Some concepts of the fuzzy multicommodity flow problem and their application in fuzzy network design

We treat with the Minimal Cost Multicommodity Flow Problem (MCMFP) in the setting of fuzzy sets, by forming a coherent algorithmic framework referred to as a fuzzy MCMFP. Given the character of granular information captured by fuzzy sets, the objective is to find multiple flows satisfying the demands of commodities, by using available supplies consuming the least possible cost. With this regard, the supply and demand of nodes may be presented linguistically; the travel cost and capacity of links can be defined under uncertainty as well. To solve this problem, two efficient algorithms are motivated. In the first, we utilize fuzzy shortest paths and K -shortest paths to generate preferred and absorbing paths, and then we find the flow on them by solving a classic MCMFP. The second algorithm exhibits with fuzzy supply-demand, and employs a total order on trapezoidal fuzzy numbers to reduce the fuzzy MCMFP into four classic MCMFPs. Some examples are solved to demonstrate the performance of the presented methods. Among the various applications of this scheme in providing a suitable interface between the model and physical world, we focus on network design under fuzziness. The granular nature of the description of the future travel demand contributes to the generality of the planning model, and determines a certain perspective from which we will looking at the network.