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.
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 (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 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.
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 Figure 2.
1.4. Notations and Abbreviations
Some abbreviations and notations are used throughout the paper. Their full names are given in Table 2 to facilitate the readers for proper understanding of the concept.
2. Preliminaries
In this section, some fundamental definitions from literature are presented for the vivid understanding of the proposed study.
Definition 1 (see [1]). Let be a fuzzy set over that can be written as such thatwhere is the membership degree of .
Definition 2 (see [43]). A complex fuzzy set can be written as follows:where M-function of is with as A-term and as P-term and .
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]). Let be set of parameters; then soft set over is given bywhere and is subset of .
Definition 4 (see [13]). The fuzzy soft set on is given bywhere , where for andis a fuzzy set over . Here, is the approximate function of , where is a fuzzy set known as -element of . If , then .
Definition 5 (see [42]). A complex fuzzy soft set over is given bywhere , where for is the complex fuzzy approximate function of , where is called -member of for all . 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 . Then complex intuitionistic fuzzy soft set over is defined as follows:whereis a complex intuitionistic fuzzy approximate function of and .
and are complex-valued M-function and complex-valued NM-function of , respectively, and all are lying within unit circle in the complex plane such that with and . The value is called -member of .
Definition 7 (see [40]). Let E be a set of attributes with . Then complex neutrosophic soft set over is defined as follows:whereis a complex neutrosophic approximate function of and .
, , and are complex-valued truth M-function, complex-valued indeterminacy M-function, and complex-valued falsity M-function of , respectively, and all are lying within unit circle in the complex plane such that with and . The value is called -member of .
Definition 8 (see [22]). is known as a hypersoft set over , if where is the Cartesian product of finite number of disjoint sets with distinct attributes , 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 set set over is given bywhere are disjoint sets having distinct attributes for , , and be a set over for all .is a complex fuzzy approximate function of and is member of set .
Definition 11 (see [30]). Let are disjoint sets having attribute values of n distinct attributes , respectively, for and be an interval-valued complex fuzzy set over for all . Then interval-valued complex fuzzy hypersoft set (IV-CFHS-set) over is defined as follows:whereis an interval-valued complex fuzzy approximate function of and . and are lower and upper bounds of the M-function of , respectively, and its value is called -member of IV-CFHS .
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. Let are disjoint sets having distinct attributes , respectively, for and be an IV-CIFS over for all . Then interval-valued complex intuitionistic fuzzy hypersoft set (IV-CIFHS-set), denoted by , over is defined as follows:whereis a approximate function of and with the following:(i) and are lower and upper bounds of the M-function of , respectively(ii) and are lower and upper bounds of the NM-function of , respectively, and its value is called -member of IV-CIFHS-set for all values of Note: The collection of all IV-CIFHS-sets is denoted by .
Example 1. Consider with interval-valued complex intuitionistic fuzzy sets given byand then IV-CIFHS-set is written by
Definition 13. Let and be two IV-CIFHS-sets over the same universe. The set is a subset of , if(i)(ii) implies , that is, , and , where amplitude and P-terms of each are given below: for for for for
Definition 14. Two IV-CIFHS-sets and over the same universe are equal if(i)(ii)
Definition 15. Let be an IV-CIFHS-set over . Then(i)It is a null IV-CIFHS-set, symbolized by if for all values of , the A-term of the M-function is given by , whereas the P-terms of is given by (ii)It is an absolute IV-CIFHS-set, symbolized by ; if for all values of , the A-terms of the M-function is given by , whereas the P-terms is given by .
Definition 16. For two IV-CIFHS-sets and over the same universe , the following definitions hold:(i)An IV-CIFHS-set is a homogeneous IV-CIFHS-set, symbolized by if is homogeneous CIF-set (ii)An IV-CIFHS-set is a completely homogeneous IV-CIFHS-set, symbolized by if is homogeneous with (iii)An IV-CIFHS-set is a completely homogeneous IV-CIFHS-set with if is a homogeneous with
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-set , denoted by , is given bywhere the M-function has the A-term given by and P-term given by and .
Proposition 1. Let be an IV-CIFHS-set over . Then .
Proof. As , so in terms of its A- and P-terms, can be expressed as follows:NowFrom equations (20) and (21), we have .
Proposition 2. Let be an IV-CIFHS-set over . Then(i)(ii)
Definition 18. For two IV-CIFHS-set and , the intersection over the same universe is the IV-CIFHS-set , where , and for all ,
Definition 19. For two IV-CIFHS-set and , the difference is defined as follows:
Definition 20. The union of two IV-CIFHS-set and over the same universe is the IV-CIFHS-set , where , and for all ,
Proposition 3. Let be an IV-CIFHS-set over . Then the following results hold true:(i)(ii)(iii)(iv)(v)(vi)
Proposition 4. Let , , and are three IV-CIFHS-sets over the same universe . Then the following commutative and associative laws hold true:(i)(ii)(iii)(iv)
Proposition 5. Let and are two IV-CIFHS-sets over the same universe . Then the following De Morgans’s laws hold true:(i)(ii)
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, and will be in accordance with Definition 12.
Definition 21. Let . Assume that and with and , each is n-tuple element of and ; the representation of can be seen in Table 3.
In Table 3, and are M-function and NM-function of , respectively, with interval-valued intuitionistic fuzzy values. If , for and , then IV-CIFHS-set is uniquely characterized by the following matrix:which is called an IV-CIFHS-set matrix.
Definition 22. If , then cardinal set of is defined as follows: