Intuitionistic fuzzy three-way formal concept analysis based attribute correlation degree

Abstract

As an important expanded of the classical formal concept, the three-way formal concept analysis integrates more information with the three-way decision theory. However, to the best of our knowledge, few scholars have studied the intuitionistic fuzzy three-way formal concept analysis. This paper proposes an intuitionistic fuzzy three-way formal concept analysis model based on the attribute correlation degree. To achieve this, we comprehensively analyze the composition of attribute correlation degree in the intuitionistic fuzzy environment, and introduce the corresponding calculation methods for different situations, as well as prove the related properties. Furthermore, we investigate the intuitionistic fuzzy three-way concept lattice ((IF3WCL) of object-induced and attribute-induced. Then, the relationship between the IF3WCL and the positive, negative and boundary domains in the three-way decision are discussed. In addition, considering the final decision problem of boundary objects, the secondary decision strategy of boundary objects is obtained for IF3WCL. Finally, a numerical example of multinational company investment illustrates the effectiveness of the proposed model. In this paper, we systematically study the IF3WCL, and give a quantitative analysis method of formal concept decision along with its connection with three-way decision, which provides new ideas for the related research.

Keywords

Intuitionistic fuzzy attribute correlation degree IF3WCL secondary decision three-way decision

1 Introduction

Three-way decision (3WD), proposed by Yao [1], is an effective tool for uncertain information processing and decision-making. The main idea of 3WD is to combine the Bayesian minimum risk decision theory to divide the discourse domain into three disjoint regions, i.e., positive domain, negative domain and boundary domain, which represent acceptance decision, rejection decision and delayed decision, respectively. In fact, we always use 3WD unconsciously in our daily lives. For example, the editor marks the manuscript as accepted, rejected, or in need of further modification based on the review comments; the doctor determines whether the patient has recovered or needs to continue treatment according to the patient’s symptoms. 3WD is an effective heuristic strategy, more in line with people’s cognitive and decision-making habits, which can be explained, depicted and motivated [2]. Due to the universality of 3WD, the theory has been widely concerned by scholars since it was proposed, and has been widely applied in the neural network [3, 4], clustering analysis [5, 6], information system [7, 8], granular computing [9, 10], etc. Nowadays, 3WD has become a part of the basic theory of artificial intelligence. The in-depth research on 3WD will help people form a more general information processing model to deal with complex decision-making problems in the uncertain environments.

Formal concept analysis (FCA), proposed by Wille [11] in 1982, is an effective mathematical tool for data mining and knowledge discovery. FCA mainly uses formal concept (concept for short) to describe the relationship between objects and attributes, and expresses the generalization and specialization of concept through Hasse diagram. In addition, the concept lattice is an important part of FCA, which consists of a pair of order pairs, namely, extension and intension. Specifically, extension is the set of attributes common to the objects and intension is the set of objects which have all attributes [12, 13]. Over the past few years, FCA has been widely combined with different disciplines and applied in many fields, such as construction of concept lattice [14, 15], extension of concept lattice [16, 17], reduction of concept lattice [18, 19], rules extraction of concept lattice [20, 21], etc. Furthermore, the generation algorithm and operator structure of concept lattice are also important research orientations.

Nevertheless, it must be pointed out that the classical FCA can only reflect the information shared by extension and intension, but ignores the information that extension and intension do not capture. From this perspective, Qi et al. [12, 22] integrates the ideas of the 3WD into FCA and proposes a three-way formal concept analysis (3WFCA) model. In 3WFCA, the three-way concept lattices are constructed by establishing a Galois connection between the power sets of the object and attribute. In addition, there are two types of concept lattices, namely, object-induced and attribute-induced. The extension and intension of the three-way concept lattices are also represented by a pair of ordered pairs, but the difference is that the intension of the attribute-induced three-way concept lattices consists of two parts: the positive domain and the negative domain, which respectively represent the attributes of the object “jointly possessed” and “jointly not possessed”. Dually, we can also get the object-induced three-way concept lattices. From this point of view, the three-way concept lattices are associated with to 3WD. It is possible to obtain the positive, negative and boundary domains of the 3WD by constructing three-way concept lattices. This is one of its attractive research points. As an important expanded form of FCA, 3WFCA has gained considerable attention since it was proposed [10 , 23–29]. For example, Ren et al. [27] investigated four kinds of attribute reduction methods for three-way concept lattices.

