There are several algorithms, in literature, for obtaining the fuzzy optimal solution of fuzzy transportation problems (FTPs). To the best of the author's knowledge, in the history of mathematics, no one has been able to solve transportation problem (TP) under four different uncertain environment using single method in the past years. So, in this chapter, the author tried to categories the TP under four different environments and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers (TFNs), and trapezoidal fuzzy numbers (TrFNs) to solve the TP. A new method, namely, PSK (P. Senthil Kumar) method for finding a fuzzy optimal solution to fuzzy transportation problem (FTP) is proposed. Practical usefulness of the PSK method over other existing methods is demonstrated with four different numerical examples. To illustrate the PSK method different types of FTP is solved by using the PSK method and the obtained results are discussed.
Top1. Introduction
Resource allocation is used to assign the available resources in an economic way. When the resources to be allocated are scarce, a well-planned action is necessary for a decision-maker (DM) to attain the optimal utility. If the supplying sources and the receiving agents are limited, the best pattern of allocation to get the maximum return or the best plan with the least cost, whichever may be applicable to the problem, is to be found out. This class of problems is termed as ‘allocation problems’ and is divided into ‘transportation problems (TPs)’ and ‘assignment problems (APs)’. The TPs and APs both are also called optimization problems.
TPs play an important role in logistics and supply chain management for reducing cost and improving service. In today’s highly competitive market, the pressure on organizations to find better ways to create and deliver products and services to customers becomes stronger. How and when to send the products to the customers in the quantities which they want in a cost-effective manner becomes more challenging. Transportation models provide a powerful framework to meet this challenge. They ensure the efficient movement and timely availability of raw materials and finished goods.
The TP is a special class of linear programming problem (LPP) which deals with the distribution of single homogeneous product from various origins (sources) to various destinations (sinks). The objective of the TP is to determine the optimal amount of a commodity to be transported from various supply points to various demand points so that the total transportation cost is minimum for a minimization problem or total transportation profit is maximum for a maximization problem.
The unit costs, that is, the cost of transporting one unit from a particular supply point to a particular demand point, the amounts available at the supply points and the amounts required at the demand points are the parameters of the TP. Efficient algorithms have been developed for solving TPs when the cost coefficients, the demand and supply quantities are known precisely.
In the history of mathematics, Hitchcock (1941) originally developed the basic TP. Charnes and Cooper (1954) developed the stepping stone method which provides an alternative way of determining the simplex method information. Appa (1973) discussed several variations of the TP. Arsham et al. (1989) proposed a simplex type algorithm for general TPs. An Introduction to Operations Research Taha (2008) deals the TP. Aljanabi and Jasim (2015) presented an approach for solving TP using modified Kruskal’s algorithm. Ahmed et al. (2016) developed a new approach to solve TPs. Akpan and Iwok (2017) presented a minimum spanning tree approach of solving a TP.
In today’s real world problems such as in corporate or in industry many of the distribution problems are imprecise in nature due to variations in the parameters. To deal quantitatively with imprecise information in making decision, Zadeh (1965) introduced the fuzzy set theory and has applied it successfully in various fields. The use of fuzzy set theory becomes very rapid in the field of optimization after the pioneering work done by Bellman and Zadeh (1970). The fuzzy set deals with the degree of membership (belongingness) of an element in the set. In a fuzzy set the membership value (level of acceptance or level of satisfaction) lies between 0 and 1 where as in crisp set the element belongs to the set represent 1 and the element not belongs to the set represent 0.