1. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-1 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Quantitative Analysis for Management Linear Programming Models:

2. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-2 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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.

3. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-3 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Basic Assumptions of Linear Programming  Certainty  Proportionality  Additivity  Divisibility  Nonnegativity

4. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-4 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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 4242 3131 240 100 Profit/unit$7$5

5. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-5 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Flair Furniture Company Data - Table 7.1 Constraints: Maximize: Objective: )varnishing & (painting    XX   XX  )(carpentry    XX STEP 1: STEP 2:

6. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-6 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Flair Furniture Company Feasible Region 120 100 80 60 40 20 0 Number of Chairs 20 40 60 80 100 Number of Tables Painting/Varnishing Carpentry Feasible Region STEP 3: Plot Constraints

7. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-7 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Flair Furniture Company Isoprofit Lines Number of Chairs 120 100 80 60 40 20 0 20 40 60 80 100 Number of Tables Painting/Varnishing Carpentry 7X 1 + 5X 2 = 210 7X 1 + 5X 2 = 420 STEP 4: Plot Objective Function

8. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-8 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Flair Furniture Company Optimal Solution Number of Chairs 120 100 80 60 40 20 0 20 40 60 80 100 Number of Tables Painting/Varnishing Carpentry Solution (X 1 = 30, X 2 = 40) Corner Points 1 2 3 4

9. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-9 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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

10. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-10 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Solve Equations Simultaneously 4X1 + 3X2 <= 240 2X1 + 1X2 <= 100 X1 = 60 - 3/4 X2 X1 = 50 - 1/2 X2 60 - 3/4 X2 = 50 - 1/2 X2 60 - 50 = 3/4 X2 - 1/2 X2 10 = 1/4 X2 40 = X2; so, 4X1 + 3(40) = 240 4X1 = 240 - 120 X1 = 30 To get X1 & X2 values for Point 3:

11. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-11 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Special Cases in LP  Infeasibility  Unbounded Solutions  Redundancy  Degeneracy  More Than One Optimal Solution

12. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-12 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 A Problem with No Feasible Solution X2X2 X1X1 8642086420 24682468 Region Satisfying 3rd Constraint Region Satisfying First 2 Constraints

13. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-13 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 A Solution Region That is Unbounded to the Right X2X2 X1X1 15 10 5 0 5 10 15 Feasible Region X 1 > 5 X 2 < 10 X 1 + 2X 2 > 10

14. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-14 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 A Problem with a Redundant Constraint X2X2 X1X1 30 25 20 15 10 5 0 510 15 20 25 30 Feasible Region 2X 1 + X 2 < 30 X 1 < 25 X 1 + X 2 < 20 Redundant Constraint

15. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 7-15 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 An Example of Alternate Optimal Solutions 876543210876543210 1234567812345678 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

16. To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-16 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Marketing Applications Media Selection - Win Big Gambling Club

17. To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-17 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Win Big Gambling Club :toSubject )spots/weekTV(max 12X 1  expense) radio(max 1800380X290X 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

18. To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-18 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Manufacturing Applications Production Mix - Fifth Avenue

19. To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-19 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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.

20. To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-20 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Manufacturing Applications Truck Loading - Goodman Shipping ItemValue ($)Weight (lbs) 122,5007,500 224,0007,500 38,0003,000 49,5003,500 511,5004,000 69,7503,500

21. To accompany Quantitative Analysis for Management, 7e by Render/Stair 8-21 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Goodman Shipping                               X X X X X X X XXXXX XX XXXX (Capacity) :toSubject :value load Maximize

22. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-22 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Flair Furniture Company 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 Maximize: Objective: )varnishing & (painting    XX   XX  )(carpentry    XX

23. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-23 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 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

24. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-24 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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 

25. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-25 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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 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

26. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-26 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 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

27. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-27 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 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

28. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-28 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 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

29. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-29 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 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 -1/2 30 11/2 0 50 (1/2)(1/2) (0) (1) (-2) (1) (40) =-x =-x =-x =-x

30. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-30 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 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

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

32. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-32 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Surplus & Artificial Variables Constraints       XX XXX       AXX ASXXX Constraints-Surplus & Artificial Variables Objective Function   XXX  :Min   MA SXXX  :Min Objective Function-Surplus & Artificial Variables

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

34. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-34 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Special Cases Infeasibility

35. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-35 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Special Cases Unboundedness Pivot Column

36. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-36 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Special Cases Degeneracy Pivot Column

37. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-37 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Special Cases Multiple Optima

38. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-38 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Sensitivity Analysis High Note Sound Company          XX XX X X :toSubject :Max

39. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-39 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Sensitivity Analysis High Note Sound Company

40. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-40 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Simplex Solution High Note Sound Company

41. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-41 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Simplex Solution High Note Sound Company

42. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-42 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Nonbasic Objective Function Coefficients

43. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-43 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Basic Objective Function Coefficients

44. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-44 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Simplex Solution High Note Sound Company Objective increases by 30 if 1 additional hour of electricians time is available.

45. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-45 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 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.

46. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-46 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Primal & Dual          XX XX XX :toSubject :Max Primal:Dual          UU UU UU :toSubject :Min

47. To accompany Quantitative Analysis for Management, 7e by Render/ Stair 9-47 © 2000 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 Comparison of the Primal and Dual Optimal Tableaus Dual’s Optimal Solution