key: cord-0049387-8upajakt
authors: Ashraf, Shahzaib; Abdullah, Saleem
title: Emergency decision support modeling for COVID‐19 based on spherical fuzzy information
date: 2020-08-18
journal: nan
DOI: 10.1002/int.22262
sha: 8a9ce4547ccc8b9abfafb7f8e27c4fd5632271f6
doc_id: 49387
cord_uid: 8upajakt

Significant emergency measures should be taken until an emergency event occurs. It is understood that the emergency is characterized by limited time and information, harmfulness and uncertainty, and decision‐makers are always critically bound by uncertainty and risk. This paper introduces many novel approaches to addressing the emergency situation of COVID‐19 under spherical fuzzy environment. Fundamentally, the paper includes six main sections to achieve appropriate and accurate measures to address the situation of emergency decision‐making. As the spherical fuzzy set (FS) is a generalized framework of fuzzy structure to handle more uncertainty and ambiguity in decision‐making problems (DMPs). First, we discuss basic algebraic operational laws (AOLs) under spherical FS. In addition, elaborate on the deficiency of existing AOLs and present three cases to address the validity of the proposed novel AOLs under spherical fuzzy settings. Second, we present a list of Einstein aggregation operators (AgOp) based on the Einstein norm to aggregate uncertain information in DMPs. Thirdly, we are introducing two techniques to demonstrate the unknown weight of the criteria. Fourthly, we develop extended TOPSIS and Gray relational analysis approaches based on AgOp with unknown weight information of the criteria. In fifth, we design three algorithms to address the uncertainty and ambiguity information in emergency DMPs. Finally, the numerical case study of the novel carnivorous (COVID‐19) situation is provided as an application for emergency decision‐making based on the proposed three algorithms. Results explore the effectiveness of our proposed methodologies and provide accurate emergency measures to address the global uncertainty of COVID‐19.

Spherical FS is acknowledged to become more general than traditional FS frameworks. Come up with an example to illustrate the notion of spherical FS, which consists of an expert giving priority relating to attribute values with positive, neutral, and negative grades 0.7, 0.5, and 0.5, so obviously 0.7 + 0.5 + 0.5 = 1.7 > 1 and So picture FS cannot handle this kind of information, but we do have spherical FS 0.7 2 + 0.5 2 + 0.5 2 = 1.7 > 1 and so spherical FS can handle such situation very effectively. The notion of spherical FS extend to present the novel aggregation operators 30 (AgOp) using algebraic norm-based operational laws (OLs) for spherical FS. These AgOp are generalized to established Dombi norm-based list of Dombi AgOp 31 and explore their applicability in DMPs. To engage the logarithm function, Jin et al. 32 proposed logarithmic OLs based logarithmic AgOp and also link the spherical fuzzy information to linguistic present linguistic AgOp 33 and discussed the applicability in DMPs. List of decision-making techniques are established to contribute to spherical FS like cosine similarity measure 34 are discussed to tackle the uncertain information in DMPs. To find out the distance measure between the spherical FSs 35 and proposed their representation to spherical fuzzy norm 36 are explore and implemented to determine the uncertain information in real-life DMPs. TOPSIS under spherical fuzzy rough set 37 are establish successfully to deal DMPs. Spherical fuzzy symmetric sum based AgOp 38 is proposed to deal with real-life DMPs more effectively.

Decision-making for people and organizations, is one of the most critical activities and it an interdisciplinary research area that involves psychologists, economists, mathematicians, computer scientists from nearly all fields. Multiple attribute decision making (MADM) has achieved great reputation as an interesting research division of decisionmaking theory. There are actually two categories of methodology involved with continuous and discrete decision making issues, respectively. And those are methodologies of decision-making with multiple objective (MO) and multiple attribute (MA). The range of alternatives in MODM problems is infinite, and the trade-offs between design requirements are usually characterized by continuous functions. MADM problems only provide for the determination of the "best" alternative by recognizing trade-offs within a set of attractive design parameters. MADM leads to determining between possible actions in the presence of several typically conflicting attributes. Emergency decision making is one of the important and crucial branch of the decision-making theory. Under this scenario, to understand human actions to provide people with efficient means of responding to emergency situations.

