On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

Abstract

This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0-1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.

On the resolution of bipolar max-min equations

“…Besides, those omitted equations corresponding to the tautologies in the characteristic boolean formula should be taken into consideration as well because the components of a solution x ∈ S(A + , A − , b) may assume the values that are not contained inx andx. It turns out that a system of integer linear inequalities is sufficient to characterize S(A + , A − , b) by applying the techniques developed in Li and Fang [7] and Li and Jin [9]. Moreover, if the nonempty elements inQ are all singletons, e. g., Example 2.6, the situation is somehow easier to deal with as is illustrated analogously in Li and Jin [9] for min-biimplication equations.…”

“…In this section, we apply some basic techniques originally developed for solving fuzzy relational equations, see, e. g., Li and Fang [7] and Li and Jin [9], to demonstrate that determining the consistency of a system of bipolar max-min equations is an NP-complete problem.…”

“…However, the method developed in Li and Fang [7] can be naturally extended to handle these general scenarios of bipolar max-min equations.…”

“…The system A − • ¬x = b can be handled analogously and its solution set, if not empty, can be characterized by a minimum solution and finitely many maximal solutions. For some detailed discussion on fuzzy relational equations, see, e. g., Di Nola et al [3], De Baets [2], Peeva and Kyosev [11], Li and Fang [7,8], Li [6], and references therein.…”

“…Besides, the identification of these maximal and minimal solution pairs itself may not be an easy problem. In this paper, by combining the techniques developed in Li and Fang [7] and Li and Jin [9], we provide a reformulation approach to bipolar max-min equations and demonstrate that a system of bipolar max-min equations can be characterized by a system of integer linear inequalities. This implies that the bipolar max-min equation constrained optimization problems may be handled within the framework of integer and combinatorial optimization and hence demand no particular solving techniques.…”