Solving Neutrosophic Multi-Dimensional Fixed Charge Transportation Problem

Abstract

In the decision making problems, transportation problem (TP) is one of the most significant applications that aim to optimize the objective function of products shipped from different sources to different destinations. When an additional charge such as toll tax, parking fee, permit fee and so on are included in the transportation cost then the problem becomes a fixed charge TP. If the shipped products are damageable, they are affected by the type of vehicle and the distance of route. So it is quite evident that the vehicle type and the distance of route play a significant role in TP. In reality, along with sources and destinations, the decision maker (DM) desires to consider the parameters such as the mode of transport and routes of transportation that are used for shipment of the products from sources to destinations in order to minimize the loss of damageable products and to maximize the profit. As a result, mode of transport and routes of transportation are added to TP, then the TP becomes a four-dimensional TP. Sometimes, the data/parameters of the problems may not be always precise due to incomplete information, inadequate data or shortage of evidence. To deal with these obstacles, the parameters of the problem can be represented as the single-valued trapezoidal neutrosophic numbers (SVTrNNs). The neutrosophic data facilitate a reasonable and practicable way for DMs to tackle decisionmaking problems by managing indeterminacy and providing an effective framework for analysis and synthesis of complex decision scenarios. In view of this, in this paper we have considered the single objective four-dimensional fixed charge transportation problem (4DFCTP) with parameters, supply, demand, conveyance, transportation cost, fixed charge as single-valued trapezoidal neutrosophic numbers and the distance of routes as real numbers. First, the score function is utilized to transform the neutrosophic parameter into its deterministic parameter in order to avoid the negative values for the decision variable. Then the deterministic problem comprises deterministic parameters such as supply, demand, conveyance, transportation cost with fixed charges and distance of routes. Secondly, a novel approach namely, min zeromin cost approach is introduced for finding the optimal solution to the equivalent deterministic problem in polynomial time. The main objective of this paper is to optimize the breakable products and routing plan of vehicles in a way to minimize the total transportation cost with fixed cost of the business organizations using the proposed min zero-min cost approach. To demonstrate the problem’s validity and relevance, two numerical examples are solved using our proposed approach. To highlight the proposed approach, comparison of the solution with the LINGO software is performed. The obtained optimal solution from the proposed approach is the same as the LINGO software. At last, conclusions as well as future work related to the study are presented.