Multi-granulation covering rough intuitionistic fuzzy sets

Abstract

In this paper, firstly we defined a new covering rough intuitionistic fuzzy set model in covering approximation space (U, C). Then, the properties of lower and upper approximation operators were discussed. Secondly, we extended covering rough intuitionistic fuzzy sets in rough sets from single-granulation to multi-granulation. From the view of granularity, we proposed three models, which are based on minimal description. Some properties of these models were proved and demonstrated with examples. These models were illustrated with examples.

Keywords

Intuitionistic fuzzy sets rough sets intuitionistic fuzzy covering rough membership minimal description multi-granulation rough sets

1 Introduction

In order to deal with vague, inexact and uncertain knowledge in information systems, a rough set model was proposed by Pawlak in 1982 [1]. Covering rough sets are a kind of important generalization of rough sets because the concept of covering is more general than the one of partition and the former exists in the real world more widely. In recent years, covering rough sets have been more and more concerned, and there are a lot of interesting and meaningful research findings [2 –4].

Fuzzy sets were proposed by Zadeh in 1965 in order to describe the concepts of vagueness and fuzziness [5]. On the basis of Zadeh’s fuzzy sets, Atanassov developed the concept of intuitionistic fuzzy sets (IFS) [6 –8]. The theory interprets the fuzziness concept with “neither this nor that” and “both this and that” by presenting three notions of the membership degree, non-membership degree and hesitation degree, respectively. A great progress was made in their theories and applications such as Data Mining, Image Processing and Pattern Recognition [9 –11], since rough sets and fuzzy sets were proposed. What is more, they are not the opposite theories for imperfection, accuracy and vagueness. The combination of the two theories is natural, but not from a formal point of view. In the fusion and innovation of the two theories, Dubois firstly combined them into rough fuzzy sets and fuzzy rough sets [12, 13], which raised the attentions of many scholars [14 –18]. Covering rough sets are more general than rough sets, while intuitionistic fuzzy sets are more flexible than fuzzy sets. The studies on the combination of covering rough sets and intuitionistic fuzzy sets are more practical, and some researches have been done based on single-granulation [19 –24].

However, as illustrated in [25, 26], in many cases it is more reasonable to describe the target concept through multiple relations on the universe according to users’ requirements or problem solving targets. To apply rough sets more widely, Qian et al. extended Pawlak’s single-granulation rough sets to multi-granulation rough sets (MGRSs) [27]. In Qian’s MGRS model, two basic models were proposed. One is the optimistic MGRSs, the other is the pessimistic MGRSs. Now, many researchers have analyzed the MGRS model more deeply [25 –30].

Though both intuitionistic fuzzy sets and covering rough sets are the important generalizations of the classical models, related work is rarely done to combine the intuitionistic fuzzy set theory with the covering rough set theory, especially at the respect of multi-granulation. Motivated by this, in this paper, we explore the issue of how to fuse covering rough sets and intuitionistic fuzzy sets. The concept of covering rough intuitionistic fuzzy sets is redefined from the perspective of granularity and the covering rough intuitionistic fuzzy sets are expanded from single-granulation to multi-granulation. Meanwhile, some basic properties of the models are obtained.

The rest of this paper is organized as follows. We briefly introduce some preliminary concepts in Section 2. In Section 3, a new kind of covering rough intuitionistic fuzzy sets based on minimal description is presented, and their basic properties are discussed. In Section 4, three main types of multi-granulation covering rough intuitionistic fuzzy sets are proposed, and their important properties are proved. Finally, we conclude our work in Section 5.

2 Basic concepts

In this section, we review some basic concepts about rough sets, covering rough sets, multi-granulation rough sets and intuitionistic fuzzy sets as well.

2.1 Rough sets

Definition 1. (Rough sets [1]) Let U be a nonempty and finite universe of discourse, and R be an equivalence relation on U. R generates a partition U/R = {K₁, K₂, …, K_n} on U. For all x ∈ U, the lower and upper approximations of X are described by the following two sets, respectively, $\begin{matrix} \underline{R} (X) & = & \cup {K_{i} | K_{i} \in U / R \land K_{i} \subseteq X}, \\ \bar{R} (X) & = & \cup {K_{i} | K_{i} \in U / R \land K_{i} \cap X \neq φ} . \end{matrix}$

If $\underline{R} (X) = \bar{R} (X)$ , then X is referred to as R-defined in U, otherwise X is referred to as R-undefined in U.

Definition 2. (Degree of rough membership [31]) Let U be a nonempty set, and R be an equivalence relation of U. ∀x ∈ U, X ∈ U. Let $μ_{X}^{R} (x) = | [x]_{R} \cap X | / | [x]_{R} |$ be degree of rough membership of x on R, and | · | is the cardinality of a set and the equivalence class of x with respect to relation R is denoted by [x] _R.

2.2 Covering approximation spaces

Definition 3. (Covering [32]) Let U be a finite and nonempty universe of discourse, C be a family of subsets of U. If none of subsets in C is empty, and ∪C = U, C is called a covering of U.

It is clear that a partition is certainly a covering of U, so the concept of the covering is an extension of the concept of the partition.

Definition 4. (Covering approximation spaces [32]) Let U be a nonempty set, and C be a covering of U. The pair (U, C) is called a covering approximation space.

Definition 5. (Minimal description [32]) Let (U, C) be a covering approximation space, for any x ∈ U, then the set family Md (x) = {K ∈ C|x ∈ K ∧ (∀ S ∈ C ∧ x ∈ S ∧ S ⊆ K ⇒ K = S)} is called the minimal description of x.

2.3 Multi-granulation rough sets

Definition 6. (Optimistic multi-granulation rough sets [26]) Let I be an information system in which A₁, A₂, ⋯ , A_m ⊆ AT, then for all X ⊆ U, the optimistic multi-granulation lower and upper approximations are denoted by $\underline{\sum_{i = 1}^{m} A_{i}^{O}} (X)$ and $\bar{\sum_{i = 1}^{m} A_{i}^{O}} (X)$ , respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i}^{O}} (X) & = & {x \in U | [x]_{A_{1}} \subseteq X \lor [x]_{A_{2}} \\ \subseteq & X \lor [x]_{A_{3}} \subseteq X \lor \dots \lor [x]_{A_{m}} \subseteq X}, \\ \bar{\sum_{i = 1}^{m} A_{i}^{O}} (X) & = & \sim \underline{(\sum_{i = 1}^{m} A_{i}^{O}} (\sim X)), \end{matrix}$ 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, ${BN}_{\sum_{i = 1}^{m} A_{i}}^{O} (x) \bar{= \sum_{i = 1}^{m} A_{i}^{O}} (X) - \underline{\sum_{i = 1}^{m} A_{i}^{O}} (X) .$

Definition 7. (Pessimistic multi-granulation rough sets [27]) Let I be an information system in which A₁, A₂, ⋯ , A_m ⊆ AT, then for all X ⊆ U, the pessimistic multi-granulation lower and upper approximations are denoted by $\underline{\sum_{i = 1}^{m} A_{i}^{P}} (X)$ and $\bar{\sum_{i = 1}^{m} A_{i}^{P}} (X)$ , respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i}^{P}} (X) & = & {x \in U | [x]_{A_{1}} \subseteq X \land [x]_{A_{2}} \\ \subseteq & X \land [x]_{A_{3}} \subseteq X \land \dots \land [x]_{A_{m}} \subseteq X}, \\ \bar{\sum_{i = 1}^{m} A_{i}^{P}} (X) & = & \sim \underline{(\sum_{i = 1}^{m} A_{i}^{P}} (\sim X)), \end{matrix}$ 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 ${BN}_{\sum_{i = 1}^{m} A_{i}}^{P} (x) \bar{= \sum_{i = 1}^{m} A_{i}^{P}} (X) - \underline{\sum_{i = 1}^{m} A_{i}^{P}} (X)$

2.4 Intuitionistic fuzzy sets

Definition 8. (Intuitionistic fuzzy sets [6 –8]) Let U be the universe of discourse, then an intuitionistic fuzzy set (IFS) A in U is an object having the form 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, respectively, called the membership and non-membership degrees of the element of x in A. Furthermore, π_A (x) =1 - μ_A (x) - ν_A (x) is called the hesitancy degree of the element x in A.

