On Some Types of Multigranulation Covering Based on Binary Relations

Nawar, Ashraf; Elsakhawy, E. A.

doi:https://doi.org/10.1155/2021/1022955

Journal of Function Spaces

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Abstract Introduction Results Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 1022955 | https://doi.org/10.1155/2021/1022955

On Some Types of Multigranulation Covering Based on Binary Relations

Ashraf Nawar¹and E. A. Elsakhawy²

Academic Editor: Serena Matucci

Received09 Jun 2021

Revised16 Jul 2021

Accepted02 Aug 2021

Published08 Nov 2021

Abstract

Recently, the notions of right and left covering rough sets were constructed by right and left neighborhoods to propose four types of multigranulation covering rough set (MGCRS) models. These models were constructed using the granulations as equivalence relations. In this paper, we introduce four types of multigranulation covering rough set models under arbitrary relations using the -minimal and -maximal descriptors of objects in a given universe. We also study the properties of these new models. Thus, we explore the relationships between these models. Then, we put forward an algorithm to illustrate the method of reduction based on the presented model. Finally, we give an illustrative example to show its efficiency and importance.

1. Introduction

The notion of rough set theory originated by Pawlak in 1982 [1, 2] to deal with uncertain information and knowledge. It is a tool concerned with the approximation of sets described by a single binary relation. In the view of granular computing suggested by Zadeh [3], a general concept described by a set is characterized via the upper and lower approximations under a single granulation (always equivalence relation) on the universe. This tool has been widely used in many subjects including machine learning, data mining, decision support, and analysis. In the past 20 years, many authors have proposed several extensions of the rough set model [4–19]. In some cases, it is important to use multiequivalence relations on the universe to describe precisely a target concept. Recently, more attention is given to multigranulation rough set (MGRS) models and, also, to multigranulation covering rough set (MGCRS) models in which a target concept is approximated by employing the maximal or minimal descriptors of objects in the given universe. In [20, 21], Qian et al. developed a multigranulation rough set (MGRS) model by using equivalence relations. Several scholars worked on MGRS such as the MGRS model through multiple tolerance relations in incomplete information systems, MGRS via the fuzzy approximation space, the hierarchical structures of MGRS, the topological and lattice-theoretic properties of MGRS, and the efficient rough feature selection algorithm with MGRS [22–30]. Moreover, Liu and Miao and Liu and Wang [31, 32] introduced the multigranulation covering via rough set (MGCRS) and fuzzy rough set (MGCFRS). Lin et al. studied two types of the neighborhood via MGRS [33] and three new types of MGCRS [34]. Also, three types of MGRS via the tolerance, ordered, and generalized relations are investigated and developed the multigranulation decision-theoretic rough set [35–38]. In addition, Liu et al. [39] proposed four new types of MGCRS using the minimal and maximal descriptions and discussed relevant characteristics. For more details about MGRS, see, for instance, [40–44].

The notions of left and right covering rough sets proposed by Abd El-Monsef et al. [45] are important tool to make an extension of Liu et al. [39]. The objective of this paper is to develop new models of MGCRS using the notions of left and right covering using the concepts of -minimal and -maximal descriptions. Also, we discuss the properties of these models. The relationships between these models are studied. Then, we present the reduction method over our proposed work and establish a numerical example to show its performance. The paper consists of six sections and is organized as follows: Section 1 deals with a brief history to the subject. Section 2 includes the preliminary concepts. Section 3 is the main core of the paper and consists of the new models. In Section 4, the properties and differences between the proposed models are introduced. Section 5 explores new criteria to make a reduction with a test example. We end up with conclusion in the last section.

2. Basic Terminologies and Results

This section provides a short survey of some notions used throughout the article.

Definition 1 [26]. Let be an universal set and . We call as a covering of , if for any. Also, is called a covering approximation space (briefly, CAS).

Definition 2 [46]. Let be a binary relation on an universe set , and for every , we have the following two classes. Define the after and fore sets as follows:

Definition 3 [45]. Let be a binary relation on an universe set . For each , define the right covering (resp., the left covering ) as follows:

Definition 4 [45]. Let be a binary relation on an universe set and be a -cover of , where . Then, is said to be covering approximation space (briefly, -CAS).

