A New Similarity Measure of Intuitionistic Fuzzy Multi Sets ...A New Similarity Measure of...

A New Similarity Measure of Intuitionistic Fuzzy Multi Sets in

Medical Diagnosis Application

N. Uma

Department of Mathematics, Sri Ramakrishna College of Arts and Science (Formerly SNR

Sons College), Coimbatore, Tamil Nadu. (INDIA). [email protected]

ABSTRACT

As Similarity Measure is an important topic in fuzzy set theory, the objective of this paper is

to introduce a new efficient Similarity measure for Intuitionistic fuzzy multi sets (IFMS). This

method considers multi membership, non-membership and hesitancy degree for the same

element. This novel Similarity measure for IFMS is the combination of MAX / MIN operators

of the membership functions and the Zhang and Fu’s measure of the IFMS. We apply this

appreciable measure to medical diagnosis as it satisfies all the properties of the Similarity

Measure.

KEY WORDS: Intuitionistic fuzzy set, Fuzzy Multi sets, Intuitionistic Fuzzy Multi sets,

Similarity measure, Max and Min operators.

I. INTRODUCTION

The Intuitionistic Fuzzy sets (IFS) proposed by Krasssimir T. Atanassov[1, 2] was

the generalisation of the Fuzzy set (FS) introduced by Lofti A. Zadeh [3]. The object,

partially belong to a set with a membership degree ( ) between 0 and 1 are represented by the

FS whereas the IFS represent the uncertainty with respect to both membership ( )

and non-membership ( ) such that . Here, the number is the hesitation

degree or intuitionistic index ( .

International Journal of Pure and Applied MathematicsVolume 119 No. 17 2018, 859-872ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

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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. Chen and Tan [4] proposed two Similarity measures for measuring the

degree of Similarity between vague sets. The Hamming, Euclidean Distance and Similarity

measures were introduced by Szmidt and Kacprzyk [5], [6], [7], [8]. The Geometric

Distance and Similarity measures were given by Xu [9]. Li et al [10] made a comparative

study for Similarity measures between IFSs from the methods of Chen, Hong, Kim, Fan,

Zhangyan, Yanhong et al., Dengfeng, Chuntian, Mitchell, Zhizhen and Pengfei. They

found that most of the Similarity measures reflect the degree of membership and non-

membership functions. They also discovered the inadequate conditions for Similarity

measures and hence the hesitation degree was introduced for Similarity measure.

Zhang and Fu [11] proposed a new Similarity measure for IFSs by considering the

hesitation degree also. Later some modifications were made by Binyamin et al [12] on

Zhang and Fu’s method for better results. Recently based on the calculation of degree of the

Similarity between IFSs, Similarity methods were established by Hung and Yang [13]

andYe[14]. Farhadinia [15] developed a new Distance method on an interval by the use of

convex combination of the end points and the property of MAX / MIN operators.

As the Multi set [16] allows the repeated occurrences of any element, the Fuzzy Multi

set (FMS) introduced by R. R. Yager [17] can occur more than once with the possibly of the

same or the different membership values. Recently T.K Shinoj and Sunil Jacob John [18]

proposed the new concept Intuitionistic Fuzzy Multi sets (IFMS) which allows the repeated

occurrences of different membership and non-membership function.

The various Distance and Similarity methods of IFS are extended for IFMS distance

and similarity measuresin [19], [20], [21], [22], [23], [24], [25], [26], [27], and [28]. Our aim

is to develop a simple and efficient Similarity measure so that it is well suited to use any

linguistic variables. In this paper, we combine the Zhang and Fu’s measure of IFMSs and the

MAX / MIN measures of IFMSs. The Medical Diagnosis example show that the developed

Similarity measure work as desired for two parameters(multi membership and non-

membership function) and three parameters(multi membership, multi non membership and

multi hesitation function).

The paper organization is as follows: In section 2, the Fuzzy Multi sets, Intuitionistic

Fuzzy Multi sets and Similarity measures of IFMS are briefed. The section 3 deals with the

International Journal of Pure and Applied Mathematics Special Issue

860

new Similarity measure for the IFMS. The significance of the proposed measure using the

Medical Diagnosis is in the section 4.

