# Applications of Hesitant Interval Neutrosophic Linguistic Schweizer-Sklar Power Aggregation Operators to MADM

Authors: Qaisar Khan; Rashad A. R. Bantan; Mohammed Elgarhy
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Hesitant interval neutrosophic linguistic sets (HINLSs) are one of the core generalization of various sets, such as neutrosophic set (NS), interval neutrosophic set (INS), and interval neutrosophic linguistic set (INLS). HINLS can represent the uncertainty, inconsistency, and reluctance of assessment specialists by expressing qualitative and quantitative information. The goal of this article is to introduce a novel MADM technique that can account for changes in the semantic environment as well as negative consequences of experts’ excessive evaluation values. First, several innovative operational rules based on Schweizer-Sklar (SS)t-norm and t-conorm and a novel comparison procedure for HINLS are established by integrating different linguistic scale functions. This allows for varied semantic settings to be handled. Then, various innovative HINL Schweizer-Sklar power aggregation operators (AOs) are suggested, containing hesitant interval neutrosophic SS power average (HINLSSPA) operator, weighted hesitant interval neutrosophic SS power average (WHINLSSPA) operator, hesitant interval neutrosophic SS power geometric average (HINLSSPGA) operator, weighted hesitant interval neutrosophic SS power geometric average (WHINLSSPGA) operator, some core properties, and various special cases of these AOs are examined. Additionally, based on the initiated AOs, a multiple attribute decision making (MADM) technique with HINL information is anticipated. Finally, a numerical example is illustrated to show the effectiveness and practicality of the anticipated MADM method. A comparison with existing approaches are also discussed.


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## Body

## 1. Introduction

The preferred information in actual decision-making situations is frequently imprecise, uncertain, and unpredictable. As a result, fuzzy decision making is a beneficial approach in a variety of fuzzy situations [1–6]. Since Smarandache’s neutrosophic set (NS) [7, 8] can adequately define imprecise, ambiguous, and inconsistent data. Several researchers have created a few subclasses of NS that may be used easily in real-world scientific and engineering problems. For instance, Wang et al. [9, 10] defined a single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS), as well as the set-theoretic operators and characteristics of SVNSs and INSs. After, the introduction of SVNSs and INSs, several researchers anticipated correlation coefficient [11–13], distance measures [14–16], and normalized Bonferroni mean [17–20] and apply these concepts to solve MADM problems under SVN and IN environments. For IN MADM problems, Zhang et al. [21] advanced the score, accuracy, and certainty functions of IN numbers (INNs) and created the INN weighted average (INNWA) operator and INN weighted geometric (INNWG) operator. Khan et al. [22] and Liu et al. [23] anticipated IN Dombi power Bonferroni mean operator, IN power Hamy mean operator and applied it to solve MADM, MAGDM problems under IN information.In many real-life decision scenarios, however, experts may prefer to convey their alternatives using linguistic information rather than numerical numbers. As a result, linguistic term sets (LTS) [24] are studied extensively and used in the decision-making process to convey expertise’ preferred alternatives [25–27]. Motivated by INS and LTS, Ye [28] initiated the concept of IN linguistic sets (INLSs) and then proposed some basic aggregation operators (AOs) to deal with MAGDM problems under INL information. Further, Ye [29] merged SVNS with LTS and initiated the idea of SVN linguistic sets (SVNLS) and introduces an extended TOPSIS method to handle MADM problems under SVNL information. Ji et al. [30] initiated combined MABAC-ELECTRE for SVNLSs and applied it to solve MADM problems under SVNL information. Wang et al. [31] initiated a series of SVNL generalized Maclaurin symmetric mean (SVNLGMSM) operator and apply these AOs to solve MADM problems. Both the linguistic variable expressed by the decision maker’s assessment of the assessed entity and the quantitative performance value expressed by an INN as the credibility of the provided linguistic variable are contained in an INLS. When decision makers offer their judgments on characteristics in the form of INLNs in complicated decision-making situations, though, they might hesitate between a variety of possible interval values. To deal with such scenario, Ye [32] initiated the idea of hesitant interval neutrosophic sets (HINLSs) by merging INLS with HFS [33, 34]. Ye also introduced some basic operational laws and some weighted AOs to deal with MADM problems under HINL information.AOs are extremely helpful tools for combining expert opinions in order to calculate the total value of each option. The power average (PA) operator, which was first developed by Yager [35], can decrease the detrimental influence of high expert evaluation values on final decision results. The power geometric (PG) operator and its weighted form were created by Xu and Yager [36], who were inspired by the notion of PA operators. Zhou et al. [37] merged PA operator with generalized average operator and initiated a new type of AOs, that is, generalized power average (GPA) operators. After the introduction of PA, PG, and GPA operators, several scholars extended these AOs for different types of fuzzy extensions. However, mostly these AOs are based on traditional operational laws, and it is unable to fulfill the various semantic needs of various experts. They cannot be used to aggregate HINLNs; as a result, the goal of this paper is to offer a number of novel generalized power AOs for integrating HINL data. The processing of language information is an essential topic that requires consideration in the research of linguistic decision-making methods. Several linguistic information processing methods have been suggested thus far, including the membership function transformation method [38, 39], the symbolic calculation method based on the subscripts of linguistic words [40–42], the cloud model transformation method, and the 2-tuple linguistic representation approach [43–45]. As the abovementioned decision making have certain advantages, but it cannot deal all types of decision-making problems. When evaluating an object, decision makers may believe that the semantic divergence between “acceptable” and “somewhat acceptable” is more or smaller than the semantic difference between “acceptable” and “completely acceptable.” That is, when the number of linguistic subscripts grows, the semantic divergence between adjacent linguistic words does not necessarily remain constant [46]. Decision makers may have various semantic criteria for established linguistic terms in numerous actual decision-making scenarios. Clearly, the existing linguistic technique fails to handle identical decision-making difficulties in the presence of HINL data. So, to overcome such drawbacks, in this article, we utilized LSF to redefine the operational laws for LTs and Schweizer-Sklar t-norm and t-conorm [47] for HINLNs. Then, we further initiate four generalized HINL PA operators to solve MADM problems.As a result of the foregoing research inspirations, the following are the article’s aims and offerings:(1)
To define some novel operational laws for HINLNs based on Schweizer-Sklart-norm, Schweizer-Sklar t-conorm and linguistic scale function(2)
Anticipating four types of generalized power aggregation operators based on these novel operational laws for HINLNs(3)
Inspecting core properties and specific cases of these generalized power aggregation operators with respect to generalized parameters(4)
Presenting a MADM technique under HINL environment which can not only remove the bad impact of high assessment values on the decision making results but also adjust to distinct semantic environment, fulfill semantic requirements of distinct experts, and make decision-making process flexibleTo do so, the rest of the article is organized as follows: in Section2, some basic ideas are examined briefly. In Section 3, based on LSF and SS t-norm and t-conorm, some core operational laws are initiated for HINLNs. In Section 4, based on these operational laws, various GPA are developed to aggregate HINNs, and various core properties and special cases are investigated. In Section 5, a MADM model is presented to deal with HINL information. In Section 6, a numerical example is given to show the effectiveness and practicality of the developed MADM approach. Finally, comparison with the existing approach is also discussed.


## 2. Preliminaries

### 2.1. The Interval Neutrosophic Linguistic Set

Definition 1 (see [28]).
LetΩ be the finite set. An INLS in Ω is identified by
(1)INL=a¯¯,sϑa¯¯,T¯¯INLa¯¯,I¯¯INLa¯¯,F¯¯INLa¯¯a¯¯∈Ω,where sϑa¯¯∈S,T¯¯INLa¯¯=infT¯¯INLa¯¯,supT¯¯INLa¯¯⊆0,1,I¯¯INLa¯¯=infI¯¯INLa¯¯,supI¯¯INLa¯¯⊆0,1 and F¯¯INLa¯¯=infF¯¯INLa¯¯,supF¯¯INLa¯¯⊆0,1, respectively, signifying the TMED, IND, and FLMD of an element a¯¯ in Ω to the linguistic variable sϑa with the constraint 0≤SupT¯¯INLa¯¯+SupI¯¯INLa¯¯+SupF¯¯INLa¯¯≤3 for any a¯¯∈Ω.For simplicity, the INLN is signified bywe¯¯=sϑwe¯¯,T¯¯Lwe¯¯,T¯¯Uwe¯¯,I¯¯Lwe¯¯,I¯¯Uwe¯¯,F¯¯Lwe¯¯,F¯¯Uwe¯¯..Definition 2 (see [28]).
Letwe¯¯=sϑwe¯¯,T¯¯Lwe¯¯,T¯¯Uwe¯¯,I¯¯Lwe¯¯,I¯¯Uwe¯¯,F¯¯Lwe¯¯,F¯¯Uwe¯¯,we¯¯1=sϑwe¯¯1,T¯¯Lwe¯¯1,T¯¯Uwe¯¯1,I¯¯Lwe¯¯1,I¯¯Uwe¯¯1,F¯¯Lwe¯¯1,F¯¯Uwe¯¯1, and we¯¯2=sϑwe¯¯2,T¯¯Lwe¯¯2,T¯¯Uwe¯¯2,I¯¯Lwe¯¯2,I¯¯Uwe¯¯2,F¯¯Lwe¯¯2,F¯¯Uwe¯¯2 be three INLNs and ξ¯¯≥0, then, the core operations for INLNs are defined as follows:
(2)1we¯¯1+we¯¯2=sϑwe¯¯1+sϑwe¯¯1,T¯¯Lwe¯¯1+T¯¯Lwe¯¯2−T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1+T¯¯Uwe¯¯2−T¯¯Uwe¯¯1T¯¯Uwe¯¯2,I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1F¯¯Uwe¯¯2;(3)2we¯¯1×we¯¯2=sϑwe¯¯1×ϑwe¯¯1,,T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1T¯¯Uwe¯¯2I¯¯Lwe¯¯1+I¯¯Lwe¯¯2−I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1+I¯¯Uwe¯¯2−I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1+F¯¯Lwe¯¯2−F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1+F¯¯Uwe¯¯2−F¯¯Uwe¯¯1F¯¯Uwe¯¯2;(4)3ξ¯¯we¯¯=sξ¯¯ϑwe¯¯1−1−T¯¯Lwe¯¯ξ¯¯,1−1−T¯¯Uwe¯¯ξ¯¯,I¯¯Lwe¯¯ξ¯¯,I¯¯Uwe¯¯ξ¯¯,F¯¯Lwe¯¯ξ¯¯,F¯¯Uwe¯¯ξ¯¯;(5)4we¯¯ξ¯¯=sϑξ¯¯we¯¯T¯¯Lwe¯¯ξ¯¯,T¯¯Uwe¯¯ξ¯¯,1−1−I¯¯Lwe¯¯ξ¯¯,1−1−I¯¯Uwe¯¯ξ¯¯,1−1−F¯¯Lwe¯¯ξ¯¯,1−1−F¯¯Uwe¯¯ξ¯¯.


### 2.2. Hesitant Fuzzy Set

Definition 3 (see [33, 34]).
LetΩ be a fixed set, a HFS HF on Ω is an object of the form
(6)HF=v,hfHFv∣v∈Ω,where hfHFv=∪ΘHFv∈hfHFvΘHFv is a group of finite values in [0, 1], signifying the possible MED of an element v∈Ω to HF. For ease, we shall inscribe hf as a replacement for hfHFv=∪ΘHFv∈hfHFvΘHFv and is called a hesitant fuzzy element.
Lethf,hf1 and hf2 be HFEs, then, the core operational laws for HFEs are identified below:
(7)hfξ=∪Θ∈hfΘξ,ξ>0;ξhf=∪Θ∈hf1−1−Θξ,ξ>0;hf1⊕hf2=∪Θ1∈hf1,Θ2∈hf2Θ1+Θ2−Θ1Θ2;hf1⊗hf2=∪Θ1∈hf1,Θ2∈hf2Θ1Θ2.


### 2.3. Linguistic Scale Function

Linguistic scale functions (LSFs) apply various semantic values to linguistic scale under different conditions to make data more effective and to describe semantics more flexibly [46]. In practice, these functions are preferred because they are more versatile and may produce more predictable outputs based on varied meanings.Definition 4 (see [46]).
Ifϑ¯¯z¯¯∈0,1 is a numeric value, then, the LSF f∗ that demeanor the mapping from Sz¯¯ to ϑ¯¯z¯¯z¯¯=0,1,⋯,2r can be identified as follows: f∗:sz¯¯⟶ϑ¯¯z¯¯z¯¯=1,⋯,2r,(i)
Consider(8)f1∗sz¯¯=ϑz¯¯=z¯¯2r.
where0≤ϑ¯¯0<ϑ¯¯1<.⋯<ϑ¯¯2r. Evidently, the symbol ϑ¯¯z¯¯z¯¯=0,1,⋯,2r reflects the preference of the decision makers while they are using the linguistic term sz¯¯∈Sz¯¯=0,1,⋯,2r.. Consequently, the function value in fact denotes the semantics of the linguistic terms.
On average, the assessment scale for the linguistic information presented above is divided.(ii)
Consider(9)f2∗sz¯¯=ϑ¯¯z¯¯=ϕ¯¯r−ϕ¯¯r−z¯¯2ϕ¯¯r−2z¯¯=0,1,⋯,rϕ¯¯r−ϕ¯¯z¯¯−r−22ϕ¯¯r−2z¯¯=r+1,r+2⋯,2r.
The absolute deviation between neighboring linguistic subscripts rises as the length of the supplied linguistic term set is extended from the middle to both ends.(iii)
Consider(10)f3∗sz¯¯=ϑ¯¯z¯¯=rα−r−z¯¯α2rαz¯¯=0,1,⋯,rrβ−z¯¯−rβ2rβz¯¯=r+1,r+2⋯,2r.The absolute deviation between consecutive linguistic subscripts will decrease when the extension from the centre of the supplied linguistic term to both ends is increased.The above function may be extended to keep all of the provided data and make the computation easierf∗:S⟶R+R+=c∣c≥0,c∈R, which satisfies f∗sz¯¯=ϑ¯¯z¯¯ and is a strictly monotonically increasing and continuous function. Therefore, the mapping from StoR+ is one-to-one because of its monotonicity, and the inverse function of f∗ exists and is denoted by f∗−1.


