On Some Types of Covering-Based ( I , T ) -Fuzzy Rough Sets and Their Applications

,

A widely studied extension of RST is a covering through rough sets (CRSs).
Pomykala [16,17] generated a dual approximation for two pairs of operators. e definitions of granularity and neighborhood presented further insights in the field of approximation operators by Yao [18]. Using the assumption of incomplete knowledge, Bonikowski et al. [19] introduced a model of CRS, which depends on the of minimal description concept. Couso and Dubois [20] deliberated both pairs of operators as well. Other CRS models and relationships among them are the subjects of Zhu [21] and of Zhu and Wang [22][23][24]. e additional models of CRS were developed by Tsang et al. [25] and Xu and Zhang [26]. Moreover, Liu and Sai [27] compared the CRS model of Zhu and the CRS models of Xu and Zhang. Based on the concepts of neighborhood and complementary neighborhood, Ma [28] established some neighborhood-related forms covering rough sets in 2012. On the other direction, Dubois et al. [29] defined the fuzzy rough set (FRS) and rough fuzzy set (RFS) which assist the authors to a built a new way of CRS called fuzzy covering rough sets (FCRSs). ere are many scholars working on this idea such as Atef and El Atik [30] presented some types of covering-based multigranulation fuzzy rough sets, Ma et al. [31] proposed new fuzzy rough coverings models using fuzzy α-neighborhood and presented some of its application, and the multigranulation fuzzy rough sets notion over two universes with its decision-making application studied by [32] and Zhan et al. [33] established the concept of a covering-based multigranulation fuzzy rough sets. In this direction, the meaning of fuzzy β-neighborhood was discovered by Ma [34] to create a fuzzy β-covering approximation space (FCAS). en, Yang and Hu [35][36][37] investigated the notions of a fuzzy β-complementary neighborhood, a fuzzy β-minimal description, and a fuzzy β-maximal description to generate new models of FCAS. Deer et al. [38] studied fuzzy neighborhoods based on fuzzy coverings. Further, Atef and Azzam [39] and Jiang et al. [40] introduced covering fuzzy rough sets through variable precision. In [41][42][43], Hu and Wong have constructed generalized interval-valued fuzzy variable precision rough sets and generalized interval-valued fuzzy rough sets by using the operators of fuzzy logical. Meanwhile, L-fuzzy rough sets have been studied by the constructive method and axiomatic method [44][45][46]. Yeung et al. [47] have proposed a number of definitions for upper and lower fuzzy sets approximation operators by the means of arbitrary fuzzy relations and studied the relationship among them from the viewpoint of a constructive approach. In an axiomatic approach, they have characterized different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. After that, Zhang et al. [48] set up the FCITFRS paradigms in 2019. In addition, Jiang et al. [49] established CVPITFRS with its applications to multiattribute-based decision-making. Furthermore, between 2019 and 2021, Zhan et al. introduced some types of decision-making problems [33,50]. e later extension of the former model takes advantage of the generality provided by a fuzzy logical implication I and a t-norm T. Both works provided applications of the models to multiattribute group decision-making. e emphasis of the FCITFRS models is on granularity, which here is understood in a fuzzy sense. Other settings require alternative models. For example, the Takagi-Sugeno fuzzy models [51] show a higher ability to capture nonlinear behavior. While the former insists on approximation operators, be they defined by equivalence relations, (fuzzy) coverings, or otherwise, and the latter relies on fuzzy rules. For this reason, the Takagi-Sugeno fuzzy models have also become popular in many other applied fields [52][53][54][55][56].
Based on these recent developments and to extend and generalize the last studies by [48,49] (i.e., to raise the lower approximation and lowering the previous works' upper approximation), in this paper, we contribute to generating eight types of FCITFRS models through the descriptions of fuzzy β-minimal and fuzzy β-maximal notions and their complementary.
Contrary to the inspiring cases of Zhan et al. [33,50] and Ma et al. [57], our attention is restricted to one single source of granularity although we keep the general setting that allows for a t-norm and a fuzzy logical implication. e properties of these models as well as their relationships are introduced. Further, eight types of FCITFRSs and their related properties are considered. An application to a real issue clarifies the capacity to assist the practitioner for making the decisions. e structure and body of the paper are as follows. Our next section contains various technical preliminary concepts. en, Section 3 describes the eight paradigms of FCITFRSs that we propose for investigation. Section 4 introduces the corresponding eight types of CVPITFRSs.
e relations between these different models are set in Section 5. Section 6 gives the applicable example of this theoretical study. Section 7 puts an end to this paper.

