# Hybrid Decision-Making Frameworks under Complex Spherical FuzzyN-Soft Sets

Authors: Muhammad Akram; Maria Shabir; Ahmad N. Al-Kenani; José Carlos R. Alcantud
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper presents the novel concept of complex spherical fuzzyN-soft set (CSFNSfS) which is capable of handling two-dimensional vague information with parameterized ranking systems. First, we propose the basic notions for a theoretical development of CSFNSfSs, including ranking functions, comparison rule, and fundamental operations (complement, union, intersection, sum, and product). Furthermore, we look into some properties of CSFNSfSs. We then produce three algorithms for multiattribute decision-making that take advantage of these elements. We demonstrate their applicability with the assistance of a numerical problem (selection of best third-party app of the year). A comparison with the performance of Pythagorean N-soft sets speaks for the superiority of our approach. Moreover, with an aim to expand the range of techniques for multiattribute group decision-making problems, we design a CSFNSf-TOPSIS method. We use a complex spherical fuzzy N-soft weighted average operator in order to aggregate the decisions of all experts according to the power of the attributes and features of alternatives. We present normalized-Euclidean distances (from the alternatives to both the CSFNSf positive and negative ideal solutions, respectively) and revised closeness index in order to produce a best feasible alternative. As an illustration, we design a mathematical model for the selection of the best physiotherapist doctor of Mayo hospital, Lahore. We conduct a comparison with the existing complex spherical fuzzy TOPSIS method that confirms the stability of the proposed model and the reliability of its results.


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## Body

## 1. Introduction

Multiattribute decision-making (MADM) and multiattribute group decision-making (MAGDM) methods are broad sections in the field of decision-making. Researchers and practitioners have resorted to them in order to evaluate optimal solutions among a finite number of choices under several attributes. For the purpose of improving the flexibility of the evaluations that support the decision-making process, Zadeh [1] proposed fuzzy set (FS) theory that reshaped the field of decision-making and related disciplines such as mathematical social sciences [2, 3]. In FS theory, a membership degree belongs to the interval 0,1; thus, when assigned to an object, it represents its degree of belongingness to a mathematical object (a fuzzy set); in formal logic, it means a degree of truth. This means an extension of binary valuations, which is, henceforth, referred to as crisp evaluations. Concerning its use for solving MADM and MAGDM problems in fuzzy environments, Song et al. [4] gave an algorithm based on arithmetic operators, and Chen [5] built up a theory for a fuzzy-TOPSIS method. No doubt, FS theory produced a turn of direction in the field of decision-making. However, it was not designed to look at the dissatisfaction nature of humans in decision-making. This drawback prompted Atanassov [6] to present intuitionistic fuzzy sets (IFS) in 1986. They allocate both degrees of satisfaction and dissatisfaction to an object.Other extensions soon followed. Yager [7] further extended the IFS to Pythagorean fuzzy set (PyFS) in which the sums of the squares of degree of satisfaction and dissatisfaction should be within the closed unit interval. Later on, Cuong [8] introduced picture fuzzy set (PFS) keeping in view the existence of neutral positions under natural circumstances. For example, in case of voting systems a candidate could be either satisfied, remain neutral, and disagree with any given participant [9]. Many researchers chose this environment for solving decision-making problems, but others pointed out that PFS has the limitation that it is not applicable in situations where the sum of the degrees of satisfaction, neutrality, and dissatisfaction exceeds 1. This is the origin of spherical fuzzy sets and the spherical fuzzy TOPSIS method presented by Gundogdu and Kahraman [10] in 2019. Similarly and also motivated by spherical fuzzy sets, Kahraman et al. [11] used a spherical TOPSIS method to find the best location for hospital. Later on, Mahmood et al. [12] proposed T-spherical fuzzy sets, a generalization of spherical fuzzy sets, which are less restrictive. They overcame all the limitations of the existing models except in the presence of 2-dimensional problems. Such 2-dimensional problems in MADM and MAGDM can now be analyzed with the tool developed by Ramot et al. [13], who introduced complex fuzzy set in which the degree of satisfaction belongs to the complex unit circle and consists of a periodic term as well as the amplitude term which belong to the unit closed interval. Akram and Bashir [14] extended the averaging operators in the framework of complex fuzzy sets. Alkouri and Salleh [15] presented the idea of complex intuitionistic fuzzy set (CIFS), which describes both degree of satisfaction and dissatisfaction within the complex unit circle, where the sum of amplitude and periodic terms of the satisfaction and dissatisfaction degrees should be within the unit interval 0,1.Recently, Akram et al. [16] introduced the concept of complex spherical fuzzy set and extended the TOPSIS method to that setting. As an application, a model for the selection of best water supply strategy for Nohoor village in Iran was considered. This novel concept contains degrees of satisfaction, neutrality, and dissatisfaction which lie in the complex unit circle. They are further restricted by the condition that the sum of the squares of their amplitude and phase terms should be less than or equal to 1.There is a widespread handicap in the aforementioned models and methods: they discard the frameworks that are characterized by the satisfaction of certain attributes or the fulfilment of properties. Soft set theory, launched in 1999, accommodates all type of parameters [17]. Alkouri and Salleh [15] introduced some new operators on soft set theory which soon found applications in the fields of operations research, game theory, stability, regularization, medicine, and obviously in decision-making. Following this trend, researchers brought up many models and methods for soft sets and its extensions, inclusive of a new decision-making method for valuation fuzzy soft sets introduced by Alcantud et al. [18]. Despite these improvements there were still problems in real life that could not be solved using the existing MADM and MAGDM methods, for example, because the objects are evaluated using a ranking system or a nonbinary scale. When we check out from hotels, hotel staff ask for our feedback, which we give, for example, in the form of 4 stars, 3 stars, 2 stars, 1 star, and big dot: 4 stars mean “outstanding,” 3 stars mean “superb,” 2 stars mean “good,” 1 star means “satisfactory,” and big dot means “unacceptable.” Similarly, nonbinary rates are given to third-party apps, whether we use a transportation service (Uber, Cabify, etc.) or online shopping facilities. As technology improved and extended, people have become accostumed to such types of ranking systems due to their ease of use and widespread utilization. For this reason, many researchers have become interested in formal models for nonbinary evaluations. The idea presented by Fatima et al. [19], namely, N-soft set and their decision-making methods, stirred up new decision-making methodologies. Very soon and keeping in view the possible fuzziness of the parameters, Akram et al. [20] combined the concept of N-soft with a fuzzy definition of the attributes thus producing fuzzy N-soft sets FNSfS. This novel prescription involves a finite number of ordered grades as well as fuzziness in the conception of the attributes that are used for decision-making. Still another hybrid model called hesitant N-soft set was introduced by Akram et al. [21] in order to allow for hesitancy in the allocation of grades. Hesitant fuzzy N-soft sets [22] combine the features of these two models. Akram et al. [23] extended the idea of fuzzy N-soft set in another direction. They conceived intuitionistic fuzzy N-soft sets IFNSfS that describe the dissatisfactory part separately, with the usual constraint that the sums of the degrees of membership and nonmembership always belong to 0,1. Finally, so far, Zhang et al. [24] extended IFNSfS to Pythagorean fuzzy N-soft set (PFNSfS) which is more flexible than the existing models.The motivation of this article depends on the following facts:(1)
The existing modelsIFNSfS and PFNSfS make decisions based on degrees of membership and nonmembership; however, they are unable to incorporate a neutral part of judgement.(2)
The decision-making techniques based on existing modelsFNSfS, IFNSfS, and PFNSfS can solve only problems of the 1-dimensional type. Neither of these models can operate in the presence of a periodic term or 2-dimensional type problems.(3)
Although CSFSs deal with 2-dimensional problems of real life, they are unable to describe parameterized information as well as finitely many ranked grades of association of the alternatives with the pertinent parameters.(4)
These limitations motivated us to put forward a new model calledCSFNSfS which efficiently deals with abstention (together with degrees of satisfaction and dissatisfaction) as well as the periodic term of 2-dimensional decision-making problems. At the same time, CSFNSfS competently handles the ordered grades of the alternatives according to the different attributes.The main contributions of this article are as follows:(1)
The proposed model,CSFNSfS, allows for neutral opinions in the framework of 2-dimensional problems. In this way, it can manipulate conditions on amplitude and periodic terms with more flexibility.(2)
This model establishes a modern theory that captures a new perspective of decision-making. It is based on ratings or ranking systems including ordered grades of elements according to related attributes.(3)
The algorithms andCSFNSf-TOPSIS method defined in this article solve MADM and MAGDM problems, respectively. They apply to more general situations than the existing algorithms and TOPSIS Method. These methods for decision-making under the framework of CSFNSf are illustrated with numerical examples.(4)
The comparative study withPFNSf algorithms and the CSF-TOPSIS method shows their ability and significance.The rest of the paper is organized as follows. Section2 contains some definitions from existing models. In Section 3, we propose the novel concept of CSFNSfS which is then followed by the operations on CSFNSfSs and CSFNSfNs. Section 3 describes three algorithms for making decisions and performs a comparison with a PFNSf method. In Section 4, we develop a theatrical foundation for the CSFNSfS-TOPSIS method. In Section 5, we present the mathematical algorithms of these decision-making mechanisms that are applied to some numerical examples. Section 6 describes the comparison analysis with CSF-TOPSIS method. In Section 7, we conclude the paper and provide future directions of research.


## 2. Preliminaries

Definition 1 (see [10]).
Aspherical fuzzy setSFSϒ on a universe of discourse U has the form(1)ϒ=u,μϒu,ηϒu,νϒu|u∈U,where μϒu, ηϒu, and νϒu, which lie within the unit interval, are called the grade of the positive, neutral, and negative membership, respectively; and they are restricted by the condition μϒu2+ηϒu2+νϒu2≤1, for every u∈U. The degree of refusal of u in U is defined as(2)Θϒu=1−pϒu2+vϒu2+rϒu2.
The tripletμϒu,ηϒu,νϒu is called spherical fuzzy number SFN.Definition 2 (see [25]).
AcomplexT-spherical fuzzy setCTSFSϒ on the universe U is defined as(3)ϒ=u,μϒu,ηϒu,νϒu|u∈U,where μϒu=pϒuei2πϕϒu,ηϒu=vϒuei2πδϒu, and νϒu=rϒuei2πλϒu, which denote the positive, neutral, and negative degree of membership, respectively. They are restricted by the conditions pϒuq+vϒuq+rϒuq≤1 and ϕϒuq+δϒuq+λϒuq≤1, for each u∈U, where i=−1, and pϒ, vϒ, rϒ, ϕϒ, δϒ, λϒ∈0,1. The degree of refusal of u in U is defined as(4)Πϒu=1−pϒuq+vϒuq+rϒuqei2π1−ϕϒuq+δTuq+λϒuq.
The tripletμϒ,ηϒ,νϒ=pϒei2πϕϒ,vϒei2πδϒ,rϒei2πλϒ is called CTSFN.Particular case: WhenT=2, a CTSFS becomes a complex spherical fuzzy set CSFS.Definition 3 (see [26]).
LetW be a nonempty set and R be a set of attributes and Z⊆R. A soft setSfS over W is a pair Γ,Z, where Γ is a set-valued function from Z to the set of all subsets of W, which is denoted as(5)Γ,Z=z,Γz|z∈Z,Γz∈2W.Definition 4.
LetW be a nonempty set and R be a set of attributes, Z⊆R. A complex spherical fuzzy soft setCSFSfS over W is a pair Λ,Z, where Λ is a function from Z to the set of all subsets of CSFSs of W, which is denoted as(6)Λ,Z=z,Λz|z∈Z,Λz∈CSFSW=z,w,μzw,ηzw,νzw=z,w,pzwei2πϕzw,vzwei2πδzw,rzwei2πλzw,where pz,vz,rz, ϕz,δz,λz∈0,1 are restricted by the conditions(7)0≤pzw2+vzw2+rzw2≤1,0≤ϕzw2+δzw2+λzw2≤1,∀w∈W.Definition 5 (see [19]).
LetW be a nonempty set and R be a set of attributes. Let Z⊆R and G=0,1,2,…,N−1 be a set of ordered grades with N∈2,3,…. A triple F,Z,N is called N-soft setNSfS over W if F is a mapping from Z to 2U×G, with the property that, for each z∈Z and w∈W, there exist a unique w,gzw∈W×G such that w,gzw∈Fz,w∈W,gzw∈G [27–40].


