# Decision-Making Approach with Complex Bipolar Fuzzy N-Soft Sets

Authors: Adeel Farooq; Mohammed M. Ali Al-Shamiri; Mohammed M. Khalaf; Umaira Amjad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The primary aim of this article is to extend the bipolar fuzzyN-soft sets with the concern of pursuing the periodicity involved real-world problems and introduce a new multiskilled hybrid model, namely, complex bipolar fuzzy N-soft sets. The novel model possesses the parametric characteristics of the versatile N-soft set and enjoys the distinguished attributes of a complex bipolar fuzzy set to handle the double-sided periodic vague data. We illustrate that the innovative model assists as a proficient mechanism for grading-based parameterized two-dimensional bipolar fuzzy information. We present some elementary operations and results for a complex bipolar fuzzy N-soft environment. Further, we establish the three dexterous algorithms to find the optimal solution to multiattribute decision-making problems. Moreover, the algorithms are supported with the robust assessment of a real-world application. Lastly, a comparison with existent decision-making techniques, such as choice values, weighted choice values, and D-choice of values of bipolar fuzzy N-soft sets, is also conducted to manifest the phenomenal accountability and authenticity of the presented decision-making approaches.


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## Body

## 1. Introduction

The primary concept of fuzzy setℱS theory was presented by Zadeh [1] in 1965, which is a generalization of classical set theory to handle vague and uncertain information. In ℱS, the value of membership degree MD lies in 0,1. In 1983, Atanassov [2] suggested the intuitionistic fuzzy set ℐℱS by adding the nonmembership degree (NMD) having the property that the sum of MD and NMD should be less than or equal to 1. Among many generalizations of ℱS theory, our main focus is on the bipolar fuzzy set put forward by Zhang [3].The traditional models ofℱS are incapable of dealing with the periodic information in any meaningful way. Hence, another ground-breaking model was established by Ramot et al. [4] that can tackle the two-dimensional uncertain information. Among many other extensions of CFS, our focus is on complex bipolar fuzzy sets CℬℱS, which was initiated by Akram et al. [5]. The rationale of CℬℱS is to represent the bipolar information having vagueness and periodicity in complex geometry, as shown in Figures 1 and 2. By implementing the concept of bipolar fuzzy set in complex geometry to portray diverse phenomena at various phases, the notion of complex bipolar fuzzy set expresses bipolar behavior of uncertainty and periodicity simultaneously.Figure 1
Phase term inBFS.Figure 2
Graphical representation ofCℬℱS.It has been a dilemma for researchers, as well as decision makers, to deal with the imprecision and ambiguity as it shows up in every discipline of life including social sciences, information technology, economics, business management, and so forth. Many attempts have been made to confront this concern. Molodtsov [6, 7] gave the idea of a soft set SfS in 1999. Maji et al. [8] illustrated the application of SfS for choosing the best house. Maji et al. [9] introduced the fuzzy SfS and investigated its basic properties. Maji et al. [10] presented the intuitionistic fuzzy SfS. Bipolar SfS was proposed by Shabir and Naz [11]. Later, Karaaslan and Karataş [12] redefined the bipolar SfS and worked for the decision-making method along with the application. Aslam et al. [13] put forth the idea of a hybrid model, namely, bipolar fuzzy SfS and defined its fundamental operations. Alghamdi et al. [14] used the various multicriteria decision techniques under a bipolar fuzzy environment. Later, Akram et al. [15] applied the TOPSIS and ELECTRE I approaches to diagnose medical diseases with the help of bipolar fuzzy data.SfS theory is used to evaluate the binary evaluation based information. It is not helpful for nonbinary discreet evaluation based systems. Nowadays, mostly systems are being assessed on the basis of rating. In these systems, the rating of the alternatives is done by a number of stars, check marks, dots, numbers, et cetera. To overcome these obstacles, Fatimah et al. [16] developed the stimulated concept of N-soft set NSfS along with set-theoretic operations and decision-making algorithms which are useful to capture the ordered graded information. Furthermore, Akram et al. [17] merged the novel models of ℱS and NSfS to introduce the fuzzy NSfSℱNSfS. Akram et al. also proposed the intuitionistic fuzzy NSfS [18], complex Pythagorean fuzzy NSfS [19], complex spherical fuzzy NSfS [20], complex neutrosophic NSfS [21], and bipolar fuzzy NSfSℬℱNSfS [22]. Pythagorean fuzzy NSfS was initiated by Zhang et al. [23]. Fatimah and Alcantud [24] put forward the multifuzzy NSfS. Kamaci and Petchimuthu [25] introduced the hybrid model of bipolar NSfS along with practical applications.To sum up, the motivation of this article is given as follows:Although the traditionalNSfS can capture the graded evaluation of parameters and is superior than SfS; nevertheless it can not handle the fuzziness involved in the information.TheCℬℱS is beneficial to tackle the two-dimensional bipolar information, but it can not cope with the rating/ranking based parameterized information.ℬℱNSfS can deal with graded parameterized double-sided ambiguous information, but still it has the inadequacy of phase term.Motivated by the aforementioned concerns, this research article introduced a new hybrid model with multiple characteristics, namely, complex bipolar fuzzyN-soft sets. This methodology is designed to capture and interpret the two-dimensional bipolar fuzzy graded parameterized information. This article also describes the fundamental operations of the proposed model. Moreover, complex bipolar fuzzy N-soft number along with some algebraic operations and properties is also defined. Three algorithms for decision-making have been defined and ingeniously implemented on real-life problems. The rationality and applicability have been illustrated through comparative analysis with existing methodologies.This research paper is organized as follows: Section2 defines some preliminary concepts for the development of the new hybrid model. Section 3 describes the mathematical framework of the proposed complex bipolar fuzzy N-soft sets models and develops its operations. Section 4 provides the three algorithms for decision-making purposes. Section 5 illustrates the applicability of the presented novel algorithms. Section 6 demonstrates the comparative analysis of proposed techniques with existing decision-making methods. Section 7 sums up this article with some concluding remarks.


## 2. Preliminaries

Now, we will present some fundamental definitions that are essential for further developments.Definition 2.1.
(see [1]). Let V be a nonempty set. A fuzzy setμ over V is an object of the form:(1)μ=v,αμpv|v∈V,where αμp:V⟶0,1 denotes the degree of membership.Definition 2.2.
(see [3]) Let V be a nonempty set. A bipolar fuzzy setB over V is of the form:(2)B=v,αBpv,βBnv|v∈V,where αBp:V⟶0,1 and βBn:V⟶−1,0 represent the satisfaction function and dissatisfaction function, respectively. The satisfaction value αBpv indicates the strength of belongingness of element v to a certain property and the dissatisfaction value βBnv indicates the belongingness of element v to some counter property of bipolar fuzzy set B.Definition 2.3.
(see [26]). Let b¨=αp,βn be a bipolar fuzzy number. Then, the bipolar fuzzy score function f and accuracy function g are formulated as follows:(3)fb¨=1+αp+βn2,gb¨=αp−βn2,where fb¨,gb¨∈0,1.Definition 2.4.
(see [5]). Let V be a universal set. A CℬℱSCb on a nonempty set V is an object of the form:(4)Cb=v,αCbpveiωCbv,βCbnveiψCbv|v∈V,where i=−1,αCbp:V⟶0,1 and βCbn:V⟶−1,0 are mappings, ωCbv∈0,π and ψCbv∈−π,0. For any element v∈V,αCbpv and βCbnv are known to be amplitude terms; ωCbv and ψCbv are phase terms.Definition 2.5.
(see [6]). Let V be a universe of discourse under consideration and A be the set of all attributes, L⊆A. A pair ϑ,L is called SfS over V if ϑ:L⟶PV where ϑ is a set-valued function.Definition 2.6.
(see [16]). Let V be a universe of discourse and A be the set of all attributes, L⊆A. Consider ℛ=0,1,…,N−1 be a set of ordered grades where N∈2,3,…. A triple U,L,N is an NSfS on V if U:L⟶2V×ℛ, with the property that for each kj∈L, there exists a unique vt,rjt∈V×ℛ such that vt,rjt∈Ukj,vt∈V,rjt∈ℛ.Definition 2.7.
(see [13]). Let V be a universe of discourse under consideration and A be the set of all attributes, L⊆A. A pair ℘,L is called ℬℱSfS over V if ℘:L⟶BFV, where BFV is the collection of all bipolar fuzzy subsets of V. It is defined as follows:(5)℘,L=v,αlpv,βlnv|∀v∈Vandl∈L.Definition 2.8.
(see [22]). Let V be a universe of discourse and A be the set of all attributes under consideration, L⊆A. Let ℛ=0,1,2,…,N−1 be a set of ordered grades where N∈2,3,…. A triple ℑ,P,N is called a ℬℱNSfS, when P=U,L,N is an NSfS on V where U:L⟶2V×ℛ and ℑ is a mapping such that ℑ:L⟶2V×ℛ×ℬℱV, which is as follows:(6)ℑ,P,N=l,Wl,Zl|l∈L,Wl,Zl∈2V×ℛ×ℬℱV=l,v,rlv,αlp,βln|l∈L,v∈V,rlv∈ℛ,where(7)Z,W:K⟶ℬℱV,and ℬℱV represents the collection of all bipolar fuzzy values on V.
Here,αlp∈0,1 and βln∈−1,0 for all v∈V.