### 2.4. Hesitant Interval Neutrosophic Linguistic Set

Definition 5 (see [32]).
LetΩ be the domain set. Then, a HINLS in Ω is characterized by the following mathematical form:
(11)HN¯¯=v,hn¯¯v∣v∈Ω,where hn¯¯v=∪we¯¯v∈hn¯¯vwe¯¯v is a group of INLNs, representing the possible INLNs of the element v∈Ω to the set HN¯¯ is an INLN. For ease, we shall inscribe hn¯¯=∪we¯¯∈hn¯¯we¯¯ as a replacement for hn¯¯v=∪we¯¯v∈hn¯¯vwe¯¯v in HN¯¯ . Here we identify hn¯¯ a HINULE and we¯¯=sϑwe¯¯Tr¯¯Lwe¯¯,Tr¯¯Uwe¯¯,In¯¯Lwe¯¯,In¯¯Uwe¯¯,Fl¯¯Lwe¯¯,Fl¯¯Uwe¯¯ is called an INLN. Then, HN¯¯ is the group of all INLNs.Definition 6 (see [32]).
Lethn¯¯1,hn¯¯2 and hn¯¯3 be any three HINLNs and ξ¯¯≥0. Then, some core operational rules for HINLNs are described as follows:
(12)1hn¯¯1⊕hn¯¯2=∪we¯¯1∈hn¯¯1,u¯¯we∈hn¯¯2sϑwe¯¯1+ϑwe¯¯2,T¯¯Lwe¯¯1+T¯¯Lwe¯¯2−T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1+T¯¯Uwe¯¯2−T¯¯Uwe¯¯1T¯¯Uwe¯¯2,I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1F¯¯Uwe¯¯2;(13)2hn¯¯1⊗hn¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2sϑwe¯¯1×ϑwe¯¯2,T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1T¯¯Uwe¯¯2,I¯¯Lwe¯¯1+I¯¯Lwe¯¯2−I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1+I¯¯Uwe¯¯2−I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1+F¯¯Lwe¯¯2−F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1+F¯¯Uwe¯¯2−F¯¯Uwe¯¯1F¯¯Uwe¯¯2(14)3ξ¯¯hn¯¯=∪we¯¯∈hn¯¯sϑwe¯¯,1−1−T¯¯Lwe¯¯ξ¯¯,1−1−T¯¯Uwe¯¯ξ¯¯,I¯¯Lwe¯¯ξ¯¯,I¯¯Uwe¯¯ξ¯¯,F¯¯Lwe¯¯ξ¯¯,F¯¯Uwe¯¯ξ¯¯;(15)4hn¯¯ξ¯¯=∪we¯¯∈hn¯¯sϑwe¯¯,T¯¯Lwe¯¯ξ¯¯,T¯¯Uwe¯¯ξ¯¯,1−1−I¯¯Lwe¯¯ξ¯¯,1−1−I¯¯Uwe¯¯ξ¯¯,1−1−F¯¯Lwe¯¯ξ¯¯,1−1−F¯¯Uwe¯¯ξ¯¯.Definition 7 (see [32]).
Lethn¯¯ be a HINLN. Then, the score function is signified as follows:
(16)Srehn¯¯=1≠hn¯¯∑we¯¯∈hn¯¯4+T¯¯Lwe¯¯−I¯¯Lwe¯¯−F¯¯Lwe¯¯+T¯¯Uwe¯¯−I¯¯Uwe¯¯−F¯¯Uwe¯¯we¯¯ϑ6l,
where≠hn¯¯ is the number INLNs in hn¯¯, ând l+1 is the cardinality of the linguistic term set S.
The score and accuracy function identified by Ye [32] have some limitation in some special cases for comparing two HINLNs, and this can be shown in an example below.Example 1.
LetS=s0,s1,s2,s3,s4,s5,s6=verypoor,slightlypoor,poor,fair,slightlygood,good,verygood,hn¯¯1=s4,0.6,0.7,0,0,0.4,0.5,s5,0.5,0.6,0,0,0.3,0.4, and hn¯¯2=s4,0.6,0.7,0.4,0.5,0,0,s5,0.5,0.6,0.3,0.4,,0,0 be two HINLNs. Then, by utilizing the above score function defined by Ye [32], we have
(17)Srehn¯¯1=Srehn¯¯2=0.5500.
Which shows thathn¯¯1=hn¯¯2. However, hn¯¯2 is greater than hn¯¯1.
To overcome the above existing limitation identified in an example, we signified new score function based on linguistic scale functions for comparing HINLNs.Definition 8.
Lethn¯¯ be a HINLN. Then, the improved score function can be signified as
(18)Scrhn¯¯=1≠hn¯¯∑we¯¯∈hn¯¯χf∗sϑu0.5T¯¯Uwe¯¯+1−F¯¯Lwe¯¯+χI¯¯Uwe¯¯+1−χf∗sϑu0.5T¯¯Lwe¯¯+1−F¯¯Uwe¯¯+χI¯¯Lwe¯¯,where the values of a χ∈0,1 indicate the decision-makers’ views, and a χ>0.5, χ=0.5, and χ<0.5 denote the decision-makers’ levels of optimist, temperance, and pessimist. Furthermore, by using various linguistic scale functions, alternative scoring functions can be produced.Definition 9.
Lethn¯¯1 and hn¯¯2 be two HINLNs. Then, the comparison rules for comparing two HINLNs are identified as follows:
(1)
IfScrhn¯¯1>Scrhn¯¯2, then hn¯¯1>hn¯¯2(2)
IfScrhn¯¯1<Scrhn¯¯2, then hn¯¯1<hn¯¯2(3)
IfScrhn¯¯1=Scrhn¯¯2, then hn¯¯1=hn¯¯2
Now, utilizing the improve score function to solve Example1 and assume that f∗sϑ=i/2l,χ=0.5, we have Scrhn¯¯1=0.4500andScrhn¯¯2=0.8146.
From the score values, we can observe thathn¯¯2 is greater than hn¯¯1.Definition 10.
Lethn¯¯1 and hn¯¯2 be any two HINLNs. Then, the Hamming distance between hn¯¯1 and hn¯¯2 can be described as
(19)dis¯¯hn¯¯1,hn¯¯2=1≠len∑i=1lf∗sϑwe¯¯1−f∗sϑwe¯¯2+16f∗sϑwe¯¯1T¯¯Lwe¯¯1−f∗sϑwe¯¯2T¯¯Lwe¯¯2+f∗sϑwe¯¯1T¯¯Uwe¯¯1−f∗sϑwe¯¯2T¯¯Uwe¯¯2+f∗sϑwe¯¯11−I¯¯Lwe¯¯1−f∗sϑwe¯¯21−I¯¯Lwe¯¯2+f∗sϑwe¯¯11−I¯¯Uwe¯¯1−f∗sϑwe¯¯21−I¯¯Uwe¯¯2+f∗sϑwe¯¯11−F¯¯Lwe¯¯1−f∗sϑwe¯¯21−F¯¯Lwe¯¯2+f∗sϑwe¯¯11−F¯¯Uwe¯¯1−f∗sϑwe¯¯21−F¯¯Uwe¯¯2


### 2.5. The PA Operator

PA operator initiated by Yager [35] is one of the imperative AOs. The PA operator reduces a number of unconstructive influences of unreasonably high or unreasonably low arguments given by DMs. The conservative PA operator can only contract with real numbers and is identified as follows.Definition 11 (see [35]).
Letwe¯¯1,we¯¯2,⋯,we¯¯z be a group of positive real numbers. A PA operator is classified as follows:
(20)PAwe¯¯1,we¯¯2,⋯,we¯¯z=∑i=1z1+Twe¯¯iwe¯¯i∑i=1z1+Twe¯¯i,
whereTwe¯¯i=∑i=1j≠izSupwe¯¯i,we¯¯j, and Supwe¯¯i,we¯¯j are the support degree (SPD) for we¯¯i from we¯¯j satisfying the following axioms:
(1)
Supwe¯¯i,we¯¯j∈0,1;(2)
Supwe¯¯i,we¯¯j=Supwe¯¯j,we¯¯i;(3)
Supwe¯¯i,we¯¯j≥Supwe¯¯p,we¯¯q, if we¯¯i,we¯¯j<we¯¯p,we¯¯qDefinition 12 (see [36]).
Letwe¯¯1,we¯¯2,⋯,we¯¯z be a set of positive real numbers. A PG operator is described as follows:
(21)PGwe¯¯1,we¯¯2,⋯,we¯¯z=∏i=1zwe¯¯i1+Twe¯¯i∑i=1z1+Twe¯¯i,
whereTwe¯¯i=∑i=1j≠izSupwe¯¯i,we¯¯j, and Supwe¯¯i,we¯¯j are the SPD for we¯¯i from we¯¯j satisfying the above axioms.Definition 13 (see [37]).
Letwe¯¯1,we¯¯2,⋯,we¯¯Z be a group of positive real numbers. A WGPA operator is described as follows:
(22)WGPAwe¯¯1,we¯¯2,⋯,we¯¯z=∑i=1z1+Twe¯¯iwe¯¯iA¯¯∑i=1z1+Twe¯¯i1A¯¯,
whereTwe¯¯G=∑G=1H≠GZSupwe¯¯G,we¯¯H, and Supwe¯¯G,we¯¯H are the SPD for we¯¯G from we¯¯H satisfying the following axioms.


## 2.1. The Interval Neutrosophic Linguistic Set

Definition 1 (see [28]).
LetΩ be the finite set. An INLS in Ω is identified by
(1)INL=a¯¯,sϑa¯¯,T¯¯INLa¯¯,I¯¯INLa¯¯,F¯¯INLa¯¯a¯¯∈Ω,where sϑa¯¯∈S,T¯¯INLa¯¯=infT¯¯INLa¯¯,supT¯¯INLa¯¯⊆0,1,I¯¯INLa¯¯=infI¯¯INLa¯¯,supI¯¯INLa¯¯⊆0,1 and F¯¯INLa¯¯=infF¯¯INLa¯¯,supF¯¯INLa¯¯⊆0,1, respectively, signifying the TMED, IND, and FLMD of an element a¯¯ in Ω to the linguistic variable sϑa with the constraint 0≤SupT¯¯INLa¯¯+SupI¯¯INLa¯¯+SupF¯¯INLa¯¯≤3 for any a¯¯∈Ω.For simplicity, the INLN is signified bywe¯¯=sϑwe¯¯,T¯¯Lwe¯¯,T¯¯Uwe¯¯,I¯¯Lwe¯¯,I¯¯Uwe¯¯,F¯¯Lwe¯¯,F¯¯Uwe¯¯..Definition 2 (see [28]).
Letwe¯¯=sϑwe¯¯,T¯¯Lwe¯¯,T¯¯Uwe¯¯,I¯¯Lwe¯¯,I¯¯Uwe¯¯,F¯¯Lwe¯¯,F¯¯Uwe¯¯,we¯¯1=sϑwe¯¯1,T¯¯Lwe¯¯1,T¯¯Uwe¯¯1,I¯¯Lwe¯¯1,I¯¯Uwe¯¯1,F¯¯Lwe¯¯1,F¯¯Uwe¯¯1, and we¯¯2=sϑwe¯¯2,T¯¯Lwe¯¯2,T¯¯Uwe¯¯2,I¯¯Lwe¯¯2,I¯¯Uwe¯¯2,F¯¯Lwe¯¯2,F¯¯Uwe¯¯2 be three INLNs and ξ¯¯≥0, then, the core operations for INLNs are defined as follows:
(2)1we¯¯1+we¯¯2=sϑwe¯¯1+sϑwe¯¯1,T¯¯Lwe¯¯1+T¯¯Lwe¯¯2−T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1+T¯¯Uwe¯¯2−T¯¯Uwe¯¯1T¯¯Uwe¯¯2,I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1F¯¯Uwe¯¯2;(3)2we¯¯1×we¯¯2=sϑwe¯¯1×ϑwe¯¯1,,T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1T¯¯Uwe¯¯2I¯¯Lwe¯¯1+I¯¯Lwe¯¯2−I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1+I¯¯Uwe¯¯2−I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1+F¯¯Lwe¯¯2−F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1+F¯¯Uwe¯¯2−F¯¯Uwe¯¯1F¯¯Uwe¯¯2;(4)3ξ¯¯we¯¯=sξ¯¯ϑwe¯¯1−1−T¯¯Lwe¯¯ξ¯¯,1−1−T¯¯Uwe¯¯ξ¯¯,I¯¯Lwe¯¯ξ¯¯,I¯¯Uwe¯¯ξ¯¯,F¯¯Lwe¯¯ξ¯¯,F¯¯Uwe¯¯ξ¯¯;(5)4we¯¯ξ¯¯=sϑξ¯¯we¯¯T¯¯Lwe¯¯ξ¯¯,T¯¯Uwe¯¯ξ¯¯,1−1−I¯¯Lwe¯¯ξ¯¯,1−1−I¯¯Uwe¯¯ξ¯¯,1−1−F¯¯Lwe¯¯ξ¯¯,1−1−F¯¯Uwe¯¯ξ¯¯.


## 2.2. Hesitant Fuzzy Set

Definition 3 (see [33, 34]).
LetΩ be a fixed set, a HFS HF on Ω is an object of the form
(6)HF=v,hfHFv∣v∈Ω,where hfHFv=∪ΘHFv∈hfHFvΘHFv is a group of finite values in [0, 1], signifying the possible MED of an element v∈Ω to HF. For ease, we shall inscribe hf as a replacement for hfHFv=∪ΘHFv∈hfHFvΘHFv and is called a hesitant fuzzy element.
Lethf,hf1 and hf2 be HFEs, then, the core operational laws for HFEs are identified below:
(7)hfξ=∪Θ∈hfΘξ,ξ>0;ξhf=∪Θ∈hf1−1−Θξ,ξ>0;hf1⊕hf2=∪Θ1∈hf1,Θ2∈hf2Θ1+Θ2−Θ1Θ2;hf1⊗hf2=∪Θ1∈hf1,Θ2∈hf2Θ1Θ2.


## 2.3. Linguistic Scale Function

Linguistic scale functions (LSFs) apply various semantic values to linguistic scale under different conditions to make data more effective and to describe semantics more flexibly [46]. In practice, these functions are preferred because they are more versatile and may produce more predictable outputs based on varied meanings.Definition 4 (see [46]).
Ifϑ¯¯z¯¯∈0,1 is a numeric value, then, the LSF f∗ that demeanor the mapping from Sz¯¯ to ϑ¯¯z¯¯z¯¯=0,1,⋯,2r can be identified as follows: f∗:sz¯¯⟶ϑ¯¯z¯¯z¯¯=1,⋯,2r,(i)
Consider(8)f1∗sz¯¯=ϑz¯¯=z¯¯2r.
where0≤ϑ¯¯0<ϑ¯¯1<.⋯<ϑ¯¯2r. Evidently, the symbol ϑ¯¯z¯¯z¯¯=0,1,⋯,2r reflects the preference of the decision makers while they are using the linguistic term sz¯¯∈Sz¯¯=0,1,⋯,2r.. Consequently, the function value in fact denotes the semantics of the linguistic terms.
On average, the assessment scale for the linguistic information presented above is divided.(ii)
Consider(9)f2∗sz¯¯=ϑ¯¯z¯¯=ϕ¯¯r−ϕ¯¯r−z¯¯2ϕ¯¯r−2z¯¯=0,1,⋯,rϕ¯¯r−ϕ¯¯z¯¯−r−22ϕ¯¯r−2z¯¯=r+1,r+2⋯,2r.
The absolute deviation between neighboring linguistic subscripts rises as the length of the supplied linguistic term set is extended from the middle to both ends.(iii)
Consider(10)f3∗sz¯¯=ϑ¯¯z¯¯=rα−r−z¯¯α2rαz¯¯=0,1,⋯,rrβ−z¯¯−rβ2rβz¯¯=r+1,r+2⋯,2r.The absolute deviation between consecutive linguistic subscripts will decrease when the extension from the centre of the supplied linguistic term to both ends is increased.The above function may be extended to keep all of the provided data and make the computation easierf∗:S⟶R+R+=c∣c≥0,c∈R, which satisfies f∗sz¯¯=ϑ¯¯z¯¯ and is a strictly monotonically increasing and continuous function. Therefore, the mapping from StoR+ is one-to-one because of its monotonicity, and the inverse function of f∗ exists and is denoted by f∗−1.


