Emergency decision support modeling under generalized spherical fuzzy Einstein aggregation information.

Dominant emergency action should be adopted in the case of an emergency situation. Emergency is interpreted as limited time and information, harmfulness and uncertainty, and decision-makers are often critically bound by uncertainty and risk. This framework implements an emergency decision-making approach to address the emergency situation of COVID-19 in a spherical fuzzy environment. As the spherical fuzzy set (SFS) is a generalized framework of fuzzy structure to handle more uncertainty and ambiguity in decision-making problems (DMPs). Keeping in view the features of the SFSs, the purpose of this paper is to present some robust generalized operating laws in accordance with the Einstein norms. In addition, list of propose aggregation operators using Einstein operational laws under spherical fuzzy environment are developed. Furthermore, we design the algorithm based on the proposed aggregation operators to tackle the uncertainty in emergency decision making problems. Finally, numerical case study of COVID-19 as an emergency decision making is presented to demonstrate the applicability and validity of the proposed technique. Besides, the comparison of the existing and the proposed technique is established to show the effectiveness and validity of the established technique.

Introduction

In the 21st century, with the rapid economic globalization growth and the acceleration of industrialization, environmental problems, people are facing with several disasters such as epidemic, extremist attacks, earthquakes, storms and other natural disasters that are quite sensitive to human beings. An epidemic of respirational virus triggered by a novel coronavirus disease (COVID-19) that was first spotted in Wuhan City, Hubei Province, China and which has now spotted in above 200 places globally, including cases in the United States, Italy, England, Germany, Iran and Pakistan . It has become dominated headline all over the world. The virus is new and rapidly scattering all over the world. There are various uncertainties regarding its origins, nature, and development. The number of people infected with the COVID-19 is increasing day by day. People are being isolated. Clinical masks and gloves, frequently used as a fence to epidemiologic spread, are selling out, however fitness authorities for example the World Health Organization (WHO) and the Centers for Disease Control and Prevention (Centers for Disease Control and Prevention 2019) recommend people that wearing masks and gloves are not beneficial or essential for avoiding infection in healthy people. The anxiety of COVID-19 is expected to be due to its novelty and the uncertainties about how bad the recent outbreak might become. The anxiety of COVID-19 is much larger than the anxiety of seasonal influenza, however, the latter has killed considerably more people. In such type of emergencies, decision makers must be split with some arrangements to lessen and release the infected lives. To date mostly affected countries (WHO COVID-19 Dashboard 2021) are from America, Europe and South-East Asia, because they make mistake of not locked down at early stage. Till date, it has been spread to most of the countries as per WHO report. Around a million population is infected by this disease in which around 1,305,164 deaths reported (Johns Hopkins CSSE 2020). The worst affected part of the world for this outbreak is America, where around 22,960,102 confirmed cases are reported. Among all countries around the world United States, Italy, Spain are in the top list of affected countries (WHO COVID-19 Dashboard 2021). The numbers of COVID-19 patients increasing very rapidly and till date it is going to touch the figure of a million, in which mostly affected countries are from United States, Europe and Asia. Although it started from china even though US and Europe are in top of the list of infected patients.

The COVID-19 is affecting 209 countries and territories around the globe and 2 international conveyances. As of November 15, 2020, Global update; more than 53,507,282 (Johns Hopkins CSSE 2020) cases have been confirmed globally. At least 1,305,164 (Pakistan COVID-19 Dashboard 2019) deaths have been attributed to the disease, most in mainland china, Italy, Iran, USA, UK etc. with more than 1000 of deaths in other countries. More than 36769792 (Pakistan COVID-19 Dashboard 2019) people have recovered. The risk of it spreading further is very high. The pandemic has resulted in travel restrictions and nationwide lockdowns in several countries.

In the case of an emergency, decision makers or disaster response departments should implement strategies or select an appropriate emergency strategy to avoid further escalation of the crisis. It is a matter for the evolving strategy sector to take rapid and effective decisions. Emergency decision-making as an integral part of the disaster response has been a significant task for many governments and a subject of discussion in academic circles. Under this scenario, when making decisions, people are usually bound logical rather than completely reasonable. It is therefore important to establish decision-making approaches that understand human actions in order to provide people with efficient means of responding to emergency situations. However, it is usually difficult to quantify the considered attributes in crisp values due to ambiguity, incompleteness, and uncertainty of the criterion information. To this end, fuzzy sets (FSs) were introduced to tackle multi criteria group decision making (MCGDM) and have become one of the most effective tools for quantification of the considered criteria/attributes. So far, many generalization of fuzzy sets (Zadeh 1965), such as intuitionistic FS (IFS) (Atanassov 1986), Pythagorean FS (PyFS) (Yager 2013), picture FS (PFS) (Cuong and Kreinovich 2013), and spherical FS (SFS) (Ashraf and Abdullah 2019a; Ashraf et al. 2019b), have been presented within academia. The SFSs, which was presented by Ashraf and Abdullah (2019a), is considered as one of the most recent and important type.

