Optimal granulation selection for similarity measure-based multigranulation intuitionistic fuzzy decision-theoretic rough sets

Abstract

Similarity measure is an important uncertainty measurement in intuitionistic fuzzy set (IFS) theory. In this study, a novel similarity measure is presented by the combination of the information carried by hesitancy degree and the endpoint distance of membership and nonmembership, respectively. Moreover, a numerical example is used to verify the reasonable of the proposed similarity measure. After that, the similarity measure is applied to construct the IF decision-theoretic rough set (IF-DTRS) model and multigranulation IF decision-theoretic rough set (MG-IF-DTRS) model. Some properties of IF-DTRS and MG-IF-DTRS are also investigated. Thirdly, based on granular significance, a novel approach of optimal granulation selection is formulated. Finally, a heuristic algorithm is designed and the effectiveness of this algorithm is demonstrated by an illustrative example.

Keywords

1 Introduction

In fuzzy set theory [3], a membership function is used to describe the uncertainty. And now it has revolutionized many research fields. Based on this theory, several extensions of fuzzy set for various practical applications have been proposed. Atanassov [4, 5] proposed the concept of IFS, in which two values, called membership and nonmembership, are used to characterize the uncertainty of the objective world. The study found that the concept described by IFSs is similar as the essence of human induction and cognition. Therefore, the theoretical and application of IFS have been attentively studied by many scholars [51] and have been widely used in many fields, such as pattern recognition [6, 8], medicine diagnosis [12], decision making [1 , 50], and clusteranalysis [11, 13].

In many real world application problems, people often need to compare two or more objects to determine the similarity or proximity between them, which is usually called similarity measure. Many scholars have made great efforts in developing new similarity measures between IFSs and studying their applications. Chen [14, 15] firstly presented a series of similarity measures in 1997. After that, Hong and Kim [16] indicated that the measures were not applicable in some cases and then modified the similarity measures. Based on an axiomatic approach, Li [7] proposed several similarity measures in handlingpattern recognition problems. In [10], Liang and Shi presented some counterintuitive examples to show the weakness of the similarity measures introduced by Li. Mitchell [9] modified the axiomatic definition and introduced a new similarity measure. Using geometrical representation of IFS, Liu [17] proposed some similarity measures. By means of the medians of intervals, Chen [18] presented a similarity measure between IFSs. Depending on Hausdorff distance, Hung [6] defined the distance between IFSs, which was used to generate several similarity measures. Hwang et.al [19] presented a cosine similarity measure for IFSs induced by the Sugeno integral. Considering the Frank t-norms family, Iancu [20] extended some crisp cardinality measures to calculate the similarity of IFSs. According to the uncertainty and hesitancy of IFS simultaneously, Beliakov et al., [21] introduced vector valued similarity measures for IFS. In recent years, some scholars continue to study similarity measurement between intuitionistic fuzzy sets from different perspectives [22 , 32]. However, the existing similarity measures between IFSs may exist some counter-intuitive, it is necessary to generate more reasonable and effective similarity measure method.

Rough set [25] is an important tool for handing the vagueness and uncertainty of data. According to equivalence relation in approximation space, the lower and upper approximation sets were calculated to approximate the target concept. However, equivalence relation is still restrictive. In order to extend the application areas, tolerance relation [26 , 34], dominance relation [28], neighborhood relation [29] and fuzzy relation [31] were introduced into the original rough set theory. By considering the tolerance of classification error, the decision-theoretic rough set (DTRS) model [35, 36], the Bayesian rough set model and the probabilistic rough set model were proposed. Those works make rough set theory more realistic and useful. By the view of granular computing [37, 38], an equivalence relation can be considered to be a granule. Hence, in Pawlak rough set, the target concept has been approximated by a single granule structure. If the relationship between attributes are independent, the intersection of knowledge obtained from all attributes is clearly irrational.

To address the issue above, Qian [39, 40] proposed multigranulation rough sets(MGRSs) in 2010, where multiple equivalent relations(multiple granular structures) have been used to approximate a target set. And now, many researchers have extended the MGRSs to the generalized MGRSs. Lin [42] introduced neighborhood-based MGRSs model. Combination with fuzzy set theory, Xu [41] proposed multigranulation fuzzy rough set model by considering crisp to fuzzy case. Li [44] developed the MGRSs in ordered information system. Feng [43] illuminated variable precision MGRSs model. Tan el at. [45] characterized the approximations operators and attribute reductions by means of belief structure.

Although IFS theory and MG-DTRSs theory have significant advantages for information expression and processing, respectively. Few works have focused on combining the two theories together to construct MG-IF-DTRSs. Huang [46] introduced the inclusion measure-based MG-IF-DTRSs model, in which the approximated sets were IFSs on the universe of discourse. However, this model can not process labeled data.

According to the above mentioned analysis, we investigate the MG-IF-DTRSs by using a novel similarity measure. The main contributions of this paper can be summarized as follows: (1) A novel similarity measure between IFSs is introduced which is more reasonable and effective. (2) We extend MG-DTRSs in multi-source IFDS and put forward new similarity measure-based MG-IF-DTRSs model based on the proposed similarity measure. (3) We focus on a fundamental issue on how to acquire optimal granulation reduction for MG-IF-DTRSs.

This study is constituted as follows. In Section 2, we review some basic notions in order to facilitate the subsequent discussions. And a novel similarity measure is proposed, we also compare the results with existing similarity measures in Section 3. Section 4 presents the definition of the similarity measure-based IF-DTRSs. Section 5, MG-IF-DTRSs are considered in multi-source IF information systems and mathematical properties are also investigated. In Section 6, we propose a heuristic algorithm to compute the optimistic optimal granulation. After that, the processing of the algorithm is illustrated by an example. We conclude this paper in Section 7.

2 Preliminaries

In this section, some basic notions are introduced including IFS, DTRS and MG-DTRSs.

2.1 IFS and IFIS

Definition 1. [4, 5] Let U be a universe of discourse. An IFS A on U is defined by A = {〈μ_A (x) , ν_A (x) 〉 : x ∈ U} , where 0 ≤ μ_A (x) + ν_A (x) ≤1, ∀x ∈ U. The mappings μ_A : U → [0, 1] and ν_A : U → [0, 1] represent the membership and nonmembership degrees of the element x ∈ U to A (namely μ_A (x) and ν_A (x)) respectively. 1 - μ_A (x) - ν_A (x) can be interpreted as the hesitancy degree of x ∈ U to A. All IFS subsets of U is IFS(U).

Definition 2. [4, 5] For two IFSs A = {〈μ_A (x) , ν_A (x) 〉 : x ∈ U}, B = {〈μ_B (x) , ν_B (x) 〉 : x ∈ U}. The operational laws are defined as follows:

A ⊆ B ⇔ ∀ x ∈ U, μ_A (x) ≤ μ_B (x), ν_A (x) ≥ ν_B (x);

A = B ⇔ A ⊆ B and B ⊆ A;

∼A = {〈ν_A (x) , μ_A (x) 〉 : x ∈ U};

A ∩ B = {〈μ_A (x) ∧ μ_B (x) , ν_A (x) ∨ ν_B (x) 〉 : x ∈ U};

A ∪ B = {〈μ_A (x) ∨ μ_B (x) , ν_A (x) ∧ ν_B (x) 〉 : x ∈ U}.

Definition 3. An intuitionistic fuzzy information system (IFIS) is defined as IFIS = (U, AT, {V_a : a ∈ AT} , {f_a : a ∈ AT}), in which U and AT are nonempty, finite object set and attribute set respectively. V_a consists of all IF values of a ∈ AT, f_a is an information function, such that f_a(x)= 〈 μ_a(x),ν_a(x)〉 , and ∀x ∈ U, ∀a ∈ AT, 0 ≤ μ_a (x) + ν_a (x) ≤1.

Moveover, if C and D are conditional and decision attribute sets with C∩ D = ∅ and AT = C ∪ D, then IFIS is denoted as intuitionistic fuzzy decision information system (short for IFDS).

2.2 DTRS and MG-DTRSs

Since considering the risk and enhance the robustness, Yao [47] proposed the DTRS model in 1992.

Definition 4. [47] Let R be an equivalence relation on U, and 0 ≤ β < α ≤ 1. The lower and upper approximations of X ⊆ U are defined by, respectively, $\begin{matrix} {\underline{R}}^{α} (X) = {x \in U : P (X | [x]_{R}) \geq α}; \\ {\bar{R}}^{β} (X) = {x \in U : P (X | [x]_{R}) > β}, \end{matrix}$ where α, β are two probability thresholds, P (X| [x] _R) is the conditional probability of [x] _R with respect to X. We refere the pair $({\underline{R}}^{α} (X), {\bar{R}}^{β} (X))$ as DTRS.

After that, new DTRS models called the MG-DTRSs model were proposed by combining the MGRSs and DTRS.

Definition 5. [48] Let $R = {R_{1}, R_{2}, \dots, R_{m}}$ be m equivalence relations on U and 0 ≤ β < α ≤ 1. The optimistic and pessimistic multigranulation lower and upper approximations of X ⊆ U are defined by, respectively, $\begin{matrix} \underline{R_{α}^{O}} (X) = {x \in U : \lor (P (X | [x]_{R_{i}}) \geq α), i \leq m}; \\ \bar{R_{β}^{O}} (X) = {x \in U : \land (P (X | [x]_{R_{i}}) > β), i \leq m}, \\ \underline{R_{α}^{P}} (X) = {x \in U : \land (P (X | [x]_{R_{i}}) \geq α), i \leq m}; \\ \bar{R_{β}^{P}} (X) = {x \in U : \lor (P (X | [x]_{R_{i}}) > β), i \leq m}, \end{matrix}$ where ∨ and ∧ mean the logical operator “or” and “and”, respectively.

The order pairs $(\underline{R_{α}^{O}} (X), \bar{R_{β}^{O}} (X))$ and $(\underline{R_{α}^{P}} (X), \bar{R_{β}^{P}} (X))$ are denoted as the optimistic and pessimistic MG-DTRSs, respectively.

3 A new similarity measure between IFSs

In this section, we consider similarity measures between IFSs starting from reminding the axiomatic definition of similarity measures.

