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

• Published in: Fuzzy optimization and decision making Vol. 7; no. 2; pp. 169 - 214
• Main Authors: Li, Pingke, Fang, Shu-Cherng
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.06.2008; Springer Nature B.V
• Subjects: Artificial Intelligence; Calculus of Variations and Optimal Control; Optimization; Decision making; Fuzzy logic; Fuzzy sets; Integer programming; Investigations; Mathematical Logic and Foundations; Mathematical models; Mathematics; Mathematics and Statistics; Norms; Operations Research/Decision Theory; Optimization; Probability Theory and Stochastic Processes; Statistical analysis; Studies; Integer programming; Fuzzy relational equations; Triangular norms; Fuzzy optimization
• ISSN: 1568-4539; 1573-2908
• DOI: 10.1007/s10700-008-9029-y

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.