# Multiple-Attribute Decision-Making Method Based on Normalized Geometric Aggregation Operators of Single-Valued Neutrosophic Hesitant Fuzzy Information

Authors: Li Wang; Yan-Ling Bao
Journal: Complexity
 (2021)

---

## Abstract

As a generalization of both single-valued neutrosophic element and hesitant fuzzy element, single-valued neutrosophic hesitant fuzzy element (SVNHFE) is an efficient tool for describing uncertain and imprecise information. Thus, it is of great significance to deal with single-valued neutrosophic hesitant fuzzy information for many practical problems. In this paper, we study the aggregation of SVNHFEs based on some normalized operations from geometric viewpoint. Firstly, two normalized operations are defined for processing SVNHFEs. Then, a series of normalized aggregation operators which fulfill some basic conditions of a valid aggregation operator are proposed. Additionally, a decision-making method is developed for resolving multiattribute decision-making problems based on the proposed operators. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the method.


---

## Body

## 1. Introduction

Being different from the fuzzy set which assigns one value from [0, 1] for the membership degree of an element, the neutrosophic set [1, 2] is composed of three independent functions, i.e., truth-membership function, indeterminacy-membership function, and falsity-membership function. Neutrosophic set can describe the indeterminacy of information data independently which conforms to human beings’ recognition mode better actually. Therefore, many scholars focused their attention to promote its development. Wang et al. [3] presented the single-valued neutrosophic set (SVNS) in which all the three membership degrees belong to unit interval 0,1 which brings about convenience to adopt neutrosophic theory in many real-life situations. Combining the single-valued neutrosophic set with the rough set, Yang et al. [4] introduced the single-valued neutrosophic rough set. Furthermore, Bao et al. [5] studied the characterization of the single-valued neutrosophic rough set from logic point of view. Besides, Bao et al. [6] put forward the single-valued neutrosophic refined rough set model. By means of the single-valued refined neutrosophic set, Vasantha et al. [7] did some meaningful research on imaginative play of children. In addition, the single-valued neutrosophic set contributes a lot to decision-making problems due to its flexibility and practicability. In particular, Ye [8] introduced cross-entropy in single-valued neutrosophic environment for solving decision-making problems. Liu and Wang [9] developed the decision-making method under the single-valued neutrosophic framework by using normalized weighted Bonferroni mean operator. Subsequently, Ye [10] also explored the single-valued neutrosophic decision-making method based on the correlation coefficient. Biswas et al. [11] studied the single-valued neutrosophic TOPSIS method for multiattribute group decision-making. Yang et al. [12] analyzed triangular single-valued neutrosophic data envelopment and applied it to hospital performance measurement.In an era of information explosion, people find it difficult to determine the specific membership degree of an element to a set due to various reasons. To solve this problem, Torra [13] proposed the hesitant fuzzy set (HFS) in which the membership degree of an element to a set can be some different values rather than a single one. Furthermore, Xia and Xu [14] characterized the hesitant fuzzy set through a mathematical symbol and defined some basic operations on it. Since presented, the hesitant fuzzy set has contributed a lot to decision-making problems by combining with aggregation operators. Firstly, Xia and Xu [14] put forward a number of hesitant fuzzy aggregation operators from arithmetic and geometric viewpoint, respectively. In addition, Xia et al. [15] also came up with some aggregation operators for hesitant fuzzy information based on quasi-arithmetic means. Meanwhile, Wei [16] developed hesitant fuzzy prioritized operators and applied them to resolve multiple attribute decision-making problems. Zhu et al. [17] put forward hesitant fuzzy geometric Bonferroni means which take full advantage of geometric as well as the Bonferroni aggregation operator. Later, Zhang [18] introduced power aggregation in hesitant fuzzy framework and proposed the hesitant fuzzy power aggregation operator. Wang et al. [19] gave some aggregation operators under dual hesitant fuzzy set environment and explored their application to multiple attribute decision-making.As described above, both SVNS and HFS have contributed a lot to decision-making problems. Nevertheless, there is only one truth-membership hesitant function in the hesitant fuzzy set which cannot describe indeterminacy-membership degree and falsity-membership degree effectively. On the contrary, an SVNS cannot describe the three membership degrees with different values, which maybe usual in real life due to hesitancy of decision makers. Therefore, Ye [20] first introduced the single-valued neutrosophic hesitant fuzzy set and developed a series of aggregation operators of single-valued neutrosophic hesitant fuzzy elements. Then, Şahin and Liu [21] explored correlation coefficient of the single-valued neutrosophic hesitant fuzzy set as well as its applications to decision-making. Additionally, Liu and Zhang [22] investigated neutrosophic hesitant fuzzy elements aggregation by the aid of Heronian mean aggregation operators. Liu and Luo [23] presented ordered weighted arithmetic and hybrid weighted arithmetic operator under single-valued neutrosophic hesitant fuzzy environment. However, Mishra and Kumar [24] identified the problem that the aggregation operators proposed in [23] do not satisfy monotonicity actually. Wang and Bao [25] also pointed out that the aggregation operators in [23] do not fulfill idempotency either. In fact, all the existing aggregation operators concerned with SVNHFEs do not satisfy the basic properties of a valid aggregation operator such as idempotency and monotonicity. Hence, it is necessary to give some novel aggregation operators to improve earlier results. In this paper, we focus on defining some normalized operations for SVNHFEs and developing a series of normalized geometric single-valued neutrosophic hesitant fuzzy geometric aggregation operators to provide theoretical foundation for decision-making problems.To achieve the above goal, we design the rest of paper as follows. In Section2, some basic concepts about the hesitant fuzzy set, single-valued neutrosophic hesitant fuzzy set, and several existing single-valued neutrosophic hesitant fuzzy aggregation operators are provided. In Section 3, we put forward a number of normalized single-valued neutrosophic hesitant fuzzy aggregation operators and explore some basic properties. In Section 4, a method is developed for solving multiattribute decision-making problems. Additionally, a numerical example demonstrates specific process of the method. Finally, we draw a conclusion in Section 5.