## 2.4. Hesitant Interval Neutrosophic Linguistic Set

Definition 5 (see [32]).
LetΩ be the domain set. Then, a HINLS in Ω is characterized by the following mathematical form:
(11)HN¯¯=v,hn¯¯v∣v∈Ω,where hn¯¯v=∪we¯¯v∈hn¯¯vwe¯¯v is a group of INLNs, representing the possible INLNs of the element v∈Ω to the set HN¯¯ is an INLN. For ease, we shall inscribe hn¯¯=∪we¯¯∈hn¯¯we¯¯ as a replacement for hn¯¯v=∪we¯¯v∈hn¯¯vwe¯¯v in HN¯¯ . Here we identify hn¯¯ a HINULE and we¯¯=sϑwe¯¯Tr¯¯Lwe¯¯,Tr¯¯Uwe¯¯,In¯¯Lwe¯¯,In¯¯Uwe¯¯,Fl¯¯Lwe¯¯,Fl¯¯Uwe¯¯ is called an INLN. Then, HN¯¯ is the group of all INLNs.Definition 6 (see [32]).
Lethn¯¯1,hn¯¯2 and hn¯¯3 be any three HINLNs and ξ¯¯≥0. Then, some core operational rules for HINLNs are described as follows:
(12)1hn¯¯1⊕hn¯¯2=∪we¯¯1∈hn¯¯1,u¯¯we∈hn¯¯2sϑwe¯¯1+ϑwe¯¯2,T¯¯Lwe¯¯1+T¯¯Lwe¯¯2−T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1+T¯¯Uwe¯¯2−T¯¯Uwe¯¯1T¯¯Uwe¯¯2,I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1F¯¯Uwe¯¯2;(13)2hn¯¯1⊗hn¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2sϑwe¯¯1×ϑwe¯¯2,T¯¯Lwe¯¯1T¯¯Lwe¯¯2,T¯¯Uwe¯¯1T¯¯Uwe¯¯2,I¯¯Lwe¯¯1+I¯¯Lwe¯¯2−I¯¯Lwe¯¯1I¯¯Lwe¯¯2,I¯¯Uwe¯¯1+I¯¯Uwe¯¯2−I¯¯Uwe¯¯1I¯¯Uwe¯¯2,F¯¯Lwe¯¯1+F¯¯Lwe¯¯2−F¯¯Lwe¯¯1F¯¯Lwe¯¯2,F¯¯Uwe¯¯1+F¯¯Uwe¯¯2−F¯¯Uwe¯¯1F¯¯Uwe¯¯2(14)3ξ¯¯hn¯¯=∪we¯¯∈hn¯¯sϑwe¯¯,1−1−T¯¯Lwe¯¯ξ¯¯,1−1−T¯¯Uwe¯¯ξ¯¯,I¯¯Lwe¯¯ξ¯¯,I¯¯Uwe¯¯ξ¯¯,F¯¯Lwe¯¯ξ¯¯,F¯¯Uwe¯¯ξ¯¯;(15)4hn¯¯ξ¯¯=∪we¯¯∈hn¯¯sϑwe¯¯,T¯¯Lwe¯¯ξ¯¯,T¯¯Uwe¯¯ξ¯¯,1−1−I¯¯Lwe¯¯ξ¯¯,1−1−I¯¯Uwe¯¯ξ¯¯,1−1−F¯¯Lwe¯¯ξ¯¯,1−1−F¯¯Uwe¯¯ξ¯¯.Definition 7 (see [32]).
Lethn¯¯ be a HINLN. Then, the score function is signified as follows:
(16)Srehn¯¯=1≠hn¯¯∑we¯¯∈hn¯¯4+T¯¯Lwe¯¯−I¯¯Lwe¯¯−F¯¯Lwe¯¯+T¯¯Uwe¯¯−I¯¯Uwe¯¯−F¯¯Uwe¯¯we¯¯ϑ6l,
where≠hn¯¯ is the number INLNs in hn¯¯, ând l+1 is the cardinality of the linguistic term set S.
The score and accuracy function identified by Ye [32] have some limitation in some special cases for comparing two HINLNs, and this can be shown in an example below.Example 1.
LetS=s0,s1,s2,s3,s4,s5,s6=verypoor,slightlypoor,poor,fair,slightlygood,good,verygood,hn¯¯1=s4,0.6,0.7,0,0,0.4,0.5,s5,0.5,0.6,0,0,0.3,0.4, and hn¯¯2=s4,0.6,0.7,0.4,0.5,0,0,s5,0.5,0.6,0.3,0.4,,0,0 be two HINLNs. Then, by utilizing the above score function defined by Ye [32], we have
(17)Srehn¯¯1=Srehn¯¯2=0.5500.
Which shows thathn¯¯1=hn¯¯2. However, hn¯¯2 is greater than hn¯¯1.
To overcome the above existing limitation identified in an example, we signified new score function based on linguistic scale functions for comparing HINLNs.Definition 8.
Lethn¯¯ be a HINLN. Then, the improved score function can be signified as
(18)Scrhn¯¯=1≠hn¯¯∑we¯¯∈hn¯¯χf∗sϑu0.5T¯¯Uwe¯¯+1−F¯¯Lwe¯¯+χI¯¯Uwe¯¯+1−χf∗sϑu0.5T¯¯Lwe¯¯+1−F¯¯Uwe¯¯+χI¯¯Lwe¯¯,where the values of a χ∈0,1 indicate the decision-makers’ views, and a χ>0.5, χ=0.5, and χ<0.5 denote the decision-makers’ levels of optimist, temperance, and pessimist. Furthermore, by using various linguistic scale functions, alternative scoring functions can be produced.Definition 9.
Lethn¯¯1 and hn¯¯2 be two HINLNs. Then, the comparison rules for comparing two HINLNs are identified as follows:
(1)
IfScrhn¯¯1>Scrhn¯¯2, then hn¯¯1>hn¯¯2(2)
IfScrhn¯¯1<Scrhn¯¯2, then hn¯¯1<hn¯¯2(3)
IfScrhn¯¯1=Scrhn¯¯2, then hn¯¯1=hn¯¯2
Now, utilizing the improve score function to solve Example1 and assume that f∗sϑ=i/2l,χ=0.5, we have Scrhn¯¯1=0.4500andScrhn¯¯2=0.8146.
From the score values, we can observe thathn¯¯2 is greater than hn¯¯1.Definition 10.
Lethn¯¯1 and hn¯¯2 be any two HINLNs. Then, the Hamming distance between hn¯¯1 and hn¯¯2 can be described as
(19)dis¯¯hn¯¯1,hn¯¯2=1≠len∑i=1lf∗sϑwe¯¯1−f∗sϑwe¯¯2+16f∗sϑwe¯¯1T¯¯Lwe¯¯1−f∗sϑwe¯¯2T¯¯Lwe¯¯2+f∗sϑwe¯¯1T¯¯Uwe¯¯1−f∗sϑwe¯¯2T¯¯Uwe¯¯2+f∗sϑwe¯¯11−I¯¯Lwe¯¯1−f∗sϑwe¯¯21−I¯¯Lwe¯¯2+f∗sϑwe¯¯11−I¯¯Uwe¯¯1−f∗sϑwe¯¯21−I¯¯Uwe¯¯2+f∗sϑwe¯¯11−F¯¯Lwe¯¯1−f∗sϑwe¯¯21−F¯¯Lwe¯¯2+f∗sϑwe¯¯11−F¯¯Uwe¯¯1−f∗sϑwe¯¯21−F¯¯Uwe¯¯2


## 2.5. The PA Operator

PA operator initiated by Yager [35] is one of the imperative AOs. The PA operator reduces a number of unconstructive influences of unreasonably high or unreasonably low arguments given by DMs. The conservative PA operator can only contract with real numbers and is identified as follows.Definition 11 (see [35]).
Letwe¯¯1,we¯¯2,⋯,we¯¯z be a group of positive real numbers. A PA operator is classified as follows:
(20)PAwe¯¯1,we¯¯2,⋯,we¯¯z=∑i=1z1+Twe¯¯iwe¯¯i∑i=1z1+Twe¯¯i,
whereTwe¯¯i=∑i=1j≠izSupwe¯¯i,we¯¯j, and Supwe¯¯i,we¯¯j are the support degree (SPD) for we¯¯i from we¯¯j satisfying the following axioms:
(1)
Supwe¯¯i,we¯¯j∈0,1;(2)
Supwe¯¯i,we¯¯j=Supwe¯¯j,we¯¯i;(3)
Supwe¯¯i,we¯¯j≥Supwe¯¯p,we¯¯q, if we¯¯i,we¯¯j<we¯¯p,we¯¯qDefinition 12 (see [36]).
Letwe¯¯1,we¯¯2,⋯,we¯¯z be a set of positive real numbers. A PG operator is described as follows:
(21)PGwe¯¯1,we¯¯2,⋯,we¯¯z=∏i=1zwe¯¯i1+Twe¯¯i∑i=1z1+Twe¯¯i,
whereTwe¯¯i=∑i=1j≠izSupwe¯¯i,we¯¯j, and Supwe¯¯i,we¯¯j are the SPD for we¯¯i from we¯¯j satisfying the above axioms.Definition 13 (see [37]).
Letwe¯¯1,we¯¯2,⋯,we¯¯Z be a group of positive real numbers. A WGPA operator is described as follows:
(22)WGPAwe¯¯1,we¯¯2,⋯,we¯¯z=∑i=1z1+Twe¯¯iwe¯¯iA¯¯∑i=1z1+Twe¯¯i1A¯¯,
whereTwe¯¯G=∑G=1H≠GZSupwe¯¯G,we¯¯H, and Supwe¯¯G,we¯¯H are the SPD for we¯¯G from we¯¯H satisfying the following axioms.


## 3. SS Operational Laws for HINLNs

The SS operations [47] contain SS product and SS addition, which are exacting cases of Archemedean t-norm and t-conorm.The SSt-norm and t-conorm are elucidated as follows:
(23)Θ¯¯SS,ςα¯¯,β¯¯=α¯¯ς+β¯¯ς−11/ς,Θ¯¯SS,ς∗α¯¯,β¯¯=1−1−α¯¯ς+1−β¯¯ς−11/ς,where ς<0,α¯¯,β¯¯∈0,1.Moreover, whenς=0, SS t-norm and SS t-conorm degenerate into algebraic t-normn and t-conorm.Based ont-norm and t-conorm of SS operations, we can provide the following definition about SS operations for HINLNs.Definition 14.
Lethn,hn1,hn2 be any three HINLNS, and ξ>0. Then, we initiate some core operational laws for HINLNs based on Schweizer-Sklar t-norm and t-co-norm.
(24)1hn¯¯1⊕hn¯¯2=∪we¯¯1∈hn1,we¯¯2∈hn2f∗−1f∗sϑwe¯¯1+f∗sϑwe¯¯2,1−1−T¯¯1Lς+1−T¯¯2Lς−11/ς,1−1−T¯¯1Uς+1−T¯¯2Uς−11/ς,I¯¯1Lς+I¯¯2Lς−11/ς,I¯¯1Uς+I¯¯2Uς−11/ς,F¯¯1Lς+F¯¯2Lς−11/ς,F¯¯1Uς+F¯¯2Uς−11/ς;(25)2hn¯¯1⊕hn¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1f∗sϑwe¯¯1+f∗sϑwe¯¯2,T¯¯1Lς+T¯¯2Lς−11/ς,T¯¯1Uς+T¯¯2Uς−11/ς,1−1−I¯¯1Lς+1−I¯¯2Lς−11/ς1−1−I¯¯1Uς+1−I¯¯2Uς−11/ς,1−1−F¯¯1Lς+1−F¯¯2Lς−11/ς,1−1−F¯¯1Uς+1−F¯¯2Uς−11/ς;(26)3ξhn¯¯=∪we¯¯∈hn¯¯f∗−1ξf∗sϑwe¯¯,1−ξ1−T¯¯Lς−ξ−11/ς,1−ξ1−T¯¯Uς−ξ−11/ς,ξI¯¯Lς−ξ−11/ς,ξI¯¯Uς−ξ−11/ς,ξF¯¯Lς−ξ−11/ς,ξF¯¯Uς−ξ−11/ς,ξ>0,(27)4hn¯¯ξ=∪we¯¯∈hn¯¯f∗−1f∗sϑwe¯¯ξ,ξT¯¯Lς−ξ−11/ς,ξT¯¯Uς−ξ−11/ς,1−ξ1−I¯¯Lς−ξ−11/ς,1−ξ1−I¯¯Uς−ξ−11/ς,1−ξ1−F¯¯Lς−ξ−11/ς,1−ξ1−F¯¯Lς−ξ−11/ς,ξ>0.Theorem 15.
Lethn¯¯,hn¯¯1 and hn¯¯2 be any three HINLNs. Then
(28)hn¯¯1⊕SShn¯¯2=hn¯¯2⊕SShn¯¯1,hn¯¯1⊗SShn¯¯2=hn¯¯2⊗SShn¯¯1,ξhn¯¯1⊕SShn¯¯2=ξhn¯¯1⊕SSξhn¯¯2,ξ≥0,ξ1hn¯¯⊕SSξ2hn¯¯=ξ1+ξ2hn¯¯,ξ1,ξ2≥0,hn¯¯ξ1⊗SShn¯¯ξ2=hn¯¯ξ1+ξ2,ξ1,ξ2≥0,hn¯¯1ξ⊗SShn¯¯2ξ=hn¯¯1⊗SShn¯¯2ξ,ξ>0.


## 4. Some Generalized Power Aggregation Operators for HINLNs

In this part, we develop some generalized power AOs established on the initiated operational rules for HINLNs.


### 4.1. Weighted Generalized Hesitant Interval Neutrosophic Linguistic Schweizer-Sklar Power Aggregation Operator

In this subpart, we initiate generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power average (GHINLSSPA) operator, weighted (WGHINLSSPA) operator and examine their enviable properties and various particular cases.Definition 16.
For a collection of HINLNshn¯¯gg=1,2,⋯,s, GHINLSSPA operator is a function ℵs⟶ℵ,(29)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1s1+Thn¯¯ghn¯¯gζ⊕g=1s1+Thn¯¯g1/ζ,where Thn¯¯g=⊕h=1,g≠hssuphn¯¯g,hn¯¯h, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the SPD for hn¯¯g from hn¯¯h with the following constraint:
(1)
Suphn¯¯g,hn¯¯h∈0,1;(2)
Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;(3)
Suphn¯¯g,if¯¯h≥Suphn¯¯u,hn¯¯v if dis¯¯hn¯¯g,hn¯¯h<ds¯¯hn¯¯u,hn¯¯v, where dis¯¯ is the distance measure among two HINLNs.To write Equation (29) in unsophisticated form, we have
(30)pw¯¯g=1+Thn¯¯g⊕~g=1s1+Thn¯¯g.So, from Equation (30), Equation (29) becomes
(31)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1spw¯¯ghn¯¯gζ1/ζ,Theorem 17.
Lethn¯¯gg=1,2,⋯,s be a set of HINLNs, then the value aggregated utilizing Definition 16 is still HINLN, and we have
(32)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,..⋯,we¯¯s∈hn¯¯sf∗−1⊕g=1spw¯¯gf∗sϑwe¯¯gζ1/ζ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ.Proof.
In the following, first, we prove(33)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯shn¯¯sζ=∪u1∈hn¯¯1,u2∈hn¯¯2,..⋯,us∈hn¯¯sf∗−1⊕g=1spw¯¯gf∗sϑugζ,1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,by exploiting mathematical induction on s.Fors=2.From the operational laws explained for HINLNs in Definition14, we have
(34)hn¯¯1ζ=∪u1∈hn¯¯1f∗−1f∗sϑu1ζ,ζT¯¯1Lς−ζ−11/ς,ζT¯¯1Uς−ζ−11/ς,1−ζ1−I¯¯1Lς−ζ−11/ς,1−ζ1−I¯¯1Uς−ζ−11/ς,1−ζ1−F¯¯1Lς−ζ−11/ς,1−ζ1−F¯¯1Uς−ζ−11/ς,and
(35)pw¯¯1hn¯¯1ζ=∪we¯¯1∈hn¯¯1f∗−1Ω¯¯1f∗sϑwe¯¯1ζ,1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ς,1−pw¯¯11−ζT¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−I¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−I¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−F¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ς,Similarly,(36)pw¯¯2hn¯¯2ζ=∪we¯¯2∈hn¯¯2f∗−1pw¯¯2f∗sϑwe¯¯2ζ,1−pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ς,1−pw¯¯21−ζT¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−I¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−I¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−F¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ς.Then,(37)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1pw¯¯1f∗sϑwe¯¯1ζ⊕pw¯¯1f∗sϑwe¯¯1ζ,1−1−1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+1−1−pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,1−1−1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+1−1−pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−I¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−I¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−F¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−F¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς;=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊕g=12pw¯¯gf∗sϑwe¯¯gζ,1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−I¯¯1Lς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−I¯¯2Lς−ζ−11/ςς−pw¯¯2−1−11/ς,pw¯¯11−ζ1−I¯¯1Uς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−I¯¯2Uς−ζ−11/ςς−pw¯¯2−1−11/ς,pw¯¯11−ζ1−F¯¯1Lς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−F¯¯2Lς−ζ−11/ςς−pw¯¯2−1−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−1−11/ς;pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊕g=12pw¯¯gf∗sϑwe¯¯gζ,1−⊕g=12pw¯¯g1−ζT¯¯gLς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,1−⊕g=12pw¯¯g1−ζT¯¯gUς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,⊕g=12pw¯¯g1−ζ1−I¯¯gLς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,⊕g=12pw¯¯g1−ζ1−I¯¯gUς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς⊕g=12pw¯¯g1−ζ1−F¯¯gLς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,⊕g=12pw¯¯g1−ζ1−F¯¯gUς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς.If Equation (33) holds for s=m(38)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯mhn¯¯mζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊕g=1mpw¯¯gf∗sϑwe¯¯gζ,1−⊕g=1mpw¯¯g1−ζT¯¯gLς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,1−⊕g=1mpw¯¯g1−ζT¯¯gUς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gLς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gUς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gLς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gUς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς.Then, whens=m+1, by the operational laws explained in Definition 14, we have
(39)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯m+1hn¯¯m+1ζ=pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯mhn¯¯mζ⊕pw¯¯m+1hn¯¯m+1ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊕g=1mpw¯¯gf∗sϑwe¯¯gζ,1−⊕g=1mpw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς⊕∪we¯¯m+1∈hn¯¯m+1f∗sϑwe¯¯m+1,1−pw¯¯m+11−ζT¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζT¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−I¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−I¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−F¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−F¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯m+1hn¯¯m+1ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯m+1∈hn¯¯m+1f∗−1⊕g=1m+1pw¯¯gf∗sϑwe¯¯gζ,1−⊕g=1m+1pw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς.That is, Equation (33) is true for g=m+1. So Equation (33) is true for all g. Then,
(40)⊕g=1spw¯¯ghn¯¯gζ1ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯m+1∈hn¯¯m+1f∗−1⊕g=1spw¯¯gf∗sϑwe¯¯gζ1ξ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς.Therefore,(41)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯m+1∈hn¯¯m+1f∗−1⊕~g=1spw¯¯gf∗sϑwe¯¯gζ1ξ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς.Which completes the proof of the Theorem17.Theorem 18.
Commutativity: lethn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗ be any permutation of hn¯¯1,hn¯¯2,⋯,hn¯¯s, then
(42)GHINLSSPAζhn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗=GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s.Proof.
Suppose thathn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗ is any permutation of hn¯¯1,hn¯¯2,⋯,hn¯¯s, then for each hn¯¯g, there exists one and only one hn¯¯h∗ such that hn¯¯g=hn¯¯h∗ and vice versa. And Thn¯¯g=Thn¯¯h∗.. Then based on Theorem 17, we have
(43)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spw¯¯ghn¯¯gζ1/ζ=⊕h=1spw¯¯h∗hn¯¯h∗ζ1/ζ=GHINLSSPAζhn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗.
Note that the GHINLSSPA operators are neither idempotent nor monotonic.Further, we shall discuss a few cases of the initiated AOs with respect to the parameterζ, which are listed below:
(1)
Ifζ=1, then, the GHINLSSPA operator trim down to HINLSS power average (PA) operator:(44)GHINLSSPAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spw¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,us∈hn¯¯sf∗−1⊕g=1mpw¯¯gf∗sϑug,1−⊕g=1spw¯¯g1−T¯¯gL−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−T¯¯gL−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gI¯¯gLς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gI¯¯gUς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gF¯¯gLς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gF¯¯gUς−⊕g=1spw¯¯g+11ς.(2)
Ifζ=1 and ς=0, then, the GHINLSSPA operator trim down to HINL PA operator based on algebraic operation. That is(45)GHINLSSPAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spw¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,us∈hn¯¯sf∗−1⊕g=1m+1pw¯¯gf∗sϑug,1−⊗g=1s1−T¯¯gLpw¯¯g,1−⊗g=1s1−T¯¯gUpw¯¯g,⊗g=1sI¯¯gLpw¯¯g,⊗g=1sI¯¯gUpw¯¯g,⊗g=1sF¯¯gLpw¯¯g,⊗g=1sF¯¯gUpw¯¯g.(3)
Ifζ=1, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the GHINLSSPA operator trim down to HINL average operator based on algebraic operation. That is(46)WSSHINLPAς⟶0hn¯¯1,hn¯¯2,⋯,hn¯¯s=1s⊕g=1shn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,us∈hn¯¯sf∗−1⊕g=1m1sf∗sϑug1−⊗g=1s1−T¯¯gL1s,1−⊗g=1s1−T¯¯gU1s,⊗g=1sI¯¯gL1s,⊗g=1sI¯¯gU1s,⊗g=1sF¯¯gL1s,⊗g=1sF¯¯gU1s.Definition 19.
For a group of HINLNshn¯¯gg=1,2,⋯,s, a WGHINLSSPA operator is a function ℵs⟶ℵ,(47)WGHINLSSPAϕ,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1sWe¯¯g1+Thn¯¯ghn¯¯gζ⊕g=1sWe¯¯g1+Thn¯¯g1/ζ,where Thn¯¯g=⊕h=1,g≠hsSuphn¯¯g,hn¯¯h,We¯¯=We¯¯1,We¯¯2,⋯,We¯¯sT is the weight vector for hn¯¯gg=1,2,⋯,s such that We¯¯g∈0,1 and ⊕g=1sWe¯¯g=1, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the support for hn¯¯g from hn¯¯h with the following constraint:
(48)Suphn¯¯g,hn¯¯h∈0,1;Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;To write Equation (47) in uncomplicated way, we have
(49)pow¯¯g=Weg1+Thn¯¯g⊕g=1sWeg1+Thn¯¯g,so, from Equation (49), Equation (47) becomes
(50)WGHILNSSPAϕ,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1spow¯¯ghn¯¯gζ1/ζ,Theorem 20.
Lethn¯¯gg=1,2,⋯,s be a group of HINLNs, then the value aggregated utilizing Definition 19 is still HINLN, and we have
(51)WGHINLSSPAζ,ϕhn¯¯1,hn¯¯2,⋯,hn¯¯s∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑugζ1ζ,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−FgLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−FgUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς.Proof.
Proof of the Theorem20 is same as Theorem 17.Theorem 21.
For a group of HINLNshn¯¯gg=1,2,⋯,s, ζ is a parameter and ζ>0. We¯¯=We¯¯1,We¯¯2,⋯,We¯¯sT is weighting vector for hn¯¯gg=1,2,⋯,s, We¯¯g∈0,1, and ⊕g=1sWe¯¯g=1. If We¯¯=1/s,1/s,⋯,1/sT, then WGHINLSSPA trims down to generalized HINLSS PA operator:
(52)WGHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spow¯¯ghn¯¯gζ1ζ=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑugζ1ζ,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−1,1−1ζ1−⊕g=1spow¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,wherepow¯¯g=1+Thn¯¯g/⊕g=1s1+Thn¯¯g is the power weight vector. Further, we shall discuss a few cases of the initiated AOs with respect to the parameter ζ and We¯¯, which are listed below:
(1)
Ifζ=1, We¯¯=1/s,1/s,⋯,1/sT, then, the WGHINLSSPA operator reduces into HINLSS power average (PA) operator:(53)HINLSSPAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spow¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑug,1−⊕g=1spow¯¯g1−T¯¯gLς−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−T¯¯gUς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gI¯¯gLς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gI¯¯gUς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gF¯¯gLς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gFgUς−⊕g=1spow¯¯g+11ς.(2)
Ifζ=1, We¯¯=1/s,1/s,⋯,1/sT, and ς=0, then, the WGHINLSSPA operator trims down to HINL PA operator based on algebraic operation. That is(54)WGHINLSSPAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spow¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑug,=1−⊗g=1s1−T¯¯gLpow¯¯g,1−⊗g=1s1−T¯¯gUpow¯¯g,⊗g=1sI¯¯gLpow¯¯g,⊗g=1sI¯¯gUpow¯¯g,⊗g=1sF¯¯gLpow¯¯g,⊗g=1sF¯¯gUpow¯¯g.(3)
Ifζ=1, We¯¯=1/s,1/s,⋯,1/sT, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the WGHINLSSPA operator reduces into HINL average operator based on algebraic operation. That is(55)WGHINLSSPAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=1s⊕g=1shn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−11s⊕g=1sf∗sϑug,1−⊗g=1s1−T¯¯gL1s,1−⊗g=1s1−T¯¯gU1s,⊗g=1sI¯¯gL1s,⊗g=1sI¯¯gU1s,⊗g=1sF¯¯gL1s,⊗g=1sF¯¯gU1s.


