Aplikom03 excel or
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This document discusses various optimization models that can be solved using linear programming, including linear programming, transportation, assignment, minimal cost network, maximal flow network, and shortest path network. It provides examples of the objective functions and constraints for each model and explains how they can be set up and solved using the Solver tool in Microsoft Excel.
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Aplikom03 excel or
- 1. 1 Aplikasi Riset Operasional Dalam Spread Sheet Arif Rahman, ST MT
- 2. 2 Linear Programming Linear Programming merupakan pemodelan matematika optimasi pada permasalahan minimasi dan maksimasi satu fungsi tujuan linier dengan satu atau beberapa persamaan dan atau pertidaksamaan fungsi kendala linier. Pada tahun 1939, L.V. Kantorovich (Soviet) merumuskan permasalahan dalam formula linear programming. Metode simplex sebagai algoritma pemecahan permasalahan linear programming dikembangkan oleh George B. Dantzig (Amerika Serikat) pada 1947. Istilah linear programming dicetuskan oleh T.C.Koopmans pada 1948.
- 3. 3 Linear Programming Variables Decision variables Structural variables Auxiliary variables Slack variables Artificial variables Coefficients Cost coefficients Technological coefficients Constraint parameter or Right Hand Side value Function Objective or criterion function Restriction or functional constraints Nonnegativity constraints
- 4. 4 Linear Programming Maximize z = c1x1 + c2x2 + … + cnxn Subject to a11x1 + a12x2 + … + a1nxn ≤ b1 a21x1 + a22x2 + … + a2nxn ≤ b2 am1x1 + am2x2 + … + amnxn ≤ bm and x1 ≥ 0; x2 ≥ 0; … ; xn ≥ 0 Maximize or Minimize ≤ or ≥ or = ≤ 0 or ≥0 or unrestricted
- 5. 5 Linear Programming Metode Simplex Metode dua fase Metode Big-M Metode Revised Simplex Metode Primal-Dual Special variables Bounded variables Unrestricted variables Integer variables Goal Programming
- 6. 6 Linear Programming Asumsi Proportionality Additivity Divisibility Deterministic or certainty Variabel basis dan nonbasis Solusi Solusi optimal unik Solusi optimal alternatif Solusi unbounded Solusi infeasible Analisis Sensitivitas Perubahan cost coefficient Perubahan constraint parameter Perubahan technological coefficient Penambahan decision variable Penambahan restriction constraint
- 7. 7 Program Solver dalam Excel MS Excel 2003 & sebelumnya Pilih menu Tools Pilih pulldown submenu Add-Ins Aktifkan Solver Add-in
- 8. 8 Program Solver dalam Excel MS Excel 2007 & sesudahnya Buka Excel Options Pilih Add-Ins Manage Excell Add-Ins, & Go Aktifkan Solver Add-in
- 9. 9 Program Solver dalam Excel
- 10. 10 Linear Programming Maximize z = 100 x1 + 200 x2 + 300 x3 Subject to 2x1 + 1x2 + 3x3 ≤ 5000 -1x1 + 2x2 + 1x3 ≤ 2000 4x1 + 2x2 + 2x3 ≥ 100 3x1 + 0x2 + 1x3 = 500
- 11. 11 Linear Programming Formula pada Z di objective function = ( c1*x1 ) + ( c2*x2 ) + … + ( cn*xn ) Atau = Sumproduct (cost coefficients array ; decision variables array) Formula pada Value di constraints = ( a11*x1 ) + ( a12*x2 ) + … + ( a1n*xn ) Atau = Sumproduct (technological coefficients array ; decision variables array) Formula pada Slack Variables = RHS value - value
- 12. 12 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables Add Constraint meliputi restriction constraints dan nonnegativity constraints
- 13. 13 Solusi Answer Report Sensitivity AnalysisReport Variables Limits Report
- 14. 14 Integer Linear Programming Add Constraint dengan range dari decision variables sebagai integer
- 15. 15 Goal Programming Factor Contribution 1 2 3 Goal Penalty Profit 12 9 15 ≥ 120 5 (-) Employment 5 3 4 = 40 2 (+), 4 (-) Investment 5 7 8 ≤ 60 3 (+)
- 16. 16 Goal Programming Minimize z = 5 y1 - + 2 y2 + + 4 y2 - + 3 y3 + Subject to 12x1 + 9x2 + 15x3 – ( y1 + - y1 - )= 120 5x1 + 3x2 + 4x3 – ( y2 + - y2 - )= 40 5x1 + 7x2 + 8x3 – ( y3 + - y3 - )= 60
- 17. 17 Goal Programming Formula pada Z di objective function = Sumproduct (cost coefficients array ; auxiliary variables array) Formula pada Value di constraints = Sumproduct (technological coefficients array ; overall variables array) Formula pada Slack Variables = RHS value - value
- 18. 18 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables dan auxiliary variables Add Constraint meliputi restriction constraints dan nonnegativity constraints
- 19. 19 Transportation Minimize Subject to for each i = 1, 2, … , m for each j = 1, 2, … , n and xij ≥ 0 where i = 1, 2, … , m; j = 1, 2, … , n ∑∑= = = m i n j ijij xcZ 1 1 . i n j ij sx ≤∑=1 j m i ij dx ≥∑=1
- 20. 20 Transportation Destination Sup- 1 2 3 4 ply Source 464 513 654 867 1 75 352 416 690 791 2 125 995 682 388 685 3 100 Demand 80 65 70 85
