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

,


Introduction
e 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. e 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. e researchers [5][6][7][8][9][10][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][18][19][20][21] are more prominent relevant to these models. ey 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. e 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. e rudiments and elementary axioms of HS.Sets have been discussed in [23] and elaborated with numerical examples. Rahman et al. [24][25][26][27][28][29][30] investigated the hybridized properties of HS.Sets under the environments of complex set, convexity and concavity, parameterization, and bijection. ey employed decision-making algorithmic approaches to solve real-world problems. Saeed et al. [31][32][33] developed the theories of neutrosophic hypersoft mappings and complex multi-fuzzy hypersoft sets with applications in decision-making and clinical diagnosis.

Research Gap and Motivation.
e 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. e literature review of the most relevant models [34][35][36][37][38][39][40][41][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. ey 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 ( Figure 1 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 setenvironment, 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.

Main
Contributions. e following are the possible objectives of this study: (1) e existing relevant models, that is, [34][35][36][37][38][39][40][41][42], are made adequate with the entitlement of multi-argument approximate function through development of IV-CIFHS-set and IV-CNHS-set (2) e 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) e 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) e generalization and particular cases of proposed models are discussed with the pictorial depiction (7) e advantages and future directions of the proposed study are presented

Paper
Organization. e organization of the remaining paper is given in Figure 2.

Notations and
Abbreviations. Some abbreviations and notations are used throughout the paper. eir full names are given in Table 2 to facilitate the readers for proper understanding of the concept.

Preliminaries
In this section, some fundamental definitions from literature are presented for the vivid understanding of the proposed study.
Definition 1 (see [1]). Let U be a fuzzy set over Z that can be written as U � (], α U (]))|] ∈ Z such that where α U (]) is the membership degree of ] ∈ U.
Definition 2 (see [43]). A complex fuzzy set F C can be written as follows: where M-function of Buckley [44][45][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]). Let E be set of parameters; then soft set S 0 over Z is given by where ζ S 0 : E 1 ⟶ P(Z) and E 1 is subset of E.
Definition 4 (see [13]). e fuzzy soft set Ω E 1 on Z is given by where ω E 1 : Ali et al. [34] Complex intuitionistic fuzzy soft set Set of parameters Complex intuitionistic fuzzy set Al-Quran et al. [35] Complex neutrosophic soft expert set Soft expert set Complex neutrosophic set Al-Sharqi et al. [36] Interval-valued complex neutrosophic soft set Set of parameters Interval-valued complex neutrosophic set Fan et al. [37] Interval-valued complex fuzzy soft set Set of parameters Interval-valued complex fuzzy set Khan et al. [38] Complex intuitionistic fuzzy soft set Set of parameters Complex intuitionistic fuzzy set Kumar et al. [39] Complex intuitionistic fuzzy soft set Set of parameters Complex intuitionistic fuzzy set Smarandache et al. [40] Complex neutrosophic soft set Set of parameters Complex neutrosophic set Selvachandran et al. [41] Interval-valued complex fuzzy soft set Set of parameters Interval-valued complex fuzzy set irunavukarasu et al. [42] Complex fuzzy soft set Set of parameters Complex fuzzy set  Applied Computational Intelligence and Soft Computing 3 is a fuzzy set over Z. Here, ω E 1 is the approximate function of Ω E 1 , where ω E 1 (δ) is a fuzzy set known as δ-element of Definition 5 (see [42]). A complex fuzzy soft set ξ E 1 over Z is given by where χ E 1 : Set operations of complex fuzzy set and complex fuzzy soft set have been described in [42,47], respectively.
Definition 6 (see [39]). Let E be a set of attributes with A⊆E. en complex intuitionistic fuzzy soft set ξ A � (Ψ, A) over Z is defined as follows: where is a complex intuitionistic fuzzy approximate function of ξ A and Ψ(a) � 〈Ψ T (a), Ψ F (a)〉. Ψ T (a) � p T e iθ T and Ψ F (a) � p F e iθ F are complex-valued M-function and complex-valued NM-function of ξ A , respectively, and all are lying within unit circle in the complex plane such that Definition 7 (see [40]). Let E be a set of attributes with A⊆E. en complex neutrosophic soft set ξ A � (Ψ, A) over Z is defined as follows: where Some fundamental definitions are presented from existing literature in Section 2 eory of Inteval-Valued Complex Intuitionistic Fuzzy Hyperso set is developed and decision-support system is constructed with the help of its aggregation in Section 3 e advantages and generalization of the proposed structures are discussed in Section 6 e paper is summarized with future directions in Conclusion Section. relevant models in Section 5 A comparison of proposed structures is presented with some existing eory of Inteval-Valued Complex Neutrosophic Hyperso set is developed and decision-support system is constructed with the help of its aggregation in Section 4 is a complex neutrosophic approximate function of ξ A and Ψ(a) � 〈Ψ T (a), Ψ I (a), Ψ F (a)〉. Ψ T (a) � p T e iθ T , Ψ I (a) � p I e iθ I , and Ψ F (a) � p F e iθ F are complex-valued truth M-function, complex-valued indeterminacy M-function, and complex-valued falsity M-function of ξ A , respectively, and all are lying within unit circle in the complex plane such that p T , p I , Definition 8 (see [22] 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. e fundamental properties and set theoretic operations of hypersoft set are discussed in [23].

