Novel Decision Making Methodology under Pythagorean Probabilistic Hesitant Fuzzy Einstein Aggregation Information

1  Introduction

Coronavirus disease 2019 (COVID-19) is a highly contagious sickness that can range from mild to severe. Fever, cough, and shortness of breath are common symptoms of COVID-19, which can escalate to acute respiratory distress syndrome, which causes breathing problems, a low ratio of the partial pressure arterial oxygen to the fraction of inspired oxygen (PaO2/FiO2), and multi-organ failure [1,2]. MSCs were first found in the bone marrow in 1991, but they can now be extracted from a variety of tissues, including the umbilical cord and adipose tissue [3]. MSCs have anti-inflammatory characteristics due to their ability to modify cellular immune responses. MSCs can be used to treat a variety of disorders due to their regeneration capabilities and immunoregulatory function [4]. Finding and testing acceptable MSC methods with effective utilization may be difficult since MSCs must first meet the selection and general requirements. Additionally, dealing with very precious MSCs and a huge number of patients needs an intelligent transfusion approach for COVID-19 patients with varying degrees of sickness severity, making prioritization problematic. This method falls under a multicriteria decision making (MCDM) problem since the needs in the second stage should follow the ranking of emergency COVID-19 patients in each severity level according to the evaluation criteria. As a result, choosing the finest MSC preparation process that conforms with national health laws, as well as selecting the most suited patients for the treatment, is critical for delivering effective cell-based therapy. To assist medical teams in solving the complicated obstacles outlined above, intelligent computing concerns should be addressed in order to present a fully intelligent real-time MSC transfusion framework based on the emergency severity of COVID-19 patients. Triage and prioritization are used in the medical field to either classify patients into distinct emergency categories or to prioritize them in the hereafter. Triage is an emergency medical procedure that classifies the severity of a patient’s condition and defines the treatment sequence. Automating this task could considerably enhance the quality of patient treatment and hospitalization, allowing for the greatest number of lives to be saved. Patients with chronic heart disease were previously classified into distinct emergency levels using an automated triage approach.

In fact, a system’s complexity grows more and more, making it impossible for DMs to select the best option among a wide range of viable options. It is difficult to put into words how challenging achieving a single objective is, yet it is not unattainable. Several corporations are faced with encouraging workers, identifying aims, and creating viewpoint. As a result, whether affecting individuals or panels, organizational decisions must consider a number of goals simultaneously. In this context, each DM should be restricted to finding the optimum solution for each factor associated with real problems, based on criteria that can be handled consciously. For decision-makers, establishing suitable ways for deciding on the appropriate option has become increasingly vital.

Classic or crisp procedures may not necessarily be the most successful when dealing with unstructured situations in decision-making circumstances. Zadeh [5] developed fuzzy sets (FS) in 1965 as a technique to manage such inconsistency. In FSs, Zadeh assigns membership grades to a set of components in the interval [0,1]. Zadeh’s work on this topic is notable since many of the set theoretic elements of crisp circumstances were described for fuzzy sets and it plays an important role in decision making [6,7].

Atanassov [8] investigated the intuitionistic fuzzy set (IFS), a better variation of the fuzzy set (FS) that includes both membership and non-membership degrees. IFSs have been shown to be useful and commonly used by researchers to evaluate uncertainty and instability in data throughout the previous few decades. Several researchers work on it, such as Yue [9] proposed the bilateral matching decision-making for knowledge innovation management taking into account matching willingness by using the intuitionistic fuzzy set environment, group decision making [10], aggregation operators [11], geometric aggregation operators [12], etc.