## 3. Theoretical Structure of Complex Bipolar FuzzyN-Soft Set

Definition 3.1.
LetV be the universal set and A be the set of parameters under examination, L⊆A. Let ℛ=0,1,2,…,N−1 be the set of ordered grades with N∈2,3,…. A triple χ=ΥΦ,T,N is called a complex bipolar fuzzyN-soft setCℬℱNSfS on V, if T=Φ,L,N possessing Φ:L⟶2V×ℛ is an N-soft set NSfS on V and ΥΦ:L⟶2V×ℛ×CℬℱV, which is described as follows:(8)χ=Φli,Υli|li∈L=l,v,rlv,μl,νl|l∈L,v∈V,rlv∈ℛ=l,v,rlv,αlpeiωl,βlneiψl|l∈L,v∈V,rlv∈ℛ,where(9)Υ:L⟶CℬℱV,CℬℱV indicates the collection of all complex bipolar fuzzy values on V.
Here,i=−1,αlp∈0,1,βln∈−1,0,ωl∈0,π and ψ∈−π,0 for all v∈V.Remark.
Letχli=vy,riy,αyipeiωyi,βyineψyi be a CℬℱNSfS. Then, the complex bipolar fuzzy N-soft number CℬℱNSfN is defined as follows:(10)χyi=riy,αyipeiωyi,βyineψyi.
The following example of decision-making demonstrates the importance and significance of the proposed hybrid model. This example elaborates the shortcomings of the hybrid modelℬℱNSfS, put forward by Akram et al. [22], and illustrates the proficiency of the presented hybrid model in decision-making problems.Example 3.2.
Multiattribute decision-making procedures are considered as a primary tool for examining the patient’s medical history in the medical symptomatic system and suggest the required tests. But most of the time, syndromes are not apparent and show bipolar behavior. In such circumstances, bipolar fuzzy sets are more capable of applying because it deals with positive along with negative behavior of an object towards a certain property. Suppose that a person with brain disorder visits a neurologist. A neurologist examines the symptoms first. The symptoms are as follows:l1: headachel2: seizuresl3: fatiguel4: mood swings
The doctor examines the patient’s history, but still, there is confusion about whether it is a traumatic brain injury or brain tumor. Hence, the doctor wants to suggest one of the following brain scans for clear examination.v1: computed tomography (CT) scanv2: magnetic resonance imaging (MRI)v3: positron emission tomography (PET)v4: single photon emission computed tomography (SPECT)
The neurologist will assign the4Sf grades to the alternatives in order to find out the best option for brain scanning according to the symptoms. The rating and associated 4SfS of alternatives are given in Table 1, whereThree• represent outstandingTwo• represent superbOne• represents good° represents average
The corresponding8SfS can be associated as follows:<°’ indicates 0<•’ indicates 1<••’ indicates 2<•••’ indicates 3
Corresponding to the grades,Cℬℱ8SfNs is assigned to the criteria of vaccines by utilizing the following grading criteria:(11)0.0≤fb¨yi<0.25when grade0,0.25≤fb¨yi<0.50when grade1,0.50≤fb¨yi<0.75when grade2,0.75≤fb¨yi≤1.00when grade3.where b¨yi=αyip,βyin,fb¨yi=αyip+βyin+1/2 and y=1,2,3,4;i=1,2,3,4.
According to the aforementioned criteria, the corresponding grading criteria are represented in Table2.
Thereby, theℬℱ4SfS can be defined as follows:(12)ΥΦl1=v1,2,0.55,−0.33,v2,3,0.81,−0.15,v3,1,0.46,−0.52,v4,2,0.65,−0.29,ΥΦl2=v1,2,0.71,−0.44,v2,3,0.89,−0.13,v3,2,0.68,−0.31,v4,3,0.91,−0.17,ΥΦl3=v1,1,0.42,−0.57,v2,3,0.86,−0.08,v3,1,0.39,−0.64,v4,0,0.15,−0.89,ΥΦl4=v1,2,0.69,−0.45,v2,3,0.79,−0.05,v3,1,0.31,−0.59,v4,1,0.27,−0.68.
The tabular representation ofℬℱ4Sf decision matrix is given in Table 3.
The doctor assigns theℬℱNs corresponding to each grade value of the alternatives. Now, suppose that the doctor observes “Initially, almost 3 months the headache was bearable, but from the last 8 months it is severe.” Then the bipolar fuzzy framework is ambiguous and fails to handle the complete situation. Hence, it is necessary to use the complex bipolar fuzzy environment in order to deal with the periodicity of the data. Therefore, we set up the CℬℱNSfS instead of ℬℱNSfS for the assessment of such problems. The phase term in data will indicate the information related to the reference frame of time under consideration. Now, the grading criteria are redefined according to the framework of CℬℱS.(13)0.0≤Sℬyi<0.50when grade0,0.50≤Sℬyi<1.00when grade1,1.00≤Sℬyi<1.50when grade2,1.50≤Sℬyi≤2.00when grade3.where ℬyi=αyipeiωyiπ,βyineψyi,Sℬyi=αyip+βyin+1/2+1/2ωyi/π+ψyi/π+1 and y=1,2,3,4;i=1,2,3,4.
According to the aforementioned criteria, the corresponding grading criteria are represented in Table4.
Hence,Cℬℱ4SfS can be obtained by applying Definition 3.1 as follows:(14)ΥΦl1=v1,2,0.55ei0.62π,−0.33e−i0.29π,v2,3,0.81ei0.89π,−0.15e−i0.12π,v3,1,0.46ei0.43π,−0.52e−i0.62π,v4,2,0.65ei0.58π,−0.29e−i0.34π,ΥΦl2=v1,2,0.71ei0.63π,−0.44e−i0.38π,v2,3,0.89ei0.81π,−0.13e−i0.07π,v3,2,0.68ei0.66π,−0.31e−i0.37π,v4,3,0.91ei0.78π,−0.17e−i0.11π,ΥΦl3=v1,1,0.42ei0.38π,−0.57e−i0.65π,v2,3,0.86ei0.93π,−0.08e−i0.17π,v3,1,0.39ei0.34π,−0.64e−i0.57π,v4,0,0.15ei0.21π,−0.89e−i0.92π,ΥΦl4=v1,2,0.69ei0.65π,−0.45e−i0.42π,v2,3,0.79ei0.82π,−0.05e−i0.21π,v3,1,0.31ei0.29π,−0.59e−i0.72π,v4,1,0.27ei0.31π,−0.68e−i0.61π.
The tabular representation ofCℬℱ4Sf decision matrix is depicted in Table 5.Table 1
Rating of brain scanning tests.
V/Ll1l2l3l4v12=••2=••1=•2=••v23=•••3=•••3=•••3=•••v31=•2=••1=•1=•v42=••3=•••0=°1=•Table 2
Grading criteria.
GradesPositive membershipNegative membershipriyαyipβyinriy=0[0, 0.25)[−1,−0.75)riy=1[0.25, 0.50)[−0.75,−0.50)riy=2[0.50, 0.75)[−0.50,−0.25)riy=3[0.75, 1.00][−0.25,−0.00]Table 3
Tabular form of theℬℱ4SfS.
ΥΦ,T,4l1l2l3l4v12,0.55,−0.332,0.71,−0.441,0.42,−0.572,0.69,−0.45v23,0.81,−0.153,0.89,−0.133,0.86,−0.083,0.79,−0.05v31,0.46,−0.522,0.68,−0.311,0.39,−0.641,0.31,−0.59v42,0.65,−0.293,0.91,−0.170,0.15,−0.891,0.27,−0.68Table 4
Grading criteria.
GradesAmplitude termsPhase termsriyαyipβyinωyiψyiriy=0[0, 0.25)[−1,−0.75)[0,0.25π)[−π,−0.75π)riy=1[0.25, 0.50)[−0.75,−0.50)[0.25π,0.50π)[−0.75π,−0.50π)riy=2[0.50, 0.75)[−0.50,−0.25)[0.50π,0.75π)[−0.50π,−0.25π)riy=3[0.75, 1.00][−0.25,−0.00][0.75π,π][−0.25π,−0.00]Table 5
Tabular representation of theCℬℱ4SfS.
ΥΦ,T,4l1l2l3l4v12,0.55ei0.62π,−0.33e−i0.29π2,0.71ei0.63π,−0.44e−i0.38π1,0.42ei0.38π,−0.57e−i0.65π2,0.69ei0.65π,−0.45e−i0.42πv23,0.81ei0.89π,−0.15e−i0.12π3,0.89ei0.81π,−0.13e−i0.07π3,0.86ei0.93π,−0.08e−i0.17π3,0.79ei0.82π,−0.05e−i0.21πv31,0.46ei0.43π,−0.52e−i0.62π2,0.68ei0.66π,−0.31e−i0.37π1,0.39ei0.34π,−0.64e−i0.57π1,0.31ei0.29π,−0.59e−i0.72πv42,0.65ei0.58π,−0.29e−i0.34π3,0.91ei0.78π,−0.17e−i0.11π0,0.15ei0.21π,−0.89e−i0.92π1,0.27ei0.31π,−0.68e−i0.61πDefinition 3.3.
Letχyi=riy,αyipeiωyi,βyineψyi be a CℬℱNSfN over V. The score function is defined as follows:(15)Sχyi=riyN−1+αyip+βyin+12+12ωyiπ+ψyiπ+1,where Sχyi∈0,3.Definition 3.4.
Letχyi=riy,αyipeiωyi,βyineψyi be a CℬℱNSfN over V. The accuracy function is defined as follows:(16)Aχyi=riyN−1+αyip−βyin+ωyiπ−ψyiπ,where Aχyi∈0,4.Definition 3.5.
For any two distinctCℬℱNSfNsχyi=riy,αyipeiωyi,βyineψyi and χxi=rix,αxipeiωxi,βxineψxi, we have the following:(1)
IfSχyi<Sχxi, then χyi≺χxiχxiis superior toχyi(2)
IfSχyi>Sχxi, then χyi≻χxiχxiis inferior toχyi(3)
IfSχyi=Sχxi, then(a)
IfAχyi>Aχxi, then χyi≺χxiχxiis superior toχyi(b)
IfAχyi<Aχxi, then χyi≻χxiχxiis inferior toχyi(c)
IfAχyi=Aχxi, then χyi∼χxiχyiis equivalent toχxiRemark.
It can be observed that(1)
ForN=2, CℬℱNSfS becomes CℬℱSfS.(2)
WhenN=2 and L=1, CℬℱNSfS becomes CℬℱS.
Now, we will investigate the notions of complementarities ofCℬℱNSfS.Definition 3.6.
LetΥΦ,T,N be a CℬℱNSfS over nonempty set V, where T=Φ,L,N is an NSfS, then ΥΦ,Tc,N is called weak complement if Tc=Φc,L,N is a weak complement of T=Φ,L,N. In consequence, Φcli∩Φli=∅ for all li∈L.
