key: cord-0057401-t4i4kq32
authors: Kadian, Ratika; Kumar, Satish
title: A new picture fuzzy divergence measure based on Jensen–Tsallis information measure and its application to multicriteria decision making
date: 2021-03-14
journal: Granul
DOI: 10.1007/s41066-021-00254-6
sha: 5431fa9597ea5c9644b9aa7617ca3c373d324966
doc_id: 57401
cord_uid: t4i4kq32

Picture Fuzzy Sets (PFSs) originated by Cuong and Kreinovich are more capable to capture uncertain, inconsistent and vague information in multi-criteria decision making. In this paper, we propose a new picture fuzzy divergence measure based on Jensen-Tsallis function between PFSs. Further, the concept has been extended from fuzzy sets to novel picture fuzzy divergence measure. Besides the validation of the proposed measure, some of its key properties with specific cases are additionally talked about. The performance of the proposed measure is compared with other existing measures in the literature. Some illustrative examples are provided in the context of novel rapacious COVID-19 and pattern recognition which demonstrate the adequacy and practicality of the proposed approach in solving real-life problems.

1 Introduction Zadeh (1965) introduced the idea of the fuzzy set to quantify the vague and uncertain information and applied them in decision making problems from different angles (Chen et al. 2012; Chen and Chen 2014; Kadian and Kumar 2020a) . Atanassov (1986) introduced a significant generalization of the fuzzy set known as intuitionistic fuzzy set (IFS). Fuzzy set theory is applied in various areas, but in real life, few circumstances happen that cannot be dealt with by a fuzzy set. The idea of the intuitionistic fuzzy set for a component is a membership degree (s), non-membership degree (m) and the third component 'hesitancy degree' (p) respectively, satisfying s þ m þ p ¼ 1, improved the capability of FSs to handle the uncertain information. The third factor namely 'hesitancy degree' (p) was introduced by Atanassov (1999) in the existing structure of FSs. Various researchers gave their contribution in this field such as Szmidt and Kacprzyk (2000) developed a similarity measure between IFSs based on the Hamming distance, Xu and Xia (2010) characterized the geometric distance and similarity measures of IFSs for group decision-making problems. Due to their flexibility and successful applications, various researchers have been started research work on Atanassov intuitionistic fuzzy theory. More interesting applications have been developed in various fields, for example, image processing, risk analysis, medical diagnosis (Son and Phong 2016; Srivastava and Maheshwari 2016) , decision making (Chen and Chang 2016; Zeng et al. 2019; Joshi 2020; Kadian and Kumar 2020b) .

The transmission of COVID-19 and different mediations additionally had an extraordinary contrary impact on the normal existences of individuals and the solid working of society. Numerous researchers contribute to prevent and control the crisis circumstance of COVID-19. Togaçar et al. (2020) introduced the deducting technique of COVID-19 using fuzzy color and stacking approaches, Tuite et al. (2020) propose the COVID-19 transmission and relief techniques based numerical model among Canada's population, Sohail and Nutini (2020) presented the novel numerical model of determining the time period of COVID-19.

Tackling ambiguous and unsure information in real-life circumstances have consistently been a trouble. A few methodologies have been investigated to address the unpredictability and uncertainty found in real-life measures, for example the theory of fuzzy set (FS) . Despite the fact that the IFSs find their applications in various fields, yet at the same time there are numerous circumstances where IFSs can not be applied. Again, this can be better understood by an example on voting where we face such sort of problem: yes, abstain, no and refusal. This limits the area of applications of IFSs. To conquer this circumstance, a new generalized fuzzy set namely picture fuzzy set (PFS) has been proposed by Cuong and Kreinovich (2014) . It is a generalization of fuzzy set and intuitionistic fuzzy set with the introduction of the positive ðsÞ, the negative ðmÞ, the neutral ðpÞ and the refusal degrees (n ¼ 1 À s À m À p) demonstrating different possibilities of an element to a given set. The main requirement is that the sum of the degrees must not exceed 1. The picture fuzzy set theory is utilized in many day to day life problems such as voting problems, clustering, decision making, fuzzy inference. Son and Thong (2017) presented a few novel fuzzy clustering algorithm depicting the advantages of utilizing PFSs, proposed picture fuzzy cross-entropy model for multiple attribute decision-making problems, acquaints various methods to multi-criteria group decision-making problems for picture fuzzy environment.

