Quantitative Analysis for Management Linear Programming Models:

Transcription

1 Quantitative Analysis for Management Linear Programming Models: by Prentice Hall, Inc., Upper Saddle River,

2 Linear Programming Problem. Tujuan adalah maximize or minimize variabel dependen dari beberapa kuantitas variabel independen (fungsi tujuan).. Batasan-batasan yang diperlukan guna mencapai tujuan. Tujuan dan Batasan dinyatakan dalam persamaan linear by Prentice Hall, Inc., Upper Saddle River,

3 Basic Assumptions of Linear Programming Certainty Proportionality Additivity Divisibility Nonnegativity by Prentice Hall, Inc., Upper Saddle River,

4 Flair Furniture Company Data - Table 7. Department Hours Required to Produce One Unit X Tables X Chairs Available Hours This Week Carpentry Painting/Varnishing Profit/unit $7 $ by Prentice Hall, Inc., Upper Saddle River,

5 Flair Furniture Company Data - Table 7. STEP : Objective: Maximize: 7 X + X 5 STEP : Constraints: X + 3 X X + X 4 40 (carpentry) 00 (painting & varnishing) by Prentice Hall, Inc., Upper Saddle River,

6 Number of Chairs Flair Furniture Company STEP 3: Plot Constraints 0 Feasible Region Painting/Varnishing Feasible Region Carpentry Number of Tables by Prentice Hall, Inc., Upper Saddle River,

7 Number of Chairs Flair Furniture Company Isoprofit Lines Painting/Varnishing 7X + 5X = 0 7X + 5X = 40 STEP 4: Plot Objective Function 40 0 Carpentry Number of Tables by Prentice Hall, Inc., Upper Saddle River,

8 Number of Chairs Flair Furniture Company Optimal Solution 0 Corner Points Painting/Varnishing 3 Solution (X = 30, X = 40) Carpentry 4 Number of Tables by Prentice Hall, Inc., Upper Saddle River,

9 Test Corner Point Solutions Point ) (0,0) => 7(0) + 5(0) = $0 Point ) (0,00) => 7(0) + 5(80) = $400 Point 3) (30,40) => 7(30) + 5(40) = $40 Point 4) (50,0) => 7(50) + 5(0) = $ by Prentice Hall, Inc., Upper Saddle River,

10 Solve Equations Simultaneously To get X & X values for Point 3: 4X + 3X <= 40 X + X <= 00 X = 60-3/4 X X = 50 - / X 60-3/4 X = 50 - / X = 3/4 X - / X 0 = /4 X 40 = X; so, 4X + 3(40) = 40 4X = 40-0 X = by Prentice Hall, Inc., Upper Saddle River,

11 Special Cases in LP Infeasibility Unbounded Solutions Redundancy Degeneracy More Than One Optimal Solution by Prentice Hall, Inc., Upper Saddle River,

12 A Problem with No Feasible Solution X Region Satisfying 3rd Constraint Region Satisfying First Constraints X by Prentice Hall, Inc., Upper Saddle River,

13 A Solution Region That is Unbounded to the Right X 5 X > 5 X < Feasible Region X + X > X by Prentice Hall, Inc., Upper Saddle River,

14 A Problem with a Redundant Constraint X X + X < 30 Redundant Constraint X < Feasible Region X + X < by Prentice Hall, Inc., Upper Saddle River, X

15 An Example of Alternate Optimal Solutions A Optimal Solution Consists of All Combinations of X and X Along the AB Segment B Isoprofit Line for $8 Isoprofit Line for $ Overlays Line Segment AB by Prentice Hall, Inc., Upper Saddle River,

16 Marketing Applications Media Selection - Win Big Gambling Club Medium Audience Reached Per Ad Cost Per Ad($) Maximum Ads Per Week TV spot ( minute) 5, Daily newspaper (full-page ad) Radio spot (30 seconds, prime time) Radio spot ( minute, afternoon) 8, , , Management, 7e by Render/Stair by Prentice Hall, Inc., Upper Saddle River,

17 Win Big Gambling Club Maximize : 5000 X X X X4 Subject to: X (max TV spots/week) X 5 ( max newspaper ads/week) X 3 5 ( max 30- sec. radio spots/week) X 4 0 (max - min. radio spots/week) 800 X + 95 X + 90 X X ( ad weekly budget) X 3 + X 4 5 (min radio spots/week) 90X + 380X (max radio expense) Management, 7e by Render/Stair by Prentice Hall, Inc., Upper Saddle River,