### 4.2. Some Generalized Hesitant Interval Neutrosophic Schweizer-Sklar Power Geometric Aggregation Operators

In this subpart, we initiate generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power geometric average (GHINLSSPWGA) operator, weighted GHINLSSPGA, discuss their desirable properties and some particular cases.Definition 22.
For a group of HINLNshn¯¯gg=1,2,⋯,s, GHINLSSPGA operator is a function ℵs⟶ℵ,(56)GHINLSSPGAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗~g=1sζhn¯¯g1+Thn¯¯g⊕h=1s1+Thn¯¯h,where Thn¯¯g=⊕h=1,g≠hsSuphn¯¯g,hn¯¯h, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the SPD for hn¯¯g from hn¯¯h with the following constraint:
(1)
Suphn¯¯g,hn¯¯h≥Suphn¯¯u,hn¯¯vif dis¯¯hn¯¯g,hn¯¯h<dis¯¯hn¯¯u,hn¯¯v, where dis¯¯ is the distance measure among two HINLNs(57)Suphn¯¯g,hn¯¯h∈0,1;Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;To write Equation (56) in an uncomplicated way, we have
(58)pw¯¯g=1+Thn¯¯g⊕g=1s1+Thn¯¯g.So, from Equation (58), Equation (56) becomes
(59)GHINLSSPGAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗g=1sζhn¯¯gpw¯¯g.Theorem 23.
Lethn¯¯gg=1,2,⋯,s be a group of HINLNs, then, the value aggregated utilizing Definition 22 is still HINLN, and we have
(60)GHINLSSPGζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpw¯¯g1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς.Proof.
In the following, first, we prove(61)ζhn¯¯1pw¯¯1×ζhn¯¯2pw¯¯2×⋯×ζhn¯¯spw¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sζf∗sϑwe¯¯gpw¯¯g⋅⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζIgUς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ς.
To prove Equation (61), we utilize mathematical induction on s.
Fors=2.
From the operational rules explained for HINLNs in Definition14, we have
(62)ζhn¯¯1=∪we¯¯1∈hn¯¯1f∗−1ζf∗sϑwe¯¯g,1−ζ1−T¯¯1Lς−ζ−11ς,1−ζ1−T¯¯1Uς−ζ−11ς,ζI¯¯1Lς−ζ−11ς,ζI¯¯1Lς−ζ−11ς,ζF¯¯1Lς−ζ−11ς,ζF¯¯1Uς−ζ−11ςζhn¯¯1pw¯¯1=∪we¯¯1∈hn¯¯1f∗−1ζf∗sϑwe¯¯2pw¯¯1,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−11ς,pw¯¯11−ζ1−T¯¯1Uς−ζ−11ςς−pw¯¯1−11ς,1−pw¯¯11−ζI¯¯1Lς−ζ−11ςς−pw¯¯1−11ς,pw¯¯11−ζI¯¯1Uς−ζ−11ςς−pw¯¯1−11ς,1−pw¯¯11−ζF¯¯1Lς−ζ−11ςς−pw¯¯1−11ς,pw¯¯11−ζF¯¯1Uς−ζ−11ςς−pw¯¯1−11ς;
Similarly,(63)ζhn¯¯2pw¯¯2=∪we¯¯2∈hn¯¯2f∗−1ζf∗sϑwe¯¯2pw¯¯2,pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−11ς,pw¯¯21−ζ1−T¯¯2Uς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζI¯¯2Lς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζI¯¯2Uς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζF¯¯2Lς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζF¯¯2Uς−ζ−11ςς−pw¯¯2−11ς.
Then,(64)ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1ζf∗sϑwe¯¯1pw¯¯1ζf∗sϑwe¯¯2pw¯¯2,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζI¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζI¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζI¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζI¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζF¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζF¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζF¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζF¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς;=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊗g=12ζf∗sϑwe¯¯gpw¯¯g,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−1+pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−1−11ς,pw¯¯11−ζ1−T¯¯1Uς−ζ−11ςς−pw¯¯1−1+pw¯¯21−ζ1−T¯¯2Uς−ζ−11ςς−pw¯¯2−1−11ς,1−pw¯¯11−ζI¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζI¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−pw¯¯11−ζI¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζI¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−pw¯¯11−ζF¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζF¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−pw¯¯11−ζF¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζF¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς;ζhn¯¯1pw¯¯1×ζhn¯¯2pw¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊗g=12ζf∗sϑwe¯¯gpw¯¯g,=⊕g=12pw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=12pw¯¯g+11ς,⊕g=12pw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=12pw¯¯g+11ς;
If Equation (50) holds for s=m(65)ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯mpw¯¯m=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊗g=1mζf∗sϑwe¯¯gpw¯¯g⊕g=1mpw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+1,1−⊕g=1mpw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+1.
Then, whens=m+1, by the operational rules given in Definition 14, we have
(66)ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯m+1pw¯¯m+1=ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯mpw¯¯m⊗ζhn¯¯m+1pw¯¯m+1=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊗g=1mζf∗sϑwe¯¯gpw¯¯g⊕g=1mpw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς⊕∪we¯¯m+1∈hn¯¯m+1f∗−1ζf∗sϑwe¯¯m+1pw¯¯m+1,pw¯¯m+11−ζ1−T¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−T¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζI¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζI¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζF¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζF¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯m+1pw¯¯m+1=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯m+1f∗−1⊗g=1m+1ζf∗sϑwe¯¯gpw¯¯g,⊕g=1m+1pw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζI¯¯gLgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζI¯¯gUgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζF¯¯gLgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζF¯¯gUgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς.
That is, Equation (61) is true for g=m+1. So Equation (61) is true for all g. Then,
(67)1ζ⊗g=1sζhn¯¯gpw¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpw¯¯g1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,
Therefore,(68)GHINLSSPGAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpw¯¯g1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,
This completes the proof of Theorem23.Further, we shall examine a few cases of the initiated AOs with respect to the parameterζ and pw¯¯, which are listed below:
(1)
Ifζ=1, then, the GHINLSSPGA operator trims down to HINLSS PGA operator:(69)GHINLSSPGAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpw¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpw¯¯g⊕g=1spw¯¯gT¯¯gLς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gT¯¯gUς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−I¯¯gL−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−I¯¯gU−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−F¯¯gL−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−F¯¯gU−⊕g=1spw¯¯g+11ς.(2)
ζ=1 and ς=0, then, the GHINLSSPGA operator reduces into HINL PGA operator based on algebraic operation. That is(70)GHINLSSPGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpw¯¯g∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpw¯¯g,⊗g=1sT¯¯gLpw¯¯g,⊗g=1sT¯¯gUpw¯¯g,1−⊗g=1s1−I¯¯gLpw¯¯g,1−⊗g=1s1−I¯¯gUpw¯¯g,1−⊗g=1s1−F¯¯gLpw¯¯g,1−⊗g=1s1−F¯¯gUpw¯¯g.(3)
Ifζ=1, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the GHINLSSPGA operator reduces into HINL GA operator based on algebraic operation. That is(71)GHIINLSSPGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯g1s∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯g1s,⊗g=1sT¯¯gL1s,⊗g=1sT¯¯gU1s,1−⊗g=1s1−I¯¯gL1s,1−⊗g=1s1−I¯¯gU1s,1−⊗g=1s1−F¯¯gL1s,1−⊗g=1s1−F¯¯gU1s.The GIFSSPGA operator has the property of commutativity.Definition 24.
For a group of HINLNshn¯¯gg=1,2,⋯,s, WGHINLSSPGA operator is a function ℵs⟶ℵ,(72)WGHINLSSPGAwt¯¯,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗g=1sζhn¯¯gwt¯¯g1+Thn¯¯g⊕h=1swt¯¯h1+Thn¯¯h,where Tif¯¯g=⊕h=1,g≠hsSuphn¯¯g,hn¯¯h is the weight vector for hn¯¯gg=1,2,⋯,s such that Wt¯¯g∈0,1 and ⊕g=1sWt¯¯g=1, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the support for hn¯¯g from hn¯¯h with the following constraint:
(1)
Suphn¯¯g,hn¯¯h≥Suphn¯¯u,hn¯¯v if dis¯¯hn¯¯g,hn¯¯h<dis¯¯hn¯¯u,hn¯¯v, where dis¯¯ is the distance measure among two HINLNs(73)Suphn¯¯g,hn¯¯h∈0,1;Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;To write Equation (72) in unsophisticated way, we have
(74)pow¯¯g=Wt¯¯g1+Thn¯¯g⊕g=1sWt¯¯g1+Thn¯¯g,So, from Equation (74), Equation (72) becomes
(75)WGHINLSSPGAwt¯¯,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗g=1sζhn¯¯gpow¯¯g.Theorem 25.
Lethn¯¯gg=1,2,⋯,s be a group of HINLNs, then, the value aggregated utilizing Definition 24 is still HINLN, and we have
(76)WGHINLSSPGAζ,Wt¯¯hn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,wt¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpow¯¯g,1−⊕g=1spow¯¯g1ζ1−1−T¯¯gLς−1ζ−11ς,1−⊕g=1spow¯¯g1ζ1−1−T¯¯gUς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−I¯¯gLς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−I¯¯gUς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−F¯¯gLς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−F¯¯gUς−1ζ−11ς.Further, we shall discuss a few cases of the initiated AO with respect to the parameterζ and Wt¯¯, which are listed below:
(1)
Ifζ=1, Wt¯¯=1/s,1/s,⋯,1/sT, then, the WGHINLSSPGA operator trims down to WHINLSS PA operator:(77)WGHINLSSPGAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpow¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpow¯¯g,⊕g=1spow¯¯gT¯¯gLς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gT¯¯gUς−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−I¯¯gL−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−I¯¯gU−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−F¯¯gL−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−F¯¯gU−⊕g=1spow¯¯g+11ς.(2)
Ifζ=1, Wt¯¯=1/s,1/s,⋯,1/sT, and ς=0, then, the WGHINLSSPGA operator reduces into HINL PGA operator based on algebraic operation. That is(78)WGHINLSSPGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpow¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpow¯¯g,⊗g=1sT¯¯gLpow¯¯g,⊗g=1sT¯¯gUpow¯¯g,1−⊗g=1s1−I¯¯gLpow¯¯g,1−⊗g=1s1−I¯¯gUpow¯¯g,1−⊗g=1s1−F¯¯gLpow¯¯g,1−⊗g=1s1−F¯¯gUpow¯¯g.(3)
Ifζ=1, Wt¯¯=1/s,1/s,⋯,1/sT, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the WGHINLSSPGA operator trims down to HINL GA operator based on algebraic operation. That is(79)WGHINLPSSGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯g1s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯g1s,⊗g=1sT¯¯gL1s,⊗g=1sT¯¯gU1s,1−⊗g=1s1−I¯¯gL1s,1−⊗g=1s1−I¯¯gU1s,1−⊗g=1s1−F¯¯gL1s,1−⊗g=1s1−F¯¯gU1s.


