Intuitionistic Fuzzy Multi Similarity Measure Based on Cotangent Function

Intuitionistic Fuzzy Multi Similarity Measure Based on Cotangent Function

* P. Rajarajeswari, ** N. Uma

* Department of Mathematics, Chikkanna Arts College, Tirupur, Tamil Nadu. (INDIA).

** Department of Mathematics, SNR Sons College, Coimbatore, Tamil Nadu. (INDIA).

ABSTRACT

In this paper, the Similarity measure of Intuitionistic Fuzzy Multi sets (IFMS) based on Cotangent function is presented and analyzed. The properties of the similarity measure are proved and verified using the numerical evaluation of the proposed similarity measure. The unique feature of this proposed method is that it considers multi membership and non membership for the same element. As the proposed method is mathematically valid, it can be applied to any decision making problems, medical diagnosis, engineering problems, pattern recognition, etc. Finally, the application of medical diagnosis shows that the proposed similarity measures are much simpler, well suited one to use with linguistic variables.

KEY WORDS: Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Similarity measure, Cotangent Function

INTRODUCTION

Krasssimir T. Atanassov [1], [ 2] proposed the Intuitionistic Fuzzy sets (IFS) as the generalisation of the Fuzzy set (FS) introduced by Lofti A. Zadeh [3]. The FS allows the object to partially belong to a set with a membership degree () between 0 and 1 whereas IFS represent the uncertainty with respect to both membership ( [0,1]) and non membership ( [0,1]) such that + 1 . The number = 1 is called the hesitiation degree or intuitionistic index.

The Multi set [4] allows the repeated occurrences of any element and hence the Fuzzy Multi set (FMS) can occur more than once with the possibly of the same or the different membership values was introduced by R. R. Yager [5]. Recently, the new concept Intuitionistic Fuzzy Multi sets (IFMS) was proposed by T.K Shinoj and Sunil Jacob John [6].

The study of distance and similarity measure of IFSs gives lots of measures, each representing specific properties and behaviour in real-life decision making and pattern recognition works. For measuring the degree of similarity between vague sets, Chen and Tan [7] proposed two similarity measures. The Hamming, Euclidean distance and similarity measures were introduced by Szmidt and Kacprzyk [8], [9], [10], [11]. The Geometric distance and similarity measures were given by Xu (12). Using the Cotangent function, a new similarity measure was proposed by Wang et al (13). Later a new fuzzy cotangent similarity measure for IFSs was introduced by Tian Maoying (14).

As the extension of the distance and similarity measure of IFSs to IFMSs [15], [16] are possible; in this paper we extend the fuzzy cotangent similarity measure of IFSs to IFMSs. The numerical results of the examples show that the developed similarity measures are well suited to use any linguistic variables.

The organization of this paper is as follows: In section 2, the Fuzzy Multi sets, Intuitionistic Fuzzy Multi sets and similarity measures of IFMS are presented. The section 3 deals with the proposed Cotangent Similarity measure for the IFMS, along with the numerical evaluation. The application of medical diagnosis using IFMS is explained in detail in section 4.

Definition: 2.1

II. PRELIMINARIES

Let X be a nonempty set. A fuzzy set A in X is given by A = , / — (2.1)

where : X [0, 1] is the membership function of the fuzzy set A (i.e.) 0,1 is the membership of in A. The generalizations of fuzzy sets are the Intuitionistic fuzzy (IFS) set proposed by Atanassov [1], [2] is with independent memberships and non memberships.

Definition: 2.2

An Intuitionistic fuzzy set (IFS), A in X is given by A = , , / — (2.2)

where : X [0,1] and : X [0,1] with the condition 0 + 1 , Here

[0,1] denote the membership and the non membership functions of the fuzzy set A; For each Intuitionistic fuzzy set in X, = 1 1 = 0 for all that is

= 1 is the hesitancy degree of in A. Always 0 1, . The complementary set of A is defined as = , , / — (2.3) Definition: 2.3

Let X be a nonempty set. A Fuzzy Multi set (FMS) A in X is characterized by the count membership function Mc such that Mc : X Q where Q is the set of all crisp multi sets in [0,1]. Hence, for any , Mc(x) is the crisp multi set from [0, 1]. The membership sequence is defined as

( 1 , 2 , ) where 1 2 .