Note that the things we encounter and deal with in our daily life are usually vague and uncertain, such as natural language. To better describe and process complex fuzzy problems more delicately, Atanassov [30] proposed the theory of intuitionistic fuzzy sets in 1986. Intuitionistic fuzzy sets describe the fuzzy nature of the objective world by using two parameters of membership degree and non-membership degree. It is much more exquisite than the single membership of the fuzzy set. As a new mathematical method, intuitionistic fuzzy set theory has been applied to various fields [31 –35]. The research on intuitionistic fuzzy sets can be categorized into the following two aspects, namely, theory and application. In terms of theoretical research, it mainly studies the flexible properties of intuitionistic fuzzy set, such as operator [31], correlation coefficient [34], distance measure [35], etc., while the applied research mainly focuses on the decision-making problem of uncertain environment [7 , 31–33]. For example, Garg et al. [31] proposed novel aggregation operators and ranking method for complex intuitionistic fuzzy sets. Subsequently, they studied decision-making in intuitionistic fuzzy environment from different perspectives [32, 33]. Moreover, intuitionistic fuzzy set combined with 3WD theory can be extended to address the uncertainty of information systems. Yang et al. [36] studied the 3WD model in the multi-granularity space through intuitionistic fuzzy numbers. Liang et al. [37] proposed an intuitionistic fuzzy decision rough set model based on point operator and used to construct the 3WD model. In practice, the theory of intuitionistic fuzzy set has greatly promoted the development of analyzing and solving uncertain problems. Therefore, scholars began to study the concept lattice in uncertain environments, such as fuzzy and intuitionistic fuzzy concept lattices [16 , 38–40] and rough concept lattices [41, 42]. Intuitionistic fuzzy sets provide a better theoretical basis for the study of three-way concept lattices of uncertain problems, but so far, there is rarely research on the fusion between them.

Three-way concept lattice analyzes the relationship between the generalization and specialization of concepts from the perspective of attributes and objects, which is more in line with human thinking process. On the one hand, the membership degree, non-membership degree and hesitation degree of intuitionistic fuzzy set are nicely complementary to 3WD’s positive domain, negative domain and boundary domains. Thus, if one can use the intuitionistic fuzzy set theory to construct three-way concept lattices and use them to analyze 3WD, there will be unexpected gains. On the other hand, the degree of attribute correlation is also a problem worthy of attention, because it is closely related to the changes of the constructed concept lattice. Their combination can not only expand the research scope of 3WFCA to intuitionistic fuzzy environment, but also quantitatively analyze the correlation degree of each concept.

In practice, by introducing positive domain, negative domain and boundary domain, the relationship between 3WFCA and 3WD can be more clearly analyzed, and it also provides a new method and direction for studying intuitionistic fuzzy decision-making problems. Hence, it is a subject worthy of further intensive research. The main contributions of this paper are as follows.

The concepts of membership correlation degree, non-membership correlation degree and hesitation degree perturbation correlation coefficient (abbreviated as disturbance coefficient) are defined, and their related properties are discussed. Then, the correlation degree between any two intuitionistic fuzzy attributes is extended to multiple attributes.

We extend * operator and $\bar{*}$ operator to intuitionistic fuzzy environment, and construct intuitionistic fuzzy concept lattice based on attribute correlation degree. On this basis, we obtain the object-induced IF3WCL (OI-IF3WCL) and attribute-induced IF3WCL (AI-IF3WCL).

Combining the positive domain, negative domain and boundary domain of 3WD, we analyze the relationship between IF3WCL and 3WD. For the boundary domain objects in OI-IF3WCL and AI-IF3WCL, we propose secondary decision strategies and corresponding algorithms, respectively.

This paper mainly combines intuitionistic fuzzy set, 3WFCA and 3WD ideas to construct IF3WCL. This paper is organized as follows. In Section 2, we simply review some basic knowledge of classical concept lattice, three-way concept lattice and intuitionistic fuzzy set. In Section 3, we propose a method for calculating the intuitionistic fuzzy attribute correlation degree and analyze its related properties. On the basis of it, the IF3WCL of object-induced and attribute-induced are given. In Section 4, we deeply analyze the relationship between IF3WCL and 3WD, and give a secondary decision strategy for the boundary domain of IF3WCL. In Section 5, the paper ends with conclusions.