## 4.1. Weighted Generalized Hesitant Interval Neutrosophic Linguistic Schweizer-Sklar Power Aggregation Operator

In this subpart, we initiate generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power average (GHINLSSPA) operator, weighted (WGHINLSSPA) operator and examine their enviable properties and various particular cases.Definition 16.
For a collection of HINLNshn¯¯gg=1,2,⋯,s, GHINLSSPA operator is a function ℵs⟶ℵ,(29)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1s1+Thn¯¯ghn¯¯gζ⊕g=1s1+Thn¯¯g1/ζ,where Thn¯¯g=⊕h=1,g≠hssuphn¯¯g,hn¯¯h, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the SPD for hn¯¯g from hn¯¯h with the following constraint:
(1)
Suphn¯¯g,hn¯¯h∈0,1;(2)
Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;(3)
Suphn¯¯g,if¯¯h≥Suphn¯¯u,hn¯¯v if dis¯¯hn¯¯g,hn¯¯h<ds¯¯hn¯¯u,hn¯¯v, where dis¯¯ is the distance measure among two HINLNs.To write Equation (29) in unsophisticated form, we have
(30)pw¯¯g=1+Thn¯¯g⊕~g=1s1+Thn¯¯g.So, from Equation (30), Equation (29) becomes
(31)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1spw¯¯ghn¯¯gζ1/ζ,Theorem 17.
Lethn¯¯gg=1,2,⋯,s be a set of HINLNs, then the value aggregated utilizing Definition 16 is still HINLN, and we have
(32)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,..⋯,we¯¯s∈hn¯¯sf∗−1⊕g=1spw¯¯gf∗sϑwe¯¯gζ1/ζ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11/ζς−⊕g=1spw¯¯g+11/ζς−1ζ−11/ζ.Proof.
In the following, first, we prove(33)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯shn¯¯sζ=∪u1∈hn¯¯1,u2∈hn¯¯2,..⋯,us∈hn¯¯sf∗−1⊕g=1spw¯¯gf∗sϑugζ,1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11/ςς−⊕g=1spw¯¯g+11/ς,by exploiting mathematical induction on s.Fors=2.From the operational laws explained for HINLNs in Definition14, we have
(34)hn¯¯1ζ=∪u1∈hn¯¯1f∗−1f∗sϑu1ζ,ζT¯¯1Lς−ζ−11/ς,ζT¯¯1Uς−ζ−11/ς,1−ζ1−I¯¯1Lς−ζ−11/ς,1−ζ1−I¯¯1Uς−ζ−11/ς,1−ζ1−F¯¯1Lς−ζ−11/ς,1−ζ1−F¯¯1Uς−ζ−11/ς,and
(35)pw¯¯1hn¯¯1ζ=∪we¯¯1∈hn¯¯1f∗−1Ω¯¯1f∗sϑwe¯¯1ζ,1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ς,1−pw¯¯11−ζT¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−I¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−I¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−F¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ς,Similarly,(36)pw¯¯2hn¯¯2ζ=∪we¯¯2∈hn¯¯2f∗−1pw¯¯2f∗sϑwe¯¯2ζ,1−pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ς,1−pw¯¯21−ζT¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−I¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−I¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−F¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ς,pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ς.Then,(37)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1pw¯¯1f∗sϑwe¯¯1ζ⊕pw¯¯1f∗sϑwe¯¯1ζ,1−1−1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+1−1−pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,1−1−1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+1−1−pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−I¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−I¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−F¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−F¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς;=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊕g=12pw¯¯gf∗sϑwe¯¯gζ,1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,1−pw¯¯11−ζT¯¯1Lς−ζ−11/ςς−pw¯¯1−11/ςς+pw¯¯21−ζT¯¯2Lς−ζ−11/ςς−pw¯¯2−11/ςς−11/ς,pw¯¯11−ζ1−I¯¯1Lς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−I¯¯2Lς−ζ−11/ςς−pw¯¯2−1−11/ς,pw¯¯11−ζ1−I¯¯1Uς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−I¯¯2Uς−ζ−11/ςς−pw¯¯2−1−11/ς,pw¯¯11−ζ1−F¯¯1Lς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−F¯¯2Lς−ζ−11/ςς−pw¯¯2−1−11/ς,pw¯¯11−ζ1−F¯¯1Uς−ζ−11/ςς−pw¯¯1−1+pw¯¯21−ζ1−F¯¯2Uς−ζ−11/ςς−pw¯¯2−1−11/ς;pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊕g=12pw¯¯gf∗sϑwe¯¯gζ,1−⊕g=12pw¯¯g1−ζT¯¯gLς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,1−⊕g=12pw¯¯g1−ζT¯¯gUς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,⊕g=12pw¯¯g1−ζ1−I¯¯gLς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,⊕g=12pw¯¯g1−ζ1−I¯¯gUς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς⊕g=12pw¯¯g1−ζ1−F¯¯gLς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς,⊕g=12pw¯¯g1−ζ1−F¯¯gUς−ζ−11/ςς−⊕g=12pw¯¯g+11/ς.If Equation (33) holds for s=m(38)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯mhn¯¯mζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊕g=1mpw¯¯gf∗sϑwe¯¯gζ,1−⊕g=1mpw¯¯g1−ζT¯¯gLς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,1−⊕g=1mpw¯¯g1−ζT¯¯gUς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gLς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gUς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gLς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gUς−ζ−11/ςς−⊕g=1mpw¯¯g+11/ς.Then, whens=m+1, by the operational laws explained in Definition 14, we have
(39)pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯m+1hn¯¯m+1ζ=pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯mhn¯¯mζ⊕pw¯¯m+1hn¯¯m+1ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊕g=1mpw¯¯gf∗sϑwe¯¯gζ,1−⊕g=1mpw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς⊕∪we¯¯m+1∈hn¯¯m+1f∗sϑwe¯¯m+1,1−pw¯¯m+11−ζT¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζT¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−I¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−I¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−F¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−F¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯1hn¯¯1ζ⊕pw¯¯2hn¯¯2ζ⊕⋯⊕pw¯¯m+1hn¯¯m+1ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯m+1∈hn¯¯m+1f∗−1⊕g=1m+1pw¯¯gf∗sϑwe¯¯gζ,1−⊕g=1m+1pw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς.That is, Equation (33) is true for g=m+1. So Equation (33) is true for all g. Then,
(40)⊕g=1spw¯¯ghn¯¯gζ1ζ=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯m+1∈hn¯¯m+1f∗−1⊕g=1spw¯¯gf∗sϑwe¯¯gζ1ξ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς.Therefore,(41)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯m+1∈hn¯¯m+1f∗−1⊕~g=1spw¯¯gf∗sϑwe¯¯gζ1ξ,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς.Which completes the proof of the Theorem17.Theorem 18.
Commutativity: lethn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗ be any permutation of hn¯¯1,hn¯¯2,⋯,hn¯¯s, then
(42)GHINLSSPAζhn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗=GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s.Proof.
Suppose thathn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗ is any permutation of hn¯¯1,hn¯¯2,⋯,hn¯¯s, then for each hn¯¯g, there exists one and only one hn¯¯h∗ such that hn¯¯g=hn¯¯h∗ and vice versa. And Thn¯¯g=Thn¯¯h∗.. Then based on Theorem 17, we have
(43)GHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spw¯¯ghn¯¯gζ1/ζ=⊕h=1spw¯¯h∗hn¯¯h∗ζ1/ζ=GHINLSSPAζhn¯¯1∗,hn¯¯2∗,⋯,hn¯¯s∗.
Note that the GHINLSSPA operators are neither idempotent nor monotonic.Further, we shall discuss a few cases of the initiated AOs with respect to the parameterζ, which are listed below:
(1)
Ifζ=1, then, the GHINLSSPA operator trim down to HINLSS power average (PA) operator:(44)GHINLSSPAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spw¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,us∈hn¯¯sf∗−1⊕g=1mpw¯¯gf∗sϑug,1−⊕g=1spw¯¯g1−T¯¯gL−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−T¯¯gL−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gI¯¯gLς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gI¯¯gUς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gF¯¯gLς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gF¯¯gUς−⊕g=1spw¯¯g+11ς.(2)
Ifζ=1 and ς=0, then, the GHINLSSPA operator trim down to HINL PA operator based on algebraic operation. That is(45)GHINLSSPAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spw¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,us∈hn¯¯sf∗−1⊕g=1m+1pw¯¯gf∗sϑug,1−⊗g=1s1−T¯¯gLpw¯¯g,1−⊗g=1s1−T¯¯gUpw¯¯g,⊗g=1sI¯¯gLpw¯¯g,⊗g=1sI¯¯gUpw¯¯g,⊗g=1sF¯¯gLpw¯¯g,⊗g=1sF¯¯gUpw¯¯g.(3)
Ifζ=1, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the GHINLSSPA operator trim down to HINL average operator based on algebraic operation. That is(46)WSSHINLPAς⟶0hn¯¯1,hn¯¯2,⋯,hn¯¯s=1s⊕g=1shn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,us∈hn¯¯sf∗−1⊕g=1m1sf∗sϑug1−⊗g=1s1−T¯¯gL1s,1−⊗g=1s1−T¯¯gU1s,⊗g=1sI¯¯gL1s,⊗g=1sI¯¯gU1s,⊗g=1sF¯¯gL1s,⊗g=1sF¯¯gU1s.Definition 19.
For a group of HINLNshn¯¯gg=1,2,⋯,s, a WGHINLSSPA operator is a function ℵs⟶ℵ,(47)WGHINLSSPAϕ,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1sWe¯¯g1+Thn¯¯ghn¯¯gζ⊕g=1sWe¯¯g1+Thn¯¯g1/ζ,where Thn¯¯g=⊕h=1,g≠hsSuphn¯¯g,hn¯¯h,We¯¯=We¯¯1,We¯¯2,⋯,We¯¯sT is the weight vector for hn¯¯gg=1,2,⋯,s such that We¯¯g∈0,1 and ⊕g=1sWe¯¯g=1, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the support for hn¯¯g from hn¯¯h with the following constraint:
(48)Suphn¯¯g,hn¯¯h∈0,1;Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;To write Equation (47) in uncomplicated way, we have
(49)pow¯¯g=Weg1+Thn¯¯g⊕g=1sWeg1+Thn¯¯g,so, from Equation (49), Equation (47) becomes
(50)WGHILNSSPAϕ,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕SSg=1spow¯¯ghn¯¯gζ1/ζ,Theorem 20.
Lethn¯¯gg=1,2,⋯,s be a group of HINLNs, then the value aggregated utilizing Definition 19 is still HINLN, and we have
(51)WGHINLSSPAζ,ϕhn¯¯1,hn¯¯2,⋯,hn¯¯s∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑugζ1ζ,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−FgLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−FgUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς.Proof.
Proof of the Theorem20 is same as Theorem 17.Theorem 21.
For a group of HINLNshn¯¯gg=1,2,⋯,s, ζ is a parameter and ζ>0. We¯¯=We¯¯1,We¯¯2,⋯,We¯¯sT is weighting vector for hn¯¯gg=1,2,⋯,s, We¯¯g∈0,1, and ⊕g=1sWe¯¯g=1. If We¯¯=1/s,1/s,⋯,1/sT, then WGHINLSSPA trims down to generalized HINLSS PA operator:
(52)WGHINLSSPAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spow¯¯ghn¯¯gζ1ζ=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑugζ1ζ,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spow¯¯g1−ζT¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−1,1−1ζ1−⊕g=1spow¯¯g1−ζ1−I¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−I¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gLς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spow¯¯g1−ζ1−F¯¯gUς−ζ−11ςς−⊕g=1spow¯¯g+11ςς−1ζ−11ς,wherepow¯¯g=1+Thn¯¯g/⊕g=1s1+Thn¯¯g is the power weight vector. Further, we shall discuss a few cases of the initiated AOs with respect to the parameter ζ and We¯¯, which are listed below:
(1)
Ifζ=1, We¯¯=1/s,1/s,⋯,1/sT, then, the WGHINLSSPA operator reduces into HINLSS power average (PA) operator:(53)HINLSSPAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spow¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑug,1−⊕g=1spow¯¯g1−T¯¯gLς−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−T¯¯gUς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gI¯¯gLς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gI¯¯gUς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gF¯¯gLς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gFgUς−⊕g=1spow¯¯g+11ς.(2)
Ifζ=1, We¯¯=1/s,1/s,⋯,1/sT, and ς=0, then, the WGHINLSSPA operator trims down to HINL PA operator based on algebraic operation. That is(54)WGHINLSSPAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊕g=1spow¯¯ghn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕g=1spow¯¯gf∗sϑug,=1−⊗g=1s1−T¯¯gLpow¯¯g,1−⊗g=1s1−T¯¯gUpow¯¯g,⊗g=1sI¯¯gLpow¯¯g,⊗g=1sI¯¯gUpow¯¯g,⊗g=1sF¯¯gLpow¯¯g,⊗g=1sF¯¯gUpow¯¯g.(3)
Ifζ=1, We¯¯=1/s,1/s,⋯,1/sT, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the WGHINLSSPA operator reduces into HINL average operator based on algebraic operation. That is(55)WGHINLSSPAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=1s⊕g=1shn¯¯g=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−11s⊕g=1sf∗sϑug,1−⊗g=1s1−T¯¯gL1s,1−⊗g=1s1−T¯¯gU1s,⊗g=1sI¯¯gL1s,⊗g=1sI¯¯gU1s,⊗g=1sF¯¯gL1s,⊗g=1sF¯¯gU1s.


