# Multi-Attribute Decision-Support System Based on Aggregations of Interval-Valued Complex Neutrosophic Hypersoft Set

Authors: Atiqe Ur Rahman; Muhammad Saeed; Muhammad Arshad; Salwa El-Morsy
Journal: Applied Computational Intelligence and Soft Computing
 (2021)

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## Abstract

Hypersoft set is an emerging field of study that is meant to address the insufficiency and the limitation of existing soft-set-like models regarding the consideration and the entitlement of multi-argument approximate function. This type of function maps the multi-subparametric tuples to the power set of the universe. It focuses on the partitioning of each attribute into its attribute-valued set that is missing in existing soft-set-like structures. This study aims to introduce novel concepts of complex intuitionistic fuzzy set and complex neutrosophic set under the hypersoft set environment with interval-valued settings. Two novel structures, that is, interval-valued complex intuitionistic hypersoft set (IV-CIFHS-set) and interval-valued complex neutrosophic hypersoft set (IV-CNHS-set), are developed via employing theoretic, axiomatic, graphical, and algorithmic approaches. After conceptual characterization of essential elementary notions of these structures, decision-support systems are presented with the proposal of algorithms to assist the decision-making process. The proposed algorithms are validated with the help of real-world applications. A comprehensive inter-cum-intra comparison of proposed structures is discussed with the existing relevant models, and their generalization is elaborated under certain evaluating features.


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## Body

## 1. Introduction

The traditional logic (i.e., Boolean logic) is not always pertinent in real-world scenarios, where the available data is vague or imprecise. To deal with such kinds of situations, a particular class of sets known as fuzzy sets (F.Sets) that were proposed by Zadeh [1] is considered appropriate. In these sets, every member of the universe is specified by a membership grade in a unit closed interval. However, to tackle scenarios having more complexity and uncertainty, it was observed that the concept of F.Sets is not sufficient, and therefore, these concepts were expanded with few extensions. Intuitionistic fuzzy sets (IF.Sets) by Atanassov [2] was one of such major developments. Due to the consideration of nonmembership grade, IF.Sets are more effective in tackling with the vagueness of data. Moreover, IF.Sets are proficient to emulate the available information more precisely and rationally. As far as the consideration of the degree of indeterminacy was concerned, both F.Sets and IF.Set were inadequate for such kind of grade, so neutrosophic sets (N.Sets) were initiated by Smarandache [3] to cope with such shortcoming. N.Sets are more capable to maintain impreciseness in the contents of information and may facilitate approximate reasoning behavior diligently. Although the descriptive capability of N.Sets is higher than that of the traditional F.Sets and IF.Sets due to their additional presence of nonmembership and indeterminant graded functions; however, they have fairly higher computational complexity over F.Sets and IF.Sets.The models such as F.Sets, IF.Sets, and N.Sets depicted some sort of limitation regarding the validation for some parameterization tools. To address this scarcity, Molodtsov [4] characterized soft sets (S.Sets) as a new mathematical parameterized model. In S.Sets, every parameter in a set of parameters maps to power set of the universe of discourse while defining single-argument approximate function. The researchers [5–11] studied the basic properties, elementary set theoretic operations, relations, and functions of S.Sets with illustrative numerical examples. To hybridize the characteristics of F.Sets, IF.Sets, and N.Sets with S.Sets, fuzzy soft sets (FS.Sets) [12, 13], intuitionistic fuzzy soft sets (IFS.Sets) [14, 15], and neutrosophic soft sets (NS.Sets) [16] were conceptualized. Although there are many researchers who contributed a lot towards the expansion and extension of these hybridized structures with the interval-valued setting, the contributions of researchers [17–21] are more prominent relevant to these models. They not only discussed the fundamentals of interval-valued fuzzy soft-set-like models but also employed certain techniques for their applications in different situations.In certain real-world scenarios, the classification of attributes into subattributive values in the form of sets is necessary. The existing concept of S.Sets is not sufficient and incompatible with such scenarios so Smarandache [22] introduced the concept of hypersoft sets (HS.Sets) to address the insufficiency of S.Sets and to cope with the situations with multi-argument approximate function. The rudiments and elementary axioms of HS.Sets have been discussed in [23] and elaborated with numerical examples. Rahman et al. [24–30] investigated the hybridized properties of HS.Sets under the environments of complex set, convexity and concavity, parameterization, and bijection. They employed decision-making algorithmic approaches to solve real-world problems. Saeed et al. [31–33] developed the theories of neutrosophic hypersoft mappings and complex multi-fuzzy hypersoft sets with applications in decision-making and clinical diagnosis.


### 1.1. Research Gap and Motivation

The following points depict the need and motivation behind the proposed study:(1)
Many researchers discussed the hybridized structures of a complex set with fuzzy set, intuitionistic fuzzy set, and neutrosophic set under soft set environments. The literature review of the most relevant models [34–42] is presented in Table 1.(2)
It is vivid that these structures consider only one set of parameters and use the single-argument approximate function. They depict some kind of insufficiency to tackle the scenarios (recruitment process, product selection, medical diagnosis, etc.) where further classification of parameters into their subparametric values in the form of disjoint sets is necessary for deep learning and observation in decision-support systems.(3)
Along these lines, another construction requests its place in writing for tending to such obstacle, so the hypersoft set is conceptualized to handle such situations (Figure1 depicts the vivid comparison of soft set model and hypersoft set model. It presents the optimal selection of a mobile with the help of suitable parameters in the case of soft set and suitable subparametric values in the case of hypersoft set). It has made the decision-making process more flexible and reliable. Also, it not only fulfills the requirements of existing soft set-like literature for multi-argument approximate functions but also supports the decision-makers to make decisions with the deep inspection.(4)
Although the models IV-CNSS [36] and IV-CFSS [37, 41] have been developed to tackle the scenarios with periodic and interval type data under soft set-environment, these are inadequate to deal subattribute values in the form of disjoint sets as a collective domain of multi-argument approximate function.(5)
Inspiring from the above literature in general and from [36, 37, 41] in specific, this study aims to characterize novel structures of IV-CIFHS-set and IV-CNHS-set that not only generalize the existing relevant models but also address their limitations.Table 1
Literature review on complex fuzzy-set-like models under soft set.
AuthorsStructureDomain of approximate functionRange settingAli et al. [34]Complex intuitionistic fuzzy soft setSet of parametersComplex intuitionistic fuzzy setAl-Quran et al. [35]Complex neutrosophic soft expert setSoft expert setComplex neutrosophic setAl-Sharqi et al. [36]Interval-valued complex neutrosophic soft setSet of parametersInterval-valued complex neutrosophic setFan et al. [37]Interval-valued complex fuzzy soft setSet of parametersInterval-valued complex fuzzy setKhan et al. [38]Complex intuitionistic fuzzy soft setSet of parametersComplex intuitionistic fuzzy setKumar et al. [39]Complex intuitionistic fuzzy soft setSet of parametersComplex intuitionistic fuzzy setSmarandache et al. [40]Complex neutrosophic soft setSet of parametersComplex neutrosophic setSelvachandran et al. [41]Interval-valued complex fuzzy soft setSet of parametersInterval-valued complex fuzzy setThirunavukarasu et al. [42]Complex fuzzy soft setSet of parametersComplex fuzzy setFigure 1
Comparison of soft set and hypersoft set.


### 1.2. Main Contributions

The following are the possible objectives of this study:(1)
The existing relevant models, that is, [34–42], are made adequate with the entitlement of multi-argument approximate function through development of IV-CIFHS-set and IV-CNHS-set(2)
The scenarios where parameters are further partitioned into subparametric values in the form of sets are tackled by using IV-CIFHS-set and IV-CNHS-set(3)
Some essential rudiments, that is, properties, elementary laws, and set theoretic operations of IV-CIFHS-set and IV-CNHS-set, are characterized(4)
Two algorithms based on IV-CIFHS-set and IV-CNHS-set are proposed to deal with daily-life decision-making problems having periodic and interval type data/information(5)
The proposed study is compared with some existing relevant models by considering some important evaluating indicators so that the advantageous aspect of the proposed study may be depicted(6)
The generalization and particular cases of proposed models are discussed with the pictorial depiction(7)
The advantages and future directions of the proposed study are presented


### 1.3. Paper Organization

The organization of the remaining paper is given in Figure2.Figure 2
Organization of paper.


### 1.4. Notations and Abbreviations

Some abbreviations and notations are used throughout the paper. Their full names are given in Table2 to facilitate the readers for proper understanding of the concept.Table 2
Notations and abbreviations.
NotationUsed forNotationUsed forℤUniverse of discoursePℤPower set ofℤFℤCollection of fuzzy setsIFℤCollection of intuitionistic fuzzy setsℕℤCollection of neutrosophic setsCℤCollection of complex fuzzy setsCIntℤCollection of complex intuitionistic fuzzy setsCNeuUCollection of complex neutrosophic setsA-termAmplitude termM-functionMembership functionNM-functionNonmembership functionID-functionIndeterminacy functionP-termPhase termCIVℤCollection of interval-valued complex fuzzy sets


## 1.1. Research Gap and Motivation

The following points depict the need and motivation behind the proposed study:(1)
Many researchers discussed the hybridized structures of a complex set with fuzzy set, intuitionistic fuzzy set, and neutrosophic set under soft set environments. The literature review of the most relevant models [34–42] is presented in Table 1.(2)
It is vivid that these structures consider only one set of parameters and use the single-argument approximate function. They depict some kind of insufficiency to tackle the scenarios (recruitment process, product selection, medical diagnosis, etc.) where further classification of parameters into their subparametric values in the form of disjoint sets is necessary for deep learning and observation in decision-support systems.(3)
Along these lines, another construction requests its place in writing for tending to such obstacle, so the hypersoft set is conceptualized to handle such situations (Figure1 depicts the vivid comparison of soft set model and hypersoft set model. It presents the optimal selection of a mobile with the help of suitable parameters in the case of soft set and suitable subparametric values in the case of hypersoft set). It has made the decision-making process more flexible and reliable. Also, it not only fulfills the requirements of existing soft set-like literature for multi-argument approximate functions but also supports the decision-makers to make decisions with the deep inspection.(4)
Although the models IV-CNSS [36] and IV-CFSS [37, 41] have been developed to tackle the scenarios with periodic and interval type data under soft set-environment, these are inadequate to deal subattribute values in the form of disjoint sets as a collective domain of multi-argument approximate function.(5)
Inspiring from the above literature in general and from [36, 37, 41] in specific, this study aims to characterize novel structures of IV-CIFHS-set and IV-CNHS-set that not only generalize the existing relevant models but also address their limitations.Table 1
Literature review on complex fuzzy-set-like models under soft set.
AuthorsStructureDomain of approximate functionRange settingAli et al. [34]Complex intuitionistic fuzzy soft setSet of parametersComplex intuitionistic fuzzy setAl-Quran et al. [35]Complex neutrosophic soft expert setSoft expert setComplex neutrosophic setAl-Sharqi et al. [36]Interval-valued complex neutrosophic soft setSet of parametersInterval-valued complex neutrosophic setFan et al. [37]Interval-valued complex fuzzy soft setSet of parametersInterval-valued complex fuzzy setKhan et al. [38]Complex intuitionistic fuzzy soft setSet of parametersComplex intuitionistic fuzzy setKumar et al. [39]Complex intuitionistic fuzzy soft setSet of parametersComplex intuitionistic fuzzy setSmarandache et al. [40]Complex neutrosophic soft setSet of parametersComplex neutrosophic setSelvachandran et al. [41]Interval-valued complex fuzzy soft setSet of parametersInterval-valued complex fuzzy setThirunavukarasu et al. [42]Complex fuzzy soft setSet of parametersComplex fuzzy setFigure 1
Comparison of soft set and hypersoft set.


## 1.2. Main Contributions

The following are the possible objectives of this study:(1)
The existing relevant models, that is, [34–42], are made adequate with the entitlement of multi-argument approximate function through development of IV-CIFHS-set and IV-CNHS-set(2)
The scenarios where parameters are further partitioned into subparametric values in the form of sets are tackled by using IV-CIFHS-set and IV-CNHS-set(3)
Some essential rudiments, that is, properties, elementary laws, and set theoretic operations of IV-CIFHS-set and IV-CNHS-set, are characterized(4)
Two algorithms based on IV-CIFHS-set and IV-CNHS-set are proposed to deal with daily-life decision-making problems having periodic and interval type data/information(5)
The proposed study is compared with some existing relevant models by considering some important evaluating indicators so that the advantageous aspect of the proposed study may be depicted(6)
The generalization and particular cases of proposed models are discussed with the pictorial depiction(7)
The advantages and future directions of the proposed study are presented


## 1.3. Paper Organization

The organization of the remaining paper is given in Figure2.Figure 2
Organization of paper.


## 1.4. Notations and Abbreviations

Some abbreviations and notations are used throughout the paper. Their full names are given in Table2 to facilitate the readers for proper understanding of the concept.Table 2
Notations and abbreviations.
NotationUsed forNotationUsed forℤUniverse of discoursePℤPower set ofℤFℤCollection of fuzzy setsIFℤCollection of intuitionistic fuzzy setsℕℤCollection of neutrosophic setsCℤCollection of complex fuzzy setsCIntℤCollection of complex intuitionistic fuzzy setsCNeuUCollection of complex neutrosophic setsA-termAmplitude termM-functionMembership functionNM-functionNonmembership functionID-functionIndeterminacy functionP-termPhase termCIVℤCollection of interval-valued complex fuzzy sets