## 3. Complex Spherical FuzzyN-Soft Sets

Definition 6.
LetW be a nonempty set and R be a set of attributes. Let Z⊆R and G=0,1,2,…,N−1 be a set of ordered grades with N∈2,3,…. A triple FJ,Z,N is called a complex spherical fuzzyN-soft setCSFNSfS on Z, when F,Z,NF:Z⟶2W×G is an NSfS on W, if FJ:Z⟶2W×G×CSFN is a mapping, which is defined as(8)FJ,Z,N=z,Fz,Jz|z∈Z,Fz,Jz∈2W×G×CSFN=z,w,gzw,μzw,ηzw,νzw=z,w,gzw,pzwei2πϕzw,vzwei2πδzw,rzwei2πλzw,where J:Z⟶CSFN, CSFN denotes the collection of all complex spherical fuzzy numbers of W, gzw denotes the level of attribute for the element w and pz,vz,rz, ϕz,δz,λz∈0,1, restricted with conditions(9)0≤pzw2+vzw2+rzw2≤1,0≤ϕzw2+δzw2+λzw2≤1,for allwbelongs toW.Definition 7.
LetFJzk = wj,gkj,pkjei2πϕkj,vkjei2πδkjrkjei2πλkj be a CSFNSfS. Then, the complex spherical fuzzyN-soft numberCSFNSfN is defined as(10)ϒkj=gkj,pkjei2πϕkj,vkjei2πδkj,rkjei2πλkj,Ωϒkj=1−pkj2+wkj2+rkj2ei2π1−ϕkj2+δkj2+λkj2,is the hesitancy degree, where pkj,vkj,rkj,ϕkj,δkj, and λkj represent pzkwj,vzkwj,rzkwj,ϕzkwj,δzkwj, and λzkwj, respectively.Definition 8.
Consider aCSFNSfNϒkj=gkj,pkjei2πϕkj,vkjei2πδkjrkjei2πλkj. The score function Sϒkj is(11)Sϒkj=gkjN−12+pkj2−wkj2−rkj2+ϕkj2−δkj2−λkj2,where Sϒkj∈−2,3. The accuracy function Aϒkj is(12)Aϒkj=gkjN−12+pkj2+wkj2+rkj2+ϕkj2+δkj2+λkj2,where Aϒkj∈0,3, respectively.Definition 9.
Letϒlj=glj,pljei2πϕlj,vljei2πδlj,rljei2πλlj and ϒkj=gkj,pkjei2πϕkj,vkjei2πδkj,rkjei2πλkj be two CSFNSfNs:(1)
IfSϒlj<Sϒkj, then ϒlj≺ϒkj (ϒlj is inferior to ϒkj).(2)
IfSϒlj>Sϒkj, then ϒlj≻ϒkj (ϒlj is superior to ϒkj).(3)
IfSϒlj=Sϒkj, then(i)
Aϒlj<Aϒkj, then ϒlj≺ϒkj (ϒlj is inferior to ϒkj)(ii)
Aϒlj>Aϒkj, then ϒlj≻ϒkj (ϒlj is superior to ϒkj)(iii)
Aϒlj=Aϒkj, then ϒlj∼ϒkj (ϒlj is equivalent to ϒkj)Remark 1.
We see that(1)
ForN=2, CSFNSfS becomes complex spherical fuzzy soft set(2)
WhenZ=1, CSFNSfS becomes complex spherical fuzzy set(3)
Whenϕz=δz=λz=0, CSFNSfS becomes spherical fuzzy N-soft setExample 1.
In a city, a parent wants to choose the best school for their child. It is necessary to go after the advice of experts, for the selection of a school based on rankings and ratings. LetW=w1,w2,w3 be the family of three schools under consideration and Z = {z1 = size of school, z2 = location, z3 = academic performance, z4 = services} be the attributes which are used to assign rankings to schools by the experts. In a relation to these parameters, a 5-soft set is given in Table 1, whereFour diamonds means “Outstanding”Three diamonds means “Super”Two diamonds means “Good”One diamond means “Satisfactory”Big dot means “Acceptable”
This level assessment by diamonds can be represented by numbers asG=0,1,2,3,4, where0 means “•”1 means “⋄”2 means “⋄⋄”3 means “⋄⋄⋄”4 means “⋄⋄⋄⋄”
Table2 can be adopted as natural convention of 5-soft set model.
By Definition6, when the data is vague and uncertain, we need CSFNSfSs which provides us information on how these grades are given to schools. The evaluation of schools by experts follows the following grading:(13)whengzw=0,−2.00≤SJ<−1.85,whengzw=1,−1.85≤SJ<−1.30,whengzw=2,−1.30≤SJ<0.15,whengzw=3,0.15≤SJ<1.30,whengzw=4,1.30≤SJ<2.00.
According to the above criteria, we can obtain Table3.
At last,CSF5SfS is defined as(14)μz1,ηz1,νz1=w1,2,0.4ei0.82π,0.017ei0.0356π,0.6ei1.22π,w2,0,0.02ei0.06π,0.012ei0.026π,0.98ei1.962π,w3,4,1,0,0,μz2,ηz2,νz2=w1,3,0.65ei1.32π,0.018ei0.038π,0.28ei0.58π,w2,3,0.7ei1.42π,0.019ei0.04π,0.3ei0.56π,w3,1,0.16ei0.34π,0.1ei0.204π,0.89e1.784π,μz3,ηz3,νz3=w1,0,0.1ei0.24π,0.012ei0.022π,0.985ei1.964π,w2,1,0.2ei0.36π,0.027ei0.05π,0.91ei1.824π,w3,3,0.69ei1.384π,0.101ei0.204π,0.32e0.62π,μz4,ηz4,νz4=w1,2,0.5ei1.1π,0.1ei0.18π,0.59e1.28iπ,w2,1,0.3ei0.56π,0.019ei0.042π,0.885e1.72iπ,w3,2,0.45ei0.86π,0.015ei0.022π,0.78ei1.566π.
The tabular representation ofCSF5SfS is shown by Table 4.Table 1
Evaluation data provided by the experts.
W/Zz1z2z3z4w1⋄⋄⋄⋄⋄•⋄⋄w2•⋄⋄⋄⋄⋄w3⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄Table 2
Tabular representation of 5-soft set.
W/Zz1z2z3z4w12302w20311w34132Table 3
Grading criteria.
gzw/JPositive membershipNeutral membershipNegative membershipGradespz2πϕzvz2πδzrz2πλzgzw=00,0.150,0.3π0,0.0170,0.034π0.98,11.96π,2πgzw=10.15,0.350.3π,0.7π0.017,0.10310.034π,0.2062π0.88,0.981.76π,1.96πgzw=20.35,0.650.7π,1.3π0,0.0170,0.034π0.58,0.881.16π,1.76πgzw=30.65,0.851.3π,1.7π0.017,0.10310.034π,0.2062π0.28,0.580.56π,1.16πgzw=40.85,11.7π,2π0,0.0170,0.034π0,0.280,0.56πTable 4
Tabular representation of theCSF5SfSFJ,Z,5.
FJ,Z,5z1z2z3z4w12,0.4ei0.82π,0.017ei0.0356π,0.6ei1.22π3,0.65ei1.32π,0.018ei0.038π,0.28ei0.58π0,0.1ei0.24π,0.012ei0.022π,0.985ei1.964π2,0.5ei1.1π,0.1ei0.18π,0.59e1.28iπw20,0.02ei0.06π,0.012ei0.026π,0.98ei1.962π3,0.7ei1.42π,0.019ei0.04π,0.3ei0.56π1,0.2ei0.36π,0.027ei0.05π,0.91ei1.824π1,0.3ei0.56π,0.019ei0.042π,0.885e1.72iπw34,1,0,01,0.16ei0.34π,0.1ei0.204π,0.89e1.784π3,0.69ei1.384π,0.101ei0.204π,0.32e0.62π2,0.45ei0.86π,0.015ei0.022π,0.78ei1.566πDefinition 10.
ACSFNSfSFJ,Z,N over a nonempty set W is said to be efficient, where F,Z,N is an NSfS if FJz=w,N−1,1,0,0 for some z∈Z,w∈W.Example 2.
LetFJ,Z,5 be CSF5SfS, as in Example 1. It is easy to check from Table 4 that FJz1=w3,4,1,0,0, i.e., Example 1 is efficient.Definition 11.
LetFJ,Z,N1 and HA,B,N2 be two CSFNSfSs on a universe of discourse W. Then, they are said to be equal if and only if F=H, J=A, Z=B, and N1=N2.Definition 12.
LetFJ,Z,N be CSFNSfS on W. The weak complement of CSFNSfS is defined as the weak complement of the N-soft set F,Z,N, that is, any N-soft set such that Fcz∩Fz=∅ for all z∈Z. The weak complement of CSFNSfS of FJ,Z,N is represented as FJc,Z,N.Example 3.
LetFJ,Z,5 be CSF5SfS, as in Example 1. The weak complement FJc,Z,N is given in Table 5.Table 5
A weak complement of theCSF5SfSFJ,Z,5 in Example 1.
FJc,Z,5z1z2z3z4w13,0.4ei0.82π,0.017ei0.0356π,0.6ei1.22π0,0.65ei1.32π,0.018ei0.038π,0.28ei0.58π1,0.1ei0.24π,0.012ei0.022π,0.985ei1.964π3,0.5ei1.1π,0.1ei0.18π,0.59e1.28iπw23,0.02ei0.06π,0.012ei0.026π,0.98ei1.962π1,0.7ei1.42π,0.019ei0.04π,0.3ei0.56π4,0.69ei1.384π,0.101ei0.204π,0.32e0.62π2,0.3ei0.56π,0.019ei0.042π,0.885e1.72iπw32,1,0,03,0.16ei0.34π,0.1ei0.204π,0.89e1.784π1,0.69ei1.384π,0.101ei0.204π,0.32e0.62π4,0.45ei0.86π,0.015ei0.022π,0.78ei1.566πDefinition 13.
LetFJ,Z,N be CSFNSfS on W. The complex spherical fuzzy complement of CSFNSfS is denoted as FJc,Z,N and is defined as(15)FJcz=νzw,ηzw,μzw=w,gzwrzwei2πλzw,vzwei2πδzw,pzwei2πϕzw.Example 4.
LetFJ,Z,5 be CSF5SfS, as in Example 1. The complex spherical fuzzy complement FJc,Z,N is given in Table 6.Table 6
The complex spherical fuzzy complementFJc,Z,N of the CSF5SfS in Example 1.
FJc,Z,5z1z2z3z4w12,0.6ei1.22π,0.017ei0.0356π,0.4ei0.82π3,0.28ei0.58π,0.018ei0.038π,0.65ei1.32π0,0.985ei1.964π,0.012ei0.022π,0.1ei0.24π2,0.59e1.28iπ,0.1ei0.18π,0.5ei1.1πw20,0.98ei1.962π,0.012ei0.026π,0.02ei0.06π3,0.3ei0.56π,0.019ei0.04π,0.7ei1.42π1,0.32e0.62π,0.101ei0.204π,0.69ei1.384π1,0.885e1.72iπ,0.019ei0.042π,0.3ei0.56πw34,0,0,11,0.89e1.784π,0.1ei0.204π,0.16ei0.34π3,0.32e0.62π,0.101ei0.204π,0.69ei1.384π2,0.78ei1.566π,0.015ei0.022π,0.45ei0.86πDefinition 14.
LetFJ,Z,N be a CSFNSfS on W. FJcc,Z,N is referred to as a weak complex spherical fuzzy complement of FJ,Z,N if and only if FJc,Z,N is a weak complement and FJc,Z,N is a complex spherical fuzzy complement of FJ,Z,N.Example 5.
LetFJ,Z,5 be CSF5SfS, as in Example 1. The weak complex spherical fuzzy complement FJcc,Z,N is given in Table 7.Table 7
The weak complex spherical fuzzy complementFJcc,Z,N of the CSF5SfS in Example 1.
FJcc,Z,5z1z2z3z4w13,0.6ei1.22π,0.017ei0.0356π,0.4ei0.82π0,0.28ei0.58π,0.018ei0.038π,0.65ei1.32π1,0.985ei1.964π,0.012ei0.022π,0.1ei0.24π3,0.59e1.28iπ,0.1ei0.18π,0.5ei1.1πw23,0.98ei1.962π,0.012ei0.026π,0.02ei0.06π1,0.3ei0.56π,0.019ei0.04π,0.7ei1.42π4,0.32e0.62π,0.101ei0.204π,0.69ei1.384π2,0.885e1.72iπ,0.019ei0.042π,0.3ei0.56πw32,0,0,13,0.89e1.784π,0.1ei0.204π,0.16ei0.34π1,0.32e0.62π,0.101ei0.204π,0.69ei1.384π4,0.78ei1.566π,0.015ei0.022π,0.45ei0.86πDefinition 15.
LetFJ,Z,N be a CSFNSfS on W; then, the top weak complex spherical fuzzy complementFJ>,Z,N is defined as(16)FJ>,Z,N=FJzk=wj,N−1,rzkwjei2πλzkwj,vzkwjei2πδzkwjpzkwjei2πϕzkwj,ifgkj<N−1,FJzk=wj,0,rzkwjei2πλzkwj,vzkwjei2πδzkwjpzkwjei2πϕzkwj,ifgkj=N−1.Example 6.
LetFJ,Z,5 be CSF5SfS, as in Example 1. The top weak complex spherical fuzzy complement FJ>,Z,N, is given in Table 8.Table 8
The top weak complex spherical fuzzy complementFJ>,Z,N of the CSF5SfS set in Example 1.
FJ>,Z,5z1z2z3z4w14,0.6ei1.22π,0.017ei0.0356π,0.4ei0.82π4,0.28ei0.58π,0.018ei0.038π,0.65ei1.32π4,0.985ei1.964π,0.012ei0.022π,0.1ei0.24π4,0.59e1.28iπ,0.1ei0.18π,0.5ei1.1πw24,0.98ei1.962π,0.012ei0.026π,0.02ei0.06π4,0.3ei0.56π,0.019ei0.04π,0.7ei1.42π4,0.32e0.62π,0.101ei0.204π,0.69ei1.384π4,0.885e1.72iπ,0.019ei0.042π,0.3ei0.56πw30,0,0,14,0.89e1.784π,0.1ei0.204π,0.16ei0.34π4,0.32e0.62π,0.101ei0.204π,0.69ei1.384π4,0.78ei1.566π,0.015ei0.022π,0.45ei0.86πDefinition 16.
LetFJ,Z,N be CSFNSfS on W; then, the bottom weak complex spherical fuzzy complementFJ<,Z,N is defined as(17)FJ<,Z,N=FJzk=wj,0,rzkwjei2πλzkwj,vzkwjei2πδzkwjpzkwjei2πϕzkwj,ifgkj>0,FJzk=wj,N−1,rzkwjei2πλzkwj,vzkwjei2πδzkwjpzkwjei2πϕzkwj,ifgkj=0.Example 7.
LetFJ,Z,5 be CSF5SfS, as in Example 1. The bottom weak complex spherical fuzzy complement FJ<,Z,N is given in Table 9.Table 9
The bottom weak complex spherical fuzzy complementFJ<,Z,N of the CSF5SfS in Example 1.
FJ<,Z,5z1z2z3z4w10,0.6ei1.22π,0.017ei0.0356π,0.4ei0.82π0,0.28ei0.58π,0.018ei0.038π,0.65ei1.32π4,0.985ei1.964π,0.012ei0.022π,0.1ei0.24π0,0.59e1.28iπ,0.1ei0.18π,0.5ei1.1πw24,0.98ei1.962π,0.012ei0.026π,0.02ei0.06π0,0.3ei0.56π,0.019ei0.04π,0.7ei1.42π0,0.32e0.62π,0.101ei0.204π,0.69ei1.384π0,0.885e1.72iπ,0.019ei0.042π,0.3ei0.56πw30,0,0,10,0.89e1.784π,0.1ei0.204π,0.16ei0.34π0,0.32e0.62π,0.101ei0.204π,0.69ei1.384π0,0.78ei1.566π,0.015ei0.022π,0.45ei0.86πDefinition 17.
LetW be a nonempty set and FJ,Z,N1 and HA,B,N2 be CSFN1SfS and CSFN2SfSs on W, respectively, and their restricted intersection is defined as KL,M,S = FJ,Z,N1∩RHA,B,N2, where KL=FJ∩RHA, M=Z∩B, S=minN1,N2, i.e., ∀uk∈M and wJ∈W, gkj,μij,ηij,νij∈KLuj, gkj=mingk1,gk2, μijuj=minμij1uk1,μij2uk2 = minpij1uk1,pij2uk2ei2πminϕij1uk1,ϕij2uk2, ηijuj=maxηij1uk1,ηij2uk2 = maxvij1uk1,vij2uk2ei2πmaxδij1uk1,δij2uk2, νijuj=maxνij1uk1,νij2uk2 = maxrij1uk1,rij2uk2ei2πmaxλij1uk1,λij2uk2, where, gk1,μij1uk1,ηij1uk1,νij1uk1∈μFJuk1,ηFJuk1,νFJuk1, and gk2,μij2uk2,ηij2uk2,νij2uk2∈μHAuk2,ηHAuk2,νHAuk2, with uk1∈Z and uk2∈B.Example 8.
LetEP,Z,5 and HA,B,6 be two CSF5SfS and CSF6SfS, given in Tables 10 and 11, respectively. Their restricted intersection KL,M,5 = EP,Z,5∩RHA,B,6 is shown in Table 12.Table 10
Tabular representation of theCSF5SfSEP,Z,5.
EP,Z,5z1z2z3w12,0.4ei0.82π,0.017ei0.0356π,0.6ei1.22π3,0.65ei1.32π,0.018ei0.038π,0.28ei0.58π0,0.1ei0.24π,0.012ei0.022π,0.985ei1.964πw20,0.02ei0.06π,0.012ei0.026π,0.98ei1.962π3,0.7ei1.42π,0.019ei0.04π,0.3ei0.56π1,0.2ei0.36π,0.027ei0.05π,0.91ei1.824πw34,1,0,01,0.16ei0.34π,0.1ei0.204π,0.89e1.784π3,0.69ei1.384π,0.101ei0.204π,0.32e0.62πTable 11
Tabular representation ofCSF6SfSHA,B,6.
HA,B,6z1z2z6w11,0.23ei0.44π,0.019ei0.036π,0.92ei1.85π5,0.95ei1.88π,0.03ei0.062π,0.14ei0.29π3,0.69ei1.39π,0.04ei0.084π,0.65ei1.36πw20,0.03ei0.1π,0.015ei0.032π,0.983ei1.968π2,0.35ei0.66π,0.014ei0.029π,0.8ei1.6π4,0.87e1.72π,0.035ei0.068π,0.93ei1.862πw32,0.4ei0.82π,0.012ei0.026π,0.77ei1.52π3,0.6e1.22π,0.02ei0.06π,0.49eiπ1,0.17e0.344π,0.035ei0.068π,0.93ei1.862πTable 12
Tabular representation of restricted intersectionKL,M,5.
KL,M,5z1z2w11,0.23ei0.44π,0.019ei0.036π,0.92ei1.85π3,0.65ei1.32π,0.03ei0.062π,0.28ei0.58πw20,0.02ei0.06π,0.015ei0.032π,0.983ei1.968π2,0.35ei0.66π,0.019ei0.04π,0.8ei1.6πw32,0.4ei0.82π,0.012ei0.026π,0.77ei1.52π1,0.16ei0.34π,0.1ei0.204π,0.89e1.784πDefinition 18.
LetW be a nonempty set and FJ,Z,N1 and HA,B,N2 be two CSFNSfSs on W; their extended intersection is defined as QD,C,Y = FJ,Z,N1∩EHA,B,N2, where QD=FJ∩EHA, C=Z∪B, and Y=maxN1,N2, that is, ∀uk∈C and wj∈W, gkj,μkj,ηkj,νkj∈QDuk, with(18)QDuk=gk1,μkj1,ηkj1,νkj1,ifuk∈Z−B,gk2,μkj2,ηkj2,νkj2,ifuk∈B−Z,gkj,μkj,ηkj,νkj,such thatgkj=mingk1,gk2,μkjuk=minμkj1uk1,μkj2uk2=minpkj1uk1,pkj2uk2ei2πminϕkj1uk1,ϕkj2uk2,ηkjuk=maxηkj1uk1,ηkj2uk2=maxvkj1uk1,vkj2uk2ei2πmaxδkj1uk1,δkj2uk2,=νijuk=maxνkj1uk1,νkj2uk2=maxrkj1uk1,rkj2uk2ei2πminλkj1uk1,λkj2uk2,wheregk1,μkj1uk1,ηkj1uk1,νkj1uk1∈μFJuk1,ηFJuk1,νFJuk1,andgk2,μkj2uk2,ηkj2uk2,νkj2uk2∈μHAuk2,ηHAuk2,νHAuk2,withuk1∈Zanduk2∈B.Example 9.
LetEP,Z,5 and HA,B,6 be two CSF5SfS and CSF6SfS, given in Tables 10 and 11, respectively. Their extended intersection QD,C,Y = EP,Z,N1∩EHA,B,N2 is shown in Table 13.Table 13
Tabular representation of extended intersectionQD,C,6.
QD,C,6z1z2z3z6w11,0.23ei0.44π,0.019ei0.036π,0.92ei1.85π3,0.65ei1.32π,0.03ei0.062π,0.28ei0.58π0,0.1ei0.24π,0.04ei0.084π,0.985ei1.964π3,0.69ei1.39π,0.04ei0.084π,0.65ei1.36πw20,0.02ei0.06π,0.015ei0.032π,0.983ei1.968π2,0.35ei0.66π,0.019ei0.04π,0.8ei1.6π1,0.2ei0.36π,0.027ei0.05π,0.91ei1.824π4,0.87e1.72π,0.035ei0.068π,0.93ei1.862πw32,0.4ei0.82π,0.012ei0.026π,0.77ei1.52π1,0.16ei0.34π,0.1ei0.204π,0.89e1.784π3,0.69ei1.384π,0.101ei0.204π,0.32e0.62π1,0.17e0.344π,0.035ei0.068π,0.93ei1.862πDefinition 19.
LetW be a nonempty set and FJ,Z,N1 and HA,B,N2 be two CSFNSfSs on W; their restricted union is defined as ℛT,M,S = FJ,Z,N1∪RHA,B,N2, where ℛT=FJ∪RHA, M=Z∩B, S=maxN1,N2, i.e., ∀uk∈M and wj∈W, gkj,μkj,ηkj,νkj∈ℛTuk, gkj=maxgk1,gk2,μkjuk=maxμkj1uk1,μkj2uk2 = maxpkj1uk1,pkj2uk2ei2πmaxϕkj1uk1,ϕkj2uk2, ηkjuk=minηkj1uk1,ηkj2uk2 = minvkj1uk1,vkj2uk2ei2πminδkj1uk1,δkj2uk2, νkjuk=minνkj1uk1,νkj2uk2 = minrkj1uk1,rkj2uk2ei2πminλkj1uk1,λkj2uk2, where, gk1,μkj1uk1,ηkj1uk1,νkj1uk1∈μFJuk1,ηFJuk1,νFJuk1, and gk2,μkj2uk2,ηkj2uk2,νkj2uk2∈μHAuk2,ηHAuk2,νHAuk2, with uk1∈Z and uk2∈B.Example 10.
LetEP,Z,5 and HA,B,6 be two CSF5SfS and CSF6SfS, given in Tables 10 and 11, respectively. Their restricted union QD,C,Y = EP,Z,N1∩EHA,B,N2 is shown in Table 14.Table 14
Tabular representation of restricted unionℛT,M,6.
ℛT,M,6z1z2w12,0.4ei0.82π,0.017ei0.0356π,0.6ei1.22π5,0.95ei1.88π,0.018ei0.038π,0.14ei0.29πw20,0.03ei0.1π,0.012ei0.026π,0.98ei1.962π3,0.7ei1.42π,0.014ei0.029π,0.3ei0.56πw34,1,0,03,0.6e1.22π,0.02ei0.06π,0.49eiπDefinition 20.
LetW be a nonempty set FJ,Z,N1 and HA,B,N2 be CSFN1SfSs and CSFN2SfSs on W; their extended union is defined as OX,C,Y = FJ,Z,N1∪EHA,B,N2, where OX=FJ∪EHA, C=Z∪B,Y=maxN1,N2, that is, ∀uk∈C and wj∈W, gkj,μkj,ηkj,νkj∈OXuk, with(19)OXuk=gk1,μkj1,ηkj1,νkj1,ifuk∈Z−B,gk2,μkj2,ηkj2,νkj2,ifuk∈B−Z,gkj,μkj,ηkj,νkj,such thatgkj=maxgk1,gk2,μkjuk=maxμkj1uk1,μkj2uk2=maxpkj1uk1,pkj2uk2eimaxϕkj1uk1,ϕkj2uk2,ηkjuk=minηkj1uk1,ηkj2uk2=minvkj1uk1,vkj2uk2eiminδkj1uk1,δkj2uk2,νkjuk=minνkj1uk1,νkj2uk2=minrkj1uk1,rkj2uk2eiminλkj1uk1,λkj2uk2,wheregk1,μkj1uk1,ηkj1uk1,νkj1uk1∈μFJuk1,ηFJuk1,νFJuk1,andgk2,μkj2uk2,ηkj2uk2,νkj2uk2∈μHAuk2,ηHAuk2,νHAuk2,withuk1∈Zanduk2∈B.Example 11.
LetEP,Z,5 and HA,B,6 be two CSF5SfS and CSF6SfS, given in Tables 10 and 11, respectively. Their restricted union QD,C,Y = EP,Z,N1∩EHA,B,N2 is shown in Table 15.
We state the following properties without their proofs.Table 15
Tabular representation of extended unionOX,C,6.
OX,C,6z1z2z3z6w12,0.4ei0.82π,0.017ei0.0356π,0.6ei1.22π5,0.95ei1.88π,0.018ei0.038π,0.14ei0.29π0,0.1ei0.24π,0.04ei0.084π,0.985ei1.964π3,0.69ei1.39π,0.04ei0.084π,0.65ei1.36πw20,0.03ei0.1π,0.012ei0.026π,0.98ei1.962π3,0.7ei1.42π,0.014ei0.029π,0.3ei0.56π1,0.2ei0.36π,0.027ei0.05π,0.91ei1.824π4,0.87e1.72π,0.035ei0.068π,0.93ei1.862πw34,1,0,03,0.6e1.22π,0.02ei0.06π,0.49eiπ3,0.69ei1.384π,0.101ei0.204π,0.32e0.62π1,0.17e0.344π,0.035ei0.068π,0.93ei1.862πTheorem 1.
LetFJ,Z,NbeCSFNSfS over a nonempty set W. Then,(1)
FJ,Z,N∩RFJ,Z,N=FJ,Z,N(2)
FJ,Z,N∩EFJ,Z,N=FJ,Z,N(3)
FJ,Z,N∪RFJ,Z,N=FJ,Z,N(4)
FJ,Z,N∪EFJ,Z,N=FJ,Z,N
We state the following properties without their proofs.Theorem 2.
LetFJ,Z,N1andHA,B,N2be twoCSFNSfSsover the same universeW; then, the absorption properties hold:(1)
FJ,Z,N1∪EHA,B,N2∩RFJ,Z,N1=FJ,Z,N1(2)
FJ,Z,N1∪EHA,B,N2∩RFJ,Z,N1=FJ,Z,N1(3)
FJ,Z,N1∩RHA,B,N2∪EFJ,Z,N1=FJ,Z,N1(4)
FJ,Z,N1∩RHA,B,N2∪EFJ,Z,N1=FJ,Z,N1
We state the following properties without their proofs.Theorem 3.
LetFJ,Z,N1,HA,B,N2andD℘,ℜ,N2be any threeCSFNSfSsover the same universeW; then, the following properties hold:(1)
FJ,Z,N1∪EHA,B,N2=HA,B,N2∪EFJ,Z,N1(2)
FJ,Z,N1∪RHA,B,N2=HA,B,N2∪RFJ,Z,N1(3)
FJ,Z,N1∩EHA,B,N2=HA,B,N2∩EFJ,Z,N1(4)
FJ,Z,N1∩RHA,B,N2=HA,B,N2∩RFJ,Z,N1(5)
FJ,Z,N1∪EHA,B,N2∪ED℘,ℜ,N3=FJ,Z,N1∪EHA,B,N2∪ED℘,ℜ,N3(6)
FJ,Z,N1∪RHA,B,N2∪RD℘,ℜ,N3=FJ,Z,N1∪RHA,B,N2∪RD℘,ℜ,N3(7)
FJ,Z,N1∩EHA,B,N2∩ED℘,ℜ,N3=FJ,Z,N1∩EHA,B,N2∩ED℘,ℜ,N3(8)
FJ,Z,N1∩RHA,B,N2∩RD℘,ℜ,N3=FJ,Z,N1∩RHA,B,N2∩RD℘,ℜ,N3(9)
FJ,Z,N1∪EHA,B,N2∩RD℘,ℜ,N3=FJ,Z,N1∪EHA,B,N2∩RFJ,Z,N1∪ED℘,ℜ,N3(10)
FJ,Z,N1∩EHA,B,N2∪RD℘,ℜ,N3=FJ,Z,N1∩EHA,B,N2∪RFJ,Z,N1∪ED℘,ℜ,N3(11)
FJ,Z,N1∪RHA,B,N2∩ED℘,ℜ,N3=FJ,Z,N1∪RHA,B,N2∩EFJ,Z,N1∪RD℘,ℜ,N3(12)
FJ,Z,N1∩RHA,B,N2∪ED℘,ℜ,N3=FJ,Z,N1∩RHA,B,N2∪EFJ,Z,N1∩RD℘,ℜ,N3Definition 21.
LetFJ,Z,N be a CSFNSfS, where F,Z,N is NSfS over the universe W, and 0<L<N be a threshold. CSFSfS over W associated with F,Z,N and L, denoted by FJL,Z, is defined as follows:(20)FJLz=μkjz,ηkjz,νkjz,ifw,gzw∈Fzandgzw≥L,0,0,1,otherwise.Example 12.
LetEJ,Z,N be a CSF5SfS given in Table 10. Then, CSFSfS associated with the thresholds 1, 2, 3, and 4 are shown in Tables 16–19, respectively.Table 16
CSFSfS related with E,Z,5 and threshold 1.
E1,Z,5z1z2z3w10.4ei0.82π,0.017ei0.0356π,0.6ei1.22π0.65ei1.32π,0.018ei0.038π,0.28ei0.58π0,0,1w20,0,10.7ei1.42π,0.019ei0.04π,0.3ei0.56π0.2ei0.36π,0.027ei0.05π,0.91ei1.824πw31,0,00.16ei0.34π,0.1ei0.204π,0.89e1.784π0.69ei1.384π,0.101ei0.204π,0.32e0.62πTable 17
CSFSfS related with E,Z,5 and threshold 2.
E2,Z,5z1z2z3w10.4ei0.82π,0.017ei0.0356π,0.6ei1.22π0.65ei1.32π,0.018ei0.038π,0.28ei0.58π0,0,1w20,0,10.7ei1.42π,0.019ei0.04π,0.3ei0.56π0,0,1w31,0,00,0,10,0,1Table 18
CSFSfS related with EJ,Z,5 and threshold 3.
E3,Z,5z1z2z3w10,0,10.65ei1.32π,0.018ei0.038π,0.28ei0.58π0,0,1w20,0,10.7ei1.42π,0.019ei0.04π,0.3ei0.56π0,0,1w31,0,00,0,10,0,1Table 19
CSFSfS related with E,Z,5 and threshold 4.
E4,Z,5z1z2z3w10,0,10,0,10,0,1w20,0,10,0,10,0,1w31,0,00,0,10,0,1Definition 22.
LetFJ,Z,N be CSFNSfS, where F,Z,N is NSfS over the universe W. Let 0<L<N and α∈−2,2 be two thresholds. SfS over W associated with FJ,Z,N and L,α, denoted by FJL,α,Z, is defined as follows:(21)FJL,αz=SzFJLw>α:w∈W,∀z∈Z,where SzFJL represents the score function of FJLz.Definition 23.
LetTlj=glj,pljei2πϕlj,vljei2πδlj,rljei2πλlj and Tkj=gkj,pkjei2πϕkj,vkjei2πδkj,rkjei2πλkj be two CSFNSfNs and σ>0. Some operation for CSFNSfNs are(22)σTlj=glj,1−1−plj2σei2π1−1−ϕlj2σ,vljσei2πδljσ,rljσei2πλljσ,Tljσ=glj,pljσei2πϕljσ,1−1−vlj2σei2π1−1−δlj2σ,1−1−rlj2σei2π1−1−λlj2σ,Tlj⊕Tkj=maxglj,gkj,plj2+pkj2−plj2pkj2ei2πϕlj2+ϕkj2−ϕlj2ϕkj2,vljvkjei2πδljδkj,rljrkjei2πλljλkj,Tlj⊗Tkj=minglj,gkj,pljpkjei2πϕljϕkj,vlj2+vkj2−vlj2vkj2ei2πδlj2+δkj2−δlj2δkj2,rlj2+rkj2−rlj2rkj2ei2πλlj2+λkj2−λlj2λkj2.


