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Article

Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description

1
School of Computer and Information Engineering, Institute for Artificial Intelligence, Shanghai Polytechnic University, Shanghai 201209, China
2
School of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(8), 1217; https://doi.org/10.3390/sym17081217 (registering DOI)
Submission received: 9 June 2025 / Revised: 18 July 2025 / Accepted: 21 July 2025 / Published: 1 August 2025
(This article belongs to the Section Mathematics)

Abstract

Rough sets and fuzzy sets are two complementary approaches for modeling uncertainty and imprecision. Their integration enables a more comprehensive representation of complex, uncertain systems. However, existing rough fuzzy sets models lack the expressive power to fully capture the interactions among structural uncertainty, cognitive hesitation, and multi-level granular information. To address these limitations, we achieve the following: (1) We propose intuitionistic fuzzy covering rough membership and non-membership degrees based on maximal description and construct a new single-granulation model that more effectively captures both the structural relationships among elements and the semantics of fuzzy information. (2) We further extend the model to a multi-granulation framework by defining optimistic and pessimistic approximation operators and analyzing their properties. Additionally, we propose a neutral multi-granulation covering rough intuitionistic fuzzy sets based on aggregated membership and non-membership degrees. Compared with single-granulation models, the multi-granulation models integrate multiple levels of information, allowing for more fine-grained and robust representations of uncertainty. Finally, a case study on real estate investment was conducted to validate the effectiveness of the proposed models. The results show that our models can more precisely represent uncertainty and granularity in complex data, providing a flexible tool for knowledge representation in decision-making scenarios.

1. Introduction

In research fields such as expert decision-making systems and multi-source data analysis, data often exhibit characteristics such as fuzziness, incompleteness, and uncertainty [1,2,3,4]. These complex uncertainty issues have seriously affected the accuracy of information understanding and modeling. How to model such uncertain information simultaneously from the structural and cognitive levels has become an important research topic in artificial intelligence and data mining [5,6,7].
Rough set theory provides a solid mathematical foundation for the processing of uncertain information [8]. Its core idea is to approximate the target set through lower and upper approximations without relying on any prior knowledge and to model the target concept through approximation. Therefore, it has been widely applied in data mining, pattern recognition, and decision support [3,4,5,6,7]. However, the conventional rough set theory is based on a partition structure and has difficulty handling the common asymmetric, incomplete, and overlapping phenomena in actual data. To overcome this limitation, the covering rough sets model was proposed, which replaces partitions with coverings, offering greater flexibility for modeling non-partitioned or imprecise spaces [9]. Covering rough sets is an important generalization of rough sets because the concept of a covering is more general than the concept of a partition. Coverings are a useful form of data structure, and covering rough sets provides an effective tool to handle restrictive data. In recent years, covering rough sets have received widespread attention from researchers, leading to numerous significant and insightful research findings [10,11,12,13].
Moreover, in real-world problems, there is also cognitive uncertainty arising from expert knowledge or subjective experience. This type of uncertainty is often difficult to express through structural approximation alone. To characterize this cognitive uncertainty, fuzzy sets [14] and their extended form, i.e., intuitionistic fuzzy sets [15], have been extensively employed. On the basis of fuzzy sets, Atanassov developed the concept of intuitionistic fuzzy sets for more efficiently describing the concepts of vagueness and fuzziness by proposing three notions, namely, the membership degree, non-membership degree, and hesitation degree [16,17].
In fact, rough sets and fuzzy sets are not opposite theories for imperfection, accuracy, and vagueness, and these two theories complement each other. It is natural to combine rough sets and fuzzy sets. In the fusion and innovation of these two theories, Dubois first combined rough and fuzzy sets into rough fuzzy sets and fuzzy rough sets [18,19], which has attracted the attention of many scholars [20,21,22,23,24,25]. As covering rough sets are more general than rough sets, and intuitionistic fuzzy sets are more flexible than fuzzy sets, studies on the combination of covering rough sets and intuitionistic fuzzy sets are more practical. Extensive research has focused on single-granulation models [26,27,28,29]. However, in real-world scenarios, information often presents multi-source, multi-scale, and multi-granular characteristics. For example, in intelligent transportation systems, trajectory data are collected from various sensors and span multiple temporal granularities; in medical image analysis, disease diagnosis typically involves the integration of multi-modal data. Faced with such complex and hierarchical information structures, rough set models based on a single granularity are no longer sufficient to meet practical requirements. To address the limitations of single-granularity analysis, Qian et al. extended Pawlak’s single-granulation rough sets model to a multi-granulation framework (MGRS) [30,31,32,33]. Recently, MGRS have attracted considerable attention [34,35,36,37]. MGRS has been successfully applied in multi-view learning, classifier fusion, and hierarchical decision-making [38,39,40,41].
Although relatively good results have been obtained by the above-mentioned methods, there is still room for improvement. Most existing MGRS models are based on conventional rough sets and lack the capability to model cognitive uncertainty or structural overlaps. While some studies have introduced rough sets into multi-granulation frameworks, these efforts typically remain at a basic level and do not fully integrate covering-based approximations with intuitionistic fuzzy representations. These limitations restrict their applicability to real-world data, particularly in complex scenarios characterized by overlapping structures, uncertain boundaries, and incomplete information. Enhancing the modeling capability for uncertainty can significantly improve performance in real-world applications such as decision-making, classification tasks, and uncertainty data analysis.
To alleviate the above concerns, we propose a novel approach to covering rough intuitionistic fuzzy sets based on maximal description. By defining the intuitionistic fuzzy covering rough membership and non-membership degrees, a more precise single-granulation model is established. The proposed model effectively captures the structural relationships among elements and better represents the semantics of fuzzy information, thereby significantly enhancing the ability to characterize uncertainty and vagueness in complex data. Furthermore, we extend the model from single-granulation to a multi-granulation framework, which enables the integration of multi-source or multi-level uncertain information within a unified approximation space. Multi-granulation covering rough intuitionistic fuzzy sets provides a robust and flexible solution for real-world applications that involve hierarchical data and complex uncertainty.
The main contributions of this paper are summarized as follows:
1. We propose a novel model of covering rough intuitionistic fuzzy sets (CRIFS) that jointly captures structural overlaps and cognitive hesitation in uncertain environments.
2. We extend the CRIFS to a multi-granular framework and construct three types of multi-granulation covering rough intuitionistic fuzzy sets, which incorporate multiple granular structures for improved adaptability in complex systems.
3. We demonstrate the effectiveness and interpretability of the proposed model in real-world scenarios of real estate investment, highlighting its theoretical value and practical applicability.
The remainder of this paper is organized as follows: Section 2 presents a comprehensive literature review. Section 3 provides a brief introduction to some preliminary concepts. In Section 4, first, we propose a new model of covering rough intuitionistic fuzzy sets and discuss its basic properties. Second, we introduce three main types of multi-granulation covering rough intuitionistic fuzzy sets and prove several important properties. Section 5 presents a case study to demonstrate the effectiveness of the proposed model. Finally, Section 6 concludes the paper.

2. Literature Review

Rough sets theory, introduced by Pawlak in 1982 [1], is an important mathematical tool for handling uncertainty and incomplete information and has been widely applied in data mining, pattern recognition, and decision analysis [2,3,4,5].
Covering rough sets represent a prominent extension of rough set theory and have been extensively investigated in recent years [8]. To address the limitations of conventional rough sets that rely on strict equivalence relations, covering rough sets were proposed as a generalization. By replacing partitions with more flexible coverings, this model allows overlapping subsets and better captures the uncertainty in complex or real-world data. Many researchers have investigated covering rough sets from various perspectives, leading to a series of theoretical extensions and practical applications. Zhu et al. [42,43] proposed three types of covering-based rough sets and summarized the relationships among three types of covering rough sets. Dai et al. [44] presented a new type of partial order for coverings to evaluate uncertainty measures in covering rough sets. On the basis of this framework, they further investigated uncertainty measures such as roughness, accuracy, entropy, and granularity, particularly in models defined by neighborhood and friend relations. Cai et al. [45] and Lang et al. [46] explored knowledge acquisition and attribute reduction in dynamic covering decision information systems and proposed incremental attribute reduction algorithms to handle variations in object sets, coverings, and attribute sets. Ma et al. [47] investigated a covering-based variable precision rough set model defined via boundary regions and proposed an attribute reduction approach that preserves approximations. Huang et al. [48] proposed a multi-scale data analysis model by extending traditional partition-based rough sets to covering-based structures. They introduced multi-scale covering decision tables for knowledge representation and developed optimal scale selection methods for consistent and inconsistent tables.
In 1965, Zadeh proposed the theory of fuzzy sets, which characterizes the degree to which an element belongs to a set by assigning it a membership value [14]. This theory effectively addresses the limitations of conventional set theory in handling continuous and vague information. Since its proposal, fuzzy set theory has been successfully applied in various fields, including control systems, decision analysis, pattern recognition, image processing, and natural language processing [20,21,22,23,24,25]. Intuitionistic fuzzy sets, as an extension of fuzzy set theory, introduce an additional concept of non-membership degree to describe the extent to which an element does not belong to a set [14]. This enables simultaneous characterization of membership, non-membership, and hesitation degrees, leading to a more refined and comprehensive representation of uncertainty. Since then, this field has attracted increasing attention from researchers worldwide. Zhang [49] introduced linguistic intuitionistic fuzzy sets (LIFSs), where membership and non-membership degrees are represented by linguistic terms. On the basis of the t-norm and t-conform, several aggregation operators were developed and applied to multiple attribute group decision-making. Sahoo [50] investigated the intuitionistic fuzzy competition graph and its extensions, such as intuitionistic fuzzy k-competition graphs and p-competition intuitionistic fuzzy graphs, and explored their applications in ecological systems. Kumar [51] proposed algorithms for solving optimization problems under crisp, fuzzy, and intuitionistic fuzzy settings. The research demonstrated the effectiveness of the approach through type-2 models and real-world examples. Alcantud et al. [52] proposed the first method for aggregating infinite sequences of intuitionistic fuzzy sets and introduced a temporal aggregation tool to handle indefinitely long decision-making processes for intuitionistic fuzzy information.
Fuzzy sets and rough sets each have their own advantages and characteristics in addressing uncertainty in information systems. By combining their strengths, they can be jointly used to study specific types of problems. As a result, researchers have proposed hybrid models such as fuzzy rough sets and rough fuzzy sets [18,19]. In fuzzy rough sets, fuzzy concepts are incorporated into the rough approximation process to handle continuous or imprecise data. Conversely, rough fuzzy sets apply rough approximations to fuzzy membership functions, enhancing the ability to capture uncertainty in vague linguistic terms. These hybrid models have been widely applied in areas such as data mining, classification, feature selection, and decision-making [21,22,23,24,25].
From the perspective of particle computation, the classical Pawlak rough sets model describes the target concept in a single particle size space. While single-granulation models based on a fixed granularity structure are effective for certain tasks, they often fail to capture the complexity and diversity of real-world data, which may require reasoning at multiple levels of granularity. To address this limitation, the concept of multi-granulation rough sets was introduced, allowing the integration of multiple granularity spaces to improve approximation quality and enhance model robustness. To handle multi-source fuzzy information systems, Yang et al. [39] proposed a multi-granulation rough set model for multi-source fuzzy information systems using tolerance relations and analyzed different approximation types and uncertainty measures. Kang et al. [40] introduced a gray multi-granulation rough set model by integrating gray relational relations into the MGRS and proposed two approximation strategies for knowledge acquisition and attribute reduction. Yu et al. [41] developed the VMFS, a multi-label feature selection method that combines multi-granulation rough sets and three-way decisions to address uncertainty and inter-feature redundancy.