Preliminary Concepts
roughout this section, we introduce several fundamental notions of fuzzy logical operators, CFRSs, FCITFRSs, and CVPITFRSs. In this paper, F(Ω) denotes the set of all fuzzy sets on a crisp set Ω. Also, we use T M (x, y) � x∧y and I M (x, y) � (1 − x)∨y. Further, we say that N is involutive when N(N(x)) � x for every x ∈ [0, 1]. e standard negator operator is defined as N(x) � 1 − x, for any x ∈ [0, 1]. For more data, see [58].
For any x ∈ Ω, define the first type of the variable precision (I, T)-fuzzy lower approximation (for short I-VPITFLA) (resp., II-VPITFLA, III-VPITFLA, and IV-VPITFLA) and the first type of the variable precision (I, T)-fuzzy upper approximation (for short I-VPITFUA) (resp., II-VPITFUA, III-VPITFUA, and IV-VPITFUA) as follows: x (y, X(y)) , x (y, X(y)) , If I L I vp (X) (resp., II L I vp (X), III L I vp (X), and IV L I vp (X) ≠ I U T vp (X) (resp., II U T vp (X), III U T vp (X), and IV U T vp (X)), then X is called a I-CVPITFRS (resp., II-CVPITFRS, III-CVPITFRS, and IV-CVPITFRS)); otherwise, it is variable precision fuzzy definable.

Eight Types of Covering-Based (I, T)-Fuzzy
Rough Sets is section contains the definitions of eight types of FCITFRS models. Also, the related characteristics are discussed. e study by Zhan et al. [50] is a direct inspiration by their utilization of a fuzzy logical implication and a t-norm in a similar setting. However, we refrain from using various sources of granularity, which opens the door for additional modelizations.
Let us fix I m and T M . en, we have the following. en, we can calculate the Md β x i and MD β x i (i � 1, 2, 3, 4, 5, 6), respectively. See Table 2 for the results of the corresponding computations. Now, if X � (0.6/x 1 ) + (0.8/x 2 ) + (0.5/x 3 ) + (0.7/x 4 ) + (0.8/x 5 ) + (0.9/x 6 ) then we introduce fuzzy β-neighborhood and the union of the fuzzy β-minimal description for Γ � C 1 as n Tables 3 and 4.   4 Journal of Mathematics So, we have the following results: We proceed to calculate the fuzzy β-complementary for x i and is set in Table 5. So, we obtain the following values: en, we calculate ∨Md x as shown in Table 6. us, we have Let us now calculate ∨M d x , which produces Table 7.
en, we get the following outcomes: eorem 1 satisfies all kinds of operators in Definition 5. Here, we prove it for the first kind only and the other kinds similarly. Table 1: A fuzzy β-covering approximation space. Table 3: Fuzzy β-neighborhood of Γ. Table 4: Journal of Mathematics Theorem 1. Presume that (Ω, Γ) be an FCAS for a given β ∈ (0, 1]. Select X ∈ F(Ω) and ∀x ∈ Ω. en, the following affirmations hold true: . By the same manner, we also obtain (2) As I is a left monotonic, then we have Also, we have Table 5: Journal of Mathematics (3) As I is a right monotonic, if X⊆Y, then we have us, 1 L I (X)⊆ 1 L I (Y) holds, and similarly 1 U I (X)⊆ 1 U I (Y) holds. (4) As I is a right monotonic, then we have Because X ∩ Y⊆X and X ∩ Y⊆Y, we deduce from (3), Also, we obtain
So, the following results are obtained: en, we calculate ∧Md Table 11. erefore, we get the following outcomes: eorem 2 satisfies all types of operators in Definition 6. Next, we prove it for the fifth model only and the other models similarly.
(2) As I is a left monotonic, then we have Also, we have         As X ∩ Y⊆X and X ∩ Y⊆Y, from (3), we obtain from (3), we have 5 L I (X)⊆ 5 L I (X ∪ Y) and 5 L I (Y)⊆ 5 L I (X ∪ Y). us, 5 L I (X ∪ Y)⊇ 5 L I (X) ∪ 5 L I (Y). Also, we obtain

Example 3 (in continuation of
Next, we have the relevant properties of Definition 7. e proof of each point in eorem 3 is easily obtained, so we omit it. Let us select j ∈ 1, 2, 3, . . . , Theorem 3. Presume that (Ω, Γ) be an FCAS for a given β ∈ (0, 1]. Select X ∈ F(Ω) and ∀x ∈ Ω. en, the following affirmations hold true: when I is a right monotonicity

The Relationships among Our Models
Now, we explain some relationships among the paradigms proposed previously.

Proof.
e proof is obtained by using Definition 5 and Proposition 1. Proof. e proof is obtained by using Definition 6 and Proposition 2.
Proof. e proof is obtained by using Definitions 3 and 5. □

Journal of Mathematics
Proof. It is obvious from Definition 7.

