#### Abstract

In our current era, a new rapidly spreading pandemic disease called coronavirus disease (COVID-19), caused by a virus identified as a novel coronavirus (SARS-CoV-2), is becoming a crucial threat for the whole world. Currently, the number of patients infected by the virus is expanding exponentially, but there is no commercially available COVID-19 medication for this pandemic. However, numerous antiviral drugs are utilized for the treatment of the COVID-19 disease. Identification of the appropriate antivirus medicine to treat the infection of COVID-19 is still a complicated and uncertain decision. This study’s key objective is to develop a novel approach called *q*-rung orthopair probabilistic hesitant fuzzy rough set (*q*-ROPHFRS), which incorporates the *q*-rung orthopair fuzzy set, probabilistic hesitant fuzzy set, and rough set structures. New *q*-ROPHFR aggregation operators have been established: the *q*-ROPHFR Einstein weighted averaging (q-ROPHFREWA) operator and the *q*-ROPHFR Einstein weighted geometric (q-ROPHFREWG) operator. In this study, we explored some basic features of the developed operators. Afterward, to demonstrate the viability and feasibility of the established decision-making approach in real-world applications, a case study related to selecting drugs for COVID-19 pandemic is addressed. Furthermore, a comprehensive comparison with the *q*-rung orthopair probabilistic hesitant fuzzy rough TOPSIS technique is also presented to illustrate the benefits of the new framework. The obtained results confirm the reliability and effectiveness of the proposed approach for finding uncertainty in real-world decision-making.

#### 1. Introduction

Wuhan, China, was faced with a dangerous challenge in December 2019, which distorted the health of humans and created global instability. The pneumonia cases were caused by a new virus known as coronavirus 2019 (COVID-19). COVID-19 was unknown; therefore, the government of China controlled Wuhan’s traffic to prevent the spread of the infection [1]. Russia, the United States, Brazil, India, and France are the most infected countries in terms of the number of confirmed COVID-19 cases. The World Health Organization (WHO) recognized COVID-19 as a pandemic by March 2020. Several governments and organizations have been closed down and have implemented strict social distancing processes to prevent virus proliferation. According to a WHO report released on June 13, 2021, more than 176,396,104 cases of COVID-19 have been reported around the world, resulting in more than 3,810,989 deaths, and a total of 160,398,032 people have been recovered [2]. The virus that causes COVID-19 is primarily spread via the droplets created when someone infected with COVID-19 sneezes, coughs, or exhales. Coronavirus is more harmful to those who have a low immune system, are elderly, have diabetes, or have medical problems, especially for those involving the lungs problem [3–6]. Virus propagation can be influenced by various factors, including population density, medical care facilities, climate, and others [7]. Coronaviruses are a vast family of viruses that can cause various diseases in both animals and humans. They mainly cause respiratory tract infections in humans, varying from an ordinary cold towards more severe illness disorders such as Middle East respiratory syndrome (MERS) and severe acute respiratory syndrome (SARS) [8, 9]. Phylogenetic and sequencing analyses have shown that COVID-19 is closely related to a collection of human and bat SARS-like coronaviruses [4, 10, 11]. COVID-19 is believed to have evolved from bats to a greater level of life chains [12–14]. The statistic is shown in Figure 1. Doctors, experts, or medical sections should implement an ideal plan, tests, or techniques for the COVID-19 treatment process to avoid further crisis expansion. The department in the process of establishing strategy must make quick and effective decisions. While making decisions in this situation, individuals are often bound logically instead of entirely reasonable. As a result, it is essential to identify appropriate multicriteria decision-making (MCDM) models that recognize human activities to provide individuals with practical ways of responding to emergencies. Dealing with uncertainty and unpredictable information in realistic circumstances has always been challenging. Several tools have been developed to address the complexities and conflicts encountered in real-life activities. Zadeh [15] explored a solution to such problems by establishing the foundations of fuzzy set (FS) theory, in which each element is assigned a membership degree ranging between 0 and 1. Atanassov [16] extended the idea of FS into intuitionistic FS (IFS) by introducing nonmembership () to the membership () of the FS, with the restriction that .

Yager [17] introduced the Pythagorean FS (PFS) theory, which relaxes the previously mentioned IFS condition to . PF expressions are undoubtedly raising the interest of many researchers, especially in terms of their applications to DM. For example, Huang et al. [18] described a PF MULTIMOORA approach that utilizes a novel distance measure and a score function. They used this approach to evaluate disk productions and energy projects. Zhang and Xu [19] established the TOPSIS approach in a Pythagorean fuzzy environment and used it to assess the efficiency of private airline services.

Hesitancy is a natural phenomenon in the universe. Identifying the better alternatives having the same characteristics in daily life is complicated. Due to the uncertainty and hesitancy of the results, professional experts are experiencing difficulty in DM. To tackle hesitancy, Torra [20] developed the concept of hesitant FS (HFS). The HFS can be used to solve a variety of DM problems. Many authors used HFS to solve issues by aggregating operators (AOPs) in group DM (for detailed information, see [21–24]). Liao and Xu [25] identified generalized forms of the HF hybrid weighted averaging (HFHWA) operator, the HF hybrid weighted geometric (HFHWG) operator, generalized form of the quasi-HFHWA operator, and the generalized form of the quasi-HFHWG. Khan et al. [26] introduced the concept of Pythagorean HFS (PyHFS). They established an evaluation method and identified operators for data aggregation. Xu and Zhou [27] identified a novel concept of probabilistic HF sets (PHFSs). Inspired by the power of PHFSs, researchers extensively investigated the idea of multiattribute decision-making (MADM) (see [28–30] for detailed information). Yager [31] established a new idea called *q*-rung orthopair FSs (q-ROFSs), in which the number of the *q*th exponent of support for membership and the *q*th exponent of support nonmembership is restricted to one, i.e., , and demonstrated that the *q*-ROFS is more general than the IFS and PFS. The *q*-ROFSs provide a broader range of fuzzy information and are the versatile and appropriate approach to deal with unpredictable situations. Yager and Alajlan [32] explored the fundamental properties of these *q*-ROFSs and discussed how they can be used in information representation. Subsequently, the authors in [33] put forward the notation of *q*-rung orthopair HF set (*q*-ROHFS) and explored the operational laws which exist for any two *q*-ROHFSs. Wang et al. [34] investigated the Heronian mean operators in MADM in a *q*-ROHFS framework. They also proposed the Hamacher norm-based AOPs under dual hesitant *q*-ROFSs and discussed their usefulness in DM problems. Wang et al. [35] established the AOPs based on Muirhead mean under dual hesitant *q*-rung orthopair fuzzy information. Hussain and Yang [36] measured the entropy for HF information using the Hausdorff metric and the structure of HF TOPSIS. The TOPSIS is a valuable information analysis tool developed by Hwang and Yoon [37]; it is also known as the approximate ideal solution. It investigates the appropriate approach in terms of relative closeness based on their distances from the positive ideal solution (PIS) and the negative ideal solution (NIS), ensuring that the shortest distance from the PIS and the farthest distance from the NIS are satisfied. This analysis method effectively eliminates decision information uncertainty while maintaining the validity and precision of decision-making by simply measuring the distance between PIS and NIS and ranking them accordingly. TOPSIS method is straightforward and simple to understand and analyze as compared to the ELECTRE method, VIKOR method, and other conventional methods, so it has been extensively studied and implemented by researchers.

In recent years, several authors have presented TOPSIS in various fuzzy information. For example, Boran et al. [38] used TOPSIS to identify the best supplier by using IF information. Chen and Tsao [39] suggested the TOPSIS technique based on interval-valued fuzzy information and addressed the experimental results. The authors in [40] established the extended TOPSIS method for *q*-ROHFSs and addressed their significance in DM. Li [41] proposed a TOPSIS-based nonlinear programming technique for MADM with interval-valued IFs in order to deal with uncertainty in real-world DM problems. The TOPSIS model for DM problems in interval-valued IF information was introduced by Park et al. [42]. The Dombi-based AOPs for PF information is formulated in [43]. Barukab et al. [44] proposed the extended fuzzy TOPSIS method for spherical fuzzy information, which is based on the entropy measure. The aforesaid approach has been used by many other researchers; see [45–47] for more information. However, there are many research findings in applying the fuzzy TOPSIS method to solve MADM problems; the decision information used by these approaches is too old and restricted to manage increasingly challenging decision environments.

Pawlak [48] was the founder of exploring the dominating concept of rough set (RS) theory. The classical set theory which deals with inconsistent and imprecise information is extended by rough set theory. Recently, research on the rough set has progressed significantly, both in terms of theoretical implementations and theory itself. In recent decades, research has demonstrated the TOPSIS technique in a number of RS information. Su et al. [49] studied RS theory based on fuzzy TOPSIS on the serious game design assessment procedure. Khan et al. [50] implemented a rough set strategy and the TOPSIS method for selection of sites for food distribution. Lu and Zhao [51] investigated the improved TOPSIS method based on RS theory for selection of suppliers. A rough set model and its applications to DM using the TOPSIS approach have been discussed in [52]. The concept of RS has been expanded by several researchers around the world in different directions. Using the fuzzy relation rather than the crisp binary relation, Dubois and Prade [53] initiated the notion of fuzzy rough sets (FRSs). The hybrid structure of IFSs and RS, intuitionistic RS (IFR set), was introduced by Cornelis et al. [54]. Zhou and Wu [55] established a novel DM technique under the IFR environment to address their constructive and axiomatic analysis in detail by utilizing IFR approximations. Zhan et al. [56] presented the DM methodology under the IFR environment and explored their applications in real-world problems. Different extensions of the IFRS are being investigated [57, 58] to tackle the uncertainty in MCGDM problems. Chinram et al. [59] established the algebraic norm-based AOPs based on the EDAS technique under IFR information and discussed their applications in MAGDM.

