# Decision-Making Approach Based on Generalized Aggregation Operators with Complex Single-Valued Neutrosophic Hesitant Fuzzy Set Information

Authors: Harish Garg; Zeeshan Ali; Ibrahim M. Hezam; Jeonghwan Gwak
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A strategic decision-making technique can help the decision maker to accomplish and analyze the information in an efficient manner. However, in our real life, an uncertainty will play a dominant role during the information collection phase. To handle such uncertainties in the data, we present a decision-making algorithm under the single-valued neutrosophic (SVN) environment. The SVN is a powerful way to deal the information in terms of three degrees, namely, “truth,” “falsity,” and “indeterminacy,” which all are considered independent. The main objective of this study is divided into three folds. In the first fold, we state the novel concept of complex SVN hesitant fuzzy (CSVNHF) set by incorporating the features of the SVN, complex numbers, and the hesitant element. The various fundamental and algebraic laws of the proposed CSVNHF set are described in details. The second fold is to state the various aggregation operators to obtain the aggregated values of the considered CSVNHF information. For this, we stated several generalized averaging operators, namely, CSVNHF generalized weighted averaging, ordered weighted average, and hybrid average. The various properties of these operators are also stated. Finally, we discuss a multiattribute decision-making (MADM) algorithm based on the proposed operators to address the problems under the CSVNHF environment. A numerical example is given to illustrate the work and compare the results with the existing studies’ results. Also, the sensitivity analysis and advantages of the stated algorithm are given in the work to verify and strengthen the study.


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## Body

## 1. Introduction

The multiattribute decision-making (MADM) method is one of the efficient methods to solve the decision-making problems by considering the different experts, their preferences, and alternatives. The chief objective of this problem is to address the best alternatives, when the information related to them is accessed under the vague and imprecise information. In other words, the decision-making strategy aims to grow the chance of the benefits and reduce the chance of the cost during the decision-making procedure for simplifying genuine life dilemmas. Since its appearance, a huge number of people have worked on decision-making strategies under the presence of a crisp set. However, in several situations, it is very complicated to provide the information related to the objects in terms of precise number, due to the involvement of the uncertainties in the data. To reduce the loss of data during the process, in 1965, Zadeh [1] firstly put forward the theory of fuzzy set (FS), by extending the range of the crisp set (which is {0, 1}) to the unit interval. Due to this beneficial work, a lot of space was created for a decision maker to make a beneficial decision from the family of alternatives. After the successful presentation of the FS theory, a huge number of individuals have described it in the circumstance of different places [2]. As ambiguity and complexity are involved in every region of life, in the presence of these dilemmas, it is very complicated for FS to survive with the old mathematical structure (covered only truth grade (TG)). In several cases, several experts have faced a lot of data in the arrangement of “yes” or “no,” which is very complex for FS to resolve. To reduce the level of the deficiencies and worries, Atanassov [3] changed the shape of FS and put forward the well-known shape, called intuitionistic FS (IFS). IFS is the modified technique of FS, which includes two different terms, called TG Ŧu and falsity grade (FG) Fu with a satiable and strong character in the shape of 0≤Ŧu+Fu≤1. IFS is a different structure from the mathematical structure of FS to switch uncertain data. By taking advantage of the IFSs, several studies have been conducted by various scholars such as interval-valued IFSs [4], distance measures [5], circular IFS [6], and so on.In the IFSs, each element is characterized with two degrees, truth and falsity, to access the information. However, in several real-life situations, very complex ambiguity is encountered during processing the information, and hence under the consideration of these dilemmas, it is very complicated for IFS to survive with the old mathematical structure (in terms of TG and FG only). In other words, sometimes several experts have faced a lot of data in the arrangement of “yes,” “abstinence,” and “no,” which is very complex for IFS to resolve. To reduce the level of such deficiencies, the fundamental mathematical structure of the neutrosophic set (NS) was put forward by Smarandache [7]. NS is one of the massive dominant and reliable techniques which can easily determine the solution to every complicated problem that occurs in genuine life dilemmas The concept of NS is extended to the single-valued NS (SVNS) and its corresponding operators [8] by the researchers. Since its appearance, scholars have studied it under different environments. For instance, in [9], the authors have defined the Dombi weighted aggregation operators for the collections of SVNSs. In [10], the scholars put forward the Bonferroni mean operators for SVNS. In [11], the authors put forward the COPRAS method for SVNS. For more details about the study on NSs, we refer the readers to [12–17] and their corresponding references.In all the studies listed above, almost all the studies were conducted by considering only the real component of the grades of the element. However, the periodic nature of the rating of the expert is not considered in the decision-making process. To address it completely, there is a need to express the rating of the expert from real interval [0, 1] to the unit disc in the complex plane. This idea was highlighted by Ramot et al. [18] in 2002 who presented the concept of complex FS (CFS). In CFS, each object is identified with two degrees TG and FG under complex domain such as t′ei2πθt′ where t′,θt′∈0,1 represent the real and amplitude terms of the expert rating. It is clearly seen that CFS can handle the vague information with one or two sorts of data in the shape of singleton terms. Some application of the CFS towards the decision-making process is summarized in [19]. Again, the scope of the CFS is limited as it considers only the truth degree and fails to consider the falsity degree at the time of the execution. For instance, if some expert diagnosed data like “yes” or “no” and each has two possibilities, then CFS is very complicated for diagnosing the solution of the above scenario. To reduce the above complications, Alkouri and Salleh [20] proposed the complex IFS (CIFS), which includes the two different terms, called TG (t′ei2πθt′) and FG (f′ei2πθf′) in the shape of complex numbers with proficient and well-known characteristics 0≤t′+f′≤1 and 0≤θt′+θf′≤1. To handle problematic and unseen situations, a huge number of people have employed the above theory in different regions, for illustration, the study in [21] includes the distance measures constructed under the CIFSs, while the study in [22] includes the information measures constructed under the CIFS. Further, CIFS theory has been widely applied in different categories such as aggregation operators [23], group theory [24], and generalized geometric operators [25].Since CIFS theory is able to deal only with “yes” or “no” decision in the form of degrees TG and FG, it is unable to deal with the term “abstinence.” For this, a structure of complex NS (CNS) was proposed by Ali and Smarandache [26] by considering the independent membership grades of “yes,” “abstinence,” and “no” over the unit disc of complex plane. The structure of CNS is easily implemented in every region of life which includes ambiguity and awkward sort of data. In order to flexibly share preferences, Torra [27] came up with the idea of hesitant fuzzy set (HFS), which allowed agents to provide multiple membership grades for a specific alternative-criterion pair. By this, the issue of hesitation was handled effectively. Related to MADM problems, several researchers have addressed the problem by using HFS features. For instance, Rodriguez et al. [28] investigated an interesting review on HFS models and its usage in MADM models. Xu and Zhou [29] identified a problem with HFS and designed a consensus building model by considering multiple experts for a specific alternative-criterion pair. In [30], the authors defined the similarity measures based on complex HFS and stated their application to pattern recognition.From the above listed literature, we noted that the several researchers have utilized the advantages of CIFS, HFS, NS, and CNS to address the problems related to the MADM. However, it is noted that all these theories are unable to handle some uncertain cases which occur during accessing the decision-making problems. For instance, if a person made committee, for laptop enterprise, which consists of ten members, the head of this committee would like to choose the suitable laptop according to the feasibility and suitability. To get the best one, each committee member provides their opinions about different laptops in terms of their prices and name of the model. As the model and price of the laptop change frequently over time, there exist a lot of uncertainties during the execution. Under such circumstances, it is difficult to access the information using several existing sets. To address it completely, in this article, we have presented an extension of the NSs by keeping the features of hesitant set and complex membership degree and defined the novel set named as complex single-valued neutrosophic hesitant fuzzy set (CSVNHFS). The idea behind this set is to address the ambiguity in the data when it is arranged in the form of “yes,” “abstinence,” and “no” under the complex domain. In the presented set, each element is characterized with three independent hesitant degrees, namely, TG (t′eiθt′), abstinence (a′eiθa′), and FG (f′eiθf′), over the unit disc of complex plane with the conditions 0≤t′+a′+f′≤3 and + where 0≤t′,a′,f′≤1 and 0≤θt′,θa′,θf′≤2. After managing the information under such features and to state more information about it, we define various operational laws and study their characteristics. To explore about the laws, we stated several weighted averaging operators to aggregate the collective information into a single one. Additionally, we state a MADM algorithm to explain the working of the proposed work and demonstrate it with the help of numerical examples. The major advantages of the proposed set are that several existing theories are considered as a special case of the proposed one. For instance, by removing the components θt′,θa′,θf′ during the information phase, the proposed set reduces to SVNS. On the other hand, when we set θa′=θf′=0, then the set reduces to CHFS. Similarly, when we set θa′=0 and all other degrees as a single number, then it reduces to CIFS. Finally, when we consider all the degrees in the form of singleton set, then the proposed CSVNHFS reduces to CSVNS, while when we set θt′=θa′=θf′=0, then the set reduces to SVN hesitant fuzzy set.In this paper, the main contribution of the present work is summarized as follows:(1)
To present a new concept named as CSVNHFS to address the uncertainties in the data and hence describe their algebraic and operational laws.(2)
To initiate several generalized averaging operators, namely, CSVNHF generalized weighted averaging, ordered weighted average, and hybrid average, denoted by CSVNHFGWA, CSVNHFGOWA, and CSVNHFGHWA, respectively(3)
To discuss the MADM technique under the presence of stated work. Also, to show the flexibility of the stated operators, several important results and their properties are also elaborated.(4)
A numerical example is given to illustrate the work and compare the results with the existing studies’ results. Also, the sensitivity analysis and advantages of the stated algorithm are given in the work to verify and strengthen the study.The rest of the work is organized as follows. In Section2, we revise various prevailing concepts like FSs, CFSs, NSs, SVNSs, CNSs, HFSs, generalized weighted averaging (GWA), generalized ordered weighted averaging (GOWA), generalized hybrid averaging (GHA) operators, and their operational laws. In Section 3, we analyze the fundamental theory of the CSVNHF setting and described its algebraic laws. In Section 4, we define the various generalized operators, namely, CSVNHFGWA, CSVNHFGOWA, and CSVNHFGHWA. To show the flexibility of the diagnosed operators, several important results and their properties are also elaborated. In Section 5, a MADM algorithm is stated and illustrated with numerical example. Sensitivity analysis and advantages of the work are also presented to verify and feasibility of the theory. Section 6 draws the conclusion of our study.