2 Preliminaries

In this section, we review some necessary notions in classical FCA, three-way concept lattice and intuitionistic fuzzy set. Throughout this paper, we assume that the domain of discourse U and the attribute set A are both non-empty finite sets, P (U) and P (A) are power sets of U and A, respectively. X^C is the complementary set of X.

2.1 Three-way decision

Definition 1. [43]. Let U be the universe of discourse; R is an equivalent relationship defined on U, and apr = (U,) is an approximate space. The division of the universe U under the equivalence relation R can be described as U/R = {[x] _R|x ∈ U}, where [x] _R is an equivalence class containing x. For any X ⊆ U, its upper and lower approximate sets can be expressed as: $\begin{matrix} {\bar{apr}}_{R} (X) = {x \in U | [x]_{R} \cap X \neq \emptyset}, \\ {\underline{apr}}_{R} (X) = {x \in U | [x]_{R} \subseteq X} . \end{matrix}$

The discourse domain can be divided into three disjointed parts by upper and lower approximations, namely, the positive domain POS(X), the negative domain NEG(X), and the boundary domain BND(X) are as follows: $\begin{matrix} POS (X) = {\underline{apr}}_{R} (X), \\ NEG (X) = U - {\bar{apr}}_{R} (X), \\ BND (X) = {\bar{apr}}_{R} (X) - {\underline{apr}}_{R} (X) . \end{matrix}$

Definition 2. [1]. Let Θ = {X, ¬ X} be a set representing two complementary states, i.e., element x belongs to object X or does not belong to object X. Λ = {a_P, a_B, a_N} represents three different decisions, where a_P is the acceptance decision, a_B is the delay decision, and a_N is the rejection decision. The decision losses caused by actions taken in different states are shown in Table 1.

Table 1
Cost function matrix

Action Cost function

X ¬X

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

Action	Cost function
a _P	λ _PP	λ _PN
a _B	λ _BP	λ _BN
a _N	λ _NP	λ _NN

In Table 1, λ_PP, λ_BP, and λ_NP represent the cost losses of adopting a_P, a_B, and a_N decisions when in state X. Similarly, λ_PN, λ_BN, and λ_NN denote the decision loss under state ¬X. Therefore, the expected loss of object x under different actions is (a_i| [x]) (i = P, B, N), which can be expressed as follows: $\begin{matrix} R (a_{P} | [x]_{R}) = λ_{PP} Pr (X | [x]_{R}) + λ_{PN} Pr (\neg X | [x]_{R}), \\ R (a_{B} | [x]_{R}) = λ_{BP} Pr (X | [x]_{R}) + λ_{BN} Pr (\neg X | [x]_{R}), \\ R (a_{N} | [x]_{R}) = λ_{NP} Pr (X | [x]_{R}) + λ_{NN} Pr (\neg X | [x]_{R}) . \end{matrix}$

Pr(X| [x] ₎ and Pr(¬ X| [x] _R) represent the probabilities that the equivalence class belongs to X and ¬X. By a reasonable hypothesis,λ_PP ≤ λ_BP < λ_NP and λ_NN ≤ λ_BN < λ_PN; if the BND domain exists, then (λ_BP - λ_PP) (λ_BP - λ_NN) < (λ_PN - λ_BN) (λ_NP - λ_BP), and condition Pr(X| [x] _R) + Pr(¬ X| [x] _R) =1 holds. We can get the following rules: $\begin{matrix} If Pr (X | [x]_{R}) \geq α, then x \in POS (X), \\ If β < Pr (X | [x]_{R}) < α, then x \in BND (X), \\ If Pr (X | [x]_{R}) \leq β, then x \in NEG (X) . \end{matrix}$

Here, α and βrepresent: $\begin{matrix} α = \frac{λ_{PN} - λ_{BN}}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})}, \\ β = \frac{λ_{BP} - λ_{NN}}{(λ_{BP} - λ_{NN}) + (λ_{NP} - λ_{BP})} . \end{matrix}$

2.2 Formal concept analysis

Definition 3. [13]. Let (P, ≤) and (Q, ≤)be two partially ordered sets, and a pair of mappings φ : (P, ≤) → (Q, ≤) and ψ : (Q, ≤) → (P, ≤) satisfy the following conditions:

p₁ ≤ p₂ ⇒ φ (p₂) ≤ φ (p₁).

q₁ ≤ q₂ ⇒ ψ (q₂) ≤ ψ (q₁).

p ≤ ψ (φ (p) , q ≤ φ (ψ (q).

