On the global optimal solutions of continuous FRE programming problems

This paper presents some novel theoretical results as well as practical algorithms and computational procedures on continuous fuzzy relational equations programming problems. The fuzzy relational programming problem is a minimization (maximization) problem with a linear objective function subject to fuzzy relational equalities or inequalities defined with certain algebraic operations. In the literature, the commonly seen frameworks for such optimization models are to assume that the operation takes minimum t-norm, strict continuous t-norms (e.g., product t-norm), nilpotent continuous t-norms (e.g., Lukasiewicz t-norm) or Archimedean continuous t-norms. Based on new concepts called partial solution sets, the current paper considers this problem\textbf{ }in the most general case where the fuzzy relational equality constraints are defined by an arbitrary continuous t-norm and capture some special characteristics of its feasible domain and the optimal solutions. It is shown that the current generalized results are automatically reduced to (apparently) different ones that hold for special operators when continuous t-norm is replaced by strict, nilpotent or Archimedean continuous t-norm. Also, the relationship between the results derived here and those of previous publications regarding this subject is also discussed. Finally, the proposed algorithm is outlined and illustrated by a numerical example where the continuous fuzzy relational equations is defined by Mayor-Torrens operator that is not an Archimedean t-norm (and then, neither strict nor nilpotent).