Ranking function-based solutions of fully fuzzified minimal cost flow problem

The aim of minimal cost flow problem (MCFP) is to find the least transportation cost of a single commodity through a capacitated network. This paper presents a model to deal with one particular group of such problems in which the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. For easier reference, hereafter, we refer to this group of problems as fully fuzzified MCFP. To represent our model, Hukuhara’s difference and approximated multiplication are used. Thereafter, we sort fuzzy numbers by an order using a ranking function and show that it is a total order, i.e., a reflexive, anti-symmetric, transitive and complete binary relation. Utilizing the proposed ranking function, we transform the fully fuzzified MCFP into three crisp problems solvable in polynomial time. From this standpoint, combinatorial algorithms are provided to solve the above-mentioned problem and find the fuzzy optimal flow. Furthermore, the proposed order is related to the importance weights of the center, the left spread and the right spread of each fuzzy number. Thus, this method is capable of handling the decision maker’s risk taking. By comparing some previous ranking function-based works with our method, the efficiency of the latter is revealed. Finally, an application of our proposed method to petroleum industry is presented.