Mixed-Discrete Fuzzy Multiobjective Programming for Engineering Optimization Using Hybrid Genetic Algorithm

Abstract

Although much attention has been focused on the development and applications of fuzzy optimization, multiobjective programming, and mixed-discrete optimization methods separately, fuzzy multiobjective optimization problems in mixed-discrete design space have not been addressed in the literature. It is mainly because of the lack of mature and robust theories of mixed-discrete and multiobjective optimization. In most practical applications, designers often encounter problems involving imprecise or fuzzy information, multiple objectives, and mixed-discrete design variables. A new method is presented in which the fuzzy λ formulation and game theory techniques are combined with a mixed-discrete hybrid genetic algorithm for solving mixed-dixcrete fuzzy multiobjective programming problems. Three example problems, dealing with the optimal designs of a two-bar truss, a conical convective spine, and a 25-bar truss, demonstrate that the method can be flexibly and effectively applied to various kinds of engineering design problems to obtain more realistic and satisfactory results in an imprecise environment. Nomenclature D = fuzzy feasible solution domain d =f avorable search direction Fiti(X) = fitness function fi(X) = ith objective function f max i = upper bound of ith objective function f min i =l ower bound of ith objective function G j = allowable interval of the constraint function g j g j(X) = jth inequality constraint function k = number of objective functions M = size of the population m = number of inequality constraints n = total number of design variables nd = number of discrete design variables nq = number of discrete design variables with equal spacing X = design vector X c = centroid of the composite polygon Xw =v ertex with the lowest fitness value xi = ith design variable µD(X) = membership function of the design vector µfi(X) = membership function of the ith objective µgj(X) = membership function of the jth constraint