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

Farhadinia, Bahram

doi:https://doi.org/10.1155/2022/7685033

Mathematical Problems in Engineering

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Applications of Fuzzy Sets and their Extensions in Engineering 2021

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Research Article | Open Access

Volume 2022 | Article ID 7685033 | https://doi.org/10.1155/2022/7685033

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

Bahram Farhadinia¹

Academic Editor: Ali Ahmadian

Received03 Dec 2021

Revised12 Jan 2022

Accepted18 Jan 2022

Published11 Feb 2022

Abstract

This study is concerned with introducing a class of parametric and symmetric divergence measures under hesitant fuzzy environment. The 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. There, we notify that the other existing divergence measures may not provide satisfactory results.

1. Introduction

These days, the COVID-19 pandemic is drastically impacting healthcare systems [1–3] 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 Internet-based 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.

The 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.

The 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. There 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. Then, 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. Therefore, 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)This contribution reviews and explores counter-intuitive cases of existing divergence measures under hesitant fuzzy environment(4)The experimental results demonstrate that the parametric hesitant fuzzy divergence measure is more effective than the existing ones in decision-making situations

This 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.

2. Preliminaries to Hesitant Fuzzy Sets (HFSs)

In this section, we review some basic notions and well-known results about HFSs that are used in the next discussion.

Let be the reference set. A hesitant fuzzy set (HFS) on is defined by Torra [21] in terms of a function that when it is applied to , it returns a subset of .

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 ofwhere stands for all possible membership degrees of belonging to the set , and it is afterwards named as the hesitant fuzzy element (HFE) of .

Adding to the latter presented concept are the following set and arithmetic operations. Let and be two HFEs. Then, it is defined (i.e., [23]).(i)Complement: (ii)Union: (iii)Intersection: (iv)Addition: (v)Multiplication: (vi)Multiplication by scalar: (vii)Power:

We explain below how the total ordering on HFEs was proposed and introduced. This was achieved by keeping in mind the score function of given by [22]and its variance function [24] is defined by the following formulation:

Indeed, the total ordering of HFEs and could be defined by using the following comparison scheme:(i)If we have , then it is concluded that (ii)If we have , then(i)For the relation , we get (ii)For , we conclude that

Now, we are in a position to explain the unified length scale of HFSs as follows: in most situations, we observe that . In order for comparing and correctly, we may extend the shorter HFE until the length of both HFEs are the same [25–28]. Suppose that . Then, the shorter HFE is extended by appending the same value repeatedly. The 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 and are two length-unified HFEs on . The elementwise ordering of HFEs is defined by (i.e., [29])for any .

Eventually, we represent the definition of two widely used aggregation operators of HFEs [25, 29]. Let , , and be a set of HFEs with the corresponding weights (). Then, it is defined.(i)The hesitant fuzzy weighted averaging (HFWA) operator:(ii)The hesitant fuzzy weighted geometric (HFWG) operator:

3. 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. The 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 and be two length-unified HFEs as described above.

The following formula introduces a class of innovative divergence measures for HFSs:where is a real convex function, and () are the positive and real numbers.

Among all the real convex functions, may be chosen as follows:(i) for (affine function)(ii) for (exponential function)(iii) for (power function)(iv) for (absolute-value function)(v) (logarithmic function)(vi) (combinatorial function)

To simplify the next discussion, hereafter, we only restrict by its power type for together with . In view of this, we attain the following class of divergence measures for HFSs:

Remark 1. It is interesting to note that the measure given by (8) will be the divergence measure of Mishra et al. [10] if we set .
Before presenting the main properties of divergence measure given by (8), we are going to state the following lemma.

Lemma 1. Assume that for any . Then, it holds that

Proof. To prove the left-hand inequality, we set and . Then, we easily find that together with . Now, from the fact that for any , we conclude that . This implies that , and therefore, .
To prove the right-hand inequality, we apply Jensen’s inequalityto with . Thus, we conclude that , which implies that .
Now, we establish the fundamental aim of this study which is given by the following theorem.

Theorem 1. Suppose that and are two length-unified HFEs. Then, the formula given by (8) presents a divergence measure for any .

Proof. It needs to show that the formula satisfies the two items given in the beginning of this section, that is, for any two HFEs and , it must be held thatThe proof of relation (11): from Lemma 1, we find that is true for any . Equivalently,holds true for any , and hence,The proof of relation (12): assume that which means that for any . Therefore,Conversely, we suppose that , that is,for any . This implies thatfor any .
The latter equality is possible if and only if the equalities () hold true. This finding implies that .

Theorem 2. Suppose that , , and are three length-unified HFEs, and the formula given by (8) presents a divergence measure. Then, for any , the following inequalities hold true:

Proof. Referring to the definition of elementwise ordering of HFEs given by (4), we observe that is valid if and only if for any . Therefore, for any , we haveand

Theorem 3. Suppose that , , and are three length-unified HFEs, and the formula given by (8) presents a divergence measure. Then, the following equalities hold true:

Proof. For any and , we propose the following proofs.
The proof of relation (21):In the case where , we easily conclude thatand hence,For the other case, that is, , we conclude again the latter result. Therefore, we haveThe proof of relation (22):In the case where , the following result is obtained:which gives rise toFor the other case, that is, , we achieve thatNow, it follows from (30) and (31) thatThe proof of relation (23): the justification of relation (23) is similar to that of relation (22).