Then, the pair of mappings is called Galois connections.

Definition 4. [13]. A formal context K is a triple K = (U, A, I), where the element x of U is called object, the element a of A is called attribute and I is a binary relationship between U and A, i.e., I ⊆ U × A.

Concretely, for a formal context, any pair elements (x, a) ∈ I or xIa means that the object x has the attribute a. Dually, the meaning of the relationship (x, a) ∉ I or xI^Ca is that the object x does not possess the attribute a.

For a formal context K = (U, A, I), a pair of dual operators* : (U) → (A) and * : (A) → (U) for any X ⊆ U and B ⊆ A are defined as follows: $\begin{matrix} X^{*} = {a \in A | \forall x \in X, (x, a) \in I}, \\ B^{*} = {x \in U | \forall a \in B, (x, a) \in I} . \end{matrix}$

As understood by Definition 4, X^* is the set of attributes common to the objects in A, and B^* is the set of objects which have all attributes in B.

Proposition 1. [13]. For a formal context K = (U, A, I), ∀X₁, X₂, X ⊆ U, ∀B₁, B₂, B ⊆ A, we have the following conclusions:

$X_{1} \subseteq X_{2} \Rightarrow X_{2}^{*} \subseteq X_{1}^{*}, B_{1} \subseteq B_{2} \Rightarrow B_{2}^{*} \subseteq B_{1}^{*} .$

X ⊆ X^**, B ⊆ B^** .

X^* = X^***, B = B^*** .

X ⊆ B^* ⇔ B ⊆ X^* .

$(X_{1} \cup X_{2})^{*} = X_{1}^{*} \cap X_{2}^{*}, (B_{1} \cup B_{2})^{*} = B_{1}^{*} \cap B_{2}^{*} .$

$(X_{1} \cap X_{2})^{*} \supseteq X_{1}^{*} \cup X_{2}^{*}, (B_{1} \cap B_{2})^{*} \supseteq B_{1}^{*} \cup B_{2}^{*} .$

Definition 5. [13]. Let K = (U, A, I) be a formal context. A formal concept (concept for short) of K is a pair (X, B) with B ⊆ A and X ⊆ U. If X^* = B and B^* = A, then X and B are referred to as the extension and intension of (X, B), respectively. If C₁ = (X₁, B₁) and C₂ = (X₂, B₂) are two concepts on K, provided that X₁ ⊆ X₂ (⇔ B₂ ⊆ B₁), then C₁ is called a sub-concept of AE₂, and x₁ ∈ POSis called a super-concept of AE₁. Obviously, a partial order relationship is formed between the sub-concept and the super-concept, which can be expressed as (X₁, B₁) ≤ (X₂, B₂). All concepts on K are denoted as $\underline{B} (K)$ .

Moreover, the infimum and supremum of C₁ and C₂ are defined by: $\begin{matrix} (X_{1}, B_{1}) \land (X_{2}, B_{2}) = (X_{1} \cap X_{2}, (B_{1} \cup B_{2})^{* *}), \\ (X_{1}, B_{1}) \lor (X_{2}, B_{2}) = ((X_{1} \cup X_{2})^{* *}, B_{1} \cap B_{2}) . \end{matrix},$

Thus, the $\underline{B} (K)$ is a complete lattice. A complete lattice L is isomorphic to $\underline{B} (K)$ if and only if there are mappings γ : U → L and λ : A → L such that γ (U) is supremum-dense in L, and λ (A) is infimum-dense in L, and xIa is equivalent to γx ≤ λa for all x ∈ U and all a ∈ A.

Example 1. Let K = (U, A, I) be a formal context, where U = {x₁, x₂, x₃, x₄, x₅} is a collection of objects and A = {a, b, c, d} is a collection of attributes, as shown in Table 2.

