A fuzzy dual decomposition method for large-scale multiobjective nonlinear programming problems
Fuzzy Sets and Systems - Special issue on operations research
Connection between fuzzy theory, simulated annealing, and convex duality
Fuzzy Sets and Systems
On duality in linear programming under fuzzy environment
Fuzzy Sets and Systems - Theme: Decision and optimization
Matrix Games with Fuzzy Goals and Fuzzy Linear Programming Duality
Fuzzy Optimization and Decision Making
Duality Theory in Fuzzy Optimization Problems
Fuzzy Optimization and Decision Making
Duality in Fuzzy Linear Programming: Some New Concepts and Results
Fuzzy Optimization and Decision Making
Duality theory in fuzzy mathematical programming problems with fuzzy coefficients
Computers & Mathematics with Applications
Convex fuzzy mapping and operations of convex fuzzy mappings
Computers & Mathematics with Applications
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Under a general setting of partial ordering defined on the set of all fuzzy numbers, the duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on two solution concepts for primal problem and three solution concepts for dual problem are derived in this paper. Those solution concepts are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among fuzzy numbers introduced in this paper is a partial ordering, not a total ordering. We also provide a concept of fuzzy scalar (inner) product for fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function are proposed via the concept of fuzzy scalar product. Under these settings, the dual problem is formulated, and the objective function of this dual problem is a point-to-set fuzzy-valued function. In this case, three solution concepts for dual problem are proposed. The duality theorems and saddle point optimality conditions can be naturally elicited based on these three solution concepts.