, Several

Similarity


Introduction
After the introduction of Fuzzy Set Theory by by Prof. L. A. Zadeh in 1965 [27], several researchers have extended this concept in many directions.The traditional fuzzy sets is characterized by the membership value or the grade of membership value.Some times it may be very difficult to assign the membership value for a fuzzy set.Consequently the concept of interval valued fuzzy sets [28] was proposed to capture the uncertainty of grade of membership value.In some real life problems in expert system, belief system, information fusion and so on, we must consider the truth-membership as well as the falsitymembership for proper description of an object in uncertain, ambiguous environment.Neither the fuzzy sets nor the interval valued fuzzy sets is appropriate for such a situation.Intuitionistic fuzzy sets [1] introduced by Atanassov in 1986 and interval valued intuitionistic fuzzy sets [2] introduced by K. Atanassov and G. Gargov in 1989 are appropriate for such a situation.The intuitionistic fuzzy sets can only handle the incomplete information considering both the truth-membership (or simply membership) and falsity-membership (or non-membership) values.But it does not handle the indeterminate and inconsistent information which exists in belief system.F. Smarandache in 1995 introduced the concept of neutrosophic set [19], which is a mathematical tool for han-dling problems involving imprecise, indeterminacy and inconsistent data.Soft set theory [11,14] has enriched its potentiality since its introduction by Molodtsov in 1999.Using the concept of soft set theory P. K. Maji in 2013 introduced neutrosophic soft set [15] and Irfan Deli in 2014 introduced the concept of interval valued neutrosophic soft sets [8] .Neutrosophic sets and neutrosophic soft sets now become the most useful mathematical tools to deal with the problems which involves the indeterminate and inconsistent informations.

Anjan Mukherjee and Sadhan Sarkar, Several Similarity Measures of Interval Valued Neutrosophic Soft Sets and Their
Application in Pattern Recognition Problems intuitionistic fuzzy soft sets [4], similarity measures of interval-valued fuzzy soft sets have been studied by several researchers.Recently Said Broumi and Florentin Smarandache introduced the concept of several similarity measures of neutrosophic sets [3] , Jun Ye introduced the concept of similarity measures between interval neutrosophic sets [26] and A. Mukherjee and S. Sarkar intoduced similarity measures for neutrosophic soft sets [18].In this paper the Hamming and Euclidean distances between two interval valued neutrosophic soft sets(IVNS sets) are defined and similarity measures between two IVNS sets based on distances are proposed.Similarity measures between two IVNS sets based on set theoretic approach also proposed in this paper.A decision making method is established based on the proposed similarity measures.An illustrative example demonstrates the application of proposed decision making method in pattern recognition problem.
The rest of the paper is organized as ---section 2: some preliminary basic definitions are given in this section.In section 3 similarity measures between two IVNS sets is defined with example.In section 4 similarity measures between two IVNS sets based on set theoretic approach is defined with example, weighted distances , similarity measures based on weighted distances is defined.Also some properties of similarity measures are studied.In section 5 a decision making method is established with an with an example in pattern recognition problem.In Section 6 a comparative study of similarity measures is given.Finally in section 7 some conclusions of the similarity measures between IVNS sets and the proposed decision making method are given.

Preliminaries
In this section we briefly review some basic definitions related to interval-valued neutrosophic soft sets which will be used in the rest of the paper.

Definition 2.3[1]
Let X be a non empty set.Then an intuitionistic fuzzy set (IFS for short) A is a set having the form A={(x,  A (x),  A (x)): xX} where the functions  A : X[0,1] and  A : X[0,1] represents the degree of membership and the degree of non-membership respectively of each element xX and 0 A (x)+ A (x)1 for each xX.

Definition 2.4[2]
An interval valued intuitionistic fuzzy set A over a universe set U is defined as the object of the form A={<x,  A (x),  A (x)>: xU)}, where  A (x): UD[0,1] and  A (x): UD[0,1] are functions such that the condition: xU, sup A (x)+sup A (x)1 is satisfied( where D[0,1] is the set of all closed subintervals of [0,1]).Definition 2.5 [11,14] Let U be an initial universe and E be a set of parameters.Let P(U) denotes the power set of U and A . Then the pair (F,A) is called a soft set over U, where F is a mapping given by F:A→P(U).

