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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter 9 Linear Programming: The Simplex Method The Simplex Method

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Maximize: Objective: XX Hours Required to Produce One Unit Department X 1 Tables X 2 Chairs Available Hours This Week Carpentry Painting/Varnishing 4242 3131 240 100 Profit/unit Constraints: $7$5 )varnishing & (painting XX )(carpentry XX

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company's Feasible Region & Corner Points Number of Chairs 100 80 60 40 20 020406080100X X2X2 Number of Tables B = (0,80) C = (30,40) D = (50,0) Feasible Region XX XX

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture - Adding Slack Variables )varnishing & (painting XX )(carpentry XX Constraints: Constraints with Slack Variables ) varnishing & (painting )(carpentry S XX SXX XX Objective Function Objective Function with Slack Variables SSXX

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture’s Initial Simplex Tableau Profit per Unit Column Prod. Mix Column Real Variables Columns Slack Variables Columns Constant Column CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 Profit per unit row 2110 4301 $0 $7$5$0 S1S1 S2S2 ZjZj C j - Z j 100 240 $0 Constraint equation rows Gross Profit row Net Profit row

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Pivot Row, Pivot Number Identified in the Initial Simplex Tableau CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 2110 4301 $7$5$0 S1S1 S2S2 ZjZj C j - Z j 100 240 $0 Pivot row Pivot number Pivot column

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Completed Second Simplex Tableau for Flair Furniture CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 11/2 0 01-21 $7$7/2 $0 $3/2-$7/2$0 $7 $0 X1X1 S2S2 ZjZj C j - Z j 50 40 $350

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Pivot Row, Column, and Number Identified in Second Simplex Tableau CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 11/2 0 01-21 $7$7/2 $0 $3/2-$7/2$0 $7 $0 X1X1 S2S2 ZjZj C j - Z j 50 40 $350 (Total Profit) Pivot row Pivot number Pivot column

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Calculating the New X 1 Row for Flair’s Third Tableau =-x 1 0 3/2 -1/2 30 11/2 0 50 (1/2)(1/2) (0) (1) (-2) (1) (40) =-x =-x =-x =-x

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Final Simplex Tableau for the Flair Furniture Problem CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 103/2-1/2 01-21 $75$1/2$3/2 $0 -$1/2-$3/2 $7 $5 X1X1 X2X2 ZjZj C j - Z j 30 40 $410

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Simplex Steps for Maximization 1. Choose the variable with the greatest positive C j - Z j 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 C j and C j - Z j values for this tableau. If there are any C j - Z j values greater than zero, return to Step 1.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Surplus & Artificial Variables Constraints Constraints-Surplus & Artificial Variables XX XXX AXX ASXXX Objective Function XXX :Min MA SXXX :Min Objective Function-Surplus & Artificial Variables

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Simplex Steps for Minimization 1. Choose the variable with the greatest negative C j - Z j 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 C j and C j - Z j values for this tableau. If there are any C j - Z j values less than zero, return to Step 1.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Special Cases Infeasibility 02M +21 M - 31 200C j - Z j 180 0+2 M 0- 21- M 31 - M -285ZjZj 201 000A2A2 M 10002110X2X2 8 20003-201X1X1 5 QtyA2A2 A1A1 S2S2 S1S1 X2X2 X1X1 Sol Mix MM0085CjCj

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Special Cases Unboundedness Pivot Column CjCj 6900 Sol Mix X1X1 X2X2 S1S1 S2S2 Qty X1X1 12030 S1S1 -20110 ZjZj -99180270 C j - Z j 150-180

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Special Cases Degeneracy Pivot Column C j 582000 Solution Mix X 1 X 2 X 3 S 1 S 2 S 3 Qty 8X 2 1/411-20010 0S 2 401/31020 0S 3 2022/50110 Z j 288160080 C j -Z j 3061600

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Special Cases Multiple Optima CjCj 3200 Sol Mix X1X1 X2X2 S1S1 S2S2 Qty 2X1X1 3/21106 0S2S2 101/213 ZjZj 322012 C j - Z j 00-20

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Sensitivity Analysis High Note Sound Company XX XX X X :toSubject :Max

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Sensitivity Analysis High Note Sound Company

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Simplex Solution High Note Sound Company CjCj 5012000 Sol Mix X1X1 X2X2 S1S1 S2S2 Qty 120X2X2 1/211/4020 0S2S2 5/20-1/4140 ZjZj 601203002400 C j - Z j -100-300

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Nonbasic Objective Function Coefficients CjCj 5012000 Sol Mix X1X1 X2X2 S1S1 S2S2 Qty 120X2X2 1/211/4020 0S2S2 5/20-1/4140 ZjZj 601203002400 C j – Z j -100-300

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Basic Objective Function Coefficients CjCj 5012000 Sol Mix X1X1 X2X2 S1S1 S2S2 Qty 120 + X1X1 1/211/4020 0S2S2 5/20-1/4140 ZjZj 60+ /2 120 + 30+ /4 02400 +20 C j - Z j -10- /2 0-30- /4 0

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Simplex Solution High Note Sound Company Objective increases by 30 if 1 additional hour of electricians time is available. CjCj 5012000 Sol Mix X1X1 X2X2 S1S1 S2S2 Qty X1X1 ½11/4020 S2S2 5/20- 1/4 140 ZjZj 6012030040 C j - Z j 00-3002400

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-28 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-29 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-30 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Primal & Dual Primal:Dual XX XX Subject to: UU UU Subject to: XX :Max UU :Min

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 9-31 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Comparison of the Primal and Dual Optimal Tableaus Primal’s Optimal Solution

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