Remember that a weak compliment is not peculiar.Definition 3.7.
LetΥΦ,T,N be a CℬℱNSfS over V, such that T=Φ,L,N is an NSfS, then a complex bipolar fuzzy complement is represented by ΥΦc,T,N, such that ΥΦc:L⟶CℬℱVV×ℛ, which is conveyed by the following:(17)ΥΦcli=vy,riy,1−αyipei1−ωyi/ππ,−1−βyinei−1−ψyi/ππ.Definition 3.8.
LetΥΦ,T,N be a CℬℱNSfS over V, such that T=Φ,L,N is an NSfS, then ΥΦc,Tc,N is stated as weak complex bipolar fuzzy complement⇔ΥΦ,Tc,N is a weak complement and ΥΦc,T,N is a complex bipolar fuzzy complement.Example 3.9.
Consider aCℬℱ4SfSΥΦ,T,4 as arranged in Example 3.2. A weak complement ΥΦ,Tc,4, complex bipolar fuzzy complement ΥΦc,T,4 , and weak complex bipolar fuzzy complement ΥΦc,Tc,4 are calculated and organized by Tables 6–8, respectively.Table 6
A weak complement of.ΥΦ,T,4.
ΥΦ,Tc,4l1l2l3l4v13,0.55ei0.62π,−0.33e−i0.29π3,0.71ei0.63π,−0.44e−i0.38π2,0.42ei0.38π,−0.57e−i0.65π1,0.69ei0.65π,−0.45e−i0.42πv20,0.81ei0.89π,−0.15e−i0.12π1,0.89ei0.81π,−0.13e−i0.07π1,0.86ei0.93π,−0.08e−i0.17π0,0.79ei0.82π,−0.05e−i0.21πv32,0.46ei0.43π,−0.52e−i0.62π0,0.68ei0.66π,−0.31e−i0.37π0,0.39ei0.34π,−0.64e−i0.57π3,0.31ei0.29π,−0.59e−i0.72πv41,0.65ei0.58π,−0.29e−i0.34π2,0.91ei0.78π,−0.17e−i0.11π3,0.15ei0.21π,−0.89e−i0.92π2,0.27ei0.31π,−0.68e−i0.61πTable 7
Complex bipolar fuzzy complement.
ΥΦc,T,4l1l2l3l4v12,0.45ei0.38π,−0.67e−i0.71π2,0.29ei0.37π,−0.56e−i0.62π1,0.58ei0.62π,−0.43e−i0.35π2,0.31ei0.35π,−0.55e−i0.58πv23,0.19ei0.11π,−0.85e−i0.88π3,0.11ei0.19π,−0.87e−i0.93π3,0.14ei0.07π,−0.92e−i0.83π3,0.21ei0.18π,−0.95e−i0.79πv31,0.54ei0.57π,−0.48e−i0.38π2,0.32ei0.34π,−0.69e−i0.63π1,0.61ei0.66π,−0.36e−i0.43π1,0.69ei0.71π,−0.41e−i0.28πv42,0.35ei0.42π,−0.71e−i0.66π3,0.09ei0.22π,−0.83e−i0.89π0,0.85ei0.79π,−0.11e−i0.08π1,0.73ei0.69π,−0.32e−i0.39πTable 8
Weak complex bipolar fuzzy complement.
ΥΦc,Tc,4l1l2l3l4v13,0.45ei0.38π,−0.67e−i0.71π3,0.29ei0.37π,−0.56e−i0.62π2,0.58ei0.62π,−0.43e−i0.35π1,0.31ei0.35π,−0.55e−i0.58πv20,0.19ei0.11π,−0.85e−i0.88π1,0.11ei0.19π,−0.87e−i0.93π1,0.14ei0.07π,−0.92e−i0.83π0,0.21ei0.18π,−0.95e−i0.79πv32,0.54ei0.57π,−0.48e−i0.38π0,0.32ei0.34π,−0.69e−i0.63π0,0.61ei0.66π,−0.36e−i0.43π3,0.69ei0.71π,−0.41e−i0.28πv41,0.35ei0.42π,−0.71e−i0.66π2,0.09ei0.22π,−0.83e−i0.89π3,0.85ei0.79π,−0.11e−i0.08π2,0.73ei0.69π,−0.32e−i0.39πDefinition 3.10.
For aCℬℱNSfSΥΦ,T,N, where T=Φ,L,N is an NSfS, the top weak complement of ΥΦ,T,N is ΥΦ,Tt,N, also the top weak complex bipolar fuzzy complement of ΥΦ,T,N is ΥΦc,Tt,N and defined as follows:(18)ΥΦ,Tt,N=ΥΦli=vy,N−1,αyipeiωyi,βyineiψyi,ifriy<N−1,ΥΦli=vy,0,αyipeiωyi,βyineiψyi,ifriy=N−1.ΥΦc,Tt,N=ΥΦli=vy,N−1,1−αyipei1−ωyi/ππ,−1−βyinei−1−ψyi/ππ,ifriy<N−1,ΥΦli=vy,0,1−αyipei1−ωyi/ππ,−1−βyinei−1−ψyi/ππ,ifriy=N−1.Example 3.11.
The top weak complement and the top weak complex bipolar fuzzy complement of theCℬℱ4SfS, defined by Table 5 in Example 3.2, are shown in Tables 9 and 10, respectively.Table 9
Top weak complement of theCℬℱ4SfS.
ΥΦ,Tt,4l1l2l3l4v13,0.55ei0.62π,−0.33e−i0.29π3,0.71ei0.63π,−0.44e−i0.38π3,0.42ei0.38π,−0.57e−i0.65π3,0.69ei0.65π,−0.45e−i0.42πv20,0.81ei0.89π,−0.15e−i0.12π0,0.89ei0.81π,−0.13e−i0.07π0,0.86ei0.93π,−0.08e−i0.17π0,0.79ei0.82π,−0.05e−i0.21πv33,0.46ei0.43π,−0.52e−i0.62π3,0.68ei0.66π,−0.31e−i0.37π3,0.39ei0.34π,−0.64e−i0.57π3,0.31ei0.29π,−0.59e−i0.72πv43,0.65ei0.58π,−0.29e−i0.34π0,0.91ei0.78π,−0.17e−i0.11π3,0.15ei0.21π,−0.89e−i0.92π3,0.27ei0.31π,−0.68e−i0.61πTable 10
Top weak complex bipolar fuzzy complement.
ΥΦc,Tt,4l1l2l3l4v13,0.45ei0.38π,−0.67e−i0.71π3,0.29ei0.37π,−0.56e−i0.62π3,0.58ei0.62π,−0.43e−i0.35π3,0.31ei0.35π,−0.55e−i0.58πv20,0.19ei0.11π,−0.85e−i0.88π0,0.11ei0.19π,−0.87e−i0.93π0,0.14ei0.07π,−0.92e−i0.83π0,0.21ei0.18π,−0.95e−i0.79πv33,0.54ei0.57π,−0.48e−i0.38π3,0.32ei0.34π,−0.69e−i0.63π3,0.61ei0.66π,−0.36e−i0.43π3,0.69ei0.71π,−0.41e−i0.28πv43,0.35ei0.42π,−0.71e−i0.66π0,0.09ei0.22π,−0.83e−i0.89π3,0.85ei0.79π,−0.11e−i0.08π3,0.73ei0.69π,−0.32e−i0.39πDefinition 3.12.
For aCℬℱNSfSΥΦ,T,N, where T=Φ,L,N is an NSfS, the bottom weak complement of ΥΦ,T,N is ΥΦ,Tb,N also the bottom weak complex bipolar fuzzy complement of ΥΦ,T,N is ΥΦc,Tb,N and defined as follows:(19)ΥΦ,Tb,N=ΥΦli=vy,0,αyipeiωyi,βyineiψyi,ifriy>0,ΥΦli=vy,N−1,αyipeiωyi,βyineiψyi,ifriy=0.ΥΦc,Tb,N=ΥΦli=vy,0,1−αyipei1−ωyi/ππ,−1−βyinei−1−ψyi/ππ,ifriy>0,ΥΦli=vy,N−1,1−αyipei1−ωyi/ππ,−1−βyinei−1−ψyi/ππifriy=0.Example 3.13.
The bottom weak complement and the bottom weak complex bipolar fuzzy complement of theCℬℱ4SfS, defined by Table 5 in Example 3.2, are given by Tables 11 and 12, respectively.Table 11
Bottom weak complement of theCℬℱ4SfS.
ΥΦ,Tb,4l1l2l3l4v10,0.55ei0.62π,−0.33e−i0.29π0,0.71ei0.63π,−0.44e−i0.38π0,0.42ei0.38π,−0.57e−i0.65π0,0.69ei0.65π,−0.45e−i0.42πv20,0.81ei0.89π,−0.15e−i0.12π0,0.89ei0.81π,−0.13e−i0.07π0,0.86ei0.93π,−0.08e−i0.17π0,0.79ei0.82π,−0.05e−i0.21πv30,0.46ei0.43π,−0.52e−i0.62π0,0.68ei0.66π,−0.31e−i0.37π0,0.39ei0.34π,−0.64e−i0.57π0,0.31ei0.29π,−0.59e−i0.72πv40,0.65ei0.58π,−0.29e−i0.34π0,0.91ei0.78π,−0.17e−i0.11π3,0.15ei0.21π,−0.89e−i0.92π0,0.27ei0.31π,−0.68e−i0.61πTable 12
Bottom weak complex bipolar fuzzy complement.
ΥΦc,Tb,4l1l2l3l4v10,0.45ei0.38π,−0.67e−i0.71π0,0.29ei0.37π,−0.56e−i0.62π0,0.58ei0.62π,−0.43e−i0.35π0,0.31ei0.35π,−0.55e−i0.58πv20,0.19ei0.11π,−0.85e−i0.88π0,0.11ei0.19π,−0.87e−i0.93π0,0.14ei0.07π,−0.92e−i0.83π0,0.21ei0.18π,−0.95e−i0.79πv30,0.54ei0.57π,−0.48e−i0.38π0,0.32ei0.34π,−0.69e−i0.63π0,0.61ei0.66π,−0.36e−i0.43π0,0.69ei0.71π,−0.41e−i0.28πv40,0.35ei0.42π,−0.71e−i0.66π0,0.09ei0.22π,−0.83e−i0.89π3,0.85ei0.79π,−0.11e−i0.08π0,0.73ei0.69π,−0.32e−i0.39πDefinition 3.14.
LetV be a nonempty set and ΥΦ1,T1,N1 and ΥΦ2,T2,N2 be two CℬℱNSfSs over V, where T1=Φ1,L1,N1 and T2=Φ2,L2,N2 are NSfSs over V, then their restricted intersection is defined as follows:(20)ϱ,T1∩rT2,N3=ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2,where, N3=minN1,N2 and T1∩rT2=A,L1∩L2,N3 i.e. ∀li∈L1∩L2,vy∈V,vy,riy,a,b∈ϱli⇔riy=minriv1,riv2,a=minαEp,αℱpeiminωE,ωℱ,b=maxβEn,βℱneimaxψE,ψℱ, if vy,riv1,αEpeiωE,βEneiψE∈L1li and vy,riv2,αℱpeiωℱ,βℱneiψℱ∈L2li,E and ℱ are complex bipolar fuzzy sets on Φ1li and Φ2li, respectively.Example 3.15.
LetΥΦ1,T1,5 be a Cℬℱ5SfS and ΥΦ2,T2,4 be a Cℬℱ4SfS given by Tables 13 and 14, respectively, then their restricted intersection ϱ,T1∩rT2,4=ΥΦ1,T1,5∩ℜΥΦ2,T2,4, is presented in Table 15.Table 13
Tabular representation ofCℬℱ5SfSΥΦ1,T1,5.
ΥΦ1,T1,5l1l2l3v13,0.65ei0.72π,−0.33e−i0.29π4,0.92ei0.83π,−0.03e−i0.15π1,0.25ei0.32π,−0.63e−i0.72πv22,0.43ei0.52π,−0.52e−i0.43π4,0.87ei0.93π,−0.13e−i0.09π2,0.57ei0.45π,−0.45e−i0.51πv31,0.27ei0.21π,−0.65e−i0.75π3,0.75ei0.69π,−0.25e−i0.33π0,0.12ei0.07π,−0.92e−i0.83πTable 14
Tabular representation ofCℬℱ4SfSΥΦ2,T2,4.
ΥΦ2,T2,4l1l2l4v13,0.81ei0.92π,−0.17e−i0.09π2,0.56ei0.67π,−0.29e−i0.32π1,0.31ei0.37π,−0.65e−i0.59πv22,0.61ei0.57π,−0.35e−i0.41π3,0.89ei0.95π,−0.13e−i0.19π2,0.71ei0.63π,−0.38e−i0.42πv30,0.09ei0.21π,−0.89e−i0.91π1,0.43ei0.47π,−0.58e−i0.61π0,0.12ei0.17π,−0.86e−i0.79πTable 15
Restricted intersection.
ϱ,T1∩rT2,4l1l2v13,0.65ei0.72π,−0.17e−i0.09π2,0.56ei0.67π,−0.03e−i0.15πv22,0.43ei0.52π,−0.35e−i0.41π3,0.87ei0.93π,−0.13e−i0.09πv30,0.09ei0.21π,−0.65e−i0.75π1,0.43ei0.47π,−0.25e−i0.33πDefinition 3.16.
LetV be a universe of discourse and ΥΦ1,T1,N1 and ΥΦ2,T2,N1 be two CℬℱNSfSs over V, where T1=Φ1,L1,N1 and T2=Φ2,L2,N2 are NSfSs on V; then their extended intersection is given as η,T1∩eT2,N4=ΥΦ1,T1,N1∩EΥΦ2,T2,N2, where N4=maxN1,N2,T1∩eT2=B,L1∪L2,N4 defined by the following:(21)ηli=ΥΦ1li,ifli∈L1−L2,ΥΦ2li,ifli∈L2−L1,vy,riy,a,b,such thatriy=minr1v1,r1v2,a=minαEp,αℱpeiminωE,ωℱ,b=maxβEn,βℱneimaxψE,ψℱ,wherevy,r1v1,αEpeiωE,βEneiψE∈L1li,andvy,r1v2,αℱpeiωℱ,βℱneiψℱ∈L2li,Eandℱare complex bipolar fuzzy sets onΦ1liandΦ2li,respectively.Example 3.17.