18 Manufacturing Applications Production Mix - Fifth Avenue Variety of Tie Selling Price per Tie ($) Monthly Contract Minimum Monthly Demand Material Required per Tie (Yds) Material Require ments All silk % silk All polyester % polyester Polycotton blend % poly/50% cotton Polycotton - blend % poly/70% cotton Management, 7e by Render/Stair by Prentice Hall, Inc., Upper Saddle River,

19 Fifth Avenue Maximize:4.08X X X X4 Subject to: 0. 5 X 800 (yards of X X X X X4 X 6000 (contract min,silk) X 000 (contract min,all polyester) X 4000 (contract max, silk) 600 (yards cotton) X (contract min,blend) X (contract min,blend) 3000 (yards all polyester) polyester) X 7000 (contract max,silk) X (contract max,blend) X (contract max, blend ) Management, 7e by Render/Stair by Prentice Hall, Inc., Upper Saddle River,

20 3,500 Manufacturing Applications Truck Loading - Goodman Shipping Item Value ($) Weight (lbs),500 7,500 4,000 7, ,000 3, ,500 3,500 5,500 4, ,750 3,500 Management, 7e by Render/Stair by Prentice Hall, Inc., Upper Saddle River,

21 Goodman Shipping Maximize Subject to : 7500 X X X X X X X load X value Management, 7e by Render/Stair X X 6 : 500 X X X X 3 (Capacity) X X X X by Prentice Hall, Inc., Upper Saddle River,

22 Flair Furniture Company Department Hours Required to Produce One Unit X Tables X Chairs Available Hours This Week Carpentry Painting/Varnishing Profit/unit Constraints: Objective: $7 $5 4 X + 3 X 40 X + X 00 Maximize: (carpentry) (painting & varnishing) 5 7 X + X by Prentice Hall, Inc., Upper Saddle River,

23 Number of Chairs Flair Furniture Company's Feasible Region & Corner Points X B = (0,80) Feasible Region X 4 X + 3 X 40 C = (30,40) X + X D = (50,0) Number of Tables by Prentice Hall, Inc., Upper Saddle River,

24 Flair Furniture - Adding Constraints: 4 X + 3 X X + X Slack Variables (carpentry) (painting & varnishing) Constraints with Slack Variables 4 X X + 3 X + X + S 7 X + X 5 + S Objective Function = 40 (carpentry = 00 (painting &varnishing Objective Function with Slack Variables X + S X S by Prentice Hall, Inc., Upper Saddle River, ) )

25 Flair Furniture s Initial Simplex Tableau Profit per Production Unit Mix Column Column C j $0 $0 Real Variables Columns Slack Variables Columns Constant Column $7 $5 $0 $0 Solution Mix X X S S Quantity S S Z j C j - Z j $0 $0 $0 $0 $7 $5 $0 $ $0 $0 Profit per unit row Constraint equation rows Gross profit row Net profit row by Prentice Hall, Inc., Upper Saddle River,

26 Pivot Row, Pivot Number Identified in the Initial Simplex Tableau C j $0 $0 $7 $5 $0 $0 Solution Mix X X S S Quantity S S Z j C j - Z j Pivot number $0 $0 $0 $0 $7 $5 $0 $0 Pivot column $0 $0 Pivot row by Prentice Hall, Inc., Upper Saddle River,

27 Completed Second Simplex Tableau for Flair Furniture C j $7 $0 $7 $5 $0 $0 Solution Mix X X S S Quantity X S / / Z j C j - Z j $7 $7/ $7/ $0 $350 $0 $3/ -$7/ $ by Prentice Hall, Inc., Upper Saddle River,

28 Pivot Row, Column, and Number Identified in Second Simplex Tableau C j $7 $0 $7 $5 $0 $0 Solution Mix X X S S Quantity X S Z j C j - Z j / / Pivot number $7 $7/ $7/ $0 $0 $3/ -$7/ $0 Pivot column $350 (Total Profit) Pivot row by Prentice Hall, Inc., Upper Saddle River,

29 Calculating the New X Row for Flair s Third Tableau ( Number in new X row) Number in old X row 0 3/ -/ 30 = ( ) ( Number ) Corresponding - x ( ) number in above pivot number = - (/) x = / - (/) x = / - (/) x = 0 - (/) x = 50 - (/) x new X row (0) () (-) () (40) by Prentice Hall, Inc., Upper Saddle River,

