A Divergence-Based Medical Decision-Making Process of COVID-19 Diagnosis

(is study is concerned with introducing a class of parametric and symmetric divergence measures under hesitant fuzzy environment. (e proposed divergence measures have several interesting properties which make their use attractive. In order for exploring the features of proposed divergence measures for hesitant fuzzy sets (HFSs), we compare them with other existing ones in terms of divergence-initiated weighs and counter-intuitive cases. In the process of comparison, we first modify the conventional framework of hesitant fuzzy additive ratio assessment (HFARAS) using the proposed divergence measures, and then, the superiority of proposed measures is further demonstrated in a COVID-19 case study. (ere, we notify that the other existing divergence measures may not provide satisfactory results.


Introduction
ese days, the COVID-19 pandemic is drastically impacting healthcare systems [1][2][3] worldwide. To solve the problems of this pandemic, many medical scientists are focusing their research on that, and for recognizing and diminishing the COVID-19 effects, a large number of researchers have prepared a variant of workable models. Fouladi et al. [4] considered ResNet, OxfordNet, convolutional neural network, convolutional autoencoder neural network, and machine learning methods in order to classify chest CT images of COVID-19. Melin et al. [5] predicted successfully the consequence of COVID-19 time series by the help of a multiple collaborative convolutional neural network tool which is described thoroughly by encountering the concept of fuzzy set. Abdel-Basst et al. [6] merged two techniques, namely, the Best-Worst Method (BWM) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method to explore the association between COVID-19 and different viral chest diseases in uncertainty environment. By implementation of a model of Internetbased reporting, Bonilla-Aldana et al. [7] gathered the data on COVID-19 to increase its effectiveness during the pandemic. Ashraf and Abdullah [8] proposed a number of tools for dealing with the emergency condition of COVID-19 using the concept of spherical fuzzy set. In the period of COVID-19 pandemic in China, Wu et al. [9] introduced a technique to plan several emergency production procedures for producing a proper medical mask. Mishra et al. [10] enhanced the additive ratio assessment model by encountering the divergence measure to evaluate the medicine being used to treat those patients involving the mild symptoms of COVID- 19. e additive ratio assessment (ARAS) technique [11] implements the concept of optimality degree for extracting a ranking. In brief, this technique is described as a fraction of two values: the sum of normalized weighted values for criteria corresponding to each alternative and the sum of normalized weighted values for the best alternative. Indeed, the ARAS framework has intuitive procedures yielding relatively exact outcomes in the process of choosing diversified alternatives.
Up to now, there exist a large number of fuzzy-based contributions dedicated to the ARAS technique. Following, Zavadskas and Turskis [11] who first argued that a complicated phenomenon in the real world could be realized by the help of simple comparisons, Turskis and Zavadskas [12] tried to select the logistic center location based on the combination of AHP and ARAS for data in the form of fuzzy sets. In the sequel, Stanujkic [13] generalized the ARAS framework to that of interval-valued fuzzy sets. Büyüközkan and Göçer [14] developed the ARAS framework to that of interval-valued intuitionistic fuzzy sets for evaluating the digital supply chains. Büyüközkan and Göçer [15] assessed the digital maturity scores of the firms on the basis of hesitant fuzzy ARAS framework. Iordache et al. [16] suggested an interval type II hesitant fuzzy ARAS framework for choosing the location of underground hydrogen storage. Liao et al. [17] offered an ARAS framework encountering the hesitant fuzzy linguistic term data to choose a digital finance supplier selection. e technique of ASAS yields benefits which are association with criteria weights proportionally and straightly [11], scalability and flexibility [13,14], and adaptability to various fuzzy environments [14]. It also yields weaknesses which are behaviour dependency on the different levels of knowledge elicited by decision makers [14] and behaviour dependency on given data-type of participants [18].
Divergence measure is generally used to quantify the distance between two distributions by evaluating the amount of their discrimination. ere exist a set of diverse contributions which deal with the divergence applications in the context of research framework, especially the field of multiple criteria decision making.
In order to show the applicability of divergence measure under a hesitant fuzzy environment in which the criteria weights are to be computed in terms of the Shapley function, Mishra et al. [19] investigated the problem of service quality decision making. en, Mishra et al. [19] offered another exponential HFS divergence measure to assess the green supplier problem. Furthermore, Mishra et al. [10] developed an ARAS technique by encountering a divergence-based procedure for assessing rationally the relative importance of criteria. In the sequel, Mishra et al. [20] defined a parametric hesitant fuzzy-based divergence measure for evaluating the criteria weights.
In any way, the weight determination process of criteria has a remarkable impact on the decision outcomes, and the divergence measure is a factor which plays an important role in the determination of criteria weight. As shown in Section 5, the abovementioned HFS divergence measures are limited in nature. erefore, we have been in search of new divergence measures with fewer drawbacks.
In summary, the major distinctive features of the study are as follows: (1) It introduces an innovative class of divergence measures for HFSs which are parametric and symmetric (2) A number of interesting properties of proposed divergence measure are proved and discussed (3) is contribution reviews and explores counter-intuitive cases of existing divergence measures under hesitant fuzzy environment (4) e experimental results demonstrate that the parametric hesitant fuzzy divergence measure is more effective than the existing ones in decisionmaking situations is contribution is set up as follows. We first recall the concept of HFS, and a brief review of some preliminaries is given in Section 2. In Section 3, an innovative class of hesitant fuzzy divergence measures is introduced parametrically and symmetrically. We modify the existing framework of hesitant fuzzy ARAS (HFARAS) using the proposed divergence measures in Section 4. Section 5 is devoted to present the application of proposed divergence measures in a case study of COVID-19 coronavirus. Finally, several conclusions are drawn in Section 6.

