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. 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

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