Chapter 9 Linear Programming The Simplex Method To

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Chapter 9 Linear Programming: The Simplex Method To accompany Quantitative Analysis for Management, 8

Chapter 9 Linear Programming: The Simplex Method To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -1 1

Learning Objectives Students will be able to • Convert LP constraints to equalities with

Learning Objectives Students will be able to • Convert LP constraints to equalities with slack, surplus, and artificial variables. • Set up and solve both maximization and minimization LP problems with simplex tableaus. • Interpret the meaning of every number in a simplex tableau. To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -2 2

Learning Objectives continued Students will be able to • Recognize cases of infeasibility, unboundedness,

Learning Objectives continued Students will be able to • Recognize cases of infeasibility, unboundedness, degeneracy, and multiple optimal solutions in a simplex output. • Understand the relationship between the primal and dual and when to formulate and use the dual. To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -3 3

Chapter Outline 9. 1 Introduction 9. 2 How to Set Up the Initial Solution

Chapter Outline 9. 1 Introduction 9. 2 How to Set Up the Initial Solution 9. 3 Simplex Solution Procedures 9. 4 The Second Simplex Tableau 9. 5 Developing the Third Simplex Tableau To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -4 4

Chapter Outline continued 9. 6 Review of Procedures for Solving LP Maximization Problems 9.

Chapter Outline continued 9. 6 Review of Procedures for Solving LP Maximization Problems 9. 7 Surplus and Artificial Variables 9. 8 Solving Minimization Problems 9. 9 Review of Procedures for Solving LP Minimization Problems 9. 10 Special Cases To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -5 5

Flair Furniture Company Hours Required to Produce One Unit X 2 Chairs Available Hours

Flair Furniture Company Hours Required to Produce One Unit X 2 Chairs Available Hours This Week Carpentry 4 Painting/Varnishing 2 3 1 240 100 Profit/unit $5 X 1 Tables Department Constraints: $7 4 X 1 + 3 X 2 £ 240 (carpentry ) 2 X 1 + 1 X 2 £ 100 (painting & varnishing ) Objective: Maximize: 7 X 1 +5 X 2 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -6 6

Flair Furniture Company's Feasible Region & Corner Points X 2 100 Number of Chairs

Flair Furniture Company's Feasible Region & Corner Points X 2 100 Number of Chairs 80 B = (0, 80) 60 4 X 1 +3 X 2 £ 240 40 C = (30, 40) 20 Feasible Region 2 X 1+1 X 1£ 100 D = (50, 0) 0 20 40 60 80 Number of Tables To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -7 7 100 X

Flair Furniture Adding Slack Variables Constraints: 4 X 1 + 3 X 2 £

Flair Furniture Adding Slack Variables Constraints: 4 X 1 + 3 X 2 £ 240 (carpentry ) 2 X 1 + 1 X 2 £ 100 (painting & varnishing ) Constraints with Slack Variables 4 X 1 + 3 X 2 + S 1 2 X 1 + 1 X 2 = 240 (carpentry ) + S 2 = 100 (painting & varnishing ) Objective Function 7 X 1 +5 X 2 Objective Function with Slack Variables 7 X 1 +5 X 2 +0 S 1 +0 S 2 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -8 8

Flair Furniture’s Initial Simplex Tableau Real Profit Variables per Unit Prod. Columns Slack Constant

Flair Furniture’s Initial Simplex Tableau Real Profit Variables per Unit Prod. Columns Slack Constant Column Mix Variables Columns Column Cj $7 $5 $0 $0 Solution Mix X 1 X 2 S 1 S 2 $0 S 1 2 1 1 0 $0 S 2 4 3 0 1 Zj $0 $0 Cj $7 $5 Z To accompanyj. Quantitative Analysis $0 $0 for Management, 8 e by Render/Stair/Hanna 9 -9 9 Profit per unit row Quantity Constraint 100 equation rows 240 Gross Profit $0 row Net $0 Profit row

Pivot Row, Pivot Number Identified in the Initial Simplex Tableau Cj $7 $5 $0

Pivot Row, Pivot Number Identified in the Initial Simplex Tableau Cj $7 $5 $0 $0 Solution Mix X 1 X 2 S 1 S 2 Quantity 1 1 0 100 3 0 1 Pivot number 240 $0 S 1 2 $0 S 2 4 Zj $0 $0 $0 Cj Zj $7 $5 $0 $0 $0 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna Pivot column 9 -10 10 Pivot row

Completed Second Simplex Tableau for Flair Furniture Cj $7 $5 $0 $0 Solution Mix

Completed Second Simplex Tableau for Flair Furniture Cj $7 $5 $0 $0 Solution Mix X 1 X 2 S 1 S 2 Quantity $7 X 1 1 1/2 0 50 $0 S 2 0 1 -2 1 40 Zj $7 $7/2 $0 Cj Zj $0 $3/2 -$7/2 $0 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -11 11 $350

