Fuzzy relation equations: new trends and applications

EDITORIAL Fuzzy relation equations were initially studied in 1976 by E. Sanchez for modeling of medical diagnosis realized in the context of complete Brouwerian lattices. Successively they were widely studied from a mathematical and application points of view by many authors: e.g., refer to comprehensive bibliographies in the monography of the 1989 [3] and to those more recent contained in [1,2,4,5]. This special issue reflects upon the state of art of the theory of fuzzy relation equations and its applications. The diversity of the area is captured through a series of the papers forming this special issue: – in the paper of S. Freson, B. De Baets and H. De Meyer a generalized linear optimization problem with max-min fuzzy relational (in)equality constraints is considered by using n bipolar max-min constraints, i.e. constraints in which not only independent variables but also their negations occur. A feasible domain of solutions is algebraically characterized; – in the paper of M. Stepnicka and B. De Baets, the authors deal with the interpolative features of a fuzzy inference system. If the fuzzy rule base expresses a monotone relationship, it is desirable that this property is preserved from the defuzzified resulting function. This goal is often reached through using suitable modifiers to the antecedent and consequent fuzzy sets of the system. The authors propose some solutions (arising from two different points of view) to the case of single- input single-output fuzzy rules; – in the paper of I. Perfilieva, two types of systems of n-fuzzy relation equations (those with sup-⁄ and inf-? compositions, to be specific) are studied in detail. The author propose new solvability criteria characterized from a relationship between a skeleton square matrix of sizes n n and a vector of the right-hand side; – in the paper of K. Peeva, the inverse problem resolution of fuzzy linear systems of equations in BL-algebras (Godel algebra in case of maxmin and minmax compositions, and Goguen algebra in case of maxproduct composition) is studied. Two software packages, already known in literature, are presented with applications to the optimization of fuzzy linear systems of equation constraint, fuzzy machines and covering problem; – in the paper of M. Gavalec and K. Zimmermann, the problem of maximizing an objective function f(x) defined as maximum of strictly increasing real functions of single variable xj, x = (x1, . . ., xn) satisfying max-min relation inequalities constraints, is presented. A similar dual problem is also considered and globally solved; – in the paper of C.-W. Chang and B.-S. Shieh, a minimization problem of an objective function subject to a max–min fuzzy relation equation constraint is solved, significantly improving results present in some previous papers: indeed the authors show an improved upper bound on the optimal objective value, improved rules for simplifying the problem and a rule for reducing the tree of the solutions; – in the paper of X.-P. Wang and S. Zhao, the whole set of solutions of a finite max–min fuzzy relation equation over a bounded Brouwerian lattice is totally characterized: the authors give a vector representation of any solution; – in the paper of A.A. Molai, an algorithm is proposed in order to find the so-called LU-factorization (lower-upper) of a fuzzy square matrix with respect to the max–product composition. An algorithm is also proposed to find the solution set of a square system of fuzzy relation equations using the LU-factorization. Both algorithms present polynomial time complexity. We sincerely thank all the referees for their thorough and constructive reviews. In particular, we express our thanks to Prof. W. Pedrycz, the Editor-in-Chief, for giving us the opportunity of organizing this special issue. References [1] R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, 2002. [2] B. De Baets, Analytical solution methods for fuzzy relational equations, in: D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, vol. 1, Kluwer Academic Publishers, 2000, pp. 291–340. [3] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic Press, Boston, 1989. [4] K. Peeva, Y. Kyosev, Fuzzy Relational Calculus-Theory, Applications and Software (with CD-ROM), Advances in Fuzzy Systems – Applications and Theory, vol. 22, World Scientific Publishing Company, 2004. [5] P. Li, S.-C. Fang, A survey on fuzzy relational equations, part I: classification and solvability, Fuzzy Optimization and Decision Making 8 (2) (2009) 179– 229.