Solving a tri-criteria best path problem using the fuzzy decision making

Abstract

In this paper we restrict our attention to formulating and solving a tri-criterion nonlinear combinatorial problem on a network with crisp arc costs, fuzzy arc times, and a fuzzy goal on total traversing time. Here, arc times are discrete fuzzy sets and goal is a trapezoidal number. We called it tri-criteria best path problem. The main contribution of this model is an actual interpretation of given fuzzy time goal, as quality of delivered commodities. Since presented problem has a fuzzy structure, one of fuzzy decision making criteria, i.e. Bellman and Zadeh's max-min criterion, can be used to treat it as a single-criterion nonlinear programming problem. Then, special structure of model enables us to reformulate this problem as a mixed integer linear programming problem. However, this linearization process increases size of problem. To reduce size of it, a relaxation strategy, instead of exploitation of well-known methods, can be employed in solving such problems. Correspondingly, a new named the best shipping pattern algorithm is proposed to get best path. An illustrative example is solved, to explain presented details.