An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality

Modified Vogel’s approximation method for transportation problem under uncertain environment

The fuzzy transportation problem is a very popular, well-known optimization problem in the area of fuzzy set and system. In most of the cases, researchers use type 1 fuzzy set as the cost of the transportation problem. Type 1 fuzzy number is unable to handle the uncertainty due to the description of human perception. Interval type 2 fuzzy set is an extended version of type 1 fuzzy set which can handle this ambiguity. In this paper, the interval type 2 fuzzy set is used in a fuzzy transportation problem to represent the transportation cost, demand, and supply. We define this transportation problem as interval type 2 fuzzy transportation problems. The utility of this type of fuzzy set as costs in transportation problem and its application in different real-world scenarios are described in this paper. Here, we have modified the classical Vogel’s approximation method for solved this fuzzy transportation problem. To the best of our information, there exists no algorithm based on Vogel’s approximation method in the literature for fuzzy transportation problem with interval type 2 fuzzy set as transportation cost, demand, and supply. We have used two Numerical examples to describe the efficiency of the proposed algorithm.

A Multiobjective Multi-Product Solid Transportation Model with Rough Fuzzy Coefficients

Transportation management is one of the key success factors to keep an organization competitive, sustain its growth pace, and raise profits not only at a local but also a global scale. Therefore, planning and designing a transport system are prerequisite and vital topics for achieving these goals. In this paper, a multiobjective multi-product solid transportation problem (MOMPSTP) under uncertainty is formulated and solved by two different methods of multiobjective optimization problems (MOPs). The system parameters namely unit transportation cost, availability of products at source points, demands of products at destinations and the capacity of transportation mode all are taken as rough fuzzy variables (RFVs). A chance constraint programming model for MOP with RFVs is developed in order to obtain satisfactory solutions when decision makers (DMs) aim to optimize multiple objectives (cost, time, profit, etc.) simultaneously. For given credibility (Cr) and trust (Tr) levels of RFVs, Cr-Tr constraint programming technique is used to reduce the uncertain transportation problem into equivalent deterministic form. Two classical solution techniques-weighted sum method (WSM) and ideal point method (IPM) are utilized to solve the problem. Finally, a numerical example is provided to illustrate the usefulness of our proposed model and then a sensitivity analysis is performed to verify different solutions due to different level of satisfaction.