# Complex Spherical Fuzzy Decision Support System Based on Entropy Measure and Power Operator

Authors: Muhammad Naeem; Muhammad Qiyas; Thongchi Botmart; Saleem Abdullah; Neelam Khan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The objectives of this paper are to define novel aggregation operators (AOs) for aggregating different complex spherical fuzzy numbers (CSFNs) under the influence of their membership grades. The uncertainties included in the information are dealt with in contemporary studies of the fuzzy set and its extensions by membership grades, which are a subset of real numbers that lose some relevant information and hence alter the decision results. The conversion to these complex spherical fuzzy sets addresses the classes’ uncertainty, whose ranges differ from the specific subset of the complex subset of the unit disk. For this purpose, we defined new CSF power AOs. Some of the desirable properties of these operators have also been investigated. A multiattribute group decision-making (MAGDM) approach is implemented in the structure developed by the CSFNs on the basis of these operators. A numerical example concerning the selection of the best alternatives is given to demonstrate the effectiveness of the defined method and is tested by comparing its results with the various methods.


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## Body

## 1. Introduction

Zadeh [1] defined fuzzy set (FS) theory in 1965 FS is a remarkable achievement having applications in a wide range of industries. An FS is focused on a characteristic function with a membership degree for each number of the X (universal set) on the closed interval from 0 to 1 for each number of the X (universal set) on the closed interval from 0 to 1. Some FS extensions already exist, including Zadeh’s [2] interval-valued fuzzy set. Coupland and John [3] also invented a geometric type-1 and type-2 fuzzy logic systems, which they used in real-world decision-making (DM) situations. Since fuzzy sets have various advantages, as previously stated, there are some situations in which relying solely on the membership function to solve the problem is difficult or impossible.Atanassov’s [4] definition of intuitionistic fuzzy set was developed to overcome this problem (IFS). An IFS has two functions, positive and negative membership grades indicated by μ and ν, for each element of the universal set X of the closed-interval from 0 to 1. Additionally, the sum of their values must fall inside the range 0,1, i.e., sum μ,ν∈0,1. It is fine to use μ=0.7 and ν=0.4, respectively. In IFS, we know that the values of its characteristic functions cannot change on their own. Xue and Deng [5] developed the conjunction of possibility measures under IF sets. Garg and Arora [6] proposed a new MADM approach based on nonlinear programming (NLP), TOPSIS method, and interval-valued intuitionistic fuzzy values. Furthermore, this concept has received a lot of attention and has been successfully employed in problems involving mathematics, engineering, information sciences, and MADM [6–8]. Chen et al. [9] defined an expertise-based bid evaluation for construction-contractor selection with generalized comparative linguistic ELECTRE III. Chen et al. [10] suggested a large-scale group DM based on online-review analysis for determining passenger requests and assessing passenger satisfaction: a case study of China’s high-speed rail system. Yager [11] IFNs have a BM aggregation operator that combines the concept of the Bonferroni mean (BM) with IFS. Verma [12] proposed new generalized BM operators for IFNs. Kaur and Garg [13] suggest a BM operator in a cubic IFS system. Li et al. [14] proposed the generalized BM operators for IFNs. Garg and Arora [15] are for intuitionistic fuzzy soft set theory. BM operators were seen. The main advantage of the BM operator is that it can analyze two conflicts at once; therefore, it is better to estimate the analysis rather than use operational AOs.The criteria that their sum must be in0,1 cannot be met when a decision maker provides such values for membership grade and nonmembership grade. This condition is difficult for IFS to describe. Yager and Abbasov [16] devised the Pythagorean fuzzy set (PyFS) to address this issue by broadening the scope of IFS. A Pythagorean fuzzy set (PyFS) has two functions, positive and negative membership degree and for each element of the set X (universal set), sum μ2,ν2∈0,1 is the sum of squares of the numbers belongs to the closed interval 0,1. PyFS is thus a generalization of IFS, as PyFS has a larger scope than IFS. Xue and Deng [17] proposed revised expected value decision rules within the orthopair fuzzy framework. Gao et al. [18] defined quantum PyF evidence theory is a theory that is characterized as quantum Pythagorean fuzzy evidence. Gao and Deng [19] built a quantum mass function model. Pan et al. [20] proposed a PyFS with a constrained similarity measure. Khan et al. [21] discussed new Pythagorean fuzzy Dombi AOs and their application in decision support system.Despite the fact that IFSs and PyFSs can accurately express uncertain information, there are still issues that IFSs and PyFSs cannot solve. When a decision maker gives a membership degree of0.6 and a nonmembership degree of 0.9, they have not satisfied the PyFs requirement of 0.36+0.81=1.08>1. So, to cope with complicated and uncertain information in the context of fuzzy set theory, Yager [22] created the notion of q-rung orthopair fuzzy set (q-ROFS), which is more powerful and general than IFS and PyFS. Liu and Wang [23] have proposed the q-ROF aggregation operators for averaging assessment data. Liu and Liu [24] studied the q-ROF Bonferroni mean operators for q-ROFS information. The exponential operations and aggregation operator for q-ROFS were introduced by Peng et al. [25], and further researches for the q-ROFS were also produced [26–28].In the case of IFS, we know that human opinion is limited to yes or no phenomena. Cuong [29] introduced the concept of the picture fuzzy set (PFS) and explained its basic operation and properties as a result of this limitation. In this situation, human opinion is divided into four categories: yes, no, abstinence, and denial. A PFS has three categories of functions: membership, abstinence, and nonmembership grades, denoted by μ,η, and ν, and for each element of the set X of the closed-interval 0,1. Their sum must be in the closed interval 0,1 in the PFS, i.e., sum μ,η,ν∈0,1. As a result, it was demonstrated that PFS is a simple generalization of FS and IFS. Khan et al. [30] proposed some 2-tuple picture fuzzy linguistic AOs and used for the analysis of robot selection. Qiyas et al. [31] extended the concept of Yager operators for PFS and proposed some AOs.Cuong expanded Zadeh’s FS definition and Atanassov’s IFS concept, but there is still a drawback to this framework, such as if we takeμ=0.5,η=0.3, and ν=0.4, respectively, it is enough. Since, we know that the values of its characteristic functions cannot be allowed to change independently in PFS. As a result of this restriction, Mahmood et al. [32] specified SFS by expanding the scope of PFS. For each element of the universal set X on the closed-interval 0,1, an SFS has three functions: positive, abstinence, and negative grades, denoted by, and for each element of the universal set X of the closed-interval 0,1. The sum of squares that is, sum μ2,η2,ν2∈0,1, and belongs to the closed-interval 0,1. As a result, since the domain of SFS is greater than that of PFS, it is the generalization of PFS. However, in times when taking μ=0.6,η=0.7, and ν=0.8 is sufficient since the number of their squares is greater than 1. For some related work on SFS, we may refer to [33, 34]. Abdullah et al. [35] defined some AOs based on 2-tuple spherical fuzzy linguistic numbers. Qiyas and Abdullah [36] defined sine trigonometric spherical fuzzy AOs and discussed their application in DM.Inspire from the performance of the power operator, several power aggregation operators (PAOs) are introduced for different fuzzy information, such as the generalized PAOs [37], intuitionistic fuzzy power AO [38], the interval-valued intuitionistic fuzzy power AO [39], Pythagorean fuzzy power AO [40], neutrosophic fuzzy power AO [41, 42], and hesitant fuzzy PAO [43]. Garg and Rani [44, 45] proposed weighted and powerful AOs to solve the multicriteria decision making (MCDM) problem. Zhou et al. suggested generalized PAOs [46], and Zhou and Chen also introduced linguistic generalized power AOs [47]. Xu [48] has defined the intuitionistic fuzzy power aggregation operators and interval-valued intuitionistic fuzzy power AOs. In addition, Xu and Cai [49] established an uncertain weighted power ordered averaging (UPOWA) operator on the basis of the PA (power average) operator and the UOWA operator. Xu and Yager [50] have implemented an uncertain ordered weighted geometric (UOWG) operator using the PG (power geometric) operator and the UOW averaging operator. The PA operator has been widely used in a variety of disciplines, including software quality evaluation [51], MAGDM [52], and green product development [53]. Chen et al. [54] defined power-average-operator-based hybrid multiattribute online product recommendation model for consumer decision-making.It should be remembered that there are other scholars, like as Buckly [55], Zhang et al. [56], and Nguyen et al. [57] who mixed complex numbers and fuzzy sets. Further, Ramot et al. [58] defined a new notion that is somewhat different from other studies, in which they expanded the range of positive membership function of the unit circle in the complex plane, unlike the others that were limited to, and the CFS paradigm, which is a generalization of Zadah fuzzy sets (FSs). The CFS is represented by μIx=rIx.e2πi✠rIx and satisfied the condition: 0≤rIx,✠rIx≤1.The distinction between CFSs and FSs is that the CFS range is not limited to0,1, but spread in a complex plan to a unit disk. In the FS theory, setting the CFSs has earned more attention. Yazdanbakhsh and Dick [59] have suggested the time series forecasting utilizing the CF logic and a systematic analysis of CFSs. Bi et al. [60] recently proposed complex fuzzy geometrical AOs. Due to its merits and benefits, CFSs have been extensively applied to problems in DM and other fields [61]. Since FSs and CFSs can define only the grade of positive membership function and their complex-valued grade and cannot express the grade of negative membership and complex-valued negative membership degree. Dick et al. [62] discussed several CFSs, also Liu and Zhang [63] developed the results of Dick et al. [62]. Greenfield et al. [64] suggested a novel definition of complex interval-valued fuzzy set (CIVFS) that is definitely the development of the concept of CFSs and extended the interval-valued fuzzy sets paradigm.However, in some theories like as FSs [1], IFSs [4], complex fuzzy sets [58], and complex interval-evaluated intuitionistic fuzzy set [65] are generally used to treat data imprecision. Each element in these type sets is represented an ordered pair of positive grade and negative grade with the ultimate aim of limiting their total to one [66]. Kumar and Bajaj [67] defined some distance measures for CIF soft sets. Singh et al. [68] proposed the interval-valued lattices of CFS and their granular decompositions. Selvachandran et al. [69] developed and analyzed the idea of a complex vague soft set and the entropy measures. Hu et al. [70] suggested new distance steps for the CFS and investigated the consistency of CFS operations. Selvachandran et al. [71] suggest several similarity tests of CFS, and their applications are studied in pattern recognition. In [72], Quek et al. researched group-associated CIFS algebraic structures, while [73] addressed the application of CFS in e-commerce. The idea of CF lattice and CF interval-valued soft set is discussed in [74, 75], respectively. Garg and Rani [76, 77] established new generalized Bonferroni mean (BM) operators and robust averaging and geometric AOs for CIFS. Using Heronian mean operators, Liu et al. [78] developed a method for the MAGDM problem under the Cq-ROF linguistic information. AOs for complex q-rung orthopair fuzzy set (Cq-ROFS) were defined by Liu et al. [79], and their applications in MCGDM were studied. Garg et al. [80] explored the novel Cq-ROFS approach for dealing with challenging and sophisticated material, which is a combination of q-ROFS and CFS. Yang et al. [81] defined some complex q-rung orthopair uncertain linguistic partitioned Bonferroni mean operators and their application in antivirus mask selection. Liu et al. [82] defined a new group decision-making approach based on Cq-ROF Bonferroni mean operator. Ali et al. [83] proposed TOPSIS method using complex spherical fuzzy sets (CSFSs) with Bonferroni mean operators. Akram et al. [84] developed a group decision-making based on the CSFSs VIKOR method. Akram et al. [85] defined a hybrid decision-making framework under CSF prioritized weighted aggregation operators.In this article, we present the theory of PA operators among the CSFS, maintaining the benefits of this collection and focusing on the value of aggregation operators. Based on our knowledge, operators cannot be used in the aforementioned studies to manage CSFS information. In order to do this, we first describe certain operating laws between pairs of CSFS and analyze their properties. Next, some PA operators called CSF power average, CSF power geometric, CSF weighted power average, and CSF weighted power geometric as well as their respective ordered weighted operators are proposed to aggregate the various complex spherical fuzzy numbers (CSFNs). The basic properties of these operators will be discussed in detail. In addition, we suggest a MAGDM approach using the proposed operators. Both the viability and the effectiveness of the strategy have been illustrated by an illustrative example of real-life problem (green suppler selection). The advantages and comparative analysis of the established work are also discussed.The remainder of the manuscript is listed accordingly. In Section2, we summarize briefly the definitions of IFSs, PyFSs, CFSs, and CPFSs. In Section 3, we define some simple operational laws for CSFNs. In Section 4, we proposed some series of averaging AOs based on the defined operational laws. In Section 5, we proposed some series of geometric AOs based on the defined operational laws. In Section 5, we proposed an MAGDM algorithm based on the developed operators with the CSFS information, where CSFNs are characterized by each element of the set. An illustrative example is described in Section 6 to show the functionality of the suggested method and compare its results with some of the current results of the approaches, and finally this analysis is summarized in Section 7.