30 Final Simplex Tableau for the Flair Furniture Problem C j $7 $5 $7 $5 $0 $0 Solution Mix X X S S Quantity X X 0 3/ -/ Z j C j - Z j $7 5 $/ $3/ $0 $0 -$/ -$3/ $ by Prentice Hall, Inc., Upper Saddle River,

31 Simplex Steps for Maximization. Choose the variable with the greatest positive C j - Z j to enter the solution.. Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-topivot-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 by Prentice Hall, Inc., Upper Saddle River,

32 Surplus & Artificial Variables Constraints 5 X 5 X 5 X 5 X + 0 X + 30 X + 0X + 30X + 8 X + 8 X 3 3 = Constraints-Surplus & Artificial Variables - S Objective Function + 9 Min: X 7 5 X + X A A + 9 X + 7 X3 + 0 = 0 = 900 Objective Function-Surplus & Artificial Variables Min : 5 X S + MA + MA by Prentice Hall, Inc., Upper Saddle River,

33 Simplex Steps for Minimization. Choose the variable with the greatest negative C j - Z j to enter the solution.. Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-topivot-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 by Prentice Hall, Inc., Upper Saddle River,

34 Special Cases Infeasibility C j M M Solution X X S S A A Qty Mix 5 X X M A Z j M --M M 800+0M C j -Z j 0 0 M-3 M by Prentice Hall, Inc., Upper Saddle River,

35 Special Cases Unboundedness C j Solution Mix X X S S Qty 9 X S Z j C j -Z j Pivot Column by Prentice Hall, Inc., Upper Saddle River,

36 Special Cases Degeneracy C j Solution Mix X X X 3 S S S 3 Qty 8 X / S 4 0 / S 3 0 /5 0 0 Z j C j -Z j Pivot Column by Prentice Hall, Inc., Upper Saddle River,

37 Special Cases Multiple Optima C j Solution Mix X X S S Qty X 3/ S 0 / 3 Z j 3 0 C j -Z j by Prentice Hall, Inc., Upper Saddle River,

38 Sensitivity Analysis High Note Sound Company Max : 50 X X 3 X Subject to : + 0 X + 4 X + X by Prentice Hall, Inc., Upper Saddle River,

39 Sensitivity Analysis High Note Sound Company by Prentice Hall, Inc., Upper Saddle River,

40 Simplex Solution High Note Sound Company C j Solution Mix X X S S Qty 0 X / ¼ S 5/ 0 -/4 40 Z j C j -Z j by Prentice Hall, Inc., Upper Saddle River,

41 Simplex Solution High Note Sound Company C j Solution Mix X X S S Qty 0 X / ¼ S 5/ 0 -/4 40 Z j C j -Z j by Prentice Hall, Inc., Upper Saddle River,

42 Nonbasic Objective Function Coefficients C j Solution Mix X X S S Qty 0 X / ¼ S 5/ 0 -/4 40 Z j C j -Z j by Prentice Hall, Inc., Upper Saddle River,

43 Basic Objective Function Coefficients C j Solution Mix X X S S Qty 0+ X / ¼ S 5/ 0 -/4 40 Z j 60+/ / C j -Z j -0-/ 0-30-/ by Prentice Hall, Inc., Upper Saddle River,

44 Simplex Solution High Note Sound Company C j Solution Mix X X S S Qty 0 X / ¼ S 5/ 0 -/4 40 Z j C j -Z j Objective increases by 30 if additional hour of electricians time is available by Prentice Hall, Inc., Upper Saddle River,

45 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 by Prentice Hall, Inc., Upper Saddle River,

46 Primal & Dual Max : Primal: 50 X Subject to : X 3 X + 0 X + 4 X + X Dual Min : 80 U Subject to : U 4 U + 60 U U U by Prentice Hall, Inc., Upper Saddle River,

47 Dual s Optimal Solution Primal s Optimal Solution Comparison of the Primal and Dual Optimal Tableaus C j $7 $5 Solution Mix X S Z j C j - Z j Quantity 0 40 $,400 $50 $0 $0 $0 X X S S / /4 0 5/ 0 -/ C j $7 $5 Solution Mix U S Z j C j - Z j Quantity 30 0 $, $0 $0 X X S S /4 0 -/4 0-5/ -/ $ M M A A 0 / - / 0 40 M M by Prentice Hall, Inc., Upper Saddle River,