A SFS contains three degrees of membership grades, namely positive, neutral and negative, such that the square sum of membership grades is less than or equal to 1. For the aggregation of the criterion information to sort alternatives, traditional decision-making methods are used by many researcher (Ashraf et al. 2020b, c; Ashraf and Abdullah 2020d; Jin et al. 2019c; Qiyas et al. 2021) to contribute the spherical fuzzy set theory. The algebraic aggregation operators (Ashraf et al. 2019b) proposed by using the algebraic norms to address the problem of ambiguity in decision making (DM). Ashraf et al. (2020a) presented Dombi norm based some novel aggregation operators (AOp) for spherical fuzzy environment. Jin et al. (2019a) introduced the logarithmic operational laws based novel AOp and also give brief study of their implementation in real world DM problems. Ashraf et al. (2019c) developed the DM algorithm utilizing distance measure of SFSs and discussed their implementations in DM. Ashraf et al. (2019d) presented the idea of the representation of spherical fuzzy t-norms and t-conorms and explained the technique of TOPSIS to deal with uncertainty to aggregate the criteria for sorting the alternative among the list of alternatives. GRA method for spherical fuzzy data based on linguistic SFSs are given by Ashraf et al. (2018). Jan et al. (2021) presented the analysis of double domination by using the concept of spherical fuzzy information and discussed their application in decision making problems.

It is evident that the above-mentioned AOp is focused on the algebraic operating laws of the SFSs for the implementation of the combination process. Algebraic product and sum are not only fundamental SFS operations that describe the union and the intersection of any two SFSs. A general union and intersection under SF information can be developed from a generalized norm, i.e., instances of deferent-norms families may be used to execute the respective intersections and unions under SF environment. The Einstein product is a good replacement of the algebraic product for an intersection and is capable of delivering smooth estimates of the algebraic product. Equally, a good alternative to the algebraic sum for an intersection is the Einstein sum. However, there seems to be little work in the literature on aggregation approaches that use the Einstein operations on FSs and IFSs to aggregate the intuitionistic fuzzy values. Geometric interaction operators based on Einstein operations for IFS proposed by Garg (2016a). In (Garg 2016a, 2017) Garg presented the Einstein aggregation operators for PyFS and addressed their implementations in DM problems. In (Rahman et al. 2019a, b, 2020) proposed the generalized Einstein aggregation operators for interval values PyFS and addressed their implementations in DM. Khan et al. (2019) proposed picture fuzzy Einstein aggregation (PFEA) operators and discussed their implementation (application) in decision making. The main aim of this paper is therefore to establish some AOp based on Einstein’s operational laws under spherical fuzzy environment.

From the above discussion we note that in many practical applications, various aggregation operators have been put forward and implemented. Although, in practical problems many existing AOp are not capable to address such specific cases. In some circumstances, many of these may result in unreasonable or counter-intuitive results. Certain new regulations built without a simple function may have a complicated description. But generalized aggregation operators for SFSs continue to be an open subject that attracts many researchers.

WHO is working with global expert networks and partnerships for laboratory, infection prevention and control, clinical management and mathematical modelling. In such cases, it is essential to provide an efficient way in emergency response for avoiding additional losses and to save the lives of the people. Preventive and mitigation measures play a vital role in both health care and community settings. In this research, a new way of aggregating the SF information is built. The novel generalized spherical fuzzy Einstein aggregation operators are established employing Einstein t-norm and t-conorm. This is an attempt to propose generalized Einstein aggregation operators from a different perspective. We note that, generalized aggregation operators for dealing the uncertainty in practical problems using novel and emerging spherical fuzzy sets do not exist in the literature. The properties of the proposed Einstein operators are discussed. The performance of the proposed generalized Einstein aggregation operators is illustrated by the emergency decision making for COVID-19, as a decision making problem, where the weights of the attributes are unknown. In this regards, spherical fuzzy entropy measure is used to evaluate the weight vector of the attributes.

The motivation of developed AOp is summarized as below. Therefore, some generalized SF Einstein AOp are developed to choose the best option in different emergency decision-making situations like COVID-19. The developed operators have some advantages over other approaches which are given as below: The contributions of this paper are mainly reflected in the following aspects: The rest of this article is organized as set out below. Section 2 provides information concerning IFSs, FSs, PFSs, PyFSs, and SFSs. Section 3 describes the algebraic operations of SFSs. In Sect. 4, novel Einstein basic operational laws are proposed based on the Einstein norm, along with associated proof of its properties. The generalized AOp and their properties are proposed in Sect. 5, is the cornerstone of this work. Section 6 introduces the novel methodology for interacting with the ambiguity in DM problems in order to pick the finest alternative according to the list of attributes. Section 7 provides an overview of the evolved MCDM approach and a comparative analysis with some existing frameworks to MCDM. The article is concluded in Sect. 8 (Table 1).

Table 1

List of abbreviation

Abbreviation	Description	Abbreviation	Description
COVID-19	Coronavirus disease	MCGDM	Multi criteria group decision making
DMPs	Decision making problems	WHO	World Health Organization
FSs	Fuzzy sets	IFSs	Intuitionistic fuzzy sets
PyFSs	Pythagorean fuzzy sets	PFSs	Picture fuzzy sets
SFSs	Spherical fuzzy sets	DM	Decision making
AOp	Aggregation operators	PPE	Personal protective equipment

A very difficult MCGDM problem is the estimation of the supreme option in spherical fuzzy environment due to the involvement of several imprecise factors. Assessment of information in different MCGDM techniques is simply depicted through existing fuzzy numbers which may not consider all the data in a real-world problem.

As a general theory, spherical fuzzy numbers describe efficient execution in the assessment process about uncertain, imprecise and vague information. Thus, spherical fuzzy set theory provides an excellent approach for the assessment of objects under multinary data.

In view of the fact that generalized Einstein AOp are simple but provide a pioneering tool for solving MCDM problems when combine with other powerful mathematical tools, this article aims to develop Einstein AOs in spherical fuzzy environment to handle complex emergency problems.