Definition 6. [7] ∀A, B, C ∈ IFS (U), a similarity measure S on IFS (U) is a real function S : IFS (U) × IFS (U) → [0, 1], which satisfies the following conditions:

0 ≤ S (A, B) ≤1;

S (A, B) =1, if A = B;

S (A, B) = S (B, A);

S (A, C) ≤ S (A, B) and S (A, C) ≤ S (B, C) if A ⊆ B ⊆ C;

Mitchell [9] pointed out that condition (2) should be replaced by: S (A, B) =1, if and only if A = B; Li [49] claimed that the following condition should be rejoined: (5) S (A, B) =0, if and only if ∀x ∈ U, μ_A (x) =1, ν_A (x) =0 and B = ∼ A, or μ_B (x) =1, ν_B (x) =0 and A = ∼ B.

Now, we review some existing similarity measures, which were widely used. A, B ∈ IFS (U) are two IFSs.

Chen’s Similarity Measure [14, 15]:

\begin{matrix} S_{C} (A, B) = 1 - \frac{\sum_{i = 1}^{n} | {Score}_{A} (x_{i}) - {Score}_{B} (x_{i}) |}{2 n}, \end{matrix}

where

\begin{matrix} {Score}_{A} (x_{i}) = μ_{A} (x_{i}) - ν_{A} (x_{i}) . \end{matrix}

Hong’s Similarity Measure [16]:

\begin{matrix} S_{H} (A, B) = 1 \\ - \frac{\sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{B} (x_{i}) - ν_{B} (x_{i}) |)}{2 n} . \end{matrix}

Li and Xu’s Similarity Measure [40]:

$\begin{matrix} S_{LX} (A, B) \\ = 1 - \frac{\sum_{i = 1}^{n} | {Score}_{A} (x_{i}) - {Score}_{B} (x_{i}) |}{4 n} \\ - \frac{\sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | μ_{B} (x_{i}) - ν_{B} (x_{i}) |)}{4 n} . \end{matrix}$

Li Dongfeng’s Similarity Measure [7]:

$\begin{matrix} S_{DF} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} | ψ_{A} (x_{i}) - ψ_{B} (x_{i}) |^{p}}{n}}, \end{matrix}$ where $\begin{matrix} ψ_{A} (x_{i}) = \frac{1 + μ_{A} (x_{i}) - ν_{A} (x_{i})}{2}, \\ ψ_{B} (x_{i}) = \frac{1 + μ_{B} (x_{i}) - ν_{B} (x_{i})}{2}, \end{matrix}$ and p⪡ ∞.

Mitchell’s Similarity Measure [9]:

\begin{matrix} S_{M} (A, B) = \frac{ρ_{μ} (A, B) + ρ_{ν} (A, B)}{2}, \end{matrix}

where

$ρ_{μ} (A, B) = 1 - \sqrt[p]{\sum_{i = 1}^{n} | μ_{A} (x_{i}) - μ_{B} (x_{i}) |^{p} / n}$ ,

$ρ_{ν} (A, B) = 1 - \sqrt[p]{\sum_{i = 1}^{n} | ν_{A} (x_{i}) - ν_{B} (x_{i}) |^{p} / n}$

and p⪡ ∞.

Szmidt’s Similarity Measure [23]:

\begin{matrix} S_{Sz} (A, B) = 1 - \frac{l (A, B)}{l (A, B) + l (A, \sim B)}, \end{matrix}

where

$l (A, B) = \frac{1}{2 n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | + | π_{A} (x_{i}) - π_{B} (x_{i}) |)$ induced by Hamming distance formula.

Song and Wang’s Similarity Measure [24]:

\begin{matrix} S_{SW} (A, B) & = & \frac{1}{2 n} \sum_{i = 1}^{n} (\sqrt{μ_{A} (x_{i}) μ_{B} (x_{i})} \\ + 2 \sqrt{ν_{A} (x_{i}) ν_{B} (x_{i})} \\ + \sqrt{π_{A} (x_{i}) π_{B} (x_{i})} \\ + \sqrt{H_{A} (x_{i}) H_{B} (x_{i})}), \end{matrix}

where

\begin{matrix} H_{A} (x_{i}) = 1 - ν_{A} (x_{i}) and H_{B} (x_{i}) = 1 - ν_{B} (x_{i}) . \end{matrix}

In fact, in order to calculate the similarity of two IFSs, we should comprehensively consider the membership, non-membership and hesitation. There may be three kinds of problems with the similarity calculated using the above formula. First, Comparing the same IFS A with two different IFSs B and C, one might get S_* (A, B) = S_* (A, C). However, it is counter-intuitive. Second, S_* (A, A) =1, ∀A ∈ IFS (U), and we will get S_* (〈0, 0〉, 〈0, 0〉) = S_* (〈0.2, 0.5〉, 〈0.2, 0.5〉) = S_* (〈0.2, 0.8〉, 〈0.2, 0.8〉) =1. However, the hesitance of the three groups of IFVs are not equal, and the degree of similarity should not be the same. Third, let A = 〈0, 0〉 and B = 〈a, b〉, if a + b = 1, then S_* (A, B) =0.5 for many formulas above. This is obviously unreasonable.

In order to resolve some unreasonable results presented above, we modify the condition (2) as (2″) S (A, B) =1 if and only if A = B, and π_A (x_i) = π_B (x_i) =0, ∀ x_i ∈ U. Thus we define a new similarity measure.

Definition 7. Let A, B ∈ IFS (U), a new measure is defined by S_LM (A, B), in which $\begin{matrix} S_{LM} (A, B) & = & 1 - \frac{1}{4 n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | ν_{A} (x_{i}) - ν_{B} (x_{i}) | + | μ_{A} (x_{i}) \\ - ν_{A} (x_{i}) - μ_{B} (x_{i}) + ν_{B} (x_{i}) | \\ + | π_{A} (x_{i}) - π_{B} (x_{i}) | \\ + (π_{A} (x_{i}) + π_{B} (x_{i})) / 2) \end{matrix}$

Theorem 1. For A, B ∈ IFS (U), S_LM (A, B) is a similarity measure on U.

Proof. Let A = {〈μ_A (x) , ν_A (x) 〉 : x ∈ U}, B = {〈μ_B (x) , ν_B (x) 〉 : x ∈ U}.

(1) It is obviously that S_LM (A, B) ≤1.

If 〈a + x, b - y〉 and 〈a, b〉 are two IFVs, thena + x, b - y ∈[0, 1], and 0 ≤ a + x + b - y ≤ 1.

Denote f (x, y) = |a - (a + x) | + |b - (b - y) | + | (a - b) - (a + x - b + y) | + |(1 - a - b) - (1 - a - x - b + y) | + ((1 - a - b) + (1 - a - x - b + y))/2 = |x| + |y| + |x + y| + |x - y| + (y - x)/2 + (1 - a - b).

Let x ∈ [0, 1 - a] and y = 0, $f (x, y) = \frac{5}{2} x + (1 - a - b)$ . If y = 0, x ∈ [0, 1 - a], f is an increasing function on the first argument.

Let x ∈ [- a, 0] and y = 0, $f (x, y) = - \frac{7}{2} x + (1 - a - b)$ . If y = 0, x ∈ [- a, 0], f is a decreasing function of the first argument.

Similarly, if x = 0, y ∈ [0, b], f is an increasing function of the second argument. If x = 0, y ∈ [a - 1, 0], f is a decreasing function of the second argument.

Therefore, given x, y ≥ 0, the maximum value of f is $f_{\max} (a, 1 - b) = \frac{1}{2} a + \frac{7}{2} (1 - b) + 1 - a - b = \frac{9}{2} (1 - b) - \frac{1}{2} a \leq 4 (1 - b) \leq 4$ ; given x, y ≤ 0, the maximum value of f is $f_{\max} (- a, b - 1) = \frac{3}{2} a + \frac{5}{2} b + 1 - a - b = \frac{7}{2} (1 - b) + \frac{1}{2} a \leq 4 (1 - b) \leq 4$ .

Thus, S_LM (A, B) ≥0.

(2″) S_LM (A, B) =1 ⇔ ∀x_i ∈ U, |μ_A (x_i) - μ_B (x_i) | + |ν_A (x_i) - ν_B (x_i) | + |μ_A (x_i) - ν_B (x_i) - μ_B (x_i) + ν_A (x_i) | + |π_A (x_i) - π_B (x_i) | + (π_A (x_i) - π_B (x_i))/2 = 0 ⇔ μ_A (x_i) - μ_B (x_i) =0, ν_A (x_i) - ν_B (x_i) = 0, μ_A (x_i) - ν_B (x_i) - μ_B (x_i) + ν_A (x_i) =0, π_A (x_i) - π_B (x_i) =0, and (π_A (x_i) - π_B (x_i))/2 = 0 ⇔ A = B, and ∀x_i ∈ U, π_A (x_i) = π_B (x_i) =0 .

(3) It is clear that S_LM (A, B) = S_LM (B, A).

(4) According to (1), set 0 ≤ u₁ ≤ u₂ ≤ 1 - a, 0 ≤ v₂ ≤ v₁ ≤ b, let A (x_i) = 〈a, b〉, B (x_i) = 〈a + u₁, b - v₁〉, C (x_i) = 〈a + u₂, b - v₂〉, then A (x_i) ⊆ B (x_i) ⊆ C (x_i) and f (u₂, v₂) ≥ f (u₂, v₁) ≥ f (u₁, v₁); that is S_LM (A (x_i) , B (x_i)) ≥ S_LM (A (x_i) , C (x_i)), which implies that S_LM (A, B) ≥ S_LM (A, C).

we can also have S_LM (B, C) ≥ S_LM (A, C). □

Proposition 1. Let A, B ∈ IFS (U) be two IFSs, then S_LM (A, B) = S_LM (∼ A, ∼ B).

Thus, S_LM (A, B) = S_LM (∼ A, ∼ B). □

Lemma 1. Let A, B ∈ IFS (U) be two IFSs. For all x ∈ U, if μ_A (x) + ν_A (x) =1, μ_B (x) + ν_B (x) =1 and π_A (x) ≥ π_B (x), then S_LM (A, A) ≤ S_LM (B, B).

Proof. It is straightforward by Definition 7. □

In fact, hesitancy degree reflects the uncertainty of information. The greater the hesitation is, the more uncertainty there are. Thus, S (A, B) =1 if and only if A = B, and π_A (x_i) = π_B (x_i) =0, ∀ x_i ∈ U.

