Ranking Effectiveness of COVID-19 Tests Using Fuzzy Bipolar Soft Expert Sets

The theory of fuzzy bipolar soft sets is an eﬃcient extension of soft sets for depicting the bipolarity of uncertain fuzzy soft information; however, it is limited to a single expert. The present research article introduces the theory of an innovative hybrid model called the fuzzy bipolar soft expert sets, as a natural extension of two existing models (including fuzzy soft expert sets and fuzzy bipolar soft sets). The proposed model is highly suitable for describing the bipolarity of fuzzy soft information having multiple expert opinions. Some fundamental properties of the developed hybrid model are discussed, including subset, complement, union, intersection, AND operation, and OR operation. The proposed concepts are explained with detailed examples. Moreover, to demonstrate the applicability of our initiated model, an application of the proposed hybrid model is presented along with the developed algorithm to tackle the real-world group decision-making situation, that is, ranking eﬀectiveness of tests in spread analysis of COVID-19. Finally, a comparative analysis of the developed model with some existing mathematical tools such as fuzzy soft expert sets and fuzzy bipolar soft sets is provided to show the cogency and reliability of the initiated model.


Introduction
e ranking and selection of alternatives (based on the preferences of decision-makers) is an important aspect of decision sciences, but the situations become difficult when dealing with vague data and uncertainties. e conception of the modern theory of probability in the sixteenth century led to the proposal of different mathematical tools and algorithms by many computer, logics, and mathematical experts in order to deal with uncertainty and fuzziness, whether considering social sciences or economics, medical sciences, or engineering. In 1965, Zadeh [1] initiated the idea of a fuzzy set model which is capable of handling partial truth between "absolute false" and "absolute truth." In such concepts of fuzzy sets, it is declared that an element of a universal set U can have some membership degree belonging to interval [0, 1] instead of the set 0, 1 { }, thus allowing to deal with situations considering "how much" an element satisfies a criteria instead of just declaring whether it satisfies or not. is powerful concept of fuzziness was later used by various scientific researchers from almost every scientific domain. One important limitation of this model is to deal with bipolar or dual behaviour situations involving a positive and a negative side, for instance, effect and side effect, good and bad, pros and cons. To remove this limitation, Zhang [2] presented Yin-Yang bipolar fuzzy sets which are capable of handling these bipolar situations. Yin and Yang are considered as negative and positive parts, respectively, of a system in Chinese medicine. e fuzzy set theory led to the development and proposal of extensions like intuitionistic fuzzy sets [3] and bipolar fuzzy sets [2]. However, these models fail to deal with uncertain situations involving different parameters. is limitation due to lack of parameterization tools in existing models led Molodtsov [4] to initiate the notion of soft sets, which is different from all other preexisting methods and allows to deal with situations having different parameters. e soft set theory has been used in several domains including medicine, engineering, and economics. Certain applications of such a model in decision-making were offered by Maji et al. [5]. Furthermore, Ali et al. [6] studied several properties of the soft sets. Later on, such a model was combined with existing uncertain models to develop the powerful hybrid models incorporating the characteristics of both the combined models such as fuzzy soft sets [7]. e soft set model is a powerful tool, but fails to deal with two-sided information in a soft environment having interrelated parameters, where one parameter affects the other one, such as "expensive and cheap," "old and young," and "dependent and independent." To tackle this situation, the idea of bipolar soft sets (BSSs) was proposed by Shabir and Naz [8]. e BSS model considers two opposite meaning sets of parameters named the "set of parameters" and the "not set of parameters." us, dealing with bipolar parameters much efficiently, Naz and Shabir [9] combined fuzzy set theory with BSSs to introduce the fuzzy BSSs and further discussed their decision-making applications. Akram and Ali [10] introduced Pythagorean fuzzy BSSs and rough Pythagorean fuzzy BSSs and discussed their applications. Afterwards, Akram et al. [11] introduced two new decision-making models, including m-polar fuzzy BSSs and rough m-polar fuzzy BSSs. A survey on hybrid soft set models was launched by Ma et al. [12]. Later, Mahmood [13] presented a novel decision-making method for BSSs and studied some of their practical applications.
Hybrid models proved to be an important tool in dealing with group decision-making problems. Many researchers across the globe constructed novel hybrid models that give more reliable outputs, when dealing with different sorts of information pieces in decision-making problems. In the last few decades, hybrid models have been developed to tackle numerous real-world multiattribute group decision-making (MAGDM) situations containing vague information. All the abovementioned models deal with only one expert and fail to consider the opinion of multiple experts in the same place and are not considered suitable in MAGDM situations having multiple expert opinions as in the case of studies utilizing questionnaires. Alkazellah and Salleh [14] resolved this issue by introducing the notion of soft expert sets (SESs), which considers all expert opinions dealing with a MAGDM situation. Later on, Alkazellah and Salleh [15] presented fuzzy soft expert sets (FSESs).
is powerful concept inspired many researchers to solve different MAGDM problems using the SES approach, as discussed in [16][17][18][19][20][21]. For instance, Broumi and Smarandache [22] presented an intuitionistic fuzzy SES model and discussed its applications. Hassan and Alhazaymeh [23] introduced vague SESs. Shabir and Gul [24] developed modified rough BSSs. Recently, Ali and Akram [25] introduced N-SESs and fuzzy N-SESs with their use in MAGDM situations. In addition, Akram et al. [26] launched a novel hybrid model called m-polar fuzzy SESs and solved certain MAGDM problems. For more important terminologies, the readers are referred to [27][28][29][30][31][32][33][34][35][36][37][38][39]. e motivations of this research article are described as follows: (1) e ability of BSSs and fuzzy BSSs is depicting bipolar information in a soft environment, which allows to deal with bipolar parameters more effectively, but not so efficient in group decisions. (2) SESs and fuzzy SESs appear as powerful tools in MAGDM problems allowing multiple expert opinions in the same model. Much effective model is possible, if it deals with bipolarities.
e major contributions of this study are described as follows: ( e research article is organized as follows. In Section 2, we present a new hybrid mathematical tool called FBSESs and discuss some of its fundamental properties including subset, complement, agree FBSES, disagree FBSES, AND operation, and OR operation. In Section 3, we discuss a real-world MAGDM problem concerning the emerging COVID-19 using the FBSES approach. We also provide an efficient algorithm for the developed model in this section. Section 4 provides a comparative analysis of the developed method with certain preexisting models, including fuzzy BSSs and fuzzy SESs. Finally, in Section 5, we provide some concluding remarks and future orientations.

