Mathematical calculation of COVID-19 disease in Pakistan by emergency response modeling based on complex Pythagorean fuzzy information

Rahman, K.; Khan, H.; Abdullah, S.

doi:10.3233/JIFS-212160

Mathematical calculation of COVID-19 disease in Pakistan by emergency response modeling based on complex Pythagorean fuzzy information

Article type: Research Article

Authors: Rahman, K.^{a; *} | Khan, H.^a | Abdullah, S.^b

Affiliations: [a] Department of Mathematics, Shaheed Benazir Bhutto University Sheringal, Pakistan | [b] Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan

Correspondence: [*] Corresponding author. K. Rahman, Department of Mathematics Shaheed Benazir Bhutto University Sheringal, Pakistan. E-mail: khaista355@yahoo.com.

Keywords: Complex Pythagorean set, CPFWA operator, CPFOWA operator, CPFHA operator, I-CPFOWA operator, I-CPFHA operator

DOI: 10.3233/JIFS-212160

Journal: Journal of Intelligent & Fuzzy Systems, vol. 43, no. 3, pp. 3411-3427, 2022

Published: 21 July 2022

Get PDF

Abstract

The new emerged infectious disease that is known the coronavirus disease (COVID-19), which is a high contagious viral infection that started in December 2019 in China city Wuhan and spread very fast to the rest of the world. This infection caused millions of infected cases globally and still poses an alarming situation for human lives. Pakistan in Asian countries is considered the third country with higher number of cases of coronavirus with more than 649824. Recently, some mathematical models have been constructed for better understanding the coronavirus infection. Mostly, these models are based on classical integer-order derivative using real numbers which cannot capture the fading memory. So at the current position it is a challenge for the world to understand and control the spreading of COVID-19. Therefore, the aim of our paper is to develop some novel techniques, namely complex Pythagorean fuzzy weighted averaging (abbreviated as CPFWA) operator, complex Pythagorean fuzzy ordered weighted averaging (abbreviated as CPFOWA) operator, complex Pythagorean fuzzy hybrid averaging (abbreviated as CPFHA) operator, induced complex Pythagorean fuzzy ordered weighted averaging (abbreviated as I-CPFOWA) operator and induced complex Pythagorean fuzzy hybrid averaging (abbreviated as I-CPFHA) operator to analysis the spreading of COVID-19. At the end of the paper, an illustrative the emergency situation of COVID-19 is given for demonstrating the effectiveness of the suggested approach along with a sensitivity analysis, showing the feasibility and reliability of its results.

1Introduction

The emerged infectious virus that is known as the coronavirus (COVID-19), which is transmittable virus reported at the city of Wuhan in China at the end of 2019. After studies and diagnosis these cases reported as novel coronavirus (COVID-19). Actually the word “coronavirus” comes from the Latin word “corona” which means a “crown, circle of light or nimbus”. This lethal virus has infected the whole world and several people have been died due to this noxious virus. In Pakistan the first case reported in March 2020. This virus not only affected the common life of the people but also affected the healthy society. This virus affects directly to your lungs. It has some symptoms like pneumonia and influenza. Now it is a challenge for the world to spend the peaceful and normal life due the COVID-19. The COVID-19 is a viral and pandemic disease; therefore the world health organization (WHO) professed an emergency situation due their spreading. Commonly, the virus infects through respiratory droplets emitted and during close touch when people cough or sneeze. When you touch an infected person its virus can be transmitted to your face. This virus is most transferable, because it can be transmitted to your body before started the symptoms. Usually it can be appear within 5 days; however it can be extended in the human body from 2 to 14 days. The most common symptoms of the virus are cough, fever, Pneumonia, chest tightness and acute respiratory distress syndrome. The suitable and perfect treatment is not available still, but the primary care is Symptomatic, covering the mouth in the case of coughing, self-isolation from the affected people, supportive therapy, hand washing, keeping distance from other people and monitoring.

However, there are many upcoming events related with COVID-19, such as how many people can be affected tomorrow?, how many individuals people can be affected during peak time?, the appearance of the inflection point of affected people?, the current treatments, and technologies can be control COVID-19?, to understand these events what is the best mathematical models which can help us. To control and prevent the emergency situation of COVID-19, some scholars have been contributed.

1.1Emergency response modeling

Yang and Wang [1] developed a mathematical model to show the transmitted of COVID-19. Togacar et al. [2] utilized the fuzzy color and stacking methods to develop the deducting approach for COVID-19. Tuite et al. [3] developed a mathematical model for the transmission of COVID-19, for the population of Canada. Ciufolini and Paolozzi [4] developed Gauss error function using a mathematical model of COVID-19, outbreak in Italy. For predicting the timeframe of COVID-19, Sohail and Nutini [5] developed a mathematical model. Khan and Atangana [6] utilize dynamic mathematical model and develop a fractional derivative for COVID-19. Some related research to COVID-19 and some othere desease can be found in [9–17]. The COVID-19 is recently identified so information on the prevalent is still insufficient, and the medical technologies such as clinical trials are still in a critical phase of investigation.

1.2Fuzzy information and emergency response modeling

Due to the less information of COVID-19 it is hard to apply directly to the current mathematical models and questions need to be answered about how successful the ongoing emergency response is and whether health care services can be spent more scientifically in the future, and so forth. In emergency case, decision-makers should implement an appropriate emergency scheme to control the further spiraling of the crisis. Therefore it is important to develop a decision making to understand the human actions to deliver people with effective means of responding to emergency situations. In a real life it is always difficult to solve uncertain data due to their incompleteness. In decision-making process, the decision and verdict information and data is mostly and often incomplete, indeterminate and inconsistent information. Therefore Zadeh [18] introduced the notion of fuzzy set which is more powerful and controlling tool to process fuzzy information in our daily life problems. However, there are some problems in this approach, because it has only one element called membership function. To control the above limitations, Atanassov [19] developed the idea of intuitionistic fuzzy set by adding non-membership function, whose sum is less than or equal to one. Later on, several scholars developed some aggregation operators using IFNs. Xu [20], Xu and Yager [21] introduced several arithmetic and geometric aggregation operators using algebraic operational laws and applied then on group decision making. Wang and Liu [22, 23] introduced the notion of some Einstein operators, such as IFEWG operator, IFEOWG operator, IFEWA operator, IFEOWA operator, and applied them to also to group decision making.

But there are several cases where the decision maker may provide the degree of membership and non-membership of a particular attribute in such a way that their sum is greater than one. For example, suppose a man expresses his preferences towards the alternative in such a way that degree of their satisfaction is 0.6 and degree of rejection is 0.8. Clearly its sum is greater than one. Therefore, Yager [24] introduced the concept of Pythagorean fuzzy set where the square sum of membership and non-membership is less than or equal to one. Pythagorean fuzzy set is more powerful tool to solve uncertain problems. Like intuitionistic fuzzy operators, Pythagorean fuzzy operators also become an interesting area for research, after the advent of Pythagorean fuzzy sets theory. Rahman et al. [25–31] introduced several aggregation operators based on Pythagorean fuzzy numbers and applied them on group decision making problems.

Ashraf and Abdullah [7] and Ashraf, Abdullah and Almagrabi [8] developed some new mathematical models for COVID-19 using fuzzy information. Rahman [49] introduced some new mathematical models for COVID-19 based on interval-valved intuitionistic fuzzy information.