## 4.2. Some Generalized Hesitant Interval Neutrosophic Schweizer-Sklar Power Geometric Aggregation Operators

In this subpart, we initiate generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power geometric average (GHINLSSPWGA) operator, weighted GHINLSSPGA, discuss their desirable properties and some particular cases.Definition 22.
For a group of HINLNshn¯¯gg=1,2,⋯,s, GHINLSSPGA operator is a function ℵs⟶ℵ,(56)GHINLSSPGAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗~g=1sζhn¯¯g1+Thn¯¯g⊕h=1s1+Thn¯¯h,where Thn¯¯g=⊕h=1,g≠hsSuphn¯¯g,hn¯¯h, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the SPD for hn¯¯g from hn¯¯h with the following constraint:
(1)
Suphn¯¯g,hn¯¯h≥Suphn¯¯u,hn¯¯vif dis¯¯hn¯¯g,hn¯¯h<dis¯¯hn¯¯u,hn¯¯v, where dis¯¯ is the distance measure among two HINLNs(57)Suphn¯¯g,hn¯¯h∈0,1;Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;To write Equation (56) in an uncomplicated way, we have
(58)pw¯¯g=1+Thn¯¯g⊕g=1s1+Thn¯¯g.So, from Equation (58), Equation (56) becomes
(59)GHINLSSPGAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗g=1sζhn¯¯gpw¯¯g.Theorem 23.
Lethn¯¯gg=1,2,⋯,s be a group of HINLNs, then, the value aggregated utilizing Definition 22 is still HINLN, and we have
(60)GHINLSSPGζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpw¯¯g1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς.Proof.
In the following, first, we prove(61)ζhn¯¯1pw¯¯1×ζhn¯¯2pw¯¯2×⋯×ζhn¯¯spw¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sζf∗sϑwe¯¯gpw¯¯g⋅⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζIgUς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ς.
To prove Equation (61), we utilize mathematical induction on s.
Fors=2.
From the operational rules explained for HINLNs in Definition14, we have
(62)ζhn¯¯1=∪we¯¯1∈hn¯¯1f∗−1ζf∗sϑwe¯¯g,1−ζ1−T¯¯1Lς−ζ−11ς,1−ζ1−T¯¯1Uς−ζ−11ς,ζI¯¯1Lς−ζ−11ς,ζI¯¯1Lς−ζ−11ς,ζF¯¯1Lς−ζ−11ς,ζF¯¯1Uς−ζ−11ςζhn¯¯1pw¯¯1=∪we¯¯1∈hn¯¯1f∗−1ζf∗sϑwe¯¯2pw¯¯1,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−11ς,pw¯¯11−ζ1−T¯¯1Uς−ζ−11ςς−pw¯¯1−11ς,1−pw¯¯11−ζI¯¯1Lς−ζ−11ςς−pw¯¯1−11ς,pw¯¯11−ζI¯¯1Uς−ζ−11ςς−pw¯¯1−11ς,1−pw¯¯11−ζF¯¯1Lς−ζ−11ςς−pw¯¯1−11ς,pw¯¯11−ζF¯¯1Uς−ζ−11ςς−pw¯¯1−11ς;
Similarly,(63)ζhn¯¯2pw¯¯2=∪we¯¯2∈hn¯¯2f∗−1ζf∗sϑwe¯¯2pw¯¯2,pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−11ς,pw¯¯21−ζ1−T¯¯2Uς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζI¯¯2Lς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζI¯¯2Uς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζF¯¯2Lς−ζ−11ςς−pw¯¯2−11ς,1−pw¯¯21−ζF¯¯2Uς−ζ−11ςς−pw¯¯2−11ς.
Then,(64)ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1ζf∗sϑwe¯¯1pw¯¯1ζf∗sϑwe¯¯2pw¯¯2,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζI¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζI¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζI¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζI¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζF¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζF¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−1−1−pw¯¯11−ζF¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+1−1−pw¯¯21−ζF¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς;=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊗g=12ζf∗sϑwe¯¯gpw¯¯g,pw¯¯11−ζ1−T¯¯1Lς−ζ−11ςς−pw¯¯1−1+pw¯¯21−ζ1−T¯¯2Lς−ζ−11ςς−pw¯¯2−1−11ς,pw¯¯11−ζ1−T¯¯1Uς−ζ−11ςς−pw¯¯1−1+pw¯¯21−ζ1−T¯¯2Uς−ζ−11ςς−pw¯¯2−1−11ς,1−pw¯¯11−ζI¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζI¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−pw¯¯11−ζI¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζI¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−pw¯¯11−ζF¯¯1Lς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζF¯¯2Lς−ζ−11ςς−pw¯¯2−11ςς−11ς,1−pw¯¯11−ζF¯¯1Uς−ζ−11ςς−pw¯¯1−11ςς+pw¯¯21−ζF¯¯2Uς−ζ−11ςς−pw¯¯2−11ςς−11ς;ζhn¯¯1pw¯¯1×ζhn¯¯2pw¯¯2=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2f∗−1⊗g=12ζf∗sϑwe¯¯gpw¯¯g,=⊕g=12pw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=12pw¯¯g+11ς,⊕g=12pw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=12pw¯¯g+11ς,1−⊕g=12pw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=12pw¯¯g+11ς;
If Equation (50) holds for s=m(65)ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯mpw¯¯m=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊗g=1mζf∗sϑwe¯¯gpw¯¯g⊕g=1mpw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+1,1−⊕g=1mpw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+1.
Then, whens=m+1, by the operational rules given in Definition 14, we have
(66)ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯m+1pw¯¯m+1=ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯mpw¯¯m⊗ζhn¯¯m+1pw¯¯m+1=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯mf∗−1⊗g=1mζf∗sϑwe¯¯gpw¯¯g⊕g=1mpw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,⊕g=1mpw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζI¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζI¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζF¯¯gLς−ζ−11ςς−⊕g=1mpw¯¯g+11ς,1−⊕g=1mpw¯¯g1−ζF¯¯gUς−ζ−11ςς−⊕g=1mpw¯¯g+11ς⊕∪we¯¯m+1∈hn¯¯m+1f∗−1ζf∗sϑwe¯¯m+1pw¯¯m+1,pw¯¯m+11−ζ1−T¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,pw¯¯m+11−ζ1−T¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζI¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζI¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζF¯¯m+1Lς−ζ−11ςς−pw¯¯m+1−11ς,1−pw¯¯m+11−ζF¯¯m+1Uς−ζ−11ςς−pw¯¯m+1−11ς,ζhn¯¯1pw¯¯1⊗ζhn¯¯2pw¯¯2⊗⋯⊗ζhn¯¯m+1pw¯¯m+1=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯m+1f∗−1⊗g=1m+1ζf∗sϑwe¯¯gpw¯¯g,⊕g=1m+1pw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,⊕g=1m+1pw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζI¯¯gLgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζI¯¯gUgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζF¯¯gLgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς,1−⊕g=1m+1pw¯¯g1−ζF¯¯gUgς−ζ−11ςς−⊕g=1m+1pw¯¯g+11ς.
That is, Equation (61) is true for g=m+1. So Equation (61) is true for all g. Then,
(67)1ζ⊗g=1sζhn¯¯gpw¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpw¯¯g1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,
Therefore,(68)GHINLSSPGAζhn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpw¯¯g1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gLς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1−1ζ1−⊕g=1spw¯¯g1−ζ1−T¯¯gUς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζI¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gLgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,1ζ1−⊕g=1spw¯¯g1−ζF¯¯gUgς−ζ−11ςς−⊕g=1spw¯¯g+11ςς−1ζ−11ς,
This completes the proof of Theorem23.Further, we shall examine a few cases of the initiated AOs with respect to the parameterζ and pw¯¯, which are listed below:
(1)
Ifζ=1, then, the GHINLSSPGA operator trims down to HINLSS PGA operator:(69)GHINLSSPGAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpw¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpw¯¯g⊕g=1spw¯¯gT¯¯gLς−⊕g=1spw¯¯g+11ς,⊕g=1spw¯¯gT¯¯gUς−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−I¯¯gL−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−I¯¯gU−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−F¯¯gL−⊕g=1spw¯¯g+11ς,1−⊕g=1spw¯¯g1−F¯¯gU−⊕g=1spw¯¯g+11ς.(2)
ζ=1 and ς=0, then, the GHINLSSPGA operator reduces into HINL PGA operator based on algebraic operation. That is(70)GHINLSSPGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpw¯¯g∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpw¯¯g,⊗g=1sT¯¯gLpw¯¯g,⊗g=1sT¯¯gUpw¯¯g,1−⊗g=1s1−I¯¯gLpw¯¯g,1−⊗g=1s1−I¯¯gUpw¯¯g,1−⊗g=1s1−F¯¯gLpw¯¯g,1−⊗g=1s1−F¯¯gUpw¯¯g.(3)
Ifζ=1, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the GHINLSSPGA operator reduces into HINL GA operator based on algebraic operation. That is(71)GHIINLSSPGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯g1s∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯g1s,⊗g=1sT¯¯gL1s,⊗g=1sT¯¯gU1s,1−⊗g=1s1−I¯¯gL1s,1−⊗g=1s1−I¯¯gU1s,1−⊗g=1s1−F¯¯gL1s,1−⊗g=1s1−F¯¯gU1s.The GIFSSPGA operator has the property of commutativity.Definition 24.
For a group of HINLNshn¯¯gg=1,2,⋯,s, WGHINLSSPGA operator is a function ℵs⟶ℵ,(72)WGHINLSSPGAwt¯¯,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗g=1sζhn¯¯gwt¯¯g1+Thn¯¯g⊕h=1swt¯¯h1+Thn¯¯h,where Tif¯¯g=⊕h=1,g≠hsSuphn¯¯g,hn¯¯h is the weight vector for hn¯¯gg=1,2,⋯,s such that Wt¯¯g∈0,1 and ⊕g=1sWt¯¯g=1, parameter ζ∈0,+∞ and Suphn¯¯g,hn¯¯h is the support for hn¯¯g from hn¯¯h with the following constraint:
(1)
Suphn¯¯g,hn¯¯h≥Suphn¯¯u,hn¯¯v if dis¯¯hn¯¯g,hn¯¯h<dis¯¯hn¯¯u,hn¯¯v, where dis¯¯ is the distance measure among two HINLNs(73)Suphn¯¯g,hn¯¯h∈0,1;Suphn¯¯g,hn¯¯h=Suphn¯¯h,hn¯¯g;To write Equation (72) in unsophisticated way, we have
(74)pow¯¯g=Wt¯¯g1+Thn¯¯g⊕g=1sWt¯¯g1+Thn¯¯g,So, from Equation (74), Equation (72) becomes
(75)WGHINLSSPGAwt¯¯,ζhn¯¯1,hn¯¯2,⋯,hn¯¯s=1ζ⊗g=1sζhn¯¯gpow¯¯g.Theorem 25.
Lethn¯¯gg=1,2,⋯,s be a group of HINLNs, then, the value aggregated utilizing Definition 24 is still HINLN, and we have
(76)WGHINLSSPGAζ,Wt¯¯hn¯¯1,hn¯¯2,⋯,hn¯¯s=∪we¯¯1∈hn¯¯1,wt¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−11ζ⊗g=1sζf∗sϑwe¯¯gpow¯¯g,1−⊕g=1spow¯¯g1ζ1−1−T¯¯gLς−1ζ−11ς,1−⊕g=1spow¯¯g1ζ1−1−T¯¯gUς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−I¯¯gLς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−I¯¯gUς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−F¯¯gLς−1ζ−11ς,⊕g=1spow¯¯g1ζ1−F¯¯gUς−1ζ−11ς.Further, we shall discuss a few cases of the initiated AO with respect to the parameterζ and Wt¯¯, which are listed below:
(1)
Ifζ=1, Wt¯¯=1/s,1/s,⋯,1/sT, then, the WGHINLSSPGA operator trims down to WHINLSS PA operator:(77)WGHINLSSPGAhn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpow¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpow¯¯g,⊕g=1spow¯¯gT¯¯gLς−⊕g=1spow¯¯g+11ς,⊕g=1spow¯¯gT¯¯gUς−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−I¯¯gL−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−I¯¯gU−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−F¯¯gL−⊕g=1spow¯¯g+11ς,1−⊕g=1spow¯¯g1−F¯¯gU−⊕g=1spow¯¯g+11ς.(2)
Ifζ=1, Wt¯¯=1/s,1/s,⋯,1/sT, and ς=0, then, the WGHINLSSPGA operator reduces into HINL PGA operator based on algebraic operation. That is(78)WGHINLSSPGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯gpow¯¯g=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯gpow¯¯g,⊗g=1sT¯¯gLpow¯¯g,⊗g=1sT¯¯gUpow¯¯g,1−⊗g=1s1−I¯¯gLpow¯¯g,1−⊗g=1s1−I¯¯gUpow¯¯g,1−⊗g=1s1−F¯¯gLpow¯¯g,1−⊗g=1s1−F¯¯gUpow¯¯g.(3)
Ifζ=1, Wt¯¯=1/s,1/s,⋯,1/sT, suphn¯¯g,hn¯¯h=b (i.e., a constant) and ς=0, then, the WGHINLSSPGA operator trims down to HINL GA operator based on algebraic operation. That is(79)WGHINLPSSGAς=0hn¯¯1,hn¯¯2,⋯,hn¯¯s=⊗g=1shn¯¯g1s=∪we¯¯1∈hn¯¯1,we¯¯2∈hn¯¯2,⋯,we¯¯s∈hn¯¯sf∗−1⊗g=1sf∗sϑwe¯¯g1s,⊗g=1sT¯¯gL1s,⊗g=1sT¯¯gU1s,1−⊗g=1s1−I¯¯gL1s,1−⊗g=1s1−I¯¯gU1s,1−⊗g=1s1−F¯¯gL1s,1−⊗g=1s1−F¯¯gU1s.


## 5. An Application of Generalized Hesitant Interval Neutrosophic Linguistic Schweizer-Sklar Power Aggregation Operator to Group Decision Making

In this part, we pertain the aforementioned generalized hesitant interval neutrosophic linguistic Schweizer-Sklar power AOs to ascertain productive approaches for MADM under HINL environments. LetArb¯¯=Arb¯¯1,Arb¯¯2,⋯,Arb¯¯g be the group of detached alternatives, the group of attributes is articulated by Cta¯¯=Cta¯¯1,Cta¯¯2,⋯,Cta¯¯h, and the weight vector of the attributes is symbolized by WE¯¯=we¯¯1,we¯¯2,⋯,we¯¯hT such that w¯¯e∈0,1,∑e=1hw¯¯e=1. In the process of decision making, the assessment information about the alternative Arb¯¯uu=1,2,⋯,g with respect to the attribute Cta¯¯ww=1,2,⋯,h is expressed by a HINL decision matrix DT¯¯=hn¯¯deg×h, where hn¯¯de=∪ude∈hn¯¯deude,ude=sϑudeTLude,TUude,ILude,IUude,FLude,FUude is a HINLN.Then, gamble on factual decision situations where the weight vector of attributes is entirely identified in advance. For that reason, we initiate MADM approaches established on the proposed GHINLSSPA operators.


### 5.1. MADM with Known Weight Vectors of Attributes

In this subsection, to deal with real decision situations in which the weighting vectors of attributes is totally known, we apply WGHINLSSPA operator and WGHINLSSPGA operator to establish the following approach to solve MADM problems under HINL environments. To do so, immediately go behind the steps below.Step 1.
Find out supportSuphn¯¯de,hn¯¯dx by the following formula;
(80)Suphn¯¯de,hn¯¯ex=1−dis¯¯hn¯¯de,hn¯¯ex,d=1,2,⋯,g,e=1,2,⋯,h,e≠x,where dis¯¯hn¯¯de,hn¯¯ex is the distance measure and is calculated by utilizing Equation (19).Step 2.
Find out the support degreeThn¯¯de that HINLN hn¯¯de collects from other HINLNs hn¯¯dxx=1,2,⋯,h;e≠x, where
(81)Thn¯¯de=∑x=1,x≠ehsuphn¯¯de,hn¯¯dx.Step 3.
Determine weighting vectorϒded=1,2,.⋯,g,e=1,2,⋯,h associated with hn¯¯de,
(82)ϒde=w¯¯e1+Thn¯¯de∑e=1hw¯¯e1+Thn¯¯de.Step 4.
Utilize WGHINLSSPA or WHINLSSPGA operators to collective all evaluation valueshn¯¯ded=1,2,.⋯,g,e=1,2,⋯,h into overall evaluation value hn¯¯dd=1,2,.⋯,g matching to the alternatives Arb¯¯dd=1,2,⋯,g;(83)hn¯¯d=WGHINLSSPAζ,we¯¯hn¯¯d1,hn¯¯d2,⋯,hn¯¯ds=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕e=1sΨ¯¯gf∗sϑudeζ1ζ,1ζ1−⊕e=1sΨ¯¯de1−ζTdeLς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς,1ζ1−⊕e=1sΨ¯¯g1−ζTdeUς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−1,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−IdeLς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−IdeUς−ζ−11ςς−⊕g=1sΨ¯¯de+11ςς−1ζ−11ς,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−FdeLς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−FdeUς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς.
Or(84)hn¯¯d=WGHINLSSPGAζ,We¯¯hn¯¯d1,hn¯¯d2,⋯,hn¯¯ds=∪ud1∈hn¯¯d1,ud2∈hn¯¯d2,⋯,uds∈hn¯¯dsf∗−11ζ⊗e=1sζf∗sϑudeΞde,1−⊕e=1sΞde1ζ1−1−TdeLς−1ζ−11ς,1−⊕e=1sΞde1ζ1−1−TdeUς−1ζ−11ς,⊕e=1sΞde1ζ1−IdeLς−1ζ−11ς,⊕e=1sΞde1ζ1−IdeUς−1ζ−11ς,⊕e=1sΞde1ζ1−FdeLς−1ζ−11ς,⊕e=1sΞde1ζ1−FdeUς−1ζ−11ς.Step 5.
Find out the scoresScr¯¯hn¯¯d for the overall HINLN of the alternatives Arb¯¯dd=1,2,⋯,g by exploiting Definition 8.Step 6.
Rank all alternativesArb¯¯dd=1,2,⋯,g and select the best one (s) with the ranking orderhn¯¯dd=1,2,⋯,g.


## 5.1. MADM with Known Weight Vectors of Attributes

In this subsection, to deal with real decision situations in which the weighting vectors of attributes is totally known, we apply WGHINLSSPA operator and WGHINLSSPGA operator to establish the following approach to solve MADM problems under HINL environments. To do so, immediately go behind the steps below.Step 1.
Find out supportSuphn¯¯de,hn¯¯dx by the following formula;
(80)Suphn¯¯de,hn¯¯ex=1−dis¯¯hn¯¯de,hn¯¯ex,d=1,2,⋯,g,e=1,2,⋯,h,e≠x,where dis¯¯hn¯¯de,hn¯¯ex is the distance measure and is calculated by utilizing Equation (19).Step 2.
Find out the support degreeThn¯¯de that HINLN hn¯¯de collects from other HINLNs hn¯¯dxx=1,2,⋯,h;e≠x, where
(81)Thn¯¯de=∑x=1,x≠ehsuphn¯¯de,hn¯¯dx.Step 3.
Determine weighting vectorϒded=1,2,.⋯,g,e=1,2,⋯,h associated with hn¯¯de,
(82)ϒde=w¯¯e1+Thn¯¯de∑e=1hw¯¯e1+Thn¯¯de.Step 4.
Utilize WGHINLSSPA or WHINLSSPGA operators to collective all evaluation valueshn¯¯ded=1,2,.⋯,g,e=1,2,⋯,h into overall evaluation value hn¯¯dd=1,2,.⋯,g matching to the alternatives Arb¯¯dd=1,2,⋯,g;(83)hn¯¯d=WGHINLSSPAζ,we¯¯hn¯¯d1,hn¯¯d2,⋯,hn¯¯ds=∪u1∈hn¯¯1,u2∈hn¯¯2,⋯,um+1∈hn¯¯sf∗−1⊕e=1sΨ¯¯gf∗sϑudeζ1ζ,1ζ1−⊕e=1sΨ¯¯de1−ζTdeLς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς,1ζ1−⊕e=1sΨ¯¯g1−ζTdeUς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−1,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−IdeLς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−IdeUς−ζ−11ςς−⊕g=1sΨ¯¯de+11ςς−1ζ−11ς,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−FdeLς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς,1−1ζ1−⊕e=1sΨ¯¯de1−ζ1−FdeUς−ζ−11ςς−⊕e=1sΨ¯¯de+11ςς−1ζ−11ς.
Or(84)hn¯¯d=WGHINLSSPGAζ,We¯¯hn¯¯d1,hn¯¯d2,⋯,hn¯¯ds=∪ud1∈hn¯¯d1,ud2∈hn¯¯d2,⋯,uds∈hn¯¯dsf∗−11ζ⊗e=1sζf∗sϑudeΞde,1−⊕e=1sΞde1ζ1−1−TdeLς−1ζ−11ς,1−⊕e=1sΞde1ζ1−1−TdeUς−1ζ−11ς,⊕e=1sΞde1ζ1−IdeLς−1ζ−11ς,⊕e=1sΞde1ζ1−IdeUς−1ζ−11ς,⊕e=1sΞde1ζ1−FdeLς−1ζ−11ς,⊕e=1sΞde1ζ1−FdeUς−1ζ−11ς.Step 5.
Find out the scoresScr¯¯hn¯¯d for the overall HINLN of the alternatives Arb¯¯dd=1,2,⋯,g by exploiting Definition 8.Step 6.
Rank all alternativesArb¯¯dd=1,2,⋯,g and select the best one (s) with the ranking orderhn¯¯dd=1,2,⋯,g.


