Alpha-cut based fuzzy cognitive maps with applications in decision-making

Fuzzy cognitive maps (FCMs) are widely used fuzzy modeling tools for handling causal interdependencies in complex systems. In FCMs, system variables (concepts) and degree of interrelationships are quantified by means of fuzzy sets. However, these fuzzy sets are defuzzified and fuzzy singletons are processed in the inference algorithm. Defuzzification of fuzzy numbers before the FCM inference implies information loss which is an undesired situation. Moreover, the resultant concept values are fuzzy singletons that these crisp numbers do not provide any information about the range of possible outcomes. There is a research gap in the literature regarding fuzzy number representation in FCMs that the both of the inputs and outputs of FCMs being fuzzy sets. This study proposes alpha-cut based computational procedures for simulating FCMs in which concepts and degree of relationships are represented via fuzzy numbers. The proposed model is tested on well-known problems adopted from the literature by using type-1 fuzzy numbers and the results are compared with the extension principle-based approach. Moreover, the proposed model is extended to interval type-2 (IT2) fuzzy sets and computational details regarding IT2 FCMs are given. Because relative importance of criteria is usually modeled via fuzzy sets in multiple-attribute decision making problems, a new FCM-based objective weighting method is proposed in order to demonstrate the usefulness of fuzzy number representation in FCMs. The proposed model is implemented in the real-life third-party logistics service provider selection problem