- 21. 21 Transportation Minimize z = 464 x11 + 513 x12 + … + 685 x34 Supply constraints x11 + x12 + x13 + x14 ≤ 75 x21 + x22 + x23 + x24 ≤ 125 x31 + x32 + x33 + x34 ≤ 100 Demand constraints x11 + x21 + x31 ≥ 80 x12 + x22 + x32 ≥ 65 x13 + x23 + x33 ≥ 70 x14 + x24 + x34 ≥ 85
- 23. 23 Transportation Formula pada Z di objective function = Sumproduct (cost coefficients array ; decision variables array) Formula pada Value di constraints = Sumproduct (technological coefficients array ; decision variables array) Formula pada Slack Variables = RHS value - value
- 24. 24 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables Add Constraint meliputi supply constraints dan demand constraints
- 25. 25 Assignment Minimize Subject to for each i = 1, 2, … , m for each j = 1, 2, … , n and xij ≥ 0 where i = 1, 2, … , m; j = 1, 2, … , n ∑∑= = = m i n j ijij xcZ 1 1 . 1 1 =∑= n j ijx 1 1 =∑= m i ijx
- 26. 26 Assignment Job 1 2 3 Operator 13 12 11 1 15 13 20 2 5 10 6 3
- 27. 27 Assignment Minimize z = 13 x11 + 12 x12 + … + 6 x33 Operator constraints x11 + x12 + x13 = 1 x21 + x22 + x23 = 1 x31 + x32 + x33 = 1 Job constraints x11 + x21 + x31 = 1 x12 + x22 + x32 = 1 x13 + x23 + x33 = 1
- 28. 28 Assignment
- 29. 29 Assignment Formula pada Z di objective function = Sumproduct (cost coefficients array ; decision variables array) Formula pada Value di constraints = Sumproduct (technological coefficients array ; decision variables array) Formula pada Slack Variables = RHS value - value
- 30. 30 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables Add Constraint meliputi restriction constraints, nonnegativity constraint dan Integer constraints
- 31. 31 Minimal Cost Network Minimize Subject to for each i = 1, 2, … , m and xij ≥ 0 where i , j = 1, 2, … , m ∑∑= = = m i m j ijij xcZ 1 1 . i m k ki m j ij bxx =− ∑∑ == 11
- 32. 32 Minimal Cost Network 4 2 -5 -1 7 -5 3 6 -1 2 4
- 33. 33 Minimal Cost Network Minimize z = 2 x12 - 5 x13 + … + 7 x41 Subject to ( x12 + x13 ) - ( x41 ) = 4 ( x23 + x24 )- ( x12 + x32 ) = 2 ( x32 + x34 )- ( x13 + x23 ) = -1 ( x41 )- ( x24 + x34 ) = -5
- 34. 34 Minimal Cost Network Formula pada Z di objective function = Sumproduct (cost coefficients array ; decision variables array) Formula pada Value di constraints = Sumproduct (technological coefficients array ; decision variables array) Formula pada Slack Variables = RHS value - value
- 35. 35 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables Add Constraint meliputi restriction constraints dan nonnegativity constraint
- 36. 36 Maximal Flow Network Minimize Z = flow Subject to xij ≤ uij where i = 1, 2, … , m and xij ≥ 0 where i , j = 1, 2, … , m =− << = =− ∑∑ == miflow mi iflow xx m k ki m j ij if 1if0 1if 11
- 37. 37 Maximal Flow Network 4 2 2 1 3
- 38. 38 Maximal Flow Network Maximize z = x12 + x13 Subject to ( x12 + x13 ) = z ( x23 + x24 )- ( x12 ) = 0 ( x34 )- ( x13 + x23 ) = 0 - ( x24 + x34 ) = -z x12 ≤ 1 ; x13 ≤ 4 ; x23 ≤ 2 ; x24 ≤ 3 ; x34 ≤ 2
- 39. 39 Maximal Flow Network Formula pada Z di objective function = Sum (from node 1 decision variables array) Formula pada RHS value = Z untuk node 1 = 0 untuk node selain 1 atau m = -Z untuk node m Formula pada Value di constraints = Sumproduct (technological coefficients array ; decision variables array) Formula pada Slack Variables = RHS value - value
- 40. 40 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables Add Constraint meliputi restriction constraints dan nonnegativity constraint
- 41. 41 Shortest Path Network Minimize Subject to for each i = 1, 2, … , m and xij ≥ 0 where i , j = 1, 2, … , m =− << = =− ∑∑ == mi mi i xx m k ki m j ij if1 1if0 1if1 11 ∑∑= = = m i m j ijij xcZ 1 1 .
- 42. 42 Shortest Path Network -1 -6 -4 2 3
- 43. 43 Shortest Path Network Minimize z = 2 x12 - 1 x13 - 4 x23 + 3 x24 - 6 x34 Subject to ( x12 + x13 ) = 1 ( x23 + x24 )- ( x12 ) = 0 ( x34 )- ( x13 + x23 ) = 0 - ( x24 + x34 ) = -1
- 44. 44 Shortest Path Network Formula pada Z di objective function = Sumproduct (cost coefficients array ; decision variables array) Formula pada RHS value = 1 untuk node 1 = 0 untuk node selain 1 atau m = -1 untuk node m Formula pada Value di constraints = Sumproduct (technological coefficients array ; decision variables array) Formula pada Slack Variables = RHS value - value
- 45. 45 Tools Solver… Set Target Cell pada cell dari Z Changing Cells pada range dari decision variables Add Constraint meliputi restriction constraints, nonnegativity constraint dan Integer constraints
- 46. 46 Akhir Perkuliahan…Akhir Perkuliahan… …… Ada Yang DitanyakanAda Yang Ditanyakan