Preliminaries to Hesitant Fuzzy Sets (HFSs)
In this section, we review some basic notions and wellknown results about HFSs that are used in the next discussion.
Let X be the reference set. A hesitant fuzzy set (HFS) on X is defined by Torra [21] in terms of a function that when it is applied to X, it returns a subset of [0, 1].
In fact, the notion of HFS is employed for handling a class of decision-making problems where the belongingness degree of an element to a set includes a variety of values.
Toward a better understanding, Xia and Xu [22] reconsidered the concept of HFS in the form of where h A (x) stands for all possible membership degrees of x ∈ X belonging to the set H A , and it is afterwards named as the hesitant fuzzy element (HFE) of H A . Adding to the latter presented concept are the following set and arithmetic operations.
(vi) Multiplication by scalar: We explain below how the total ordering on HFEs was proposed and introduced. is was achieved by keeping in mind the score function of h � h δ(i) | i � 1, . . . , l h given by [22] and its variance function [24] is defined by the following formulation: Indeed, the total ordering of HFEs .., l h 2 could be defined by using the following comparison scheme: Now, we are in a position to explain the unified length scale of HFSs as follows: in most situations, we observe that l h 1 ≠ l h 2 . In order for comparing h 1 and h 2 correctly, we may extend the shorter HFE until the length of both HFEs are the same [25][26][27][28]. Suppose that l � max l h 1 , l h 2 .
en, the shorter HFE is extended by appending the same value repeatedly. e repeated value depends on the risk preference of the decision makers, that is, if we consider (i) the pessimistic case, then the repeated value is the shortest one; (ii) if the optimistic case is considered, then the largest value will be repeated, and (iii) in the general case, we consider the convex combination of maximum and minimum values of a HFE.
Suppose that .., l be a set of m HFEs with the corresponding weights ϖ i (i � 1, . . . , m). en, it is defined.

A New Class of HFS Divergence Measures
Axiomatically, a divergence measure of two HFSs satisfies the following items similar to that of fuzzy sets [30] and intuitionistic fuzzy sets [31]: (i) It is nonnegative and symmetric (ii) It returns the zero value whenever the two sets coincide In this contribution, we are going to develop a procedure that estimates the objective weights of criteria by using the concept of divergence measure, needless to say that the criteria weights are computed subjectively or objectively. e former technique computes the criteria weights by taking the thought of decision makers, while the latter technique characterizes the criteria weights by considering the mathematical assessments. Let .., l be two length-unified HFEs as described above. e following formula introduces a class of innovative divergence measures for HFSs: where Γ is a real convex function, and c k (k � 1, 2) are the positive and real numbers. Among all the real convex functions, Γ may be chosen as follows: To simplify the next discussion, hereafter, we only restrict Γ by its power type Γ(h) � h p for p ≥ 1 together with c 1 � c 2 � 1. In view of this, we attain the following class of divergence measures for HFSs: Remark 1. It is interesting to note that the measure Div Γ given by (8) will be the divergence measure of Mishra et al. [10] if we set p � 2. Before presenting the main properties of divergence measure Div Γ given by (8), we are going to state the following lemma.
Proof. To prove the left-hand inequality, we set H: � n j�1 h j and H j : � h j /H. en, we easily find that To prove the right-hand inequality, we apply Jensen's inequality to Γ(x) � x p with c 1 � · · · � c n � 1/n. us, we conclude that (1/n n j�1 h j ) p ≤ n j�1 1/n(h j ) p , which implies that ( n j�1 h j ) p ≤ n p− 1 n j�1 (h j ) p . Now, we establish the fundamental aim of this study which is given by the following theorem.
Proof. It needs to show that the formula Div Γ (h 1 , h 2 ) satisfies the two items given in the beginning of this section, that is, for any two HFEs e proof of relation (11): from Lemma 1, we find that holds true for any j � 1, . . . , l, and hence, e proof of relation (12): assume that h: Conversely, we suppose that Div Γ (h 1 , h 2 ) � 0, that is, for any p ≥ 1. is implies that for any j � 1, . . . , l.
e latter equality is possible if and only if the equalities h .., l are three length-unified HFEs, and the formula Div Γ (h 1 , h 2 ) given by (8) presents a divergence measure. en, for any h 1 ≺ E h 2 ≺ E h 3 , the following inequalities hold true: Proof. Referring to the definition of elementwise ordering of HFEs given by (4), we observe that for any j � 1, . . . , l. erefore, for any p ≥ 1, we have and .., l are three length-unified HFEs, and the formula Div Γ (h 1 , h 2 ) given by (8) presents a divergence measure. en, the following equalities hold true: Proof. For any j � 1, . . . , l and p ≥ 1, we propose the following proofs. e proof of relation (21): In the case where h and hence, For the other case, that is, h 2 , we conclude again the latter result. erefore, we have e proof of relation (22): Mathematical Problems in Engineering In the case where h 2 , the following result is obtained: which gives rise to For the other case, that is, h Now, it follows from (30) and (31) that e proof of relation (23): the justification of relation (23) is similar to that of relation (22).
| j � 1, ..., l are three length-unified HFEs, and the formula Div Γ (h 1 , h 2 ) given by (8) presents a divergence measure. en, the following inequalities hold true: Proof. For any j � 1, . . . , l and p ≥ 1, we offer the following proofs. e proof of relation (33): Accordingly, all the possible cases are as follows: From the first case to the last case which are labelled by (36)-(41), we conclude, respectively, that 6 Mathematical Problems in Engineering Clearly, the first case and the second case (i.e, equations (36) and (37)) give rise to Div Γ (h 3 , h 3 ) ≤ Div Γ (h 1 , h 2 ), and moreover, the third case and the fourth case (i.e, equations (38) and erefore, by taking all the above results into consideration, we find that