Fuzzy Bipolar Soft Expert Sets
is section reviews some basic terminologies and provides the notion of FBSESs with necessary properties of these concepts along with detailed supporting examples.
Definition 1 (see [8]). Let U be a universe and V be the universe of parameters. For every B ⊆ V, a triplet (F, G, B) is said to be a bipolar soft set or BSS on U, where F and G are functions defined as follows: 2 Mathematical Problems in Engineering such that F(ϑ) ∩ G(ϑ) � ∅, ∀ ϑ ∈ B, ϑ ∈B. Here, P(U) represents the power set of U.
Notice that B(Not B) is the set containing attributes opposite to those contained in set B.
Definition 2 (see [15]). Let U be the universe, V the universe of parameters, and E the set of experts. Let � agree represents the set of opinions. en, for A ⊆ X, a pair (c, A) is said to be a fuzzy soft expert set over the universe U, where c is the function defined as follows: where F(U) serves as the power set of fuzzy subsets over U. Now, we are ready to define the fuzzy bipolar soft expert sets.
Definition 3. Let U be the universal set and V be the universe of parameters. Let E be the set of experts, O � 0 � disagree, 1 � agree the set of opinions, and X � V × E × O. en, the triplet (c, χ, A) is called a fuzzy bipolar soft expert set or FBSES on U, where c and χ are functions defined as follows: with A ⊆ X and A ⊆ X, such that 0 ≤ (c(α))(u) +(χ(α)) (u) ≤ 1 for all α ∈ A and u ∈ U.

