Cmjv03i01p0063

J. Comp. & Math. Sci. Vol.3 (1), 63-69 (2012)

Similarity Measure for Intuitionistic Fuzzy Sets: An Intuitive Approach SOURIAR SEBASTIAN1 and JAMES PHILIP2 1

Department of Mathematics, St. Albert’s College, Ernakulam, India.

2 Department of Mathematics, St. Dominic’s College, Kanjirapally, India.

(Received on: 28th December, 2011) ABSTRACT Several measures are available in the literature on fuzzy mathematics to compare Intuitionistic Fuzzy Sets. Measures due to Chen, Dengfeng and Chuntian, Szmidt and Kacprzyk are some of the important ones. We briefly review some of the existing measures and introduce a new measure which is simple, intuitively obvious and overcomes the defects of the existing ones.

Keywords: Fuzzy Sets, Intuitionistic Fuzzy Sets, Hesitancy Grade, Similarity Measure, Singleton IFS, Derived Singleton IFS, Null IFS, Pseudo-null IFS.

1. INTRODUCTION The concept of a Fuzzy Set (FS) introduced by Lotfi A. Zadeh10 in 1965 generalised the concept of (crisp) sets developed by Cantor. This was further generalized by Krassimir T. Atanassov2,3 into Intuitionistic Fuzzy Set (IFS). He introduced a new component degree of nonmembership and studied the properties of the new object so defined. Since then, the notion of IFSs has been explored by researchers and a number of theoretical and practical results have appeared. We have investigated the relation between fuzzy and intuitionistic fuzzy sets1 by proposing a

method of fuzzification of IFSs. Different IFSs defined on the same universe are compared using similarity measures. In this paper, we introduce a new measure of similarity between IFSs. 2. PRELIMINARY CONCEPTS 2.1 Definition10 Let X be a collection of objects regarded as the universal set (universe). Then, a Fuzzy Set (FS) A on X is defined as the set of all ordered pairs (x, µA(x)) where µA: X→ [0, 1]. i.e., A = {(x, µA(x))/x∈X, 0 ≤ µA(x) ≤ 1}.

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The function µA is called the membership function for the FS A. The membership function maps each element of X to a membership grade between 0 and 1, both inclusive. If the membership grade is restricted to {0, 1} the FS becomes an ordinary crisp set. This shows that the concept of a FS is a generalization of the concept of a crisp set.

The concept of FSs is useful in dealing with real world vagueness.

However, the ordinary FSs consider only the membership grade in a given set. Since the membership is not total, we need to consider also non-membership. In addition, there is also an aspect of indeterminacy. The notion of IFSs given below takes into consideration all these aspects.

2.2 Definition3 An Intuitionistic Fuzzy Set (IFS) A in a universe X is an object A = {<x, µA(x), νA(x)>: x ∈ X }, with 0 ≤ µA(x) ≤ 1, 0 ≤ νA(x) ≤ 1, and µA(x) + νA(x) ≤ 1, for each x ∈ X. Both µA and νA are functions from X to [0, 1]. For each x∈X, µA(x) and νA(x) are respectively called membership degree and non-membership degree of x in A. 2.3 Definition2 Let A be an IFS on X and x ∈ X. Then, hA(x) = 1- (µA(x) + νA(x)) is called the hesitancy grade or simply, the hesitancy of x. It is the degree of uncertainty of x as a member of the set A. The hesitancy may also be called the indeterminacy.

2.4 Notation In this paper, we adopt the following notations. µA(x) = A+(x), νA(x) = A-(x) and 0 hA(x) = A (x) Also, the collection of all IFSs on X will be denoted by IFS(X)

2.5 Definition3 Let A, B ∈ IFS(X). We say that A is a subset of B, denoted by A⊆B, if and only if A+(x) ≤ B+(x) and A-(x) ≥ B-(x), ∀ x∈ X. 2.6 Definition2 The complement of an IFS A = {<x, A+(x), A-(x)>: x ∈ X} is the IFS Ā = {<x, Ā+(x), Ā-(x)>: x ∈ X}, where Ā+(x) = A-(x) and Ā-(x) = A+(x), ∀ x∈ X. That is, Ā = {<x, A-(x), A+(x)>: x ∈ X} As obvious consequences of the above definitions and notations, we have, ∀ x∈ X: (i) 0 ≤ Ā+ (x) ≤ 1, 0 ≤ Ā- (x) ≤ 1, and Ā+ (x) + Ā- (x) ≤ 1 (ii) Ā0(x) = A0(x) (iii) ‫ܣ‬Ӗ = A 2.7 Definition5 An IFS A is called a Null IFS if and only if A-(x) = 1, ∀ x∈ X. It may be denoted by φ.