The reasonability of the proposed method is illustrated by a numerical example.

Example 1. The result of comprehensive comparison is shown in Table 1.

We can see that S_C (A, B) = S_DF (A, B) =1 for two different IFSs in cases 3,4,8,9, which means that the axiom (2) is not satisfied.

In case 2, S_SW (A, B) =0, which means that the axiom (5) is not satisfied.

In case 7, S_SZ (A, B) = non is unreasonable.

In case 8,9, None of these methods can distinguish between the two pairs of IFSs except S_LM.

As for S_C, S_H, S_LX, S_DF, S_Sz, S_M, S_SW, one may get identical results from different pairs of A, B, which cannot satisfy the application of pattern recognition. According to the voting model, the existing voting results will affect the abusers. If π_A (x_i) = π_B (x_i) ≠0, due to the uncertainty of description, even if A = B, their similarity should not equal to 1.

∥Overall, the proposed similarity measure is more reasonable and has non counterintuitive cases.

Table 1
The result of comprehensive comparison(counterintuitive cases are in bold type)

1 2 3 4 5 6 7 8 9 10

A 〈 0,1〉〈 0,0〉〈 0,0〉〈 0.3,0.3〉〈 0.3,0.7〉〈 0.3,0.5〉〈 0.2,0.2〉〈 0.2,0.3〉〈 0.2,0.3〉〈 0.2,0.6〉

B 〈 1,0 〉〈 0,1〉〈 0.5,0.5〉〈 0.4,0.4〉〈 0.3,0.7〉〈 0.3,0.5〉〈 0.2,0.2〉〈 0.1,0.2〉〈 0.3,0.4〉〈 0.1,0.6〉

S_C 0 0.50 1 1 1 1 1 1 1 0.90

S_H 0 0.50 0.50 0.90 1 1 1 0.90 0.90 0.90

S_LX 0 0.50 0.75 0.95 1 1 1 0.95 0.95 0.90

S_DF(p=1) 0 0.50 1 1 1 1 1 1 1 0.90

S_Sz 0 0.50 0.50 0.50 1 1 non 0.5 0.5 0.81

S_M 0 0.50 0.50 0.90 1 1 1 0.90 0.90 0.90

S_SW 0 0 0.35 0.99 1 1 1 0.99 0.99 0.99

S_LM 0 0.13 0.38 0.83 1 0.95 0.85 0.75 0.80 0.85

	1	2	3	4	5	6	7	8	9	10
A	〈 0,1〉	〈 0,0〉	〈 0,0〉	〈 0.3,0.3〉	〈 0.3,0.7〉	〈 0.3,0.5〉	〈 0.2,0.2〉	〈 0.2,0.3〉	〈 0.2,0.3〉	〈 0.2,0.6〉
B	〈 1,0 〉	〈 0,1〉	〈 0.5,0.5〉	〈 0.4,0.4〉	〈 0.3,0.7〉	〈 0.3,0.5〉	〈 0.2,0.2〉	〈 0.1,0.2〉	〈 0.3,0.4〉	〈 0.1,0.6〉
S_C	0	0.50	1	1	1	1	1	1	1	0.90
S_H	0	0.50	0.50	0.90	1	1	1	0.90	0.90	0.90
S_LX	0	0.50	0.75	0.95	1	1	1	0.95	0.95	0.90
S_DF(p=1)	0	0.50	1	1	1	1	1	1	1	0.90
S_Sz	0	0.50	0.50	0.50	1	1	non	0.5	0.5	0.81
S_M	0	0.50	0.50	0.90	1	1	1	0.90	0.90	0.90
S_SW	0	0	0.35	0.99	1	1	1	0.99	0.99	0.99
S_LM	0	0.13	0.38	0.83	1	0.95	0.85	0.75	0.80	0.85

4 Similarity measure-based IF-DTRS

In this section, the similarity measure is applied to define approximation operators (lower approximation,upper approximation).

Definition 8. Let IFIS = (U, AT, {V_a : a ∈ AT} , {f_a : a ∈ AT}), the mapping between two objects x, y ∈ U w.r.t. attribute set A ⊆ AT is defined as $\begin{matrix} S_{LM (A)} (x, y) & = & 1 - \frac{1}{4 | A |} \sum_{a \in A} (| μ_{a} (x) - μ_{a} (y) | \\ + | ν_{a} (x) - ν_{a} (y) | + | μ_{a} (x) \\ - ν_{a} (x) - μ_{a} (y) + ν_{a} (y) | \\ + | π_{a} (x) - π_{a} (y) | \\ + (π_{a} (x) + π_{a} (y)) / 2) . \end{matrix}$

In order to improve readability, we write S_A (x, y) in stead of S_LM(A) (x, y).

Definition 9. Let 0 ≤ λ ≤ 1, A ⊆ AT. A similarity relation $S_{A}^{λ}$ on U is defined as: $\begin{matrix} S_{A}^{λ} = {(x, y) \in U \times U : S_{A} (x, y) \geq λ} . \end{matrix}$

It is obviously that $S_{A}^{λ}$ is reflexive and symmetric. $[x]_{S_{A}^{λ}} = {y \in U : (x, y) \in S_{A}^{λ}}$ is denoted as the similarity class of x w.r.t. A in IFIS. The similarity relation above forms a covering of U.

Proposition 2. ∀x ∈ U, if 0 ≤ λ₁ ≤ λ₂ ≤ 1, then $[x]_{S_{A}^{λ_{2}}} \subseteq [x]_{S_{A}^{λ_{1}}}$ .

Proof. It is straightforward.□

Definition 10. Let 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. For any A ⊆ AT, the similarity measure-based α-lower and β-upper approximations of X ⊆ U are denoted by $\underline{A_{λ}^{α}} (X)$ and $\bar{A_{λ}^{β}} (X)$ , respectively, in which $\begin{matrix} \underline{A_{λ}^{α}} (X) = {x \in U : P (X | [x]_{S_{A}^{λ}}) \geq α}; \\ \bar{A_{λ}^{β}} (X) = {x \in U : P (X | [x]_{S_{A}^{λ}}) > β} . \end{matrix}$

We call the order pair $({\underline{A}}_{λ}^{α} (X), {\bar{A}}_{λ}^{β} (X))$ as similarity measure-based IF-DTRS of X w.r.t. attribute set A in IFIS. If ${\underline{A}}_{λ}^{α} (X) = {\bar{A}}_{λ}^{α} (X)$ , then X is definable according to A in IFIS; otherwise, X is undefinable inIFIS.

Let IFDS = (U, AT = C ∪ D, {V_a : a ∈ C} , {f_a : a ∈ C}), R_D is an indiscernibility relation on U, and U/R_D = {D₁, D₂, ⋯ , D_r}. Therefore, the similarity measure-based α-IF positive is defined by ${POS}_{A_{λ}}^{α} (U / R_{D}) = ⋃_{i = 1}^{r} \underline{A_{λ}^{α}} (D_{i})$ .

Now the dependent degree in similarity measure-based IF-DTRS can be defined as follows, $\begin{matrix} Γ_{A_{λ}}^{α} (U / R_{D}) = \frac{| {POS}_{A_{λ}}^{α} (U / R_{D}) |}{| U |} . \end{matrix}$

Proposition 3. Let 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. For any A ⊆ AT, the following properties hold:

$\underline{A_{λ}^{α}} (\emptyset) = \emptyset$ and $\bar{A_{λ}^{β}} (\emptyset) = \emptyset$ ;

$\underline{A_{λ}^{α}} (U) = U$ and $\bar{A_{λ}^{β}} (U) = U$ ;

If X ⊆ U, then $\underline{A_{λ}^{α}} (X) \subseteq \bar{A_{λ}^{β}} (X)$ ;

if X₁ ⊆ X₂ ⊆ U, then $\underline{A_{λ}^{α}} (X_{1}) \subseteq \underline{A_{λ}^{α}} (X_{2})$ , $\bar{A_{λ}^{β}} (X_{1}) \subseteq \bar{A_{λ}^{β}} (X_{2})$ ;

if 0 ≤ β₁ ≤ β₂ < α₁ ≤ α₂ ≤ 1, then $\underline{A_{λ}^{α_{2}}} (X) \subseteq \underline{A_{λ}^{α_{1}}} (X)$ , $\bar{A_{λ}^{β_{2}}} (X) \subseteq \bar{A_{λ}^{β_{1}}} (X)$ .

Proof. They are straightforward. □

From (4) of Proposition 3, if X, Y ⊆ U, then it is clear that: $\begin{matrix} \underline{A_{λ}^{α}} (X \cap Y) \subseteq \underline{A_{λ}^{α}} (X) \cap \underline{A_{λ}^{α}} (Y), \\ \underline{A_{λ}^{α}} (X \cup Y) \supseteq \underline{A_{λ}^{α}} (X) \cup \underline{A_{λ}^{α}} (Y); \\ \bar{A_{λ}^{β}} (X \cap Y) \subseteq \bar{A_{λ}^{β}} (X) \cap \bar{A_{λ}^{β}} (Y), \\ \bar{A_{λ}^{β}} (X \cup Y) \supseteq \bar{A_{λ}^{β}} (X) \cup \bar{A_{λ}^{β}} (Y) . \end{matrix}$ Example 2. Let IFIS = (U, AT, {V_a : a ∈ AT} , {f_a : a ∈ AT}) as shown in Table 2, where U = {x₁, x₂, ⋯ , x₁₄}, A = {a₁, a₂, a₃, a₄}.