## 2. Preliminaries

In this section, some prevailing concepts are revised. LetX, Ŧu,Ąu, and Fu, be fixed set, TG, abstinence, and FG, respectively.Definition 1 (see [1]).
The FS is initiated by(1)F=u,Ŧuu∈X,where 0≤Ŧu≤1.Definition 2 (see [18]).
The CFS is initiated by(2)N=u,Ŧuu∈X,where Ŧu=t′eiθt′ with the conditions 0≤t′≤1 and 0≤θt′≤2.Definition 3 (see [7]).
The NS is initiated by(3)N=u,Ŧu,Ąu,Fuu∈X,with the conditions 0−£Ŧu+Ąu+Fu£3+ and 0−£Ŧu,Ąu,Fu£1+. Further, n=Ŧu,Ąu,Fu represents the NN (neutrosophic number).Definition 4 (see [8]).
The SVNS is initiated by(4)N=u,Ŧu,Ąu,Fuu∈X,with the conditions 0≤Ŧu+Ąu+Fu≤3 and 0≤Ŧu,Ąu,Fu≤1. Further, n=Ŧu,Ąu,Fu represents the single-valued neutrosophic number (SVNN); simply, we write n=Ŧ,Ą,F.Definition 5 (see [26]).
The CNS is initiated by(5)N=u,Ŧu,Ąu,Fuu∈X,where Ŧu=t′eiθt′,Ąu=a′eiθa′, and Fu=f′eiθf′ with the conditions 0−≤t′+a′+f′≤3+ and 0−≤θt′+θa′+θf′≤6+, where 0−≤t′,a′,f′≤1+ and 0−≤θt′,θa′,θf′≤2+. Further, n=Ŧu,Ąu,Fu represents the complex neutrosophic number (CNN); simply, we write n=Ŧ,Ą,F=t′eiθt′,a′eiθa′,f′eiθf′.Definition 6.
(see [27]). A HFS is initiated by(6)E=u,hEu:wherehEuis a finite subset of0,1,is called HFS, where h=hEu is called hesitant fuzzy element (HFE).Definition 7 (see [27]).
Leth,h1, and h2 be three HFEs with γ>0. Then,(1)
h1⊕h2=∐t1∈h1,t2∈h2t1+t2−t1t2.(2)
h1⊗h2=∐t1∈h1,t2∈h2t1t2.(3)
hγ=∏t∈htγ.(4)
γh=∐t∈h1−1−tγ.Definition 8 (see [8]).
The generalized weighted average (GWA) operator is given byGWA:Ωn⟶Ω:(7)GWAn1,n2,…,nn=∑i=1nωiniγ1/γ,where Ω represents the family of all positive integers with γ>0. Further, the weighted vector is denoted and defined by ω=ω1,ω2,…,ωnŦ,ωi∈0,1, where ∑i=1nωi=1.Definition 9 (see [8]).
The generalized ordered weighted average (GOWA) operator is given byGOWA:Ωn⟶Ω:(8)GOWAn1,n2,…,nn=∑i=1nωinoi′γ1/γ,where Ω represents the family of all positive integers with γ>0 and noi′ is the ith largest term of ni, i.e., noi′≤noi−1′. Further, the weighted vector is denoted and defined by ω=ω1,ω2,…,ωnŦ,ωi∈0,1, where ∑i=1nωi=1.Definition 10 (see [8]).
The generalized hybrid weighted average (GHWA) operator is given byGHWA:Ωn⟶Ω:(9)GHWAn1,n2,…,nn=∑i=1nωinoi′γ1/γ,where Ω represents the family of all positive integers with γ>0 and n0i′ is the ith largest term of ni, i.e., noi′≤noi−1′, where ni′=nώini. Further, the weighted vector is denoted and defined by ώ=ώ1,ώ2,…,ώnŦ,ώi∈0,1, where ∑i=1nώi=1, and ω=ω1,ω2,…,ωnŦ,ωi∈0,1, ∑i=1nωi=1.


## 3. Proposed CSVNHFS

In this study, we explored two sets named as CSVNSs and CSVNHFSs and their algebraic laws.


### 3.1. Complex Single-Valued Neutrosophic Fuzzy Set (CSVNFS)

Definition 11.
The CSVNFS is initiated by(10)N=u,Ŧu,Ąu,Fuu∈X,where Ŧu=t′eiθt′,Ąu=a′eiθa′, and Fu=f′eiθf′ with the conditions 0≤t′+a′+f′≤3 and 0≤θt′+θa′+θf′≤6, where 0≤t′,a′,f′≤1 and 0≤θt′,θa′,θf′≤2. Further, u=Ŧu,Ąu,Fu represents the complex single-valued neutrosophic fuzzy number (CSVNFN). Symbolically, n=Ŧ,Ą,F=t′eiθt′,a′eiθa′,f′eiθf′.Definition 12.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNFNs with γ>0. Then,(1)
n1⊕n2=t1′+t2′−t1′t2′eiθt′1+θt′2−θt1′θt2′/2π,a1′a2′eiθa1′θa2′/2πf1′f2′eiθf1′θf2′/2π.(2)
n1⊗n2=t1′t2′eiθt1′θt2′/2π,a1′+a2′−a1′a2′eiθa1′+θa2′−θa1′θa2′/2πf1′+f2′−f1′f2′eiθf1′+θf2′−θf1′θf2′/2π.(3)
γn1=1−1−t1′γei2π1−1−θt1′/2πγ,a1′γeiθa1′γ,f1′γeiθf1′γ.(4)
n1γ=t1′γeiθt′1γ,1−1−a1′γei2π1−1−θa1′/2πγ,1−1−f1′γei2π1−1−θf1′/2πγ.Theorem 1.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNFNs with γ,γ1,γ2>0. Then,(1)
n1⊕n2=n2⊕n1.(2)
n1⊗n2=n2⊗n1.(3)
γn1⊕n2=γn2⊕γn1.(4)
γ1n1⊕γ2n1=γ1+γ2n1.(5)
n1γ⊗n2γ=n1⊗n2γ.(6)
n1γ1⊗n1γ2=n1γ1+γ2.Proof.
It can be easily derived, so we omit it here.Definition 13.
Letn=ti′eiθti′,ai′eiθai′,fi′eiθfi′ be CSVNFN. Then,(11)Şn=16ti′+1−ai′+1−fi′+12θti′+1−θai′+1−θfi′is called the score function (SF), and the accuracy function (AF) is defined as(12)Hn=161−ti′+ai′+fi′+121−θti′+θai′+θfi′.
If we considered the two CSVNFNsn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt′2,a2′eiθa′2,f2′eiθf′2, then(1)
IfŞn1>Şn2, then n1>n2.(2)
IfŞn1<Şn2, then n1<n2.(3)
IfŞn1=Şn2, then n1=n2.(1)
IfHn1>Hn2, then n1>n2.(2)
IfHn1<Hn2, then n1<n2.(3)
IfHn1=Hn2, then n1=n2.