## 2. Preliminaries

In this section, we mainly recall some basic notions and operations of the hesitant fuzzy set and single-valued neutrosophic hesitant fuzzy set which are necessary for understanding the article.Definition 1.
(see [13]). Let U be a fixed set, and a hesitant fuzzy set A on U is defined in terms of function hA that returns a set of several values in 0,1 when applied to U.
For convenience and directness, Xia and Xu [14] characterized the hesitant fuzzy set as A=x,hAx|x∈U, where hAx is a point subset of unit interval 0,1, representing the possible membership degrees of the element x∈U to A. For any x∈U, hAx is termed as a hesitant fuzzy element, and the set of all hesitant fuzzy elements is denoted by H.Definition 2.
(see [14]). For a hesitant fuzzy element (HFE) h, sh=∑γ∈hγ/δh is termed as the score of h, where δh is the number of elements in h. For any two HFEs, h1 and h2, if sh1>sh2, then h1>h2; if sh1=sh2, then h1=h2.
Let HFEsh1=0.5,0.4andh2=0.1,0.8, and it is obvious that sh1=sh2=0.45, which implies h1=h2. However, from the data itself we can find that h1 is much more stable than h2; thus, it is not reasonable enough to judge the order between HFEs only by the score of element. In the following, we introduce a novel comparison rule to improve Definition 2. First, we need to introduce a creative method to extend an HFE to a fixed length. For convenience, given two HFEs, h1andh2, and δh1=n1<n2=δh2, Xia and Xu [14] suggested h1 should be extended by adding the minimum value in it until it reaches the same length with h2. Zhang [18] pointed out the selection of the appended value depends primarily on the decision makers’ risk preferences. Optimists would append the maximum value, while pessimists would append the minimum value. However, both of the methods cannot guarantee the steady of data. In fact, the best choice to extend an HFE is the closest number to the given data, and it is merited to add the score of the HFE repeatedly until it reaches the fixed length. In the present paper, we adopt this method to extend an HFE to a fixed length if without other explanation.Example 1.
Leth1=0.2,0.5andh2=0.1,0.3,0.7,0.8; in order to extend h1 to reach the same length with h2, we need to calculate sh1=0.35; then, h1 should be extended as h1′=0.2,0.5,0.35,0.35.Definition 3.
For an HFEh, we define sh=∑γ∈hγ/δh as the score of h, ah=maxγ|γ∈h−minγ|γ∈h as the amplitude of h, and vh=∑γ∈hγ−sh2/δh as the variance of h.Definition 4.
For any two HFEs,h1 and h2, the order relation is defined as follows:(1)
Ifsh1<sh2, then h1 is smaller than h2, denoted by h1≺h2(2)
Ifsh1=sh2 and ah1>ah2, then h1 is smaller than h2, denoted by h1≺h2(3)
Ifsh1=sh2,ah1=ah2 and vh1>vh2, then h1 is smaller than h2, denoted by h1≺h2(4)
Ifsh1=sh2,ah1=ah2 and vh1=vh2, then h1 is equivalent to h2, denoted by h1∼h2(5)
Ifγ1|γ1∈h1=γ2|γ2∈h2, then h1 is equal to h2, denoted by h1=h2(6)
Supposehi=γij|j=1,2,…,nii=1,2 with n1≤n2, if γ1σj<γ2σjj=1,2,…,n2, then h1 is strictly smaller than h2, denoted by h1≺sh2, where γiσj is the j th largest element of hi, and it should be pointed that there are n2−n1 elements sh1 inserted in h1 to ensure the lengths of h1 and h2 are the same in the process of comparisonExample 2.
Leth1=0.1,0.5,0.6,h2=0.3,0.5,h3=0.1,0.5,0.6,h4=0.2,0.3,0.4, and h5=0.2,0.3,0.7; then, we can obtain that(1)sh1=0.4000,sh2=0.4000,sh3=0.4000,sh4=0.3000,sh5=0.4000,ah1=0.5000,ah2=0.2000,ah3=0.5000,ah5=0.5000,vh1=0.0467,vh3=0.0467,vh5=0.0467,which indicates h4≺h1=h3∼h5≺h2 and h4≺sh2.
For any three hesitant fuzzy elements,h,h1, and h2, Torra [13] and Xia and Xu [14] gave the operations between them as follows:(i)
hc=∪γ∈h1−γ(ii)
h1∪h2=∪γ1∈h1,γ2∈h2maxγ1,γ2(iii)
h1∩h2=∪γ1∈h1,γ2∈h2minγ1,γ2(iv)
hλ=∪γ∈hγλ,λ>0(v)
λh=∪γ∈h1−1−γλ,λ>0(vi)
h1⊕h2=∪γ1∈h1,γ2∈h2γ1+γ2−γ1γ2(vii)
h1⊗h2=∪γ1∈h1,γ2∈h2γ1γ2
When defining some new operation rules, people always expect they are convenient to implement and satisfy some basic properties, such as distributive law and associative law. Whereas, in the aforementioned definition, we can find out that some desirable properties do not hold. For instance, let an HFEh=0.2,0.3; then,(2)h⊕h=0.2+0.2−0.2×0.2,0.2+0.3−0.2×0.3,0.3+0.2−0.3×0.2,0.3+0.3−0.3×0.3=0.36,0.44,0.44,0.51,whereas 2h=1−0.82,1−0.72=0.36,0.51, which means that h⊕h≠2h. In addition, h⊗h=0.2×0.2,0.2×0.3,0.3×0.2,0.3×0.3=0.04,0.06,0.06,0.09 and h2=0.22,0.32=0.04,0.09, and it is obvious that h⊗h≠h2.
In what follows, we give some new normalized operations which turn out to satisfy a number of basic desirable properties.Definition 5.
Given HFEsh1=∪i=1n1ξi and h2=∪i=1n2ηi with n1≤n2, normalized sum ⊕N and normalized product ⊗N are defined as follows:(1)
h1⊕Nh2=∪i=1n2ξσi+ησi−ξσiησi(2)
h1⊗Nh2=∪i=1n2ξσiησiwhere ξσi is the ith largest element of h1, ησi is the ith largest element of h2, and there are n2−n1 elements sh1 inserted in h1 such that the lengths of two HHEs are the same.Proposition 1.
Leth,h1,andh2 be three HFEs and λ,λ1,λ2>0; then, the following operation rules hold:(1)
h1⊕Nh2=h2⊕Nh1,h1⊗Nh2=h2⊗Nh1(2)
h⊕Nh1⊕Nh2=h⊕Nh1⊕Nh2,h⊗Nh1⊗Nh2=h⊗Nh1⊗Nh2(3)
λh1⊕Nh2=λh1⊕Nλh2(4)
λ1+λ2h=λ1h⊕Nλ2h(5)
h1⊗Nh2λ=h1λ⊗Nh2λ(6)
hλ1+λ2=hλ1⊗Nhλ2Proof.
(1)
and(2)
can be quickly proved by Definition5. Next, we detail the rest. Suppose h1=∪i=1n1ξi,h2=∪i=1n2ηi,n1≤n2,h=∪i=1nγi; then, we have(3)
h1⊕Nh2=∪i=1n2ξσi+ησi−ξσiησi,(3)λh1⊕Nh2=∪i=1n21−1−ξσi−ησi+ξσiησiλ=∪i=1n21−1−ξσiλ1−ησiλ.On the contrary,λh1=∪i=1n11−1−ξiλ,λh2=∪i=1n21−1−ηiλ,(4)λh1⊕Nλh2=∪i=1n21−1−ξσiλ+1−1−ησiλ−1−1−ξσiλ1−1−ησiλ=∪i=1n21−1−ξσiλ1−ησiλ.Therefore, (3) is proved.(4)
λ1+λ2h=∪i=1n1−1−γiλ1+λ2,λ1h=∪i=1n1−1−γiλ1,λ2h=∪i=1n1−1−γiλ2,(5)λ1h⊕Nλ2h=∪i=1n1−1−γσiλ1+1−1−γσiλ2−1−1−γσiλ11−1−γσiλ2=∪i=1n1−1−γσiλ11−γσiλ2=∪i=1n1−1−γiλ1+λ2.(5)
h1⊗Nh2=∪i=1n2ξσiησi,h1⊗Nh2λ=∪i=1n2ξσiησiλ=∪i=1n2ξσiλησiλ,(6)h1λ⊗Nh2λ=∪i=1n1ξiλ⊗N∪i=1n2ηiλ=∪i=1n2ξσiλησiλ.(6)
hλ1+λ2=∪i=1nγiλ1+λ2, and hλ1=∪i=1nγiλ1,hλ2=∪i=1nγiλ2; then,(7)hλ1⊗Nhλ2=∪i=1nγσiλ1γσiλ2=∪i=1nγiλ1+λ2.Definition 6.