Table 2
Intuitionistic fuzzy information system

U a ₁ a ₂ a ₃ a ₄ a ₅ a ₆

x ₁ 〈0.40, 0.50〉〈0.30, 0.70〉〈0.80, 0.20〉〈0.40, 0.50〉〈0.70, 0.10〉〈0.10, 0.70〉

x ₂ 〈0.30, 0.50〉〈0.40, 0.50〉〈0.60, 0.10〉〈0.40, 0.50〉〈0.70, 0.30〉〈0.30, 0.70〉

x ₃ 〈0.30, 0.50〉〈0.10, 0.80〉〈0.80, 0.10〉〈0.40, 0.50〉〈0.80, 0.20〉〈0.20, 0.80〉

x ₄ 〈0.10, 0.80〉〈0.10, 0.80〉〈0.40, 0.50〉〈0.10, 0.80〉〈0.80, 0.10〉〈0.10, 0.80〉

x ₅ 〈0.70, 0.30〉〈0.40, 0.50〉〈0.90, 0.10〉〈0.40, 0.60〉〈0.90, 0.00〉〈0.00, 0.90〉

x ₆ 〈0.30, 0.60〉〈0.40, 0.60〉〈0.70, 0.20〉〈0.50, 0.50〉〈0.40, 0.50〉〈0.50, 0.40〉

x ₇ 〈0.50, 0.40〉〈0.50, 0.00〉〈0.50, 0.40〉〈0.50, 0.40〉〈0.50, 0.40〉〈0.40, 0.50〉

x ₈ 〈0.70, 0.00〉〈0.70, 0.10〉〈0.60, 0.10〉〈0.60, 0.20〉〈0.60, 0.10〉〈0.10, 0.60〉

x ₉ 〈0.60, 0.40〉〈0.40, 0.40〉〈0.60, 0.10〉〈0.40, 0.40〉〈0.40, 0.50〉〈0.50, 0.40〉

x ₁₀ 〈0.30, 0.60〉〈0.30, 0.60〉〈0.30, 0.70〉〈0.30, 0.60〉〈0.30, 0.60〉〈0.60, 0.30〉

x ₁₁ 〈0.20, 0.00〉〈0.70, 0.00〉〈0.60, 0.10〉〈0.20, 0.20〉〈0.20, 0.70〉〈0.70, 0.20〉

x ₁₂ 〈0.50, 0.30〉〈0.50, 0.00〉〈0.70, 0.20〉〈0.20, 0.00〉〈0.60, 0.30〉〈0.30, 0.60〉

x ₁₃ 〈0.50, 0.40〉〈0.60, 0.40〉〈0.50, 0.00〉〈0.40, 0.10〉〈0.50, 0.50〉〈0.40, 0.50〉

x ₁₄ 〈0.70, 0.20〉〈0.50, 0.50〉〈0.60, 0.00〉〈0.60, 0.00〉〈0.60, 0.10〉〈0.10, 0.60〉

U	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
x ₁	〈0.40, 0.50〉	〈0.30, 0.70〉	〈0.80, 0.20〉	〈0.40, 0.50〉	〈0.70, 0.10〉	〈0.10, 0.70〉
x ₂	〈0.30, 0.50〉	〈0.40, 0.50〉	〈0.60, 0.10〉	〈0.40, 0.50〉	〈0.70, 0.30〉	〈0.30, 0.70〉
x ₃	〈0.30, 0.50〉	〈0.10, 0.80〉	〈0.80, 0.10〉	〈0.40, 0.50〉	〈0.80, 0.20〉	〈0.20, 0.80〉
x ₄	〈0.10, 0.80〉	〈0.10, 0.80〉	〈0.40, 0.50〉	〈0.10, 0.80〉	〈0.80, 0.10〉	〈0.10, 0.80〉
x ₅	〈0.70, 0.30〉	〈0.40, 0.50〉	〈0.90, 0.10〉	〈0.40, 0.60〉	〈0.90, 0.00〉	〈0.00, 0.90〉
x ₆	〈0.30, 0.60〉	〈0.40, 0.60〉	〈0.70, 0.20〉	〈0.50, 0.50〉	〈0.40, 0.50〉	〈0.50, 0.40〉
x ₇	〈0.50, 0.40〉	〈0.50, 0.00〉	〈0.50, 0.40〉	〈0.50, 0.40〉	〈0.50, 0.40〉	〈0.40, 0.50〉
x ₈	〈0.70, 0.00〉	〈0.70, 0.10〉	〈0.60, 0.10〉	〈0.60, 0.20〉	〈0.60, 0.10〉	〈0.10, 0.60〉
x ₉	〈0.60, 0.40〉	〈0.40, 0.40〉	〈0.60, 0.10〉	〈0.40, 0.40〉	〈0.40, 0.50〉	〈0.50, 0.40〉
x ₁₀	〈0.30, 0.60〉	〈0.30, 0.60〉	〈0.30, 0.70〉	〈0.30, 0.60〉	〈0.30, 0.60〉	〈0.60, 0.30〉
x ₁₁	〈0.20, 0.00〉	〈0.70, 0.00〉	〈0.60, 0.10〉	〈0.20, 0.20〉	〈0.20, 0.70〉	〈0.70, 0.20〉
x ₁₂	〈0.50, 0.30〉	〈0.50, 0.00〉	〈0.70, 0.20〉	〈0.20, 0.00〉	〈0.60, 0.30〉	〈0.30, 0.60〉
x ₁₃	〈0.50, 0.40〉	〈0.60, 0.40〉	〈0.50, 0.00〉	〈0.40, 0.10〉	〈0.50, 0.50〉	〈0.40, 0.50〉
x ₁₄	〈0.70, 0.20〉	〈0.50, 0.50〉	〈0.60, 0.00〉	〈0.60, 0.00〉	〈0.60, 0.10〉	〈0.10, 0.60〉

Setting λ = 0.85, the similarity measure classes based on similarity measure defined by Definition 9 for attribute set A are $\begin{matrix} [x_{1}]_{S_{A}^{0.85}} = {x_{1}, x_{2}, x_{3}, x_{6}}, \\ [x_{2}]_{S_{A}^{0.85}} = {x_{1}, x_{2}, x_{3}, x_{6}, x_{9}}, \\ [x_{3}]_{S_{A}^{0.85}} = {x_{1}, x_{2}, x_{3}}, [x_{4}]_{S_{A}^{0.85}} = {x_{4}}, \\ [x_{5}]_{S_{A}^{0.85}} = {x_{5}}, [x_{6}]_{S_{A}^{0.85}} = {x_{1}, x_{2}, x_{6}}, \\ [x_{7}]_{S_{A}^{0.85}} = {x_{7}}, [x_{8}]_{S_{A}^{0.85}} = {x_{8}}, \\ [x_{9}]_{S_{A}^{0.85}} = {x_{2}, x_{9}}, [x_{10}]_{S_{A}^{0.85}} = {x_{10}}, \\ [x_{11}]_{S_{A}^{0.85}} = {x_{11}}, [x_{12}]_{S_{A}^{0.85}} = {x_{12}}, \\ [x_{13}]_{S_{A}^{0.85}} = {x_{13}}, [x_{14}]_{S_{A}^{0.85}} = {x_{14}} . \end{matrix}$

Let X = {x₂, x₃, x₄, x₆, x₇, x₉, x₁₂, x₁₃} and α = 0.80, β = 0.50. On the basis of A, the similarity measure-based α-lower and β-upper approximations of X can be obtained by $\begin{matrix} \underline{A_{0.85}^{0.8}} (X) = {x_{2}, x_{4}, x_{7}, x_{9}, x_{12}, x_{13}}, \\ \bar{A_{0.85}^{0.5}} (X) = {x_{1}, x_{2}, x_{3}, x_{4}, x_{7}, x_{9}, x_{12}, x_{13}} . \end{matrix}$

Therefore, the positive region and the dependency degree of X w.r.t. A are calculated as $\begin{matrix} {POS}_{A_{0.85}}^{0.8} (X) = {x_{2}, x_{4}, x_{7}, x_{9}, x_{12}, x_{13}}, \\ Γ_{A_{0.85}}^{0.8} (X) = \frac{6}{8} = \frac{3}{4} . \end{matrix}$

5 Similarity measure-based MG-IF-DTRSs

In practical problems, information data is often obtained from different sources with the same domain set. Inspired by the idea of multi-granulation, multiple granular structures induced by IF similarity measure relations will be used to approximate any target concept in this section.

Let MIFIS = {IFIS_i : IFIS_i = (U, A_i, {V_a : a ∈ A_i} , {f_a : a ∈ AT_i} , 1 ≤ i ≤ m)} be a multi-source IFIS, briefly written as MIFIS = (U, A), where A = {A₁, A₂, ⋯ , A_m}. In particular, when a decision attribute is added to MIFIS, we refer it as the multi-source IFDS, denoted as MIFDS = {IFDS_i : IFDS_i = (U, A_i, {V_a : a ∈ A_i} , {f_a : a ∈ AT_i} , 1 ≤ i ≤ m, D)}, briefly written as MIFDS = (U, A, D).

Definition 11. Let MIFIS = (U, A), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1 and X ⊆ U, the optimistic similarity measure-based α-lower and β-upper approximations of X w.r.t. A in MIFIS, are defined asfollows $\begin{matrix} \underline{A_{λ}^{O, α}} (X) = {x \in U : \lor (P (X | [x]_{S_{A_{i}}^{λ}}) \geq α), i \leq m}; \\ \bar{A_{λ}^{O, β}} (X) = {x \in U : \land (P (X | [x]_{S_{A_{i}}^{λ}}) > β), i \leq m}, \end{matrix}$ where ∨ and ∧ mean the logical operator “or” and “and”, respectively.

If $\underline{A_{λ}^{O, α}} (X) = \bar{A_{λ}^{O, β}} (X)$ , then X is definable according to MIFIS; otherwise, X is undefinable in MIFIS. The order pair $(\underline{A_{λ}^{O, α}} (X), \bar{A_{λ}^{O, β}} (X))$ is referred to similarity measure-based optimistic MG-IF-DTRS of X w.r.t. A in MIFIS.