## 2. Preliminaries

In this portion, we present along with some other concepts, a short literature survey of preexisting notions like as IFS, PyFS, SPS, CFS, CIFS, and CPyFS.Definition 1 (see [4]).
LetX be a universal set. Then, a IFS I=x,μIx,νIxx∈X, where μIx and νIx show the degree of positive and negative membership, correspondingly, provided that 0≤sumμIx,νIx≤1. Further, πIx=1−sumμIx,νIx is referred to as the refusal degree of x∈X in I.Definition 2 (see [16]).
LetX be a universal set. Then, a PyFS P=x,μPx,νPxx∈X, where μPx and νPx show the degree of positive and negative membership correspondingly, provided that 0≤sumμP2x,νP2x≤1. Further, πPx=1−sumμP2x,νP2x is referred as the refusal degree of x∈X in P.Definition 3 (see [86]).
LetX be a universal set. Then, a SFS S=x,μSx,ηSx,νSxx∈X, where μSx,ηSx, and νSx show the degree of positive, neutral, and negative membership, respectively, provided that 0≤sumμS2x,ηS2x,νS2x≤1. Further, πSx=1−sumμS2x,ηS2x,νS2x is referred to as the refusal degree of x∈X in S.Definition 4 (see [58]).
LetX be a universal set. Then, a CFS C=x,μCxx∈X, where μC:U⟶z:z∈C,z≤1 and μCx=a+ib=κCx.e2πi✠Cx. Here, κCx=a2+b2∈R and κCx,✠Cx∈0,1, where i=−1.Definition 5 (see [61]).
LetX be a universal set. Then, a CIFS I=x,μIx,νIxx∈X, where μI:U⟶z1:z1∈I,z1≤1,νI:U⟶z2:z2∈I,z2≤1 such that μIx=z1=a1+ib1 and νIx=z2=a2+ib2 provided that 0≤z1+z2≤1orμIx=κIx.e2πi✠κIx and νix=ξix.e2πi✠ξIx satisfying the conditions; 0≤κIx+ξIx≤1 and 0≤✠κIx+✠ξIx≤1. The term HIx=R.e2πi✠R, such that R=1−z1+z2 and ✠Rx=1−✠κIx+✠ξIx are considered as hesitancy degree of x. Furthermore, I=κ.e2πi✠κ,ξ.e2πi✠ξ is referred to as the complex intuitionistic fuzzy number (CIFN).Definition 6 (see [86]).
LetX be a universal set. Then, a CPyFS P=x,μPx,νPxx∈X, where μP:U⟶z1:z1∈P,z1≤1,νP:U⟶z2:z2∈P,z2≤1, such that μPx=z1=a1+ib1 and νPx=z2=a2+ib2 provided that 0≤z12+z22≤1orμPx=κPx.e2πi✠κPx and νPx=ξPx.e2πi✠ξPx satisfying the conditions, 0≤κP2x+ξP2x≤1 and 0≤✠κPx2+✠ξPx2≤1. The term HPx=R.e2πi✠R, such that R=1−κP2x+ξP2x and ✠Rx=1−✠κPx2+✠ξPx2 is considered as a hesitancy degree of x. Furthermore, P=κ.e2πi✠κ,ξ.e2πi✠ξ is referred as complex Pythagorean fuzzy number (CPyFN). CPyFS positive and negative membership degree is clearly polar or Cartesian numbers in complex form. The following two types of concepts are inter convertible;
(1)μPx=κPx.e2πi✠κPx=κPxcos2π✠κPx+isin2π✠κPx=a1+ib1=z1,(2)νPx=ξPx.e2πi✠ξPx=ξPxcos2π✠ξPx+isin2π✠ξPx=a2+ib2=z2.Definition 7 (see [37]).
For a family ofψii=1,⋯,n, the power averaging operator is defined as
(3)PAψ1,⋯,ψn=∑i=1n1+Tψiψi∑i=1n1+Tψi,where Tψi=∑i=1nSupψi,ψs and Supψi,ψs are the support of ψi from ψs called as similarity index and satisfies the following properties;
(1)
Supψi,ψs∈0,1(2)
Supψi,ψs=Supψs,ψi(3)
Supψi,ψs≥Supψk,ψl, if ψi−ψs≤ψk−ψl


## 3. Complex Spherical Fuzzy Set

In this portion, we introduce CSFS and their basic operation laws for the collection of CSFNs and their corresponding PA operators.Definition 8 (see [83]).
LetX be a universal set. Then, a CSFS S=x,μSx,ηSx,νSxx∈X, where μS:U⟶z1:z1∈S,z1≤1,ηS:U⟶z2:z2∈S,z2≤1,νS:U⟶z3:z3∈S,z3≤1 such that μSx=z1=a1+ib1,ηSx=z2=a2+ib2 and νSx=z3=a3+ib3 provided that 0≤z12+z22+z32≤1orμSx=κSx.e2πi✠κSx,ηSx=δSx.e2πi✠δSx and νSx=ξSx.e2πi✠ξSx satisfying the conditions: 0≤κS2x+δS2x+ξS2x≤1 and 0≤✠κSx2+✠δSx2+✠ξSx2≤1. The term HSx=R.e2πi✠R, such that R=1−κS2x+δS2x+ξS2x and ✠Rx=1−✠κSx2+✠δSx2+✠ξSx2 are considered as a hesitancy degree of x. Furthermore, S=κ.e2πi✠κ,δ.e2πi✠δ,ξ.e2πi✠ξ is referred as complex spherical fuzzy number (CSFN). The positive, neutral, and negative membership degrees are clearly polar or Cartesian numbers in complex form. The following three types of notations are interconvertible;
(4)μSx=κSx.e2πi✠κSx=κSxcos2π✠κSx+isin2π✠κSx=a1+ib1=z1,(5)ηSx=δSx.e2πi✠δSx=δSxcos2π✠δSx+isin2π✠δSx=a2+ib2=z2,(6)νSx=ξSx.e2πi✠ξSx=ξSxcos2π✠ξSx+isin2π✠ξSx=a3+ib3=z3.


### 3.1. Operational Laws of CSFNs

In this subsection, we defined the score and accuracy function. Also, the operational laws of the CSFNs are defined.Definition 9.
For any CSFNR=κR,✠κR,δR,✠δR,ξR,✠ξR, the score function Sc is defined as
(7)ScR=κR2+δR2−ξR2+12π✠κR2+✠δR2−✠ξR2,and accuracy function Ac is defined as
(8)AcR=κR2+δR2+ξR2+12π✠κR2+✠δR2+✠ξR2,where ScR∈−2,2 and AcR∈0,2.Definition 10.
For two CSFNsR1=κR1,✠κR1,δR1,✠δR1,ξR1,✠ξR1 and R2=κR2,✠κR2,δR2,✠δR2,ξR2,✠ξR2, the distance measure between them is defined as
(9)dR1,R2=16κR1−κR2+δR1−δR2+ξR1−ξR2+12π✠κR1−✠κR2+✠δR1−✠δR2+✠ξR1−✠ξR2.Definition 11.
The following comparison rules between two CSFNsR1 and R2 are defined as
(1)
IfScR1>ScR2, then R1>R2; similarly, if AcR1>AcR2, then R1>R2(2)
IfScR1=ScR2, then R1=R2; similarly, if AcR1=AcR2, then R1=R2Definition 12.
For two CSFNsR1=κR1,✠κR1,δR1,✠δR1,ξR1,✠ξR1 and R2=κR2,✠κR2,δR2,✠δR2,ξR2,✠ξR2 and λ (positive real number). Then, the basic operation laws are defined as
(1)
R1⊕R2=1−∏i=121−κRi2,2π1−∏i=121−✠κRi2/2π,∏i=12δRi,2π∏i=12✠δRi/2π,∏i=12ξRi,2π∏i=12✠ξRi/2π(2)
R1⊗R2=∏i=12κRi,2π∏i=12✠κRi/2π,1−∏i=121−δRi2,2π1−∏i=121−✠δRi2/2π,1−∏i=121−ξRi2,2π1−∏i=121−✠ξRi2/2π(3)
λR1=1−1−κR12λ,2π1−1−✠κR12/2πλ,δR1λ,2π✠δR1/2πλ,ξR1λ,2π✠ξR1/2πλ(4)
R1λ=κR1λ,2π✠κR1/2πλ,1−1−δR12λ,2π1−1−✠δR12/2πλ,1−1−ξR12λ,2π1−1−✠ξR12/2πλTheorem 13.
For any three CSFNsR1,R2,R3, we have
(1)
R1⊕R2=R2⊕R1(2)
R1⊗R2=R2⊗R1(3)
R1⊕R2⊕R3=R1⊕R2⊕R3(4)
R1⊗R2⊗R3=R1⊗R2⊗R3Proof.
For proof, see AppendixA.Theorem 14.
For any two CSFNsR1 and R2 and a positive real numbers λ,λ1,λ2, we have
(1)
λR1⊕R2=λR1⊕λR2(2)
R1⊗R2λ=R1λ⊕R2λ(3)
λ1+λ2R1=λ1R1⊕λ2R1(4)
R1λ1+λ2=R1λ1⊗R1λ2Proof.
For proof, see AppendixB.


## 3.1. Operational Laws of CSFNs

In this subsection, we defined the score and accuracy function. Also, the operational laws of the CSFNs are defined.Definition 9.
For any CSFNR=κR,✠κR,δR,✠δR,ξR,✠ξR, the score function Sc is defined as
(7)ScR=κR2+δR2−ξR2+12π✠κR2+✠δR2−✠ξR2,and accuracy function Ac is defined as
(8)AcR=κR2+δR2+ξR2+12π✠κR2+✠δR2+✠ξR2,where ScR∈−2,2 and AcR∈0,2.Definition 10.
For two CSFNsR1=κR1,✠κR1,δR1,✠δR1,ξR1,✠ξR1 and R2=κR2,✠κR2,δR2,✠δR2,ξR2,✠ξR2, the distance measure between them is defined as
(9)dR1,R2=16κR1−κR2+δR1−δR2+ξR1−ξR2+12π✠κR1−✠κR2+✠δR1−✠δR2+✠ξR1−✠ξR2.Definition 11.
The following comparison rules between two CSFNsR1 and R2 are defined as
(1)
IfScR1>ScR2, then R1>R2; similarly, if AcR1>AcR2, then R1>R2(2)
IfScR1=ScR2, then R1=R2; similarly, if AcR1=AcR2, then R1=R2Definition 12.
For two CSFNsR1=κR1,✠κR1,δR1,✠δR1,ξR1,✠ξR1 and R2=κR2,✠κR2,δR2,✠δR2,ξR2,✠ξR2 and λ (positive real number). Then, the basic operation laws are defined as
(1)
R1⊕R2=1−∏i=121−κRi2,2π1−∏i=121−✠κRi2/2π,∏i=12δRi,2π∏i=12✠δRi/2π,∏i=12ξRi,2π∏i=12✠ξRi/2π(2)
R1⊗R2=∏i=12κRi,2π∏i=12✠κRi/2π,1−∏i=121−δRi2,2π1−∏i=121−✠δRi2/2π,1−∏i=121−ξRi2,2π1−∏i=121−✠ξRi2/2π(3)
λR1=1−1−κR12λ,2π1−1−✠κR12/2πλ,δR1λ,2π✠δR1/2πλ,ξR1λ,2π✠ξR1/2πλ(4)
R1λ=κR1λ,2π✠κR1/2πλ,1−1−δR12λ,2π1−1−✠δR12/2πλ,1−1−ξR12λ,2π1−1−✠ξR12/2πλTheorem 13.
For any three CSFNsR1,R2,R3, we have
(1)
R1⊕R2=R2⊕R1(2)
R1⊗R2=R2⊗R1(3)
R1⊕R2⊕R3=R1⊕R2⊕R3(4)
R1⊗R2⊗R3=R1⊗R2⊗R3Proof.
For proof, see AppendixA.Theorem 14.
For any two CSFNsR1 and R2 and a positive real numbers λ,λ1,λ2, we have
(1)
λR1⊕R2=λR1⊕λR2(2)
R1⊗R2λ=R1λ⊕R2λ(3)
λ1+λ2R1=λ1R1⊕λ2R1(4)
R1λ1+λ2=R1λ1⊗R1λ2Proof.
For proof, see AppendixB.


## 4. Power Averaging Aggregation Operators

In this section, using the defined operational laws of the CSFNs, several PA operators are introduced with the CSF information in order to aggregate the CSFNs.