Definition 5 [45]. Let be -CAS. For every , define the right neighborhood , the left neighborhood , the intersection neighborhood , and the union neighborhood , respectively, as follows:

Definition 6 [45]. Let be -CAS and and . Define the -lower approximation, -upper approximation, -boundary, -positive, -negative, and -accuracy of , respectively, as follows:Pawlak’s [1, 2] rough set properties are given as follows:
(L₁)
(L₂)
(L₃)
(L₄)
(L₅)
(L₆)
(L₇)
(L₈)
(L₉)

Definition 7 [47]. Let be a CAS and . For any , define the minimal and maximal descriptions of , respectively, as follows:

Definition 8 [39]. Let be MGCAS and . For any , define four types of the lower and upper approximations, respectively, as follows: If (resp., , , and ) (resp., , , and ), then is called the first kind of a multigranulation covering rough set (briefly, type 1-MGCRS) (resp., type 2-MGCRS, type 3-MGCRS, and type 4-MGCRS), else it is definable.

Definition 9 [48]. Let be a covering information system. For any and , define the first type of optimistic multigranulation covering lower approximation (briefly, 1-OMGCLA) and the first type of optimistic multigranulation covering upper approximation (briefly, 1-OMGCUA) as follows:

Definition 10 [48]. Let be a covering information system. For any and , define the first type of pessimistic multigranulation covering lower approximation (briefly, 1-PMGCLA) and the first type of pessimistic multigranulation covering upper approximation (briefly, 1-PMGCUA) as follows:

Next, we have the following definitions using the notion of -CAS.

Definition 11. Let be -CA and . For any , define the -minimal and -maximal descriptions of , respectively, as follows:

We give the following example to illustrate the above definition.

Example 1. Let be -CAS, and . Then, we have the following results:

Definition 12. Let be -CAS and . For any , define the lower and upper approximations, respectively, as follows:

To explain the above definition, we give the following example.

Example 2. Consider Example 1, if , then we have the following results.

3. Multi--Covering Approximation Space

Presume that is an universal set, is a family of binary relations on , and is -cover of depending on , where . Thus, is called a multi--covering approximation space (briefly, MCAS).

Definition 13. Assume that is a MCAS and , for any and . Then, we have four novel kinds of lower and upper approximations written as follows:
Style 1
The 1-MCLA and the 1-MCUA are shown as follows:If , then is said to be the first kind of -covering multigranulation rough set (briefly, 1-MGCRS), else it is definable.
Style 2
The 2-MCLA and the 2-MCUA are seen as follows:If , then is said to be the second kind of -covering multigranulation rough set (briefly, 2-MGCRS), else it is definable.
Style 3
The 3-MCLA and the 3-MCUA are seen as follows:If , then is said to be the third kind of -covering multigranulation rough set (briefly, 3-MGCRS), else it is definable.
Style 4
The 4-MCLA and the 4-MCUA are seen as follows:If , then is said to be the fourth kind of -covering multigranulation rough set (briefly, 4-MGCRS), else it is definable.

Example 3. Consider is a MCAS, and , where and . Take ; then, we have the presented outcomes:
(1_r) .
(2_r) .
(3_r) .
(4_r) .

Theorem 14. Suppose that is a MCAS. For any , we get the following properties:(1)(2)(3)(4)

Proof. Here, we want to set (1) only.(1)Also, it is easy to see

Proposition 15. Suppose that is a MCAS. For any , we get the following properties:(1)(2)(3)

Proof. Here, we want to set (1) only.(1)It is obvious that . On the other hand, we have . Thus, we get thatAlso, it is clear that . Consequently, we have . So, we haveHence, and .

The above Proposition 15 is not true for 4-MGCRS as in the following example.

Example 4. Consider that is a MCAS, and , where and
Take and ; then, we have the presented outcomes.
(1_r) Then,
(2_r) and Then,
(1_l) and Then,
(2_l) and Then,

Next, we will establish new properties in Proposition 16. These characteristics are done for 1-MGCRS, 2-MGCRS, 3-MGCRS, and 4-MGCRS, though we demonstrate it in the case of 1-MGCRS and others are similar.

