# Novel Development to the Theory of Dombi Exponential Aggregation Operators in Neutrosophic Cubic Hesitant Fuzzy Sets: Applications to Solid Waste Disposal Site Selection

Authors: Ateeq Ur Rehman; Muhammad Gulistan; Nasreen Kausar; Sajida Kousar; Mohammed M. Al-Shamiri; Rashad Ismail
Journal: Complexity
 (2022)

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## Abstract

The neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co-norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi exponential operational laws in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision-making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision-making problems.


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## Body

## 1. Introduction

Decision-making (DM) is one of the crucial problems in real life. Due to population increase and urbanization, the proper disposal of waste is a challenging problem nowadays. The proper disposal of waste is necessary for prevention of viral diseases like typhoid, dengue, and tuberculosis. Aggregation operators are fundamental tools in DM. Different sets and their generalizations like fuzzy set (FS), interval-valued fuzzy set (IVFS), intuitionistic fuzzy set (IFS), hesitant fuzzy set (HFS), neutrosophic set (NS), neutrosophic cubic set (NCS), and several aggregation operators have been defined so far [1–4]. Different researchers [5–8] established similarity measures and other important concepts and successfully applied their models to medical diagnosis and selection criteria. Krohling et al. established different useful techniques to sort out MADM problems [9–11]. Jun et al. combined IVFS and FS to form a cubic set (CS). The CS is a combination of FS and IVFS. CS has become a vital tool to deal with vague data. Several researchers explored algebraic aspects and apparently defined ideal theory in CS [12–15]. Smarandache initiated the concept of indeterminacy and described the notion of the neutrosophic set (NS) [16]. NS consist of three independent components, truth, indeterminacy, and falsehood. This characteristic of NS enables researchers to deal smoothly with inconsistent and vague data. For engineering purposes, the neutrosophic set is strict to [0, 1] and is called the single-valued neutrosophic set (SVNS) [17] presented by Wang et al. The NS was further extended to the interval neutrosophic set (INS) [18]. After the appearance of NS, researchers put their contributions to theoretical as well as technological developments of the set [19–22]. Several researchers use these sets to construct multiattribute decision-making MADM [23–27]. Zhan et al. define aggregation operators and furnished some applications in MADM [28]. Torra defines a hesitant fuzzy set (HFS) [29]. An HFS is basically a function set on X that, when applied to X, returns a subset of [0, 1]. Jun et al. introduced the concept of neutrosophic cubic set (NCS) [30] which consists of both interval-valued neutrosophic and neutrosophic set. These characteristics of neutrosophic cubic sets make it a powerful tool to deal the vague and inconsistent data more efficiently. Soon after its exploration, it attracted researchers to work in many fields like medicine, algebra, engineering, and decision-making theory. Later, the idea of a cubic hesitant fuzzy set was introduced by Mehmood et al. [31]. Ye [32] proposed the concept of the single-valued neutrosophic hesitant fuzzy set (SVNHFS) by combining the advantages of the NS and HFS, which permits its membership functions to have sets of possible values, known as truth, indeterminacy, and falsity membership hesitant functions. Liu and Shi [33] proposed hybrid geometric aggregation operators in INHFS and discuss its presentations in MADM. Zhu et al. [34] proposed the method of β-normalization to add some elements to an HFE, which is a useful technique in case of different cardinalities. Lu and Ye [35] introduced exponential laws in single-valued neutrosophic numbers. Later, the exponential aggregation operators were introduced and applied in typhoon disaster evaluation by Tan et al. [36]. Wang and Li proposed some aggregation operators in pictures hesitant fuzzy sets and compared these operators with some existing decision-making methods [37, 38]. Tan and Zhang introduced trapezoidal fuzzy neutrosophic number arithmetic averaging and hybrid arithmetic averaging for multicriteria decision-making [39]. Feng et al. define type-2 hesitant fuzzy sets and explore some important properties of these sets [40]. Turkarslan et al. proposed the similarity measures in fuzzy multiset with application in medical diagnosis [41].Dombi [42] established a general class of fuzzy operators. The Dombi T-operators are based on an operational parameter, and hence, the operations based on Dombi operators are more flexible than the other algebraic operations. Shi and Ye [43] established Dombi arithmetic and geometric aggregation operators in neutrosophic cubic sets. Liu et al. [44] established Dombi Bonferroni mean operators of intuitionistic fuzzy sets and discuss their applications in MADM problems. Chen and Ye [45] proposed the Dombi weighted aggregation operators in SVNSs and discuss their applications in MADM problems. Bhattacharya et al. [46] proposed the machine learning model-based approach to discussing the demographic influence of risk behavior of urban investors. Biswas et al. [47] establish an ensemble approach for portfolio selection in a multicriteria decision-making framework. Shi and Ye [43] proposed Dombi aggregation operators in the neutrosophic cubic environments for MADM. Rehman et al. [48] established neutrosophic cubic hesitant fuzzy geometric aggregation operators for MADM problems. Ashraf et al. [49] proposed interval-valued the picture fuzzy Maclaurin symmetric mean operators for MADM problems. Riaz and Farid [50] established picture fuzzy aggregation approach and applied it to the third-party logistic provider selection process. Several researchers [51–54], proposed useful techniques and successfully applied their models to solve decision-making problems.Motivation: the Dombi exponential laws and Dombi exponential aggregation operators are not defined in the intuitionistic fuzzy set, neutrosophic set, and their extensions. The Dombi T-operators are more flexible than the usual T-operators, and exponential operators are an essential family of aggregation operators. This motivates the researchers to define Dombi exponential aggregation operators so that the inconsistent, hesitant, and vague data can be efficiently handled in the complex work frame where the base is crisp value.Organization: the rest of this paper is organized as follows: Section2 deals with some basic definitions used later on. Section 3 discusses Dombi algebraic operational laws and score functions in NCHFSs. In Section 4, we introduced distance and similarity measures in NCHFSs. Exponential operating rules and some useful results in NCHFSs. Section 5 deals with establishing Dombi exponential aggregation operators as NCHFSs. In Section 6, we demonstrate a MADM method based on NCHFDEA operators and use this method in the MADM problem. Section 7 presents the conclusion of the proposed study.


