A New Similarity Measure of Intuitionistic Fuzzy Multi Sets in Medical Diagnosis Application

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1 Volume 119 No , ISSN: (on-line version) url: 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). umasnr@gmail.com 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 (. 859

2 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 nonmembership 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 nonmembership 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 860

3 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) 861

4 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 an IFMS, as is said to be the similarity measure between A and B, where A, B satisfies the following properties X and X 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. 862

5 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 ) = 863

6 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, nonmembership and hesitation function then the new IFMS Similarity measure becomes Where ) = ) = and ) = 864

7 PROPOSITION : 3.4 The defined Similarity measure properties between IFMS A and B satisfies the following 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 nonmembership 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 = { P 1, P 2, P 3, P 4 } 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. 865

8 TABLE : 4.1 IFMs Q : The Relation between Patient and Symptoms Q Temperature Cough Throat Pain Head Ache Body Pain P 1 P 2 P 3 (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) (0.4, 0.5) (0.7, 0.2) (0.6, 0.3) (0.3, 0.7) (0.8, 0.1) (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) (0.2, 0.6) (0.2, 0 ) (0.7, 0.1 ) (0.2, 0.7) (0.3, 0.7) (0.1, 0.9) (0.1, 0.7) (0.8, 0.1) (0.2, 0.6) (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 Viral Fever Tuberculosis Typhoid Throat disease measure P P P The highest similarity measure from the table 4.3 gives the proper medical diagnosis. 866

9 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 Viral Fever Tuberculosis Typhoid Throat disease Similarity measure P P P 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 Viral Fever Tuberculosis Typhoid Throat disease Similarity Measure P P P 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. 867

10 On comparison of the three tables (4.3, 4.4 and 4.5), the diagnosis of the patients P 2 and P 3 are the same. The diagnosis differs for patient P 1. In Zhang and Fu s Similarity measure, the patient P 1 suffers from Typhoid.From the Table (4.4), the similarity value of patient P 1 is comparatively higher only for Typhoid In MAX/MIN operators Similarity measure, the patient P 1 suffers from Viral Fever.From the Table (4.5), the similarity value of patient P 1 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 P 1 suffers fromviral fever. From the Table (4.3), the similarity value of patient P 1 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, nonmembership 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) [2] Atanassov K., More on Intuitionistic fuzzy sets,fuzzy Sets and Systems 33 (1989) [3] Zadeh L. A., Fuzzy sets, Information and Control 8 (1965) [4] Chen S.M., Tan J.M., Handling multi-criteria fuzzy decision-making problems based on vague sets. Fuzzy sets and systems, 67 (1994),

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