(see [20]). Let X be a fixed set; then, a single-valued neutrosophic hesitant fuzzy set (SVNHFS) N on X is defined as follows:(8)N=x,t˜x,i˜x,f˜x|x∈X,in which t˜x,i˜x,andf˜x are three point subsets of 0,1, denoting the possible truth hesitant membership degree, indeterminacy hesitant membership degree, and falsity hesitant membership degree of the element x to N, respectively, with the condition 0≤γ,δ,η≤1 and 0≤γ++δ++η+≤3, where γ∈t˜x,δ∈i˜x,η∈f˜x,γ+=maxt˜x,δ+=maxi˜x,η+=maxf˜x. For each x∈X, the triplet nx=t˜x,i˜x,f˜x is termed as a single-valued neutrosophic hesitant fuzzy element (SVNHFE), which can be denoted by the simplified symbol n=t˜,i˜,f˜, and the set of all SVNHFEs is represented by Ω.
It should be pointed out that the single-valued neutrosophic hesitant fuzzy is the same with the hesitant neutrosophic set essentially in literature [26]. In order to compare SVNHFEs, we give the following concept.Definition 7.
For an SVNHFEn=t˜,i˜,f˜, we define sn=1/32+st˜−si˜−sf˜ as the score of n, an=1/3at˜+ai˜+af˜ as the amplitude of n, and vn=1/3vt˜+vi˜+vf˜ as the variance of n.Definition 8.
For any two SVNHFEsn1=t˜1,i˜1,f˜1 and n2=t˜2,i˜2,f˜2, the order relation is defined as follows:(1)
Ifsn1<sn2, then n1 is smaller than n2, denoted by n1≺n2(2)
Ifsn1=sn2 and an1>an2, then n1 is smaller than n2, denoted by n1≺n2(3)
Ifsn1=sn2,an1=an2 and vn1>vn2, then n1 is smaller than n2, denoted by n1≺n2(4)
Ifsn1=sn2,an1=an2 and vn1=vn2, then n1 is equivalent to n2, denoted by n1∼n2(5)
Ift˜1=t˜2,i˜1=i˜2 and f˜1=f˜2 then n1 is equal to n2, denoted by n1=n2(6)
Ift˜1≺st˜2,i˜2≺si˜1 and f˜2≺sf˜1, then n1 is strictly smaller than n2, denoted by n1≺sn2Example 3.
Suppose SVNHFEsn1=0.1,0.4,0.5,0.1,0.3,0.5,0.2,0.3,0.4, n2=0.1,0.5,0.2,0.3,0.7,0.3,0.5,0.6, n3=0.1,0.4,0.5,0.2,0.3,0.4,0.2,0.3,0.4, and n4=0.3,0.5,0.1,0.3,0.7,0.2,0.3,0.4; then, we can calculate that sn1=0.5778,sn2=0.4778,sn3=0.5778,andsn4=0.5778 which indicates n2≺n3,n2≺n4,andn2≺n1. In addition, an3=0.2667<0.3333=an1=an4 means n1≺n3andn4≺n3. Furthermore, vn1=0.02963andvn4=0.0263 imply that n1≺n4. Therefore, n2≺n1≺n4≺n3.
For any two single-valued neutrosophic hesitant fuzzy elementsn1=t˜1,i˜1,f˜1 and n2=t˜2,i˜2,f˜2, some operations between them are given as follows [20]:(i)
n1⊕n2=t˜1⊕t˜2,i˜1⊗i˜2,f˜1⊗f˜2=∪γ1∈t˜1,δ1∈i˜1,η1∈f˜1γ2∈t˜2,δ2∈i˜2,η2∈f˜2γ1+γ2−γ1γ2,δ1δ2,η1η2(ii)
n1⊗n2=t˜1⊗t˜2,i˜1⊕i˜2,f˜1⊕f˜2=∪γ1∈t˜1,δ1∈i˜1,η1∈f˜1γ2∈t˜2,δ2∈i˜2,η2∈f˜2γ1γ2,δ1+δ2−δ1δ2,η1+η2−η1η2(iii)
λn1=λt˜1,i˜1λ,f˜iλ=∪γ1∈t˜1,δ1∈i˜1,η1∈f˜11−1−γ1λ,δ1λ,ηiλ,λ>0(iv)
n1λ=t1λ,λi˜1,λf˜1=∪γ1∈t˜1,δ1∈i˜1,η1∈f˜1γ1λ,1−1−δ1λ,1−1−η1λ,λ>0
For the aforementioned operations, we can find out that some desirable properties do not hold. For instance, let an SVNHFE ben=0.2,0.3,0.2,0.3; then,(9)n⊕n=0.2+0.2−0.2×0.2,0.2+0.3−0.2×0.3,0.3+0.2−0.3×0.2,0.3+0.3−0.3×0.3,0.2×0.2,0.3×0.3=0.36,0.44,0.44,0.51,0.04,0.09.
On the contrary,2n=1−0.82,1−0.72,0.22,032=0.36,0.51,0.04,0.09, which means that n⊕n≠2n. In addition,(10)n⊗n=0.2×0.2,0.2×0.3,0.3×0.2,0.3×0.3,0.2+0.2−0.2×0.2,0.3+0.3−0.3×0.3=0.04,0.06,0.06,0.09,0.36,0.51,n2=0.22,0.32,1−0.82,1−0.72=0.04,0.09,0.36,0.51,and it is obvious that n⊗n≠n2.
In what follows, we introduce two normalized single-valued neutrosophic hesitant fuzzy operations which obviously satisfy a number of basic operational rules.Definition 9.
Given SVNHFEsn1=t˜1,i˜1,f˜1=γ1i|i=1,2,…,δt˜1,δ1i|i=1,2,…,δi˜1,η1i|i=1,2,…,δf˜1 and n2=t˜2,i˜2,f˜2=γ2i|i=1,2,…,δt˜2,δ2i|i=1,2,…,δi˜2,η2i|i=1,2,…,δf˜2 with l=maxδt˜1,δt˜2,p=maxδi˜1,δi˜2,q=maxδf˜1,δf˜2, then normalized sum ⊕N and normalized product ⊗N are defined as follows:(1)
n1⊕Nn2=t˜1⊕Nt˜2,i˜1⊗Ni˜2,f˜1⊗Nf˜2=∪i=1lγ1σi+γ2σi−γ1σiγ2σi,∪i=1pδ1σiδ2σi,∪i=1qη1σiη2σi(2)
n1⊗Nn2=t˜1⊗Nt˜2,i˜1⊕Ni˜2,f˜1⊕Nf˜2=∪i=1lγ1σiγ2σi,∪i=1pδ1σi+δ2σi−δ1σiδ2σi,∪i=1qη1σi+η2σi−η1σiη2σi
γjσi is the ith largest element of t˜j and there are l−δt˜j element st˜j inserted in t˜j. Similarly, δjσi is the ith largest element of i˜j and there are p−δi˜j elements si˜j inserted in i˜j and ηjσi is the i th largest elements of f˜j and there are q−δf˜j elements sf˜j inserted in f˜j.Example 4.
Given two SVNHFEsn1=0.1,0.4,0.5,0.3,0.2,0.3,0.4andn2=0.1,0.5,0.2,0.3,0.7,0.3,0.5,0.6, then(11)n1⊕Nn2=0.5+0.5−0.5×0.5,0.3+0.4−0.3×0.4,0.1+0.1−0.1×0.1,0.3×0.7,0.3×0.3,0.3×0.2,⋅0.4×0.6,0.3×0.5,0.2×0.3=0.75,0.58,0.19,0.21,0.09,0.06,0.24,0.15,0.06,n1⊗Nn2=0.5×0.5,0.3×0.4,0.1×0.1,0.3+0.2−0.3×0.2,0.3+0.3−0.3×0.3,0.3+0.7−0.3×0.7,⋅0.2+0.3−0.2×0.3,0.3+0.5−0.3×0.5,0.4+0.6−0.4×0.6=0.01,0.12,0.15,0.44,0.51,0.79,0.44,0.65,0.76.Proposition 2.
Letn,n1,andn2 be three SVNHFEs and λ,λ1,λ2>0; then, the following properties hold:(1)
n1⊕Nn2=n2⊕Nn1,n1⊗Nn2=n2⊗Nn1(2)
n⊕Nn1⊕Nn2=n⊕Nn1⊕Nn2,n⊗Nn1⊗Nn2=n⊗Nn1⊗Nn2(3)
λn1⊕Nn2=λn1⊕Nλn2(4)
λ1+λ2n=λ1n⊕Nλ2n(5)
n1⊗Nn2λ=n1λ⊗Nn2λ(6)
nλ1+λ2=nλ1⊗Nnλ2Proof.
Supposen1=t˜1,i˜1,f˜1,n2=t˜2,i˜2,f˜2,andn=t˜,i˜,f˜; then, we have(1)
n1⊕Nn2=t˜1⊕Nt˜2,i˜1⊗Ni˜2,f˜1⊗Nf˜2=t˜2⊕Nt˜1,i˜2⊗Ni˜1,f˜2⊗Nf˜1=n2⊕Nn1,n1⊗Nn2=t˜1⊗Nt˜2,i˜1⊕Ni˜2,f˜1⊕Nf˜2=t˜2⊗Nt˜1,i˜2⊕Ni˜1,f˜2⊕Nf˜1=n2⊗Nn1(2)
n⊕Nn1⊕Nn2=t˜1⊕Nt˜1⊕Nt˜2,i˜1⊗Ni˜1⊗Ni˜2,f˜⊗Nf˜1⊗Nf˜2=t˜⊕Nt˜1⊕Nt˜2,i˜⊗Ni˜1⊗Ni˜2,f˜⊗Nf˜1⊗Nf˜2=n⊕Nn1⊕Nn2,n⊗Nn1⊗Nn2=t˜⊗Nt˜1⊗Nt˜2,i˜⊕Ni˜1⊕Ni˜2,f˜⊕Nf˜1⊕Nf˜2=t˜⊗Nt˜1⊗Nt˜2,i˜⊕Ni˜1⊕Ni˜2,f˜⊕Nf˜1⊕Nf˜2=n⊗Nn1⊗Nn2(3)
λn1⊕Nn2=λt˜1⊕Nt˜2,i˜1⊗Ni˜2,f˜1⊗Nf˜2=λt˜1⊕Nt˜2,i˜1⊗Ni˜2λ,f˜1⊗Nf˜2λ=λt˜1⊕Nλt˜2,i˜1λ⊗Ni˜2λ,f˜1λ⊗Nf˜2λ=λt˜1,i˜1λ,f˜1λ⊕Nλt˜2,i˜2λ,f˜2λ=λn1⊕Nλn2(4)
λ1+λ2n=λ1+λ2t˜,i˜λ1+λ2,f˜λ1+λ2=λ1t˜⊕Nλ2t˜,i˜λ1⊗Ni˜λ2,f˜λ1⊗Nf˜λ2=λ1n⊕Nλ2n(5)
n1⊗Nn2λ=t˜1⊗Nt˜2λ,λi˜1⊕Ni˜2,λf˜1⊕Nf˜2(6)
nλ1+λ2=t˜λ1+λ2,λ1+λ2i˜,λ1+λ2f˜=t˜λ1⊗Nt˜λ2,λ1i˜⊕Nλ2i˜,λ1f˜⊕Nλ2f˜=nλ1⊕Nnλ2Definition 10.
(see [14]). For a collection of HFEs hjj=1,2,…,n, some hesitant fuzzy aggregation operators are defined as follows:(1)
The hesitant fuzzy weighted geometric operator HFWG:(12)HFWGh1,h2,…,hn=⊗j=1nhjwj=∪γj∈hj∏j=1nγjwj,wherew=w1,w2,…,wnT is the weight vector of h1,h2,…,hn with wj∈0,1j=1,2,…,n and ∑j=1nwj=1.(2)
The hesitant fuzzy ordered weighted geometric operator HFOWG:(13)HFOWGh1,h2,…,hn=⊗j=1nhσjωj=∪γσj∈hσj∏j=1nγσjωj,wherehσj is the jth largest element of hii=1,2,…,n and ω=ω1,ω2,…,ωnT is the aggregation-associated vector such that ωj∈0,1j=1,2,…,n and ∑j=1nωj=1.(3)
The hesitant fuzzy hybrid geometric operator HFHG:(14)HFHGh1,h2,…,hn=⊗j=1nh˙σjωj=∪γ˙σj∈h˙σj∏j=1nγ˙σjωj,where h˙σj is the jth largest element of h˙i=hinwii=1,2,…,n,w=w1,w2,…,wnT is the weight vector of h1,h2,…,hn with wj∈0,1j=1,2,…,n and ∑j=1nwj=1, and ω=ω1,ω2,…,ωnT is the aggregation-associated vector such that ωj∈0,1j=1,2,…,n and ∑j=1nωj=1.