## 2. Preliminaries

Definition 1.
(see [1]). Let E≠∅ be a set. An FS on E is a function ƛ:E⟶0,1.Definition 2.
(see [55]). Let E≠∅ be a set. The CS on E is defined by Ψ=℘;I℘,ƛ℘a|℘∈E, where I℘ is an IVFS on E and ƛ℘ is a FS on E.Definition 3.
(see [17]). An NS on nonempty set E is a set of the form ℘;ΓT℘,ΓI℘,ΓF℘|℘∈E, where ΓT,ΓI,ΓF are mappings from set E to closed interval [0, 1].Definition 4.
(see [30]). An NCS in E≠∅ is a pair β=B,ℏ where B=℘;BT℘,BI℘,BF℘/℘∈E is an INS in E and ℏ=℘;ℏT℘,ℏI℘,ℏF℘/℘∈E is an NS in E.Definition 5.
(see [32]). A neutrosophic hesitant fuzzy set (NHFS) on nonempty set E is described by ℏ=℘;ℏT℘,ℏI℘,ℏF℘/℘∈E, where ℏT℘,ℏI℘,ℏF℘ are three hesitant fuzzy sets such that ℏT℘+ℏI℘,+ℏF℘≤3.Definition 6.
(see [33]). Consider a nonempty set E. An interval neutrosophic hesitant fuzzy set (INHFS) on E is given by the object B=℘;BT℘,BI℘,BF℘/℘∈E, where BT℘,BI℘,BF℘ are intervalued hesitant fuzzy sets.
The following technique ofβ-normalization proposed by Zhu et al. is a useful technique in case of different cardinalities of two hesitant fuzzy sets.Definition 7.
(see [34]). Let ϖ+ and ϖ− be respectively the maximum and minimum values in a hesitant fuzzy set H and Θ0≤Θ≤1 be an optimized parameter; then, an element can be added to H as ϖ=ϖ++1−Θϖ.Definition 8.
(see [42]). The Dombi norms and dual conorms are given, respectively, as(1)Ds,t=11+1−s/sδ+1−t/tδ1/δ,D∗s,t=1−11+s/1−tδ+t/1−tδ1/δ,where a,b∈0,1×0,1 and δ∈R is an operational parameter. Furthermore, the values of D and D∗ for boundaries are defined as D1,0=D0,1=D0,0=0,D1,1=1,D∗1,1=D∗1,0=D∗0,1=1 and D∗0,0=0.
In the next section, first, we define Dombi algebraic operational laws by using Dombi T-norms and T-conorms in NCHFSs. Second, the score functions are defined for the comparison of two NCHFSs.


