Aplikom03 excel or
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Aplikasi Riset Operasional
Dalam Spread SheetArif Rahman, ST MT
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Linear ProgrammingLinear 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.
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Linear ProgrammingVariables
Decision variablesStructural variablesAuxiliary variablesSlack variablesArtificial variables
CoefficientsCost coefficientsTechnological coefficientsConstraint parameter or Right Hand Side value
FunctionObjective or criterion functionRestriction or functional constraintsNonnegativity constraints
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Linear ProgrammingMaximize z = c1x1 + c2x2 + … + cnxn
Subject toa11x1 + a12x2 + … + a1nxn b1
a21x1 + a22x2 + … + a2nxn b2
am1x1 + am2x2 + … + amnxn bm
andx1 0; x2 0; … ; xn 0
Maximize or Minimize
or or =
0 or 0 or unrestricted
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Linear ProgrammingMetode SimplexMetode dua faseMetode Big-MMetode Revised SimplexMetode Primal-DualSpecial variables
Bounded variablesUnrestricted variablesInteger variables
Goal Programming
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Linear ProgrammingAsumsi
ProportionalityAdditivityDivisibilityDeterministic or certainty
Variabel basis dan nonbasisSolusi
Solusi optimal unikSolusi optimal alternatifSolusi unboundedSolusi infeasible
Analisis SensitivitasPerubahan cost coefficientPerubahan constraint parameterPerubahan technological coefficientPenambahan decision variablePenambahan restriction constraint
7Program Solver dalam ExcelMS Excel 2003 &
sebelumnyaPilih menu ToolsPilih pulldown submenu Add-InsAktifkan Solver Add-in
8Program Solver dalam ExcelMS Excel 2007 &
sesudahnyaBuka Excel OptionsPilih Add-InsManage Excell Add-Ins, & GoAktifkan Solver Add-in
9Program Solver dalam Excel
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Linear ProgrammingMaximize 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
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Linear ProgrammingFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variables Add Constraint meliputi restriction constraints dan nonnegativity constraints
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SolusiAnswer ReportSensitivity AnalysisReportVariables Limits Report
14Integer Linear Programming
Add Constraint dengan range dari decision variables sebagai integer
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Goal ProgrammingFactor
Contribution1 2 3
Goal Penalty
Profit 12 9 15 120 5 (-)
Employment 5 3 4 = 40 2 (+), 4 (-)
Investment 5 7 8 60 3 (+)
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Goal ProgrammingMinimize z = 5 y1
- + 2 y2+ + 4 y2
- + 3 y3+
Subject to12x1 + 9x2 + 15x3 – ( y1
+ - y1- )= 120
5x1 + 3x2 + 4x3 – ( y2+ - y2
- )= 40
5x1 + 7x2 + 8x3 – ( y3+ - y3
- )= 60
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Goal ProgrammingFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variables dan auxiliary variablesAdd Constraint meliputi restriction constraints dan nonnegativity constraints
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TransportationMinimize
Subject to
for each i = 1, 2, … , m
for each j = 1, 2, … , n
andxij 0 where i = 1, 2, … , m; j = 1, 2, … , n
m
i
n
jijij xcZ
1 1
.
i
n
jij sx
1
j
m
iij dx
1
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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
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TransportationMinimize z = 464 x11 + 513 x12 + … + 685 x34
Supply constraintsx11 + x12 + x13 + x14 75
x21 + x22 + x23 + x24 125
x31 + x32 + x33 + x34 100
Demand constraintsx11 + x21 + x31 80
x12 + x22 + x32 65
x13 + x23 + x33 70
x14 + x24 + x34 85
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Transportation
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TransportationFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variablesAdd Constraint meliputi supply constraints dan demand constraints
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AssignmentMinimize
Subject to
for each i = 1, 2, … , m
for each j = 1, 2, … , n
andxij 0 where i = 1, 2, … , m; j = 1, 2, … , n
m
i
n
jijij xcZ
1 1
.
11
n
jijx
11
m
iijx
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Assignment Job 1 2 3
Operator
13 12 11
1
15 13 20
2
5 10 6
3
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AssignmentMinimize z = 13 x11 + 12 x12 + … + 6 x33
Operator constraintsx11 + x12 + x13 = 1
x21 + x22 + x23 = 1
x31 + x32 + x33 = 1
Job constraintsx11 + x21 + x31 = 1
x12 + x22 + x32 = 1
x13 + x23 + x33 = 1
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Assignment
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AssignmentFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variablesAdd Constraint meliputi restriction constraints, nonnegativity constraint dan Integer constraints
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Minimal Cost NetworkMinimize
Subject to
for each i = 1, 2, … , m
andxij 0 where i , j = 1, 2, … , m
m
i
m
jijij xcZ
1 1
.
i
m
kki
m
jij bxx
11
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Minimal Cost Network
4
2
-5
-1
7
-5 3
6 -1
2 4
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Minimal Cost NetworkMinimize 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
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Minimal Cost NetworkFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variablesAdd Constraint meliputi restriction constraints dan nonnegativity constraint
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Maximal Flow NetworkMinimize Z = flowSubject to
xij uij where i = 1, 2, … , m
andxij 0 where i , j = 1, 2, … , m
miflowmi
iflowxx
m
kki
m
jij
if1 if0
1 if
11
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Maximal Flow Network
4 2
2
1 3
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Maximal Flow NetworkMaximize z = x12 + x13
Subject to( x12 + x13 ) = z( x23 + x24 ) - ( x12 ) = 0( x34 ) - ( x13 + x23 ) = 0- ( x24 + x34 ) = -zx12 1 ; x13 4 ; x23 2 ; x24 3 ; x34 2
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Maximal Flow NetworkFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variablesAdd Constraint meliputi restriction constraints dan nonnegativity constraint
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Shortest Path NetworkMinimize
Subject to
for each i = 1, 2, … , m
andxij 0 where i , j = 1, 2, … , m
mimi
ixx
m
kki
m
jij
if11 if0
1 if1
11
m
i
m
jijij xcZ
1 1
.
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Shortest Path Network
-1 -6
-4
2 3
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Shortest Path NetworkMinimize 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
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Shortest Path NetworkFormula 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
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Tools Solver…Set Target Cell pada cell dari ZChanging Cells pada range dari decision variablesAdd Constraint meliputi restriction constraints, nonnegativity constraint dan Integer constraints
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