## 4.CSFNSf-TOPSIS Method for MAGDM

In this section, we combineCSFNSfS with the TOPSIS method. The main idea of this methodology is the selection of a best alternative using both the positive ideal solution (PIS) and the negative ideal solution (NIS). Therefore, we present the corresponding CSFNSf-TOPSIS method in order to solve MAGDM problems in a CSFNSf environment under such methodology. The elements and steps of this algorithm for MAGDM are as follows.LetW=w1,w2,w3,…,wq denote the set of alternatives that are evaluated by s experts E˜1,E˜2,E˜3,…,E˜s. According to the needs of MAGDM problems, set of m attributes Z=z1,z2,z3,…,zm are assigned to these alternatives by the experts. Let σ=σ1,σ2,σ3,…,σsT be the weight vector, which represents the weightage of experts such that ∑d=1sσd=1, where σd∈0,1. The step by step Algorithm 1 of CSFNSf-TOPSIS method is presented in Section 5.1, and its theoretical description is as follows:Step 1: according to the MAGDM problem and attributes related to the alternatives, each expert assigns ratings to them. There is a linguistic term corresponding with each rating, which could be a number of stars (such as “three stars,” “two stars,” and “one star” in MAGDM), numerical labels (such as 3 as a label for “high,” 2 for “medium,” and 0 for “low”). In such a way,NSfSFd,Z,N is found corresponding to each expert E˜d with G=0,1,2,3,…,N−1 as the set of grades, where N∈1,2,3,… and d∈1,2,3,…,s. Now, CSFNSfN is assigned by the dth expert E˜d, corresponding to each rank in the NSfSFd,Z,N, according to the grading criteria defined for the MAGDM problem. Similarly, we get sCSFNSfSs by s experts, respectively. The complex spherical fuzzy N-soft decision matrix CSFNSfDM of the dth expert E˜d is as follows:(23)Pd=g11d,μ11d,η11d,ν11dg21d,μ12d,η12d,ν12d…gm1d,μ1md,η1md,ν1mdg12d,μ21d,η21d,ν21dg22d,μ22d,η22d,ν22d…gm2d,μ2md,η2md,ν2md⋮⋮⋱⋮g1qd,μq1d,ηq1d,νq1dg2qd,μq2d,ηq2d,νq2d…gmqd,μqmd,ηqmd,νqmd,wherePjkd=gkjd,μjkd,ηjkd,νjkd=gkjd,pjkdei2πϕjkd,vjkdei2πδjkd,rjkdei2πλjkd, j=1,2,3,…,q, k=1,2,3,…,m, and d=1,2,3,…,s.Step 2: to formulate the aggregate complex spherical fuzzyN-soft decision matrix ACSFNSfDM, P=Pjkq×m, CSFNSfDM of all experts are assembled with the help of CSFNSfWA operator. For ACSFNSfDM, CSFNSfWA operator is defined as(24)Pjk=CSFNSfWAPjk1,Pjk2,…,Pjks=σ1Pjk1⊕σ2Pjk2⊕⋯⊕σsPjks=max1d=sgkjd,1−∏1d=s1−pjkd2σdei2π1−∏1d=s1−ϕjkd2σd,∏1d=svjkdei2π∏1d=sδjkd,∏1d=srjkdei2π∏1d=sλjkd=gkj,pjkei2πϕjk,vjkei2πδjk,rjkei2πλjk.Using these entities, we can formACSFNSfDM as(25)P=g11,μ11,η11,ν11g21,μ12,η12,ν12…gm1,μ1m,η1m,ν1mg12,μ21,η21,ν21g22,μ22,η22,ν22…gm2,μ2m,η2m,ν2m⋮⋮⋱⋮g1q,μq1,ηq1,νq1g2q,μq2,ηq2,νq2…gmq,μqm,ηqm,νqm.Step 3: in MAGDM problem, each attribute has it is own worth. Therefore, each expertE˜d assigns rank as weightage of each attribute zk relative to their importance in MAGDM problem. Furthermore, CSFNSfNs are assigned to the weights, according to the grading criteria, by the experts. Let χkd=gkd,pkdei2πϕkd,vkdei2πδkd,rkdei2πλkd be the weightage of kth attribute given by the dth expert. To find out the weight vector χ=χ1,χ2,…,χmT, we aggregated them, as follows:(26)χk=CSFNSfWAχ11,χ22,…,χms=σ1χk1⊕σ2χk2⊕⋯⊕σsχks=maxd=1sgkjd,1−∏1d=s1−pkd2σdei2π1−∏1d=s1−ϕkd2σd,∏1d=svkdei2π∏1d=sδkd,∏1d=srkdei2π∏1d=sλkd=gk,pkei2πϕk,vkei2πδk,rkei2πλk.Step 4: calculate the aggregated weighted complex spherical fuzzyN-soft decision matrix AWCSFNSfDM using ACSFNSfDMYjk and the weight vector of attribute χk as follows:(27)Pjk¯=Pjk⊗χk=mingkj,gk,pjkpkei2πϕjkϕk,vjk2+vk2−vjk2vk2ei2πδjk2+δk2−δjk2δk2,rjk2+rk2−rjk2rk2ei2πλjk2+λk2−λjk2λk2=g¯kj,μ¯jk,η¯jk,ν¯jk=g¯kj,p¯jkei2πϕ¯jk,v¯jkei2πδ¯jk,r¯jkei2πλ¯jk.Using these entities, we can formAWCSFNSfDM as(28)P¯=g¯11,μ¯11,η¯11,ν¯11g¯121,μ¯12,η¯12,ν¯12…g¯m1,μ¯1m,η¯1m,ν¯1mg¯12,μ¯21,η¯21,ν¯21g¯22,μ¯22,η¯22,ν¯22…g¯m2,μ¯2m,η¯2m,ν¯2m⋮⋮⋱⋮g¯1q,μ¯q1,η¯q1,ν¯q1g¯2q,μ¯q2,η¯q2,ν¯q2…g¯mq,μ¯qm,η¯qm,ν¯qm.Step 5: letPB and PC be the collection of benefit-type attribute and cost-type attribute, respectively. CSFNSf-PIS related to the attribute zk can be taken as follows:(29)P^k=maxj=1qP¯jk,ifzk∈PB,minj=1qP¯jk,ifzk∈PC.Now,CSFNSf-NIS related to the attribute zk can be taken as follows:(30)Pˇk=maxj=1qP¯jk,ifzk∈PC,minj=1qP¯jk,ifzk∈PB.To evaluatemaxP¯jk and minP¯jk, we use the score value and accuracy value of CSFNSfN. CSFNSf-PIS and CSFNSf-NIS are denoted as follows: P^k=g^k,μ^k,η^k,ν^k=g^k,p^kei2πϕ^k,v^kei2πδ^k,r^kei2πλ^k and Pˇk=gˇk,μˇk,ηˇk,νˇk=gˇk,pˇkei2πϕˇk,vˇkei2πδˇk,rˇkei2πλˇk, respectively.Step 6: calculate the normalized Euclidean distance of each alternativewj from CSFNSf-PIS and CSFNSf-NIS. In this way, we get the best alternative that is nearer to CSFNSf-PIS and far from CSFNSf-NIS. The normalized Euclidean distance between CSFNSf-PIS and any of the alternative wj can be formulated as follows:(31)dP^k,wj=14k∑k=1mg^kN−12−g¯kjN−122+p^k2−p¯jk22+v^k2−v¯jk22+r^k2−r¯jk22+ϕ^k2−ϕ¯jk22+δ^k2−δ¯jk22+λ^k2−λ¯jk221/2.Similarly, the normalized Euclidean distance betweenCSFNSf-NIS and any of the alternative wj, can be formulated as follows:(32)dPˇk,wj=14k∑k=1mgˇkN−12−g¯kjN−122+pˇk2−p¯jk22+vˇk2−v¯jk22+rˇk2−r¯jk22+ϕˇk2−ϕ¯jk22+δˇk2−δ¯jk22+λˇk2−λ¯jk221/2.Step 7: to chose one of the most appropriate alternative, we have to use some ranking index. For this purpose, the revised closeness index corresponding to the alternativewk is evaluated using the formula [10](33)ℑwj=dP^k,wjminjdP^k,wj−dPˇk,wjmaxjdPˇk,wj,wherek=1,2,…,m.Step 8: the alternative with the minimum value of revised closeness index would be the best solution for the MAGDM problem. Therefore, the ascending order of the revised closeness index gives the ranking of the alternatives.Algorithm 1: The algorithm of CSFNSf-TOPSIS method.
Input: W=w1,w2,w3,…,wq as universal element.Z=z1,z2,z3,…,zm as set of attributes.NSfSF,Z,N with G=0,1,2,3,…,N−1, N∈1,2,3,…, σ1,σ2,σ3,…,σsT as weight vector of experts E˜d.(1)
ConstructCSFNSfDMPd, corresponding to each level of attribute for the element wj.(2)
ComputeACSFNSfDM using equation Pjk=max1d=sgkjd,1−∏1d=s1−pjkd2σdei2π1−∏1d=s1−ϕjkd2σd,∏1d=svjkdei2π∏1d=sδjkd,∏1d=srjkdei2π∏1d=sλjkd.(3)
Evaluate the weight vectorχ=χ1,χ2,…,χmT as follows: χk=maxd=1sgkjd,1−∏1d=s1−pkd2σdei2π1−∏1d=s1−ϕkd2σd,∏1d=svkdei2π∏1d=sδkd,∏1d=srkdei2π∏1d=sλkd.(4)
CalculateAWCSFNSfDM using ACSFNSfDM and the weight vector of attributes as follows: Pjk¯=mingkj,gk,pjkpkei2πϕjkϕk,vjk2+vk2−vjk2vk2ei2πδjk2+δk2−δjk2δk2,rjk2+rk2−rjk2rk2ei2πλjk2+λk2−λjk2λk2(5)
CalculateCSFNSf PIS and CSFNSf NIS, using equations (29) and (30).(6)
Calculate the normalized Euclidean distance ofCSFNSf PIS and CSFNSf NIS from each alternative, by utilizing equations (31) and (32), respectively.(7)
Calculate the revised closeness index corresponding to each alternative using the formula from [10] ℑwj=dP^k,wj/minjdP^k,wj−dPˇk,wj/maxjdPˇk,wj.(8)
Identify the alternative with minimum revised closeness index.


## 5. Development of Algorithms and Numerical Examples

In this section, we describe multiattribute decision-making (MADM) methods that work on models to identify the best alternative. Therefore, we characterize respective algorithms for the MADM problems inCSFNSf environment, as well as we present Algorithm 1 for CSFNSf-TOPSIS method described in Section 4. Let W=w1,w2,w3,…,wq be a set, representing the available alternatives with a set of attributes Z=z1,z2,z3,…,zm having weight vector σ=σ1,σ2,σ3,…,σmT describing the worth of attributes according to the MADM problem, where ∑k=1mσk=1 and σk∈0,1.The algorithm forCSFNSf-TOPSIS method is described in Algorithm 1.Let us now introduce some explicit MADM and MAGDM problems and solve them using Algorithms1– 4, respectively. We apply Algorithms 2– 4 to solve the MADM problem defined in Section 5.1 and Algorithm 1 is used to solve the MAGDM problem defined in Section 5.2 which show their importance and feasibility in the field of decision-making.Algorithm 2: The algorithm of choice values of CSFNSfSs.
(1)
h!Input: W=w1,w2,w3,…,wq as universal element.Z=z1,z2,z3,…,zm as set of attributes.NSfSF,Z,N with G=0,1,2,3,…,N−1, N∈1,2,3,….(2)
Construct theCSFNSfNΥkj, corresponding to each level of attribute for the element wj.(3)
ComputeXj=⊕k=1mϒkj using equation (22), where ϒkj=gkj,pkjei2πϕkj,vkjei2πδkj,rkjei2πλkj, wj,gkj∈Fz.(4)
Calculate the score functionSXj, using equation (11) for all j=1,2,3,…,q.(5)
ifSj=Sp, for somej,p∈1,2,3,…,q,then(6)
Use accuracy degree and identify alternative with maximum accuracy value(7)
else(8)
Identify the alternative with maximum score value.(9)Algorithm 3: The algorithm of weighted choice values of CSFNSfSs.
Input: W=w1,w2,w3,…,wq as universal element.Z=z1,z2,z3,…,zm as set of attributes.NSfSF,Z,N with G=0,1,2,3,…,N−1, N∈1,2,3,…, σ1,σ2,σ3,…,σkT as weight vector.(1)
ConstructCSFNSfNϒkj, corresponding to each level of attribute for the element wj.(2)
ComputeXjσ=⊕k=1mσkϒkj, where ϒkj=gkj,pkjei2πϕkj,vkjei2πδkj,rkjei2πλkj, wj,gkj∈Fz.(3)
Calculate the score functionSXj, for all j=1,2,3,…,q. Calculate all the indices j for which Sj=maxjSXjσ.(4)
ifSj=Sp, for somej,p∈1,2,3,…,q,then(5)
Use accuracy degree and identify alternative with maximum accuracy value(6)
else(7)
Identify the alternative with maximum score value.(8)Algorithm 4: The algorithm of L-choice values of CSFNSfSs.
Input: W=w1,w2,w3,…,wq as universal element.Z=z1,z2,z3,…,zm as set of attributes.NSfSF,Z,N with G=0,1,2,3,…,N−1,N∈1,2,3,…,Input0<L<N−1, threshold.(1)
Work out forFJLz=μkjz,ηkjz,νkjz,ifw,gzw∈Fzandgzw≥L,0,0,1,otherwise.(2)
First, computeXjL=⊕k=1mϒkjL, where ϒkjL=pkjei2πϕkj,vkjei2πδkj,rkjei2πλkj, wj,gkj∈Fz.(3)
Calculate the score functionSXjL, for all j=1,2,3,…,q.(4)
Calculate all the indicesj for which SjL=maxjSXjL.(5)
ifSj=Sp, for somej,p∈1,2,3,…,q,then(6)
Use accuracy degree and identify alternative with maximum accuracy value(7)
else(8)
Identify the alternative with maximum score value.(9)