### 4.1. Complex Spherical Fuzzy Power Averaging Operator

In this subsection, we proposed the complex spherical fuzzy power averaging operator and discussed the basic properties of the defined operator.Definition 15.
For a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n, complex spherical fuzzy power averaging (CSFPA) aggregation operator is a function CSFPA:Ωn⟶Ω defined by
(10)CSFPAR1,⋯,Rn=ρ1R1⊕⋯⊕ρnRn,where ρi=1+TRi/∑i=1n1+TRi and TRi=∑s=1s≠inSupRi,Rsi=1,⋯,n. Here, SupRi,Rs is the support of Ri from Rs satisfying the aforementioned properties and SupRi,Rs=1−dRi,Rs, d be the distance measure defined in Definition 10.Theorem 16.
Let a family of CSFNs areRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value obtained by using CSFPA operator is also a CSFN and is given as
(11)CSFPAR1,⋯,Rn=1−∏i=1n1−κRi2ρi,2π1−∏i=1n1−✠κRi22πρi,∏i=1nδRiρi,2π∏i=1n✠δRi2πρi,∏i=1nξRiρi,2π∏i=1n✠ξRi2πρi.Proof.
For proof, see AppendixC.Example 1.
LetR1=0.3,2π0.2,0.5,2π0.3,0.2,2π0.1,R2=0.5,2π0.4,0.2,2π0.1,0.3,2π0.1,R3=0.6,2π0.5,0.1,2π0.1,0.2,2π0.2, and R4=0.3,2π0.3,0.4,2π0.3,0.3,2π0.1 be the four CSFNs. First, we calculate dRi,Rs∀i,s∈1,⋯,4,i≠s, using equation (9), we have
(12)dR1,R2=16κR1−κR2+δR1−δR2+ξR1−ξR2+12π✠κR1−✠κR2+✠δR1−✠δR2+✠ξR1−✠ξR2=160.3−0.5+0.5−0.2+0.2−0.3+12π2π0.2−2π0.4+2π0.3−2π0.1+2π0.1−2π0.1=0.167.
Which give us, SupR1,R2=1−dR1,R2=0.984. Similarly, we can obtain SupR1,R3=0.978, SupR1,R4=0.950, SupR2,R3=0.916, SupR2,R4=0.883, SupR3,R4=0.980.
(13)TR1=∑s=1s≠inSupRi,Rs=SupR1,R2+SupR1,R3+SupR1,R4=0.984+0.978+0.950=2.912.
Similarly, we obtainTR2=2.783,TR3=2.874 and TR4=2.813, which have
(14)∑i=141+TRi=1+TR1+1+TR2+1+TR3+1+TR4=1+2.912+1+2.783+1+2.874+1+2.813=15.382.
Hence,(15)ρ1=1+TR1∑i=1n1+TRi=1+2.91215.382=0.254.
Similarly, we getρ2=0.246,ρ3=0.252, and ρ4=0.248. Further,
(16)∏i=141−κRi2ρi=1−κR12ρ1×1−κR22ρ2×1−κR32ρ3×1−κR42ρ4=1−0.320.254×1−0.520.246×1−0.620.252×1−0.320.248=0.784,(17)∏i=141−✠κRi22πρi=1−✠κR122πρ1×1−✠κR222πρ2×1−✠κR322πρ3×1−✠κR422πρ4=1−0.220.254×1−0.420.246×1−0.520.252×1−0.320.248=0.845,(18)∏i=14δRiρi=δR1ρ1×δR2ρ2×δR3ρ3×δR4ρ4=0.50.254×0.20.246×0.10.252×0.40.248=0.1971,(19)∏i=14✠δRi2πρi=✠δR12πρ1×✠δR22πρ2×✠δR32πρ3×✠δR42πρ4=0.30.254×0.10.246×0.10.252×0.30.248=0.1731,(20)∏i=14ξRiρi=ξR1ρ1×ξR2ρ2×ξR3ρ3×ξR4ρ4=0.20.254×0.30.246×0.20.252×0.30.248=0.2437,(21)∏i=14✠ξRi2πρi=✠ξR12πρ1×✠ξR22πρ2×✠ξR32πρ3×✠ξR42πρ4=0.10.254×0.10.246×0.20.252×0.10.248=0.119.
Now, by using equation (11), we have
(22)CSFPAR1,⋯,R4=1−0.7841,2π1−0.8458,0.1971,2π0.1731,0.2437,2π0.119=0.4642,2π0.3926,0.1971,2π0.1731,0.2437,2π0.119.
Which shows that the aggregated value is again a CSFN.Based on Theorem (16), it is noted that the CSFPA operator satisfies some basic properties defined as follows.Property 17 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(23)CSFPAR1,⋯,Rn=R0.Proof.
For proof, see AppendixD.Property 18 (boundedness).
LetRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n be the family of CSFNs, and R−=miniκRi,mini✠κRi,miniδRi,mini✠δRi,maxiξRi,maxi✠ξRi and R+=maxiκRi,maxi✠κRi,miniδRi,mini✠δRi,miniξRi,mini✠ξRi. Then,
(24)R−≤CSFPAR1,⋯,Rn≤R+.Proof.
For proof, see AppendixE.Property 19 (commutavity).
LetRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n be the family of CSFNs, if R1∗,⋯,Rn∗ are the permutation of R1,⋯,Rn. Then,
(25)CSFPAR1,⋯,Rn=CSFPAR1∗,⋯,Rn∗.Proof.
For proof, see AppendixF.


### 4.2. Complex Spherical Fuzzy Weighted Power Averaging Operator

In this subsection, as compared to the abovementioned CSFPA operator, we take into account the different weighting factor of the CSFNsRii=1,⋯,n during the aggregation process and therefore introduced a modified CSF weighted power aggregation operator (CSFWPA) as follows.Definition 20.
For a family of CSFNsRii=1,⋯,n, CSFWPA operator is a function CSFWPA:Ωn⟶Ω defined by
(26)CSFWPAR1,⋯,Rn=ϕ1R1⊕⋯⊕ϕnRn,where ϕi=ψi1+T/Ri/∑i=1nψi1+T/Ri,T/Ri=∑s=1s≠inψsSupRi,Rs and ψ=ψ1,⋯,ψnT be the weight vector assigned to CSFNs Ri, such as ψi>0 and ∑i=1nψi=1.Theorem 21.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n, with their corresponding weight vector ψ=ψ1,⋯,ψnT, such as ψi>0 and ∑i=1nψi=1, the aggregated value using CSFWPA operator is again a CSFN and given as
(27)CSFWPAR1,⋯,Rn=1−∏i=1n1−κRi2ϕi,2π1−∏i=1n1−✠κRi22πϕi,∏i=1nδRiϕi,2π∏i=1n✠δRi2πϕi,∏i=1nξRiϕi,2π∏i=1n✠ξRi2πϕi.Proof.
The proof of theorem can be derived by using the mathematical induction principal as close to that of Theorem (16), so the proof are omitted here.Example 2.
LetR1=0.4,2π0.3,0.3,2π0.3,0.2,2π0.1,R2=0.1,2π0.1,0.6,2π0.5,0.3,2π0.1,R3=0.2,2π0.2,0.4,2π0.3,0.4,2π0.4, and R4=0.5,2π0.4,0.2,2π0.2,0.1,2π0.1 be the four CSFNs and ϕ=0.3,0.25,0.15,0.3 are the corresponding weights of Rii=1,⋯,n. Then, as the proceeding Example (1), we can obtain SupR1,R2=0.981, SupR1,R3=0.850, SupR1,R4=0.916, SupR2,R3=0.988, SupR2,R4=0.973, and SupR3,R4=0.976. Thus,
(28)T/R1=∑s=1s≠inϕsSupRi,Rs=ϕ2SupR1,R2+ϕ3SupR1,R3+ϕ4SupR1,R4=0.25×0.981+0.15×0.850+0.3×0.916=0.647.Similarly, we getT/R2=0.734,T/R3=0.795, and T/R4=0.664, which gives
(29)∑i=14ϕi1+T/Ri=ϕ11+T/R1+ϕ21+T/R2+ϕ31+T/R3+ϕ41+T/R4=0.3×1+0.647+0.25×1+0.734+0.15×1+0.795+0.3×1+0.664=1.642.Thus,(30)ϕ1=ϕ11+T/R1∑i=1nϕi1+T/Ri=0.31+0.6471.642=0.3.Similarly, we obtainϕ2=0.25,ϕ3=0.15, and ϕ4=0.3. Further,
(31)∏i=141−κRi2ϕi=1−κR12ϕ1×1−κR22ϕ2×1−κR32ϕ3×1−κR42ϕ4=1−0.420.30×1−0.120.25×1−0.220.15×1−0.520.3=0.8451,(32)∏i=141−✠κRi22πϕi=1−✠κR122πϕ1×1−✠κR222πϕ2×1−✠κR322πϕ3×1−✠κR422πϕ4=1−0.320.3×1−0.120.25×1−0.220.15×1−0.420.3=0.9115,(33)∏i=14δRiϕi=δR1ϕ1×δR2ϕ2×δR3ϕ3×δR4ϕ4=0.30.3×0.60.25×0.40.15×0.20.30=0.328,(34)∏i=14✠δRi2πϕi=✠δR12πϕ1×✠δR22πϕ2×✠δR32πϕ3×✠δR42πϕ4=0.30.3×0.50.25×0.30.15×0.20.3=0.299,(35)∏i=14ξRiϕi=ξR1ϕ1×ξR2ϕ2×ξR3ϕ3×ξR4ϕ4=0.20.3×0.30.25×0.40.15×0.10.3=0.199,(36)∏i=14✠ξRi2πϕi=✠ξR12πϕ1×✠ξR22πϕ2×✠ξR32πϕ3×✠ξR42πϕ4=0.10.3×0.10.25×0.40.15×0.10.3=0.122.Now, using equation (27), we have
(37)CSFWPAR1,⋯,R4=1−0.845,2π1−0.911,0.328,2π0.299,0.199,2π0.122=0.394,2π0.298,0.1971,2π0.1731,0.2437,2π0.119.For a family of CSFNsRii=1,⋯,n and their weights are ψ=ψ1,⋯,ψnT such as ψi>0 and ∑i=1nψi=1. CSFWPA operator also fulfills the same properties as the CSFPA operator which are described, without proof, as shown below.Property 22 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(38)CSFWPAR1,⋯,Rn=R0.Property 23 (boundedness).
LetR− and R+ are the lower bound and upper bound of the CSFNs Rii=1,⋯,n, respectively. Then, we have
(39)R−≤CSFWPAR1,⋯,Rn≤R+.Property 24 (commutavity).
LetR1∗,⋯,Rn∗ be a permutation of CSFNs R1,⋯,Rn with the corresponding permutation weight vector ψ∗=ψ1∗,⋯,ψn∗T of the ψ=ψ1,⋯,ψnT. Then,
(40)CSFWPAR1,⋯,Rn=CSFWPAR1∗,⋯,Rn∗.


### 4.3. Complex Spherical Fuzzy Ordered Weighted Power Averaging Operator

In this portion, the defined AO has been extended to its ordered weighted:Definition 25.
For a family of CSFNsRii=1,⋯,n, a complex spherical fuzzy ordered weighted power averaging (CSFOWPA) operator is a function CSFOWPA:Ωn⟶Ω defined as
(41)CSFOWPAR1,⋯,Rn=ζ1Rμ1⊕⋯⊕ζnRμn,where Ω denoted the set of CSFNs and μ1,⋯,μn is the permutation of 1,⋯,n satisfy that Rμi−1≥Rμi for i=2,⋯,n. Also, ζi is defined as
(42)ζi=gBiTV−gBi−1TV,where Bi=∑s=1iVμs,Vμi=1+∑s=1s≠inSupRi,Rs,TV=∑i=1nVμi, and the mapping g:0,1⟶0,1 is a basic unit-interval monotonic function satisfy the three properties g0=0,g1=1 and if x≤y then gx≤gy.Theorem 26.
Let a family of CSFNs areRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value obtained by using CSFOWPA operator is again a CSFN and given as
(43)CSFOWPAR1,⋯,Rn=1−∏i=1n1−κRμi2ζi,2π1−∏i=1n1−✠κRμi22πζi,∏i=1nδRμiζi,2π∏i=1n✠δRμi2πζi,∏i=1nξRμiζi,2π∏i=1n✠ξRμi2πζi,where ζi is defined in equation (42).Proof.
The proof of theorem is close to that of Theorem (16), so the proof is omitted here.Example 3.
LetR1=0.2,2π0.1,0.4,2π0.3,0.4,2π0.4,R2=0.6,2π0.5,0.2,2π0.2,0.1,2π0.1,R3=0.3,2π0.2,0.1,2π0.1,0.5,2π0.5, and R4=0.1,2π0.1,0.3,2π0.3,0.4,2π0.3 be the four CSFNs. We determine the scores of these CSFN by using equation ??, and obtain as ScR1=0.2,ScR2=1.3,ScR3=−0.1 and ScR4=0.1. Thus, ScR2>ScR1>ScR4>ScR4, therefore, we get Rμ1=0.6,2π0.5,0.2,2π0.2,0.1,2π0.1,Rμ2=0.2,2π0.1,0.4,2π0.3,0.4,2π0.4,Rμ3=0.1,2π0.1,0.3,2π0.3,0.4,2π0.3, and Rμ4=0.3,2π0.2,0.1,2π0.1,0.5,2π0.5. Now, using equation (9) to find the distance measure value between the CSFNs as, dR1,R2=0.0283. Hence, SupR1,R2=1−0.028=0.9717. Similarly, we can calculate SupR1,R3=0.9734, SupR1,R4=0.9734, SupR2,R3=0.950, SupR2,R4=0.850, and SupR3,R4=0.9834. Now, using the equation Vμi=1+∑s=1s≠inSupRi,Rs, we have Vμ1=3.9185,Vμ2=3.771,Vμ3=3.9068, and Vμ4=3.8068. Thus, B1=3.9185,B2=7.6895,B3=11.5963, and B4=15.403. Also, TV=∑i=1nVμi=15.5963. Now, take the value of gx=x2 and using equation (42), we obtain
(44)ζ1=gB1TV=g3.918515.5963=g0.25122=0.0631,(45)ζ2=gB2TV−gB1TV=g7.689515.5963−g3.918515.5963=0.49302−0.25122=0.1798,(46)ζ3=gB3TV−gB2TV=g11.596315.5963−g7.689515.5963=0.74352−0.49302=0.3097,(47)ζ4=gB4TV−gB3TV=g15.596615.5963−g11.596315.5963=12−0.74352=0.4472.
Using these information, we get(48)∏i=141−κRμi2ζi=1−0.620.0631×1−0.220.1798×1−0.120.3097×1−0.320.4472=0.9302,(49)∏i=141−✠κRμi22πζi=1−0.520.0631×1−0.120.1798×1−0.120.3097×1−0.220.4472=0.9431,(50)∏i=14δRiζi=0.20.0631×0.40.1798×0.30.3097×0.10.4472=0.1884,(51)∏i=14✠δRi2πζi=0.20.0631×0.30.1798×0.30.3097×0.10.4472=0.1789,(52)∏i=14ξRiζi=0.10.0631×0.40.1798×0.40.3097×0.50.4472=0.3704,(53)∏i=14✠ξRi2πζi=0.10.0631×0.40.1798×0.30.3097×0.50.4472=0.3008.
Now, using equation (43), we have
(54)CSFOWPAR1,⋯,R4=1−0.9302,2π1−0.9431,0.1884,2π0.1789,0.3704,2π0.3008=0.2641,2π0.2385,0.1884,2π0.1789,0.3704,2π0.3008.
Furthermore, with regard to the set of CSFNs, it is noted that the CSFOWPA operator also satisfies the properties, idempotency, commutativity, and boundedness.