3. Basic Concepts

In this section, we review some basic notions concerning rough sets, covering rough sets, multi-granulation rough sets, and intuitionistic fuzzy sets. Table 1 summarizes the key notations and explanations used throughout this paper.

3.1. Rough Sets

Definition 1.
[8].  Let U be a nonempty and finite universe of discourse, and let R be an equivalence relation in U. R generates a partition  U / R = { K 1 , K 2 , , K n }   on U. For all  X U , the lower and upper approximations of X are described by the following two sets:
R _ ( X ) = { K i | K i U / R K i X } , R ¯ ( X ) = { K i | K i U / R K i X ϕ } .
If  R _ ( X ) = R ¯ ( X ) , then X is referred to as R-defined in U; otherwise X is referred to as R-undefined in U.
Definition 2.
[53].  Let U be a nonempty set, and let R be an equivalence relation of U.
  • x U , X U . Let  μ X R ( x ) = | [ x ] R X | / | [ x ] R |  be the degree of rough membership of x in R.  | |  is the cardinality of a set, and the equivalence class of x with respect to relation R is denoted as  [ x ] R .

3.2. Covering Approximation Spaces

Definition 3.
[9].  Let U be a finite and nonempty universe of discourse, and let C be a family of subsets of U. If none of the subsets in C are empty and C = U , C is called a covering of U.
Definition 4.
[9].  Let U be a nonempty set, and let C be a covering of U. The pair (U, C) is called the covering approximation space.
Definition 5.
[54]. Let (U, C) be a covering approximation space, for any x U . The set family  M a x d ( x ) = { K C | x K ( S C x S S K K = S ) } is called the maximal description of x.

3.3. Multi-Granulation Rough Sets

Definition 6.
[31,32,33].  Let I be an information system where A 1 , A 2 , , A m A T . For all  X U , the optimistic multi-granulation lower and upper approximations are denoted as i = 1 m A i O ¯ ( X ) and  i = 1 m A i O ¯ ( X ) , respectively,
i = 1 m A i O ¯ ( X ) = { x U | [ x ] A 1 X [ x ] A 2 X [ x ] A 3 X [ x ] A m X } , i = 1 m A i O ¯ ( X ) = ~ ( i = 1 m A i O ¯ ( ~ X ) ) ,
where  x A i   1     i     m  is the equivalence class of x in terms of the equivalence relation  A i 1     i     m  and  ~ X  is the complement of X.
By the optimistic multi-granulation lower and upper approximations, the optimistic multi-granulation boundary region of X is expressed as follows:
B N i = 1 m A i O ( x ) = i = 1 m A i O ¯ ( X ) i = 1 m A i O ¯ ( X ) .
Definition 7.
[31,32,33].  Let I be an information system where A 1 , A 2 , , A m A T . Then, for all  X U , the pessimistic multi-granulation lower and upper approximations are denoted as i = 1 m A i P ¯ ( X ) and  i = 1 m A i P ¯ ( X ) , respectively,
i = 1 m A i P ¯ ( X ) = { x U | [ x ] A 1 X [ x ] A 2 X [ x ] A 3 X [ x ] A m X } , i = 1 m A i P ¯ ( X ) = ~ ( i = 1 m A i P ¯ ( ~ X ) ) ,
where    x A i   1     i     m  is the equivalence class of x in terms of the equivalence relation  A i 1     i     m  and  ~ X  is the complement of X.
By the pessimistic multi-granulation lower and upper approximations, the pessimistic multi-granulation boundary region of X is expressed as follows:
B N i = 1 m A i P ( x ) = i = 1 m A i P ¯ ( X ) i = 1 m A i P ¯ ( X ) .

3.4. Intuitionistic Fuzzy Sets

Definition 8.
[15,16].  Let U be the universe of discourse. Then, an intuitionistic fuzzy set (IFS) of A in U is an object having the form of A = { ( x , μ A ( x ) , ν A ( x ) ) | x U } , where μ A ( x ) : U [ 0 , 1 ] and ν A ( x ) : U [ 0 , 1 ] satisfy 0 μ A ( x ) + ν A ( x ) 1 for all  x U , and μ A ( x ) and  ν A ( x )  are the membership and non-membership degrees of the element of  x  in A, respectively. Furthermore, π A ( x ) = 1 μ A ( x ) ν A ( x ) is the hesitancy degrees of the element  x  in A.
The family of all IF sets in U is denoted as IFS(U).
Definition 9.
[15,16].  Let A = { ( x , μ A ( x ) , ν A ( x ) ) | x U } , B = { ( x , μ B ( x ) , ν B ( x ) ) | x U } , and A , B I F S ( U ) .  Some basic operations performed on IFS(U) are defined as follows:
(1) 
A = B μ A ( x ) = μ B ( x ) ν A ( x ) = ν B ( x ) .
(2) 
A B μ A ( x ) μ B ( x ) ν A ( x ) ν B ( x ) .
(3) 
A B = { ( x , min { μ A ( x ) , μ B ( x ) } , max { ν A ( x ) , ν B ( x ) } ) } .
(4) 
A B = { ( x , max { μ A ( x ) , μ B ( x ) } , min { ν A ( x ) , ν B ( x ) } ) } .
(5) 
The complementary set of A is denoted as follows:
~ A = { ( x , ν A ( x ) , μ A ( x ) ) | x U } I F ( U ) .
(6) 
A B = A ~ B .

4. Methodology

In this section, first, we define the definitions of the intuitionistic fuzzy covering rough membership and non-membership degrees and a new kind of covering rough intuitionistic fuzzy sets based on maximal description (in brief, the CRIFS). Some related theorems are then discussed. Within the multi-granulation framework, we construct optimistic, pessimistic, and neutral multi-granulation covering rough intuitionistic fuzzy sets and systematically define and investigate their fundamental theorems and operation properties. These theoretical results provide solid support for the rationality and applicability of the proposed models. To further validate the effectiveness of the models, several examples are presented to illustrate their computational procedures in detail. Finally, a comparative analysis with existing approaches is conducted to demonstrate the advantages of the proposed model.

