On the resolution and optimization of a system of fuzzy relational equations with sup-T composition
On the resolution and optimization of a system of fuzzy relational equations with sup-T composition
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DOI：
10.1007/s10700-008-9029-y
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发表时间：
2008-06
影响因子：
4.7
通讯作者：
Pingke Li;S. Fang
中科院分区：
文献类型：
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作者：
Pingke Li;S. Fang
本文研究了一类具有连续三角范数的模糊关系方程有限系统的解析问题。当这样的系统是一致的，尽管我们知道解集可以由一个最大解和有限多个最小解来表征，但如何有效地找到所有的最小解仍然是一项具有挑战性的任务。利用连续三角范数的表示定理，证明了根据所涉及的三角范数的不同，超方程系统可以分为两类。当三角范数为阿基米德时，最小解与集覆盖问题的无冗余覆盖一一对应。当其为非阿基米德时，它们仅对应于集合覆盖问题的约束无冗余覆盖的一个子集。在此基础上，我们证明了一类线性目标函数的最小化问题可以在多项式时间内简化为一个0-1整数规划问题。这项工作推广了大多数（如果不是全部的话）已知结果，并提供了一个统一的框架来处理超方程系统的求解和优化问题。进一步的概括和相关问题也包括讨论。
This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-Tcomposition, whereTis 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-Tequations 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 aset covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of aset covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-Tequations 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-Tequations. Further generalizations and related issues are also included for discussion.