A New Divergence Measure for Intuitionistic Fuzzy Matrices

Distance / similarity measures facilitate in decision-making by identifying the discrimination or similarity between two or more course of actions to identify the best decision. Data available in the real world may not be in a crisp format. Intuitionistic fuzzy matrices are applicable managing uncertainty and useful in decision making, relational equation, clustering, etc. To take decision the divergence between two intuitionistic fuzzy matrices identifies dissimilarity between matrices. This paper presents a new divergence measure for intuitionistic fuzzy matrices with the verification of its validity. The fundamental properties are demonstrated for the new intuitionistic fuzzy divergence measure. A technique to solve multi-criteria decision-making problems is developed by utilizing the proposed intuitionistic fuzzy divergence measure. Finally, application in the medical diagnosis of this intuitionistic fuzzy divergence measure to decision making is shown using real data


Introduction
In real-world data involved in medical sciences, social sciences, engineering, and management are usually not all crisp, meticulous, and deterministic due to various uncertainties lying in the problem.Ambiguity may not be handled only dealing with fuzziness to reduce uncertainty of fuzzy values.Fuzzy sets use only crisp number for the membership value.However, in some situations the exact value of the function cannot be easily obtained.To tackle this issue, we should consider intuitionistic fuzzy data for linguistic evaluation for more reliable results than fuzzy data which allocates the degrees of membership and nonmembership and provide more degree of flexibility to the expert for expressing his judgment.In recent times, due to increasing complexity of socio-economic environment, intuitionistic fuzzy sets are more eminent than the fuzzy sets in maintaining the ambiguity and exaggeration of reallife problems.In information theory, while studying the set of objects we must differentiate or discriminate two objects on the basis of some parameters associated with them.Divergence measure is a significant tool for evaluating the amount of discrimination.Shannon (1948) defined entropy measure for probability distribution.Kullback and Leibler (1951) firstly investigated divergence measure which assumed probability distribution deviates from the original one found applications in many studies.Bhandari and Pal (1993) give a fuzzy divergence information measure for discrimination of a fuzzy set A relative to some other fuzzy set B which has found wide applications in many fields such as image processing, signal processing, Fuzzy clustering, and pattern recognition etc. Atanassov (1986) proposed a generalization of fuzzy set, an intuitionistic fuzzy set, is characterized by two functions expressing the degree of membership and nonmembership and a hesitation index.IFSs have widely implemented by the researchers to examine problems with uncertainties.Intuitionistic fuzzy measures are more appropriate in decision making such as in medical diagnosis (Wei et al. (2011) and Wu and Zhang (2011)), engineering, speech recognition, pattern recognition (Li and Cheng, (2002); Mitchell (2003); Vlachos and Sergiadis (2007); Wei and Ye (2010)) and in many more areas.Wei and Ye (2010) proposed an improved version of Vlachos and Sergiadis (2007) intuitionistic fuzzy divergence and applied in pattern recognition.Later, various research scholars have paid attention on divergence measure for IFSs (Li and Cheng (2002); Vlachos and Sergiadis (2007); Verma and Maheshwari (2017)).Yue and Jia (2017) investigated a GDM method based on projection measurement for IFSs to solve complex decision-making problem.He et al. (2020) proposed distance measures on IFSs based on IF dissimilarity functions and successfully applied on pattern recognition.Taruna et al. (2021) studied fuzzy distance measure and applied successfully in medical diagnosis.Taruna et al. (2021) proposed a new parametric generalized exponential entropy measure on Intuitionistic vague sets.Arora and Naithani (2021) proposed logarithmic entropy measures under PFSs and demonstrate application proposed measures in detecting disease related to Post Covid-19 implications through TOPSIS method.Mishra et al. (2018) provides me direction to propose this work, they proposed an intuitionistic fuzzy divergence measure based on ELECTRE method for the performance of cellular mobile telephone service providers.The concept of intuitionistic fuzzy matrix was studied by Pal et al. (2002).Intuitionistic fuzzy matrices (IFM) generalize the fuzzy matrix presented by Thomson (2005) and has been more applicable in multi-criteria decision, bioinformatics, aircraft control and many related fields.It also helps in generalization of various results on fuzzy matrices and can be used in discussion in intuitionistic fuzzy relations.IFM can be used in linear intuitionistic fuzzy transformations whereas it can be helpful in nonlinear transformations.It is noticeable that the strength of a measure lies in its properties.The proposed measure has elegant properties proved in the paper, to enhance the applicability of this measure.Inspired by the above-mentioned work, we propose a divergence measure for intuitionistic fuzzy matrices for applying in multi criteria decision making in every field of real world like in medical, engineering, business where decision have to be taken on the basis of some criteria.The aim of the measure is to evaluate the optimal alternative under the set of the different ones.The remainder of the paper is organized as follows.Section 2 is devoted to introducing some well-known concepts, and notions related to fuzzy set theory, intuitionistic fuzzy set theory and intuitionistic fuzzy matrix theory.In section 3, we proposed a new intuitionistic fuzzy divergence measure for intuitionistic fuzzy matrix corresponding to Mishra et al. (2018).Section 4 provides more elegant properties of the proposed measure.It is followed by the applications of the proposed intuitionistic fuzzy divergence measure for the IFM to medical diagnosis and a method of multi-criteria decision making in section 5. Finally, some concluding remarks are drawn in section 6.