## 3. Dombi Operational Laws in Neutrosophic Cubic Hesitant Fuzzy Set

Definition 9.
A neutrosophic cubic hesitant fuzzy set (NCHFS) on a nonempty setE is a pair α=Α,ℏ, where Α=e;ΑTe,ΑIe,ΑFe is an INHFS in E and ℏ=e;ℏTe,ℏIe,ℏFe is an NHFS in E. Furthermore eT=ejTL,ejTU;j=1,…,le,Ie=ejTL,ejTU;j=1,…,me,andAFe=ejTL,ejFU;i=1,…,ne are some interval values in [0, 1] with the property that for each j0≤xjTU+xjIU+xjFU≤3, and eT=ejT;j=1,…,re,eI=ejI;j=1,…,se,eF=ejF;j=1,…,te are some values in [0, 1] satisfying 0≤e+jIe+jFe≤3.Example 1.
LetE=e1,e2,e3. The pair α=A,λ with(2)ΑTe1=0.1,0.5,0.2,0.7,ℏTe1=0.3,0.5,0.7,ΑIe1=0.2,0.4,0.3,0.6,ℏIe1=0.1,0.4,0.7,ΑFe1=0.1,0.4,0,0.3,0.6,0.8,ℏFe1=0.4,0.6,ΑTe2=0.1,0.5,0.2,0.7,ℏTe2=0.3,0.5,ΑIe2=0.2,0.3,0.1,0.6,ℏIe2=0.7,0.8,ΑFe2=0.1,0.4,0,0.3,ℏFe2=0.4,0.6,ΑTe3=0.1,0.5,0.2,0.7,ℏTe3=0.3,0.5,ΑIe3=0.2,0.3,0.1,0.6,ℏIe3=0.7,0.8,ΑFe3=0.1,0.4,0,0.3,ℏFe3=0.4,0.6,is a NCHFS.Definition 10.
For two NCHF valuesα=A,λ,β=B,μ and scalars q,σ>0, we define, respectively, the Dombi sum, Dombi product, Dombi scalar multiplication, and the Dombi qth power of α as(3)α⊕β=1−1+AjTL1−AjTLσ+BjTL1−BjTLσ1/σ−1,1−1+AjTU1−AjTUσ+BjTU1−BjTUσ1/σ−1,1−1+AjIL1−AjILσ+BjIL1−BjILσ1/σ−1,1−1+AjTU1−AjTUσ+BjTU1−BjTUσ1/σ−1,1+1−AjFLAjFLσ+1−BjFLBjFLσ1/σ−1,1+1−AjFUAjFUσ+1−BjFUBjFUσ1/σ−1,1−1+λjT1−λjTσ+μjT1−μjTσ1/σ−1,1−1+λjI1−λjIσ+μjI1−μjIσ1/σ−1,1+1−λjFλjFσ+1−μjFμjFσ1/σ−1,qα=1−1+qAjTL1−AjTLσ1/σ−1,1−1+qAjTU1−AjTUσ1/σ−1,1−1+qAjIL1−AjILσ1/σ−1,1−1+qAjTU1−AjTUσ1/σ−1,1+q1−AjFLAjFLσ1/σ−1,1+q1−AjFUAjFUσ1/σ−1,1−1+qλjT1−λjTσ1/σ−1,1−1+qλjI1−λjIσ1/σ−1,1+q1−λjFλjFσ1/σ−1,αq=1+q1−AjTLAjTLσ1/σ−1,1+q1−AjTUAjTUσ1/σ−1,1+q1−AjILAjILσ1/σ−1,1+q1−AjTUAjTUσ1/σ−1,1−1+qAjFL1−AjFLσ1/σ−1,1−1+qAjFU1−AjFUσ1/σ−1,1+q1−λjTλjTσ1/σ−1,1+q1−λjIλjIσ1/σ−1,1−1+qλjF1−λjFσ1/σ−1.
Moreover, theβ-normalization is used in case of different cardinalities.Example 11.
Ifα=0.1,0.2,0.3,0.4,0,.1,0.3,0,0.2,0.6,0.7,0.3,0.5,0.1,0.2,0.2,0.3,0.6,0.7 and β=0.6,0.8,0.5,0.7,0.5,0.6,0.6,0.7,0.1,0.2,0,0.2,0.6,0.7,0.5,0.6,0.2,0.3, then for σ=2,(4)α⊕β=0.600656,0.800312,0.521065,0.708174,0.501534,0.609379,0.6,0.701197,0.099754,0.199088,0,0.296528,0.600656,0.701197,0.507577,0.609379,0.197817,0.296528,α⊗β=0.099754,0.199688,0.282598,0.390621,0.099449,0.291826,0,0.199088,0.600656,0.701197,0.3,0.507577,0.099754,0.199088,0.195194,0.291826,0.603283,0.703472,,3α=0.16139,0.302169,0.426048,0.535898,0.16139,0.426048,0,0.302169,0.464102,0.573952,0.198356,0.366025,0.16139,0.302169,0.302169,0.426048,0.464102,0.573952,α3=0.060283,0.126132,0.198356,0.277926,0.060283,0.198356,0,0.126132,0.722074,0.801644,0.426048,0.633975,0.060283,0.126132,0.126132,0.198356,0.722074,0.801644.
It is very important to discuss the comparison of NCHF values. For comparison of NCHF values, we propose score functions in next section.Definition 12.
Score, accuracy, and certainty.
The score, accuracy, and certainty of a NCHF valueα=A,λ, where A=AT,AI,AF,AT=AjTL,AjTU;j=1,…,l,AI=AjIL,AjIU;j=1,…,m,AF=AjFL,AjFU;i=1,…,n and λ=λT,λI,λF,λT=λjT;j=1,…,r,λI=λjI;j=1,…,s,λF=λjF;j=1,…,t are defined as(5)Sα=12161l∑j=1lAjTL+AjTU+1m∑j=1mAjIL+AjIU+1n∑j=1n2−AjFL+AjFU+131r∑j=1rλjT+1s∑j=1sλjI+1t∑j=1t1−λjF,Hα=191l∑j=1lAjTL+AjTU−1n∑j=1nAjFL+AjFU+1r∑j=1rλjT−1t∑j=1tλjF,Cα=131l∑j=1lAjTL+AjTU+1r∑j=1rλjT.
The score function is a useful tool for ranking the NCHF values. An NCHF value is greater if the interval hesitant and hesitant degrees of truth and indeterminacy membership grades are bigger and the interval hesitant and hesitant degrees of falsity membership grades are smaller. The greater the difference between truth and falsity memberships, the accuracy is higher and the corresponding NCHF value becomes greater. The certainty of an NCHF value depends on the interval hesitant and hesitant degrees of truth membership grades. The bigger the interval between hesitant and hesitant degrees of truth and indeterminacy membership grades, the greater the NCHF value is.
The following technique is use for ranking the values.
Ifα=A,λ,β=B,μ be two NCHF values then we say that α≻β if and only if Sα≻Sβ.
IfSα=Sβ, we compare them on the basis of accuracy. If they have same accuracies too, we define the above comparison on the basis of certainty.Example 2.
Ifα=0.1,0.2,0.3,0.4,0,.1,0.3,0,0.2,0.6,0.7,0.3,0.5,0.1,0.2,0.2,0.3,0.6,0.7 and β=0.6,0.8,0.5,0.7,0.5,0.6,0.6,0.7,0.1,0.2,0,0.2,0.6,0.7,0.5,0.6,0.2,0.3, then(6)Sα=0.270833,Sβ=0.679167,Hα=0.322222,Hβ=0.466667,Cα=0.216667,Cβ=0.65.
The distances and similarities are important notions in theory of sets. These two concepts are important tools to describe the degree of closeness in sets. Keeping in mind the importance of these two notions, we define these in the neutrosophic cubic hesitant fuzzy structure.