1.3Complex fuzzy information

From the above study, it has been investigated that they have explored decision making problems under FS, IFS and PFS, which are able only to handle the ambiguity and fuzziness of the exists data. These models are unable to present the fractional ignorance of the data and its variations at a given phase of time. But in the sets of complex data, ambiguity and vagueness of the data occur concurrently with changes to the phase of the given data. Therefore, Remote et al. [32] presented the notion of complex fuzzy set (CFS), where the range of membership function can be extended from the [0, 1] to the unit disc. In CFS membership H = ℏ e^iα has two elements such as ℏ and e^iα, where the amplitude term ℏ ∈ [0, 1] and the phase term e^iα ∈ [0, 2π]. Later on, several scholars [33–35] worked on the field of CFSs. Alkouri and Salleh [36, 37] generalized the notion of CFS to complex intuitionistic fuzzy set (CIFS), by adding non-membership Q = qe^iβ with some conditions such as, ℏ + q ⩽ 1 and α + β ⩽ 2π. Rani and Garg [38, 39] developed the idea of distance measures between two CIFSs and also develop some complex power operators. Garg and Rani [40, 41] presented the some generalized operators using CIFNs. Garg and Rani [42] introduced the notion of complex interval-valued intuitionistic fuzzy set (CIVIFS), which is the generalization of complex intuitionistic fuzzy set. Ullah et al. [43] introduced the notion of complex Pythagorean fuzzy set (CPFS), which is the generalization of FSs, CFSs, IFSs and CIFSs. In CPFS H = ℏ e^iα and Q = qe^iβ are the membership and non-membership with conditions such as ℏ² + q² ⩽ 1 and α² + β² ⩽ 4π². Akram and Naz [44], Akram, Garg et al. [45] explored the graph theory and extended the Electre-1 and Topsis methods respectively. Mahmood et al [46] developed some new mathematical models for COVID-19 using fuzzy information. Akram et al. [47, 48] developed many aggregation operators using complex fuzzy information.

This paper having the following main approaches.

i) To control and reduce the spreading rate of the COVID-19 using emergency group decision-making.
ii) To develop some new operational laws base on complex fuzzy environment.
iii) To develop several operators namely, CPFWA operator, CPFOWA operator, CPFHA operator, I-CPFOWA operator, and I-CPFHA operator.
iv) To design an algorithm for emergency decision making for COVID-19.

The rest of the paper constructed as follows: In section 2, we present some basic information. In section 3, we present some properties of CPFS. In section 4, we develop several operators. In section 5, we present emergency decision-making. In section 6, we develop defferent techniques for emergency decision-making. In section 7, we construct a practice example. Section 8, havin limitation. In section 9, we develop the conclusion.

	Notations	Name	Abbreviation
ℏ	Membership	Complex fuzzy set	CFS
ƛ	Non-membership	Complex intuitionistic fuzzy set	CIFS
Λ	CPFNs	Complex Pythagorean fuzzyset	CPFS
s	Score function	Complex Pythagorean fuzzy number	CPFN
h	Accuracy degree	World health organization	WHO
Q	CFS	Health care system	HCS
I	CIFS	Risk communication	RC
M	CPFS	Lock down	LD
		Research need	RN

2Preliminaries

Definition 1. A Complex fuzzy set Q on a fixed set Z is the set of ordered pairs presented by the following equation:

(1)

Q={y,ℏQ(y)eiαQ(y)|y∈Z},

where ℏ_Q : Q→ { r : r ∈ C, |r| ≺ 1 } is called the membership degree, in the complex plane it lies in a unit circle, i=-1 , ℏ_Q (y) ∈ [0, 1] and e^iα_Q(y) is a real valued function.

Definition 2. [36] A complex intuitionistic fuzzy set I on a fixed set Z is the set of ordered pairs presented by the following equation:

(2)

I={〈y,ℏI(y)eiαI(y),ƛI(y)eiβI(y)〉|y∈Z},

where i=-1 , ℏ_I (y), ƛ_I (y) ∈ [0, 1] are called complex membership and non-membership function respectively, and α_I (y), β_I (y) ∈ [0, 2π] with conditions 0 ⩽ ℏ _I (y) + ƛ_I (y) ⩽ 1 and 0⩽αI(y)2π+βI(y)2π⩽1 , ∀ y ∈ Z.

Definition 3. [43] A Complex Pythagorean fuzzy set M on a fixed set Z is the set of ordered pairs presented by the following equation:

(3)

M={〈y,ℏM(y)eiαM(y),ƛM(y)eiβM(y)〉|y∈Z},

where i=-1 , ℏM(z) , ƛM(z)∈[0,1] are called complex membership and non-membership function respectively, and αM(z) , βM(z)∈[0,2π] with some conditions such as, 0⩽(ℏM(y))2+(ƛM(y))2⩽1 and 0⩽(αM(y)2π)2+(βM(y)2π)2⩽1 , ∀ y ∈ Z.

Definition 4. [43] If Λ_j (j = 1, 2, 3), then conditions hold:

i) Λ1⊕Λ2=(ℏΛ12+ℏΛ22-ℏΛ12ℏΛ22ei2π(αΛ12π)2+(αΛ22π)2-(αΛ12π)2(αΛ22π)2,(ƛΛ1ƛΛ2)e(βΛ12π)(βΛ22π)) ,
ii) Λ1⊗Λ2=((ℏΛ1ℏΛ2)e(αΛ12π)(αΛ22π),ƛƛΛ12+ƛƛΛ22-ƛƛΛ12ƛƛΛ22ei2π(βΛ12π)2+(βΛ22π)2-(βΛ12π)2(βΛ22π)2) ,
iii) ∂(Λ)=(1-(1-ℏΛ2)∂e1-(1-(αΛ2π)2)∂i2π,ƛƛΛ∂ei2π(βΛ2π)∂) ,
iv) Λ∂=(ℏΛ∂ei2π(αΛ2π)∂,1-(1-ƛƛΛ2)∂e1-(1-(βΛ2π)2)∂i2π) .

Definition 5. [44] If Λ = (ℏ _Λe^iα_Λ, ƛ_Λe^iβ_Λ) be aCPFN, then its score function and accuracy degreecan be defined as follows: s(Λ)=(ℏ2-ƛƛ2)+14π2(α2-β2) and h(Λ)=(ℏ2+ƛƛ2)+14π2(α2+β2) with s (Λ) ∈ [- 2, 2] and h (Λ) ∈ [0, 2] respectively.

Definition 6. [44] If Λ₁ = (ℏ _Λ₁e^iα_Λ₁, ƛ_Λ₁e^iβ_Λ₁) and Λ₂ = (ℏ _Λ₂e^iα_Λ₂, ƛ_Λ₂e^iβ_Λ₂) are two CPFNs, then the following conditions hold:

i) If s (Λ₁) ≻ s (Λ₂), then Λ₁ ≻ Λ₂.
ii) If s (Λ₁) = s (Λ₂) then we have three conditions
- a) If h (Λ₁) = h (Λ₂), then Λ₁ = Λ₂.
- b) If h (Λ₁) ≻ h (Λ₂), then Λ₁ ≻ Λ₂.

3Some operations on complex Pythagorean fuzzy set

Definition 7. If Λ_j = { y, ℏ _{Λ_j} (y) e^{iα_{Λ_j}(y)}, ƛ_{Λ_j} (y) e^{iβ_{Λ_j}(y)}|y ∈ Z } (j = 1, 2) are two CPFNs, then some basic operations on Λ₁ and Λ₂ are defined as follows:

i) Λ₁∪ Λ₂ = { y, ℏ _{Λ₁∪Λ₂} (y) e^{iα_{Λ₁∪Λ₂}(y)}, ƛ_{Λ₁∪Λ₂} (y) e^{iβ_{Λ₁∪Λ₂}(y)}|y ∈ Z },
where ℏ_{Λ₁∪Λ₂} (y) = max{ ℏ _Λ₁ (y) , ℏ _Λ₂ (y) } , ƛ_{Λ₁∪Λ₂} (y) = min { ƛ_Λ₁ (y) , ƛ_Λ₂ (y) }
and α_{Λ₁∪Λ₂} (y) = max{ α_Λ₁ (y) , α_Λ₂ (y) } , β_{Λ₁∪Λ₂} (y) = min { β_Λ₁ (y) , β_Λ₂ (y) }.
ii) Λ₁∩ Λ₂ = { y, ℏ _{Λ₁∩Λ₂} (y) e^{iα_{Λ₁∩Λ₂}(y)}, ƛ_{Λ₁∩Λ₂} (y) e^{iβ_{Λ₁∩Λ₂}(y)}|y ∈ Z },
where ℏ_{Λ₁∩Λ₂} (y) = min{ ℏ _Λ₁ (y) , ℏ _Λ₂ (y) } , ƛ_{Λ₁∩Λ₂} (y) = max { ƛ_Λ₁ (y) , ƛ_Λ₂ (y) }
and α_{Λ₁∩Λ₂} (y) = min{ α_Λ₁ (y) , α_Λ₂ (y) } , β_{Λ₁∩Λ₂} (y) = max { β_Λ₁ (y) , β_Λ₂ (y) }.
iii) Λ1c={y,ƛΛ1(y)eiβΛ1(y),ℏΛ1(y)eiαΛ1(y)|y∈Z} .

