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

In the framework of (fuzzy) Multi-Criteria Decision-Making, we propose a method that1 allows the decision maker to subjectively approach the problem by suitably modifying the decision matrix. We consider a decision problem related to a random quantity X with 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, 2, . . . , m, are seen as the probabilities of the events “Cj is relevant with respect to the decision problem”. For each i= 1, . . . , n and j= 1, 2, . . . , m, we interpret the scores aij as membership functions representing “how much alternative Ai satisfies criterion Cj”. By adopting the interpretation of membership functions as suitable conditional probabilities, together with the theory of logical operations among conditional events, we allow logical operations among 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.