The positive, boundary and negative regions of X w.r.t. A in optimistic similarity measure-basedMG-IF-DTRS are defined by $\begin{matrix} {POS}_{A_{λ}^{(α, β)}}^{O} (X) = \underline{A_{λ}^{O, α}} (X); \\ {BN}_{A_{λ}^{(α, β)}}^{O} (X) = \bar{A_{λ}^{O, β}} (X)) - \underline{A_{λ}^{O, α}} (X); \\ {NEG}_{A_{λ}^{(α, β)}}^{O} (X) = U - \bar{A_{λ}^{O, β}} (X)) . \end{matrix}$

Objects in the three regions are changed corresponding to the value of α and β in optimistic similarity measure-based MG-IF-DTRS. And the degree of dependence in optimistic similarity measure-based MG-IF-DTRS is defined by $\begin{matrix} Γ_{A_{λ}^{(α, β)}} (X) = \frac{| {POS}_{A_{λ}^{(α, β)}}^{O} (X) |}{| U |} . \end{matrix}$

Proposition 4. Let MIFIS = (U, A), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1 and X ⊆ U, then:

$\underline{A_{λ}^{O, α}} (U) = \bar{A_{λ}^{O, β}} (U) = U$ , $\underline{A_{λ}^{O, α}} (\emptyset) = \bar{A_{λ}^{O, β}} (\emptyset) = \emptyset$ ;

$\underline{A_{λ}^{O, α}} (X) \subseteq \bar{A_{λ}^{O, β}} (X)$ ;

$\underline{A_{λ}^{O, α}} (X) = ⋃_{i = 1}^{m} {\underline{A}}_{i, λ}^{α} (X)$ , $\bar{A_{λ}^{O, β}} (X) = ⋃_{i = 1}^{m} {\bar{A}}_{i, λ}^{β} (X)$ ;

If X ⊆ Y ⊆ U, then $\underline{A_{λ}^{O, α}} (X) \subseteq \underline{A_{λ}^{O, α}} (Y)$ and $\bar{A_{λ}^{O, β}} (X) \subseteq \bar{A_{λ}^{O, β}} (Y)$ ;

if 0 ≤ β₁ ≤ β₂ < α₁ ≤ α₂ ≤ 1, then $\underline{A_{λ}^{O, α}} (X) \subseteq \underline{A_{λ}^{O, α}} (X)$ , and $\bar{A_{λ}^{O, β}} (X) \subseteq \bar{A_{λ}^{O, β}} (X)$ .

Proof. They are straightforward. □

Proposition 5. Let MIFIS = (U, A), 0 ≤ λ ≤ 1, and 0 ≤ β < α ≤ 1. ∀X ⊆ U, if B ⊆ A, then:

$\underline{B_{λ}^{O, α}} (X) \subseteq \underline{A_{λ}^{O, α}} (X)$ ;

$\bar{B_{λ}^{O, β}} (X) \subseteq \bar{A_{λ}^{O, β}} (X)$ .

Proof. (1) $\forall x \in \underline{B_{λ}^{O, α}} (X) \Leftrightarrow \exists A_{j} \in B$ such that $P (X | [x]_{S_{A_{j}}^{λ}}) \geq α$ . Since B ⊆ A, from Definition 11, we have $x \in \underline{A_{λ}^{O, α}} (X)$ .

(2) It can be obtained similarly. □

Definition 12. Let MIFIS = (U, A), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1 and X ⊆ U, the pessimistic similarity measure-based α-lower and β-upper approximations of X w.r.t. A in MIFIS are defined as follows $\begin{matrix} \underline{A_{λ}^{P, α}} (X) = {x \in U : \land (P (X | [x]_{S_{A_{i}}^{λ}}) \geq α), i \leq m}; \\ \bar{A_{λ}^{P, β}} (X) = {x \in U : \lor (P (X | [x]_{S_{A_{i}}^{λ}}) > β), i \leq m}, \end{matrix}$ where ∨ and ∧ mean the logical operator “or” and “and”, respectively.

If $\underline{A_{λ}^{P, α}} (X) = \bar{A_{λ}^{P, β}} (X)$ , then X is definable according to MIFIS; otherwise, X is undefinable in MIFIS. The order pair $(\underline{A_{λ}^{P, α}} (X), \bar{A_{λ}^{P, β}} (X))$ is denoted as pessimistic similarity measure-based MG-IF-DTRS of X w.r.t. A in MIFIS.

The positive, boundary and negative regions of X w.r.t. A in pessimistic similarity measure-based MG-IF-DTRS are defined, respectively, by $\begin{matrix} {POS}_{A_{λ}^{(α, β)}}^{P} (X) = \underline{A_{λ}^{P, α}} (X); \\ {BN}_{A_{λ}^{(α, β)}}^{P} (X) = \bar{A_{λ}^{P, β}} (X) - \underline{A_{λ}^{P, α}} (X); \\ {NEG}_{A_{λ}^{(α, β)}}^{P} (X) = U - \bar{A_{λ}^{P, β}} (X)) . \end{matrix}$

Objects in the three regions are changed corresponding to the value of α and β in similarity measure-based pessimistic MG-IF-DTRS. And the dependent degree in similarity measure-based pessimistic MG-IF-DTRS is defined by $\begin{matrix} Γ_{A_{λ}}^{α} (X) = \frac{| {POS}_{A_{λ}^{(α, β)}}^{P} (X) |}{| U |} . \end{matrix}$

Proposition 6. Let MIFIS = (U, A), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1 and X ⊆ U, then

$\underline{A_{λ}^{P, α}} (U) = \bar{A_{λ}^{P, β}} (U) = U$ , $\underline{A_{λ}^{P, α}} (\emptyset) = \bar{A_{λ}^{P, β}} (\emptyset) = \emptyset$ ;

$\underline{A_{λ}^{P, α}} (X) \subseteq \bar{A_{λ}^{P, β}} (X)$ ;

$\underline{A_{λ}^{P, α}} (X) = ⋂_{i = 1}^{m} \underline{A_{λ}^{α}} (X)$ , $\bar{A_{i, λ}^{P, β}} (X) = ⋂_{i = 1}^{m} \bar{A_{λ}^{β}} (X)$ ;

If X ⊆ Y ⊆ U, then $\underline{A_{λ}^{P, α}} (X) \subseteq \underline{A_{λ}^{P, α}} (Y)$ and $\bar{A_{λ}^{P, β}} (X) \subseteq \bar{A_{λ}^{P, β}} (Y)$ ;

if 0 ≤ β₁ ≤ β₂ < α₁ ≤ α₂ ≤ 1, then $\underline{A_{λ}^{P, α_{2}}} (X) \subseteq \underline{A_{λ}^{P, α_{1}}} (X)$ and $\bar{A_{λ}^{P, β_{2}}} (X) \subseteq \bar{A_{λ}^{P, β_{1}}} (X)$ .

Proof. They are straightforward. □

Proposition 7. Let MIFIS = (U, A), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1 and X ⊆ U. If B ⊆ A, then:

$\underline{A_{λ}^{P, α}} (X) \subseteq \underline{B_{λ}^{P, α}} (X)$ ;

$\bar{A_{λ}^{P, β}} (X) \subseteq \bar{B_{λ}^{P, β}} (X)$ .

Proof. They are obtained similarly as Proposition 5.

□

Example 3. In Example 2, let A₁ = {a₁, a₂}, A₂ = {a₃, a₄}, A₃ = {a₅}, A₄ = {a₆}, the IFIS is changed into the MIFIS. Setting λ = 0.85, the similarity measure classes based on similarity measure defined by Definition 9 for attribute set A_i are as in Table 3.

Table 3
The similarity measure based similarity measure classes

x $[x]_{S_{A_{1}}^{λ}}$ $[x]_{S_{A_{2}}^{λ}}$ $[x]_{S_{A_{3}}^{λ}}$ $[x]_{S_{A_{4}}^{λ}}$

x ₁ {x₁, x₂, x₃, x₆, x₁₀} {x₁, x₂, x₃, x₅, x₆} {x₁, x₃, x₄, x₈, x₁₄} {x₁, x₃, x₄, x₈, x₁₄}

x ₂ {x₁, x₂, x₆, x₁₀} {x₁, x₂, x₃, x₆, x₉} {x₂, x₃, x₁₂} {x₂, x₃, x₁₂}

x ₃ {x₁, x₃} {x₁, x₂, x₃, x₅, x₆} {x₁, x₂, x₃, x₄} {x₁, x₂, x₃, x₄}

x ₄ {x₄} {x₄} {x₁, x₃, x₄, x₅} {x₁, x₃, x₄, x₅}

x ₅ {x₅, x₉, x₁₄} {x₁, x₃, x₅, x₆} {x₄, x₅} {x₄, x₅}

x ₆ {x₁, x₂, x₆, x₁₀} {x₁, x₂, x₃, x₅, x₆, x₉} {x₆, x₇, x₉, x₁₀, x₁₃} {x₆, x₇, x₉, x₁₀, x₁₃}

x ₇ {x₇, x₁₂} {x₇} {x₆, x₉, x₁₃} {x₆, x₉, x₁₃}

x ₈ {x₈} {x₈} {x₁, x₈, x₁₄} {x₁, x₈, x₁₄}

x ₉ {x₅, x₉, x₁₃} {x₂, x₆, x₉} {x₆, x₇, x₉, x₁₀, x₁₃} {x₆, x₇, x₉, x₁₀, x₁₃}

x ₁₀ {x₁, x₂, x₆, x₁₀} {x₁₀} {x₆, x₉, x₁₀, x₁₁} {x₆, x₉, x₁₀, x₁₁}

x ₁₁ {x₁₁} {x₁₁} {x₁₀, x₁₁} {x₁₀, x₁₁}

x ₁₂ {x₇, x₁₂} {x₁₂} {x₂, x₇, x₁₂, x₁₃} {x₂, x₇, x₁₂, x₁₃}

x ₁₃ {x₉, x₁₃} {x₁₃} {x₆, x₇, x₉, x₁₂, x₁₃} {x₆, x₇, x₉, x₁₂, x₁₃}

x ₁₄ {x₅, x₁₄} {x₁₄} {x₁, x₄, x₈, x₁₄} {x₁, x₄, x₈, x₁₄}

x	$[x]_{S_{A_{1}}^{λ}}$	$[x]_{S_{A_{2}}^{λ}}$	$[x]_{S_{A_{3}}^{λ}}$	$[x]_{S_{A_{4}}^{λ}}$
x ₁	{x₁, x₂, x₃, x₆, x₁₀}	{x₁, x₂, x₃, x₅, x₆}	{x₁, x₃, x₄, x₈, x₁₄}	{x₁, x₃, x₄, x₈, x₁₄}
x ₂	{x₁, x₂, x₆, x₁₀}	{x₁, x₂, x₃, x₆, x₉}	{x₂, x₃, x₁₂}	{x₂, x₃, x₁₂}
x ₃	{x₁, x₃}	{x₁, x₂, x₃, x₅, x₆}	{x₁, x₂, x₃, x₄}	{x₁, x₂, x₃, x₄}
x ₄	{x₄}	{x₄}	{x₁, x₃, x₄, x₅}	{x₁, x₃, x₄, x₅}
x ₅	{x₅, x₉, x₁₄}	{x₁, x₃, x₅, x₆}	{x₄, x₅}	{x₄, x₅}
x ₆	{x₁, x₂, x₆, x₁₀}	{x₁, x₂, x₃, x₅, x₆, x₉}	{x₆, x₇, x₉, x₁₀, x₁₃}	{x₆, x₇, x₉, x₁₀, x₁₃}
x ₇	{x₇, x₁₂}	{x₇}	{x₆, x₉, x₁₃}	{x₆, x₉, x₁₃}
x ₈	{x₈}	{x₈}	{x₁, x₈, x₁₄}	{x₁, x₈, x₁₄}
x ₉	{x₅, x₉, x₁₃}	{x₂, x₆, x₉}	{x₆, x₇, x₉, x₁₀, x₁₃}	{x₆, x₇, x₉, x₁₀, x₁₃}
x ₁₀	{x₁, x₂, x₆, x₁₀}	{x₁₀}	{x₆, x₉, x₁₀, x₁₁}	{x₆, x₉, x₁₀, x₁₁}
x ₁₁	{x₁₁}	{x₁₁}	{x₁₀, x₁₁}	{x₁₀, x₁₁}
x ₁₂	{x₇, x₁₂}	{x₁₂}	{x₂, x₇, x₁₂, x₁₃}	{x₂, x₇, x₁₂, x₁₃}
x ₁₃	{x₉, x₁₃}	{x₁₃}	{x₆, x₇, x₉, x₁₂, x₁₃}	{x₆, x₇, x₉, x₁₂, x₁₃}
x ₁₄	{x₅, x₁₄}	{x₁₄}	{x₁, x₄, x₈, x₁₄}	{x₁, x₄, x₈, x₁₄}