II. PRELIMINARIES

The generalizations of fuzzy sets are the Intuitionistic fuzzy (IFS) set proposed by Atanassov

[1], [2] is with independent memberships and non-memberships.

Definition: 2.1

An Intuitionistic fuzzy set (IFS), A in X is given by

A = -- (2.1)

: X → [0,1] and : X → [0,1] with the condition 0

Here [0,1] denote the membership and the non-membership functions

of the fuzzy set A; For each Intuitionistic fuzzy set in X,

= 1 is the hesitancy degree of in A.

Always 0

Definition: 2.2

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( , where

andthe corresponding non membership sequence NMc

(x) is defined as( , where the non-membership can be either

decreasing or increasing function. such that

0

Therefore, AnIFMS A is given by

- (2.3)


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where

Definition: 2.3

The Cardinality of the membership function Mc(x) and the non-membershipfunctionNMc

(x) is the length of an element xin an IFMS A denoted as defined as

η = =

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

Definition: 2.4

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

5. if A is a crisp set.

IFMS SIMILARITY MEASURE USING MAX / MIN OPERATORS

The Extended Similarity measure of the Intuitionistic Fuzzy Multi Sets from the Intuitionistic

Fuzzy Sets was developed in [26] was

Where A and B are two different IFMSs, consisting the multi membership

and non membership functions . This measure focuses the property of

MAX/MIN operators which avoids all the cons cases exists usually in other Distance and

Similarity measures.


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ZHANG AND FU’S SIMILARITY MEASURE OF IFMSs

In IFMS, because of their multi membership and non membership functions, the

considerations are combined together by means of Summation concept based on their

cardinality explain in depth in [27].

of the membership and non membership functions

Also this similarity measure becomes

Where ) = and

) =

And for three parameters like membership, non-membership and hesitation function the

similarity measure becomes

Where ) =

) = and

) =


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III. PROPOSED NEW SIMILARITY MEASURE FOR IFMS

As the ZHANG AND FU’S Similarity Measure and Similarity Measure Using MAX /

MIN Operators of IFMS are efficient in determination, our new proposed method, the

combination these two measures surely will leads to a best similarity measure of IFMS

Hence the new developed Similarity measure consisting of multi

membership and non membershipfunctions is as follows

Where ) = and

) =

And if there are three parameters like multi membership , non-

membership and hesitation function then the new IFMS Similarity measure

becomes

Where ) =

) = and

) =


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PROPOSITION : 3.4

The defined Similarity measure between IFMS A and B satisfies the following

properties

D1.

D2. = 1if and only if A = B

D3.

D4. If for A, B, C are IFMS

then and

D5. if A is a crisp set.

IV. MEDICAL DIAGNOSIS USINGIFMS – NEW SIMILARITY MEASURE

As Medical diagnosis contains lots of uncertainties, they are the most interesting and

fruitful areas of application for Intuitionistic fuzzy set theory. 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. In some practical

situations, there is the possibility of each element having different membership and non-

membership functions. The proposed 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 patient.


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

Q Temperature Cough Throat Pain Head Ache Body Pain

P1

(0.6, 0.2)

(0.7, 0.1)

(0.5, 0.4)

(0.4, 0.3)

(0.3, 0.6)

(0.4, 0.4)

(0.1, 0.7)

(0.2, 0.7)

(0, 0.8)

(0.5, 0.4)

(0.6, 0.3)

(0.7, 0.2)

(0.2, 0.6)

(0.3, 0.4)

(0.4, 0.4)

P2

(0.4, 0.5)

(0.3, 0.4)

(0.5, 0.4)

(0.7, 0.2)

(0.6, 0.2)

(0.8, 0.1)

(0.6, 0.3)

(0.5, 0.3)

(0.4, 0.4)

(0.3, 0.7)

(0.6, 0.3)

(0.2, 0.7)

(0.8, 0.1)

(0.7, 0.2)

(0.5, 0.3)

P3

(0.1, 0.7)

(0.2, 0.6)

(0.1, 0.9)

(0.3, 0.6)

(0.2, 0 )

(0.1, 0.7)

(0.8, 0)

(0.7, 0.1 )

(0.8, 0.1)

(0.3, 0.6)

(0.2, 0.7)

(0.2, 0.6)

(0.4, 0.4)

(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 New Similarity Measure between IFMs Q and R :

New Similarity

measure

Viral Fever Tuberculosis Typhoid Throat disease

P1 0.7166 0.3932 0.6930 0.3678

P2 0.4914 0.6727 0.5309 0.4168

P3 0.3997 0.4734 0.5006 0.7883

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


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The diagnosis refers that Patient P1 suffers from Viral Fever, Patient P2 suffers from

Tuberculosis and Patient P3 suffers from Throat disease.