### 3.2. Complex Single-Valued Neutrosophic Hesitant Fuzzy Set (CSVNHFS)

Definition 14.
The CSVNHFS is denoted and defined by(13)N=u,Ŧu,Ąu,Fuu∈X,where Ŧu=t=t′eiθt′/t∈Ŧu,Ąu=a=a′eiθa′/a∈Ąu, and Fu=f=f′eiθf′/f∈Fu with the conditions 0≤maxt′+maxa′+maxf′≤3 and 0≤maxθt′+maxθa′+maxθf′≤6, where 0≤t′,a′,f′≤1 and 0≤θt′,θa′,θf′≤2. Further, n=Ŧu,Ąu,Fu represents the CSVNHFN; simply, we write n=Ŧ,Ą,F=t′eiθt′,a′eiθa′,f′eiθf′.Definition 15.
Letn1=Ŧ1,Ą1,F1=t1′eiθt′1,a1′eiθa′1,f1′eiθf′1 and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNHFNs. Then,(1)
n1∪n2=Ŧ1∪Ŧ2,Ą1∩Ą2,F1∩F2=t1′∨t2′eiθt1′∨θt2′,a1′∧a2′eiθa1′∧θa2′f1′∧f2′eiθf1′∧θf2′.(2)
n1∩n2=Ŧ1∩Ŧ2,Ą1∪Ą2,F1∪F2=t1′∧t2′eiθt1′∧θt2′a1′∨a2′eiθa1′∨θa2′f1′∨f2′eiθf1′∨θf2′.Definition 16.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNHFNs with γ>0. Then,(1)
n1⊕n2=Ŧ1⊕Ŧ2,Ą1⊕Ą2,F1⊕F2=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F1,t2′∈Ŧ2,a2′∈Ą2,f2′∈n2t1′+t2′−t1′t2′eiθt1′+θt2′−θt1′θt2′/2π,a1′a2′eiθa1′θa2′/2πf1′f2′eiθf1′θf2′/2π.(2)
n1⊗n2=Ŧ1⊗Ŧ2,Ą1⊗Ą2,F1⊗F2=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F1t2′∈Ŧ2,a2′∈Ą2,f2′∈F2t1′t2′eiθt1′θt2′/2π,a1′+a2′−a1′a2′eiθa1′+θa2′−θa1′θa2′/2π,f1′+f2′−f1′f2′eiθf1′+θf2′−θf1′θf2′/2π.(3)
γn1=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F11−1−t1′γei2π1−1−θt1′/2πγ,a1′γeiθa1′γ,f1′γeiθf1′γ.(4)
n1γ=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F1t1′γeiθt1′γ,1−1−a1′γei2π1−1−θa1′/2πγ1−1−f1′γei2π1−1−θf1′/2πγ.Theorem 2.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNHFNs with a positive real number γ,γ1,γ2>0. Then,(1)
n1⊕n2=n2⊕n1.(2)
n1⊗n2=n2⊗n1.(3)
γn1⊕n2=γn2⊕γn11/2.(4)
γ1n1⊕n2γ1=γ1+γ2n1.(5)
n1γ⊗n2γ=n1⊗n2γ.(6)
n1γ1⊗n1γ2=n1γ1+γ2.Definition 17.
Letn=ti′eiθti′,ai′eiθai′,fi′eiθfi′ be CSVNHFN. Then,(14)Şn=161α∑i=1αti′+1β∑i=1β1−ai′+1γ∑i=1γ1−fi′+121α∑i=1αθti′+1β∑i=1β2−θai′+1γ∑i=1γ2−θfi′Is called the SF, and the AF is denoted and defined by(15)Hn=161α∑i=1α1−ti′+1β∑i=1βai′+1γ∑i=1γfi′+121α∑i=1α1−θti′+1β∑i=1βθai′+1γ∑i=1γθfi′.
If we considered the two CSVNHFNsn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt′2,a2′eiθa′2,f2′eiθf′2, then(1)
IfŞn1>Şn2, then n1>n2.(2)
IfŞn1<Şn2, then n1<n2.(3)
IfŞn1=Şn2, then n1=n2.(1)
IfHn1>Hn2, then n1>n2.(2)
IfHn1<Hn2, then n1<n2.(3)
IfHn1=Hn2, then n1=n2.


## 3.1. Complex Single-Valued Neutrosophic Fuzzy Set (CSVNFS)

Definition 11.
The CSVNFS is initiated by(10)N=u,Ŧu,Ąu,Fuu∈X,where Ŧu=t′eiθt′,Ąu=a′eiθa′, and Fu=f′eiθf′ with the conditions 0≤t′+a′+f′≤3 and 0≤θt′+θa′+θf′≤6, where 0≤t′,a′,f′≤1 and 0≤θt′,θa′,θf′≤2. Further, u=Ŧu,Ąu,Fu represents the complex single-valued neutrosophic fuzzy number (CSVNFN). Symbolically, n=Ŧ,Ą,F=t′eiθt′,a′eiθa′,f′eiθf′.Definition 12.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNFNs with γ>0. Then,(1)
n1⊕n2=t1′+t2′−t1′t2′eiθt′1+θt′2−θt1′θt2′/2π,a1′a2′eiθa1′θa2′/2πf1′f2′eiθf1′θf2′/2π.(2)
n1⊗n2=t1′t2′eiθt1′θt2′/2π,a1′+a2′−a1′a2′eiθa1′+θa2′−θa1′θa2′/2πf1′+f2′−f1′f2′eiθf1′+θf2′−θf1′θf2′/2π.(3)
γn1=1−1−t1′γei2π1−1−θt1′/2πγ,a1′γeiθa1′γ,f1′γeiθf1′γ.(4)
n1γ=t1′γeiθt′1γ,1−1−a1′γei2π1−1−θa1′/2πγ,1−1−f1′γei2π1−1−θf1′/2πγ.Theorem 1.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNFNs with γ,γ1,γ2>0. Then,(1)
n1⊕n2=n2⊕n1.(2)
n1⊗n2=n2⊗n1.(3)
γn1⊕n2=γn2⊕γn1.(4)
γ1n1⊕γ2n1=γ1+γ2n1.(5)
n1γ⊗n2γ=n1⊗n2γ.(6)
n1γ1⊗n1γ2=n1γ1+γ2.Proof.
It can be easily derived, so we omit it here.Definition 13.
Letn=ti′eiθti′,ai′eiθai′,fi′eiθfi′ be CSVNFN. Then,(11)Şn=16ti′+1−ai′+1−fi′+12θti′+1−θai′+1−θfi′is called the score function (SF), and the accuracy function (AF) is defined as(12)Hn=161−ti′+ai′+fi′+121−θti′+θai′+θfi′.
If we considered the two CSVNFNsn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt′2,a2′eiθa′2,f2′eiθf′2, then(1)
IfŞn1>Şn2, then n1>n2.(2)
IfŞn1<Şn2, then n1<n2.(3)
IfŞn1=Şn2, then n1=n2.(1)
IfHn1>Hn2, then n1>n2.(2)
IfHn1<Hn2, then n1<n2.(3)
IfHn1=Hn2, then n1=n2.


## 3.2. Complex Single-Valued Neutrosophic Hesitant Fuzzy Set (CSVNHFS)

Definition 14.
The CSVNHFS is denoted and defined by(13)N=u,Ŧu,Ąu,Fuu∈X,where Ŧu=t=t′eiθt′/t∈Ŧu,Ąu=a=a′eiθa′/a∈Ąu, and Fu=f=f′eiθf′/f∈Fu with the conditions 0≤maxt′+maxa′+maxf′≤3 and 0≤maxθt′+maxθa′+maxθf′≤6, where 0≤t′,a′,f′≤1 and 0≤θt′,θa′,θf′≤2. Further, n=Ŧu,Ąu,Fu represents the CSVNHFN; simply, we write n=Ŧ,Ą,F=t′eiθt′,a′eiθa′,f′eiθf′.Definition 15.
Letn1=Ŧ1,Ą1,F1=t1′eiθt′1,a1′eiθa′1,f1′eiθf′1 and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNHFNs. Then,(1)
n1∪n2=Ŧ1∪Ŧ2,Ą1∩Ą2,F1∩F2=t1′∨t2′eiθt1′∨θt2′,a1′∧a2′eiθa1′∧θa2′f1′∧f2′eiθf1′∧θf2′.(2)
n1∩n2=Ŧ1∩Ŧ2,Ą1∪Ą2,F1∪F2=t1′∧t2′eiθt1′∧θt2′a1′∨a2′eiθa1′∨θa2′f1′∨f2′eiθf1′∨θf2′.Definition 16.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNHFNs with γ>0. Then,(1)
n1⊕n2=Ŧ1⊕Ŧ2,Ą1⊕Ą2,F1⊕F2=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F1,t2′∈Ŧ2,a2′∈Ą2,f2′∈n2t1′+t2′−t1′t2′eiθt1′+θt2′−θt1′θt2′/2π,a1′a2′eiθa1′θa2′/2πf1′f2′eiθf1′θf2′/2π.(2)
n1⊗n2=Ŧ1⊗Ŧ2,Ą1⊗Ą2,F1⊗F2=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F1t2′∈Ŧ2,a2′∈Ą2,f2′∈F2t1′t2′eiθt1′θt2′/2π,a1′+a2′−a1′a2′eiθa1′+θa2′−θa1′θa2′/2π,f1′+f2′−f1′f2′eiθf1′+θf2′−θf1′θf2′/2π.(3)
γn1=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F11−1−t1′γei2π1−1−θt1′/2πγ,a1′γeiθa1′γ,f1′γeiθf1′γ.(4)
n1γ=∐t1′∈Ŧ1,a1′∈Ą1,f1′∈F1t1′γeiθt1′γ,1−1−a1′γei2π1−1−θa1′/2πγ1−1−f1′γei2π1−1−θf1′/2πγ.Theorem 2.
Letn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt2′,a2′eiθa2′,f2′eiθf2′ be two CSVNHFNs with a positive real number γ,γ1,γ2>0. Then,(1)
n1⊕n2=n2⊕n1.(2)
n1⊗n2=n2⊗n1.(3)
γn1⊕n2=γn2⊕γn11/2.(4)
γ1n1⊕n2γ1=γ1+γ2n1.(5)
n1γ⊗n2γ=n1⊗n2γ.(6)
n1γ1⊗n1γ2=n1γ1+γ2.Definition 17.
Letn=ti′eiθti′,ai′eiθai′,fi′eiθfi′ be CSVNHFN. Then,(14)Şn=161α∑i=1αti′+1β∑i=1β1−ai′+1γ∑i=1γ1−fi′+121α∑i=1αθti′+1β∑i=1β2−θai′+1γ∑i=1γ2−θfi′Is called the SF, and the AF is denoted and defined by(15)Hn=161α∑i=1α1−ti′+1β∑i=1βai′+1γ∑i=1γfi′+121α∑i=1α1−θti′+1β∑i=1βθai′+1γ∑i=1γθfi′.
If we considered the two CSVNHFNsn1=Ŧ1,Ą1,F1=t1′eiθt1′,a1′eiθa1′,f1′eiθf1′ and n2=Ŧ2,Ą2,F2=t2′eiθt′2,a2′eiθa′2,f2′eiθf′2, then(1)
IfŞn1>Şn2, then n1>n2.(2)
IfŞn1<Şn2, then n1<n2.(3)
IfŞn1=Şn2, then n1=n2.(1)
IfHn1>Hn2, then n1>n2.(2)
IfHn1<Hn2, then n1<n2.(3)
IfHn1=Hn2, then n1=n2.