## 3. Normalized Single-Valued Neutrosophic Hesitant Fuzzy Geometric Aggregation Operators

In this part, we propose some normalized aggregation operators based on the normalized product operation.Definition 11.
For a collection of SVNHFEsnj=t˜j,i˜j,f˜j∈Ωj=1,2,…,k, a normalized single-valued neutrosophic hesitant fuzzy geometric mean aggregation operator NSVNHFG:Ωk⟶Ω is defined as(15)NSVNHFGn1,n2,…,nk=⊗j=1Nknj1/k=⊗j=1Nkt˜j1/k,1k⊕j=1Nki˜j,1k⊕j=1Nkf˜j.Definition 12.
For a collection of SVNHFEsnj=t˜j,i˜j,f˜j∈Ωj=1,2,…,k and w=w1,w2,…,wkT which is the weight vector of n1,n2,…,nk with wj∈0,1, ∑j=ikwj=1, a normalized single-valued neutrosophic hesitant fuzzy weighted geometric aggregation operator NSVNHFWG: Ωk⟶Ω is a mapping such that(16)NSVNHFWGn1,n2,…,nk=⊗j=1Nknjwj=⊗j=1Nkt˜jwj,⊕j=1Nkwji˜j,⊕j=1Nkwjf˜j.
It can be observed that ifw=1/k,1/k,…,1/kT, then the operator NSVNHFWG reduces to the operator NSVNHFG.Theorem 1.
Letnj=t˜j,i˜j,f˜j=γji|i=1,2,…,lj,δji|i=1,2,…,pj,ηji|i=1,2,…,qjj=1,2,…,k be a collection of SVNHFEs; if the weight vector is w=w1,w2,…,wnT, then the aggregated result by operator NSVNHFWG can be expressed as(17)NSVNHFWGn1,n2,…,nk=∪i=1l∏j=1kγjσiwj,∪i=1p1−∏j=1k1−δjσiwj,∪i=1q1−∏j=1k1−ηjσiwj,where γjσi,δjσi,andηjσi are the ith largest element of t˜j,i˜j, and f˜j, respectively, l=maxj=1,…,klj, p=maxj=1,…,kpj, and q=maxj=1,…,kqj.Proof.
We prove the result by mathematical induction onk.
First, we demonstrate (17) holds for k=2. Since(18)n1w1=t˜1w1,w1i˜1,w1f˜1=∪i=1l1γ1iw1,∪i=1p11−1−δ1iw1,∪i=1q11−1−η1iw1,n2w2=t˜2w2,w2i˜2,w2f˜2=∪i=1l2γ2iw2,∪i=1p21−1−δ2iw2,∪i=1q21−1−η2iw2,then(19)NSVNHFWGn1,n2=n1w1⊗Nn2w2=∪i=1maxl1,l2γ1σiw1γ2σiw2,∪i=1maxp1,p21−1−δ1σiw1+1−1−δ2σiw2−1−1−δ1σiw11−1−δ2σiw2,∪i=1maxq1,q21−1−η1σiw1+1−1−η2σiw2−1−1−η1σiw11−1−η2σiw2=∪i=1maxl1,l2∏j=12γjσiwj,∪i=1maxp1,p21−1−δ1σiw11−δ2σiw2,∪i=1maxq1,q21−1−η1σiw11−η2σiw2=∪i=1maxl1,l2∏j=12γjσiwj,∪i=1maxp1,p21−∏j=121−δjσiwj,∪i=1maxq1,q21−∏j=121−ηjσiwj.
If (17) holds for k=m, that is,(20)NSVNHFWGn1,n2,…,nm=∪i=1l′∏j=1mγjσiwj,∪i=1p′1−∏j=1m1−δjσiwj,∪i=1q′1−∏j=1m1−ηjσiwj,where l′=maxi=1,…,mli,p′=maxi=1,…,mpi, and q′=maxi=1,…,mqi, then when k=m+1, let l=maxl′,lm+1,p=maxp′,pm+1, and q=maxq′,qm+1, by Proposition 2, we have(21)NSVNHFWGn1,n2,…,nm,nm+1=∪i=1l′∏j=1mγjσiwj,∪i=1p′1−∏j=1m1−δjσiwj,∪i=1q′1−∏j=1m1−ηjσiwj⊗Nt˜m+1wm+1,wm+1i˜m+1,wm+1f˜m+1=∪i=1l′∏j=1mγjσiwj,∪i=1p′1−∏j=1m1−δjσiwj,∪i=1q′1−∏j=1m1−ηjσiwj⊗N∪i=1lm+1γm+1iwm+1,∪i=1pm+11−1−δm+1iwm+1,∪i=1qm+11−1−ηm+1iwm+1=∪i=1l∏j=1m+1γjσiwj,∪i=1p1−∏j=1m1−δjσiwj+1−1−δm+1σiwm+1−1−∏j=1m1−δjσiwj1−1−δm+1σiwm+1,∪i=1q1−∏j=1m1−ηjσiwj+1−1−ηm+1σiwm+1−1−∏j=1m1−ηjσiwj1−1−ηm+1σiwm+1,=∪i=1l∏j=1m+1γjσiwj,∪i=1p1−∏j=1m+11−δjσiwj,∪i=1q1−∏j=1m+11−ηjσiwj.
That is to say, (17) holds for k=m+1.
Therefore, (17) holds for all k∈ℕ, which completes the proof.Theorem 2.
Letnjj=1,2,…,k be a collection of SVNHFEs; then, for the proposed aggregation operator NSVNHFWG, the following properties always hold:(1)
Idempotency: if allnjj=1,2,…,k are equal, i.e., nj=nj=1,2,…,k; then,(22)NSVNHFWGn1,n2,…,nk=n.(2)
Boundary: if there is a pair of SVNHFEsnc and nd such that nc≺snjj=1,2,…,k,j≠c and nj≺sndj=1,2,…,k,j≠d, then(23)nc≺sNSVNHFWGn1,n2,…,nk≺snd.(3)
Monotonicity: if there are a collection of SVNHFEsnj∗j=1,2,…,k such that nj≺snj∗, then(24)NSVNHFWGn1,n2,…,nk≺sNSVNHFWGn1∗,n2∗,…,nk∗.Proof.
(1)
It is not difficult to achieve the above results from Definition9, herein we omit it.(2)
Supposenc=t˜c,i˜c,f˜c and nd=t˜d,i˜d,f˜d. Since nc≺snj=t˜j,i˜j,f˜j≺snd for any j≠c,d, we have t˜c≺st˜j≺st˜d,i˜d≺si˜j≺si˜c,andf˜d≺sf˜j≺sf˜c; furthermore, t˜c≺s⊗j=1Nkt˜jwj≺st˜d,i˜d≺s⊗j=1Nki˜jwj≺si˜c,andf˜d≺s⊗j=1Nkf˜jwj≺sf˜c, where wj is the weight of nj.Hence,nc≺s⊗j=1Nkt˜jwj,⊕j=1Nkwji˜j,⊕j=1Nkwjf˜j≺snd.On the contrary,NSVNHFWGn1,n2,…,nk=⊗j=1Nkt˜jwj,⊕j=1Nkwji˜j,⊕j=1Nkwjf˜j.Therefore,nc≺sNSVNHFWGn1,n2,…,nk≺snd.(3)
Supposenj=t˜j,i˜j,f˜j and nj∗=t˜j∗,i˜j∗,f˜j∗j=1,2,…,k. nj≺snj∗ implies t˜j≺st˜j∗,i˜j∗≺si˜j,andf˜j∗≺sf˜j; hence, ⊗j=1Nkt˜jwj,⊕j=1Nkwji˜j,⊕j=1Nkwjf˜j≺s⊗j=1Nkt˜j∗wj,⊕j=1Nkwji˜j∗,⊕j=1Nkwjf˜j∗. Therefore, NSVNHFWGn1,n2,…,nk≺sNSVNHFWGn1∗,n2∗,…,nk∗.Definition 13.
Letnj=t˜j,i˜j,f˜j∈Ωj=1,2,…,k be a collection of SVNHFEs, and a normalized single-valued neutrosophic hesitant fuzzy ordered weighted geometric operator NSVNHFOWG is defined as(25)NSVNHFOWGn1,n2,…,nk=⊗j=1Nknσjωj=⊗j=1Nkt˜σjωj,⊕j=1Nkωji˜σj,⊕j=1Nkωjf˜σj,where nσj=t˜σj,i˜σj,f˜σj is the jth largest element of njj=1,2,…,k and ω=ω1,ω2,…,ωkT is the aggregation-associated vector such that ωj∈0,1j=1,2,…,k and ∑j=1kωj=1.
Especially, ifω=1/k,1/k,…,1/kT, then by the commutativity of ⊗N, the operator NSVNHFOWG is also reduced to operator NSVNHFG. Similar to Theorem 1, we can give the following result.Theorem 3.