## 6. Illustrative Example

In this section, an example of alternative selection taken from Ye [32] is utilized to demonstrate the usefulness of the anticipated decision-making process under a hesitant interval neutrosophic linguistic environment. An investment firm would like to put money into the best reasonable choice. A panel with four investment options (alternatives) is available to spend the money. The available options are, Arb¯¯1 a car firm, Arb¯¯2 is a food firm, Arb¯¯3 is a computer firm, and Arb¯¯4 is an arms firm. The investment firm must make a decision based on the three attributes, the risk cta¯¯1, the growth cta¯¯2, and the environmental impact cta¯¯3. The weight vector of the attributes is We¯¯=0.35,0.25,0.4T. The possible four alternatives are assess with respect to three attributes by three decision maker and provide the assessment values in the form of HINLNs under the linguistic term set S=s0=extremelypoor,s1=verypoor,s2=poor,s3=medium,s4=good,s5=verygood,s6=extremelygood. Thus, when the possible four alternatives are assessed by three decision makers with respect to the three attributes, and the HINL decision matrix are constructed as given in Table 1.Step 1.
Find out the supportsSuphn¯¯de,hn¯¯dxd=1,2,3,4,e,x=1,2,3,e≠x by utilizing formula (80). For simplicity, we shall denote Suphn¯¯de,hn¯¯dz4×1 by Sez which means the supports between the eth and the zth columns of DT¯¯.
(85)S12=S21=0.85690.83610.81940.7544,S13=S31=0.68190.75810.78400.7379,S23=S32=0.53890.81680.75560.7438.Step 2.
Utilizing Equation (81) to find out the weighted support degree Thn¯¯de that HINLN hn¯¯de collects from other HINLNs hn¯¯dxx=1,2,3. We express Thn¯¯de4×3 by T.
(86)T=1.53891.39581.22081.59421.65291.57481.60351.57501.53961.49221.49811.4816.Step 3.
Utilize Equation (82) to get the weights ϒded=1,2,3,4;e=1,2,3 associated with hn¯¯de. This is revealed as follows.
(87)V=0.37400.25210.37390.34910.25500.39600.35450.25040.39510.35040.25090.3987.Step 4.
Utilize Equation (83) to amalgamate all the evaluation values Arb¯¯ded=1,2,3,4;e=1,2,3 in the dth row of DT¯¯ and acquire the inclusive evaluation value Arb¯¯dd=1,2,3,4. This is revealed as follows. Considerς=−3andζ=2.
(88)Arb¯¯1=s3.9562,0.4677,0.5690,0.3358,0.3832,0.3995,0.4703,s4.2742,0.4241,0.5311,0.3359,0.3832,0.4136,0.4796,s4.36093,0.4005,0.5037,0.3474,0.3995,0.4265,0.5055,s4.6513,0.3410,0.4481,0.3474,0.3995,0.4534,0.5200(89)Arb¯¯2=s3.7636,0.6810,0.7835,0.2973,0.3381,0.3465,0.3984,s4,0.6439,0.7483,0.3000,0.3533,0.3465,0.3984,s4.4231,0.6200,0.7296,0.3000,0.3533,0.3753,0.4265,s4.0460,0.6489,0.7532,0.2973,0.3381,0.3359,0.3984,s4.3751,0.6000,0.7000,0.3000,0.3533,0.3359,0.3984,s4.7650,0.5670,0.6690,0.3000,0.3533,0.3485,0.4265(90)Arb¯¯3=s3.6379,0.6033,0.8349,0.3465,0.4419,0.3508,0.4065,s3.9354,0.5837,0.8331,0.3383,0.4168,0.3519,0.4098,s4.0527,0.5000,0.6000,0.3498,0.4419,0.3831,0.4369,s4.3218,0.4637,0.5800,0.3404,0.4168,0.3868,0.4431(91)Arb¯¯4=s3.4338,0.6199,0.7296,0.2989,0.3484,0.3358,0.3831,s3.6807,0.6024,0.7156,0.2989,0.3484,0.3370,0.3868,s3.9756,0.5908,0.7076,0.2990,0.3509,0.3310,0.3753,s3.9217,0.5911,0.7183,0.2970,0.3484,0.3358,0.3831,s4.1396,0.5698,0.7027,0.2970,0.3484,0.3370,0.3868,s4.4038,0.5555,0.6937,0.2970,0.3510,0.3310,0.3753
Or utilize Equation (84) to amalgamate all the evaluation values Arb¯¯ded=1,2,3,4;e=1,2,3 in the dth row of DT¯¯ and acquire the inclusive evaluation value Arb¯¯dd=1,2,3,4. This is revealed as follows. Considerς=−3andζ=2.
(92)Arb¯¯1=s3.7999,0.5220,0.5951,0.1278,0.2282,0.2662,0.4153,s4.2315,0.4263,0.4997,0.1278,0.2282,0.3711,0.4789,s4.1307,0.4657,0.5334,0.1657,0.2662,0.3000,0.4428,s4.6000,0.4085,0.4715,0.1657,0.2662,0.3948,0.5000(93)Arb¯¯2=s3.5694,0.6873,0.7772,0.0436,0.1380,0.1635,0.2641,s4,0.6537,0.7407,0.0788,0.1769,0.1635,0.2641,s4.3695,0.6129,0.6976,0.0788,0.1769,0.2000,0.3000,s3.8585,0.6574,0.7447,0.0436,0.1380,0.1281,0.2641,s4.3240,0.6287,0.7126,0.0788,0.1769,0.1281,0.2640,s4.7235,0.5934,0.6746,0.0788,0.1769,0.1680,0.3000(94)Arb¯¯3=s3.5702,0.5922,0.6921,0.1636,0.3398,0.1982,0.2998,s3.7754,0.5212,0.6591,0.1385,0.3205,0.2372,0.34203,s3.8640,0.5528,0.6286,0.2067,0.3398,0.2281,0.3287,s4.0861,0.4990,0.6054,0.1845,0.3205,0.2639,0.3670(95)Arb¯¯4=s3.3620,0.6128,0.6974,0.0706,0.1679,0.1277,0.2281,s3.6131,0.5917,0.6743,0.0706,0.1679,0.1617,0.2640,s3.8207,0.5626,0.6440,0.1002,0.1986,0.1000,0.2000,s3.6725,0.5026,0.6461,0.0344,0.1679,0.1277,0.2281,s3.9468,0.4943,0.6285,0.0344,0.1679,0.1617,0.2640,s4.1737,0.4820,0.6052,0.0666,0.1986,0.1000,0.2000Step 5.
Exploiting Definition8 to find out the scores Scr¯¯hn¯¯dd=1,2,3,4. This is revealed as follows:
(96)Sre¯¯Arb¯¯1=0.4338,Sre¯¯Arb¯¯2=0.5276,Sre¯¯Arb¯¯3=0.4863,Sre¯¯Arb¯¯4=0.4928.
Or(97)Sre¯¯Arb¯¯1=0.3922,Sre¯¯Arb¯¯2=0.5182,Sre¯¯Arb¯¯3=0.4447,Sre¯¯Arb¯¯4=0.4501.Step 6.
According to the score values the ranking order of the alternativesArb¯¯dd=1,2,3,4.
Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1, or Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1.
So, the best alternative isArb¯¯2, and the worst alternative is Arb¯¯1.Table 1
HINL decision matrixDT¯¯.
cta¯¯1cta¯¯2cta¯¯3Arb¯¯1s4,0.5,0.6,0.1,0.2,0.2,0.3,s5,0.3,0.4,0.2,0.3,0.3,0.4s5,0.5,0.6,0.2,0.3,0.3,0.4s4,0.4,0.5,0.1,0.2,0.3,0.5,s4,0.2,0.3,0.1,0.2,0.5,0.6Arb¯¯2s4,0.7,0.8,0.1,0.2,0.2,0.3,s5,0.6,0.7,0.1,0.2,0.1,0.3s4,0.6,0.7,0.01,0.1,0.2,0.3s3,0.7,0.8,0.01,0.1,0.1,0.2,s4,0.6,0.7,0.1,0.2,0.1,0.2,s5,0.5,0.6,0.1,0.2,0.2,0.3Arb¯¯3s4,0.7,0.9,0.2,0.4,0.1,0.2,s5,0.5,0.6,0.3,0.4,0.2,0.3s4,0.5,0.6,0.2,0.3,0.3,0.4,s5,0.3,0.5,0.1,0.2,0.4,0.5s3,0.5,0.6,0.1,0.3,0.2,0.3Arb¯¯4s3,0.7,0.8,0.01,0.1,0.1,0.2s3,0.6,0.7,0.1,0.2,0.2,0.3,s4,0.5,0.6,0.1,0.2,0.3,0.4,s5,0.4,0.5,0.2,0.3,0.1,0.2s4,0.5,0.6,0.1,0.2,0.1,0.2,s5,0.3,0.5,0.01,0.2,0.1,0.2


### 6.1. The Effect of the LSFs on Ranking Results

In this subsection, other different kinds of LSFs are also used to the abovementioned decision-making process to obtain the ranking results to demonstrate the effect from other LSFs on the ranking results. The score values and final ranking orders are shown in Table2.Table 2
Score values and ranking orders of alternatives utilizing different LSFs.
LSFHINSSPWA operatorHINSSPWGA operatorRanking orderf∗2Sre¯¯Arb¯¯1=0.3959,Sre¯¯Arb¯¯2=0.4793,Sre¯¯Arb¯¯3=0.4434,Sre¯¯Arb¯¯4=0.4496.Sre¯¯Arb¯¯1=0.3887,Sre¯¯Arb¯¯2=0.4749,Sre¯¯Arb¯¯3=0.3847,Sre¯¯Arb¯¯4=0.4429.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯1>Arb¯¯3f∗3Sre¯¯Arb¯¯1=0.3486,Sre¯¯Arb¯¯2=0.4592,Sre¯¯Arb¯¯3=0.3902,Sre¯¯Arb¯¯4=0.4154.Sre¯¯Arb¯¯1=0.3677,Sre¯¯Arb¯¯2=0.5022,Sre¯¯Arb¯¯3=0.4018,Sre¯¯Arb¯¯4=0.4266.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1From Table2, we can observe that when the LSF is utilized the ranking orders gained from both the aggregation operators remain the same as that gained from the first LSF. But when the second LSF is used, the ranking order acquired from the HINLSSPWA operator is the same as the ranking order gained from the first LSF; however, when the HINLSSPWA operator is used, the ranking order is modified. That is, the best alternative remains the same but the worst one is changed, which is Arb¯¯3. The major explanation for this variation is that three distinct forms of LSFs affect three different sorts of semantic circumstances. This might lead to a variety of semantic preferences and semantic deviations, resulting in a variety of ranking results. As a result, one of the benefits of our suggested technique is that it can adapt to various semantic decision-making environments and fulfill the semantic needs of various experts. So, experts can choose the suitable LSF in real-time decision-making based on their linguistic preferences.


### 6.2.Effect of the Parameterςon Final Ranking Order

From Table3, one can observe that for different values of the parameter ς, different score values are obtained, while utilizing WHINLSSPWA and WHINLSSPGA operators. We can also observe from Table 3, when the values of the parameter ς increases while exploiting WHINLSSPWA operator, the score values of the alternatives increases. Similarly, when utilizing WHINLSSPWG operator, the score values of the alternatives decreases, while the final ranking order remains the same at both the cases. This makes the decision-making process more flexible, so, the decision makers may choose the value of the parameter ς according to the actual need of the situations.Table 3
Effect of the parameterς on final ranking order.
ParameterWSSHINPWA operatorWSSHINPWG operatorRanking orderς=−1Sre¯¯Arb¯¯1=0.3938,Sre¯¯Arb¯¯2=0.4730,Sre¯¯Arb¯¯3=0.4266,Sre¯¯Arb¯¯4=0.4295.Sre¯¯Arb¯¯1=0.4857,Sre¯¯Arb¯¯2=0.5579,Sre¯¯Arb¯¯3=0.5030,Sre¯¯Arb¯¯4=0.5074.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−2Sre¯¯Arb¯¯1=0.4246,Sre¯¯Arb¯¯2=0.5143,Sre¯¯Arb¯¯3=0.4705,Sre¯¯Arb¯¯4=0.4751.Sre¯¯Arb¯¯1=0.4147,Sre¯¯Arb¯¯2=0.5242,Sre¯¯Arb¯¯3=0.4570,Sre¯¯Arb¯¯4=0.4622.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−7Sre¯¯Arb¯¯1=0.4481,Sre¯¯Arb¯¯2=0.5409,Sre¯¯Arb¯¯3=0.5043,Sre¯¯Arb¯¯4=0.5167.Sre¯¯Arb¯¯1=0.3649,Sre¯¯Arb¯¯2=0.5135,Sre¯¯Arb¯¯3=0.4276,Sre¯¯Arb¯¯4=0.4342.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−15Sre¯¯Arb¯¯1=0.4608,Sre¯¯Arb¯¯2=0.5468,Sre¯¯Arb¯¯3=0.5113,Sre¯¯Arb¯¯4=0.5269.Sre¯¯Arb¯¯1=0.3490,Sre¯¯Arb¯¯2=0.5083,Sre¯¯Arb¯¯3=0.4123,Sre¯¯Arb¯¯4=0.4221.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−25Sre¯¯Arb¯¯1=0.4665,Sre¯¯Arb¯¯2=0.5489,Sre¯¯Arb¯¯3=0.5134,Sre¯¯Arb¯¯4=0.5296.Sre¯¯Arb¯¯1=0.3434,Sre¯¯Arb¯¯2=0.5044,Sre¯¯Arb¯¯3=0.4049,Sre¯¯Arb¯¯4=0.4163.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−50Sre¯¯Arb¯¯1=0.4710,Sre¯¯Arb¯¯2=0.5502,Sre¯¯Arb¯¯3=0.5148,Sre¯¯Arb¯¯4=0.5311.Sre¯¯Arb¯¯1=0.3400,Sre¯¯Arb¯¯2=0.5004,Sre¯¯Arb¯¯3=0.3990,Sre¯¯Arb¯¯4=0.4119.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−70Sre¯¯Arb¯¯1=0.4722,Sre¯¯Arb¯¯2=0.5505,Sre¯¯Arb¯¯3=0.5152,Sre¯¯Arb¯¯4=0.5314.Sre¯¯Arb¯¯1=0.3392,Sre¯¯Arb¯¯2=0.4992,Sre¯¯Arb¯¯3=0.3974,Sre¯¯Arb¯¯4=0.4106.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−100Sre¯¯Arb¯¯1=0.4731,Sre¯¯Arb¯¯2=0.5507,Sre¯¯Arb¯¯3=0.5155,Sre¯¯Arb¯¯4=0.5317.Sre¯¯Arb¯¯1=0.3386,Sre¯¯Arb¯¯2=0.4984,Sre¯¯Arb¯¯3=0.3961,Sre¯¯Arb¯¯4=0.4097.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1


### 6.3.Effect of the Parameterζon Decision Result

From Table4, we can see that for different values of the parameter ζ different score values are obtained, while utilizing WHINLSSPA and WHINLSSPGA operators. One can also observe from Table 4, when utilizing WHINLSSPA operator the ranking order remains the same, but when we utilized WHINLSSPGA operator different ranking orders are obtained. This makes the decision-making process more flexible, and the makers may use the value of the parameter ζ according to the need of the actual situations.Table 4
Effect of the parameterζ on decision result.
ParameterWSSHINPWA operatorWSSHINPGA operatorRanking orderζ=1Sre¯¯Arb¯¯1=0.4309,Sre¯¯Arb¯¯2=0.5244,Sre¯¯Arb¯¯3=0.4737,Sre¯¯Arb¯¯4=0.4817.Sre¯¯Arb¯¯1=0.3950,Sre¯¯Arb¯¯2=0.5184,Sre¯¯Arb¯¯3=0.4437,Sre¯¯Arb¯¯4=0.4508.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=3Sre¯¯Arb¯¯1=0.4374,Sre¯¯Arb¯¯2=0.5311,Sre¯¯Arb¯¯3=0.4957,Sre¯¯Arb¯¯4=0.5015.Sre¯¯Arb¯¯1=0.3920,Sre¯¯Arb¯¯2=0.5183,Sre¯¯Arb¯¯3=0.4456,Sre¯¯Arb¯¯4=0.4502.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=5Sre¯¯Arb¯¯1=0.4446,Sre¯¯Arb¯¯2=0.5379,Sre¯¯Arb¯¯3=0.5106,Sre¯¯Arb¯¯4=0.5158.Sre¯¯Arb¯¯1=0.3924,Sre¯¯Arb¯¯2=0.5184,Sre¯¯Arb¯¯3=0.4469,Sre¯¯Arb¯¯4=0.4506.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=10Sre¯¯Arb¯¯1=0.4584,Sre¯¯Arb¯¯2=0.5511,Sre¯¯Arb¯¯3=0.5335,Sre¯¯Arb¯¯4=0.5382.Sre¯¯Arb¯¯1=0.3936,Sre¯¯Arb¯¯2=0.5186,Sre¯¯Arb¯¯3=0.4485,Sre¯¯Arb¯¯4=0.4511.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=17Sre¯¯Arb¯¯1=0.4695,Sre¯¯Arb¯¯2=0.5618,Sre¯¯Arb¯¯3=0.5484,Sre¯¯Arb¯¯4=0.5526.Sre¯¯Arb¯¯1=0.3948,Sre¯¯Arb¯¯2=0.5188,Sre¯¯Arb¯¯3=0.4496,Sre¯¯Arb¯¯4=0.4514.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=25Sre¯¯Arb¯¯1=0.4762,Sre¯¯Arb¯¯2=0.5681,Sre¯¯Arb¯¯3=0.5561,Sre¯¯Arb¯¯4=0.5600Sre¯¯Arb¯¯1=0.3956,Sre¯¯Arb¯¯2=0.5190,Sre¯¯Arb¯¯3=0.4504,Sre¯¯Arb¯¯4=0.4515.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=60Sre¯¯Arb¯¯1=0.4848,Sre¯¯Arb¯¯2=0.5766,Sre¯¯Arb¯¯3=0.5653,Sre¯¯Arb¯¯4=0.5689.Sre¯¯Arb¯¯1=0.3971,Sre¯¯Arb¯¯2=0.5192,Sre¯¯Arb¯¯3=0.4518,Sre¯¯Arb¯¯4=0.4515.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯3>Arb¯¯4>Arb¯¯1ζ=100Sre¯¯Arb¯¯1=0.4870,Sre¯¯Arb¯¯2=0.5790,Sre¯¯Arb¯¯3=0.5677,Sre¯¯Arb¯¯4=0.5712.Sre¯¯Arb¯¯1=0.3978,Sre¯¯Arb¯¯2=0.5193,Sre¯¯Arb¯¯3=0.4525,Sre¯¯Arb¯¯4=0.4513.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯3>Arb¯¯4>Arb¯¯1