## 4. Some Aggregation Operators Based on CSVNHFSs

In this section, we propose new aggregation operators called CSVNHFGWA operator, CSVNHFGOWA operator, and CSVNHFGHWA operator to aggregate the CSVNHFNs effectively. Throughout the paper,X represents the fixed set and the weighted vector is denoted and defined by ω=ω1,ω2,…,ωnŦ,ωi∈0,1, where ∑i=1nωi=1.Definition 18.
The CSVNHFGWA operator is given byCSVNHFGWA:Ωn⟶Ω:(16)CSVNHFGWAn1,n2,…,nn=∑i=1nωiniγ1/γ,where Ω represents the family of all CSVNHFNs with γ>0. The CSVNHFN is of the form ni=ti′eiθti′,ai′eiθai′,fi′eiθfi′i=1,2,…,n.Theorem 3.
Letni=ti′eiθti′,ai′eiθai′,fi′eiθai′i=1,2,…,n be the family of CSVNHFNs with γ>0. Then, consider the concept of CSVNHFGWA operator, and we get CSVNHFGWAn1,n2,…,nn.(17)1−∏i=1n∐ti′1−ti′γωiei2π1−∏i=1n∐ti′∈Ti1−θti′/2πγωi1/γ1−1−∏i=1n∐ai′∈Ai1−1−ai′γωi1/γei2π1−1−∏i=1n∐ti′∈Ti1−1−θai′/2πγωi1/γ1−1−∏i=1n∐fi′∈Fi1−1−fi′γωi1/γ.Proof.
(1) First, we have proven that(18)∑i=1nωiniγ=1−∏i=1n∐ti′∈Ti1−ti′γωiei2π1−∏i=1n∐ti′1−θfi′/2πγωi∏i=1n∐ai′∈Ai1−1−ai′γωiei2π∏i=1n∐ai′∈Ai1−1−θai′/2πγωi∏i=1n∐fi′∈fi1−1−fi′γωiei2π∏i=1n∐fi′∈fi1−1−θfi′/2πγωi.
We utilize the mathematical induction onn to proof equation (18).Case 1.
If we consideredn=1,(19)n1γ=∐t1′∈T1,a1′∈A1,f1′∈f1t1′γeiθt1′γ,1−1−a1′γei2π1−1−θa1′/2π1−1−f1′γei2π1−1−θf1′/2π,ω1n1γ=∐t1′∈T1,a1′∈A1,f1′∈f11−1−t1′γω1ei2π1−1−θt1′γω11−1−a1′γω1ei2π1−1−θa1′/2πγω11−1−f1′γω1ei2π1−1−θf1′/2πγω1.
It is true forn=1.Case 2.
Ifn=k is right, then(20)∑i=1kωiniγ=1−∏i=1k∐ti′∈Ti1−ti′γωiei2π1−∏i=1k∐ti′∈Ti1−θti′/2πγωi∏i=1k∐ai′∈Ai1−1−ai′γωiei2π∏i=1k∐ai′∈Ai1−1−θai′/2πγωi∏i=1k∐fi′∈fi1−1−fi′γωiei2π∏i=1k∐fi′∈fi1−1−θfi′/2πγωi.
Then, we checked forn=k+1, and we get(21)ωk+1nk+1γ=∐tk+1′∈Tk+1′,ak+1′∈Ak+1′,fk+1′∈Fk+1′1−1−tk+1′γωk+1ei2π1−1−θtk+1′/2πγωk+11−1−ak+1′γωk+1ei2π1−1−θak+1′/2πγωk+11−1−fk+1′γωk+1ei2π1−1−θfk+1′/2πγωk+1,and(22)∑i=1k+1ωiniγ=∏i=1k∐ai′∈Ąi1−1−ai′γωiei2π∏i=1k∐ai′∈Αi˜1−1−θai′/2πγωi,∏i=1k∐fi′∈Fi1−1−fi′γωiei2π∏i=1k∐fi′∈Fi1−1−θfi′/2πγωi1−∏i=1k∐ti′∈Ti1−ti′γωiei2π1−∏i=1k∐ti′∈Ti1−θti′/2πγωi,∐tk+1′∈Τk+1,ak+1′∈Ąk+1,fk+1′∈Fk+11−1−ak+1′γωk+1ei1−1−θa′k+1/2πγωk+1,1−1−fk+1′γωk+1ei1−1−θf′k+1/2πγωk+1,1−1−tk+1′γωk+1ei1−1−θt′k+1/2πγωk+1,=1−∏i=1k+1∐ai′∈Ti1−1−ai′γωiei2π1−∏i=1k+1∐ai′∈Ąi1−θai′/2πγωi1−∏i=1k+1∐ti′∈Ti1−fi′γωiei2π1−∏i=1k+1∐fi′∈∉Fi1−1−θfi′/2πγωi1−∏i=1k+1∐ti′∈Ti1−ti′γωiei2π1−∏i=1k+1∐ti′∈Ti1−θti′/2πγωi.
It is true also forn=k+1, so it is true for all n.
Now, we have(23)CSVNHFGWAn1,n2,…,nn=∑i=0nωini=1−∏i=1n∐ti′∉Τi1−ti′ωiei2π1−∏i=1n∐ti′∉Τi1−θti′/2πωi,1−∏i=1n∐ai′∉Ąiai′ωiei2π1−∏i=1n∐ai′∉Ąiθai′/2πωi,1−∏i=1n∐fi′∉Ffi′ωiei2π1−∏i=1n∐fi′∉Fiθfi′/2πωi,.
Hence, the result is completed.
Next, we state some properties for CSVNHFGWA operator.Theorem 4.
Letn=nii=1,2,…,n be the family of CSVNHFNs with γ>0. Then, CSVNHFGWAn1,n2,…,nn=n.Proof.
Ifn=ni, then(24)CSVNHFGWAn1,n2,…,nn=∑i=1nωiniγ1/γ=∑i=1nωinγ1/γ=nγ∑i=1nγωi1/γ=n.
Hence, the result is completed.Theorem 5.
Letni=ti′eiθti′,ai′eiθai′,fi′eiθfi′i=1,2,…,n be the family of CSVNHFNs with γ>0. If ni≥nj′, then(25)CSVNHFGWAn1,n2,…,nn≥CSVNHFGWAn1′,n2′,…,nn′.Proof.
We consideredni≥ni′, that is, ti′≥ti″,ai′≤ai″,fi′≤fi″ and θti′≥θti′,θai′≤θai′,θfi′≤θfi′ for all i; then, firstly we prove for membership grades such that(26)1−ti′γωiei2π1−θti′/2πγωi≤1−ti″γωiei2π1−θti″′/2πγωi∏i=1n∐ti′∈Ŧi1−ti′γωiei2π∏i=1n∐ti′∈Ŧi1−θti′/2πγωi≤∏i=1n∐ti′∈Ŧi1−ti″γωiei2π∏i=1n∐ti′∈Ŧi1−θti′′/2πγωi1−∏i=1n∐ti′∈Ŧi1−ti′γωi1/γei2π1−∏i=1n∐ti′∈Ŧi1−θti′/2πγωi1/γ≥1−∏i=1n∐ti′∈Ŧi1−ti″γωi1/γei2π1−∏i=1n∐ti′∈Ŧi1−θti′′/2πγωi1/γ.
Similarly, for falsity and non-membership grades, we get(27)1−1−∏i=1n∐ai′∈Ąi1−1−ai′γωi1/γei2π1−1−∏i=1n∐ai′∈Ąi1−1−θai′/2πγωi1/γ≤1−1−∏i=1n∐ai′∈Ąi1−1−ai″γωi1/γei2π1−1−∏i=1n∐ai′∈Ąi1−1−θai′′/2πγωi1/γ,and(28)1−1−∏i=1n∐fi′∈Fi1−1−fi′γωi1/γei2π1−1−∏i=1n∐fi′∈Fi1−1−θfi′/2πγωi1/γ≤1−1−∏i=1n∐fi′∈Fi1−1−fi″γωi1/γei2π1−1−∏i=1n∐fi′∈Fi1−1−θfi′′/2πγωi1/γ.
Hence, we combine the above equations such that(29)1−∏i=1n∐ti′∉Τi1−ti′γωi1/γei2π1−∏i=1n∐ti′∉Τi1−θti′/2πγωi1/γ,1−1−∏i=1n∐ai′∉Ąi1−ai′γωi1/γei2π1−∏i=1n∐ai′∉Ąi1−θai′/2πγωi1/γ,1−1−∏i=1n∐fi′∉Fi1−fi′γωi1/γei2π1−∏i=1n∐fi′∉Fi1−θfi′/2πγωi1/γ,≥1−∏i=1n∐ti′∉Τi1−ti′′γωi1/γei2π1−∏i=1n∐ti′∉Τi1−θ′ti′/2πγωi1/γ,1−1−∏i=1n∐ai′∉Ąi1−ai′′γωi1/γei2π1−∏i=1n∐ai′∉Ąi1−θ′ai′/2πγωi1/γ,1−1−∏i=1n∐fi′∉Fi1−fi′′γωi1/γei2π1−∏i=1n∐fi′∉Fi1−θ′fi′/2πγωi1/γ,
So,(30)CSVNHFGWAn1,n2,…,nn≥CSVNHFGWAn1′,n2′,…,nn′.
Hence, the result is completed.Theorem 6.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n be the family of CSVNHFNs which lies between max and min operators with γ>0. Then,(31)minn1,n2,…,nn≤CSVNHFGWAn1,n2,…,nn≤maxn1,n2,…,nn.Proof.
We know thatα=minn1,n2,…,nn and β=maxn1,n2,…,nn; then,(32)α≤CSVNHFGWAn1,n2,…,nn≤β.
Then, we get(33)∑i=1nωiαγ1/γ≤∑i=1nωiniγ1/γ≤∑i=1nωiβγ1/γ.
This implies that(34)α≤∑i=1nωiniγ1/γ≤β,i.e.,(35)minn1,n2,…,nn≤CSVNHFGWAn1,n2,…,nn≤maxn1,n2,…,nn.
Hence, the result is completed.Remark 1.
The aim of this study is to discover the particular cases of the presented approach.(1)
Ifγ⟶0, our proposed work will be reduced to CSVNHF weighted geometric operator:(36)CSVNHFWGn1,n2,…,nn=∏i=1nniωi=∏i=1n∐ti′∈Ŧiti′ωiei2π∏i=1n∐ti′∈Ŧiθti′/2πωi1−∏i=1n∐ai′∈Ąi1−ai′ωiei2π1−∏i=1n∐ai′∈Ąi1−θai′/2πωi1−∏i=1n∐fi′∈Fi1−fi′ωiei2π1−∏i=1n∐fi′∈Fi1−θfi′/2πωi.(2)
Ifγ=1, our proposed work will be reduced to CSVNHF weighted averaging operator:(37)CSVNHFGWAn1,n2,…,nn=∑i=0nωini=1−∏i=1n∐ti′∉Τi1−ti′ωiei2π1−∏i=1n∐ti′∉Τi1−θti′/2πωi,1−∏i=1n∐ai′∉Ąiai′ωiei2π1−∏i=1n∐ai′∉Ąiθai′/2πωi,1−∏i=1n∐fi′∉Fifi′ωiei2π1−∏i=1n∐fi′∉Fiθfi′/2πωi,.