Let X = {x₂, x₃, x₄, x₆, x₇, x₉, x₁₂, x₁₃} and α = 0.80, β = 0.50. On the basis of A, the similarity measure-based α-lower and β-upper approximations of X for both optimistic and pessimistic MG-IF-DTRSs can be calculated as $\begin{matrix} \underline{A_{0.85}^{O, 0.8}} (X) & = & {x_{2}, x_{4}, x_{7}, x_{9}, x_{12}, x_{13}}, \\ \bar{A_{0.85}^{O, 0.5}} (X) & = & {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, \\ x_{7}, x_{9}, x_{10}, x_{12}, x_{13}} . \\ \underline{A_{0.85}^{P, 0.8}} (X) & = & {x_{7}, x_{12}, x_{13}}, \\ \bar{A_{0.85}^{P, 0.5}} (X) & = & {x_{2}, x_{3}, x_{4}, x_{7}, x_{12}, x_{13}} . \end{matrix}$

The positive, boundary and negative regions of X w.r.t A in MIFIS are $\begin{matrix} {POS}_{A_{0.85}^{(0.8, 0.5)}}^{O} (X) & = & {x_{2}, x_{4}, x_{7}, x_{9}, x_{12}, x_{13}}; \\ {BN}_{A_{0.85}^{(0.8, 0.5)}}^{O} (X) & = & {x_{1}, x_{3}, x_{5}, x_{6}, x_{10}}; \\ {NEG}_{A_{0.85}^{(0.8, 0.5)}}^{O} (X) & = & {x_{7}, x_{12}, x_{13}}; \\ Γ_{A_{0.85}^{0.8}}^{O} (X) & = & \frac{3}{7} . \\ {POS}_{A_{0.85}^{(0.8, 0.5)}}^{P} (X) & = & {x_{7}, x_{12}, x_{13}}; \\ {BN}_{A_{0.85}^{(0.8, 0.5)}}^{P} (X) & = & {x_{2}, x_{3}, x_{4}}; \\ {NEG}_{A_{0.85}^{(0.8, 0.5)}}^{P} (X) & = & {x_{1}, x_{5}, x_{6}, x_{8}, \\ x_{9}, x_{10}, x_{11}, x_{14}}; \\ Γ_{A_{0.85}^{0.8}}^{P} (X) & = & \frac{1}{2} . \end{matrix}$

6 Optimal granulation selection in MG-IF-DTRSs

Granulation selection is an important issue in knowledge representation and data mining in rough set theory [52]. In this section, we investigated optimal granulation selection with different requirements in optimistic and pessimistic similarity measure-based MG-IF-DTRSs.

Let MIFDS = (U, A, D), the similarity measure-based optimistic and pessimistic α-lower and β-upper approximations distribution functions of MIFDS are defined as follows: $\begin{matrix} f (\underline{A_{λ}^{O, α}}) = (\underline{A_{λ}^{O, α}} (D_{1}), \underline{A_{λ}^{O, α}} (D_{2}), \dots \underline{A_{λ}^{O, α}} (D_{r})); \\ f (\bar{A_{λ}^{O, β}}) = (\bar{A_{λ}^{O, β}} (D_{1}), \bar{A_{λ}^{O, β}} (D_{2}), \dots \bar{A_{λ}^{O, β}} (D_{r})); \\ f (\underline{A_{λ}^{P, α}}) = (\underline{A_{λ}^{P, α}} (D_{1}), \underline{A_{λ}^{P, α}} (D_{2}), \dots \underline{A_{λ}^{P, α}} (D_{r})); \\ f (\bar{A_{λ}^{P, β}}) = (\bar{A_{λ}^{P, β}} (D_{1}), \bar{A_{λ}^{P, β}} (D_{2}), \dots \bar{A_{λ}^{P, β}} (D_{r})) . \end{matrix}$

Definition 13. Let MIFIS = (U, A), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1, B ⊆ A = {A₁, A₂, ⋯ , A_m}.

If $f (\underline{B_{λ}^{O, α}}) = f (\underline{A_{λ}^{O, α}})$ ( $f (\bar{B_{λ}^{O, β}}) = f (\bar{A_{λ}^{O, β}})$ ), B is an optimistic α-lower (β-upper)distribution granulation selection of MIFDS. Moveover, ∀B′ ⊂ B is not an optimistic α-lower (β-upper) distribution granulation selection of MIFDS, B is an optimistic α-lower (β-upper) distribution optimal granulation selection of MIFDS.

If $f (\underline{B_{λ}^{P, α}}) = f (\underline{A_{λ}^{P, α}})$ ( $f (\bar{B_{λ}^{P, β}}) = f (\bar{A_{λ}^{P, β}})$ ), B is a pessimistic α-lower (β-upper) distribution granulation selection of MIFDS. Moveover, ∀B′ ⊂ B is not a pessimistic α-lower (β-upper) distribution granulation selection of MIFDS, B is a pessimistic α-lower (β-upper) distribution optimal granulation selection of MIFDS.

In order to select the optimal granulation of MIFDS, some measures are discussed in MG-IF-DTRS.

The optimistic and pessimistic similarity measure-based α-lower and β-upper approximations quality functions are denoted as follows: $\begin{matrix} Θ (A_{λ}^{O, α}) = \frac{\sum_{j = 1}^{r} | \underline{A_{i, λ}^{O, α}} (D_{j}) |}{| U |}, \\ Θ (A_{λ}^{O, β}) = \frac{\sum_{j = 1}^{r} | \bar{A_{i, λ}^{O, β}} (D_{j}) |}{| U |}; \\ Θ (A_{λ}^{P, α}) = \frac{\sum_{j = 1}^{r} | \underline{A_{i, λ}^{P, α}} (D_{j}) |}{| U |}, \\ Θ (A_{λ}^{P, β}) = \frac{\sum_{j = 1}^{r} | \bar{A_{i, λ}^{P, β}} (D_{j}) |}{| U |} . \end{matrix}$

Proposition 8. Setting 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1, let B ⊆ A = {A₁, A₂, ⋯ , A_m}. Then we have:

$Θ (B_{λ}^{O, α}) \leq Θ (A_{λ}^{O, α})$ , $Θ (B_{λ}^{O, β}) \leq Θ (A_{λ}^{O, β})$ ;

$Θ (B_{λ}^{P, α}) \geq Θ (A_{λ}^{P, α})$ , $Θ (B_{β}^{P, α}) \geq Θ (A_{λ}^{P, β})$ .

Proof. They are obtained by Propositions 5and 7. □

Definition 14. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1, B ⊆ A = {A₁, A₂, ⋯ , A_m}.

If $Θ (B_{λ}^{O, α}) = Θ (A_{λ}^{O, α})$ ( $Θ (\bar{B_{i, λ}^{O, β}}) = Θ (\bar{A_{i, λ}^{O, β}})$ ), B is an optimistic α-lower (β-upper)quality granulation selection of MIFDS. Moveover, ∀B′ ⊂ B is not an optimistic α-lower (β-upper) distribution granulation selection of MIFDS, B is an optimistic α-lower (β-upper) quality optimal granulation selection of MIFDS.

If $Θ (\underline{B_{i, λ}^{P, α}}) = Θ (\underline{A_{i, λ}^{P, α}})$ ( $Θ (\bar{B_{i, λ}^{O, β}}) = Θ (\bar{A_{i, λ}^{O, β}})$ ), B is a pessimistic α-lower (β-upper) quality granulation selection of MIFDS. Moveover, ∀B′ ⊂ B is not a pessimistic α-lower (β-upper) quality granulation selection of MIFDS, B is a pessimistic α-lower (β-upper) quality optimal granulation selection ofMIFDS.

Theorem 2. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1 and B ⊆ A = {A₁, A₂, ⋯ , A_m.

B is an optimistic α-lower (β-upper) distribution optimal granulation selection ⇔ B is an optimistic α-lower (β-upper) quality optimal granulation selection of MIFDS.

B is a pessimistic α-lower (β-upper) distribution optimal granulation selection ⇔ B is a pessimistic α-lower (β-upper) quality optimal granulation selection of MIFDS.

Proof. (1) For optimistic α-lower distribution optimal granulation selection, $f (\underline{B_{λ}^{O, α}}) = f (\underline{A_{λ}^{O, α}}) \Rightarrow Θ (\underline{B_{λ}^{O, α}}) = Θ (\underline{A_{λ}^{O, α}})$ is obvious.