## 2. Preliminaries

In this section, some fundamental definitions from literature are presented for the vivid understanding of the proposed study.Definition 1 (see [1]).
LetU be a fuzzy set over ℤ that can be written as U=ν,αUν|ν∈ℤ such that(1)αU:ℤ⟶0,1,where αUν is the membership degree of ν∈U.Definition 2 (see [43]).
A complex fuzzy setFC can be written as follows:(2)FC=δ,ηFCδ:δ∈ℤ=δ,kFCδeiλFCδ:δ∈ℤ,where M-function of FC is ηFCδ with kFCδ∈0,1 as A-term and λFCδ∈0,2π as P-term and i=−1.Buckley [44–46] and Zhang et al. [47] presented fuzzy complex sets and numbers in a different way. Amplitude terms and P-terms in the form of fuzzy sets are discussed in [43, 48].Definition 3 (see [4]).
LetE be set of parameters; then soft set S0 over ℤ is given by(3)S0=δ,ζS0δ:δ∈E1,where ζS0:E1⟶Pℤ and E1 is subset of E.Definition 4 (see [13]).
The fuzzy soft setΩE1 on ℤ is given by(4)ΩE1=δ,ωE1δ:δ∈E1,ωE1δ∈Fℤ,where ωE1:E1⟶Fℤ, where ωE1δ=∅ for δ∉E1 and(5)ωE1δ=ηωE1δv/v:v∈ℤ,ηωE1δv∈Iis a fuzzy set over ℤ. Here, ωE1 is the approximate function of ΩE1, where ωE1δ is a fuzzy set known as δ-element of ΩE1. If ωE1δ=∅, then δ,ωE1δ∉ΩE1.Definition 5 (see [42]).
A complex fuzzy soft setξE1 over ℤ is given by(6)ξE1=δ,χE1δ:δ∈E1,χE1δ∈Cℤ,where χE1:E1⟶Cℤ, where χE1δ=∅ for δ∉E1 is the complex fuzzy approximate function of ξE1, where χE1δ is called δ-member of ξE1 for all δ∈E1. Set operations of complex fuzzy set and complex fuzzy soft set have been described in [42, 47], respectively.Definition 6 (see [39]).
LetE be a set of attributes with A⊆E. Then complex intuitionistic fuzzy soft set ξA=Ψ,A over ℤ is defined as follows:(7)ξA=a,Ψa:a∈A,Ψa∈CIntℤ,where(8)Ψ:A⟶CIntℤ,Ψa=∅ifa∉A,is a complex intuitionistic fuzzy approximate function of ξA and Ψa=ΨTa,ΨFa.
ΨTa=pTeiθT and ΨFa=pFeiθF are complex-valued M-function and complex-valued NM-function of ξA, respectively, and all are lying within unit circle in the complex plane such that pT,pF∈0,1 with 0≤pT+pF≤1or0≤pT+pF≤1 and θT,θF∈0,2π. The value Ψa is called a-member of ξA=Ψ,A∀a∈A.Definition 7 (see [40]).
LetE be a set of attributes with A⊆E. Then complex neutrosophic soft set ξA=Ψ,A over ℤ is defined as follows:(9)ξA=a,Ψa:a∈A,Ψa∈CNeuU,where(10)Ψ:A⟶CNeuℤ,Ψa=∅ifa∉A,is a complex neutrosophic approximate function of ξA and Ψa=ΨTa,ΨIa,ΨFa.
ΨTa=pTeiθT, ΨIa=pIeiθI, and ΨFa=pFeiθF are complex-valued truth M-function, complex-valued indeterminacy M-function, and complex-valued falsity M-function of ξA, respectively, and all are lying within unit circle in the complex plane such that pT,pI,pF∈0,1 with 0−≤pT+pI+pF≤3+or0≤pT+pI+pF≤3 and θT,θI,θF∈0,2π. The value Ψa is called a-member of ξA=Ψ,A∀a∈A.Definition 8 (see [22]).
Hs,D is known as a hypersoft set over ℤ, if Hs:D⟶Pℤ where D is the Cartesian product of finite number of disjoint sets Hs1,Hs2,Hs3,…,Hsn with n distinct attributes hs1,hs2,hs3,…,hsn, respectively.Definition 9 (see [22]).
Fuzzy hypersoft set, intuitionistic fuzzy hypersoft set, and neutrosophic hypersoft set are hypersoft sets defined over fuzzy universe, intuitionistic fuzzy universe, and neutrosophic universe, respectively.The fundamental properties and set theoretic operations of hypersoft set are discussed in [23].Definition 10 (see [24]).
The complex fuzzy hypersoft setCFHS− set ξD over ℤ is given by(11)ξD=δ¯,χδ¯:δ¯∈D,χδ¯∈Cℤ,where ℒ1,ℒ2,ℒ3,…,ℒn are disjoint sets having n distinct attributes l1,l2,l3,…,ln for n≥1, D=ℒ1×ℒ2×ℒ3×⋯×ℒn, and χx⌣ be a CF− set over ℤ for all δ¯=d1,d2,d3,…,dn∈D.(12)χ:D⟶Cℤ,χδ¯=∅ifδ¯∉D,is a complex fuzzy approximate function of ξG and χδ¯ is δ¯− member of CFHS− set ∀δ¯∈D.Definition 11 (see [30]).
LetW1,W2,W3,…,Wn are disjoint sets having attribute values of n distinct attributes w1,w2,w3,…,wn, respectively, for n≥1,W=W1×W2×W3×⋯×Wn and Ψω¯ be an interval-valued complex fuzzy set over ℤ for all ω¯=b1,b2,b3,…,bn∈W. Then interval-valued complex fuzzy hypersoft set (IV-CFHS-set) ΩW=Ψ,W over ℤ is defined as follows:(13)ΩW=ω¯,Ψω¯:ω¯∈W,Ψω¯∈CIVℤwhere(14)Ψ:W⟶CIVℤ,Ψω¯=∅ifω¯∉W,is an interval-valued complex fuzzy approximate function of ΩW and Ψω¯=Ψ←ω¯,Ψ⟶ω¯. Ψ←ω¯=r←eiθ← and Ψ⟶ω¯=r⟶eiθ⟶ are lower and upper bounds of the M-function of ΩW, respectively, and its value Ψω¯ is called ω¯-member of IV-CFHS ∀ω¯∈W.


## 3. Interval-Valued Complex Intuitionistic Fuzzy Hypersoft Set (IV-CIFHS-Set)

Consider the daily-life scenario of the clinical study to diagnose heart diseases in patients, doctors (decision-makers) usually prefer chest pain type, resting blood pressure, serum cholesterol, and so on as diagnostic parameters. After keen analysis, it is vivid that these parameters are required to be further partitioning into their subparametric values, that is, chest pain type (typical angina, atypical angina, etc.), resting blood pressure (110 mmHg, 150 mmHg, 180 mmHg, etc.), and serum cholesterol (210 mg/dl, 320 mg/dl, 430 mg/dl, etc.). Patients are advised to visit medical laboratories for test reports regarding indicated parameters. As the efficiency of medical instruments in laboratories varies that leads to different observations (data) for each patient. This may be categorized in the form of the set having a range of data from the minimum value (lower bounds) to the maximum (upper bounds) that is treated as interval data. Sometimes test reports have repeated values corresponding to these prescribed parameters. This can be of either lab-to-lab basis or day-to-day basis. Such type of data is treated as periodic data. The existing fuzzy set-like literature has no suitable model to deal with (i) subattribute values in the form of disjoint sets, (ii) interval-type data, and (iii) periodic nature of data collectively. In order to meet the demand of literature, the models IV-CIFHS-set and IV-CNHS-set are being characterized. Case (i) is addressed by considering multi-argument approximate function that considers the Cartesian product of attribute-valued disjoint sets as its domain and then maps it to power set of the initial universe (collection of intuitionistic fuzzy sets or neutrosophic sets). Case (ii) is tackled by considering lower and upper limits of reported intervals, and case (iii) is dealt with the introduction of amplitude and phase terms in the Argand plane.Now we develop the theory of complex intuitionistic fuzzy hypersoft set with interval settings in the remaining part of this section.Definition 12.
Letℳ1,ℳ2,ℳ3,…,ℳn are disjoint sets having n distinct attributes m1,m2,m3,…,mn, respectively, for n≥1,ℳ=ℳ1×ℳ2×ℳ3×⋯×ℳn and Λλ¯ be an IV-CIFS over ℤ for all λ¯=b1,b2,b3,…,bn∈ℳ. Then interval-valued complex intuitionistic fuzzy hypersoft set (IV-CIFHS-set), denoted by Ωℳ=Λ,ℳ, over ℤ is defined as follows:(15)Ωℳ=λ¯,Λλ¯:λ¯∈ℳ,Λλ¯∈CIVℤ,where(16)Λ:ℳ⟶CIVℤ,Λλ¯=∅ifλ¯∉ℳ,is a IV−CIF approximate function of Ωℳ and Λλ¯=Λ←1λ¯,Λ⟶1λ¯,Λ←2λ¯,Λ⟶2λ¯ with the following:(i)
Λ←1λ¯=γ←eiθ← and Λ⟶1λ¯=γ⟶eiθ⟶ are lower and upper bounds of the M-function of Ωℳ, respectively(ii)
Λ←2λ¯=γ←eiθ← and Λ⟶2λ¯=γ⟶eiθ⟶ are lower and upper bounds of the NM-function of Ωℳ, respectively, and its value Λλ¯ is called λ¯-member of IV-CIFHS-set for all values of λ¯∈ℳ
Note: The collection of all IV-CIFHS-sets is denoted by⊎IVCIFHS.Example 1.
Considerℳ=e1,e2,e3,…,e8 with interval-valued complex intuitionistic fuzzy sets Λℳe1,Λℳe2,…,Λℳe8 given by(17)Λℳe1=0.4,0.5,0.3,0.4ei0.3,0.5,0.3,0.4πt1,0.1,0.7,0.2,0.2ei0.3,0.5,0.1,0.3πt20.3,0.3,0.3,0.5ei0.1,0.8,0.1,0.1πt3,0.3,0.4,0.2,0.5ei0.2,0.6,0.2,0.3πt4,Λℳe2=0.3,0.6,0.2,0.3ei0.1,0.7,0.1,0.2πt1,0.3,0.4,0.1,0.2ei0.2,0.6,0.1,0.3πt20.2,0.5,0.3,0.4ei0.1,0.5,0.3,0.4πt3,0.4,0.5,0.1,0.4ei0.2,0.5,0.2,0.3πt4,Λℳe3=0.4,0.5,0.2,0.3ei0.2,0.6,0.1,0.2πt1,0.1,0.8,0.1,0.1ei0.2,0.3,0.2,0.5πt20.1,0.4,0.1,0.2ei0.2,0.5,0.3,0.4πt3,0.2,0.6,0.1,0.3ei0.1,0.5,0.3,0.4πt4,Λℳe4=0.2,0.3,0.3,0.6ei0.1,0.7,0.2,0.2πt1,0.2,0.4,0.2,0.3ei0.1,0.5,0.2,0.4πt20.4,0.5,0.3,0.4ei0.1,0.4,0.4,0.5πt3,0.4,0.5,0.2,0.4ei0.2,0.5,0.1,0.3πt4,Λℳe5=0.1,0.2,0.4,0.5ei0.4,0.5,0.1,0.3πt1,0.2,0.2,0.2,0.7ei0.1,0.4,0.3,0.5πt20.2,0.6,0.1,0.3ei0.1,0.2,0.2,0.3πt3,0.3,0.5,0.2,0.3ei0.3,0.4,0.2,0.3πt4,Λℳe6=0.4,0.5,0.1,0.4ei0.2,0.3,0.3,0.6πt1,0.1,0.3,0.4,0.4ei0.2,0.5,0.2,0.3πt20.2,0.7,0.1,0.2ei0.1,0.8,0.1,0.1πt3,0.2,0.5,0.2,0.5ei0.1,0.6,0.2,0.3πt4,Λℳe7=0.2,0.4,0.1,0.2ei0.2,0.3,0.1,0.6πt1,0.1,0.3,0.3,0.5ei0.2,0.3,0.2,0.6πt20.1,0.5,0.3,0.4ei0.1,0.6,0.2,0.3πt3,0.4,0.5,0.3,0.4ei0.2,0.5,0.1,0.2πt4,Λℳe8=0.3,0.3,0.1,0.4ei0.4,0.5,0.3,0.4πt1,0.2,0.3,0.2,0.6ei0.3,0.6,0.1,0.3πt20.1,0.2,0.4,0.5ei0.3,0.5,0.3,0.4πt3,0.2,0.5,0.1,0.4ei0.4,0.5,0.3,0.4πt4,and then IV-CIFHS-set Ωℳ is written by(18)Ωℳ=e1,0.4,0.5,0.3,0.4ei0.3,0.5,0.3,0.4πt1,0.1,0.7,0.2,0.2ei0.3,0.5,0.1,0.3πt2,0.3,0.3,0.3,0.5ei0.1,0.8,0.1,0.1πt3,0.3,0.4,0.2,0.5ei0.2,0.6,0.2,0.3πt4,e2,0.3,0.6,0.2,0.3ei0.1,0.7,0.1,0.2πt1,0.3,0.4,0.1,0.2ei0.2,0.6,0.1,0.3πt20.2,0.5,0.3,0.4ei0.1,0.5,0.3,0.4πt3,0.4,0.5,0.1,0.4ei0.2,0.5,0.2,0.3πt4,e3,0.4,0.5,0.2,0.3ei0.2,0.6,0.1,0.2πt1,0.1,0.8,0.1,0.1ei0.2,0.3,0.2,0.5πt20.1,0.4,0.1,0.2ei0.2,0.5,0.3,0.4πt3,0.2,0.6,0.1,0.3ei0.1,0.5,0.3,0.4πt4,e4,0.2,0.3,0.3,0.6ei0.1,0.7,0.2,0.2πt1,0.2,0.4,0.2,0.3ei0.1,0.5,0.2,0.4πt2,0.4,0.5,0.3,0.4ei0.1,0.4,0.4,0.5πt3,0.4,0.5,0.2,0.4ei0.2,0.5,0.1,0.3πt4,e5,0.1,0.2,0.4,0.5ei0.4,0.5,0.1,0.3πt1,0.2,0.2,0.2,0.7ei0.1,0.4,0.3,0.5πt2,0.2,0.6,0.1,0.3ei0.1,0.2,0.2,0.3πt3,0.3,0.5,0.2,0.3ei0.3,0.4,0.2,0.3πt4,e6,0.4,0.5,0.1,0.4ei0.2,0.3,0.3,0.6πt1,0.1,0.3,0.4,0.4ei0.2,0.5,0.2,0.3πt2,0.2,0.7,0.1,0.2ei0.1,0.8,0.1,0.1πt3,0.2,0.5,0.2,0.5ei0.1,0.6,0.2,0.3πt4,e7,0.2,0.4,0.1,0.2ei0.2,0.3,0.1,0.6πt1,0.1,0.3,0.3,0.5ei0.2,0.3,0.2,0.6πt2,0.1,0.5,0.3,0.4ei0.1,0.6,0.2,0.3πt3,0.4,0.5,0.3,0.4ei0.2,0.5,0.1,0.2πt4e8,0.3,0.3,0.1,0.4ei0.4,0.5,0.3,0.4πt1,0.2,0.3,0.2,0.6ei0.3,0.6,0.1,0.3πt2,0.1,0.2,0.4,0.5ei0.3,0.5,0.3,0.4πt3,0.2,0.5,0.1,0.4ei0.4,0.5,0.3,0.4πt4.Definition 13.
LetΩℳ1=Λ1,ℳ1 and Ωℳ2=Λ2,ℳ2 be two IV-CIFHS-sets over the same universe. The set Ωℳ1=Λ1,ℳ1 is a subset of Ωℳ2=Λ2,ℳ2, if(i)
ℳ1⊆ℳ2(ii)
∀x⌣∈ℳ1,Λ1x⌣⊆Λ2x⌣ implies Λ←1x⌣⊆Λ←2x⌣,Λ⟶1x⌣⊆Λ⟶2x⌣, that is,γ←ℳ1x⌣≤γ←ℳ2x⌣,γ⟶ℳ1x⌣≤γ⟶ℳ2x⌣,θ←ℳ1x⌣≤θ←ℳ2x⌣, and θ⟶ℳ1x⌣≤θ⟶ℳ2x⌣,where amplitude and P-terms of eachΛi,i=1,2 are given below:γ←ℳ1x⌣,θ←ℳ1x⌣ for Λ←1x⌣γ←ℳ1x⌣,θ←ℳ1x⌣ for Λ←1x⌣γ←ℳ2x⌣,θ←ℳ2x⌣ for Λ←2x⌣γ←ℳ2x⌣,θ←ℳ2x⌣ for Λ←2x⌣Definition 14.
Two IV-CIFHS-setsΩℳ1=Λ1,ℳ1 and Ωℳ2=Λ2,ℳ2 over the same universe are equal if(i)
Λ1,ℳ1⊆Λ2,ℳ2(ii)
Λ2,ℳ2⊆Λ1,ℳ1Definition 15.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then(i)
It is a null IV-CIFHS-set, symbolized byΛ,ℳΦ if for all values of x⌣∈ℳ, the A-term of the M-function is given by γ←ℳx⌣=γ⟶ℳx⌣=0, whereas the P-terms of is given by θ←ℳx⌣=θ⟶ℳx⌣=0π(ii)
It is an absolute IV-CIFHS-set, symbolized byΛ,ℳΔ; if for all values of x⌣∈ℳ, the A-terms of the M-function is given by γ←ℳx⌣=γ⟶ℳx⌣=1, whereas the P-terms is given by θ←ℳx⌣=θ⟶ℳx⌣=2π.Definition 16.
For two IV-CIFHS-setsΛ1,ℳ1 and Λ2,ℳ2 over the same universe ℤ, the following definitions hold:(i)
An IV-CIFHS-setΛ1,ℳ1 is a homogeneous IV-CIFHS-set, symbolized by Λ1,ℳ1Hom if Λ1x⌣ is homogeneous CIF-set ∀x⌣∈ℳ1(ii)
An IV-CIFHS-setΛ1,ℳ1 is a completely homogeneous IV-CIFHS-set, symbolized by Λ1,ℳ1CHom if Λ1x⌣ is homogeneous with Λ1y¯∀x⌣,y¯∈ℳ1(iii)
An IV-CIFHS-setΛ1,ℳ1 is a completely homogeneous IV-CIFHS-set with Λ2,ℳ2 if Λ1x⌣ is a homogeneous with Λ2x⌣∀x⌣∈ℳ1∏ℳ2