The family of all IFS in U is denoted by IFS (U).

Definition 9. [6, 8] Let A = {(x, μ_A (x) , ν_A (x)) |x ∈ U}, B = {(x, μ_B (x) , ν_B (x)) |x ∈ U}, A, B ∈ IFS (U) . Some basic operations on IFS (U) are defined as follows:

A = B ⇔ μ_A (x) = μ_B (x) ∧ ν_A (x) = ν_B (x).

A ⊆ B ⇔ μ_A (x) ≤ μ_B (x) ∧ ν_A (x) ≥ ν_B (x).

A ∩ B = {(x, min {μ_A (x) , μ_B (x)} , max {ν_A (x) , ν_B (x)})}.

A ∪ B = {(x, max {μ_A (x) , μ_B (x)} , min {ν_A (x) , ν_B (x)})}.

The complementary set of A is denoted by ∼A = {(x, ν_A (x) , μ_A (x)) |x ∈ U} ∈ IF (U).

A - B = A ∩ ∼ B.

3 A new kind of covering rough intuitionistic fuzzy sets

3.1 Analysis of an existing model

For the sake of contrastive analysis, firstly, we introduce briefly an existing covering rough intuitionistic fuzzy set model which is defined in reference [22], then we analyze its some limitations.

Definition 10. (Covering rough intuitionistic fuzzy sets [22]) Let (U, C) be a covering approximation space, and U be a finite and nonempty universe of discourse, C be a covering of U. For A ∈ IFS (U), x ∈ U, the lower and upper approximations of A with respect to the approximation space (U, C), donated by $\underline{C} (A)$ and $\bar{C} (A)$ , are defined as follows, $\begin{matrix} \underline{C} (A) & = & {(x, sup_{K \in Md (x)} {inf μ_{A} (y)}, \\ inf_{K \in Md (x)} {sup ν_{A} (y)}) | y \in K}, \\ \bar{C} (A) & = & {(x, inf_{K \in Md (x)} {sup μ_{A} (y)}, \\ sup_{K \in Md (x)} {inf ν_{A} (y)}) | y \in K} . \end{matrix}$

For better readability, we call the above model Type-I covering rough intuitionistic fuzzy sets (in brief Type-I CRIFS). When A is a fuzzy set, this model is degenerated into a covering rough fuzzy set [23].

Although the Type-I CRIFS describes the concept of intuitionistic fuzziness on the domain of discourse, it is inaccurate. As the maximal membership degree, the minimal membership degree, the minimal non-membership degree and the maximal non-membership degree that are described in the minimal description sets of the element of x are determined once, the rest elements have little effect. The calculation shows that the model does not correspond to reality. In order to solve the membership and non-membership degrees of x, which only rely on the maximal and minimal membership degrees or the minimal and maximal non-membership degrees of its minimal description, we elucidate that the Type-I CRIFS is inconsistent in the following case.

Example 1. Suppose there is a domain of discourse which consists of nine credit card applicants U = {x₁, x₂, x₃, …, x₉}, their educational achievements are evaluated by multiple experts, we have a covering, $\begin{matrix} C & = & {good, average, poor} \\ = & {{x_{1}, x_{2}, x_{3}}, {x_{3}, x_{4}, x_{5}, x_{6}, x_{7}}, \\ {x_{6}, x_{7}, x_{8}, x_{9}}} . \end{matrix}$

The intuitionistic fuzzy sets A and B are applicants’ education level as follows, respectively, $\begin{matrix} A & = & {(x_{1}, 0.9, 0.1), (x_{2}, 0.1, 0.8), (x_{3}, 0.1, 0.8), \\ (x_{4}, 0.9, 0.1), (x_{5}, 0, 0.9), (x_{6}, 0, 0.8), \\ (x_{7}, 0, 0.9), (x_{8}, 0.9, 0.1), (x_{9}, 0, 0.9)} . \\ B & = & {(x_{1}, 0.9, 0.1), (x_{2}, 0.9, 0.1), (x_{3}, 0.1, 0.8), \\ (x_{4}, 0.9, 0.1), (x_{5}, 0.9, 0.1), (x_{6}, 0, 0.9), \\ (x_{7}, 0, 0.8), (x_{8}, 0.9, 0.1), (x_{9}, 0.9, 0.1)} . \end{matrix}$

As follows, we calculate the lower and upper approximations of A and B according to the model we proposed in Definition 10.

Step 1. To calculate the minimal description of the element x, we can obtain, $\begin{matrix} Md (x_{1}) & = & Md (x_{2}) = {{x_{1}, x_{2}, x_{3}}}, \\ Md (x_{3}) & = & {{x_{1}, x_{2}, x_{3}}, {x_{3}, x_{4}, x_{5}, x_{6}, x_{7}}}, \\ Md (x_{4}) & = & Md (x_{5}) = {{x_{3}, x_{4}, x_{5}, x_{6}, x_{7}}}, \\ Md (x_{6}) & = & Md (x_{7}) = {{x_{3}, x_{4}, x_{5}, x_{6}, x_{7}}, \\ {x_{6}, x_{7}, x_{8}, x_{9}}}, \\ Md (x_{8}) & = & Md (x_{9}) = {{x_{6}, x_{7}, x_{8}, x_{9}}} . \end{matrix}$

Step 2. By Definition 10, we compute the lower and upper approximations of A and B with Type-I CRIFS. $\begin{matrix} \underline{C} (A) & = & {(x_{1}, 0.1, 0.8), (x_{2}, 0.1, 0.8), (x_{3}, 0.1, 0.8), \\ (x_{4}, 0, 0.9), (x_{5}, 0, 0.9), (x_{6}, 0, 0.9), \\ (x_{7}, 0, 0.9), (x_{8}, 0, 0.9), (x_{9}, 0, 0.9)} . \\ \bar{C} (A) & = & {(x_{1}, 0.9, 0.1), (x_{2}, 0.9, 0.1), (x_{3}, 0.9, 0.1), \\ (x_{4}, 0.9, 0.1), (x_{5}, 0.9, 0.1), (x_{6}, 0.9, 0.1), \\ (x_{7}, 0.9, 0.1), (x_{8}, 0.9, 0.1), (x_{9}, 0.9, 0.1)} . \\ \underline{C} (B) & = & {(x_{1}, 0.1, 0.8), (x_{2}, 0.1, 0.8), (x_{3}, 0.1, 0.8), \\ (x_{4}, 0, 0.9), (x_{5}, 0, 0.9), (x_{6}, 0, 0.9), \\ (x_{7}, 0, 0.9), (x_{8}, 0, 0.9), (x_{9}, 0, 0.9)} . \\ \bar{C} (B) & = & {(x_{1}, 0.9, 0.1), (x_{2}, 0.9, 0.1), (x_{3}, 0.9, 0.1), \\ (x_{4}, 0.9, 0.1), (x_{5}, 0.9, 0.1), (x_{6}, 0.9, 0.1), \\ (x_{7}, 0.9, 0.1), (x_{8}, 0.9, 0.1), (x_{9}, 0.9, 0.1)} . \end{matrix}$

There exists great difference between A and B in Example 1, even the membership and the non-membership degrees of these elements are almost completely opposite. We calculate the lower and upper approximations of covering rough intuitionistic fuzzy sets of A and B, and the results are identical when using the Type-I CRIFS. Although this model is simple and easy to understand, it can’t reflect the real situation effectively.