Table 2
A formal context K

I a b c d

x ₁ × × ×

x ₂ × ×

x ₃ × × ×

x ₄ × × ×

x ₅ ×

I	a	b	c	d
x ₁	×	×	×
x ₂	×			×
x ₃		×	×	×
x ₄		×	×	×
x ₅	×

For the convenience of description, we use the symbol “×” to represent xIa and the blank to indicate the relationship xI^Ca.

The concept lattice corresponding to the formal context K is shown in Fig. 1.

Fig. 1

Concept lattice for the formal context K.

2.3 Three-way formal concept analysis

To reflect the information that the FCA actually contains but not represented, that is, objects (or attributes) does not have any elements in the intension (or the extension) of a formal concept, Qi et al. [12, 22] regards the * operator in the classical FCA as a positive operator and gives the definition of negative operator as follows.

Definition 6. [12]. For a formal context K = (U, A, I), a pair of dual operators $\bar{*}$ : P (U) → P (A) and $\bar{*}$ : P (A) → P (U) for any X ⊆ U and B ⊆ A, we have: $\begin{matrix} X^{\bar{*}} = {a \in A | \forall_{x} \in X (\neg (x I a))} \\ = {a \in A | \forall_{x} \in X ((x I^{C} a))} \\ = {a \in A | X \subseteq I^{C} a}, \\ B^{\bar{*}} = {x \in U | \forall_{a} \in B (\neg (x I a))} \\ = {a \in B | \forall_{x} \in U ((x I^{C} a))} \\ = {x \in U | X \subseteq I^{C}} . \end{matrix}$

∀X ⊆ U and ∀B ⊆ A, the three-way operators are composed of operator * and operator $\bar{*}$ , which are defined as follows: $X^{⋖} = (X^{*}, X^{\bar{*}}), B^{⋖} = (B^{*}, B^{\bar{*}}) .$

In relation to the three-way operators, a pair of inverse operators is defined on X, Y ⊆ U and B, C ⊆ A respectively [12]: $\begin{matrix} (X, Y)^{⋗} = {a \in A | a \in X^{*} and a \in Y^{\bar{*}}} = X^{*} \cap Y^{\bar{*}}, \\ (B, C)^{⋗} = {x \in U | x \in A^{*} and x \in B^{\bar{*}}} = A^{*} \cap B^{\bar{*}} . \end{matrix}$

Definition 7. [12, 22]. Let K = (U, A, I) be a formal context. ∀X, Y ⊆ U, ∀B, C ⊆ A, if and only if X^⋖ = (B, C) and (B, C) ^⋗ = X, then (X, (B, C)) is called the object-induced three-way concept, abbreviated as OE-concept, where X is called extension and (B, C) is the intension of (X, (B, C)). The set of all OE-concepts in the formal context K is denoted as OEL (U, A, I).

If OE₁ = (X, (B, C)) and OE₂ = (Y, (D, E)) are both OE-concepts, the partial order relationship between them can be expressed as: $(X, (B, C)) \leq (Y, (D, E)) \Leftrightarrow X \subseteq Y \Leftrightarrow (B, C) \supseteq (D, E) .$

OE₁ is called a sub-concept of OE₂, and OE₂is called a super-concept of OE₁. Obviously, OEL (U, A, I) is a complete lattice, and the infimum and supremum are defined as: $\begin{matrix} (X, (B, C)) \land (Y, (D, E)) = (X \cap Y, (B, C) \cup (D, E)^{⋗ ⋖}), \\ (X, (B, C)) \lor (Y, (D, E)) = ((X \cup Y)^{⋖ ⋗}, (B, C) \cap (D, E)) . \end{matrix},$

Example 2. In the formal context K shown in Table 2, the corresponding object-induced three-way concepts are shown in Fig. 2.

Fig. 2

The OE-concept lattice of Table 2.

Similarly, ((X, Y) , B) is called attribute-induced three-way concept (AE-concept for short). If and only if X^⋖ = (B, C) and (B, C) ^⋗ = X, then (X, Y) is called extension and B is the intension of ((X, Y) , B). The set of all AE-concepts in the formal context K is denoted as AEL (U, A, I).