LetΥΦ1,T1,5 and ΥΦ2,T2,4 be two Cℬℱ5SfS and Cℬℱ4SfS, respectively, arranged by Tables 13 and 14, respectively. Then their extended intersection η,T1∩eT2,5=ΥΦ1,T1,5∩EΥΦ2,T2,4 is given in Table 16.Table 16
Extended intersection.
η,T1∩rT2,5l1l2v13,0.65ei0.72π,−0.17e−i0.09π2,0.56ei0.67π,−0.03e−i0.15πv22,0.43ei0.52π,−0.35e−i0.41π3,0.87ei0.93π,−0.13e−i0.09πv30,0.09ei0.21π,−0.65e−i0.75π1,0.43ei0.47π,−0.25e−i0.33πl3l4v11,0.25ei0.32π,−0.63e−i0.72π1,0.31ei0.37π,−0.65e−i0.59πv22,0.57ei0.45π,−0.45e−i0.51π2,0.71ei0.63π,−0.38e−i0.42πv30,0.12ei0.07π,−0.92e−i0.83π0,0.12ei0.17π,−0.86e−i0.79πDefinition 3.18.
LetV be a universal set and ΥΦ1,T1,N1 and ΥΦ2,T2,N2 be two CℬℱNSfSs over V, where T1=Φ1,L1,N1 and T2=Φ2,L2,N2 are NSfSs on V; then their restricted union is defined as follows:(22)ΥΦ1,T1,N1∪ℜΥΦ2,T2,N2=σ,T1∪rT2,N4,where N4=maxN1,N2,T1∪rT2=E,L1∩L2,N4, i.e. ∀li∈L1∩L2,vy∈V,vy,riy,a,b∈σli⇔riy=maxr1v1,r1v2,a=maxαEp,αℱpeimaxωE,ωℱ,b=minβEn,βℱneiminωE,ωℱ, if vy,riv1,αEpeiωE,βEneiψE∈L1li and vy,riv2,αℱpeiωℱ,βℱneiψℱ∈L2li,E and ℱ are complex bipolar fuzzy sets on Φ1li and Φ2li, respectively.Example 3.19.
LetΥΦ1,T1,5 and ΥΦ2,T2,4 be two Cℬℱ5SfS and Cℬℱ4SfS, respectively, defined by Tables 13 and 14, respectively; then their restricted union ΥΦ1,T1,5∪ℜΥΦ2,T2,4=σ,T1∪rT2,5 is defined in Table 17.Table 17
Restricted union.
σ,T1∩rT2,5l1l2v13,0.81ei0.92π,−0.33e−i0.29π4,0.92ei0.83π,−0.29e−i0.32πv22,0.61ei0.57π,−0.52e−i0.43π4,0.89ei0.95π,−0.13e−i0.19πv31,0.27ei0.21π,−0.89e−i0.91π3,0.75ei0.69π,−0.58e−i0.61πDefinition 3.20.
LetV be a nonempty and ΥΦ1,T1,N1 and ΥΦ2,T2,N1 be two CℬℱNSfSs on V, where T1=Φ1,L1,N1 and T2=Φ2,L2,N2 are NSfSs over V; then their extended union is described as ΥΦ1,T1,N1∪EΥΦ2,T2,N2=ϑ,T1∪ϵT2,N4, where N4=maxN1,N2,T1∪εT2=F,L1∪L2,N4, and ϑli is given by the following:(23)ϑli=ΥΦ1li,ifli∈L1−L2,ΥΦ2li,ifli∈L2−L1,vy,riy,a,b,such thatriy=maxr1v1,r1v2,a=maxαEp,αℱpeimaxωE,ωℱ,b=minβEn,βℱneiminψE,ψℱ,wherevy,r1v1,αEpeiωE,βEneiψE∈L1li,andvy,r1v2,αℱpeiωℱ,βℱneiψℱ∈L2li,Eandℱare complex bipolar fuzzy sets onΦ1liandΦ2li,respectively.Example 3.21.
LetΥΦ1,T1,5 and ΥΦ2,T2,4 be two Cℬℱ5SfS and Cℬℱ4SfS, respectively, shown by Tables 13 and 14, respectively; then their extended union ΥΦ1,T1,5∪EΥΦ2,T2,4=ϑ,T1∪ϵT2,5 is given in Table 18.Table 18
Extended union.
ϑ,T1∩rT2,5l1l2v13,0.81ei0.92π,−0.33e−i0.29π4,0.92ei0.83π,−0.29e−i0.32πv22,0.61ei0.57π,−0.52e−i0.43π4,0.89ei0.95π,−0.13e−i0.19πv31,0.27ei0.21π,−0.89e−i0.91π3,0.75ei0.69π,−0.58e−i0.61πl3l4v11,0.25ei0.32π,−0.63e−i0.72π1,0.31ei0.37π,−0.65e−i0.59πv22,0.57ei0.45π,−0.45e−i0.51π2,0.71ei0.63π,−0.38e−i0.42πv30,0.12ei0.07π,−0.92e−i0.83π0,0.12ei0.17π,−0.86e−i0.79πDefinition 3.22.
Consider thatΥΦ,T,N be a CℬℱNSfS over nonempty set V, where T=Φ,L,N is an NSfS on V. Let 0<D<N be a threshold. A CℬℱS_fS related with ΥΦ,T,N and D, denoted by ΥΦD,L, is defined as follows:(24)ΥΦDli=αyipeiωyi,βyineiψyi,ifΥΦli=vy,riy,αyipeiωyi,βyineiψyi,riy≥D,0.0ei0π,−1.0e−iπ,otherwise.
Particularly,ΥΦ1,L, and ΥΦN−1,L, are known as bottom CℬℱS_fS and top CℬℱS_fS, respectively.Definition 3.23.
Let0<D<N and ρ∈0,2 be the threshold. Then the SfS on V associated with ΥΦ,T,N and D,ρ denoted by ΥΦD,ρ,L is defined as follows:(25)ΥΦD,ρli=y∈V:SΥΦDli>ρ,∀li∈L,where, SΥΦDli is represented as the score value of ΥΦDli=αyipDeiωyiD,βyinDeiψyiD.Example 3.24.
Consider theCℬℱ4SfS defined in Example 3.2. By employing Definition 3.22, the associated CℬℱSfSs with Cℬℱ4SfS can be found out. Let 0<D<4 be the thresholds. The possible associated CℬℱSfSs with threshold values 1,2 and 3 are given by Tables 19–21. Moreover, by taking D,ρ=2,0.9, associated SfSΥΦ2,0.9,L is arranged by Table 22.
It is clear from above analysis thatCℬℱNSfS can be converted into CℬℱSfS and SfS. Hence, it is a generalization of both these models. The following properties are stated without proofs.Table 19
CℬℱSfS associated with Cℬℱ4SfS and threshold D=1.
ΥΦ1,L,l1l2l3l4v10.55ei0.62π,−0.33e−i0.29π0.71ei0.63π,−0.44e−i0.38π0.42ei0.38π,−0.57e−i0.65π0.69ei0.65π,−0.45e−i0.42πv20.81ei0.89π,−0.15e−i0.12π0.89ei0.81π,−0.13e−i0.07π0.86ei0.93π,−0.08e−i0.17π0.79ei0.82π,−0.05e−i0.21πv30.46ei0.43π,−0.52e−i0.62π0.68ei0.66π,−0.31e−i0.37π0.39ei0.34π,−0.64e−i0.57π0.31ei0.29π,−0.59e−i0.72πv40.65ei0.58π,−0.29e−i0.34π0.91ei0.78π,−0.17e−i0.11π0ei0π,−1e−iπ0.27ei0.31π,−0.68e−i0.61πTable 20
CℬℱSfS associated with Cℬℱ4SfS and threshold D=2.
ΥΦ2,L,l1l2l3l4v10.55ei0.62π,−0.33e−i0.29π0.71ei0.63π,−0.44e−i0.38π0ei0π,−1e−iπ0.69ei0.65π,−0.45e−i0.42πv20.81ei0.89π,−0.15e−i0.12π0.89ei0.81π,−0.13e−i0.07π0.86ei0.93π,−0.08e−i0.17π0.79ei0.82π,−0.05e−i0.21πv30ei0π,−1e−iπ0.68ei0.66π,−0.31e−i0.37π0ei0π,−1e−iπ0ei0π,−1e−iπv40.65ei0.58π,−0.29e−i0.34π0.91ei0.78π,−0.17e−i0.11π0ei0π,−1e−iπ0ei0π,−1e−iπTable 21
CℬℱSfS associated with Cℬℱ4SfS and threshold D=3.
ΥΦ3,L,l1l2l3l4v10ei0π,−1e−iπ0ei0π,−1e−iπ0ei0π,−1e−iπ0ei0π,−1e−iπv20.81ei0.89π,−0.15e−i0.12π0.89ei0.81π,−0.13e−i0.07π0.86ei0.93π,−0.08e−i0.17π0.79ei0.82π,−0.05e−i0.21πv30ei0π,−1e−iπ0ei0π,−1e−iπ0ei0π,−1e−iπ0ei0π,−1e−iπv40ei0π,−1e−iπ0.91ei0.78π,−0.17e−i0.11π0ei0π,−1e−iπ0ei0π,−1e−iπTable 22
SfS associated with Cℬℱ4SfS and thresholds D=2,ρ=0.9.
ΥΦ2,L,l1l2l3l4v11101v21111v30100v41100Theorem 3.25.
LetΥΦ,T,Nbe aCℬℱNSfSoverV. Then:(1)
ΥΦ,T,N∩ℜΥΦ,T,N=ΥΦ,T,N(2)
ΥΦ,T,N∩EΥΦ,T,N=ΥΦ,T,N(3)
ΥΦ,T,N∪ℜΥΦ,T,N=ΥΦ,T,N(4)
ΥΦ,T,N∪EΥΦ,T,N=ΥΦ,T,NTheorem 3.26.
LetΥΦ1,T1,N1 and ΥΦ2,T2,N2 be two CℬℱNSfSs over V. Then the absorption properties are preserved:(1)
ΥΦ1,T1,N1∪EΥΦ2,T2,N2∩ℜΥΦ1,T1,N1=ΥΦ1,T1,N1(2)
ΥΦ1,T1,N1∪EΥΦ2,T2,N2∩ℜΥΦ1,T1,N1=ΥΦ1,T1,N1(3)
ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2∪EΥΦ1,T1,N1=ΥΦ1,T1,N1(4)
ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2∪EΥΦ1,T1,N1=ΥΦ1,T1,N1Theorem 3.27.
LetΥΦ1,T1,N1,ΥΦ2,T2,N2andΥΦ3,T3,N3be threeCℬℱNSfSsoverV, then the following properties hold:(1)
ΥΦ1,T1,N1∪ℜΥΦ2,T2,N2=ΥΦ2,T2,N2∪ℜΥΦ1,T1,N1(2)
ΥΦ1,T1,N1∪EΥΦ2,T2,N2=ΥΦ2,T2,N2∪EΥΦ1,T1,N1(3)
ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2=ΥΦ2,T2,N2∩ℜΥΦ1,T1,N1(4)
ΥΦ1,T1,N1∩EΥΦ2,T2,N2=ΥΦ2,T2,N2∩EΥΦ1,T1,N1(5)
ΥΦ1,T1,N1∪ℜΥΦ2,T2,N2∪ℜΥΦ3,T3,N3=ΥΦ1,T1,N1∪ℜΥΦ2,T2,N2∪ℜΥΦ3,T3,N3(6)
ΥΦ1,T1,N1∪EΥΦ2,T2,N2∪EΥΦ3,T3,N3=ΥΦ1,T1,N1∪EΥΦ2,T2,N2∪EΥΦ3,T3,N3(7)
ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2∩ℜΥΦ3,T3,N3=ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2∩ℜΥΦ3,T3,N3(8)
ΥΦ1,T1,N1∩EΥΦ2,T2,N2∩EΥΦ3,T3,N3=ΥΦ1,T1,N1∩EΥΦ2,T2,N2∩EΥΦ3,T3,N3(9)
ΥΦ1,T1,N1∪EΥΦ2,T2,N2∩ℜΥΦ3,T3,N3=ΥΦ1,T1,N1∪EΥΦ2,T2,N2∩ℜΥΦ1,T1,N1∪EΥΦ3,T3,N3(10)
ΥΦ1,T1,N1∩EΥΦ2,T2,N2∪ℜΥΦ3,T3,N3=ΥΦ1,T1,N1∩EΥΦ2,T2,N2∪ℜΥΦ1,T1,N1∩EΥΦ3,T3,N3(11)
ΥΦ1,T1,N1∪ℜΥΦ2,T2,N2∩EΥΦ3,T3,N3=ΥΦ1,T1,N1∪ℜΥΦ2,T2,N2∩EΥΦ1,T1,N1∪ℜΥΦ3,T3,N3(12)
ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2∪EΥΦ3,T3,N3=ΥΦ1,T1,N1∩ℜΥΦ2,T2,N2∪EΥΦ1,T1,N1∩ℜΥΦ3,T3,N3
Now, we will define some fundamental operations on complex bipolar fuzzyN-soft numbers.Definition 3.28.
Letχyi=riy,αyipeiωyi,βyineψyi and χxi=rix,αxip,βxin be two CℬℱNSfNs and τ>0. Then, some operations for CℬℱNSfNs are as follows:(26)τχyi=riy,1−1−αyipτei1−1−ωyi/πτπ,−βyinτe−iψyi/πτπ,χyiτ=riy,αyipτeiωyi/πτπ,−1+1+βyinτe−i−1+1+ψyi/πτπ,χyi⊗χxi=maxriy,rix,αyip+αxip−αyipαxipeiωyi/π+ωxi/π−ωyi/πωxi/ππ,−βyinβxine−iψyi/πψxi/ππ,minriy,rix,αyipαxipeiωyi/πωxi/ππ,−βyin−βxin+βyinβxinei−ψyi/π−ψxi/π+ψyi/πψxi/ππ.Theorem 3.29.
Letχ=g,αp,βn,χyi=riy,αyipeiωyi,βyineψyiandχxi=rix,αxip,βxinbe threeCℬℱNSfNsandτ,τy,τx>0be any real numbers; then(1)
χyi⊕χxi=χxi⊕χyi(2)
χyi⊗χxi=χxi⊗χyi(3)
τχyi⊕τχxi=τχyi⊕χxi(4)
χyiτ⊗χxiτ=χyi⊗χxiτ(5)
τyχ⊕τxχ=τy+τxχ(6)
χτy⊗χτx=χτy+τx