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. is 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. is can be of either lab-to-lab basis or day-to-day basis. Such type of data is treated as periodic data. e 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 multiargument 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.
en interval-valued complex intuitionistic fuzzy hypersoft set (IV-CIFHS-set), denoted by Ω M � (Λ, M), over Z is defined as follows: and then IV-CIFHS-set Ω M is written by Applied Computational Intelligence and Soft Computing 7 Definition 16. For two IV-CIFHS-sets (Λ 1 , M 1 ) and (Λ 2 , M 2 ) over the same universe Z, the following definitions hold:

Set Operations and Laws on IV-CIFHS-Set.
In this section, some basic set theoretic operations and laws are discussed on IV-CIFHS-set.
where the M-function (Λ(x ⌣ )) c has the A-term given by Proof. As Λ(x ⌣ ) ∈ C IV (Z), so in terms of its A-and P-terms, (Λ, M) can be expressed as follows: Now 8 Applied Computational Intelligence and Soft Computing From equations (20) and (21), Definition 19. For two IV-CIFHS-set (Λ 1 , M 1 ) and (Λ 2 , M 2 ), the difference is defined as follows: Applied Computational Intelligence and Soft Computing Proposition 3. Let (Λ, M) be an IV-CIFHS-set over Z. en the following results hold true: are three IV-CIFHS-sets over the same universe Z. en the following commutative and associative laws hold true: Proposition 5. Let (Λ 1 , M 1 ) and (Λ 2 , M 2 ) are two IV-CIFHS-sets over the same universe Z. en the following De Morgans's laws hold true:

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. e approximate functions of an IV-CIFHS-set are fuzzy. Here, D, E, ξ D and ⊎ IVCIFHS will be in accordance with Definition 12.  Table 3.
In Table 3, η 1 χ D (x) and η 2 χ D (x) are M-function and NMfunction of χ D , respectively, with interval-valued intuitionistic fuzzy values. If α ij � (η 1 1 and j � N r 1 , then IV-CIFHS-set ξ D is uniquely characterized by the following matrix: which is called an m × r IV-CIFHS-set matrix.

