Certain Types of Covering-Based Multigranulation ( I , T )-Fuzzy Rough Sets with Application to Decision-Making

As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to deﬁne the family of complementary fuzzy β -neighborhoods and thus three kinds of covering-based multigranulation ( I , T )-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation ( I , T )-fuzzy rough sets are deﬁned and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.


Introduction
Group decision-making aims at aggregating individual judgments to construct a composite group decision, which must be a true representative of individual preferences. e MAGDM methods choose among a discrete set of alternatives evaluated on multiple attributes and overall utility of the decision makers. MAGDM have some of the popular methods such as the weighted sum and the weighted product method (see, e.g., [1][2][3][4][5][6][7]). e theory of rough set was founded by Pawlak [8,9] for dealing with the vagueness and granularity in information systems and data analysis. is theory has been applied to many different fields (see, e.g., [10][11][12][13][14][15][16][17][18][19][20]). Furthermore, we have noticed a wide range of generalized rough set models (see, e.g., [21][22][23]). Coveringbased rough sets (CRSs) are considered to be one of the most studied generalized models. Pomykala [24,25] obtained two pairs of dual approximation operators. Yao [26] studied these approximation operators by the concepts of neighborhood and granularity. Couso and Dubois [27] examined the two pairs within the context of incomplete information. Bonikowski et al. [28] established a CRS model based on the notion of minimal description. Zhu and Wang [29][30][31][32] presented several CRS models and discussed their relationships. Tsang et al. [33] and Xu and Zhang [34] proposed additional CRS models. Liu and Sai [35] compared Zhu′s CRS models and Xu and Zhang ′ s CRS models. Ma [36] constructed some types of neighborhood-related covering rough sets by using the definitions of the neighborhood and complementary neighborhood. In 2016, Ma [37] introduced the definition of fuzzy β-neighborhood. In 2017, Yang and Hu [38] constructed the definition of the fuzzy β-complementary neighborhood to establish some types of fuzzy covering-based rough sets. Zhang et al. [39], in 2019, established the fuzzy covering-based (I, T)-fuzzy rough set models and applications to multiattribute decision-making. Also, in 2019, Zhan et al. [40] proposed covering-based multigranulation (I, T)-fuzzy rough set models and applications in multiattribute group decision-making. e concept of a family fuzzy β-neighborhoods was defined and their properties were studied by Zhan et al. [40]. Hence, to increase the lower approximation and decrease the upper approximation of Zhan's model, this article's contribution is to introduce three kinds of covering-based multigranulation (I, T)-fuzzy rough sets models and explore the properties of these models with their relationships. Also, six kinds of covering-based variable precision multigranulation (I, T)-fuzzy rough sets are demonstrated. An application to a practical problem illustrates their ability to help practitioners to make decisions. e outline of this paper is as follows. Section 2 gives technical preliminaries. Section 3 describes our three new types of covering-based multigranulation (I, T)-fuzzy rough sets and also we introduce variable precision in order to produce the six types of covering-based variable precision multigranulation (I, T)-fuzzy rough sets. Section 4 establishes relationships among our models. Section 5 puts forward a decisionmaking procedure that takes advantage of our theoretical framework. e conclusion is written in Section 6.

Preliminaries
In this section, we provide a brief survey of some notions used throughout the paper.
We say that T, a t-norm, and S, a t-conorm, are dual with respect to negator N, when for each x, y ∈ [0, 1], it must be the case that S(N(x), N(y)) � N(T(x, y)) and T(N(x), N(y)) � N(S(x, y)).
We say that I is hybrid monotonic when it is both left and right monotonic.
An implicator I is a border implicator when Next, we recall three relevant classes of implicator operators [1].