4.1. New Kind of Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description

To establish a formal relationship between the covering C and the membership and non-membership degrees of the elements in the intuitionistic fuzzy set A, the notions of intuitionistic fuzzy covering rough membership and non-membership degrees based on maximal description are defined.
Definition 10.
Let (U, C) be a covering approximation space, and let U be a finite and nonempty universe of discourse.  C  is a covering of U. For  A I F S ( U ) , x U , the intuitionistic fuzzy covering rough membership and non-membership degrees of x based on maximal description with respect to A are defined as follows:
μ A ( x ) = y M a x d ( x ) μ A ( y ) / | M a x d ( x ) | , ν A ( x ) = y M a x d ( x ) ν A ( y ) / | M a x d ( x ) | .
The intuitionistic fuzzy covering rough membership and non-membership degrees capture both the relationship between an element and its maximal description and the corresponding membership information within the intuitionistic fuzzy set. This provides a more comprehensive view of the degree to which each element belongs to set A. On this basis, we propose a new model of covering rough intuitionistic fuzzy sets.
Definition 11.
Let (U, C) be a covering approximation space, let U be a finite and nonempty universe of discourse, and let C be a covering of U. For  A I F S ( U ) , x U , the lower and upper approximations of A with respect to (U, C), denoted C X ¯ ( A ) and  C X ¯ ( A ) , respectively, are defined as follows:
C X ¯ ( A ) = { ( x , μ C X ¯ ( A ) ( x ) , ν C X ¯ ( A ) ( x ) ) | x U } , C X ¯ ( A ) = { ( x , μ C X ¯ ( A ) ( x ) , ν C X ¯ ( A ) ( x ) ) | x U } ,
where
μ C X ¯ ( A ) ( x ) = min ( μ A ( x ) , μ A ( x ) ) , ν C X ¯ ( A ) ( x ) = max ( ν A ( x ) , ν A ( x ) ) , μ C X ¯ ( A ) ( x ) = max ( μ A ( x ) , μ A ( x ) ) , ν C X ¯ ( A ) ( x ) = min ( ν A ( x ) , ν A ( x ) ) .
This model is called the covering rough intuitionistic fuzzy sets (CRIFS).
On the basis of the preceding definitions, we proceed to formalize a series of fundamental properties in order to rigorously characterize the theoretical structure of the proposed model.
Theorem 1.
Let (U, C) be a covering approximation space, let U be a finite and nonempty universe of discourse, and let C be a covering of U.  A , B I F S ( U ) , x U . Then, CRIFS has the following properties:
(1) 
C X ¯ ( U ) = U , C X ¯ ( U ) = U ,
(2) 
C X ¯ ( ϕ ) = ϕ , C X ¯ ( ϕ ) = ϕ ,
(3) 
C X ¯ ( A ) A C X ¯ ( A ) ,
(4) 
If  A B , then  C X ¯ ( A ) C X ¯ ( B ) , C X ¯ ( A ) C X ¯ ( B )
Proof
(1) When the intuitionistic fuzzy set A is the universe U, for all x U , there exists μ U ( x ) = 1 , ν U ( x ) = 0 . By Definition 10, μ U ( x ) = 1 , ν U ( x ) = 0 .
From Definition 11, μ C X ¯ ( U ) ( x ) = 1 , ν C X ¯ ( U ) ( x ) = 0 , and μ C X ¯ ( U ) ( x ) = 1 , ν C X ¯ ( U ) ( x ) = 0 ; hence, C X ¯ ( U ) = C X ¯ ( U ) = U .
(2) The proving process of (2) is similar to (1).
(3) By Definition 12, for all x U , μ A ( x ) μ A ( x ) or μ A ( x ) > μ A ( x ) , and ν A ( x ) ν A ( x ) or ν A ( x ) > ν A ( x ) .
There are four different situations:
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
We prove the processes of ① and ② as follows:
①When μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) , and μ C X ¯ ( A ) ( x ) = μ A ( x ) , μ C X ¯ ( A ) ( x ) = μ A ( x ) ; ν C X ¯ ( A ) ( x ) = ν A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) . Then, μ C X ¯ ( A ) ( x ) μ A ( x ) μ C X ¯ ( A ) ( x ) ,   ν C X ¯ ( A ) ( x ) ν A ( x ) ν C X ¯ ( A ) ( x ) .
② When μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) , and μ C X ¯ ( A ) ( x ) = μ A ( x ) , μ C X ¯ ( A ) ( x ) = μ A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) . Then, ν C X ¯ ( A ) ( x ) ν A ( x ) ν C X ( A ) ( x ) .
Similarly, we prove ③ and ④.
Hence, C X ¯ ( A ) A C X ¯ ( A ) .
(4) Since A B , x U , μ A ( x ) μ B ( x ) , ν A ( x ) ν B ( x ) ,   μ A ( x ) μ B ( x ) , ν A ( x ) ν B ( x ) . Then, there are four kinds of circumstances for the membership and non-membership degrees of the lower and upper approximations.
① When μ C X ¯ ( A ) ( x ) = μ A ( x ) ,   μ C X ¯ ( A ) ( x ) = μ A ( x ) , and
ν C X ¯ ( A ) ( x ) = ν A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) .
② When μ C X ¯ ( A ) ( x ) = μ A ( x ) ,   μ C X ¯ ( A ) ( x ) = μ A ( x ) , and
ν C X ¯ ( A ) ( x ) = ν A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) .
③ When μ C X ¯ ( A ) ( x ) = μ A ( x ) , μ C X ¯ ( A ) ( x ) = μ A ( x ) , and
ν C X ¯ ( A ) ( x ) = ν A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) .
④ When μ C X ¯ ( A ) ( x ) = μ A ( x ) , μ C X ¯ ( A ) ( x ) = μ A ( x ) , and
ν C X ¯ ( A ) ( x ) = ν A ( x ) , ν C X ¯ ( A ) ( x ) = ν A ( x ) .
Here, only ① is proven.
Case 1. If μ C X ¯ ( B ) ( x ) = μ B ( x ) , μ C X ¯ ( B ) ( x ) = μ B ( x ) , and
ν C X ¯ ( B ) ( x ) = ν B ( x ) , ν C X ¯ ( B ) ( x ) = ν B ( x ) .
Then, μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) , μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) , and
ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) , ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) .
Case 2. If μ C X ¯ ( B ) ( x ) = μ B ( x ) , μ C X ¯ ( B ) ( x ) = μ B ( x ) , and
ν C X ¯ ( B ) ( x ) = ν B ( x ) , ν C X ¯ ( B ) ( x ) = ν B ( x ) .
Since μ A ( x ) μ A ( x ) μ B ( x ) ,   μ A ( x ) μ B ( x ) μ B ( x ) , then μ A ( x ) μ B ( x ) , μ A ( x ) μ B ( x ) .
Hence, μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) , μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) , and
ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) , ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) .
Case 3. If μ C X ¯ ( B ) ( x ) = μ B ( x ) , μ C X ¯ ( B ) ( x ) = μ B ( x ) , and
ν C X ¯ ( B ) ( x ) = ν B ( x ) , ν C X ¯ ( B ) ( x ) = ν B ( x ) .
Then, μ C X ¯ ( A ) ( x ) μ C K ¯ ( B ) ( x ) , μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) , and
ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) , ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) .
Since ν A ( x ) ν A ( x ) ν B ( x ) , ν A ( x ) ν B ( x ) ν B ( x ) .
Hence, ν A ( x ) ν B ( x ) , ν A ( x ) ν B ( x ) .
Then, μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) , μ C X ¯ ( A ) ( x ) μ C X ¯ ( B ) ( x ) ,
ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) , ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) .
Case 4. If μ C X ¯ ( B ) ( x ) = μ B ( x ) , μ C X ¯ ( B ) ( x ) = μ B ( x ) , and
ν C X ¯ ( B ) ( x ) = ν B ( x ) , ν C X ¯ ( B ) ( x ) = ν B ( x ) .
Then, μ C X ¯ ( A ) ( x ) μ C X ( B ) ¯ ( x ) , μ C X ¯ ( A ) ( x ) μ C X ( B ) ¯ ( x ) , and
ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) , ν C X ¯ ( A ) ( x ) ν C X ¯ ( B ) ( x ) .
The proving processes of ②, ③, and ④ are similar to ①.
Then, C X ¯ ( A ) C X ¯ ( B ) , C X ¯ ( A ) C X ¯ ( B ) . □
We use an example to illustrate the model of CRIFS in order to better understand the obtained results.
Example 1.
Let  U = { x 1 , x 2 , x 3 , , x 7 } , let C be a covering of U, and let  C = { { x 1 , x 2 , x 3 } , { x 1 , x 2 } , { x 3 , x 4 } , { x 2 , x 3 , x 4 , x 5 , x 7 } , { x 3 , x 4 , x 6 } , { x 5 , x 7 } , { x 5 , x 6 , x 7 } } . Then, the intuitionistic fuzzy set A is expressed as follows:
A = { ( x 1 , 1 , 0 ) , ( x 2 , 0.8 , 0.1 ) , ( x 3 , 0.5 , 0.4 ) , ( x 4 , 0.3 , 0.4 ) ,                 ( x 5 , 0.1 , 0.8 ) , ( x 6 , 0 , 0.7 ) , ( x 7 , 0.2 , 0.7 ) } .
The lower and upper approximations of A can be calculated with CRIFS.
Step 1. This step calculates the maximal description of x.
M a x d ( x 1 ) = { { x 1 , x 2 , x 3 } } , M a x d ( x 2 ) = { { x 1 , x 2 , x 3 } , { x 2 , x 3 , x 4 , x 5 , x 7 } } , M a x d ( x 3 ) = { { x 1 , x 2 , x 3 } , { x 2 , x 3 , x 4 , x 5 , x 7 } , { x 3 , x 4 , x 6 } } , M a x d ( x 4 ) = { { x 2 , x 3 , x 4 , x 5 , x 7 } , { x 3 , x 4 , x 6 } } , M a x d ( x 5 ) = { { x 2 , x 3 , x 4 , x 5 , x 7 } , { x 5 , x 6 , x 7 } } , M a x d ( x 6 ) = { { x 3 , x 4 , x 6 } , { x 5 , x 6 , x 7 } } , M a x d ( x 7 ) = { { x 2 , x 3 , x 4 , x 5 , x 7 } , { x 5 , x 6 , x 7 } } .
Step 2. According to Definition 10, the intuitionistic fuzzy covering rough membership and non-membership degrees based on maximal description are calculated as follows:
μ A ( x 1 ) = 2.3 3 0.77 , μ A ( x 2 ) = 2.9 6 0.48 , μ A ( x 3 ) = 2.9 7 0.41 , μ A ( x 4 ) = 1.9 6 0.32 , μ A ( x 5 ) = 1.9 6 0.32 , μ A ( x 6 ) = 1.1 5 = 0.22 , μ A ( x 7 ) = 1.9 6 0.32 .
The intuitionistic fuzzy covering rough non-membership degrees based on maximal description are similar to intuitionistic fuzzy covering rough membership degrees based on maximal description.
Step 3. According to Definition 11, we calculate the membership and non-membership degrees of x.
μ C X ¯ ( x 1 ) = min { 1 , 0.77 } = 0.77 , μ C X ¯ ( x 1 ) = max { 1 , 0.77 } = 1 ; μ C X ¯ ( x 2 ) = min { 0.8 , 0.48 } = 0.48 , μ C X ¯ ( x 2 ) = max { 0.8 , 0.48 } = 0.8 ; μ C X ¯ ( x 3 ) = min { 0.5 , 0.41 } = 0.41 , μ C X ¯ ( x 3 ) = max { 0.5 , 0.41 } = 0.5 ; μ C X ¯ ( x 4 ) = min { 0.3 , 0.32 } = 0.3 , μ C X ¯ ( x 4 ) = max { 0.3 , 0.32 } = 0.32 ; μ C X ¯ ( x 5 ) = min { 0.1 , 0.32 } = 0.1 , μ C X ¯ ( x 5 ) = max { 0.1 , 0.32 } = 0.32 ; μ C X ¯ ( x 6 ) = min { 0 , 0.22 } = 0 , μ C X ¯ ( x 6 ) = max { 0 , 0.22 } = 0.22 ; μ C X ¯ ( x 7 ) = min { 0.2 , 0.32 } = 0.2 , μ C X ¯ ( x 7 ) = max { 0.2 , 0.32 } = 0.32 .
By using the same method to calculate the non-membership degrees, the lower and upper approximations can be calculated with the CRIFS.
C X ¯ ( A ) ( x ) = { ( x 1 , 0.77 , 0.17 ) , ( x 2 , 0.48 , 0.4 ) , ( x 3 , 0.41 , 0.44 ) ,                                               ( x 4 , 0.3 , 0.52 ) , ( x 5 , 0.1 , 0.8 ) , ( x 6 , 0 , 0.7 ) , ( x 7 , 0.2 , 0.7 ) } . C X ¯ ( A ) ( x ) = { ( x 1 , 1 , 0 ) , ( x 2 , 0.8 , 0.1 ) , ( x 3 , 0.5 , 0.4 ) , ( x 4 , 0.32 , 0.4 ) ,                                               ( x 5 , 0.32 , 0.52 ) , ( x 6 , 0.22 , 0.6 ) , ( x 7 , 0.32 , 0.52 ) } .

4.2. Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description

In this section, we propose three types of multi-granulation covering rough intuitionistic fuzzy sets. After that, several important theorems are proved.
We extend the CRIFS from single-granulation to multi-granulation and propose the optimistic and pessimistic multi-granulation covering rough intuitionistic fuzzy sets (optimistic and pessimistic MGCRIFS). Furthermore, from the perspective of multi-granularity, we introduce maximal description and define the corresponding membership and non-membership degrees of elements under a multi-granulation framework. On the basis of these definitions, we construct a novel model of neutral multi-granulation covering rough intuitionistic fuzzy sets (neutral MGCRIFS).