In some real-life circumstances, there exist numerous cases when decision makers (DMs) have their strong points of view about ranking or rating of plans, projects, or political statements of a government. For example, let the administration of a university start megaprojects of the football ground to render his accomplishment and performance. The members of the university administration may rate their project highly by assigning positive membership, whereas the others may rate the same project as a wastage of money and try to defame it by providing strongly opposite points of view. So, they assign negative membership. In this situation, their sum and but for so that it is neither IFN nor PFN but it is *q*-ROFN. Thus, Yager’s *q*-ROFNs are efficient to deal with vagueness in the data. *q*-rung orthopair probabilistic hesitant fuzzy rough set (q-ROHFRS), a hybrid intelligent structure of rough sets and *q*-ROPHFS, is an advanced classification strategy that has attracted researchers to address ambiguous and incomplete data. From the analysis, it is concluded that, in decision-making, AOP plays a significant role in aggregating the collective data from different sources to a single value. In accordance with the best available knowledge to date, the development of the AOP with the hybridization of the *q*-ROPHFS with a rough set is not observed in the *q*-ROF setting. As a result, the current *q*-ROPHF rough structure is inspired, and we define a list of Einstein aggregation operators depending on rough data, such as *q*-rung orthopair probabilistic hesitant fuzzy Einstein weighted averaging, Einstein ordered weighted averaging, Einstein hybrid weighted averaging, Einstein weighted geometric, Einstein ordered weighted geometric, and Einstein hybrid weighted geometric aggregation operators, under the Einstein *t*-norm and *s*-norm.

The description of the main objectives of the present work is as follows:(1)To introduce a novel idea of *q*-rung orthopair probabilistic hesitant fuzzy rough sets (*q*-ROPHFRSs) and investigate their basic operational laws.(2)Establish a list of AOPs based on Einstein *t*-norm and *t*-conorm and comprehensively explore the relevant properties.(3)To develop a DM strategy for aggregating unpredictability in real-world DM problems employing suggested aggregation operators.(4)In addition, a case study of drug selection for mild COVID-19 symptoms is described to demonstrate the applicability and utility of the established operators.(5)Finally, a comparison with the *q*-ROPHFR-TOPSIS method is made to interpret the outcomes. The ranking of the obtained results is presented graphically.

#### 2. Basic Terminologies

This section covers a variety of significant and fundamental concepts, i.e., fuzzy set (FS), intuitionistic FS (IFS), -rung orthopair FS (-ROFS), hesitant FS (HFS), *q*-rung orthopair HFS (-ROHFS), -rung orthopair probabilistic HFS (-ROPHFS), rough sets (RSs), and -rung orthopair FRS (-ROFRS).

*Definition 1. *(see [15]). For a universal set an FS is presented asfor each , and the function belongs to [0, 1] that represent the degree of membership.

*Definition 2. *(see [16]). For a universal set an IFS over is described asFor each , the functions ] and ] represent the membership and nonmembership, respectively, which must satisfy the property

*Definition 3. *(see [60]). For a universal set , an HFS in is represented mathematically aswhere is a set of some values in representing the degree of membership for the element of the set .

*Definition 4. *For a universal set , a probabilistic HF set (PHFS) in is described mathematically aswhere is a subset of and shows a membership grade of the element to the set . And shows the possibilities with the property that .

*Definition 5. *(see [31]). For a universal set a *q*-ROFS over is defined asfor each ; the functions ] and ] denote the membership and nonmembership, respectively, which must satisfy , (Figure 2).

*Definition 6. *(see [33]). For a universal set the mathematical representation of -ROHFS is as follows:where and are sets of some values in . It is required to satisfy the following properties: , , with and . For simplicity, we will use a pair to mean -ROHF number (*q*-ROHFN).