In this paper, we propose a new picture fuzzy divergence measure based on the generalization of Jensen-Tsallis function. Numerical examples illustrate that the proposed measure is reasonably measuring the degree of dissimilarity between PFSs.

The main objectives and motivation of this paper are:

• A new picture fuzzy divergence measure is proposed based on joint representation of Jensen-Tsallis measure and studied some of its properties. • A new picture fuzzy divergence measure is introduced to measure the fuzziness degree for PFSs. • The reliability and flexibility of the proposed measure are proved with the help of examples in the context of COVID-19 and pattern recognition. • The proposed divergence measure is applied to solve the MCDM problems under the picture fuzzy condition.

The remainder of the paper is designed as follows: Sect. 1 contains the work done by the earlier researchers in the field. In Sect. 2, we introduce a new generalization of Jensen-Tsallis divergence measure and its properties. In Sect. 3, a new picture fuzzy divergence measure based on Jensen-Tsallis entropy is proposed with basic definitions and few major properties. Along the way, Sect. 4 proposes the numerical examples to demonstrate the applicability and reliability of the proposed technique. Lastly, the conclusion and future scope are drawn in Sect. 5.

In the present section, we introduce the Jensen-Tsallis divergence along with their properties for the probabilistic view point.

. . .; e k Þ; e p ! 0; P k p¼1 e p ¼ 1g; k ! 2 be the complete probability distribution set. For some E 2 M k , Shannon entropy is

For some E 1 ; E 2 2 M k with respect to the coefficient weights Lin (1991) define the Jensen-Shannon divergence measure as

The quantity (2) is non negative and disappear if and only if E 1 ¼ E 2 and also a convex function for E 1 and E 2 . So, it is called Jensen difference arising out of the convex function due to negative of the Shannon entropy. Further, Tsallis (1988) introduced a generalization of Shannon entropy as

when a ! 1, then Eq. (3) recovers a Shannon entropy and is a concave function of E for a [ 0ð6 ¼ 1Þ. Now, we introduce a new concept called Jensen-Tsallis divergence based on Eq. (3) and also studied by Kumar and Joshi (2019) .

where d 1 and d 2 are the weight coefficients with d 1 ; d 2 ! 0 and d 1 þ d 2 ¼ 1. Particular Cases: Case 1. If a ! 1 , then Eq. (4) becomes

Granular Computing which becomes generalization of Jensen-Shannon J-divergence measure with coefficient weights d 1 ; d 2 , which is slightly different from Lin (1991) .

which becomes generalization of Jensen-Shannon J-divergence measure.

which is called Jensen-Tsallis J-divergence measure, which is studied by Burbea and Rao (1982) .

Definition 2.1 The Hessian matrix of a function Cðe p ; f p Þ of two variables is defined as

) UðE 1 ; E 2 Þ ! 0: 3. Now, we have to show the convexity of function C, where

Taking derivative partially w.r.t. e and f, we get oC oe ¼aðd 1 e þ d 2 f Þ aÀ1 d 1 À ad 1 e aÀ1 oC of ¼aðd 1 e þ d 2 f Þ aÀ1 d 2 À ad 2 f aÀ1 Put oC oe ¼ oC of ¼ 0 to obtain the critical point. Now, calculating the Hessian(C) from definition (2.1) at e ¼ f and utilizing d 1 þ d 2 ¼ 1, we get

which becomes a positive semi definite. This ensures that the function (4) is convex. h 3 Generalized Jensen-Tsallis picture fuzzy divergence measure

This section provides the definitions and concepts related to PFSs.

Definition 3.1 (Zadeh 1965) A fuzzy set in X (fixed set) is defined by

in which s L : X ! ½0; 1 is indicated a membership function in X and s L ðtÞ 2 ½0; 1. Next, we introduce a new generalized Jensen-Tsallis picture fuzzy divergence measure. First we take X ¼ ftg (single element). Then, for some L; M 2 PFSsðXÞ, for simplicity, we take L ¼ ðs L ; m L ; p L ; n L Þ and

L ; m L ; p L ; n L Þ and ðs M ; m M ; p M ; n M Þ might be regarded as two probability distributions. We define a dissimilarity measure on PFSs(X) corresponding to (2) as

where H a T ðÁÞ is Tsallis entropy for PFS ðÁÞ, a [ 0ð6 ¼ 1Þ,

So, the Eq. (15) can be written as

Now, we discuss a few properties of proposed measure introducing in Eq. (17).