4.2.1. Optimistic and Pessimistic Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets

Building upon the definitions of intuitionistic fuzzy covering rough membership and non-membership degrees and the single-granulation covering rough intuitionistic fuzzy sets, we further extend the model to a multi-granulation framework. In this extended setting, we introduce the lower and upper approximations of optimistic and pessimistic MGCRIFS, which are defined on the basis of maximal descriptions. This extension enables a more comprehensive representation of uncertainty by integrating information from multiple granular levels.
Definition 12.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A I F S ( U ) , x U . Then, the optimistic multi-granulation lower and upper approximation intuitionistic fuzzy operators of A based on maximal description with respect to   ( U , C ) , denoted i = 1 m C X i O ¯ ( A ) and  i = 1 m C X i O ¯ ( A ) , respectively, are defined as follows:
                  i = 1 m C X i O ¯ ( A ) = { x , μ i = 1 m C X i ¯ ( A ) O ( x ) , ν i = 1 m C X i ¯ ( A ) O ( x ) | x U } , i = 1 m C X i O ¯ ( A ) = { x , μ i = 1 m C X i ¯ ( A ) O ( x ) , ν i = 1 m C X i ¯ ( A ) O ( x ) | x U } ,
where
μ i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { min ( μ A ( x ) , μ A ( x ) ) } = i = 1 m μ C X i ¯ ( A ) , ν i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { max ( ν A ( x ) , ν A ( x ) ) } = i = 1 m ν C X i ¯ ( A ) , μ i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { max ( μ A ( x ) , μ A ( x ) ) } = i = 1 m μ C X i ¯ ( A ) , ν i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { min ( ν A ( x ) , ν A ( x ) ) } = i = 1 m ν C X i ¯ ( A ) .
We call this model the optimistic multi-granulation covering rough intuitionistic fuzzy sets (optimistic MGCRIFS).
Definition 13.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m ,   A I F S ( U ) , x U .  Then, the pessimistic multi-granulation lower and upper approximation intuitionistic fuzzy operators of A based on maximal description with respect to   ( U , C ) , denoted i = 1 m C X i P ¯ ( A ) and  i = 1 m C X i P ¯ ( A ) , respectively, are defined as follows:
                  i = 1 m C X i P ¯ ( A ) = { ( x , μ i = 1 m C X i ¯ ( A ) P ( x ) , ν i = 1 m C X i ¯ ( A ) P ( x ) ) | x U } , i = 1 m C X i P ¯ ( A ) = { ( x , μ i = 1 m C X i ¯ ( A ) P ( x ) , ν i = 1 m C X i ¯ ( A ) P ( x ) ) | x U } ,
where
μ i = 1 m C X i ¯ ( A ) P ( x ) = i = 1 m { min ( μ A ( x ) , μ A ( x ) ) } = i = 1 m μ C X i ¯ ( A ) , ν i = 1 m C X i ¯ ( A ) P ( x ) = i = 1 m { max ( ν A ( x ) , ν A ( x ) ) } = i = 1 m ν C X i ¯ ( A ) . μ i = 1 m C X i ¯ ( A ) P ( x ) = i = 1 m { max ( μ A ( x ) , μ A ( x ) ) } = i = 1 m μ C X i ¯ ( A ) , ν i = 1 m C X i ¯ ( A ) P ( x ) = i = 1 m { min ( ν A ( x ) , ν A ( x ) ) } = i = 1 m ν C X i ¯ ( A ) .
We call this model the pessimistic multi-granulation covering rough intuitionistic fuzzy sets (pessimistic MGCRIFS).
When m = 1, the optimistic and pessimistic MGCRIFS models degenerate into the CRIFS model presented in Section 4.1.
When A is an exact set, the optimistic and pessimistic MGCRIFS models degenerate into optimistic and pessimistic multi-granulation covering rough sets models.
When A is a fuzzy set, the optimistic and pessimistic MGCRIFS models degenerate into the optimistic and pessimistic multi-granulation covering rough fuzzy sets models.
After presenting the formal definitions of the optimistic and pessimistic MGCRIFS, we now turn to the investigation of their fundamental operational properties. These properties provide essential theoretical support for the models and help to further clarify their behavior under different operations.
Definition 14.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A I F S ( U ) , x U . i = 1 m C X i O ¯ and i = 1 m C X i O ¯ are the optimistic lower and upper approximation operators of the intuitionistic fuzzy set.
( 1 ) ( i = 1 m C X i O ¯ ( A ) , i = 1 m C X i O ¯ ( A ) ) = ( i = 1 m C X i O ¯ ( A ) , - i = 1 m C X i O ¯ ( A ) ) = ( i = 1 m C X i O ¯ ( A ) , i = 1 m C X i O ¯ ( A ) ) . ( 2 ) ( i = 1 m C X i O ¯ ( A ) , i = 1 m C X i O ¯ ( A ) ) ( i = 1 m C X i O ¯ ( B ) , i = 1 m C X i O ¯ ( B ) ) = ( i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) , i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) ) = ( μ i = 1 m C X i ¯ ( A ) O ( x ) μ i = 1 m C X i ¯ ( B ) O ( x ) , ν i = 1 m C X i ¯ ( A ) O ( x ) ν i = 1 m C X i ¯ ( B ) O ( x ) ) . ( 3 ) ( i = 1 m C X i O ¯ ( A ) , i = 1 m C X i O ¯ ( A ) ) ( i = 1 m C X i O ¯ ( B ) , i = 1 m C X i O ¯ ( B ) ) = ( i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) , i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) ) = ( μ i = 1 m C X i ¯ ( A ) O ( x ) μ i = 1 m C X i ¯ ( B ) O ( x ) , ν i = 1 m C X i ¯ ( A ) O ( x ) ν i = 1 m C X i ¯ ( B ) O ( x ) ) .
Definition 15.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A I F S ( U ) , x U . i = 1 m C X i P ¯  and i = 1 m C X i P ¯ are the pessimistic lower and upper approximation operators of the intuitionistic fuzzy set. Then,
( 1 ) ( i = 1 m C X i P ¯ ( A ) , i = 1 m C X i P ¯ ( A ) ) = ( i = 1 m C X i P ¯ ( A ) , - i = 1 m C X i P ¯ ( A ) ) = ( i = 1 m C X i P ¯ ( A ) , i = 1 m C X i P ¯ ( A ) ) . ( 2 ) ( i = 1 m C X i P ¯ ( A ) , i = 1 m C X i P ¯ ( A ) ) ( i = 1 m C X i P ¯ ( B ) , i = 1 m C X i P ¯ ( B ) ) = ( i = 1 m C X i P ¯ ( A ) i = 1 m C X i P ¯ ( B ) , i = 1 m C X i P ¯ ( A ) i = 1 m C X i P ¯ ( B ) ) = ( μ i = 1 m C X i ¯ ( A ) P ( x ) μ i = 1 m C X i ¯ ( B ) P ( x ) , ν i = 1 m C X i ¯ ( A ) P ( x ) ν i = 1 m C X i ¯ ( B ) P ( x ) ) . ( 3 ) ( i = 1 m C X i P ¯ ( A ) , i = 1 m C X i P ¯ ( A ) ) ( i = 1 m C X i P ¯ ( B ) , i = 1 m C X i P ¯ ( B ) ) = ( i = 1 m C X i P ¯ ( A ) i = 1 m C X i P ¯ ( B ) , i = 1 m C X i P ¯ ( A ) i = 1 m C X i P ¯ ( B ) ) = ( μ i = 1 m C X i ¯ ( A ) P ( x ) μ i = 1 m C X i ¯ ( B ) P ( x ) , ν i = 1 m C X i ¯ ( A ) P ( x ) ν i = 1 m C X i ¯ ( B ) P ( x ) ) .
Following the definitions of optimistic and pessimistic MGCRIFS models, we now investigate their fundamental properties in order to reveal the theoretical characteristics of the model.
Theorem 2.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A I F S ( U ) , x U . Then,
(1) 
i = 1 m C X i O ¯ ( A ) = i = 1 m C X i O ¯ ( A ) ,
(2) 
i = 1 m C X i O ¯ ( A ) = i = 1 m C X i O ¯ ( A ) ,
(3) 
i = 1 m C X i P ¯ ( A ) = i = 1 m C X i P ¯ ( A ) ,
(4) 
i = 1 m C X i P ¯ ( A ) = i = 1 m C X i P ¯ ( A ) .
Proof
This is easy to prove by Definitions 10, 11, 12, and 13. □
Theorem 3.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . i = 1 m C X i O ¯ and i = 1 m C X i O ¯ are the optimistic lower and upper approximation operators of the intuitionistic fuzzy set.  i = 1 m C X i P ¯  and i = 1 m C X i P ¯ are the pessimistic lower and upper approximation operators of the intuitionistic fuzzy set, respectively. Then,
(1) 
i = 1 m C X i O ¯ ( U ) = i = 1 m C X i O ¯ ( U ) = U ,
(2) 
i = 1 m C X i P ¯ ( U ) = i = 1 m C X i P ¯ ( U ) = U ,
(3) 
i = 1 m C X i O ¯ ( ϕ ) = i = 1 m C X i O ¯ ( ϕ ) = ϕ ,
