Fuzzy Decision Making in Medical Diagnosis Using an Advanced Distance Measure on Intuitionistic Fuzzy Sets

Received: June 20, 2018 Revised: August 8, 2018 Accepted: August 15, 2018 Abstract: Background: Decision making in medical diagnosis is becoming a vast research field in medical science. In the last few decades, the role of distance measure in decision science is significant. Various distance measures on IFSs have been developed by many researchers and used in medical diagnosis in last few decades.


INTRODUCTION
Initially, Zadeh in 1965, introduced the concept of Fuzzy Set Theory (FST) [1].Later, in 1986, Atanassov developed the system of IFS [2] as an extension of classic fuzzy set theory.In fuzzy set theory, an element in the universal set is assigned to a membership degree.But, in IFSs theory, an element in the universal set is assigned to each element a membership degree, a non-membership degree and a hesitation degree.This is one of the main reason why IFS has been treated as a more effective and efficient concept than the FST to deal with the uncertainty in decision making.In recent years, significant effort has been made by researchers to show efficiency of IFS in the uncertainty modelling problems and applicability in a wide range of areas, such as decision making, fuzzy optimization, pattern recognition, medical diagnosis etc.
The similarity measure is important tools that can be used in decision-making problem to deal with uncertainty through IFS theory.Various distance measures have been proposed by different researchers.It has been observed that different distance measure produces different values while measuring the distance degree between two IFSs.Also, sometimes existing distance measures are not able to give an appropriate and convenient result for a pair of IFSs.For this reason, it is always necessary to derive advanced measures for better decision making.
Several similarity measures have been introduced on IFS by many researchers.Li et al. in 2002 [3], Liang et al. in 2003 [4], Li et al. 2007 [5], Xu in 2007 [6] and in 2008 [7] developed some similarity measures on IFSs.Li et al. introduced a new axiomatic definition of a similarity measure for IFSs.Later, Mitchell in 2003 developed a statistical approach as an extension of Li's similarity measure and introduced some examples [8].Hung and Yang in 2004 generalized the Hausdorff distance and proposed three new similarity measures on IFSs [9].Using the cosine function, Ye in 2011 proposed a new similarity measure and a weighted similarity measure [10].Hwang et al. in 2012, presented a new similarity measure by utilizing Sugeno integral and used in clustering analysis [11].Xu in 2007 developed some similarity measures and used it in multi-criteria decision-making problems [6].A general type of similarity measure [12] was developed by Boran and Akay in 2014 that connects two parameters norm and the level of uncertainty.An axiomatic approach of the similarity measure of IFS [13] was developed by Li et al. in 2012.Distance measures and similarity measures have some significance and importance in decision-making situations.Many researchers are also studied and developed several distance measures and used in various field of decisionmaking situations.Szmidt and Kacprzyk in 2000 developed the Hamming distance measure and Euclidean distance measure extending the original Hamming and Euclidean distance measure [14].Later, Grzegorzewski in 2004 developed the Hausdorff distance measure between two IFSs [15].Then, Szmidt and Kacprzyk, in 2004 proposed a distance measure for IFSs [16].Wang and Xin in 2005 developed a new distance measure [17] using the Hausdorff distance measure developed by Grzegorzewski.In a similar way, Park et al. in 2009 also developed a new distance measure [18] as an extension of distance measure developed by Wang and Xin.Then, Maheshwari and Srivastava in 2016 proposed a new distance measure [19] using the logarithmic function.Ngan et al., in 2018 proposed the HMaxdistance measure between two IFSs [20].

Motivation for this Study
There are several studies carried out to make a decision support system for medical diagnosis using fuzzy similarity and fuzzy distance measure.Though in literature, there are several numbers of distance measures, it is always necessary to have a sufficient number of distance measures to check and validate the results by applying different distance measures to obtain more reliable and convenient result.In addition, the results obtained by some earlier studies [21,20,16] are not relevant and contradicts with each other.Also, a recent study [20] shows more appropriate and relevant result which are different from the aforementioned studies.These are the main motivation for us to study in this particular topic.
In this paper, a new distance measure on Intuitionistic fuzzy set has been introduced and used in an existing data set of medical diagnosis studies to show the usability of the proposed distance measure.In the introduction section, the general view of uncertainty in medical diagnosis and the literature review on existing studies of medical diagnosis using IFS, similarity and distance measure has been given.In the preliminary section, the basic concepts and definitions of Intuitionistic fuzzy set theory are explained.The explanations and definitions of existing and well-known distance measures are given in section 3.In the next section, i.e. section 4, a new distance measure on Intuitionistic fuzzy sets has been proposed and proved that the proposed distance measure satisfies all necessary conditions.In section 5, a methodology has been discussed to determine the possible disease suffered by any patient using intuitionistic fuzzy set and the proposed distance measure.In section 6, a case study is carried out on existing data sets taken by some earlier researchers.In section 7, a discussion on the obtained result has been elaborated and compared with existing and recent studies and found that the proposed distance measure produces more relevant results.

