Matakuliah Tahun : K0414 / Riset Operasi Bisnis dan Industri : 2008 / 2009 Fuzzy Linear Programming Pertemuan 8 (GSLC) Learning Outcomes • Mahasiswa dapat menyelesaiakan masalah Fuzzy Linear Programming untuk berbagai masalah. Bina Nusantara University 3 Outline Materi: • • • • Pengertian Fuzzy LP Kasus Maksimalisasi Kasus Minimalisasi Contoh pemakaian Bina Nusantara University 4 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. Bina Nusantara University 5 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. Bina Nusantara University 6 Fuzzy Linear Programming • • • • 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. • …….. Bina Nusantara University 7 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} Bina Nusantara University 8 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 Bina Nusantara University if BiX d, if di BiX di Pi if di pi BiX 9 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. Bina Nusantara University 10 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 Bina Nusantara University 11 Example A cont’d • Fuzzy Objective function ( keep constraints crisp) z1 ( x) 3 0 z1 ( x) (3) - 3 z1 ( x) 14 U1 ( x) 14 (3) z1 ( x) 14 1 z 2 ( x) 7 0 z 2 ( x) 7 7 z1 ( x) 21 U 2 ( x) 21 7 z1 ( x) 21 1 Bina Nusantara University 12 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) Bina Nusantara University 4 x1 3x2 45 3x1 x2 30 x1 , x2 0 13 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) Bina Nusantara University 14 Example B cont’d • • • • • 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. Bina Nusantara University 15 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) Bina Nusantara University 16 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. Bina Nusantara University 17 Conclusion Cont’d • 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. Bina Nusantara University 18 Bina Nusantara University 19