Letnj=t˜j,i˜j,f˜j=γji|i=1,2,…,lj,δji|i=1,2,…,pj,ηji|i=1,2,…,qjj=1,2,…,k be a collection of SVNHFEs; if the aggregation-associated vector is ω=ω1,ω2,…,ωnT and l=maxj=1,…,klj, p=maxj=1,…,kpj, and q=maxj=1,…,kqj, then the aggregated result by operator NSVNHFOWG can be expressed as(26)NSVNHFOWGn1,n2,…,nk=∪i=1l∏j=1kγσjσiωj,∪i=1p1−∏j=1k1−δσjσiωj,∪i=1q1−∏j=1k1−ησjσiωj,where γσjσi,δσjσi,andησjσi are the ith largest element of t˜σj,i˜σj, and f˜σj, respectively, and nσj=t˜σj,i˜σj,f˜σjj=1,2,…,k is a permutation of nj=t˜j,i˜j,f˜jj=1,2,…,k such that nσj−1≻nσj.Proof.
It can be proved similar to Theorem1, herein we omit it.Example 5.
For three SVNHFEsn1=0.1,0.4,0.3,0.2,0.6, n2=0.6,0.5,0.3,0.5, and n3=0.2,0.3,0.6,0.7 with aggregation-associated vector ω=0.3,0.2,0.5T, then sn1=0.5167,sn2=0.5667,andsn3=0.3167, which means n2≻n1≻n3, i.e., nσ1=n2,nσ2=n1,andnσ3=n3; thus,(27)NSVNHFOWGn1,n2,n3=∪i=12∏j=13γσjσiωj,1−∏j=131−δσjσ1ωj,∪i=121−∏j=131−ησjσiωj=0.60.30.40.20.30.5,0.60.30.10.20.20.5,1−1−0.50.31−0.30.21−0.60.5,1−1−0.50.31−0.60.21−0.70.5,1−1−0.30.31−0.20.21−0.70.5=0.3912,0.2421,0.5217,0.6296,0.5293.Theorem 4.
Letnjj=1,2,…,k be a collection of SVNHFEs; then, for the aggregation operator NSVNHFOWG with aggregation-associated weight vector ω=ω1,ω2,…,ωnT, and the following properties always hold.(1)
Idempotency: if allnjj=1,2,…,k are equal, i.e., nj=nj=1,2,…,k, then(28)NSVNHFOWGn1,n2,…,nk=n.(2)
Boundary: if there are a pair of SVNHFEsnc and nd such that nc≺snjj=1,2,…,k,j≠c and nj≺sndj=1,2,…,k,j≠d, then(29)nc≺sNSVNHFOWGn1,n2,…,nk≺snd.(3)
Monotonicity: if there are a collection of SVNHFEsnj∗j=1,2,…,k such that nj≺snj∗, then(30)NSVNHFOWGn1,n2,…,nk≺sNSVNHFOWGn1∗,n2∗,…,nk∗.Proof.
It can be proved similar to Theorem2, herein we omit it.
In what follows, we develop a sort of hybrid aggregation operator which weights the given arguments as well as their ordered positions simultaneously.Definition 14.
Letnj∈Ωj=1,2,…,k be a collection of SVNHFEs; then, a normalized single-valued neutrosophic hesitant fuzzy hybrid weighted geometric operator NSVNHFHWG which has an aggregation-associated vector ω=ω1,ω2,…,ωkT with ωj∈0,1,∑j=1kωj=1 is defined as(31)NSVNHFHWGn1,n2,…,nk=⊗j=1Nkn˙σjωj=⊗j=1Nkt˜˙σjωj,⊕j=1Nkωji˜˙σj,⊕j=1Nkωjf˜˙σj,where n˙σj is the jth largest element of n˙j=njkwjj=1,2,…,k, w=w1,w2,…,wkT is the weight vector of njj=1,2,…,k such that wj∈0,1,∑j=1kwj=1, and k is the balancing coefficient.
Especially, ifω=ω1,ω2,…,ωkT=1/k,1/k,…,1/kT, then the aggregation operator NSVNHFHWG can be reduced to operator NSVNHFWG. On the contrary, if w=w1,w2,…,wkT=1/k,1/k,…,1/kT, then the aggregation operator NSVNHFHWG can be reduced to operator NSVNHFOWG. Furthermore, if w1,w2,…,wkT=ω1,ω2,…,ωkT=1/k,1/k,…,1/kT, then the aggregation operator NSVNHFHWG can be reduced to operator NSVNHFG. Put it another way, the hybrid operator NSVNHFHG is a generalization of aggregation operators NSVNHFG, NSVNHFWG, and NSVNHFOWG.Theorem 5.
Letnj=t˜j,i˜j,f˜j=γji|i=1,2,…,lj,δji|i=1,2,…,pj,ηji|i=1,2,…,qjj=1,2,…,k be a collection of SVNHFEs with weight vector w=w1,w2,…,wnT and l=maxj=1,…,klj, p=maxj=1,…,kpj, and q=maxj=1,…,kqj; then, the aggregated result by using operator NSVNHFHWG with aggregation-associated vector ω=ω1,ω2,…,ωnT can be expressed as(32)NSVNHFHWGn1,n2,…,nk=∪i=1l∏j=1kγ˙σjσiωj,∪i=1p1−∏j=1k1−δ˙σjσiωj,∪i=1q1−∏j=1k1−η˙σjσiωj,where γ˙σjσi,δ˙σjσi,andη˙σjσi are the ith largest element of t˜˙σj,i˜˙σj, and f˜˙σj, respectively, and n˙σj=t˜˙σj,i˜˙σj,f˜˙σjj=1,2,…,k is a permutation of n˙j=njkwj=t˜jkwj,kwji˜j,kwjf˜jj=1,2,…,k such that n˙σj−1≻n˙σj.Proof.
It can be proved similar to Theorem1, herein we omit it.Example 6.
Suppose SVNHFEsn1=0.1,0.4,0.3,0.2,0.6,n2=0.6,0.5,0.3,0.5,andn3=0.2,0.3,0.6,0.7 with the weight vector w=0.1,0.3,0.6T; then, their aggregated result by operator NSVNHFHG with aggregation-associated vector ω=0.7,0.2,0.1T can be calculated through the following process:(1)
n˙1=n13×0.1=0.1,0.4,0.3,0.2,0.60.3=0.5012,0.7597,0.1015,0.0648,0.2403,n˙2=n23×0.3=0.6,0.5,0.3,0.50.9=0.6314,0.4641,0.2746,0.4641,n˙3=n33×0.6=0.2,0.3,0.6,0.71.8=0.0552,0.1145,0.8078,0.8855(2)
sn˙1=0.7921,sn˙2=0.5993,sn˙3=0.1305(3)
n˙σ1=n˙1,n˙σ2=n˙2,n˙σ3=n˙3(4)
NSVNHFHWGn1,n2,n3=∪i=12∏j=13γ˙σjσiωj,1−∏j=131−δ˙σjσ1ωj,∪i=121−∏j=131−η˙σjσiωj=0.75970.70.63140.20.11450.1,0.50120.70.63140.20.05520.1,1−1−0.10150.71−0.46410.21−0.80780.1,1−1−0.24030.71−0.46410.21−0.88550.1,1−1−0.06480.71−0.27460.21−0.88550.1=0.6059,0.4210,0.3055,0.4137,0.2795Remark 1.
From Theorems2 and 4, we can conclude that the aggregation operator NSVNHFHWG also satisfy idempotency, boundary, and monotonicity.