If AE₁ = ((X, Y) , B) and AE₂ = ((Z, W) , C) are both AE-concepts, the partial order relationship between them can be expressed as: $((X, Y), B) \leq ((Z, W), C) \Leftrightarrow (X, Y) \subseteq ((Z, W) \Leftrightarrow B \supseteq C .$

AE₁ is called a sub-concept of AF₂, and AF₂is called a super-concept of AE₁. Obviously, AEL (U, A, I) is a complete lattice, and the infimum and supremum are defined as: $\begin{matrix} ((X, Y), B) \land ((Z, W), C) = ((X, Y) \cap (Z, W), (B \cup C)^{⋖ ⋗}), \\ ((X, Y), B) \lor ((Z, W), C) = (((X, Y) \cup (Z, W)^{⋗ ⋖}, B \cap C) . \end{matrix}$

Example 3. In the formal context K shown in Table 2, the corresponding attribute-induced three-way concepts are shown in Fig. 3.

Fig. 3

The AE-concept lattice of Table 2.

2.4 Intuitionistic fuzzy set

Definition 8. [30] Let U be a non-empty finite universe, $\tilde{S}$ be a non-empty subset of U, with two mappings: $μ_{\tilde{S}} (x) : U \to [0, 1]$ and $ν_{\tilde{S}} (x) : U \to [0, 1]$ , and satisfy $0 \leq μ_{\tilde{S}} (x) + ν_{\tilde{S}} (x) \leq 1$ , then $\tilde{S} = {< x, μ_{\tilde{S}} (x), ν_{\tilde{S}} (x) > | x \in U}$ is called an intuitionistic fuzzy set of U. Further, $μ_{\tilde{S}}$ and $ν_{\tilde{S}}$ are the membership function and non-membership function of $\tilde{S}$ , respectively. $μ_{\tilde{S}} (x)$ and $ν_{\tilde{S}} (x)$ indicate that the element x belongs to the membership degree and non- membership degree of S. The family of the intuitionistic fuzzy sets of the domain U is denoted as IF(U).

In addition, $η_{\tilde{S}} (x) = 1 - μ_{\tilde{S}} (x) - ν_{\tilde{S}} (x) (0 \leq η_{\tilde{S}} (x) \leq 1)$ is called the hesitation degree of intuitionistic fuzzy set S. It can be seen that when hesitation degree is equal to zero, the intuitionistic fuzzy set degenerates into a fuzzy set, that is, the fuzzy set is a special case of the intuitionistic fuzzy set.

Definition 9. [45] Let S₁ and S₂ be two intuitionistic fuzzy sets, then the normalized Euclidean distance between them is as follows: $\begin{matrix} d (S_{1}, S_{2}) = \\ \sqrt{\frac{1}{2 n} \sum_{i = 1}^{n} ((μ_{S_{1}} (x_{i}) - μ_{S_{2}} (x_{i}))^{2} + (ν_{S_{1}} (x_{i}) - ν_{S_{2}} (x_{i}))^{2} + (π_{S_{1}} (x_{i}) - π_{S_{2}} (x_{i}))^{2}} \end{matrix}$

Definition 10. [30] Let $\tilde{A} = {< x, μ_{\tilde{A}} (x), v_{\tilde{A}} (x) > | x \in U}$ and $\tilde{B} = {< x, μ_{\tilde{B}} (x), v_{\tilde{B}} (x) > | x \in U}$ be two intuitionistic fuzzy sets. The operations and relations related to intuitionistic fuzzy sets are given below.

$\tilde{A} = \tilde{B} \Leftrightarrow {(μ_{\tilde{A}} (x) = μ_{\tilde{B}} (x)) \land (ν_{\tilde{A}} (x) = ν_{\tilde{B}} (x)) | \forall x \in U}$ .

$\tilde{A} \subseteq \tilde{B} \Leftrightarrow {(μ_{\tilde{A}} (x) \leq μ_{\tilde{B}} (x)) \land (ν_{\tilde{A}} (x) \geq ν_{\tilde{B}} (x)) | \forall x \in U}$ .

$\tilde{A} \cup \tilde{B} = {< x, \lor {μ_{\tilde{A}} (x), μ_{\tilde{B}} (x)}, \land {ν_{\tilde{A}} (x),$ $ν_{\tilde{B}} (x)} > | x \in U}$ .