Theorem 1. Let Λ₁ and Λ₂ are any two CPFNs and a real number ∂≻0, then the resulting values of: Λ₁ ⊕ Λ₂, Λ₁ ⊗ Λ₂, (Λ₁) ^∂ and ∂ (Λ₁) are also CPFN.

Proof. First we show that Λ₁ ⊕ Λ₂ is CPFN. Let Λ₃ = Λ₁ + Λ₂ = (ℏ _Λ₃e^iα_Λ₃, ƛ_Λ₃e^iβ_Λ₃). Since Λ₁ = (ℏ _Λ₁e^iα_Λ₁, ƛ_Λ₁e^iβ_Λ₁) and Λ₂ = (ℏ _Λ₂e^iα_Λ₂, ƛ_Λ₂ e^iβ_Λ₂) are two CPFNs. By Definition 3, we have ℏ_{Λ_j}, ƛ_{Λ_j} ∈ [0, 1] (j = 1, 2) with some conditions such as, 0 ⩽ (ℏ _{Λ_j}) ² + (ƛ_{Λ_j}) ² ⩽ 1 and α_{Λ_j}, β_{Λ_j} ∈ [0, 2π] (j = 1, 2) with 0⩽(αΛj2π)2+(βΛj2π)2⩽1 . As 0 ⩽ ℏ _{Λ_j} ⩽ 1, this implies that 0⩽1-∏j=12(1-ℏΛj2)⩽1 . Hence, 0 ⩽ ℏ _Λ₃ ⩽ 1. On the other hand 0 ⩽ ƛ_{Λ_j} ⩽ 1, this implies that 0⩽∏j=12ƛΛj⩽1 . This implies that 0 ⩽ ƛ_Λ₃ ⩽ 1. Furthermore (ℏ _{Λ_j}) ² + (ƛ_{Λ_j}) ² ⩽ 1, this implies that (ƛ_{Λ_j}) ² ⩽ 1 - (ℏ _{Λ_j}) ². Thus we have, (∏j=12(ƛΛj)2)2⩽(∏j=121-(ℏΛj)2)2 and hence (ℏΛj)2+(ƛΛj)2=(1-∏j=12(1-ℏΛj2))2+(∏j=12(ƛΛj)2)2⩽(1-∏j=12(1-ℏΛj2))2+(∏j=12(1-ℏΛj2))2 =1-∏j=12(1-ℏΛj2)+∏j=12(1-ℏΛj2)=1 . Now ℏ_{Λ_j} ⩾ 0, ƛ_{Λ_j} ⩾ 0, this implies that (ℏ _{Λ_j}) ² ⩾ 0, (ƛ_{Λ_j}) ² ⩾ 0. Hence, (ℏ _{Λ_j}) ² + (ƛ_{Λ_j}) ² ⩾ 0. Thus 0 ⩽ (ℏ _{Λ_j}) ² + (ƛ_{Λ_j}) ² ⩽ 1. On the same way we can prove that α_{Λ_j}, β_{Λ_j} ∈ [0, 2π] with 0⩽(αΛj2π)2+(βΛj2π)2⩽1 . Thus, we obtain Λ₃ = Λ₁ + Λ₂ is also CPFN.

Now we are going show that (Λ₁) ^∂ is CPFN, for this we have, Λ₁ = (ℏ _Λ₁e^iα_Λ₁, ƛ_Λ₁e^iβ_Λ₁), then by Definition 3, we have ℏ_Λ₁, ƛ_Λ₁ ∈ [0, 1], α_Λ₁, β_Λ₁ ∈ [0, 2π] with 0 ⩽ (ℏ _Λ₁) ² + (ƛ_Λ₁) ² ⩽ 1 and 0⩽(αΛ12π)2+(βΛ12π)2⩽1 . Since ℏ_Λ₁ ⩾ 0, it implies that (ℏ _Λ₁) ^∂ ⩾ 0, 1-(1-ℏΛ12)∂⩾0 . On the other hand ƛ_Λ ⩾ 0, then (ƛ_Λ) ^∂ ⩾ 0. Further 1-(1-ℏΛ2)∂⩾0 and (ƛƛΛ2)∂⩾0 , then we have (1-(1-ℏΛ2)∂)2+((ƛƛΛ2)∂)2⩽(1-(1-ℏΛ2)∂)2+((1-ℏΛ2)∂)2= 1-(1-ℏΛ2)∂+(1-ℏΛ2)∂=1 . On the same way we can prove that α_Λ₁, β_Λ₁ ∈ [0, 2π] with 0⩽(αΛ12π)2+(βΛ12π)2⩽1 . Thus, (Λ₁) ^∂ is CPFN. On the same way we can prove the remaining parts.

Theorem 2. Let Λ₁, Λ₂ and Λ₃ are any three CPFNs, then the following hold:

1) Λ₁ ∪ Λ₂ ∪ Λ₃ = Λ₁ ∪ Λ₃ ∪ Λ₂,
2) Λ₁ ∩ Λ₂ ∩ Λ₃ = Λ₁ ∩ Λ₃ ∩ Λ₂,
3) Λ₁ ⊕ Λ₂ ⊕ Λ₃ = Λ₁ ⊕ Λ₃ ⊕ Λ₂,
4) Λ₁ ⊗ Λ₂ ⊗ Λ₃ = Λ₁ ⊗ Λ₃ ⊗ Λ₂.

Proof. the proof is easy, so it is omitted here

4Some complex Pythagorean fuzzy aggregation operators

In this section, we introduce some new operators namely, CPFWA operator, CPFOWA operator, CPFHA operator, I-CPFOWA operator, and I-CPFHA operator with their properties.

Definition 8. Let Λ_j = (ℏ _{Λ_j}e^iα_{Λ_j}, ƛ_{Λ_j}e^iβ_{Λ_j}) (j = 1, . . . , n) be a collection of CPFVs, where Φ = (Φ₁, Φ₂, . . . , Φ_n) ^T be the weighted vector of Λ_j with conditions Φ_j ∈ [0, 1] and ∑j=1nΦj=1 . Then the complex Pythagorean fuzzy weighted averaging (CPFWA) aggregation operator can be defined as:

(4)

CPFWAΦ(Λ1,Λ2,...Λn)=(1-∏j=1n(1-ℏΛj2)Φjei2π1-∏j=1n(1-(αΛj2π)2)Φj,∏j=1n(ƛΛj)Φjei2π∏j=1n(βΛj2π)Φj),

Theorem 3. Let Λ_j = (ℏ _{Λ_j}e^iα_{Λ_j}, ƛ_{Λ_j}e^iβ_{Λ_j}) (j = 1, . . . , n) be a collection of CPFVs, then their aggregated value by using the CPFWA operator is still CPFV, and

(5)

CPFWAΦ(Λ1,Λ2,...Λn)=(1-∏j=1n(1-ℏΛj2)Φjei2π1-∏j=1n(1-(αΛj2π)2)Φj,∏j=1n(ƛΛj)Φjei2π∏j=1n(βΛj2π)Φj),

Proof. Since Λ_j is a CPFN and for any real number Φ_j ≻ 0. Then by Theorem 1, Φ_jΛ_j is also a CPFN. So the first part of Theorem is proved. By mathematical induction principle.

Step 1: For n = 2, we have Λ₁ = (ℏ _Λ₁e^iα_Λ₁, ƛ_Λ₁e^iβ_Λ₁) and Λ₂ = (ℏ _Λ₂e^iα_Λ₂, ƛ_Λ₂e^iβ_Λ₂), so by the operational laws of CPFNs, we get and Φ1Λ1=(1-(1-(ℏΛ1)2)Φ1ei2π1-(1-(αΛ12π)2)Φ1,(ƛΛ1)Φ1ei2π(βΛ12π)Φ1) , Φ2Λ2=(1-(1-(ℏΛ2)2)Φ2ei2π1-(1-(αΛ22π)2)Φ2,(ƛΛ2)Φ2ei2π(βΛ22π)Φ2) .