Theorem 4. Suppose that , , and are three length-unified HFEs, and the formula given by (8) presents a divergence measure. Then, the following inequalities hold true:

Proof. For any and , we offer the following proofs.
The 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, thatClearly, the first case and the second case (i.e, equations (36) and (37)) give rise to , and moreover, the third case and the fourth case (i.e, equations (38) and (39)) result in . Therefore, by taking all the above results into consideration, we find thatThe proof of relation (34): it is proved in a similar way as the proof of inequality (33).

4. 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]. They used mainly the concept of divergence measure for developing the complex multiple criteria decision-making methodology. This work concentrates more on the methodology of Mishra et al. [10], in which a class of fruitful divergence measures for HFSs is employed instead. The resulted methodology is constituted by the following steps:

Step 1. We initially form each individual decision matrix corresponding to the evaluation of experts as follows:in which the HFS array (, ) indicates the rating of alternative corresponding to the criterion with the weight of .
By the way, the degree of significant for each expert is computed by the use ofwhere . Furthermore, it is easily seen that and .

Step 2. We aggregate all the individual decision matrices into the aggregated matrix:in whichfor any and .

4.1. The 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.

4.2. The Rationality Degree

(1)If we employ a divergence measure , we then find the support degree between the criteria and as(2)Using the support degree , we are able to calculate the total support degree, for any over the criteria .(3)The utilization of total support degree leads to the rationality degree in which .(4)By using the rationality degree , we attain the overall rationality degree where .

4.3. The Importance Degree

(1)We calculate the individual importance degree matrix as follows: where denotes the importance degree of criterion given by the ^th expert. Now, all the individual importance degree matrices can be aggregated into the matrix: in which for any .(2)The utilization of aggregated degree leads to the overall importance degree: where denotes the score function given by (2), and moreover, .

Now, with the parameters of overall rationality degree and overall importance degree given, respectively, by (51) and (55), it derives the subjective weights of criteria aswhere indicates the adjustment coefficient.

It is worthwhile to mention that the coefficient is chosen in accordance with the actual demand of decision maker, that is, the maximum value of stands for the superior influence of rationality degree of criteria in the assessment, and the minimum value of indicates the lesser influence of importance degree of criteria.

Step 3. We evaluate the ^th element of optimal significance rating by the help offor , which results in the optimal significance rating .

Step 4. We are able to normalize each array of aggregated hesitant fuzzy decision matrix by using the transformationwhere denotes the score function given by (2).

Step 5. We calculate the weighted normalized form of decision matrix asfor any .

Step 6. We obtain the overall performance rating in terms ofin which stands for the score function given by (2).
With the help of parameter , we can estimate the preference of options. That means that the greatest value of 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 byThe largest value of determines the desirable one.

5. Case Study of the COVID-19 Coronavirus

COVID-19 is the most recognized and thoroughly known virus by humans in the recent times. According to the World Health Organization report on November 29, 2020, more than 62,570,316 cases of COVID-19 across the world have been estimated which cause more than 1,466,426 deaths and 44,671,725 recovered persons [32]. Using data from the aforementioned report, most people with COVID-19 are associated with the symptoms and signs including fever (83%–99%), cough (59%–82%), fatigue (44%–70%), anorexia (40%–84%), shortness of breath (31%–40%), sputum production (28%–33%), and myalgias (11%–35%) [33, 34].

In this contribution, we have selected five medicines to manage the critical care of COVID-19 patients [35] including LPV/RTV-IFNb (), favipiravir (), LPV/RTV (), remdesivir (), and hydroxychloroquine ().

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 (), cough (), fatigue (), fever (), myalgia (), shortness of breath (), and sputum production () [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 for each expert () asMoreover, Table 2 provides the evaluation of five drugs performance in accordance with the seven criteria for each of three experts.

Step 8. The opinions of three experts are aggregated using formula (47), and this leads to results expressible in the form given in Table 3.

5.1. 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 and be two length-unified HFEs. Mishra et al. [19] and Mishra et al. [36] introduced, respectively, the exponential form of HFE divergence measures:

Before going more deeply into the definition of next existing divergence measures, we here point out that the latter measures cannot discriminate different HFEs correctly in some situations. This happens especially when a HFE contains elements with the conditionfor any and . In this situation,which are not logical and give rise to inconsistent and inaccurate outcomes.

Bearing the abovementioned shortcoming of and in mind, we only examine in detail the next exiting divergence measures together with the proposed ones in the current contribution.

Now, we are going to review the following HFE divergence measures, which were, respectively, introduced by Mishra et al. [10] and Mishra et al. [20]:and

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 , , and given, respectively, by (67), (68), and (8) into (48). Then, to save more space for convenient storage, we present only the combined criteria weights () 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 ofwhich is extracted from Table 3.

Step 10. Since all the criteria are cost-based criteria, therefore, we do not need to normalize them.

Step 11. The weighted normalized form of decision matrices corresponding to the divergence measures , , and are, respectively, given in Tables 6, 7, and 8.

Step 12. The preference orders for the drug options in accordance with the considered divergence measures , , and 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 ().
Although the final outcome of two existing divergence measures and is coincided with that of the proposed divergence measure , two major issues need to be addressed here:(i)The parametric divergence measure provides us with a class of divergence values (based on the parameter ) wider than that of the nonparametric divergence measure , which is contained in the former as a special case;(ii)Both divergence measures and are parametric, but the latter one is meaningless when , and this shortcoming is not going to be visible in the former one.

6. Conclusions

This contribution offers an ARAS framework being based on the HFS divergence measure for evaluating mainly the criteria weights. The 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

The 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

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

Conflicts of Interest

The author declares that there are no conflicts of interest.

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Copyright © 2022 Bahram Farhadinia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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