Preliminaries
This section is devoted to introducing some well-known concepts and the notions of fuzzy set theory and intuitionistic fuzzy set theory.Then, we recall the concepts and the notions related to fuzzy set theory and the axiomatic definition of intuitionistic fuzzy divergence measure.

Fuzzy set
The linguistic values of the alternative's assessment are usually symbolized by fuzzy sets for dealing with fuzziness of real -world problem.Fuzzy sets are the sets whose elements have a degree of membership.Zadeh (1965) acquainted the fuzzy sets as the extension of the classical notion of sets i.e. crisp sets.

Intuitionistic fuzzy set
In some situation, fuzzy sets are not efficient for linguistic evaluation.Atanassov (1986) introduced generalization of fuzzy sets which can be applied in such situation.Intuitionistic fuzzy sets (IFSs) allocate degrees of membership and non-membership which give more degrees of flexibility to the decision expert for express her/ his evaluation.They can be implemented to model situations in which fuzzy sets do not provide all the available information.
Definition (Atanassov, 1986) Let Ü = {  1 ,  2 , … ,   } be a discourse set, then an IFS Ġ on Ü is defined asĠ show the membership degree and non-membership degree of μ i to Ġ in Ü, respectively, with the condition 0 Here, the intuitionistic index (hesitancy degree) of an element

Intuitionistic fuzzy matrix theory
The intuitionistic fuzzy matrix theory is very simple and easily applicable in circumstances where agreeness is not sufficient we have to consider disagreeness for the same.Example 2.3.1:Suppose a person wants to buy a mobile phone.He selected three companies to purchase a mobile phone, but he wants only one mobile phone of any of the following (Samsung, Mi, Vivo) three companies which suited him best according to features provided by manufacturer.There are three features (cost, camera, storage capacity) on the basis he can select best alternative for him out of those he has chosen.He assigns membership and non-membership values as follows:  = [ (0.1,0.5) (0.3,0.7) (0.4,0.6) (0.7,0.2 (0.4,02) (0.5,0.3) (0.2,0.6) (0.1,0.7) (0.9,0.0) ] is3 × 3 intuitionistic fuzzy matrix.

2.1.4
Boolean intuitionistic fuzzy matrix Boolean intuitionistic fuzzy matrix of order  ×  if all the elements of  are either 0  1 maintaining the condition 0 ≤   +   ≤ 1.For example, is2 × 3 boolean intuitionistic fuzzy matrix.
An intuitionistic fuzzy matrix is triangular if it is either intuitionistic fuzzy lower or intuitionistic fuzzy upper triangular matrix.

Definition:
[ÏƑ(M)] × ,where  = (  ,   ),then the elements  11 ,  22 , … ,   are the diagonal elements and which they lie along the line is called the principal diagonal of the intuitionistic fuzzy matrix.

Operations on two intuitionistic fuzzy matrices
Here we performed some operation on Intuitionistic fuzzy matrices.Let us define two intuitionistic fuzzy matrices  and  of order 3 × 3 as

Addition and subtraction of two fuzzy matrices
The addition of two intuitionistic fuzzy matrices is possible only when they have same order.Let  = Then we define addition and subtraction of intuitionistic fuzzy matrices of   as: +  = {max(  ,   ) , min(  ,   )} ∀ and .
The addition of above two matrices given by (2.4.1) and (2.4.2) is possible because they have same order and Clearly  +  is alsoan intuitionistic fuzzy matrix.However, addition of two standard IFMs is an intuitionistic fuzzy matrix.

Product of two fuzzy matrices
We need to define an operation corresponding to multiplication of two intuitionistic fuzzy matrices so that the product again happens to be an intuitionistic fuzzy matrix.The two types of operation which we have are max-min operation and min-max operation.We define multiplication of  and  as The product of two intuitionistic fuzzy matrices  and  given by (2.4.1) and (2.4.2) is possible because they are square matrices of same order.and so on.