### 5.1. Selection of Best Third-Party App of the Year

A third-party app is a software application made by someone other than the manufacturer of a mobile device or its operating system. This world is full of gadgets and gadgets are full of apps. We can access the world if we have these apps. Therefore, selecting one of the best third-party app of the year and keeping in view the priorties of people is a very difficult task. For this purpose, the data has been collected from the websiteshttp://www.makeawebsitehub.com and http://www.trustraduis.com regarding to each third-party app. To find out the best app of the year, we will use CSFNSfS.LetA = {a1 = Facebook, a2 = Skype, a3 = Viber, a4 = Twitter, a5 = Whatsapp be universe of third-party apps and Z = {z1 = telecom framework, z2 = reliability, z3 = worldwide contact, z4 = data usage be the attributes. According to these attributes, a 6-soft set is modeled in Table 20, whereFive diamonds mean “Marvellous”Four diamonds mean “Outstanding”Three diamonds mean “Super”Two diamonds mean “Good”One diamond means “Satisfactory”Big dot means “Acceptable.”Table 20
Evaluation of data from the websites.
A/Zz1z2z3z4a1⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄a2⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄a3⋄⋄⋄⋄⋄⋄⋄⋄⋄a4⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄a5⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄This level assessment by diamonds can be represented by numbers asG=0,1,2,3,4,5, where0 means “•”1 means “⋄”2 means “⋄⋄”3 means “⋄⋄⋄,”4 means “⋄⋄⋄⋄”5 means “⋄⋄⋄⋄⋄”Thus, tabular representation of 6-soft set is shown in Table21 and the tabular representation of CSF6SfSFJ,Z,6 is shown in Table 22.Table 21
Tabular representation of 6-soft set.
A/Zz1z2z3z4a15442a23233a32223a42354a53432Table 22
Tabular representation of theCSF6SfSFJ,Z,6.
FJ,Z,6z1z2z3z4a15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπa23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπa32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67ei1.38πa42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπa53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ


#### 5.1.1. Choice Values ofCSF6SfS

The choice values ofCSF6SfS is evaluated using the steps defined in Algorithm 2. Table 23 presents the calculated choice values of CSF6SfS for the selection of the third-party app. We can observe from Table 23 that, according to the choice values, the ranking of third-party apps is as follows: a1>a4>a5>a2>a3, which shows that a1 = Facebook has maximum choice value. Therefore, Facebook is selected as best third-party app of the year.Table 23
Tabular representation of the choice values ofFJ,Z,6.
FJ,Z,6z1z2z3z4XjSXja15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπ5,0.9979ei1.9958π,2.47×10−8ei6.34×10−8π,4.21×10−3e5.62×10−3iπ2.6860a23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπ3,0.9053ei1.8024π,1.66×10−6ei5.02×10−7π,0.22e0.44iπ1.78227a32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67e1.38iπ3,0.7295ei1.53π,2.25×10−7ei18.93×10−7π,0.3857e0.802iπ1.0606a42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ5,0.9853ei1.9704π,1.149×10−6ei18.78×10−6π,0.033e0.064iπ2.6337a53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π.2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ4,0.9723ei1.9506π,1.493×10−7ei3.08×10−7π,0.053e0.106iπ2.3353


#### 5.1.2. Weighted Choice Values ofCSF6SfS

Letσ1=0.4,σ2=0.3,σ3=0.2, and σ4=0.1 be the weights for each attribute zk,k=1,2,3,4. Using these weights in Algorithm 3, we can compute weighted choice values of CSF6SfS, which are given by Table 24.Table 24
Tabular representation of the weighted choice values ofFJ,Z,6.
FJ,Z,6z1z2z3z4XjσSXjσa15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπ5,0.9019ei1.7974π,0.0131ei0.0278π,0.195e0.3402iπ2.2482a23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπ3,0.5963ei1.168π,0.0263ei0.052π,0.655e1.319iπ0.081a32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67e1.38iπ3,0.3925ei0.813π,0.015ei0.032π,0.807e1.61iπ−0.7378a42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ5,0.7235ei1.443π,0.0319ei0.121π,0.5229e1.049iπ1.185a53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ4,0.7570ei1.532π,0.0195ei0.0394π,0.5503e0.843iπ1.123It is clear from Table24 that G4 has maximum score; therefore, a4 = Facebook is selected as best third-party app of the year. According to the weighted choice values, ranking of third-party apps is as follows: a1>a4>a5>a2>a3.


#### 5.1.3.L-Choice Values of CSF6SfS

TheL-choice values of CSF6SfS are evaluated using Algorithm 4 to find out the best alternative for the proposed MADM problem. Let L=4 be threshold; then, 4-choice values of CSFSfS is shown in Table 25. We can observe that, from Table 25, the ranking of third-party apps according to 4-choice values is as follows: a1>a4>a5>a2≥a3, which shows that a1 = Facebook has maximum choice value so that Facebook is selected as the best third-party app of the year.Table 25
Tabular representation of the 4-choice values ofFJ,Z,6.
FJ4,Z,6z1z2z3z4XjSXja10.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π0.87ei1.76π,0.01ei0.024π,0.21ei0.38π0.88ei1.78π,0.012ei0.026π,0.2ei0.394π0,0,10.9973ei1.9944π,2.064×10−6ei1.45×10−11π,5.46×10−3e2.8×10−5iπ1.989a20,0,10,0,10,0,10,0,10,0,1−1a30,0,10,0,10,0,10,0,10,0,1−1a40,0,10,0,10.9ei1.82π,0.08ei0.158π,0.18e0.358π0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ0.9605ei1.926π,1.36×10−3ei3.18×10−7π,0.063ei6.978×10−3π1.8459a50,0,10.89ei1.784π,0.013ei0.0278π,0.145ei0.4π0,0,10,0,10.89ei1.78π,0.013ei0.0278π,0.145e0.4iπ1.5263


### 5.2. Selection of the Best Physiotherapist Doctor of Mayo Hospital in Lahore

Physiotherapy helps to restore movement and function when people are affected by injury or disability. A physiotherapist treats such kind of people and helps them through exercise, manual therapy, education, and advice. A physiotherapist is very helpful in maintaining the health of people of all ages as well as encourages them for happy life. A physiotherapist must have patience, communication skills, and ability to establish a good relationship with patients and their families. The motive of this study is to select the best physiotherapist doctor in Lahore relative to their attributes under the environment ofCSFNSf. For this purpose, the data has been collected from the students of Mayo Hospital, Lahore, enact here as experts E˜1, E˜2, E˜3, and E˜4 whose weight vectors are σ=0.4,0.2,0.1,0.3T. The following physiotherapists of Mayo Hospital are treated as alternatives in this MAGDM problem:w1: Dr. Amnaw2: Dr. Rizwanw3: Dr. Akmalw4: Dr. Sidraw5: Dr. SaleemFive attributes considered as key factors for a physiotherapist are as follows:z1: knowledge and experience.z2: behavioral (positivity, patience, and humbleness).z3: availability and flexibility.z4: master of skills (communication, organizational, or problem-solving skills).z5: session fee.We solve this MAGDM problem by following theCSFNSf-TOPSIS method.Step 1: according to these attributes, each expert model 6-soft set is in Table26, whereFive stars mean “Marvellous”Four stars mean “Outstanding”Three stars mean “Super”Two stars mean “Good”One star mean “Satisfactory”Big dot means “Acceptable”Table3 represents the grading criteria, used for assigning the CSFNSfN corresponding to each rank by the expert E˜1, E˜2, E˜3, and E˜4 tabulated in Tables 27–30, respectively.Step 2: using equation (24), we can put together the opinions of all experts. ACSFNSfDM formed by aggregation is given in Table 31.Step 3: to demonstrate the importance of attributes in the MAGDM problem, experts rank them and associateCSFNSfN to each attribute which are arranged in Table 32. We cumulated the weights given by experts using equation (26) to form CSFNSf weight vector χ of attributes, i.e.,(34)χ=4,0.89ei1.72π,0.017ei0.034π,0.23e0.5iπ4,0.91ei1.86π,0.016ei0.034π,0.09e0.2iπ3,0.62ei1.24π,0.016ei0.028π,0.53e1.1iπ3,0.53ei1.06π,0.02ei0.04π,0.73e1.42iπ2,0.55ei1.14π,0.019ei0.042π,0.67e1.34iπ.Step 4: by utilizingACSFNSfDM and weight vector χ of attribute in equation (34), AWCSFNSfDM is evaluated and summarized in Table 33.Step 5: in the MAGDM problem, all the attributes’ knowledge and experience, behavior, availability and flexibility, and master of skills are benefit-type attributes except the session fee, which is a cost-type attribute. According to the nature of attributes and applying equation (29) and (30), CSFNSf-PIS and CSFNSf-NIS are evaluated and arranged in Table 34Step 6: Table35 represents the normalized Euclidean distance from each alternative to CSFNSf-PIS and CSFNSf-NIS using equations (31) and (32), respectively.Step 7: the revised closeness index of each alternative is calculated by utilizing equation (33) and given in Table 36.Step 8: sincew1 has least revised closeness index, therefore, Dr. Amna is the best physiotherapist in Mayo Hospital, Lahore. The ranking of alternatives is shown in Table 37.Table 26
Experts’ rating according to attributes.
AttributesAlternativesE˜1E˜2E˜3E˜4z1w1∗∗∗∗=4∗∗∗=3∗∗∗=3∗∗∗∗=4w2∗∗∗=3∗∗∗=3∗∗∗=3∗∗∗=3w3∗=1•=0∗∗=2∗=1w4∗∗=2∗∗∗=3∗∗∗∗=4∗∗=2w5∗∗=2∗=1∗∗=2•=0z2w1∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4∗∗∗=3w2∗∗=2∗∗∗=3∗∗=2∗=1w3∗=1∗=1∗=1∗∗=2w4∗∗∗=3∗∗=2∗∗∗=3∗∗∗∗=4w5∗=1∗∗=2∗∗=2∗=1z3w1∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4w2∗=1∗∗=2∗∗=2•=0w3∗∗=2∗∗=2∗∗=2∗∗=2w4∗∗∗=3∗=1•=0∗∗=2w5∗∗∗=3∗∗=2∗∗∗∗=4∗=1z4w1∗∗∗=3∗∗∗=3∗∗=2∗∗=2w2∗∗∗∗=4∗∗∗∗=4∗∗∗=3∗∗∗=3w3∗=1∗∗∗=3∗∗∗∗=4∗∗=2w4∗∗=2∗∗∗=3∗∗∗=3∗∗∗=3w5∗=1∗=1∗∗=2∗∗=2z5w1∗∗∗∗=4∗∗∗=3∗∗∗=3∗∗=2w2∗∗=2∗∗=2∗∗∗=3∗∗=2w3∗=1∗∗=2∗=1∗=1w4∗∗∗=3∗∗=2∗=1•=0w5∗∗=2•=0∗∗∗=3∗=1Table 27
Tabular representation ofCSFNSfDM of expert E˜1.
FJ1,Z,5z1z2z3z4z5w14,0.98ei1.98π,0.01ei0.024π,0.02ei0.06π4,0.99ei1.94π,0.012ei0.02π,0.1ei0.26π4,0.97ei1.96π,0.013ei0.02π,0.05ei0.08π3,0.84ei1.66π,0.019ei0.04π,0.29e0.6iπ4,0.98ei1.92π,0.012ei0.026π,0.03e0.01iπw23,0.84ei1.66π,0.02ei0.042π,0.29ei0.62π2,0.37ei0.78π,0.015ei0.026π,0.87ei1.72π1,0.16ei0.34π,0.018ei0.038π,0.89e1.78π4,0.96ei1.94π,0.013ei0.024π,0.04e0.06iπ2,0.36ei0.74π,0.016ei0.03π,0.59e1.2iπw31,0.17ei0.36π,0.02ei0.038π,0.91ei1.84π1,0.19ei0.36π,0.021ei0.04π,0.93e1.82π2,0.59ei0.12π,0.0155ei0.032π,0.82e1.6π1,0.2ei0.38π,0.022ei0.042π,0.92e1.86iπ1,0.21ei0.44π,0.024ei0.05π,0.93e1.88iπw42,0.58ei1.12π,0.015ei0.032π,0.82ei1.66π3,0.66ei1.34π,0.1ei0.18π,0.56ei1.14π3,0.67ei1.36π,0.021ei0.044π,0.31e0.64π2,0.55ei1.14π,0.014ei0.028π,0.8e1.62iπ3,0.67ei1.38π,0.025ei0.048π,0.33e0.68iπw52,0.52ei1.02π,0.012ei0.026π,0.74ei1.5π1,0.21ei0.4π,0.021ei0.046π,0.95ei1.88π1,0.7ei1.36π,0.026ei0.05π,0.35ei0.68π1,0.16ei0.42π,0.018ei0.038π,0.89e1.88iπ2,0.58ei1.14π,0.01ei0.024π,0.82e1.62iπTable 28
Tabular representation ofCSFNSfDM of expert E˜2.
FJ2,Z,5z1z2z3z4z5w13,0.83ei1.64π,0.019ei0.036π,0.3ei0.58π4,0.98ei1.94π,0.013ei0.024π,0.1ei0.26π4,0.98ei1.98π,0.01ei0.024π,0.02ei0.06π3,0.84ei1.66π,0.02ei0.036π,0.31e0.63iπ3,0.8ei1.62π,0.022ei0.042π,0.32e0.6iπw23,0.71ei1.38π,0.028ei0.058π,0.37ei0.72π3,0.72ei1.42π,0.03ei0.062π,0.38ei0.76π2,0.57ei1.1π,0.012ei0.028π,0.81e1.6π4,0.86ei1.74π,0.0169ei0.032π,0.027e0.052iπ2,0.56ei1.1π,0.015ei0.07π,0.8e1.58iπw30,0.05ei0.012π,0.01ei0.06π,0.985ei1.972π1,0.21ei0.46π,0.024ei0.52π,0.94e1.86π2,0.5ei1.02π,0.01ei0.02π,0.7e1.4π3,0.73ei1.48π,0.032ei0.062π,0.39e0.76iπ2,0.55ei1.08π,0.01ei0.02π,0.79e1.56iπw43,0.74ei1.5π,0.035ei0.068π,0.4ei0.78π2,0.54ei1.04π,0.015ei0.026π,0.78ei1.54π1,0.24ei0.44π,0.03ei0.056π,0.94e1.84π3,0.71ei1.38π,0.028ei0.058π,0.37e0.72iπ2,0.53ei1.04π,0.013ei0.026π,0.77e1.52iπw51,0.24ei0.42π,0.028ei0.062π,0.91ei1.84π2,0.51ei1.04π,0.013ei0.06π,0.76ei1.5π2,0.52ei1.02π,0.013ei0.06π,0.75ei1.52π1,0.23ei0.5π,0.035ei0.066π,0.89e1.82iπ0,0.1ei0.24π,0.012ei0.026π,0.99e1.972iπTable 29
Tabular representation ofCSFNSfDM of expert E˜3.
FJ3,Z,5z1z2z3z4z5w13,0.84ei1.66π,0.019ei0.04π,0.29ei0.6π4,0.97ei1.98π,0.013ei0.028π,0.14ei0.26π4,0.96ei1.98π,0.01ei0.04π,0.04ei0.04π2,0.5ei1.02π,0.014ei0.026π,0.74e1.46iπ3,0.8ei1.62π,0.018ei0.038π,0.29e0.6iπw23,0.72ei1.46π,0.036ei0.068π,0.41ei0.8π2,0.49eiπ,0.013ei0.028π,0.71ei1.44π2,0.48ei0.98π,0.015ei0.032π,0.7e1.36π3,0.74ei1.44π,0.034ei0.07π,0.42e0.82iπ3,0.75ei1.48π,0.035ei0.072π,0.43e0.82iπw32,0.47ei0.96π,0.015ei0.032π,0.7ei1.36π1,0.27ei0.5π,0.034ei0.066π,0.94e1.92π2,0.46ei0.94π,0.014ei0.03π,0.68e1.38π4,0.88ei1.78π,0.015ei0.032π,0.25e0.46iπ1,0.29ei0.56π,0.04ei0.078π,0.96e1.9iπw44,0.98ei1.94π,0.013ei0.06π,0.1ei0.26π3,0.76ei1.5π,0.036ei0.074π,0.44ei0.86π0,0.13ei0.6π,0.014ei0.03π,0.986e1.97π3,0.77ei1.52π,0.038ei0.078π,0.45e0.88iπ1,0.3ei0.58π,0.042ei0.082π,0.925e1.86iπw52,0.44ei0.92π,0.012ei0.028π,0.66ei1.34π2,0.43ei0.88π,0.015ei0.028π,0.66ei1.34π4,0.87ei1.78π,0.016ei0.028π,0.21ei0.43π2,0.42ei0.86π,0.012ei0.02π,0.65e1.32iπ3,0.78ei1.54π,0.04ei0.076π,0.46e0.9iπTable 30
Tabular representation ofCSFNSfDM of expert E˜4.
FJ4,Z,5z1z2z3z4z5w14,0.95ei1.94π,0.012ei0.02π,0.13ei0.2π3,0.8ei1.62π,0.018ei0.038π,0.31ei0.6π4,0.97ei1.96π,0.01ei0.02π,0.15ei0.32π2,0.63ei1.28π,0.012ei0.02π,0.6e1.18iπ2,0.64ei1.26π,0.01ei0.024pi,0.59e1.2iπw23,0.77ei1.56π,0.041ei0.08π,0.47ei0.92π1,0.31ei0.64π,0.022ei0.04π,0.89ei1.8π0,0.06ei0.14π,0.012ei0.02π,0.986e1.97π3,0.8ei1.58π,0.05ei0.102π,0.48e0.94iπ2,0.37ei1.78π,0.014ei0.024π,0.86e1.74iπw31,0.3ei0.64π,0.05ei0.012π,0.91ei1.84π2,0.41ei0.84π,0.01ei0.06π,0.64e1.3π2,0.4ei0.84π,0.015ei0.032π,0.63e1.28π2,0.39ei0.8π,0.014ei0.03π,0.62e1.26iπ1,0.33ei0.64π,0.07ei0.1π,0.93e1.88iπw42,0.38ei0.78π,0.013ei0.07π,0.61ei1.24π4,0.9ei1.84π,0.014ei0.03π,0.19ei0.36π2,0.37ei0.76π,0.012ei0.026π,0.6e1.22π3,0.81ei1.6π,0.051ei0.104π,0.49eiπ0,0.13ei0.28π,0.014ei0.026π,0.982e1.966iπw50,0.07ei0.012π,0.016ei0.03π,0.991ei1.984π1,0.34ei0.64π,0.065ei0.132π,0.95ei1.92π1,0.3ei0.58π,0.014ei0.03π,0.93ei1.88π2,0.36ei0.74π,0.01ei0.024π,0.59e1.2iπ1,0.21ei0.46π,0.024ei0.052π,0.94e1.86iπTable 31
Tabular representation ofACSFNSfDM.
FJ1,Z,5z1z2z3z4z5w14,0.95ei1.94π,0.013ei0.026π,0.07ei0.18π4,0.97ei1.9π,0.013ei0.026π,0.15ei0.34π4,0.97ei1.96π,0.01ei0.02π,0.05ei0.1π3,0.77ei1.54π,0.016ei0.03π,0.4e0.82iπ4,0.91ei1.76π,0.013ei0.03π,0.15e0.36iπw23,0.79ei1.56π,0.028ei0.058π,0.36ei0.74π3,0.48ei0.96π,0.019ei0.034π,0.73ei1.46π2,0.33ei0.66π,0.01ei0.02π,0.88e1.76π4,0.9ei1.82π,0.02ei0.04π,0.15e0.26iπ3,0.47ei0.96π,0.016ei0.03π,0.68e1.36iπw32,0.25ei0.52π,0.022ei0.048π,0.9ei1.74π2,0.29ei0.58π,0.018ei0.038π,0.83e1.66π2,0.5ei1.04π,0.013ei0.028π,0.72e1.42π4,0.55ei1.12π,0.029ei0.038π,0.6e1.2iπ2,0.35ei0.7π,0.029ei0.054π,0.9e1.82iπw44,0.7ei1.18π,0.016ei0.034π,0.63ei1.28π4,0.77ei1.56π,0.034ei0.064π,0.42ei0.84π3,0.51ei1.02π,0.01ei0.02π,0.53e1.06π3,0.71ei1.4π,0.026ei0.054π,0.55e1.12iπ3,0.52ei1.06π,0.019ei0.038π,0.6e1.22iπw52,0.39ei0.74π,0.015ei0.032π,0.83ei1.68π2,0.35ei0.7π,0.025ei0.052π,0.87ei1.74π4,0.63ei1.26π,0.02ei0.034π,0.51ei1.02π2,0.28ei0.6π,0.016ei0.034π,0.76e1.58iπ3,0.49ei0.96π,0.015ei0.034π,0.83e1.6iπTable 32
Experts opinion related to each attribute.
FJ,Z,6E˜1E˜2E˜3E˜4z13,0.85ei1.66π,0.019ei0.042π,0.29ei0.6π4,0.99ei1.94π,0.012ei0.02π,0.1ei0.26π4,0.88ei1.78π,0.015ei0.032π,0.25ei0.46π3,0.67ei1.36π,0.021ei0.044π,0.31e0.64iπz23,0.66ei1.34π,0.021ei0.044π,0.31ei0.64π4,0.96ei1.94π,0.013ei0.024π,0.04ei0.06π3,0.78ei1.54π,0.04ei0.076π,0.46e0.9π4,0.98ei1.98π,0.01ei0.024π,0.02e0.06iπz32,0.63ei1.28π,0.012ei0.02π,0.59ei1.2π3,0.72ei1.42π,0.03ei0.062π,0.38e0.78π3,0.75ei1.48π,0.035ei0.072π,0.43e0.82π2,0.42ei0.86π,0.012ei0.02π,0.65ei1.32πz41,0.29ei0.6π,0.014ei0.03π,0.94ei1.86π1,0.21ei0.46π,0.024ei0.052π,0.94ei1.86π2,0.37ei0.76π,0.012ei0.026π,0.6e1.22π3,0.78ei1.54π,0.04ei0.076π,0.46e0.9iπz52,0.62ei1.28π,0.01ei0.024π,0.58ei1.2π2,0.66ei1.34π,0.1ei0.18π,0.56ei1.14π2,0.64ei1.26π,0.01ei0.024π,0.59ei1.2π1,0.21ei0.4π,0.021ei0.046π,0.95e1.88iπTable 33
Tabular representation ofAWCSFNSfDM.
z1z2z3z4z5w14,0.85ei1.66π,0.021ei0.042π,0.23ei0.52π4,0.88ei1.76π,0.021ei0.042π,0.174ei0.38π3,0.601ei1.22π,0.019ei0.034π,0.053ei1.1π3,0.41ei0,82π,0.025ei0.05π,0.77e1.54iπ2,0.5eiπ,0.023ei0.05π,0.67e1.36iπw23,0.7ei1.34π,0.03ei0.066π,0.41ei0.86π3,0.43ei0.9π,0.025ei0.048π,0.73ei1.46π2,0.2ei0.4π,0.019ei0.034π,0.91e1.82π3,0.47ei0.96π,0.028ei0.056π,0.73e1.42iπ2,0.25ei0.54π,0.025ei0.05π,0.83e1.66iπw32,0.22ei0.44π,0.027ei0.058π,0.91ei1.74π2,0.26ei0.54π,0.024ei0.025π,0.83e1.66π2,0.32ei0.64π,0.021ei0.038π,0.8e1.62π3,0.29ei0.58π,0.035ei0.068π,0.73e1.42iπ2,0.19ei0.34π,0.035ei0.068π,0.95e1.9iπw44,0.62ei1.02π,0.023ei0.048π,0.65ei1.34π4,0.70ei1.46π,0.037ei0.072π,0.43ei0.86π3,0.32ei0.64π,0.019ei0.034π,0.69e1.42π3,0.37ei0.74π,0.033ei0.066π,0.82e1.62iπ2,0.29ei0.6π,0.027ei0.056π,0.8e1.62iπw52,0.35ei0.62π,0.022ei0.046π,0.83ei1.7π2,0.31ei0.66π,0.029ei0.06π,0.87ei1.74π3,0.39ei0.78π,0.025ei0.044π,0.68ei1.38π2,0.15ei0.318π,0.025ei0.052π,0.89e1.8iπ2,0.27ei0.54π,0.024ei0.042π,0.91e1.82iπTable 34
Tabular representation ofCSFNSf-PIS and CSFNSf-NIS.
AttributeCSFNSf-PISCSFNSf-NISz14,0.85ei1.66π,0.021ei0.042π,0.23ei0.52π2,0.22ei0.44π,0.027ei0.058π,0.91ei1.74πz24,0.88ei1.76π,0.021ei0.042π,0.174ei0.038π2,0.31ei0.66π,0.029ei0.06π,0.87ei1.74πz33,0.601ei1.22π,0.019ei0.034π,0.53ei1.1π2,0.2ei0.4π,0.019ei0.034π,0.91e1.82πz43,0.47ei0.96π,0.028ei0.056π,0.73e1.42π2,0.15ei0.318π,0.025ei0.052π,0.89e1.8πz52,0.19ei0.38π,0.035ei0.068π,0.95ei1.9π2,0.5eiπ,0.023ei0.046π,0.67ei1.36πTable 35
Tabular representation of normalized Euclidean distance from ideal solution.
AlternativedP^k,wjdPˇk,wjw10.205590.55439w20.366770.311911w30.5281060.18778w40.2546490.396793w50.506490.1936Table 36
Tabular representation of revised closeness index of each alternative.
Alternativeℑwjw10w21.22136w32.236816w40.485238w52.1143Table 37
Tabular representation of revised closeness index of each alternative.
AlternativeRankingw11w23w35w42w54