## 4.1. Complex Spherical Fuzzy Power Averaging Operator

In this subsection, we proposed the complex spherical fuzzy power averaging operator and discussed the basic properties of the defined operator.Definition 15.
For a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n, complex spherical fuzzy power averaging (CSFPA) aggregation operator is a function CSFPA:Ωn⟶Ω defined by
(10)CSFPAR1,⋯,Rn=ρ1R1⊕⋯⊕ρnRn,where ρi=1+TRi/∑i=1n1+TRi and TRi=∑s=1s≠inSupRi,Rsi=1,⋯,n. Here, SupRi,Rs is the support of Ri from Rs satisfying the aforementioned properties and SupRi,Rs=1−dRi,Rs, d be the distance measure defined in Definition 10.Theorem 16.
Let a family of CSFNs areRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value obtained by using CSFPA operator is also a CSFN and is given as
(11)CSFPAR1,⋯,Rn=1−∏i=1n1−κRi2ρi,2π1−∏i=1n1−✠κRi22πρi,∏i=1nδRiρi,2π∏i=1n✠δRi2πρi,∏i=1nξRiρi,2π∏i=1n✠ξRi2πρi.Proof.
For proof, see AppendixC.Example 1.
LetR1=0.3,2π0.2,0.5,2π0.3,0.2,2π0.1,R2=0.5,2π0.4,0.2,2π0.1,0.3,2π0.1,R3=0.6,2π0.5,0.1,2π0.1,0.2,2π0.2, and R4=0.3,2π0.3,0.4,2π0.3,0.3,2π0.1 be the four CSFNs. First, we calculate dRi,Rs∀i,s∈1,⋯,4,i≠s, using equation (9), we have
(12)dR1,R2=16κR1−κR2+δR1−δR2+ξR1−ξR2+12π✠κR1−✠κR2+✠δR1−✠δR2+✠ξR1−✠ξR2=160.3−0.5+0.5−0.2+0.2−0.3+12π2π0.2−2π0.4+2π0.3−2π0.1+2π0.1−2π0.1=0.167.
Which give us, SupR1,R2=1−dR1,R2=0.984. Similarly, we can obtain SupR1,R3=0.978, SupR1,R4=0.950, SupR2,R3=0.916, SupR2,R4=0.883, SupR3,R4=0.980.
(13)TR1=∑s=1s≠inSupRi,Rs=SupR1,R2+SupR1,R3+SupR1,R4=0.984+0.978+0.950=2.912.
Similarly, we obtainTR2=2.783,TR3=2.874 and TR4=2.813, which have
(14)∑i=141+TRi=1+TR1+1+TR2+1+TR3+1+TR4=1+2.912+1+2.783+1+2.874+1+2.813=15.382.
Hence,(15)ρ1=1+TR1∑i=1n1+TRi=1+2.91215.382=0.254.
Similarly, we getρ2=0.246,ρ3=0.252, and ρ4=0.248. Further,
(16)∏i=141−κRi2ρi=1−κR12ρ1×1−κR22ρ2×1−κR32ρ3×1−κR42ρ4=1−0.320.254×1−0.520.246×1−0.620.252×1−0.320.248=0.784,(17)∏i=141−✠κRi22πρi=1−✠κR122πρ1×1−✠κR222πρ2×1−✠κR322πρ3×1−✠κR422πρ4=1−0.220.254×1−0.420.246×1−0.520.252×1−0.320.248=0.845,(18)∏i=14δRiρi=δR1ρ1×δR2ρ2×δR3ρ3×δR4ρ4=0.50.254×0.20.246×0.10.252×0.40.248=0.1971,(19)∏i=14✠δRi2πρi=✠δR12πρ1×✠δR22πρ2×✠δR32πρ3×✠δR42πρ4=0.30.254×0.10.246×0.10.252×0.30.248=0.1731,(20)∏i=14ξRiρi=ξR1ρ1×ξR2ρ2×ξR3ρ3×ξR4ρ4=0.20.254×0.30.246×0.20.252×0.30.248=0.2437,(21)∏i=14✠ξRi2πρi=✠ξR12πρ1×✠ξR22πρ2×✠ξR32πρ3×✠ξR42πρ4=0.10.254×0.10.246×0.20.252×0.10.248=0.119.
Now, by using equation (11), we have
(22)CSFPAR1,⋯,R4=1−0.7841,2π1−0.8458,0.1971,2π0.1731,0.2437,2π0.119=0.4642,2π0.3926,0.1971,2π0.1731,0.2437,2π0.119.
Which shows that the aggregated value is again a CSFN.Based on Theorem (16), it is noted that the CSFPA operator satisfies some basic properties defined as follows.Property 17 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(23)CSFPAR1,⋯,Rn=R0.Proof.
For proof, see AppendixD.Property 18 (boundedness).
LetRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n be the family of CSFNs, and R−=miniκRi,mini✠κRi,miniδRi,mini✠δRi,maxiξRi,maxi✠ξRi and R+=maxiκRi,maxi✠κRi,miniδRi,mini✠δRi,miniξRi,mini✠ξRi. Then,
(24)R−≤CSFPAR1,⋯,Rn≤R+.Proof.
For proof, see AppendixE.Property 19 (commutavity).
LetRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n be the family of CSFNs, if R1∗,⋯,Rn∗ are the permutation of R1,⋯,Rn. Then,
(25)CSFPAR1,⋯,Rn=CSFPAR1∗,⋯,Rn∗.Proof.
For proof, see AppendixF.


## 4.2. Complex Spherical Fuzzy Weighted Power Averaging Operator

In this subsection, as compared to the abovementioned CSFPA operator, we take into account the different weighting factor of the CSFNsRii=1,⋯,n during the aggregation process and therefore introduced a modified CSF weighted power aggregation operator (CSFWPA) as follows.Definition 20.
For a family of CSFNsRii=1,⋯,n, CSFWPA operator is a function CSFWPA:Ωn⟶Ω defined by
(26)CSFWPAR1,⋯,Rn=ϕ1R1⊕⋯⊕ϕnRn,where ϕi=ψi1+T/Ri/∑i=1nψi1+T/Ri,T/Ri=∑s=1s≠inψsSupRi,Rs and ψ=ψ1,⋯,ψnT be the weight vector assigned to CSFNs Ri, such as ψi>0 and ∑i=1nψi=1.Theorem 21.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n, with their corresponding weight vector ψ=ψ1,⋯,ψnT, such as ψi>0 and ∑i=1nψi=1, the aggregated value using CSFWPA operator is again a CSFN and given as
(27)CSFWPAR1,⋯,Rn=1−∏i=1n1−κRi2ϕi,2π1−∏i=1n1−✠κRi22πϕi,∏i=1nδRiϕi,2π∏i=1n✠δRi2πϕi,∏i=1nξRiϕi,2π∏i=1n✠ξRi2πϕi.Proof.
The proof of theorem can be derived by using the mathematical induction principal as close to that of Theorem (16), so the proof are omitted here.Example 2.
LetR1=0.4,2π0.3,0.3,2π0.3,0.2,2π0.1,R2=0.1,2π0.1,0.6,2π0.5,0.3,2π0.1,R3=0.2,2π0.2,0.4,2π0.3,0.4,2π0.4, and R4=0.5,2π0.4,0.2,2π0.2,0.1,2π0.1 be the four CSFNs and ϕ=0.3,0.25,0.15,0.3 are the corresponding weights of Rii=1,⋯,n. Then, as the proceeding Example (1), we can obtain SupR1,R2=0.981, SupR1,R3=0.850, SupR1,R4=0.916, SupR2,R3=0.988, SupR2,R4=0.973, and SupR3,R4=0.976. Thus,
(28)T/R1=∑s=1s≠inϕsSupRi,Rs=ϕ2SupR1,R2+ϕ3SupR1,R3+ϕ4SupR1,R4=0.25×0.981+0.15×0.850+0.3×0.916=0.647.Similarly, we getT/R2=0.734,T/R3=0.795, and T/R4=0.664, which gives
(29)∑i=14ϕi1+T/Ri=ϕ11+T/R1+ϕ21+T/R2+ϕ31+T/R3+ϕ41+T/R4=0.3×1+0.647+0.25×1+0.734+0.15×1+0.795+0.3×1+0.664=1.642.Thus,(30)ϕ1=ϕ11+T/R1∑i=1nϕi1+T/Ri=0.31+0.6471.642=0.3.Similarly, we obtainϕ2=0.25,ϕ3=0.15, and ϕ4=0.3. Further,
(31)∏i=141−κRi2ϕi=1−κR12ϕ1×1−κR22ϕ2×1−κR32ϕ3×1−κR42ϕ4=1−0.420.30×1−0.120.25×1−0.220.15×1−0.520.3=0.8451,(32)∏i=141−✠κRi22πϕi=1−✠κR122πϕ1×1−✠κR222πϕ2×1−✠κR322πϕ3×1−✠κR422πϕ4=1−0.320.3×1−0.120.25×1−0.220.15×1−0.420.3=0.9115,(33)∏i=14δRiϕi=δR1ϕ1×δR2ϕ2×δR3ϕ3×δR4ϕ4=0.30.3×0.60.25×0.40.15×0.20.30=0.328,(34)∏i=14✠δRi2πϕi=✠δR12πϕ1×✠δR22πϕ2×✠δR32πϕ3×✠δR42πϕ4=0.30.3×0.50.25×0.30.15×0.20.3=0.299,(35)∏i=14ξRiϕi=ξR1ϕ1×ξR2ϕ2×ξR3ϕ3×ξR4ϕ4=0.20.3×0.30.25×0.40.15×0.10.3=0.199,(36)∏i=14✠ξRi2πϕi=✠ξR12πϕ1×✠ξR22πϕ2×✠ξR32πϕ3×✠ξR42πϕ4=0.10.3×0.10.25×0.40.15×0.10.3=0.122.Now, using equation (27), we have
(37)CSFWPAR1,⋯,R4=1−0.845,2π1−0.911,0.328,2π0.299,0.199,2π0.122=0.394,2π0.298,0.1971,2π0.1731,0.2437,2π0.119.For a family of CSFNsRii=1,⋯,n and their weights are ψ=ψ1,⋯,ψnT such as ψi>0 and ∑i=1nψi=1. CSFWPA operator also fulfills the same properties as the CSFPA operator which are described, without proof, as shown below.Property 22 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(38)CSFWPAR1,⋯,Rn=R0.Property 23 (boundedness).
LetR− and R+ are the lower bound and upper bound of the CSFNs Rii=1,⋯,n, respectively. Then, we have
(39)R−≤CSFWPAR1,⋯,Rn≤R+.Property 24 (commutavity).
LetR1∗,⋯,Rn∗ be a permutation of CSFNs R1,⋯,Rn with the corresponding permutation weight vector ψ∗=ψ1∗,⋯,ψn∗T of the ψ=ψ1,⋯,ψnT. Then,
(40)CSFWPAR1,⋯,Rn=CSFWPAR1∗,⋯,Rn∗.


## 4.3. Complex Spherical Fuzzy Ordered Weighted Power Averaging Operator

In this portion, the defined AO has been extended to its ordered weighted:Definition 25.
For a family of CSFNsRii=1,⋯,n, a complex spherical fuzzy ordered weighted power averaging (CSFOWPA) operator is a function CSFOWPA:Ωn⟶Ω defined as
(41)CSFOWPAR1,⋯,Rn=ζ1Rμ1⊕⋯⊕ζnRμn,where Ω denoted the set of CSFNs and μ1,⋯,μn is the permutation of 1,⋯,n satisfy that Rμi−1≥Rμi for i=2,⋯,n. Also, ζi is defined as
(42)ζi=gBiTV−gBi−1TV,where Bi=∑s=1iVμs,Vμi=1+∑s=1s≠inSupRi,Rs,TV=∑i=1nVμi, and the mapping g:0,1⟶0,1 is a basic unit-interval monotonic function satisfy the three properties g0=0,g1=1 and if x≤y then gx≤gy.Theorem 26.
Let a family of CSFNs areRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value obtained by using CSFOWPA operator is again a CSFN and given as
(43)CSFOWPAR1,⋯,Rn=1−∏i=1n1−κRμi2ζi,2π1−∏i=1n1−✠κRμi22πζi,∏i=1nδRμiζi,2π∏i=1n✠δRμi2πζi,∏i=1nξRμiζi,2π∏i=1n✠ξRμi2πζi,where ζi is defined in equation (42).Proof.
The proof of theorem is close to that of Theorem (16), so the proof is omitted here.Example 3.
LetR1=0.2,2π0.1,0.4,2π0.3,0.4,2π0.4,R2=0.6,2π0.5,0.2,2π0.2,0.1,2π0.1,R3=0.3,2π0.2,0.1,2π0.1,0.5,2π0.5, and R4=0.1,2π0.1,0.3,2π0.3,0.4,2π0.3 be the four CSFNs. We determine the scores of these CSFN by using equation ??, and obtain as ScR1=0.2,ScR2=1.3,ScR3=−0.1 and ScR4=0.1. Thus, ScR2>ScR1>ScR4>ScR4, therefore, we get Rμ1=0.6,2π0.5,0.2,2π0.2,0.1,2π0.1,Rμ2=0.2,2π0.1,0.4,2π0.3,0.4,2π0.4,Rμ3=0.1,2π0.1,0.3,2π0.3,0.4,2π0.3, and Rμ4=0.3,2π0.2,0.1,2π0.1,0.5,2π0.5. Now, using equation (9) to find the distance measure value between the CSFNs as, dR1,R2=0.0283. Hence, SupR1,R2=1−0.028=0.9717. Similarly, we can calculate SupR1,R3=0.9734, SupR1,R4=0.9734, SupR2,R3=0.950, SupR2,R4=0.850, and SupR3,R4=0.9834. Now, using the equation Vμi=1+∑s=1s≠inSupRi,Rs, we have Vμ1=3.9185,Vμ2=3.771,Vμ3=3.9068, and Vμ4=3.8068. Thus, B1=3.9185,B2=7.6895,B3=11.5963, and B4=15.403. Also, TV=∑i=1nVμi=15.5963. Now, take the value of gx=x2 and using equation (42), we obtain
(44)ζ1=gB1TV=g3.918515.5963=g0.25122=0.0631,(45)ζ2=gB2TV−gB1TV=g7.689515.5963−g3.918515.5963=0.49302−0.25122=0.1798,(46)ζ3=gB3TV−gB2TV=g11.596315.5963−g7.689515.5963=0.74352−0.49302=0.3097,(47)ζ4=gB4TV−gB3TV=g15.596615.5963−g11.596315.5963=12−0.74352=0.4472.
Using these information, we get(48)∏i=141−κRμi2ζi=1−0.620.0631×1−0.220.1798×1−0.120.3097×1−0.320.4472=0.9302,(49)∏i=141−✠κRμi22πζi=1−0.520.0631×1−0.120.1798×1−0.120.3097×1−0.220.4472=0.9431,(50)∏i=14δRiζi=0.20.0631×0.40.1798×0.30.3097×0.10.4472=0.1884,(51)∏i=14✠δRi2πζi=0.20.0631×0.30.1798×0.30.3097×0.10.4472=0.1789,(52)∏i=14ξRiζi=0.10.0631×0.40.1798×0.40.3097×0.50.4472=0.3704,(53)∏i=14✠ξRi2πζi=0.10.0631×0.40.1798×0.30.3097×0.50.4472=0.3008.
Now, using equation (43), we have
(54)CSFOWPAR1,⋯,R4=1−0.9302,2π1−0.9431,0.1884,2π0.1789,0.3704,2π0.3008=0.2641,2π0.2385,0.1884,2π0.1789,0.3704,2π0.3008.
Furthermore, with regard to the set of CSFNs, it is noted that the CSFOWPA operator also satisfies the properties, idempotency, commutativity, and boundedness.


