Fuzzy relational equations: Resolution and optimization.

摘要

Fuzzy relational equations play an important role as a platform in various applications of fuzzy sets and systems. The resolution and optimization of fuzzy relational equations are of our particular interests from both of the theoretical and applicational viewpoints. In this dissertation, fuzzy relational equations are treated in a unified framework and classified according to different aspects of their composite operations.;For a given finite system of fuzzy relational equations with a specific composite operation, the consistency of the system can be verified in polynomial time by constructing a potential maximum/minimum solution and characteristic matrix. The solution set of a consistent system can be characterized by a unique maximum solution and finitely many minimal solutions, or dually, by a unique minimum solution and finitely many maximal solutions. The determination of all minimal/maximal solutions is closely related to the detection of all irredundant coverings of a set covering problem defined by the characteristic matrix, which may involve additional constraints. In particular, for fuzzy relational equations with sup- T composition where T is a continuous triangular norm, the existence of the additional constraints depends on whether T is Archimedean or not.;Fuzzy relational equation constrained optimization problems are investigated as well in this dissertation. It is shown that the problem of minimizing an objective function subject to a system of fuzzy relational equations can be reduced in general to a 0-1 mixed integer programming problem. If the objective function is linear, or more generally, separable and monotone in each variable, then it can be further reduced to a set covering problem. Moreover, when the objective function is linear fractional, it can be reduced to a 0-1 linear fractional optimization problem and then solved via parameterization methods. However, if the objective function is max-separable with continuous monotone or unimodal components, then the problem can be solved efficiently, and its optimal solution set can be well characterized.