Logic and Probabilistic Operations on a Decision Matrix in a Fuzzy Multi-Criteria Decision-Making Problem

In the framework of (fuzzy) Multi-Criteria Decision-Making, we propose a method that allows decision-makers to subjectively approach problems by suitably modifying a decision matrix. We consider a decision problem related to a random quantity X with a set of values {x1,x2,…,xn} and a set of properties {C1,C2,…,Cm} of X. In this setting, the properties Cj are the criteria of the decision problem, the alternatives represent the events Ai=(X=xi), for i=1,…,n, and the criteria’s weights wj, for j=1,…,m, are seen as the probabilities for the event that “Cj is relevant with respect to the decision problem”. For each i=1,…,n and j=1,…,m, we interpret the scores aij as membership functions representing “how much alternative Ai satisfies criterion Cj”. By adopting an interpretation of membership functions as suitable conditional probabilities together with the theory of logical operations between conditional events, we allow logical operations between criteria and consistently apply this interpretation to the corresponding scores. In particular, when considering the complement, conjunction, and disjunction of criteria, the resulting scores are the (coherent) previsions of the respective compound conditionals within the framework of conditional random quantities. To illustrate our approach, we present an example concerning career choices.