A new approach for ranking of intuitionistic fuzzy numbers using a centroid concept

Ranking of intuitionistic fuzzy numbers is a difficult task. Many methods have been proposed for ranking of intuitionistic fuzzy numbers. In this paper we have ranked both trapezoidal intuitionistic fuzzy numbers and triangular intuitionistic fuzzy numbers using the centroid concept. Some of the properties of the ranking function have been studied. Also, comparative examples are given to show the effectiveness of the proposed method.

trapezoidal fuzzy number. Based on expected values, score functions and accuracy function of intuitionistic trapezoidal fuzzy numbers a new kind of ranking was proposed by Wang et al. in 2009. They also developed the Hamming distance of intuitionistic trapezoidal fuzzy numbers and Intuitionistic Trapezoidal Fuzzy Weighted Arithmetic Averaging (ITFWAA) operator, then proposed multi-criteria decision-making method with incomplete certain information based on intuitionistic trapezoidal fuzzy number.
In 2011, Salim Rezvani defined a new ranking technique for trapezoidal intuitionistic fuzzy numbers based on value-index and ambiguity -index of trapezoidal intuitionistic fuzzy numbers. Similar valueindex and ambiguityindex based ranking method for triangular intuitionistic fuzzy numbers was given by Li et al. [7] in 2010. Li [8] proposed a ranking order relation of TIFN using lexicographic technique. Nayagam et al. [12] introduced TIFNs of special type and described a method to rank them which seems to be unrealistic. Nehi [11] put forward a new ordering method for TIFNs in which two characteristic values for IFN.
Symmetric trapezoidal intuitionistic fuzzy numbers are ranked with a special ranking function which has been applied to solve a class of linear programming problems in which the data parameters are symmetric trapezoidal intuitionistic fuzzy number by Parvathi et al. [14] in 2012. Dubey et al. in 2011 developed a ranking technique for special form of triangular intuitionistic fuzzy numbers. This paper is organized as follows. In Section 2 some preliminary definitions and concepts regarding intuitionistic fuzzy numbers were presented. In Section 3, we define the magnitude of different forms of trapezoidal and triangular intuitionistic fuzzy numbers. Section 4 is devoted to the illustration of some numerical examples for the concepts defined in the Section 3 and also contains the comparative study of results obtained by the proposed method with other existing ranking methods. Section 5 concludes the paper by giving some advantages of the proposed method over other methods.

Preliminaries Definition 1. [1] An IFS A in X is given by
where the functions  1] define, respectively, the degree of membership and degree of non-membership of the element ∈ to the set A, which is a subset of X, and for every ∈ , 0 ≤ ( ) + ( ) ≤ 1.
In the above definition, if we let 2 = 3 ( ℎ Fig. 1. Trapezoidal intuitionistic fuzzy number. Definition 6. [7]. A TIFN ̃= ( , , ;̃,̃) is a special IF set on the real number set R, whose membership function and non-membership function are defined as follows: Where the values ̃ and ̃ represent the maximum degree of membership and the minimum degree of non-membership, respectively, such that they satisfy the conditions 0 ≤̃≤ 1,0 ≤̃≤ 1,0 ≤ +̃≤ 1.

New Approach for Ranking of Intuitionistic Fuzzy Numbers
In this section we define the concept of magnitude of an intuitionistic fuzzy number and discussed various methods for ranking the different forms of triangular intuitionistic fuzzy numbers and trapezoidal intuitionistic fuzzy numbers by means of magnitude.
Definition 11. Let = ( 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 ) be a Trapezoidal intuitionistic fuzzy number we define magnitude as follows: where ( ) is a non-negative and increasing weighting function on In this paper we assume ( ) = for our convenience, we get magnitude of A as Using this definition of ( ), we define the ranking procedure of any two trapezoidal intuitionistic fuzzy numbers as follows:  . In [11], Nehi used characteristic values of membership or non-membership functions to rank trapezoidal intuitionistic fuzzy numbers. The ranking procedure depends on the value of 'k'. As 'k' varies in the interval(0, ∞), the ranking also varies which leads to an unreasonable result. This can be seen from the following example. ( ) Mag(A) = 1 12 {a 1 + a 3 + 6a 2 + 2(a 1 ′ + a 3 ′ )} ( ) Table 1. Calculation of ( ).  [15]. Here the ranking of STIFNs are obtained by a special ranking function by considering all the parameters of both membership and non-membership functions of given STIFNs. The values obtained by this method are similar to the proposed method.  2) as given in [7]. In the paper [7] Li used ratio ranking method to rank triangular intuitionistic fuzzy numbers and applied it to multi attribute decision making problem In the case of ration ranking method, the raking differs on the choice of . For the above IFN's we have So this leads to a conflicted state which yields an unreasonable result.

Example 5.
Consider the same IFN's as in example 4 and ranking developed in [8]. Here the ranking is done by the extended additive weighted method using the value-index and ambiguity-index. For the above numbers, we have the following ranking results as tabulated below from [8].  [17]. If we use (̃) to rank these numbers we obtain ̃<̃. But when we rank in terms of (̃), we get ̃>̃. Hence the ranking of generalized triangular intuitionistic fuzzy numbers varies with the use of membership and non-membership value in ranking. This is an unreasonable result. Therefore the proposed method which uses both membership and non-membership values as a whole is suitable for ranking such GTIFN's.  [4]. In this paper, Dubey used the concept of value and ambiguity of a triangular intuitionistic fuzzy numbers to rank the above numbers. The ranking obtained in [4] is similar to the proposed method.  [13]. In this paper ranking is done by using the score function and the result obtained is similar to the proposed method.
The following table gives a comparative analysis of various ranking methods so far defined in intuitionistic fuzzy setting with the proposed method.

Conclusions
In many of the existing ranking methods, ranking is done either by considering the membership or nonmembership values only. But in the newly proposed method the ranking is done directly by taking both membership and non-membership values in a single formula. This ranking procedure is very simple and time consuming compared to the existing methods. We also illustrated the advantages of our method by means of suitable examples. The proposed ranking technique can be applied to multi-criteria decision making problems, linear programming problems, assignment problems, transportation, some management problems and industrial problems which are our future research works.  (23,25,1,1;23,25,3,3) B= (5,7,2,2;5,7,4,4) ℜ( ) = 49 ℜ( ) = 13, ≻ [15] ( ) = 24; ( ) = 6, ≻ 3