## 5. Complex Spherical Fuzzy Power Geometric Aggregation Operator

In this section, the aggregation operators described above are extended to geometric aggregation operators with complex spherical fuzzy information, including CSF power geometric (CSFPG), CSF weighted power geometric (CSFWPG), and CSF ordered weighted power geometric (CSFOWPG) operators.Definition 27.
For a family of CSFNsRii=1,⋯,n, complex spherical fuzzy power geometric (CSFPG) operator is a function CSFPG:Ωn⟶Ω defined by
(55)CSFPG=R1,⋯,Rn=R1ρ1⊗⋯⊗Rnρn,where ρi=1+TRi/∑i=1n1+TRi and TRi=∑s=1s≠inSupRi,Rsi=1,⋯,n. Here, SupRi,Rs is the support of Ri from Rs satisfying the aforementioned properties and SupRi,Rs=1−dRi,Rs, where d is the distance measure.Theorem 28.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value is obtained by using CSFPG operator is again a CSFN and given as
(56)CSFPGR1,⋯,Rn=∏i=1nκRiρi,2π∏i=1n✠κRi2πρi,1−∏i=1n1−δRi2ρi,2π1−∏i=1n1−✠δRi22πρi,1−∏i=1n1−ξRi2ρi,2π1−∏i=1n1−✠ξRi22πρi.Proof.
For proof, see AppendixG.On the basis of the Theorem (28), it is noted that CSFPG operator satisfies certain properties as follows.Property 29 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(57)CSFPGR1,⋯,Rn=R0.Property 30 (boundedness).
LetRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n be the family of CSFNs, and R−=miniκRi,mini✠κRi,miniδRi,mini✠δRi,maxiξRi,maxi✠ξRi and R+=maxiκRi,maxi✠κRi,miniδRi,mini✠δRi,miniξRi,mini✠ξRi. Then,
(58)R−≤CSFPGR1,⋯,Rn≤R+.Property 31 (commutavity).
LetRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n be the family of CSFNs, if R1∗,⋯,Rn∗ are the permutation of R1,⋯,Rn. Then,
(59)CSFPGR1,⋯,Rn=CSFPGR1∗,⋯,Rn∗.


### 5.1. Complex Spherical Fuzzy Weighted Power Geometric Operator

In this portion, as compared to the above CSFPG operator, we take into account the different weighting factor of the CSFNsRii=1,⋯,n during the aggregation process and therefore propose a new CSF weighted power geometric (CSFWPG) aggregation operator are below.Definition 32.
For a family of CSFNsRii=1,⋯,n, CSFWPG operator is a function CSFWPG:Ωn⟶Ω defined by
(60)CSFWPGR1,⋯,Rn=R1ϕ1⊗⋯⊗Rnϕn,where ϕi=ψi1+T/Ri/∑i=1nψi1+T/Ri,T/Ri=∑s=1s≠inψsSupRi,Rs and ψ=ψ1,⋯,ψnT be the weight vector assigned to CSFNs Ri, such as ψi>0 and ∑i=1nψi=1.Theorem 33.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n, with the respective weight vector ψ=ψ1,⋯,ψnT, such that ψi>0 and ∑i=1nψi=1, the aggregated value calculated by using CSFWPG operator is again a CSFN and given as
(61)CSFWPGR1,⋯,Rn=∏i=1nκRiϕi,2π∏i=1n✠κRi2πϕi,1−∏i=1n1−δRi2ϕi,2π1−∏i=1n1−✠δRi22πϕi,1−∏i=1n1−ξRi2ϕi,2π1−∏i=1n1−✠ξRi22πϕi.Proof.
The proof of this theorem can be achieved by the mathematical induction principal as close to that of Theorem (28), so here, we omit the proof.For a family of CSFNsRii=1,⋯,n and their weights are ψ=ψ1,⋯,ψnT such that ψi>0 and ∑i=1nψi=1. CSFWPG operator also fulfills the same properties as the CSFPG operator which are described, without proof, as shown below.Property 34 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(62)CSFWPGR1,⋯,Rn=R0.Property 35 (boundedness).
LetR− and R+ are the lower bound and upper bound of the CSFNs Rii=1,⋯,n, respectively. Then, we have
(63)R_≤CSFWPGR1,⋯,Rn≤R+.Property 36 (commutavity).
LetR1∗,⋯,Rn∗ be a permutation of CSFNs R1,⋯,Rn with the corresponding permutation weight vector ψ∗=ψ1∗,⋯,ψn∗T of the ψ=ψ1,⋯,ψnT. Then,
(64)CSFWPGR1,⋯,Rn=CSFWPGR1∗,⋯,Rn∗.


### 5.2. Complex Spherical Fuzzy Ordered Weighted Power Geometric Operator

In this section, the existing aggregation operator was extended to their ordered weighted aggregation operator.Definition 37.
For a family of CSFNsRii=1,⋯,n, a complex spherical fuzzy ordered weighted power geometric (CSFOWPG) operator is a function CSFOWPG:Ωn⟶Ω defined by
(65)CSFOWPGR1,⋯,Rn=Rμ1ζ1⊗⋯⊗Rμnζn,where Ω denoted the set of CSFNs and μ1,⋯,μn is the permutation of 1,⋯,n satisfy that Rμi−1≥Rμi for i=2,⋯,n. Also, ζi is defined as
(66)ζi=gBiTV−gBi−1TV,where Bi=∑s=1iVμs,Vμi=1+∑s=1s≠inSupRi,Rs,TV=∑i=1nVμi, and the mapping g:0,1⟶0,1 is a basic unit-interval monotonic function satisfy these three properties g0=0,g1=1 and if x≤y then gx≤gy.Theorem 38.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value calculated by using CSFOWPG operator is also a CSFN and given as
(67)CSFOWPGR1,⋯,Rn=∏i=1nκRμiζi,2π∏i=1n✠κRμi2πζi,1−∏i=1n1−δRμi2ζi,2π1−∏i=1n1−✠δRμi22πζi,1−∏i=1n1−ξRμi2ζi,2π1−∏i=1n1−✠ξRμi22πζi,where ζi is defined in equation (66).Proof.
The proof of the theorem can be derived by using the mathematical induction principal as close to that of Theorem (28), so the proof is omitted here.Furthermore, with regard to the set of CSFNs, it is noted that the CSFOWPA operator also satisfies the properties, idempotency, commutativity, and boundedness.


## 5.1. Complex Spherical Fuzzy Weighted Power Geometric Operator

In this portion, as compared to the above CSFPG operator, we take into account the different weighting factor of the CSFNsRii=1,⋯,n during the aggregation process and therefore propose a new CSF weighted power geometric (CSFWPG) aggregation operator are below.Definition 32.
For a family of CSFNsRii=1,⋯,n, CSFWPG operator is a function CSFWPG:Ωn⟶Ω defined by
(60)CSFWPGR1,⋯,Rn=R1ϕ1⊗⋯⊗Rnϕn,where ϕi=ψi1+T/Ri/∑i=1nψi1+T/Ri,T/Ri=∑s=1s≠inψsSupRi,Rs and ψ=ψ1,⋯,ψnT be the weight vector assigned to CSFNs Ri, such as ψi>0 and ∑i=1nψi=1.Theorem 33.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n, with the respective weight vector ψ=ψ1,⋯,ψnT, such that ψi>0 and ∑i=1nψi=1, the aggregated value calculated by using CSFWPG operator is again a CSFN and given as
(61)CSFWPGR1,⋯,Rn=∏i=1nκRiϕi,2π∏i=1n✠κRi2πϕi,1−∏i=1n1−δRi2ϕi,2π1−∏i=1n1−✠δRi22πϕi,1−∏i=1n1−ξRi2ϕi,2π1−∏i=1n1−✠ξRi22πϕi.Proof.
The proof of this theorem can be achieved by the mathematical induction principal as close to that of Theorem (28), so here, we omit the proof.For a family of CSFNsRii=1,⋯,n and their weights are ψ=ψ1,⋯,ψnT such that ψi>0 and ∑i=1nψi=1. CSFWPG operator also fulfills the same properties as the CSFPG operator which are described, without proof, as shown below.Property 34 (idempotency).
LetR0 be a CSFN, if Ri=R0 for all i=1,⋯,n. Then,
(62)CSFWPGR1,⋯,Rn=R0.Property 35 (boundedness).
LetR− and R+ are the lower bound and upper bound of the CSFNs Rii=1,⋯,n, respectively. Then, we have
(63)R_≤CSFWPGR1,⋯,Rn≤R+.Property 36 (commutavity).
LetR1∗,⋯,Rn∗ be a permutation of CSFNs R1,⋯,Rn with the corresponding permutation weight vector ψ∗=ψ1∗,⋯,ψn∗T of the ψ=ψ1,⋯,ψnT. Then,
(64)CSFWPGR1,⋯,Rn=CSFWPGR1∗,⋯,Rn∗.


## 5.2. Complex Spherical Fuzzy Ordered Weighted Power Geometric Operator

In this section, the existing aggregation operator was extended to their ordered weighted aggregation operator.Definition 37.
For a family of CSFNsRii=1,⋯,n, a complex spherical fuzzy ordered weighted power geometric (CSFOWPG) operator is a function CSFOWPG:Ωn⟶Ω defined by
(65)CSFOWPGR1,⋯,Rn=Rμ1ζ1⊗⋯⊗Rμnζn,where Ω denoted the set of CSFNs and μ1,⋯,μn is the permutation of 1,⋯,n satisfy that Rμi−1≥Rμi for i=2,⋯,n. Also, ζi is defined as
(66)ζi=gBiTV−gBi−1TV,where Bi=∑s=1iVμs,Vμi=1+∑s=1s≠inSupRi,Rs,TV=∑i=1nVμi, and the mapping g:0,1⟶0,1 is a basic unit-interval monotonic function satisfy these three properties g0=0,g1=1 and if x≤y then gx≤gy.Theorem 38.
Let a family of CSFNsRi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,n. Then, the aggregated value calculated by using CSFOWPG operator is also a CSFN and given as
(67)CSFOWPGR1,⋯,Rn=∏i=1nκRμiζi,2π∏i=1n✠κRμi2πζi,1−∏i=1n1−δRμi2ζi,2π1−∏i=1n1−✠δRμi22πζi,1−∏i=1n1−ξRμi2ζi,2π1−∏i=1n1−✠ξRμi22πζi,where ζi is defined in equation (66).Proof.
The proof of the theorem can be derived by using the mathematical induction principal as close to that of Theorem (28), so the proof is omitted here.Furthermore, with regard to the set of CSFNs, it is noted that the CSFOWPA operator also satisfies the properties, idempotency, commutativity, and boundedness.