Ifγ=2, our proposed work will be reduced to CSVNHF weighted quadratic averaging operator:(38)CSVNHFGWQAn1,n2,…,nn=∑i=0nωini1/2=1−∏i=1n∐ti′∉Τi1−ti′21/2ei2π1−∏i=1n∐ti′∉Τi1−θti′/2π2ωi1/2,1−1−∏i=1n∐ai′∉Ąi1−1−ai′2ωi1/2ei2π1−1−∏i=1n∐ai′∉Ąi1−θai′/2π2ωi1/2,1−1−∏i=1n∐fi′∉Fi1−1−fi′2ωi1/2ei2π1−1−∏i=1n∐fi′∉Fi1−θfi′/2π2ωi1/2,.Definition 19.
The CSVNHFGOWA operator is given byCSVNHFGOWA:Ωn⟶Ω:(39)CSVNHFGOWAn1,n2,…,nn=∑i=1nωinoiγ1/γ,where Ω represents the family of all CSVNHFNs with γ>0 and noi is the ordered CSVNHFNs which is an ascending ordered (AO) or descending ordered (DO) i.e., noi≤noi−1. The CSVNHFN is of the form ni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n.Theorem 7.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,..,n be the family of CNHFNs with γ>0. Then, considering the concept of CSVNHFGOWA, we get(40)CSVNHFGOWAn1,n2,…,nn,CSVNHF/GOWAn1,n2,…,nn=1−∏i=1n∐ti′∈Ti1−ti′γωi1/γei2π1−∏i=1n∐ti′∈Ti1−θti′/2πγωi1/γ,1−∏i=1n∐ai′∈Ai1−ai′γωi1/γei2π1−∏i=1n∐ai′∈Ai1−θai′/2πγωi1/γ,1−∏i=1n∐fi′∈Fi1−fi′γωi1/γei2π1−∏i=1n∐fi′∈Fi1−θfi′/2πγωi1/γ,.Proof.
Straightforward.Theorem 8.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n be the family of CSVNHFNs with γ>0. Then,(41)CSVNHFGOWAn1,n2,…,nn=n.Proof.
Straightforward.Theorem 9.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,..,n be the family of CSVNHFNs with γ>0. If ni≥nj′, then(42)CSVNHFGOWAn1,n2,…,nn≥CSVNHFGOWAn1′,n2′,…,nn′.Proof.
Straightforward.Theorem 10.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n be the family of CSVNHFNs which lies between max and min operators with γ>0. Then,(43)minn1,n2,..,nn≤CSVNHFGOWAn1,n2,…,nn≤maxn1,n2,…,nn.Proof.
We know thatα=minn1,n2,…,nn and β=maxn1,n2,…,nn; then,(44)α≤CSVNHFGOWAn1,n2,…,nn≤β.
Then, we get(45)∑i=1nωiαγ1/γ≤∑i=1nωinoiγ1/γ≤∑i=1nωiβγ1/γ.
This implies that(46)α≤∑i=1nωinoiγ1/γ≤β.
That is,(47)minn1,n2,..,nn≤CSVNHFGOWAn1,n2,…,nn≤maxn1,n2,…,nn.
Hence, the result is completed.Remark 2.
The aim of this study is to discover the particular cases of the presented approach.(1)
Ifγ⟶0, our proposed work will be reduced to CSVNHF ordered weighted geometric operator:(48)CSVNHFGOWAn1,n2,…,nn=∏i=1nn0iωi=∏i=1n∐ti′∈Titoi′ωiei2π∏i=1n∐ti′∈Tiθtoi/2πωi∏i=1n∐ai′∈Aiaoi′ωiei2π∏i=1n∐ai′∈Aiθaoi/2πωi∏i=1n∐fi′∈Fifoi′ωiei2π∏i=1n∐fi′∈Fiθfoi/2πωi.(2)
Ifγ=1, our proposed work will be reduced to CSVNHF ordered weighted averaging operator:(49)CSVNHFGOWAn1,n2,…,nn=∏i=1nn0iωi=∏i=1n∐ti′∈Titoi′ωiei2π∏i=1n∐ti′∈Tiθtoi/2πωi∏i=1n∐ai′∈Aiaoi′ωiei2π∏i=1n∐ai′∈Aiθaoi/2πωi∏i=1n∐fi′∈Fifoi′ωiei2π∏i=1n∐fi′∈Fiθfoi/2πωi.(3)
Ifγ=2, the proposed work will be reduced to CSVNHF ordered weighted quadratic averaging operator:(50)CSVNHFGOWAn1,n2,…,nn=∑i=1nωinoi=1−∏i=1n∐ti′∈Titoi′ωiei2π∏i=1n∐ti′∈Tiθtoi/2πωi,1−∏i=1n∐ai′∈Aiaoi′ωiei2π∏i=1n∐ai′∈Aiθaoi/2πωi,1−∏i=1n∐fi′∈Fifoi′ωiei2π∏i=1n∐fi′∈Fiθfoi/2πωi,.Definition 20.
The CSVNHFGHWA operator is given byCSVNHFGHWA:Ωn⟶Ω:(51)CSVNHFGHWAn1,n2,…,nn=∑i=1nωin˙oiγ1/γ,where Ω represents the family of all CSVNHFNs with γ>0 and n˙oi=nώini i.e. non≤noi−1. The CSVNHFN is of the form ni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n. Where ∑i=1nώi=1.Theorem 11.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n be the family of CSVNHFNs with γ>0. Then, considering the concept of CSVNHFGHWA, we get(52)CSVNHFGHWAn1,n2,…,nn=1−∏i=1n∐ti′∈Ti1−toi′γwi1/γei2π1−∏i=1n∐ti′∈Ti1−θt′oi/2πγwi1/γ,1−1−∏i=1n∐ai′∈Ai1−1−a˙oi′γwi1/γei2π1−1−∏i=1n∐ai′∈Ai1−1−θa˙oi′/2πγwi1/γ,1−1−∏i=1n∐fi′∈Fi1−1−f˙oi′γwi1/γei2π1−1−∏i=1n∐fi′∈Fi1−1−θf˙oi′/2πγwi1/γ.Proof.
Straightforward.Theorem 12.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n be the family of CSVNHFNs with γ>0. If ni≥nj′, then(53)CSVNHFGHWAn1,n2,…,nn≥CSVNHFGHWAn1′,n2′,…,nn′.Proof.
Straightforward.Theorem 13.
Letni=ti′eiθti′,ai′eiθti′,fi′eiθti′i=1,2,…,n be the family of CSVNHFNs which lies between max and min operators with γ>0. Then,(54)minn1,n2,…,nn≤CSVNHFGHWAn1,n2,…,nn≤maxn1,n2,…,nn.Proof.
We know thatα=minn1,n2,…,nn and β=maxn1,n2,…,nn; then,(55)α≤CSVNHFGHWAn1,n2,…,nn≤β.
Then, we get(56)∑i=1nωiαγ1/γ≤∑i=1nωin˙oiγ1/γ≤∑i=1nωiβγ1/γ.
This implies that(57)α≤∑i=1nωin˙oiγ1/γ≤β.
That is,(58)minn1,n2,..,nn≤CSVNHFGHWAn1,n2,…,nn≤maxn1,n2,…,nn.
Hence, the result is completed.Remark 3.
The aim of this study is to discover the cases of the presented approach.(1)
Ifγ⟶0, our proposed work will be reduced to CSVNHF hybrid weighted geometric (CSVNHFHWG) operator:(59)CSVNHFGHWAn1,n2,…,nn=∏i=1nn˙oiwi=∏i=1n∐ti′∈Tit˙oi′wiei2π1−∏i=1n∐ti′∈Tiθt′oi/2πwi,1−∏i=1n∐ai′∈Ai1−a˙oi′wiei2π1−∏i=1n∐ai′∈Ai1−θa˙′oi/2πwi,1−∏i=1n∐fi′∈Fi1−f˙oi′wiei2π1−∏i=1n∐fi′∈Fi1−θf˙′oi/2πwi.(2)
Ifγ=1, our proposed work will be reduced to CSVNHF hybrid weighted averaging (CSVNHFHWA) operator:(60)CSVNHFGHWAn1,n2,…,nn=∑i=1nWin˙oi=1−∏i=1n∐ti′∈Ti1−t˙oi′wiei2π1−∏i=1n∐ti′∈Ti1−θt′oi/2πwi,∏i=1n∐ai′∈Ai1−a˙oi′wiei2π∏i=1n∐ai′∈Aiθa˙′oi/2πwi,∏i=1n∐fi′∈Ff˙oi′wiei2π∏i=1n∐fi′∈Fiθf˙oi′/2πwi.(3)
Ifγ=2, our proposed work will be reduced to CSVNHF hybrid weighted quadratic averaging (CSVNHFHWQA) operator:(61)CSVNHFGHWQAn1,n2,…,nn=∑i=1nWin˙oi21/2=1−∏i=1n∐ti′∈Ti1−toi′2wi1/2ei2π1−∏i=1n∐ti′∈Ti1−θt′oi/2π2wi1/21−1−∏i=1n∐ai′∈Ai1−1−a˙oi′2wi1/2ei2π1−1−∏i=1n∐ai′∈Ai1−1−θa˙oi′/2π2wi1/21−1−∏i=1n∐fi′∈F1−1−f˙oi′2wi1/2ei2π1−1−∏i=1n∐fi′∈Fi1−1−θf˙oi′/2π2wi1/2.