## 4. Distance and Similarity Measures in Neutrosophic Cubic Hesitant Fuzzy Sets

Definition 13.
A distance measure is a mapping such that(7)i0≤dα,β≤1,iidα,β=0⇔α=β,iiidβ,α=dα,β,ivIfα⊆β⊂γ,thendα,γ≥dα,βanddα,γ≥dβ,γ.
Ifα=A,λ,β=B,μ are two NCHFS on a set X=x1,….,xt, then we define the following.
Normalized Hamming Distance:(8)dHα,β=19t∑i=1t1li∑j=1liAjTUxi−BjTUxi+AjTLxi−BjTLxi+1mi∑j=1miAjIUxi−BjIUxi+AjILxi−BjILxi+1ni∑j=1niAjFUxi−BjFUxi+AjFLxi−BjFLxi+1pi∑j=1piλjTxi−μjTxi+1qi∑j=1qiλjIxi−μjIxi+1ri∑j=1riλjFxi−μjFxi.
Normalized Euclidean Distance:(9)dEα,β=19t∑i=1t1li∑j=1liAjTUxi−BjTUxi2+AjTLxi−BjTLxi2+1mi∑j=1miAjIUxi−BjIUxi2+AjILxi−BjILxi2+1ni∑j=1niAjFUxi−BjFUxi2+AjFLxi−BjFLxi2+1pi∑j=1piλjTxi−μjTxi2+1qi∑j=1qiλjIxi−μjIxi2+1ri∑j=1riλjFxi−μjFxi21/2.
Normalized Generalized Distance:(10)dGα,β=19t∑i=1t1li∑j=1liAjTUxi−BjTUxiq+AjTLxi−BjTLxiq+1mi∑j=1miAjIUxi−BjIUxiq+AjILxi−BjILxiq+1ni∑j=1niAjFUxi−BjFUxiq+AjFLxi−BjFLxiq+1pi∑j=1piλjTxi−μjTxiq+1qi∑j=1qiλjIxi−μjIxiq+1ri∑j=1riλjFxi−μjFxiq1/q,q≻0.Definition 14.
A similarity measure is mappingsm:NCHFS×NCHFS⟶0,1, such that(11)i0≤smα,β≤1,iismα,β=1⇔α=β,iiismα,β=smβ,α,ivIfα⊆β⊂γ,thensmα,γ≤smα,βandsmα,γ≤smβ,γ.Theorem 1.
Ifdα,β is a distance measure, then smα,β=1−dα,β is a similarity measure.
Using the above theorem, the similarity measures based on distance measures defined in section 3.5.1 section 3.5.1 are as follows:(12)smHα,β=1−dHα,β,smEα,β=1−dEα,β,smGα,β=1−dGα,β.Remark 1.
We can order the NCHF values by using similarity measures as follows:
Consider the NCHF valuesα=A,λ, β=B,μ and corresponding maximum ideal NCHF value Ω=1,1,1,10,0,1,1,0.(13)Ifsmα,Ω≻smβ,Ω,thenα≻β,Ifsmα,Ω=smβ,Ω,thenα=β.
In the next section, we define novel Dombi exponential laws in which bases are crisp values and Table1 exponents are NCHF values. Using these Dombi exponential laws, we define Dombi exponential aggregation operators in NCHFSs.Table 1
Similarities of given values with maximum ideal.
smGα,ΩsmGβ,ΩRanking (using Equation (12))q=10.277778 0688889β>αq=20.259880.660065β>αq=30.2448910.640545β>αq=40.2318610.625411β>αq=50.2202230.613145β>αExample 3.
Ifα=0.1,0.2,0.3,0.4,0,.1,0.3,0,0.2,0.6,0.7,0.3,0.5,0.1,0.2,0.2,0.3,0.6,0.7 and β=0.6,0.8,0.5,0.7,0.5,0.6,0.6,0.7,0.1,0.2,0,0.2,0.6,0.7,0.5,0.6,0.2,0.3, then using (10) and (12), we have Table 1 of similarities:


## 5. Dombi Exponential Laws and Aggregation Operators

Definition 15.
For an NCHFSα=A,λ, scalar q∈0,1 and σ>0, we define(14)qα=1+1−AjTL1−qqσ1/σ−1,1+1−AjTU1−qqσ1/σ−1,1+1−AjIL1−qqσ1/σ−1,1+1−AjTU1−qqσ1/σ−1,1−1+AjFL1−qqσ1/σ−1,1−1+AjFU1−qqσ1/σ−1,1+1−λjT1−qqσ1/σ−1,1+1−λjI1−qqσ1/σ−1,1−1+λjF1−qqσ1/σ−1.
Note that ifα>β, then qα>qβ.Theorem 2.
For an NCHFNα=A,λ and a scalar q>0, qα is a NCHFN.Theorem 3.
For two NCHFNsα=A,λ,β=B,μ and a scalar ϖ∈0,1, the following commutative laws hold:(15)iϖα⊕ϖβ=ϖβ⊕ϖα,iiϖα⊗ϖβ=ϖβ⊗ϖα,Theorem 4.
For three NCHFNsα=A,λ,β=B,μ,γ=C,ν and a scalar ϖ∈0,1, the following associative laws hold:(16)iϖα⊕ϖβ⊕ϖγ=ϖβ⊕ϖα⊕ϖγ,iiϖα⊗ϖβ⊗ϖγ=ϖβ⊗ϖα⊗ϖγ.Definition 16.
Dombi exponential aggregation operators
Letαk=Ak,λk be a collection of NCHFNs and qk∈0,1 be real numbers, then neutrosophic cubic hesitant fuzzy Dombi weighted exponential operator is defined as(17)NCHDWEα1,…,αn=⊗j=1nqα,where αk=Ak,λk is the exponential weight of attribute values qk∈0,1.Theorem 5.
Letαk=Ak,λk be a collection of NCHFS and qk∈0,1 be real numbers, then neutrosophic cubic hesitant fuzzy Dombi weighted exponential operator is(18)NCHFDWEα1,α2,…,αn=1+∑r=1n1−AjTrL1−qrqrσ1/σ−1,1+∑r=1n1−AjTrU1−qrqrσ1/σ−1,1+∑r=1n1−AjIrL1−qrqrσ1/σ−1,1+∑r=1n1−AjIrU1−qrqrσ1/σ−1,1−1+∑r=1nAjFrL1−qrqrσ1/σ−1,1−1+∑r=1nAjFrU1−qrqrσ1/σ−1,1+∑r=1n1−λjTr1−qrqrσ1/σ−1,1+∑r=1n1−λjIr1−qrqrσ1/σ−1,1−1+∑r=1nλjFr1−qrqrσ1/σ−1,and αk=Ak,λk are the exponential weights of qk∈0,1. Furthermore, NCHFDWEα1,α2,...,αm is also a neutrosophic cubic hesitant fuzzy value.Proof.
We use induction. Forn = 2, we have(19)NCHFDWEα1,α2=1+1−AjT1L1−q1q1σ+1−AjT2L1−q2q2σ1/σ−1,1+1−AjT1U1−q1q1σ+1−AjT2U1−q2q2σ1/σ−1,1+1−AjI1L1−q1q1σ+1−AjI2L1−q2q2σ1/σ−1,1+1−AjI1U1−q1q1σ+1−AjI2U1−q2q2σ1/σ−1,1−1+AjF1L1−q1q1σ+AjF2L1−q2q2σ1/σ−1,1−1+AjF1L1−q1q1σ+AjF2L1−q2q2σ1/σ−1,1+1−λjT11−q1q1σ+1−λjT21−q2q2σ1/σ−1,1+1−λjI11−q1q1σ+1−λjI21−q2q2σ1/σ−1,1−1+λjF11−q1q1σ+λjF21−q2q2σ1/σ−1.
Considering the result to be true forn = m we prove it for n = m+1 as follows:(20)NCHDWEα1,…,αm+1=⊗j=1mqαj⊗qα.
Using (2) and (14), we have(21)NCHFDWEα1,α2,…,αm+1=1+∑r=1m+11−AjTrL1−qrqrσ1/σ−1,1+∑r=1m+11−AjTrU1−qrqrσ1/σ−1,1+∑r=1m+11−AjIrL1−qrqrσ1/σ−1,1+∑r=1m+11−AjIrU1−qrqrσ1/σ−1,1−1+∑r=1m+1AjFrL1−qrqrσ1/σ−1,1−1+∑r=1m+1AjFrU1−qrqrσ1/σ−1,1+∑r=1m+11−λjTr1−qrqrσ1/σ−1,1+∑r=1m+11−λjTr1−qrqrσ1/σ−1,1−1+∑r=1m+1λjFr1−qrqrσ1/σ−1.


