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Fuzzy Linear Programming

Fuzzy Linear Programming. Wang YU Iowa State University 12/07/2001. 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: Ã= A fuzzy set is represented solely by stating its membership function. Linear Programming.

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Fuzzy Linear Programming

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  1. Fuzzy Linear Programming Wang YU Iowa State University 12/07/2001

  2. 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: • Ã= • A fuzzy set is represented solely by stating its membership function.

  3. 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.

  4. 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. • ……..

  5. 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

  6. Our model for fuzzy LP Cont’d

  7. 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.

  8. Example A • Crisp LP

  9. Example A cont’d • Fuzzy Objective function ( keep constraints crisp)

  10. Example A cont’d • Example A cont’d

  11. Example B • Crisp LP

  12. 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.

  13. Example B cont’d • Now d=(3700000,170,1300,6) • P=(500000,10,100,6)

  14. 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.

  15. 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.

  16. Reference • [1] Fuzzy set theory and its applications • H.-J. Zimmermann 1991 • [2] Fuzzy set and decision analysis • H.-J. Zimmermann, L.A.Zadeh, B.R.Gaines • 1983

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