Wherever the MG and NMG numbers are more than one, the DMs are frequently confronted with certain qualities. In that circumstance, IFSs have little chance of producing a satisfactory result. To handle such a circumstance, Yager [13] proposed the Pythagorean fuzzy set (PyFS), which is a broad form of IFS. For PyFS, the sum of the squares of MG and NMG equals one. Wan et al. [14] introduced a three-step process for managing haze using the Pythagorean fuzzy multi-attribute group decision making, Yager et al. [15,16] also presented decision making procedures by using PyFS, Peng et al. [17] proposed some PyFS results. Torra [18] created the hesitant fuzzy set (HFS) to demonstrate the FS more firmly than the previous classical fuzzy set modifications, which needed the membership to have a set of specific values. Yue et al. [19] introduced the methodology for matching two-sided under hesitant fuzzy information and discussed their application related to smart intelligence, Xia et al. [20] presented hesitant fuzzy aggregation information, Liu et al. [21] introduced generalized power aggregation operator, Yu et al. [22] presented the MCDM under choquet integral based on HFS, Zhang [23] introduced the power aggregation operator, Liao et al. [24,25] introduced the hybrid and extended hybrid weighted aggregation operator on HFS, Joshi et al. [26] introduced TOPSIS method based on interval valued IHF choquet integral and similarly many other researchers also worked on HFS.

The concept of Pythagorean HFS (PyHFS) was suggested by Khan et al. [27]. They developed operators to aggregate the data and proposed a comparison mechanism. Wan et al. [28,29] presented the decision methodologies under Pythagorean fuzzy information and Garg [30] presented hesitant PyF (HPyF) maclaurin symmetric mean operator, Garg et al. [31,32] introduced aggregation operators on HPyFS and Khan et al. [33] introduced hybrid aggregation operators on PyHFS. Zulqarnain et al. [34] developed some aggregation operators under interval-valued Pythagorean fuzzy soft information. Batool et al. [35] proposed the notion of a Pythagorean probabilistic hesitant fuzzy set and investigated its application in decision-making approaches. PyPHFS has both positive and negative hesitant membership degrees with the restriction that the square sum of the positive and negative hesitant membership degrees is either less than or equal to one. The DMs in PHFS is constrained to a single domain and are unaware of the negative membership degree or its possible occurrence probabilities. Each hesitantly negative membership level also has a preference over others. If one DM includes values 0.3, 0.4, and 0.6 for positive membership degree with their associated preference values 0.1 and 0.9, other DM may reject. As an example, the DMs may offer their opinions on DM difficulties in the form of multiple alternative values. Under the PyPHFS idea, the possibility of rejection level with hesitation is taken into account. Despite HFSs and PHFSs, the information about chances will reduce. The probability of occurrence with positive and negative membership degrees provides extra information on the degree of disagreement among the DMs. Batool et al. [35] also used its application for Fog-Haze Factor Assessment Problem.

Different priority levels are frequently assigned to each criterion in each emergency situation, adding to the task’s complexity. Ultimately, a COVID-19 patient prioritization process inside each emergency/triage situation necessitates a synchronized evaluation of the inverse relationship among the criteria, resulting in a trade-off. To address the first issue, an accurate automated triage guideline must be developed and implemented into the proposed MSC transfusion system. By prioritizing COVID-19 patients within each triage level, MCDM approaches are critical for overcoming the aforementioned problems. MCDM is a multi-objective decision model that is an extension of decision theory. It can address multi-criteria decision-making problems by constructing a decision matrix based on the intersection of each triage level’s evaluation criteria. The fundamental purpose of the MCDM is to rank/prioritize a group of alternatives based on a variety of evaluation criteria. For the MCDM technique Pythagorean Probabilistic Hesitant Fuzzy Information Aggregation when dealing with difficulties like this, Einstein Operations can be quite useful.

In this research work, we administered the Einstein aggregation operators (AOs) to PyPHFS environment, i.e., PyPHFEWA, PyPHFEOWA, PyPHFEHWA, PyPHFEWG, PyPHFEOWG and PyPHFEHWG operator. Idempotency, boundedness, and monotonicity are among the properties of the recommended operators that are established. The PyPHF AOs are taken into consideration by such operators in probabilistic scenarios, which is their main benefit. In the case of probabilistic material, the lack of PyPHFE AOs could lead to a scarcity of probabilistic information.