### 3.1. Set Operations and Laws on IV-CIFHS-Set

In this section, some basic set theoretic operations and laws are discussed on IV-CIFHS-set.Definition 17.
The complement of IV-CIFHS-setΛ,ℳ, denoted by Λ,ℳc, is given by(19)Λ,ℳc=x⌣,Λx⌣c:x⌣∈ℳ,Λx⌣c∈CIVℤ,where the M-function Λx⌣c has the A-term given by γ←ℳx⌣c=1−γ←ℳx⌣γ⟶ℳx⌣c=1−γ⟶ℳx⌣ and P-term given by θ←ℳx⌣c=2π−θ←ℳx⌣ and θ⟶ℳx⌣c=2π−θ⟶ℳx⌣.Proposition 1.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then Λ,ℳcc=Λ,ℳ.Proof.
AsΛx⌣∈CIVℤ, so in terms of its A- and P-terms, Λ,ℳ can be expressed as follows:(20)Λ,ℳ=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Gx⌣:x⌣∈ℳ.
Now(21)Λ,ℳcx⌣=x⌣,γ←ℳx⌣ceiθ←ℳx⌣c,γ⟶ℳx⌣ceiθ⟶Dx⌣c:x⌣∈ℳ,Λ,ℳcx⌣=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Dx⌣c:x⌣∈ℳ,χ,Dcc=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Gx⌣c:x⌣∈ℳ,Λ,ℳcc=x⌣,1−1−γ←ℳx⌣ei2π−2π−θ←ℳx⌣,1−1−γ⟶ℳx⌣ei2π−2π−θ⟶Gx⌣:x⌣∈ℳ,Λ,ℳcc=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Gx⌣:x⌣∈ℳ.From equations (20) and (21), we have Λ,ℳcc=Λ,ℳ.Proposition 2.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then(i)
Λ,ℳΦc=Λ,ℳΔ(ii)
Λ,ℳΔc=Λ,ℳΦDefinition 18.
For two IV-CIFHS-setΛ1,ℳ1 and Λ2,ℳ2, the intersection Λ1,ℳ1∏Λ2,ℳ2 over the same universe ℤ is the IV-CIFHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(22)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifxˇ∈ℳ1ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifxˇ∈ℳ2ℳ1,minγ←ℳ1x⌣,γ←ℳ2x⌣eiminθ←ℳ1x⌣,θ←ℳ2x⌣,ifxˇ∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,minγ⟶ℳ1x⌣,γ⟶ℳ2x⌣eiminθ⟶ℳ1x⌣,θ⟶ℳ2x⌣ifx⌣∈ℳ1∏ℳ2.Definition 19.
For two IV-CIFHS-setΛ1,ℳ1 and Λ2,ℳ2, the difference is defined as follows:(23)Λ1,ℳ1∖Λ2,ℳ2=Λ1,ℳ1∏Λ2,ℳ2c.Definition 20.
The unionΛ1,ℳ1∐Λ2,ℳ2 of two IV-CIFHS-set Λ1,ℳ1 and Λ2,ℳ2 over the same universe ℤ is the IV-CIFHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(24)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifx⌣∈ℳ1\ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifx⌣∈ℳ2\ℳ1,maxγ←ℳ1x⌣,γ←ℳ2x⌣eimaxθ←ℳ1x⌣,θ←ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,maxγ⟶ℳ1x⌣γ⟶ℳ2x⌣eimaxθ⟶ℳ1x⌣,θ⟶ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2.Proposition 3.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then the following results hold true:(i)
Λ,ℳ∐Λ,ℳΦ=Λ,ℳ(ii)
Λ,ℳ∐Λ,ℳΔ=Λ,ℳΔ(iii)
Λ,ℳ∏Λ,ℳΦ=Λ,ℳΦ(iv)
Λ,ℳ∏Λ,ℳΔ=Λ,ℳ(v)
Λ,ℳΦ∐Λ,ℳΔ=Λ,ℳΔ(vi)
Λ,ℳΦ∏Λ,ℳΔ=Λ,ℳΦProposition 4.
LetΛ1,ℳ1, Λ2,ℳ2, and Λ3,ℳ3 are three IV-CIFHS-sets over the same universe ℤ. Then the following commutative and associative laws hold true:(i)
Λ1,ℳ1∏Λ2,ℳ2=Λ2,ℳ2∏Λ1,ℳ1(ii)
Λ1,ℳ1∐Λ2,ℳ2=Λ2,ℳ2∐Λ1,ℳ1(iii)
Λ1,ℳ1∏Λ2,ℳ2∏Λ3,ℳ3=Λ1,ℳ1∏Λ2,ℳ2∏Λ3,ℳ3(iv)
Λ1,ℳ1∐Λ2,ℳ2∐Λ3,ℳ3=Λ1,ℳ1∐Λ2,ℳ2∐Λ3,ℳ3Proposition 5.
LetΛ1,ℳ1 and Λ2,ℳ2 are two IV-CIFHS-sets over the same universe ℤ. Then the following De Morgans’s laws hold true:(i)
Λ1,ℳ1∏Λ2,ℳ2c=Λ1,ℳ1c∐Λ2,ℳ2c(ii)
Λ1,ℳ1∐Λ2,ℳ2c=Λ1,ℳ1c∏Λ2,ℳ2c


### 3.2. Aggregation of Interval-Valued Complex Intuitionistic Fuzzy Hypersoft Set

In this section, we define an aggregation operator on IV-CIFHS-set that produces an aggregate fuzzy set from an IV-CIFHS-set and its cardinal set. The approximate functions of an IV-CIFHS-set are fuzzy. Here,D,E,ξD and ⊎IVCIFHS will be in accordance with Definition 12.Definition 21.
LetξD∈⊎IVCIFHS. Assume that ℤ=ν1,ν2,…,νm and E=ℒ1,ℒ2,…,ℒn with ℒ1=e11,e12,…,e1n,ℒ2=e21,e22,…,e2n,…,ℒn=en1,en2,…,enn and D=ℒ1×ℒ2×⋯×ℒn=x⌣1,x⌣2,…,x⌣n,…,x⌣nn=x⌣r, each x⌣i is n-tuple element of D and D=r=nn; the representation of ξD can be seen in Table 3.
In Table3, ηχDx1 and ηχDx2 are M-function and NM-function of χD, respectively, with interval-valued intuitionistic fuzzy values. If αij=ηχDx⌣j1νi,ηχDx⌣j2νi, for i=ℕ1m and j=ℕ1r, then IV-CIFHS-set ξD is uniquely characterized by the following matrix:(25)αij=α11α12⋯α1rα21α22⋯α2r⋮⋮⋱⋮αm1αm2⋯αmr,which is called an m×r IV-CIFHS-set matrix.Table 3
Tabular representation ofξD.
ξDx⌣1x⌣2…x⌣rν1ηχDx⌣11ν1,ηχDx⌣12ν1ηχDx⌣21ν1,ηχDx⌣22ν1⋯ηχDx⌣r1ν1,ηχDx⌣r2ν1ν1ηχDx⌣11ν2,ηχDx⌣12ν2ηχDx⌣21ν2,ηχDx⌣22ν2⋯ηχDx⌣r1ν2,ηχDx⌣r2ν2⋮⋮⋮⋱⋮ν1ηχDx⌣11νm,ηχDx⌣12νmηχDx⌣21νm,ηχDx⌣22νm⋯ηχDx⌣r1νm,ηχDx⌣r2νmDefinition 22.
IfξD∈⊎IVCIFHS, then cardinal set of ξD is defined as follows:(26)ξD=ηξD1x⌣,ηξD2x⌣x⌣:x⌣∈D,where ηξD1:D⟶0,1 is an M-function of ξD with ηξD1x⌣=χDx⌣/ℤ and ηξD2:D⟶0,1 is an NM-function of ξD with ηξD2x⌣=χDx⌣/ℤ. Both have interval-valued intuitionistic fuzzy values. Note that Civcifhssℤ is the collection of all cardinal sets of IV-CIFHS-sets and Civcifhssℤ⊆IVIFD.Definition 23.
LetξD∈Civcifhssℤ and ξD∈Civcifhssℤ. Consider E as in Definition 12, the representation of ξD can be seen in Table 4.
Ifα1j=ηξD1x⌣j,ηξD2x⌣j, for j=ℕ1r, then the cardinal set ξD is represented by the following matrix:(27)αij1×r=α11α12⋯α1r,which is called the cardinal matrix of ξD.Table 4
Tabular representation ofξD.
Dxˇ1xˇ2⋯xˇrηξDηξD1x⌣1,ηξD2x⌣1ηξD1x⌣2,ηξD2x⌣2⋯ηξD1x⌣r,ηξD2x⌣rDefinition 24.
LetξD∈Civcifhssℤ and ξD∈Civcifhssℤ. Then IV-CIFHS-aggregation operator is defined as follows:(28)ξD︷=AivcifhssξD,ξD,where(29)Aivcifhss:Civcifhssℤ×Civcifhssℤ⟶Fℤ,
whereξD︷ is called the aggregate fuzzy set of IV-CIFHS-set ξD.
Its M-function is given as follows:(30)ηξD︷:ℤ⟶0,1,with(31)ηξD︷ν=1D∑x⌣∈DηCardξDx⌣ηCardχDν.Definition 25.
LetξD∈Civcifhssℤ and ξD︷ be its aggregate fuzzy set. Assume that ℤ=ν1,ν2,…,νm, then ξD︷ can be presented as follows:(32)ξD⋮ηξD︷……⋮……ν1⋮ηξD︷ν1ν2⋮ηξD︷ν2⋮⋮⋮νm⋮ηξD︷νm.
Ifαi1=ηξD︷νi for i=ℕ1m, then ξD︷ is represented by the following matrix:(33)αi1m×1=α11α21⋮αm1,which is called the aggregate matrix of ξD︷ over ℤ.


### 3.3. Applications of Interval-Valued Complex Intuitionistic Fuzzy Hypersoft Set

In this section, an algorithm is presented to solve the problems in decision-making by having under consideration the concept of aggregations defined in the previous section. An example is demonstrated to explain the proposed algorithm. It is necessary to determine an aggregate fuzzy set of IV-CIFHS-set for choosing the best option (parameter) from the given set (set of choices/alternatives). The following algorithm is proposed based on the definitions given in Subsection3.2 that may help make optimal decision.Algorithm1: Decision-support algorithm based on aggregations of an IV-CIFHS-set
⊳ Start⊳ Input Stage:———–1. Considerℤ as the universe of discourse———–2. ConsiderE as a set of n parameters———–3. Classifyn parameters into disjoint parametric-valued sets ℒ1,ℒ2,ℒ3,…,ℒn⊳ Construction Stage:———–4.D=ℒ1×ℒ2×ℒ3×⋯×ℒn———–5. Construct IV-CIFHS-setξD over ℤ in accordance with Definition 12⊳ Computation Stage:———–6. ComputeξD for A-term and P-term separately by using Definition 22———–7. ComputeξD︷ for A-term and P-term separately with the help of Definition 24———–8. ComputeηξD︷ν by using Definition 24⊳ Output Stage:———–9. Find the best option by the max modulus ofηξD︷ν with the help of Definition 25⊳ EndNow, Algorithm1 is explained with the help of the following example.Example 2.
Suppose a businessman wants to buy a share from the share market. There are four same kinds of share that form the set,ℤ=s1,s2,s3,s4. The expert committee considers a set of attributes, E=e1,e2,e3. For i = 1, 2, 3, the attributes ei stand for “current trend of company performance,” “particular company’s stock price for last one year,” and “Home country inflation rate,” respectively. Corresponding to each attribute, the sets of attribute values are ℒ1=e11,e12, ℒ2=e21, and ℒ3=e31,e32. Then the set D=ℒ1×ℒ2×ℒ3=δ1,δ2,δ3,δ4, where each δi is a three-tuple. IV-CIFHS-sets χDδ1,χDδ2,χDδ3,χDδ4 are defined as follows:(34)χDδ1=0.4,0.5,0.3,0.4ei0.1,0.4,0.3,0.5πs1,0.2,0.3,0.2,0.6ei0.2,0.3,0.3,0.5πs20.1,0.4,0.4,0.5ei0.3,0.4,0.1,0.3πs3,0.4,0.5,0.2,0.4ei0.1,0.4,0.1,0.2πs4,χDδ2=0.1,0.2,0.2,0.7ei0.2,0.4,0.3,0.5πs1,0.1,0.6,0.2,0.3ei0.4,0.5,0.2,0.3πs20.2,0.3,0.1,0.6ei0.2,0.2,0.1,0.5πs3,0.2,0.4,0.3,0.4ei0.1,0.3,0.3,0.6πs4.χDδ3=0.2,0.3,0.2,0.5ei0.1,0.4,0.4,0.5πs1,0.2,0.4,0.1,0.3ei0.3,0.6,0.2,0.3πs2,0.2,0.4,0.1,0.3ei0.2,0.5,0.1,0.2πs3,0.3,0.3,0.2,0.6ei0.2,0.5,0.2,0.4πs4,χDδ4=0.3,0.5,0.3,0.4ei0.4,0.5,0.3,0.4πs1,0.1,0.3,0.3,0.5ei0.4,0.4,0.3,0.5πs20.1,0.4,0.3,0.5ei0.3,0.5,0.1,0.2πs3,0.4,0.5,0.2,0.4ei0.3,0.5,0.2,0.3πs4,Step 1.
IV-CIFHS-setξD is written as follows:(35)ξD=δ1,0.4,0.5,0.3,0.4ei0.1,0.4,0.3,0.5πs1,0.2,0.3,0.2,0.6ei0.2,0.3,0.3,0.5πs2,0.1,0.4,0.4,0.5ei0.3,0.4,0.1,0.3πs3,0.4,0.5,0.2,0.4ei0.1,0.4,0.1,0.2πs4δ2,0.1,0.2,0.2,0.7ei0.2,0.4,0.3,0.5πs1,0.1,0.6,0.2,0.3ei0.4,0.5,0.2,0.3πs20.2,0.3,0.1,0.6ei0.2,0.2,0.1,0.5πs3,0.2,0.4,0.3,0.4ei0.1,0.3,0.3,0.6πs4δ3,0.2,0.3,0.2,0.5ei0.1,0.4,0.4,0.5πs1,0.2,0.4,0.1,0.3ei0.3,0.6,0.2,0.3πs20.2,0.4,0.1,0.3ei0.2,0.5,0.1,0.2πs3,0.3,0.3,0.2,0.6ei0.2,0.5,0.2,0.4πs4δ4,0.3,0.5,0.3,0.4ei0.4,0.5,0.3,0.4πs1,0.1,0.3,0.3,0.5ei0.4,0.4,0.3,0.5πs20.1,0.4,0.3,0.5ei0.3,0.5,0.1,0.2πs3,0.4,0.5,0.2,0.4ei0.3,0.5,0.2,0.3πs4.Step 2.
The cardinal is computed as follows:(36)ξDAmplitude Term=0.275,0.425,0.275,0.475δ1,0.150,0.375,0.200,0.500δ20.225,0.350,0.150,0.425δ3,0.225,0.425,0.275,0.450δ4,ξDPhase Term=0.175,0.375,0.200,0.375δ1,0.225,0.350,0.225,0.475δ20.200,0.500,0.225,0.350δ3,0.350,0.475,0.225,0.350δ4.Step 3.
The setξD︷ can be determined as follows:(37)ξD︷Apmlitude Term=14m11m12m13m14m21m22m23m24m31m32m33m34m41m42m43m440.275,0.425,0.275,0.4750.150,0.375,0.200,0.5000.225,0.350,0.150,0.4250.225,0.425,0.275,0.450,where(38)m11=0.4,0.5,0.3,0.4,m12=0.1,0.2,0.2,0.7,m13=0.2,0.3,0.2,0.5,m14=0.3,0.5,0.3,0.4,m21=0.2,0.3,0.2,0.6,m22=0.1,0.6,0.2,0.3,m23=0.2,0.4,0.1,0.3,m24=0.1,0.3,0.3,0.5,m31=0.1,0.4,0.4,0.5,m32=0.2,0.3,0.1,0.6,m33=0.2,0.4,0.1,0.3,m34=0.1,0.4,0.3,0.5,m41=0.4,0.5,0.2,0.4,m42=0.2,0.4,0.3,0.4,m43=0.3,0.3,0.2,0.6,m44=0.4,0.5,0.2,0.4=140.20.00.10.20.10.40.30.00.00.20.30.10.30.10.10.30.1500.1750.2000.150=0.0200000.0362500.0275000.031875.
ξD︷PhaseTerm=(39)140.1,0.4,0.3,0.50.2,0.4,0.3,0.50.1,0.4,0.4,0.50.4,0.5,0.3,0.40.2,0.3,0.3,0.50.4,0.5,0.2,0.30.3,0.6,0.2,0.30.4,0.4,0.3,0.50.3,0.4,0.1,0.30.2,0.2,0.1,0.50.2,0.5,0.1,0.20.3,0.5,0.1,0.20.1,0.4,0.1,0.20.1,0.3,0.3,0.60.2,0.5,0.2,0.40.3,0.5,0.2,0.30.175,0.375,0.200,0.3750.225,0.350,0.225,0.4750.200,0.500,0.225,0.3500.350,0.475,0.225,0.350=140.10.10.00.20.00.30.40.10.30.10.40.40.30.00.30.30.1750.1250.2750.250=0.0200000.0431250.0687500.052500,ξD︷=0.020000ei0.020000πs1,0.036250ei0.043125πs2,0.027500ei0.068750πs3,0.031875ei0.052500πs4.
Consider the modulus value ofmaxηξD︷=max0.02124999361/s1,0.01624997922/s2,0.0131496449/s3,0.00749999154/s4=0.02124999361/s1. This means that s1 may be recommended for suitable investment.


