Aplikasi Komputer - 03 - Riset Operasi di Excel

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

22. 22
Transportation

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
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