Definition 5
(1) e S-implicator defined by S and N is given: If T and S are dual with respect to N, the QL-implicator defined from T, S, and N is given: , according to S L and N S (2) I KD (x, y) � (1 − x)∨y, according to S M and N S (3) I P (x, y) � 1 − x + x * y, according to S M and N S Definition 6 (see [42,43]). Let Ω be an arbitrary universal set, and F(Ω) be the fuzzy power set of Ω. We call Γ � C 1 , C 2 , . . . , C m , with C i ∈ F(Ω)(i � 1, 2, . . . , m), a fuzzy covering of Ω, if ( ∪ m i�1 C i )(x) � 1 for each x ∈ Ω. As a generalization of fuzzy covering, Ma [37] defined a fuzzy β-covering by replacing 1 with a parameter is called a fuzzy β-covering approximation space (briefly, FβCAS).
Zhan et al. [40] defined the covering-based multigranulation (I, T)-fuzzy rough set models (briefly, CMGITFRSs). So, in the following, some basic notions related to CMGITFRSs are given.

3.1.
ree Types of the Optimistic Multigranulation (I, T)-Fuzzy Rough Sets. In the following, three kinds of COMGITFRS models are given and some of their properties are presented.
Case 2 Let us fix I � I KD based on S M and N S , and T � T M . en, we have the following results: Theorem 1. Let (Ω, Γ) be a FβCAS and Γ � C 1 , C 2 , . . . , C n } be n fuzzy β-coverings of Ω for some β ∈ (0, 1], for each X ∈ F(Ω). en, we have the following properties: (1) If I is an S-implicator based on S, a continuous t-conorm, and N, an involutive negator, (2) If either I is an R-implicator based on T, a continuous t-norm, and N, a negator induced by I, or I is a QL-implicator based on T, a continuous t-norm, and N, an involutive negator, If I satisfies right monotonicity and X⊆Y, then 0.5 0.9 0.6 0.1 Table 2: Table for C 2 .
. Also, since X∩Y⊆X and X∩Y⊆Y, from Since the weakened distributivity laws are satisfied, In particular, if x ∈ Ω, we have , then X is called a covering-based optimistic multigranulation (I, T)-fuzzy rough set (briefly, 2-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.
. . , 6 for some β � 0.5, as shown in Tables 7 and 8 as follows. Now, we calculate the 2-OMGITFLA and 2-OMGIT-FUA as explored in the following two cases.
Case 1 Let us fix I � I * based on S P and N S and T � T P . So, Case 2 Let us fix I � I KD based on S M and N S and T � T M . us, Remark 1. Definition 13 satisfies eorem 1. Complexity 7 , then X is called a covering-based optimistic multigranulation (I, T)-fuzzy rough set (briefly, 3-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.
Case 1 Let us fix I � I * based on S P and N S and T � T P . So, Case 2 Let us fix I � I KD based on S M and N S and T � T M . us, Remark 2. Definition 14 satisfies eorem 1.

3.2.
ree Types of the Pessimistic Multigranulation (I, T)-Fuzzy Rough Sets. In the following, we introduce three kinds of CPMGITFRS models and study some of their properties.

The Relationships between COMGITFRS Models and CPMGITFRS Models
In this section, we explain relationships among our models. rough the proposed study, we have the following results.
From Definitions 10 and 12, we conclude the following results.

An Application to Decision-Making
In this section, we apply the proposed method to make a decision on a real-life problem.

Description and Process.
Let Ω � u 1 , u 2 , . . . , u n be n alternatives and E � e 1 , e 2 , . . . , e l be l decision makers. Suppose for each ] i is a weighted vector correspondingly to e i ,

Complexity 11
where ] i ≥ 0 for i � 1, . . . , l and l i�1 ] i � 1. Hence, C i � C i1 , C i2 , . . . , C im i , for all i � 1, 2, . . . , n is a set of attributes. A family of mappings G � g l , where g l : Ω × C i ⟶ [0, 1]. So, we construct the MAGDM with fuzzy information system (Ω, C, E, G). Based on the proposed covering methods, we present a decision-making algorithm to find the best alternative through the following steps: Step 1: Construct the decision-making object with fuzzy information of the universe of discourse. rough the rule of fuzzy TOPSIS method, we have . . , C lm , ∨ g lm , . . , C lm , ∧ g lm (22) where ∧ and ∧ denote "max" and "min," respectively.
Step 2: Compute the respective distances D and D as follows: where S(Y, tZ) �
Step 3: Calculate the lower and upper approximations of the best and worst decision-making objects with fuzzy information by Definition 13 (2-OMGITFLA and 2-OMGITFUA).
Step 4: Calculate the closeness coefficient degree by be the worst and the best decision-making objects for individual ranking function of expert l for the candidates u i , and 0 ≤ W l (u i ), W l (u i ) ≤ 1.
Step 5: Calculate the group ranking function by the following equation R(u i ) � t l�1 ] l R l (u i ), and hence rank the alternatives.
According to these steps, we give an algorithm to solve the decision-making problems based on the 2-COMGITFRS model.
e steps corresponding to it are summarized in Algorithm 1.