## 3.1. Set Operations and Laws on IV-CIFHS-Set

In this section, some basic set theoretic operations and laws are discussed on IV-CIFHS-set.Definition 17.
The complement of IV-CIFHS-setΛ,ℳ, denoted by Λ,ℳc, is given by(19)Λ,ℳc=x⌣,Λx⌣c:x⌣∈ℳ,Λx⌣c∈CIVℤ,where the M-function Λx⌣c has the A-term given by γ←ℳx⌣c=1−γ←ℳx⌣γ⟶ℳx⌣c=1−γ⟶ℳx⌣ and P-term given by θ←ℳx⌣c=2π−θ←ℳx⌣ and θ⟶ℳx⌣c=2π−θ⟶ℳx⌣.Proposition 1.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then Λ,ℳcc=Λ,ℳ.Proof.
AsΛx⌣∈CIVℤ, so in terms of its A- and P-terms, Λ,ℳ can be expressed as follows:(20)Λ,ℳ=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Gx⌣:x⌣∈ℳ.
Now(21)Λ,ℳcx⌣=x⌣,γ←ℳx⌣ceiθ←ℳx⌣c,γ⟶ℳx⌣ceiθ⟶Dx⌣c:x⌣∈ℳ,Λ,ℳcx⌣=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Dx⌣c:x⌣∈ℳ,χ,Dcc=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Gx⌣c:x⌣∈ℳ,Λ,ℳcc=x⌣,1−1−γ←ℳx⌣ei2π−2π−θ←ℳx⌣,1−1−γ⟶ℳx⌣ei2π−2π−θ⟶Gx⌣:x⌣∈ℳ,Λ,ℳcc=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Gx⌣:x⌣∈ℳ.From equations (20) and (21), we have Λ,ℳcc=Λ,ℳ.Proposition 2.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then(i)
Λ,ℳΦc=Λ,ℳΔ(ii)
Λ,ℳΔc=Λ,ℳΦDefinition 18.
For two IV-CIFHS-setΛ1,ℳ1 and Λ2,ℳ2, the intersection Λ1,ℳ1∏Λ2,ℳ2 over the same universe ℤ is the IV-CIFHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(22)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifxˇ∈ℳ1ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifxˇ∈ℳ2ℳ1,minγ←ℳ1x⌣,γ←ℳ2x⌣eiminθ←ℳ1x⌣,θ←ℳ2x⌣,ifxˇ∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,minγ⟶ℳ1x⌣,γ⟶ℳ2x⌣eiminθ⟶ℳ1x⌣,θ⟶ℳ2x⌣ifx⌣∈ℳ1∏ℳ2.Definition 19.
For two IV-CIFHS-setΛ1,ℳ1 and Λ2,ℳ2, the difference is defined as follows:(23)Λ1,ℳ1∖Λ2,ℳ2=Λ1,ℳ1∏Λ2,ℳ2c.Definition 20.
The unionΛ1,ℳ1∐Λ2,ℳ2 of two IV-CIFHS-set Λ1,ℳ1 and Λ2,ℳ2 over the same universe ℤ is the IV-CIFHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(24)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifx⌣∈ℳ1\ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifx⌣∈ℳ2\ℳ1,maxγ←ℳ1x⌣,γ←ℳ2x⌣eimaxθ←ℳ1x⌣,θ←ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,maxγ⟶ℳ1x⌣γ⟶ℳ2x⌣eimaxθ⟶ℳ1x⌣,θ⟶ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2.Proposition 3.
LetΛ,ℳ be an IV-CIFHS-set over ℤ. Then the following results hold true:(i)
Λ,ℳ∐Λ,ℳΦ=Λ,ℳ(ii)
Λ,ℳ∐Λ,ℳΔ=Λ,ℳΔ(iii)
Λ,ℳ∏Λ,ℳΦ=Λ,ℳΦ(iv)
Λ,ℳ∏Λ,ℳΔ=Λ,ℳ(v)
Λ,ℳΦ∐Λ,ℳΔ=Λ,ℳΔ(vi)
Λ,ℳΦ∏Λ,ℳΔ=Λ,ℳΦProposition 4.
LetΛ1,ℳ1, Λ2,ℳ2, and Λ3,ℳ3 are three IV-CIFHS-sets over the same universe ℤ. Then the following commutative and associative laws hold true:(i)
Λ1,ℳ1∏Λ2,ℳ2=Λ2,ℳ2∏Λ1,ℳ1(ii)
Λ1,ℳ1∐Λ2,ℳ2=Λ2,ℳ2∐Λ1,ℳ1(iii)
Λ1,ℳ1∏Λ2,ℳ2∏Λ3,ℳ3=Λ1,ℳ1∏Λ2,ℳ2∏Λ3,ℳ3(iv)
Λ1,ℳ1∐Λ2,ℳ2∐Λ3,ℳ3=Λ1,ℳ1∐Λ2,ℳ2∐Λ3,ℳ3Proposition 5.
LetΛ1,ℳ1 and Λ2,ℳ2 are two IV-CIFHS-sets over the same universe ℤ. Then the following De Morgans’s laws hold true:(i)
Λ1,ℳ1∏Λ2,ℳ2c=Λ1,ℳ1c∐Λ2,ℳ2c(ii)
Λ1,ℳ1∐Λ2,ℳ2c=Λ1,ℳ1c∏Λ2,ℳ2c


## 3.2. Aggregation of Interval-Valued Complex Intuitionistic Fuzzy Hypersoft Set

In this section, we define an aggregation operator on IV-CIFHS-set that produces an aggregate fuzzy set from an IV-CIFHS-set and its cardinal set. The approximate functions of an IV-CIFHS-set are fuzzy. Here,D,E,ξD and ⊎IVCIFHS will be in accordance with Definition 12.Definition 21.
LetξD∈⊎IVCIFHS. Assume that ℤ=ν1,ν2,…,νm and E=ℒ1,ℒ2,…,ℒn with ℒ1=e11,e12,…,e1n,ℒ2=e21,e22,…,e2n,…,ℒn=en1,en2,…,enn and D=ℒ1×ℒ2×⋯×ℒn=x⌣1,x⌣2,…,x⌣n,…,x⌣nn=x⌣r, each x⌣i is n-tuple element of D and D=r=nn; the representation of ξD can be seen in Table 3.
In Table3, ηχDx1 and ηχDx2 are M-function and NM-function of χD, respectively, with interval-valued intuitionistic fuzzy values. If αij=ηχDx⌣j1νi,ηχDx⌣j2νi, for i=ℕ1m and j=ℕ1r, then IV-CIFHS-set ξD is uniquely characterized by the following matrix:(25)αij=α11α12⋯α1rα21α22⋯α2r⋮⋮⋱⋮αm1αm2⋯αmr,which is called an m×r IV-CIFHS-set matrix.Table 3
Tabular representation ofξD.
ξDx⌣1x⌣2…x⌣rν1ηχDx⌣11ν1,ηχDx⌣12ν1ηχDx⌣21ν1,ηχDx⌣22ν1⋯ηχDx⌣r1ν1,ηχDx⌣r2ν1ν1ηχDx⌣11ν2,ηχDx⌣12ν2ηχDx⌣21ν2,ηχDx⌣22ν2⋯ηχDx⌣r1ν2,ηχDx⌣r2ν2⋮⋮⋮⋱⋮ν1ηχDx⌣11νm,ηχDx⌣12νmηχDx⌣21νm,ηχDx⌣22νm⋯ηχDx⌣r1νm,ηχDx⌣r2νmDefinition 22.
IfξD∈⊎IVCIFHS, then cardinal set of ξD is defined as follows:(26)ξD=ηξD1x⌣,ηξD2x⌣x⌣:x⌣∈D,where ηξD1:D⟶0,1 is an M-function of ξD with ηξD1x⌣=χDx⌣/ℤ and ηξD2:D⟶0,1 is an NM-function of ξD with ηξD2x⌣=χDx⌣/ℤ. Both have interval-valued intuitionistic fuzzy values. Note that Civcifhssℤ is the collection of all cardinal sets of IV-CIFHS-sets and Civcifhssℤ⊆IVIFD.Definition 23.
LetξD∈Civcifhssℤ and ξD∈Civcifhssℤ. Consider E as in Definition 12, the representation of ξD can be seen in Table 4.
Ifα1j=ηξD1x⌣j,ηξD2x⌣j, for j=ℕ1r, then the cardinal set ξD is represented by the following matrix:(27)αij1×r=α11α12⋯α1r,which is called the cardinal matrix of ξD.Table 4
Tabular representation ofξD.
Dxˇ1xˇ2⋯xˇrηξDηξD1x⌣1,ηξD2x⌣1ηξD1x⌣2,ηξD2x⌣2⋯ηξD1x⌣r,ηξD2x⌣rDefinition 24.
LetξD∈Civcifhssℤ and ξD∈Civcifhssℤ. Then IV-CIFHS-aggregation operator is defined as follows:(28)ξD︷=AivcifhssξD,ξD,where(29)Aivcifhss:Civcifhssℤ×Civcifhssℤ⟶Fℤ,
whereξD︷ is called the aggregate fuzzy set of IV-CIFHS-set ξD.
Its M-function is given as follows:(30)ηξD︷:ℤ⟶0,1,with(31)ηξD︷ν=1D∑x⌣∈DηCardξDx⌣ηCardχDν.Definition 25.
LetξD∈Civcifhssℤ and ξD︷ be its aggregate fuzzy set. Assume that ℤ=ν1,ν2,…,νm, then ξD︷ can be presented as follows:(32)ξD⋮ηξD︷……⋮……ν1⋮ηξD︷ν1ν2⋮ηξD︷ν2⋮⋮⋮νm⋮ηξD︷νm.
Ifαi1=ηξD︷νi for i=ℕ1m, then ξD︷ is represented by the following matrix:(33)αi1m×1=α11α21⋮αm1,which is called the aggregate matrix of ξD︷ over ℤ.