The Einstein AOp employed in the construction of spherical fuzzy Einstein AOp are more suitable than all other aggregation approaches to tackle the MCGDM situations as developed AOp have ability to consider all the information within the aggregation procedure.

Einstein AOp make the optimal outcomes more accurate and definite when utilized in practical MCGDM problems under spherical fuzzy environment. However, the proposed AOp handle the drawbacks of AOp present in the literature.

Our proposed methods explain the problems more accurately which involve multiple attributes because they consider spherical fuzzy numbers.

The developed AOp are more precise and efficient with single attribute.

To solve practical problems by using generalized Einstein AOp with spherical fuzzy numbers is very significant.

To present some more advanced and generalized operational laws under spherical FSs based on Einstein t-norm and t-conorm.

To present list of novel aggregation operators with the help of the defined generalized Einstein t-norm and t-conorm under spherical fuzzy numbers. Also, the several fundamental properties between the proposed aggregation operators are derived to show its significance.

To present a novel MAGDM technique based on the proposed generalized Einstein aggregation operators to address the group decision making problems under spherical fuzzy environment.

The consistency and effectiveness of the proposed method is demonstrated through a numerical illustration of emergency decision making/emergency response, as well as their detailed evaluations.

The reliability and validity of the proposed technique is demonstrated with the help of sensitivity and comparison analysis.

Preliminaries

Let’s recall the rudiments of picture FSs and spherical FSs in this section for a while. Upon review, these ideas will be used here.

Definition 1

(Cuong and Kreinovich 2013) A PFS in fixed set V is described asfor each the positive membership neutral membership and the negative membership specifies the degree of positive, neutral and negative membership grades of the element to the picture fuzzy set respectively, where (is unit interval). In addition, it is important that for each .

Definition 2

(Ashraf and Abdullah 2019a) A SFS in fixed set V is described asfor each the positive membership neutral membership and the negative membership specifies the degree of positive, neutral and negative membership grades of the element to the spherical fuzzy set respectively, where . In addition, it is important that for each .

We shall signify, for convenience, the SFN by the triplet

Let Ashraf (Ashraf and Abdullah 2019a) defined the following notions:

(1) if and only if and for each Clearly if and

(2)

(3)

(4) where

Definition 3

(Ashraf and Abdullah 2019a) Let and with Then, the operational rules are as follows:

(1)

(2)

(3)

(4)

Spherical fuzzy Einstein operators

In this segment, we shall be familiarized with generalized union and intersection for the spherical fuzzy numbers, which are as follows:We can also write:In above equations, T and S represent the t-norm and s-norm respectively. As, we know that, t-norm and s-norm are the general terms including all types of operators and also contented the necessitate of conjunction and disjunction operators, respectively. Here, we enlist some types of t-norm and s-norm in the Table 2:

Table 2

Different types of t-norm and s-norm

Name	t-norm	s-norm
Algebraic	Tˇ_ad,l=dl	S^_Ad,l=d+l-dl
Einstein	Tˇ_ed,l=dl1+1-d1-l	S^_Ed,l=d+l1+dl
Hamacher	Tˇ_hd,l=dlγ+1-γd+l-dl,γ>0	S^_Hd,l=d+l-dl-1-γdl1-1-γdl,γ>0
Frank	Tˇ_fd,l=log_γ1+γ^d-1γ^l-1γ-1	S^_Fd,l=1-log_γ1+γ^1-d-1γ^1-l-1γ-1

However, algebraic sum and algebraic product are obtained using algebraic norm. Algebraic operators can not be only norm operator which used to perform union and intersection. We have many families of norm operators, which can be used to perform corresponding union and intersection. Einstein t-norm and Einstein s-norm are one of the effective family member of norm operators. Einstein sum and product are suitable replacements, respectively, giving the same smooth approximation as the algebraic product and the sum. Einstein t-norm and s-norm for spherical fuzzy environment as follows:Where and are said to be Einstein t-norm and Einstein s-norm respectively. Also satisfies the basic properties as follows:

For unite interval the mapping is said to be t-norm iff and

is commutative, monotonic and associative,

For unite interval the mapping is said to be s-norm iff

is commutative, monotonic and associative,

Definition 4

Let and and . Then the Einstein operations for spherical fuzzy numbers are follows as:

Comparison technique for spherical fuzzy sets

In this subsection, we define the score and accuracy values of the spherical fuzzy sets (SFSs). On the basis of score and accuracy values, we can find that which spherical fuzzy set is better than other one.

Definition 5

(Ashraf and Abdullah 2019a) Let . Then

is called score.

is called the accuracy.

Definition 6

(Ashraf and Abdullah 2019a) Let and Then

If then

If then

If then

then

then

then

Spherical fuzzy Einstein aggregation (SFEA) operators

In this section, we propose the novel aggregation operators based on Spherical Einstein t-norm and Spherical Einstein s-norm under spherical fuzzy environments.

Generalized Einstein averaging aggregation operators

In this part of the section, we propose the Einstein weighted averaging and ordered weighted averaging aggregation operators under spherical fuzzy environments.

Definition 7

Consider . Then the Einstein averaging aggregation operator for is denoted by SFEWA and defined as follows:where the weights of with is

Theorem 1

Suppose and weight vectors of be related to limit Then the operator of the SFEWA is a mapping of the such that

Proof

By using Mathematical induction on to prove the Equation (4.2).