The paper is arranged in the prescribed manner. Section 2 contains the definition of FSs, IFSs, PyFSs, HFSs, and PyHFSs, PyPHFSs as well as PyPHFE aggregation operators. Section 3 is comprised of the PyPHFEWA, PyPHFEOWA, PyPHFEHWA, PyPHFEWG, PyPHFEOWG, and PyPHFEHWG operators, as well as their features such as monotonicity, idempotency, and boundedness. Section 4 presents how to use the MCDM technique. Section 5 contains the conclusion and discussion of the study.

2  Preliminaries

In this section, let’s review the fundamentals of fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets. These concepts will be used here when they have been reviewed.

Definition 2.1.[5] For a fixed set ¥. A FS O~ in ¥ is presented as

O~={⟨y¨h^,υO~(y¨h^)⟩|y¨h^∈¥},

for each y¨h^∈¥, the positive membership grade υO~:¥→Δ specifies the degree to which the element y¨h^∈O~, where Δ=[0,1] be the unit interval.

Definition 2.2.[8] For a fixed set ¥. An IFS O~ in ¥ is presented as

O~={⟨y¨h^,υO~(y¨h^),τO~(y¨h^)⟩|y¨h^∈¥},

for each y¨h^∈¥, the positive membership grade υO~:¥→Δ and the negative membership grade τO~:¥→Δ specifies the positive and negative degrees of membership of y¨h^ to the Intuitionistic fuzzy set O~, respectively, where Δ=[0,1] be the unit interval. Moreover, it is required that 0≤υO~(y¨h^)+τO~(y¨h^)≤1, for each y¨h^∈¥.

Definition 2.3.[10] For a fixed set ¥. A PyFS O~ in ¥ is presented as

O~={⟨y¨h^,υO~(y¨h^),τO~(y¨h^)⟩|y¨h^∈¥},

for each y¨h^∈¥, the positive membership grade υO~:¥→Δ and the negative membership grade τO~:¥→Δ specifies the positive and negative degrees of membership of y¨h^ to the Pythagorean fuzzy set O~, respectively, where Δ=[0,1] be the unit interval. Furthermore, it is required that 0≤υO~2(y¨h^)+τO~2(y¨h^)≤1, for each y¨h^∈¥. Conventionally, ∪χϰ=1−υO~2(y¨h^)−τO~2(y¨h^) is said to be degree of hesitancy of y¨h^ to O~.

In what follows, we symbolize by PyϝS^(¥) the group of all Pythagorean fuzzy sets in ¥. For ease, we shall symbolize the Pythagorean fuzzy number (PyFN) by the pair O~=(υO~,τO~).

Definition 2.4.[10] Let O~1,O~2∈PyϝS^(¥). Then

(1) O~1⊑O~2 if and only if υO~1(y¨h^)≤υO~2(y¨h^) and τO~1(y¨h^)≥τO~2(y¨h^) for each y¨h^∈¥. Clearly O~1=O~2 if O~1⊑O~2 and O~2⊑O~1.

(2) O~1⊓O~2={min(υO~1(y¨h^),υO~2(y¨h^)),max(τO~1(y¨h^),τO~2(y¨h^))|y¨h^∈¥},

(3) O~1⊔O~2={max(υO~1(y¨h^),υO~2(y¨h^)),min(τO~1(y¨h^),τO~2(y¨h^))|y¨h^∈¥},

(4) O~1c={τO~1(y¨h^),υO~1(y¨h^)|y¨h^∈¥}.

Definition 2.5.[18] For a fixed set ¥. A HFS O~ in ¥ is presented as

O~={⟨y¨h^,hϰ(y¨h^)⟩|y¨h^∈¥}

where hϰ(y¨h^) is in the form of set, that is contained some possible values in unit interval, i.e., [0,1] which represents the membership degree of y¨h^∈¥ in O~.