## 3.3. Applications of Interval-Valued Complex Intuitionistic Fuzzy Hypersoft Set

In this section, an algorithm is presented to solve the problems in decision-making by having under consideration the concept of aggregations defined in the previous section. An example is demonstrated to explain the proposed algorithm. It is necessary to determine an aggregate fuzzy set of IV-CIFHS-set for choosing the best option (parameter) from the given set (set of choices/alternatives). The following algorithm is proposed based on the definitions given in Subsection3.2 that may help make optimal decision.Algorithm1: Decision-support algorithm based on aggregations of an IV-CIFHS-set
⊳ Start⊳ Input Stage:———–1. Considerℤ as the universe of discourse———–2. ConsiderE as a set of n parameters———–3. Classifyn parameters into disjoint parametric-valued sets ℒ1,ℒ2,ℒ3,…,ℒn⊳ Construction Stage:———–4.D=ℒ1×ℒ2×ℒ3×⋯×ℒn———–5. Construct IV-CIFHS-setξD over ℤ in accordance with Definition 12⊳ Computation Stage:———–6. ComputeξD for A-term and P-term separately by using Definition 22———–7. ComputeξD︷ for A-term and P-term separately with the help of Definition 24———–8. ComputeηξD︷ν by using Definition 24⊳ Output Stage:———–9. Find the best option by the max modulus ofηξD︷ν with the help of Definition 25⊳ EndNow, Algorithm1 is explained with the help of the following example.Example 2.
Suppose a businessman wants to buy a share from the share market. There are four same kinds of share that form the set,ℤ=s1,s2,s3,s4. The expert committee considers a set of attributes, E=e1,e2,e3. For i = 1, 2, 3, the attributes ei stand for “current trend of company performance,” “particular company’s stock price for last one year,” and “Home country inflation rate,” respectively. Corresponding to each attribute, the sets of attribute values are ℒ1=e11,e12, ℒ2=e21, and ℒ3=e31,e32. Then the set D=ℒ1×ℒ2×ℒ3=δ1,δ2,δ3,δ4, where each δi is a three-tuple. IV-CIFHS-sets χDδ1,χDδ2,χDδ3,χDδ4 are defined as follows:(34)χDδ1=0.4,0.5,0.3,0.4ei0.1,0.4,0.3,0.5πs1,0.2,0.3,0.2,0.6ei0.2,0.3,0.3,0.5πs20.1,0.4,0.4,0.5ei0.3,0.4,0.1,0.3πs3,0.4,0.5,0.2,0.4ei0.1,0.4,0.1,0.2πs4,χDδ2=0.1,0.2,0.2,0.7ei0.2,0.4,0.3,0.5πs1,0.1,0.6,0.2,0.3ei0.4,0.5,0.2,0.3πs20.2,0.3,0.1,0.6ei0.2,0.2,0.1,0.5πs3,0.2,0.4,0.3,0.4ei0.1,0.3,0.3,0.6πs4.χDδ3=0.2,0.3,0.2,0.5ei0.1,0.4,0.4,0.5πs1,0.2,0.4,0.1,0.3ei0.3,0.6,0.2,0.3πs2,0.2,0.4,0.1,0.3ei0.2,0.5,0.1,0.2πs3,0.3,0.3,0.2,0.6ei0.2,0.5,0.2,0.4πs4,χDδ4=0.3,0.5,0.3,0.4ei0.4,0.5,0.3,0.4πs1,0.1,0.3,0.3,0.5ei0.4,0.4,0.3,0.5πs20.1,0.4,0.3,0.5ei0.3,0.5,0.1,0.2πs3,0.4,0.5,0.2,0.4ei0.3,0.5,0.2,0.3πs4,Step 1.
IV-CIFHS-setξD is written as follows:(35)ξD=δ1,0.4,0.5,0.3,0.4ei0.1,0.4,0.3,0.5πs1,0.2,0.3,0.2,0.6ei0.2,0.3,0.3,0.5πs2,0.1,0.4,0.4,0.5ei0.3,0.4,0.1,0.3πs3,0.4,0.5,0.2,0.4ei0.1,0.4,0.1,0.2πs4δ2,0.1,0.2,0.2,0.7ei0.2,0.4,0.3,0.5πs1,0.1,0.6,0.2,0.3ei0.4,0.5,0.2,0.3πs20.2,0.3,0.1,0.6ei0.2,0.2,0.1,0.5πs3,0.2,0.4,0.3,0.4ei0.1,0.3,0.3,0.6πs4δ3,0.2,0.3,0.2,0.5ei0.1,0.4,0.4,0.5πs1,0.2,0.4,0.1,0.3ei0.3,0.6,0.2,0.3πs20.2,0.4,0.1,0.3ei0.2,0.5,0.1,0.2πs3,0.3,0.3,0.2,0.6ei0.2,0.5,0.2,0.4πs4δ4,0.3,0.5,0.3,0.4ei0.4,0.5,0.3,0.4πs1,0.1,0.3,0.3,0.5ei0.4,0.4,0.3,0.5πs20.1,0.4,0.3,0.5ei0.3,0.5,0.1,0.2πs3,0.4,0.5,0.2,0.4ei0.3,0.5,0.2,0.3πs4.Step 2.
The cardinal is computed as follows:(36)ξDAmplitude Term=0.275,0.425,0.275,0.475δ1,0.150,0.375,0.200,0.500δ20.225,0.350,0.150,0.425δ3,0.225,0.425,0.275,0.450δ4,ξDPhase Term=0.175,0.375,0.200,0.375δ1,0.225,0.350,0.225,0.475δ20.200,0.500,0.225,0.350δ3,0.350,0.475,0.225,0.350δ4.Step 3.
The setξD︷ can be determined as follows:(37)ξD︷Apmlitude Term=14m11m12m13m14m21m22m23m24m31m32m33m34m41m42m43m440.275,0.425,0.275,0.4750.150,0.375,0.200,0.5000.225,0.350,0.150,0.4250.225,0.425,0.275,0.450,where(38)m11=0.4,0.5,0.3,0.4,m12=0.1,0.2,0.2,0.7,m13=0.2,0.3,0.2,0.5,m14=0.3,0.5,0.3,0.4,m21=0.2,0.3,0.2,0.6,m22=0.1,0.6,0.2,0.3,m23=0.2,0.4,0.1,0.3,m24=0.1,0.3,0.3,0.5,m31=0.1,0.4,0.4,0.5,m32=0.2,0.3,0.1,0.6,m33=0.2,0.4,0.1,0.3,m34=0.1,0.4,0.3,0.5,m41=0.4,0.5,0.2,0.4,m42=0.2,0.4,0.3,0.4,m43=0.3,0.3,0.2,0.6,m44=0.4,0.5,0.2,0.4=140.20.00.10.20.10.40.30.00.00.20.30.10.30.10.10.30.1500.1750.2000.150=0.0200000.0362500.0275000.031875.
ξD︷PhaseTerm=(39)140.1,0.4,0.3,0.50.2,0.4,0.3,0.50.1,0.4,0.4,0.50.4,0.5,0.3,0.40.2,0.3,0.3,0.50.4,0.5,0.2,0.30.3,0.6,0.2,0.30.4,0.4,0.3,0.50.3,0.4,0.1,0.30.2,0.2,0.1,0.50.2,0.5,0.1,0.20.3,0.5,0.1,0.20.1,0.4,0.1,0.20.1,0.3,0.3,0.60.2,0.5,0.2,0.40.3,0.5,0.2,0.30.175,0.375,0.200,0.3750.225,0.350,0.225,0.4750.200,0.500,0.225,0.3500.350,0.475,0.225,0.350=140.10.10.00.20.00.30.40.10.30.10.40.40.30.00.30.30.1750.1250.2750.250=0.0200000.0431250.0687500.052500,ξD︷=0.020000ei0.020000πs1,0.036250ei0.043125πs2,0.027500ei0.068750πs3,0.031875ei0.052500πs4.
Consider the modulus value ofmaxηξD︷=max0.02124999361/s1,0.01624997922/s2,0.0131496449/s3,0.00749999154/s4=0.02124999361/s1. This means that s1 may be recommended for suitable investment.


## 4. Interval-Valued Complex Neutrosophic Hypersoft Set (IV-CNHS-Set)

The basic theory of the IV-CNHS-set is developed in this section.Definition 27.
Letℳ1,ℳ2,ℳ3,…,ℳn are disjoint sets of n distinct attributes m1,m2,m3,…,mn, respectively, for n≥1,ℳ=ℳ1×ℳ2×ℳ3×…×ℳn and Λλ¯ be an IV-CNFS over ℤ for all λ¯=b1,b2,b3,…,bn∈ℳ. Then, interval-valued complex neutrosophic hypersoft set (IV-CNHS-set), denoted by Ωℳ=Λ,ℳ, over ℤ is given as follows:(40)Ωℳ=λ¯,Λλ¯:λ¯∈ℳ,Λλ¯∈CIVℤ,where(41)Λ:ℳ⟶CIVℤ,Λλ¯=∅ifλ¯∉ℳ,which is a IV−CN approximate function of Ωℳ and Λλ¯=Λ←1λ¯,Λ⟶1λ¯,Λ←2λ¯,Λ⟶2λ¯,Λ←3λ¯,Λ⟶3λ¯ with lower and upper bounds of membership, nonmembership, and indeterminacy function are given below, respectively:(i)
Λ←1λ¯=γ←eiθ←,Λ⟶1λ¯=γ⟶eiθ⟶ for the M-function of Ωℳ(ii)
Λ←2λ¯=γ←eiθ←,Λ⟶2λ¯=γ⟶eiθ⟶ for the NM-function of Ωℳ(iii)
Λ←3λ¯=γ←eiθ←,Λ⟶3λ¯=γ⟶eiθ⟶ for the ID-function of Ωℳ and its value Λλ¯ is called λ¯-member of IV-CNHS-set ∀λ¯∈ℳNote: The collection of all IV-CNHS-sets is denoted by⊎IVCNHS.Example 3.
Considerℳ=e1,e2,e3,…,e8 with IV-CNHS-sets Λℳe1,Λℳe2,…,Λℳe8 that are defined as follows:(42)Λℳe1=0.1,0.2,0.1,0.4,0.2,0.3ei0.1,0.2,0.3,0.4,0.3,0.3πt1,0.1,0.3,0.2,0.4,0.2,0.2ei0.1,0.20.3,0.4,0.1,0.3πt20.1,0.2,0.3,0.3,0.3,0.4ei0.1,0.1,0.4,0.5,0.3,0.3πt3,0.1,0.2,0.3,0.4,0.2,0.3ei0.2,0.2,0.2,0.4,0.2,0.3πt4,Λℳe2=0.2,0.3,0.3,0.3,0.1,0.3ei0.1,0.2,0.1,0.5,0.1,0.2πt1,0.3,0.4,0.3,0.4,0.1,0.1ei0.1,0.2,0.2,0.4,0.2,0.3πt20.2,0.3,0.2,0.4,0.2,0.3ei0.1,0.2,0.1,0.4,0.3,0.3πt3,0.1,0.1,0.4,0.4,0.3,0.4ei0.1,0.3,0.2,0.2,0.2,0.3πt4,Λℳe3=0.1,0.3,0.2,0.4,0.2,0.2ei0.1,0.2,0.2,0.3,0.3,0.4πt1,0.1,0.3,0.1,0.2,0.3,0.4ei0.1,0.2,0.2,0.3,0.2,0.4πt20.1,0.1,0.1,0.4,0.1,0.2ei0.1,0.2,0.2,0.4,0.3,0.3πt3,0.1,0.3,0.2,0.3,0.1,0.3ei0.2,0.3,0.1,0.2,0.3,0.4πt4,Λℳe4=0.1,0.2,0.2,0.3,0.1,0.4ei0.1,0.2,0.1,0.3,0.2,0.2πt1,0.1,0.3,0.2,0.4,0.1,0.2ei0.1,0.2,0.1,0.3,0.2,0.4πt20.4,0.4,0.1,0.2,0.1,0.1ei0.1,0.2,0.1,0.4,0.2,0.3πt3,0.1,0.1,0.2,0.3,0.2,0.4ei0.1,0.3,0.2,0.2,0.1,0.3πt4,Λℳe5=0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.3,0.2,0.3,0.1,0.3πt1,0.1,0.2,0.2,0.2,0.2,0.5ei0.1,0.1,0.1,0.4,0.3,0.4πt20.1,0.2,0.2,0.6,0.1,0.1ei0.1,0.1,0.1,0.2,0.2,0.3πt3,0.1,0.3,0.3,0.4,0.1,0.2ei0.1,0.2,0.3,0.4,0.1,0.3πt4,Λℳe6=0.2,0.2,0.3,0.4,0.1,0.2ei0.1,0.2,0.2,0.3,0.3,0.4πt1,0.1,0.1,0.1,0.3,0.4,0.4ei0.1,0.1,0.2,0.5,0.2,0.3πt2,0.1,0.2,0.2,0.4,0.1,0.3ei0.1,0.2,0.1,0.6,0.1,0.1πt3,0.1,0.1,0.2,0.5,0.2,0.3ei0.1,0.2,0.1,0.6,0.1,0.1πt4,Λℳe7=0.1,0.2,0.2,0.4,0.1,0.3ei0.1,0,1,0.2,0.3,0.1,0.5πt1,0.1,0.2,0.1,0.3,0.3,0.4ei0.1,0.2,0.2,0.2,0.2,0.5πt20.1,0.2,0.1,0.3,0.3,0.4ei0.1,0.2,0.1,0.3,0.2,0.4πt3,0.1,0.2,0.4,0.5,0.1,0.2ei0.1,0.1,0.2,0.5,0.1,0.2πt4,(43)Λℳe8=0.1,0.2,0.3,0.3,0.1,0.4ei0.1.0.2,0.4,0.5,0.1,0.2πt1,0.1,0.2,0.2,0.3,0.2,0.4ei0.1,0.2,0.1,0.2,0.1,0.3πt20.1,0.10.1,0.2,0.4,0.5ei0.1,0.2,0.1,0.3,0.3,0.4πt3,0.1,0.2,0.2,0.3,0.1,0.4ei0.1,0.3,0.4,0.5,0.1,0.1πt4.Then IV-CNHS-set Ωℳ is written by
ΩW=(44)e1,0.1,0.2,0.1,0.4,0.2,0.3ei0.1,0.2,0.3,0.4,0.3,0.3πt1,0.1,0.3,0.2,0.4,0.2,0.2ei0.1,0.20.3,0.4,0.1,0.3πt20.1,0.2,0.3,0.3,0.3,0.4ei0.1,0.1,0.4,0.5,0.3,0.3πt3,0.1,0.2,0.3,0.4,0.2,0.3ei0.2,0.2,0.2,0.4,0.2,0.3πt4e2,0.2,0.3,0.3,0.3,0.1,0.3ei0.1,0.2,0.1,0.5,0.1,0.2πt1,0.3,0.4,0.3,0.4,0.1,0.1ei0.1,0.2,0.2,0.4,0.2,0.3πt20.2,0.3,0.2,0.4,0.2,0.3ei0.1,0.2,0.1,0.4,0.3,0.3πt3,0.1,0.1,0.4,0.4,0.3,0.4ei0.1,0.3,0.2,0.2,0.2,0.3πt4e3,0.1,0.3,0.2,0.4,0.2,0.2ei0.1,0.2,0.2,0.3,0.3,0.4πt1,0.1,0.3,0.1,0.2,0.3,0.4ei0.1,0.2,0.2,0.3,0.2,0.4πt20.1,0.1,0.1,0.4,0.1,0.2ei0.1,0.2,0.2,0.4,0.3,0.3πt3,0.1,0.3,0.2,0.3,0.1,0.3ei0.2,0.3,0.1,0.2,0.3,0.4πt4e4,0.1,0.2,0.2,0.3,0.1,0.4ei0.1,0.2,0.1,0.3,0.2,0.2πt1,0.1,0.3,0.2,0.4,0.1,0.2ei0.1,0.2,0.1,0.3,0.2,0.4πt20.4,0.4,0.1,0.2,0.1,0.1ei0.1,0.2,0.1,0.4,0.2,0.3πt3,0.1,0.1,0.2,0.3,0.2,0.4ei0.1,0.3,0.2,0.2,0.1,0.3πt4e5,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.3,0.2,0.3,0.1,0.3πt1,0.1,0.2,0.2,0.2,0.2,0.5ei0.1,0.1,0.1,0.4,0.3,0.4πt20.1,0.2,0.2,0.6,0.1,0.1ei0.1,0.1,0.1,0.2,0.2,0.3πt3,0.1,0.3,0.3,0.4,0.1,0.2ei0.1,0.2,0.3,0.4,0.1,0.3πt4e6,0.2,0.2,0.3,0.4,0.1,0.2ei0.1,0.2,0.2,0.3,0.3,0.4πt1,0.1,0.1,0.1,0.3,0.4,0.4ei0.1,0.1,0.2,0.5,0.2,0.3πt2,0.1,0.2,0.2,0.4,0.1,0.3ei0.1,0.2,0.1,0.6,0.1,0.1πt3,0.1,0.1,0.2,0.5,0.2,0.3ei0.1,0.2,0.1,0.6,0.1,0.1πt4e7,0.1,0.2,0.2,0.4,0.1,0.3ei0.1,0,1,0.2,0.3,0.1,0.5πt1,0.1,0.2,0.1,0.3,0.3,0.4ei0.1,0.2,0.2,0.2,0.2,0.5πt20.1,0.2,0.1,0.3,0.3,0.4ei0.1,0.2,0.1,0.3,0.2,0.4πt3,0.1,0.2,0.4,0.5,0.1,0.2ei0.1,0.1,0.2,0.5,0.1,0.2πt4e8,0.1,0.2,0.2,0.4,0.1,0.3ei0.1,0,1,0.2,0.3,0.1,0.5πt1,0.1,0.2,0.1,0.3,0.3,0.4ei0.1,0.2,0.2,0.2,0.2,0.5πt20.1,0.2,0.1,0.3,0.3,0.4ei0.1,0.2,0.1,0.3,0.2,0.4πt3,0.1,0.2,0.4,0.5,0.1,0.2ei0.1,0.1,0.2,0.5,0.1,0.2πt4.Definition 28.
LetΩℳ1=Λ1,ℳ1 and Ωℳ2=Λ2,ℳ2 be two IV-CNHS-set over the same ℤ. The set Ωℳ1=Λ1,ℳ1 is said to be the subset of Ωℳ2=Λ2,ℳ2, if(i)
ℳ1⊆ℳ2(ii)
∀x⌣∈ℳ1,Λ1x⌣⊆Λ2x⌣ implies Λ←1x⌣⊆Λ←2x⌣,Λ⟶1x⌣⊆Λ⟶2x⌣, that is,γ←ℳ1x⌣≤γ←ℳ2x⌣,γ⟶ℳ1x⌣≤γ⟶ℳ2x⌣,θ←ℳ1x⌣≤θ←ℳ2x⌣ and θ⟶ℳ1x⌣≤θ⟶ℳ2x⌣,where A- and P-terms of eachΛi,i=1,2 are given below:γ←ℳ1x⌣,θ←ℳ1x⌣ for Λ←1x⌣γ⟶ℳ1x⌣,θ⟶ℳ1x⌣ for Λ←1x⌣γ⟶ℳ2x⌣,θ⟶ℳ2x⌣ for Λ←2x⌣γ⟶ℳ2x⌣,θ⟶ℳ2x⌣ for Λ←2x⌣Definition 29.
Two IV-CNHS-setΩℳ1=Λ1,ℳ1 and Ωℳ2=Λ2,ℳ2 over the same ℤ, are said to be equal if(i)
Λ1,ℳ1⊆Λ2,ℳ2(ii)
Λ2,ℳ2⊆Λ1,ℳ1Definition 30.
LetΛ,ℳ be a IV-CNHS-set over ℤ. Then(i)
Λ,ℳ is called a null IV-CNHS-set, presented by Λ,ℳΦ; if for all values of x⌣∈ℳ, the A-term of the M-function is given by γ←ℳx⌣=γ⟶ℳx⌣=0 and P-term by θ←ℳx⌣=θ⟶ℳx⌣=0π(ii)
Λ,ℳ is called an absolute IV-CNHS-set, represented by Λ,ℳΔ; if ∀x⌣∈ℳ, the A-term of the M-function is given by γ←ℳx⌣=γ⟶ℳx⌣=1 whereas P-term by θ←ℳx⌣=θ⟶ℳx⌣=2πDefinition 31.
For two IV-CNHS-setsΛ1,ℳ1 and Λ2,ℳ2 over the same universe ℤ, the following definitions hold:(i)
A IV-CNHS-setΛ1,ℳ1 is a homogeneous IV-CNHS-set, represented by Λ1,ℳ1Hom if and only if Λ1x⌣ is a homogeneous IV-CNHS-set ∀x⌣∈ℳ1(ii)
A IV-CNHS-setΛ1,ℳ1 is a completely homogeneous IV-CNHS-set, represented by Λ1,ℳ1CHom if and only if Λ1x⌣ is a homogeneous with Λ1y¯∀x⌣,y¯∈ℳ1(iii)
A IV-CNHS-setΛ1,ℳ1 is a completely homogeneous IV-CNHS-set with Λ2,ℳ2 if and only if Λ1x⌣ is a homogeneous with Λ2x⌣∀x⌣∈ℳ1∏ℳ2


