Deriving priority vector from pairwise comparisons matrix with fuzzy elements by solving optimization problem

Pairwise comparisons matrix with fuzzy elements (FPCM) are appropriate for the decision makers who are uncertain about the relative importance of elements. We can primarily find them in Fuzzy Analytic Hierarchy Process, PROMETHEE, TOPSIS methods, and many exact and heuristic algorithms. They are also useful in aggregating pairwise comparisons, particularly in consensus group decision making problems and they form the basis for many decision-making models as intuitionistic fuzzy relations, pythagorean, q-rung orthopair fuzzy preference relations, hesitant or interval fuzzy sets, and also stochastic judgments. Here, the decision model is formulated by investigating pairwise comparisons matrices (PCMs) with elements from abelian linearly ordered group (alo-group), which enables unifying multiplicative, additive and fuzzy PCMs. Then we define a novel concept of consistency, coherence and intensity of FPCMs, and propose a number of optimization methods for finding a consistent vector, coherent vector and intensity vector of a FPCM satisfying the desirable properties. Finally, two illustrating examples are discussed.