Solving a tri-criteria best path problem using the fuzzy decision making
Article type: Research Article
Authors: Hassasi, Hamid* | Tohidi, Ghasem
Affiliations: Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
Correspondence: [*] Corresponding author. Hamid Hassasi, Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran. Tel.: +98 2188385773; Fax: +98 2188385777; E-mail: Hamidhassasi@yahoo.com.
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 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.
Keywords: The tri-criteria best path problem, discrete fuzzy set, fuzzy decision-making, Bellman-Zadeh’s criterion, relaxation strategy
DOI: 10.3233/IFS-151845
Journal: Journal of Intelligent & Fuzzy Systems, vol. 30, no. 2, pp. 1207-1217, 2016
A fuzzy best path problem with the trapezoidal time goal
What is it about?
As an example of network flows which arises quite naturally in the real world, the following problem can be presented: consider the problem of transporting some special commodities, as banana, mango, dates and cut flowers which are picked unripe and continuation of the ripening process is done until reaching consumers. As different fruits need different times for ripening, and their quality and taste are changed over time, in order to avoid storing the fruits with storage costs, the ripening process can be done during the shipping process. On the other hands, in the real-world problems the quality of a transportation path can be measured by some special factors, such as length of the path, reliability or safety of the path, traffic and capacity of the path, etc., but all of these factors are strongly related to the transit time of the path. Consider for transporting a commodity along arcs of the network we have received some different suggestions from several companies. We are going to pay a fixed value of shipping cost along each arc. If companies’ suggestions include various quality levels corresponding to different shipping times (such as multi-modal transportation), we can describe these situations as a discrete fuzzy sets which the membership degrees of different times are quality levels.
Why is it important?
Related to the mentioned situation, we present a tri-criteria best path problem on a network with crisp arc costs, fuzzy arc times, and a fuzzy goal on the total traversing time. Arc times are discrete fuzzy sets and the goal is a trapezoidal number. The main contribution of this model is an actual interpretation of the given fuzzy time goal, as the quality of delivered commodities. Also, corresponding to each arc of the network we can define a discrete fuzzy time which its membership degrees show the various level of quality.