Mata kuliah:K0164/ Pemrograman Matematika
Tahun
:2008
Fuzzy Linear Programming
Pertemuan 10:
Learning Outcomes
• Mahasiswa dapat menyelesaiakan masalah Fuzzy
Linear Programming untuk berbagai masalah.
Outline Materi:
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Pengertian Fuzzy LP
Kasus Maksimalisasi
Kasus Minimalisasi
Contoh pemakaian
Fuzzy Sets
• If X is a collection of objects denoted generically by x,
then a fuzzy set à in X is a set of ordered pairs:
( x)) | x  X }
• Ã= {( x,
• A fuzzy set is represented solely by stating its
membership function.

Linear Programming
• Min z=c’x
• St. Ax<=b,
•
x>=0,
• Linear Programming can be solved efficiently by simplex
method and interior point method. In case of special
structures, more efficiently methods can be applied.
Fuzzy Linear Programming
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There are many ways to modify a LP into a fuzzy LP.
The objective function maybe fuzzy
The constraints maybe fuzzy
The relationship between objective function and
constraints maybe fuzzy.
• ……..
Our model for fuzzy LP
• Ĉ~fuzzy constraints {c,Uc}
• Ĝ~fuzzy goal (objective function) {g,Ug}
• Ď= Ĉ and Ĝ{d,Ud}
• Note: Here our decision Ď is fuzzy. If you want a crisp
decision, we can define:
• λ=max Ud to be the optimal decision
Ud  min{ Uc,Ug}
Our model for fuzzy LP Cont’d
 CT X  Z
AX  b
c
( A)  B
z
( b)  d
X 0
1
Ui( x)   [0,1]
1
if BiX  d,
if di  BiX  di  Pi
if di  pi  BiX
Our model for fuzzy LP Cont’d
• Maximize λ
• St. λpi+Bix<=di+pi i= 1,2,….M+1
•
x>=0
• It’s a regular LP with one more constraint and can be
solved efficiently.
Example A
• Crisp LP
 1 2 x1
max Z ( x)  (
)( )
2 1 x2
(z1 , z 2 )  (14,7) at (7,0)
 x1  3x2  21
(z1 , z 2 )  (3,21) at (3.4,0.2)
x1  3x2  27
4 x1  3x2  45
3x1  x2  30
x1 , x2  0
Example A cont’d
• Fuzzy Objective function ( keep constraints crisp)
0
z1 ( x)  3
z1 ( x)  (3)
U1 ( x) 
- 3  z1 ( x)  14
14  (3)
1
z1 ( x)  14
0
z 2 ( x)  7
z 2 ( x)  7
U 2 ( x) 
7  z1 ( x)  21
21  7
1
z1 ( x)  21
Example A cont’d
• Example A cont’d
max 
17
 1 2 x1
( )(
)( )
14
2 1 x2
 x1  3x2  21
x1  3x2  27
  0.74
(z1 , z 2 )  (17.38,4.58)
at (5.03,7.32)
4 x1  3x2  45
3x1  x2  30
x1 , x2  0
Example B
• Crisp LP
min z  41400 x1  44300 x2  48100 x3  49100 x4
0.84 x1  1.44 x2  2.16 x3  2.4 x4  170
16 x1  16 x2  16 x3  16 x4  1300
x1  6
x2 , x3 , x4  0
( x1 , x2 , x3 , x4 , z )  (6,16.29,0,58.96,3864795)
constra int s  (170,1300,6)
Example B cont’d
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Fuzzy Objective function Fuzzy Constraints
Maximize λ
St. λpi+Bix<=di+pi i= 1,2,….M+1
x>=0
Apply this to both of the objective function and
constraints.
Example B cont’d
• Now d=(3700000,170,1300,6)
• P=(500000,10,100,6)
41400 x1  44300 x2  48100 x3  49100 x4  500000  3700000
0.84 x1  1.44 x2  2.16 x3  0.24 x4  10  170
16 x1  16 x2  16 x3  16 x4  100  1300
x1  6  6
x2 , x3 , x4  0
( x1 , x2 , x3 , x4 , z )  (17.414,0,0,66.54,3988250)
constra int s  (174.33,1343.33,17.414)
Conclusion
• Here we showed two cases of fuzzy LP. Depends on the
models used, fuzzy LP can be very differently. ( The
choosing of models depends on the cases, no general
law exits.)
• In general, the solution of a fuzzy LP is efficient and give
us some advantages to be more practical.
Conclusion Cont’d
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Advantages of our models:
1. Can be calculated efficiently.
2. Symmetrical and easy to understand.
3. Allow the decision maker to give a fuzzy description of
his objectives and constraints.
• 4. Constraints are given different weights.