## 6. MAGDM Approach Using Complex Spherical Fuzzy Power Aggregation Operators

In this section, a MAGDM algorithm is proposed based on defined operators with the CSF information.Suppose that a DM problem withm alternatives ℕ1,⋯,ℕm which are evaluated with n attribute C1,⋯,Cn. Let we have p experts E=E1,⋯,Ep, who evaluated the different alternatives with the different attributes. Every expert evaluates every alternative with the CSF information and assigns its rating values in the form of CSFNs as Rijk=κRijk,✠κRijk,δRijk,✠δRijk,ξRijk,✠ξRijk, where k=1,⋯,p;i=1,⋯,m and j=1,⋯,n,0≤κRijk+δRijk+ξRijk≤1 and 0≤✠κRijk+✠δRijk+✠ξRijk≤2π. Further, assume that the weights of the attribute are ψ=ψ1,⋯,ψnT, such that ψi>0 and ∑i=1nψi=1. Then, to find the most desirable alternatives, the defined operators are used to define a MAGDM approach under the complex spherical fuzzy information, with the following steps.Step 1.
Developed a complex spherical fuzzy decision-matrixRk=Rijkm×n with the rating values of every alternative given by expert Ekk=1,⋯,p as
(68)Rk=C1C2⋅⋅⋅Cnℕ1R11kR12k⋅⋅⋅R1nkℕ2R21kR22kR2nk⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ℕmRm1kRm2k⋅⋅⋅Rmnk.Step 2.
Aggregate the rating values of each expertRkk=1,⋯,p into the total collective CSF decision matrix R=Ξij, by using CSFOWPA operator as follows;
(69)Ξij=CSFOWPARij1,⋯,Rijp=1−∏k=1p1−κRij2μkζijk,2π1−∏k=1p1−✠κRij2μk2πζijk,∏k=1pδRijμkζijk,2π∏k=1p✠δRijμk2πζijk,∏k=1nξRijμkζijk,2π∏k=1p✠ξRijμk2πζijk.Step 3.
Aggregate the rating values of each expertRkk=1,⋯,p into the total collective CSF decision matrix R=Ξij, by using the CSFOWPG operator as follows;
(70)Ξij=CSFOWPGRij1,⋯,Rijp=∏k=1pκRijμkζijk,2π∏k=1p✠κRijμk2πζijk,1−∏k=1p1−δRij2μkζijk,2π1−∏k=1p1−✠δRij2μk2πζijk,1−∏k=1p1−ξRij2μkζijk,2π1−∏k=1p1−✠ξRij2μk2πζijk,where ζij1,⋯,ζijp are the weights obtained by utilizing equation (42), and μ be the permutation mapping from 1,⋯,p to 1,⋯,p.Step 4.
We use the complex Spherical fuzzy entropy to find the weights of each attribute in this step.(71)Ej=12−1m∑j=1mκRj2+✠κRj2+δRj2+✠δRj2+ξRj2+✠ξRj2,where 1/2−1m is a constant for assuring 0≤Ej≤1.Based on the equation (71), the weights of the attribute are computed as ω=ω1,ω2,⋯,ωn, where
(72)ψj=1−Ejn−∑j=1nEj.Step 5.
Aggregate the collected rating valuesR=Ξij of the alternative ℕii=1,⋯,m into the total assessment value Ξi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRi using the power averaging or power geometric aggregation operator as defined in equation (11).We used a CSFPA operator to aggregate each value of the alternativeℕi to obtain the total evaluation value of the alternative Ξii=1,⋯,m as
(73)Ξi=CSFPAΞi1,⋯,Ξin=1−∏j=1n1−κRij2ρj,2π1−∏j=1n1−✠κRij22πρj,∏j=1nδRijρj,2π∏i=1n✠δRij2πρi,∏i=1nξRijρi,2π∏i=1n✠ξRij2πρi.Step 6.
Aggregate the collected rating valuesR=Ξij of the alternative ℕii=1,⋯,m into the total assessment value Ξi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRi using the power averaging or power geometric aggregation operator as defined in equation (27).We used a CSFWPG operator to aggregate each value of the alternativeℕi to obtain the total evaluation value of the alternative Ξii=1,⋯,m as
(74)Ξi=CSFPGΞi1,⋯,Ξin=∏j=1nκRijρj,2π∏i=1n✠κRij2πρi,1−∏j=1n1−δRij2ρj,2π1−∏j=1n1−✠δRij22πρj,1−∏j=1n1−ξRij2ρj,2π1−∏j=1n1−✠ξRij22πρj.Step 7.
Calculate the scores for the aggregated total values,Ξi=κRi,✠κRi,δRi,✠δRi,ξRi,✠ξRii=1,⋯,m.Step 8.
Give ranking to the alternativesℕii=1,⋯,m and choose the most suitable alternative(s).


## 7. Application

We will use the example of green supplier selection to demonstrate the abovementioned approach to solving MAGDM problems. In the example, we will take four attributes and five alternatives. The description of the problem is demonstrated as follows.


### 7.1. Description of the Problem

A realistic implementation of green supplier selection for some chemical processing industry is considered in this section. The chemical processing industry’s business is to alter the chemical composition of natural materials in order to extract goods of importance to other businesses or in everyday life. Chemicals are made from these raw materials, which are mostly minerals, metals, and hydrocarbons, in a series of refining phases. Further processing, such as mixing and blending, is needed to transform them into final products (e.g., paints, adhesives, drugs, and cosmetics). It is known that the supplier selection of raw materials or natural materials is very important to the chemical processing industry. Many suppliers may satisfy the requirements of the chemical processing industry in terms of quality, service, and technique capability. Green processing, however, will impact the protection and the environmental quality of the commodity and the company’s growth prospects. The information in this case study was obtained from the procurement department, which is responsible for a given chemical processing industry’s entire buying process. An interview for the purpose of data collection was held. Four manufacturers replied with a quote with their characteristics reflecting the prescribed commodity. To work with small samples, the suggested weighted grey incidence decision model has been used, and the evidence of supplier selection in the chemical processing industry is only a sample. The weighted grey occurrence decision model approach can also be used to pick the green suppliers for the process industries. In summary, due to the growing understanding of environmental sustainability and its long-term impact on company and marketing results, green supply selection has become an important focus for researchers. We have four criteria and five alternatives here.Quality (C1): management must consider quality control and process development in order to increase the quality of manufacturing. Quality assurance should satisfy consumer needs for optimum resource usage, which is compatible with the interests of an organization. Total quality control and quality-related approvals are taken into consideration, along with ISO 9000, EN 29000, and BS 5750. The standard can also be demonstrated by low toxicity and removal by clients.Service (C2): in the competitive market environment of the 21st century, businesses not only need to try to meet consumer demand for high-quality products at affordable prices, in order to enhance customer satisfaction, they should also perform at a high level of service with high stock management and design ability. Companies can accomplish this goal through fast deliveries, cheap prices, minimum waste, quick reaction, improved quality, lower inventories, no harm, few errors, high morale of employees, and so on.Technique capability (C3): technical ability is the life of an enterprise. The company will be helped by innovative production technology to be a leading partner. Manufacturing skills and the distributor’s latest technological growth strategies are therefore required to achieve the company’s current and future expectations. In addition, it is important to understand the supplier’s latest product nature, technology, flexibility, capability, and pace of production.Green product (C4): there has been a greater emphasis within manufacturers and suppliers on a green competency that has competitive importance and favours the reputation of the industry in recent years. Green packaging is a form of packaging that, with the use of environmentally sustainable materials that can be recycled or reused, attempts to preserve the environment.The data are given in the decision Tables1–3. Each entry of the decision table is of the form;
(75)Ξi=κRij,✠κRij,δRij,✠δRij,ξRij,✠ξRij,(76)i=1,⋯,n;j=1,⋯,m.Step 1.
The three experts analyze alternatives in the context of the CSFS, and their respective rating values are listed in the decision matrix shown in Tables1–3, respectively. In these tables, the rating value of expert E1 for alternative ℕ1 under C1 attribute is given as 0.6,2π0.4,0.2,2π0.2,0.1,2π0.2, which describes that the first expert agrees 60 percent on the suitability of model ℕ1 under C1,20 percent are neutral (may be agree or disagree) and disagree with 10 percent on the suitability of model ℕ1 under C1. The other data values can be viewed in the same way.Step 2.
Different assessments of the expertsRijkk=1,2,3 are aggregated into a collective one Ξiji=1,⋯,5;j=1,⋯,4 taking the function gx=x2 and using CSFOWPA operator. The obtained values are given in Table 4.Step 3.
If we used a CSFOWPG operator to aggregate different assessments of the expertsRijkk=1,2,3, aggregated into a collective one Ξiji=1,⋯,5;j=1,⋯,4 taking the function gx=x2, then the values obtained using this operator are given in Table 5.Step 4.
In this step, we used equations (71) and (72) of the entropy to obtain the attribute weights
(77)ψ=0.25,0.20,0.15,0.40T.Step 5.
Now, we have been using the CSFWPA operator to aggregate the different valuesΞijj=1,⋯,4, obtained from Table 4, with the corresponding weight vector ψ=0.25,0.20,0.15,0.40T. Then, the total values of each alternative ℕii=1,⋯,5 are
(78)ℕ1=0.46,2π0.51,0.16,2π0.22,0.28,2π0.20,(79)ℕ2=0.63,2π0.46,0.21,2π0.18,0.15,2π0.31,(80)ℕ3=0.47,2π0.43,0.24,2π0.18,0.20,2π0.37,(81)ℕ4=0.46,2π0.47,0.24,2π0.23,0.21,2π0.25,(82)ℕ5=0.43,2π0.41,0.29,2π0.19,0.23,2π0.28.Step 6.
If we used the CSFWPG operator to aggregate the different valuesΞijj=1,⋯,4, obtained from Table 5, with the corresponding weight vector ψ=0.25,0.20,0.15,0.40T. Then, the total values of each alternative ℕii=1,⋯,5 are
(83)ℕ1=0.26,2π0.27,0.36,2π0.34,0.39,2π0.42,(84)ℕ2=0.32,2π0.17,0.36,2π0.40,0.31,2π0.34,(85)ℕ3=0.30,2π0.23,0.32,2π0.32,0.40,2π0.46,(86)ℕ4=0.25,2π0.27,0.41,2π0.34,0.31,2π0.39,(87)ℕ5=0.15,2π0.24,0.43,2π0.34,0.38,2π0.40.Step 7.
The score value of the alternativeℕii=1,⋯,5 is computed on the basis of the overall assessment values of Ξii=1,⋯,5 as follows,
(88)Scℕ1=0.894,Scℕ2=1.028,Scℕ3=0.763,Scℕ4=0.949,Scℕ5=0.818.Table 1
Preferences for alternatives provided by expertE1.
C1ℕ10.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ20.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ50.3,2π0.5,0.1,2π0.2,0.4,2π0.1C2ℕ10.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ20.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C3ℕ10.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ20.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C4ℕ10.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ20.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ30.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ40.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ50.1,2π0.3,0.5,2π0.1,0.3,2π0.6Table 2
Preferences for alternatives provided by expertE2.
C1ℕ10.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ20.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ30.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ40.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ50.2,2π0.1,0.3,2π0.4,0.5,2π0.3C2ℕ10.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ20.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ30.3,2π0.2,0.4,2π0.3,0.3,2π0.4ℕ40.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ50.5,2π0.5,0.1,2π0.2,0.4,2π0.1C3ℕ10.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ20.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C4ℕ10.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ20.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ30.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ40.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ50.5,2π0.2,0.4,2π0.3,0.1,2π0.4Table 3
Preferences for alternatives provided by expertE3.
C1ℕ10.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ20.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ30.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ40.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C2ℕ10.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ20.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ30.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ40.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ50.2,2π0.1,0.3,2π0.4,0.5,2π0.3C3ℕ10.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ20.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ30.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ40.2,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ50.1,2π0.3,0.5,2π0.1,0.3,2π0.6C4ℕ10.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ20.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ30.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ40.4,2π0.4,0.2,2π0.2,0.2,2π0.2ℕ50.5,2π0.2,0.4,2π0.3,0.1,2π0.4Table 4
Aggregated values using CSFOWPA operator.
C1ℕ10.35,2π0.18,0.21,2π0.37,0.36,2π0.37ℕ20.54,2π0.21,0.22,2π0.25,0.21,2π0.35ℕ30.25,2π0.37,0.48,2π0.12,0.21,2π0.45ℕ40.46,2π0.62,0.11,2π0.16,0.38,2π0.11ℕ50.54,2π0.57,0.14,2π0.23,0.18,2π0.15C2ℕ10.54,2π0.32,0.32,2π0.24,0.13,2π0.23ℕ20.53,2π0.42,0.25,2π0.28,0.17,2π0.47ℕ30.46,2π0.62,0.12,2π0.19,0.36,2π0.21ℕ40.63,2π0.53,0.19,2π0.21,0.13,2π0.26ℕ50.36,2π0.28,0.25,2π0.38,0.37,2π0.30C3ℕ10.46,2π0.62,0.11,2π0.20,0.32,2π0.15ℕ20.73,2π0.52,0.23,2π0.18,0.10,2π0.23ℕ30.52,2π0.32,0.26,2π0.27,0.21,2π0.42ℕ40.37,2π0.31,0.37,2π0.28,0.12,2π0.34ℕ50.24,2π0.41,0.45,2π0.17,0.22,2π0.31C4ℕ10.45,2π0.62,0.12,2π0.17,0.34,2π0.15ℕ20.62,2π0.52,0.17,2π0.13,0.13,2π0.27ℕ30.53,2π0.32,0.26,2π0.21,0.15,2π0.44ℕ40.35,2π0.26,0.38,2π0.31,0.23,2π0.37ℕ50.25,2π0.47,0.46,2π0.13,0.22,2π0.35Table 5
Aggregated values using CSFOWPG operator.
C1ℕ10.26,2π0.37,0.31,2π0.38,0.43,2π0.27ℕ20.43,2π0.19,0.37,2π0.35,0.20,2π0.42ℕ30.51,2π0.32,0.31,2π0.36,0.17,2π0.28ℕ40.10,2π0.25,0.48,2π0.22,0.37,2π0.52ℕ50.24,2π0.17,0.34,2π0.45,0.40,2π0.34C2ℕ10.17,2π0.12,0.48,2π0.31,0.32,2π0.43ℕ20.45,2π0.38,0.37,2π0.22,0.29,2π0.46ℕ30.21,2π0.24,0.52,2π0.32,0.36,2π0.38ℕ40.36,2π0.32,0.21,2π0.36,0.33,2π0.32ℕ50.38,2π0.22,0.26,2π0.37,0.37,2π0.13C3ℕ10.12,2π0.34,0.46,2π0.21,0.31,2π0.53ℕ20.27,2π0.36,0.21,2π0.32,0.48,2π0.28ℕ30.31,2π0.15,0.32,2π0.27,0.20,2π0.52ℕ40.24,2π0.21,0.52,2π0.41,0.16,2π0.36ℕ50.45,2π0.31,0.31,2π0.26,0.21,2π0.40C4ℕ10.13,2π0.25,0.41,2π0.36,0.32,2π0.37ℕ20.21,2π0.18,0.27,2π0.31,0.47,2π0.51ℕ30.27,2π0.34,0.20,2π0.28,0.51,2π0.47ℕ40.41,2π0.32,0.36,2π0.34,0.24,2π0.25ℕ50.33,2π0.13,0.41,2π0.39,0.16,2π0.37And the score value of the alternativeℕii=1,⋯,5, based on the aggregated values of Step 6 as follows;
(89)Scℕ1=0.462,Scℕ2=0.614,Scℕ3=0.314,Scℕ4=0.586,Scℕ5=0.398.Step 8.
The ranking of all feasible alternativesℕii=1,⋯,5 is shown as
(90)ℕ2>ℕ4>ℕ1>ℕ5>ℕ3.Thus, the best alternative isℕ2. Hence, ℕ2 is the most optimal device.Now, in Table6, we take preferences for v alternatives provided by expert Ep.Table 6
Rating values of the transferred alternativeℕ5/ by each expert.
C1E10.1,2π0.1,0.4,2π0.3,0.4,2π0.4E20.4,2π0.2,0.3,2π0.3,0.1,2π0.4E30.2,2π0.1,0.3,2π0.4,0.5,2π0.3C2E10.3,2π0.2,0.5,2π0.3,0.1,2π0.4E20.2,2π0.1,0.3,2π0.4,0.5,2π0.3E30.3,2π0.5,0.1,2π0.2,0.4,2π0.1C3E10.4,2π0.2,0.3,2π0.3,0.1,2π0.4E20.2,2π0.3,0.6,2π0.1,0.2,2π0.5E30.3,2π0.2,0.1,2π0.4,0.2,2π0.3C4E10.4,2π0.2,0.2,2π0.3,0.2,2π0.5E20.1,2π0.3,0.3,2π0.1,0.4,2π0.6E30.6,2π0.1,0.2,2π0.5,0.1,2π0.4