Note 1. A(not A)
is the set containing attributes opposite to those contained in A.
Here, 0 ≤ (c(α))(u) + (χ(α))(u) ≤ 1 for all α ∈ A and u ∈ U acts as a condition for c and χ to be consistent with the definition of FBSESs. e functions c(α) and χ(α) are considered as the α-approximate and α-approximate elements of the FBSES (c, χ, A) for each α ∈ A.
Definition 4. Let (c, χ, A) be a FBSES over a universe U, then the hesitancy region of this FBSES is determined by a fuzzy soft expert set (H, A), such that for all α ∈ A and each u ∈ U.
Here, (H, A) represents the grey area or the uncertainty in making the decision, thus indicating the lack of knowledge of the expert in the case of a particular object with respect to a parameter under consideration. Moreover, c(α))(u) + χ(α))(u) + H(α))(u) � 1 for all α ∈ A, α ∈A, and for all u ∈ U.
Example 1. Consider that 5 patients are monitored in a mental health facility for the diagnosis of their mental state by a group of 3 psychiatrists. Let U � u 1 , u 2 , . . . , u 5 be the set of patients.
� very quiet, tearfulness or crying diffidence be the set of manic behaviours, whereas V � ϑ 1 , ϑ 2 , ϑ 3 � very quiet, tearfulness or crying diffidence be the set of respective depressive behaviours. Let E � x, y, z be the set of psychiatrists.
After monitoring the patients for a fixed period of time, the psychiatrists make data on the basis of their observations and we get the FBSES (c, χ, X) as follows: is FBSES shows the observations of psychiatrists during the diagnosis. For example, psychiatrist x declares "u 1 " to be 10% talkative, 80% quiet, and 10% normal (grey Mathematical Problems in Engineering area) during the diagnosis, whereas y declares "u 1 " to be 10% talkative, 85% quiet, while acting normal 5% of the time. On the basis of the collected data, patients having high membership degrees in manic behaviours are declared as hypomania patients, those with high membership degrees in depressive behaviours are declared to be under depression, whereas the patients with almost equal degrees of membership in both manic and depressive conditions (above 30%-35%) are declared to be suffering with bipolar disorder.
us, the above set shows that the psychiatrist x considers "u 1 " to be under depression, "u 3 " to be suffering with hypomania, "u 2 and u 5 " to be bipolar, while "u 4 " to behave like a normal person, and so on.
Tables 1 and 2 represent the FBSES in Example 1 with respect to c and χ functions, respectively, whereas Table 3 gives a single table representation of the above FBSES, such that entries a ij of the table of i-th rows and j-th columns are represented as It is denoted by (c, χ, A) ⊆ (c 1 , χ 1 , B). Similarly, (c 1 , χ 1 , B) is a fuzzy bipolar soft expert superset of (c, χ, A), and the superset relation is then denoted as (c 1 , χ 1 , B)⊇(c, χ, A). Definition 6. Any two FBSESs (c, χ, A) and (c 1 , χ 1 , B) are said to be equal over a universe U if and only if (c, χ, A) is a subset of (c 1 , χ 1 , B) and (c 1 , χ 1 , B) is a subset of (c, χ, A).
Example 2. Consider Example 1. After treating the patients for a month, the psychiatrists again diagnose the patients on the basis of their behaviours.
Here, we see that A ⊂ B. In addition, c 1 (α)⊇c(α) and

Definition 7.
e complement of a FBSES (c, χ, A) over universe U is denoted by (c, χ, A) c and is defined by where c c and χ c are functions given as c c (α) � χ(α) and χ c (α) � c(α) for all α ∈ A, α ∈A.
Definition 10. e support of a FBSES (c, χ, A) on the universe U returns the objects u ∈ U with membership degrees greater than 0. Mathematically, of a relative null FBSES Definition 11. An agree FBSES (c, χ, A) 1 on the universe U is a fuzzy bipolar soft expert subset of (c, χ, A) given as    χ 1 , B).   Table 6:

Mathematical Problems in Engineering
Definition 12. A disagree FBSES (c, χ, A) 0 over the universe U is a fuzzy bipolar soft expert subset of (c, χ, A) given by Example 4. Consider the FBSES discussed in Example 1. en, the agree FBSES and disagree FBSES are represented in Tables 7 and 8 , respectively.
where M � min(c c (α), c c 1 (β)) and N � max □ Example 6. Consider the FBSESs discussed in Example 5. By Definition 14, the fuzzy bipolar soft expert OR operation on these sets (i.e., (c, ) is given in Table 12.