Pivot Row, Column, and Number Identified in Second Simplex Tableau Cj $7 $5 $0

Pivot Row, Column, and Number Identified in Second Simplex Tableau Cj $7 $5 $0 $0 Solution Mix X 1 X 2 S 1 S 2 Quantity 1/2 0 50 $7 X 1 1 1/2 $0 S 2 0 1 Zj $7 $7/2 $0 Cj Zj $0 $3/2 -$7/2 $0 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna -2 1 40 Pivot number $350 (Total Profit) Pivot column 9 -12 12 Pivot row

Calculating the New X 1 Row for Flair’s Third Tableau 1 0 3/2 -1/2

Calculating the New X 1 Row for Flair’s Third Tableau 1 0 3/2 -1/2 30 = = 1 1/2 0 50 = To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna - x (1/2) (1/2) - x x x - x 9 -13 13 (0) (1) (-2) (1) (40)

Final Simplex Tableau for the Flair Furniture Problem Cj $7 $5 $0 $0 Solution

Final Simplex Tableau for the Flair Furniture Problem Cj $7 $5 $0 $0 Solution Mix X 1 X 2 S 1 S 2 Quantity $7 X 1 1 0 3/2 -1/2 30 $5 X 2 0 1 -2 40 Zj $7 5 Cj Zj $0 $0 -$1/2 -$3/2 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -14 1 $1/2 $3/2 14 $410

Simplex Steps for Maximization 1. Choose the variable with the greatest positive Cj -

Simplex Steps for Maximization 1. Choose the variable with the greatest positive Cj - Zj to enter the solution. 2. Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-to-pivotcolumn ratio. 3. Calculate the new values for the pivot row. 4. Calculate the new values for the other row(s). 5. Calculate the Cj and Cj - Zj values for this tableau. If there any Cj - Zj values greater than zero, return to Step 1. To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -15 15

Surplus & Artificial Variables Constraints 5 X 1 +10 X 2 +8 X 3

Surplus & Artificial Variables Constraints 5 X 1 +10 X 2 +8 X 3 ³ 210 25 X 1 +30 X 2 = 900 Constraints-Surplus & Artificial Variables 5 X 1 +10 X 2 +8 X 3 -S 1 +A 1 =210 25 X 1 +30 X 2 +A 2 =900 Objective Function Min: 5 X 1 +9 X 2 +7 X 3 Objective Function-Surplus & Artificial Variables Min: 5 X 1 +9 X 2 +7 X 3 +0 S 1 + MA 2 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -16 16

Simplex Steps for Minimization 1. Choose the variable with the greatest negative Cj -

Simplex Steps for Minimization 1. Choose the variable with the greatest negative Cj - Zj to enter the solution. 2. Determine the row to be replaced by selecting that one with the smallest (non -negative) quantity-to-pivot-column ratio. 3. Calculate the new values for the pivot row. 4. Calculate the new values for the other row(s). 5. Calculate the Cj and Cj - Zj values for this tableau. If there any Cj - Zj values less than zero, return to Step 1. To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -17 17

Special Cases Infeasibility Cj 5 8 0 0 M Sol X 1 X 2

Special Cases Infeasibility Cj 5 8 0 0 M Sol X 1 X 2 S 1 S 2 A 1 Mix 5 X 1 1 0 -2 3 -1 8 X 2 0 1 1 2 -1 M A 2 0 0 0 -1 -1 Zj 5 8 -2 31 -21 -M M Cj Zj 0 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 0 2 M- 2 M 31 +21 9 -18 18 M A 2 Qty 0 0 1 0 0 200 100 20 180 0+2 M

Special Cases Unboundedness Cj 6 9 0 0 Sol X 1 Mix X 1

Special Cases Unboundedness Cj 6 9 0 0 Sol X 1 Mix X 1 -1 S 1 -2 Zj -9 Cj - Zj 15 X 2 S 1 S 2 Qty 1 0 9 0 2 -1 18 -18 0 1 0 0 Pivot Column To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -19 19 30 10 270

Special Cases Degeneracy Cj 5 Solution X 1 Mix 8 X 2 2 X

Special Cases Degeneracy Cj 5 Solution X 1 Mix 8 X 2 2 X 3 0 S 1 0 S 2 0 S 3 Qty 8 X 2 1/4 1 1 -2 0 0 10 0 S 2 4 0 1/3 -1 1 0 20 0 S 3 2 0 2 2/5 0 1 10 Zj 2 8 8 16 0 0 80 3 0 6 16 0 0 C j-Z j Pivot Column To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -20 20