When According to Definition 4, we haveThen,Thus the Equation (4.2), is true for

Suppose that Equation 4.2, is true for we haveNext we have to prove that Equation (4.2) is true for For this we haveHence Equation (4.2) is true for Therefore, by the principal of mathematical induction the result is true for all

The following properties of SFEWA operator can be proved using definitons.

Theorem 2

1) Let , if , then .

2) Let and , . Then,

3) Let and i.e. for all g. Then

Definition 8

Let . Then the Generalized Einstein averaging aggregation operator for is denoted by GSFEWA and defined as follows:where the weights of with , is

Theorem 3

Let and weights of subject to be denoted by The GSFEWA operator is a mapping such that

Using Definition (4), we have Suppose that Therefore,By back substitution, we have Hence, proved.

In procedure of aggregating the spherical fuzzy information parameter plays the vital role. When we fixed the parameter to special number, then the GSFEWA operator can be reduced. Like, when we take , GSFEWA operator is reduced to SFEWA.The following properties of GSFEWA operator can be simply proved.

Theorem 4

1) Let , if , then .

2) Let and , . Then,

3) Let and such that for all g. Then

Definition 9

Let . Then, the Generalized Einstein ordered averaging aggregation operator for is denoted by GSFEOWA and defined as follows:where weights of with , are Where is the real number greater than zero, , SFN is the gth largest of SFN .

Theorem 5

Let and weights of subject to be The GSFEOWA operator is a mapping such that

The proof of this theorem is similar to that of Theorem 3 and hence it is omitted here.

Generalized Einstein geometric aggregation operators

In this part of the section, we propose the Einstein weighted geometric and ordered weighted geometric aggregation operators under SF environments.

Definition 10

Let . Then, the Einstein weighted geometric aggregation operator for is represented by SFEWG and defined as follows:where weights of with , is

Theorem 6

Let and weights of subject to be denoted by The SFEWG operator is a mapping such that

By using Mathematical induction on to prove the Equation 4.7 .

When According to Definition 4, we haveThen,Thus, Equation 4.7, is true for

Assume Equation 4.7, for is true, we haveThen, we have to prove Equation 4.7 for is true, for this we havethat is, when , Equation 4.7 also holds.

Hence, Equation 4.7 holds for any . The proof is completed.

The following properties of SFEWG operator can be simply proved.

Theorem 7

1) Let , if , then .

2) Let and , . Then,

3) Let and such that for all g. Then

Definition 11

Let . Then, the Generalized Einstein geometric aggregation operator for is denoted by GSFEWG and defined as follows:where weights of with , is and

Theorem 8

Let and weights of subject to with be denoted by The GSFEWG operator is a mapping such that

Since, form Definition 4 we haveSuppose that Therefore,By back substitution, we have Hence, proved.

In procedure of aggregating the spherical fuzzy information parameter plays the vital role. When we fixed the parameter to special number, then the GSFEWG operator can be reduced. Like, when we take , GSFEWG operator is reduced to SFEWG.The following properties of GSFEWG operator can be simply proved.

Theorem 9

1) Let , if , then .

2) Let and , . Then,

3) Let and such that for all g. Then

Definition 12

Let . Then, the Generalized Einstein ordered geometric aggregation operator for is denoted by GSFEOWG and defined as follows:where weights of subject to and is Where is the real number greater than zero, , SFN is the gth largest of SFN .

Theorem 10

Let and weights of subject to be represented by . The GSFEOWG operator is a mapping such that

The proof of this theorem is similar to that of Theorem 8 and hence it is omitted here.

The following properties of GSFEOWG operator can be simply proved.

Theorem 11

1) Let , if , then .

2) Let and , . Then,

3) Let and i.e. g. Then

Technique of solving MAGDM problem based on generalized Einstein aggregation operators

To tackle the MAGDM problems in a SF setting, we propose a generalized spherical fuzzy Einstein AOp-based methodology. MADM problems can be interpreted as a decision matrix (D), where columns show attributes collection and the alternatives are shown in the rows. Therefore, let consider a set of alternatives for the decision matrix , and m attributes . is presented the unknown weight of m attributes with such that Supposed the SF decision matrix (D) is denoted by where represents the degree of the alternative gratifies the criteria considered by decision maker (DM), represents the degree of the alternative is neutral for the criteria considered by DM and represents the degree of the alternative doesn’t gratify the criteria considered by DM. The following steps form the algorithm (Fig. 1);

Fig. 1

Algorithm flow chart

Data Collection: In the form of SF information, the decision maker provides the decision matrices as follows

Normalization Data: Normalize the matrix. There are usually two types of attributes/criteria in a MAGDM problem. They are benefit and cost type criteria, whose values have the opposite effects on decision making. To unify such effects, the values of cost criteria in need to be normalized and a normalized matrix is obtained as follow: where refers to “if is a benefit criterion” and refers to “if is a cost criterion”.

Aggregated information: The aggregated information of all decision-maker information is calculated in this step using the SFWA/SFWG operator, that is

Unknown weight vector determined using SF-entropy measure: To determined the unknown weight vector of the attributes using spherical fuzzy entropy measure as following formula;

Aggregate the information using entropy weight vector:

Using SFEWA (operator) to combine (aggregate) the SF information.

Using SFEWG to combine the SF information.

Using GSFEWA to combine the SF information.

Using GSFEWG to combine the SF information.

Using GSFEOWA to combine the SF information.

Using GSFEOWG to combine the SF information.

Calculate the scores values of aggregated (combined) SFNs and rank by the maximum values for the score. If the score values are the same for two and , then we must take into consideration the accuracy degrees and respectively, then we’re going to rank the maximum degree of and .