*Definition 7. *(see [33]). Let and be two *q*-ROHFNs. Then, the basic set theoretic operations are as follows:

*Definition 8. *Let be two *q*-ROHFNs where and are any real number. Then, the operational laws based on Einstein *t*-norm and *t*-conorm can be defined as

*Definition 9. *For a universal set a -ROPHFS is defined aswhere and are sets of some values in which denote the membership and nonmembership, respectively. and represent the possibilities of membership and nonmembership with the following property: and with and (*p* represents that the total elements exist in the -ROPHFS). It is required to satisfy the following properties: , and with and For simplicity, we will use a pair to mean a -ROPHF number (q-ROPHFN).

*Definition 10. *Let and be two q-ROPHFNs. Then, the basic set theoretic operations are as follows:

*Definition 11. *Let be two *q*-ROPHFNs and and be any real number. Then, the operational laws based on Einstein *t*-norm and *t*-conorm can be defined as

*Definition 12. *Let be the universal set and be a (crisp) relation. Then,(1) is reflexive if for each (3) is symmetric if and then (4) is transitive if , and implies

*Definition 13. *(see [48]). Let be a universal set and be any relation on Define a set-valued mapping by for where is called a successor neighborhood of the element with respect to relation . The pair is called the (crisp) approximation space. Now, for any set the lower and upper approximation of with respect to the approximation space is defined asThe pair is called the rough set, and both are upper and lower approximation operators.

*Definition 14. *(see [59]). Let be the universal set and be an intuitionistic fuzzy relation. Then,(1) is reflexive if and (2) is symmetric if , and (3) is transitive if

*Definition 15. *Let be the universal set. Then, any is called a *q*-rung relation. The pair is said to be *q*-rung approximation space. Now, for any , the upper and lower approximations of with respect to the *q*-RF approximation space are two *q*-RFSs, which are denoted by and and are defined aswheresuch thatAs are , are upper and lower approximation operators. The pairis known as the *q*-rung rough set. For simplicity,is represented as and is known as *q*-RFRV.

#### 3. Construction of -Rung Orthopair Hesitant Fuzzy Rough Sets

In this section, we propose the notion of -ROHFRS which is the hybrid structure of the rough set and *q*-ROFS. We also introduce the new accuracy and score functions to rank the *q*-ROHFRS and also put forward its basic operational laws.

*Definition 16. *Let be the universal set. Then, any subset is said to be a -RHF relation. The pair is called the *q*-ROHF approximation space. If for any , then the upper and lower approximations of with respect to the *q*-ROHF approximation space are two *q*-ROHFSs, which are denoted by and and defined aswheresuch thatAs are *q*-ROHFSs, are upper and lower approximation operators. The pairwill be called -ROHFRS. For simplicity,is represented as and is known as *q*-ROHFRV.

*Definition 17. *Let and be two *q*-ROHFRSs. Then,(1)(2)(3)(4)(5)(6) for (7) for (8) where and show the complement of *q*-rung fuzzy rough approximation operators and , which is (9) iff and For comparing/ranking two or more q-ROHFRVs, the score function will be utilized, whereas the accuracy function will be used when the score values are equal. The accuracy function will be used when the score values are equal.

*Definition 18. *The score function for *q*-ROHFRV is given asThe accuracy function for *q*-ROHFRV is given aswhere and are the number of elements in and , respectively.

*Definition 19. *Suppose and are two *q*-ROHFRVs. Then,(1)If then (2)If then (3)If then(a)If , then (b)If , then (c)If , then

#### 4. Construction of -Rung Orthopair Probabilistic Hesitant Fuzzy Rough Sets

This section deals with the notion of -ROPHFRS which is the hybrid structure of the rough set and *q*-ROPHFS. We also establish the new score and accuracy functions to rank the *q*-ROPHFRS and also discuss the operational laws.

*Definition 20. *Let be the universal set. Then, any subset is said to be a *q*-rung probabilistic HF relation. The pair is called the *q*-ROPHF approximation space. If for any , the upper and lower approximations of with respect to the *q*-ROPHF approximation space are two *q*-ROPHFSs, which are denoted by and and defined aswheresuch thatAs are are upper and lower approximation operators. The pairwill be called -rung orthopair HFRS. For simplicity,is represented as and known as *q*-ROPHFRV.

*Definition 21. *Let and be two *q*-ROPHFRSs. Then,(1)(2)(3)(4)(5)(6) for (7) for (8) where and show the complement of *q*-RFR approximation operators and which is (9) iff and

*Definition 22. *The score function for *q*-ROPHFRVis given asThe accuracy function for *q*-ROPHFRV is given aswhere and represent the number of elements in and , respectively.

*Definition 23. *Suppose and are two *q*-ROPHFRVs. Then,(1)If then (2)If then (3)If then(a)If , then (b)If