Special Cases Multiple Optima Cj 3 2 0 0 Sol X 1 X 2

Special Cases Multiple Optima Cj 3 2 0 0 Sol X 1 X 2 S 1 S 2 Qty Mix 2 X 1 3/2 1 1 0 6 0 S 2 1 0 1/2 1 3 Zj 3 2 2 0 12 Cj - Zj 0 0 -2 0 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -21 21

Sensitivity Analysis High Note Sound Company Max: 50 X 1 +120 X 2 Subject

Sensitivity Analysis High Note Sound Company Max: 50 X 1 +120 X 2 Subject to : 2 X 1 + 4 X 2 £ 80 3 X 1 +1 X 2 £ 60 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -22 22

Sensitivity Analysis High Note Sound Company To accompany Quantitative Analysis for Management, 8 e

Sensitivity Analysis High Note Sound Company To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -23 23

Simplex Solution High Note Sound Company Cj Sol Mix 120 X 2 0 S

Simplex Solution High Note Sound Company Cj Sol Mix 120 X 2 0 S 2 Zj Cj Zj 50 120 0 0 X 1 X 2 S 1 S 2 Qty 0 1 0 0 20 40 2400 1/2 1 1/4 5/2 0 -1/4 60 120 30 -10 0 -30 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -24 24

Nonbasic Objective Function Coefficients Cj 120 0 Sol Mix X 2 S 2 Zj

Nonbasic Objective Function Coefficients Cj 120 0 Sol Mix X 2 S 2 Zj Cj – Zj To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 50 120 0 0 X 1 X 2 S 1 S 2 Qty 1/2 5/2 60 -10 1 1/4 0 -1/4 120 30 0 -30 9 -25 25 0 20 1 40 0 2400 0

Basic Objective Function Coefficients Cj 120 + 0 Sol Mix X 1 S 2

Basic Objective Function Coefficients Cj 120 + 0 Sol Mix X 1 S 2 Zj Cj - Zj 50 120 0 0 X 1 X 2 S 1 S 2 Qty 1/2 1 1/4 0 20 -1/4 30+ /4 -30 /4 1 0 40 2400 +20 5/2 0 60+ 120 /2 + -10 - 0 /2 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -26 26 0

Simplex Solution High Note Sound Company Cj Sol Mix X 1 S 2 Zj

Simplex Solution High Note Sound Company Cj Sol Mix X 1 S 2 Zj Cj Zj 50 120 0 0 X 1 X 2 S 1 S 2 Qty ½ 5/2 1 0 1/4 120 30 0 -30 0 1 20 40 0 0 40 2400 60 0 Objective increases by 30 if 1 additional hour of electricians time is available. To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -27 27

Shadow Prices • Shadow Price: Value of One Additional Unit of a Scarce Resource

Shadow Prices • Shadow Price: Value of One Additional Unit of a Scarce Resource • Found in Final Simplex Tableau in C-Z Row • Negatives of Numbers in Slack Variable Column To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -28 28

Steps to Form the Dual To form the Dual: • If the primal is

Steps to Form the Dual To form the Dual: • If the primal is max. , the dual is min. , and vice versa. • The right-hand-side values of the primal constraints become the objective coefficients of the dual. • The primal objective function coefficients become the right-hand-side of the dual constraints. • The transpose of the primal constraint coefficients become the dual constraint coefficients. • Constraint inequality signs are reversed. To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -29 29

Primal & Dual Primal: Dual Max: 50 X 1 +120 X 2 Min :

Primal & Dual Primal: Dual Max: 50 X 1 +120 X 2 Min : 80 U 1 + 60 U 2 Subject to: 2 X 1 +4 X 2 £ 80 2 U 1 + 3 U 2 ³ 50 3 X 1 +1 X 2 £ 60 4 U 1 + 1 U 2 ³ 120 To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -30 30

Primal’s Optimal Solution Comparison of the Primal and Dual Optimal Tableaus Cj $50 $120

Primal’s Optimal Solution Comparison of the Primal and Dual Optimal Tableaus Cj $50 $120 Solution Quantity Mix X 1 $0 X 2 S 1 S 2 $7 X 2 20 1/2 1 1/4 0 $5 S 2 40 5/2 0 -1/4 1 Zj $2, 400 60 120 30 0 -10 0 -30 0 80 60 $0 $0 M M X 2 S 1 S 2 A 1 A 2 C j - Zj Dual’s Optimal Solution $0 Cj Solution Quantity Mix X 1 $7 U 1 30 1 1/4 0 -1/4 0 1/2 $5 S 1 10 0 -5/2 1 -1/2 -1 1/2 Zj $2, 400 80 20 0 -20 0 40 $0 40 0 20 M M-40 C j - Zj To accompany Quantitative Analysis for Management, 8 e by Render/Stair/Hanna 9 -31 31