Choose the optimal alternative as per the highest score value or the accuracy degree.

Numerical application of the proposed technique

A numerical application regarding the emergency decision making for COVID-19 is firstly used in this section to demonstrate the designed MAGDM process. Then a comparison is made between the proposed AOp and the existing AOp of SFNs to demonstrate the feature and benefit of the generalized Einstein AOp.

Case study

To demonstrate the applicability and validity of the proposed methods, we extend a real case study about an emergency caused by an outbreak of novel coronavirus disease (COVID-19) pandemic that occurred in China.

Actions taken by governments and organizations: The spread was first observed around December 2019 in Wuhan, Hubei, China, and reported by the World Health Organization (WHO) on March 11, 2020 as an epidemic disease. In early 2020, the novel coronavirus pushed the Chinese government to initiate the largest lockdown in human history, threatening an estimated 45 million individuals. The name of the "Novel Coronavirus (COVID-19)" virus was announced by the WHO. On January 30, 2020, the WHO Director-General announced that the outbreak was triggering a global health emergency.

The risk that it will spread further is quite high. WHO defined the outbreak as a public health emergency of international concern. This disease is undoubtedly caused by enormous economic losses, environmental pollution, lack of personal protective equipment (PPE), PPE consists of respiratory/surgical masks, gloves, face protection. The potential for an expansion of the supply of EPP is limited and it is not possible to meet the current requirements for respirators and masks, especially if the widespread and improper use of EPP continues. The WHO collaborates with public health experts and laboratory partnerships, prevention and monitoring of diseases, clinical management and mathematical modeling.

In such a situation, it is important to provide an effective means of emergency response to prevent further losses and to save the lives of individuals. In both health care and community environments, preventive and mitigation measures are key. As a result of such an emergency decision, health professionals must respond immediately, rescue them urgently to control the situation effectively and prevent more fatalities. There are three decision makers to provied their information using spherical fuzzy information to deal the uncertainty in emergency response and their weights are

There are five emergency alternatives to overcome the COVID-2019 which are given by And also four criteria. which are given by

Control vector

Food facility at Home, Lockdown

Educate the People

Disease and surveillance

Improving nutrition

Space

People

Time (4) Rescue people/Test

The expert evaluation results are listed in the Tables 3, 4 and  5:

Table 3

Expert-1 information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.84,0.34,0.400.43,0.39,0.780.67,0.50,0.300.31,0.21,0.710.60,0.11,0.530.23,0.35,0.590.72,0.31,0.410.11,0.25,0.820.79,0.19,0.390.11,0.21,0.910.71,0.41,0.130.34,0.25,0.510.63,0.51,0.130.49,0.33,0.420.61,0.43,0.450.49,0.37,0.590.57,0.36,0.290.50,0.15,0.600.70,0.32,0.400.33,0.44,0.65

Table 4

Expert-2 information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.61,0.15,0.530.16,0.35,0.620.61,0.35,0.470.55,0.17,0.740.66,0.11,0.510.43,0.23,0.770.93,0.08,0.090.02,0.06,0.990.88,0.09,0.070.05,0.06,0.890.56,0.17,0.440.43,0.13,0.610.59,0.32,0.340.24,0.48,0.510.68,0.53,0.390.34,0.21,0.610.71,0.31,0.240.35,0.41,0.690.73,0.44,0.210.22,0.49,0.74

Table 5

Expert-3 information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.85,0.25.0.150.14,0.23,0.880.78,0.38,0.180.29,0.39,0.830.94,0.04,0.070.39,0.19,0.610.63,0.18,0.350.48,0.49,0.560.73,0.13,0.460.19,0.39,0.880.87,0.35,0.180.41,0.13,0.810.82,0.12,0.430.55,0.21,0.630.53,0.33,0.470.46,0.23,0.510.61,0.33,0.290.28,0.41,0.630.74,0.34,0.140.37,0.32,0.65

The attributes and are benefits type, and are cost attributes according to the experts. Normalized matrix calculated as the formula given 5.1, and outcomes are seen in Tables 6, 7 and 8:

Table 6

Expert-1 normalized information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.84,0.34,0.400.78,0.39,0.430.67,0.50,0.300.71,0.21,0.310.60,0.11,0.530.59,0.35,0.230.72,0.31,0.410.82,0.25,0.110.79,0.19,0.390.91,0.21,0.110.71,0.41,0.130.51,0.25,0.340.63,0.51,0.130.42,0.33,0.490.61,0.43,0.450.59,0.37,0.490.57,0.36,0.290.60,0.15,0.500.70,0.32,0.400.65,0.44,0.33

Table 7

Expert-2 normalized information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.61,0.15,0.530.62,0.35,0.160.61,0.35,0.470.74,0.17,0.550.66,0.11,0.510.77,0.23,0.430.93,0.08,0.090.99,0.06,0.020.88,0.09,0.070.89,0.06,0.050.56,0.17,0.440.61,0.13,0.430.59,0.32,0.340.51,0.48,0.240.68,0.53,0.390.61,0.21,0.340.71,0.31,0.240.69,0.41,0.350.73,0.44,0.210.74,0.49,0.22

Table 8

Expert-3 normalized information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.85,0.25.0.150.88,0.23,0.140.78,0.38,0.180.83,0.39,0.290.94,0.04,0.070.61,0.19,0.390.63,0.18,0.350.56,0.49,0.480.73,0.13,0.460.88,0.39,0.190.87,0.35,0.180.81,0.13,0.410.82,0.12,0.430.63,0.21,0.550.53,0.33,0.470.51,0.23,0.460.61,0.33,0.290.63,0.41,0.280.74,0.34,0.140.65,0.32,0.37

In Table 9 aggregated SF information is calculated using the SFWA operator.

