A New Generalized Intuitionistic Fuzzy Number

In this paper, a Generalized Intuitionistic Fuzzy Number ( ) have been introduced and their desirable properties are also studied. Also we defined set of -cut, value of and gave their main properties.


Introduction
Zadeh (1965 [12]) introduced fuzzy set theory.Later on Atanassov (1986 [1]) introduced concept intuitionistic fuzzy set as a generalization of fuzzy set.The concept of fuzzy numbers and its arithmetic operations were first introduced and investigated by Chang and Zadeh (1972 [4]) and others.The notion of a fuzzy number was introduced by Dubois and Prade (1978 [5]) as a fuzzy subset of the real line.Burillo et al. (1994 [3]) proposed the definition of intuitionistic fuzzy number.Mahapatra and Roy (2009 [6]) presented triangular intuitionistic fuzzy number and used it for reliability evaluation.Also Mahapatra and Mahapatra (2010 [7]) defined trapezoidal intuitionistic fuzzy number (TrIFN) and arithmetic operations of TrIFN based on -cut method.Atanassov and Gargov (1989 [2]) proposed the notion of the intervalvalued intuitionistic fuzzy set (IVIFS).Based on IVIFS, Xu (2007 [10]) defined the notion of intervalvalued intuitionistic fuzzy number (IVIFN) and introduced some operations on IVIFNs.Wang and Zhang (2009 [9]) defined the trapezoidal intuitionistic fuzzy number (TrIFN) and their operational laws.Parvathi (2012 [11]) introduced Symmetric Trapezoidal Intuitionistic Fuzzy Numbers (STrIFNs) and discussed their desirable properties and arithmetic operations based on -cut.The aim of this paper is to introduction the some generalized from of the intuitionistic fuzzy number.Also, the properties of have been discussed.Figure1 shows membership and non-membership functions of with respectively.Figure 2 shows membership and non-membership functions of with respectively.Figure 3    ] is defined as the crisp set of elements which belong to at least to the degree and which does not belong to at most to the degree .A -cut set of a A is a crisp subset of , which defined is as According to the definition of it can be easily shown that

[ ] [ ]
Similarity a -cut set of a A is a crisp subset of , which defined is as According to the definition of it can be easily shown that .
Remark 3.1.In special case we have

Therefore the -cut of a is given by
) ]. Proof Definition3.1.Let and be two s; then, The proof is complete.The proof of (iii) is similar to (i) The proof of (iv) is similar to (ii) (v) (vi) According to the part (v) we have The proof of (vii) is similar to (v) The proof of (viii) is similar to (vi)

Conclusion
We have introduced Baloui's generalized intuitionistic fuzzy numbers.The cut sets on and the concept of value of and some their features are presented.It is worth mentioning that these results generalize the results obtained for intuitionistic fuzzy numbers.A list of open problems is as follows: definition of possibility degree, metric space, similarity measures, ranking methods, aggregation operators and etc over and study their properties.

[
According to part (ii), if then we have | | .Proof of (v) and (vi) are obvious.

Value of a
Where.By comparing value index of , a ranking of the alternatives set can be obtained.In special case we have http://www.ispacs.com/journals/jfsva/2014/jfsva-00199/International Scientific Publications and Consulting Services International Scientific Publications and Consulting Services International Scientific Publications and Consulting Services