## 5.1. Selection of Best Third-Party App of the Year

A third-party app is a software application made by someone other than the manufacturer of a mobile device or its operating system. This world is full of gadgets and gadgets are full of apps. We can access the world if we have these apps. Therefore, selecting one of the best third-party app of the year and keeping in view the priorties of people is a very difficult task. For this purpose, the data has been collected from the websiteshttp://www.makeawebsitehub.com and http://www.trustraduis.com regarding to each third-party app. To find out the best app of the year, we will use CSFNSfS.LetA = {a1 = Facebook, a2 = Skype, a3 = Viber, a4 = Twitter, a5 = Whatsapp be universe of third-party apps and Z = {z1 = telecom framework, z2 = reliability, z3 = worldwide contact, z4 = data usage be the attributes. According to these attributes, a 6-soft set is modeled in Table 20, whereFive diamonds mean “Marvellous”Four diamonds mean “Outstanding”Three diamonds mean “Super”Two diamonds mean “Good”One diamond means “Satisfactory”Big dot means “Acceptable.”Table 20
Evaluation of data from the websites.
A/Zz1z2z3z4a1⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄a2⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄a3⋄⋄⋄⋄⋄⋄⋄⋄⋄a4⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄a5⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄This level assessment by diamonds can be represented by numbers asG=0,1,2,3,4,5, where0 means “•”1 means “⋄”2 means “⋄⋄”3 means “⋄⋄⋄,”4 means “⋄⋄⋄⋄”5 means “⋄⋄⋄⋄⋄”Thus, tabular representation of 6-soft set is shown in Table21 and the tabular representation of CSF6SfSFJ,Z,6 is shown in Table 22.Table 21
Tabular representation of 6-soft set.
A/Zz1z2z3z4a15442a23233a32223a42354a53432Table 22
Tabular representation of theCSF6SfSFJ,Z,6.
FJ,Z,6z1z2z3z4a15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπa23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπa32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67ei1.38πa42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπa53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ


### 5.1.1. Choice Values ofCSF6SfS

The choice values ofCSF6SfS is evaluated using the steps defined in Algorithm 2. Table 23 presents the calculated choice values of CSF6SfS for the selection of the third-party app. We can observe from Table 23 that, according to the choice values, the ranking of third-party apps is as follows: a1>a4>a5>a2>a3, which shows that a1 = Facebook has maximum choice value. Therefore, Facebook is selected as best third-party app of the year.Table 23
Tabular representation of the choice values ofFJ,Z,6.
FJ,Z,6z1z2z3z4XjSXja15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπ5,0.9979ei1.9958π,2.47×10−8ei6.34×10−8π,4.21×10−3e5.62×10−3iπ2.6860a23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπ3,0.9053ei1.8024π,1.66×10−6ei5.02×10−7π,0.22e0.44iπ1.78227a32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67e1.38iπ3,0.7295ei1.53π,2.25×10−7ei18.93×10−7π,0.3857e0.802iπ1.0606a42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ5,0.9853ei1.9704π,1.149×10−6ei18.78×10−6π,0.033e0.064iπ2.6337a53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π.2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ4,0.9723ei1.9506π,1.493×10−7ei3.08×10−7π,0.053e0.106iπ2.3353


### 5.1.2. Weighted Choice Values ofCSF6SfS

Letσ1=0.4,σ2=0.3,σ3=0.2, and σ4=0.1 be the weights for each attribute zk,k=1,2,3,4. Using these weights in Algorithm 3, we can compute weighted choice values of CSF6SfS, which are given by Table 24.Table 24
Tabular representation of the weighted choice values ofFJ,Z,6.
FJ,Z,6z1z2z3z4XjσSXjσa15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπ5,0.9019ei1.7974π,0.0131ei0.0278π,0.195e0.3402iπ2.2482a23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπ3,0.5963ei1.168π,0.0263ei0.052π,0.655e1.319iπ0.081a32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67e1.38iπ3,0.3925ei0.813π,0.015ei0.032π,0.807e1.61iπ−0.7378a42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ5,0.7235ei1.443π,0.0319ei0.121π,0.5229e1.049iπ1.185a53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ4,0.7570ei1.532π,0.0195ei0.0394π,0.5503e0.843iπ1.123It is clear from Table24 that G4 has maximum score; therefore, a4 = Facebook is selected as best third-party app of the year. According to the weighted choice values, ranking of third-party apps is as follows: a1>a4>a5>a2>a3.


### 5.1.3.L-Choice Values of CSF6SfS

TheL-choice values of CSF6SfS are evaluated using Algorithm 4 to find out the best alternative for the proposed MADM problem. Let L=4 be threshold; then, 4-choice values of CSFSfS is shown in Table 25. We can observe that, from Table 25, the ranking of third-party apps according to 4-choice values is as follows: a1>a4>a5>a2≥a3, which shows that a1 = Facebook has maximum choice value so that Facebook is selected as the best third-party app of the year.Table 25
Tabular representation of the 4-choice values ofFJ,Z,6.
FJ4,Z,6z1z2z3z4XjSXja10.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π0.87ei1.76π,0.01ei0.024π,0.21ei0.38π0.88ei1.78π,0.012ei0.026π,0.2ei0.394π0,0,10.9973ei1.9944π,2.064×10−6ei1.45×10−11π,5.46×10−3e2.8×10−5iπ1.989a20,0,10,0,10,0,10,0,10,0,1−1a30,0,10,0,10,0,10,0,10,0,1−1a40,0,10,0,10.9ei1.82π,0.08ei0.158π,0.18e0.358π0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ0.9605ei1.926π,1.36×10−3ei3.18×10−7π,0.063ei6.978×10−3π1.8459a50,0,10.89ei1.784π,0.013ei0.0278π,0.145ei0.4π0,0,10,0,10.89ei1.78π,0.013ei0.0278π,0.145e0.4iπ1.5263


## 5.1.1. Choice Values ofCSF6SfS

The choice values ofCSF6SfS is evaluated using the steps defined in Algorithm 2. Table 23 presents the calculated choice values of CSF6SfS for the selection of the third-party app. We can observe from Table 23 that, according to the choice values, the ranking of third-party apps is as follows: a1>a4>a5>a2>a3, which shows that a1 = Facebook has maximum choice value. Therefore, Facebook is selected as best third-party app of the year.Table 23
Tabular representation of the choice values ofFJ,Z,6.
FJ,Z,6z1z2z3z4XjSXja15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπ5,0.9979ei1.9958π,2.47×10−8ei6.34×10−8π,4.21×10−3e5.62×10−3iπ2.6860a23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπ3,0.9053ei1.8024π,1.66×10−6ei5.02×10−7π,0.22e0.44iπ1.78227a32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67e1.38iπ3,0.7295ei1.53π,2.25×10−7ei18.93×10−7π,0.3857e0.802iπ1.0606a42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ5,0.9853ei1.9704π,1.149×10−6ei18.78×10−6π,0.033e0.064iπ2.6337a53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π.2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ4,0.9723ei1.9506π,1.493×10−7ei3.08×10−7π,0.053e0.106iπ2.3353


## 5.1.2. Weighted Choice Values ofCSF6SfS

Letσ1=0.4,σ2=0.3,σ3=0.2, and σ4=0.1 be the weights for each attribute zk,k=1,2,3,4. Using these weights in Algorithm 3, we can compute weighted choice values of CSF6SfS, which are given by Table 24.Table 24
Tabular representation of the weighted choice values ofFJ,Z,6.
FJ,Z,6z1z2z3z4XjσSXjσa15,0.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π4,0.87ei1.76π,0.01ei0.024π,0.21ei0.38π4,0.88ei1.78π,0.012ei0.026π,0.2ei0.394π2,0.48ei0.98π,0.012ei0.022π,0.77e1.52iπ5,0.9019ei1.7974π,0.0131ei0.0278π,0.195e0.3402iπ2.2482a23,0.7ei1.34π,0.0173ei0.0344π,0.49eiπ2,0.32ei0.66π,0.015ei0.03π,0.9ei1.78π3,0.6ei1.24π,0.081ei0.16π,0.69e1.39π3,0.62ei1.23π,0.079ei0.15π,0.73e1.48iπ3,0.5963ei1.168π,0.0263ei0.052π,0.655e1.319iπ0.081a32,0.41ei0.84π,0.014ei0.026π,0.77ei1.52π2,0.33ei0.64π,0.016ei0.028π,0.89e1.8π2,0.35ei0.8π,0.016ei0.033π,0.84e1.7π3,0.53ei1.1π,0.063ei0.13π,0.67e1.38iπ3,0.3925ei0.813π,0.015ei0.032π,0.807e1.61iπ−0.7378a42,0.47ei0.92π,0.0169ei0.0338π,0.9ei1.8π3,0.72ei1.4π,0.05ei0.88π,0.59ei1.22π5,0.9ei1.82π,0.08ei0.158π,0.18e0.358π4,0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ5,0.7235ei1.443π,0.0319ei0.121π,0.5229e1.049iπ1.185a53,0.71ei1.44π,0.0174ei0.0342π,0.51ei0.98π4,0.89ei1.784π,0.013ei0.0278π,0.145ei0.4π3,0.55ei1.2π,0.06ei0.118π,0.7ei1.42π2,0.49ei0.96π,0.011ei0.022π,0.76e1.54iπ4,0.7570ei1.532π,0.0195ei0.0394π,0.5503e0.843iπ1.123It is clear from Table24 that G4 has maximum score; therefore, a4 = Facebook is selected as best third-party app of the year. According to the weighted choice values, ranking of third-party apps is as follows: a1>a4>a5>a2>a3.


## 5.1.3.L-Choice Values of CSF6SfS

TheL-choice values of CSF6SfS are evaluated using Algorithm 4 to find out the best alternative for the proposed MADM problem. Let L=4 be threshold; then, 4-choice values of CSFSfS is shown in Table 25. We can observe that, from Table 25, the ranking of third-party apps according to 4-choice values is as follows: a1>a4>a5>a2≥a3, which shows that a1 = Facebook has maximum choice value so that Facebook is selected as the best third-party app of the year.Table 25
Tabular representation of the 4-choice values ofFJ,Z,6.
FJ4,Z,6z1z2z3z4XjSXja10.95ei1.88π,0.0172ei0.0346π,0.13ei0.2π0.87ei1.76π,0.01ei0.024π,0.21ei0.38π0.88ei1.78π,0.012ei0.026π,0.2ei0.394π0,0,10.9973ei1.9944π,2.064×10−6ei1.45×10−11π,5.46×10−3e2.8×10−5iπ1.989a20,0,10,0,10,0,10,0,10,0,1−1a30,0,10,0,10,0,10,0,10,0,1−1a40,0,10,0,10.9ei1.82π,0.08ei0.158π,0.18e0.358π0.77ei1.52π,0.017ei0.032π,0.35e0.66iπ0.9605ei1.926π,1.36×10−3ei3.18×10−7π,0.063ei6.978×10−3π1.8459a50,0,10.89ei1.784π,0.013ei0.0278π,0.145ei0.4π0,0,10,0,10.89ei1.78π,0.013ei0.0278π,0.145e0.4iπ1.5263