## 4. Decision-Making Method Based on Normalized Single-Valued Neutrosophic Hesitant Fuzzy Geometric Aggregation Operators

### 4.1. Decision-Making Method

Assume there arem alternatives Aii=1,2,…,m under consideration for a decision-making problem, and they are estimated in terms of attributes Cjj=1,2,…,k which possess weight vector w=w1,w2,…,wkT such that wj∈0,1j=1,2,…,k and ∑j=1kwj=1. Some decision makers provide their evaluation values for alternative Ai with respect to attribute Cj which is characterized by a single-valued neutrosophic hesitant fuzzy element nij=t˜ij,i˜ij,f˜ij, where t˜ij,i˜ij,andf˜ij represent the truth degree, uncertain degree, and falsity degree of alternative of Ai satisfying attribute Cj. It should be noticed that if two decision makers give the same evaluation value on one alternative, then the values cannot be merged in nij since it will affect the score of a hesitant fuzzy element actually. Next, take operator NSVNHFHWG with aggregation-associated vector ω=ω1,ω2,…,ωkT as an example, and we demonstrate the following decision-making process in detail:Step 1. Collect evaluation values from all decision makers and construct SVNHFEs information matrix N˜=nijm×k.Step 2. For any i=1,2,…,m, calculate n˙ij=nijkwjj=1,2,…,k.Step 3. For any i=1,2,…,m, rank n˙ijj=1,2,…,k according to the score, amplitude, or variance of n˙ijj=1,2,…,k and obtain n˙iσj=t˜˙iσj,i˜˙iσj,f˜˙iσj=γ˙iσj,δ˙iσj,η˙iσj.Step 4. For any i=1,2,…,m, aggregate the collection nijj=1,2,…,k by aggregation operator NSVNHFHWG and denote the result as ni, that is,(33)ni=NSVNHFHWGni1,ni2,…,nik=∪t=1l∏j=1kγ˙iσjσtωj,∪t=1p1−∏j=1k1−δ˙iσjσtωj,∪t=1q1−∏j=1k1−η˙iσjσtωj,wherel=maxγ˙iσj,p=maxδ˙iσj,q=maxη˙iσj, γ˙iσjσt is the tth largest element of t˜˙iσj, δ˙iσjσt is the tth largest element of i˜˙iσj, and η˙iσjσt is the tth largest element of f˜˙iσj.Step 5. Rank nii=1,2,…,m based on the score, amplitude, and variance of nii=1,2,…,m.Step 6. Rank the alternatives Aii=1,2,…,m based on the order of nii=1,2,…,m and obtain optimal alternative.


### 4.2. Numerical Example and Analysis

An example from [27] is utilized to illustrate the applicability and validity of the proposed MADM method. An example from [27] is utilized to illustrate the applicability and validity of the proposed MADM method. Now, there are four alternatives Aii=1,2,3,4 which were considered with respect to twelve attributes: C1: functionality, C2: reliability, C3:usability, C4: efficiency, C5: maintainability, C6: portability, C7: acquisition, C8: customization, C9: training, C10: operation, C11: maintenance, and C12: standards, which possess weight vector w=0.1,0.12,0.2,0.05,0.06,0.04,0.08,0.05,0.1,0.08,0.02,0.1T. Some decision makers estimate these alternatives and provide their evaluation information adequately that is listed in Table 1. Meanwhile, the weighted vector ω=0.08,0.12,0.1,0.06,0.02,0.06,0.06,0.04,0.06,0.1,0.2,0.1T of the operator NSVNHFHWG is given. We perform the following steps.Step 1. Collect evaluation values from all decision makers and construct SVNHFEs information matrix N˜=nij4×12 (see Table 1).Step 2. Utilize weight vector w=0.1,0.12,0.2,0.05,0.06,0.04,0.08,0.05,0.1,0.08,0.02,0.1T to obtain n˙ij=nij12wj. Take n˙21 as an example:(34)n˙21=0.5,0.1,0.412×0.1=0.51.2,1.20.1,1.20.4=0.51.2,1−1−0.11.2,1−1−0.41.2=0.4353,0.1188,0.4583.Further details are shown in Table2.Step 3. We utilize function to figure out the order relationship of n˙ijj=1,2,…,12. Take the alternative A2 as an example: according to the score, amplitude, or variance of n˙2j, we obtain(35)sn˙21=0.6194,sn˙22=0.6697,sn˙23=0.6412,sn˙24=0.5175,sn˙25=0.5854,sn˙26=0.5647,sn˙27=0.5625,sn˙28=0.6179,sn˙29=0.7203,sn˙210=0.5617,sn˙211=0.4223,sn˙212=0.7779.We rank the order asn˙212≻n˙29≻n˙22≻n˙23≻n˙21≻n˙28≻n˙25≻n˙26≻n˙27≻n˙210≻n˙24≻n˙211. Similarly, we obtain(36)n˙13≻n˙19≻n˙112≻n˙12≻n˙110≻n˙17≻n˙11≻n˙15≻n˙14≻n˙16≻n˙18≻n˙111,n˙32≻n˙39≻n˙312≻n˙38≻n˙33≻n˙310≻n˙31≻n˙34≻n˙37≻n˙36≻n˙35≻n˙311,n˙42≻n˙412≻n˙43≻n˙41≻n˙49≻n˙45≻n˙48≻n˙46≻n˙410≻n˙47≻n˙44≻n˙411.Step 4. Utilize one aggregation operator to aggregate nijj=1,2,…,12 and obtain ni, and we take operator NSVNHFHWG as an example:(37)n2=NSVNHFHWGn21,n22,n23,n24,n25,n26,n27,n28,n29,n210,n211,n212=∪t=12∏j=112γ˙2σjσtωj,∪t=121−∏j=1121−δ˙2σjσtωj,∪t=121−∏j=1121−η˙2σjσtωj=0.54170.080.54170.120.26730.10.05560.060.43530.020.48560.060.42030.060.71700.040.41490.060.21330.10.38070.20.74900.1,0.54170.080.54170.120.09850.10.05560.060.43530.020.48560.060.42030.060.71700.040.41490.060.21330.10.38070.20.74900.1,1−1−0.14500.081−0.14500.121−0.17660.11−0.02100.061−0.11880.021−0.25120.061−0.19050.061−0.46180.041−0.21330.061−0.31480.11−0.25120.21−0.67960.1,1−1−0.14500.081−0.14500.121−0.09850.11−0.02100.061−0.11880.021−0.25120.061−0.19050.061−0.46180.041−0.21330.061−0.31480.11−0.25120.21−0.67960.1,1−1−0.06310.081−0.23580.121−0.03630.11−0.11090.061−0.45830.021−0.38070.061−0.42030.061−0.56110.041−0.51410.061−0.21330.11−0.57710.21−0.80260.1=0.3329,0.3744,0.2769,0.2834,0.4268.Similarly, we can getn1,n3,andn4 as follows:(38)n1=∪t=12∏j=112γ˙1σjσtωj,∪t=121−∏j=1121−δ˙1σjσtωj,∪t=121−∏j=1121−η˙1σjσtωj=0.4975,0.5171,0.3519,0.3755,0.3326,0.3550,n3=∪t=12∏j=112γ˙1σjσtωj,1−∏j=1121−δ˙1σjσ1ωj,1−∏j=1121−η˙1σjσ1ωj=0.4682,0.5022,0.3253,0.3492,n4=∪t=12∏j=112γ˙1σjσtωj,∪t=121−∏j=1121−δ˙1σjσtωj,∪t=121−∏j=1121−η˙1σjσtωj=0.3655,0.4131,0.2738,0.2851,0.4451,0.4466.Step 5. Based on the score function of SVNHFEs, we get sn1=0.5999,sn2=0.5489,sn3=0.6036,andsn4=0.5547.Step 6. Since sn3>sn1>sn4>sn2, the ranking order of all the alternatives is A3≻A1≻A4≻A2 and the most desirable one is A3.Table 1
Single-valued neutrosophic hesitant fuzzy information.
C1C2C3C4A10.4,0.2,0.1,0.30.3,0.4,0.2,0.1,0.30.6,0.3,0.20.5,0.3,0.2A20.5,0.1,0.40.2,0.4,0.2,0.3,0.10.3,0.2,0.40.2,0.1,0.4A30.3,0.5,0.2,0.30.6,0.3,0.20.4,0.1,0.20.5,0.2,0.3A40.2,0.1,0.10.6,0.1,0.30.3,0.5,0.2,0.30.3,0.1,0.4C5C6C7C8A10.3,0.2,0.1,0.20.3,0.4,0.1,0.30.6,0.4,0.30.4,0.2,0.3,0.2A20.2,0.3,0.1,0.30.5,0.2,0.30.4,0.2,0.50.3,0.1,0.2A30.6,0.3,0.40.2,0.4,0.2,0.10.3,0.1,0.40.6,0.2,0.1A40.6,0.2,0.40.6,0.3,0.20.3,0.2,0.40.4,0.1,0.3C9C10C11C12A10.7,0.3,0.40.6,0.4,0.20.3,0.2,0.10.5,0.3,0.2A20.6,0.2,0.30.2,0.3,0.20.3,0.2,0.40.6,0.2,0.1A30.4,0.1,0.20.5,0.2,0.30.3,0.2,0.10.4,0.1,0.2A40.5,0.2,0.3,0.40.4,0.1,0.3,0.50.5,0.3,0.40.4,0.1,0.2Table 2
Weighted single-valued neutrosophic hesitant fuzzy information.
C1C2A10.3330,0.2349,0.1188,0.34820.1766,0.2673,0.2748,0.1408,0.4017A20.4353,0.1188,0.45830.0985,0.2673,0.2748,0.4017,0.1408A30.2358,0.4353,0.2349,0.34820.4792,0.4017,0.2748A40.1450,0.1188,0.11880.4792,0.1408,0.4017C3C4A10.2935,0.5752,0.41460.6598,0.1927,0.1253A20.0556,0.4146,0.70650.3807,0.0613,0.2640A30.1109,0.2234,0.41460.6598,0.1253,0.1927A40.0556,0.1895,0.4146,0.57520.4856,0.0613,0.2640C5C6A10.4203,0.1484,0.0731,0.14840.5611,0.6442,0.0493,0.1574A20.3139,0.4203,0.0731,0.22650.7170,0.1016,0.1574A30.6923,0.2265,0.30770.4618,0.6442,0.1016,0.0493A40.6923,0.1484,0.30770.7826,0.1574,0.1016C7C8A10.6124,0.3876,0.28990.5771,0.1253,0.1927,0.1253A20.4149,0.1928,0.48590.4856,0.0613,0.1253A30.3148,0.0962,0.38760.7360,0.1253,0.0613A40.3148,0.1928,0.38760.5771,0.0613,0.1927C9C10A10.6518,0.3482,0.45830.6124,0.3876,0.1928A20.5417,0.2349,0.34820.2133,0.2899,0.1928A30.3330,0.1188,0.23490.5141,0.1928,0.2899A40.4353,0.2349,0.3482,0.45830.4149,0.0962,0.2899,0.4859C11C12A10.7490,0.0521,0.02500.4353,0.3482,0.2349A20.7490,0.0521,0.11540.5417,0.2349,0.1188A30.7490,0.0521,0.02500.3330,0.1188,0.2349A40.8467,0.0820,0.11540.3330,0.1188,0.2349Comparing to the decision resultA1≻A3≻A4≻A2 in [23], we can observe that there exist a little difference from the result of the present paper A3≻A1≻A4≻A2. As Mishra and Kumar asserted in [24], it does not make any sense to apply the aggregation operator in [23] to solve decision-making problems since the aggregation operators do not fulfill monotonicity and idempotency. However, the aggregation operator proposed in the present paper definitely satisfies monotonicity and idempotency, and the normalized sum operation is completely available which means the decision-making result is convincing absolutely. Besides, the computation is less than the earlier method since we avoid crossover operation. Therefore, it is wise to apply the method to many other decision-making problems.