The main objective of this study is to establish emergency decision making with the help of FS theory. To solve the environmental, economic, and social issues, we will develop the new tool to describe the actually information and based on the proposed tool, we construct the emergency decision support system for environmental problem of COVID-19. We will apply the emergency decision support model of COVD-19 in this emergency situation. The main objective of this study is to develop the emergency decision support system for environmental and economic issue of COVD-19 under spherical fuzzy information. This study has following contributions:

(1) Discuss the deficiency of existing operational laws of spherical FS and propose novel operational laws using Algebraic t-norm and t-conorm. (2) Discussed the novel operational laws using Einstein t-norm and t-conorm using spherical fuzzy environment. (3) Introduced the list of aggregation operators based on the Einstein operational laws to aggregate the uncertainty in decision making problems. (4) Establish two techniques (1) analytical hierarchy process and (2) spherical fuzzy entropy measure to determine the unknown weight information of the considered attributes. (5) Technique for order preference by similarity to ideal solution (TOPSIS) and grey relational analysis (GRA) methods are introduce to tackle the uncertainty in emergency decisionmaking problems under spherical fuzzy information. (6) We design three different algorithm to tackle emergency decision-making problems. (7) We shall collect the exact data disaster during the COVD-19 and then construct the mathematical model of emergency decision support systems for COVD-19 under the generalized structure of spherical FSs and compare our proposed three techniques with existing techniques to shows the validity and effectiveness of the proposed techniques.

To achieve the list of goals the structure of the paper is arranged as follows: Section 2 presented the rudiments of different structures of FSs and also explore the deficiency of the existing operational laws of spherical FS. Novel operational laws of spherical FS and their important properties are established in Section 3. Sections 4 and 5 introduced the list novel spherical fuzzy Einstein aggregation operators based on Einstein operational laws. Section 6 presented the two crucial techniques to determined the weight vector of the attributes. The main contribution of this study is three algorithms to tackle the uncertainty in emergency decision-making problems are executed in Section 7. Section 8 proposes the numerical case study of the outbreak of coronavirus as an emergency decision support problem to demonstrate the applicability and reliability of the proposed techniques. Section 9 presented the comparison of the proposed and existing techniques and the conclusion of this study is drawn in Section 10.

Let us briefly recall in this segment the rudiments of FSs, Pythagorean FSs, and spherical FSs. For the following review, these definitions will be included in the here.

∼ in fixed nonempty set Ö is described as follows:

is said to be the positive membership grade of ϱ in г͠ .

Definition 2 (Yager 17 ). A Pythagorean FS B ∼ in fixed nonempty set Ö is described as follows:

Definition 3 (Ashraf and Abdullah 29 ). A spherical FS F ∼ in fixed nonempty set Ö is described as follows:

For convenience, we indicate SFS Ôˆ( " ), the list of all spherical FS in Ö. We shall signify the spherical fuzzy number (SFN) by the triplet

Ashraf 29 defined the following notions:

(1) Mahmood et al. 39 proposed the basic operation for SFNs, which are described as follows: with ŵ > 0. The operational laws then shall be as follows:

( )

Operational rules defined in Definition 4 has some deficiency. Multiplication and addition of any two SFNs, violates the basic condition of spherical FS, that is square sum of its membership degrees less or equal to one. For supporting this, let us take F = {1, 0, 0} ≰ Hence obtaining results shows the violation of the condition that 0 P (ϱ) + I (ϱ) + N (ϱ) 1

(2) Liu et al. 40 proposed the basic operation for SFNs, which are described as: with ŵ > 0. The Algebraic operational laws are described as follows:

