CALCULATION OF LINEAR FRACTIONAL FUZZY TRANSPORTATION PROBLEM USING SIMPLEX METHOD

In this research article, we implement a methodology for solving fuzzy transportation problems involving linear fractional fuzzy numbers. The main aim of this paper is to find optimum values of the fuzzy transportation problems by simplex method with the help of a triangular fuzzy number (TFN) as the costs of objective function. The outcome of this method is explained with a numerical example.


INTRODUCTION
Operation research is widely used for developing a new method in case of real-world problem. "This transportation problem was first introduced by Hitchcock in 1941Hitchcock (1941. The objective of this concept is to find an optimal solution of the transportation problem in case of economics and Mathematics". This problem was first introduced by the French mathematician Gaspard Monge. For solvtation of transportation problem, all the parameters like supply, demands and unit transportation cost are represented in a crisp value. These values can also be represented in case of fuzzy numbers. If the cost associated in transportation problem are fuzzy, then the optimal solution of the problem will be fuzzy, then this type of problem is termed as fuzzy transportation problem Charnes and Cooper (1973).
Fractional transportation problem be a unique class of mathematical technique, in which all the constraint variables are form of linear and in the objective function is upgraded into two linear functions Chandra (1968). In 1960, Hungarian mathematician B. Metros first exposed the linear fractional problem. In actual existence problem, this idea is commonly utilized in stock income real cost-preferred cost, and income cost Pandian and Jayalakshmi (2013), Bitran and Novaes (1972) and additionally it enables in finance & business etc. In this paper, we proposed a technique for finding an optimal solution of transportation problem using simplex method. The linear fractional problem i.e., Where β 0. In section 2, we discuss some basic definitions on fuzzy set theory with properties and theorems. In section 4, we anticipated an algorithm about the LFFPP. In next section, we discuss a simple numerical example on LFFPP by simplex method. In last section, we provided the concluding remarks

PRELIMINARIES
This section contains basic preliminary definitions and its some properties, which will be used in a sequel.

DEFINITIONS
1) Fuzzy set: Let X be a general set and x be a member of X, then fuzzy set on X is denoted by a membership value µ_ () (x), which identifies the function maps from every element to the interval Bitran and Novaes (1972) and it can be defined as 2) α − cut: Let be a fuzzy set on X and α be a real number between Bitran and Novaes (1972), then cut of A is determined by 3) Strong α − cut: Let A be a fuzzy set on X and α be a real number between [0, 1], then strong α − cut of fuzzy set A is defined as

4) Support of a fuzzy set:
Consider X be a crisp set and A be a fuzzy set associate on X. Then support of A is described by Where l and n are representing the lower and the upper boundaries respectively, with membership degree is 0 and m is the centre with membership degree is 1 9) Arithmetic Operations: Let A = (l_1, m_1, n1) and = (l_2, m_2, n_2) be two triangular fuzzy numbers, then  Theorem 1. Narayanamoorthy and Kalyani (2015) Suppose X t is the set of efficient points for P t , t = 1, 2, then X t is a subset of the set of all efficient points X of P Theorem 2. Hitchcock (1941) Suppose X = {x i ∈ P i } be an optimal solution of P 1 and = {x i ∈P} be an another optimal solution of P 2 ,then x = X X is an optimal solution of P, Where all elements of x 1 , x 2 . are in P 1 and P 2 .

1) Linear Fractional Fuzzy Programming Problem (LFFPP)
In this section, we expand the linear fractional problem into linear fractional fuzzy programming problem. The LFFPP can be considered as:

Equation 1
Where p(x) and q(x) are continuous linear functions and q(x) > 0, x now the set S is designated by Where S is a bounded polyhedron and A is a fuzzy matrix of order. Now n×n the above equation may be represented as

Equation 2
Then by using above theorem-2, we have the set of efficient points for p t , t = 1, 2 is a subset of p.

ALGORITHM
This section deals with a methodology for solving a linear fractional fuzzy transpotation problem (LFFTP). The LFFTP can be considered as Where the objective function is of fractional type. This problem can be decomposed into two linear fuzzy problem such as Then by simplex method, we have to find the solution of two linear fuzzy problems. Thus, the solutions min z N and min z D are determined distinctly. Then by the above theorem, we have to obtain the optimal solution of P

Equation 6
Where objective function is expressed in case of triangular fuzzy number and crisp values for case of supply and demands.

Equation 9
Where s 1 and s 2 are slack variables. Now to solve the LFTP "p 2 " and simplex method. The primary datas are given below. From the above table, we get one − ≥ 0 value is negative. So, the optimal feasible solution is not satisfied. Thus, the non-basic variable s1 will leave the basis and the basic variable x1 enter the basis. Now by using the procedure of simplex method, we have the following table.

CONCLUSION
In this research article, we anticipated a methodology for finding a solution of linear fractional fuzzy transportation problem, where objective functions are expressed by tri-angular fuzzy number. This proposed method i.e., simplex method is one of the exclusive techniques for calculating the optimal solution of any transportation problem. This additionally be prolonged into fractional quadratic problems.