## 6. Applications of Neutrosophic Cubic Hesitant Fuzzy Dombi Weighted Exponential Aggregation Operators to MADM Problems

Many methods in MADM ignore the uncertainty and hence yield the results that are unreliable. In this section, we construct algorithm using the Dombi exponential aggregation operators (NCHFDWEs) for MADM problems.


### 6.1. Solid Waste Disposal Site Selection

Due to population increase and urbanization, the proper disposal of waste is a challenging problem nowadays. The proper disposal of waste is necessary for the prevention of viral diseases like typhoid, dengue, and tuberculosis. Using the above algorithm to select the best alternative (solid waste disposal site) among the given alternatives (sites)K1,…,K5 on the basis of attributes (i) F1 water pollution; (ii) F2 slope; (iii) F3 distance from the residential area.Algorithm 1: Algorithm using the dombi exponential aggregation operators (NCHFDWE) for MADM problems.
Step 1: Identification of alternatives and attributes.LetK1,K2,…,Kr, be the set of r alternatives, F1,F2,…,Fs be s attributes. The NCHFS αj is used as weight for the attribute Fj. A decision matrix is D=dij consisting fuzzy values, where dij represent the preference of alternative Ki corresponding to attribute Fj.Step 2. Allocation of weights to attributesThe NCHF valueαj is used as weight assigned to attribute Fj.Step 3. Computation of weighted aggregated valuesUsing NCHFDWE operators (equation (18)), we compute the aggregated values dj′s (j = 1, …, r) of alternatives Kj′s.Step 4. Ranking of AlternativesWe calculate the scores and similarity measuresSdj,smdj,Ω;j=1,…,r of the alternatives Kj;i=1,…,r using equations (5), (10) and (14). Using scores and similarity measures, we rank the alternatives Ki;i=1,…,n. If scores of two alternatives are equal, then, we use accuracy function for ranking and if they have same accuracy, we use certainty.The decision matrixD=dij5×3, where entry dij represents the preference of alternative Kii=1,…,5 corresponding to attribute Fjj=1,2,3, is constructed using fuzzy values with the help of the following scale.Equal preference: 0.1, moderate preference: 0.3, strong preference: 0.5, demonstrated preference: 0.7, and extreme preference: 0.9. The mean values between the two assessments: 0.2, 0.4, 0.6, and 0.8.(22)D=0.20.20.40.80.80.70.30.30.20.80.50.60.60.70.6.The weights of the attributes are given as(23)w1=0.3,0.7,0.2,0.4,0.2,0.5,0.1,0.6,0.2,0.4,0,0.1,0.5,0.6,0.2,0.4,0.2,0.3,w2=0.5,0.7,0.2,0.5,0.2,0.3,0.1,0.6,0.1,0.4,0,0.3,0.4,0.5,0.3,0.4,0.2,0.4,w3=0.4,0.5,0.6,0.7,0.1,0.3,0.2,0.5,0.1,0.2,0.3,0.4,0.3,0.5,0.4,0.6,0.3,0.4.The explanation of weights is elaborated as follows.In the case ofw1, {[0.3, 0.7], [0.2, 0.4]} is the interval hesitant degree of preference for attribute F1, {[0.2, 0.5], [0.1, 0.6]} is the interval hesitant degree of indeterminacy (preference/nonpreference) for attribute F1, {[0.2, 0.4], [0, 0.1]} is the interval hesitant degree of nonpreference for an attribute F1, {0.1, 0.6}} is hesitant degree of preference for attribute F1, {0.2, 0.4} is hesitant degree of indeterminacy (preference/nonpreference) for an attribute F1, and {0.4, 0.6} is the hesitant degree of nonpreference for attribute F1.Aggregated values of alternatives using NCHFDWE operators ((17)) with σ=1:(24)d1=0.149254,0.240964,0.125,0.17094,0.114286,0.145985,0.106383,0.20202,0.574468,0.777778,0.310345,0.6875,0.155039,0.186916,0.126582,0.15625,0.672131,0.75,d2=0.642202,0.732984,0.636364,0.712468,0.56,0.625,0.557769,0.707071,0.105431,0.222222,0.113924,0.213483,0.634921,0.694789,0.612691,0.679612,0.186047,0.243243,d3=0.16129,0.227273,0.157895,0.20979,0.12,0.151515,0.119048,0.205479,0.52381,0.727273,0.545455,0.716981,0.157068,0.196078,0.144928,0.185185,0.680851,0.75,d4=0.481928,0.585366,0.441176,0.540541,0.384615,0.436364,0.376176,0.545455,0.178082,0.387755,0.166667,0.371728,0.456274,0.517241,0.434783,0.495868,0.310345,0.390863,d5=0.480549,0.601719,0.466667,0.551181,0.403846,0.47619,0.396975,0.564516,0.195402,0.363636,0.166667,0.315961,0.486111,0.551181,0.447761,0.519802,0.295302,0.373134.ScoresSdiare given using equation (5).(25)Sd1=0.2213,Sd2=0.7043,Sd3=0.2226,Sd4=0.5456,Sd5=0.5655.SoK2>K5>K4>K3>K1.Similarity measuressmGdi are given using (10) and (12).Figure1 is the graphical representation of score and similarity measures of aggregated values obtained in Table 2.Figure 1