## 5. Proposed MADM Method

The decision-making strategy aims to grow the chance of the benefits and reduce the chance of the cost during the decision-making procedure for simplifying genuine life dilemmas. Several people have worked on decision-making strategies under the presence of a crisp set. In several situations, it is very complicated to study the decision-making strategy considering fuzzy sets because they have covered a lot of possibilities. Inspired by the MADM technique, we will employ it here in the presence of the proposed works.


### 5.1. Decision-Making Algorithm

During the decision-making process, ambiguity and complexity always occur in our day-to-day life. This study aims to employ the decision-making strategy in the presence of proposed works. For this, consider a finite number of alternativesX=u1,u2,…,um and attributes C=c1,c2,…,cn. To evaluate each alternative under each attribute, we assign a weight vector for attribute as ω=ω1,ω2,…,ωnŦ,∑i=1nωi=1. Experts are invited to evaluate each alternative Ąi and give their preferences in terms of the CSVNHF information nij=tij,aij,fiji,j=1,2,…,n for criteria Cj where tij=tij′eiθtij′/tij∈Ŧu, aij=aij′eiθaij′/aij∈Ąu and fij=fij′eiθfij′/fij∈Fu, gives the TG, abstinence grade, and FG . Then, the following are the steps summarized to find the best alternative(s).Step 1: collect the data in the shape of the CSVNHF setting and summarize them in the form of the matrix called as decision matrix.Step 2: normalize the collective information, if needed by using the following equation.(62)Drij=tij,aij,fij,for benefit criteria,fij,aij,tij,for cost criteria.Step 3: aggregate the collective information by using stated operators. For instance, utilize equation (7) to aggregate the information rij into ri=ti′eiθti′,ai′eiθai′,fi′eiθfi′.Step 4: compute the crisp value for each numberri=ti′eiθti′,ai′eiθai′,fi′eiθfi′ by using the following equation:(63)Şri=161α∑i=1αti′+1β∑i=1β1−ai′+1γ∑i=1γ1−fi′+121α∑i=1αθti′+1β∑i=1β2−θai′+1γ∑i=1γ2−θfi′.If for any two indices,Şri values are equal, then compute accuracy degree as(64)Hri=161α∑i=1α1−ti′+1β∑i=1βai′+1γ∑i=1γfi′+121α∑i=1α1−θti′+1β∑i=1βθai′+1γ∑i=1γθfi′.Step 5: rank the alternatives based on the crisp values and get the beneficial optimal.


### 5.2. Illustrated Example

To illustrate the working of the above stated algorithm to the decision-making process, we consider a numerical example which can be read as follows.Consider a decision-making process related to the selection of the best enterprise for the investment. Since the market always shows an up and down phases at a regular interval of time, the decision maker will always look for the desired option for the investment. After deep analysis of the market scenario, an investor selects the following four potential options, considered as an alternative, which are characterized as(i)
Ą1: invest in the local market.(ii)
Ą2: invest in the southern Asian market.(iii)
Ą3: invest in the northeastern market.(iv)
Ą4: invest in the European market.To access all these alternatives deeply, we have taken the following three attributes with attribute weights taken asω=0.5,0.4,0.1Ŧ:(i)
C1C1: economic growth.(ii)
C2: profit in the long term.(iii)
C3: social political impact analysis.Then, the following steps of the stated algorithms are implemented to find the best alternative as follows.Step 1: a senior expert from the market sector is hired for accessing the given alternatives under each attribute. Their corresponding rating is summarized in Table1Step 2: as all the criteria are of benefit types, there is no need for normalization.Step 3: with attribute weights taken asω=0.5,0.4,0.1Ŧ, utilize CSVNHFGHA operator as stated in equation. (52) to aggregate the information for γ=1.Step 4: the score values of all the aggregated numbers (as obtained from Step 3) are calculated, and their results are summarized in Table2 along with the ranking order.Step 5: based on the score values, we rank the given alternatives as(65)Ą3≥Ą2≥Ą4≥Ą1,and found that Ą3 is the best one for this suitable job.Table 1
CSVNHF generalized aggregation operator decision matrix.
Data representationsC1C2C3A10.7ei.20.90.5ei.20.40.7ei.20.70.6ei.20.870.56ei.20.450.76ei.20.660.67ei.20.980.65ei.20.670.14ei.20.560.77ei.20.80.78ei.20.810.79ei.20.820.83ei.20.670.84ei.20.450.85ei.20.670.56ei.20.780.56ei.20.790.67ei.20.690.13ei.20.460.23ei.20.640.45ei.20.660.45ei.20.560.54ei.20.650.55ei.20.870.89ei.20.890.67ei.20.660.47ei.20.88Ą20.45ei.20.930.56ei.20.450.78ei.20.76,0.67ei.20.880.6ei.20.50.7ei.20.7,0.8ei.20.30.6ei.20.670.6ei.20.560.77ei.20.560.34ei.20.890.56ei.20.78,0.67ei.20.770.92ei.20.660.57ei.20.45,0.67ei.20.450.56ei.20.790.45ei.20.570.58ei.20.690.67ei.20.490.68ei.20.59,0.59ei.20.580.39ei.20.670.49ei.20.34,0.33ei.20.550.44ei.20.650.45ei.20.77Ą30.7ei.20.450.45ei.20.450.45ei.20.450.76ei.20.230.5ei.20.430.7ei.20.450.54ei.20.640.65ei.20.570.46ei.20.740.68ei.20.850.86ei.20.480.58ei.20.840.66ei.20.450.9ei.20.670.57ei.20.360.67ei.20.560.45ei.20.670.33ei.20.340.45ei.20.680.46ei.20.480.67ei.20.460.5ei.20.450.39ei.20.670.49ei.20.670.56ei.20.570.45ei.20.450.67ei.20.34Ą40.4ei.20.450.56ei.20.780.46ei.20.460.6ei.20.90.7ei.20.20.8ei.20.30.4ei.20.470.5ei.20.490.45ei.20.590.8ei.20.670.8ei.20.540.9ei.20.630.66ei.20.90.78ei.20.880.89ei.20.450.56ei.20.690.57ei.20.670.68ei.20.70.35ei.20.480.67ei.20.490.38ei.20.510.52ei.20.650.53ei.20.750.54ei.20.850.86ei.20.750.87ei.20.730.74ei.20.71Table 2
Ranking results of the CSVNHFSs.
ĄlternativesŞcore functionRankĄ1ŞĄ1=0.4674Ą2ŞĄ2=0.5152Ą3ŞĄ3=0.5241Ą4ŞĄ4=0.4783