## 4. Formation of Decision-Making Algorithms underCℬℱNSfS Framework

In this section, we will present the three Algorithms 1, 2, and 3 in order to deal with MADM problems in the framework ofCℬℱNSf model that will help to choose the best opt. Consider V=v1,v2,v3,…,vk to be the universal set of alternatives and L=l1,l2,l3,…,lr the set of criteria that will be used to solve the decision-making problems. Let τ=τ1,τ2,τ3,…,τr be the weight vector of criteria representing the importance of the parameters in the MADM problem, where ∑i=1rτi=1 and τi∈0,1.Algorithm 1:The algorithm of choice values of CℬℱNSfSs.
Input:V=v1,v2,…,vk: Universal set of objects,L=l1,l2,…,lr: Set of parameters,Φ,L,N:NSfS with ℛ=0,1,2,…,N−1 where N∈2,3,4,…,ΥΦ,T,N:CℬℱNSfS, where T=Φ,L,N.Computeϑy=⊕i=1rχyi where χyi=ryi,αyipeiωyi,βyineiψyi, and sum of twoCℬℱNSfNsχyi and χyg is calculated as follows:χyi⊕χyg=maxriy,rgy,αyip+αygp−αyipαygpeiωyi/π+ωyg/π−ωyi/πωyg/ππ,−βyinβygne−iψyi/πψyg/ππ.Compute the choice values of eachvy∈V by employing the Sϑy, using (15), ∀y=1,2,…,k.Reckon all the indicesy for which Sy=maxySϑy.ifSy=Sx, for arbitraryy,x∈1,2,3,…,k,thenapply accuracy function given in (16) and find out the alternative that has the highestaccuracy value;elseDetermine the alternative with the highest score value.output: The alternative having highest score or accuracy value will be the optimal solution.Algorithm 2: The algorithm of weighted choice values of CℬℱNSfSs.
Input:V=v1,v2,…,vk: Universal set of objects,L=l1,l2,…,lr: Set of parameters,Φ,L,N:NSfS with ℛ=0,1,2,…,N−1 where N∈2,3,4,…,ΥΦ,T,N: CℬℱNSfS, where T=Φ,L,N,τ=τ1,τ2,τ3,…,τr: weight vector for attributes.Computeϑyτ=⊕i=1rτiχyi whereτχyi=riy,1−1−αyipτei1−1−ωyi/πτπ,−βyinτe−iψyi/πτπ.Sum of two weightedCℬℱNSfNsχyiτ and χygτ is calculated as follows:τχyi⊕τχyg=maxriyτ,rgyτ,αyipτ+αygpτ−αyipταygpτeiωyiτ/π+ωygτ/π−ωyiτ/πωygτ/ππ,−βyinτβygnτe−iψyiτ/πψygτ/ππ.Compute the weighted choice values of eachvy∈V by employing the Sϑyτ, using (15), ∀y=1,2,…,k.Determine all indicesy for which Sy=maxySϑyτ.ifSyw=Sxw, for somey,x∈1,2,3,…,k,thenutilize accuracy function defined in (16) and find out the alternative that has the extreme accuracy value;elseSpecify the alternative with the greatest score value.output: The alternative having greatest score or accuracy value will be the optimum solution.Algorithm 3: The algorithm of D-choice values of CℬℱNSfSs.
Input:V=v1,v2,…,vk: Universal set of objects,L=l1,l2,…,lr: Set of parameters,Φ,L,N: NSfS with ℛ=0,1,2,…,N−1 where N∈2,3,4,…,ΥΦ,T,N: CℬℱNSfS, whereT=Φ,L,N, 0<D<N: threshold value.ComputeΥΦDli=/span/pαyipeiωyi,βyineiψyi,ifΥΦli=vy,ryi,αyipeiωyi,βyineiψyi,andryi≥D,0.0ei0π,−1.0e−iπ,otherwise.ComputeϑyD=⊕y=1kχyiD, where χyiD=αyipDeiωyiD,βyinDeiψyiD, and sum of two CℬℱNSfNsχyiD and χygD is calculated as follows:χyiD⊕χygD=αyipD+αygpD−αyipDαygpDeiωyiD/π+ωygD/π−ωyiD/πωygD/ππ,−βyinDβygnDe−iψyiD/πψygD/ππ.Compute theD-choice values of each vy∈V by employing the SϑyD, ∀y=1,2,…,k. Here, SϑyD=αyipD+βyinD+1/2+1/2ωyiD/π+ωyiD/π+1Determine all indicesy for which Sy=maxySϑyD.ifSyD=SxD, for somey,x∈1,2,3,…,k,thenutilize accuracy function and find out the alternative that has the extreme accuracy value;elseFind out the alternative with maximum score value.output: The alternative having maximum accuracy or score value will be the best solution.


## 5. Selection of the Best COVID-19 Vaccine

COVID-19 is a contagious disease that has heavily influenced globally, leading to the pandemic. COVID-19 vaccine is a vaccine designed to provide acquired immunity against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the virus that causes COVID-19. COVID-19 vaccine is being formulated in various advanced countries, but underdeveloped countries have not been able to invent their own vaccine. So they have to import it from the developed countries. Suppose that a developing country wants to import the vaccine with the highest urgency for the sake of saving the lives of citizens, but due to the low GDP and budget, the government can import only one vaccine at a time. The following are the available options of the vaccines:v1: Sinopharm,v2: AstraZeneca,v3: Sinovac,v4: Novavax,v5: SANOFI.The attributes on the basis of which the decision-maker will assess the alternatives and assign them grades are as follows:l1: age factorl2: costl3: efficacyl4: manufacturer,l5: administrationEach expert will assign the8Sf grade in order to find out the best alternative with respect to the parameters. The rating and associated 8SfS of alternatives are given in Table 23, whereSeven⋆ symbolize < Exceptional’Six⋆ symbolize < Superb’Five⋆ symbolize < Impressive’Four⋆ symbolize < Excellent’Three⋆ symbolize < Good’Two⋆ symbolize < Average’One⋆ symbolizes < Subpar’,2009⋄ symbolizes < Substandard’Table 23
Grading of vaccinations and associated8SfS.
V/Ll1l2l3l4l5v16=⋆⋆⋆⋆⋆⋆6=⋆⋆⋆⋆⋆⋆7=⋆⋆⋆⋆⋆⋆⋆6=⋆⋆⋆⋆⋆⋆5=⋆⋆⋆⋆⋆v25=⋆⋆⋆⋆⋆4=⋆⋆⋆⋆6=⋆⋆⋆⋆⋆⋆5=⋆⋆⋆⋆⋆4=⋆⋆⋆⋆v35=⋆⋆⋆⋆⋆5=⋆⋆⋆⋆⋆5=⋆⋆⋆⋆⋆3=⋆⋆⋆3=⋆⋆⋆v44=⋆⋆⋆⋆3=⋆⋆⋆3=⋆⋆⋆5=⋆⋆⋆⋆⋆4=⋆⋆⋆⋆v53=⋆⋆⋆2=⋆⋆..4=⋆⋆⋆⋆2=⋆⋆The corresponding8SfS can be associated as follows:<⋄’ refers to 0<⋆’ refers to 1<⋆⋆’ refers to 2<⋆⋆⋆’ refers to 3<⋆⋆⋆⋆’ refers to 4<⋆⋆⋆⋆⋆’ refers to 5<⋆⋆⋆⋆⋆⋆’ refers to 6<⋆⋆⋆⋆⋆⋆⋆’ refers to 7Corresponding to the grades, theCBF8SfNs are assigned to the criteria of vaccines by utilizing the following grading criteria:(27)0.0≤SByi<0.25when grade0,0.25≤SByi<0.50when grade1,0.50≤SByi<0.75when grade2,0.75≤SByi<1.00when grade3,1.00≤SByi<1.25when grade4,1.25≤SByi<1.50when grade5,1.50≤SByi<1.75when grade6,1.75≤SByi≤2.00when grade7.where Byi=αyipeiωyi,βyineψyi,SByi=αyip+βyin+1/2+1/2ωyi/π+ψyi/π+1 and y=1,2,3,4,5;i=1,2,3,4,5.According to the above-mentioned conditions, the corresponding grading criteria are represented in Table24.Table 24
Grading criteria.
GradesAmplitude termsPhase termsriyαyipβyinωyiψyiriy=0[0, 0.125)[−1,−0.875)[0,0.125π)[−π,−0.875π)riy=1[0.125, 0.250)[−0.875,−0.750)[0.125π,0.250π)[−0.875π,−0.750π)riy=2[0.250, 0.375)[−0.750,−0.625)[0.250π,0.375π)−0.750π,−0.625πriy=3[0.375, 0.500)[−0.625,−0.500)[0.375π,0.500π)[−0.625π,−0.500π)riy=3[0.500, 0.625)[−0.500,−0.375)[0.500π,0.625π)[−0.500π,−0.375π)riy=3[0.625, 0.750)[−0.375,−0.250)[0.625π,0.750π)[−0.375π,−0.250π)riy=3[0.750, 0.875)[−0.250,−0.125)[0.750π,0.875π)[−0.250π,−0.125π)riy=30.875,1.00−0.125,00.875π,π−0.125π,0Thereby, theCBF8SfS can be determined by using Definition 3.1 as follows:(28)ΥΦl1=v1,6,0.77ei0.82π,−0.13e−i0.17π,v2,5,0.71ei0.64π,−0.35e−i0.32π,v3,5,0.69ei0.72π,−0.32e−i0.26π,v4,4,0.52ei0.54π,−0.42e−i0.48π,v5,3,0.38ei0.48π,−0.53e−i0.55π,ΥΦl2=v1,6,0.79ei0.84π,−0.14e−i0.19π,v2,4,0.59ei0.55π,−0.43e−i0.47π,v3,5,0.67ei0.66π,−0.31e−i0.29π,v4,3,0.39ei0.44π,−0.55e−i0.61π,v5,2,0.26ei0.31π,−0.71e−i0.65π,ΥΦl3=v1,7,0.92ei0.95π,−0.07e−i0.05π,v2,6,0.81ei0.77π,−0.18e−i0.14π,v3,5,0.65ei0.71π,−0.34e−i0.27π,v4,3,0.45ei0.41π,−0.52e−i0.58π,v5,1,0.15ei0.21π,−0.79e−i0.81π,ΥΦl4=v1,6,0.82ei0.85π,−0.15e−i0.18π,v2,5,0.63ei0.68π,−0.35e−i0.33π,v3,3,0.43ei0.39π,−0.57e−i0.55π,v4,5,0.64ei0.69π,−0.29e−i0.28π,v5,4,0.53ei0.61π,−0.39e−i0.43π,ΥΦl5=v1,5,0.74ei0.72π,−0.28e−i0.31π,v2,4,0.61ei0.55π,−0.39e−i0.43π,v3,3,0.41ei0.39π,−0.61e−i0.59π,v4,4,0.54ei0.58π,−0.44e−i0.49π,v5,2,0.32ei0.33π,−0.71e−i0.68π.TheCBF8Sf decision matrix is organized in Table 25.Table 25
CBF8Sf decision matrix.
ΥΦ,T,8l1l2l3v16,0.77ei0.82π,−0.13e−i0.17π6,0.79ei0.84π,−0.14e−i0.19π7,0.92ei0.95π,−0.07e−i0.05πv25,0.71ei0.64π,−0.35e−i0.32π4,0.59ei0.55π,−0.43e−i0.47π6,0.81ei0.77π,−0.18e−i0.14πv35,0.69ei0.72π,−0.32e−i0.26π5,0.67ei0.66π,−0.31e−i0.29π5,0.65ei0.71π,−0.34e−i0.27πv44,0.52ei0.54π,−0.42e−i0.48π3,0.39ei0.44π,−0.55e−i0.61π3,0.45ei0.41π,−0.52e−i0.58πv53,0.38ei0.48π,−0.53e−i0.55π2,0.26ei0.31π,−0.71e−i0.65π1,0.15ei0.21π,−0.79e−i0.81πl4l5v16,0.82ei0.85π,−0.15e−i0.18π5,0.74ei0.72π,−0.28e−i0.31πv25,0.63ei0.68π,−0.35e−i0.33π4,0.61ei0.55π,−0.39e−i0.43πv33,0.43ei0.39π,−0.57e−i0.55π3,0.41ei0.39π,−0.61e−i0.59πv45,0.64ei0.69π,−0.29e−i0.28π4,0.54ei0.58π,−0.44e−i0.49πv54,0.53ei0.61π,−0.39e−i0.43π2,0.32ei0.33π,−0.71e−i0.68π