Hesitant Fuzzy Additive Ratio Assessment (HFARAS)
In this part of contribution, we modify the framework of hesitant fuzzy additive ratio assessment (HFARAS) which was initiated by Mishra et al. [10]. ey used mainly the concept of divergence measure for developing the complex multiple criteria decision-making methodology. is work concentrates more on the methodology of Mishra et al. [10], in which a class of fruitful divergence measures for HFSs is employed instead. e resulted methodology is constituted by the following steps: Step 1. We initially form each individual decision matrix corresponding to the evaluation of experts ϵ k (k � 1, . . . , r) as follows: in which the HFS array ϵ k h ij (i � 1, . . . , m, j � 1, . . . , n) indicates the rating of alternative A i corresponding to the criterion C j with the weight of ω j . By the way, the degree of significant for each expert ϵ k (k � 1, . . . , r) is computed by the use of ij | t � 1, . . . , l . Furthermore, it is easily seen that ϖ ϵ k ≥ 0 and r k�1 ϖ ϵ k � 1.
Step 2. We aggregate all the individual decision matrices into the aggregated matrix: in which Mathematical Problems in Engineering for any i � 1, . . . , m and j � 1, . . . , n.

e Intermediate
Steps. Now, if we use the intermediate steps, then the weight of criteria is to be computed based on the two parameters: rationality degree and importance degree.

e Rationality Degree
(1) If we employ a divergence measure Div, we then find the support degree between the criteria C j and C l as (2) Using the support degree S, we are able to calculate the total support degree, for any h ij over the criteria C j . (3) e utilization of total support degree TS leads to the rationality degree in which 0 ≤ R j ≤ 1. (4) By using the rationality degree R j , we attain the overall rationality degree where 0 ≤ OR j ≤ 1.

e Importance Degree
(1) We calculate the individual importance degree matrix ϵ k I as follows: where ϵ k I j denotes the importance degree of criterion C j given by the k th expert. Now, all the individual importance degree matrices can be aggregated into the matrix: in which for any j � 1, . . . , n. (2) e utilization of aggregated degree I j leads to the overall importance degree: where s denotes the score function given by (2), and moreover, 0 ≤ OI j ≤ 1. Now, with the parameters of overall rationality degree OR j and overall importance degree OI j given, respectively, by (51) and (55), it derives the subjective weights of criteria as where 0 ≤ θ ≤ 1 indicates the adjustment coefficient. It is worthwhile to mention that the coefficient θ is chosen in accordance with the actual demand of decision maker, that is, the maximum value of θ stands for the superior influence of rationality degree of criteria in the assessment, and the minimum value of θ indicates the lesser influence of importance degree of criteria.
Step 3. We evaluate the j th element of optimal significance rating by the help of for j � 1, . . . , n, which results in the optimal significance rating h O � n j�1 h j O .
Step 4. We are able to normalize each array of aggregated hesitant fuzzy decision matrix by using the transformation 8