It becomes a picture fuzzy dissimilarity measure studied by Lin (1991) .

It becomes a picture fuzzy dissimilarity measure for assigning the equal weights for each probability distribution.

It becomes a picture fuzzy dissimilarity measure corresponding to intuitionistic fuzzy dissimilarity measure studied by Hung and Yang (2006) .

which is based on Ginni index of diversity for the case of picture fuzzy sets.

where g l PFS ðL; MÞ was characterize as a distance between PFSs. Hence, we can write as

which becomes an intuitionistic fuzzy dissimilarity measure.

It becomes an intuitionistic fuzzy dissimilarity measure, which is studied by Hung and Yang (2006) .

which is based on Ginni index of diversity for the case of intuitionistic fuzzy sets.

where g l IFS ðL; MÞ was characterize as a distance between IFSs by Szmidt and Kacprzyk (2000) . Hence, we can write as

which becomes Jensen-Shannon fuzzy divergence measure corresponding to Eq. (7). Case 11. If hesitancy degree, i.e.,

which becomes a fuzzy divergence measure corresponding to Eq. (8).

Case 12. If hesitancy degree, i.e.,

which becomes a Jensen-Shannon J-divergence measure for fuzzy sets corresponding to Eq. (6). Case 13. If hesitancy degree, i.e., p L ¼ 0 and

Hence, we can write as

which is characterize as an Euclidean distance between two fuzzy sets (FSs).

Presently, the justification of Eq. (17) is establishing through a theorem. Definition 3.6 ðR a new ðL; MÞ for Finite Universal Set ): Now, we extend the concept of Jensen-Tsallis divergence measure from single element universe to finite universal set. For this, for coefficient weights d 1 ; d 2 , let X ¼ ft 1 ; t 2 ; . . .; t k g be a fixed set and for any L; M 2 PFSsðXÞ, corresponding to (17) 

Now, we give some properties of measure (31) through a theorem.

Theorem 3.2 For L; M; I 2 PFSsðXÞ, Proof Suppose X bifurcate into two parts X 1 and X 2 such that:

From (32), it is clear that for all t 2 X 1 , 

1. Proof of properties (1-2) is directly from the definition (31). h 5. Property 5 is proved as same as of the property 4. 6. Consider

7. Property 7 is proved as same as the property 6. 8. Consider 

þðd 1 p L þ d 2 p I Þ a þ ðd 1 n L þ d 2 n I Þ a Àðd 1 s a L þ d 2 s a I Þ À ðd 1 m a L þ d 2 m a I Þ Àd 1 p a L þ d 2 p a I Þ À ðd 1 n a L þ d 2 n a I Þ 11. Property 11 is proved directly from definition (31). 12. Property 12 is proved directly from definition (31). 13. Property 13 is proved directly from definition (31).

Under this section, we present the applications for an outbreak of COVID-19 and pattern recognition.

To exhibit the relevance and legitimacy of the proposed method, we apply it on an outbreak of novel coronavirus ailment .

Example 1 Now, in the circumstance of COVID-19, it is fundamental to give a productive route in crisis reaction for evading extra misfortunes and to save the lives of the people. Because of such a crisis choice, the wellbeing specialists need to make a quick reaction, desperately salvage to control the circumstance proficiently and prevent it from more deaths. There are eight fundamental public health emergency factors to diminish the overall danger of this ailment. The best preventive measures are: let us consider the situations Z={A 1 -Medical facilities, A 2 -research needs, A 3 -lock down the borders, A 4 -banned intracity transportation, A 5 -public awareness} and five emergency factor to reduce this risk D= {g 1 -clinical management (CM), g 2 -monitoring (MN), g 3 -country-level coordination (CLC), g 4 -consult experts (CE), g 5 -increase personal protective equipment (IPP)} with symptoms V={q 1 -shortness of breath, q 2 -chest pain, q 3 -loss of taste, q 4 -loss of speech, q 5 -conjunctivitis}. Every element in situations, symptoms and emergency factor is given as a picture fuzzy number. One need to find a proper preventive measure for every emergency situation in context of symptoms. The procedure is repeated for all components. At last, we recommend the measure for emergency situation whose symptoms have minimum degree of divergence measure. Let situations A 1 ; A 2 ; A 3 ; A 4 ; A 5 , w.r.t. all symptoms in the form of following PFSs: Table 1 , 2, 3, 2, 6.