Therefore, A FMS A is given by = , ( 1 , 2 , ) / — (2.4)

Definition: 2.4

Let X be a nonempty set. A Intuitionistic Fuzzy Multi set (IFMS) A in X is characterized by two functions namely count membership function Mc and count non membership function NMc such that Mc : X Q and NMc : X Q where Q is the set of all crisp multi sets in [0,1]. Hence, for any , Mc(x) is the crisp multi set from [0, 1] whose membership sequence is defined as

( 1 , 2 , ) where 1 2 and the corresponding

non membership sequence NMc (x) is defined as ( 1 , 2 , ) where the non membership can be

either decreasing or increasing function. such that 0 + 1 , = 1,2, . Therefore,

An IFMS A is given by = , 1 , 2 , , ( 1 , 2 , ) / — (2.5)

where 1 2 . The complementary set of A is defined as

= , ( 1 , 2 , ), 1 , 2 , , / (2.6)

1 2

Definition: 2.5

The Cardinality of the membership function Mc(x) and the non membership function NMc (x) is the length of an element x in an IFMS A denoted as , defined as = Mc(x) = NMc(x)

If A, B, C are the IFMS defined on X , then their cardinality = Max { (A), (B), (C) }.

Definition: 2.6

, is said to be the similarity measure between A and B , where A, B X and X is an IFMS, as ,

satisfies the following properties

1. , [0,1]

2. , = 1 if and only if A = B

3. , = ,

4. If A B C X , then , , and , ,

COTANGENT SIMILARITY MEASURE OF IFSs

The entropy measure of IFSs proposed by Wang et al [13] was as follows

=

1 1

+

()

=1

4 4 1 + ()

As there is a relationship between the Entropy and Similarity Measure, the fuzzy cotangent similarity measure of IFSs was proposed by Tian Maoying [14]

1 +

, =

=1

4

where A, B IFS(X) and E be the entropy of IFSs, consisting of the membership and non membership functions. If there are three parameters like membership, non membership and hesitation function then the fuzzy cotangent similarity measure of IFSs becomes

1 +

, =

=1

4

III PROPOSED COTANGENT SIMALRITY MEASURE OF IFMSs

In IFS, the Similarity measures are onsidered for the membership and non membership functions only once. But in IFMS, it should be considered more than once; because of their multi membership and non membership functions. And, their considerations are combined together by means of Summation concept based on their cardinality.

Definition: 3.1

1 1

+

, =

=1

=1

4

of the membership and non membership functions. And if there are three parameters like membership, non membership and hesitation function then the fuzzy multi cotangent similarity measure becomes

1 1

+

, =

=1

=1

4

PROPOSITION: 3.2

The defined similarity measure , between IFMS A and B satisfies the following properties

D1. 0 , 1

D2. A = B if and only if , = 0

D3. , = ,

D4. If , for A, B, C are IFMS then

, , and , ,

Proof

D1. ,

As the membership and the non membership functions of the IFMSs lies between 0 and 1, the similarity measure based cotangent function also lies between 0 and 1. (Always cot function lies between 0 and 1)

D2. A = B if and only if , = 0

(i) Let the two IFMS A and B be equal (i.e.) A = B. This implies for any = and =

which states that = 0. Hence , = 0

(ii) Let the , = 0

The zero distance measure is possible only if both = 0, as the Cotangent

similarity measure concerns with max operator ( ) of membership and non membership difference. This refers that

= and = for all i, j values. Hence A = B.

D3. , = ,

It is obvious that and

But = and =

Hence , = 1

1

+

=1 =1 4

= 1

1

+

= ,

=1 =1 4

D4. If for A, B, C are IFMS then , , and , ,

Let , then the assumption is and for every

Case (i) Let Then from the assumption of non membership function, we

have —- (3.4.1)

Also —- (3.4.2)

Now from the assumption of the membership, we have

and —- (3.4.3)

From (3.4.1, 3.4.2, 3.4.3) , , and , ,

Case (ii) Let Then from the assumption of membership function, we

have —- (3.4.4)

Also —- (3.4.5)

Now from the assumption of the non membership, we have

and — (3.4.6)