### 5.3. Comparative Analysis

To compare the study with the several existing studies, we compare the performance of the existing algorithms in [20, 21, 26, 31–34] under different environments. The result corresponding to each method is listed in Table 3. It is clear that from Table 3 that Ą3 is the best alternative identified by all existing methods and the proposed method. Although the ranking result is same by all the methods, the proposed MADM algorithm has several advantages over such existing studies.Table 3
Comparative study with the existing methods.
MethodScore valuesRanking resultsBest alternativesAli and Smarandache [26]Cannot be calculatedNoNoAlkouri and Salleh [20]Cannot be calculatedNoNoGarg and Rani [31, 32]Cannot be calculatedNoNoRani and Garg [21, 33]Cannot be calculatedNoNoBeg and Rashid [35]ŞĄ1=0.66,ŞĄ2=0.71, Ş3=0.72,ŞĄ4=0.67Ą3≥Ą2≥Ą4≥Ą1Ą3Torra [27]ŞĄ1=0.573,ŞĄ2=0.601, Ş3=0.603,ŞĄ4=0.579Ą3≥Ą2≥Ą4≥Ą1Ą3Beg and Rashid [35]ŞĄ1=0.47,ŞĄ2=0.52, Ş3=0.53,ŞĄ4=0.49Ą3≥Ą2≥Ą4≥Ą1Ą3Proposed methodŞĄ1=0.467,ŞĄ2=0.515, Ş3=0.524,ŞĄ4=0.478Ą3≥Ą2≥Ą4≥Ą1Ą3To highlight such things, we summarize the characteristics of the stated method over the existing ones in Table4. It is seen from this table that proposed method in this paper is more generalized than the existing methods. Also, it is clear that the proposed concept is more general and reliable than IFS [3], CIFS [20], CNS [26], and NS [7], and all are considered as special cases of CSVNHFS. For instance, by taking γ=1 in the proposed work, the stated method reduces to CSVNHFWA operators, and if γ=2, then our proposed work is reduced to CSVNHFWG.Table 4
Characteristic comparison between proposed work and existing methods.
MethodsAbility to integrate informationAbility to capture information using complex numbersAbility to handle two-dimensional informationFlexible according to decision makers’ preferencesSuperior characteristic of the ideasDealing hesitant kind of informationAtanassov [3]YesNoNoNoNoNoAlkouri and Salleh [20]YesYesYesNoNoNoGarg and Rani [31, 32]YesYesYesNoNoNoRani and Garg [21, 33]YesYesYesNoNoNoSmarandache [7]YesNoNoYesYesNoAli and Smarandache [26]YesYesYesYesYesNoProposed workYesYesYesYesYesYes


### 5.4. Sensitivity Analysis of the Parameterγ

In this section, we examined the influence of parameterγ on to the decision-making process. For this, we vary the value of the parameter γ from 1 to 20 (by taking arbitrary values) and implement the steps of the proposed algorithm on it. The final score values and the ranking order for each value of γ are recorded and listed in Table 5. Also, we have listed some special cases of the proposed operators with γ in Remarks 1–3. From this investigation, we noted that as we increase the value of γ, the score value corresponding to each alternative increases which shows the optimistic nature. Decision makers can select the suitable value as per their choice by seeing the overall score value of each alternative. From the above analysis, we conclude that the proposed strategies are more unrivaled and more broad than existing work.Table 5
Impact of the parameterγ on the ratings.
ParameterγScore valuesRankingγ=1ŞĄ1=0.467,ŞĄ2=0.515,ŞĄ3=0.524,ŞĄ4=0.478Ą3≥Ą2≥Ą4≥Ą1γ=4ŞĄ1=0.496,ŞĄ2=0.539,ŞĄ3=0.545,ŞĄ4=0.518Ą3≥Ą2≥Ą4≥Ą1γ=10ŞĄ1=0.522,ŞĄ2=0.566,ŞĄ3=0.571,ŞĄ4=0.553Ą3≥Ą2≥Ą4≥Ą1γ=15ŞĄ1=0.531,ŞĄ2=0.57,ŞĄ3=0.58,ŞĄ4=0.56Ą3≥Ą2≥Ą4≥Ą1γ=20ŞĄ1=0.536,ŞĄ2=0.584,ŞĄ3=0.588,ŞĄ4=0.57Ą3≥Ą2≥Ą4≥Ą1


## 5.1. Decision-Making Algorithm

During the decision-making process, ambiguity and complexity always occur in our day-to-day life. This study aims to employ the decision-making strategy in the presence of proposed works. For this, consider a finite number of alternativesX=u1,u2,…,um and attributes C=c1,c2,…,cn. To evaluate each alternative under each attribute, we assign a weight vector for attribute as ω=ω1,ω2,…,ωnŦ,∑i=1nωi=1. Experts are invited to evaluate each alternative Ąi and give their preferences in terms of the CSVNHF information nij=tij,aij,fiji,j=1,2,…,n for criteria Cj where tij=tij′eiθtij′/tij∈Ŧu, aij=aij′eiθaij′/aij∈Ąu and fij=fij′eiθfij′/fij∈Fu, gives the TG, abstinence grade, and FG . Then, the following are the steps summarized to find the best alternative(s).Step 1: collect the data in the shape of the CSVNHF setting and summarize them in the form of the matrix called as decision matrix.Step 2: normalize the collective information, if needed by using the following equation.(62)Drij=tij,aij,fij,for benefit criteria,fij,aij,tij,for cost criteria.Step 3: aggregate the collective information by using stated operators. For instance, utilize equation (7) to aggregate the information rij into ri=ti′eiθti′,ai′eiθai′,fi′eiθfi′.Step 4: compute the crisp value for each numberri=ti′eiθti′,ai′eiθai′,fi′eiθfi′ by using the following equation:(63)Şri=161α∑i=1αti′+1β∑i=1β1−ai′+1γ∑i=1γ1−fi′+121α∑i=1αθti′+1β∑i=1β2−θai′+1γ∑i=1γ2−θfi′.If for any two indices,Şri values are equal, then compute accuracy degree as(64)Hri=161α∑i=1α1−ti′+1β∑i=1βai′+1γ∑i=1γfi′+121α∑i=1α1−θti′+1β∑i=1βθai′+1γ∑i=1γθfi′.Step 5: rank the alternatives based on the crisp values and get the beneficial optimal.