(4) 
i = 1 m C X i P ¯ ( ϕ ) = i = 1 m C X i P ¯ ( ϕ ) = ϕ ,
(5) 
i = 1 m C X i O ¯ ( A ) A i = 1 m C X i O ¯ ( A ) ,
(6) 
i = 1 m C X i P ¯ ( A ) A i = 1 m C X i P ¯ ( A ) ,
(7) 
If  A B , then  i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) ,
i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) ,
(8) 
If  A B , then  i = 1 m C X i P ¯ ( A ) i = 1 m C X i P ¯ ( B ) ,
i = 1 m C X i P ¯ ( A ) i = 1 m C X i P ¯ ( B ) .
Proof
(1) When intuitionistic fuzzy set A is the universe U, for any x U , there exist μ U ( x ) = 1 , ν U ( x ) = 0 .
By Definition 10, μ U ( x ) = 1 , ν U ( x ) = 0 .
Then, μ i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { min ( μ A ( x ) , μ A ( x ) ) } = 1 ,   ν i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { max ( ν A ( x ) , ν A ( x ) ) } = 0 .
μ i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { max ( μ A ( x ) , μ A ( x ) ) } = 1 , ν i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { min ( ν A ( x ) , ν A ( x ) ) } = 0 .
By Definition 12,
μ i = 1 m C X i ¯ ( U ) O ( x ) = μ i = 1 m C X i ( ¯ U ) O ( x ) = 1 , ν i = 1 m C X i ( ¯ U ) O ( x ) = ν i = 1 m C X i ( ¯ U ) O ( x ) = 0 ,
Hence, i = 1 m C X i O ¯ ( U ) = i = 1 m C X i O ¯ ( U ) = U .
The proving processes of (2), (3), and (4) are similar to that of (1).
(5) By Definition 12, for all x U ,
μ i 1 ( y ) = min ( μ A ( x ) , μ A ( x ) ) , 1 i m μ i 2 ( y ) = min ( μ A ( x ) , μ A ( x ) ) , 1 i m .
Then, μ i 1 ( y ) μ A ( x ) μ i 2 ( y ) , 1 i m .
Hence, i = 1 m μ i 1 ( y ) μ A ( x ) i = 1 m μ i 2 ( y ) , 1 i m ,
and μ i = 1 m C X i ¯ ( A ) O ( x ) μ A ( x ) μ i = 1 m C X i ¯ ( A ) O ( x ) , 1 i m .
Similarly, ν i = 1 m C X i ¯ ( A ) O ( x ) ν A ( x ) ν i = 1 m C X i ¯ ( A ) O ( x ) , 1 i m .
Then, i = 1 m C X i ¯ ( A ) A i = 1 m C X i ¯ ( A ) .
(6) The proving process of (6) is similar to that of (5).
(7) Since, A B , x U , μ A ( x ) μ B ( x ) , ν A ( x ) ν B ( x ) ,
then μ i = 1 m C X i ¯ ( A ) O ( x ) = i = 1 m { min ( μ A ( x ) , μ A ( x ) ) }   i = 1 m { min ( μ B ( x ) , μ B ( x ) ) } = μ i = 1 m C X i ¯ ( B ) O ( x ) .
Similarly, ν i = 1 m C X i ¯ ( A ) O ( x ) ν i = 1 m C X i ¯ ( B ) O ( x ) .
Then, i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) .
Similarly, i = 1 m C X i O ¯ ( A ) i = 1 m C X i O ¯ ( B ) . □
In order to more explicitly illustrate the computational process of the model, a concrete example is presented below.
Example 2.
Let  ( U , C )  be a covering approximation space, and let U be a nonempty and finite domain of discourse, where  U = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 }  and  C = { C 1 , C 2 , C 3 } ,
C 1 = { { x 1 , x 2 } , { x 3 , x 6 , x 7 } , { x 3 , x 7 } , { x 3 , x 6 } , { x 6 , x 7 } , { x 2 , x 4 , x 5 } } . C 2 = { { x 1 } , { x 2 , x 3 , x 4 } , { x 3 , x 4 , x 5 } , { x 3 , x 4 } , { x 5 , x 6 , x 7 } } . C 3 = { { x 1 , x 2 } , { x 2 , x 3 , x 4 } , { x 2 , x 3 } , { x 3 , x 4 } , { x 2 , x 4 } , { x 5 , x 7 } , { x 6 } } .
The intuitionistic fuzzy set A is expressed as follows:
A = { ( x 1 , 0.5 , 0.1 ) , ( x 2 , 0.6 , 0.2 ) , ( x 3 , 1 , 0 ) , ( x 4 , 0.4 , 0.3 ) , ( x 5 , 0.3 , 0.5 ) , ( x 6 , 0.2 , 0.7 ) , ( x 7 , 0.8 , 0.2 ) } .
The lower and upper approximations of A can be calculated with optimistic and pessimistic MGCRIFS models, respectively.
Step 1. In this step, the maximal description of every covering is calculated.
M a x d 1 ( x 1 ) = { { x 1 , x 2 } } , M a x d 1 ( x 2 ) = { { x 1 , x 2 } , { x 2 , x 4 , x 5 } } , M a x d 1 ( x 3 ) = M a x d 1 ( x 7 ) = M a x d 1 ( x 6 ) = { { x 3 , x 6 , x 7 } } , M a x d 1 ( x 4 ) = M a x d 1 ( x 5 ) = { { x 2 , x 4 , x 5 } } , M a x d 2 ( x 1 ) = { { x 1 } } , M a x d 2 ( x 2 ) = { { x 2 , x 3 , x 4 } } , M a x d 2 ( x 3 ) = M a x d 2 ( x 4 ) = { { x 2 , x 3 , x 4 } , { x 3 , x 4 , x 5 } } , M a x d 2 ( x 5 ) = { { x 3 , x 4 , x 5 } , { x 5 , x 6 , x 7 } } , M a x d 2 ( x 6 ) = M a x d 2 ( x 7 ) = { { x 5 , x 6 , x 7 } } , M a x d 3 ( x 1 ) = { { x 1 , x 2 } } , M a x d 3 ( x 2 ) = { { x 1 , x 2 } , { x 2 , x 3 , x 4 } } , M a x d 3 ( x 3 ) = M a x d 3 ( x 4 ) = { { x 2 , x 3 , x 4 } } , M a x d 3 ( x 5 ) = M a x d 3 ( x 7 ) = { { x 5 , x 7 } } , M a x d 3 ( x 6 ) = { { x 6 } } .
Step 2. The membership degrees of the elements are calculated for every covering.
For the covering C 1 = { { x 1 , x 2 } , { x 3 , x 6 , x 7 } , { x 3 , x 7 } , { x 4 , x 5 , x 6 } } , the non-membership degrees can be calculated in the same way.
μ C X 1 ¯ ( A ) ( x 1 ) = min { 0.5 , 0.55 } = 0.5 , μ C X 1 ¯ ( A ) ( x 1 ) = max { 0.5 , 0.55 } = 0.55 ; μ C X 1 ¯ ( A ) ( x 2 ) = min { 0.6 , 0.45 } = 0.45 , μ C X 1 ¯ ( A ) ( x 2 ) = max { 0.6 , 0.45 } = 0.6 ; μ C X 1 ¯ ( A ) ( x 3 ) = min { 1 , 0.67 } = 0.67 , μ C X 1 ¯ ( A ) ( x 3 ) = max { 1 , 0.67 } = 1 ; μ C X 1 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.43 } = 0.4 , μ C X 1 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.43 } = 0.43 ; μ C X 1 ¯ ( A ) ( x 5 ) = min { 0.3 , 0.43 } = 0.3 , μ C X 1 ¯ ( A ) ( x 5 ) = max { 0.3 , 0.43 } = 0.43 ; μ C X 1 ¯ ( A ) ( x 6 ) = min { 0.2 , 0.67 } = 0.2 , μ C X 1 ¯ ( A ) ( x 6 ) = max { 0.2 , 0.67 } = 0.67 ; μ C X 1 ¯ ( A ) ( x 7 ) = min { 0.8 , 0.67 } = 0.67 , μ C X 1 ¯ ( A ) ( x 7 ) = max { 0.8 , 0.67 } = 0.8 . ν C X 1 ¯ ( A ) ( x 1 ) = max { 0.1 , 0.15 } = 0.15 , ν C X 1 ¯ ( A ) ( x 1 ) = min { 0.1 , 0.15 } = 0.1 ; ν C X 1 ¯ ( A ) ( x 2 ) = max { 0.2 , 0.28 } = 0.28 , ν C X 1 ¯ ( A ) ( x 2 ) = min { 0.2 , 0.28 } = 0.2 ; ν C X 1 ¯ ( A ) ( x 3 ) = max { 0.0 , 0.3 } = 0.3 , ν C X 1 ¯ ( A ) ( x 3 ) = min { 0.0 , 0.3 } = 0.0 ; ν C X 1 ¯ ( A ) ( x 4 ) = max { 0.3 , 0.33 } = 0.33 , ν C X 1 ¯ ( A ) ( x 4 ) = min { 0.3 , 0.33 } = 0.3 ; ν C X 1 ¯ ( A ) ( x 5 ) = max { 0.5 , 0.33 } = 0.5 , ν C X 1 ¯ ( A ) ( x 5 ) = min { 0.5 , 0.33 } = 0.33 ; ν C X 1 ¯ ( A ) ( x 6 ) = max { 0.7 , 0.3 } = 0.7 , ν C X 1 ¯ ( A ) ( x 6 ) = min { 0.7 , 0.3 } = 0.3 ; ν C X 1 ¯ ( A ) ( x 7 ) = max { 0.2 , 0.3 } = 0.3 , ν C X 1 ¯ ( A ) ( x 7 ) = min { 0.2 , 0.3 } = 0.2 .
For the coverings C 2 and C 3 , the membership and non-membership degrees can be calculated in the same way that C 1 is calculated.
Step 3. According to Definitions 12 and 13, the lower and upper approximations of A are calculated with the optimistic and pessimistic MGCRIFS models as follows:
i = 1 m C X i O ¯ ( A ) = { ( x 1 , 0.50 , 0.15 ) , ( x 2 , 0.45 , 0.28 ) , ( x 3 , 0.58 , 0.30 ) , ( x 4 , 0.40 , 0.33 ) , ( x 5 , 0.30 , 0.50 ) , ( x 6 , 0.20 , 0.70 ) , ( x 7 , 0.43 , 0.47 ) } i = 1 m C X i O ¯ ( A ) = { ( x 1 , 0.55 , 0.10 ) , ( x 2 , 0.67 , 0.15 ) , ( x 3 , 1.00 , 0.00 ) , ( x 4 , 0.67 , 0.17 ) , ( x 5 , 0.55 , 0.33 ) , ( x 6 , 0.67 , 0.30 ) , ( x 7 , 0.80 , 0.20 ) } i = 1 m C X i P ¯ ( A ) = { ( x 1 , 0.50 , 0.15 ) , ( x 2 , 0.60 , 0.20 ) , ( x 3 , 0.67 , 0.17 ) , ( x 4 , 0.40 , 0.30 ) , ( x 5 , 0.30 , 0.50 ) , ( x 6 , 0.20 , 0.70 ) , ( x 7 , 0.67 , 0.30 ) } i = 1 m C X i P ¯ ( A ) = { ( x 1 , 0.55 , 0.10 ) , ( x 2 , 0.60 , 0.20 ) , ( x 3 , 1.00 , 0.00 ) , ( x 4 , 0.43 , 0.30 ) , ( x 5 , 0.43 , 0.33 ) , ( x 6 , 0.43 , 0.47 ) , ( x 7 , 0.80 , 0.20 ) }

