To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-1 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Quantitative Analysis for Management Linear Programming Models:

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-2 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Linear Programming Problem  1. Tujuan adalah maximize or minimize variabel dependen dari beberapa kuantitas variabel independen (fungsi tujuan).  2. Batasan-batasan yang diperlukan guna mencapai tujuan.  Tujuan dan Batasan dinyatakan dalam persamaan linear.

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-3 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Basic Assumptions of Linear Programming  Certainty  Proportionality  Additivity  Divisibility  Nonnegativity

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-4 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company Data - Table 7.1 Hours Required to Produce One Unit Department X 1 Tables X 2 Chairs Available Hours This Week Carpentry Painting/Varnishing Profit/unit$7$5

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-5 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company Data - Table 7.1 Constraints: Maximize: Objective: )varnishing & (painting    XX   XX  )(carpentry    XX STEP 1: STEP 2:

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-6 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company Feasible Region Number of Chairs Number of Tables Painting/Varnishing Carpentry Feasible Region STEP 3: Plot Constraints

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-7 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company Isoprofit Lines Number of Chairs Number of Tables Painting/Varnishing Carpentry 7X 1 + 5X 2 = 210 7X 1 + 5X 2 = 420 STEP 4: Plot Objective Function

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-8 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company Optimal Solution Number of Chairs Number of Tables Painting/Varnishing Carpentry Solution (X 1 = 30, X 2 = 40) Corner Points

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-9 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Test Corner Point Solutions Point 1) (0,0) => 7(0) + 5(0) = $0 Point 2) (0,100) => 7(0) + 5(80) = $400 Point 3) (30,40) => 7(30) + 5(40) = $410 Point 4) (50,0) => 7(50) + 5(0) = $350

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-10 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Solve Equations Simultaneously 4X1 + 3X2 <= 240 2X1 + 1X2 <= 100 X1 = /4 X2 X1 = /2 X /4 X2 = /2 X = 3/4 X2 - 1/2 X2 10 = 1/4 X2 40 = X2; so, 4X1 + 3(40) = 240 4X1 = X1 = 30 To get X1 & X2 values for Point 3:

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-11 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Special Cases in LP  Infeasibility  Unbounded Solutions  Redundancy  Degeneracy  More Than One Optimal Solution

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-12 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J A Problem with No Feasible Solution X2X2 X1X Region Satisfying 3rd Constraint Region Satisfying First 2 Constraints

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-13 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J A Solution Region That is Unbounded to the Right X2X2 X1X Feasible Region X 1 > 5 X 2 < 10 X 1 + 2X 2 > 10

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-14 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J A Problem with a Redundant Constraint X2X2 X1X Feasible Region 2X 1 + X 2 < 30 X 1 < 25 X 1 + X 2 < 20 Redundant Constraint

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-15 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J An Example of Alternate Optimal Solutions Optimal Solution Consists of All Combinations of X 1 and X 2 Along the AB Segment Isoprofit Line for $12 Overlays Line Segment Isoprofit Line for $8 A B AB

To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-16 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Marketing Applications Media Selection - Win Big Gambling Club

To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-17 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Win Big Gambling Club :toSubject )spots/weekTV(max 12X 1  expense) radio(max X290X 43  )spots/week radio (min   XX budget)ad weekly   (XXX X )spots/week radio min.-1(max   X )spots/week radio sec.-30max   (X ads/week) newspapermax   (X :Maximize   XXXX

To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-18 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Manufacturing Applications Production Mix - Fifth Avenue

To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-19 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Fifth Avenue 4.00X3.56X3.07X4.08X :Maximize 4321  1) blend max, (contract X 3  2) blend max,(contract    X 2) blend min, (contract    X 1) blend min, (contract    X polyester) all max, (contract    X polyester) all min, (contract    X silk) max, (contract    Xsilk) min, (contract    X :toSubject cotton) (yards    X.X. polyester) (yards    X.X.X. silk) of (yards    X.

