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

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 the total traversing time. Here, arc times are discrete fuzzy sets and the goal is a trapezoidal number. We called it the tri-criteria best path problem. The main contribution of this model is an actual interpretation of the given fuzzy time goal, as the quality of delivered commodities. Since the presented problem has a fuzzy structure, one of the 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, the special structure of the model enables us to reformulate this problem as a mixed integer linear programming problem. However, this linearization process increases the size of the problem. To reduce the size of it, a relaxation strategy, instead of the exploitation of well-known methods, can be employed in solving such problems. Correspondingly, a new algorithm named "the best shipping pattern algorithm" is proposed to get the best path. An illustrative example is solved, to explain the presented details.