Operational rules defined in Definition 5 has also some deficiency. Multiplication of any two SFNs, violates the basic condition of spherical FS, that is square sum of its grades are ≤1. For supporting this, let us take F = {0, 0, 1}

Operational rules defined in Definition 6 has also some deficiency. Multiplication of any two SFNs, violates the basic condition of spherical FS, that is square sum of its grades are 1 ≤ . For supporting this, let us take F = {0, 0, 1} 

Now, square sum of its membership degree is (1) + (1) + (0) = 2 1 2 2 2

≰ . Hence obtaining results shows the violation of the condition that 0 P (ϱ) + I (ϱ) + N (ϱ) 1,

Ashraf and Abdullah presented the operational laws of SFNs. Here we present three cases to address validity of the algebraic operators to tackle spherical fuzzy settings.

Definition 7 (Ashraf and Abdullah 29 ). Let F P I N = { , , }

The operational laws based on algebraic norm are described as follows:

(3) ( )

Now we mentioned three cases to illustrate the validity of the recommended operators, which met the fundamental requirement of spherical FS that is,

in all situation. Based on multiplication rules for two spherical numbers, these extreme circumstances are indicated below. In particular, the three cases highlight the deficiency of the multiply operation in Definitions 4 and 5, The achieved result can no longer be purely determined by an SFN, which is unreasonable to some extent. Using extreme situations, we show the effectiveness and validation of the proposed operators discussed in Definition 7.

Since, we have , for each r Ǒ∈ ⋎ in all situation. The critical circumstances addressed in three cases are also satisfied. Therefore, the existing backdraw in spherical fuzzy operational laws are successfully eliminated. The operation described in this paper shall fulfill the closure property of the multiplication of the SFNs.

In this segment, we shall familiarized with generalized union and intersection for the spherical fuzzy numbers, which are as follows:

We can also write

= max( (ϱ), (ϱ)), min( (ϱ), (ϱ)), min( (ϱ), (ϱ)) = min( (ϱ), (ϱ)), min( (ϱ), (ϱ)), max( (ϱ), (ϱ)) 1 2

In above equations, T and S represents the t-norm and s-norm, respectively. As, we know well that, t-norm T ( ) and s-norm S ( ) are the general terms including all types of operators and also contented the necessitate of conjunction and disjunction operators, respectively. Here, we enlist the some types of t-norm and s-norm, respectively. Name t-norm s-norm

However, algebraic sum and algebraic product are obtain using algebraic norm. Algebraic operators are not only which can be use 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 Einstein product are to be good replacements, that characteristically give the equivalent smooth approximation as algebraic product and algebraic sum, respectively. Einstein t-norm and Einstein s-norm for spherical fuzzy environment as follows:

are said to be Einstein t-norm and Einstein s-norm, respectively. Also T d ľ ( , ) e satisfies the basic properties as follows:

and Ψ 0 ≥ . Then the Einstein operations for spherical fuzzy numbers are as follows: 

In this section, we define the score and accuracy values of the spherical FSs. On the basis of score and accuracy values, we can compare that which spherical FS is better than other one.

In this section, we propose the novel aggregation operators (AgOp) using spherical Einstein tnorm and spherical Einstein s-norm under spherical fuzzy environments.

In this section, we present Einstein norm-based averaging aggregation operators for spherical fuzzy settings.

∑ .

and the weight vector is

, . 

Proof. Mathematical induction is being used on n to prove the Equation (1). When n = 2, 

Thus, Equation (1), is true for n = 2.

Suppose that Equation (1), is true for n z = , we obtained

Then, we need to prove the Equation (1) is valid for n z = + 1, that is

that is, when n z = + 1, Equation (1) also holds. Hence, Equation (1) holds for any n. □

The following properties of SFEWA operator can be simply proved obviously by Definition 8. ≤ ∼ Next, we propose the spherical fuzzy Einstein ordered weighted averaging SFEOWA ( ) operator as following: 

∑ . 

Proof. This theorem's proof is identical to Theorem 1 and is therefore excluded here. □

The following properties of SFEOWA operator can be simply proved obviously by Definition 8. 