4.2.2. Neutral Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets

Definition 16.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . For any  x U , M a x d C i ( x ) is the maximal description of  C i , while M a x d ( x ) = { K M a x d C i ( x ) | x K ( S M a x d C i ( x ) x S   S K K = S ) }  is called the maximal description based on multi-granulation of x.
Definition 17.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U, where  | C | = m . A I F S ( U ) , for all  x U . The multi-granulation intuitionistic fuzzy covering rough membership and non-membership degrees of x based on maximal description with respect to A are defined as follows:
μ A ( x ) = y M a x d ( x ) μ A ( y ) / | M a x d ( x ) | , ν A ( x ) = y M a x d ( x ) ν A ( y ) / | M a x d ( x ) | .
Definition 18.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A I F S ( U ) , for all  x U . Then, the multi-granulation lower and upper approximations of A based on maximal description with respect to (U,C), denoted as  i = 1 m C X i ¯ ( A )  and  i = 1 m C X i ¯ ( A ) , respectively, are defined as follows:
i = 1 m C X i ¯ ( A ) = { ( x , μ i = 1 m C X i ¯ ( A ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ) | x U } , i = 1 m C X i ¯ ( A ) = { ( x , μ i = 1 m C X i ¯ ( A ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ) | x U } ,
where
μ i = 1 m C X i ¯ ( A ) ( x ) = min ( μ A ( x ) , μ A ( x ) ) , ν i = 1 m C X i ¯ ( A ) ( x ) = max ( ν A ( x ) , ν A ( x ) ) , μ i = 1 m C X i ¯ ( A ) ( x ) = max ( μ A ( x ) , μ A ( x ) ) , ν i = 1 m C X i ¯ ( A ) ( x ) = min ( ν A ( x ) , ν A ( x ) ) .
We call the above model the neutral multi-granulation covering rough intuitionistic fuzzy sets (neutral MGCRIFS).
Definition 19.
Let   ( U , C )  be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m }  be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A , B I F S ( U ) , x U . i = 1 m C X i ¯  and i = 1 m C X i ¯  are the lower and upper approximation operators of the intuitionistic fuzzy set.
( 1 ) ( i = 1 m C X i ¯ ( A ) , i = 1 m C X i ¯ ( A ) ) = ( i = 1 m C X i ¯ ( A ) , - i = 1 m C X i ¯ ( A ) ) = ( i = 1 m C X i ¯ ( A ) , i = 1 m C X i ¯ ( A ) ) . ( 2 ) ( i = 1 m C X i ¯ ( A ) , i = 1 m C X i ¯ ( A ) ) ( i = 1 m C X i ¯ ( B ) , i = 1 m C X i ¯ ( B ) ) = ( i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) , i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) ) = ( ( μ i = 1 m C X i ¯ ( A ) ( x ) μ i = 1 m C X i ¯ ( B ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ν i = 1 m C X i ¯ ( B ) ( x ) ) , ( μ i = 1 m C X i ¯ ( A ) ( x ) μ i = 1 m C X i ¯ ( B ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ν i = 1 m C X i ¯ ( B ) ( x ) ) ) ( 3 ) ( i = 1 m C X i ¯ ( A ) , i = 1 m C X i ¯ ( A ) ) ( i = 1 m C X i ¯ ( B ) , i = 1 m C X i ¯ ( B ) ) = ( i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) , i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) ) = ( μ i = 1 m C X i ¯ ( A ) ( x ) μ i = 1 m C X i ¯ ( B ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ν i = 1 m C X i ¯ ( B ) ( x ) ) . ( μ i = 1 m C X i ¯ ( A ) ( x ) μ i = 1 m C X i ¯ ( B ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ν i = 1 m C X i ¯ ( B ) ( x ) ) )
Following the formal definitions of the neutral MGCRIFS models, we present several corresponding operational properties.
Theorem 4.
Let   ( U , C ) be a covering approximation space, and let U be a finite and nonempty universe of discourse. Let C = { C i | 1 i m } be a family of coverings of U, and let  C i  be a covering of U,  | C | = m . A , B I F S ( U ) , x U . Then,
(1) 
i = 1 m C X i ¯ ( U ) = i = 1 m C X i ¯ ( U ) = U ,
(2) 
i = 1 m C X i ¯ ( ϕ ) = i = 1 m C X i ¯ ( ϕ ) = ϕ ,
(3) 
i = 1 m C X i ¯ ( A ) A i = 1 m C X i ¯ ( A ) ,
(4) 
If  A B , then  i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) ,
i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) .
Proof
(1) When the intuitionistic fuzzy set A is the universe U, for all x U , μ U ( x ) = 1 , and ν U ( x ) = 0 .
By Definition 17, μ U ( x ) = 1 , ν U ( x ) = 0 .
By Definition 18,
μ i = 1 m C X i ¯ ( U ) ( x ) = μ i = 1 m C X i ¯ ( U ) ( x ) = 1 , ν i = 1 m C X i ¯ ( U ) ( x ) = ν i = 1 m C X i ¯ ( U ) ( x ) = 0 ,
Hence, i = 1 m C X i ¯ ( U ) = i = 1 m C X i ¯ ( U ) = U .
(2) The process of proving (2) is similar to that of proving (1).
(3) By Definitions 17 and 18, for all x U ,
μ A ( x ) μ A ( x ) or μ A ( x ) > μ A ( x ) , and ν A ( x ) ν A ( x ) or ν A ( x ) > ν A ( x ) .
There can be four different situations:
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) .
We only need to prove ① and ② as follows:
① When μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) , and μ i = 1 m C X i ¯ ( A ) ( x ) = μ A ( x ) , μ i = 1 m C X i ¯ ( A ) ( x ) = μ A ( x ) ; ν i = 1 m C X i ¯ ( A ) ( x ) = ν A ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) = ν A ( x ) .
Then, μ i = 1 m C X i ¯ ( A ) ( x ) μ A ( x ) μ i = 1 m C X i ¯ ( A ) ( x )   , ν i = 1 m C X i ¯ ( A ) ( x ) ν A ( x ) ν i = 1 m C X i ¯ ( A ) ( x ) .
② When μ A ( x ) μ A ( x ) , ν A ( x ) ν A ( x ) ,
and μ i = 1 m C X i ¯ ( A ) ( x ) = μ A ( x ) ,   μ i = 1 m C X i ¯ ( A ) ( x ) = μ A ( x ) ,   ν i = 1 m C X i ¯ ( A ) ( x ) = ν A ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) = ν A ( x ) .
Then, μ i = 1 m C X i ¯ ( A ) ( x ) μ A ( x ) μ i = 1 m C X i ¯ ( A ) ( x ) , ν i = 1 m C X i ¯ ( A ) ( x ) ν A ( x ) ν i = 1 m C X i ¯ ( A ) ( x ) .
The processes of proving ③ and ④ are similar to that of proving ① and ②.
Hence, i = 1 m C X i ¯ ( A ) A i = 1 m C X i ¯ ( A ) .
(4) The process of proving (4) is similar to that of proving (3). □
Example 3.
Let    ( U , C )  be a covering approximation space, and let U be a nonempty and finite domain of discourse,  U = { x 1 , x 2 , x 3 , x 4 , x 5 } ,  C = { C 1 , C 2 } ,  C 1 = { { x 1 , x 2 , x 5 } , { x 2 , x 3 } , { x 3 , x 4 } , { x 2 , x 5 } } . C 2 = { { x 1 , x 2 } , { x 2 , x 3 , x 4 } , { x 2 , x 4 } , { x 1 , x 5 } } .
The intuitionistic fuzzy set A is expressed as follows:
A = { ( x 1 , 0.6 , 0.3 ) , ( x 2 , 0.4 , 0.3 ) , ( x 3 , 0.8 , 0.1 ) , ( x 4 , 0.2 , 0.3 ) , ( x 5 , 0.3 , 0.6 ) } .
The lower and upper approximations of A can be calculated with MGCRIFS. 
Step 1. This step is for calculating the maximal description based on multi-granulation of x.
M a x d ( x 1 ) = { { x 1 , x 2 , x 5 } } , M a x d ( x 2 ) = { { x 1 , x 2 , x 5 } , { x 2 , x 3 , x 4 } } , M a x d ( x 3 ) = { { x 2 , x 3 , x 4 } } , M a x d ( x 4 ) = { { x 2 , x 3 , x 4 } } , M a x d ( x 5 ) = { { x 1 , x 2 , x 5 } } .
Step 2. The lower and upper approximations of A are calculated with MGCRIFS as follows:
i = 1 m C X i ¯ ( A ) = { ( x 1 , 0.43 , 0.4 ) , ( x 2 , 0.4 , 0.32 ) , ( x 3 , 0.47 , 0.23 ) , ( x 4 , 0.2 , 0.3 ) , ( x 5 , 0.3 , 0.6 ) } . i = 1 m C X i ( ¯ A ) = { ( x 1 , 0.6 , 0.3 ) , ( x 2 , 0.46 , 0.3 ) , ( x 3 , 0.8 , 0.1 ) , ( x 4 , 0.47 , 0.23 ) , ( x 5 , 0.43 , 0.4 ) } .
To illustrate the above operational properties more intuitively, we now provide a concrete example demonstrating how these properties are applied in practice.
Example 4.
Let    ( U , C )  be a covering approximation space, and let U be a nonempty and finite domain of discourse,  U = { x 1 , x 2 , x 3 , x 4 , x 5 } ,  C = { C 1 , C 2 } ,  C 1 = { { x 1 , x 2 , x 5 } , { x 2 , x 3 } , { x 3 , x 4 } , { x 2 , x 5 } } .   C 2 = { { x 1 , x 2 } , { x 2 , x 3 , x 4 } , { x 2 , x 4 } , { x 1 , x 5 } } .
The intuitionistic fuzzy set A is as follows:
A = { ( x 1 , 0.6 , 0.3 ) , ( x 2 , 0.4 , 0.3 ) , ( x 3 , 0.8 , 0.1 ) , ( x 4 , 0.2 , 0.3 ) , ( x 5 , 0.3 , 0.6 ) } . B = { ( x 1 , 0.4 , 0.5 ) , ( x 2 , 0.5 , 0.2 ) , ( x 3 , 0.7 , 0.1 ) , ( x 4 , 0.3 , 0.5 ) , ( x 5 , 0.4 , 0.5 ) } . i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) = { ( x 1 , 0.4 , 0.5 ) , ( x 2 , 0.4 , 0.36 ) , ( x 3 , 0.47 , 27 ) , ( x 4 , 0.2 , 0.5 ) , ( x 5 , 0.3 , 0.6 ) } . i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) = { ( x 1 , 0.43 , 0.4 ) , ( x 2 , 0.46 , 0.3 ) , ( x 3 , 0.7 , 0.1 ) , ( x 4 , 0.47 , 0.27 ) , ( x 5 , 0.43 , 0.4 ) } . i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) = { ( x 1 , 0.43 , 0.4 ) , ( x 2 , 0.46 , 0.32 ) , ( x 3 , 0.5 , 23 ) , ( x 4 , 0.3 , 0.3 ) , ( x 5 , 0.4 , 0.5 ) } . i = 1 m C X i ¯ ( A ) i = 1 m C X i ¯ ( B ) = { ( x 1 , 0.6 , 0.3 ) , ( x 2 , 0.5 , 0.2 ) , ( x 3 , 0.8 , 0.1 ) , ( x 4 , 0.5 , 0.23 ) , ( x 5 , 0.43 , 0.4 ) } .

4.3. Comparative Analysis

This section compares the proposed models with several representative related models, including conventional rough sets, covering rough sets, fuzzy sets, intuitionistic fuzzy sets, and multi-granulation rough sets. Figure 1 illustrates the relationships between the core definitions in this study and their connections to similar models.
The covering rough set model, as an extension of the conventional Pawlak rough sets, provides a more flexible mechanism for approximating uncertain concepts by relaxing the requirement of partitions. On the other hand, the intuitionistic fuzzy sets, as an enhancement of the traditional fuzzy sets, incorporates both membership and non-membership degrees to better capture hesitation and ambiguity in information.
By integrating these two models, we propose the covering rough intuitionistic fuzzy sets (CRIFS), which inherits the strengths of both frameworks and offers a powerful tool for representing and processing uncertain, imprecise, and incomplete information.
Furthermore, to better accommodate multi-source and multi-level information commonly encountered in real-world scenarios, we extend the CRIFS model into a multi-granulation framework. Within this framework, we define the optimistic and pessimistic multi-granulation covering rough intuitionistic fuzzy sets, which, respectively, reflect the upper and lower bounds of information granularity.
In addition, we introduce the concepts of intuitionistic fuzzy covering membership and non-membership degrees under multi-granulation conditions, on which the model of neutral multi-granulation covering rough intuitionistic fuzzy sets is constructed to capture a balanced view across multiple granular perspectives.