### 7.2. Validity Test

Since different MAGDM approaches give a different assessment (ranking), when applied to the same DM problem, which contributes to uncertain results. Thus, Wang and Triantaphyllou [87] presented the following test conditions in order to examine the reliability and validity of the MAGDM approaches:Test criteria 1:the MAGDM approach is effective if the choice of the best alternative stays the same and replaces the nonoptimal alternative to the other worse alternative without changing the relative value of each decision attribute.Test criteria 2: an efficient MAGDM approach should obey transitive properties.Test criteria 3: the MAGDM approach is effective when decomposing the MAGDM problem into subproblems and applying the proposed MAGDM approach to these subproblems for the ranking of alternatives. The overall ranking of the alternatives is identical to the overall ranking of the problem.The validity of the proposed solution is checked using the following attributes.


#### 7.2.1. Validity Check with Criteria1

The nonoptimal alternativeℕ5 is replaced by the worse alternative ℕ5/ in the original decision matrix for each expert in order to test the validity of the established method with criteria 1, and the rating values are given in Table 6.Now, using the CSFOWPA operator in Step2 and the CSFWPA operator in Step 4 for these changed results, we get the computed the scores of the alternatives are Scℕ1=0.582,Scℕ2=0.761,Scℕ3=0.383,Scℕ4=0.645, and Scℕ5=0.562. Thus, the final ranking of the alternatives suggests that ℕ2 is still the best alternative and that the method built satisfies the test criteria 1.


#### 7.2.2. Validity Check with Criteria2 and 3

In order to test the defined MAGDM method with criteria2 and 3, we decomposed the original DM problem into sub-DM problems, which have included these alternatives ℕ1,ℕ2,ℕ3,ℕ4,ℕ2,ℕ3,ℕ4,ℕ5, and ℕ1,ℕ3,ℕ4,ℕ5. When we apply the suggested MAGDM approach to these subproblems, we will get the rating of alternatives as ℕ2>ℕ4>ℕ1>ℕ3,ℕ2>ℕ4>ℕ5>ℕ3, and ℕ4>ℕ1>ℕ5>ℕ3. By introducing the ranking of alternatives to these smaller problems, we obtain the final ranking order as ℕ2>ℕ4>ℕ1>ℕ5>ℕ3. This is the same as a nondecomposed problem and reveals a transitive property. The defined MAGDM method is therefore valid with criteria 2 and criteria 3.


### 7.3. Comparative Analysis

In this portion, we compare the output of the defined MAGDM method with some of the current methods to SFS, Cq-ROFS, and CSFS theory. For it, first, the priorities considered by the experts are translated into SFNs, Cq-ROFS by taking the phase terms corresponding to CSFN and Cq-ROFS as zero. We also solve a numerical case based on established operators and also for existing operators to demonstrate the effectiveness of the investigated work in the MAGDM environment. Ashraf et al. [33], Zeng et al. [34], Yang et al. [81], and Liu et al. [82] developed existing approaches for SFSs, Cq-ROFS, and CSFSs; Ali et al. [83], Akram et al. [84], Akram et al. [85], and Mahmood et al. [88] are, respectively, using various types of aggregation operators. On the basis of these information, we applied the current methods. The obtained results are given in Table 7.Table 7
Existing methods and their ranking.
MethodsRankingAshraf et al. [33]ℕ2>ℕ4>ℕ1>ℕ5>ℕ3Zeng et al. [34]ℕ2>ℕ5>ℕ4>ℕ1>ℕ3Yang et al. [81]ℕ1>ℕ2>ℕ4>ℕ5>ℕ3Liu et al. [82]ℕ2>ℕ4>ℕ1>ℕ5>ℕ3Ali et al. [83]ℕ5>ℕ3>ℕ2>ℕ4>ℕ1Akram er al. [84]ℕ2>ℕ4>ℕ5>ℕ1>ℕ3Akram et al. [85]ℕ3>ℕ2>ℕ4>ℕ5>ℕ1Mahmood et al. [88]ℕ4>ℕ2>ℕ1>ℕ3>ℕ5It is known from the above analysis that the best alternative is different from the suggested order ranking. It is due to the matter that the SFS includes information with a positive, neutral, and negative membership grades and only the amplitude term considered, which causes a loss of information during execution. Moreover, CSFS generalizes existing approaches such as CSFS, Cq-ROFS, CPyFS, CIFS, and CFS by considering more object-related information during the decision process. For example, CSFS provides information (positive, neutral, and negative membership grades are complex valued) with amplitude and phase terms as CPS (contains only complex valued positive and negative membership grades) and SFS (with real-value positive, neutral and negative membership grades and only considered amplitude term). Therefore, the developed aggregation operators in the CSFS setting are more generalized than the current operators.


## 7.1. Description of the Problem

A realistic implementation of green supplier selection for some chemical processing industry is considered in this section. The chemical processing industry’s business is to alter the chemical composition of natural materials in order to extract goods of importance to other businesses or in everyday life. Chemicals are made from these raw materials, which are mostly minerals, metals, and hydrocarbons, in a series of refining phases. Further processing, such as mixing and blending, is needed to transform them into final products (e.g., paints, adhesives, drugs, and cosmetics). It is known that the supplier selection of raw materials or natural materials is very important to the chemical processing industry. Many suppliers may satisfy the requirements of the chemical processing industry in terms of quality, service, and technique capability. Green processing, however, will impact the protection and the environmental quality of the commodity and the company’s growth prospects. The information in this case study was obtained from the procurement department, which is responsible for a given chemical processing industry’s entire buying process. An interview for the purpose of data collection was held. Four manufacturers replied with a quote with their characteristics reflecting the prescribed commodity. To work with small samples, the suggested weighted grey incidence decision model has been used, and the evidence of supplier selection in the chemical processing industry is only a sample. The weighted grey occurrence decision model approach can also be used to pick the green suppliers for the process industries. In summary, due to the growing understanding of environmental sustainability and its long-term impact on company and marketing results, green supply selection has become an important focus for researchers. We have four criteria and five alternatives here.Quality (C1): management must consider quality control and process development in order to increase the quality of manufacturing. Quality assurance should satisfy consumer needs for optimum resource usage, which is compatible with the interests of an organization. Total quality control and quality-related approvals are taken into consideration, along with ISO 9000, EN 29000, and BS 5750. The standard can also be demonstrated by low toxicity and removal by clients.Service (C2): in the competitive market environment of the 21st century, businesses not only need to try to meet consumer demand for high-quality products at affordable prices, in order to enhance customer satisfaction, they should also perform at a high level of service with high stock management and design ability. Companies can accomplish this goal through fast deliveries, cheap prices, minimum waste, quick reaction, improved quality, lower inventories, no harm, few errors, high morale of employees, and so on.Technique capability (C3): technical ability is the life of an enterprise. The company will be helped by innovative production technology to be a leading partner. Manufacturing skills and the distributor’s latest technological growth strategies are therefore required to achieve the company’s current and future expectations. In addition, it is important to understand the supplier’s latest product nature, technology, flexibility, capability, and pace of production.Green product (C4): there has been a greater emphasis within manufacturers and suppliers on a green competency that has competitive importance and favours the reputation of the industry in recent years. Green packaging is a form of packaging that, with the use of environmentally sustainable materials that can be recycled or reused, attempts to preserve the environment.The data are given in the decision Tables1–3. Each entry of the decision table is of the form;
(75)Ξi=κRij,✠κRij,δRij,✠δRij,ξRij,✠ξRij,(76)i=1,⋯,n;j=1,⋯,m.Step 1.
The three experts analyze alternatives in the context of the CSFS, and their respective rating values are listed in the decision matrix shown in Tables1–3, respectively. In these tables, the rating value of expert E1 for alternative ℕ1 under C1 attribute is given as 0.6,2π0.4,0.2,2π0.2,0.1,2π0.2, which describes that the first expert agrees 60 percent on the suitability of model ℕ1 under C1,20 percent are neutral (may be agree or disagree) and disagree with 10 percent on the suitability of model ℕ1 under C1. The other data values can be viewed in the same way.Step 2.
Different assessments of the expertsRijkk=1,2,3 are aggregated into a collective one Ξiji=1,⋯,5;j=1,⋯,4 taking the function gx=x2 and using CSFOWPA operator. The obtained values are given in Table 4.Step 3.
If we used a CSFOWPG operator to aggregate different assessments of the expertsRijkk=1,2,3, aggregated into a collective one Ξiji=1,⋯,5;j=1,⋯,4 taking the function gx=x2, then the values obtained using this operator are given in Table 5.Step 4.
In this step, we used equations (71) and (72) of the entropy to obtain the attribute weights
(77)ψ=0.25,0.20,0.15,0.40T.Step 5.
Now, we have been using the CSFWPA operator to aggregate the different valuesΞijj=1,⋯,4, obtained from Table 4, with the corresponding weight vector ψ=0.25,0.20,0.15,0.40T. Then, the total values of each alternative ℕii=1,⋯,5 are
(78)ℕ1=0.46,2π0.51,0.16,2π0.22,0.28,2π0.20,(79)ℕ2=0.63,2π0.46,0.21,2π0.18,0.15,2π0.31,(80)ℕ3=0.47,2π0.43,0.24,2π0.18,0.20,2π0.37,(81)ℕ4=0.46,2π0.47,0.24,2π0.23,0.21,2π0.25,(82)ℕ5=0.43,2π0.41,0.29,2π0.19,0.23,2π0.28.Step 6.
If we used the CSFWPG operator to aggregate the different valuesΞijj=1,⋯,4, obtained from Table 5, with the corresponding weight vector ψ=0.25,0.20,0.15,0.40T. Then, the total values of each alternative ℕii=1,⋯,5 are
(83)ℕ1=0.26,2π0.27,0.36,2π0.34,0.39,2π0.42,(84)ℕ2=0.32,2π0.17,0.36,2π0.40,0.31,2π0.34,(85)ℕ3=0.30,2π0.23,0.32,2π0.32,0.40,2π0.46,(86)ℕ4=0.25,2π0.27,0.41,2π0.34,0.31,2π0.39,(87)ℕ5=0.15,2π0.24,0.43,2π0.34,0.38,2π0.40.Step 7.
The score value of the alternativeℕii=1,⋯,5 is computed on the basis of the overall assessment values of Ξii=1,⋯,5 as follows,
(88)Scℕ1=0.894,Scℕ2=1.028,Scℕ3=0.763,Scℕ4=0.949,Scℕ5=0.818.Table 1
Preferences for alternatives provided by expertE1.
C1ℕ10.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ20.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ50.3,2π0.5,0.1,2π0.2,0.4,2π0.1C2ℕ10.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ20.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C3ℕ10.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ20.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C4ℕ10.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ20.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ30.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ40.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ50.1,2π0.3,0.5,2π0.1,0.3,2π0.6Table 2
Preferences for alternatives provided by expertE2.
C1ℕ10.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ20.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ30.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ40.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ50.2,2π0.1,0.3,2π0.4,0.5,2π0.3C2ℕ10.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ20.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ30.3,2π0.2,0.4,2π0.3,0.3,2π0.4ℕ40.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ50.5,2π0.5,0.1,2π0.2,0.4,2π0.1C3ℕ10.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ20.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ30.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ40.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C4ℕ10.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ20.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ30.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ40.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ50.5,2π0.2,0.4,2π0.3,0.1,2π0.4Table 3
Preferences for alternatives provided by expertE3.
C1ℕ10.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ20.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ30.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ40.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ50.6,2π0.4,0.2,2π0.2,0.1,2π0.2C2ℕ10.5,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ20.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ30.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ40.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ50.2,2π0.1,0.3,2π0.4,0.5,2π0.3C3ℕ10.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ20.6,2π0.4,0.2,2π0.2,0.1,2π0.2ℕ30.4,2π0.2,0.3,2π0.3,0.2,2π0.5ℕ40.2,2π0.2,0.4,2π0.3,0.1,2π0.4ℕ50.1,2π0.3,0.5,2π0.1,0.3,2π0.6C4ℕ10.1,2π0.3,0.5,2π0.1,0.3,2π0.6ℕ20.3,2π0.5,0.1,2π0.2,0.4,2π0.1ℕ30.2,2π0.1,0.3,2π0.4,0.5,2π0.3ℕ40.4,2π0.4,0.2,2π0.2,0.2,2π0.2ℕ50.5,2π0.2,0.4,2π0.3,0.1,2π0.4Table 4
Aggregated values using CSFOWPA operator.
C1ℕ10.35,2π0.18,0.21,2π0.37,0.36,2π0.37ℕ20.54,2π0.21,0.22,2π0.25,0.21,2π0.35ℕ30.25,2π0.37,0.48,2π0.12,0.21,2π0.45ℕ40.46,2π0.62,0.11,2π0.16,0.38,2π0.11ℕ50.54,2π0.57,0.14,2π0.23,0.18,2π0.15C2ℕ10.54,2π0.32,0.32,2π0.24,0.13,2π0.23ℕ20.53,2π0.42,0.25,2π0.28,0.17,2π0.47ℕ30.46,2π0.62,0.12,2π0.19,0.36,2π0.21ℕ40.63,2π0.53,0.19,2π0.21,0.13,2π0.26ℕ50.36,2π0.28,0.25,2π0.38,0.37,2π0.30C3ℕ10.46,2π0.62,0.11,2π0.20,0.32,2π0.15ℕ20.73,2π0.52,0.23,2π0.18,0.10,2π0.23ℕ30.52,2π0.32,0.26,2π0.27,0.21,2π0.42ℕ40.37,2π0.31,0.37,2π0.28,0.12,2π0.34ℕ50.24,2π0.41,0.45,2π0.17,0.22,2π0.31C4ℕ10.45,2π0.62,0.12,2π0.17,0.34,2π0.15ℕ20.62,2π0.52,0.17,2π0.13,0.13,2π0.27ℕ30.53,2π0.32,0.26,2π0.21,0.15,2π0.44ℕ40.35,2π0.26,0.38,2π0.31,0.23,2π0.37ℕ50.25,2π0.47,0.46,2π0.13,0.22,2π0.35Table 5
Aggregated values using CSFOWPG operator.
C1ℕ10.26,2π0.37,0.31,2π0.38,0.43,2π0.27ℕ20.43,2π0.19,0.37,2π0.35,0.20,2π0.42ℕ30.51,2π0.32,0.31,2π0.36,0.17,2π0.28ℕ40.10,2π0.25,0.48,2π0.22,0.37,2π0.52ℕ50.24,2π0.17,0.34,2π0.45,0.40,2π0.34C2ℕ10.17,2π0.12,0.48,2π0.31,0.32,2π0.43ℕ20.45,2π0.38,0.37,2π0.22,0.29,2π0.46ℕ30.21,2π0.24,0.52,2π0.32,0.36,2π0.38ℕ40.36,2π0.32,0.21,2π0.36,0.33,2π0.32ℕ50.38,2π0.22,0.26,2π0.37,0.37,2π0.13C3ℕ10.12,2π0.34,0.46,2π0.21,0.31,2π0.53ℕ20.27,2π0.36,0.21,2π0.32,0.48,2π0.28ℕ30.31,2π0.15,0.32,2π0.27,0.20,2π0.52ℕ40.24,2π0.21,0.52,2π0.41,0.16,2π0.36ℕ50.45,2π0.31,0.31,2π0.26,0.21,2π0.40C4ℕ10.13,2π0.25,0.41,2π0.36,0.32,2π0.37ℕ20.21,2π0.18,0.27,2π0.31,0.47,2π0.51ℕ30.27,2π0.34,0.20,2π0.28,0.51,2π0.47ℕ40.41,2π0.32,0.36,2π0.34,0.24,2π0.25ℕ50.33,2π0.13,0.41,2π0.39,0.16,2π0.37And the score value of the alternativeℕii=1,⋯,5, based on the aggregated values of Step 6 as follows;
(89)Scℕ1=0.462,Scℕ2=0.614,Scℕ3=0.314,Scℕ4=0.586,Scℕ5=0.398.Step 8.
The ranking of all feasible alternativesℕii=1,⋯,5 is shown as
(90)ℕ2>ℕ4>ℕ1>ℕ5>ℕ3.Thus, the best alternative isℕ2. Hence, ℕ2 is the most optimal device.Now, in Table6, we take preferences for v alternatives provided by expert Ep.Table 6
Rating values of the transferred alternativeℕ5/ by each expert.
C1E10.1,2π0.1,0.4,2π0.3,0.4,2π0.4E20.4,2π0.2,0.3,2π0.3,0.1,2π0.4E30.2,2π0.1,0.3,2π0.4,0.5,2π0.3C2E10.3,2π0.2,0.5,2π0.3,0.1,2π0.4E20.2,2π0.1,0.3,2π0.4,0.5,2π0.3E30.3,2π0.5,0.1,2π0.2,0.4,2π0.1C3E10.4,2π0.2,0.3,2π0.3,0.1,2π0.4E20.2,2π0.3,0.6,2π0.1,0.2,2π0.5E30.3,2π0.2,0.1,2π0.4,0.2,2π0.3C4E10.4,2π0.2,0.2,2π0.3,0.2,2π0.5E20.1,2π0.3,0.3,2π0.1,0.4,2π0.6E30.6,2π0.1,0.2,2π0.5,0.1,2π0.4