For any D_j ∈ U/R_D, form (1) of Proposition 4, we have $\underline{B_{λ}^{O, α}} (D_{j}) \subseteq \underline{A_{λ}^{O, α}} (D_{j})$ , then $Θ (\underline{B_{λ}^{O, α}}) \leq Θ (\underline{A_{λ}^{O, α}})$ . If $Θ (\underline{B_{λ}^{O, α}}) = Θ (\underline{A_{λ}^{O, α}})$ , for any j ≤ r, we have $\underline{B_{λ}^{O, α}} (D_{j}) = \underline{A_{λ}^{O, α}} (D_{j})$ . Otherwise, if there exists D_k ∈ U/R_D such that $\underline{B_{λ}^{O, α}} (D_{k}) \subset \underline{A_{λ}^{O, α}} (D_{k})$ , then $Θ (\underline{B_{λ}^{O, α}}) < Θ (\underline{A_{λ}^{O, α}})$ .

Therefore, $Θ (\underline{B_{λ}^{O, α}}) = Θ (\underline{A_{λ}^{O, α}}) \Rightarrow f (\underline{B_{λ}^{O, α}}) = f (\underline{A_{λ}^{O, α}})$

Thus, B is an optimistic α-lower distribution optimal granulation selection ⇔ B is an optimistic α-lower quality optimal granulation selection of MIFDS.

Similarly, we have B is an optimistic β-upper distribution optimal granulation selection ⇔ B is an optimistic β-upper quality optimal granulation selection of MIFDS.

(2) can be proved by similar way. □

Definition 15. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1. For 0 ≤ β < α ≤ 1, and B = A - {A_j}.

If $Θ (\underline{B_{λ}^{O, α}}) = Θ (\underline{A_{λ}^{O, α}})$ ( $Θ (\bar{B_{i, λ}^{O, β}}) = Θ (\bar{A_{i, λ}^{O, β}})$ ), we say that A_j is anoptimistic α-lower (β-upper) quality dispensable granularity of MG-IF-DTRS; otherwise, A_j is an optimistic α-lower (β-upper) quality indispensable granularity of MG-IF-DTRS.

If $Θ (\underline{B_{λ}^{P, α}}) = Θ (\underline{A_{λ}^{P, α}})$ ( $Θ (\bar{B_{i, λ}^{P, β}}) = Θ (\bar{A_{i, λ}^{P, β}})$ ), we say that A_j is a pessimistic α-lower (β-upper) quality dispensable granularity of MG-IF-DTRS; otherwise, A_j is a pessimistic α-lower (β-upper) quality indispensable granularity of MG-IF-DTRS.

In fact, if A_j is an optimistic (pessimistic) α-lower (β-upper) quality dispensable granularity of MG-IF-DTRS, after deleting A_j, the optimistic (pessimistic) α-lower (β-upper) quality will not change.

Theorem 3. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. B ⊆ A = {A₁, A₂, ⋯ , A_m} is an α-lower (β-upper) optimistic (pessimistic) quality granulation selection. Then A_j ∈ B is an indispensable granularity in B if and only if B - {A_j} is not an α-lower (β-upper) optimistic (pessimistic) quality granulation selection.

Proof. It is easily verified by Definition 14. □

The core of MIFDS is composed of all indispensable granules.

The α-lower (β-upper) optimistic quality decreases monotonically with the granular structures decrease. Keeping the α-lower (β-upper) optimistic quality unchanged, we can get the α-lower (β-upper) optimistic quality optimal granulation. Here, we use the α-lower (β-upper) optimistic quality to characterize the significance of each granularity.

Definition 16. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. The optimistic α-lower and β-upper inner significance measures of A_j are defined by $\begin{matrix} {Sig}_{inner}^{O, α} (A, A_{j}) & = & Θ (\underline{A_{λ}^{O, α}}) \\ - Θ (\underline{(A - {A_{j}})_{λ}^{O, α}}); \\ {Sig}_{inner}^{O, β} (A, A_{j}) & = & Θ (\bar{A_{λ}^{O, β}}) \\ - Θ (\bar{(A - {A_{j}})_{λ}^{O, β}}) . \end{matrix}$

${Sig}_{inner}^{O, α} (A, A_{j})$ ( ${Sig}_{inner}^{O, β} (A, A_{j})$ ) reflects the extent to which the optimistic α-lower (β-upper) quality of A respect to MIFDS decreases after removing granularity A_j.

Similarly, the optimistic α-lower and β-upper outer significance measures of A_j can also be defined.

Definition 17. MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. B ⊆ A = {A₁, A₂, ⋯ , A_m} and A_j ∉ B. The optimistic α-lower and β-upper outer significance measure of A_j according to B are defined by $\begin{matrix} {Sig}_{outer}^{O, α} (B, A_{j}) & = & Θ (\underline{(B \cup {A_{j}})_{λ}^{O, α}}) \\ - Θ (\underline{B_{λ}^{O, α}}); \\ {Sig}_{outer}^{O, β} (B, A_{j}) & = & Θ (\bar{(B \cup {A_{j}})_{λ}^{O, β}}) \\ - Θ (\bar{B_{λ}^{O, β}}) . \end{matrix}$

${Sig}_{outer}^{α, O} (B, A_{j})$ ( ${Sig}_{outer}^{O, β} (B, A_{j})$ ) reflects the extent to which the optimistic α-lower (β-upper) quality of B respect to MIFDS increases after adding granularity A_j to B.

Theorem 4. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. B ⊆ A = {A₁, A₂, ⋯ , A_m} is an optimistic α-lower (β-upper) granulation selection. Then A_j ∈ B is an optimistic α-lower (β-upper) indispensable granularity if and only if ${Sig}_{inner}^{O, α} (B, A_{j}) > 0$ ( ${Sig}_{inner}^{O, β} (B, A_{j}) > 0$ ).

Proof. ⇒ If A_j is an optimistic α-lower (β-upper) indispensable granularity in B, according to Definition 15 and Theorem 3, it follows that $Θ (B_{λ}^{O, α}) > Θ (\underline{(B - {A_{j}})_{λ}^{O, α}})$ ( $Θ (B_{λ}^{O, β}) > Θ (\underline{(B - {A_{j}})_{λ}^{O, β}})$ ), i.e., ${Sig}_{inner}^{O, α,} (B, A_{j}) > 0$ ( ${Sig}_{inner}^{O, β,} (B, A_{j}) > 0$ ).

⇐ B is an optimistic α-lower (β-upper) granulation selection. If ${Sig}_{inner}^{O, α} (B, A_{j}) > 0$ ( ${Sig}_{inner}^{O, β} (B, A_{j}) > 0$ ), then $Θ (\underline{B_{λ}^{O, α}}) > Θ (\underline{{B - {A_{j}}}_{λ}^{O, α}})$ ( $Θ (\underline{B_{λ}^{O, β}}) > Θ (\underline{{B - {A_{j}}}_{λ}^{O, β}})$ ). From Theorem 3, if B - {A_j} is not an optimistic α-lower granulation selection. According to Definition 15, A_j is an optimistic α-lower (β-upper) indispensable granularity. □

Theorem 5. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. We denote $CORE (A_{λ}^{O, α})$ and $CORE (A_{λ}^{O, β})$ as the optimistic α-lower and β-upper core granularity set, then (1) $CORE (A_{λ}^{O, α}) = {A_{j} : {Sig}_{inner}^{O, α} (A, A_{j}) > 0}$ ; (2) $CORE (A_{λ}^{O, β}) = {A_{j} : {Sig}_{inner}^{O, β} (A, A_{j}) > 0}$ .

Proof. (1) It is easy to know that A is an optimistic α-lower granulation selection. According toTheorem 3, A_j is an optimistic α-lowerindispensable granularity if and only if ${Sig}_{inner}^{O} (A, A_{j}) > 0$ . The set of all indispensable granularity in MIFDS = (U, A, D) is denoted as the core of A. Therefore, $CORE (A_{λ}^{O, α}) = {A_{j} : {Sig}_{inner}^{O, α} (A, A_{j}) > 0}$ .

(2) can be proved by similar way. □

Theorem 6. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1, B ⊆ A = {A₁, A₂, ⋯ , A_m}. If $Θ (\underline{B_{λ}^{O, α}}) = Θ (\underline{A_{λ}^{O, α}})$ ( $Θ (\bar{B_{λ}^{O, β}}) = Θ (\bar{A_{λ}^{O, β}})$ ) and $\forall A_{j} \in B, {Sig}_{inner}^{O, α} (B, A_{j}) > 0$ ( ${Sig}_{inner}^{O, β} (B, A_{j}) > 0$ ), then B is an optimistic α-lower (β-upper) optimal granulation selection.

Proof. The proof is obvious from Definition 16, Theorems 3, 4 and 5. □

The pessimistic α-lower (β-upper) quality increases monotonically with the granular structures decrease. Keeping the pessimistic α-lower (β-upper) quality unchanged, we can get the pessimistic α-lower quality optimal granulation. Here, we use the pessimistic α-lower (β-upper) quality to characterize the significance of each granularity.

Definition 18. Let MIFDS = (U, A, D), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. The pessimistic α-lower and β-upper inner significance measures of A_j are defined by $\begin{matrix} {Sig}_{inner}^{P, α} (A, A_{j}) & = & Θ (\underline{(A - {A_{j}})_{λ}^{P, α}}) \\ - Θ (\underline{A_{λ}^{P, α}}); \\ {Sig}_{inner}^{P, β} (A, A_{j}) & = & Θ (\bar{(A - {A_{j}})_{λ}^{P, β}}) \\ - Θ (\bar{A_{λ}^{P, β}}) . \end{matrix}$

${Sig}_{inner}^{P, α} (A, A_{j})$ ( ${Sig}_{inner}^{P, β} (A, A_{j})$ ) reflects the extent to which the pessimistic α-lower (β-upper) quality of A respect to MIFDS increases after removing granularity A_j.

Similarly, the pessimistic α-lower and β-upper outer significance measures of A_j can also be defined.

Definition 19. MIFIS = (U, A), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. B ⊆ A = {A₁, A₂, ⋯ , A_m} and A_j ∉ B. The pessimistic α-lower and β-upper outer significance measure of A_j w.r.t. B are defined by $\begin{matrix} {Sig}_{outer}^{O, α} (B, A_{j}) & = & Θ (\underline{B_{λ}^{P, α}}) \\ - Θ (\underline{(B \cup {A_{j}})_{λ}^{P, α}}); \\ {Sig}_{outer}^{O, β} (B, A_{j}) & = & Θ (\bar{B_{λ}^{P, β}}) \\ - Θ (\bar{(B \cup {A_{j}})_{λ}^{P, β}}) . \end{matrix}$

${Sig}_{outer}^{O, α} (B, A_{j})$ ( ${Sig}_{outer}^{O, β} (B, A_{j})$ ) reflects the extent to which the pessimistic α-lower (β-upper) quality of B respect to MIFDS decreases after adding granularity A_j to B.