### 5.1. Choice Values ofCBF8SfS

By employing Algorithm 1, choice values ofCBF8Sf decision matrix are computed for the purpose of choosing the best vaccine to import, and the results are arranged in Table 26.Table 26
Choice values ofCBF8SfS.
ΥΦ,T,8l1l2l3l4v16,0.77ei0.82π,−0.13e−i0.17π6,0.79ei0.84π,−0.14e−i0.19π7,0.92ei0.95π,−0.07e−i0.05π6,0.82ei0.85π,−0.15e−i0.18πv25,0.71ei0.64π,−0.35e−i0.32π4,0.59ei0.55π,−0.43e−i0.47π6,0.81ei0.77π,−0.18e−i0.14π5,0.63ei0.68π,−0.35e−i0.33πv35,0.69ei0.72π,−0.32e−i0.26π5,0.67ei0.66π,−0.31e−i0.29π5,0.65ei0.71π,−0.34e−i0.27π3,0.43ei0.39π,−0.57e−i0.55πv44,0.52ei0.54π,−0.42e−i0.48π3,0.39ei0.44π,−0.55e−i0.61π3,0.45ei0.41π,−0.52e−i0.58π5,0.64ei0.69π,−0.29e−i0.28πv53,0.38ei0.48π,−0.53e−i0.55π2,0.26ei0.31π,−0.71e−i0.65π1,0.15ei0.21π,−0.79e−i0.81π4,0.53ei0.61π,−0.39e−i0.43πl5ϑySϑyv15,0.74ei0.72π,−0.28e−i0.31π7,0.9998ei0.9999π,−5.35×10−5e−i9.01×10−5π2.8748v24,0.61ei0.55π,−0.39e−i0.43π6,0.9967ei0.9946π,−0.0037e−i0.0030π2.7423v33,0.41ei0.39π,−0.61e−i0.59π5,0.9880ei0.9897π,−0.0117e−i0.0066π2.6047v44,0.54ei0.58π,−0.44e−i0.49π5,0.9733ei0.9802π,−0.0153e−i0.0233π2.5825v52,0.32ei0.33π,−0.71e−i0.68π4,0.8754ei0.9259π,−0.0823e−i0.0847π2.3172According to the results of Table26, the vaccines of COVID-19 are ranked as follows:(29)v1>v2>v3>v4>v5.Hence, the government will import thev1= Sinopharm on the urgent basis.


### 5.2. Weighted Choice Values ofCBF8SfS

In order to apply Algorithm 2 on theCBF8Sf decision matrix, the expert will assign the weight vector(30)τ=0.170.230.250.150.20T,to the attributes. The results are accumulated in Table27.Table 27
Weighted choice values ofCBF8SfS.
ΥΦ,T,8l1l2l3l4v16,0.2211ei0.2529π,−0.7069e−i0.7399π6,0.3016ei0.3439π,−0.6362e−i0.6825π7,0.4682ei0.5271π,−0.5144e−i0.4729π6,0.2268ei0.2477π,−0.7523e−i0.7732πv25,0.1898ei0.1594π,−0.8365e−i0.8239π4,0.1854ei0.1678π,−0.8236e−i0.8406π6,0.3398ei0.3075π,−0.6514e−i0.6117π5,0.1385ei0.1571π,−0.8543e−i0.8468πv35,0.1805ei0.1946π,−0.8239e−i0.7953π5,0.2251ei0.2197π,−0.7639e−i0.7522π5,0.2308ei0.2662π,−0.7636e−i0.7208π3,0.0809ei0.0715π,−0.9191e−i0.9142πv44,0.1173ei0.1237π,−0.8629e−i0.8827π3,0.1075ei0.1248π,−0.8715e−i0.8925π3,0.1388ei0.1236π,−0.8492e−i0.8727π5,0.1421ei0.1611π,−0.8305e−i0.8262πv53,0.0781ei0.1052π,−0.8977e−i0.9034π2,0.0669ei0.0818π,−0.9242e−i0.9057π1,0.0398ei0.0572π,−0.9428e−i0.9487π4,0.1071ei0.1317π,−0.8683e−i0.8811πl5ϑyτSϑyτv15,0.2362ei0.2248π,−0.7752e−i0.7912π7,0.8291ei0.8648π,−0.1349e−i0.1461π2.5815v24,0.1717ei0.1476π,−0.8283e−i0.8447π6,0.6891ei0.6519π,−0.3176e−i0.3030π2.1102v33,0.1001ei0.0941π,−0.9059e−i0.8999π5,0.5960ei0.6121π,−0.4001e−i0.3548π1.8516v44,0.1438ei0.1593π,−0.8486e−i0.8670π5,0.5017ei0.5260π,−0.4501e−i0.4925π1.6675v52,0.0742ei0.0770π,−0.9338e−i0.9258π4,0.3172ei0.3792π,−0.6342e−i0.6331π1.2145According to the results of Table27, the descending order ranking of COVID-19 vaccines is given as follows:(31)v1>v2>v3>v4>v5.Hence, it is concluded that the government will import thev1= Sinopharm.


### 5.3.D-Choice Values of CBF8SfS

TheD-choice values of CBF8Sf decision matrix are evaluated by utilizing Algorithm 3, where D=5. The results are shown in Table 28.Table 28
5-Choice values ofCBF8SfS.
ΥΦ5,T,8l1l2l3l4v10.77ei0.82π,−0.13e−i0.17π0.79ei0.84π,−0.14e−i0.19π0.92ei0.95π,−0.07e−i0.05π0.82ei0.85π,−0.15e−i0.18πv20.71ei0.64π,−0.35e−i0.32π0ei0π,−1e−i1π0.81ei0.77π,−0.18e−i0.14π0.63ei0.68π,−0.35e−i0.33πv30.69ei0.72π,−0.32e−i0.26π0.67ei0.66π,−0.31e−i0.29π0.65ei0.71π,−0.34e−i0.27π0ei0π,−1e−i1πv40ei0π,−1e−i1π0ei0π,−1e−i1π0ei0π,−1e−i1π0.64ei0.69π,−0.29e−i0.28πv50ei0π,−1e−i1π0ei0π,−1e−i1π0ei0π,−1e−i1π0ei0π,−1e−i1πl5ϑyDSϑy5v10.74ei0.72π,−0.28e−i0.31π0.9998ei0.9999π,−5.35×10−5e−i9.01×10−5π1.9998v20ei0π,−1e−i1π0.9796ei0.9735π,−0.0221e−i0.0148π1.9581v30ei0π,−1e−i1π0.9642ei0.9724π,−0.0337e−i0.0204π1.9413v40ei0π,−1e−i1π0.64ei0.69π,−0.29e−i0.28π1.38v50ei0π,−1e−i1π0ei0π,−1e−i1π0From Table28, it is summarized that the ranking of COVID-19 vaccines is as follows:(32)v1>v2>v3>v4>v5.Hence, the government will importv1= Sinopharm without further delay.


## 5.1. Choice Values ofCBF8SfS

By employing Algorithm 1, choice values ofCBF8Sf decision matrix are computed for the purpose of choosing the best vaccine to import, and the results are arranged in Table 26.Table 26
Choice values ofCBF8SfS.
ΥΦ,T,8l1l2l3l4v16,0.77ei0.82π,−0.13e−i0.17π6,0.79ei0.84π,−0.14e−i0.19π7,0.92ei0.95π,−0.07e−i0.05π6,0.82ei0.85π,−0.15e−i0.18πv25,0.71ei0.64π,−0.35e−i0.32π4,0.59ei0.55π,−0.43e−i0.47π6,0.81ei0.77π,−0.18e−i0.14π5,0.63ei0.68π,−0.35e−i0.33πv35,0.69ei0.72π,−0.32e−i0.26π5,0.67ei0.66π,−0.31e−i0.29π5,0.65ei0.71π,−0.34e−i0.27π3,0.43ei0.39π,−0.57e−i0.55πv44,0.52ei0.54π,−0.42e−i0.48π3,0.39ei0.44π,−0.55e−i0.61π3,0.45ei0.41π,−0.52e−i0.58π5,0.64ei0.69π,−0.29e−i0.28πv53,0.38ei0.48π,−0.53e−i0.55π2,0.26ei0.31π,−0.71e−i0.65π1,0.15ei0.21π,−0.79e−i0.81π4,0.53ei0.61π,−0.39e−i0.43πl5ϑySϑyv15,0.74ei0.72π,−0.28e−i0.31π7,0.9998ei0.9999π,−5.35×10−5e−i9.01×10−5π2.8748v24,0.61ei0.55π,−0.39e−i0.43π6,0.9967ei0.9946π,−0.0037e−i0.0030π2.7423v33,0.41ei0.39π,−0.61e−i0.59π5,0.9880ei0.9897π,−0.0117e−i0.0066π2.6047v44,0.54ei0.58π,−0.44e−i0.49π5,0.9733ei0.9802π,−0.0153e−i0.0233π2.5825v52,0.32ei0.33π,−0.71e−i0.68π4,0.8754ei0.9259π,−0.0823e−i0.0847π2.3172According to the results of Table26, the vaccines of COVID-19 are ranked as follows:(29)v1>v2>v3>v4>v5.Hence, the government will import thev1= Sinopharm on the urgent basis.


