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.

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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.

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