Graphical representation of scores and similarity measures withσ=1.Table 2
Similarity measures of aggregated values using the NCHFDWE operators withσ=1.
smGd1,ΩsmGd2,ΩsmGd3,ΩsmGd3,ΩsmGd4,ΩRankingq=10.2281890.7061960.2272880.5494640.568689K2>K5>K4>K3>K1q=20.2157790.6907220.2208040.5312380.551364K2>K5>K4>K3>K1q=30.2068740.6784390.215190.5174840.537774K2>K5>K4>K3>K1q=40.2001160.6683990.2103110.5066180.526813K2>K5>K4>K3>K1q=50.194730.6599480.2060460.497670.51771K2>K5>K4>K3>K1Aggregated values of alternatives using the NCHFDWE operators (equation (15)) with σ=2: (26)d1=0.180727,0.233923,0.162659,0.189576,0.159848,0.179922,0.153099,0.211344,0.691515,0.784485,0.451027,0.729865,0.185911,0.202422,0.165703,0.182372,0.726768,0.764172,d2=0.699128,0.735492,0.705967,0.739685,0.660032,0.68909,0.662536,0.726421,0.161537,0.227505,0.190112,0.238848,0.692421,0.722115,0.688889,0.721856,0.220591,0.24988,d3=0.199337,0.229538,0.204612,0.233392,0.172194,0.191898,0.173794,0.221482,0.406748,0.764784,0.382966,0.688361,0.194328,0.217786,0.191753,0.217567,0.648489,0.799751,d4=0.526252,0.5762,0.496575,0.549739,0.472136,0.494814,0.466126,0.55417,0.283751,0.417541,0.267479,0.41028,0.50742,0.53636,0.497934,0.525506,0.370307,0.413368,d5=0.549964,0.609449,0.548009,0.587782,0.512821,0.551397,0.510304,0.592385,0.28031,0.368373,0.267479,0.344956,0.554875,0.587782,0.535767,0.573149,0.337256,0.377007.ScoresSdi are given using equation (5):(27)Sd1=0.2210,Sd2=0.7296,Sd3=0.2290,Sd4=0.5590,Sd5=0.5948,K2>K5>K4>K3>K1.Similarity measuressmGdi are given using (10) and (12).Figure2 is the graphical representation of score and similarity measures of aggregated values obtained in Table 3.Figure 2
Graphical representation of scores and similarity measures withσ=2.Table 3
Similarity measures of aggregated values using the NCHFDWE operators withσ=2.
smGd1,ΩsmGd2,ΩsmGd3,ΩsmGd3,ΩsmGd4,ΩRankingq=10.2462770.7308420.2309840.5578060.596572K2>K5>K4>K3>K1q=20.2364480.7267670.2295420.5518320.591733K2>K5>K4>K3>K1q=30.2282680.7231210.2281740.5467610.587395K2>K5>K4>K3>K1q=40.2094570.7198470.2268760.5424190.583509K2>K5>K4>K3>K1q=50.2178990.7168830.2256430.5386560.58002K2>K5>K4>K3>K1Aggregated values of alternatives using NCHFDWE operators (equation (15)) with σ=3:(28)d1=0.189119,0.226119,0.175473,0.194219,0.174696,0.188901,0.169392,0.0.174524,0.729242,0.788562,0.501035,0.749884,0.193244,0.20411,0.178227,0.189559,0.749095,0.773387,d2=0.712208,0.732458,0.722704,0.743899,0.68604,0.704447,0.689676,0.728379,0.188605,0.233444,0.222939,0.251172,0.705929,0.726416,0.707496,0.730469,0.236791,0.256193,d3=0.20958,0.226801,0.218287,0.237351,0.189706,0.203424,0.192323,0.223189,0.406748,0.764784,0.382966,0.688361,0.204582,0.221478,0.205815,0.22503,0.648489,0.799751,d4=0.445183,0.588428,0.391659,0.530621,0.303378,0.385061,0.291323,0.536908,0.152128,0.374173,0.142083,0.352469,0.412698,0.500953,0.379002,0.470432,0.285738,0.422203,d5=0.430671,0.597105,0.407249,0.528306,0.315454,0.422037,0.304401,0.547489,0.172143,0.361839,0.142083,0.304007,0.437361,0.528306,0.381049,0.484407,0.278734,0.39362.ScoresSdi are given using equation (5):(29)Sd1=0.2185,Sd2=0.7324,Sd3=0.2288,Sd4=0.5497,Sd5=0.5922,K2>K5>K4>K3>K1.Similarity measuressmGdi are given using (10) and (12):Figure3 is the graphical representation of score and similarity measures of aggregated values obtained in Table 4.Figure 3
Graphical representation of scores and similarity measures withσ=3.Table 4
Similarity measures of aggregated values using the NCHFDWE operators withσ=3.
smGd1,ΩsmGd2,ΩsmGd3,ΩsmGd3,ΩsmGd4,ΩRankingq=10.2223460.7333880.2300720.5518430.602963K2>K5>K4>K3>K1q=20.2189510.7315710.2294720.5486850.600807K2>K5>K4>K3>K1q=30.2162250.7298810.2288910.5458990.598787K2>K5>K4>K3>K1q=40.2140050.7283070.2283280.5434330.5969K2>K5>K4>K3>K1q=50.2121680.7268350.2277830.5412320.595128K2>K5>K4>K3>K1It can be seen from Figures1–3, that we obtain the same ranking with respect to different choices of parameters q and σ. Hence the most desirable alternative is K2.


### 6.2. Comparative Analysis

To check the validity of results, the above problem is solved by different existing techniques proposed in [36, 46]. The results obtained by these existing methods are similar to those obtained from this study in Tables 2–4.


