Causal effect analysis for fuzzy cognitive maps designed with non-singleton fuzzy numbers

In this study, a new static analysis approach is proposed for enhanced Fuzzy Cognitive Maps (FCMs), which have non-singleton fuzzy numbers in casual relation strength representation. Cognitive Maps (CMs) are proposed as a type of directed graph that offers a means to model interrelationships or causalities among concepts, and have a clear way to visually represent them. They graphically describe a system in terms of concepts, and causal beliefs, and are powerful graphical tools to represent knowledge of the experts. Fuzzy cognitive maps, which are weighted cognitive maps, are proposed also as graphical modelling technique that follows a reasoning approach similar to processes of human reasoning and human decision-making. In FCMs, the casual relations and its strengths are assigned in a unit interval with a sign. The assigned casual strengths in conventional FCMs are singleton fuzzy (crisp) numbers, and only allow to interpret the effects linguistically but do not represent the uncertainty or ambiguity in causality. In this paper, a new analysis is presented for finding the indirect effects and total effects between the concepts of enhanced FCMs that are represented with non-singleton fuzzy numbers, especially for triangular or trapezoidal fuzzy numbers. Firstly, the mathematical approach about fuzzy numbers and the proposed analysis is presented, then secondly an experimental study on modelling ERP maintenance risks via FCM is presented. The results of the proposed causal effect analysis are discussed for this model and the outcomes are compared with a conventional FCM model where the casual strengths are singleton fuzzy numbers. The results of the experiment show the benefit of using triangular fuzzy numbers when a group of experts are involved in modelling. The uncertainty and varieties between the experts' knowledge are easily captured and the casual effect between the concepts are successfully shown with the presented static analysis.