Now by Definition 8, we have the following:

CPFWAΦ(Λ1,Λ2)=(1-∏j=12(1-(ℏΛj)2)Φjei2π1-∏j=12(1-(αΛj2π)2)Φj,∏j=12(ƛΛj)Φjei2π∏j=12(βΛj2π)Φj)

Hence, for n = 2, the result in (5) is true.

Step 2: Suppose that (5) is true for n = s, where s is any positive integer.

CPFWAΦ(Λ1,Λ2,...,Λs)=(1-∏j=1s(1-(ℏΛj)2)Φjei2π1-∏j=1s(1-(αΛj2π)2)Φj,∏j=1s(ƛΛj)Φjei2π∏j=1s(βΛj2π)Φj).

Step 3: Assume that (5) true for n = s, we show that it is true for n = s+1, for this we have

CPFWAΦ(Λ1,Λ2,...,Λs+1)=(1-∏j=1s(1-(ℏΛj)2)Φjei2π1-∏j=1s(1-(αΛj2π)2)Φj,∏j=1s(ƛΛj)Φjei2π∏j=1s(βΛj2π)Φj)⊕(1-(1-(ℏΛs+1)2)Φs+1ei2π1-(1-(αΛs+12π)2)Φs+1,(ƛΛs+1)Φs+1ei2π(βΛs+12π)Φs+1)=(1-∏j=1s+1(1-(ℏΛj)2)Φjei2π1-∏j=1s+1(1-(αΛj2π)2)Φj,∏j=1s+1(ƛΛj)Φjei2π∏j=1s+1(βΛj2π)Φj)

Thus (5) is true for n = s+1. By mathematical induction (5) holds for all positive integer.

Property 1 (Idempotency). If Λ_j (j = 1, . . . , n) be a collection of CPFVs, and let Λ_O be another CPFV such tha Λ_j = Λ_O, then

CPFWAΦ(Λ1,Λ2,...,Λn)=(1-∏j=1n(1-(ℏΛO)2)Φjei2π1-∏j=1n(1-(αΛO2π)2)Φj,∏j=1n(ƛΛO)Φjei2π∏j=1n(βΛO2π)Φj)=(ℏΛOeiαΛO,ƛΛOeiβΛO)=ΛO

Property II (Boundedness). If Λ_j = (ℏ _{Λ_j}e^iα_{Λ_j}, ƛ_{Λ_j}e^iβ_{Λ_j}) (j = 1, . . . , n) be a collection of CPFVssuch as: Λ_min = (ℏ _mine^iα_min, ƛ_mine^iβ_min) and Λ_max = (ℏ _maxe^iα_max, ƛ_maxe^iβ_max), where ℏmin=minj{ℏj} , αmin=minj{αj},ƛmin=minj{ƛj},βmin = minj{βj} , ℏmax=minj{ℏj} , αmax=minj{αj},ƛmax=minj{ƛj} and βmax=minj{βj} , then

(6)

Λmin⩽CPFWAΦ(Λ1,Λ2,...,Λn)⩽Λmax,

Proof. Since Λ = CPFWA (Λ₁, Λ₂, . . . , Λ_n) = (ℏ _{Λ_j}e^iα_{Λ_j}, ƛ_{Λ_j}e^iβ_{Λ_j}), then for any Λ_j, we have

(minj{Λmin})2⩽(Λj)2⩽(maxj{Λmax})2⇔∏j=1n(1-(maxj{Λmax})2)Φj⩽∏j=1n(1-(Λj)2)Φj⩽∏j=1n(1-(minj{Λmin})2)Φj⇔(minj{Λmin})2⩽1-∏j=1n(1-(Λj)2)Φj⩽(maxj{Λmax})2⇔minj{Λmin}⩽1-∏j=1n(1-(Λj)2)Φj⩽maxj{Λmax}

Thus minj{ℏj}⩽ℏj⩽maxj{ℏj} . On the same way minj{αj}⩽αj⩽maxj{αj} . Similarly

minj{ƛj}⩽ƛj⩽maxj{ƛj}⇔coprodj=1n(minj{ƛj})Φj⩽coprodj=1n(ƛj)Φj⩽coprodj=1n(maxj{ƛj})Φj⇔(minj{ƛj})∑j=1nΦj⩽coprodj=1n(ƛj)Φj

⩽(maxj{ƛj})∑j=1nΦj⇔minj{ƛj}⩽coprodj=1n(ƛj)Φj⩽maxj{ƛj}

Thus minj{ƛj}⩽ƛj⩽maxj{ƛj} . On the same way minj{βj}⩽βj⩽maxj{βj} . Then we have

(7)

Λmin⩽CPFWAΦ(Λ1,Λ2,Λ3,...,Λn)⩽Λmax.

Property III (Monotonicity). If Λ_j = (ℏ _{Λ_j}e^iα_{Λ_j}, ƛ_{Λ_j}e^iβ_{Λ_j}), Δ_j = (ℏ _{Δ_j}e^iα_{Δ_j}, ƛ_{Δ_j}e^iβ_{Δ_j}) are two families of CPFVs, with ℏ_{Λ_j} ⩽ ℏ _{Δ_j}, α_{Λ_j} ⩽ α_{Δ_j}, ƛ_{Λ_j} ⩾ ƛ_{Δ_j}, β_{Λ_j} ⩾ β_{Δ_j}. Then we have

(8)

CPFWAΦ(Λ1,Λ2,...,Λn)⩽CPFWAΦ(Δ1,Δ2,...,Δn).

Proof. Proof is similar as above, so here it is omitted.

Definition 9. The complex Pythagorean fuzzy ordered weighted averaging (CPFOWA) aggregation operator can be defined as:

(9)

CPFOWAΦ(Λ1,Λ2,...Λn)=(1-∏j=1n(1-ℏΛφ(j)2)Φjei2π1-∏j=1n(1-(αΛφ(j)2π)2)Φj,∏j=1n(ƛΛφ(j))Φjei2π∏j=1n(βΛφ(j)2π)Φj),

where (φ (1) , φ (2) , . . . φ (n)) is any permutation of (1, 2, . . . , n) with condition Λ_φ(j-1) ⩾ Λ_φ(j) for all j. Φ = (Φ₁, Φ₂, . . . , Φ_n) ^T be the weight vector of Λ_j with some conditions Φ_j ∈ [0, 1] and ∑j=1nΦj=1 .

Theorem 4. Let Λ_j = (ℏ _{Λ_j}e^iα_{Λ_j}, ƛ_{Λ_j}e^iβ_{Λ_j}) (j = 1, 2, . . . , n) be a collection of CPFVs, then their aggregated value by using the CPFOWA operator is still CPFV.

Proof. Proof is similar to Theorem 3, so here we omit.

Definition 10. The complex Pythagorean fuzzy hybrid averaging (CPFHA) aggregation operator can be defined as:

(10)

CPFHAϖ,Φ(Λ1,Λ2,...Λn)=(1-∏j=1n(1-ℏΛ̇φ(j)2)Φjei2π1-∏j=1n(1-(αΛ̇φ(j)2π)2)Φj,∏j=1n(ƛΛ̇φ(j))Φjei2π∏j=1n(βΛ̇φ(j)2π)Φj),

where Λ̇φ(j) be the greatest value and Λ̇φ(j)=nϖjΛj . And n be the coefficient of balancing which play role for balances. If Φ = (Φ₁, Φ₂, . . . , Φ_n) ^T goes to (1n,1n,...,1n)T , then (nϖ₁Λ₁, nϖ₂Λ₂, . . . , nϖ_nΛ_n) ^T goes to (Λ₁, Λ₂, . . . Λ_n) ^T. (ϖ₁, ϖ₂, . . . , ϖ_n) ^T be the weighted vector and Φ = (Φ₁, Φ₂, . . . , Φ_n) ^T be the associated vector both have the same conditions, such as belong to the closed interval and their sum should be equal to one.

