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
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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
Viral Fever Tuberculosis Typhoid Throat disease
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 highest similarity measure from the table 4.3 gives the proper medical diagnosis.
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
Viral Fever Tuberculosis Typhoid Throat disease
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
The highest similarity measure from the table 4.3 gives the proper medical diagnosis.
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.
International Journal of Pure and Applied Mathematics Special Issue
868
[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.
International Journal of Pure and Applied Mathematics Special Issue
869
[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.
International Journal of Pure and Applied Mathematics Special Issue
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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