Table 9

Aggregated SF information

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.788,0.229,0.3190.785,0.315,0.2080.696,0.402,0.2970.767,0.239,0.3710.807,0.078,0.2790.674,0.246,0.3420.818,0.160,0.2270.919,0.188,0.0970.814,0.128,0.2230.893,0.165,0.0990.748,0.284,0.2230.677,0.159,0.3930.702,0.267,0.2710.533,0.324,0.3950.615,0.424,0.4330.573,0.258,0.4210.639,0.331,0.2710.644,0.298,0.3630.724,0.365,0.2240.685,0.411,0.296

To computed the unknown weight vector of the attributes using spherical fuzzy entropy measure 5.2 as follows

Used SFEWA operator to combine (aggregate) the SF information as given in Table 10:

Table 10

Aggregate using SFEWA operator

S_1S_2S_3S_4S_5

0.3094,0.7936,0.09790.3264,0.7120,0.02470.3192,0.7261,0.02140.2635,0.8043,0.17950.2834,0.8195,0.0879

Used SFEWG operator to combine the SF information as given in Table 11:

Table 11

Aggregate using SFEWG

S_1S_2S_3S_4S_5

0.60005,0.79366,0.146150.66340,0.71208,0.122770.63264,0.72614,0.124850.41620,0.80431,0.180100.49192,0.81953,0.14080

Used GSFEWA operator to combine the SF information with as given in Table 12:

Table 12

Aggregate using GSFEWA

S_1S_2S_3S_4S_5

0.44445,0.79575,0.111290.46454,0.71761,0.061740.45682,0.72979,0.059630.38654,0.80591,0.170350.41154,0.82002,0.10427

Used GSFEWG operator to combine the SF information with as given in Table 13:

Table 13

Aggregate using GSFEWG

S_1S_2S_3S_4S_5

0.56223,0.79575,0.216320.60791,0.71761,0.189880.58129,0.72979,0.202670.37139,0.80591,0.265580.44870,0.82002,0.20767

Used GSFEOWA operator to combine the SF information with as given in Table 14:

Table 14

Aggregate using GSFEOWA

S_1S_2S_3S_4S_5

0.444296,0.795641,0.1105450.464678,0.716829,0.0614790.457115,0.729603,0.0592620.386712,0.805925,0.1704650.412008,0.820026,0.103837

Used GSFEOWG operator to combine the SF information with as given in Table 15:

Table 15

Aggregate using GSFEOWG

S_1S_2S_3S_4S_5

0.56141,0.79564,0.215410.60776,0.71682,0.189820.58205,0.72960,0.202520.37184,0.80592,0.265720.44997,0.82002,0.20715

Calculated the scores values of combined (aggregated) SFNs and ranked as follows according to the maximum score values in Table 16;

Table 16

Score and ranking of SFNs

Operators	Score					Ranking
	sˇcˇS_1	sˇcˇS_2	sˇcˇS_3	sˇcˇS_4	sˇcˇS_5
SFEWA	0.472	0.529	0.523	0.426	0.458	S_2>S_3>S_1>S_5>S_4
SFEWG	0.553	0.609	0.593	0.477	0.510	S_2>S_3>S_1>S_5>S_4
GSFEWA	0.512	0.561	0.555	0.470	0.495	S_2>S_3>S_1>S_5>S_4
GSFEWG	0.516	0.566	0.549	0.433	0.473	S_2>S_3>S_1>S_5>S_4
GSFEOWA	0.512	0.562	0.556	0.470	0.496	S_2>S_3>S_1>S_5>S_4
GSFEOWG	0.516	0.567	0.549	0.433	0.474	S_2>S_3>S_1>S_5>S_4

Under all the suggested generalized Einstein aggregation operators, Machine has the highest score value, therefore (Lockdown and provide food facility at home) is our best alternative with respect to offer attributes listing to tackle and prevent to novel coronavirus (COVID-2019).

Sensitivity analysis

In this subsection, we discuss the sensitivity analysis in variation of the value of the parameter from 0 to 30 on the alternatives by using the proposed generalized Einstein aggregation operators to examine the distinct trends of the scores and ranking of the alternatives. Results obtained from proposed GSFEWA, GSFEWG, GSFEOWA and GSFEOWG aggregation operators are summarized in Table 16. These results obtained show the decision makers (DM,s) that they are able to select the values of according to their preferences. If , then the GSFEWA/GSFWG reduces to the SFEWA/SFEWG respectively. Also, it has been noted that means that the attitude of DM,s is neutral, and it is transpired that the overall score values of different alternates are increasing with an increase in . Thus the management meaning of is that the DM’s different preferences had effects on the score values of alternatives, which lead to the different optimal alternatives. (can be seen in Table 17)