Notations: g 1 -clinical management, g 2 -monitoring, g 3country level coordination, g 4 -consult experts, g 5 -increase personal protective equipment.

Abbreviations: In Table 7 , some abbreviations are used like: CM-clinical management, MN-monitoring, CLCcountry level coordination, CE-consult experts, IPP-increase personal protective equipment.

We conclude the Tables 1, 2, 3, 4, 5, 6 in Table 7 . In Table 7 , proposed picture fuzzy divergence measure (31) suggests that the factors CLC, IPP, CM, CLC, MN are the Results:

Bold indicates minimum value of each situation with respect to symptoms essential measures for the situations A 1 ; A 2 ; A 3 ; A 4 ; A 5 respectively for various values of parameters d 1 ; d 2 , according to the principle of the minimum degree of divergence measure between PFSs. Thus, we have come to the conclusion that according to our proposed measure, above mentioned factors are the main preventive measures that decrease the overall risk of this infection. The results of proposed measure which are obtained on the different value of d 1 ; d 2 are depicted by Fig. 1 . In Fig. 1 , different colors show the different factors as first light blue color presents country-level coordination (CLC), orange color presents increase personal protective equipment (IPP), gray color presents clinical management (CM), yellow color presents country level coordination (CLC) and blue color presents monitoring. From Fig. 1 , we can seen that the factors CLC, IPP, CM, CLC, MN are consistently occurring in a manner for various values of parameters d 1 ; d 2 . This shows the consistency of the proposed measure.

Under this subsection, we present the comparative study for demonstrating the usefulness of the proposed measure. Comparisons are taken among the picture fuzzy divergence measures suggested by: Hung and Yang (2006) Table 8 .

Comparative Analysis: From Table 8 , we can easily clear that the proposed picture fuzzy divergence measure gives predictable outcome according to the principal of minimum degree of divergence measure between PFSs, while other divergence measures gave no consistency regardless. They do not ready to perceive the two factors because of the uniformity of divergence measure values. Table 8 gives the results produced by different methods. Two methods are not able to show the clarity about the factor to overcome the situation A 1 and A 5 . For example-Nei et al. (2017) is not able to suggest for control the situation A 4 , i.e., not able to suggest which factor either clinical management (CM) or monitoring (MN) is suitable to control the situation A 4 . In the same way, Son (2017) is disabled to suggest for handling the situation A 1 , i.e., not able to suggest which preventive factor either clinical management (CM) or country-level coordination (CLC) is suitable to control the situation A 1 .

Further, the proposed measure generates the same results with Wang et al. (2017) , which demonstrate that the proposed measure is practical in coronavirus diagnosis. Therefore, our proposed method has strong discrimination and can provide more useful results as contrasted with other divergence measures. Results:

Bold indicates minimum value of each situation with respect to symptoms Results:

Bold indicates minimum value of each situation with respect to symptoms Results:

Bold indicates minimum value of each situation with respect to symptoms Our point is to characterize B is similar to one of the samples D 1 , D 2 ,..., D p . As indicated by the principle of minimum degree of divergence measure between picture fuzzy sets, the way of assigning B to D Ã l is described by

Example 2 Suppose there be three known patterns D 1 , D 2 , D 3 with classifications C 1 , C 2 , C 3 respectively and an unknown pattern B is also given. The patterns are characterized by picture fuzzy sets in X ¼ fz 1 ; z 2 ; z 3 ; z 4 ; z 5 g: Their values are displayed in Table 9 . Our objective is to recognize the known pattern which look likes the most with unknown pattern B.

Our target is to determine the sample B to one of the patterns D 1 ; D 2 ; D 3 , respectively.

Using the proposed algorithm, one can identify the pattern which closely resembles with the given pattern.

Step 1: The normalized distances for B from the patterns D 1 ; D 2 ; D 3 are measured by Eq. 31, which are presented in Table 10 .

Step 2: The minimum degree of distances between B and D p ; p ¼ 1; 2; 3 are obtained by Eq. 36. The minimum distances are darked in Table 10 .