## 6.1. Solid Waste Disposal Site Selection

Due to population increase and urbanization, the proper disposal of waste is a challenging problem nowadays. The proper disposal of waste is necessary for the prevention of viral diseases like typhoid, dengue, and tuberculosis. Using the above algorithm to select the best alternative (solid waste disposal site) among the given alternatives (sites)K1,…,K5 on the basis of attributes (i) F1 water pollution; (ii) F2 slope; (iii) F3 distance from the residential area.Algorithm 1: Algorithm using the dombi exponential aggregation operators (NCHFDWE) for MADM problems.
Step 1: Identification of alternatives and attributes.LetK1,K2,…,Kr, be the set of r alternatives, F1,F2,…,Fs be s attributes. The NCHFS αj is used as weight for the attribute Fj. A decision matrix is D=dij consisting fuzzy values, where dij represent the preference of alternative Ki corresponding to attribute Fj.Step 2. Allocation of weights to attributesThe NCHF valueαj is used as weight assigned to attribute Fj.Step 3. Computation of weighted aggregated valuesUsing NCHFDWE operators (equation (18)), we compute the aggregated values dj′s (j = 1, …, r) of alternatives Kj′s.Step 4. Ranking of AlternativesWe calculate the scores and similarity measuresSdj,smdj,Ω;j=1,…,r of the alternatives Kj;i=1,…,r using equations (5), (10) and (14). Using scores and similarity measures, we rank the alternatives Ki;i=1,…,n. If scores of two alternatives are equal, then, we use accuracy function for ranking and if they have same accuracy, we use certainty.The decision matrixD=dij5×3, where entry dij represents the preference of alternative Kii=1,…,5 corresponding to attribute Fjj=1,2,3, is constructed using fuzzy values with the help of the following scale.Equal preference: 0.1, moderate preference: 0.3, strong preference: 0.5, demonstrated preference: 0.7, and extreme preference: 0.9. The mean values between the two assessments: 0.2, 0.4, 0.6, and 0.8.(22)D=0.20.20.40.80.80.70.30.30.20.80.50.60.60.70.6.The weights of the attributes are given as(23)w1=0.3,0.7,0.2,0.4,0.2,0.5,0.1,0.6,0.2,0.4,0,0.1,0.5,0.6,0.2,0.4,0.2,0.3,w2=0.5,0.7,0.2,0.5,0.2,0.3,0.1,0.6,0.1,0.4,0,0.3,0.4,0.5,0.3,0.4,0.2,0.4,w3=0.4,0.5,0.6,0.7,0.1,0.3,0.2,0.5,0.1,0.2,0.3,0.4,0.3,0.5,0.4,0.6,0.3,0.4.The explanation of weights is elaborated as follows.In the case ofw1, {[0.3, 0.7], [0.2, 0.4]} is the interval hesitant degree of preference for attribute F1, {[0.2, 0.5], [0.1, 0.6]} is the interval hesitant degree of indeterminacy (preference/nonpreference) for attribute F1, {[0.2, 0.4], [0, 0.1]} is the interval hesitant degree of nonpreference for an attribute F1, {0.1, 0.6}} is hesitant degree of preference for attribute F1, {0.2, 0.4} is hesitant degree of indeterminacy (preference/nonpreference) for an attribute F1, and {0.4, 0.6} is the hesitant degree of nonpreference for attribute F1.Aggregated values of alternatives using NCHFDWE operators ((17)) with σ=1:(24)d1=0.149254,0.240964,0.125,0.17094,0.114286,0.145985,0.106383,0.20202,0.574468,0.777778,0.310345,0.6875,0.155039,0.186916,0.126582,0.15625,0.672131,0.75,d2=0.642202,0.732984,0.636364,0.712468,0.56,0.625,0.557769,0.707071,0.105431,0.222222,0.113924,0.213483,0.634921,0.694789,0.612691,0.679612,0.186047,0.243243,d3=0.16129,0.227273,0.157895,0.20979,0.12,0.151515,0.119048,0.205479,0.52381,0.727273,0.545455,0.716981,0.157068,0.196078,0.144928,0.185185,0.680851,0.75,d4=0.481928,0.585366,0.441176,0.540541,0.384615,0.436364,0.376176,0.545455,0.178082,0.387755,0.166667,0.371728,0.456274,0.517241,0.434783,0.495868,0.310345,0.390863,d5=0.480549,0.601719,0.466667,0.551181,0.403846,0.47619,0.396975,0.564516,0.195402,0.363636,0.166667,0.315961,0.486111,0.551181,0.447761,0.519802,0.295302,0.373134.ScoresSdiare given using equation (5).(25)Sd1=0.2213,Sd2=0.7043,Sd3=0.2226,Sd4=0.5456,Sd5=0.5655.SoK2>K5>K4>K3>K1.Similarity measuressmGdi are given using (10) and (12).Figure1 is the graphical representation of score and similarity measures of aggregated values obtained in Table 2.Figure 1
Graphical representation of scores and similarity measures withσ=1.Table 2
Similarity measures of aggregated values using the NCHFDWE operators withσ=1.
smGd1,ΩsmGd2,ΩsmGd3,ΩsmGd3,ΩsmGd4,ΩRankingq=10.2281890.7061960.2272880.5494640.568689K2>K5>K4>K3>K1q=20.2157790.6907220.2208040.5312380.551364K2>K5>K4>K3>K1q=30.2068740.6784390.215190.5174840.537774K2>K5>K4>K3>K1q=40.2001160.6683990.2103110.5066180.526813K2>K5>K4>K3>K1q=50.194730.6599480.2060460.497670.51771K2>K5>K4>K3>K1Aggregated values of alternatives using the NCHFDWE operators (equation (15)) with σ=2: (26)d1=0.180727,0.233923,0.162659,0.189576,0.159848,0.179922,0.153099,0.211344,0.691515,0.784485,0.451027,0.729865,0.185911,0.202422,0.165703,0.182372,0.726768,0.764172,d2=0.699128,0.735492,0.705967,0.739685,0.660032,0.68909,0.662536,0.726421,0.161537,0.227505,0.190112,0.238848,0.692421,0.722115,0.688889,0.721856,0.220591,0.24988,d3=0.199337,0.229538,0.204612,0.233392,0.172194,0.191898,0.173794,0.221482,0.406748,0.764784,0.382966,0.688361,0.194328,0.217786,0.191753,0.217567,0.648489,0.799751,d4=0.526252,0.5762,0.496575,0.549739,0.472136,0.494814,0.466126,0.55417,0.283751,0.417541,0.267479,0.41028,0.50742,0.53636,0.497934,0.525506,0.370307,0.413368,d5=0.549964,0.609449,0.548009,0.587782,0.512821,0.551397,0.510304,0.592385,0.28031,0.368373,0.267479,0.344956,0.554875,0.587782,0.535767,0.573149,0.337256,0.377007.ScoresSdi are given using equation (5):(27)Sd1=0.2210,Sd2=0.7296,Sd3=0.2290,Sd4=0.5590,Sd5=0.5948,K2>K5>K4>K3>K1.Similarity measuressmGdi are given using (10) and (12).Figure2 is the graphical representation of score and similarity measures of aggregated values obtained in Table 3.Figure 2
Graphical representation of scores and similarity measures withσ=2.Table 3
Similarity measures of aggregated values using the NCHFDWE operators withσ=2.
smGd1,ΩsmGd2,ΩsmGd3,ΩsmGd3,ΩsmGd4,ΩRankingq=10.2462770.7308420.2309840.5578060.596572K2>K5>K4>K3>K1q=20.2364480.7267670.2295420.5518320.591733K2>K5>K4>K3>K1q=30.2282680.7231210.2281740.5467610.587395K2>K5>K4>K3>K1q=40.2094570.7198470.2268760.5424190.583509K2>K5>K4>K3>K1q=50.2178990.7168830.2256430.5386560.58002K2>K5>K4>K3>K1Aggregated values of alternatives using NCHFDWE operators (equation (15)) with σ=3:(28)d1=0.189119,0.226119,0.175473,0.194219,0.174696,0.188901,0.169392,0.0.174524,0.729242,0.788562,0.501035,0.749884,0.193244,0.20411,0.178227,0.189559,0.749095,0.773387,d2=0.712208,0.732458,0.722704,0.743899,0.68604,0.704447,0.689676,0.728379,0.188605,0.233444,0.222939,0.251172,0.705929,0.726416,0.707496,0.730469,0.236791,0.256193,d3=0.20958,0.226801,0.218287,0.237351,0.189706,0.203424,0.192323,0.223189,0.406748,0.764784,0.382966,0.688361,0.204582,0.221478,0.205815,0.22503,0.648489,0.799751,d4=0.445183,0.588428,0.391659,0.530621,0.303378,0.385061,0.291323,0.536908,0.152128,0.374173,0.142083,0.352469,0.412698,0.500953,0.379002,0.470432,0.285738,0.422203,d5=0.430671,0.597105,0.407249,0.528306,0.315454,0.422037,0.304401,0.547489,0.172143,0.361839,0.142083,0.304007,0.437361,0.528306,0.381049,0.484407,0.278734,0.39362.ScoresSdi are given using equation (5):(29)Sd1=0.2185,Sd2=0.7324,Sd3=0.2288,Sd4=0.5497,Sd5=0.5922,K2>K5>K4>K3>K1.Similarity measuressmGdi are given using (10) and (12):Figure3 is the graphical representation of score and similarity measures of aggregated values obtained in Table 4.Figure 3
Graphical representation of scores and similarity measures withσ=3.Table 4
Similarity measures of aggregated values using the NCHFDWE operators withσ=3.
smGd1,ΩsmGd2,ΩsmGd3,ΩsmGd3,ΩsmGd4,ΩRankingq=10.2223460.7333880.2300720.5518430.602963K2>K5>K4>K3>K1q=20.2189510.7315710.2294720.5486850.600807K2>K5>K4>K3>K1q=30.2162250.7298810.2288910.5458990.598787K2>K5>K4>K3>K1q=40.2140050.7283070.2283280.5434330.5969K2>K5>K4>K3>K1q=50.2121680.7268350.2277830.5412320.595128K2>K5>K4>K3>K1It can be seen from Figures1–3, that we obtain the same ranking with respect to different choices of parameters q and σ. Hence the most desirable alternative is K2.