So, we know that the Type-I CRIFS is inaccurate. When using the simple method to compute the maximal and minimal values, one can obtain the same values that represent the membership and non-membership degrees of all elements in a set. That is to say, the maximal and minimal values of membership degrees of elements and the minimal and maximal values of non-membership degrees of elements in minimal description can define the two boundaries of the fuzzy concept. The different elements have the same membership and non-membership degrees, if they belong to a minimal description. The membership and non-membership degrees are determined by some specific elements, which are not related to the elements themselves and the relationship between the elements. There is a big difference from the actual situation of intuitionistic fuzzy sets.

To analyze the limitations of the Type-I CRIFS model, the Section 3.2 will study the problem from a new angle. The following examples are based on this application background, not in detail.

3.2. Type-II covering rough intuitionistic fuzzy sets

Definition 11. Let (U, C) be a covering approximation space, and U be a finite and nonempty universe of discourse, C be a covering of U. For A ∈ IFS (U), x ∈ U, the intuitionistic fuzzy covering rough membership and non-membership degrees of x based on minimal description with respect to A are defined as follows, respectively, $\begin{matrix} μ_{A}^{'} (x) & = & \sum_{y \in \cup Md (x)} μ_{A} (y) / | \cup Md (x) |, \\ ν_{A}^{'} (x) & = & \sum_{y \in \cup Md (x)} ν_{A} (y) / | \cup Md (x) | . \end{matrix}$

The intuitionistic fuzzy covering rough membership and non-membership degrees based on minimal description not only reflect the relation of the elements and minimal description, but also reflect the degrees of every element belonging to the intuitionistic fuzzy set A from a new perspective.

Definition 12. Let (U, C) be a covering approximation space, and U be a finite and nonempty universe of discourse, C be a covering of U. For A ∈ IFS (U), x ∈ U, the lower and upper approximations of A with respect to (U, C), donated by $\underline{CK} (A)$ and $\bar{CK} (A)$ , are defined as follows, respectively, $\begin{matrix} \underline{CK} (A) & = & {(x, μ_{\underline{CK} (A)} (x), ν_{\underline{CK} (A)} (x)) | x \in U}, \\ \bar{CK} (A) & = & {(x, μ_{\bar{CK} (A)} (x), ν_{\bar{CK} (A)} (x)) | x \in U}, \end{matrix}$ where, $\begin{matrix} μ_{\underline{CK} (A)} (x) & = & min (μ_{A} (x), μ_{A}^{'} (x)), ν_{\underline{CK} (A)} (x) \\ = & max (ν_{A} (x), ν_{A}^{'} (x)) . \\ μ_{\bar{CK} (A)} (x) & = & max (μ_{A} (x), μ_{A}^{'} (x)), ν_{\bar{CK} (A)} (x) \\ = & min (ν_{A} (x), ν_{A}^{'} (x)) . \end{matrix}$

For the sake of convenience, the model is called Type-II covering rough intuitionistic fuzzy sets (in brief Type-II CRIFS) in this paper.

We illustrate the aforementioned model by using the following Example 2 for better understanding the results showed above.

Example 2. Let U = {x₁, x₂, … x₉}, C be a covering of U, and C = {{x₁, x₂} , {x₃, x₄, x₅} , {x₄, x₅, x₇} {x₆, x₇, x₈, x₉}}, the intuitionistic fuzzy set A is, $\begin{matrix} A & = & {(x_{1}, 0 . 2, 0 . 3), (x_{2}, 0 . 7, 0 . 2), (x_{3}, 0 . 5, 0 . 2), \\ (x_{4}, 0 . 3, 0 . 6), (x_{5}, 0 . 2, 0 . 7), (x_{6}, 0, 0 . 6), \\ (x_{7}, 0 . 5, 0 . 4), (x_{8}, 0 . 1, 0 . 5), (x_{9}, 0 . 3, 0 . 2)} . \end{matrix}$

With the Type-I CRIFS in [22] and Type-II CRIFS presented in this paper, we have the lower and upper approximations of A.

① By Definition 10, one can calculate the lower and upper approximations of A with the Type-I CRIFS. $\begin{matrix} \underline{C} (A) (x) & = & {(x_{1}, 0.2, 0.3), (x_{2}, 0.2, 0.3), \\ (x_{3}, 0.2, 0.7), (x_{4}, 0.2, 0.7), \\ (x_{5}, 0.2, 0.7), (x_{6}, 0, 0.6), (x_{7}, 0.2, 0.6), \\ (x_{8}, 0, 0.6), (x_{9}, 0, 0.6)} . \\ \bar{C} (A) (x) & = & {(x_{1}, 0.7, 0.2), (x_{2}, 0.7, 0.2), \\ (x_{3}, 0.5, 0.2), (x_{4}, 0.5, 04), \\ (x_{5}, 0.5, 0.4), (x_{6}, 0.5, 0.2), \\ (x_{7}, 0.5, 0.4), (x_{8}, 0.5, 0.2), \\ (x_{9}, 0.5, 0.2)} . \end{matrix}$

② We calculate the lower and upper approximations of A with the Type-II CRIFS.

Step 1. This step is to calculate the minimal descripttion of x. $\begin{matrix} Md (x_{1}) & = & Md (x_{2}) = {{x_{1}, x_{2}}}, Md (x_{3}) \\ = & {{x_{3}, x_{4}, x_{5}}}, \\ Md (x_{4}) & = & Md (x_{5}) = {{x_{3}, x_{4}, x_{5}}, {x_{4}, x_{5}, x_{7}}}, \\ Md (x_{6}) & = & {{x_{6}, x_{7}, x_{8}, x_{9}}}, \\ Md (x_{7}) & = & {{x_{4}, x_{5}, x_{7}}, {x_{6}, x_{7}, x_{8}, x_{9}}}, \\ Md (x_{8}) & = & Md (x_{9}) = {{x_{6}, x_{7}, x_{8}, x_{9}}} . \end{matrix}$

Step 2. By Definition 11, the intuitionistic fuzzy covering rough membership and non-membership degrees based on minimal description are calculated as follows, $\begin{matrix} μ_{A}^{'} (x_{1}) & = & μ_{A}^{'} (x_{2}) = \frac{0.2 + 0.7}{2} = 0.45, \\ μ_{A}^{'} (x_{3}) & = & \frac{0.5 + 0.3 + 0.2}{3} \approx 0.33, \\ μ_{A}^{'} (x_{4}) & = & μ_{A}^{'} (x_{5}) = \frac{0.5 + 0.3 + 0.2 + 0.5}{4} \\ \approx & 0.38, \\ μ_{A}^{'} (x_{6}) & = & \frac{0 + 0.5 + 0.1 + 0.3}{4} \approx 0.23, \\ μ_{A}^{'} (x_{7}) & = & \frac{0.3 + 0.2 + 0.5 + 0 + 0.1 + 0.3}{7} \\ \approx & 0.23, \\ μ_{A}^{'} (x_{8}) & = & μ_{A}^{'} (x_{9}) \approx 0.23 . \end{matrix}$

The intuitionistic fuzzy covering rough non-membership degrees based on minimal description are similar to the intuitionistic fuzzy covering rough membership degrees based on minimal description.