## 5.2. Illustrated Example

To illustrate the working of the above stated algorithm to the decision-making process, we consider a numerical example which can be read as follows.Consider a decision-making process related to the selection of the best enterprise for the investment. Since the market always shows an up and down phases at a regular interval of time, the decision maker will always look for the desired option for the investment. After deep analysis of the market scenario, an investor selects the following four potential options, considered as an alternative, which are characterized as(i)
Ą1: invest in the local market.(ii)
Ą2: invest in the southern Asian market.(iii)
Ą3: invest in the northeastern market.(iv)
Ą4: invest in the European market.To access all these alternatives deeply, we have taken the following three attributes with attribute weights taken asω=0.5,0.4,0.1Ŧ:(i)
C1C1: economic growth.(ii)
C2: profit in the long term.(iii)
C3: social political impact analysis.Then, the following steps of the stated algorithms are implemented to find the best alternative as follows.Step 1: a senior expert from the market sector is hired for accessing the given alternatives under each attribute. Their corresponding rating is summarized in Table1Step 2: as all the criteria are of benefit types, there is no need for normalization.Step 3: with attribute weights taken asω=0.5,0.4,0.1Ŧ, utilize CSVNHFGHA operator as stated in equation. (52) to aggregate the information for γ=1.Step 4: the score values of all the aggregated numbers (as obtained from Step 3) are calculated, and their results are summarized in Table2 along with the ranking order.Step 5: based on the score values, we rank the given alternatives as(65)Ą3≥Ą2≥Ą4≥Ą1,and found that Ą3 is the best one for this suitable job.Table 1
CSVNHF generalized aggregation operator decision matrix.
Data representationsC1C2C3A10.7ei.20.90.5ei.20.40.7ei.20.70.6ei.20.870.56ei.20.450.76ei.20.660.67ei.20.980.65ei.20.670.14ei.20.560.77ei.20.80.78ei.20.810.79ei.20.820.83ei.20.670.84ei.20.450.85ei.20.670.56ei.20.780.56ei.20.790.67ei.20.690.13ei.20.460.23ei.20.640.45ei.20.660.45ei.20.560.54ei.20.650.55ei.20.870.89ei.20.890.67ei.20.660.47ei.20.88Ą20.45ei.20.930.56ei.20.450.78ei.20.76,0.67ei.20.880.6ei.20.50.7ei.20.7,0.8ei.20.30.6ei.20.670.6ei.20.560.77ei.20.560.34ei.20.890.56ei.20.78,0.67ei.20.770.92ei.20.660.57ei.20.45,0.67ei.20.450.56ei.20.790.45ei.20.570.58ei.20.690.67ei.20.490.68ei.20.59,0.59ei.20.580.39ei.20.670.49ei.20.34,0.33ei.20.550.44ei.20.650.45ei.20.77Ą30.7ei.20.450.45ei.20.450.45ei.20.450.76ei.20.230.5ei.20.430.7ei.20.450.54ei.20.640.65ei.20.570.46ei.20.740.68ei.20.850.86ei.20.480.58ei.20.840.66ei.20.450.9ei.20.670.57ei.20.360.67ei.20.560.45ei.20.670.33ei.20.340.45ei.20.680.46ei.20.480.67ei.20.460.5ei.20.450.39ei.20.670.49ei.20.670.56ei.20.570.45ei.20.450.67ei.20.34Ą40.4ei.20.450.56ei.20.780.46ei.20.460.6ei.20.90.7ei.20.20.8ei.20.30.4ei.20.470.5ei.20.490.45ei.20.590.8ei.20.670.8ei.20.540.9ei.20.630.66ei.20.90.78ei.20.880.89ei.20.450.56ei.20.690.57ei.20.670.68ei.20.70.35ei.20.480.67ei.20.490.38ei.20.510.52ei.20.650.53ei.20.750.54ei.20.850.86ei.20.750.87ei.20.730.74ei.20.71Table 2
Ranking results of the CSVNHFSs.
ĄlternativesŞcore functionRankĄ1ŞĄ1=0.4674Ą2ŞĄ2=0.5152Ą3ŞĄ3=0.5241Ą4ŞĄ4=0.4783


## 5.3. Comparative Analysis

To compare the study with the several existing studies, we compare the performance of the existing algorithms in [20, 21, 26, 31–34] under different environments. The result corresponding to each method is listed in Table 3. It is clear that from Table 3 that Ą3 is the best alternative identified by all existing methods and the proposed method. Although the ranking result is same by all the methods, the proposed MADM algorithm has several advantages over such existing studies.Table 3
Comparative study with the existing methods.
MethodScore valuesRanking resultsBest alternativesAli and Smarandache [26]Cannot be calculatedNoNoAlkouri and Salleh [20]Cannot be calculatedNoNoGarg and Rani [31, 32]Cannot be calculatedNoNoRani and Garg [21, 33]Cannot be calculatedNoNoBeg and Rashid [35]ŞĄ1=0.66,ŞĄ2=0.71, Ş3=0.72,ŞĄ4=0.67Ą3≥Ą2≥Ą4≥Ą1Ą3Torra [27]ŞĄ1=0.573,ŞĄ2=0.601, Ş3=0.603,ŞĄ4=0.579Ą3≥Ą2≥Ą4≥Ą1Ą3Beg and Rashid [35]ŞĄ1=0.47,ŞĄ2=0.52, Ş3=0.53,ŞĄ4=0.49Ą3≥Ą2≥Ą4≥Ą1Ą3Proposed methodŞĄ1=0.467,ŞĄ2=0.515, Ş3=0.524,ŞĄ4=0.478Ą3≥Ą2≥Ą4≥Ą1Ą3To highlight such things, we summarize the characteristics of the stated method over the existing ones in Table4. It is seen from this table that proposed method in this paper is more generalized than the existing methods. Also, it is clear that the proposed concept is more general and reliable than IFS [3], CIFS [20], CNS [26], and NS [7], and all are considered as special cases of CSVNHFS. For instance, by taking γ=1 in the proposed work, the stated method reduces to CSVNHFWA operators, and if γ=2, then our proposed work is reduced to CSVNHFWG.Table 4
Characteristic comparison between proposed work and existing methods.
MethodsAbility to integrate informationAbility to capture information using complex numbersAbility to handle two-dimensional informationFlexible according to decision makers’ preferencesSuperior characteristic of the ideasDealing hesitant kind of informationAtanassov [3]YesNoNoNoNoNoAlkouri and Salleh [20]YesYesYesNoNoNoGarg and Rani [31, 32]YesYesYesNoNoNoRani and Garg [21, 33]YesYesYesNoNoNoSmarandache [7]YesNoNoYesYesNoAli and Smarandache [26]YesYesYesYesYesNoProposed workYesYesYesYesYesYes


## 5.4. Sensitivity Analysis of the Parameterγ

In this section, we examined the influence of parameterγ on to the decision-making process. For this, we vary the value of the parameter γ from 1 to 20 (by taking arbitrary values) and implement the steps of the proposed algorithm on it. The final score values and the ranking order for each value of γ are recorded and listed in Table 5. Also, we have listed some special cases of the proposed operators with γ in Remarks 1–3. From this investigation, we noted that as we increase the value of γ, the score value corresponding to each alternative increases which shows the optimistic nature. Decision makers can select the suitable value as per their choice by seeing the overall score value of each alternative. From the above analysis, we conclude that the proposed strategies are more unrivaled and more broad than existing work.Table 5
Impact of the parameterγ on the ratings.
ParameterγScore valuesRankingγ=1ŞĄ1=0.467,ŞĄ2=0.515,ŞĄ3=0.524,ŞĄ4=0.478Ą3≥Ą2≥Ą4≥Ą1γ=4ŞĄ1=0.496,ŞĄ2=0.539,ŞĄ3=0.545,ŞĄ4=0.518Ą3≥Ą2≥Ą4≥Ą1γ=10ŞĄ1=0.522,ŞĄ2=0.566,ŞĄ3=0.571,ŞĄ4=0.553Ą3≥Ą2≥Ą4≥Ą1γ=15ŞĄ1=0.531,ŞĄ2=0.57,ŞĄ3=0.58,ŞĄ4=0.56Ą3≥Ą2≥Ą4≥Ą1γ=20ŞĄ1=0.536,ŞĄ2=0.584,ŞĄ3=0.588,ŞĄ4=0.57Ą3≥Ą2≥Ą4≥Ą1


## 6. Conclusion

The main contribution of this paper is discussed below.(1)
A novel concept of CSVNHFS is defined in the paper by incorporating the features of SVN, hesitant, and complex sets. The idea behind this set is to address the ambiguity in the data when it is arranged in the form of “yes,” “abstinence,” and “no” under the complex domain. In the presented set, each element is characterized with three independent hesitant degrees, namely, TG (t′eiθt′), abstinence (a′eiθa′), and FG (f′eiθf′), over the unit disc of complex plane with the conditions 0≤t′+a′+f′≤3 and 0≤θt′+θa′+θf′≤6 where 0≤t′,a′,f′≤1 and 0≤θt′,θa′,θf′≤2.(2)
The fundamental properties and the operations of the stated set are investigated. Also, it is analyzed that the proposed CSVNHFS is the generalization of the existing sets such as IFS [3], CIFS [20], CNS [26], and NS [7].(3)
By taking the features of the CSVNHFS, we defined generalized weighted average (CSVNHFGWA), ordered weighted (CSVNHFGOWA), and hybrid weighted averaging (CSVNHFGHWA) operators to aggregate different pairs of the given information. Some of their basic properties are also discussed.(4)
A MADM algorithm based on the stated operators is presented by using the features of CSVNHFS and illustrated with the numerical examples.(5)
To determine the supremacy and efficiency of investigated operators, we utilized the advantages, sensitive analysis, and geometrical expressions of the proposed work to discover the dominance of the elaborated approaches.In the future, we will try to implement the application of the stated algorithm and extend it to different fuzzy environments [35–42].



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*Source: 9164735-2022-01-10.xml*