### 4.1. Set Theoretic Operations and Laws on IV-CNHS-Set

Some basic set operations and laws are discussed on IV-CNHS-set in this section.Definition 32.
The complement of IV-CNHS-setΛ,ℳ, denoted by Λ,ℳc, is stated as follows:(45)Λ,ℳc=x⌣,Λx⌣c:x⌣∈ℳ,Λx⌣c∈CIVℤ,where the A- and P-terms of the M-function Λx⌣c are given by γ←ℳx⌣c=1−γ←ℳx⌣ and γ⟶ℳx⌣c=1−γ⟶ℳx⌣ and θ←ℳx⌣c=2π−θ←ℳx⌣ and θ⟶ℳx⌣c=2π−θ⟶ℳx⌣, respectively.Proposition 6.
LetΛ,ℳ be a IV-CNHS-set over ℤ. Then Λ,ℳcc=Λ,ℳ.Proof.
SinceΛx⌣∈CIVℤ, therefore, in terms of its A- and P-terms, Λ,ℳ can be written as follows:(46)Λ,ℳ=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Dx⌣:x⌣∈ℳ.
Now,(47)Λ,ℳcx⌣=x⌣,γ←ℳx⌣ceiθ←ℳx⌣c,γ⟶ℳx⌣ceiθ⟶Dx⌣c:x⌣∈ℳ,Λ,ℳcx⌣=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Dx⌣c:x⌣∈ℳ,ψ,Gcc=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Dx⌣c:x⌣∈ℳ,Λ,ℳcc=x⌣,1−1−γ←ℳx⌣ei2π−2π−θ←ℳx⌣,1−1−γ⟶ℳx⌣ei2π−2π−θ⟶Dx⌣:x⌣∈ℳ,Λ,ℳcc=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Dx⌣:x⌣∈ℳ.From equations (46) and (47), we have Λ,ℳcc=Λ,ℳ.Proposition 7.
LetΛ,ℳ be a IV-CNHS-set over ℤ. Then(i)
Λ,ℳΦc=Λ,ℳΔ(ii)
Λ,ℳΔc=Λ,ℳΦDefinition 33.
For two IV-CNHS-setΛ1,ℳ1 and Λ2,ℳ2, the intersection Λ1,ℳ1∏Λ2,ℳ2 over the same universe ℤ is the IV-CNHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(48)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifx⌣∈ℳ1\ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifx⌣∈ℳ2ℳ1,minγ←ℳ1x⌣,γ←ℳ2x⌣eiminθ←ℳ1x⌣,θ←ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1\ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,minγ⟶ℳ1x⌣,γ⟶ℳ2x⌣eiminθ⟶ℳ1x⌣,θ⟶ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2.Definition 34.
The difference between two IV-CNHS-setΛ1,ℳ1 and Λ2,ℳ2 is defined as follows:(49)Λ1,ℳ1∖Λ2,ℳ2=Λ1,ℳ1∏Λ2,ℳ2c.Definition 35.
For two IV-CNHS-setΛ1,ℳ1 and Λ2,ℳ2, the union Λ1,ℳ1∐Λ2,ℳ2 over the same universe ℤ is the IV-CNHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(50)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifx⌣∈ℳ2ℳ1,maxγ←ℳ1x⌣,γ←ℳ2x⌣eimaxθ←ℳ1x⌣,θ←ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,maxγ⟶ℳ1x⌣,γ⟶ℳ2x⌣eimaxθ⟶ℳ1x⌣,θ⟶ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2.Note: Propositions3–5 are also valid for SV-NHS-sets.


### 4.2. Aggregation of Interval-Valued Complex Neutrosophic Hypersoft Set

In this part of the paper, aggregation operations and their decisive system for IV-CNHS-set are established that results an aggregate fuzzy set from a CNHS-set and its cardinal set and has fuzzy-like nature. The termsD,E,ξD and ⊎IVCNHS are in accordance with Definition 27.Definition 36.
LetξD∈⊎IVCNHS. Assume that ℤ=ν1,ν2,…,νm and E=ℒ1,ℒ2,…,ℒn with ℒ1=e11,e12,…,e1n,ℒ2=e21,e22,…,e2n,…,ℒn=en1,en2,…,enn and D=ℒ1×ℒ2×⋯×ℒn=x⌣1,x⌣2,…,x⌣n,…,x⌣nn=x⌣r, each x⌣i is n-tuple element of D and D=r=nn; the representation of ξD can be seen in Table 5.
In Table5, ηχDx1,ηχDx2, and ηχDx2 are M-function, ID-function, and NM-function of χD, respectively, with interval-valued neutrosophic values. If αij=ηχDx⌣j1νi,ηχDx⌣j2νi,ηχDx⌣j3νi, for i=ℕ1m and j=ℕ1r, then IV-CNHS-set ξD is uniquely characterized by the following matrix:(51)αij=α11α12⋯α1rα21α22⋯α2r⋮⋮⋱⋮αm1αm2⋯αmr,Which is called an m×r IV-CNHS-set matrix.Table 5
Tabular representation ofξD.
ξDx⌣1x⌣2…x⌣rν1ηχDx⌣11ν1,ηχDx⌣12ν1,ηχDx⌣13ν1ηχDx⌣21ν1,ηχDx⌣22ν1,ηχDx⌣23ν1⋯ηχDx⌣r1ν1,ηχDx⌣r2ν1,ηχDx⌣r3ν1ν2ηχDx⌣11ν2,ηχDx⌣12ν2,ηχDx⌣13ν2ηχDx⌣21ν2,ηχDx⌣22ν2,ηχDx⌣23ν2⋯ηχDx⌣r1ν2,ηχDx⌣r2ν2,ηχDx⌣r3ν2⋮⋮⋮⋱⋮νmηχDx⌣11νm,ηχDx⌣12νm,ηχDx⌣13νmηχDx⌣21νm,ηχDx⌣22νm,ηχDx⌣23νm⋯ηχDx⌣r1νm,ηχDx⌣r2νm,ηχDx⌣r3νmDefinition 37.
IfξD∈⊎IVCNHS, then cardinal set of ξD is defined as follows:(52)ξD=ηξD1x⌣,ηξD2x⌣,ηξD3x⌣x⌣:x⌣∈D,where ηξD1,ηξD2,ηξD3:D⟶0,1 are M-function, ID-function, and NM-function of ξD with(53)ηξD1x⌣,ηξD2x⌣,ηξD3x⌣=χDx⌣ℤ,respectively. These have interval-valued neutrosophic values. Note that Civcnhssℤ is the collection of all cardinal sets of IV-CNHS-sets and Civcnhssℤ⊆IVℕD.Definition 38.
LetξD∈Civcnhssℤ and ξD∈Civcnhssℤ. Consider E as in Definition 27; the tabular representation of ξD can be seen in Table 6.
Ifα1j=ηξD1x⌣j,ηξD2x⌣j,ηξD3x⌣j, for j=ℕ1r. then the cardinal set ξD is represented by the following matrix:(54)αij1×r=α11α12⋯α1r,which is called the cardinal matrix of ξD.Table 6
Tabular representation ofξD.
Dx⌣1x⌣2⋯x⌣rηξDηξD1x⌣1,ηξD2x⌣1,ηξD3x⌣1,ηξD1x⌣2,ηξD2x⌣2,ηξD3x⌣2,⋯ηξD1x⌣r,ηξD2x⌣r,ηξD3x⌣r,Definition 39.
LetξD∈Civcnhssℤ and ξD∈Civcnhssℤ. Then IV-CNHS-aggregation operator is defined as follows:(55)ξD︷=AivcifhssξD,ξD,where(56)Aivcnhss:Civcnhssℤ×Civcifhssℤ⟶Fℤ,where ξD︷ is called the aggregate fuzzy set of IV-CNHS-set ξD.
Its M-function is given as follows:(57)ηξD︷:ℤ⟶0,1,with(58)ηξD︷ν=1D∑x⌣∈DηCardξDx⌣ηCardχDν.Definition 40.
LetξD∈Civcnhssℤ and ξD︷ be its aggregate fuzzy set. Assume that ℤ=ν1,ν2,…,νm, then ξD︷ can be presented as follows:(59)ξD⋮ηξD︷……⋮……ν1⋮ηξD︷ν1ν2⋮ηξD︷ν2⋮⋮⋮νm⋮ηξD︷νm.
Ifαi1=ηξD︷νi for i=ℕ1m, then ξD︷ is represented by the following matrix:(60)αi1m×1=α11α21⋮αm1,which is called the aggregate matrix of ξD︷ over ℤ.


