Find the range of optimality for c₁, c₂ , c₃.c₄, c₅ , and c₆. b.Find the range of feasibility for b₁, and b₂.

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Question 14

The dual variable represents

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Question 15

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

(Multiple Choice)

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Question 16

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.

(True/False)

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Question 17

The primal problem is Min 2x₁ + 5x₂ + 4x₃ s.t. x₁ + 3x₂ + 3x₃ $\ge$ 30 3x₁ + 7x₂ + 5x₃ $\ge$ 70 x₁, x₂, x₃ $\ge$ 0 The final tableau for its dual problem is Basis 30 70 0 0 0 70 0 1 3/4 0 -1/4 1/2 0 0 0 -3/2 1 -1/2 0 30 1 0 -5/4 0 3/4 1/2 30 70 15 0 5 50 - 0 0 -15 0 -5 Give the complete solution to the primal problem.

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Question 18

For an objective function coefficient change outside the range of optimality, explain how to calculate the new optimal solution. Must you return to the (revised) initial tableau?

(Short Answer)

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Question 19

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

(Multiple Choice)

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Question 20

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

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.

The range of feasibility indicates right-hand side values for which

Write the dual of the following problem Min Z = 2x₁ -3x₂ + 5x₃ s.t. -3x₁ + 2x₂ + 5x₃ $\ge$ 7 2x₁ -x₃ $\ge$ 5 4x₂ + 3x₃ $\ge$ 8.

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

Explain the simplex tableau location of the dual constraint for each type of constraint.

If the dual price for b₁ is 2.7, the range of feasibility is 20 $\le$ b₁ $\le$ 50, and the original value of b₁ was 30, which of the following is true?

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.

When sensitivity calculations yield several potential upper bounds and several lower bounds, how is the range determined?

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 $\le$ c₂ $\le$ 12, the value of Z

The dual variable represents

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

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.

For an objective function coefficient change outside the range of optimality, explain how to calculate the new optimal solution. Must you return to the (revised) initial tableau?

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

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

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

Exam 18: Simplex-Based Sensitivity Analysis and Duality

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

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.

The range of feasibility indicates right-hand side values for which

Write the dual of the following problem Min Z = 2x1 -3x2 + 5x3 s.t. -3x1 + 2x2 + 5x3 ≥\ge≥ 7 2x1 -x3 ≥\ge≥ 5 4x 2 + 3x3 ≥\ge≥ 8.

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

Explain the simplex tableau location of the dual constraint for each type of constraint.

If the dual price for b1 is 2.7, the range of feasibility is 20 ≤\le≤ b1 ≤\le≤ 50, and the original value of b1 was 30, which of the following is true?

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.

When sensitivity calculations yield several potential upper bounds and several lower bounds, how is the range determined?

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 ≤\le≤ c2 ≤\le≤ 12, the value of Z

The dual variable represents

Dual prices and ranges for objective function coefficients and right-hand side values are found by considering

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.

For an objective function coefficient change outside the range of optimality, explain how to calculate the new optimal solution. Must you return to the (revised) initial tableau?

The improvement in the value of the optimal solution per-unit increase in a constraint's right-hand side is

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

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

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

Write the dual of the following problem Min Z = 2x₁ -3x₂ + 5x₃ s.t. -3x₁ + 2x₂ + 5x₃ $\ge$ 7 2x₁ -x₃ $\ge$ 5 4x₂ + 3x₃ $\ge$ 8.

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

If the dual price for b₁ is 2.7, the range of feasibility is 20 $\le$ b₁ $\le$ 50, and the original value of b₁ was 30, which of the following is true?

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 $\le$ c₂ $\le$ 12, the value of Z