In this section, we present Einstein norm-based geometric aggregation operators for spherical fuzzy settings. ℶ g ∏ and the weight vector is

∑ .

and the weight vector is

The SFEWG AgOp is a mapping G G n ⟶ defined as 

Proof. Mathematical induction is being used on n to prove the Equation (2). When n = 2,

Utilizing Definition 8, we obtained 

Then, 

Thus, Equation (2), is true for n = 2. Suppose that Equation (2), is true for n z = , we obtained 

Then, we need to prove the Equation (2) is valid for n z = + 1, that is 

that is, when n z = + 1, Equation (2) also holds. Hence, Equation (2) holds for any n. □

The following properties of SFEWG operator can be simply proved obviously by Definition 8. ≤ ∼ Next, we propose the Spherical fuzzy Einstein ordered weighted geometric SFEOWG ( ) operator as following:

The Einstein ordered weighted geometric operator for SFN Ô ( " ) is characterized as

and the weight vector is

The SFEOWG AgOp is a mapping G G n ⟶ defined as 

Proof. This theorem's proof is identical to Theorem 3 and is therefore excluded here. □

The following properties of SFEOWG operator can be simply proved obviously by Definition 8. 

In 1980, Saaty initiated the analytical hierarchy process (AHP), 42 a powerful decision-making mechanism. AHP facilitates decision-makers in setting goals for various attributes to make the right decision. In pairwise comparisons, the AHP shall determine the weight of each assessment attribute according to the attribute of the decision-maker. AHP starts to build a matrix for a pairwise comparisons г to evaluate the weights for the various attributes. 

The matrix F is an l l × real matrix, where l is the list of assessment attributes described. Each element ς jg of the matrix г represents the importance of the jth criterion relative to the gth attribute. If ς jg > 1, then the jth attribute is more important than the gth criterion, whereas if ς jg < 1, then the jth attribute is less important than the gth attribute. When two attributes are of the same significance then the component ς jg is 1.

Obviously, ς = 1 jg for all g. The relative significance of the two attributes is assessed on a scale of 1 to 9, as shown in Table 1 .

If the matrix F is constructed, the normalized pairwise comparison matrix F can be extracted by keeping sum of the entities in each column equivalent to 1. Elements can be obtained as 

In 2019, Jin et al., 32 presents the method to determine the weights of the attribute using the fixed spherical fuzzy information presented in Table 2 .

To measure the weights for the various attributes, first, the decision-maker give the linguistic decision matrix ϒ as follows: 

The matrix ϒ is an m l × rank matrix, where m is the number of evaluation alternative considered and l is the number of evaluation attribute considered. All the linguistic term elements 

where P i , I i , and N i are the positive, neutral and negative membership degrees, respectively. Measure the closeness coefficient 7 | DIFFERENT DECISION MAKING TECHNIQUES

We proposed a method for spherical fuzzy Einstein aggregation operators to resolve MADM difficulties in the context of spherical fuzzy settings. The basic steps contain the algorithm are as follows:

Step 1. The decision maker provide the evaluation of emergency measures to reduced the outbreak of COVID-19 in linguistic form. After converting the linguistic term to spherical fuzzy number in the decision matrix elements are as follows: 

Step 2. In this step dealing with application of aggregation operators to decision matrix to aggregate the spherical fuzzy information for alternative. Attributes weights are calculated using AHP and Spherical fuzzy entropy methods. 

Step 5. Distance measurement of PIS Z + and NIS Z − with each alternative are defined as

Step 6. Measure the closeness coefficient to the PIS Z + and NIS Z − is defined as

Rank the ϱ i according to descending order. Choose the larger ϱ i for best alternative.

We provided a decision-making framework of GRA 44 using spherical fuzzy settings to address MADM challenges. The basic steps contain the algorithm are as follows: 

Step 5. Determine of Grey coefficients matrices using following formulas:

where I m j l , ∈ ∈ and ρ = 0.5 be a fixed coefficient. Step 6. Attributes weights are calculated using AHP and spherical fuzzy entropy methods.