Definition 11. The I-CPFOWA operator can be defined as follow:

(11)

I-CPFOWAΦ(〈u1,Λ1〉,〈u2,Λ2〉,...,〈un,Λn〉)=(1-∏j=1n(1-ℏΛφ(j)2)Φjei2π1-∏j=1n(1-(αΛφ(j)2π)2)Φj,∏j=1n(ƛΛφ(j))Φjei2π∏j=1n(βΛφ(j)2π)Φj),

where Φ = (Φ₁, Φ₂, . . . , Φ_n) ^T be the weighted vector of Λ_j with conditions such as Φ_j ∈ [0, 1] and ∑j=1nΦj=1 , Λ_φ(j) be the value of CPFOWA pair (u_j, Λ_j) having the jth largest u_j in (u_j, Λ_j) is referred to as the order inducing variable and Λ_j as the complex Pythagorean fuzzy argument.

Theorem 5. Let 〈u_j, Λ_j 〉 (j = 1, 2, . . . , n) be a collection of 2-tuples, then their aggregated value by using the I-CPFOWA operator is still CPFV.

Proof. Proof is similar to Theorem 3, so here we omit.

Definition 12. The I-CPFHA operator can be defined as:

(12)

I-CPFHAϖ,Φ(〈u1,Λ1〉,〈u2,Λ2〉,...,〈un,Λn〉)=(1-∏j=1n(1-ℏΛ̇φ(j)2)Φjei2π1-∏j=1n(1-(αΛ̇φ(j)2π)2)Φj,∏j=1n(ƛΛ̇φ(j))Φjei2π∏j=1n(βΛ̇φ(j)2π)Φj),

5Application of the new techniques for emergency decision-making

In this section, we develop some new methods such as CPFWA operator, CPFOWA operator, CPFHA operator, I-CPFOWA operator and I-CPFHA operator to control the outbreak of COVID-19. The spreading of coronavirus, is so fast, therefore the WHO collaborates with public health experts and laboratory organizations, clinical management. This problem is taken from [49], where the author used generalized interval-valed Einstein aggregation operators, but here we used complex algebraic aggregation operators.

Algorithm: Let A ={ A₁, A₂, . . . , A_n } be a finite set of n alternatives and Ç = { Ç₁, Ç₂, . . . , Ç_m } be a finite set of m different criteria whose weight vector is Φ = (Φ₁, Φ₂, Φ₃, . . . , Φ_m) ^T such that Φ_j ∈ [0, 1] and ∑j=1mΦj=1 . To determine the more suitable alternative for selection, the proposed approaches are used to develop a MAGDM problrm under the complex Pythagorean fuzzy environment. The major steps are the following:

Step 1: Collect information of the all experts about the alternatives under the different criteria.

Step 2: Using the proposed operators to calculate all the preference values.

Step 3: To determine the scores of all preference values.

Step 4: Based on the score function select the most suitable alternative.

6Different techniques for emergency decision making

In this section, the experts provide their ideas in the terms of linguistic variables:

Based of Table 1, the health experts evaluated fuzzy linguistic information and their corresponding complex Pythagorean fuzzy numbers presented in Table 2.

Table 1

Linguistic variables

	Ç₁	Ç₂	Ç₃	Ç₄	Ç₅	Ç₆	Ç₇	Ç₈
HCS	G	G	E	M	VVB	VB	E	VG
RC	M	VB	VG	G	G	E	B	VG
LD	B	G	E	M	VB	G	VVB	G
RN	G	G	M	VB	G	VG	E	VVB

Health care system (HCS), Risk communication (RC), Lock down (LD), Research need (RN).

Table 2

Linguistic variables and their corresponding CPF-numbers

Linguistic variables	Corresponding CPF-numbers
Very Very Bad (VVB)	(0.2e^i2π(0.2), 0.9e^i2π(0.9))
Very Bad (VB)	(0.2e^i2π(0.3), 0.9e^i2π(0.8))
Bad (B)	(0.4e^i2π(0.4), 0.8e^i2π(0.8))
Medium (M)	(0.6e^i2π(0.5), 0.6e^i2π(0.5))
Good (G)	(0.8e^i2π(0.8), 0.5e^i2π(0.4))
Very Good (VG)	(0.8e^i2π(0.8), 0.3e^i2π(0.2))
Excellent (E)	(0.9e^i2π(0.9), 0.2e^i2π(0.2))

6.1Some algebraic aggregation operators

In this section, we develop some algebraic operators for emergency decision making to select the best alternative from all alternatives under consideration. The major steps of the proposed methods are the following.

Step 1: In this step, we provide the evauation of emergency measures to reduce the outbreak of coronavirus in linguistic terms. After that we convert the linguistic terms into CPFNs.

Step 2: Utilize CPFWA operators to calculate all the preference values.

Step 3: Calculate the scores of all preference values.

Step 4: Ranking the alternatives and select that having the highest score function.

6.2Some induced aggregation operators

In this section, we develop some induced operators for emergency decision making. The major steps are the following.

Step 1: In this step, we convert the linguistic terms in the terms of I-CPFNs.

Step 2: Utilize I-CPFWA operator to calculate all the preference values.

Step 3: Calculate the score of preference values

Step 4: Select that alternative having the highest score function.

7Example for the proposed methods

In this section, we present a real case such as people health emergency decision making for outbreak of coronavirus, reported in China.

Case study: In this section, we develop an application of the proposed methods for COVID-19, reported in China. The first case was reported December 2019, in Wuhan, China and appeared in the whole world in March 2020 as a prevalent and widespread disease. The Chinese government implemented the largest lockdown in human history in the beginning of 2020, due to this millions of people were distressed. The ratio of spreading of COVID-19, is very high, therefore the World health Organization (WHO) collaborates with public health experts and laboratory organizations, prevention and checking of diseases, clinical management and mathematical modeling. To show the validity and rationality of the new methods, we develop a real case that is COVID-19. Generally the following eight factors reduce the spreading of COVID-19.

Clinical Management (Ç₁): Currently there are no suitable treatments for COVID-19, however Clinical management requires timely adoption of appropriate creativities for infection avoidance and control for complication management, providing strategic organ care where necessary.

First -aid Training (Ç₂): The spread rate of COVID-19 is very high, so avoid and sidestep from those people having the symptoms of this disease. So it is recommended that attend a fully supervised online or practical first aid course persons to learn how to get out medical emergency.

Increased Personal Protective Equipment (Ç₃): The shortage of testing kits is a main factor, in this situation we have to increase the testing kits production. It is the responsibility of local government to provide some necessary equipment’s such as face masks, gloves, ventilators and gowns in every region in the country, because the best way is personal hygiene.

Expert Technician (Ç₄): The spread rate of this disease is very high, therefore the expert technician are working on potential and suitable vaccines.

Banned intra-city Transportation (Ç₅): It is necessary for local government that takes step for banned intra-city transportation and stopped all flights for local people safety.

Global Uncertainty (Ç₆): Economic effect: The whole world depends on the Chinese growth. But due to the coronavirus transport sectors will damage the economic and also harm trade and consumption, so coronavirus directly affected the global market in the current environment.

Planning and cooperation country level (Ç₇): It is necessary for government to cooperation and collaboration with its provinces, to reduce the uncertainty of COVID-19 in has country.

Monitoring (Ç₈): It is necessary for government to appoint the experts to track the current situation and provide the positive suggestion on how to control the current situation.