5. An Application Example

To evaluate the applicability and effectiveness of the proposed model, this section presents a case study focused on the real estate investment problem. Real estate investment typically involves multiple conflicting criteria—such as price, location, risk, and return—along with uncertain or vague information. These characteristics make it an ideal scenario for applying rough fuzzy set theory. By modeling the imprecision and structural uncertainty inherent in investment decision-making, the proposed approach is used to support more informed and flexible evaluations of candidate properties.
In this section, a representative case of the real estate investment problem is presented to illustrate the algorithm of the proposed model. Table 2 presents the attribute information table for real estate investment problems. Let U = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 } denote seven houses. C1, C2, and C3 constitute the attribute set C = { C 1 , C 2 , C 3 } , representing the house’s location, average price, and supporting facilities, respectively, each evaluated at three levels—high, moderate, and low. The decision attribute A is an intuitionistic fuzzy set indicating the comprehensive assessment of seven real estate properties conducted by a specific institution.
A = { ( x 1 , 0.1 , 0.8 ) , ( x 2 , 0.1 , 0.8 ) , ( x 3 , 0.2 , 0.6 ) , ( x 4 , 0.4 , 0.4 ) , ( x 5 , 0.5 , 0.4 ) , ( x 6 , 0.6 , 0.1 ) ( x 7 , 0.9 , 0.1 ) } .
For attribute C 1 , the table format is first transformed into a set format, namely,
C 1 = { H i g h , M o d e r a t e , L o w } , H i g h = { x 1 , x 2 , x 3 , x 4 } , M i d d l e = { x 3 , x 4 , x 5 , x 6 } , and L o w = { x 5 , x 6 , x 7 } .
For C 1 , the obtained covering is C 1   = { { x 1 , x 2 , x 3 , x 4 } , { x 3 , x 4 , x 5 , x 6 } , { x 5 , x 6 , x 7 } } .
Similarly, the covering representations of C 2 and C 3 can be obtained as follows:
C 2 = { { x 1 , x 2 , x 3 } , { x 3 , x 4 , x 5 , x 6 } , { x 6 , x 7 } } C 3 = { { x 1 , x 2 } , { x 3 , x 4 , x 5 } , { x 4 , x 5 , x 6 , x 7 } }
Step 1. By Definition 5, calculate the maximal description M a x d i ( x i ) of each element in coverings C = { C 1 , C 2 , C 3 } .
In C 1 :
M a x d 1 ( x 1 ) = M a x d 1 ( x 2 ) = { { x 1 , x 2 , x 3 , x 4 } } , M a x d 1 ( x 3 ) = M a x d 1 ( x 4 ) = { { x 1 , x 2 , x 3 , x 4 } , { x 3 , x 4 , x 5 , x 6 } } , M a x d 1 ( x 5 ) = M a x d 1 ( x 6 ) = { { x 3 , x 4 , x 5 , x 6 } , { x 5 , x 6 , x 7 } } , M a x d 1 ( x 7 ) = { { x 5 , x 6 , x 7 } } ,
In C 2 :
M a x d 2 ( x 1 ) = M a x d 2 ( x 2 ) = { { x 1 , x 2 , x 3 } } , M a x d 2 ( x 3 ) = { { x 1 , x 2 , x 3 } , { x 4 , x 5 , x 6 } } , M a x d 2 ( x 4 ) = M a x d 2 ( x 5 ) = { x 3 , x 4 , x 5 , x 6 } , M a x d 2 ( x 6 ) = { { x 3 , x 4 , x 5 , x 6 } , { x 6 , x 7 } } , M a x d 2 ( x 7 ) = { x 5 , x 6 , x 7 } ,
In C 3 :
M a x d 3 ( x 1 ) = M a x d 3 ( x 2 ) = { x 1 , x 2 } , M a x d 3 ( x 3 ) = { x 3 , x 4 , x 5 } , M a x d 3 ( x 4 ) = M a x d 3 ( x 5 ) = { { x 3 , x 4 , x 5 } , { x 4 , x 5 , x 6 , x 7 } } , M a x d 3 ( x 6 ) = M a x d 3 ( x 7 ) = { { x 4 , x 5 , x 6 , x 7 } }
Step 2. By Definition 10, the intuitionistic fuzzy covering rough membership and non-membership degrees based on maximal description are calculated as follows:
In C 1 :
μ A ( x 1 ) = μ A ( x 2 ) = 0.2 , μ A ( x 3 ) = μ A ( x 4 ) 0.32 , μ A ( x 5 ) = μ A ( x 6 ) = 0.52 , μ A ( x 7 ) 0.67 , ν A ( x 1 ) = ν A ( x 2 ) = 0.65 , ν A ( x 3 ) = ν A ( x 4 ) 0.26 , ν A ( x 5 ) = ν A ( x 6 ) = 0.32 , ν A ( x 7 ) = 0.2
In C 2 :
μ A ( x 1 ) = μ A ( x 2 ) 0.14 , μ A ( x 3 ) = 0.32 , μ A ( x 4 ) = μ A ( x 5 ) 0.43 , μ A ( x 6 ) = 0.52 , μ A ( x 7 ) = 0.75 , ν A ( x 1 ) = ν A ( x 2 ) 0.73 , ν A ( x 3 ) 0.52 , ν A ( x 4 ) = ν A ( x 5 ) 0.38 , ν A ( x 6 ) = 0.32 , ν A ( x 7 ) = 0.1
In C 3 :
μ A ( x 1 ) = μ A ( x 2 ) = 0.1 , μ A ( x 3 ) 0.37 , μ A ( x 4 ) = μ A ( x 5 ) = 0.52 , μ A ( x 6 ) = μ A ( x 7 ) = 0.6 , ν A ( x 1 ) = ν A ( x 2 ) = 0.8 , ν A ( x 3 ) 0.47 , ν A ( x 4 ) = ν A ( x 5 ) = 0.32 , ν A ( x 6 ) = ν A ( x 7 ) = 0.25 ,
Step 3. By Definition 11, the lower and upper approximations of A with CRIFS are calculated as follows:
In C 1 :
μ C X 1 ¯ ( A ) ( x 1 ) = min { 0.1 , 0.2 } = 0.1 , μ C X 1 ¯ ( A ) ( x 1 ) = max { 0.1 , 0.2 } = 0.2 ; μ C X 1 ¯ ( A ) ( x 2 ) = min { 0.1 , 0.2 } = 0.1 , μ C X 1 ¯ ( A ) ( x 2 ) = max { 0.1 , 0.2 } = 0.2 ; μ C X 1 ¯ ( A ) ( x 3 ) = min { 0.2 , 0.32 } = 0.2 , μ C X 1 ¯ ( A ) ( x 3 ) = max { 0.2 , 0.32 } = 0.32 ; μ C X 1 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.32 } = 0.32 , μ C X 1 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.32 } = 0.4 ; μ C X 1 ¯ ( A ) ( x 5 ) = min { 0.5 , 0.52 } = 0.5 , μ C X 1 ¯ ( A ) ( x 5 ) = max { 0.5 , 0.52 } = 0.52 ; μ C X 1 ¯ ( A ) ( x 6 ) = min { 0.6 , 0.52 } = 0.52 , μ C X 1 ¯ ( A ) ( x 6 ) = max { 0.6 , 0.52 } = 0.6 ; μ C X 1 ¯ ( A ) ( x 7 ) = min { 0.9 , 0.67 } = 0.67 , μ C X 1 ¯ ( A ) ( x 7 ) = max { 0.9 , 0.67 } = 0.9 . ν C X 1 ¯ ( A ) ( x 1 ) = max { 0.8 , 0.65 } = 0.8 , ν C X 1 ¯ ( A ) ( x 1 ) = min { 0.8 , 0.65 } = 0.65 ; ν C X 1 ¯ ( A ) ( x 2 ) = max { 0.8 , 0.65 } = 0.8 , ν C X 1 ¯ ( A ) ( x 2 ) = min { 0.8 , 0.65 } = 0.65 ; ν C X 1 ¯ ( A ) ( x 3 ) = max { 0.6 , 0.26 } = 0.6 , ν C X 1 ¯ ( A ) ( x 3 ) = min { 0.6 , 0.26 } = 0.26 ; ν C X 1 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.26 } = 0.4 , ν C X 1 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.26 } = 0.26 ; ν C X 1 ¯ ( A ) ( x 5 ) = max { 0.4 , 0.32 } = 0.4 , ν C X 1 ¯ ( A ) ( x 5 ) = min { 0.4 , 0.32 } = 0.35 ; ν C X 1 ¯ ( A ) ( x 6 ) = max { 0.1 , 0.32 } = 0.32 , ν C X 1 ¯ ( A ) ( x 6 ) = min { 0.1 , 0.32 } = 0.1 ; ν C X 1 ¯ ( A ) ( x 7 ) = max { 0.1 , 0.2 } = 0.2 , ν C X 1 ¯ ( A ) ( x 7 ) = min { 0.1 , 0.2 } = 0.1 .
In C 2 :
μ C X 1 ¯ ( A ) ( x 1 ) = min { 0.1 , 0.14 } = 0.1 , μ C X 1 ¯ ( A ) ( x 1 ) = max { 0.1 , 0.14 } = 0.14 ; μ C X 1 ¯ ( A ) ( x 2 ) = min { 0.1 , 0.14 } = 0.1 , μ C X 1 ¯ ( A ) ( x 2 ) = max { 0.1 , 0.14 } = 0.14 ; μ C X 1 ¯ ( A ) ( x 3 ) = min { 0.2 , 0.32 } = 0.2 , μ C X 1 ¯ ( A ) ( x 3 ) = max { 0.2 , 0.32 } = 0.32 ; μ C X 1 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.43 } = 0.4 , μ C X 1 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.43 } = 0.43 ; μ C X 1 ¯ ( A ) ( x 5 ) = min { 0.5 , 0.43 } = 0.43 , μ C X 1 ¯ ( A ) ( x 5 ) = max { 0.5 , 0.43 } = 0.52 ; μ C X 1 ¯ ( A ) ( x 6 ) = min { 0.6 , 0.52 } = 0.52 , μ C X 1 ¯ ( A ) ( x 6 ) = max { 0.6 , 0.52 } = 0.6 ; μ C X 1 ¯ ( A ) ( x 7 ) = min { 0.9 , 0.75 } = 0.75 , μ C X 1 ¯ ( A ) ( x 7 ) = max { 0.9 , 0.75 } = 0.9 . ν C X 2 ¯ ( A ) ( x 1 ) = max { 0.8 , 0.73 } = 0.8 , ν C X 2 ¯ ( A ) ( x 1 ) = min { 0.8 , 0.73 } = 0.73 ; ν C X 2 ¯ ( A ) ( x 2 ) = max { 0.8 , 0.73 } = 0.8 , ν C X 2 ¯ ( A ) ( x 2 ) = min { 0.8 , 0.73 } = 0.73 ; ν C X 2 ¯ ( A ) ( x 3 ) = max { 0.6 , 0.52 } = 0.6 , ν C X 2 ¯ ( A ) ( x 3 ) = min { 0.6 , 0.52 } = 0.52 ; ν C X 2 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.38 } = 0.4 , ν C X 2 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.38 } = 0.38 ; ν C X 2 ¯ ( A ) ( x 5 ) = max { 0.4 , 0.38 } = 0.4 , ν C X 2 ¯ ( A ) ( x 5 ) = min { 0.4 , 0.38 } = 0.38 ; ν C X 2 ¯ ( A ) ( x 6 ) = max { 0.1 , 0.32 } = 0.32 , ν C X 2 ¯ ( A ) ( x 6 ) = min { 0.1 , 0.32 } = 0.1 ; ν C X 2 ¯ ( A ) ( x 7 ) = max { 0.1 , 0.1 } = 0.1 , ν C X 2 ¯ ( A ) ( x 7 ) = min { 0.1 , 0.1 } = 0.1 .
In C 3 :
μ C X 1 ¯ ( A ) ( x 1 ) = min { 0.1 , 0.1 } = 0.1 , μ C X 1 ¯ ( A ) ( x 1 ) = max { 0.1 , 0.1 } = 0.1 ; μ C X 1 ¯ ( A ) ( x 2 ) = min { 0.1 , 0.1 } = 0.1 , μ C X 1 ¯ ( A ) ( x 2 ) = max { 0.1 , 0.1 } = 0.1 ; μ C X 1 ¯ ( A ) ( x 3 ) = min { 0.2 , 0.37 } = 0.2 , μ C X 1 ¯ ( A ) ( x 3 ) = max { 0.2 , 0.37 } = 0.37 ; μ C X 1 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.52 } = 0.4 , μ C X 1 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.52 } = 0.52 ; μ C X 1 ¯ ( A ) ( x 5 ) = min { 0.5 , 0.52 } = 0.5 , μ C X 1 ¯ ( A ) ( x 5 ) = max { 0.5 , 0.52 } = 0.52 ; μ C X 1 ¯ ( A ) ( x 6 ) = min { 0.6 , 0.6 } = 0.6 , μ C X 1 ¯ ( A ) ( x 6 ) = max { 0.6 , 0.6 } = 0.6 ; μ C X 1 ¯ ( A ) ( x 7 ) = min { 0.9 , 0.6 } = 0.6 , μ C X 1 ¯ ( A ) ( x 7 ) = max { 0.9 , 0.6 } = 0.9 . ν C X 2 ¯ ( A ) ( x 1 ) = max { 0.8 , 0.8 } = 0.8 , ν C X 2 ¯ ( A ) ( x 1 ) = min { 0.8 , 0.8 } = 0.8 ; ν C X 2 ¯ ( A ) ( x 2 ) = max { 0.8 , 0.8 } = 0.8 , ν C X 2 ¯ ( A ) ( x 2 ) = min { 0.8 , 0.8 } = 0.8 ; ν C X 2 ¯ ( A ) ( x 3 ) = max { 0.6 , 0.47 } = 0.6 , ν C X 2 ¯ ( A ) ( x 3 ) = min { 0.6 , 0.47 } = 0.47 ; ν C X 2 ¯ ( A ) ( x 4 ) = max { 0.4 , 0.32 } = 0.4 , ν C X 2 ¯ ( A ) ( x 4 ) = min { 0.4 , 0.32 } = 0.32 ; ν C X 2 ¯ ( A ) ( x 5 ) = max { 0.4 , 0.32 } = 0.4 , ν C X 2 ¯ ( A ) ( x 5 ) = min { 0.4 , 0.32 } = 0.32 ; ν C X 2 ¯ ( A ) ( x 6 ) = max { 0.1 , 0.25 } = 0.25 , ν C X 2 ¯ ( A ) ( x 6 ) = min { 0.1 , 0.25 } = 0.1 ; ν C X 2 ¯ ( A ) ( x 7 ) = max { 0.1 , 0.25 } = 0.25 , ν C X 2 ¯ ( A ) ( x 7 ) = min { 0.1 , 0.25 } = 0.1 .
Step 4. By Definitions 12 and 13, the lower and upper approximations of A with the optimistic and pessimistic MGCRIFS are calculated as follows:
              i = 1 m C X i O ¯ ( A ) = { ( x 1 , 0.1 , 0.8 ) , ( x 2 , 0.1 , 0.8 ) , ( x 3 , 0.2 , 0.6 ) , ( x 4 , 0.4 , 0.4 ) , ( x 5 , 0.5 , 0.4 ) , ( x 6 , 0.6 , 0.25 ) , ( x 7 , 0.75 , 0.1 ) } . i = 1 m C X i O ¯ ( A ) = { ( x 1 , 0.1 , 0.65 ) , ( x 2 , 0.1 , 0.65 ) , ( x 3 , 0.32 , 0.52 ) , ( x 4 , 0.4 , 0.38 ) , ( x 5 , 0.52 , 0.38 ) , ( x 6 , 0.6 , 0.1 ) , ( x 7 , 0.8 , 0.1 ) } .               i = 1 m C X i P ¯ ( A ) = { ( x 1 , 0.1 , 0.8 ) , ( x 2 , 0.1 , 0.8 ) , ( x 3 , 0.2 , 0.6 ) , ( x 4 , 0.4 , 0.4 ) , ( x 5 , 0.43 , 0.4 ) , ( x 6 , 0.52 , 0.32 ) , ( x 7 , 0.6 , 0.25 ) } . i = 1 m C X i P ¯ ( A ) = { ( x 1 , 0.2 , 0.65 ) , ( x 2 , 0.2 , 0.65 ) , ( x 3 , 0.37 , 0.26 ) , ( x 4 , 0.5 , 0.26 ) , ( x 5 , 0.52 , 0.32 ) , ( x 6 , 0.6 , 0.1 ) , ( x 7 , 0.9 , 0.1 ) } .
Step 5. By Definition 16, the maximal description based on multi-granulation of x are calculated as follows:
M a x d ( x 1 ) = M a x d ( x 2 ) = { { x 1 , x 2 , x 3 , x 4 } } , M a x d ( x 3 ) = { { x 1 , x 2 , x 3 , x 4 } , { x 3 , x 4 , x 5 , x 6 } } , M a x d ( x 4 ) = M a x d ( x 5 ) = { { x 3 , x 4 , x 5 , x 6 } , { x 4 , x 5 , x 6 , x 7 } } , M a x d ( x 6 ) = M a x d ( x 7 ) = { { x 4 , x 5 , x 6 , x 7 } } ,
Step 6. By Definitions 17 and 18, the lower and upper approximations of A with the neutral MGCRIFS are calculated as follows:
i = 1 m C X i ¯ ( A ) = { ( x 1 , 0.1 , 0.8 ) , ( x 2 , 0.1 , 0.8 ) , ( x 3 , 0.2 , 0.6 ) , ( x 4 , 0.4 , 0.4 ) , ( x 5 , 0.5 , 0.4 ) , ( x 6 , 0.6 , 0.25 ) , ( x 7 , 0.6 , 0.1 ) } . i = 1 m C X i ( ¯ A ) = { ( x 1 , 0.2 , 0.65 ) , ( x 2 , 0.2 , 0.65 ) , ( x 3 , 0.32 , 0.52 ) , ( x 4 , 0.52 , 0.32 ) , ( x 5 , 0.52 , 0.32 ) , ( x 6 , 0.6 , 0.1 ) , ( x 7 , 0.6 , 0.1 ) }