### 6.4. Comparison of the Proposed MADM Method with Existing Method

In this subpart, comparison of the anticipated MADM method initiated on the newly proposed AOs with existing method is discussed.From Table5, we can see that the ranking order obtained from Ye [32] is the same as the ranking order obtained from the proposed approach. This shows that the initiated approach is valid. The initiated approach has several advantages over the approach developed by Ye [32]. The initiated approach can remove the bad impact of unreasonable data by power weight vector and also make the decision making process more flexible due to general parameters and fit in with distinct semantic scenarios. Therefore, the proposed technique for solving MADM problems is more practical then the existing one.Table 5
Comparison with other approach.
ApproachScore valuesRanking orderYe [32] HINLWA operatorSre¯¯Arb¯¯1=0.4413,Sre¯¯Arb¯¯2=0.5519,Sre¯¯Arb¯¯3=0.4549,Sre¯¯Arb¯¯4=0.5051.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1Ye [32] HINLWG operatorSre¯¯Arb¯¯1=0.4231,Sre¯¯Arb¯¯2=0.5363,Sre¯¯Arb¯¯3=0.4325,Sre¯¯Arb¯¯4=0.4774.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1Proposed HINLSSGPWA operatorConsiderς=−3andζ=2Sre¯¯Arb¯¯1=0.4338,Sre¯¯Arb¯¯2=0.5276,Sre¯¯Arb¯¯3=0.4863,Sre¯¯Arb¯¯4=0.4928.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1Proposed HINLSSGPWA operatorConsiderς=−3andζ=2Sre¯¯Arb¯¯1=0.3922,Sre¯¯Arb¯¯2=0.5182,Sre¯¯Arb¯¯3=0.4447,Sre¯¯Arb¯¯4=0.4501.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1


## 6.1. The Effect of the LSFs on Ranking Results

In this subsection, other different kinds of LSFs are also used to the abovementioned decision-making process to obtain the ranking results to demonstrate the effect from other LSFs on the ranking results. The score values and final ranking orders are shown in Table2.Table 2
Score values and ranking orders of alternatives utilizing different LSFs.
LSFHINSSPWA operatorHINSSPWGA operatorRanking orderf∗2Sre¯¯Arb¯¯1=0.3959,Sre¯¯Arb¯¯2=0.4793,Sre¯¯Arb¯¯3=0.4434,Sre¯¯Arb¯¯4=0.4496.Sre¯¯Arb¯¯1=0.3887,Sre¯¯Arb¯¯2=0.4749,Sre¯¯Arb¯¯3=0.3847,Sre¯¯Arb¯¯4=0.4429.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯1>Arb¯¯3f∗3Sre¯¯Arb¯¯1=0.3486,Sre¯¯Arb¯¯2=0.4592,Sre¯¯Arb¯¯3=0.3902,Sre¯¯Arb¯¯4=0.4154.Sre¯¯Arb¯¯1=0.3677,Sre¯¯Arb¯¯2=0.5022,Sre¯¯Arb¯¯3=0.4018,Sre¯¯Arb¯¯4=0.4266.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1From Table2, we can observe that when the LSF is utilized the ranking orders gained from both the aggregation operators remain the same as that gained from the first LSF. But when the second LSF is used, the ranking order acquired from the HINLSSPWA operator is the same as the ranking order gained from the first LSF; however, when the HINLSSPWA operator is used, the ranking order is modified. That is, the best alternative remains the same but the worst one is changed, which is Arb¯¯3. The major explanation for this variation is that three distinct forms of LSFs affect three different sorts of semantic circumstances. This might lead to a variety of semantic preferences and semantic deviations, resulting in a variety of ranking results. As a result, one of the benefits of our suggested technique is that it can adapt to various semantic decision-making environments and fulfill the semantic needs of various experts. So, experts can choose the suitable LSF in real-time decision-making based on their linguistic preferences.


## 6.2.Effect of the Parameterςon Final Ranking Order

From Table3, one can observe that for different values of the parameter ς, different score values are obtained, while utilizing WHINLSSPWA and WHINLSSPGA operators. We can also observe from Table 3, when the values of the parameter ς increases while exploiting WHINLSSPWA operator, the score values of the alternatives increases. Similarly, when utilizing WHINLSSPWG operator, the score values of the alternatives decreases, while the final ranking order remains the same at both the cases. This makes the decision-making process more flexible, so, the decision makers may choose the value of the parameter ς according to the actual need of the situations.Table 3
Effect of the parameterς on final ranking order.
ParameterWSSHINPWA operatorWSSHINPWG operatorRanking orderς=−1Sre¯¯Arb¯¯1=0.3938,Sre¯¯Arb¯¯2=0.4730,Sre¯¯Arb¯¯3=0.4266,Sre¯¯Arb¯¯4=0.4295.Sre¯¯Arb¯¯1=0.4857,Sre¯¯Arb¯¯2=0.5579,Sre¯¯Arb¯¯3=0.5030,Sre¯¯Arb¯¯4=0.5074.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−2Sre¯¯Arb¯¯1=0.4246,Sre¯¯Arb¯¯2=0.5143,Sre¯¯Arb¯¯3=0.4705,Sre¯¯Arb¯¯4=0.4751.Sre¯¯Arb¯¯1=0.4147,Sre¯¯Arb¯¯2=0.5242,Sre¯¯Arb¯¯3=0.4570,Sre¯¯Arb¯¯4=0.4622.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−7Sre¯¯Arb¯¯1=0.4481,Sre¯¯Arb¯¯2=0.5409,Sre¯¯Arb¯¯3=0.5043,Sre¯¯Arb¯¯4=0.5167.Sre¯¯Arb¯¯1=0.3649,Sre¯¯Arb¯¯2=0.5135,Sre¯¯Arb¯¯3=0.4276,Sre¯¯Arb¯¯4=0.4342.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−15Sre¯¯Arb¯¯1=0.4608,Sre¯¯Arb¯¯2=0.5468,Sre¯¯Arb¯¯3=0.5113,Sre¯¯Arb¯¯4=0.5269.Sre¯¯Arb¯¯1=0.3490,Sre¯¯Arb¯¯2=0.5083,Sre¯¯Arb¯¯3=0.4123,Sre¯¯Arb¯¯4=0.4221.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−25Sre¯¯Arb¯¯1=0.4665,Sre¯¯Arb¯¯2=0.5489,Sre¯¯Arb¯¯3=0.5134,Sre¯¯Arb¯¯4=0.5296.Sre¯¯Arb¯¯1=0.3434,Sre¯¯Arb¯¯2=0.5044,Sre¯¯Arb¯¯3=0.4049,Sre¯¯Arb¯¯4=0.4163.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−50Sre¯¯Arb¯¯1=0.4710,Sre¯¯Arb¯¯2=0.5502,Sre¯¯Arb¯¯3=0.5148,Sre¯¯Arb¯¯4=0.5311.Sre¯¯Arb¯¯1=0.3400,Sre¯¯Arb¯¯2=0.5004,Sre¯¯Arb¯¯3=0.3990,Sre¯¯Arb¯¯4=0.4119.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−70Sre¯¯Arb¯¯1=0.4722,Sre¯¯Arb¯¯2=0.5505,Sre¯¯Arb¯¯3=0.5152,Sre¯¯Arb¯¯4=0.5314.Sre¯¯Arb¯¯1=0.3392,Sre¯¯Arb¯¯2=0.4992,Sre¯¯Arb¯¯3=0.3974,Sre¯¯Arb¯¯4=0.4106.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ς=−100Sre¯¯Arb¯¯1=0.4731,Sre¯¯Arb¯¯2=0.5507,Sre¯¯Arb¯¯3=0.5155,Sre¯¯Arb¯¯4=0.5317.Sre¯¯Arb¯¯1=0.3386,Sre¯¯Arb¯¯2=0.4984,Sre¯¯Arb¯¯3=0.3961,Sre¯¯Arb¯¯4=0.4097.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1


## 6.3.Effect of the Parameterζon Decision Result

From Table4, we can see that for different values of the parameter ζ different score values are obtained, while utilizing WHINLSSPA and WHINLSSPGA operators. One can also observe from Table 4, when utilizing WHINLSSPA operator the ranking order remains the same, but when we utilized WHINLSSPGA operator different ranking orders are obtained. This makes the decision-making process more flexible, and the makers may use the value of the parameter ζ according to the need of the actual situations.Table 4
Effect of the parameterζ on decision result.
ParameterWSSHINPWA operatorWSSHINPGA operatorRanking orderζ=1Sre¯¯Arb¯¯1=0.4309,Sre¯¯Arb¯¯2=0.5244,Sre¯¯Arb¯¯3=0.4737,Sre¯¯Arb¯¯4=0.4817.Sre¯¯Arb¯¯1=0.3950,Sre¯¯Arb¯¯2=0.5184,Sre¯¯Arb¯¯3=0.4437,Sre¯¯Arb¯¯4=0.4508.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=3Sre¯¯Arb¯¯1=0.4374,Sre¯¯Arb¯¯2=0.5311,Sre¯¯Arb¯¯3=0.4957,Sre¯¯Arb¯¯4=0.5015.Sre¯¯Arb¯¯1=0.3920,Sre¯¯Arb¯¯2=0.5183,Sre¯¯Arb¯¯3=0.4456,Sre¯¯Arb¯¯4=0.4502.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=5Sre¯¯Arb¯¯1=0.4446,Sre¯¯Arb¯¯2=0.5379,Sre¯¯Arb¯¯3=0.5106,Sre¯¯Arb¯¯4=0.5158.Sre¯¯Arb¯¯1=0.3924,Sre¯¯Arb¯¯2=0.5184,Sre¯¯Arb¯¯3=0.4469,Sre¯¯Arb¯¯4=0.4506.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=10Sre¯¯Arb¯¯1=0.4584,Sre¯¯Arb¯¯2=0.5511,Sre¯¯Arb¯¯3=0.5335,Sre¯¯Arb¯¯4=0.5382.Sre¯¯Arb¯¯1=0.3936,Sre¯¯Arb¯¯2=0.5186,Sre¯¯Arb¯¯3=0.4485,Sre¯¯Arb¯¯4=0.4511.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=17Sre¯¯Arb¯¯1=0.4695,Sre¯¯Arb¯¯2=0.5618,Sre¯¯Arb¯¯3=0.5484,Sre¯¯Arb¯¯4=0.5526.Sre¯¯Arb¯¯1=0.3948,Sre¯¯Arb¯¯2=0.5188,Sre¯¯Arb¯¯3=0.4496,Sre¯¯Arb¯¯4=0.4514.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=25Sre¯¯Arb¯¯1=0.4762,Sre¯¯Arb¯¯2=0.5681,Sre¯¯Arb¯¯3=0.5561,Sre¯¯Arb¯¯4=0.5600Sre¯¯Arb¯¯1=0.3956,Sre¯¯Arb¯¯2=0.5190,Sre¯¯Arb¯¯3=0.4504,Sre¯¯Arb¯¯4=0.4515.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1ζ=60Sre¯¯Arb¯¯1=0.4848,Sre¯¯Arb¯¯2=0.5766,Sre¯¯Arb¯¯3=0.5653,Sre¯¯Arb¯¯4=0.5689.Sre¯¯Arb¯¯1=0.3971,Sre¯¯Arb¯¯2=0.5192,Sre¯¯Arb¯¯3=0.4518,Sre¯¯Arb¯¯4=0.4515.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯3>Arb¯¯4>Arb¯¯1ζ=100Sre¯¯Arb¯¯1=0.4870,Sre¯¯Arb¯¯2=0.5790,Sre¯¯Arb¯¯3=0.5677,Sre¯¯Arb¯¯4=0.5712.Sre¯¯Arb¯¯1=0.3978,Sre¯¯Arb¯¯2=0.5193,Sre¯¯Arb¯¯3=0.4525,Sre¯¯Arb¯¯4=0.4513.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1orArb¯¯2>Arb¯¯3>Arb¯¯4>Arb¯¯1


## 6.4. Comparison of the Proposed MADM Method with Existing Method

In this subpart, comparison of the anticipated MADM method initiated on the newly proposed AOs with existing method is discussed.From Table5, we can see that the ranking order obtained from Ye [32] is the same as the ranking order obtained from the proposed approach. This shows that the initiated approach is valid. The initiated approach has several advantages over the approach developed by Ye [32]. The initiated approach can remove the bad impact of unreasonable data by power weight vector and also make the decision making process more flexible due to general parameters and fit in with distinct semantic scenarios. Therefore, the proposed technique for solving MADM problems is more practical then the existing one.Table 5
Comparison with other approach.
ApproachScore valuesRanking orderYe [32] HINLWA operatorSre¯¯Arb¯¯1=0.4413,Sre¯¯Arb¯¯2=0.5519,Sre¯¯Arb¯¯3=0.4549,Sre¯¯Arb¯¯4=0.5051.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1Ye [32] HINLWG operatorSre¯¯Arb¯¯1=0.4231,Sre¯¯Arb¯¯2=0.5363,Sre¯¯Arb¯¯3=0.4325,Sre¯¯Arb¯¯4=0.4774.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1Proposed HINLSSGPWA operatorConsiderς=−3andζ=2Sre¯¯Arb¯¯1=0.4338,Sre¯¯Arb¯¯2=0.5276,Sre¯¯Arb¯¯3=0.4863,Sre¯¯Arb¯¯4=0.4928.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1Proposed HINLSSGPWA operatorConsiderς=−3andζ=2Sre¯¯Arb¯¯1=0.3922,Sre¯¯Arb¯¯2=0.5182,Sre¯¯Arb¯¯3=0.4447,Sre¯¯Arb¯¯4=0.4501.Arb¯¯2>Arb¯¯4>Arb¯¯3>Arb¯¯1


## 7. Conclusion

Accessible information is frequently incomplete and incompatible in real decision-making, and the HINLS is a superior tool for indicating such information. In this article, merging LSFs, SS operational laws, and GPA operators, a technique is initiated to deal with HINL MADM problems and fit in with distinct semantic scenarios. Initially, a number of core operational laws for HINLNs are initiated based on LSF, SSt-norm, and SS t-conorm and some of its core properties are investigated. Second, limitations of the existing score function are discussed, and a new score function and distance measure are anticipated based on LSFs. Then, as standard PA operators cannot handle scenarios when expert assessment values are HINLNs, several novel generalized power AOs are proposed to aggregate HINLNs. The most significant characteristic of these operators are that they can also adapt to a variety of semantic situations while also reducing the detrimental impact of unreseasonably high or unreseasonably low evaluation values. Additionally, utilizing the newly instigated AOs, a novel MADM technique is suggested. Lastly, a numerical example is offered to reveal the potency of the initiated technique, along with comprehensive comparison with the existing approaches.In future, we will explore LSFs and SS operational laws for other generalizations of INL and SVNL sets, such as hesitant bipolar valued neutrosophic sets [48], single valued spherical hesitant neutrosophic sets [49], interval valued neutrosophic vague sets [50], refined single valued neutrosophic sets [51], and initiate different AOs such as MSM operator, Muirhead mean operators, Hamy Mean operators and apply these AOs to solve MADM and MAGDM problems in different fields.



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*Source: 1654820-2022-03-19.xml*