## 5.2. Selection of the Best Physiotherapist Doctor of Mayo Hospital in Lahore

Physiotherapy helps to restore movement and function when people are affected by injury or disability. A physiotherapist treats such kind of people and helps them through exercise, manual therapy, education, and advice. A physiotherapist is very helpful in maintaining the health of people of all ages as well as encourages them for happy life. A physiotherapist must have patience, communication skills, and ability to establish a good relationship with patients and their families. The motive of this study is to select the best physiotherapist doctor in Lahore relative to their attributes under the environment ofCSFNSf. For this purpose, the data has been collected from the students of Mayo Hospital, Lahore, enact here as experts E˜1, E˜2, E˜3, and E˜4 whose weight vectors are σ=0.4,0.2,0.1,0.3T. The following physiotherapists of Mayo Hospital are treated as alternatives in this MAGDM problem:w1: Dr. Amnaw2: Dr. Rizwanw3: Dr. Akmalw4: Dr. Sidraw5: Dr. SaleemFive attributes considered as key factors for a physiotherapist are as follows:z1: knowledge and experience.z2: behavioral (positivity, patience, and humbleness).z3: availability and flexibility.z4: master of skills (communication, organizational, or problem-solving skills).z5: session fee.We solve this MAGDM problem by following theCSFNSf-TOPSIS method.Step 1: according to these attributes, each expert model 6-soft set is in Table26, whereFive stars mean “Marvellous”Four stars mean “Outstanding”Three stars mean “Super”Two stars mean “Good”One star mean “Satisfactory”Big dot means “Acceptable”Table3 represents the grading criteria, used for assigning the CSFNSfN corresponding to each rank by the expert E˜1, E˜2, E˜3, and E˜4 tabulated in Tables 27–30, respectively.Step 2: using equation (24), we can put together the opinions of all experts. ACSFNSfDM formed by aggregation is given in Table 31.Step 3: to demonstrate the importance of attributes in the MAGDM problem, experts rank them and associateCSFNSfN to each attribute which are arranged in Table 32. We cumulated the weights given by experts using equation (26) to form CSFNSf weight vector χ of attributes, i.e.,(34)χ=4,0.89ei1.72π,0.017ei0.034π,0.23e0.5iπ4,0.91ei1.86π,0.016ei0.034π,0.09e0.2iπ3,0.62ei1.24π,0.016ei0.028π,0.53e1.1iπ3,0.53ei1.06π,0.02ei0.04π,0.73e1.42iπ2,0.55ei1.14π,0.019ei0.042π,0.67e1.34iπ.Step 4: by utilizingACSFNSfDM and weight vector χ of attribute in equation (34), AWCSFNSfDM is evaluated and summarized in Table 33.Step 5: in the MAGDM problem, all the attributes’ knowledge and experience, behavior, availability and flexibility, and master of skills are benefit-type attributes except the session fee, which is a cost-type attribute. According to the nature of attributes and applying equation (29) and (30), CSFNSf-PIS and CSFNSf-NIS are evaluated and arranged in Table 34Step 6: Table35 represents the normalized Euclidean distance from each alternative to CSFNSf-PIS and CSFNSf-NIS using equations (31) and (32), respectively.Step 7: the revised closeness index of each alternative is calculated by utilizing equation (33) and given in Table 36.Step 8: sincew1 has least revised closeness index, therefore, Dr. Amna is the best physiotherapist in Mayo Hospital, Lahore. The ranking of alternatives is shown in Table 37.Table 26
Experts’ rating according to attributes.
AttributesAlternativesE˜1E˜2E˜3E˜4z1w1∗∗∗∗=4∗∗∗=3∗∗∗=3∗∗∗∗=4w2∗∗∗=3∗∗∗=3∗∗∗=3∗∗∗=3w3∗=1•=0∗∗=2∗=1w4∗∗=2∗∗∗=3∗∗∗∗=4∗∗=2w5∗∗=2∗=1∗∗=2•=0z2w1∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4∗∗∗=3w2∗∗=2∗∗∗=3∗∗=2∗=1w3∗=1∗=1∗=1∗∗=2w4∗∗∗=3∗∗=2∗∗∗=3∗∗∗∗=4w5∗=1∗∗=2∗∗=2∗=1z3w1∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4∗∗∗∗=4w2∗=1∗∗=2∗∗=2•=0w3∗∗=2∗∗=2∗∗=2∗∗=2w4∗∗∗=3∗=1•=0∗∗=2w5∗∗∗=3∗∗=2∗∗∗∗=4∗=1z4w1∗∗∗=3∗∗∗=3∗∗=2∗∗=2w2∗∗∗∗=4∗∗∗∗=4∗∗∗=3∗∗∗=3w3∗=1∗∗∗=3∗∗∗∗=4∗∗=2w4∗∗=2∗∗∗=3∗∗∗=3∗∗∗=3w5∗=1∗=1∗∗=2∗∗=2z5w1∗∗∗∗=4∗∗∗=3∗∗∗=3∗∗=2w2∗∗=2∗∗=2∗∗∗=3∗∗=2w3∗=1∗∗=2∗=1∗=1w4∗∗∗=3∗∗=2∗=1•=0w5∗∗=2•=0∗∗∗=3∗=1Table 27
Tabular representation ofCSFNSfDM of expert E˜1.
FJ1,Z,5z1z2z3z4z5w14,0.98ei1.98π,0.01ei0.024π,0.02ei0.06π4,0.99ei1.94π,0.012ei0.02π,0.1ei0.26π4,0.97ei1.96π,0.013ei0.02π,0.05ei0.08π3,0.84ei1.66π,0.019ei0.04π,0.29e0.6iπ4,0.98ei1.92π,0.012ei0.026π,0.03e0.01iπw23,0.84ei1.66π,0.02ei0.042π,0.29ei0.62π2,0.37ei0.78π,0.015ei0.026π,0.87ei1.72π1,0.16ei0.34π,0.018ei0.038π,0.89e1.78π4,0.96ei1.94π,0.013ei0.024π,0.04e0.06iπ2,0.36ei0.74π,0.016ei0.03π,0.59e1.2iπw31,0.17ei0.36π,0.02ei0.038π,0.91ei1.84π1,0.19ei0.36π,0.021ei0.04π,0.93e1.82π2,0.59ei0.12π,0.0155ei0.032π,0.82e1.6π1,0.2ei0.38π,0.022ei0.042π,0.92e1.86iπ1,0.21ei0.44π,0.024ei0.05π,0.93e1.88iπw42,0.58ei1.12π,0.015ei0.032π,0.82ei1.66π3,0.66ei1.34π,0.1ei0.18π,0.56ei1.14π3,0.67ei1.36π,0.021ei0.044π,0.31e0.64π2,0.55ei1.14π,0.014ei0.028π,0.8e1.62iπ3,0.67ei1.38π,0.025ei0.048π,0.33e0.68iπw52,0.52ei1.02π,0.012ei0.026π,0.74ei1.5π1,0.21ei0.4π,0.021ei0.046π,0.95ei1.88π1,0.7ei1.36π,0.026ei0.05π,0.35ei0.68π1,0.16ei0.42π,0.018ei0.038π,0.89e1.88iπ2,0.58ei1.14π,0.01ei0.024π,0.82e1.62iπTable 28
Tabular representation ofCSFNSfDM of expert E˜2.
FJ2,Z,5z1z2z3z4z5w13,0.83ei1.64π,0.019ei0.036π,0.3ei0.58π4,0.98ei1.94π,0.013ei0.024π,0.1ei0.26π4,0.98ei1.98π,0.01ei0.024π,0.02ei0.06π3,0.84ei1.66π,0.02ei0.036π,0.31e0.63iπ3,0.8ei1.62π,0.022ei0.042π,0.32e0.6iπw23,0.71ei1.38π,0.028ei0.058π,0.37ei0.72π3,0.72ei1.42π,0.03ei0.062π,0.38ei0.76π2,0.57ei1.1π,0.012ei0.028π,0.81e1.6π4,0.86ei1.74π,0.0169ei0.032π,0.027e0.052iπ2,0.56ei1.1π,0.015ei0.07π,0.8e1.58iπw30,0.05ei0.012π,0.01ei0.06π,0.985ei1.972π1,0.21ei0.46π,0.024ei0.52π,0.94e1.86π2,0.5ei1.02π,0.01ei0.02π,0.7e1.4π3,0.73ei1.48π,0.032ei0.062π,0.39e0.76iπ2,0.55ei1.08π,0.01ei0.02π,0.79e1.56iπw43,0.74ei1.5π,0.035ei0.068π,0.4ei0.78π2,0.54ei1.04π,0.015ei0.026π,0.78ei1.54π1,0.24ei0.44π,0.03ei0.056π,0.94e1.84π3,0.71ei1.38π,0.028ei0.058π,0.37e0.72iπ2,0.53ei1.04π,0.013ei0.026π,0.77e1.52iπw51,0.24ei0.42π,0.028ei0.062π,0.91ei1.84π2,0.51ei1.04π,0.013ei0.06π,0.76ei1.5π2,0.52ei1.02π,0.013ei0.06π,0.75ei1.52π1,0.23ei0.5π,0.035ei0.066π,0.89e1.82iπ0,0.1ei0.24π,0.012ei0.026π,0.99e1.972iπTable 29
Tabular representation ofCSFNSfDM of expert E˜3.
FJ3,Z,5z1z2z3z4z5w13,0.84ei1.66π,0.019ei0.04π,0.29ei0.6π4,0.97ei1.98π,0.013ei0.028π,0.14ei0.26π4,0.96ei1.98π,0.01ei0.04π,0.04ei0.04π2,0.5ei1.02π,0.014ei0.026π,0.74e1.46iπ3,0.8ei1.62π,0.018ei0.038π,0.29e0.6iπw23,0.72ei1.46π,0.036ei0.068π,0.41ei0.8π2,0.49eiπ,0.013ei0.028π,0.71ei1.44π2,0.48ei0.98π,0.015ei0.032π,0.7e1.36π3,0.74ei1.44π,0.034ei0.07π,0.42e0.82iπ3,0.75ei1.48π,0.035ei0.072π,0.43e0.82iπw32,0.47ei0.96π,0.015ei0.032π,0.7ei1.36π1,0.27ei0.5π,0.034ei0.066π,0.94e1.92π2,0.46ei0.94π,0.014ei0.03π,0.68e1.38π4,0.88ei1.78π,0.015ei0.032π,0.25e0.46iπ1,0.29ei0.56π,0.04ei0.078π,0.96e1.9iπw44,0.98ei1.94π,0.013ei0.06π,0.1ei0.26π3,0.76ei1.5π,0.036ei0.074π,0.44ei0.86π0,0.13ei0.6π,0.014ei0.03π,0.986e1.97π3,0.77ei1.52π,0.038ei0.078π,0.45e0.88iπ1,0.3ei0.58π,0.042ei0.082π,0.925e1.86iπw52,0.44ei0.92π,0.012ei0.028π,0.66ei1.34π2,0.43ei0.88π,0.015ei0.028π,0.66ei1.34π4,0.87ei1.78π,0.016ei0.028π,0.21ei0.43π2,0.42ei0.86π,0.012ei0.02π,0.65e1.32iπ3,0.78ei1.54π,0.04ei0.076π,0.46e0.9iπTable 30
Tabular representation ofCSFNSfDM of expert E˜4.
FJ4,Z,5z1z2z3z4z5w14,0.95ei1.94π,0.012ei0.02π,0.13ei0.2π3,0.8ei1.62π,0.018ei0.038π,0.31ei0.6π4,0.97ei1.96π,0.01ei0.02π,0.15ei0.32π2,0.63ei1.28π,0.012ei0.02π,0.6e1.18iπ2,0.64ei1.26π,0.01ei0.024pi,0.59e1.2iπw23,0.77ei1.56π,0.041ei0.08π,0.47ei0.92π1,0.31ei0.64π,0.022ei0.04π,0.89ei1.8π0,0.06ei0.14π,0.012ei0.02π,0.986e1.97π3,0.8ei1.58π,0.05ei0.102π,0.48e0.94iπ2,0.37ei1.78π,0.014ei0.024π,0.86e1.74iπw31,0.3ei0.64π,0.05ei0.012π,0.91ei1.84π2,0.41ei0.84π,0.01ei0.06π,0.64e1.3π2,0.4ei0.84π,0.015ei0.032π,0.63e1.28π2,0.39ei0.8π,0.014ei0.03π,0.62e1.26iπ1,0.33ei0.64π,0.07ei0.1π,0.93e1.88iπw42,0.38ei0.78π,0.013ei0.07π,0.61ei1.24π4,0.9ei1.84π,0.014ei0.03π,0.19ei0.36π2,0.37ei0.76π,0.012ei0.026π,0.6e1.22π3,0.81ei1.6π,0.051ei0.104π,0.49eiπ0,0.13ei0.28π,0.014ei0.026π,0.982e1.966iπw50,0.07ei0.012π,0.016ei0.03π,0.991ei1.984π1,0.34ei0.64π,0.065ei0.132π,0.95ei1.92π1,0.3ei0.58π,0.014ei0.03π,0.93ei1.88π2,0.36ei0.74π,0.01ei0.024π,0.59e1.2iπ1,0.21ei0.46π,0.024ei0.052π,0.94e1.86iπTable 31
Tabular representation ofACSFNSfDM.
FJ1,Z,5z1z2z3z4z5w14,0.95ei1.94π,0.013ei0.026π,0.07ei0.18π4,0.97ei1.9π,0.013ei0.026π,0.15ei0.34π4,0.97ei1.96π,0.01ei0.02π,0.05ei0.1π3,0.77ei1.54π,0.016ei0.03π,0.4e0.82iπ4,0.91ei1.76π,0.013ei0.03π,0.15e0.36iπw23,0.79ei1.56π,0.028ei0.058π,0.36ei0.74π3,0.48ei0.96π,0.019ei0.034π,0.73ei1.46π2,0.33ei0.66π,0.01ei0.02π,0.88e1.76π4,0.9ei1.82π,0.02ei0.04π,0.15e0.26iπ3,0.47ei0.96π,0.016ei0.03π,0.68e1.36iπw32,0.25ei0.52π,0.022ei0.048π,0.9ei1.74π2,0.29ei0.58π,0.018ei0.038π,0.83e1.66π2,0.5ei1.04π,0.013ei0.028π,0.72e1.42π4,0.55ei1.12π,0.029ei0.038π,0.6e1.2iπ2,0.35ei0.7π,0.029ei0.054π,0.9e1.82iπw44,0.7ei1.18π,0.016ei0.034π,0.63ei1.28π4,0.77ei1.56π,0.034ei0.064π,0.42ei0.84π3,0.51ei1.02π,0.01ei0.02π,0.53e1.06π3,0.71ei1.4π,0.026ei0.054π,0.55e1.12iπ3,0.52ei1.06π,0.019ei0.038π,0.6e1.22iπw52,0.39ei0.74π,0.015ei0.032π,0.83ei1.68π2,0.35ei0.7π,0.025ei0.052π,0.87ei1.74π4,0.63ei1.26π,0.02ei0.034π,0.51ei1.02π2,0.28ei0.6π,0.016ei0.034π,0.76e1.58iπ3,0.49ei0.96π,0.015ei0.034π,0.83e1.6iπTable 32
Experts opinion related to each attribute.
FJ,Z,6E˜1E˜2E˜3E˜4z13,0.85ei1.66π,0.019ei0.042π,0.29ei0.6π4,0.99ei1.94π,0.012ei0.02π,0.1ei0.26π4,0.88ei1.78π,0.015ei0.032π,0.25ei0.46π3,0.67ei1.36π,0.021ei0.044π,0.31e0.64iπz23,0.66ei1.34π,0.021ei0.044π,0.31ei0.64π4,0.96ei1.94π,0.013ei0.024π,0.04ei0.06π3,0.78ei1.54π,0.04ei0.076π,0.46e0.9π4,0.98ei1.98π,0.01ei0.024π,0.02e0.06iπz32,0.63ei1.28π,0.012ei0.02π,0.59ei1.2π3,0.72ei1.42π,0.03ei0.062π,0.38e0.78π3,0.75ei1.48π,0.035ei0.072π,0.43e0.82π2,0.42ei0.86π,0.012ei0.02π,0.65ei1.32πz41,0.29ei0.6π,0.014ei0.03π,0.94ei1.86π1,0.21ei0.46π,0.024ei0.052π,0.94ei1.86π2,0.37ei0.76π,0.012ei0.026π,0.6e1.22π3,0.78ei1.54π,0.04ei0.076π,0.46e0.9iπz52,0.62ei1.28π,0.01ei0.024π,0.58ei1.2π2,0.66ei1.34π,0.1ei0.18π,0.56ei1.14π2,0.64ei1.26π,0.01ei0.024π,0.59ei1.2π1,0.21ei0.4π,0.021ei0.046π,0.95e1.88iπTable 33
Tabular representation ofAWCSFNSfDM.
z1z2z3z4z5w14,0.85ei1.66π,0.021ei0.042π,0.23ei0.52π4,0.88ei1.76π,0.021ei0.042π,0.174ei0.38π3,0.601ei1.22π,0.019ei0.034π,0.053ei1.1π3,0.41ei0,82π,0.025ei0.05π,0.77e1.54iπ2,0.5eiπ,0.023ei0.05π,0.67e1.36iπw23,0.7ei1.34π,0.03ei0.066π,0.41ei0.86π3,0.43ei0.9π,0.025ei0.048π,0.73ei1.46π2,0.2ei0.4π,0.019ei0.034π,0.91e1.82π3,0.47ei0.96π,0.028ei0.056π,0.73e1.42iπ2,0.25ei0.54π,0.025ei0.05π,0.83e1.66iπw32,0.22ei0.44π,0.027ei0.058π,0.91ei1.74π2,0.26ei0.54π,0.024ei0.025π,0.83e1.66π2,0.32ei0.64π,0.021ei0.038π,0.8e1.62π3,0.29ei0.58π,0.035ei0.068π,0.73e1.42iπ2,0.19ei0.34π,0.035ei0.068π,0.95e1.9iπw44,0.62ei1.02π,0.023ei0.048π,0.65ei1.34π4,0.70ei1.46π,0.037ei0.072π,0.43ei0.86π3,0.32ei0.64π,0.019ei0.034π,0.69e1.42π3,0.37ei0.74π,0.033ei0.066π,0.82e1.62iπ2,0.29ei0.6π,0.027ei0.056π,0.8e1.62iπw52,0.35ei0.62π,0.022ei0.046π,0.83ei1.7π2,0.31ei0.66π,0.029ei0.06π,0.87ei1.74π3,0.39ei0.78π,0.025ei0.044π,0.68ei1.38π2,0.15ei0.318π,0.025ei0.052π,0.89e1.8iπ2,0.27ei0.54π,0.024ei0.042π,0.91e1.82iπTable 34
Tabular representation ofCSFNSf-PIS and CSFNSf-NIS.
AttributeCSFNSf-PISCSFNSf-NISz14,0.85ei1.66π,0.021ei0.042π,0.23ei0.52π2,0.22ei0.44π,0.027ei0.058π,0.91ei1.74πz24,0.88ei1.76π,0.021ei0.042π,0.174ei0.038π2,0.31ei0.66π,0.029ei0.06π,0.87ei1.74πz33,0.601ei1.22π,0.019ei0.034π,0.53ei1.1π2,0.2ei0.4π,0.019ei0.034π,0.91e1.82πz43,0.47ei0.96π,0.028ei0.056π,0.73e1.42π2,0.15ei0.318π,0.025ei0.052π,0.89e1.8πz52,0.19ei0.38π,0.035ei0.068π,0.95ei1.9π2,0.5eiπ,0.023ei0.046π,0.67ei1.36πTable 35
Tabular representation of normalized Euclidean distance from ideal solution.
AlternativedP^k,wjdPˇk,wjw10.205590.55439w20.366770.311911w30.5281060.18778w40.2546490.396793w50.506490.1936Table 36
Tabular representation of revised closeness index of each alternative.
Alternativeℑwjw10w21.22136w32.236816w40.485238w52.1143Table 37
Tabular representation of revised closeness index of each alternative.
AlternativeRankingw11w23w35w42w54


## 6. Comparative Analysis

We now compare our proposed model with Pythagorean fuzzyN-soft set (PFNSfS) that was discussed by Zhang et al. [24].(1)
Table38 represents the ratings of MADM problem as shown in Section5.1. in PFNSfNs.(2)
Table39 presents the calculated choice values of PF6SfS using the algorithm defined in [24] for the selection of the third-party app. Clearly, from Table 39, a1 is the best choice, and the ranking of third-party apps is as follows: a1>a4>a5>a2>a3.(3)
LetL=4 be threshold; then, 4-choice values of PFSfS are shown in Table 40. The ranking of third-party apps according to 4-choice values is as follows: a1>a4>a5>a2≥a3, which further shows that a1 = Facebook has maximum choice value.(4)
We conclude the same results from both choice values andL-choice values of PFNSfS [24], which shows the reliability of our proposed method, and it can be applied to any MADM problem.(5)
The data arranged in Table23 is able to handle more real-life problems compared to Pythagorean N-soft set and intuitionistic N-soft set as it includes the neutral membership degree as well as it could deal with 2-dimensional data.(6)
The proposed model would provide the same results under spherical fuzzyN-soft environment by taking the periodic terms equal to zero.Table 38
Tabular representation of thePF6SfS defined from the problem proposed in Section 5.1.
z1z2z3z4a15,0.95,0.134,0.87,0.214,0.88,0.22,0.48,0.77a23,0.7,0.492,0.32,0.93,0.6,0.693,0.62,0.73a32,0.41,0.772,0.33,0.892,0.35,0.843,0.53,0.67a42,0.4,0.93,0.72,0.595,0.9,0.184,0.77,0.35a53,0.71,0.514,0.89,0.1453,0.55,0.72,0.49,0.76Table 39
Tabular representation of Choice value ofPF6SfS in Section 5.1.
z1z2z3z4Hia15,0.95,0.134,0.87,0.214,0.88,0.22,0.48,0.7715,2.83162a23,0.7,0.492,0.32,0.93,0.6,0.693,0.62,0.7311,1.6754702a32,0.41,0.772,0.33,0.892,0.35,0.843,0.53,0.679,1.3498273a42,0.4,0.93,0.72,0.595,0.9,0.184,0.77,0.3514,2.30656384a53,0.71,0.514,0.89,0.1453,0.55,0.72,0.49,0.7612,2.231012Table 40
Tabular representation of 4 choice value ofPF6SfS Section 5.1.
z1z2z3z4Hi4a10.95,0.130.87,0.210.88,0.20,10.33532a20,0.50,10,0.50,0.5−0.4375a30,10,10,10,0.5−0.8125a40,10,0.50.9,0.180.77,0.35−0.0005a50,0.50.89,0.1450,0.50,1−0.18223125


