A NEW AVERAGE METHOD FOR SOLVING INTUITIONISTIC FUZZY TRANSPORTATION PROBLEM

In this paper a new average method is proposed for finding an optimal solution for an intuitionistic fuzzy transportation problem. The main feature of this method is that it requires very simple arithmetical calculations and avoids large number of iterations. An accuracy function to defuzzify Triangular Intuitionistic Fuzzy Number is also used. Based on this new approach, the optimal solution of Intuitionistic Fuzzy transportation problem is obtained. Finally, an illustrative example is given to verify the developed approach. AMS Subject Classification: 03E72, 03F55, 90B06


Introduction
Transportation problem is a particular class of linear programming.It deals with the transportation of a single product manufactured at different plants (supply origins) to a number of different warehouses (demand destinations).
The objective of the transportation model is to determine the amount to be shipped from each source to each destination so as to maintain the supply and demand requirements at the lowest transportation cost.The objective is to satisfy the demand at destination from the supply constraint at the minimum transportation cost possible.
Atanassov [1] introduced the concept of Intuitionistic Fuzzy Sets (IFS), which is a Generalization of the concept of fuzzy set [1].Basic arithmetic operations of TIFNs is defined by Deng-Feng Li in [2] using membership and non-membership values.Basic arithmetic operations of TIFNs such as addition, subtraction and multiplication are defined by S.Mahapatra and T.K.Roy in [4], by considering the six tuple number itself.Most of the authors used the membership and non − membership values of TIFNs for ranking.In this paper we have defined ranking of TIFNs using integral value by considering six tuple TIFNs introduced in [6].
The aim of this paper is to find an optimal solution for an intuitionistic fuzzy transportation problem.Here we have considered TIFN and integral value for ranking.This ranking is applied to solve Intuitionistic Fuzzy Transportation Problem.An accuracy function is developed to defuzzify TIFN.

Preliminaries
Definition 1. (Fuzzy set) A fuzzy set Ã is defined by Ã= {(x, µ Ã(x)): x∈ A,µ Ã(x) ∈ [0, 1]}.In the pair (x, µ Ã(x)), the first element x belong to the classical set A, the second element µ Ã(x), belong to the interval [0, 1], called Membership function Definition 2. (Triangular fuzzy number) It is a fuzzy number represented with three points as follows: Ã = (a 1 , a 2 , a 3 ).This representation in interpreted as membership functions ) is a Triangular Fuzzy Number, then the defuzzified value or the ordinary (crisp) number of Ã, A is given below: Definition 4. (The total integral value of triangular fuzzy number) The total integral value of Ã is An Intutionistic fuzzy set(IFS) ÃI in X is given by a set of ordered triples: For each x the number µ ÃI (X) and ν ÃI (X) represent the degree of membership and degree of non membership of the element x ∈ X to A ⊂ X, respectively.

Definition 7. (Intutionistic fuzzy number) An intuitionistic fuzzy number ÃI is:
(i) An intuitionistic fuzzy subset of the real line; (ii) Normal.i.e., there is any Definition 8.An Triangular intutionistic fuzzy number(TIFN) µ ÃI is an intuitionistic fuzzy set in R with the following membership function µ ÃI (X) and non membership function ν ÃI (X) : ,an accuracy function of ÃI ,to defuzzify the given number.

Operations on Triangular Intuitionistic Fuzzy Number
) two triangular intuitionistic fuzzy number then the arithmetic operations on ãI and bI as follows:

Ranking of Triangular Intuitionistic Fuzzy Number
A Triangular Intuitionistic Fuzzy Number ÃI = {(a, b, c); (e, b, f )} is completely defined by its membership and non-membership function as follows The inverse functions L −1 and R −1 can be analytically express as given below:

Total Integral Value of Triangular Intuitionistic Fuzzy Numbers
(i) Left integral value of the membership function of ÃI is given by (ii) Left integral value of the non -membership function of ÃI is given by (iii) Right integral value of the membership function of ÃI is given by (iv)Right integral value of the non -membership function of ÃI is given by The total integral value of the membership function of ÃI is given by The total integral value of the non -membership function of ÃI is given by

Solution Algorithm for New Average Method
Step 1: Construct the transportation model (Table ) from the given transportation problem.
Step 2: Subtract each row entries of the transportation table from the respective row minimum and then subtract each column entries of the transportation table from the respective column minimum, so that each row and column will have least one zero.
Step 3: Now there will be at least one zero in each row and column in the reduced cost matrix.Select the first zero (row wise) occurring in the cost matrix.
Count the total number of zeros excluding the selected one in the corresponding row and column.Then find the average of corresponding demand and supply for this zero.Repeat the procedure for all zeros in the matrix.
Step 4: Now choose a zero for which the minimum demand supply average and allocate the maximum possible to that cell.
Step 5: After performing step 4 delete the row or column (where supply or demand becomes zero) for further calculation.
Step 6: Check whether the resultant matrix possesses at least one zero in each column and in each row.If not repeat step2, otherwise go to step 7.
Step 7: Repeat step 3 to step 6 until and unless all the demands are satisfied and all the supplies are exhausted.
Step 8: For the allocated values calculate the optimal cost.

Transportation Problem in Fuzzy Environment
Step 1: Mathematical Formulation: The fuzzy optimal solution for the given fuzzy transportation problem is (−11, 412, 797), and the crisp value is 405.67.
xI j ≥ 0 for all i and j.
The above IFTP can be stated in the below tabular form.Step 8: The optimal solution for the given intuitionistic fuzzy transportation problem is (−11, 412, 797; −233, 412, 1051), and the crisp value is 406.5.

Conclusions
The new average method provides an optimal solution in less number of iterations, directly for a given intuitionistic fuzzy transportation problem.Also this method required less number of time and is very easy to understand and apply.So it will be very helpful for decision makers who are dealing with logistic and supply chain problem.

6 .
Intuitionistic Fuzzy Transportation Problem (IFTP) Consider a transportation with m Intuitionistic Fuzzy (IF) origins and n IF destination.Let C ij (i = 1, 2, • • • , m, j = 1, 2, • • • , n) be the cost of transporting one unit of the product form ith origin to jth destination .Let ãI i (i = 1, 2, • • • , m) be the quantity of commodity available at IF origin i .Let bI j When applying the above algorithm and ranking procedure, we get the optimum solution for intuitionistic fuzzy transportation problem.