PRELIMINARY
In this section, the basic concepts of fuzzy set, intuitionistic fuzzy sets, and intuitionistic fuzzy distance measures have been discussed.

Definition: Fuzzy Set [1]
A fuzzy set is one which assigns grades of membership between 0 and 1 to an object within its universe of discourse.If X is a universal set then a fuzzy set A is defined by its membership function.

Definition: Intuitionistic Fuzzy Set [2]
An Intuitionistic fuzzy set A on a universe of discourse X is defined as Where is called the "degree of membership" of x in A, is called the "degree of non-membership" of x in A, and 0 ≤ µ A (x) + µ A (x) ≤ 1.
Here, π A (x) = 1 -µ A (x) -v A (x) is called hesitancy of x. π A (x) can be considered as the degree of lack of commitment or uncertainty associated with the membership or non-membership in A.
In IFS, each element is assigned to a membership degree and a non-membership degree, whereas FST only assigns each element a membership degree.For this reason, IFS has been used more than the FST to deal with the uncertainty efficiently.

SOME WELL-KNOWN DISTANCE MEASURES
Let be two IFSs, then some well-known distance measures are given below: The normalized Hamming distance [14]: a.
  WX-Hmax distance measures of intuitionistic fuzzy sets is given by [20]: i.
The distance measure of two IFSs [16]: c.
The Hausdorff distance measure of two IFSs [15]: d.
The distance measure of two IFSs is proposed by [17]: e.
The distance measure of two IFSs is proposed by [18]: f.The divergence measure of two IFSs is proposed by [19]: g.
The H-max distance measures of intuitionistic fuzzy sets is given by [20]

A NOVEL DISTANCE MEASURE ON INTUITIONISTIC FUZZY SETS
Let X = {x 1 , x 2 ,....x n } be the universe of discourse.Let and be two IFSs.Then the new distance measure can be defined as: All the necessary four conditions to be a distance measure are satisfied by the new distance measure and the proof is given below: Proof (a): As we know, for degree of membership 0 ≤ µ A (x) ≤ 1 and degree of non-membership 0 ≤ v A (x) ≤ 1.

This implies for (1) (2)
Therefore, from Eq. (1) and Eq. ( 2) To prove the triangular inequality we have to use the following trigonometric inequality: Adding Eq.( 3) and Eq.( 4) we get    The relation between symptom and disease and relation between patient and symptom is given in Tables 1 and 2, respectively.

METHODOLOGY
Let us consider {D 1 , D 2 ,..., D m } be a set of number of possible diseases and {P 1 , P 2 ,..., P n } be a set of n number of patients.As stated earlier, symptoms for different diseases as well as symptoms of the patients are expressed in linguistic expressions most of the time.For this reason, uncertainty and fuzziness exist for the expression of the symptoms of the patients and disease.So, to express the symptoms, IFSs has been used.

Let
be the symptoms the disease D i and be the symptoms of patient P j expressed in IFSs.Therefore, the distance between the symptoms the disease D i and symptoms of patient P j can be evaluated as follows using the proposed distance measure: The distances between each pair of disease and patients can be represented with the help of the following matrix: Now, distance measures can be used to select the disease suffered by patient.Using the fact that, less distance between two IFSs implies more similarity between them, it can be said that for the patient P j the disease most possibly suffered by him or her is the disease corresponding to .

For, this
reason Hence, d (A, B) = 0 if and only if A = B. (c) d (A, B) = d (B, A) Proof (c): ) + sin (B) -sin (A + B) = sin A + sin B -sin A cos B -sin B cos A = sin A (1 -= cos B) + sin B (1 -= cos A) ≥ 0 Since sin A, sin B, 1 -cos A ≥ 0 for A, A ≤ .sin (A) + sin (B) -sin (A + B) ≥ 0 sin (A + B) ≤ sin (A) + sin (B) when A, B ≤ (d) If A, BandC be three intuitionistic fuzzy sets, then the distance measure satisfies the triangular inequality.Ie. d(A, C) ≤ d(A, B) + d(B, C).Proof (d): Consider, and be three intuitionistic fuzzy sets.Now, from the basic inequality of real numbers we know that: (A + B) ≤ sin (A) + sin (B) when A, B ≤ ; from above trigonometric inequality)Similarly, , ) ( , ) d A C d A B d B C d 6.A CASE STUDY IN MEDICAL DIAGNOSISIn this case study, the proposed distance measure has been applied in a decision-making problem of medical diagnosis.The data are taken from the study carried out byDe et al., in 2001 [21].In this case study, we consider a set of symptoms S, a set of diagnosis D and a set of patients P. Let P = {Arman, Bijit, Chandra, Deva}, S = {Temperature, Headache, Stomach pain, Cough, Chest pain} and D = {Viral fever, Malaria, Typhoid Stomach pain, Chest problem}.Our objective is to carry out the right decision of diagnosis for each patient, from the set of symptoms, for each disease.