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

  • Authors:

  • Pingke Li;Shu-Cherng Fang

  • Affiliations:

  • Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, USA 27695-7906;Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, USA 27695-7906 and Department of Mathematical Sciences, Tsinghua University, Beijing, Ch ...

  • Venue:
  • Fuzzy Optimization and Decision Making
  • Year:
  • 2008

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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.