Example 7.
A business corporation decides to boost its business by using an AI virtual chatbot to reply the queries of their customers more efficiently and make better integration with all forums. e corporation assigns the task to two experienced virtual assistants to compare the available AI chatbots in order to find the best one for their needs. e set U � u 1 , u 2 , u 3 , u 4 , u 5 represents the available competing AI virtual chatbots. e set V � {ϑ 1 � quick response, ϑ 2 � powerful integration, ϑ 3 � comprehension} represents the suitable parameters; then, V � {ϑ 1 � loose response, ϑ 2 � weak integration, ϑ 3 � misinterpretation}. Let E � x, y represent the set of experts and O � 1 � agree, 0 � disagree represent the set of opinions.

Proposition 4. If
We consider only the nontrivial case when α ∈ A ∩ B, and then we have Similarly,

Mathematical Problems in Engineering
Hence, it proves that (c 1 , χ 1 , A).
Proofs of other parts are similar to part 1.   Table 17.
Proofs of other parts are similar to part 1. Proof.
(1) ere are 3 cases for α ∈ A ∪ B: In all three cases, it is obvious that Similarly, we can prove the remaining parts.   χ 1 , A).

Proof
(1) Consider α ∈ A ∩ (B ∪ C). en, three possibilities exist: possibilities. Moreover, all these values remain the same for any sample. For instance, we considered a sample of 10,000 people in our assumptions. e agencies analyze the tests and make data depending on the various kits and labs available in their respective areas and we get the following.
Tables 19 and 20 represent the information collected by the agencies regarding the tests in the form of agree FBSES and disagree FBSES, respectively.
Algorithm 1 based on FBSES is used to compare and rank the tests.
Using Tables 19 and 20, we calculate the focus agree and focus disagree or simply f-agree and f-disagree scores, which are displayed in Tables 21 and 22, respectively.  e final score table (see Table 23) indicates that the best available test is the rRT-PCR test. On the basis of the above calculations and assumptions, the tests are then ranked in their effectiveness as rRT − PCR > CBNAAT > antibody test > antigen test.
e above section gives a detailed comparison of the four widely used COVID-19 diagnosis tests, but the above details may vary with the original situation depending on the quality of kits, personnel, and availability of resources. Moreover, different techniques can prove to be much effective than others in different scenarios. For instance, RT-PCR takes more time; thus, for rapid testing, CBNAAT and antigen tests are more useful (though errorsome). For postdisease analysis, we use the antibody tests that detect the antibodies against COVID-19 created by the immune system of the body. Figure 1 indicates how the sensitivity and specificity of tests vary with each other and with themselves in different conditions. In order to maximize the effectiveness, the quality of the tests must not be compromised as a slight (1) Input the FBSES (c, χ, X).
(2) Find the agree and disagree FBSES tables with entries a ij � (c ij , χ ij ).
(3) Find the focus agree and focus disagree tables with entries f ij � c ij − χ ij . (4) Determine the f-agree score � ξ ∈ D 1 : � ξ j � i f ij as the last row in the focus agree FBSES table. (5) Determine the f-disagree score � η ∈ D 0 : � η j � i f ij as the last row in the focus disagree FBSES table.
misinterpretation can prove to be disastrous on a large scale. Figure 2 shows the dependence of number of cases on the effectiveness of tests. Tests with the same specificity and sensitivity can show major differences with varying prevalences. With a slight increase or decrease in prevalence percentage, the number of false positives and false negatives can change significantly. Keeping this in view, prevalencedependent factors have been analyzed at three different levels. Despite the high test load, the rRT-PCR technique appeared to be the most effective in all situations and is therefore considered as the "gold standard" for testing. In addition to the above diagnosis tests, contact tracing can also prove to be an effective strategy if implemented wisely. Countries such as China and South Korea used mobile tracing apps and geolocalization technology to announce high infection areas and track those in contact with the infected, thus getting a firm control on the spread of this disease efficiently.