Step 3: Using the principle of minimum degree of divergence between PFSs, the known sample which gives the minimum distance with B is determined.

We have calculated the values of R a new ðD 1 ; BÞ, R a new ðD 2 ; BÞ and R a new ðD 3 ; BÞ at different values of a; d 1 ; d 2 . It is preassigned or adjustable number related to individual systems. Let us take an example related to the problem of the environment. Different environmental features such as temperature, pressure and humidity may be denoted as distinct parametric values. Thus, with the existence of parameters, information measure becomes more preferable from application point of view. We may select any value of a within 0 and 1 to achieve the consistent and efficient proposed measure by employing the concept of minimum divergence measure. From Table 10 , it is clear that for any parametric value a, R a new ðD 2 ; BÞ is the minimum value among R a new ðD 1 ; BÞ, R a new ðD 2 ; BÞ, R a new ðD 3 ; BÞ. Hence, the sample B look likes the same with D 2 , i.e., B D 2 . Now, we prove the proficiency of the proposed picture fuzzy divergence measure in example 2 by comparing with other existing measures: Hung and Yang (2006) , Son and Thong (2017) Comparative Analysis: From Table 11 , obviously all divergence measures support the sample D 2 look likes the same with B. The results coincide with the proposed measure (3.17). In Fig. 2 , blue color presents the distance Results:

Bold indicates minimum value of each situation with respect to symptoms between D 1 and B, red color presents the distance between D 2 and B and green color presents the distance between D 3 and B. Also, Fig. 2 reveals that all existed picture fuzzy divergence measures support the sample D 2 according to a minimum degree of divergence between two picture fuzzy sets. Therefore, with the presence of parameter proposed measure becomes more compatible and broaden its extent of implementation as compare with other methods. This reveals that execution of proposed measure is considerably superior.

The following advantages have been taken into account from the proposed pictur fuzzy divergence measure:

• As mentioned above, the PFS is one of the generalization of the classic set, fuzzy set, intuitionistic fuzzy set. Numerous circumstances occur which are not properly controlled in IFS. To overcome this situation, Picture fuzzy set theory is one of the more broad and can handle indeterminate information, which exists commonly in real-life situations. Hence, Picture fuzzy divergence measure is more suitable in real scientific applications. • Various interpretations to a; d 1 ; d 2 is given for consistency and efficiency of the proposed measure. The presence of the parameters in the proposed measure makes it progressively reliable and widens its extent of applications as contrasted with other measures. This shows that the performance of the proposed picture fuzzy divergence measure is significantly flexible. • The proposed picture fuzzy divergence measure is more generalized and appropriate to solve financial, medical and other multicriteria decision-making problems.

In this article, we have developed a new picture fuzzy divergence measure based on Jensen-Tsallis information measure. The idea has been extended of Tsallis entropy and Jensen inequality to present a new picture fuzzy divergence Table 9 Known and unknown pattern values Samples z 1 z 2 z 3 z 4 z 5 ðs z1 ; m z1 ; p z1 ; n z1 Þ ðs z2 ; m z2 ; p z2 ; n z2 Þ ðs z3 ; m z3 ; p z3 ; n z3 Þ ðs z4 ; m z4 ; p z4 ; n z4 Þ ðs z5 ; m z5 ; p z5 ; n z5 Þ Bold indicates minimum value of each situation with respect to symptoms measure. Furthermore, a few key properties and particular cases are also studied. The outcome of the proposed measure is compared to other existing methods in literature. To build up the sufficiency, flexibility of proposed measure, some counter examples are solved in the context of COVID-19 and pattern recognition. It is summarized that the developed method can provide extraordinary evaluation results because of the presence of the parameter. The main advantages of the proposed method are the computation simplicity for picture fuzzy sets.

To discuss the future prospects of the proposed research, it can be further extended to more general sets such as interval-valued PFSs, and complex-valued fuzzy sets. Apart from this, we can also apply the applications of proposed measure in other MCDM problems such as remote sensing, speech recognition, risk analysis, weather forecasting and so on.

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Acknowledgements The authors are thankful to the editor and the anonymous reviewers for their precious suggestions and comments which improved this manuscript and enhanced our knowledge.

Conflict of Interest The authors declare that they have no conflict of interest.