## 5.2. Weighted Choice Values ofCBF8SfS

In order to apply Algorithm 2 on theCBF8Sf decision matrix, the expert will assign the weight vector(30)τ=0.170.230.250.150.20T,to the attributes. The results are accumulated in Table27.Table 27
Weighted choice values ofCBF8SfS.
ΥΦ,T,8l1l2l3l4v16,0.2211ei0.2529π,−0.7069e−i0.7399π6,0.3016ei0.3439π,−0.6362e−i0.6825π7,0.4682ei0.5271π,−0.5144e−i0.4729π6,0.2268ei0.2477π,−0.7523e−i0.7732πv25,0.1898ei0.1594π,−0.8365e−i0.8239π4,0.1854ei0.1678π,−0.8236e−i0.8406π6,0.3398ei0.3075π,−0.6514e−i0.6117π5,0.1385ei0.1571π,−0.8543e−i0.8468πv35,0.1805ei0.1946π,−0.8239e−i0.7953π5,0.2251ei0.2197π,−0.7639e−i0.7522π5,0.2308ei0.2662π,−0.7636e−i0.7208π3,0.0809ei0.0715π,−0.9191e−i0.9142πv44,0.1173ei0.1237π,−0.8629e−i0.8827π3,0.1075ei0.1248π,−0.8715e−i0.8925π3,0.1388ei0.1236π,−0.8492e−i0.8727π5,0.1421ei0.1611π,−0.8305e−i0.8262πv53,0.0781ei0.1052π,−0.8977e−i0.9034π2,0.0669ei0.0818π,−0.9242e−i0.9057π1,0.0398ei0.0572π,−0.9428e−i0.9487π4,0.1071ei0.1317π,−0.8683e−i0.8811πl5ϑyτSϑyτv15,0.2362ei0.2248π,−0.7752e−i0.7912π7,0.8291ei0.8648π,−0.1349e−i0.1461π2.5815v24,0.1717ei0.1476π,−0.8283e−i0.8447π6,0.6891ei0.6519π,−0.3176e−i0.3030π2.1102v33,0.1001ei0.0941π,−0.9059e−i0.8999π5,0.5960ei0.6121π,−0.4001e−i0.3548π1.8516v44,0.1438ei0.1593π,−0.8486e−i0.8670π5,0.5017ei0.5260π,−0.4501e−i0.4925π1.6675v52,0.0742ei0.0770π,−0.9338e−i0.9258π4,0.3172ei0.3792π,−0.6342e−i0.6331π1.2145According to the results of Table27, the descending order ranking of COVID-19 vaccines is given as follows:(31)v1>v2>v3>v4>v5.Hence, it is concluded that the government will import thev1= Sinopharm.


## 5.3.D-Choice Values of CBF8SfS

TheD-choice values of CBF8Sf decision matrix are evaluated by utilizing Algorithm 3, where D=5. The results are shown in Table 28.Table 28
5-Choice values ofCBF8SfS.
ΥΦ5,T,8l1l2l3l4v10.77ei0.82π,−0.13e−i0.17π0.79ei0.84π,−0.14e−i0.19π0.92ei0.95π,−0.07e−i0.05π0.82ei0.85π,−0.15e−i0.18πv20.71ei0.64π,−0.35e−i0.32π0ei0π,−1e−i1π0.81ei0.77π,−0.18e−i0.14π0.63ei0.68π,−0.35e−i0.33πv30.69ei0.72π,−0.32e−i0.26π0.67ei0.66π,−0.31e−i0.29π0.65ei0.71π,−0.34e−i0.27π0ei0π,−1e−i1πv40ei0π,−1e−i1π0ei0π,−1e−i1π0ei0π,−1e−i1π0.64ei0.69π,−0.29e−i0.28πv50ei0π,−1e−i1π0ei0π,−1e−i1π0ei0π,−1e−i1π0ei0π,−1e−i1πl5ϑyDSϑy5v10.74ei0.72π,−0.28e−i0.31π0.9998ei0.9999π,−5.35×10−5e−i9.01×10−5π1.9998v20ei0π,−1e−i1π0.9796ei0.9735π,−0.0221e−i0.0148π1.9581v30ei0π,−1e−i1π0.9642ei0.9724π,−0.0337e−i0.0204π1.9413v40ei0π,−1e−i1π0.64ei0.69π,−0.29e−i0.28π1.38v50ei0π,−1e−i1π0ei0π,−1e−i1π0From Table28, it is summarized that the ranking of COVID-19 vaccines is as follows:(32)v1>v2>v3>v4>v5.Hence, the government will importv1= Sinopharm without further delay.


## 6. Comparison

In this section, a comparison of the proposed MADM techniques with previous methodologies, namely, choice values, weighted choices values, andD-choice values of BFNSfS is demonstrated, which is presented by Akram et al. [22]. This comparison will demonstrate the proficiency and authenticity of our proposed techniques by examining the numerical application of “Selection of the best COVID-19 vaccine” in the environment of BFNSfS.


### 6.1. Choice Value ofBF8SfS

In order to apply the Choice value ofBFNSfS, firstly, we arranged the BF8SfS from Table 23 by taking all phase terms of CBFNSfNs equal to zero. The grading criteria and the grades given by the expert will remain the same. The BF8Sf decision matrix is arranged in Table 29.Table 29
BF8Sf decision matrix.
ℑ,T,8l1l2l3l4v16,0.77,−0.136,0.79,−0.147,0.92,−0.076,0.82,−0.15v25,0.71,−0.354,0.59,−0.436,0.81,−0.185,0.63,−0.35v35,0.69,−0.325,0.67,−0.315,0.65,−0.343,0.43,−0.57v44,0.52,−0.423,0.39,−0.553,0.45,−0.525,0.64,−0.29v53,0.38,−0.532,0.26,−0.711,0.15,−0.794,0.53,−0.39l5v15,0.74,−0.28v24,0.61,−0.39v33,0.41,−0.61v44,0.54,−0.44v52,0.32,−0.71Moreover, compute theδy as follows:(33)δy=⊕i=1rχyi,where χyi=ryi,αyip,βyin, and sum of two BFNSfNsχyi and χyg is calculated as follows:(34)χyi⊕χyg=maxriy,rgy,αyip+αygp−αyipαygp,−βyinβygn.Further, the choice value ofBFNSfNs is calculated by using the score function as follows:(35)fδy=riyN−1+αyip+βyin+12.The results are given by Table30.Table 30
Choice values ofBF8SfS.
ℑ,T,8l1l2l3l4v16,0.77,−0.136,0.79,−0.147,0.92,−0.076,0.82,−0.15v25,0.71,−0.354,0.59,−0.436,0.81,−0.185,0.63,−0.35v35,0.69,−0.325,0.67,−0.315,0.65,−0.343,0.43,−0.57v44,0.52,−0.423,0.39,−0.553,0.45,−0.525,0.64,−0.29v53,0.38,−0.532,0.26,−0.711,0.15,−0.794,0.53,−0.39l5δyfδyv15,0.74,−0.287,0.9998,−5.35×10−51.8749v24,0.61,−0.396,0.9967,−0.00371.7465v33,0.41,−0.615,0.9880,−0.01171.6131v44,0.54,−0.445,0.9733,−0.01531.6040v52,0.32,−0.714,0.8754,−0.08231.3965According to the results of Table30, the vaccines of COVID-19 are ranked as follows:(36)v1>v2>v3>v4>v5.Hence, the government will import the v1= Sinopharm on an urgent basis.


### 6.2. Weighted Choice Values ofBF8SfS

In order to apply the weighted choice values ofBFNSfS, we used the BF8Sf decision matrix given by Table 29, where phase terms are taken zero. The weight vector of criteria is given below:(37)τ=0.170.230.250.150.20T.Further, compute theδyτ as follows:(38)δyτ=⊕i=1rτχyi,where χyi=ryi,αyip,βyin,τχyi=riy,1−1−αyipτ,−βyinτ., and the sum of two weighted BFNSfNsχyiτ and χygτ is calculated as follows:(39)τχyi⊕τχyg=maxriyτ,rgyτ,αyipτ+αygpτ−αyipταygpτ,−βyinτβygnτ.Further, the weighted choice values ofBFNSfNs is calculated by using the score function as follows:(40)fδyτ=riyN−1+αyip+βyin+12.The results are arranged by Table31.Table 31
Weighted choice value ofBF8SfS.
ℑ,T,8l1l2l3l4v16,0.2211,−0.70696,0.3016,−0.63627,0.4682,−0.51446,0.2268,−0.7523v25,0.1898,−0.83654,0.1854,−0.82366,0.3398,−0.65145,0.1385,−0.8543v35,0.1805,−0.82395,0.2251,−0.76395,0.2308,−0.76363,0.0809,−0.9191v44,0.1173,−0.86293,0.1075,−0.87153,0.1388,−0.84925,0.1421,−0.8305v53,0.0781,−0.89772,0.0669,−0.92421,0.0398,−0.94284,0.1071,−0.8683l5δyτfδyτv15,0.2362,−0.77527,0.8291,−0.13491.7221v24,0.1717,−0.82836,0.6891,−0.31761.4357v33,0.1001,−0.90595,0.5960,−0.40011.2229v44,0.1438,−0.84865,0.5017,−0.45011.1508v52,0.0742,−0.93384,0.3172,−0.63420.8415According to the results of Table31, the descending order ranking of COVID-19 vaccines is given as follows:(41)v1>v2>v3>v4>v5.Hence, it is concluded that the government will import thev1= Sinopharm.


### 6.3.D-Choice Values of BF8SfS

In order to apply theD-choice values of BFNSfS, we used the BF8Sf decision matrix, given by Table 29, where phase terms are taken zero. The threshold D=5.Further, calculate(42)ℑDli=αyip,βyin,ifΥΦli=vy,ryi,αyip,βyin,andryi≥D,0.0,−1.0,otherwise.Moreover, compute theδyD as follows:(43)δyD=⊕i=1rχyiD,where, χyiD=αyipD,βyinD, and the sum of two BFNSfNsχyiD and χygD is calculated as follows:(44)χyiD⊕χygD=αyipD+αygpD−αyipDαygpD,−βyinDβygnD.Further, the 5-choice values ofBFNSfNs is calculated by using the score function as follows:(45)fδyD=αyip+βyin+12.The results are arranged by Table32.Table 32
5-choice value ofBF8SfS.
ℑ,T,8l1l2l3l4v10.77,−0.130.79,−0.140.92,−0.070.82,−0.15v20.71,−0.350,−10.81,−0.180.63,−0.35v30.69,−0.320.67,−0.310.65,−0.340,−1v40,−10,−10,−10.64,−0.29v50,−10,−10,−10,−1l5δyDfδyDv10.74,−0.280.9998,−5.35×10−50.9999v20,−10.9796,−0.02210.9788v30,−10.9642,−0.03370.9652v40,−10.64,−0.290.6750v50,−10,−10From Table32, it is summarized that the ranking of COVID-19 vaccines is as follows:(46)v1>v2>v3>v4>v5.Hence, the government will importv1= Sinopharm without further delay.Table 33
Comparative analysis.
MethodsRanking of COVID-19 vaccinesBest vaccineChoice value ofCBFNSfS (proposed)v1≻v2≻v3≻v4≻v5v1Weighted choice value ofCBFNSfS (proposed)v1≻v2≻v3≻v4≻v5v15-Choice value ofCBFNSfS (proposed)v1≻v2≻v3≻v4≻v5v1Choice value ofBFNSfS [22]v1≻v2≻v3≻v4≻v5v1Weighted choice value ofBFNSfS [22]v1≻v2≻v3≻v4≻v5v15-Choice value ofBFNSfS [22]v1≻v2≻v3≻v4≻v5v1Choice value ofFNSf S [17]v1≻v2≻v3≻v4≻v5v13-Choice value ofFNSf S [17]v1≻v2≻v3≻v4≻v5v1Figure 3
Comparative analysis.