Applied Example.
e abovementioned steps have been illustrated with a numerical example as shown next.
Example 7 (see [40]). Let Ω � u 1 , u 2 , . . . , u 6 be six system analysis engineers and Γ � { emotional steadiness (C 1 ), oral communication skill (C 2 ), personality (C 3 ), past experience (C 4 ), self-confidence (C 5 ) } be the attribute set of the basic description of the candidates. Suppose that three experts e 1 , e 2 , and e 3 are invited to evaluate the system analysis engineers according to their specialized knowledge. e weights of every expert are ] 1 � 0.4, ] 2 � 0.1, and ] 3 � 0.5. e following steps of the stated algorithm are implemented here.
Step 1: Experts evaluate each candidate under the set of the attribute and present their judgments with the real values. ese values are summarized in Tables 11-13. Step 2: According to the importance of these five attributes, we give the following results for each expert:
Step 4: Calculate the distances D l and D l as follows: Step 5: Calculate the lower and upper approximations of the best and worst decision-making objects as follows.
Take e � e 1 , and we have

Complexity 13
Take e � e 2 , and we have Take e � e 3 , and we have Step 6 Based on the importance of these five attributes, we give the worst and the best decision-making objects as follows: Step 7 Based on these results, we calculate the group optimal index as follows. R � (0.503757/u 1 ) + (0.50474/u 2 ) + (0.486027/ u 3 ) + (0.527955/u 4 ) + (0.508771/u 5 ) + (0.486722/ u 6 ), and hence get the ranking order as u 4 ≥ u 5 ≥ u 2 ≥ u 1 ≥ u 6 ≥ u 3 . From the calculations, we conclude that the 4th system analysis engineer is the best alternative among the others. Furthermore, we get the solution for Case 2 by the same analysis in Case 1. erefore, we have the group optimal index as follows: and hence get the ranking order as u 4 ≥ u 1 ≥ u 6 ≥ u 2 ≥ u 3 ≥ u 5 . rough the previous computation, we obtain the 4th system analysis engineer is the best alternative among the others.

Comparative Analysis.
e main aim of the current work is to present a method that increases the lower approximation and decreases the upper approximation of Zhan's methods in [40]. is can be seen easily from Examples 2-4. Moreover and by looking at Tables 23 and 24, we can see that the ranking results of the two decisionmaking models. It is obvious that the optimal selected alternative is the same, although there exist some differences in the ranking results because we choose different decisionmaking methods.
An easy way to see the effectiveness of our method and the differences between the four models (i.e., our three proposed models and Zhan's model) are shown in Figures 1  and 2. Figure 1 explained the comparisons between the lower approximations for the four models (i.e., 0-OMGITFLA, 1-OMGITFLA, 2-OMGITFLA, and 3-OMGITFLA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). is figure justifies that the 2-OMGITFLA is better than the others. Figure 2 clarified the differences between the upper approximations for the four models (i.e., 0-OMGITFUA, 1-OMGITFUA, 2-OMGITFUA, and 3-OMGITFUA) for the               two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). is figure illustrates that the 2-OMGITFUA is lower than the others.

Two models
Obtain a decision Zhan model Two models Obtain a decision Zhan model 18 Complexity Also, an illustrative example with algorithm is given. erefore, it is clear to see that 2-COMGITFRS is better than the other models (i.e., 0-COMGITFRS, 1-COMGITFRS, and 3-COMGITFRS).
In future research, we plan to further investigate along with the following: (1) topological properties of the presented methods [44,45], (2) combination with the soft set and the proposed methods [46,47], and (3) combination with the neutrosophic set and the current methods [48].

Data Availability
No data were used to support this study.

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