Definition 2.6.[18] Let O~1,O~2∈HFı~(¥). Then

(1) O~1c=⋃α∈hO~1(y¨h^){1−α};

(2) O~1⊔O~2=hO~1(y¨h^)⊻hO~2(y¨h^)=⋃α1∈hO~1(y¨h^)α2∈hO~2(y¨h^)max{α1,α2};

(3) O~1⊓O~2=hO~1(y¨h^)⊼hO~2(y¨h^)=⋃α1∈hO~1(y¨h^)α2∈hO~2(y¨h^)min{α1,α2};

Definition 2.7.[32] For a fixed set ¥. A PyHFS O~ in ¥ is presented as

O~={⟨y¨h^,υhϰ(y¨h^),τhϰ(y¨h^)⟩|y¨h^∈¥},

for each y¨h^∈¥, the positive membership grade υO~ and the negative membership grade τϰ are sets in some values in [0,1], specifies the possible positive and negative degrees of membership of y¨h^ to the Pythagorean hesitant fuzzy set O~, respectively. Furthermore, it is required that (max(υhϰ(y¨h^)))2+(min(τhϰ(y¨h^)))2≤1 and (min(υhϰ(y¨h^)))2+(max(τhϰ(y¨h^)))2≤1. The PyHF Number (PyHFN) will be denoted by the pair O~=(υhϰ,τhϰ) for convenience.

Definition 2.8.[32] Let O~1=(υhh^1,τhh^1) and O~2=(υhh^2,τhh^2) be two PyHFNs. The basic operational laws defined as

(1) O~1∪O~2={⋃y¨1∈υhh^1(lh^)y¨2∈υhh^2(lh^)(max(y¨1,y¨2)),⋃κ1∈τhh^1(lh^)κ2∈τhh^2(lh^)(min(κ1,κ2))};

(2) O~1∩O~2={⋃κ1∈τhh^1(lh^)κ2∈τhh^2(lh^)(min(κ1,κ2)),⋃y¨1∈υhh^1(lh^)y¨2∈υhh^2(lh^)(max(y¨1,y¨2))};

(3) O~1c={τhϰ,υhϰ}

Definition 2.9.[36] For a fixed set ¥. A PHFS O~ in ¥ is presented as

O~={⟨y¨h^,hϰ(y¨h^)/℘h^⟩|y¨h^∈¥}

where hϰ(y¨h^) is subset of [0,1] and hϰ(y¨h^)/℘h^ represents the membership degree of y¨h^∈¥ in O~. And ℘h^ represents the possibilities of hϰ(y¨h^), with the constraint that ∑h^℘h^=1.

2.1 Pythagorean Probabilistic Hesitant Fuzzy Set

Definition 2.10.For a fixed set ¥. A PyPHFS O~ in ¥ is presented as

O~={⟨y¨h^,υhϰ(y¨h^)/℘h^,τhϰ(y¨h^)/ch^⟩|y¨h^∈¥},

for all y¨h^∈¥, υhϰ(y¨h^) and τhϰ(y¨h^) are sets of some values in [0,1]. Where υhϰ(y¨h^)/℘h^ & τhϰ(y¨h^)/ch^ specifies the possible positive and negative degrees of membership of y¨h^ to the Pythagorean probabilistic hesitant fuzzy set O~, respectively. ℘h^ and ch^ represent the possibilities of membership grades Also, there is 0≤y¨i,κȷ^≤1 and 0≤℘i,cȷ^≤1 with ∑i=1L℘i≤1,∑ȷ^=1Lcȷ^≤1 (L is a positive integer to describe the number of elements contained in PyPHFS), where y¨i∈υhϰ(y¨h^),κȷ^∈τhϰ(y¨h^),℘i∈℘h^,cȷ^∈ch^. Furthermore, it is required that (max(υhϰ(y¨h^)))2+(min(τhϰ(y¨h^)))2≤1 and (min(υhϰ(y¨h^)))2+(max(τhϰ(y¨h^)))2≤1. For ease, we shall symbolize the Pythagorean Probabilistic Hesitant Fuzzy Number (PyPHFN) by the pair O~=(υhϰ/℘h^,τhϰ/ch^). The group of all Pythagorean probabilistic hesitant fuzzy sets in ¥ is symbolized by PyPHϝS^(¥).