7.1By algebraic aggregation operators

Step 1: On the based of Table 1: The health experts weighed their information in the form fuzzy linguistic variables shown in Table 3:

Table 3

Corresponding linguistic information

	HCS	RC	LD	RN
Ç₁	G	M	B	G
Ç₂	G	VB	G	G
Ç₃	E	VG	E	M
Ç₄	M	G	M	VB
Ç₅	*VVB*	G	VB	G
Ç₆	VB	E	G	VG
Ç₇	E	B	*VVB*	E
Ç₈	VG	VG	G	*VVB*

Convert linguistic terms into fuzzy information:

Table 4

Complex Pythagorean fuzzy information

	HCS	RC	LD	RN
Ç₁	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.6e^i2π(0.5), 0.6e^i2π(0.5))	(0.4e^i2π(0.4), 0.8e^i2π(0.8))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))
Ç₂	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.2e^i2π(0.3), 0.9e^i2π(0.8))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))
Ç₃	(0.9e^i2π(0.9), 0.2e^i2π(0.2))	(0.8e^i2π(0.8), 0.3e^i2π(0.2))	(0.9e^i2π(0.9), 0.2e^i2π(0.2))	(0.6e^i2π(0.5), 0.6e^i2π(0.5))
Ç₄	(0.6e^i2π(0.5), 0.6e^i2π(0.5))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.6e^i2π(0.5), 0.6e^i2π(0.5))	(0.2e^i2π(0.3), 0.9e^i2π(0.8))
Ç₅	(0.2e^i2π(0.2), 0.9e^i2π(0.9))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.2e^i2π(0.3), 0.9e^i2π(0.8))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))
Ç₆	(0.2e^i2π(0.3), 0.9e^i2π(0.8))	(0.9e^i2π(0.9), 0.2e^i2π(0.2))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.8e^i2π(0.8), 0.3e^i2π(0.2))
Ç₇	(0.9e^i2π(0.9), 0.2e^i2π(0.2))	(0.4e^i2π(0.4), 0.8e^i2π(0.8))	(0.2e^i2π(0.2), 0.9e^i2π(0.9))	(0.9e^i2π(0.9), 0.2e^i2π(0.2))
Ç₈	(0.8e^i2π(0.8), 0.3e^i2π(0.2))	(0.8e^i2π(0.8), 0.3e^i2π(0.2))	(0.8e^i2π(0.8), 0.5e^i2π(0.4))	(0.2e^i2π(0.2), 0.9e^i2π(0.9))

Step 2: Using the CPFWA aggregation operators to calculate the all preference values, where Φ = (0.10, 0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.15) ^T be their weighted vector. Thus we get

ℑ1=(0.757ei2π(0.754),0.437ei2π(0.378)),ℑ2=(0.750ei2π(0.747),0.446ei2π(0.368))ℑ3=(0.696ei2π(0.691),0.570ei2π(0.504)),ℑ4=(0.738ei2π(0.735),0.485ei2π(0.411))

Step 3: In this step, we calculate the scores of all preference values as follows:

s(ℑ1)=((0.757)2-(0.437)2)+14π2((0.754)2-(0.378)2)=0.807s(ℑ2)=((0.750)2-(0.446)2)+14π2((0.747)2-(0.368)2)=0.786

s(ℑ3)=((0.696)2-(0.570)2)+14π2((0.691)2-(0.504)2)=0.382s(ℑ4)=((0.738)2-(0.485)2)+14π2((0.735)2-(0.411)2)=0.680

Step 4: Based on the score, we have A₁ ≻ A₂ ≻ A₄ ≻ A₃. Thus the best alternative is A₁.

7.2By induced aggregation operators

Step 1: On the based of Table 1 the health experts weighed their information in the form of inducing fuzzy linguistic variables shown in Table 5:

Table 5

Inducing complex Pythagorean fuzzy information

	HCS	RC	LD	RN
Ç₁	〈0.9,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.8,(0.6ei2π(0.5),0.6ei2π(0.5))〉	〈0.6,(0.4ei2π(0.4),0.8ei2π(0.8))〉	〈0.9,(0.8ei2π(0.8),0.5ei2π(0.4))〉
Ç₂	〈0.8,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.7,(0.2ei2π(0.3),0.9ei2π(0.8))〉	〈0.7,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.8,(0.8ei2π(0.8),0.5ei2π(0.4))〉
Ç₃	〈0.6,(0.9ei2π(0.9),0.2ei2π(0.2))〉	〈0.6,(0.8ei2π(0.8),0.3ei2π(0.2))〉	〈0.5,(0.9ei2π(0.9),0.2ei2π(0.2))〉	〈0.7,(0.6ei2π(0.5),0.6ei2π(0.5))〉
Ç₄	〈0.5,(0.6ei2π(0.5),0.6ei2π(0.5))〉	〈0.5,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.4,(0.6ei2π(0.5),0.6ei2π(0.5))〉	〈0.6,(0.2ei2π(0.3),0.9ei2π(0.8))〉
Ç₅	〈0.7,(0.2ei2π(0.2),0.9ei2π(0.9))〉	〈0.4,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.8,(0.2ei2π(0.3),0.9ei2π(0.8))〉	〈0.5,(0.8ei2π(0.8),0.5ei2π(0.4))〉
Ç₆	〈0.3,(0.2ei2π(0.3),0.9ei2π(0.8))〉	〈0.3,(0.9ei2π(0.9),0.2ei2π(0.2))〉	〈0.9,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.3,(0.8ei2π(0.8),0.3ei2π(0.2))〉
Ç₇	〈0.4,(0.9ei2π(0.9),0.2ei2π(0.2))〉	〈0.2,(0.4ei2π(0.4),0.8ei2π(0.8))〉	〈0.2,(0.2ei2π(0.2),0.9ei2π(0.9))〉	〈0.2,(0.9ei2π(0.9),0.2ei2π(0.2))〉
Ç₈	〈0.2,(0.8ei2π(0.8),0.3ei2π(0.2))〉	〈0.1,(0.8ei2π(0.8),0.3ei2π(0.2))〉	〈0.3,(0.8ei2π(0.8),0.5ei2π(0.4))〉	〈0.1,(0.2ei2π(0.2),0.9ei2π(0.9))〉

Step 2: Using the I-CPFWA qggregation operators to calculate all the preference values, where Φ = (0.10, 0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.15) ^T be their weighted vector, then we have

ℑ1=(0.758ei2π(0.755),0.435ei2π(0.375)),ℑ2=(0.750ei2π(0.747),0.446ei2π(0.368))ℑ3=(0.701ei2π(0.753),0.558ei2π(0.496)),ℑ4=(0.738ei2π(0.735),0.485ei2π(0.411))

Step 3: In this step, we calculate the scores of all preference values as follows:

s(ℑ1)=0.814,s(ℑ2)=0.786,s(ℑ3)=0.501,s(ℑ4)=0.680

Step 4: Based on the score, we have A₁ ≻ A₂ ≻ A₄ ≻ A₃. Thus the best alternative is A₁.

Here we, aggregated information of all the proposed methods as follows:

Table 6

Aggregated CPF information by CPFWA operator

Alternatives	Aggregated values
HCS	(0 . 758e^i2π(0.755), 0 . 435e^i2π(0.375))
RC	(0.750e^i2π(0.747), 0 . 446e^i2π(0.368))
LD	(0 . 701e^i2π(0.753), 0 . 558e^i2π(0.496))
RN	(0 . 738e^i2π(0.735), 0 . 485e^i2π(0.411))

Table 7

Aggregated CPF information by CPFOWA operator

Alternatives	Aggregated values
HCS	(0 . 764e^i2π(0.756), 0 . 434e^i2π(0.382))
RC	(0.761e^i2π(0.746), 0 . 432e^i2π(0.367))
LD	(0 . 702e^i2π(0.752), 0 . 559e^i2π(0.494))
RN	(0 . 740e^i2π(0.744), 0 . 484e^i2π(0.413))

Table 8

Aggregated CPF information by CPFHA operator

Alternatives	Aggregated values
HCS	(0 . 774e^i2π(0.764), 0 . 421e^i2π(0.379))
RC	(0.771e^i2π(0.756), 0 . 442e^i2π(0.377))
LD	(0 . 712e^i2π(0.753), 0 . 560e^i2π(0.492))
RN	(0 . 741e^i2π(0.754), 0 . 485e^i2π(0.423))

Table 9

Aggregated CPF information by I-CPFOWA operator

Alternatives	Aggregated values
HCS	(0 . 758e^i2π(0.755), 0 . 435e^i2π(0.375))
RC	(0.750e^i2π(0.747), 0 . 446e^i2π(0.368))
LD	(0 . 701e^i2π(0.753), 0 . 558e^i2π(0.496))
RN	(0 . 738e^i2π(0.735), 0 . 485e^i2π(0.411))

Table 10

Aggregated CPF information by I-CPFHA operator

Alternatives	Aggregated values
HCS	(0 . 756e^i2π(0.756), 0 . 432e^i2π(0.365))
RC	(0.751e^i2π(0.757), 0 . 447e^i2π(0.378))
LD	(0 . 721e^i2π(0.743), 0 . 568e^i2π(0.492))
RN	(0 . 748e^i2π(0.725), 0 . 495e^i2π(0.402))