## 4.1. Decision-Making Method

Assume there arem alternatives Aii=1,2,…,m under consideration for a decision-making problem, and they are estimated in terms of attributes Cjj=1,2,…,k which possess weight vector w=w1,w2,…,wkT such that wj∈0,1j=1,2,…,k and ∑j=1kwj=1. Some decision makers provide their evaluation values for alternative Ai with respect to attribute Cj which is characterized by a single-valued neutrosophic hesitant fuzzy element nij=t˜ij,i˜ij,f˜ij, where t˜ij,i˜ij,andf˜ij represent the truth degree, uncertain degree, and falsity degree of alternative of Ai satisfying attribute Cj. It should be noticed that if two decision makers give the same evaluation value on one alternative, then the values cannot be merged in nij since it will affect the score of a hesitant fuzzy element actually. Next, take operator NSVNHFHWG with aggregation-associated vector ω=ω1,ω2,…,ωkT as an example, and we demonstrate the following decision-making process in detail:Step 1. Collect evaluation values from all decision makers and construct SVNHFEs information matrix N˜=nijm×k.Step 2. For any i=1,2,…,m, calculate n˙ij=nijkwjj=1,2,…,k.Step 3. For any i=1,2,…,m, rank n˙ijj=1,2,…,k according to the score, amplitude, or variance of n˙ijj=1,2,…,k and obtain n˙iσj=t˜˙iσj,i˜˙iσj,f˜˙iσj=γ˙iσj,δ˙iσj,η˙iσj.Step 4. For any i=1,2,…,m, aggregate the collection nijj=1,2,…,k by aggregation operator NSVNHFHWG and denote the result as ni, that is,(33)ni=NSVNHFHWGni1,ni2,…,nik=∪t=1l∏j=1kγ˙iσjσtωj,∪t=1p1−∏j=1k1−δ˙iσjσtωj,∪t=1q1−∏j=1k1−η˙iσjσtωj,wherel=maxγ˙iσj,p=maxδ˙iσj,q=maxη˙iσj, γ˙iσjσt is the tth largest element of t˜˙iσj, δ˙iσjσt is the tth largest element of i˜˙iσj, and η˙iσjσt is the tth largest element of f˜˙iσj.Step 5. Rank nii=1,2,…,m based on the score, amplitude, and variance of nii=1,2,…,m.Step 6. Rank the alternatives Aii=1,2,…,m based on the order of nii=1,2,…,m and obtain optimal alternative.