Definition 2.11.Let O~1=(υhh^1/℘h^1,τhh^1/ch^1) and O~2=(υhh^2/℘h^2,τhh^2/ch^2) be two PyPHFNs. The basic operational laws defined as

(1) O~1∪O~2={⋃y¨1∈υhh^1(lh^),℘1∈℘h^1y¨2∈υhh^2(lh^),℘2∈℘h^2(max(y¨1/℘1,y¨2/℘2)),⋃κ1∈τhh^1(lh^),c1∈ch^1κ2∈τhh^2(lh^),c2∈ch^2(min(κ1/c1,κ2/c2))};

(2) O~1∩O~2={⋃y¨1∈υhh^1(lh^),℘1∈℘h^1y¨2∈υhh^2(lh^),℘2∈℘h^2(min(y¨1/℘1,y¨2/℘2)),⋃κ1∈τhh^1(lh^),c1∈ch^1κ2∈τhh^2(lh^),c2∈ch^2(max(κ1/c1,κ2/c2))};

(3) O~1c={τhϰ/ch^,υhϰ/℘h^}

Definition 2.12.Let O~1=(υhh^1/℘h^1,τhh^1/ch^1) and O~2=(υhh^2/℘h^2,τhh^2/ch^2) be two PyPHFNs and η>0(∈R), then their operations are presented as

(1) O~1⊕O~2={⋃y¨1∈υhh^1(lh^),y¨2∈υhh^2(lh^)℘1∈℘h^1,℘2∈℘h^2(y¨12+y¨22−y¨12y¨22/℘1℘2),⋃κ1∈τhh^1(lh^),κ2∈τhh^2(lh^)c1∈ch^1,c2∈ch^2(κ1κ2/c1c2)};

(2) O~1⊗O~2={⋃y¨1∈υhh^1(lh^),y¨2∈υhh^2(lh^)℘1∈℘h^1,℘2∈℘h^2(y¨1y¨2/℘1℘2),⋃κ1∈τhh^1(lh^),κ2∈τhh^2(lh^)c1∈ch^1,c2∈ch^2(κ12+κ22−κ12κ22/c1c2)};

(3) ηO~1={⋃y¨1∈υhh^1(lh^),℘1∈℘h^1(1−(1−y¨12)η/℘1),⋃κ1∈τhh^1(lh^),c1∈ch^1(κ1η/c1)};

(4) O~1η={⋃y¨1∈υhh^1(lh^),℘1∈℘h^1(y¨1η/℘1),⋃κ1∈τhh^1(lh^),c1∈ch^1(1−(1−κ12)η/c1)}.

Definition 2.13.For any PyPHFN O~=(υhϰ/℘h^,τhϰ/ch^), a score function be defined as

ı~(O~)=(1MO~∑y¨i∈υhh^,℘i∈℘hh^(y¨i⋅℘i))2−(1NO~∑κi∈τhϰ,ci∈chh^(κi⋅ci))2

where MO~ represents the number of elements in υhϰ and NO~ represents the number of elements in τhϰ.

Definition 2.14.For any PyPHFN O~=(υhϰ/℘h^,τhϰ/ch^), an accuracy function is defined as

h(O~)=(1MO~∑y¨i∈υhh^,℘i∈℘hh^(y¨i⋅℘i))2+(1NO~∑κi∈τhϰ,ci∈chh^(κi⋅ci))2

where MO~ represents the number of elements in υhϰ and NO~ represents the number of elements in τhϰ.