Table 11

Score function of the proposed methods

	CPFWA	CPFOWA	CPFHA	I-CPFOWA	I-CPFHA
HCS	0.807	0.820	0.861	0.814	0.823
RC	0.786	0.814	0.828	0.786	0.794
LD	0.382	0.501	0.518	0.501	0.507
RN	0.680	0.696	0.703	0.680	0.678

Table 12

Comparative analysis of the proposed methods

Methods	Scores	Ranking
CPFWA	s (ℑ ₁) ≻ s (ℑ ₂) ≻ s (ℑ ₄) ≻ s (ℑ ₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
CPFOWA	s (ℑ ₁) ≻ s (ℑ ₂) ≻ s (ℑ ₄) ≻ s (ℑ ₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
CPFHA	s (ℑ ₁) ≻ s (ℑ ₂) ≻ s (ℑ ₄) ≻ s (ℑ ₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
I-CPFWA	s (ℑ ₁) ≻ s (ℑ ₂) ≻ s (ℑ ₄) ≻ s (ℑ ₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃
I-CPFHA	s (ℑ ₁) ≻ s (ℑ ₂) ≻ s (ℑ ₄) ≻ s (ℑ ₃)	A₁ ≻ A₂ ≻ A₄ ≻ A₃

Coronavirus information in Pakistan

Fig. 1

Fig. 2

Fig. 3

8Limitations

Complex Pythagorean fuzzy set is the generalization of the existing studies such as fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, complex fuzzy set, and complex intuitionistic fuzzy set by considering much more information related to an object during the process and to handle the two-dimensional information in a single set.

However, these novel methods have some limitations such as: If M=(ℏeiα,ƛƛeiβ) be complex Pythagorean fuzzy numbers, where i=-1 , ℏ, ƛ∈ [0, 1] are called complex membership and non-membership function respectively, and α, β ∈ [0, 2π] with some conditions such as, 0 ⩽ ℏ ² + ƛ² ⩽ 1 and 0⩽(α2π)2+(β2π)2⩽1 . Therefore throughout this paper we consider those complex Pythagorean fuzzy numbers which satisfy the above restriction.

8.1Comparative analysis

Complex Pythagorean fuzzy set is the generalization of the existing studies such as fuzzy set [18], intuitionistic fuzzy set [19], Pythagorean fuzzy set [24], complex fuzzy set [32], and complex intuitionistic fuzzy set [36, 37] by considering much more information related to an object during the process and to handle the two-dimensional information in a single set. For instance fuzzy set (with only crisp membership degrees with amplitude term only), intuitionistic fuzzy set and Pythagorean fuzzy set (with a real-valued membership and non-membership degrees and only considered amplitude term), complex fuzzy set (contains only complex-valued membership degree), complex intuitionistic fuzzy set contains information (both the membership and non-membership degrees are complex valued with condition their sum less than or equal to one), but complex Pythagorean fuzzy set contains information (both the membership and non-membership degrees are complex valued with condition sum of their squares less than or equal to one). Thus, the proposed aggregation operators under CPFSs environment are more generalizations than the existing operators.

Mahmood et al. [46], Akram et al. [47, 48] developed some aggregation operators using complex fuzzy information, but these papers having no concept of inducing aggregation operators and averaging aggregation operators which are more powerful tools for decision making problems. Thus in this paper we introduce the notion of some novel aggregation operators such as CPFWA operator, CPFOWA operator, CPFHA operator, I-CPFOWA operator, I-CPFHA operator. Hence our proposed methods are more flexible and effectiveness as compared to their existing methods.

The proposed methods not only perform for CPF data rather they can be successfully applied to one dimensional phenomena for Pythagorean fuzzy information and intuitionistic fuzzy information by taking their phase terms equal to zero. So, the proposed methods are more flexible to overcome the limitations of the existing methods.

Table 13

Stated the sensitivity analysis

Model	Uncertainty	Falsity	Hesitation	Periodicity	2-D information	Square in power
FSs	Yes	No	No	No	No	No
IFSs	Yes	Yes	Yes	No	No	No
PFSs	Yes	Yes	Yes	No	No	No
CFSs	Yes	No	No	Yes	Yes	No
CIFSs	Yes	Yes	Yes	Yes	Yes	No
CPFSs	Yes	Yes	Yes	Yes	Yes	Yes

9Conclusion

COVID-19 is an infectious disease that can be spread through touch and is believed to be spread throughout the entire population through direct contact between people. Outbreak prevention measures aimed at reducing the number of mixed populations can slow the peak and reduce the final extent of the epidemic. So at the current position it is a challenge for the world to control the spreading of COVID-19. Therefore in this paper we have developed a complex Pythagorean fuzzy model to control and analysis the infection of COVID-19. For this, we presented some methods such as CPFWA operator, CPFOWA operator, CPFHA operator, I-CPFOWA operator and I-CPFHA operator. At the end of the paper, we have developed an illustrative emergency situation of COVID-19 to show the effectiveness along with a sensitivity analysis.

In further research, it is necessary to give the applications of these operators to the other domains such as, inducing variables, pattern recognition, Confidence levels, Hamacher operators, Power operators, Symmetric operator, Logarithmic operators, Dombi operators, Linguistic terms, trigonometric operation, ranking method for normal intuitionistic sets, etc.

Acknowledgments

This research work was supported by Higher Education Commission (HEC) under the project “Startup Research Grant Program (SRGP), Project title, “Complex Intelligent Decision Support Models for-Decision-Making Problem in COVID19” (SRGP/NAHE/HEC/2020/225).

