Creative Kitchen Tools manufactures a wide line of gourmet cooking tools from stainless steel. For the coming production period, there is demand of 1200 for 8 quart stock pots, and unlimited demand for 3 quart mixing bowls and large slotted spoons. In the following model, the three variables measure the number of pots, bowls, and spoons to make. The objective function measures profit. Constraint 1 measures steel, constraint 2 measures manufacturing time, constraint 3 measures finishing time, and constraint 4 measures the stock pot demand. Max 5x1 + 3x2 + 6x3 s.t. 3x1 + 1x2 + 2x3 $\le$ 15000 4x1 + 4x2 + 5x3 $\le$18000 2x1 + 1x2 + 2x3 $\le$ 10000 x1 $\le$ 1200 x1, x2, x3 $\ge$ 0 The final tableau is: \[\begin{array} { c c | c c c ccc c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 }& \mathrm {~s} _ { 4 } \\ \text { Basis } & \mathrm { C } _ { \mathrm { B } } &5&3&6&0&0&0&0& \\ \hline \mathrm {~s} _ { 1 } &0&0&-2&-1.75&0&-.75&0&0&1500\\ \mathrm {~s} _ { 4 }&0&0&1&1.25&0&.25&0&1&3300\\ \mathrm {~s} _ { 3 }&0&0&-1&-.5&0&-.5&1&0&1000\\ \mathrm { x } _ { 1 } &5&1&1&1.25&0&.25&0&0&4500\\\hline & \mathrm { z } _ { j }&5&5&6.25&0&1.25&0&0&22500\\ &\mathrm{C}_{\mathrm{j}}-z_{\mathrm{j}}&0&-2&-.25&0&-1.25&0&0& \end{array}\] a.Calculate the range of optimality for c1, c2, and c3. b.Calculate the range of feasibility for b1, b2, b3, and b4. c.Suppose that the inventory records were incorrect and the company really has only 14000 units of steel.What effect will this have on your solution? d.Suppose that a cost increase will change the profit on the pots to $4.62.What effect will this have on your solution? e.Assume that the cost of time in production and finishing is relevant.Would you be willing to pay a $1.00 premium over the normal cost for 1000 more hours in the production department? What would this do to your solution?

a.4.80 @#LAT-DLM& c 1 2 @#LAT-DLM& 5 -@#LAT-DLM& 3 @#LAT-DLM&6.25 b.13500

Given the following linear programming problem Max Z 0.5x1 + 6x2 + 5x3 s.t. 4x1 + 6x2 + 3x3 $\le$24 1x1 + 1.5x2 + 3x3 $\le$ 12 3x1 + x2 $\le$ 12 and the final tableau is \[\begin{array} { c c | c c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\ \text { Basis } & \mathrm { c } _ { \mathrm { B } } & .5 & 6 & 5 & 0 & 0 & 0 & \\ \hline \mathrm { x } _ { 2 } & 6 & 1 & 1 & 0 & .22 & - .22 & 0 & 2.67 \\ \mathrm { x } _ { 3 } & 5 & 0 & 0 & 1 & .11 & - .44 & 0 & 2.67 \\ \mathrm {~s} _ { 3 } & 0 & 2.33 & 0 & 0 & - .22 & .22 & 1 & 9.33 \\ \hline & \mathrm { z } _ { \mathrm { j } } & 4 & 6 & 5 & .77 & .88 & 0 & 29.33 \\ & \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & .5 & 0 & 0 & - .77 & - .88 & 0 & \end{array}\] a.Find the range of optimality for c1, c2, c3, c4, c5, and c6. b.Find the range of feasibility for b1, b2, and b3.

a.c 1 @#LAT-DLM& 4 2.5 2 @#LAT-DLM& 10 3

Exam 18: Simplex-Based Sensitivity Analysis and Duality

Creative Kitchen Tools manufactures a wide line of gourmet cooking tools from stainless steel. For the coming production period, there is demand of 1200 for 8 quart stock pots, and unlimited demand for 3 quart mixing bowls and large slotted spoons. In the following model, the three variables measure the number of pots, bowls, and spoons to make. The objective function measures profit. Constraint 1 measures steel, constraint 2 measures manufacturing time, constraint 3 measures finishing time, and constraint 4 measures the stock pot demand. Max 5x₁ + 3x₂ + 6x₃ s.t. 3x₁ + 1x₂ + 2x₃ $\le$ 15000 4x₁ + 4x₂ + 5x₃ $\le$ 18000 2x₁ + 1x₂ + 2x₃ $\le$ 10000 x₁ $\le$ 1200 x₁, x₂, x₃ $\ge$ 0 The final tableau is: Basis 5 3 6 0 0 0 0 0 0 -2 -1.75 0 -.75 0 0 1500 0 0 1 1.25 0 .25 0 1 3300 0 0 -1 -.5 0 -.5 1 0 1000 5 1 1 1.25 0 .25 0 0 4500 5 5 6.25 0 1.25 0 0 22500 - 0 -2 -.25 0 -1.25 0 0 a.Calculate the range of optimality for c₁, c₂, and c₃. b.Calculate the range of feasibility for b₁, b₂, b₃, and b₄. c.Suppose that the inventory records were incorrect and the company really has only 14000 units of steel.What effect will this have on your solution? d.Suppose that a cost increase will change the profit on the pots to $4.62.What effect will this have on your solution? e.Assume that the cost of time in production and finishing is relevant.Would you be willing to pay a $1.00 premium over the normal cost for 1000 more hours in the production department? What would this do to your solution?