Definition 2.6[19,20]
A neutrosophic set A on the universe of discourse X is defined as

{( , ( ), ( ), ( )), }
From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ] 0,1 [  .But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of ] 0,1 [  .Hence we consider the neutrosophic set which takes the value from the subset of [0,1] that is 0 ( ) ( ) ( ) 3 Where T A (x) is called truth-membership function, I A (x) is called an indeterminacy-membership function and F A (x) is called a falsity membership function Definition 2.7[15] Let U be the universe set and E be the set of parameters.Also let A E and P(U) be the set of all neutrosophic sets of U. Then the collection (F, A) is called neutrosophic soft set over U, where F is a mapping given by F: A  P(U).Thus a IVN-set A over U is represented as

Anjan Mukherjee and Sadhan Sarkar, Several Similarity Measures of Interval Valued Neutrosophic Soft Sets and Their
Application in Pattern Recognition Problems Definition 2.9 [8] Let U be an initial universe set, E be a set of parameters and A E. Let IVNS(U) denotes the set of all interval value neutrosophic subsets of U. The collection (F,A) is termed to be the interval valued neutrosophic soft set over U, where F is a mapping given by F: A → IVNS(U).

Similarity measure between two IVNS sets based on distances
In this section we define Hamming and Euclidean distances between two interval valued neutrosophic soft sets and proposed similarity measures based on theses distances.Another similarity measure of (N 1 ,E) and (N 2 ,E) can also be defined as Where D(N 1 ,N 2 ) is the distance between the interval valued neutrosophic soft sets (N 1 ,E) and (N 2 ,E) and  is a positive real number, called steepness measure.

Definition 3.3
Let U be universe and E be the set of parameters and (N 1 ,E) , (N 2 ,E) be two interval valued neutrosophic soft sets over U. Then we define the following distances between (N 1 ,E) , (N 2 ,E) as follows:

Anjan Mukherjee and Sadhan Sarkar, Several Similarity Measures of Interval Valued Neutrosophic Soft Sets and Their Application in Pattern Recognition Problems
The weighted distance is defined as   where N 1 and N 2 are mappings given by N 1 ,N 2 : E → IVNS(U).We define similarity measure SM(N 1 ,N 2 ) between (N 1 ,E) and (N 2 ,E) based on set theoretic approach as follows: In this section we developed an algorithm based on similarity measures of two interval valued neutrosophic soft sets based on distances for possible application in pattern recognition problems.In this method we assume that if similarity between the ideal pattern and sample pattern is greater than or equal to 0.7(which may vary for The algorithm of this method is as follows: Step 1: construct an ideal IVNSS (A, E) over the universe U.
Step 2: construct IVNS Sets (A i , E), i = 1, 2, 3, … , n, over the universe U for the sample patterns which are to recognized.
Step 3: calculate the distances of (A, E) and (A i , E).
Step 4: calculate similarity measure SM(A, A i ) between (A, E) and (A i , E).
Step 5: If SM(A, A i ) ≥ 0.7 then the pattern A i is to be recognized to belong to the ideal Pattern A and if SM(A, A i ) < 0.7 then the pattern A i is to be recognized not to belong to the ideal Pattern A. Example 5.1 Here a fictitious numerical example is given to illustrate the application of similarity measures between two interval valued neutrosophic soft sets in pattern recognition problem.In this example we take three sample patterns which are to be recognized.Let U = {x 1 ,x 2 ,x 3 } be the universe and E = {e 1 ,e 2 ,e 3 } be the set of parameters.Also let (A,E) be IVNS set of the ideal pattern and (A 1 ,E), (A 2 ,E), (A 3 ,E) be the IVNS sets of three sample patterns.
Step 1: Construct an ideal IVNS Set (A,E) over the universe U.