Step 3. By Definition 12, one can calculate the membership and non-membership degrees of x. $\begin{matrix} μ_{\underline{CK} (x_{1})} & = & min {0.2, 0.45} = 0.2, \\ μ_{\bar{CK} (x_{1})} & = & max {0.2, 045} = 0.45; \\ μ_{\underline{CK} (x_{2})} & = & min {0.7, 0.45} = 0.45, \\ μ_{\bar{CK} (x_{2})} & = & max {0.7, 0.45} = 0.7; \\ μ_{\underline{CK} (x_{3})} & = & min {0.5, 0.33} = 0.33, \\ μ_{\bar{CK} (x_{3})} & = & max {0.5, 0.33} = 0.5; \\ μ_{\underline{CK} (x_{4})} & = & min {0.3, 0.38} = 0.3, \\ μ_{\bar{CK} (x_{4})} & = & max {0.3, 0.38} = 0.38; \\ μ_{\underline{CK} (x_{5})} & = & min {0.2, 0.38} = 0.2, \\ μ_{\bar{CK} (x_{5})} & = & max {0.2, 0.38} = 0.38; \\ μ_{\underline{CK} (x_{6})} & = & min {0, 0.23} = 0, \\ μ_{\bar{CK} (x_{6})} & = & max {0, 0.23} = 0.23; \\ μ_{\underline{CK} (x_{7})} & = & min {0.5, 0.23} = 0.23, \\ μ_{\bar{CK} (x_{7})} & = & max {0.5, 0.23} = 0.5; \\ μ_{\underline{CK} (x_{8})} & = & min {0.1, 0.23} = 0, \\ μ_{\bar{CK} (x_{8})} & = & max {0.1, 0.23} = 0.23; \\ μ_{\underline{CK} (x_{9})} & = & min {0.3, 0.23} = 0.23, \\ μ_{\bar{CK} (x_{9})} & = & max {0.3, 0.23} = 0.3 . \end{matrix}$

Using the same way to calculate the non-membership degrees, one can calculate the lower and upper approximations of A with the Type-II CRIFS. $\begin{matrix} \underline{CK} (A) (x) & = & {(x_{1}, 0 . 2, 0 . 3), (x_{2}, 0 . 45, 0 . 25), \\ (x_{3}, 0 . 33, 0 . 5), (x_{4}, 0 . 3, 0 . 6), \\ (x_{5}, 0 . 2, 0 . 7), (x_{6}, 0, 0 . 6), \\ (x_{7}, 0 . 23, 0 . 5), \\ (x_{8}, 0 . 1, 0 . 5), (x_{9}, 0 . 23, 43)} . \\ \bar{CK} (A) (x) & = & {(x_{1}, 0.45, 0 . 25), (x_{2}, 0 . 7, 0 . 2), \\ (x_{3}, 0 . 5, 0 . 2), (x_{4}, 0 . 38, 0 . 48), \\ (x_{5}, 0 . 36, 0 . 48), (x_{6}, 0 . 23, 0 . 43), \\ (x_{7}, 0 . 5, 0 . 4), (x_{8}, 0 . 23, 0 . 43), \\ (x_{9}, 0 . 3, 0 . 2)} . \end{matrix}$

The result shows that we can find out the result of the Type-II CRIFS more precise than the Type-I CRIFS from Example 2 easily and clearly. Hence, the Type-II CRIFS can response information that is closer to the actual situation.

From Definition 12, we can obtain some properties.

Theorem 1. Let (U, C) be a covering approximation space, and U be a finite and nonempty universe of discourse. Let C be a covering of U. For A, B ∈ IFS (U) , x ∈ U, the Type-II CRIFS has the following properties,

$\underline{CK} (U) = U, \bar{CK} (U) = U$ ,

$\underline{CK} (φ) = φ, \bar{CK} (φ) = φ$ ,

$\underline{CK} (A) \subseteq A \subseteq \bar{CK} (A)$ ,

If A ⊆ B, then $\underline{CK} (A) \subseteq \underline{CK} (B), \bar{CK} (A) \subseteq \bar{CK} (B)$ .

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 11, $μ_{U}^{'} (x) = 1, ν_{U}^{'} (x) = 0$ .

From Definition 12, $μ_{\underline{CK} (U)} (x) = 1, ν_{\underline{CK} (U)} (x) = 0,$ and $μ_{\bar{CK} (U)} (x) = 1, ν_{\bar{CK} (U)} (x) = 0$ , hence $\underline{CK} (U) = \bar{CK} (U) = U$ .

(2) The proving process of (2) is similar to (1).

(3) By Definition 11, for all x ∈ U, $μ_{A} (x) \leq μ_{A}^{'} (x)$ or $μ_{A} (x) > μ_{A}^{'} (x)$ , and $ν_{A} (x) \leq ν_{A}^{'} (x)$ or $ν_{A} (x) > ν_{A}^{'} (x)$ .

There are four different situations:

$μ_{A} (x) \leq μ_{A}^{'} (x)$ , $ν_{A} (x) \leq ν_{A}^{'} (x)$ .

$μ_{A} (x) \geq μ_{A}^{'} (x)$ , $ν_{A} (x) \leq ν_{A}^{'} (x)$ .

$μ_{A} (x) \leq μ_{A}^{'} (x)$ , $ν_{A} (x) \geq ν_{A}^{'} (x)$ .

$μ_{A} (x) \geq μ_{A}^{'} (x)$ , $ν_{A} (x) \geq ν_{A}^{'} (x)$ .

We prove the processes of ① and ② as follows.

① When $μ_{A} (x) \leq μ_{A}^{'} (x)$ , $ν_{A} (x) \leq ν_{A}^{'} (x)$ , and $μ_{\underline{CK} (A)} (x) = μ_{A} (x)$ , $μ_{\bar{CK} (A)} (x) = μ_{A}^{'} (x)$ ; $ν_{\underline{CK} (A)} (x) = ν_{A}^{'} (x)$ , $ν_{\bar{CK} (A)} (x) = ν_{A} (x)$ . Then, $μ_{\underline{CK} (A)} (x) \leq μ_{A} (x) \leq μ_{\bar{CK} (A)} (x), ν_{\underline{CK} (A)} (x) \geq ν_{A} (x) \geq ν_{\bar{CK} (A)} (x)$ .

② When $μ_{A} (x) \geq μ_{A}^{'} (x)$ , $ν_{A} (x) \leq ν_{A}^{'} (x)$ , and $μ_{\underline{CK} (A)} (x) = μ_{A}^{'} (x)$ , $μ_{\bar{CK} (A)} (x) = μ_{A} (x)$ , $ν_{\underline{CK} (A)} (x) = ν_{A}^{'} (x)$ , $ν_{\bar{CK} (A)} (x) = ν_{A} (x) .$ Then, $ν_{\underline{CK} (A)} (x) \geq ν_{A} (x) \geq ν_{\bar{CK} (A)} (x)$ .

Similarly, we can prove ③ and ④.

Hence, $\underline{CK} (A) \subseteq A \subseteq \bar{CK} (A)$ .

(4) Since A ⊆ B, ∀ x ∈ U, μ_A (x) ≤ μ_B (x) , ν_A (x) ≥ ν_B (x) , $μ_{A}^{'} (x) \leq μ_{B}^{'} (x), ν_{A}^{'} (x) \geq ν_{B}^{'} (x) .$ Then, there are four kinds of circumstances for the membership and non-membership degrees of the lower and upper approximations.