2.8 Definition An IFS A is called a Pseudo Null IFS if and only if A0(x) = 1, ∀ x∈ X. It may be denoted by φ*.

2.9 Definition Let X be a universe consisting of a single element and let A ∈ IFS(X). Then we say that A is a singleton IFS. 2.10 Example Let X = {x1} and A = {<x1, 0.4, 0.5>}. Then A is a singleton IFS. Here A+(x1) = 0.4, A-(x1) = 0.5 and A0(x1) = 0.1.

Its complement is Ā = {< x1, 0.5, 0.4>}.

Note that Ā+ (x1) = 0.5 = A-(x1), Ā- (x1) = 0.4 = A+(x1) and Ā0(x1) = 0.1 = A0(x1) 2.11 Definition Let A ∈ IFS(X) where X ={x1, x2, …, xn} is a finite universe. Then, we say that A is a finite IFS.

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2.12 Example Let X={x1, x2, x3} and A = {<x1, 0.4, 0.6>, <x2, 0.3, 0.5>, <x3, 0.8, 0.1>}. Then A is a finite IFS with A+(x1) = 0.4, A-(x1) = 0.6, A0(x1) = 0, A+(x2) = 0.3, A-(x2) = 0.5, A0(x2) = 0.2 and A+(x3) = 0.8, A-(x3) = 0.1, A0(x3) = 0.1.

The complement of A is Ā = {<x1, 0.6, 0.4>, <x2, 0.5, 0.3>, <x3, 0.1, 0.8>}. It can be easily observed that Ā+ (xi) = A-(xi), Ā- (xi) = A+(xi) and Ā0(xi) = A0(xi), ∀ i=1,2,3.

2.13 Geometric Representation9 An IFS A is uniquely represented by the three membership grades: A+(x), A-(x) and A0(x) for each x ∈ X. Hence, components of an IFS A correspond to points in the unit cube [0, 1]3. Further, since A+ + A- + A0 = 1, these points belong to the triangular region LMN, which we denote by ∆LMN (figure 1).

Hence, the triple <A+, A-, A0> representing the IFS A, can be considered as the co-ordinates of a point belonging to ∆LMN.

3. SOME EXISTING SIMILARITY MEASURES Li et al.7 observe that a similarity measure is used for estimating the degree of similarity between two IFSs. Dengfeng and Chuntian6 introduced a way of comparing IFSs defined over the same universe X. We give below the definition of similarity measure as proposed by them.

3.1 Definition6 Let S: IFS(X) x IFS(X) → [0, 1] be a map. Then S(A, B) is said to be the degree of similarity between A and B, where A,B ∈ IFS(X), if S(A,B) satisfies the following properties: p.1: p.2: p.3: p.4: p.5:

S(A, B) ∈ [0, 1].

S(A, B) = 1 if and only if A = B. S(A, B) = S(B, A).

If A⊆B⊆C ∈ IFS(X), then S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C).

S(A, B) = 0, if and only if A = φ and B =A, or A =B and B = φ.

3.2 Remark The above definition does not take into consideration the IFS φ* and so we modify property 5 as follows: P*.5: S(A, B) = 0, if and only if (A = ϕ and B = φ or φ*), or (B = ϕ and A = φ or φ*) 3.3 The Measure SC(A, B)4 Let A and B be finite IFSs defined on the same universe X = {x1, x2, ..., xn} . Then the Chen measure of similarity SC(A, B) is defined by

L(1, 0, 0) Figure 1

n

It may be noted that the IFSs φ, φ and ϕ correspond to the vertices M, N and L of the triangle LMN.

∑S

*

SC(A, B) = 1-

i =1

A

( x i ) − S B ( xi ) 2n

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,

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Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 63-69 (2012)

where SK(xi) = K+(xi) - K-(xi) for every K∈IFS(X).

Obviously, SK(xi) ∈ [-1, 1] for every K∈IFS(X). In fact, SC(A, B) is not a proper measure of similarity, as it does not satisfy the first axiom. Li et al.7 observe that it only expresses a degree of support. For example, taking X={x}, A={<x,0,0>} and B={<x,0.5,0.5>}, we get SC(A, B) = 1, though A≠B. Hence the second axiom also fails.

3.4 The Measure SDC(A, B)6 Let X = {x1, x2, ..., xn} and A, B ∈ IFS(X). Then the Dengfeng and Chuntian measure of similarity SDC(A, B) is defined by

 n p  ∑ ψ A ( xi ) −ψ B ( xi )  i =1  SDC(A, B) = 1-  2n where, ψ K ( x i ) =

K + ( xi ) − K − ( xi ) 2

1

p

Let P and Q be IFSs. Then, the measure Sim is defined by 1, for a = 0, 0 ≤ b ≤ 1  Sim(P, Q) = 1 - a/b, for 0 < a < b    0, for a = b; a, b ≠ 0   Undefined for b = 0, b > a

where, a is the distance from P to Q and b is the distance from P to Q , (P, Q and Q being considered as points on ∆LMN in figure 1, and the distance is the Euclidean distance between points in the plane).