Max-min
Here we find max-min and min-max products of membership and non-membership value of the matrices respectively to define the product of matrices  and  thus the product matrix  is Later, Fan and Xie (1999) developed exponential divergence measure for fuzzy set is as follows: Vlachos and Sergiadis (2007) introduced intuitionistic fuzzy divergence measure is as follows: ) (  (  ) +   (  ))) ) (  (  ) +   (  ))) )] ) (  (  ) +   (  ))

Divergence measure for intuitionistic fuzzy matrix
In this section, a divergence measure for intuitionistic fuzzy matrices  &  is proposed which is based on Vlachos and Sergiadis (2007).The validity of the proposed measure is also verified.
Here first we define a non-probabilistic divergence measure of intuitionistic fuzzy matrices: Here we proposed a divergence measure for intuitionistic fuzzy matrices  and  of order  ×  which is logarithmic in nature as follows:

Ɗ(A⋃B, A⋂B) = Ɗ( A⋃B | A⋂B ) + Ɗ(A⋂B|A⋃ B)
By using the definition of (1), first we have Finally, by adding these two equations, we get

≥0
This proves the theorem.

Application in medical diagnosis
The divergence measure for IFM can be utilized to quantify the significance of attribute in a specified organized task.
In the study, the proposed divergence measure is applied on a real data taken from a specialist (decision maker).
The case study consists of four patients having six symptoms and decision-maker provide information about possibility of disease patient may suffer in intuitionistic fuzzy form.He also provides us Intuitionistic Fuzzy values for symptoms corresponding to disease as well as about patient.
Our main motive is to find closeness of patient and disease based on symptoms.The application of above measure as follows.

Algorithm for application in multiplecriteria decision-making
IFM is a suitable tool for better demonstrating the imperfectly defined facts and data, as well as imprecise knowledge than fuzzy matrix (FM).In this section, we present a five-stage technique to solve a multiple criteria decision-making problem under an intuitionistic fuzzy environment.

Method:
Let  = (  1 ,  Step 4: Find the minimum value from the divergence which shows the best alternative out of other.

A case study
There are four patients of a Doctor Suraj, Raunak, Aditya and Akash in a clinic.They are suffering from a particular disease having some symptoms.The doctor needs to take decision which patient is suffering from which disease.Our proposed measure helps the doctor to take decision for the same.To solve this multiple criterion decisionmaking problem, we considered some symptoms corresponding to diseases and patients.
The set of symptoms is considered as: The tabular representation (Table 1) of these four sets is as follows: The Matrix Representation of Patient -Symptom is as follows: The Tabular Representation (Table 2) of Disease corresponding to Symptoms is: Now, we find the divergence between these two matrices (  ) or say patient-symptom matrix and diseasesymptom matrix by using the proposed measure eq.(4.1): Now, we find the divergence between  1 to each   by using proposed measure as: ( 1 :  1 ) = 3.539078, ( 1 :  2 ) = ., ( 1 :  3 ) = 3.577851, ( 1 :  4 ) = 3.037955, ( 1 :  5 ) = 2.701628, ( 1 :  6 ) = 2.887842 We find the smallest difference is 2.06004 corresponding to  2 .Thus, we conclude that patient one is suffering from disease  2 .Similarly, we find the divergence for each patient from the considered set of disease.Divergence for each patient from the considered set of disease are given in the following table 3representing minimum value of divergence which shows patient is suffering from the disease.Here, divergence value between pattern C and X is minimum.Therefore, we can say that Pattern C is more like X than others.

Conclusion
In this paper, we have recommended an adaptable technique for solving multi-criteria decision-making problem.Using the idea of Mishra et al. (2018) a new divergence measure for intuitionistic fuzzy matrix has developed.Proposed measure is a valid measure as it satisfies all the axioms.Some operations which may be applied on intuitionistic fuzzy matrices are defined.Its fundamental advantage is the capacity to combine the subjective information and attitude character of the decision maker in estimating the interaction of divergence degree.The proposed measure satisfies properties which are valuable in tacking any dynamic issue.The intuitionistic fuzzy divergence measure can be utilized in circumstances relying on the importance of decision makers.It can be utilized in situations where data should be in degree of membership and non-membership values.However, in many real-life situations, semantic factors are utilized to address subjective information.An application of multiple criteria decision-making in medical diagnosis is also demonstrated by taking information from a doctor.This divergence measure in medical diagnosis to take decision which disease based on symptoms the patient follows.We may select the best lternative in preference to other available alternatives in multi-criteria decision-making problem.Using the similar methodology different information measures such as entropy, inclusion, similarity etc. can be derived.Also, further hybridization of intuitionistic fuzzy with vague, soft and rough sets can be done to extend the applicability of the same.
The algorithms and algebra for intuitionistic fuzzy matrix theory are applicable for data related problems.Social scientists apply this approach to analyze interactions between attributes and to analyze other analytical tools.

Table 1 :
Intuitionistic fuzzy values for patients having Symptoms

Table 2 :
Intuitionistic fuzzy values for symptoms for disease

Table 3 :
Patient suffering from disease From table 3 we can conclude that Suraj is suffering from Chikungunya, Rounak is suffering from stomach infection, Aditya is suffering from respiratory problem, Radhika is suffering from respiratory problem.Divergence Measure are convenient tool to estimate dissimilarity between two probability functions and are therefore applied in various field.