References

[1]	Yang C. and Wang J. , A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math Biosci Eng 17: ((2020) ), 2708–2724.
[2]	Toğaçar M. , Ergen B. and Cömert Z. , COVID-19 detection using deep learning models to exploit Social Mimic Optimization and structured chest X-ray images using fuzzy color and stacking approaches, Comput Biol Med 122: ((2020) ), 103805.
[3]	Tuite A.R. , Fisman D.N. and Greer A.L. , Mathematical Modeling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada, Can Med Assoc J 192: ((2020) ), E497–E505.
[4]	Ciufolini I. and Paolozzi A. , Mathematical prediction of the time evolution of the COVID-19 pandemic in Italy by a Gauss error function and Monte Carlo simulations, Eur Phys J Plus ((2020) ), 135–355.
[5]	Sohail A. and Nutini A. , Forecasting the timeframe of coronavirus and human cells interaction with reverse engineering, Prog Biophys Mol Bio 155: ((2020) ), 29–35.
[6]	Khan M.A. and Atangana A. , Modeling the dynamics of novel coronavirus -nCov) with fractional derivative, Alex Eng J 59: ((2020) ), 2379–2389.
[7]	Ashraf S. and Abdullah S. , Emergency decision support modeling for COVID-19 based on spherical fuzzy information, Int J Intell Syst 35: ((2020) ), 1601–1645.
[8]	Ashraf S. , Abdullah S. and Almagrabi A.O. , A new emergency response of spherical intelligent fuzzy decision process to diagnose of COVID19, Soft Computing. https://doi.org/10.1007/s00500-020-05287-8
[9]	Elsonbaty A. , Sabir Z. , Ramaswamy R. and Adel W. , Dynamical analysis of a novel discrete fractional sitrs model for Covid-19, Fractals 29: ((2021) ), 1–15.
[10]	Sanchez Y.G. , Sabir Z. and Guirao J.L.G. , Design of a nonlinear sitr fractal model based on the dynamics of a novel coronavirus (COVID-19), Fractals 28: ((2020) ), 1–6.
[11]	Umar M. , Sabir Z. , Raja M.A.Z. , Amin F. , Saeed T. and Sanchez Y.G. , Integrated neuro-swarm heuristic with interior-point for nonlinear SITR model for dynamics of novel COVID-19, Alexandria Engineering Journal 60: ((2021) ), 2811–2824.
[12]	Umar M. , Sabir Z. , Raja M.A.Z. , Shoaib M. , Gupta M. and Sanchez Y.G. , A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19, Dynamics 12: ((2020) ), 1628 1–17.
[13]	Umar M. , Sabir Z. , Raja M.A.Z. , Baskonus H.M. , Yao S.W. and Ilhan E. , A novel study of Morlet neural networks to solve the nonlinear HIV infection system of latently infected cells, Results in Physics 25: ((2021) ), 104235.
[14]	Sanchez Y.G. , Sabir Z. , Gunerhan H. and Baskonus H.M. , Analytical and Approximate Solutions of a Novel Nervous Stomach Mathematical Model, Discrete Dynamics in Nature and Society ((2020) ), 5063271 1–9.
[15]	Sanchez Y.G. , Umar M. , Sabir Z. , Guirao J.L.G. and Raja M.A.Z. , Solving a class of biological HIV infection model of latently infected cells using heuristic approach, Discrete and Continuous Dynamical Systems Series 14: ((2021) ), 611–3628. doi:10.3934/dcdss.2020431.
[16]	Sabir Z. , Sakar M.G. , Yeskindirova M. and Saldir O. , Numerical investigations to design a novel model based on the fifth order system of Emden–Fowler equations, Theoretical & Applied Mechanics Letters 10: ((2020) ), 333–342.
[17]	Sabir Z. , G”unerhan H. and Guirao Juan L.G. , On a New Model Based on Third-Order Nonlinear Multisingular Functional Differential Equations, Mathematical Problems in Engineering ((2020) ), 1683961, 1–9.
[18]	Zadeh L.A. , Fuzzy sets, Information Control 8: ((1965) ), 338–353.
[19]	Atanassov K.T. , Intuitionistic fuzzy sets, Fuzzy Sets Syst 20: ((1986) ), 87–96.
[20]	Xu Z.S. , Intuitionistic fuzzy aggregation operators, IEEE Trans Fuzzy Syst 15: ((2007) ), 1179–1187.
[21]	Xu Z.S. and Yager R.R. , Some geometric aggregation operators based on intuitionistic fuzzy sets, Int J Gen Syst 35: ((2006) ), 417–433.
[22]	Wang W. and Liu X. , Intuitionistic Fuzzy Geometric Aggregation Operators Based on Einstein Operations, International Journal of Intelligent Systems 26: ((2011) ), 1049–1075.
[23]	Wang W. and Liu X. , Intuitionistic Fuzzy Information Aggregation Using Einstein Operations, IEEE Trans Fuzzy Systems 20: ((2012) ), 923–938.
[24]	Yager R.R. , Pythagorean fuzzy subsets, In Edmonton, Canada, Proc Joint IFSA World Congress and NAFIPS Annual Meeting ((2013) ), 57–61.
[25]	Rahman K. , Abdullah S. , Ahmed R. and Ullah M. , Pythagorean fuzzy Einstein weighted geometric aggregation operator and their application to multiple attribute group decision making, Journal of Intelligent & Fuzzy Systems 33: ((2017) ), 635–647.
[26]	Rahman K. , Abdullah S. , Jamil M. and Khan M.Y. , Some Generalized Intuitionistic Fuzzy Einstein Hybrid Aggregation Operators and Their Application to Multiple Attribute Pythagorean Fuzzy Aggregation Operators Using Einstein-Norm and-Conorm, Journal of Intelligent & Fuzzy Systems 37: ((2019) ), 3721–3742.
[27]	Rahman K. , Ali A. , Abdullah S. and Amin F. , Approaches to Multi-Attribute Group Decision Making Based on Induced Interval-Valued Pythagorean Fuzzy Einstein Aggregation Operator, New Mathematics and Natural Computation 14: ((2018) ), 343–361.
[28]	Rahman K. , Abdullah S. , Ali A. and Amin F. , Pythagorean fuzzy ordered weighted averaging aggregation operator and their application to multiple attribute group decision making, EURO Journal on Decision Processes 8: ((2020) ), 61–77.
[29]	Rahman K. , A Series of Generalized Induced Einstein Aggregation Operators and Their Application to Group Decision-Making Process Based on Pythagorean Fuzzy Numbers, Granular Computing 6: ((2021) ), 241–254.
[30]	Rahman K. , Abdullah S. and Ghani F. , Some New Generalized Interval-Valued Pythagorean Fuzzy Aggregation Operators Using Einstein t-Norm and t-Conorm, Journal of Intelligent & Fuzzy Systems 37: ((2019) ), 3721–3742.
[31]	Rahman K. and Abdullah S. , Some Induced Generalized Geometric Aggregation Operators Based on Interval-Valued Pythagorean Fuzzy Numbers, Math Sci 14: ((2020) ), 397–407.
[32]	Ramot D. , Milo R. , Fiedman M. and Kandel A. , Complex fuzzy sets, IEEE Trans Fuzzy Syst 10: ((2002) ), 171–186.
[33]	Yazdanbakhsh O. and Dick S. , A systematic review of complex fuzzy sets and logic, Fuzzy Sets Syst 338: ((2018) ), 1–22.
[34]	Greenfield S. , Chiclana F. and Dick S. , Interval-valued complex fuzzy logic, In Vancouver, BC, Canada, Proceedings of the IEEE International Conference on Fuzzy Systems ((2016) ), 1–6.
[35]	Ramot D. , Friedman M. , Langholz G. and Kandel A. , Complex fuzzy logic, IEEE Trans Fuzzy Syst 11: ((2003) ), 450–461.
[36]	Alkouri A. and Salleh A. , Complex intuitionistic fuzzy sets, AIP Conf Proc 14: ((2012) ), 464–470.
[37]	Alkouri A.U.M. and Salleh A.R. , Complex Atanassov intuitionistic fuzzy relation, Abstr Appl Anal ((2013) ), 1–18.
[38]	Rani D. and Garg H. , Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision-making process, Int J Uncertain Quantif 7: ((2017) ), 423–439.
[39]	Rani D. and Garg H. , Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision making, Expert Syst 35: ((2018) ), e12325.
[40]	Garg H. and Rani D. , Some generalized complex intuitionistic fuzzy aggregation operators and their application to multi criteria decision-making process, Arab J Sci Eng 44: ((2019) ), 2679–2698.
[41]	Garg H. and Rani D. , Generalized Geometric Aggregation Operators Based on T-Norm Operations for Complex Intuitionistic Fuzzy Sets and Their Application to Decision-making, Cognitive Computation. https://doi.org/10.1007/s12559-019-09678-4
[42]	Garg H. and Rani D. , Complex Interval-valued Intuitionistic Fuzzy Sets and their Aggregation Operators, Fundamenta Informaticae 164: ((2019) ), 61–101.
[43]	Ullah K. , Mahmood T. , Ali Z. and Jan N. , On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition, Complex Intell Syst (2019) .
[44]	Akram M. and Naz S. , A Novel Decision-Making Approach under Complex Pythagorean Fuzzy Environment, Math Comput Appl 24: ((2019) ). doi:10.3390/mca24030073.
[45]	Akram M. , Garg H. and Zahid K. , Extensions of ELECTRE-I and TOPSIS methods for group decision-making under complex Pythagorean fuzzy environment, Iranian Journal of Fuzzy Systems 17: ((2020) ), 147–164.
[46]	Mahmood T. , Ali Z. , Ullah K. , Khan Q. , AL Salman H. , Gumaei A. and Rahman S.M.M. , Complex Pythagorean fuzzy aggregation operators based on confidence levels and their applications, Mathematical Biosciences and Engineering 19: ((2022) ), 1078–1107.
[47]	Akram M. , Alsulami S. and Zahid K. , A Hybrid Method for Complex Pythagorean Fuzzy Decision Making, Mathematical Problems in Engineering. doi.org/10.1155/2021/9915432.
[48]	Akram M. , Kahraman C. and Zahid K. , Group decision-making based on complex spherical fuzzy VIKOR approach, Knowledge-Based Systems 216: ((2021) ), Article ID 106793.
[49]	Rahman K. , Mathematical Calculation of the COVID-19 Disease in Pakistan by Emergency Response Modeling based on Intuitionistic Fuzzy Decision Process, New Mathematics and Natural Computation 17: ((2022) ), 1–41.