Proposition 16. Suppose that is a MCAS. For any , we get the following properties:(1)If , then (2)If , then (3)(4)(5)(6)

Proof. Now, we just need to show (1) and (2).(1)From Definition 13 and since , then, we obtain the following:(2)From Definition 13 and since , then, we have the following:

Example 5. Consider Example 4. Then, we have the following:
(1_r) Take and , then we have and . Thus,
(1_l) Take and , then we have and . Thus,
(2_r) Take and , then we have and . Thus,
(2_l) Take and , then we have and . Thus,
Take and , then we have and . Thus,
Take and , then we have and . Thus,
Take and , then we have and . Thus,
Take and , then we have and . Thus,

4. Relationships among Different Proposed Models

Next, we present the relationships between the proposed MCAS models.

By using Definition 13, we obtain the following properties.

Proposition 17. Let be a MCAS and . Then, we have the following results: (1)(2)(3)(4)

Remark 18. Let be a MCAS and . Then, we have the following results: (1)(2)This means that 2-MGCRS and 3-MGCRS are independent.

Proposition 19. Let be a MCAS and . Then, we have the following results: (1)(2)

To illustrate the above characteristic, we give the following example.

Example 6. Consider Example 4 and let . Then, we have the following outcomes:(1)For , we have(2)For , we haveSo, you can find the following: (1)(2)(3)(4)(5)(6)(7)(8)

Tables 1 and 2 show the Pawlak characteristics for the lower and upper approximations which are given in Definition 13.

5. Relative Reduction of a MCAS

This section is aimed at discussing a relative reduction of a pessimistic multigranulation -covering rough sets (briefly, PMCRS). First, we give the following couple of definitions.

Definition 20. Let be a MCAS and . For any and , define the pessimistic multigranulation -covering lower approximation (briefly, PMGCLA) and pessimistic multigranulation -covering lower approximation (briefly, PMGCLA) as follows:

Definition 21. Let be a MCAS and . Suppose that is a decision partition of . Then,(i) and , but , for ; then, is a reduction of PMCLA(ii) and , but , for , then, is a reduction of PMCUA

We can illustrate the method of reduction as the following Algorithm 1.

Input: with information system.
Output: Reduction of PMCLA.
1: Calculate .
2: Remove , and
3: Remove a covering in again and get . If , return ; else, go to Step 2.
4: : Repeat the Steps 2 and 3 for each covering in to get all the relative reduce of the covering family.

Example 7. Presume that is a set of six houses, is a set of attributes, and is a set of decisions. The values of equally shared area could be {large, ordinary, small}. The values of color could be {excellent, good, bad}. The values of price could be {high, middle, low}. The values of surroundings could be {quiet, noisy, very noisy}. The decision values of purchase opinions could be {support, oppose}, which is randomly chosen from experts. The evaluation results are shown in Table 3.

As for the attribute set , the binary relation is obtained as follows :

It is easy to see that the is reflexive and transitive but not symmetric.

If is the decision set, then the nonequivalence relation is defined as follows:

Then, we can construct the following two covers:(i)Right covering (-cover for short)(ii)Left covering (-cover for short)

So, we have the following results:

Thus, we can establish Tables 4 and 5 for the neighborhood of as follows.

Now, we can apply Algorithm 1 as follows.

Step 1.

Step 2. , Therefore, is a reduction of the PMCRS.

Also, we can get the following outcomes of the left covering:

Therefore, is a reduction of the PMCRS.

6. Conclusion

In this article, we present a notion called multi--covering approximation space (MCAS) by using the concept of -minimal and -maximal descriptions. Based on these notions, we establish four new types of multigranulation covering rough sets, denoted MCAS. We also study the properties of these new models. Further, we put forward a new methodology to make a reduction by the presented work. Then, we demonstrate the reduction method with the help of an illustrative example which shows its effectiveness and reliability. The main differences between our proposed work and the previous one in [39] are that the authors in [39] introduced four types of MGCRSs using the minimal and maximal description based on equivalence relations and here we used the notions of right (resp., left) covering rough sets to investigate four kinds of multigranulation right (resp., left) covering rough sets using the right (resp., left) minimal and right (resp., left) maximal description induced by binary relations. In further research, we hope to use this approach in fuzzy rough covering-based fuzzy neighborhoods [49], fuzzy soft covering-based rough sets [50], and soft fuzzy covering-based rough sets [51].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Funding

The second author is responsible for funding this work.

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Copyright © 2021 Ashraf Nawar and E. A. Elsakhawy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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