Comparative Analysis
To reveal the authenticity and viability of the developed FBSES model, in this section we discuss its advantages, limitations, and comparative analysis with fuzzy SESs and fuzzy BSSs.
(i) Advantages: last few decades have proved to be very productive in the decision sciences providing numerous powerful uncertain models to deal with uncertainties and vague information. More efficient uncertain models and their hybrid structures are to be proposed in order to deal with the uncertainties and novel problems arising in the world. e existing fuzzy BSSs [9] and fuzzy SESs [15] have proved their importance and effectiveness in dealing difficult uncertain situations, but a more viable model having mutual characteristics of both these models is required. For this reason in this study, we developed a novel hybrid model called FBSESs which is capable of dealing with information involving opinions of multiple experts in the fuzzy bipolar environment. Hence, the developed model is much authentic and viable in dealing MAGDM situations comparatively. It can be readily seen that the fuzzy BSSs cannot handle multiple expert opinions. Similarly, fuzzy SESs fail to handle parameters efficiently in a bipolar environment. Meanwhile, the proposed FBSES model is capable of handling both fuzzy bipolar soft information and fuzzy soft expert information collectively as well as individually.
(ii) Comparison: when dealing with MAGDM problems, models such as fuzzy SESs [15] and fuzzy BSSs [9] happen to be very productive in making correct decisions. However, they are restricted to be used in their respective environments because they can only deal with the information supported by their respective structures. is issue can be solved by combining two or more models, where the new hybrid model is formed by the combination of its parent models, which is more general and reliable than the former ones. e fuzzy SES model [15] is best suitable to deal with fuzzy soft expert information, but fails to handle fuzzy bipolar soft data. In a similar way, fuzzy BSSs [9] are capable of dealing with fuzzy bipolar soft information efficiently, but are inadequate in dealing with information concerning multiple expert opinions. is leads to the need of the FBSES model which can deal with fuzzy bipolar soft information under the opinion of multiple experts in one place. In order to proclaim the efficacy and reliability of the developed model, its comparative analysis with the fuzzy SESs [15] and fuzzy BSSs [9] is presented in Tables 24 and 25. For further clarification, Figure 3 is also provided to represent the comparison. (iii) Limitations: the computational process of the developed model can be slow in the case of some MAGDM problems due to an increased number of parameters, as a result of the two oppositely defined sets of parameters viewed under the opinions of multiple experts. is limitation can be tackled if properly coded algorithm of the proposed model is practiced in different mathematical software such as MATLAB, which allows to deal with large datasets quickly and efficiently. Another important restriction is that the FBSES model is not very durable in the case of increased alternatives and/or parameters. Any increase or decrease in the alternatives can change the ranking order of the objects in a given MAGDM situation. e same is for the respective parameters. is is due to the independent behaviour of alternatives and parameters.

Conclusions
Numerous real-world MAGDM problems from various domains, including medical sciences and artificial intelligence, have been solved efficiently by the fuzzy SES model and its generalizations. e fuzzy BSS model is another effective model inspiring many researchers in dealing situations with fuzzy bipolar soft information, and emerging as a key component in many hybrid models, including rough m-polar fuzzy BSSs, to tackle different uncertain situations. Despite their effectiveness, the abovementioned theories have their own limitations. e existing fuzzy SESs are not capable of handling situations concerning bipolar soft information, whereas the fuzzy BSS model is inadequate when considering multiple expert opinions. To overcome these limitations, this study presented the concept of FBSESs by combining fuzzy SESs and fuzzy BSSs which is more useful and reliable than its components. An important characteristic of this initiated model is its capability of dealing with fuzzy SES and fuzzy BSS information collectively as well as individually. In this research article, some fundamental properties of the proposed model, including subset, complement, extended union, extended intersection, restricted union, restricted intersection, AND operation, and OR operation have been discussed in detail. Moreover, a detailed comparison of the different types of COVID-19 tests and the ranking of their effectiveness in analyzing the spread of COVID-19 has been done under the novel FBSES model, which is supported by an efficient algorithm. Finally, we have provided a comparison of our initiated model with existing models such as fuzzy SESs [15] and fuzzy BSSs [9] to show the authenticity and supremacy of the developed hybrid model. In future, our research work can be expanded to (1) intuitionistic fuzzy BSESs, (2) m-polar fuzzy BSESs, (3) spherical fuzzy BSESs, (4) picture fuzzy BSESs, and (5) q-rung orthopair fuzzy BSESs.

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest
e authors declare that they have no conflicts of interest.