When $μ_{\underline{CK} (A)} (x) = μ_{A} (x), μ_{\bar{CK} (A)} (x) = μ_{A}^{'} (x)$ , and $ν_{\underline{CK} (A)} (x) = ν_{A} (x), ν_{\bar{CK} (A)} (x) = ν_{A}^{'} (x) .$

When $μ_{\underline{CK} (A)} (x) = μ_{A}^{'} (x), μ_{\bar{CK} (A)} (x) = μ_{A} (x)$ , and $ν_{\underline{CK} (A)} (x) = ν_{A} (x), ν_{\bar{CK} (A)} (x) = ν_{A}^{'} (x) .$

When $μ_{\underline{CK} (A)} (x) = μ_{A} (x), μ_{\bar{CK} (A)} (x) = μ_{A}^{'} (x)$ , and $ν_{\underline{CK} (A)} (x) = ν_{A}^{'} (x), ν_{\bar{CK} (A)} (x) = ν_{A} (x) .$

When $μ_{\underline{CK} (A)} (x) = μ_{A}^{'} (x), μ_{\bar{CK} (A)} (x) = μ_{A} (x)$ , and $ν_{\underline{CK} (A)} (x) = ν_{A}^{'} (x), ν_{\bar{CK} (A)} (x) = ν_{A} (x) .$

Here only ① will be proved.

Case 1. If $μ_{\underline{CK} (B)} (x) = μ_{B} (x), μ_{\bar{CK} (B)} (x) = μ_{B}^{'} (x)$ , and $ν_{\underline{CK} (B)} (x) = ν_{B} (x), ν_{\bar{CK} (B)} (x) = ν_{B}^{'} (x) .$

Then, $μ_{\underline{CK} (A)} (x) \leq μ_{\underline{CK} (B)} (x), μ_{\bar{CK} (A)} (x) \leq μ_{\bar{CK} (B)} (x)$ , and $ν_{\underline{CK} (A)} (x) \geq ν_{\underline{CK} (B)} (x), ν_{\bar{CK} (A)} (x) \geq ν_{\bar{CK} (B)} (x) .$

Case 2. If $μ_{\underline{CK} (B)} (x) = μ_{B}^{'} (x), μ_{\bar{CK} (B)} (x) = μ_{B} (x)$ , and $ν_{\underline{CK} (B)} (x) = ν_{B} (x), ν_{\bar{CK} (B)} (x) = ν_{B}^{'} (x) .$

Since $μ_{A} (x) \leq μ_{A}^{'} (x) \leq μ_{B}^{'} (x),$ $μ_{A}^{'} (x) \leq μ_{B}^{'} (x) \leq μ_{B} (x)$ , then $μ_{A} (x) \leq μ_{B}^{'} (x), μ_{A}^{'} (x) \leq μ_{B} (x)$ .

So $μ_{\underline{CK} (A)} (x) \leq μ_{\underline{CK} (B)} (x), μ_{\bar{CK} (A)} (x) \leq μ_{\bar{CK} (B)} (x),$ and $ν_{\underline{CK} (A)} (x) \geq ν_{\underline{CK} (B)} (x), ν_{\bar{CK} (A)} (x) \geq ν_{\bar{CK} (B)} (x) .$

Case 3. If $μ_{\underline{CK} (B)} (x) = μ_{B} (x), μ_{\bar{CK} (B)} (x) = μ_{B}^{'} (x)$ , and $ν_{\underline{CK} (B)} (x) = ν_{B}^{'} (x), ν_{\bar{CK} (B)} (x) = ν_{B} (x) .$

Since $ν_{A} (x) \geq ν_{A}^{'} (x) \geq ν_{B}^{'} (x)$ , $ν_{A}^{'} (x) \geq ν_{B}^{'} (x) \geq ν_{B} (x)$ .

So $ν_{A} (x) \geq ν_{B}^{'} (x), ν_{A}^{'} (x) \geq ν_{B} (x)$ .

Then $μ_{\underline{CK} (A)} (x) \leq μ_{\underline{CK} (B)} (x), μ_{\bar{CK} (A)} (x) \leq μ_{\bar{CK} (B)} (x)$ , $ν_{\underline{CK} (A)} (x) \geq ν_{\underline{CK} (B)} (x), ν_{\bar{CK} (A)} (x) \geq ν_{\bar{CK} (B)} (x) .$

Case 4. If $μ_{\underline{CK} (B)} (x) = μ_{B}^{'} (x), μ_{\bar{CK} (B)} (x) = μ_{B} (x),$ and $ν_{\underline{CK} (B)} (x) = ν_{B}^{'} (x), ν_{\bar{CK} (B)} (x) = ν_{B} (x) .$

Then, $μ_{\underline{CK} (A)} (x) \leq μ_{\underline{CK (B)}} (x), μ_{\bar{CK} (A)} (x) \leq μ_{\bar{CK (B)}} (x)$ , and $ν_{\underline{CK} (A)} (x) \geq ν_{\underline{CK} (B)} (x), ν_{\bar{CK} (A)} (x) \geq ν_{\bar{CK} (B)} (x) .$

The proving processes of ②, ③ and ④ are similar to ①.

Then, $\underline{CK} (A) \subseteq \underline{CK} (B), \bar{CK} (A) \subseteq \bar{CK} (B)$ .

4 Multi-granulation covering rough intuitionistic fuzzy sets

In this section, three main types of multi-granulation covering rough intuitionistic fuzzy sets are proposed. After that, some important theorems are proved.

This subsection extends the Type-II CRIFS from single-granulation to multi-granulations. We propose three main types of multi-granulation covering rough intuitionistic fuzzy sets, one of which is the multi-granulation covering rough intuitionistic fuzzy set based on minimal description (in brief Type-I MGCRIFS), and the rest are optimistic and pessimistic multi-granulation covering rough intuitionistic fuzzy sets based on minimal description (in brief Type-II optimistic and pessimistic MGCRIFS). Then, we discuss some important properties of the Type-I MGCRIFS, Type-II optimistic and pessimistic MGCRIFS.

Definition 13. Let (U, C) be a covering approximation space, and U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, C_i be a covering of U, |C| = m. For all x ∈ U, Md_{C
_i} (x) is the minimal description of C_i, and Md^∧ (x) = {K′∈ Md_{C
_i} (x) |x ∈ K′ ∧(∀ S′ ∈ Md_{C
_i} (x) ∧ x ∈ S′ ∧ S′ ⊆ K′ ⇒ K′ = S′)} is called the minimal description based on multi-granulation of x.

Definition 14. Let (U, C) be a covering approximation space, U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m. A ∈ IFS (U) , x ∈ U. The multi-granulation intuitionistic fuzzy covering rough membership and non-membership degrees of x based on minimal description with respect to A are defined as follows, respectively, $\begin{matrix} μ_{A}^{\land} (x) & = & \sum_{y \in \cup {Md}^{\land} (x)} μ_{A} (y) / | \cup {Md}^{\land} (x) |, \\ ν_{A}^{\land} (x) & = & \sum_{y \in \cup {Md}^{\land} (x)} ν_{A} (y) / | \cup {Md}^{\land} (x) | . \end{matrix}$

Definition 15. Let (U, C) be a covering approximation space, U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m. A ∈ IFS (U) , x ∈ U. Then, the multi-granulation lower and upper approximations of A based on minimal description with respect to (U, C), donated by $\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)$ and $\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)$ , are defined as follows, respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} {CK}_{i}} (A) & = & {(x, μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x), \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x)) | x \in U}, \\ \bar{\sum_{i = 1}^{m} {CK}_{i}} (A) & = & {(x, μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x), \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x)) | x \in U}, \end{matrix}$ where, $\begin{matrix} μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) & = & min (μ_{A} (x), μ_{A}^{\land} (x)), \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) & = & max (ν_{A} (x), ν_{A}^{\land} (x)) . \\ μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) & = & max (μ_{A} (x), μ_{A}^{\land} (x)), \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) & = & min (ν_{A} (x), ν_{A}^{\land} (x)) . \end{matrix}$

We call the above model the multi-granulation covering rough intuitionistic fuzzy set based on minimal description (in brief Type-I MGCRIFS).