From (3.4.4, 3.4.5, 3.4.6) , , and , ,

NUMERICAL EVUALATION: 3.3

EXAMPLE: 4.1

Let X = {A1, A2, A3, A4…….. An } with A = { A1, A2, A3, A4, A5} and B ={ A6, A7, A8, A9, A10} are the IFMS defined as

A = { 1 0.6,0.4 , 0.5, 0.5 , 2 0.5,0.3 , 0.4, 0.5 , 3 , 0.5, 0.2 , 0.4, 0.4 ,

4 0.3,0.2 , 0.3, 0.2 , 5 0.2,0.1 , 0.2, 0.2 }

B = { 6 0.8,0.1 , 0.4, 0.6 , 7 0.7,0.3 , 0.4, 0.2 , 8 , 0.4, 0.5 , 0.3, 0.3

9 0.2,0.7 , 0.1, 0.8 , 10 0.2,0.6 , 0, 0.6 }

Here, the cardinality = 5 as Mc(A) = NMc(A ) = 5 and Mc(B) = NMc(B) = 5 and the cotangent similarity

measure is

1 5

1 2

+

= 0.5936

5 =1 2 =1 4

EXAMPLE: 4.2

Let X = {A1, A2, A3, A4…….. An } with A = { A1, A2 } and B ={ A9, A10} are the IFMS defined as

A = { 1 0.1,0.2 , 2 0.3,0.3 } , B = { 9 0.1,0.2 , 10 0.2,0.3 } Here, the cardinality = 2 as Mc(A) = NMc(A ) = 2 and Mc(B) = NMc(B) = 2 and the cotangent similarity measure is

1 2

1 1

+

= 0.9271

2 =1 1 =1 4

EXAMPLE: 4.3

Let X = {A1, A2, A3, A4…….. An } with A = { A1, A2 } and B ={ A3, A4 } are the IFMS defined as A = { 1 0.4,0.2,0.1 , 0.3, 0.1, 0.2 , 0.2, 0.1, 0.2 , 0.1, 0.4, 0.3 ,

2 0.6,0.3,0 , 0.4, 0.5, 0.1 , 0.4, 0.3, 0.2 , 0.2, 0.6, 0.2 }

B = { 3 0. 5,0.2,0.3 , 0.4, 0.2, 0.3 , 0.4, 0.1, 0.2 , 0.1, 0.1, 0.6

4 0.4,0.6,0.2 , 0.4, 0.5, 0 , 0.3, 0.4, 0.2 , 0.2, 0.4, 0.1 }

The cardinality = 2 as Mc(A) = NMc(A ) = Hc(A) = 2 and Mc(B) = NMc(B) = Hc(B) = 2 Hence, the cotangent similarity measure is

1 2

1 4

+

= 0.7450

2 =1 4 =1 4

EXAMPLE: 4.4

Let X = {A1, A2, A3, A4….. An } with A = { A1, A2 } and B = {A6} such that the IFMS A and B are A = { 1 0.6,0.2,0.2 , 0.4, 0.3, 0.3 , 0.1, 0.7, 0.2 ,

2 0.7,0.1,0.2 , 0.3, 0.6, 0.1 , 0.2, 0.7, 0.1 }

B = { 6 0.8,0.1,0.1 , 0.2, 0.7, 0.1 , 0.3, 0.5, 0.2 }

As Mc(A) = NMc(A ) = Hc(A) =2 and Mc(B) = NMc(B) = Hc(B) =1,

their cardinality = Max { (A), (B) } = max {2,1} = 2. Therefore the cotangent similarity measure is

1 2

1 3

+

= 0.7329

2 =1 3 =1 4

IV MEDICAL DIAGNOSIS USING IFMS – COTANGENT MEASURE

As Medical diagnosis contains lots of uncertainties, they are the most interesting and fruitful areas of application for fuzzy set theory. In some situations, terms of membership function alone is not adequate. Hence, the Intuitionistic fuzzy set theory consisting of both the terms like membership and non membership function is considered to be the better one. Due to the increased volume of information available to physicians from new medical technologies, the process of classifying different set of symptoms under a single name of disease becomes difficult. Recently, there are various models of medical diagnosis under the general framework of fuzzy sets are proposed. In some practical situations, there is the possibility of each element having different membership and non membership functions. The proposed distance and similarity measure among the Patients Vs Symptoms and Symptoms Vs diseases gives the proper medical diagnosis. The unique feature of this proposed method is that it considers multi membership and non membership. By taking one

time inspection, there may be error in diagnosis. Hence, this multi time inspection, by taking the samples of the same patient at different times gives best diagnosis.