To show the significance of the proposed measure, the same example of Medical

Diagnosis considered under Zhang and Fu’s IMFS Similarity measure and MAX/MIN

operator’s similarity measure is as follows.

TABLE : 4.4 – The Similarity Measure between IFMs Q and R :

Zhang And Fu’s

Similarity

measure


P1 0.7473 0.5650 0.8157 0.5287

P2 0.6580 0.7977 0.7167 0.5880

P3 0.5827 0.6413 0.6553 0.8407


The diagnosis refers that Patient P1 suffers from Typhoid, Patient P2 suffers from

Tuberculosis and Patient P3 suffers from Throat disease.

TABLE : 4.5 – The MAX/MIN operators Similarity Measure between IFMs Q and R :

Proposed

Similarity

Measure


P1 0.7243 0.5067 0.7224 0.3416

P2 0.5830 0.7828 0.5989 0.4446

P3 0.4558 0.5714 0.5019 0.7212


Patient P1 suffers from Viral Fever, Patient P2 suffers from Tuberculosis andPatient P3

suffers from Throat disease.


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On comparison of the three tables (4.3, 4.4 and 4.5), the diagnosis of the patients P2 and P3

are the same. The diagnosis differs for patient P1.

In Zhang and Fu’s Similarity measure, the patient P1 suffers from Typhoid.From the Table

(4.4), the similarity value of patient P1 is comparatively higher only for Typhoid

In MAX/MIN operators Similarity measure, the patient P1 suffers from Viral Fever.From

the Table (4.5), the similarity value of patient P1 is more or less the same for Typhoid and

Viral Fever. Also, the Viral Fever similarity value is marginally larger and hence the

diagnosis refers the Viral Fever.

In New Similarity measure, the patient P1 suffers fromViral fever. From the Table (4.3), the

similarity value of patient P1 is comparatively higher only for Viral fever.

VII. CONCLUSION

A new similarity measure of IFMS from IFS theory is extended. The prominent

characteristic of this method is that it performs well in case of two parameters (multi

membership and non-membership functions) and three parameters (multi membership, non-

membership and hesitation functions). This Similarity measure guarantees the best result in

the application of Medical Diagnosis problems.

REFERENCES

[1] Atanassov K., Intuitionistic fuzzy sets, Fuzzy Sets and System 20 (1986) 87-96.

[2] Atanassov K., More on Intuitionistic fuzzy sets,Fuzzy Sets and Systems 33 (1989)

37-46.

[3] Zadeh L. A., Fuzzy sets, Information and Control 8 (1965) 338-353.

[4] Chen S.M., Tan J.M., Handling multi-criteria fuzzy decision-making problems based

on vague sets.

Fuzzy sets and systems, 67 (1994), 163-172.


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[5] Szmidt E., Kacprzyk J., On measuring distances between Intuitionistic fuzzy sets,

Notes on IFS, Vol.3(1997) 1- 13.

[6] Szmidt E., Kacprzyk J., Distances between Intuitionistic fuzzy sets. Fuzzy Sets

System, 114 (2000) 505-518.

[7] Szmidt E., Kacprzyk J., Entropy for Intuitionistic fuzzy sets. Fuzzy Sets and

Systems,118 (2001) 467-477.

[8] SzmidtE., Kacprzyk J., A similarity measure for Intuitionistic fuzzy sets and its

application in supporting medical diagnostic reasoning. ICAISC 2004, Vol. LNAI 3070 (2004)

388-393.

[9] Xu Z. S., Xia M.M., Some new similarity measures for Intuitionistic fuzzy values

and their application in group decision making, Journal of System Science Engin, 19 (2010)

430-452.

