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

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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.

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Authors and Affiliations

  1. Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695-7906, USA

    Pingke Li & Shu-Cherng Fang

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Shu-Cherng Fang

  3. College of Management, Dalian University of Technology, Dalian, 116024, China

    Shu-Cherng Fang

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  1. Pingke Li
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Correspondence to Pingke Li.

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Li, P., Fang, SC. On the resolution and optimization of a system of fuzzy relational equations with sup-T composition. Fuzzy Optim Decis Making 7, 169–214 (2008). https://doi.org/10.1007/s10700-008-9029-y

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