Theorem 7. Let MIFIS = (U, A), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. B ⊆ A = {A₁, A₂, ⋯ , A_m} is a pessimistic α-lower (β-upper) granulation selection. Then pessimistic A_j ∈ B is an α-lower (β-upper) indispensable granularity if and only if ${Sig}_{inner}^{P, α} (B, A_{j}) > 0$ ( ${Sig}_{inner}^{P, β} (B, A_{j}) > 0$ ).

Proof. According to Theorem 4, similarly, we have Theorem 7. □

Theorem 8. Let MIFIS = (U, A), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1. We denote $CORE (A_{λ}^{O, α})$ and $CORE (A_{λ}^{O, β})$ as the pessimistic α-lower and β-upper core granularity set, then

$CORE (A_{λ}^{P, α}) = {A_{j} : {Sig}_{inner}^{P, α} (A, A_{j}) > 0}$ ;

$CORE (A_{λ}^{P, β}) = {A_{j} : {Sig}_{inner}^{P, β} (A, A_{j}) > 0}$ .

Proof. According to Theorem 5, similarly, we have Theorem 8. □

Theorem 9. Let MIFIS = (U, A), 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1, B ⊆ A = {A₁, A₂, ⋯ , A_m}. If $Θ (\underline{B_{λ}^{P, α}}) = Θ (\underline{A_{λ}^{P, α}})$ ( $Θ (\bar{B_{λ}^{P, β}}) = Θ (\bar{A_{λ}^{P, β}})$ ) and $\forall A_{j} \in B, {Sig}_{inner}^{P, α} (B, A_{j}) > 0$ ( ${Sig}_{inner}^{P, β} (B, A_{j}) > 0$ ),then B is an optimistic α-lower (β-upper) optimal granulation selection.

Proof. According to Theorem 6, similarly, we have Theorem 8. □

Algorithm 1 A heuristic algorithm to select the optimistic α-lower optimal granulation

Input: MIFDS = {IFDS_i : IFDS_i = (U, A_i, D)}, 0 ≤ λ ≤ 1 and 0 ≤ β < α ≤ 1, B = {A_i : i ≤ m}.

Output: The optimistic α-lower optimal granulation selection(short for LOG_α).

1: set S : =∅, LOG_α : =∅, CORE (B) : =∅ and t : =1

for A_j ∈ B

3: compute ${Sig}_{inner}^{α, O} (B, A_{j})$

4: if ${Sig}_{inner}^{α, O} (B, A_{j}) > 0$

5: set S : = CORE (B) ∪ {B_j

end if

end for

set S : = CORE (B)

While $Θ (\underline{S_{λ}^{O, α}}) \neq Θ (\underline{B_{λ}^{O, α}})$

choose any $A_{j}^{*} \in B - S$

compute ${Sig}_{outer}^{α, O} (B, A_{j}^{*})$

For A_j ∈ B - S

compute ${Sig}_{outer}^{α, O} (B, A_{j})$

If ${Sig}_{outer}^{α, O} (B, A_{j}) > {Sig}_{outer}^{α, O} (B, A_{j}^{*})$

set $A_{j}^{*} = A_{j}$

end if

end for

set $S = S \cup {A_{j}^{*}}$

End While

set LOG_α : = S

While t do

set t : =0

for A_k∈ LOG_α

if ${Sig}_{inner}^{α, O} ({LOG}_{α} :, A_{k}) = 0$ set t : =1 set LOG_α : = LOG_α - {A_k

break

end if

end for

end while

Return LOG_α

The process of obtaining the four kinds of optimal granulation is basically the same. Thus, we only consider the optimistic α-lower quality optimal granulation selection of MG-IF-DTRS in the followingdiscussion.

According to the above discussion, we present a heuristic algorithm to select the optimistic α-lower optimal granulation. The process is outlined inAlgorithm 1.

In Algorithm 1, steps 2–7 is to compute CORE (B), the time complexity is O (|U||B|). Steps 8–19, set S = CORE (B), the granules that have higher outer significance are heuristically added to S until S becomes the α-lower optimistic granulation selection, the time complexity is O (|U||B - S||S|). After that, steps 20–31 remove the dispensable granules from LOG_α, the time complexity is O (|U||LOG_α|²). Finally, the optimistic α-lower optimal granulation is obtained, and the total time complexity of Algorithm 1 isO (|U||B|²).

Example 4. In Example 3, we discussed the optimistic 0.8-lower optimal granulation in MIFDS = {IFDS_i : IFDS_i = (U, A_i, D)} which considers a decision attribute d in MIFIS = (U, A). Let $\begin{matrix} D_{1} & = & {x_{1}, x_{5}, x_{8}, x_{10}, x_{11}, x_{14}}, \\ D_{2} & = & {x_{2}, x_{3}, x_{4}, x_{6}, x_{7}, x_{9}, x_{12}, x_{13}} . \end{matrix}$

Step 2–3: Computing the optimistic 0.8-lower inner significance of A_j ∈ B according to Definition 17: $\begin{matrix} {Sig}_{inner}^{O, 0.8} (B, A_{1}) = 0, {Sig}_{inner}^{O, 0.8} (B, A_{2}) = \frac{2}{14}, \\ {Sig}_{inner}^{O, 0.8} (B, A_{3}) = 0, {Sig}_{inner}^{O, 0.8} (B, A_{4}) = 0 . \end{matrix}$

Step 4–8: Computing the $CORE (B_{0.85}^{O, 0.8})$ according to Theorem 5:

Since ${Sig}_{inner}^{O, 0.8} (B, A_{2}) = 0.07 > 0$ , then $CORE (B_{0.85}^{O, 0.8}) = {A_{2}}$ . Set $S = CORE (B_{0.85}^{O, 0.8}) = {A_{2}}$ .

Step 9: Computing the values of $Θ (\underline{S_{λ}^{O, α}})$ and $Θ (\underline{B_{0.85}^{O, 0.8}})$ : $\begin{matrix} Θ (\underline{S_{0.85}^{O, 0.8}}) = \frac{10}{14}, Θ (\underline{B_{0.85}^{O, 0.8}}) = \frac{12}{14}, \\ Θ (\underline{S_{0.85}^{O, 0.8}}) \neq Θ (\underline{B_{0.85}^{O, 0.8}}) . \end{matrix}$

Step 10–13: Computing the optimistic 0.8-lower outer significance measures of A_j, j = 1, 3, 4 according to {A₂} by Definition 19:

${Sig}_{outer}^{O, 0.8} ({A_{2}}, A_{1}) = 0$ , ${Sig}_{outer}^{O, 0.8} ({A_{2}}, A_{3}) = \frac{3}{14}$ , ${Sig}_{outer}^{O, 0.8} ({A_{2}}, A_{4}) = \frac{3}{14}$ .

Step 14–17: Ranking the granule by the optimistic 0.8-lower outer significance: $\begin{matrix} {Sig}_{outer}^{O, 0.8} ({A_{2}}, A_{3}) & = & {Sig}_{outer}^{O, 0.8} ({A_{2}}, A_{4}) \\ > {Sig}_{outer}^{O, 0.8} ({A_{2}}, A_{1}) . \end{matrix}$

Step 18–19: Adding the granule which has higher 0.8-lower outer significance to S until S becomes the optimistic granulation selection:

Adding A₃ to {A₂}, $Θ (\underline{{A_{2}, A_{3}}_{0.85}^{O, 0.8}}) = Θ (\underline{B_{0.85}^{O, 0.8}})$ , and adding A₄ to {A₂}, $Θ (\underline{{A_{2}, A_{4}}_{0.85}^{O, 0.8}}) = Θ (\underline{B_{0.85}^{O, 0.8}})$ .

Step 20: Set S₁ = {A₂, A₃} , S₂ = {A₂, A₄}. According to Definition 15, S₁, S₂ are two optimistic 0.8-lower granulation selections.

Step 21–30: Computing the optimistic 0.8-lower inner significance of A_j ∈ S₁ according to S₁ and A_j ∈ S₂ according to S₂, removing the dispensable granules form the two optimistic 0.8-lower granulation selections S₁ and S₂:

Since $Θ (\underline{S_{1, 0.85}^{O, 0.8}}) = Θ (\underline{B_{0.85}^{O, 0.8}})$ , ${Sig}_{inner}^{O, 0.85} (S_{1}, A_{2}) = \frac{2}{14} > 0$ , ${Sig}_{inner}^{O, 0.85} (S_{1}, A_{3}) = \frac{3}{14} > 0$ ;

Since $Θ (\underline{S_{2, 0.85}^{O, 0.8}}) = Θ (\underline{B_{0.85}^{O, 0.8}})$ , ${Sig}_{inner}^{O, 0.85} (S_{2}, A_{2}) = \frac{2}{14} > 0$ , ${Sig}_{inner}^{O, 0.85} (S_{2}, A_{4}) = \frac{3}{14} > 0$ .

According to Theorem 5, A_j ∈ S₁ are all optimistic 0.8-lower indispensable granularity, the same asA_j ∈ S₂.

Step 31: Return the optimistic optimal granulation:

According to Theorem 9, S₁, S₂ are two optimistic 0.8-lower optimal granulation selections.

Similarly, B is a pessimistic 0.5-upper optimal granulation of MIFDS = {IFDS_i : IFDS_i = (U, A_i, D)}.

7 Conclusion

For the defects of the existing similarity measures in IFSs, this paper has improved the similarity measures by considering the hesitancy degree and the endpoint distance of membership and non-membership. The numerical example shows that the results of the proposed method are more reasonable and elaborate than other methods. According to similarity measure, a novel similarity measure-based IF-DTRSs and two corresponding MG-IF-DTRSs have been proposed. This study can be considered as generalization model of DTRS and MGRSs in multi-source intuitionistic fuzzy information, which make the models more meaningful in practical applications. Finally, we have established a heuristic algorithm to explore the optimal granulation selection in MIFDS, and an example has been employed to illustrate the processing of the algorithm above.

As future work, we plan to consider the rule extraction in similarity measure-based MG-IF-DTRSs.

Footnotes

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (No. 61573127), Natural Science Foundation of Hebei Province (No. A2018210120).

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