Proposed Decision-Making Approach
Now, we use the presented study to obtain a decision on a realistic issue.

Description and
Process. Suppose that Ω � u 1 , u 2 , . . . , u n be n candidates and Γ � Γ 1 , Γ 2 , . . . , Γ m be a set of features. us, Γ(u j ) denotes the experts rating value related to the candidate u j to the attribute Γ, and we assume that for a fixed β ∈ (0, 1], (Ω, Γ) is an FCAS. Based on the explained covering methods, we submit an algorithm that seeks to obtain the best choice as in the following procedures: Step 1. Produce a matrix D M as D M � e 11 e 12 · · · e 1m e 21 e 22 · · · e 2m ⋮ ⋮ ⋱ ⋮ e n1 e n2 · · · e nm where e ij is the experts rating value of alternative u i .
Step 2. Count the favorable ideal I ⊕ and the unfavorable ideal I ⊖ fuzzy sets with the help of the following equations: . . , u n , ∨ 1≤j≤m e nj , where ∨ and ∧ symbolize to "maximum" and "minimum," respectively.
Step 3. Count the lower and upper approximations of I ⊕ , I ⊖ , and N I as defined by the 7-FCFCITFRS model.
Step 4. Count the respective distances D and D between the lower approximation of N I and the lower approximation of I ⊕ and also the distance among their parallel upper approximations as follows: where ] is a controlling result which is determined by the experts and D(x, y) � |x − y|.
Step 5. Count the closeness coefficient grade by and hence order the candidates. Based on these procedures, we establish an algorithm to fix the decision-making issues according to 7-FCITFRS paradigm. e steps corresponding to it are summarized in Algorithm 1.

Illustrative Example.
e above procedures have been clarified with a test example as follows.
Step 1. e evaluation for each nominee through the decision-makers is summarized in Table 12.
Step 2. Based on the significance of these five features, we have W V � 0.1, 0.3, 0.25, 0.15, 0.2 { }, we calculate I ⊕ , I ⊖ , and N I , and outcomes are listed in Table 13.
Step 3. Using I m and T M , through 7-FCITFRS, we obtain the the following values: So, we evaluate ∧MD  Table 16. Based on these information, we get Step 4. Count D i and D i indicated as follows: Step 5. According to the results on Step 4, we count R l as follows: and hence get the ranking order as 14 Journal of Mathematics By the above procedures and counting, you can say that the 5th engineer is the best nominee among the others.

Comparative Analysis.
is section aims to explain the merit of the proposed method by comparing it to the last study in the area which was given by Zhang et al. [48]. e purpose here is it is able to raise the lower approximation and down the upper approximation of Zhang et al.'s models as you see in Example 4. We established Tables 17 and 18 to explain the differences between Zhang et al.'s model and our models.
To view another way to illustrate this comparison between Zhang et al.'s [48] and our's, we present Figures 1 and 2. Figure 1 displays the relations among Zhang-lower positive ideal, Zhang-lower negative ideal, and Zhang-lower integrated ideal and our-lower positive ideal, our-lower negative ideal, and our-lower integrated ideal. From this point of view, you can see that our-lower (positive, negative, and integrated ideal) is superior to Zhang-lower (positive, negative, and integrated ideal). is means that our-lower is better than Zhang-lower. Figure 2 discusses the links among Zhang-upper positive ideal, Zhang-upper negative ideal, and Zhang-upper integrated ideal and our-upper positive ideal, our-upper negative ideal, and our-upper integrated ideal. us, you can see that our-upper (positive, negative, and integrated ideal) is down than Zhang-upper (positive, negative, and integrated ideal).
is means that our-upper is lower than Zhang-upper. rough Tables 17 and 18 and Figures 1 and 2, it is easy to say that our work is considered as an improvement of Zhang et al.'s [48] work via raising the lower approximation and down the upper approximation (for instance, see Example 4).
is means that our method is effective and reliable.         Different processes Obtain a decision Zhang's process [48] u 4 ≥ u 6 ≥ u 3 ≥ u 5 ≥ u 1 ≥ u 2 Our process u 5 ≥ u 4 ≥ u 6 ≥ u 1 ≥ u 2

Conclusion
e major purpose of the presented work is to improve Zhang et al.'s model [48] and Jiang et al.'s model [49]. In this paper, we considered the issue of the neighborhood-related β covering the classification of approximation spaces. Eight new paradigms of FCITFRSs models can be viewed as an improvement of Zhang et al. [48] model through concepts of fuzzy β-minimal description and fuzzy β-maximal description. Also, we introduce eight kinds CVPITFRSs as generalizations of Jiang et al. [49] model. Further, we study the relations between our models and the previous models in [48,49]. After that, we place the present study on a real issue to show its effectiveness and reliability.

Data Availability
No data were used to support this study.

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