Suppose that attributes weights W ρ ρ ρ ρ = { , , , …, }

. Grey coefficient are obtained as follows:

Step 7. Measure the closeness coefficients are obtained as follows:

Rank the ϱ i according to descending order. Choose the larger ϱ i for best alternative.

Under this section, a real case on public health emergency decision-making for an outbreak of COVID-19 that occurred in China is presented to demonstrate the application of the proposed techniques.

To demonstrate the applicability and validity of the proposed methods, we extant a real case study about an emergency caused by an outbreak of novel coronavirus disease 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. The novel coronavirus pushed the Chinese government to implement the largest lockdown in human history in early 2020, threatening an approximate 45 million population. The WHO has announced the name of the virus "Novel Coronavirus (COVID- 19) ." On January 30, 2020 the WHO Director-General disclosed that the epidemic is causing an international public health emergency.

As of March 25, 2020, more than 436 481 confirmed cases and 19 643 confirmed deaths are reported in 196 countries, areas, or territories. The infected and deaths cases graph are as follows in Figure 1 .

Also the statistics of active and closed cases are as follows in Figure 2 . The risk of it spreading further is very high. The outbreak has been defined by WHO as a public Health Emergency of international concern. There was no doubt that this disease caused huge economic losses, environmental pollution, insufficient of personal protective equipment (PPE), PPE consists of respiratory/surgical masks, gloves, face protection. The potential for extending the supply of PPE is restricted, and the current requirement for respirators and masks can not be achieved, especially if the widespread, improper use of PPE continues. The WHO collaborates with public health experts and laboratory partnerships, prevention and monitoring of diseases, clinical management and mathematical modeling.

In such situation, 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 are key in both health care and community settings. Due to such an emergency decision, the health experts have to make an immediate response, urgently rescue to control the situation efficiently and stop it from more deaths. There are eight basic public health emergency factor to reduce the general risk of this disease. The most effective preventive measure in the community include the following:

Clinical management ( 1 ): Vaccination after dispersal of the virus is a massively effective means of reducing these deadly diseases. Vaccines are actually quite effective, and severe side effects are rare. There are currently no clear treatments suitable for COVID-19. Clinical management requires timely adoption of approved initiatives for disease prevention and control, and support for complication management, providing strategic organ care where necessary. First-aid training ( 2 ): This disease spread very quickly, so to control on this virus first, trained, or avoid people of this disease symptoms. It is therefore highly recommended that individuals attend a fully supervised practical or online first aid course to learn how to get out of medical emergency.

Increased personal protective equipment ( 3 ): The lack of testing kits is another factor, the situation will be improved with increased production of testing kits, the loosing of confirmation criteria and the local governments decision to threat and finally quarantine all suspected cases. Masks, gloves, respirators and gowns to countries in every region. Face masks provide limited protection in preventing some one infected from spreading the virus. Therefore, the easiest way to prevent spread, is by good personal hygiene. However, the world is facing severve disruption in the marked for PPE.

Trained technician ( 4 ): It is extremely quick to share the genetic makeup of the virus to enable the rest of the world begin developing specific screening and start working on potential vaccines.

Banned intra-city transportation ( 5 ): The disease caused by the virus is serious. For safety of local people it is necessary for local government that take step or announced to banned intra-city transportation to force patients to local community clinics. And also suspended all flights and train services from and to Wuhan and cancelled their lunar new year celebrations, and you must also keep a distance of at least 1 m (3 feet) both yourself and somebody who coughs or sneezes.

Global uncertainty ( 6 ): The economic fallout from coronavirus: Rapid reduction in the transport and hospitality sectors will damage the economy in the short term and will also harm consumption and trade in the first quarter. It may significantly affect the country's overall economic situation. It has implications, not just for China, but for the entire world. The world depend on Chinese growth. The novel coronavirus has directly affected global oil market. Factory closure delays delivery of goods and parts from China impacting companies worldwide, along with Apple and Nissan.

Country-level coordination and planning ( 7 ): Each government needs the best level of cooperation with its province/states to prepare to overcome the novel uncertainty in the form of COVID-19.

Monitoring ( 8 ): Every government should appoint health and emergency decision-making experts to assess and track the current situation of every country and provide advice on how to improve the situation.

Suppose that there are five emergency alternatives namely risk communication (RC), lock down the borders and cities (LD), healthcare system (HCS), research needs (RN), and consulted experts (CE), respectively.

Health expert evaluated their information using fuzzy linguistic variable terms (given in Table 2 ) are shown in the Table 3 .

Based on the Table 2 , health expert evaluated fuzzy linguistic information and their corresponding spherical fuzzy numbers are given in Table 4 . 

First, we need to compute the pairwise comparison matrix г. In this regarding, the pairwise comparison matrix г is computed using Table 3 as follows:

        

The weights of the attribute can be calculate as following formula and also Tables 2 and 4 

The algorithmic steps are calculated as follows:

Step 1. First, we convert the linguistic information of Table 3 is converted in to the Table 4 as a spherical fuzzy information.

Step 2. The aggregated data is summarized in Table 4 using proposed Einstein aggregation operators.

Step 2(a). The aggregated data is summarized in Table 5A using SFEWA AgOp as follows:

Step 2(b) The aggregated data is summarized in Table 5B Step 5. Distance measurement of PIS Z + with each alternative are computed as follows Table 13A .

Distance measurement of NIS Z − with each alternative are computed as follows Table 13B .

Step 6. Measure the closeness coefficients are computed as follows Table 14 .

Hence, LD is the best choice according to attributes. Step 4. Distance measurement of PIS Z + with each element of the alternative to determine the spherical fuzzy positive-ideal separation matrix H + computed in Table 15 as follows.

Distance measurement of NIS Z − with each element of the alternative to determine the spherical fuzzy negative-ideal separation matrix H − computed in Table 16 as follows.

Step 5. Determine of Grey coefficient in Tables 17 and 18 as follows.

Step 6. Weighted Grey coefficient are computed in Table 19A as follows Table 19B .

Step 7. Measure the closeness coefficients are computed in Table 20 as follows. Hence, LD is the best choice according to attributes. The weight vector is applied to the proposed spherical fuzzy Einstein aggregation operators and the outcomes of decision making methods are shown in Figures 5 and 6 . Here, we can seen that LD is the best alternative under given attributes.

(Ranking using TOPSIS and GRA methods) According to the proposed TOPSIS and GRA approach using weights under AHP and spherical fuzzy entropy, we computed the finest (best) alternative (scheme) under given eights attributes.

The weight vector is applied to the proposed TOPSIS and GRA approach and the outcomes of decision making methods are shown in Figures 7 and 8 . Here, we can seen that LD is the best alternative under given attributes. 10 | CONCLUSION COVID-19, an infectious disease transmissible to the touch, is believed to spread across a population through direct contact between people. Outbreak prevention initiatives intended to decrease the amount of population mixing have the capability to slow down the peak and decrease the final extent of the epidemic. In this situation, an emergency decision makers or disaster response departments should implement strategies or select an appropriate emergency strategy to avoid further escalation of the crisis. For these concerns, the paper focus on emergency decision making to enhance the medical treatment and indorsed the living qualities of peoples. For this purpose, we proposed an emergency decision support techniques under the spherical fuzzy environment. First, give some discussion on the existing operations of spherical FS that's how existing operations have deficiency to fulfill the basic condition of a spherical FS. The novel basic operations are defined with three cases and proved that's the novel operations satisfied the square sum of positive, neutral and negative membership degrees values oscillate between 0 and 1. Next, to aggregate the spherical fuzzy information, proposed the novel Einstein aggregation operators namely, spherical fuzzy Einstein weighted average, spherical fuzzy Einstein weighted geometric, spherical fuzzy Einstein order weighted average, and spherical fuzzy Einstein order weighted geometric aggregation operators. Also, to determine the attributes weights, proposed two techniques first one is AHP method and other on is spherical fuzzy entropy method. Furthermore, to resolve the emergency situation of COVID-19 as emergency decision making, we introduced two spherical fuzzy decision support techniques namely are TOPSIS and GRA methods for spherical fuzzy information. Based on these three different techniques, we designed three algorithms to tackle emergency situation of COVID-19 effectively by the physicians or administrators. Validation and effectiveness of the proposed designed algorithms are tested over existing techniques. Results shows that the proposed techniques are 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.