### 6.1. Comparison with Complex Spherical Fuzzy TOPSIS Method

In this section, we solve the MAGDM problem “selection of best physiotherapist doctor of Mayo Hospital in Lahore” by complex spherical fuzzy TOPSIS method, proposed by Akram et al. [16], to demonstrate the importance and superiority of the proposed model. The solution by the complex spherical fuzzy TOPSIS method is as follows:Step 1: the linguistic term corresponding to each rank assessed by the experts are the same as given in Table26. To apply the CSF TOPSIS method, the grading part is excluded from CSFNSfN and CSFNs are assigned by each expert E˜1,E˜2, E˜3, and E˜4, which are arranged in Tables 41–44, respectively, according to the grading criteria defined in Table 3.Step 2: using the weight vector of expertsσ=0.4,0.2,0.1,0.3T and complex spherical fuzzy weighted average CSFWA operator [16], we can calculate the aggregated complex spherical fuzzy decision matrix ACSFDM, whose entries are evaluated by the formula defined as follows [16]:(35)Pjk=1−∏1d=s1−pjkd2σdei2π1−∏1d=s1−ϕjkd2σd,∏1d=svjkdei2π∏1d=sδjkd,∏1d=srjkdei2π∏1d=sλjkd.ACSFDM is summarized in Table 45.Step 3: the experts’ opinion about the importance of attributes are given in Table46. The experts’ opinion are combined using CSFWA operator [16], to formulate the weight vector χ for the attributes, and are defined as follows:(36)χk=1−∏1d=s1−pkd2σdei2π1−∏1d=s1−ϕkd2σd,∏1d=svkdei2π∏1d=sδkd,∏1d=srkdei2π∏1d=sλkd.Thus, we have(37)χ=0.89ei1.72π,0.017ei0.034π,0.23e0.5iπ0.91ei1.86π,0.016ei0.034π,0.09e0.2iπ0.62ei1.24π,0.016ei0.028π,0.53e1.1iπ0.53ei1.06π,0.02ei0.04π,0.73e1.42iπ0.55ei1.14π,0.019ei0.042π,0.67e1.34iπ.Step 4: the aggregated weighted complex spherical decision matrix (AWCSFDM) is arranged in Table47, where the entries of AWCSFDM are calculated using the formula [16](38)Pjk¯=pjkpkei2πϕjkϕk,vjk2+vk2−vjk2vk2ei2πδjk2+δk2−δjk2δk2,rjk2+rk2−rjk2rk2ei2πλjk2+λk2−λjk2λk2.Step 5: to compute the complex spherical fuzzy positive ideal solution (CSF-PIS) and negative ideal solution (CSF-NIS), we evaluate the score degree of all CSFNs in AWCSFDM, using the formula(39)ScPjk¯=pkj2−wkj2−rkj2+ϕkj2−δkj2−λkj2.Table48 represents the CSF-PIS and CSF-NIS with the help of equation (26).Step 6: the complex spherical fuzzy normalized Euclidean distance of each alternative is given in Table49 and computed by the formula [16], from CSF-PIS:(40)dP^k,wj=13k∑k=1mp^k2−p¯jk22+v^k2−v¯jk22+r^k2−r¯jk22+ϕ^k2−ϕ¯jk22+δ^k2−δ¯jk22+λ^k2−λ¯jk221/2.Similarly, the normalized Euclidean distance between the CSF-NIS and any of the alternativewj can be formulated as follows:(41)dPˇk,wj=13k∑k=1mpˇk2−p¯jk22+vˇk2−v¯jk22+rˇk2−r¯jk22+ϕˇk2−ϕ¯jk22+δˇk2−δ¯jk22+λˇk2−λ¯jk221/2.Step 7: Equation (33) is used to calculate the revised closeness index of each alternative, as given in Table 50.Step 8: revised closeness index in Table50 reveals that w1 is the best alternative within the ranking w1>w4>w2>w5>w3.Table 41
Tabular representation ofCSFDM of expert E˜1.
z1z2z3z4z5w10.98ei1.98π,0.01ei0.024π,0.02ei0.06π0.99ei1.94π,0.012ei0.02π,0.1ei0.26π0.97ei1.96π,0.013ei0.02π,0.05ei0.08π0.84ei1.66π,0.019ei0.04π,0.29e0.6iπ0.98ei1.92π,0.012ei0.026π,0.03e0.01iπw20.84ei1.66π,0.02ei0.042π,0.29ei0.62π0.37ei0.78π,0.015ei0.026π,0.87ei1.72π0.16ei0.34π,0.018ei0.038π,0.89e1.78π0.96ei1.94π,0.013ei0.024π,0.04e0.06iπ0.36ei0.74π,0.016ei0.03π,0.59e1.2iπw30.17ei0.36π,0.02ei0.038π,0.91ei1.84π0.19ei0.36π,0.021ei0.04π,0.93e1.82π0.59ei0.12π,0.0155ei0.032π,0.82e1.6π0.2ei0.38π,0.022ei0.042π,0.92e1.86iπ0.21ei0.44π,0.024ei0.05π,0.93e1.88iπw40.58ei1.12π,0.015ei0.032π,0.82ei1.66π0.66ei1.34π,0.1ei0.18π,0.56ei1.14π0.67ei1.36π,0.021ei0.044π,0.31e0.64π0.55ei1.14π,0.014ei0.028π,0.8e1.62iπ0.67ei1.38π,0.025ei0.048π,0.33e0.68iπw50.52ei1.02π,0.012ei0.026π,0.74ei1.5π0.21ei0.4π,0.021ei0.046π,0.95ei1.88π0.7ei1.36π,0.026ei0.05π,0.35ei0.68π0.16ei0.42π,0.018ei0.038π,0.89e1.88iπ0.58ei1.14π,0.01ei0.024π,0.82e1.62iπTable 42
Tabular representation ofCSFDM of expert E˜2.
z1z2z3z4z5w10.83ei1.64π,0.019ei0.036π,0.3ei0.58π0.98ei1.94π,0.013ei0.024π,0.1ei0.26π0.98ei1.98π,0.01ei0.024π,0.02ei0.06π0.84ei1.66π,0.02ei0.036π,0.31e0.63iπ0.8ei1.62π,0.022ei0.042π,0.32e0.6iπw20.71ei1.38π,0.028ei0.058π,0.37ei0.72π0.72ei1.42π,0.03ei0.062π,0.38ei0.76π0.57ei1.1π,0.012ei0.028π,0.81e1.6π0.86ei1.74π,0.0169ei0.032π,0.027e0.052iπ2,0.56ei1.1π,0.015ei0.07π,0.8e1.58iπw30.05ei0.012π,0.01ei0.06π,0.985ei1.972π0.21ei0.46π,0.024ei0.52π,0.94e1.86π0.5ei1.02π,0.01ei0.02π,0.7e1.4π0.73ei1.48π,0.032ei0.062π,0.39e0.76iπ2,0.55ei1.08π,0.01ei0.02π,0.79e1.56iπw40.74ei1.5π,0.035ei0.068π,0.4ei0.78π0.54ei1.04π,0.015ei0.026π,0.78ei1.54π1,0.24ei0.44π,0.03ei0.056π,0.94e1.84π0.71ei1.38π,0.028ei0.058π,0.37e0.72iπ2,0.53ei1.04π,0.013ei0.026π,0.77e1.52iπw50.24ei0.42π,0.028ei0.062π,0.91ei1.84π0.51ei1.04π,0.013ei0.06π,0.76ei1.5π0.52ei1.02π,0.013ei0.06π,0.75ei1.52π0.23ei0.5π,0.035ei0.066π,0.89e1.82iπ0.1ei0.24π,0.012ei0.026π,0.99e1.972iπTable 43
Tabular representation ofCSFDM of expert E˜3.
z1z2z3z4z5w10.84ei1.66π,0.019ei0.04π,0.29ei0.6π0.97ei1.98π,0.013ei0.028π,0.14ei0.26π0.96ei1.98π,0.01ei0.04π,0.04ei0.04π0.5ei1.02π,0.014ei0.026π,0.74e1.46iπ0.8ei1.62π,0.018ei0.038π,0.29e0.6iπw20.72ei1.46π,0.036ei0.068π,0.41ei0.8π0.49eiπ,0.013ei0.028π,0.71ei1.44π0.48ei0.98π,0.015ei0.032π,0.7e1.36π0.74ei1.44π,0.034ei0.07π,0.42e0.82iπ0.75ei1.48π,0.035ei0.072π,0.43e0.82iπw30.47ei0.96π,0.015ei0.032π,0.7ei1.36π0.27ei0.5π,0.034ei0.066π,0.94e1.92π0.46ei0.94π,0.014ei0.03π,0.68e1.38π0.88ei1.78π,0.015ei0.032π,0.25e0.46iπ0.29ei0.56π,0.04ei0.078π,0.96e1.9iπw40.98ei1.94π,0.013ei0.06π,0.1ei0.26π0.76ei1.5π,0.036ei0.074π,0.44ei0.86π0.13ei0.6π,0.014ei0.03π,0.986e1.97π0.77ei1.52π,0.038ei0.078π,0.45e0.88iπ0.3ei0.58π,0.042ei0.082π,0.925e1.86iπw50.44ei0.92π,0.012ei0.028π,0.66ei1.34π0.43ei0.88π,0.015ei0.028π,0.66ei1.34π0.87ei1.78π,0.016ei0.028π,0.21ei0.43π0.42ei0.86π,0.012ei0.02π,0.65e1.32iπ0.78ei1.54π,0.04ei0.076π,0.46e0.9iπTable 44
Tabular representation ofCSFDM of expert E˜4.
z1z2z3z4z5w10.95ei1.94π,0.012ei0.02π,0.13ei0.2π0.8ei1.62π,0.018ei0.038π,0.31ei0.6π0.97ei1.96π,0.01ei0.02π,0.15ei0.32π0.63ei1.28π,0.012ei0.02π,0.6e1.18iπ0.64ei1.26π,0.01ei0.024pi,0.59e1.2iπw20.77ei1.56π,0.041ei0.08π,0.47ei0.92π0.31ei0.64π,0.022ei0.04π,0.89ei1.8π0.06ei0.14π,0.012ei0.02π,0.986e1.97π0.8ei1.58π,0.05ei0.102π,0.48e0.94iπ0.37ei1.78π,0.014ei0.024π,0.86e1.74iπw30.3ei0.64π,0.05ei0.012π,0.91ei1.84π0.41ei0.84π,0.01ei0.06π,0.64e1.3π0.4ei0.84π,0.015ei0.032π,0.63e1.28π0.39ei0.8π,0.014ei0.03π,0.62e1.26iπ1,0.33ei0.64π,0.07ei0.1π,0.93e1.88iπw40.38ei0.78π,0.013ei0.07π,0.61ei1.24π0.9ei1.84π,0.014ei0.03π,0.19ei0.36π0.37ei0.76π,0.012ei0.026π,0.6e1.22π0.81ei1.6π,0.051ei0.104π,0.49eiπ0.13ei0.28π,0.014ei0.026π,0.982e1.966iπw50.07ei0.012π,0.016ei0.03π,0.991ei1.984π0.34ei0.64π,0.065ei0.132π,0.95ei1.92π0.3ei0.58π,0.014ei0.03π,0.93ei1.88π0.36ei0.74π,0.01ei0.024π,0.59e1.2iπ0.21ei0.46π,0.024ei0.052π,0.94e1.86iπTable 45
Tabular representation ofACSFDM.
z1z2z3z4z5w10.95ei1.94π,0.013ei0.026π,0.07ei0.18π0.97ei1.9π,0.013ei0.026π,0.15ei0.34π0.97ei1.96π,0.01ei0.02π,0.05ei0.1π0.77ei1.54π,0.016ei0.03π,0.4e0.82iπ0.91ei1.76π,0.013ei0.03π,0.15e0.36iπw20.79ei1.56π,0.028ei0.058π,0.36ei0.74π0.48ei0.96π,0.019ei0.034π,0.73ei1.46π0.33ei0.66π,0.01ei0.02π,0.88e1.76π0.9ei1.82π,0.02ei0.04π,0.15e0.26iπ0.47ei0.96π,0.016ei0.03π,0.68e1.36iπw30.25ei0.52π,0.022ei0.048π,0.9ei1.74π0.29ei0.58π,0.018ei0.038π,0.83e1.66π0.5ei1.04π,0.013ei0.028π,0.72e1.42π0.55ei1.12π,0.029ei0.038π,0.6e1.2iπ0.35ei0.7π,0.029ei0.054π,0.9e1.82iπw40.7ei1.18π,0.016ei0.034π,0.63ei1.28π0.77ei1.56π,0.034ei0.064π,0.42ei0.84π0.51ei1.02π,0.01ei0.02π,0.53e1.06π0.71ei1.4π,0.026ei0.054π,0.55e1.12iπ0.52ei1.06π,0.019ei0.038π,0.6e1.22iπw50.39ei0.74π,0.015ei0.032π,0.83ei1.68π0.35ei0.7π,0.025ei0.052π,0.87ei1.74π0.63ei1.26π,0.02ei0.034π,0.51ei1.02π0.28ei0.6π,0.016ei0.034π,0.76e1.58iπ0.49ei0.96π,0.015ei0.034π,0.83e1.6iπTable 46
Experts opinion related to each attribute.
E˜1E˜2E˜3E˜4z10.85ei1.66π,0.019ei0.042π,0.29ei0.6π0.99ei1.94π,0.012ei0.02π,0.1ei0.26π0.88ei1.78π,0.015ei0.032π,0.25ei0.46π0.67ei1.36π,0.021ei0.044π,0.31e0.64iπz20.66ei1.34π,0.021ei0.044π,0.31ei0.64π0.96ei1.94π,0.013ei0.024π,0.04ei0.06π0.78ei1.54π,0.04ei0.076π,0.46e0.9π0.98ei1.98π,0.01ei0.024π,0.02e0.06iπz30.63ei1.28π,0.012ei0.02π,0.59ei1.2π0.72ei1.42π,0.03ei0.062π,0.38e0.78π0.75ei1.48π,0.035ei0.072π,0.43e0.82π0.42ei0.86π,0.012ei0.02π,0.65ei1.32πz40.29ei0.6π,0.014ei0.03π,0.94ei1.86π0.21ei0.46π,0.024ei0.052π,0.94ei1.86π0.37ei0.76π,0.012ei0.026π,0.6e1.22π0.78ei1.54π,0.04ei0.076π,0.46e0.9iπz50.62ei1.28π,0.01ei0.024π,0.58ei1.2π0.66ei1.34π,0.1ei0.18π,0.56ei1.14π0.64ei1.26π,0.01ei0.024π,0.59ei1.2π0.21ei0.4π,0.021ei0.046π,0.95e1.88iπTable 47
Tabular representation ofAWCSFDM.
z1z2z3z4z5w10.85ei1.66π,0.021ei0.042π,0.23ei0.52π0.88ei1.76π,0.021ei0.042π,0.174ei0.38π0.601ei1.22π,0.019ei0.034π,0.053ei1.1π0.41ei0,82π,0.025ei0.05π,0.77e1.54iπ0.5eiπ,0.023ei0.05π,0.67e1.36iπw20.7ei1.34π,0.03ei0.066π,0.41ei0.86π0.43ei0.9π,0.025ei0.048π,0.73ei1.46π0.2ei0.4π,0.019ei0.034π,0.91e1.82π0.47ei0.96π,0.028ei0.056π,0.73e1.42iπ0.25ei0.54π,0.025ei0.05π,0.83e1.66iπw30.22ei0.44π,0.027ei0.058π,0.91ei1.74π0.26ei0.54π,0.024ei0.025π,0.83e1.66π0.32ei0.64π,0.021ei0.038π,0.8e1.62π0.29ei0.58π,0.035ei0.068π,0.73e1.42iπ0.19ei0.34π,0.035ei0.068π,0.95e1.9iπw40.62ei1.02π,0.023ei0.048π,0.65ei1.34π0.70ei1.46π,0.037ei0.072π,0.43ei0.86π0.32ei0.64π,0.019ei0.034π,0.69e1.42π0.37ei0.74π,0.033ei0.066π,0.82e1.62iπ0.29ei0.6π,0.027ei0.056π,0.8e1.62iπw52,0.35ei0.62π,0.022ei0.046π,0.83ei1.7π0.31ei0.66π,0.029ei0.06π,0.87ei1.74π0.39ei0.78π,0.025ei0.044π,0.68ei1.38π0.15ei0.318π,0.025ei0.052π,0.89e1.8iπ0.27ei0.54π,0.024ei0.042π,0.91e1.82iπTable 48
Tabular representation of CSF-PIS and CSF-NIS.
AttributeCSF-PISCSFN-NISz10.85ei1.66π,0.021ei0.042π,0.23ei0.52π0.22ei0.44π,0.027ei0.058π,0.91ei1.74πz20.88ei1.76π,0.021ei0.042π,0.174ei0.038π0.31ei0.66π,0.029ei0.06π,0.87ei1.74πz30.601ei1.22π,0.019ei0.034π,0.53ei1.1π0.2ei0.4π,0.019ei0.034π,0.91e1.82πz40.47ei0.96π,0.028ei0.056π,0.73e1.42π0.15ei0.318π,0.025ei0.052π,0.89e1.8πz50.19ei0.38π,0.035ei0.068π,0.95ei1.9π0.5eiπ,0.023ei0.046π,0.67ei1.36πTable 49
Normalized Euclidean distance from ideal solution.
AlternativedP^k,wjdPˇk,wjw10.1829470.565764w20.3838390.330875w30.53884220.200388w40.2663700.349150w50.5104550.208933Table 50
Revised closeness index of each alternative.
Alternativeℑwjw10w21.5132w32.5911w40.838853w52.42088Table 51
Comparison.
MethodRankingBest physiotherapistCSFNSf-TOPSISw1>w4>w2>w5>w3w1CSF-TOPSIS [16]w1>w4>w2>w5>w3w1SF-TOPSIS [10]w1>w4>w2>w5>w3w1SF-TOPSIS [11]w1>w4>w2>w5>w3w1


#### 6.1.1. Discussion

(1)
We now compare the proposed modelCSFNSf-TOPSIS method with the existing technique CSF-TOPSIS method to evaluate the accuracy of the result. The same result concludes from both methods as well as the ranking of the alternatives also same.(2)
We also apply technique on SF-TOPSIS methods [10, 11], to select the most appropriate physiotherapist. The same results including the ranking and best solution are organized in Table 51, which enhance the credibility of the proposed method.(3)
The proposed model deals not only with 2-dimensional uncertainties but also with the level of attribute for the alternative. The existing models are unable to handle MAGDM problems, but the proposed model has the ability to tackle those real-life DM problems having ranking system and parameterized information.(4)
The proposed model,CSFNSf-TOPSIS method, could be efficiently applied to the environment of SFNSf, CSFSf, and SFSf, by substituting periodic terms equal to zero and N=2.