Table 17

Sensitivity analysis on the different values of parameter

Υ	Operators	Score					Ranking
		sˇcˇS_1	sˇcˇS_2	sˇcˇS_3	sˇcˇS_4	sˇcˇS_5
→0.2	GSFEWA	0.546	0.630	0.626	0.471	0.543	2,3,1,5,4
	GSFEWG	0.628	0.687	0.668	0.521	0.565
	GSFEOWA	0.546	0.629	0.626	0.471	0.543
	GSFEOWG	0.628	0.687	0.669	0.521	0.565
→1	GSFEWA	0.472	0.529	0.523	0.426	0.458	2,3,1,5,4
	GSFEWG	0.553	0.609	0.593	0.477	0.510
	GSFEOWA	0.472	0.529	0.524	0.426	0.458
	GSFEOWG	0.553	0.609	0.594	0.477	0.511
→2	GSFEWA	0.512	0.561	0.555	0.470	0.495	2,3,1,5,4
	GSFEWG	0.516	0.566	0.549	0.433	0.473
	GSFEOWA	0.512	0.562	0.556	0.470	0.496
	GSFEOWG	0.516	0.567	0.549	0.433	0.474
→5	GSFEWA	0.551	0.596	0.590	0.512	0.533	2,3,1,5,4
	GSFEWG	0.485	0.518	0.499	0.395	0.444
	GSFEOWA	0.552	0.597	0.591	0.512	0.533
	GSFEOWG	0.485	0.518	0.499	0.395	0.444
→10	GSFEWA	0.566	0.612	0.604	0.531	0.550	2,3,1,5,4
	GSFEWG	0.464	0.488	0.470	0.376	0.429
	GSFEOWA	0.567	0.613	0.605	0.532	0.550
	GSFEOWG	0.465	0.489	0.470	0.377	0.429
→15	GSFEWA	0.571	0.617	0.609	0.539	0.557	2,3,1,5,4
	GSFEWG	0.562	0.577	0.571	0.368	0.528
	GSFEOWA	0.573	0.618	0.609	0.540	0.557
	GSFEOWG	0.562	0.577	0.571	0.368	0.529
→30	GSFEWA	0.578	0.624	0.615	0.548	0.564	2,3,1,5,4
	GSFEWG	0.553	0.568	0.561	0.486	0.524
	GSFEOWA	0.579	0.625	0.615	0.550	0.565
	GSFEOWG	0.554	0.568	0.561	0.487	0.525

Comparison analysis

A comparison of the features of these proposed generalized Einstein aggregation operators with the designed MAGDM method is presented in this section to demonstrate the benefits of the developed methodology. The comparison is provided by comparing the characteristics of the different aggregation operators which are described in different methods. The logarithmic based aggregation operators were identified in the approach (Jin et al. 2019a). The method presented in (Ashraf and Abdullah 2019a) discussed the algebraic aggregation operators.

We give the good comparison of the proposed six Einstein aggregation with existing aggregation operators presented in (Ashraf and Abdullah 2019a; Ashraf et al. 2020a; Jin et al. 2019a), showing the strength to handle real-life DMPs with uncertainty. The results are seen in below tables. It is explained as follows:

(1) Comparison with Jin et al. (2019a) logarithmic operators

Collective SF information of (Jin et al. 2019a) is shown in Table 18.

Table 18

Aggregated SF information matrix

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.788,0.229,0.3190.785,0.315,0.2080.696,0.402,0.2970.767,0.239,0.3710.807,0.078,0.2790.674,0.246,0.3420.818,0.160,0.2270.919,0.188,0.0970.814,0.128,0.2230.893,0.165,0.0990.748,0.284,0.2230.677,0.159,0.3930.702,0.267,0.2710.533,0.324,0.3950.615,0.424,0.4330.573,0.258,0.4210.639,0.331,0.2710.644,0.298,0.3630.724,0.365,0.2240.685,0.411,0.296

The matrix for comparison after aggregation by proposed Einstein aggregation operators is shown in Table 19:

Table 19

Score and ranking of SFNs

Log. Aggregation Operators (Jin et al. 2019a)	Score					Ranking
	sˇcˇS_1	sˇcˇS_2	sˇcˇS_3	sˇcˇS_4	sˇcˇS_5
L-SFWA	0.982	0.998	0.984	0.737	0.934	S_2>S_3>S_1>S_5>S_4
L-SFOWA	0.980	0.993	0.987	0.613	0.903	S_2>S_3>S_1>S_5>S_4
L-SFHWA	0.9995	0.9999	0.9997	0.646	0.984	S_2>S_3>S_1>S_5>S_4
L-SFWG	0.979	0.995	0.972	0.622	0.926	S_2>S_1>S_3>S_5>S_4
L-SFOWG	0.976	0.979	0.973	0.330	0.892	S_2>S_1>S_3>S_5>S_4
L-SFHWG	0.9998	0.9999	0.9991	0.822	0.998	S_2>S_1>S_3>S_5>S_4

(2) Comparison with Ashraf and Abdullah (2019a) algebraic aggregation operators

Collective SF information of (Ashraf and Abdullah 2019a) is shown in Table 20.

Table 20

Aggregated SF information matrix

	f_1f_2f_3
S_1S_2S_3S_4	0.658,0.427,0.2940.574,0.361,0.3390.492,0.548,0.4360.733,0.489,0.2900.452,0.677,0.2490.658,0.307,0.4990.388,0.663,0.4410.684,0.276,0.2730.443,0.266,0.6700.765,0.332,0.4430.571,0.564,0.3670.314,0.349,0.632

The matrix for comparison after aggregation by proposed Einstein aggregation operators is shown in Table 21:

Table 21

Score and ranking of SFNs

Algebraic Aggregation Operators ( Ashraf and Abdullah 2019a)	Score				Ranking
	sˇcˇS_1	sˇcˇS_2	sˇcˇS_3	sˇcˇS_4
textitSFWA	0.982	0.998	0.984	0.737	S_2>S_3>S_1>S_4
SFOWA	0.980	0.993	0.987	0.613	S_2>S_3>S_1>S_4
SFHWA	0.9995	0.9999	0.9997	0.646	S_2>S_3>S_1>S_4
SFWG	0.979	0.995	0.972	0.622	S_2>S_1>S_3>S_4
SFOWG	0.976	0.979	0.973	0.330	S_2>S_1>S_3>S_4
SFHWG	0.9998	0.9999	0.9991	0.822	S_2>S_1>S_3>S_4

Hence, is the best alternative which is computed under list of attributes.