Utilization of SFEWG AgOp to accumulate spherical information. Step 2(d)

The aggregated data is summarized in Table 8 using SFEOWA AgOp as follows

Modeling the epidemic trend of the 2019 novel coronavirus outbreak in China

World Health Organization (WHO). Coronavirus

National Health Commission of the People's Republic of China

A mathematical model for the novel coronavirus epidemic in Wuhan, China

COVID-19 detection using deep learning models to exploit Social Mimic Optimization and structured chest X-ray images using fuzzy color and stacking approaches

Mathematical modelling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada

Mathematical prediction of the time evolution of the COVID-19 pandemic in Italy by a Gauss error function and Monte Carlo simulations

Forecasting the timeframe of coronavirus and human cells interaction with reverse engineering

Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative

Fuzzy sets

Intuitionistic fuzzy sets

Entropy based pythagorean probabilistic hesitant fuzzy decision making technique and its application for fog-haze factor assessment problem

A risk assessment methodology using intuitionistic fuzzy set in FMEA

An application of intuitionistic fuzzy sets in medical diagnosis

Intuitionistic fuzzy geometric aggregation operators based on Einstein operations

Clustering algorithm for intuitionistic fuzzy sets

Pythagorean fuzzy subsets

Pythagorean fuzzy Dombi aggregation operators and their application in decision support system

Pythagorean membership grades, complex numbers, and decision making

Pythagorean fuzzy multiattribute group decision making with probabilistic information and OWA approach

A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making

Picture fuzzy sets-a new concept for computational intelligence problems

Different approaches to multi-criteria group decision making problems for picture fuzzy environment

Cleaner production evaluation in gold mines using novel distance measure method with cubic picture fuzzy numbers

Utilizing linguistic picture fuzzy aggregation operators for multiple-attribute decision-making problems

Generalized picture distance measure and applications to picture fuzzy clustering

Application of exponential jensen picture fuzzy divergence measure in multi-criteria group decision making

GRA method based on spherical linguistic fuzzy Choquet integral environment and its application in multi-attribute decision-making problems

Spherical aggregation operators and their application in multi-attribute group decision-making

Spherical fuzzy sets and their applications in multi-attribute decision making problems

Spherical fuzzy Dombi aggregation operators and their application in group decision making problems

Spherical fuzzy logarithmic aggregation operators based on entropy and their application in decision support systems

Linguistic spherical fuzzy aggregation operators and their applications in multi-attribute decision making problems

The cosine similarity measures of spherical fuzzy sets and their applications in decision making

Child development influence environmental factors determined using spherical fuzzy distance measures

Spherical fuzzy sets and its representation of spherical fuzzy t-norms and t-conorms

Covering-based spherical fuzzy rough set model hybrid with topsis formulti-attribute decision-making

Symmetric sum based aggregation operators for spherical fuzzy information: application in multi-attribute group decision making problem

approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets

T-spherical fuzzy power Muirhead mean operator based on novel operational laws and their application in multi-attribute group decision making

Algorithm for T-spherical fuzzy multi-attribute decision making based on improved interactive aggregation operators. Symmetry

Decision making with the analytic hierarchy process

Methods for multiple attribute decision making. Multiple Attribute Decision Making