Definition 2.15.Let O~1=(υhh^1/℘h^1,τhh^1/ch^1) and O~2=(υhh^2/℘h^2,τhh^2/ch^2) be two PyPHFNs. Then by using above definition, comparison of PyPHFNs can be described as

(1) If ı~(O~1)>ı~(O~2), then O~1>O~2.

(2) If ı~(O~1)=ı~(O~2), and h(O~1)>h(O~2) then O~1>O~2.

Definition 2.16.Let O~ȷ^=(υhh^ȷ^/℘h^ȷ^,τhh^ȷ^/ch^ȷ^) (ȷ^=1,2,…,ı~) be any group of PyPHFNs, ℱ=(ℱ1,ℱ2,…,ℱı~)T are the weights of O~ȷ^∈[0,1] with ∑ȷ^=1ı~ℱȷ^=1. Then

(1) the PyPHFWA operator can be described as

PyPHFWA(O~1,O~2,….,O~ı~)=(⋃y¨ȷ^∈υO~ȷ^℘O~ȷ^∈℘O~ȷ^1−Πȷ^=1ı~(1−(y¨O~ȷ^)2)ℱȷ^/Πȷ^=1ı~℘O~ȷ^,⋃κO~ȷ^∈τO~ȷ^cO~ȷ^∈cO~ȷ^Πȷ^=1ı~(κO~ȷ^)ℱȷ^/Πȷ^=1ı~cO~ȷ^)

(2) the PyPHFWG operator can be described as

PyPHFWG(O~1,O~2,….,O~ı~)=(⋃κO~ȷ^∈τO~ȷ^cO~ȷ^∈cO~ȷ^Πȷ^=1ı~(κO~ȷ^)ℱȷ^/Πȷ^=1ı~cO~ȷ^,⋃y¨ȷ^∈υO~ȷ^℘O~ȷ^∈℘O~ȷ^1−Πȷ^=1ı~(1−(y¨O~ȷ^)2)ℱȷ^/Πȷ^=1ı~℘O~ȷ^)

Definition 2.17.Let O~ȷ^=(υhh^ȷ^/℘h^ȷ^,τhh^ȷ^/ch^ȷ^) (ȷ^=1,2,…,ı~) be any group of PyPHFNs, O~κ(ȷ^) be the jth largest in them and ℱ=(ℱ1,ℱ2,…,ℱı~)T are the weights of O~ȷ^∈[0,1] with ∑ȷ^=1ı~ℱȷ^=1. Then

(1) PyPHFOWA operator can be described as

PyPHFOWA(O~1,O~2,…,O~ı~)=ℱ1O~κ(1)⊕ℱ2O~κ(2)⊕…⊕ℱı~O~κ(ı~)=(⋃y¨O~κ(ȷ^)∈υO~κ(ȷ^)℘O~κ(ȷ^)∈℘O~κ(ȷ^)1−Πȷ^=1ı~(1−(y¨O~κ(ȷ^))2)ℱȷ^/Πȷ^=1ı~℘O~κ(ȷ^),⋃κO~κ(ȷ^)∈τO~κ(ȷ^)cO~κ(ȷ^)∈cO~κ(ȷ^)Πȷ^=1ı~(κO~κ(ȷ^))ℱȷ^/Πȷ^=1ı~cO~κ(ȷ^))

(2) PyPHFOWG operator can be described as

PyPHFOWG(O~1,O~2,…,O~ı~)=(⋃κO~κ(ȷ^)∈τO~κ(ȷ^)cO~κ(ȷ^)∈cO~κ(ȷ^)Πȷ^=1ı~(κO~κ(ȷ^))ℱȷ^/Πȷ^=1ı~cO~κ(ȷ^),⋃y¨O~κ(ȷ^)∈υO~κ(ȷ^)℘O~κ(ȷ^)∈℘O~κ(ȷ^)1−Πȷ^=1ı~(1−(y¨O~κ(ȷ^))2)ℱȷ^/Πȷ^=1ı~℘O~κ(ȷ^))where O~κ(ȷ^) is the jth largest in them