### 4.3. Applications of Interval-Valued Complex Neutrosophic Hypersoft Set

Now considering the definitions stated in Subsection4.2, an algorithm is presented in this segment of the paper to assist the decision-support system, and the presented algorithm will be validated with the help of an example from a daily-life scenario.Now, Algorithm2 is explained with the help of the following example.Algorithm2: Decision-support algorithm based on aggregations of IV-CNHS-set.
⊳ Start⊳ Input Stage:———–1. Considerℤ as the universe of discourse———–2. ConsiderE as a set of n parameters———–3. Classifyn parameters into disjoint parametric-valued sets ℒ1,ℒ2,ℒ3,…,ℒn⊳ Construction Stage:———–4.D=ℒ1×ℒ2×ℒ3×⋯×ℒn———–5. Construct IV-CNHS-setχD over ℤ, in accordance with Definition 27⊳ Computation Stage:———–6. ComputeξD for A-term and P-term separately by using Definition 37———–7. ComputeξD︷ for A-term and P-term separately with the help of Definition 39———–8. ComputeηξD︷ν by using Definition 39⊳ Output Stage:———–9. Find the best option by the max modulus ofηξD︷ν with the help of Definition 40⊳ EndExample 4.
Considering the problem statement stated in Example 2, we construct IV-CNHS-setsψDδ1,ψDδ2 and ψDδ3,ψDδ4 that are defined as follows:(61)ψDδ1=0.4,0.5,0.1,0.2,0.1,0.2ei0.1,0.2,0.1,0.4,0.3,0.3πs1,0.1,0.1,0.2,0.3,0.3,0.4ei0.1,0.3,0.2,0.3,0.3,0.3πs20.1,0.2,0.1,0.4,0.2,0.2ei0.3,0.4,0.3,0.4,0.1,0.2πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.1,0.4,0.1,0.2πs4,ψDδ2=0.1,0.2,0.3,0.5,0.2,0.2ei0.1,0.2,0.2,0.40.2,0.2πs1,0.1,0.2,0.1,0.5,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs20.1,0.1,0.2,0.3,0.1,0.5ei0.3,0.4,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.2,0.4,0.3,0.3ei0.2,0.3,0.1,0.3,0.3,0.3πs4,ψDδ3=0.3,0.4,0.2,0.3,0.1,0.2ei0.1,0.2,0.1,0.4,0.2,0.3πs1,0.1,0.2,0.2,0.3,0.1,0.3ei0.2,0.3,0.3,0.4,0.2,0.2πs20.1,0.2,0.2,0.4,0.1,0.3ei0.3,0.3,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.3,0.3,0.2,0.3ei0.1,0.2,0.2,0.5,0.2,0.2πs4,ψDδ4=0.1,0.3,0.3,0.4,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs1,0.4,0.5,0.1,0.3,0.1,0.1ei0.1,0.2,0.1,0.4,0.2,0.3πs20.1,0.1,0.1,0.4,0.1,0.2ei0.2,0.3,0.3,0.3,0.3,0.3πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.2,0.4,0.2,0.3πs4.Step 1: IV-CNH-setχD is written as follows:χD=(62)δ1,0.4,0.5,0.1,0.2,0.1,0.2ei0.1,0.2,0.1,0.4,0.3,0.3πs1,0.1,0.1,0.2,0.3,0.3,0.4ei0.1,0.3,0.2,0.3,0.3,0.3πs20.1,0.2,0.1,0.4,0.2,0.2ei0.3,0.4,0.3,0.4,0.1,0.2πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.1,0.4,0.1,0.2πs4δ2,0.1,0.2,0.3,0.5,0.2,0.2ei0.1,0.2,0.2,0.4,0.2,0.2πs1,0.1,0.2,0.1,0.5,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs20.1,0.1,0.2,0.3,0.1,0.5ei0.3,0.4,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.2,0.4,0.3,0.3ei0.2,0.3,0.1,0.3,0.3,0.3πs4δ3,0.3,0.4,0.2,0.3,0.1,0.2ei0.1,0.2,0.1,0.4,0.2,0.3πs1,0.1,0.2,0.2,0.3,0.1,0.3ei0.2,0.3,0.3,0.4,0.2,0.2πs20.1,0.2,0.2,0.4,0.1,0.3ei0.3,0.3,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.3,0.3,0.2,0.3ei0.1,0.2,0.2,0.5,0.2,0.2πs4δ4,0.1,0.3,0.3,0.4,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs1,0.4,0.5,0.1,0.3,0.1,0.1ei0.1,0.2,0.1,0.4,0.2,0.3πs20.1,0.1,0.1,0.4,0.1,0.2ei0.2,0.3,0.3,0.3,0.3,0.3πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.2,0.4,0.2,0.3πs4.Step 2: the cardinal is computed as follows:χDAmplitude Term=(63)0.175,0.250,0.200,0.350,0.200,0.250δ1,0.100,0.175,0.200,0.425,0.175,0.300δ20.150,0.250,0.225,0.325,0.125,0.275δ3,0.175,0.275,0.225,0.400,0.125,0.175δ4.χDPhase Term=(64)0.150,0.275,0.175,0.375,0.200,0.250δ1,0.175,0.275,0.225,0.375,0.200,0.225δ20.175,0.250,0.200,0.400,0.175,0.225δ3,0.125,0.225,0.250,0.425,0.225,0.275δ4.Step 3: the setχD︷ can be determined as follows:χD︷Apmlitude Term=(65)140.4,0.5,0.1,0.2,0.1,0.20.1,0.2,0.3,0.5,0.2,0.20.3,0.4,0.2,0.3,0.1,0.20.1,0.3,0.3,0.4,0.1,0.20.1,0.1,0.2,0.3,0.3,0.40.1,0.2,0.1,0.5,0.1,0.20.1,0.2,0.2,0.3,0.1,0.30.4,0.5,0.1,0.3,0.1,0.10.1,0.2,0.1,0.4,0.2,0.20.1,0.1,0.2,0.3,0.1,0.50.1,0.2,0.2,0.4,0.1,0.30.1,0.1,0.1,0.4,0.1,0.20.1,0.2,0.4,0.5,0.2,0.20.1,0.2,0.2,0.4,0.3,0.30.1,0.2,0.3,0.3,0.2,0.30.1,0.2,0.4,0.5,0.2,0.2×0.175,0.250,0.200,0.350,0.200,0.2500.100,0.175,0.200,0.425,0.175,0.3000.150,0.250,0.225,0.325,0.125,0.2750.175,0.275,0.225,0.400,0.125,0.175=140.30.30.10.10.40.00.10.30.10.20.10.10.40.30.30.40.1500.2000.1000.075=0.0306250.0231250.0181250.045000.χD︷Phase Term=(66)140.1,0.2,0.1,0.4,0.3,0.30.1,0.2,0.2,0.4,0.2,0.20.1,0.2,0.1,0.4,0.2,0.30.1,0.2,0.4,0.5,0.2,0.20.1,0.3,0.2,0.3,0.3,0.30.1,0.2,0.4,0.5,0.2,0.20.2,0.3,0.3,0.4,0.2,0.20.1,0.2,0.1,0.4,0.2,0.30.3,0.4,0.3,0.4,0.1,0.20.3,0.4,0.2,0.3,0.1,0.20.3,0.3,0.2,0.3,0.1,0.20.2,0.3,0.3,0.3,0.3,0.30.1,0.2,0.1,0.4,0.1,0.20.2,0.3,0.1,0.3,0.3,0.30.1,0.2,0.2,0.5,0.2,0.20.1,0.2,0.2,0.4,0.2,0.3×0.150,0.275,0.175,0.375,0.200,0.2500.175,0.275,0.225,0.375,0.200,0.2250.175,0.250,0.200,0.400,0.175,0.2250.125,0.225,0.250,0.425,0.225,0.275=140.20.20.10.40.20.40.20.10.00.10.00.30.00.10.20.20.1000.3000.1250.250=0.0481250.0475000.0262500.026250χD︷=0.030625ei0.048135πs1,0.023125ei0.047500πs2,0.018125ei0.026250πs3,0.045000ei0.026250πs4.
Consider the modulus value ofmaxηξD︷=max0.08562297237/s1,0.08312368641/s2,0.0812493106/s3,0.0793734965/s4=0.08562297237/s1. This means that s1 may be recommended for suitable investment.


## 4.1. Set Theoretic Operations and Laws on IV-CNHS-Set

Some basic set operations and laws are discussed on IV-CNHS-set in this section.Definition 32.
The complement of IV-CNHS-setΛ,ℳ, denoted by Λ,ℳc, is stated as follows:(45)Λ,ℳc=x⌣,Λx⌣c:x⌣∈ℳ,Λx⌣c∈CIVℤ,where the A- and P-terms of the M-function Λx⌣c are given by γ←ℳx⌣c=1−γ←ℳx⌣ and γ⟶ℳx⌣c=1−γ⟶ℳx⌣ and θ←ℳx⌣c=2π−θ←ℳx⌣ and θ⟶ℳx⌣c=2π−θ⟶ℳx⌣, respectively.Proposition 6.
LetΛ,ℳ be a IV-CNHS-set over ℤ. Then Λ,ℳcc=Λ,ℳ.Proof.
SinceΛx⌣∈CIVℤ, therefore, in terms of its A- and P-terms, Λ,ℳ can be written as follows:(46)Λ,ℳ=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Dx⌣:x⌣∈ℳ.
Now,(47)Λ,ℳcx⌣=x⌣,γ←ℳx⌣ceiθ←ℳx⌣c,γ⟶ℳx⌣ceiθ⟶Dx⌣c:x⌣∈ℳ,Λ,ℳcx⌣=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Dx⌣c:x⌣∈ℳ,ψ,Gcc=x⌣,1−γ←ℳx⌣cei2π−θ←ℳx⌣c,1−γ⟶ℳx⌣cei2π−θ⟶Dx⌣c:x⌣∈ℳ,Λ,ℳcc=x⌣,1−1−γ←ℳx⌣ei2π−2π−θ←ℳx⌣,1−1−γ⟶ℳx⌣ei2π−2π−θ⟶Dx⌣:x⌣∈ℳ,Λ,ℳcc=x⌣,γ←ℳx⌣eiθ←ℳx⌣,γ⟶ℳx⌣eiθ⟶Dx⌣:x⌣∈ℳ.From equations (46) and (47), we have Λ,ℳcc=Λ,ℳ.Proposition 7.
LetΛ,ℳ be a IV-CNHS-set over ℤ. Then(i)
Λ,ℳΦc=Λ,ℳΔ(ii)
Λ,ℳΔc=Λ,ℳΦDefinition 33.
For two IV-CNHS-setΛ1,ℳ1 and Λ2,ℳ2, the intersection Λ1,ℳ1∏Λ2,ℳ2 over the same universe ℤ is the IV-CNHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(48)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifx⌣∈ℳ1\ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifx⌣∈ℳ2ℳ1,minγ←ℳ1x⌣,γ←ℳ2x⌣eiminθ←ℳ1x⌣,θ←ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1\ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,minγ⟶ℳ1x⌣,γ⟶ℳ2x⌣eiminθ⟶ℳ1x⌣,θ⟶ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2.Definition 34.
The difference between two IV-CNHS-setΛ1,ℳ1 and Λ2,ℳ2 is defined as follows:(49)Λ1,ℳ1∖Λ2,ℳ2=Λ1,ℳ1∏Λ2,ℳ2c.Definition 35.
For two IV-CNHS-setΛ1,ℳ1 and Λ2,ℳ2, the union Λ1,ℳ1∐Λ2,ℳ2 over the same universe ℤ is the IV-CNHS-set Λ3,ℳ3, where ℳ3=ℳ1∐ℳ2, and for all x⌣∈ℳ3,(50)Λ←3x⌣=γ←ℳ1x⌣eiθ←ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ←ℳ2x⌣eiθ←ℳ2x⌣,ifx⌣∈ℳ2ℳ1,maxγ←ℳ1x⌣,γ←ℳ2x⌣eimaxθ←ℳ1x⌣,θ←ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2,Λ⟶3x⌣=γ⟶ℳ1x⌣eiθ⟶ℳ1x⌣,ifx⌣∈ℳ1ℳ2,γ⟶ℳ2x⌣eiθ⟶ℳ2x⌣,ifx⌣∈ℳ2ℳ1,maxγ⟶ℳ1x⌣,γ⟶ℳ2x⌣eimaxθ⟶ℳ1x⌣,θ⟶ℳ2x⌣,ifx⌣∈ℳ1∏ℳ2.Note: Propositions3–5 are also valid for SV-NHS-sets.


## 4.2. Aggregation of Interval-Valued Complex Neutrosophic Hypersoft Set

In this part of the paper, aggregation operations and their decisive system for IV-CNHS-set are established that results an aggregate fuzzy set from a CNHS-set and its cardinal set and has fuzzy-like nature. The termsD,E,ξD and ⊎IVCNHS are in accordance with Definition 27.Definition 36.
LetξD∈⊎IVCNHS. Assume that ℤ=ν1,ν2,…,νm and E=ℒ1,ℒ2,…,ℒn with ℒ1=e11,e12,…,e1n,ℒ2=e21,e22,…,e2n,…,ℒn=en1,en2,…,enn and D=ℒ1×ℒ2×⋯×ℒn=x⌣1,x⌣2,…,x⌣n,…,x⌣nn=x⌣r, each x⌣i is n-tuple element of D and D=r=nn; the representation of ξD can be seen in Table 5.
In Table5, ηχDx1,ηχDx2, and ηχDx2 are M-function, ID-function, and NM-function of χD, respectively, with interval-valued neutrosophic values. If αij=ηχDx⌣j1νi,ηχDx⌣j2νi,ηχDx⌣j3νi, for i=ℕ1m and j=ℕ1r, then IV-CNHS-set ξD is uniquely characterized by the following matrix:(51)αij=α11α12⋯α1rα21α22⋯α2r⋮⋮⋱⋮αm1αm2⋯αmr,Which is called an m×r IV-CNHS-set matrix.Table 5
Tabular representation ofξD.
ξDx⌣1x⌣2…x⌣rν1ηχDx⌣11ν1,ηχDx⌣12ν1,ηχDx⌣13ν1ηχDx⌣21ν1,ηχDx⌣22ν1,ηχDx⌣23ν1⋯ηχDx⌣r1ν1,ηχDx⌣r2ν1,ηχDx⌣r3ν1ν2ηχDx⌣11ν2,ηχDx⌣12ν2,ηχDx⌣13ν2ηχDx⌣21ν2,ηχDx⌣22ν2,ηχDx⌣23ν2⋯ηχDx⌣r1ν2,ηχDx⌣r2ν2,ηχDx⌣r3ν2⋮⋮⋮⋱⋮νmηχDx⌣11νm,ηχDx⌣12νm,ηχDx⌣13νmηχDx⌣21νm,ηχDx⌣22νm,ηχDx⌣23νm⋯ηχDx⌣r1νm,ηχDx⌣r2νm,ηχDx⌣r3νmDefinition 37.
IfξD∈⊎IVCNHS, then cardinal set of ξD is defined as follows:(52)ξD=ηξD1x⌣,ηξD2x⌣,ηξD3x⌣x⌣:x⌣∈D,where ηξD1,ηξD2,ηξD3:D⟶0,1 are M-function, ID-function, and NM-function of ξD with(53)ηξD1x⌣,ηξD2x⌣,ηξD3x⌣=χDx⌣ℤ,respectively. These have interval-valued neutrosophic values. Note that Civcnhssℤ is the collection of all cardinal sets of IV-CNHS-sets and Civcnhssℤ⊆IVℕD.Definition 38.
LetξD∈Civcnhssℤ and ξD∈Civcnhssℤ. Consider E as in Definition 27; the tabular representation of ξD can be seen in Table 6.
Ifα1j=ηξD1x⌣j,ηξD2x⌣j,ηξD3x⌣j, for j=ℕ1r. then the cardinal set ξD is represented by the following matrix:(54)αij1×r=α11α12⋯α1r,which is called the cardinal matrix of ξD.Table 6
Tabular representation ofξD.
Dx⌣1x⌣2⋯x⌣rηξDηξD1x⌣1,ηξD2x⌣1,ηξD3x⌣1,ηξD1x⌣2,ηξD2x⌣2,ηξD3x⌣2,⋯ηξD1x⌣r,ηξD2x⌣r,ηξD3x⌣r,Definition 39.
LetξD∈Civcnhssℤ and ξD∈Civcnhssℤ. Then IV-CNHS-aggregation operator is defined as follows:(55)ξD︷=AivcifhssξD,ξD,where(56)Aivcnhss:Civcnhssℤ×Civcifhssℤ⟶Fℤ,where ξD︷ is called the aggregate fuzzy set of IV-CNHS-set ξD.
Its M-function is given as follows:(57)ηξD︷:ℤ⟶0,1,with(58)ηξD︷ν=1D∑x⌣∈DηCardξDx⌣ηCardχDν.Definition 40.
LetξD∈Civcnhssℤ and ξD︷ be its aggregate fuzzy set. Assume that ℤ=ν1,ν2,…,νm, then ξD︷ can be presented as follows:(59)ξD⋮ηξD︷……⋮……ν1⋮ηξD︷ν1ν2⋮ηξD︷ν2⋮⋮⋮νm⋮ηξD︷νm.
Ifαi1=ηξD︷νi for i=ℕ1m, then ξD︷ is represented by the following matrix:(60)αi1m×1=α11α21⋮αm1,which is called the aggregate matrix of ξD︷ over ℤ.