## 7.2. Validity Test

Since different MAGDM approaches give a different assessment (ranking), when applied to the same DM problem, which contributes to uncertain results. Thus, Wang and Triantaphyllou [87] presented the following test conditions in order to examine the reliability and validity of the MAGDM approaches:Test criteria 1:the MAGDM approach is effective if the choice of the best alternative stays the same and replaces the nonoptimal alternative to the other worse alternative without changing the relative value of each decision attribute.Test criteria 2: an efficient MAGDM approach should obey transitive properties.Test criteria 3: the MAGDM approach is effective when decomposing the MAGDM problem into subproblems and applying the proposed MAGDM approach to these subproblems for the ranking of alternatives. The overall ranking of the alternatives is identical to the overall ranking of the problem.The validity of the proposed solution is checked using the following attributes.


### 7.2.1. Validity Check with Criteria1

The nonoptimal alternativeℕ5 is replaced by the worse alternative ℕ5/ in the original decision matrix for each expert in order to test the validity of the established method with criteria 1, and the rating values are given in Table 6.Now, using the CSFOWPA operator in Step2 and the CSFWPA operator in Step 4 for these changed results, we get the computed the scores of the alternatives are Scℕ1=0.582,Scℕ2=0.761,Scℕ3=0.383,Scℕ4=0.645, and Scℕ5=0.562. Thus, the final ranking of the alternatives suggests that ℕ2 is still the best alternative and that the method built satisfies the test criteria 1.


### 7.2.2. Validity Check with Criteria2 and 3

In order to test the defined MAGDM method with criteria2 and 3, we decomposed the original DM problem into sub-DM problems, which have included these alternatives ℕ1,ℕ2,ℕ3,ℕ4,ℕ2,ℕ3,ℕ4,ℕ5, and ℕ1,ℕ3,ℕ4,ℕ5. When we apply the suggested MAGDM approach to these subproblems, we will get the rating of alternatives as ℕ2>ℕ4>ℕ1>ℕ3,ℕ2>ℕ4>ℕ5>ℕ3, and ℕ4>ℕ1>ℕ5>ℕ3. By introducing the ranking of alternatives to these smaller problems, we obtain the final ranking order as ℕ2>ℕ4>ℕ1>ℕ5>ℕ3. This is the same as a nondecomposed problem and reveals a transitive property. The defined MAGDM method is therefore valid with criteria 2 and criteria 3.


## 7.2.1. Validity Check with Criteria1

The nonoptimal alternativeℕ5 is replaced by the worse alternative ℕ5/ in the original decision matrix for each expert in order to test the validity of the established method with criteria 1, and the rating values are given in Table 6.Now, using the CSFOWPA operator in Step2 and the CSFWPA operator in Step 4 for these changed results, we get the computed the scores of the alternatives are Scℕ1=0.582,Scℕ2=0.761,Scℕ3=0.383,Scℕ4=0.645, and Scℕ5=0.562. Thus, the final ranking of the alternatives suggests that ℕ2 is still the best alternative and that the method built satisfies the test criteria 1.


## 7.2.2. Validity Check with Criteria2 and 3

In order to test the defined MAGDM method with criteria2 and 3, we decomposed the original DM problem into sub-DM problems, which have included these alternatives ℕ1,ℕ2,ℕ3,ℕ4,ℕ2,ℕ3,ℕ4,ℕ5, and ℕ1,ℕ3,ℕ4,ℕ5. When we apply the suggested MAGDM approach to these subproblems, we will get the rating of alternatives as ℕ2>ℕ4>ℕ1>ℕ3,ℕ2>ℕ4>ℕ5>ℕ3, and ℕ4>ℕ1>ℕ5>ℕ3. By introducing the ranking of alternatives to these smaller problems, we obtain the final ranking order as ℕ2>ℕ4>ℕ1>ℕ5>ℕ3. This is the same as a nondecomposed problem and reveals a transitive property. The defined MAGDM method is therefore valid with criteria 2 and criteria 3.


## 7.3. Comparative Analysis

In this portion, we compare the output of the defined MAGDM method with some of the current methods to SFS, Cq-ROFS, and CSFS theory. For it, first, the priorities considered by the experts are translated into SFNs, Cq-ROFS by taking the phase terms corresponding to CSFN and Cq-ROFS as zero. We also solve a numerical case based on established operators and also for existing operators to demonstrate the effectiveness of the investigated work in the MAGDM environment. Ashraf et al. [33], Zeng et al. [34], Yang et al. [81], and Liu et al. [82] developed existing approaches for SFSs, Cq-ROFS, and CSFSs; Ali et al. [83], Akram et al. [84], Akram et al. [85], and Mahmood et al. [88] are, respectively, using various types of aggregation operators. On the basis of these information, we applied the current methods. The obtained results are given in Table 7.Table 7
Existing methods and their ranking.
MethodsRankingAshraf et al. [33]ℕ2>ℕ4>ℕ1>ℕ5>ℕ3Zeng et al. [34]ℕ2>ℕ5>ℕ4>ℕ1>ℕ3Yang et al. [81]ℕ1>ℕ2>ℕ4>ℕ5>ℕ3Liu et al. [82]ℕ2>ℕ4>ℕ1>ℕ5>ℕ3Ali et al. [83]ℕ5>ℕ3>ℕ2>ℕ4>ℕ1Akram er al. [84]ℕ2>ℕ4>ℕ5>ℕ1>ℕ3Akram et al. [85]ℕ3>ℕ2>ℕ4>ℕ5>ℕ1Mahmood et al. [88]ℕ4>ℕ2>ℕ1>ℕ3>ℕ5It is known from the above analysis that the best alternative is different from the suggested order ranking. It is due to the matter that the SFS includes information with a positive, neutral, and negative membership grades and only the amplitude term considered, which causes a loss of information during execution. Moreover, CSFS generalizes existing approaches such as CSFS, Cq-ROFS, CPyFS, CIFS, and CFS by considering more object-related information during the decision process. For example, CSFS provides information (positive, neutral, and negative membership grades are complex valued) with amplitude and phase terms as CPS (contains only complex valued positive and negative membership grades) and SFS (with real-value positive, neutral and negative membership grades and only considered amplitude term). Therefore, the developed aggregation operators in the CSFS setting are more generalized than the current operators.


## 8. Conclusion

The purpose of this article is to provide information on the CSFS for various types of PA operators involved in the DM process. The complex Pythagorean fuzzy sets and the complex q-rung orthopair fuzzy sets are inferior to the CSFSs in terms of expressing uncertain information. Previously, various existing aggregation operators were introduced to SFSs where the collection of positive, neutral, and negative degree corresponds to the subset of real numbers. However, because the real part (also for imaginary part) of the truth and real part (also for imaginary part) of the falsity grades are limited to the unit interval, and CSFS is more general, allowing it to deal with difficult and complicated information. This condition was clarified in the presented work by considering the CSFSs, where the range of positive, neutral, and negative degree extends from the real numbers to the complex numbers with the unit disk. Keeping these points in mind with the CSFS details, we present a number of power averaging and geometric AOs, such as CSFPA, CSFWPA, CSFOWPA, CSFPG, CSFWPG, and CSFOWPG and study the basic properties of these operators. For obtaining objective weights of decision attributes, entropy measures are used. In addition, using these operators, a DM algorithm is introduced to find the best alternative under CSFN information. A practical example of green supplier selection is given to explain the defined method, and its results are in contrast with some of the other current methods to demonstrate the validity of the approach. It is also noted that existing operators can be considered as a special case of the developed measure under the CSFS information.In the future, the result of this work may be extended to many other fields of fuzzy set such as Archimedean Bonferroni Operators, covering-based spherical fuzzy rough set model, complex spherical fuzzy sets with Bonferroni mean operators, and complex spherical fuzzy VIKOR method.



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*Source: 8315733-2022-03-08.xml*