6. Conclusions

The theories of covering rough sets, intuitionistic fuzzy sets, and multi-granulation rough sets play crucial roles in addressing problems involving uncertainty and imprecision. In this work, we further investigated their integration by proposing a novel model of covering rough intuitionistic fuzzy sets based on maximal description. Furthermore, from the perspective of granularity, we extended the proposed single-granulation model to a multi-granulation framework. three types of multi-granulation covering rough intuitionistic fuzzy set models were constructed. Several theoretical properties were proved and illustrated with examples. Finally, the effectiveness of the proposed model was validated through a real estate investment case study. These contributions are meaningful for enriching the theoretical system of rough fuzzy sets integration.
Despite these promising results, this study has several limitations. The validation of the proposed models is limited to a single case study, which may not fully capture their performance in broader applications. Further empirical evaluation across various domains—particularly on real-world, large-scale datasets—is necessary to assess the models’ generalizability, robustness, and practical effectiveness. In addition, the proposed model may suffer from high computational complexity when applied to large-scale or high-dimensional datasets. As the number of granules and covering increases, time and memory costs rise rapidly, which can limit its efficiency in practical applications.
Future work will focus on applying the proposed models to more real-world applications, such as medical diagnosis, recommendation systems, and fault detection, particularly on large-scale and heterogeneous datasets. Key challenges include increasing computational efficiency, developing adaptive granulation mechanisms, and enhancing robustness under conditions with noisy or incomplete data. Furthermore, model interpretability and integration with intelligent systems are important considerations for practical deployment. Addressing these issues is essential to ensure the real-world applicability and scalability of the proposed framework.

Author Contributions

Conceptualization, X.-M.S.; Methodology, X.-M.S.; Validation, X.-M.S.; Formal analysis, X.-M.S.; Investigation, X.-M.S.; Resources, Z.-A.X.; Writing—original draft, X.-M.S.; Writing—review & editing, Z.-A.X.; Supervision, Z.-A.X.; Funding acquisition, Z.-A.X. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Young Teachers Funding Project of Shanghai (ZZEGD202413).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Schematic conceptual diagram of the proposed model.
Figure 1. Schematic conceptual diagram of the proposed model.
Symmetry 17 01217 g001
Table 1. Notations and explanations.
Table 1. Notations and explanations.
NotationsExplanations
UNonempty and finite universe of discourse
REquivalence relation
CCovering relation
X X U
μ X R ( x ) The degree of rough membership of x in R
R _ ( X ) , R ¯ ( X ) The lower and upper approximations of X in the rough sets
M a x d ( x ) The maximal description of x
i = 1 m A i O ¯ ( X ) , i = 1 m A i O ¯ ( X ) The lower and upper approximations of A with optimistic multi-granulation rough sets
i = 1 m A i P ¯ ( X ) ,   i = 1 m A i P ¯ ( X ) The lower and upper approximations of A with pessimistic multi-granulation rough sets
AThe intuitionistic fuzzy set (IFS)
μ A ( x ) , ν A ( x ) The membership and non-membership degrees of the element in x
μ A ( x ) , ν A ( x ) The intuitionistic fuzzy covering rough membership and non-membership degrees of x based on maximal description
C X ¯ ( A ) ,   C X ¯ ( A ) The lower and upper approximations of A with covering rough intuitionistic fuzzy sets
i = 1 m C X i O ¯ ( A ) ,   i = 1 m C X i O ¯ ( A ) The lower and upper approximation of A with optimistic multi-granulation covering rough intuitionistic fuzzy sets
i = 1 m C X i P ¯ ( A ) ,   i = 1 m C X i P ¯ ( A ) The lower and upper approximation of A with pessimistic multi-granulation covering rough intuitionistic fuzzy sets
M a x d ( x ) The maximal description based on multi-granulation of x
μ A ( x ) ,   ν A ( x ) The multi-granulation intuitionistic fuzzy covering rough membership and non-membership degrees of x based on maximal description
i = 1 m C X i ¯ ( A ) ,   i = 1 m C X i ¯ ( A ) The lower and upper approximation of A with neutral multi-granulation covering rough intuitionistic fuzzy sets
Table 2. Attribute information table for the real estate investment problems.
Table 2. Attribute information table for the real estate investment problems.
UC1C2C3
x1HighHighHigh
x2HighHighHigh
x3High/ModerateHigh/ModerateModerate
x4High/ModerateModerateModerate/Low
x5Moderate/LowModerateModerate/Low
x6Moderate/LowModerate/LowLow
x7LowLowLow
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Si, X.-M.; Xue, Z.-A. Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description. Symmetry 2025, 17, 1217. https://doi.org/10.3390/sym17081217

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Si X-M, Xue Z-A. Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description. Symmetry. 2025; 17(8):1217. https://doi.org/10.3390/sym17081217

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Si, Xiao-Meng, and Zhan-Ao Xue. 2025. "Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description" Symmetry 17, no. 8: 1217. https://doi.org/10.3390/sym17081217

APA Style

Si, X.-M., & Xue, Z.-A. (2025). Multi-Granulation Covering Rough Intuitionistic Fuzzy Sets Based on Maximal Description. Symmetry, 17(8), 1217. https://doi.org/10.3390/sym17081217

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