## 4.3. Applications of Interval-Valued Complex Neutrosophic Hypersoft Set

Now considering the definitions stated in Subsection4.2, an algorithm is presented in this segment of the paper to assist the decision-support system, and the presented algorithm will be validated with the help of an example from a daily-life scenario.Now, Algorithm2 is explained with the help of the following example.Algorithm2: Decision-support algorithm based on aggregations of IV-CNHS-set.
⊳ Start⊳ Input Stage:———–1. Considerℤ as the universe of discourse———–2. ConsiderE as a set of n parameters———–3. Classifyn parameters into disjoint parametric-valued sets ℒ1,ℒ2,ℒ3,…,ℒn⊳ Construction Stage:———–4.D=ℒ1×ℒ2×ℒ3×⋯×ℒn———–5. Construct IV-CNHS-setχD over ℤ, in accordance with Definition 27⊳ Computation Stage:———–6. ComputeξD for A-term and P-term separately by using Definition 37———–7. ComputeξD︷ for A-term and P-term separately with the help of Definition 39———–8. ComputeηξD︷ν by using Definition 39⊳ Output Stage:———–9. Find the best option by the max modulus ofηξD︷ν with the help of Definition 40⊳ EndExample 4.
Considering the problem statement stated in Example 2, we construct IV-CNHS-setsψDδ1,ψDδ2 and ψDδ3,ψDδ4 that are defined as follows:(61)ψDδ1=0.4,0.5,0.1,0.2,0.1,0.2ei0.1,0.2,0.1,0.4,0.3,0.3πs1,0.1,0.1,0.2,0.3,0.3,0.4ei0.1,0.3,0.2,0.3,0.3,0.3πs20.1,0.2,0.1,0.4,0.2,0.2ei0.3,0.4,0.3,0.4,0.1,0.2πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.1,0.4,0.1,0.2πs4,ψDδ2=0.1,0.2,0.3,0.5,0.2,0.2ei0.1,0.2,0.2,0.40.2,0.2πs1,0.1,0.2,0.1,0.5,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs20.1,0.1,0.2,0.3,0.1,0.5ei0.3,0.4,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.2,0.4,0.3,0.3ei0.2,0.3,0.1,0.3,0.3,0.3πs4,ψDδ3=0.3,0.4,0.2,0.3,0.1,0.2ei0.1,0.2,0.1,0.4,0.2,0.3πs1,0.1,0.2,0.2,0.3,0.1,0.3ei0.2,0.3,0.3,0.4,0.2,0.2πs20.1,0.2,0.2,0.4,0.1,0.3ei0.3,0.3,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.3,0.3,0.2,0.3ei0.1,0.2,0.2,0.5,0.2,0.2πs4,ψDδ4=0.1,0.3,0.3,0.4,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs1,0.4,0.5,0.1,0.3,0.1,0.1ei0.1,0.2,0.1,0.4,0.2,0.3πs20.1,0.1,0.1,0.4,0.1,0.2ei0.2,0.3,0.3,0.3,0.3,0.3πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.2,0.4,0.2,0.3πs4.Step 1: IV-CNH-setχD is written as follows:χD=(62)δ1,0.4,0.5,0.1,0.2,0.1,0.2ei0.1,0.2,0.1,0.4,0.3,0.3πs1,0.1,0.1,0.2,0.3,0.3,0.4ei0.1,0.3,0.2,0.3,0.3,0.3πs20.1,0.2,0.1,0.4,0.2,0.2ei0.3,0.4,0.3,0.4,0.1,0.2πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.1,0.4,0.1,0.2πs4δ2,0.1,0.2,0.3,0.5,0.2,0.2ei0.1,0.2,0.2,0.4,0.2,0.2πs1,0.1,0.2,0.1,0.5,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs20.1,0.1,0.2,0.3,0.1,0.5ei0.3,0.4,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.2,0.4,0.3,0.3ei0.2,0.3,0.1,0.3,0.3,0.3πs4δ3,0.3,0.4,0.2,0.3,0.1,0.2ei0.1,0.2,0.1,0.4,0.2,0.3πs1,0.1,0.2,0.2,0.3,0.1,0.3ei0.2,0.3,0.3,0.4,0.2,0.2πs20.1,0.2,0.2,0.4,0.1,0.3ei0.3,0.3,0.2,0.3,0.1,0.2πs3,0.1,0.2,0.3,0.3,0.2,0.3ei0.1,0.2,0.2,0.5,0.2,0.2πs4δ4,0.1,0.3,0.3,0.4,0.1,0.2ei0.1,0.2,0.4,0.5,0.2,0.2πs1,0.4,0.5,0.1,0.3,0.1,0.1ei0.1,0.2,0.1,0.4,0.2,0.3πs20.1,0.1,0.1,0.4,0.1,0.2ei0.2,0.3,0.3,0.3,0.3,0.3πs3,0.1,0.2,0.4,0.5,0.2,0.2ei0.1,0.2,0.2,0.4,0.2,0.3πs4.Step 2: the cardinal is computed as follows:χDAmplitude Term=(63)0.175,0.250,0.200,0.350,0.200,0.250δ1,0.100,0.175,0.200,0.425,0.175,0.300δ20.150,0.250,0.225,0.325,0.125,0.275δ3,0.175,0.275,0.225,0.400,0.125,0.175δ4.χDPhase Term=(64)0.150,0.275,0.175,0.375,0.200,0.250δ1,0.175,0.275,0.225,0.375,0.200,0.225δ20.175,0.250,0.200,0.400,0.175,0.225δ3,0.125,0.225,0.250,0.425,0.225,0.275δ4.Step 3: the setχD︷ can be determined as follows:χD︷Apmlitude Term=(65)140.4,0.5,0.1,0.2,0.1,0.20.1,0.2,0.3,0.5,0.2,0.20.3,0.4,0.2,0.3,0.1,0.20.1,0.3,0.3,0.4,0.1,0.20.1,0.1,0.2,0.3,0.3,0.40.1,0.2,0.1,0.5,0.1,0.20.1,0.2,0.2,0.3,0.1,0.30.4,0.5,0.1,0.3,0.1,0.10.1,0.2,0.1,0.4,0.2,0.20.1,0.1,0.2,0.3,0.1,0.50.1,0.2,0.2,0.4,0.1,0.30.1,0.1,0.1,0.4,0.1,0.20.1,0.2,0.4,0.5,0.2,0.20.1,0.2,0.2,0.4,0.3,0.30.1,0.2,0.3,0.3,0.2,0.30.1,0.2,0.4,0.5,0.2,0.2×0.175,0.250,0.200,0.350,0.200,0.2500.100,0.175,0.200,0.425,0.175,0.3000.150,0.250,0.225,0.325,0.125,0.2750.175,0.275,0.225,0.400,0.125,0.175=140.30.30.10.10.40.00.10.30.10.20.10.10.40.30.30.40.1500.2000.1000.075=0.0306250.0231250.0181250.045000.χD︷Phase Term=(66)140.1,0.2,0.1,0.4,0.3,0.30.1,0.2,0.2,0.4,0.2,0.20.1,0.2,0.1,0.4,0.2,0.30.1,0.2,0.4,0.5,0.2,0.20.1,0.3,0.2,0.3,0.3,0.30.1,0.2,0.4,0.5,0.2,0.20.2,0.3,0.3,0.4,0.2,0.20.1,0.2,0.1,0.4,0.2,0.30.3,0.4,0.3,0.4,0.1,0.20.3,0.4,0.2,0.3,0.1,0.20.3,0.3,0.2,0.3,0.1,0.20.2,0.3,0.3,0.3,0.3,0.30.1,0.2,0.1,0.4,0.1,0.20.2,0.3,0.1,0.3,0.3,0.30.1,0.2,0.2,0.5,0.2,0.20.1,0.2,0.2,0.4,0.2,0.3×0.150,0.275,0.175,0.375,0.200,0.2500.175,0.275,0.225,0.375,0.200,0.2250.175,0.250,0.200,0.400,0.175,0.2250.125,0.225,0.250,0.425,0.225,0.275=140.20.20.10.40.20.40.20.10.00.10.00.30.00.10.20.20.1000.3000.1250.250=0.0481250.0475000.0262500.026250χD︷=0.030625ei0.048135πs1,0.023125ei0.047500πs2,0.018125ei0.026250πs3,0.045000ei0.026250πs4.
Consider the modulus value ofmaxηξD︷=max0.08562297237/s1,0.08312368641/s2,0.0812493106/s3,0.0793734965/s4=0.08562297237/s1. This means that s1 may be recommended for suitable investment.


## 5. Comparison Analysis

In literature, various decision-making algorithmic approaches have already been discussed by [24, 34, 36–42] that are based on hybridized structures of complex set with fuzzy set, intuitionistic fuzzy set, and neutrosophic set under soft set environments. Decision-making is badly affected due to the omission of some features with a key role. For example, in stock-exchange share market-based scenario, it is insufficient to consider “current trend of company performance,” “particular company’s stock price for last one year,” and “home country inflation rate” as only attributes because these indicators may have different values, so it is much appropriate to further classify these parameters into their disjoint attributive sets as we have done in Example 3.22. The above-mentioned existing decision-making models are insufficient either for interval-valued data or for multi-argument approximate function, but in the proposed model, the inadequacies of these models have been addressed. The consideration of multi-argument approximate function will make the decision-making process more reliable and trustworthy. We present a comparison analysis of our proposed structure with the above relevant existing structures in Table 7 and 8.Table 7
Comparison of proposed structures with some existing relevant structures.
AuthorsStructureApprox. Func.RemarksAli et al. [34]CIFSSSAAFInsufficient for interval-valued data, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint setsAl-Sharqi et al. [36]IV-CNSSSAAFInadequacy for further partitioning of attributes into attribute-valued disjoint setsFan et al. [37]IV-CFSSSAAFInadequacy for the degree of nonmembership, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint setsKhan et al. [38]CIFSSSAAFInsufficient for interval-valued data, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint setsKumar et al. [39]CIFSSSAAFInsufficient for interval-valued data, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint setsSmarandache et al. [40]CNSSSAAFInsufficient for interval-valued data and further partitioning of attributes into attribute-valued disjoint setsTable 8
Comparison of proposed structures with some existing relevant structures.
AuthorsStructureApprox. Func.RemarksSelvachandran et al. [41]IV-CFSSSAAFInsufficient for the degree of nonmembership, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint setsThirunavukarasu et al. [42]CFSSSAAFInsufficient for interval-valued data, degree of nonmembership, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint setsRahman et al. [24]CFHSSMAAFInsufficient for interval-valued data, degree of nonmembership, and degree of indeterminacyRahman et al. [24]CIFHSSMAAFInsufficient for interval-valued data and degree of indeterminacyRahman et al. [24]CNHSSMAAFInsufficient for interval-valued dataRahman et al. [30]IV-CFHSSMAAFInsufficient for the degree of nonmembership and degree of indeterminacyProposed structure 1IV-CIFHSSMAAFIt has limitations regarding the consideration of degree of indeterminacyProposed structure 2IV-CNHSSMAAFAddresses the limitations and shortcomings of the above structures


## 6. Discussion

In this section, we show that our proposed structure IV-CNHS-set is a more generalized and flexible structure as compared to existing relevant models in the sense that the existing relevant models [24, 34, 36–42] are its particular cases by omitting one or more features among MD (membership degree), NMG (nonmembership degree), ID (indeterminacy degree), SAAF (single argument approximate function), MAAF (multiargument approximate function), PND (periodic nature of data), and IVD (interval-valued data). Figure 3 presents the pictorial version of this generalization of our proposed structure.Figure 3
Generalization of the proposed structure.


### 6.1. Merits of Proposed Study

In this subsection, some merits of the proposed study are highlighted, which are given below:(i)
The introduced approach took the significance of the idea of IV-CIFHS-set and IV-CNHS-set to deal with current decision-making issues. The presented idea enables the researchers to deal with the real-world scenario where the periodicity of data in the form of intervals is involved; along these lines, this association has tremendous potential in the genuine depiction inside the space of computational incursions.(ii)
As the proposed structure emphasizes on an in-depth study of attributes (i.e., further partitioning of attributes) rather than focusing on attributes merely, therefore, it makes the decision-making process better, flexible, and more reliable.(iii)
It covers the characteristics and properties of the existing relevant structures, that is, IV-CFHS-set, CFHS-set, CIFHS-set, CNHS-set, IV-CFSS-set, IV-CIFSS-set, IV-CNSS-set, CFSS-set, CIFSS-set, CNSS-set, and so on, so it is not unreasonable to call it the generalized form of all these structures.The advantage of the proposed study can easily be judged from Tables7–9. The comparison is evaluated on the basis of two different aspects as follows:(1)
Main features discussed in the study (see Tables7 and 8)(2)
Features such as MD, NMG, ID, SAAF, MAAF, PND, and IVD (see Table9)Table 9
Comparison with existing models under appropriate features.
AuthorsStructureMDNMDIDSAAFMAAFPNDIVDAli et al. [34]CIFSS√√×√×√×Al-Sharqi et al. [36]IV-CNSS√√√√×√√Fan et al. [37]IV-CFSS√××√×√√Khan et al. [38]CIFSS√√×√×√×Kumar et al. [39]CIFSS√√×√×√×Smarandache et al. [40]CNSS√√√√×√×Selvachandran et al. [41]IV-CFSS√××√×√√Thirunavukarasu et al. [42]CFSS√××√×√×Rahman et al. [24]CFHSS√××√√√×Rahman et al. [24]CIFHSS√√×√√√×Rahman et al. [24]CNHSS√√√√√√×Rahman et al. [30]IV-CFHSS√××√√√√Proposed structure 1IV-CIFHSS√√×√√√√Proposed structure 2IV-CNHSS√√√√√√√


## 6.1. Merits of Proposed Study

In this subsection, some merits of the proposed study are highlighted, which are given below:(i)
The introduced approach took the significance of the idea of IV-CIFHS-set and IV-CNHS-set to deal with current decision-making issues. The presented idea enables the researchers to deal with the real-world scenario where the periodicity of data in the form of intervals is involved; along these lines, this association has tremendous potential in the genuine depiction inside the space of computational incursions.(ii)
As the proposed structure emphasizes on an in-depth study of attributes (i.e., further partitioning of attributes) rather than focusing on attributes merely, therefore, it makes the decision-making process better, flexible, and more reliable.(iii)
It covers the characteristics and properties of the existing relevant structures, that is, IV-CFHS-set, CFHS-set, CIFHS-set, CNHS-set, IV-CFSS-set, IV-CIFSS-set, IV-CNSS-set, CFSS-set, CIFSS-set, CNSS-set, and so on, so it is not unreasonable to call it the generalized form of all these structures.The advantage of the proposed study can easily be judged from Tables7–9. The comparison is evaluated on the basis of two different aspects as follows:(1)
Main features discussed in the study (see Tables7 and 8)(2)
Features such as MD, NMG, ID, SAAF, MAAF, PND, and IVD (see Table9)Table 9
Comparison with existing models under appropriate features.
AuthorsStructureMDNMDIDSAAFMAAFPNDIVDAli et al. [34]CIFSS√√×√×√×Al-Sharqi et al. [36]IV-CNSS√√√√×√√Fan et al. [37]IV-CFSS√××√×√√Khan et al. [38]CIFSS√√×√×√×Kumar et al. [39]CIFSS√√×√×√×Smarandache et al. [40]CNSS√√√√×√×Selvachandran et al. [41]IV-CFSS√××√×√√Thirunavukarasu et al. [42]CFSS√××√×√×Rahman et al. [24]CFHSS√××√√√×Rahman et al. [24]CIFHSS√√×√√√×Rahman et al. [24]CNHSS√√√√√√×Rahman et al. [30]IV-CFHSS√××√√√√Proposed structure 1IV-CIFHSS√√×√√√√Proposed structure 2IV-CNHSS√√√√√√√


## 7. Conclusion

The key features of this work can be summarized as follows:(1)
The novel notion of IV-CIFHS-set and IV-CNHS-set are characterized, and some of their elementary properties, that is, subset, null set, equal set, absolute set, homogeneous set, and complete homogeneous set are discussed with illustrated numerical examples.(2)
The set theoretic operations of IV-CIFHS-set and IV-CNHS-set, that is, complement, difference, union, and intersection, are characterized with the help of elaborated examples. Their fundamental laws and properties are also discussed.(3)
Aggregation operators, that is, aggregate matrix, cardinal set, cardinal matrix, and aggregate fuzzy set, of IV-CIFHS-set and IV-CNHS-set are developed.(4)
Two decision-making algorithms are proposed that are based on aggregation operators of IV-CIFHS-set and IV-CNHS-set. These are explained with the help of daily-life problems.(5)
The proposed structures along with their decision-support systems are compared with some existing relevant literature.(6)
Authors have carved out a conceptual framework for a generalized model, that is, IV-CNHS-set to deal with decision-making real-life problems by considering hypothetical data. The authors are committed to discussing some case studies based on IV-CNHS-set by using real data.(7)
Furthermore, it may also be extended to develop hybridized structures with expert sets, possibility fuzzy-set-like models, and fuzzy-set-like parameterized family and introduce algebraic structures.



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*Source: 4368770-2021-12-25.xml*