Theorem 2. Let (U, C) be a covering approximation space, U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m. A, B ∈ IFS (U) , x ∈ U. Then,

$\underline{\sum_{i = 1}^{m} {CK}_{i}} (U) = \bar{\sum_{i = 1}^{m} {CK}_{i}} (U) = U$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}} (φ) = \bar{\sum_{i = 1}^{m} {CK}_{i}} (φ) = φ$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}} (A) \subseteq A \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}} (A)$ ,

If A ⊆ B, then $\underline{\sum_{i = 1}^{m} {CK}_{i}} (A) \subseteq \underline{\sum_{i = 1}^{m} {CK}_{i}} (B)$ , $\bar{\sum_{i = 1}^{m} {CK}_{i}} (A) \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}} (B)$ .

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 14, $μ_{U}^{\land} (x) = 1, ν_{U}^{\land} (x) = 0$ .

By Definition 15, $\begin{matrix} μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (U)} (x) & = & μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (U)} (x) = 1, \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (U)} (x) & = & ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (U)} (x) = 0, \end{matrix}$

Hence, $\underline{\sum_{i = 1}^{m} {CK}_{i}} (U) = \bar{\sum_{i = 1}^{m} {CK}_{i}} (U) = U$ .

(2) The proving process of (2) is similar to (1).

(3) By Definition 14 and 15, for all x ∈ U, $μ_{A} (x) \leq μ_{A}^{\land} (x)$ or $μ_{A} (x) > μ_{A}^{\land} (x)$ , and $ν_{A} (x) \leq ν_{A}^{\land} (x)$ or $ν_{A} (x) > ν_{A}^{\land} (x)$ .

There can be four different situations:

$μ_{A} (x) \leq μ_{A}^{\land} (x)$ , $ν_{A} (x) \leq ν_{A}^{\land} (x)$ .

$μ_{A} (x) \geq μ_{A}^{\land} (x)$ , $ν_{A} (x) \leq ν_{A}^{\land} (x)$ .

$μ_{A} (x) \leq μ_{A}^{\land} (x)$ , $ν_{A} (x) \geq ν_{A}^{\land} (x)$ .

$μ_{A} (x) \geq μ_{A}^{\land} (x)$ , $ν_{A} (x) \geq ν_{A}^{\land} (x)$ .

We only need to prove ① and ② as follows.

① When $μ_{A} (x) \leq μ_{A}^{\land} (x)$ , $ν_{A} (x) \leq ν_{A}^{\land} (x)$ , and $μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = μ_{A} (x)$ , $μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = μ_{A}^{\land} (x)$ , $ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = ν_{A}^{\land} (x)$ , $ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = ν_{A} (x)$ .

Then, $μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) \leq μ_{A} (x) \leq μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x),$ $ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) \geq ν_{A} (x) \geq ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) .$

② When $μ_{A} (x) \geq μ_{A}^{\land} (x)$ , $ν_{A} (x) \leq ν_{A}^{\land} (x)$ ,

and,

$μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = μ_{A}^{\land} (x),$ $μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = μ_{A} (x),$

$ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = ν_{A}^{\land} (x)$ , $ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)} (x) = ν_{A} (x)$ .

The proving processes of ③ and ④ are similar to ① and ②.

Hence, $\underline{\sum_{i = 1}^{m} {CK}_{i}} (A) \subseteq A \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}} (A)$ .

(4) The proving process of (4) is similar to (3).

Example 3. Let (U, C) be a covering approximation space, and U be a nonempty and finite domain of discourse, U = {x₁, x₂, x₃, x₄, x₅}, C = {C₁, C₂}, $\begin{matrix} C_{1} & = & {{x_{1}, x_{2}, x_{3}}, {x_{2}, x_{3}}, {x_{4}, x_{5}}}, \\ C_{2} & = & {{x_{1}}, {x_{1}, x_{2}, x_{3}}, {x_{3}, x_{4}, x_{5}}} . \end{matrix}$

The intuitionistic fuzzy set A is: $\begin{matrix} A & = & {(x_{1}, 0.6, 0.3), (x_{2}, 0.3, 0.4), (x_{3}, 0.7, 0.2), \\ (x_{4}, 0, 0.8), (x_{5}, 0.1, 0.3)} . \end{matrix}$

One can calculate the lower and upper approximations of A with Type-I MGCRIFS.

Step 1. This step is to calculate the minimal description based on multi-granulation of x. $\begin{matrix} {Md}^{\land} (x_{1}) & = & {{x_{1}}}, {Md}^{\land} (x_{2}) = {{x_{2}, x_{3}}}, \\ {Md}^{\land} (x_{3}) & = & {{x_{2}, x_{3}}, {x_{3}, x_{4}, x_{5}}}, \\ {Md}^{\land} (x_{4}) & = & {Md}^{\land} (x_{5}) = {{x_{4}, x_{5}}} . \end{matrix}$

Step 2. The lower and upper approximations of A are calculated with Type-I MGCRIFS as follows, $\begin{matrix} \underline{\sum_{i = 1}^{m} {CK}_{i}} (A) & = & {(x_{1}, 0 . 6, 0 . 3), (x_{2}, 0 . 3, 0 . 4), \\ (x_{3}, 0 . 28, 0 . 43), (x_{4}, 0, 0 . 8), \\ (x_{5}, 0.05, 0 . 55)} . \\ \bar{\sum_{i = 1}^{m} {CK}_{i}} (A) & = & {(x_{1}, 0 . 6, 0 . 3), (x_{2}, 0 . 5, 0 . 3), \\ (x_{3}, 0 . 7, 0 . 2), (x_{4}, 0 . 05, 0 . 55), \\ (x_{5}, 0 . 1, 0 . 3)} . \end{matrix}$

Definition 16. Let (U, C) be a covering approximation space, U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m . A ∈ IFS (U) , for all x ∈ U . Then, the optimistic multi-granulation lower and upper approximations of A based on minimal description with respect to (U, C), donated by $\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A)$ and $\bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A)$ , are defined as follows, respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) & = & {(x, μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x), \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x)) | x \in U}, \\ \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) & = & {(x, μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x), \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x)) | x \in U}, \end{matrix}$ where, $\begin{matrix} μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \lor_{i = 1}^{m} {min (μ_{A} (x), μ_{A}^{'} (x))} \\ = & \lor_{i = 1}^{m} μ_{\underline{{CK}_{i}} (A)}, \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \land_{i = 1}^{m} {max (ν_{A} (x), ν_{A}^{'} (x))} \\ = & \land_{i = 1}^{m} ν_{\underline{{CK}_{i}} (A)} \\ μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \land_{i = 1}^{m} {max (μ_{A} (x), μ_{A}^{'} (x))} \\ = & \land_{i = 1}^{m} μ_{\bar{{CK}_{i}} (A)}, \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \lor_{i = 1}^{m} {min (ν_{A} (x), ν_{A}^{'} (x))} \\ = & \lor_{i = 1}^{m} ν_{\bar{{CK}_{i}} (A)} . \end{matrix}$

We call this model optimistic multi-granulations covering rough intuitionistic fuzzy set based on minimal description (in brief Type-II optimistic MGCRIFS).