## 4.2. Numerical Example and Analysis

An example from [27] is utilized to illustrate the applicability and validity of the proposed MADM method. An example from [27] is utilized to illustrate the applicability and validity of the proposed MADM method. Now, there are four alternatives Aii=1,2,3,4 which were considered with respect to twelve attributes: C1: functionality, C2: reliability, C3:usability, C4: efficiency, C5: maintainability, C6: portability, C7: acquisition, C8: customization, C9: training, C10: operation, C11: maintenance, and C12: standards, which possess weight vector w=0.1,0.12,0.2,0.05,0.06,0.04,0.08,0.05,0.1,0.08,0.02,0.1T. Some decision makers estimate these alternatives and provide their evaluation information adequately that is listed in Table 1. Meanwhile, the weighted vector ω=0.08,0.12,0.1,0.06,0.02,0.06,0.06,0.04,0.06,0.1,0.2,0.1T of the operator NSVNHFHWG is given. We perform the following steps.Step 1. Collect evaluation values from all decision makers and construct SVNHFEs information matrix N˜=nij4×12 (see Table 1).Step 2. Utilize weight vector w=0.1,0.12,0.2,0.05,0.06,0.04,0.08,0.05,0.1,0.08,0.02,0.1T to obtain n˙ij=nij12wj. Take n˙21 as an example:(34)n˙21=0.5,0.1,0.412×0.1=0.51.2,1.20.1,1.20.4=0.51.2,1−1−0.11.2,1−1−0.41.2=0.4353,0.1188,0.4583.Further details are shown in Table2.Step 3. We utilize function to figure out the order relationship of n˙ijj=1,2,…,12. Take the alternative A2 as an example: according to the score, amplitude, or variance of n˙2j, we obtain(35)sn˙21=0.6194,sn˙22=0.6697,sn˙23=0.6412,sn˙24=0.5175,sn˙25=0.5854,sn˙26=0.5647,sn˙27=0.5625,sn˙28=0.6179,sn˙29=0.7203,sn˙210=0.5617,sn˙211=0.4223,sn˙212=0.7779.We rank the order asn˙212≻n˙29≻n˙22≻n˙23≻n˙21≻n˙28≻n˙25≻n˙26≻n˙27≻n˙210≻n˙24≻n˙211. Similarly, we obtain(36)n˙13≻n˙19≻n˙112≻n˙12≻n˙110≻n˙17≻n˙11≻n˙15≻n˙14≻n˙16≻n˙18≻n˙111,n˙32≻n˙39≻n˙312≻n˙38≻n˙33≻n˙310≻n˙31≻n˙34≻n˙37≻n˙36≻n˙35≻n˙311,n˙42≻n˙412≻n˙43≻n˙41≻n˙49≻n˙45≻n˙48≻n˙46≻n˙410≻n˙47≻n˙44≻n˙411.Step 4. Utilize one aggregation operator to aggregate nijj=1,2,…,12 and obtain ni, and we take operator NSVNHFHWG as an example:(37)n2=NSVNHFHWGn21,n22,n23,n24,n25,n26,n27,n28,n29,n210,n211,n212=∪t=12∏j=112γ˙2σjσtωj,∪t=121−∏j=1121−δ˙2σjσtωj,∪t=121−∏j=1121−η˙2σjσtωj=0.54170.080.54170.120.26730.10.05560.060.43530.020.48560.060.42030.060.71700.040.41490.060.21330.10.38070.20.74900.1,0.54170.080.54170.120.09850.10.05560.060.43530.020.48560.060.42030.060.71700.040.41490.060.21330.10.38070.20.74900.1,1−1−0.14500.081−0.14500.121−0.17660.11−0.02100.061−0.11880.021−0.25120.061−0.19050.061−0.46180.041−0.21330.061−0.31480.11−0.25120.21−0.67960.1,1−1−0.14500.081−0.14500.121−0.09850.11−0.02100.061−0.11880.021−0.25120.061−0.19050.061−0.46180.041−0.21330.061−0.31480.11−0.25120.21−0.67960.1,1−1−0.06310.081−0.23580.121−0.03630.11−0.11090.061−0.45830.021−0.38070.061−0.42030.061−0.56110.041−0.51410.061−0.21330.11−0.57710.21−0.80260.1=0.3329,0.3744,0.2769,0.2834,0.4268.Similarly, we can getn1,n3,andn4 as follows:(38)n1=∪t=12∏j=112γ˙1σjσtωj,∪t=121−∏j=1121−δ˙1σjσtωj,∪t=121−∏j=1121−η˙1σjσtωj=0.4975,0.5171,0.3519,0.3755,0.3326,0.3550,n3=∪t=12∏j=112γ˙1σjσtωj,1−∏j=1121−δ˙1σjσ1ωj,1−∏j=1121−η˙1σjσ1ωj=0.4682,0.5022,0.3253,0.3492,n4=∪t=12∏j=112γ˙1σjσtωj,∪t=121−∏j=1121−δ˙1σjσtωj,∪t=121−∏j=1121−η˙1σjσtωj=0.3655,0.4131,0.2738,0.2851,0.4451,0.4466.Step 5. Based on the score function of SVNHFEs, we get sn1=0.5999,sn2=0.5489,sn3=0.6036,andsn4=0.5547.Step 6. Since sn3>sn1>sn4>sn2, the ranking order of all the alternatives is A3≻A1≻A4≻A2 and the most desirable one is A3.Table 1
Single-valued neutrosophic hesitant fuzzy information.
C1C2C3C4A10.4,0.2,0.1,0.30.3,0.4,0.2,0.1,0.30.6,0.3,0.20.5,0.3,0.2A20.5,0.1,0.40.2,0.4,0.2,0.3,0.10.3,0.2,0.40.2,0.1,0.4A30.3,0.5,0.2,0.30.6,0.3,0.20.4,0.1,0.20.5,0.2,0.3A40.2,0.1,0.10.6,0.1,0.30.3,0.5,0.2,0.30.3,0.1,0.4C5C6C7C8A10.3,0.2,0.1,0.20.3,0.4,0.1,0.30.6,0.4,0.30.4,0.2,0.3,0.2A20.2,0.3,0.1,0.30.5,0.2,0.30.4,0.2,0.50.3,0.1,0.2A30.6,0.3,0.40.2,0.4,0.2,0.10.3,0.1,0.40.6,0.2,0.1A40.6,0.2,0.40.6,0.3,0.20.3,0.2,0.40.4,0.1,0.3C9C10C11C12A10.7,0.3,0.40.6,0.4,0.20.3,0.2,0.10.5,0.3,0.2A20.6,0.2,0.30.2,0.3,0.20.3,0.2,0.40.6,0.2,0.1A30.4,0.1,0.20.5,0.2,0.30.3,0.2,0.10.4,0.1,0.2A40.5,0.2,0.3,0.40.4,0.1,0.3,0.50.5,0.3,0.40.4,0.1,0.2Table 2
Weighted single-valued neutrosophic hesitant fuzzy information.
C1C2A10.3330,0.2349,0.1188,0.34820.1766,0.2673,0.2748,0.1408,0.4017A20.4353,0.1188,0.45830.0985,0.2673,0.2748,0.4017,0.1408A30.2358,0.4353,0.2349,0.34820.4792,0.4017,0.2748A40.1450,0.1188,0.11880.4792,0.1408,0.4017C3C4A10.2935,0.5752,0.41460.6598,0.1927,0.1253A20.0556,0.4146,0.70650.3807,0.0613,0.2640A30.1109,0.2234,0.41460.6598,0.1253,0.1927A40.0556,0.1895,0.4146,0.57520.4856,0.0613,0.2640C5C6A10.4203,0.1484,0.0731,0.14840.5611,0.6442,0.0493,0.1574A20.3139,0.4203,0.0731,0.22650.7170,0.1016,0.1574A30.6923,0.2265,0.30770.4618,0.6442,0.1016,0.0493A40.6923,0.1484,0.30770.7826,0.1574,0.1016C7C8A10.6124,0.3876,0.28990.5771,0.1253,0.1927,0.1253A20.4149,0.1928,0.48590.4856,0.0613,0.1253A30.3148,0.0962,0.38760.7360,0.1253,0.0613A40.3148,0.1928,0.38760.5771,0.0613,0.1927C9C10A10.6518,0.3482,0.45830.6124,0.3876,0.1928A20.5417,0.2349,0.34820.2133,0.2899,0.1928A30.3330,0.1188,0.23490.5141,0.1928,0.2899A40.4353,0.2349,0.3482,0.45830.4149,0.0962,0.2899,0.4859C11C12A10.7490,0.0521,0.02500.4353,0.3482,0.2349A20.7490,0.0521,0.11540.5417,0.2349,0.1188A30.7490,0.0521,0.02500.3330,0.1188,0.2349A40.8467,0.0820,0.11540.3330,0.1188,0.2349Comparing to the decision resultA1≻A3≻A4≻A2 in [23], we can observe that there exist a little difference from the result of the present paper A3≻A1≻A4≻A2. As Mishra and Kumar asserted in [24], it does not make any sense to apply the aggregation operator in [23] to solve decision-making problems since the aggregation operators do not fulfill monotonicity and idempotency. However, the aggregation operator proposed in the present paper definitely satisfies monotonicity and idempotency, and the normalized sum operation is completely available which means the decision-making result is convincing absolutely. Besides, the computation is less than the earlier method since we avoid crossover operation. Therefore, it is wise to apply the method to many other decision-making problems.


## 5. Conclusion

In this paper, we have defined two normalized single-valued neutrosophic hesitant fuzzy operations which are indeed meaningful for the processing of single-valued neutrosophic hesitant fuzzy elements, since it turned out that the operations satisfy some basic desirable operation rules such as associative law and distributive law. Moreover, a series of normalized geometric aggregation operators possessing all the basic properties of a valid aggregation operator such as indempotency, boundary, and monotonicity are proposed from the geometric point of view. Furthermore, a decision-making method based on the aggregation operators is developed to resolve multiattribute group decision-making problems, and its feasibility and validity have been illustrated with the help of a practical example. However, the information to be aggregated is mutually connected sometimes and the weight vector can be affected by other evaluation values. Thus, how to handle the relationship between single-valued neutrosophic hesitant fuzzy elements is a critical problem, and it is a research direction we will focus on in the future.



---

*Source: 5580761-2021-05-06.xml*