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**On the resolution and optimization of a system of fuzzy relational equations with sup-\(T\) composition.**
*(English)*
Zbl 1169.90493

Summary: 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.

### MSC:

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90C10 | Integer programming |

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\textit{P. Li} and \textit{S.-C. Fang}, Fuzzy Optim. Decis. Mak. 7, No. 2, 169--214 (2008; Zbl 1169.90493)

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