Definition 17. Let (U, C) be a covering approximation space, U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m . A ∈ IFS (U) , for all x ∈ U . Then, the pessimistic multi-granulation lower and upper approximations of A based on minimal description with respect to (U, C), donated by $\underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A)$ and $\bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A)$ , are defined as follows, respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) & = & {(x, μ_{\underline{\sum_{i = 1}^{m} {CK}_{i} (} A)}^{P} (x), \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x)) | x \in U}, \\ \bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) & = & {(x, μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x), \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x)) | x \in U}, \end{matrix}$ where, $\begin{matrix} μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x) & = & \land_{i = 1}^{m} {min (μ_{A} (x), μ_{A}^{'} (x))} \\ = & \land_{i = 1}^{m} μ_{\underline{{CK}_{i}} (A)}, \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x) & = & \lor_{i = 1}^{m} {max (ν_{A} (x), ν_{A}^{'} (x))} \\ = & \lor_{i = 1}^{m} ν_{\underline{{CK}_{i}} (A)} . \\ μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x) & = & \lor_{i = 1}^{m} {max (μ_{A} (x), μ_{A}^{'} (x))} \\ = & \lor_{i = 1}^{m} μ_{\bar{{CK}_{i}} (A)}, \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{P} (x) & = & \land_{i = 1}^{m} {min (ν_{A} (x), ν_{A}^{'} (x))} \\ = & \land_{i = 1}^{m} ν_{\bar{{CK}_{i}} (A)} . \end{matrix}$

We call this model pessimistic multi-granulations covering rough intuitionistic fuzzy set based on minimal description (in brief Type-II pessimistic MGCRIFS).

When m = 1, the Type-II optimistic and pessimistic MGCRIFS models degenerate into the Type-II CRIFS model presented in Section 3.2.

When A is an exact set, the Type-II optimistic and pessimistic MGCRIFS models degenerate into the optimistic and pessimistic multi-granulation covering rough sets based on minimal description models.

When A is a fuzzy set, Type-II optimistic and pessimistic MGCRIFS models degenerate into the optimistic and pessimistic multi-granulations covering rough fuzzy sets based on minimal description models.

Theorem 3. Let (U, C) be a covering approximation space, U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m. A ∈ IFS (U) , x ∈ U. Then,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) = ⋃_{i = 1}^{m} \underline{{CK}_{i}^{O}} (A)$ ,

$\bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) = ⋂_{i = 1}^{m} \bar{{CK}_{i}^{O}} (A)$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) = ⋂_{i = 1}^{m} \bar{{CK}_{i}^{P}} (A)$ ,

$\bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) = ⋃_{i = 1}^{m} \bar{{CK}_{i}^{P}} (A)$ .

Proof. It directly follows from Definitions 11, 12, 16 and 17.

Theorem 4. Let (U, C) be a covering approximation space, and U be a finite and nonempty universe of discourse. Let C = {C_i|1 ≤ i ≤ m} be a family of coverings of U, and C_i be a covering of U, |C| = m. A, B ∈ IFS (U) , x ∈ U. Then,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (U) = \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (U) = U$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (U) = \bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (U) = U$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (φ) = \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (φ) = φ$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (φ) = \bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (φ) = φ$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) \subseteq A \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A)$ ,

$\underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) \subseteq A \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A)$ ,

If A ⊆ B, then $\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) \subseteq \underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (B)$ , $\bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (B)$ ,

If A ⊆ B, then $\underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) \subseteq \underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (B)$ , $\bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (B)$ .

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 11, $μ_{U}^{'} (x) = 1, ν_{U}^{'} (x) = 0$ .

By Definition 16, $\begin{matrix} μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \lor_{i = 1}^{m} {min (μ_{A} (x), μ_{A}^{'} (x))} = 1, \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \land_{i = 1}^{m} {max (ν_{A} (x), ν_{A}^{'} (x))} = 0 . \\ μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \land_{i = 1}^{m} {max (μ_{A} (x), μ_{A}^{'} (x))} = 1, \\ ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) & = & \lor_{i = 1}^{m} {min (ν_{A} (x), ν_{A}^{'} (x))} = 0 . \end{matrix}$

Then, $\begin{matrix} μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (U)}^{O} (x) & = & μ_{\bar{\sum_{i = 1}^{m} {CK}_{i} (} U)}^{O} (x) = 1, \\ ν_{\underline{\sum_{i = 1}^{m} {CK}_{i} (} U)}^{O} (x) & = & ν_{\bar{\sum_{i = 1}^{m} {CK}_{i} (} U)}^{O} (x) = 0 . \end{matrix}$

Hence, $\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (U) = \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (U) = U$ .

The proving processes of (2), (3), (4) are similar to the (1).

(5) By Definition 11, for all x ∈ U, $\begin{matrix} μ_{i 1} (y) & = & min (μ_{A} (x), μ_{A}^{'} (x)), 1 \leq i \leq m, \\ μ_{i 2} (y) & = & min (μ_{A} (x), μ_{A}^{'} (x)), 1 \leq i \leq m . \end{matrix}$

Then μ_i1 (y) ≤ μ_A (x) ≤ μ_i2 (y) , 1 ≤ i ≤ m.

So, $\lor_{i = 1}^{m} μ_{i 1} (y) \leq μ_{A} (x) \leq \land_{i = 1}^{m} μ_{i 2} (y), 1 \leq i \leq m$ , and $μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) \leq μ_{A} (x) \leq μ_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x), 1 \leq i \leq m$ .

Similarly, $ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) \geq ν_{A} (x) \geq ν_{\bar{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x), 1 \leq i \leq m$ .

Then $\underline{\sum_{i = 1}^{m} {CK}_{i}} (A) \subseteq A \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}} (A)$ .

(6) The proving process of (6) is similar to (5).

(7) Since, A ⊆ B, ∀ x ∈ U, μ_A (x) ≤ μ_B (x) , ν_A (x) ≥ ν_B (x), then $μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) = \lor_{i = 1}^{m} {min (μ_{A} (x), μ_{A}^{'} (x))} \leq \lor_{i = 1}^{m} {min (μ_{B} (x), μ_{B}^{'} (x))} = μ_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (B)}^{O} (x) .$

Similarly, $ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (A)}^{O} (x) \geq ν_{\underline{\sum_{i = 1}^{m} {CK}_{i}} (B)}^{O} (x)$ .

Then, $\underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) \subseteq \underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (B)$ .

Similarly, $\bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) \subseteq \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (B)$ .

(8) The proving process of (8) is similar to (7).

Example 4. Let (U, C) be a covering approximation space, and U be a nonempty and finite domain of discourse, U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇}, C = {C₁, C₂, C₃}, $\begin{matrix} C_{1} & = & {{x_{1}, x_{2}}, {x_{3}, x_{6}, x_{7}}, {x_{3}, x_{7}}, {x_{4}, x_{5}, x_{6}}}, \\ C_{2} & = & {{x_{1}}, {x_{2}}, {x_{2}, x_{4}}, {x_{3}, x_{4}, x_{6}}, {x_{5}, x_{7}}}, \\ C_{3} & = & {{x_{1}, x_{2}}, {x_{4}, x_{6}}, {x_{5}, x_{6}}, {x_{4}, x_{5}}, \\ {x_{4}, x_{5}, x_{6}}, {x_{3}, x_{7}}} . \end{matrix}$

The intuitionistic fuzzy set A is: $\begin{matrix} A & = & {(x_{1}, 0.5, 0.4), (x_{2}, 0.6, 0.1), (x_{3}, 1, 0), \\ (x_{4}, 0.4, 0.3), (x_{5}, 0.3, 0.5), \\ (x_{6}, 0.2, 0.7), (x_{7}, 0.8, 0.2)} . \end{matrix}$

One can calculate the lower and upper approximations of A with Type-II optimistic and pessimistic MGCRIFS.

