Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

Authors:
Cheng Zhang;Xue-hai Yuan;E. Stanley Lee
Affiliations:
College of Information Engineering Dalian University, Dalian 116622, P.R. China;Department of Mathematics Liaoning Normal University, Dalian 116029, P.R. China;Department of Industrial & Manufacturing Systems Engineering Kansas State University, Manhattan, KS 66506, U.S.A.
Venue:
Computers & Mathematics with Applications
Year:
2005

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Abstract

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the ''perturbed'' convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.