Let P = { P1, P2, P3, P4 } be a set of Patients.

D = { Fever, Tuberculosis, Typhoid, Throat disease } be the set of diseases

and S = { Temperature, Cough, Throat pain, Headache, Body pain } be the set of symptoms.

Our solution is to examine the patient at different time intervals (three times a day), which in turn give arise to different membership and non membership function for each ptient.

TABLE : 4.1 IFMs Q : The Relation between Patient and Symptoms

Q	Temperature	Cough	Throat Pain	Head Ache	Body Pain
	(0.6, 0.2)	(0.4, 0.3)	(0.1, 0.7)	(0.5, 0.4)	(0.2, 0.6)
P1	(0.7, 0.1) (0.5, 0.4)	(0.3, 0.6) (0.4, 0.4)	(0.2, 0.7) (0, 0.8)	(0.6, 0.3) (0.7, 0.2)	(0.3, 0.4) (0.4, 0.4)
	(0.4, 0.5)	(0.7, 0.2)	(0.6, 0.3)	(0.3, 0.7)	(0.8, 0.1)
P2	(0.3, 0.4)	(0.6, 0.2)	(0.5, 0.3)	(0.6, 0.3)	(0.7, 0.2)
	(0.5, 0.4)	(0.8, 0.1)	(0.4, 0.4)	(0.2, 0.7)	(0.5, 0.3)
	(0.1, 0.7)	(0.3, 0.6)	(0.8, 0)	(0.3, 0.6)	(0.4, 0.4)
P3	(0.2, 0.6) (0.1, 0.9)	(0.2, 0 ) (0.1, 0.7)	(0.7, 0.1 ) (0.8, 0.1)	(0.2, 0.7) (0.2, 0.6)	(0.3, 0.7) (0.2, 0.7)

Let the samples be taken at three different timings in a day (morning, noon and night)

TABLE : 4.2 IFMs R : The Relation among Symptoms and Diseases

R	Viral Fever	Tuberculosis	Typhoid	Throat disease
Temperature	(0.8, 0.1)	(0.2, 0.7)	(0.5, 0.3)	(0.1, 0.7)
Cough	(0.2, 0,7)	(0.9, 0)	(0.3, 0,5)	(0.3, 0,6)
Throat Pain	(0.3, 0.5)	(0.7, 0.2)	(0.2, 0.7)	(0.8, 0.1)
Head ache	(0.5, 0.3)	(0.6, 0.3)	(0.2, 0.6)	(0.1, 0.8)
Body ache	(0.5, 0.4)	(0.7, 0.2)	(0.4, 0.4)	(0.1, 0.8)

TABLE : 4.3 The Cotangent Similarity Measure between IFMs Q and R :

Cotangent Similarity measure	Viral Fever	Tuberculosis	Typhoid	Throat disease
P1	0.7242	0.5152	0.7711	0.4868
P2	0.5967	0.7292	0.6446	0.5327
P3	0.5271	0.5777	0.6177	0.7825

The highest similarity measure from the table 4.3 gives the proper medical diagnosis.

Patient P1 suffers from Typhoid, Patient P2 suffers from Tuberculosis and Patient P3 suffers from Throat disease.

V. CONCLUSION

In this paper, we have developed the method for deriving the cotangent similarity measure of IFMS from IFS theory. The prominent characteristic of this method is that it considers multi membership, non membership functions and this similarity measure also guarantee that the cotangent function of any two IFMSs equals to

one if and only if the two IFMSs are the same. The example 3.1, 3.2 shows that the new measure perform well in the case of membership and non membership function and example 3.3, 3.4 depicts that the proposed measure is effective with three representatives of IFMSs membership, non membership and hesitation functions. Hence, from the analysis and numerical evaluation, it is clear that this proposed method can be applied to any pattern recognitions and decision making problems. Finally, an illustrative example, the medical diagnosis has been given to show the efficiency of the developed cotangent similarity measure.

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