### 6.4. Discussion

(1)
Now, the results of the presented MADM methodologies in the framework ofCBFNSfS are compared with MADM methods, namely, choice values of BFNSfS, weighted choice values of BFNSfS and D-choice values of BFNSfS, presented by Akram et al. [22], to show the authenticity and veracity of the proposed decision-making algorithms.(2)
Moreover, we have also applied the decision-making methods ofFNSf S [17] in order to find the best available COVID-19 vaccine to import. The results of the proposed and existing methods, including the final ranking and best alternatives, are arranged in Table 33 as follows:(3)
It is clear from Table33 that the v1 is the best vaccine in all environments. Moreover, the ranking of vaccines is also similar in all methodologies, which illustrates the reliability and accuracy of the presented MADM techniques.(4)
The comparative analysis of the results is also demonstrated in Figure3 through the bar chart between the ranking and COVID-19 vaccines, which shows the proficiency of the proposed decision-making techniques.(5)
The proposed hybrid model ofCBFNSfS can effectively handle the information in the environment of CBFNSfS, BFSfS and BFS by taking N=2 and L=1 respectively.(6)
The proposed decision-making methods ofCBFNSfS can handle the environment of FNSf S by taking negative membership values and phase terms equal to zero.(7)
The presentedCBFNSfS is a parameterized framework that is used to deal with the bipolar fuzziness of the information. It has the capability to handle two-dimensional vague information; i.e., it can deal with the periodicity involved in the bipolar information. Particularly, it can efficiently deal with the ranking based assessment of imprecise information that involves effects along with side effects.


## 6.1. Choice Value ofBF8SfS

In order to apply the Choice value ofBFNSfS, firstly, we arranged the BF8SfS from Table 23 by taking all phase terms of CBFNSfNs equal to zero. The grading criteria and the grades given by the expert will remain the same. The BF8Sf decision matrix is arranged in Table 29.Table 29
BF8Sf decision matrix.
ℑ,T,8l1l2l3l4v16,0.77,−0.136,0.79,−0.147,0.92,−0.076,0.82,−0.15v25,0.71,−0.354,0.59,−0.436,0.81,−0.185,0.63,−0.35v35,0.69,−0.325,0.67,−0.315,0.65,−0.343,0.43,−0.57v44,0.52,−0.423,0.39,−0.553,0.45,−0.525,0.64,−0.29v53,0.38,−0.532,0.26,−0.711,0.15,−0.794,0.53,−0.39l5v15,0.74,−0.28v24,0.61,−0.39v33,0.41,−0.61v44,0.54,−0.44v52,0.32,−0.71Moreover, compute theδy as follows:(33)δy=⊕i=1rχyi,where χyi=ryi,αyip,βyin, and sum of two BFNSfNsχyi and χyg is calculated as follows:(34)χyi⊕χyg=maxriy,rgy,αyip+αygp−αyipαygp,−βyinβygn.Further, the choice value ofBFNSfNs is calculated by using the score function as follows:(35)fδy=riyN−1+αyip+βyin+12.The results are given by Table30.Table 30
Choice values ofBF8SfS.
ℑ,T,8l1l2l3l4v16,0.77,−0.136,0.79,−0.147,0.92,−0.076,0.82,−0.15v25,0.71,−0.354,0.59,−0.436,0.81,−0.185,0.63,−0.35v35,0.69,−0.325,0.67,−0.315,0.65,−0.343,0.43,−0.57v44,0.52,−0.423,0.39,−0.553,0.45,−0.525,0.64,−0.29v53,0.38,−0.532,0.26,−0.711,0.15,−0.794,0.53,−0.39l5δyfδyv15,0.74,−0.287,0.9998,−5.35×10−51.8749v24,0.61,−0.396,0.9967,−0.00371.7465v33,0.41,−0.615,0.9880,−0.01171.6131v44,0.54,−0.445,0.9733,−0.01531.6040v52,0.32,−0.714,0.8754,−0.08231.3965According to the results of Table30, the vaccines of COVID-19 are ranked as follows:(36)v1>v2>v3>v4>v5.Hence, the government will import the v1= Sinopharm on an urgent basis.


## 6.2. Weighted Choice Values ofBF8SfS

In order to apply the weighted choice values ofBFNSfS, we used the BF8Sf decision matrix given by Table 29, where phase terms are taken zero. The weight vector of criteria is given below:(37)τ=0.170.230.250.150.20T.Further, compute theδyτ as follows:(38)δyτ=⊕i=1rτχyi,where χyi=ryi,αyip,βyin,τχyi=riy,1−1−αyipτ,−βyinτ., and the sum of two weighted BFNSfNsχyiτ and χygτ is calculated as follows:(39)τχyi⊕τχyg=maxriyτ,rgyτ,αyipτ+αygpτ−αyipταygpτ,−βyinτβygnτ.Further, the weighted choice values ofBFNSfNs is calculated by using the score function as follows:(40)fδyτ=riyN−1+αyip+βyin+12.The results are arranged by Table31.Table 31
Weighted choice value ofBF8SfS.
ℑ,T,8l1l2l3l4v16,0.2211,−0.70696,0.3016,−0.63627,0.4682,−0.51446,0.2268,−0.7523v25,0.1898,−0.83654,0.1854,−0.82366,0.3398,−0.65145,0.1385,−0.8543v35,0.1805,−0.82395,0.2251,−0.76395,0.2308,−0.76363,0.0809,−0.9191v44,0.1173,−0.86293,0.1075,−0.87153,0.1388,−0.84925,0.1421,−0.8305v53,0.0781,−0.89772,0.0669,−0.92421,0.0398,−0.94284,0.1071,−0.8683l5δyτfδyτv15,0.2362,−0.77527,0.8291,−0.13491.7221v24,0.1717,−0.82836,0.6891,−0.31761.4357v33,0.1001,−0.90595,0.5960,−0.40011.2229v44,0.1438,−0.84865,0.5017,−0.45011.1508v52,0.0742,−0.93384,0.3172,−0.63420.8415According to the results of Table31, the descending order ranking of COVID-19 vaccines is given as follows:(41)v1>v2>v3>v4>v5.Hence, it is concluded that the government will import thev1= Sinopharm.


## 6.3.D-Choice Values of BF8SfS

In order to apply theD-choice values of BFNSfS, we used the BF8Sf decision matrix, given by Table 29, where phase terms are taken zero. The threshold D=5.Further, calculate(42)ℑDli=αyip,βyin,ifΥΦli=vy,ryi,αyip,βyin,andryi≥D,0.0,−1.0,otherwise.Moreover, compute theδyD as follows:(43)δyD=⊕i=1rχyiD,where, χyiD=αyipD,βyinD, and the sum of two BFNSfNsχyiD and χygD is calculated as follows:(44)χyiD⊕χygD=αyipD+αygpD−αyipDαygpD,−βyinDβygnD.Further, the 5-choice values ofBFNSfNs is calculated by using the score function as follows:(45)fδyD=αyip+βyin+12.The results are arranged by Table32.Table 32
5-choice value ofBF8SfS.
ℑ,T,8l1l2l3l4v10.77,−0.130.79,−0.140.92,−0.070.82,−0.15v20.71,−0.350,−10.81,−0.180.63,−0.35v30.69,−0.320.67,−0.310.65,−0.340,−1v40,−10,−10,−10.64,−0.29v50,−10,−10,−10,−1l5δyDfδyDv10.74,−0.280.9998,−5.35×10−50.9999v20,−10.9796,−0.02210.9788v30,−10.9642,−0.03370.9652v40,−10.64,−0.290.6750v50,−10,−10From Table32, it is summarized that the ranking of COVID-19 vaccines is as follows:(46)v1>v2>v3>v4>v5.Hence, the government will importv1= Sinopharm without further delay.Table 33
Comparative analysis.
MethodsRanking of COVID-19 vaccinesBest vaccineChoice value ofCBFNSfS (proposed)v1≻v2≻v3≻v4≻v5v1Weighted choice value ofCBFNSfS (proposed)v1≻v2≻v3≻v4≻v5v15-Choice value ofCBFNSfS (proposed)v1≻v2≻v3≻v4≻v5v1Choice value ofBFNSfS [22]v1≻v2≻v3≻v4≻v5v1Weighted choice value ofBFNSfS [22]v1≻v2≻v3≻v4≻v5v15-Choice value ofBFNSfS [22]v1≻v2≻v3≻v4≻v5v1Choice value ofFNSf S [17]v1≻v2≻v3≻v4≻v5v13-Choice value ofFNSf S [17]v1≻v2≻v3≻v4≻v5v1Figure 3
Comparative analysis.


## 6.4. Discussion

(1)
Now, the results of the presented MADM methodologies in the framework ofCBFNSfS are compared with MADM methods, namely, choice values of BFNSfS, weighted choice values of BFNSfS and D-choice values of BFNSfS, presented by Akram et al. [22], to show the authenticity and veracity of the proposed decision-making algorithms.(2)
Moreover, we have also applied the decision-making methods ofFNSf S [17] in order to find the best available COVID-19 vaccine to import. The results of the proposed and existing methods, including the final ranking and best alternatives, are arranged in Table 33 as follows:(3)
It is clear from Table33 that the v1 is the best vaccine in all environments. Moreover, the ranking of vaccines is also similar in all methodologies, which illustrates the reliability and accuracy of the presented MADM techniques.(4)
The comparative analysis of the results is also demonstrated in Figure3 through the bar chart between the ranking and COVID-19 vaccines, which shows the proficiency of the proposed decision-making techniques.(5)
The proposed hybrid model ofCBFNSfS can effectively handle the information in the environment of CBFNSfS, BFSfS and BFS by taking N=2 and L=1 respectively.(6)
The proposed decision-making methods ofCBFNSfS can handle the environment of FNSf S by taking negative membership values and phase terms equal to zero.(7)
The presentedCBFNSfS is a parameterized framework that is used to deal with the bipolar fuzziness of the information. It has the capability to handle two-dimensional vague information; i.e., it can deal with the periodicity involved in the bipolar information. Particularly, it can efficiently deal with the ranking based assessment of imprecise information that involves effects along with side effects.


## 7. Conclusion

In this paper, an innovative hybrid model, namely, complex bipolar fuzzyN-soft sets has been set up by integrating the CBFSs with NSfSs. The generalized, more efficient theory is useful to handle the two-dimensional bipolar fuzzy information. It is superior to the existing BFNSfS as it deals with periodic information. Firstly, we have presented the conventional definition of CBFNSfSs in addition to its fundamental operations and related results. We have defined elementary algebraic operations for CBFNSfNs. We have also developed three algorithms for MADM problems in order to choose the favorable parameter under the environment of CBFNSfS. We have illustrated the significance of proposed algorithms by applying them to practical applications. Finally, to demonstrate their validity and applicability, we have shown a comparison with existing MADM approaches. In the future, we intend to establish more decision-making approaches, including ELECTRE I, ELECTRE II, and TOPSIS methods under the aforementioned abundant and multifaceted model of CBFNSfS.



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*Source: 2635568-2022-02-21.xml*