## 6.1. Comparison with Complex Spherical Fuzzy TOPSIS Method

In this section, we solve the MAGDM problem “selection of best physiotherapist doctor of Mayo Hospital in Lahore” by complex spherical fuzzy TOPSIS method, proposed by Akram et al. [16], to demonstrate the importance and superiority of the proposed model. The solution by the complex spherical fuzzy TOPSIS method is as follows:Step 1: the linguistic term corresponding to each rank assessed by the experts are the same as given in Table26. To apply the CSF TOPSIS method, the grading part is excluded from CSFNSfN and CSFNs are assigned by each expert E˜1,E˜2, E˜3, and E˜4, which are arranged in Tables 41–44, respectively, according to the grading criteria defined in Table 3.Step 2: using the weight vector of expertsσ=0.4,0.2,0.1,0.3T and complex spherical fuzzy weighted average CSFWA operator [16], we can calculate the aggregated complex spherical fuzzy decision matrix ACSFDM, whose entries are evaluated by the formula defined as follows [16]:(35)Pjk=1−∏1d=s1−pjkd2σdei2π1−∏1d=s1−ϕjkd2σd,∏1d=svjkdei2π∏1d=sδjkd,∏1d=srjkdei2π∏1d=sλjkd.ACSFDM is summarized in Table 45.Step 3: the experts’ opinion about the importance of attributes are given in Table46. The experts’ opinion are combined using CSFWA operator [16], to formulate the weight vector χ for the attributes, and are defined as follows:(36)χk=1−∏1d=s1−pkd2σdei2π1−∏1d=s1−ϕkd2σd,∏1d=svkdei2π∏1d=sδkd,∏1d=srkdei2π∏1d=sλkd.Thus, we have(37)χ=0.89ei1.72π,0.017ei0.034π,0.23e0.5iπ0.91ei1.86π,0.016ei0.034π,0.09e0.2iπ0.62ei1.24π,0.016ei0.028π,0.53e1.1iπ0.53ei1.06π,0.02ei0.04π,0.73e1.42iπ0.55ei1.14π,0.019ei0.042π,0.67e1.34iπ.Step 4: the aggregated weighted complex spherical decision matrix (AWCSFDM) is arranged in Table47, where the entries of AWCSFDM are calculated using the formula [16](38)Pjk¯=pjkpkei2πϕjkϕk,vjk2+vk2−vjk2vk2ei2πδjk2+δk2−δjk2δk2,rjk2+rk2−rjk2rk2ei2πλjk2+λk2−λjk2λk2.Step 5: to compute the complex spherical fuzzy positive ideal solution (CSF-PIS) and negative ideal solution (CSF-NIS), we evaluate the score degree of all CSFNs in AWCSFDM, using the formula(39)ScPjk¯=pkj2−wkj2−rkj2+ϕkj2−δkj2−λkj2.Table48 represents the CSF-PIS and CSF-NIS with the help of equation (26).Step 6: the complex spherical fuzzy normalized Euclidean distance of each alternative is given in Table49 and computed by the formula [16], from CSF-PIS:(40)dP^k,wj=13k∑k=1mp^k2−p¯jk22+v^k2−v¯jk22+r^k2−r¯jk22+ϕ^k2−ϕ¯jk22+δ^k2−δ¯jk22+λ^k2−λ¯jk221/2.Similarly, the normalized Euclidean distance between the CSF-NIS and any of the alternativewj can be formulated as follows:(41)dPˇk,wj=13k∑k=1mpˇk2−p¯jk22+vˇk2−v¯jk22+rˇk2−r¯jk22+ϕˇk2−ϕ¯jk22+δˇk2−δ¯jk22+λˇk2−λ¯jk221/2.Step 7: Equation (33) is used to calculate the revised closeness index of each alternative, as given in Table 50.Step 8: revised closeness index in Table50 reveals that w1 is the best alternative within the ranking w1>w4>w2>w5>w3.Table 41
Tabular representation ofCSFDM of expert E˜1.
z1z2z3z4z5w10.98ei1.98π,0.01ei0.024π,0.02ei0.06π0.99ei1.94π,0.012ei0.02π,0.1ei0.26π0.97ei1.96π,0.013ei0.02π,0.05ei0.08π0.84ei1.66π,0.019ei0.04π,0.29e0.6iπ0.98ei1.92π,0.012ei0.026π,0.03e0.01iπw20.84ei1.66π,0.02ei0.042π,0.29ei0.62π0.37ei0.78π,0.015ei0.026π,0.87ei1.72π0.16ei0.34π,0.018ei0.038π,0.89e1.78π0.96ei1.94π,0.013ei0.024π,0.04e0.06iπ0.36ei0.74π,0.016ei0.03π,0.59e1.2iπw30.17ei0.36π,0.02ei0.038π,0.91ei1.84π0.19ei0.36π,0.021ei0.04π,0.93e1.82π0.59ei0.12π,0.0155ei0.032π,0.82e1.6π0.2ei0.38π,0.022ei0.042π,0.92e1.86iπ0.21ei0.44π,0.024ei0.05π,0.93e1.88iπw40.58ei1.12π,0.015ei0.032π,0.82ei1.66π0.66ei1.34π,0.1ei0.18π,0.56ei1.14π0.67ei1.36π,0.021ei0.044π,0.31e0.64π0.55ei1.14π,0.014ei0.028π,0.8e1.62iπ0.67ei1.38π,0.025ei0.048π,0.33e0.68iπw50.52ei1.02π,0.012ei0.026π,0.74ei1.5π0.21ei0.4π,0.021ei0.046π,0.95ei1.88π0.7ei1.36π,0.026ei0.05π,0.35ei0.68π0.16ei0.42π,0.018ei0.038π,0.89e1.88iπ0.58ei1.14π,0.01ei0.024π,0.82e1.62iπTable 42
Tabular representation ofCSFDM of expert E˜2.
z1z2z3z4z5w10.83ei1.64π,0.019ei0.036π,0.3ei0.58π0.98ei1.94π,0.013ei0.024π,0.1ei0.26π0.98ei1.98π,0.01ei0.024π,0.02ei0.06π0.84ei1.66π,0.02ei0.036π,0.31e0.63iπ0.8ei1.62π,0.022ei0.042π,0.32e0.6iπw20.71ei1.38π,0.028ei0.058π,0.37ei0.72π0.72ei1.42π,0.03ei0.062π,0.38ei0.76π0.57ei1.1π,0.012ei0.028π,0.81e1.6π0.86ei1.74π,0.0169ei0.032π,0.027e0.052iπ2,0.56ei1.1π,0.015ei0.07π,0.8e1.58iπw30.05ei0.012π,0.01ei0.06π,0.985ei1.972π0.21ei0.46π,0.024ei0.52π,0.94e1.86π0.5ei1.02π,0.01ei0.02π,0.7e1.4π0.73ei1.48π,0.032ei0.062π,0.39e0.76iπ2,0.55ei1.08π,0.01ei0.02π,0.79e1.56iπw40.74ei1.5π,0.035ei0.068π,0.4ei0.78π0.54ei1.04π,0.015ei0.026π,0.78ei1.54π1,0.24ei0.44π,0.03ei0.056π,0.94e1.84π0.71ei1.38π,0.028ei0.058π,0.37e0.72iπ2,0.53ei1.04π,0.013ei0.026π,0.77e1.52iπw50.24ei0.42π,0.028ei0.062π,0.91ei1.84π0.51ei1.04π,0.013ei0.06π,0.76ei1.5π0.52ei1.02π,0.013ei0.06π,0.75ei1.52π0.23ei0.5π,0.035ei0.066π,0.89e1.82iπ0.1ei0.24π,0.012ei0.026π,0.99e1.972iπTable 43
Tabular representation ofCSFDM of expert E˜3.
z1z2z3z4z5w10.84ei1.66π,0.019ei0.04π,0.29ei0.6π0.97ei1.98π,0.013ei0.028π,0.14ei0.26π0.96ei1.98π,0.01ei0.04π,0.04ei0.04π0.5ei1.02π,0.014ei0.026π,0.74e1.46iπ0.8ei1.62π,0.018ei0.038π,0.29e0.6iπw20.72ei1.46π,0.036ei0.068π,0.41ei0.8π0.49eiπ,0.013ei0.028π,0.71ei1.44π0.48ei0.98π,0.015ei0.032π,0.7e1.36π0.74ei1.44π,0.034ei0.07π,0.42e0.82iπ0.75ei1.48π,0.035ei0.072π,0.43e0.82iπw30.47ei0.96π,0.015ei0.032π,0.7ei1.36π0.27ei0.5π,0.034ei0.066π,0.94e1.92π0.46ei0.94π,0.014ei0.03π,0.68e1.38π0.88ei1.78π,0.015ei0.032π,0.25e0.46iπ0.29ei0.56π,0.04ei0.078π,0.96e1.9iπw40.98ei1.94π,0.013ei0.06π,0.1ei0.26π0.76ei1.5π,0.036ei0.074π,0.44ei0.86π0.13ei0.6π,0.014ei0.03π,0.986e1.97π0.77ei1.52π,0.038ei0.078π,0.45e0.88iπ0.3ei0.58π,0.042ei0.082π,0.925e1.86iπw50.44ei0.92π,0.012ei0.028π,0.66ei1.34π0.43ei0.88π,0.015ei0.028π,0.66ei1.34π0.87ei1.78π,0.016ei0.028π,0.21ei0.43π0.42ei0.86π,0.012ei0.02π,0.65e1.32iπ0.78ei1.54π,0.04ei0.076π,0.46e0.9iπTable 44
Tabular representation ofCSFDM of expert E˜4.
z1z2z3z4z5w10.95ei1.94π,0.012ei0.02π,0.13ei0.2π0.8ei1.62π,0.018ei0.038π,0.31ei0.6π0.97ei1.96π,0.01ei0.02π,0.15ei0.32π0.63ei1.28π,0.012ei0.02π,0.6e1.18iπ0.64ei1.26π,0.01ei0.024pi,0.59e1.2iπw20.77ei1.56π,0.041ei0.08π,0.47ei0.92π0.31ei0.64π,0.022ei0.04π,0.89ei1.8π0.06ei0.14π,0.012ei0.02π,0.986e1.97π0.8ei1.58π,0.05ei0.102π,0.48e0.94iπ0.37ei1.78π,0.014ei0.024π,0.86e1.74iπw30.3ei0.64π,0.05ei0.012π,0.91ei1.84π0.41ei0.84π,0.01ei0.06π,0.64e1.3π0.4ei0.84π,0.015ei0.032π,0.63e1.28π0.39ei0.8π,0.014ei0.03π,0.62e1.26iπ1,0.33ei0.64π,0.07ei0.1π,0.93e1.88iπw40.38ei0.78π,0.013ei0.07π,0.61ei1.24π0.9ei1.84π,0.014ei0.03π,0.19ei0.36π0.37ei0.76π,0.012ei0.026π,0.6e1.22π0.81ei1.6π,0.051ei0.104π,0.49eiπ0.13ei0.28π,0.014ei0.026π,0.982e1.966iπw50.07ei0.012π,0.016ei0.03π,0.991ei1.984π0.34ei0.64π,0.065ei0.132π,0.95ei1.92π0.3ei0.58π,0.014ei0.03π,0.93ei1.88π0.36ei0.74π,0.01ei0.024π,0.59e1.2iπ0.21ei0.46π,0.024ei0.052π,0.94e1.86iπTable 45
Tabular representation ofACSFDM.
z1z2z3z4z5w10.95ei1.94π,0.013ei0.026π,0.07ei0.18π0.97ei1.9π,0.013ei0.026π,0.15ei0.34π0.97ei1.96π,0.01ei0.02π,0.05ei0.1π0.77ei1.54π,0.016ei0.03π,0.4e0.82iπ0.91ei1.76π,0.013ei0.03π,0.15e0.36iπw20.79ei1.56π,0.028ei0.058π,0.36ei0.74π0.48ei0.96π,0.019ei0.034π,0.73ei1.46π0.33ei0.66π,0.01ei0.02π,0.88e1.76π0.9ei1.82π,0.02ei0.04π,0.15e0.26iπ0.47ei0.96π,0.016ei0.03π,0.68e1.36iπw30.25ei0.52π,0.022ei0.048π,0.9ei1.74π0.29ei0.58π,0.018ei0.038π,0.83e1.66π0.5ei1.04π,0.013ei0.028π,0.72e1.42π0.55ei1.12π,0.029ei0.038π,0.6e1.2iπ0.35ei0.7π,0.029ei0.054π,0.9e1.82iπw40.7ei1.18π,0.016ei0.034π,0.63ei1.28π0.77ei1.56π,0.034ei0.064π,0.42ei0.84π0.51ei1.02π,0.01ei0.02π,0.53e1.06π0.71ei1.4π,0.026ei0.054π,0.55e1.12iπ0.52ei1.06π,0.019ei0.038π,0.6e1.22iπw50.39ei0.74π,0.015ei0.032π,0.83ei1.68π0.35ei0.7π,0.025ei0.052π,0.87ei1.74π0.63ei1.26π,0.02ei0.034π,0.51ei1.02π0.28ei0.6π,0.016ei0.034π,0.76e1.58iπ0.49ei0.96π,0.015ei0.034π,0.83e1.6iπTable 46
Experts opinion related to each attribute.
E˜1E˜2E˜3E˜4z10.85ei1.66π,0.019ei0.042π,0.29ei0.6π0.99ei1.94π,0.012ei0.02π,0.1ei0.26π0.88ei1.78π,0.015ei0.032π,0.25ei0.46π0.67ei1.36π,0.021ei0.044π,0.31e0.64iπz20.66ei1.34π,0.021ei0.044π,0.31ei0.64π0.96ei1.94π,0.013ei0.024π,0.04ei0.06π0.78ei1.54π,0.04ei0.076π,0.46e0.9π0.98ei1.98π,0.01ei0.024π,0.02e0.06iπz30.63ei1.28π,0.012ei0.02π,0.59ei1.2π0.72ei1.42π,0.03ei0.062π,0.38e0.78π0.75ei1.48π,0.035ei0.072π,0.43e0.82π0.42ei0.86π,0.012ei0.02π,0.65ei1.32πz40.29ei0.6π,0.014ei0.03π,0.94ei1.86π0.21ei0.46π,0.024ei0.052π,0.94ei1.86π0.37ei0.76π,0.012ei0.026π,0.6e1.22π0.78ei1.54π,0.04ei0.076π,0.46e0.9iπz50.62ei1.28π,0.01ei0.024π,0.58ei1.2π0.66ei1.34π,0.1ei0.18π,0.56ei1.14π0.64ei1.26π,0.01ei0.024π,0.59ei1.2π0.21ei0.4π,0.021ei0.046π,0.95e1.88iπTable 47
Tabular representation ofAWCSFDM.
z1z2z3z4z5w10.85ei1.66π,0.021ei0.042π,0.23ei0.52π0.88ei1.76π,0.021ei0.042π,0.174ei0.38π0.601ei1.22π,0.019ei0.034π,0.053ei1.1π0.41ei0,82π,0.025ei0.05π,0.77e1.54iπ0.5eiπ,0.023ei0.05π,0.67e1.36iπw20.7ei1.34π,0.03ei0.066π,0.41ei0.86π0.43ei0.9π,0.025ei0.048π,0.73ei1.46π0.2ei0.4π,0.019ei0.034π,0.91e1.82π0.47ei0.96π,0.028ei0.056π,0.73e1.42iπ0.25ei0.54π,0.025ei0.05π,0.83e1.66iπw30.22ei0.44π,0.027ei0.058π,0.91ei1.74π0.26ei0.54π,0.024ei0.025π,0.83e1.66π0.32ei0.64π,0.021ei0.038π,0.8e1.62π0.29ei0.58π,0.035ei0.068π,0.73e1.42iπ0.19ei0.34π,0.035ei0.068π,0.95e1.9iπw40.62ei1.02π,0.023ei0.048π,0.65ei1.34π0.70ei1.46π,0.037ei0.072π,0.43ei0.86π0.32ei0.64π,0.019ei0.034π,0.69e1.42π0.37ei0.74π,0.033ei0.066π,0.82e1.62iπ0.29ei0.6π,0.027ei0.056π,0.8e1.62iπw52,0.35ei0.62π,0.022ei0.046π,0.83ei1.7π0.31ei0.66π,0.029ei0.06π,0.87ei1.74π0.39ei0.78π,0.025ei0.044π,0.68ei1.38π0.15ei0.318π,0.025ei0.052π,0.89e1.8iπ0.27ei0.54π,0.024ei0.042π,0.91e1.82iπTable 48
Tabular representation of CSF-PIS and CSF-NIS.
AttributeCSF-PISCSFN-NISz10.85ei1.66π,0.021ei0.042π,0.23ei0.52π0.22ei0.44π,0.027ei0.058π,0.91ei1.74πz20.88ei1.76π,0.021ei0.042π,0.174ei0.038π0.31ei0.66π,0.029ei0.06π,0.87ei1.74πz30.601ei1.22π,0.019ei0.034π,0.53ei1.1π0.2ei0.4π,0.019ei0.034π,0.91e1.82πz40.47ei0.96π,0.028ei0.056π,0.73e1.42π0.15ei0.318π,0.025ei0.052π,0.89e1.8πz50.19ei0.38π,0.035ei0.068π,0.95ei1.9π0.5eiπ,0.023ei0.046π,0.67ei1.36πTable 49
Normalized Euclidean distance from ideal solution.
AlternativedP^k,wjdPˇk,wjw10.1829470.565764w20.3838390.330875w30.53884220.200388w40.2663700.349150w50.5104550.208933Table 50
Revised closeness index of each alternative.
Alternativeℑwjw10w21.5132w32.5911w40.838853w52.42088Table 51
Comparison.
MethodRankingBest physiotherapistCSFNSf-TOPSISw1>w4>w2>w5>w3w1CSF-TOPSIS [16]w1>w4>w2>w5>w3w1SF-TOPSIS [10]w1>w4>w2>w5>w3w1SF-TOPSIS [11]w1>w4>w2>w5>w3w1


### 6.1.1. Discussion

(1)
We now compare the proposed modelCSFNSf-TOPSIS method with the existing technique CSF-TOPSIS method to evaluate the accuracy of the result. The same result concludes from both methods as well as the ranking of the alternatives also same.(2)
We also apply technique on SF-TOPSIS methods [10, 11], to select the most appropriate physiotherapist. The same results including the ranking and best solution are organized in Table 51, which enhance the credibility of the proposed method.(3)
The proposed model deals not only with 2-dimensional uncertainties but also with the level of attribute for the alternative. The existing models are unable to handle MAGDM problems, but the proposed model has the ability to tackle those real-life DM problems having ranking system and parameterized information.(4)
The proposed model,CSFNSf-TOPSIS method, could be efficiently applied to the environment of SFNSf, CSFSf, and SFSf, by substituting periodic terms equal to zero and N=2.


## 6.1.1. Discussion

(1)
We now compare the proposed modelCSFNSf-TOPSIS method with the existing technique CSF-TOPSIS method to evaluate the accuracy of the result. The same result concludes from both methods as well as the ranking of the alternatives also same.(2)
We also apply technique on SF-TOPSIS methods [10, 11], to select the most appropriate physiotherapist. The same results including the ranking and best solution are organized in Table 51, which enhance the credibility of the proposed method.(3)
The proposed model deals not only with 2-dimensional uncertainties but also with the level of attribute for the alternative. The existing models are unable to handle MAGDM problems, but the proposed model has the ability to tackle those real-life DM problems having ranking system and parameterized information.(4)
The proposed model,CSFNSf-TOPSIS method, could be efficiently applied to the environment of SFNSf, CSFSf, and SFSf, by substituting periodic terms equal to zero and N=2.


## 7. Conclusions

Complex spherical fuzzyN-soft sets (CSFNSfS) broaden the families of both fuzzy sets and N-soft sets. This novel concept has allowed us to propose techniques in a wide environment that have a large ability of solving real-life MADM and MAGDM problems. The model of CSFNSfS described in this paper copes with 2-dimensional fuzziness, parameterized information, and ordinal ranking systems. In addition to the notion of CSFNSfS, we have defined score and accuracy functions for the purpose of comparing two CSFNSfNs. We have defined useful operations on CSFNSfS and given relevant examples. We developed three direct algorithms and, furthermore, a CSFNSf-TOPSIS Method to solve decision-making problems. We compared them with existing methods for Pythagorean fuzzy N-soft sets and with the complex spherical fuzzy TOPSIS Method, respectively. For the purpose of extending the theoretical background of TOPSIS methods to the new CSFNSf-TOPSIS method, we have defined complex spherical fuzzy N-soft weighted averaging operator CSFNSfWA which produces an aggregate complex spherical fuzzy N-soft decision matrix ACSFNSfDM and aggregates the weight vectors of attributes given by experts. Similarly, we have defined a normalized Euclidean distance in CSFNSf environment that simultaneously evaluates the distances of alternatives from CSFNSf-PIS and CSFNSf-NIS. This is required to find a revised closeness index. The ascending order of such an index gives us a ranking of the alternatives, where the smallest revised closeness index indicates a best solution. In the future, we intend to pursue the formalization of other methods (ELECTRE I, II, and III and VIKOR methodologies), under the framework of CSFNSf. We can also extend this theory to accommodate T-spherical fuzzy soft sets, T-spherical fuzzy N-soft sets, and complex T-spherical fuzzy N-soft sets.



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