Inverse inference based on interpretable constrained solutions of fuzzy relational equations with extended max–min composition

Abstract

In this paper, we propose a method for solving the System of Fuzzy Relation Equations (SFRE) with extended max-min composition for inverse inference problems. The properties of interval and constrained solutions with granular and relational structure of the solution set are investigated. The extended max-min SFRE can be represented in the form of the max-min subsystems aggregated using the min operator or dual min-max subsystems aggregated using the max operator. When decomposing the SFRE, the set of solutions can be decomposed into the lower and upper subsets bounded by the same aggregating solutions. Each lower (upper) subset is defined by the unique greatest (least) or aggregating solution and the set of minimal (maximal) solutions. Following (Bartl et al. 2012), to avoid excessive granularity and ensure interpretability of the interval solutions when restoring causes through observed effects, the constraints in the form of linguistic modifiers are imposed on the measures of causes significances. The interval solutions are modeled by the complete crisp solutions, that is, the maximum solutions for the vectors of binary weights of the linguistic modifiers. The search for approximate solutions of the SFRE amounts to solving the optimization problem using the genetic algorithm. Due to the properties of the solution set, the genetic search for the lower and upper subsets is parallelized for each aggregating solution. The developed method makes it possible to simplify the search for the solution set based on the constraints on accuracy (interpretability) of the applied problem