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Question 21

The range of optimality is useful only for basic variables.

(True/False)

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Question 22

Write the dual to the following problem. Min 12x₁ + 15x₂ + 20x₃ + 18x₄ s.t. x₁ + x₂ + x₃ + x₄ $\ge$ 50 3x₁ + 4x₃ $\ge$ 60 2x₂ + x₃ - 2x₄ $\le$ 10 x₁, x₂, x₃, x₄ $\ge$ 0

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Question 23

Given the following linear programming problem Max Z 0.5x₁ + 6x₂+ 5x₃ s.t. 4x₁ + 6x₂+ 3x₃ $\le$ 24 1x₁ + 1.5x₂ + 3x₃ $\le$ 12 3x₁ + x₂ $\le$ 12 and the final tableau is Basis .5 6 5 0 0 0 6 1 1 0 .22 -.22 0 2.67 5 0 0 1 .11 -.44 0 2.67 0 2.33 0 0 -.22 .22 1 9.33 4 6 5 .77 .88 0 29.33 - .5 0 0 -.77 -.88 0 a.Find the range of optimality for c_1,c_2,c_3,c_4, c_5, and c₆. b.Find the range of feasibility for b₁, b₂, and b₃.

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Question 24

Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

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Question 25

Explain why the z_j value for a slack variable is the dual price.

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Question 26

The dual price for an equality constraint is the z_j value for its artificial variable.

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Question 27

A one-sided range of optimality

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Question 28

There is a dual price associated with each decision variable.

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Question 29

The dual price is the improvement in value of the optimal solution per unit increase in the value of the right-hand side associated with a linear programming problem.

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Question 30

Given the simplex tableau for the optimal primal solution

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Question 31

For the basic feasible solution to remain optimal

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Question 32

As long as the objective function coefficient remains within the range of optimality, the variable values will not change although the value of the objective function could.

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Question 33

For this optimal simplex tableau the original right-hand sides were 100 and 90. The problem was a maximization. Basis 2 4 8 0 0 8 0 .48 1 .12 -.04 8.4 2 1 .2 0 -.2 .4 16 2 4.24 8 .56 .48 99.2 - 0 -.24 0 -.56 -.48 a.What would the new solution be if there had been 150 units available in the first constraint? b.What would the new solution be if there had been 70 units available in the second constraint?

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Question 34

The range of optimality is useful only for basic variables.

Write the dual to the following problem. Min 12x₁ + 15x₂ + 20x₃ + 18x₄ s.t. x₁ + x₂ + x₃ + x₄ $\ge$ 50 3x₁ + 4x₃ $\ge$ 60 2x₂ + x₃ - 2x₄ $\le$ 10 x₁, x₂, x₃, x₄ $\ge$ 0

Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

Explain why the z_j value for a slack variable is the dual price.

The dual price for an equality constraint is the z_j value for its artificial variable.

A one-sided range of optimality

There is a dual price associated with each decision variable.

The dual price is the improvement in value of the optimal solution per unit increase in the value of the right-hand side associated with a linear programming problem.

Given the simplex tableau for the optimal primal solution

For the basic feasible solution to remain optimal

As long as the objective function coefficient remains within the range of optimality, the variable values will not change although the value of the objective function could.

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 range of optimality is useful only for basic variables.

Write the dual to the following problem. Min 12x1 + 15x2 + 20x3 + 18x4 s.t. x1 + x2 + x3 + x4 ≥\ge≥ 50 3x1 + 4x3 ≥\ge≥ 60 2x2 + x3 - 2x4 ≤\le≤ 10 x1, x2, x3, x4 ≥\ge≥ 0

Explain how to put an equality constraint into canonical form and how to calculate its dual variable value.

Explain why the zj value for a slack variable is the dual price.

The dual price for an equality constraint is the zj value for its artificial variable.

A one-sided range of optimality

There is a dual price associated with each decision variable.

The dual price is the improvement in value of the optimal solution per unit increase in the value of the right-hand side associated with a linear programming problem.

Given the simplex tableau for the optimal primal solution

For the basic feasible solution to remain optimal

As long as the objective function coefficient remains within the range of optimality, the variable values will not change although the value of the objective function could.

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 to the following problem. Min 12x₁ + 15x₂ + 20x₃ + 18x₄ s.t. x₁ + x₂ + x₃ + x₄ $\ge$ 50 3x₁ + 4x₃ $\ge$ 60 2x₂ + x₃ - 2x₄ $\le$ 10 x₁, x₂, x₃, x₄ $\ge$ 0

Explain why the z_j value for a slack variable is the dual price.

The dual price for an equality constraint is the z_j value for its artificial variable.