Resolution and simplification of Dombi-fuzzy relationalequations and latticized optimization programming on DombiFREs

In this paper, we introduce a type of latticized optimization problem whose objective function is themaximum component function and the feasible region is defined as a system of fuzzy relational equalities(FRE) defined by the Dombi t-norm. Dombi family of t-norms includes a parametric family of continuousstrict t-norms, whose members are increasing functions of the parameter. This family of t-norms coversthe whole spectrum of t-norms when the parameter is changed from zero to infinity. Since the feasiblesolutions set of FREs is non-convex and the finding of all minimal solutions is an NP-hard problem,designing an efficient solution procedure for solving such problems is not a trivial job.Some necessary and sufficient conditions are derived to determine the feasibility of the problem. Thefeasible solution set is characterized in terms of a finite number of closed convex cells. An algorithm ispresented for solving this nonlinear problem. It is proved that the algorithm can find the exact optimalsolution and an example is presented to illustrate the proposed algorithm.

Keywords:

Fuzzy relational equations , Dombi t-norm , strict t-norm , latticized objective function , nonlinear programming