A
e 1 e 2 x 1 [0.and (A i ,E) for i = 1,2,3 are given by SM(A,A 1 ) = 0.39 SM(A,A 2 ) = 0.87 SM(A,A 3 ) = 0.86 Step 5: Here we see that SM(A,A 1 ) < 0.7 , SM(A,A 2 ) > 0.7 and SM(A,A 3 ) > 0.7 .Hence the sample patterns whose corresponding IVNS sets are represented by (A 2 ,E) and (A 3 ,E) are recognized as similar patterns of the family of ideal pattern whose IVNS set is represented by (A,E) and the pattern whose IVNS set is represented by (A 1 ,E) does not belong to the family of ideal pattern (A 1 ,E).Here we see that if we use similarity measures based on set theoretic approach then also we get the same results.

Comparison of different similarity measures
In this section we make comparative study among similarity measures proposed in this paper.Table 7 shows the comparison of similarity measures between two IVNS sets based on distance (Hamming distance) and similarity measure based on set theoretic approach as obtained in example 3.5, 4.2 and 5.1 .

Table 7: comparison of similarity measures
Table 7 shows that each method has its own measuring but the results are almost same.So any method can be applied to evaluate the similarity measures between two interval valued neutrosophic soft sets.

Conclusions
In this paper we have defined several distances between two interval valued neutrosophoic soft sets and on these distances we proposed similarity measure between two interval valued neutrosophic soft sets.We also proposed similarity measure between two interval valued neutrosophic soft sets based on set theoretic approach.A decision making method based on similarity measure is developed and a numerical example is illustrated to show the possible application of similarity measures between two interval valued neutrosophic soft sets for a pattern recognition problem.Thus we can use the method to solve the problem that contain uncertainty such as problem in social, economic system, medical diagnosis, game theory, coding theory and so on.A comparative study of different similarity measures also done .

Definition 2 . 1 [ 27 ]Definition 2 . 2 [ 28 ]
Let X be a non empty collection of objects denoted by x.Then a fuzzy set (FS for short)  in X is a set of ordered pairs having the form membership function or grade of membership (also degree of compatibility or degree of truth) of x in  .The interval M = [0,1] is called membership space.Let D[0, 1] be the set of closed subintervals of the interval [0, 1].An interval-valued fuzzy set in X is an expression A given by x X  ,where M A :X→D[0,1].

Definition 2 . 8 [ 21 ]
Let U be a space of points (objects), with a generic element in U.An interval value neutrosophic set (IVN-set) A in U is characterized by truth membership function T A , a indeterminacy-membership function I A and a falsity-membership function F A .For each point u ; T A , I A and F A [ ] .

2
Let U be universe and E be the set of parameters and (N 1 ,E) , (N 2 ,E) be two interval valued neutrosophic soft sets over U. Then based on the distances defined in definition 3.1 similarity measure between (N 1 ,E) and (N 2 ,E) is defined as SM(N 1 ,N 2 0. If p = 1 then equation (3.3) and (3.4) are respectively reduced to Hamming distance and Normalized Hamming distance.Again if p = 2 then equation (3.3) and (3.4) are respectively reduced to Euclidean distance and Normalized Euclidean distance.

5 )Definition 3 . 4 Example 3 . 5
Where w = (w 1 ,w 2 ,w 3 ,….,wn ) T is the weight vector of x i (i = 1,2,3, …. ,n) and p > 0. Especially, if p = 1 then (3.5) is reduced to the weighted Hamming distance and if p = 2, then (3.5) is reduced to the weighted Euclidean distance.Based on the weighted distance between two interval valued neutrosophic soft sets (N 1 ,E) and (N 2 ,E) given by equation (3.6) , the similarity measure between (N 1 ,E) and (N 2 ,E)) is defined as SM(N 1 ,N 2 Let U={x 1 ,x 2 ,x 3 } be the universal set and E={e 1 ,e 2 } be the set of parameters.Let (N 1 ,E) and (N 2 ,E) be two interval-valued neutrosophic soft sets over U such that their tabular representations are as follows:

Anjan
Mukherjee and Sadhan Sarkar, Several Similarity Measures of Interval Valued Neutrosophic Soft Sets and TheirApplication in Pattern Recognition Problems different problem) then the pattern belongs to the family of ideal pattern in consideration.

Table 3 :
tabular representation of (A,E)

Table 6 :
tabular representation of (A 3 ,E) Anjan Mukherjee and Sadhan Sarkar, Several Similarity Measures of Interval Valued Neutrosophic Soft Sets and Their Application in Pattern Recognition Problems