# PSK Method for Solving Mixed and Type-4 Intuitionistic Fuzzy Solid Transportation Problems

P. Senthil Kumar (Department of Humanities and Sciences, Navodaya Institute of Technology, Raichur, India)
DOI: 10.4018/IJORIS.2019040102

## Abstract

In this article, the author categorises the solid transportation problem (STP) under uncertain environments. He formulates the mixed and fully intuitionistic fuzzy solid transportation problems (FIFSTPs) and utilizes the triangular intuitionistic fuzzy number (TIFN) to deal with uncertainty and hesitation. The PSK (P. Senthil Kumar) method for finding an intuitionistic fuzzy optimal solution for fully intuitionistic fuzzy transportation problem (FIFTP) is extended to solve the mixed and type-4 IFSTP and the optimal objective value of mixed and type-4 IFSTP is obtained in terms of triangular intuitionistic fuzzy number (TIFN). The main advantage of this method is that the optimal solution of mixed and type-4 IFSTP is obtained without using the basic feasible solution and the method of testing optimality. Moreover, the proposed method is computationally very simple and easy to understand. Finally, the procedure for the proposed method is illustrated with the help of numerical examples which is followed by graphical representation of the finding.
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## Introduction

The transportation problem is a special class of linear programming problem, widely used in the areas of inventory control, communication network, aggregate planning, employment scheduling, personal management and so on. In several real-life situations, there is a need for shipping the product from different origins (Factories) to different destinations (warehouses). The transportation problem deals with shipping commodities from different origins to various destinations. The objective of the transportation problem is to determine the optimum amount of a commodity to be transported from various supply points (origins) to different demand points (destinations) so that the total transportation cost is minimum or total transportation profit is maximum.

In the history of mathematics, Hitchcock (1941) originally developed a basic transportation problem. The transportation algorithm for solving transportation problems with equality constraints introduced by Dantzig (1963) is the simplex method specialized to the format of a table called transportation table. It involves two steps. First, we compute an initial basic feasible solution for the transportation problem and then, we test optimality and look at improving the basic feasible solution to the transportation problem. Swarup et al. (1997) presented tracts in operations research which deals the transportation problem when all the parameters are crisp number.

The solid transportation problem is a generalization of the classical transportation problem in which three-dimensional properties are taken into account in the objective and constraint set instead of source (origin) and destination. Shell (1955) stated an extension of well-known transportation problem is called a solid transportation problem in which bounds are given on three items, namely, supply, demand and conveyance. In many industrial problems, a homogeneous product is transported from an origin to a destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships and so on. Haley (1962) presented the solution procedure for solving solid transportation problem, which is an extension of the modified distribution method. Patel and Tripathy (1989) proposed a computationally superior method for a solid transportation problem with mixed constraints. Basu et al. (1994) developed an algorithm for finding the optimum solution of a solid fixed charge linear transportation problem.

For finding an optimal solution, the solid transportation problem requires non-negative values of the decision variables to start with a basic feasible solution. Jimenez and Verdegay (1996) investigated interval multiobjective solid transportation problem via genetic algorithms. Li et al. (1997a) designed a neural network approach for a multicriteria solid transportation problem. Roy and Mahapatra (2014) gave solving solid transportation problems with multi-choice cost and stochastic supply and demand. Efficient algorithms have been developed for solving transportation problems when the coefficient of the objective function, demand, supply and conveyance values are known precisely.

Many of the distribution problems are imprecise in nature in today’s world such as in corporate or in industry 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 but it does not consider the non-membership (non-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 in the set represent 0.

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