Comparison with TOPSIS approach

In this subsection, we propose the comparison study of the developed generalized Einstein aggregation opeators with the improved TOPSIS methodology proposed by Barukab et al. (2019). The spherical fuzzy information evaluted in Barukab et al. (2019) is given in Table 22, 23 and 24.

Table 22

Spherical fuzzy information (Barukab et al. 2019)

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.84,0.34,0.400.43,0.39,0.780.67,0.50,0.300.31,0.21,0.710.60,0.11,0.530.23,0.35,0.590.72,0.31,0.410.11,0.25,0.820.79,0.19,0.390.11,0.21,0.910.71,0.41,0.130.34,0.25,0.510.63,0.51,0.130.49,0.33,0.420.61,0.43,0.450.49,0.37,0.590.57,0.36,0.290.50,0.15,0.600.70,0.32,0.400.33,0.44,0.65

Table 23

Spherical fuzzy information (Barukab et al. 2019)

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.61,0.15,0.530.16,0.35,0.620.61,0.35,0.470.55,0.17,0.740.66,0.11,0.510.43,0.23,0.770.93,0.08,0.090.02,0.06,0.990.88,0.09,0.070.05,0.06,0.890.56,0.17,0.440.43,0.13,0.610.59,0.32,0.340.24,0.48,0.510.68,0.53,0.390.34,0.21,0.610.71,0.31,0.240.35,0.41,0.690.73,0.44,0.210.22,0.49,0.74

Table 24

Spherical fuzzy information (Barukab et al. 2019)

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.85,0.25.0.150.14,0.23,0.880.78,0.38,0.180.29,0.39,0.830.94,0.04,0.070.39,0.19,0.610.63,0.18,0.350.48,0.49,0.560.73,0.13,0.460.19,0.39,0.880.87,0.35,0.180.41,0.13,0.810.82,0.12,0.430.55,0.21,0.630.53,0.33,0.470.46,0.23,0.510.61,0.33,0.290.28,0.41,0.630.74,0.34,0.140.37,0.32,0.65

Collective spherical information using spherical fuzzy weighted averaging aggregation operator is calculated in Table 25:

Table 25

Collected spherical fuzzy information (Barukab et al. 2019)

	f_1f_2f_3f_4
S_1S_2S_3S_4S_5	0.79,0.23,0.310.78,0.31,0.210.69,0.40,0.290.76,0.24,0.360.80,0.07,0.270.67,0.24,0.340.81,0.16,0.230.91,0.19,0.100.81,0.13,0.230.89,0.17,0.100.75,0.29,0.210.67,0.16,0.390.70,0.26,0.260.53,0.32,0.400.61,0.42,0.430.57,0.26,0.420.63,0.33,0.270.64,0.29,0.360.72,0.36,0.220.68,0.41,0.29

The comparison ranking matrix after aggregation by proposed Einstein aggregation operators is shown as follows in Table 26:

Table 26

Score and ranking of spherical fuzzy information

	Final revised closeness indices					Ranking
	S_1	S_2	S_3	S_4	S_5
TOPSIS Method (Barukab et al. 2019)	0.4047	0.5641	0.5908	0.2576	0.3018	S_3>S_2>S_1>S_5>S_4

From the outcomes of the proposed operators and the existing improved TOPSIS methodology, we conclude that ranking lists obtained from both the proposed method and the compared methods are same. Hence, Proposed generalized Einstein aggregation operators under the spherical fuzzy set environment is a generalized and novel approach to tackle uncertainty in DM problems. The proposed operators with the spherical fuzzy environment are a more flexible and effective to evaluate best alternative in real word problems.

Conclusion

In this paper, generalized spherical fuzzy Einstein weighted average, weighted geometric, ordered weighted average and ordered weighted geometric aggregation operators have been presented to aggregate the uncertainty in emergency situation of COVID-19 as a real life emergency decision making problem. The formal definitions and properties of these generalized Einstein aggregation operators have been respectively provided and explored. Their specific expressions are established via the operational laws of SFNs based on the Einstein t-norm and t-conorm. Based on the specific expressions, a new method for solving the MAGDM problem has been proposed. Based on these generalized spherical fuzzy Einstein aggregation operators, we designed an algorithms to tackle emergency situation of COVID-19 effectively by the physicians or administrators. Validation and effectiveness of the proposed designed algorithm is tested over existing techniques. Results shows that the proposed technique is reliable and effective to reduce/prevent the outbreak of COVID-19.

In future research, the other techniques of spherical FSs, like VIKOR, TODAM, Electric-I, II, and III with real life problems are investigated. Future work will also focus on applying the proposed method to solve practical MAGDM problems in manufacturing domain.

Limitation: A number of included studies were limited in terms data availability and methodological quality. Therefore, the reported findings should be interpreted cautiously within that context. Furthermore, our study was limited to the articles published in English. Considering the epicenter of COVID-19, Chinese literature should be included in future systematic reviews. We will continue to monitor the literature, and this method will be updated when new evidence emerges.