The Basic Methods of Grey System

Multi-criteria group decision making based on ELECTRE I method in Pythagorean fuzzy information

Approach to fuzzy TOPSIS method based on entropy measure under spherical fuzzy information

A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory

A novel spherical fuzzy analytic hierarchy process and its renewable energy application

A novel spherical fuzzy QFD method and its application to the linear delta robot technology development

A novel fuzzy TOPSIS method using emerging interval-valued spherical fuzzy sets

Location selection of electric vehicles charging stations by using a fuzzy MCDM method: a case study in Turkey

Generalized picture fuzzy soft sets and their application in decision support systems

Picture fuzzy aggregation information based on Einstein operations and their application in decision making

Logarithmic aggregation operators of picture fuzzy numbers for multi-attribute decision making problems

Pythagorean fuzzy set: State of the art and future directions

Multiparametric similarity measures on Pythagorean fuzzy sets with applications to pattern recognition

Picture inference system: a new fuzzy inference system on picture fuzzy set

Projection models for multiple attribute decision making with picture fuzzy information

A multi-attribute decision making process with immediate probabilistic interactive averaging aggregation operators of t-spherical fuzzy sets and its application in the selection of solar cells

Multicriteria pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based rankingmethods

Emergency decision support modeling for COVID-19 based on spherical fuzzy information

i ͠ of aggregated spherical fuzzy numbers and order according to the highest value.Step 4. Choose the best alternative according to the highest score value.

We proposed a decision making methodology to solve the MADM problems under spherical fuzzy settings. The TOPSIS method introduced by Hwang and Yoon, 43 it is one of very suitable practical methods of GDM. In practice, the TOPSIS method is based on the idea that the best alternative will be at greatest distance from the negative ideal solution (NIS) and at the shortest distance from the positive ideal solution (PIS). The basic steps contain the algorithm are as follows:Step 1. The decision maker provide the evaluation of emergency measures to reduced the outbreak of COVID-19 in linguistic form. After converting the linguistic term to spherical fuzzy number in the decision matrix elements are as follows: Step 2. Basically, MADM problems have two types of attributes namely are benefit type and cost type attributes. To facilitate and dimensionless measurements of all the attributes, we proposed the normalize formulas for each attributes value c ij in decision matrix. Normalized decision matrix calculated using following formulas: Step 3. Attributes weights are calculated using AHP and spherical fuzzy entropy methods. Suppose that attributes weights W ρ ρ ρ ρ = { , , , …, }. Weighted normalized decision matrix constructed using following formula: Step 4. Identifying the PIS Z + and NIS Z − as follows:Step 1. The decision maker provide the evaluation of emergency measures to reduced the outbreak of COVID-19 in linguistic form. After converting the linguistic term to spherical fuzzy number in the decision matrix elements are as follows: Step 2. Basically, MADM problems have two types of attributes namely are benefit type and cost type attributes. To facilitate and dimensionless measurements of all the attributes, we proposed the normalize formulas for each attributes value c ij in decision matrix. Normalized decision matrix calculated using following formulas: Step 3. Identifying the PIS Z + and NIS Z − as follows:Step 4. Distance measurement of PIS Z + and NIS Z − with each element of the alternative to determine the spherical fuzzy positive-ideal separation matrix H + and Spherical fuzzy negative-ideal separation matrix H − as follows: Graphical representation is given in Figure 3 . Hence, LD is the best alternative according to give attributes. 

The algorithmic steps are calculated as follows:Step 1. In this step, we convert the linguistic information of Table 3 is converted in to the Table 4 as a spherical fuzzy information.Step 2. As, the spherical fuzzy information are benefit type. So, the normalized decision matrix computed in Table 10 as follows.Step 3. (Case 1. Using AHP weights) Weighted normalized decision matrix computed as follows; Table 11 .(Case 2: Using spherical fuzzy entropy weights) Table 12 .Step 4. Identifying the PIS Z + and NIS Z − as follows: Case 1.