[10] Li Y., Olson D.L., Qin Z., Similarity measures between Intuitionistic fuzzy (vague)

sets : A Comparative analysis, Pattern recognition Letters 28,(2007) 278-285.

[11] Zhang C., Fu H., The measure of similarity between the vague sets,Pattern recognition

Letters 27,(2007) 1307-1317.

[12] BinyaminYusoff, Imran Taib, Lazim Abdullah, Abd Fatah Wahab A new similarity

measure on Intuitionistic Fuzzy sets,World Academy of Science, Engineering and Technology

Issue 54,(2011) 36-40.

[13] Hung W.L., Yang M.S., Similarity measures of Intuitionistic fuzzy setbased on

metric, International Journal Approx Reason 46 : (2007) 120- 136..

[14] Ye J., Cosine Similarity measure for intuitionistic fuzzy sets and their application.

Math Computational Modelling (2011) 53 : 91-97.

[15] Farhadinia B., An efficient similarity measure forIntuitionistic fuzzy sets Soft

Computing (2013).

[16] Blizard W. D., Multi set Theory, Notre Dame Journal of Formal Logic, Vol. 30, No. 1

(1989)36-66.


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[17] Yager R. R., On the theory of bags,(Multi sets), Int. Jou. Of General System, 13

(1986) 23-37.

[18] Shinoj T.K., Sunil Jacob John , Intuitionistic Fuzzy Multi sets and its Application in

Medical Diagnosis, World Academy of Science, Engineering and Technology, Vol. 61

(2012).

[19] Rajarajeswari P., Uma N., On Distance and Similarity Measures of Intuitionistic

Fuzzy Multi Set, IOSR Journal of Mathematics (IOSR-JM),Vol. 5, Issue 4 (Jan. - Feb. 2013)

PP 19-23.

[20] Rajarajeswari P., Uma N.,Hausdroff Similarity measures for Intuitionistic Fuzzy

Multi Sets and Its Application in Medical diagnosis International Journal of Mathematical

Archive-4(9) (2013) 106-111.

[21] Rajarajeswari P., Uma N., A Study of Normalized Geometric and Normalized

Hamming Distance Measures in Intuitionistic Fuzzy Multi Sets, International Journal of

Science and Research (IJSR), Vol. 2, Issue 11, November 2013, 76-80.

[22] Rajarajeswari P., Uma N., Intuitionistic Fuzzy Multi Similarity Measure Based on

Cotangent Function, International Journal of Engineering Research & Technology (IJERT)

Vol. 2 Issue 11, (Nov- 2013) 1323–1329.

[23] Rajarajeswari P., Uma N., Normalized Hamming Similarity Measure for

Intuitionistic Fuzzy Multi Sets and Its Application in Medical diagnosis, International

Journal of Mathematical Trends and Technique (IJMTT), Vol. 5, Jan 2014.

[24] Rajarajeswari P., Uma N., Correlation Measure for Intuitionistic Fuzzy Multi Sets ,

International Journal of Research in Engineering and Technology (IJRET), Vol. 5, Jan 2014.

[25] Rajarajeswari P., Uma N., Intuitionistic Fuzzy Multi Similarity Measure Based on

Cosine Function, International Journal of Research in Information Technology (IJRIT), Vol.

5, Mar 2014.


870

[26] Rajarajeswari P., Uma N., A Similarity Measure Based on Min and Max operators

for Intuitionistic Fuzzy Multi Sets,International Journal of Innovative Research Development

(IJIRD), Vol. 5, Issue -3, (May 2014) 12309-12317.

[27] M.Kaviyarasu, K.Indhira , V.M.Chandrasekaran ., Fuzzy Sub algebras and Fuzzy K-

ideals in INK-Algebras, Inter national Journal of Pure and Applied Mathematics( IJPAM),

Vol 113 , Issue 6 (2017), 47 – 55.

[28] Rajarajeswari P., Uma N., The Zhang and Fu’s Similarity Measure on Intuitionistic

Fuzzy Multi Sets, International Journal of Innovative Research in Science, Engineering and

Technology (IJIRSET), Vol. 3, Issue- 5, (May 2014) 754-759.


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