Definition 22.
If ξ D ∈ ⊎ IVCIFHS , then cardinal set of ξ D is defined as follows: where Definition 23. Let ξ D ∈ C ivcifhss (Z) and ‖ξ D ‖ ∈ ‖C ivcifhss (Z)‖. Consider E as in Definition 12, the representation of ‖ξ D ‖ can be seen in Table 4.
), for j � N r 1 , then the cardinal set ‖ξ D ‖ is represented by the following matrix: which is called the cardinal matrix of ‖ξ D ‖.
Definition 24. Let ξ D ∈ C ivcifhss (Z) and ‖ξ D ‖ ∈ ‖C ivcifhss (Z)‖. en IV-CIFHS-aggregation operator is defined as follows: where 10 Applied Computational Intelligence and Soft Computing where ξ D is called the aggregate fuzzy set of IV-CIFHSset ξ D .
Its M-function is given as follows: Definition 25. Let ξ D ∈ C ivcifhss (Z) and ξ D be its aggregate fuzzy set. Assume that Z � ] 1 , ] 2 , . . . , ] m , then ξ D can be presented as follows: If α i1 � η ξ D (] i ) for i � N m 1 , then ξ D is represented by the following matrix: which is called the aggregate matrix of ξ D over Z.

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). e following algorithm is proposed based on the definitions given in Subsection 3.2 that may help make optimal decision. Now, Algorithm 1 is explained with the help of the following example.
Example 2. Suppose a businessman wants to buy a share from the share market. ere are four same kinds of share that form the set, Z � s 1 , s 2 , s 3 , s 4 . e expert committee considers a set of attributes, E � e 1 , e 2 , e 3 . For i � 1, 2, 3, the attributes e i 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 L 1 � e 11 , e 12 , L 2 � e 21 , and L 3 � e 31 , e 32 . en the set are defined as follows: Table 3: Tabular representation of ξ D . Table 4: Tabular representation of ‖ξ D ‖.

Interval-Valued Complex Neutrosophic Hypersoft Set (IV-CNHS-Set)
e basic theory of the IV-CNHS-set is developed in this section.  � (b 1 , b 2 , b 3 , . . . , b n ) ∈ M. en, interval-valued complex neutrosophic hypersoft set (IV-CNHS-set), denoted by Ω M � (Λ, M), over Z is given as follows: where which is a IV − CN approximate function of Ω M and Λ(λ) 〉 with lower and upper bounds of membership, nonmembership, and indeterminacy function are given below, respectively: en IV-CNHS-set Ω M is written by Step 3: the set χ D can be determined as follows:

Comparison Analysis
In literature, various decision-making algorithmic approaches have already been discussed by [24,34,[36][37][38][39][40][41][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. e 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. e 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.

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][37][38][39][40][41][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.

Merits of Proposed Study.
In this subsection, some merits of the proposed study are highlighted, which are given below: (i) e introduced approach took the significance of the idea of IV-CIFHS-set and IV-CNHS-set to deal with current decision-making issues. e presented idea enables the researchers to deal with the realworld scenario where the periodicity of data in the form of intervals is involved; along these lines, this 24 Applied Computational Intelligence and Soft Computing CIFSS SAAF Insufficient for interval-valued data, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint sets Smarandache et al. [40] CNSS SAAF Insufficient for interval-valued data and further partitioning of attributes into attributevalued disjoint sets Selvachandran et al. [41] IV-CFSS SAAF Insufficient for the degree of nonmembership, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint sets irunavukarasu et al. [42] CFSS SAAF Insufficient for interval-valued data, degree of nonmembership, degree of indeterminacy, and further partitioning of attributes into attribute-valued disjoint sets Rahman et al. [24] CFHSS MAAF Insufficient for interval-valued data, degree of nonmembership, and degree of indeterminacy Rahman et al. [24] CIFHSS MAAF Insufficient for interval-valued data and degree of indeterminacy Rahman et al. [24] CNHSS MAAF Insufficient for interval-valued data Rahman et al. [30] IV-CFHSS MAAF Insufficient for the degree of nonmembership and degree of indeterminacy association has tremendous potential in the genuine depiction inside the space of computational incursions. (ii) As the proposed structure emphasizes on an indepth 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.
e advantage of the proposed study can easily be judged from Tables 7-9. e comparison is evaluated on the basis of two different aspects as follows: (1) Main features discussed in the study (see Tables 7 and 8) (2) Features such as MD, NMG, ID, SAAF, MAAF, PND, and IVD (see Table 9)

Conclusion
e key features of this work can be summarized as follows: (1) e 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.  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. e authors are committed to discussing some case studies based on IV-CNHSset 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.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.