3.6 Remark This measure is also a restricted one and has the following disadvantages: 1. Sim is not defined for b = 0 and b > a.

2. If P is equidistant from Q and Q , P and Q , become totally dissimilar.

Hence, we propose the following simple measure, which does not have the defects for mentioned above.

every K∈IFS(X) and p is a fixed positive integer.

Li et al.7 observe that ψ K ( xi ) is the median value of the interval [ K + ( xi ), 1 − K − ( xi ) ], and hence, if the median values of two IFSs are equal, then the IFSs become similar.

Hence, SDC is only a rough measure of similarity. It may be further observed that SC is a particular case of SDC obtained by taking p = 1 and so in this case also the second axiom fails.

3.5 The Measure Sim8 Szmidt and Kacperzyk define a similarity measure as follows:

4. THE PROPOSED NEW SIMILARITY MEASURE First, we restrict our attention to the case where A and B are singleton IFSs only.

Each point in ∆LMN (see figure 1) represents an IFS and conversely, each IFS A∈ IFS(X) can be represented by a point in the region. This gives us a simple, intuitive measure of similarity in terms of the geometric distance between IFSs. 4.1 Definition Let A, B be singleton IFSs on the same universe. Then, the geometric distance between A and B, denoted by Dg(A, B), is defined by Dg(A, B) = [(A+ - B+)2 + (A- - B-)2 +(A0 – B0)2]1/2.

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Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 63-69 (2012)

4.2 Remark (i) Let P, Q be points in ∆LMN corresponding to the IFSs A and B then, the distance Dg(A, B) between A and B is simply the Euclidean distance between P and Q. (ii) 0≤ Dg(A, B)≤√2, ∀ A,B ∈ IFS(X) (iii) Dg(A, B) = 0 ⇔ A = B (iv) Dg(A, B) = Dg(B, A) 4.3 Definition Let Dg(A, B) be the geometric distance between two IFSs A and B. Then, the normalized geometric distance between A and B, denoted by DG(A, B), is given by DG(A, B) = (1/√2)Dg(A, B).

4.4 Remark IFS(X).

0≤DG(A, B)≤1, ∀ A,B∈

67

⇔ √2 DG(A, B) = 0 ⇔ Dg(A, B) = 0 ⇔ A = B, see remark 4.2 (iii) P.3: SSP(A, B) = SSP(B, A) SSP(A, B) = 1- DG(A, B) = 1- (1/√2)Dg(A, B) = 1-(1/√2)Dg(B,A), by remark 4.2 (iv) = 1- DG(B, A) = SSP(B, A) Next, we consider property 5.

P*.5: SSP(A, B) = 0, if and only if (A = ϕ and B = φ or φ*) or, (B = ϕ and A = φ or

It is intuitively clear that when the distance between points decreases, their similarity increases. Hence the normalized geometric distance may be used to define a similarity measure.

φ*) Suppose that A= ϕ and B = φ. Then, A+ = 1 and A- = A0 = 0 and B- = 1 and B+ = B0 = 0. Then, Dg(A, B) = √2 and DG(A, B) = 1.

Consequently, SSP(A, B) = 0. By a similar computation we get SSP(A, B) = 0, if A = ϕ and B = φ*

4.6 Definition (SSP(A, B)) We define SSP(A, B) = 1- DG(A, B).

The case when B = ϕ and A = φ or φ* is also similar.

We proceed to prove that SSP(A, B) satisfies all the five properties of similarity measure given in definition 3.1.

4.7 Proposition SSP(A, B) is a similarity measure.

Proof P.1: 0 ≤SSP(A, B)≤1.

We have, 0 ≤ DG(A, B) ≤ 1 and since SSP = 1-DG, the result is obvious.

P.2: SSP(A, B) = 1 if and only if A = B. SSP(A, B) = 1 ⇔ 1- DG(A, B) = 1 ⇔ DG(A, B) = 0

Conversely, assume that SSP(A, B) = 0.

This implies that DG(A, B) = 1 and Dg(A, B) = √2. This is possible only if A and B correspond to vertices of the triangle LMN in figure (1). When IFS A and vertex L correspond, we have A+ = 1 and B+ = 0.

Clearly, B = φ and A =B. The other cases can also be obtained by similar arguments.