Mathematical Problems in Engineering
where s denotes the score function given by (2).
Step 5. We calculate the weighted normalized form of decision matrix as for any i � 1, . . . , m.
Step 6. We obtain the overall performance rating in terms of in which s stands for the score function given by (2). With the help of parameter OP i , we can estimate the preference of options. at means that the greatest value of OP i specifies the best option, and its lowest value characterizes the worst one.
However, besides the above selection option, we may assess the optimal option in accordance with the relative impact of that option being called the utility degree and evaluated by e largest value of U i determines the desirable one.
In this contribution, we have selected five medicines to manage the critical care of COVID-19 patients [35]  However, what is to be noted here is that the antiviral drugs should be considered not only for their impact on signs but also for their probable side effects and performance. To do this task, we have chosen the following parameters: anorexia (C 1 ), cough (C 2 ), fatigue (C 3 ), fever (C 4 ), myalgia (C 5 ), shortness of breath (C 6 ), and sputum production (C 7 ) [15,17,33]. In order to select an ideal drug, Mishra et al. [10] presented the assessment values in the form of linguistic variables together with hesitant fuzzy preference degrees as those given in Table 1.
Now, if we employ the algorithm of hesitant fuzzy additive ratio assessment (HFARAS) presented thoroughly in Section 4 to the abovementioned problem, then each step of algorithm can be carried out as follows.
Step 7. On the basis of data given in Table 1 and relation (45), we get the degree of significant ϖ ϵ k for each expert ϵ k (k � 1, 2, 3) as Moreover, Table 2 provides the evaluation of five drugs performance in accordance with the seven criteria for each of three experts.
e opinions of three experts are aggregated using formula (47), and this leads to results expressible in the form given in Table 3.

Comparison of Divergence-Initiated Weights.
In this part of Section 5, we are interested to perform a comparison between the weight values concluded from the proposed and the exiting divergence measures [19,36] to demonstrate more capabilities of the proposed ones. Let .., l be two length-unified HFEs. Mishra et al. [19] and Mishra et al. [36] introduced, respectively, the exponential form of HFE divergence measures:

Mathematical Problems in Engineering
Let us return back again to the algorithm of HFARAS presented in Section 4, and we execute the steps from Step 8 to Step 12 of that framework.
In order to perform the intermediate steps and obtaining the rationality degree and the importance degree, we incorporate the divergence measures Div M3 , Div M4 , and Div Γ given, respectively, by (67), (68), and (8) into (48). en, to save more space for convenient storage, we present only the combined criteria weights ω j (j � 1, . . . , n) which are given by (56). All the results are, respectively, given in Table 4 and  Table 5, and they are correspondingly shown in Figures 1-3.
Step 9. It is needless to say that for the three different divergence-based processes, the relation (57) returns the optimal performance rating vector of drug options in the form of which is extracted from Table 3.
Step 10. Since all the criteria are cost-based criteria, therefore, we do not need to normalize them.  Figure 2: e combined criteria weights ω j corresponding to the divergence measure Div M4 . e sixth column is not preserved due to "Not-A-Number" value. Step 11.
e weighted normalized form of decision matrices corresponding to the divergence measures Div M3 , Div M4 , and Div Γ are, respectively, given in Tables 6, 7, and 8.
e preference orders for the drug options in accordance with the considered divergence measures Div M3 , Div M4 , and Div Γ are determined as those shown in Figures 4-6.
As can be observed, the preference order for the drug options corresponding to all three cases remains the same as follows: and the desirable drug option is remdesivir (A 4 ).
Although the final outcome of two existing divergence measures Div M3 and Div M4 is coincided with that of the proposed divergence measure Div Γ , two major issues need to be addressed here: (i) e parametric divergence measure Div Γ provides us with a class of divergence values (based on the parameter p ≥ 1) wider than that of the nonparametric divergence measure Div M3 , which is contained in the former as a special case; (ii) Both divergence measures Div Γ and Div M4 are parametric, but the latter one is meaningless when p � 2, and this shortcoming is not going to be visible in the former one.

Conclusions
is contribution offers an ARAS framework being based on the HFS divergence measure for evaluating mainly the criteria weights. e prominent role of HFS divergence measures are apparent in their parametrically and symmetrically properties.
Other main contributions of the present work are summarized as follows: (1) Investigating of several properties for the proposed divergence measures (2) Pointing out the counter-intuitive cases corresponding to the existing divergence measures versus the proposed ones (3) Illustrating the validity and more applicability of the proposed divergence-based decision-making methodology e direction of future work of this research may be focused on the other applications such as renewable energy technology selection, optimal selection of antiviral therapy for the mild symptoms of COVID-19, and other applications in the process of bid evaluation [37] and reverse logistics [38], [39].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.