To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-20 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Manufacturing Applications Truck Loading - Goodman Shipping ItemValue ($)Weight (lbs) 122,5007, ,0007,500 38,0003,000 49,5003, ,5004,000 69,7503,500

To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-21 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Goodman Shipping                               X X X X X X X XXXXX XX XXXX (Capacity) :toSubject :value load Maximize

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-22 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company Hours Required to Produce One Unit Department X 1 Tables X 2 Chairs Available Hours This Week Carpentry Painting/Varnishing Profit/unit Constraints: $7$5 Maximize: Objective: )varnishing & (painting    XX   XX  )(carpentry    XX

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-23 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture Company's Feasible Region & Corner Points Number of Chairs X X2X2 Number of Tables B = (0,80) C = (30,40) D = (50,0) Feasible Region    XX    XX

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-24 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture - Adding Slack Variables )varnishing & (painting    XX )(carpentry    XXConstraints: Constraints with Slack Variables )varnishing& (painting )(carpentry       S XX SXX   XX  Objective Function Objective Function with Slack Variables   SSXX 

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-25 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Flair Furniture’s Initial Simplex Tableau Profit per Unit Column Production 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 $0 $7$5$0 S1S1 S2S2 ZjZj C j - Z j $0 Constraint equation rows Gross profit row Net profit row

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-26 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Pivot Row, Pivot Number Identified in the Initial Simplex Tableau CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$ $7$5$0 S1S1 S2S2 ZjZj C j - Z j $0 Pivot row Pivot number Pivot column

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-27 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Completed Second Simplex Tableau for Flair Furniture CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 11/ $7$7/2 $0 $3/2-$7/2$0 $7 $0 X1X1 S2S2 ZjZj C j - Z j $350

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-28 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Pivot Row, Column, and Number Identified in Second Simplex Tableau CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 11/ $7$7/2 $0 $3/2-$7/2$0 $7 $0 X1X1 S2S2 ZjZj C j - Z j $350 (Total Profit) Pivot row Pivot number Pivot column

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-29 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Calculating the New X 1 Row for Flair’s Third Tableau ( Number in new X 1 row ) Number in old X 1 row ( ) Number above pivot number ( ) Corresponding number in new X 2 row ( ) = - x =-x 1 0 3/2 -1/ / (1/2)(1/2) (0) (1) (-2) (1) (40) =-x =-x =-x =-x

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-30 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Final Simplex Tableau for the Flair Furniture Problem CjCj Solution Mix X1X1 X2X2 S1S1 S2S2 Quantity $7$5$0 103/2-1/ $75$1/2$3/2 $0 -$1/2-$3/2 $7 $5 X1X1 X2X2 ZjZj C j - Z j $410

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-31 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J 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.

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-32 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Surplus & Artificial Variables Constraints       XX XXX       AXX ASXXX Constraints-Surplus & Artificial Variables Objective Function   XXX  :Min   MA SXXX  :Min Objective Function-Surplus & Artificial Variables

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-33 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J 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.

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-34 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Special Cases Infeasibility

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-35 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Special Cases Unboundedness Pivot Column

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-36 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Special Cases Degeneracy Pivot Column

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-37 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Special Cases Multiple Optima

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-38 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Sensitivity Analysis High Note Sound Company          XX XX X X :toSubject :Max

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-39 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Sensitivity Analysis High Note Sound Company

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-40 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Simplex Solution High Note Sound Company

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-41 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Simplex Solution High Note Sound Company

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-42 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Nonbasic Objective Function Coefficients

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-43 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Basic Objective Function Coefficients

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-44 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Simplex Solution High Note Sound Company Objective increases by 30 if 1 additional hour of electricians time is available.

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-45 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J 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, 7e by Render/ Stair 9-46 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Primal & Dual          XX XX XX :toSubject :Max Primal:Dual          UU UU UU :toSubject :Min

To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-47 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J Comparison of the Primal and Dual Optimal Tableaus Dual’s Optimal Solution