## 6.2. Comparative Analysis

To check the validity of results, the above problem is solved by different existing techniques proposed in [36, 46]. The results obtained by these existing methods are similar to those obtained from this study in Tables 2–4.


## 7. Conclusion

First, the Dombi algebraic operations distance and similarity measures in neutrosophic cubic hesitant fuzzy sets are defined. Then the properties of these notions are discussed. Furthermore, the novel Dombi exponential laws and Dombi exponential aggregation operators are proposed. The Dombi exponential laws are more flexible than the usual exponential laws as they involve an operational parameter. Using these aggregation operators, we established a MADM method. The proper disposal of waste is necessary for the prevention of viral diseases like typhoid, dengue, and tuberculosis. The solid waste disposal site selection problem is solved using the proposed MADM method. To discuss the effectiveness of the proposed method, different values of parameters are considered to check the efficiency of the proposed method. At last, the comparative analysis is presented for validation of proposed operators and methods.For further research, the neutrosophic cubic hesitant fuzzy Dombi arithmetic and geometric aggregation operators and multiexpert decision-making methods using these aggregation operators will be developed. The Dombi Bonferroni mean operators of neutrosophic cubic hesitant fuzzy sets will be established. The Dombi exponential laws and Dombi exponential aggregation operators for Pythagorean fuzzy sets and picture hesitant fuzzy sets will be established [51–54, 56–60].


### 7.1. Limitations of the Proposed Study

As Dombi operations and generalized similarity measures involve operational parameters, the manual calculation is very complicated and time-consuming. Hence there is a need to develop a software program for such computations. Furthermore, for MADM problems involving more alternatives and attributes handled by Dombi exponential aggregation operators, software programming is needed. We are trying to develop such programming using PYTHON.


## 7.1. Limitations of the Proposed Study

As Dombi operations and generalized similarity measures involve operational parameters, the manual calculation is very complicated and time-consuming. Hence there is a need to develop a software program for such computations. Furthermore, for MADM problems involving more alternatives and attributes handled by Dombi exponential aggregation operators, software programming is needed. We are trying to develop such programming using PYTHON.



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*Source: 3828370-2022-09-26.xml*