On the Solution of Fuzzy Equation Systems

Abstract

We consider a set of mixed linear and non-linear equations, which arise typically in operative controlling. The variables in the single equations are connected by arithmetic operations. In order to allow for imprecision, each variable is modelled as a fuzzy set with a given membership function. Given a vector of observed variables and an equation system with terms built up by fuzzy sets, a controlling decision is to be made, whether the data set is ‘consistent’ with the equation system or not.

We first give a typical example where the functional relationships link various microeconomic indicators together. Next we describe the nature of operative controlling and give some formal definitions. The fuzzy set theory is presented limited to what is needed for controlling. An algorithm is presented which has as input a real data set and a fuzzy equation system, and computes consistent values for all variables, if a simultaneous solution exists. Otherwise, it is flagged, that the data is inconsistent with the fuzzy model. We close with various scenarios illustrating the (reasonable) behaviour of the algorithm for solving such equations in the context of operative controlling.