Step 1. This step is to calculate minimal description of every covering. $\begin{matrix} {Md}_{1} (x_{1}) & = & {Md}_{1} (x_{2}) = {{x_{1}, x_{2}}}, \\ {Md}_{1} (x_{3}) & = & {Md}_{1} (x_{7}) = {{x_{3}, x_{7}}}, \\ {Md}_{1} (x_{4}) & = & {Md}_{1} (x_{5}) = {{x_{4}, x_{5}, x_{6}}}, \\ {Md}_{1} (x_{6}) & = & {{x_{3}, x_{6}, x_{7}}, {x_{4}, x_{5}, x_{6}}} . \\ {Md}_{2} (x_{1}) & = & {{x_{1}}}, {Md}_{2} (x_{2}) = {{x_{2}}}, \\ {Md}_{2} (x_{3}) & = & {Md}_{2} (x_{6}) = {{x_{3}, x_{4}, x_{6}}}, \\ {Md}_{2} (x_{4}) & = & {{x_{2}, x_{4}}, {x_{3}, x_{4}, x_{6}}}, \\ {Md}_{2} (x_{5}) & = & {Md}_{2} (x_{7}) = {{x_{5}, x_{7}}} . \\ {Md}_{3} (x_{1}) & = & {Md}_{3} (x_{2}) = {{x_{1}, x_{2}}}, \\ {Md}_{3} (x_{3}) & = & {{x_{3}, x_{7}}}, \\ {Md}_{3} (x_{4}) & = & {{x_{4}, x_{5}} {x_{4}, x_{6},}}, \\ {Md}_{3} (x_{5}) & = & {{x_{4}, x_{5}}, {x_{5}, x_{6}}}, \\ {Md}_{3} (x_{6}) & = & {{x_{4}, x_{6}}, {x_{5}, x_{6}}}, \\ {Md}_{3} (x_{7}) & = & {{x_{3}, x_{7}}} . \end{matrix}$

Step 2. The membership degrees of the elements are calculated in every covering. For the covering C₁ = {{x₁, x₂} , {x₃, x₆, x₇} , {x₃, x₇} , {x₄, x₅, x₆}}, we can calculate the non-membership degrees in the same way. $\begin{matrix} μ_{\underline{{CK}_{1}} (A)} (x_{1}) & = & min {0.5, 0.55} = 0.5, \\ μ_{\bar{{CK}_{1}} (A)} (x_{1}) & = & max {0.5, 0.55} = 0.55; \\ μ_{\underline{{CK}_{1}} (A)} (x_{2}) & = & min {0.6, 0.55} = 0.55, \\ μ_{\bar{{CK}_{1}} (A)} (x_{2}) & = & max {0.6, 0.55} = 0.6; \\ μ_{\underline{{CK}_{1}} (A)} (x_{3}) & = & min {1, 0.9} = 0.9, \\ μ_{\bar{{CK}_{1}} (A)} (x_{3}) & = & max {1, 0.9} = 1; \\ μ_{\underline{{CK}_{1}} (A)} (x_{4}) & = & min {0.4, 0.3} = 0.3, \\ μ_{\bar{{CK}_{1}} (A)} (x_{4}) & = & max {0.4, 0.3} = 0.4; \\ μ_{\underline{{CK}_{1}} (A)} (x_{5}) & = & min {0.3, 0.3} = 0.3, \\ μ_{\bar{{CK}_{1}} (A)} (x_{5}) & = & max {0.3, 0.3} = 0.3; \\ μ_{\underline{{CK}_{1}} (A)} (x_{6}) & = & min {0.2, 0.54} = 0.2, \\ μ_{\bar{{CK}_{1}} (A)} (x_{6}) & = & max {0.2, 0.54} = 0.54; \\ μ_{\underline{{CK}_{1}} (A)} (x_{7}) & = & min {0.8, 0.9} = 0.8, \\ μ_{\bar{{CK}_{1}} (A)} (x_{7}) & = & max {0.8, 0.9} = 0.9 . \end{matrix}$

For the covering C₂, C₃, one can calculate the membership and non-membership degrees in the same way as C₁.

Step 3. By Definitions 16 and 17, the lower and upper approximations of A are calculated with Type-II MGCRIFS as follows, $\begin{matrix} \underline{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) & = & {(x_{1}, 0 . 5, 0 . 4), (x_{2}, 0 . 6, 0 . 1), \\ (x_{3}, 0 . 9, 0 . 1), (x_{4}, 0 . 4, 0 . 3), \\ (x_{5}, 0 . 3, 0 . 5), (x_{6}, 0 . 2, 0 . 7), \\ (x_{7}, 0 . 8, 0 . 2)} . \\ \bar{\sum_{i = 1}^{m} {CK}_{i}^{O}} (A) & = & {(x_{1}, 0 . 5, 0 . 4), (x_{2}, 0 . 6, 0 . 1), \\ (x_{3}, 1, 0), (x_{4}, 0 . 4, 0 . 3), \\ (x_{5}, 0 . 3, 0 . 5), (x_{6}, 0 . 3, 0 . 5), \\ (x_{7}, 0 . 8, 0 . 2)} . \\ \underline{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) & = & {(x_{1}, 0 . 5, 0 . 4), (x_{2}, 0 . 55, 0 . 25), \\ (x_{3}, 0 . 53, 0 . 33), (x_{4}, 0 . 3, 0 . 5), \\ (x_{5}, 0 . 3, 0 . 5), (x_{6}, 0 . 2, 0 . 7), \\ (x_{7}, 0 . 55, 0 . 35)} . \\ \bar{\sum_{i = 1}^{m} {CK}_{i}^{P}} (A) & = & {(x_{1}, 0 . 55, 0 . 25), (x_{2}, 0 . 6, 0 . 1), \\ (x_{3}, 1, 0), (x_{4}, 0 . 55, 0 . 28), \\ (x_{5}, 0 . 55, 0 . 35), (x_{6}, 0 . 54, 0 . 33), \\ (x_{7}, 0 . 9, 0 . 1)} . \end{matrix}$

5 Conclusion

The theories of covering rough sets, intuitionistic fuzzy sets and multi-granulation rough sets are important for dealing with uncertainty and inaccuracy problems. In order to handle these uncertainty and inaccuracy problems more effectively, the combination of the covering rough and intuitionistic fuzzy sets is further researched in this paper. A new covering rough intuitionistic fuzzy set model is proposed. From the view of granularity, we extend the covering rough intuitionistic fuzzy sets in rough sets from single-granulation to multi-granulation. From the view of minimal description, the contribution of this paper is to construct three new types of multi-granulation covering rough fuzzy sets in covering approximation space (U,C). Some properties of these models are proved and demonstrated by some examples. We will investigate the applications of the presented multi-granulation covering rough intuitionistic fuzzy set, and reconstruct multi-granulation covering rough intuitionistic fuzzy set from the view of maximal description, which will be part of the future research directions considered by our group.

Footnotes

Acknowledgments

This work is supported by the national natural science foundation of China under Grant No.612730 18, and foundation and advanced technology research program of Henan Province of China under Grant No.132300410174, and the key scientific and technological project of Education Department of Henan Province of China under Grant No.14A520082, and the key scientific and technological project of Xin Xiang City of China under Grant No.ZG14020.

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