Now we prove property 4.

P.4: If A⊆B⊆C are IFSs, then SSP(A, C) ≤ SSP(A, B) and SSP(A, C) ≤ SSP(B, C) A point A in ∆LMN is represented by a triple (A+,A-,A0). This is, however, equivalent

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to (A+, A-, 1-A+-A-). That is, ∆LMN is a plane region and hence it is sufficient to consider the projection of the region onto the (A+ - A-) plane, which we denote by πLMN. AP

Q

R A+

Then, x1 < x2 < x3 => Q is to the right of P and R is to the right of Q, And, y1 > y2 > y3 => Q is below P and R is below Q (by choosing suitable orientations).

Now, consider the triangle PQR (in figure 2).

PR<PQ only if PQR is an acute angle. But, PQR is an obtuse angled triangle with Angle PQR as the obtuse angle. Hence PR ≥ PQ. The other assertion, that PR ≥ QR can be proved similarly.

■ 5. GENERALIZATION TO FINITE IFSs

Figure 2

Now, let points P, Q, R in πLMN (Figure 2) be points associated with IFSs P, Q, R such that P ⊆ Q ⊆ R. For convenience, we may choose the co-ordinates as P(x1, y1), Q(x2, y2), R(x3, y3).

Then, P ⊆ Q => x1≤ x2 and y1≥y2.

Similarly, x2≤ x3 and y2≥y3.

Therefore, x1≤ x2≤ x3 and y1≥y2 ≥y3.

Now we want to show that SSP(P, R) ≤ SSP(P, Q) and SSP(P, R) ≤ SSP(Q, R). It is enough to prove that the Euclidean distances PR, PQ and QR satisfy the relations PR ≥ PQ and PR ≥ QR. To prove PR ≥ PQ, we consider three cases:

The above measure deals with singleton IFSs only. However, the measure may be extended to finite IFSs, by taking the average of similarity of each element in the universe. First, we need a definition.

5.1 Definition Let A be an IFS defined over a finite universe with n elements x1, x2, …, xn. Now, take Ai to be the singleton IFS consisting of the element xi and the same membership and non-membership values as in A. That is, Ai = <xi, A+(xi), A-(xi)>, i=1,2, …, n. Then Ai is called a Derived Singleton IFS. 5.2 Remark Clearly, A = {A1, A2, …, An} 5.3 Computation of Similarity

Case 1: If P = Q or Q = R. In this case, the result is trivially true.

Case 2: If P, Q, R are collinear. In this case also the result is obvious.

Case 3: P, Q, R are three distinct non collinear points.

Let A and B be finite IFSs over the same finite universe X. We consider the derived singleton IFSs Ai and Bi and compute the similarity between each pair (Ai, Bi). Now, the similarity SSP(A, B) between A and B is taken as the arithmetic mean of the similarities of the above n pairs of IFSs, where n is the cardinality of X. That is,

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SSP(A, B) = (1/n)

∑

n i =1

S SP ( Ai , Bi )

5.4 Example 1 Let A={<x1,0.5, 0.2>, <x2, 0.4, 0.2>, <x3, 0.3, 0.3>}, B = {<x1, 0.4, 0.4>, <x2, 0.4, 0.3>, <x3, 0.5, 0.3>} Now, DG(A1, B1) = (1/√2)√(0.01 + 0.04 + 0.01) = 0.173 and SSP(A1, B1) = 0.827 Similarly, DG(A2, B2) = 0.1 and SSP(A2, B2) = 0.9 And, DG(A3, B3) = 0.2 and SSP(A3, B3) = 0.8 Hence, SSP(A, B) = (1/3)(0.827 + 0.9 + 0.8) = 0.842 5.5 Example 2 Let A = {<x1, 0.3, 0.5>, <x2, 0.4, 0.4>}, B = {<x1, 0.3, 0.4>, <x2, 0.4, 0.3>}, C = {<x1, 0.4, 0.3>, <x2, 0.5, 0.3>}. Here, A ⊆ B ⊆ C. Computing as above, we get, SSP(A, B) = 0.9, SSP(B, C) = 0.9 and SSP (A, C)= 0.86.

Observe that SSP(A, C) ≤ SSP(A, B) and SSP(A, C) ≤ SSP(B, C).

2.

3.

4.

5.

6.

7.

6. CONCLUSION In this paper we have introduced a new measure of similarity between singleton IFSs and extended it to IFSs over a finite universe. This measure is more intuitive and does not have the defects mentioned above for other measures. Hence, it may be used as an effective measure to compare IFSs. We also propose to extend this measure to arbitrary IFSs. 7. REFERENCES 1. Ansari, A.Q., James Philip, Shadab A.

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