$\tilde{A} \cap \tilde{B} = {< x, \land {μ_{\tilde{A}} (x), μ_{\tilde{B}} (x)}, \lor {ν_{\tilde{A}} (x),$ $ν_{\tilde{B}} (x)} > | x \in U}$ .

${\tilde{A}}^{C} = {< x, ν_{\tilde{A}} (x), ν_{\tilde{B}} (x) > | x \in U}$ .

3 Intuitionistic fuzzy 3WFCA

In order to enable the traditional 3WFCA to deal with the uncertainty problem in the intuitionistic fuzzy environment, and to quantitatively analyze the correlation degree between the concepts in OI-IF3WCL and AI-IF3WCL, this section will study them through attribute correlation degree. In addition, the fusion of 3WD and intuitionistic fuzzy 3WFCA can also provide new ideas and methods for intuitionistic fuzzy uncertain decision-making.

For the convenience of expression, if there is no special description, it is assumed that all data have been dimensionless processed, and x_i (1 ≤ i ≤ m) is an element in the universe U and a_j (1 ≤ j ≤ n) is an element in the attribute set A.

3.1 Correlation degree of intuitionistic fuzzy attributes

Definition 11. Let Ω = (U, A, ℵ) be an Intuitionistic Fuzzy Object-Attribute Association Space (Association Space for short), where U = {x₁, x₂, ·· · , x_m} and A = {a₁, a₂, ·· · , a_n}. $c_{p}^{q} (x_{i})$ is the degree of correlation between attributes a_p and a_qwith respect to object x_i. ℵ is the correlation matrix between attributes, which can be expressed as follows: $ℵ_{mn} = [\begin{matrix} c_{1}^{1} (x_{1}) & c_{1}^{2} (x_{1}) & c_{1}^{3} (x_{1}) & \dots & c_{1}^{n} (x_{1}) \\ c_{2}^{1} (x_{2}) & c_{2}^{2} (x_{2}) & c_{2}^{3} (x_{2}) & \dots & c_{2}^{n} (x_{2}) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ c_{m - 1}^{1} (x_{m - 1}) & c_{m - 1}^{2} (x_{m - 1}) & c_{m - 1}^{3} (x_{m - 1}) & \dots & c_{m - 1}^{n} (x_{m - 1}) \\ c_{m}^{1} (x_{1}) & c_{m}^{2} (x_{1}) & c_{m}^{3} (x_{1}) & \dots & c_{m}^{n} (x_{1}) \end{matrix}]$

The correlation degree of intuitionistic fuzzy attributes is mainly composed of membership correlation degree, non-membership correlation degree and hesitation degree perturbation correlation coefficient, which will be analyzed below separately.

Part 1: Membership correlation degree.

∀x_i ∈ U, $μ_{{\tilde{A}}_{j}} (x_{i})$ describes the degree to which the object x_i belongs to the attribute a_j. In other words, $μ_{{\tilde{A}}_{j}} (x_{i})$ depicts the ability of the attribute a_j to subordinate the object x_i, which indirectly reflects the membership correlation between different attributes. Therefore, we can analyze the membership description differences of objects, and calculate the membership correlation degree as follows: $c_{μ} (a_{p}, a_{q}) = - | μ_{a}^{p} (x_{i}) - μ_{a}^{q} (x_{i}) | ln \frac{| μ_{a}^{p} (x_{i}) - μ_{a}^{q} (x_{i}) |}{μ_{a}^{p} (x_{i}) + μ_{a}^{q} (x_{i})}$ (1)

It can be found that the larger value of c_μ (a_p, a_q), the greater the membership correlation degree between attributes a_p and a_q.

Remark 1. Two special cases.

Case 1. When $μ_{a}^{p} (x_{i}) = μ_{a}^{q} (x_{i}) = 0$ , it means that neither the attribute a_p nor the attribute a_q can describe any membership information of the object x_i. So the membership correlation degree between a_p and a_q is c_μ (a_p, a_q) = 0.

Case 2. When $μ_{a}^{p} (x_{i}) = μ_{a}^{q} (x_{i}) \neq 0$ , it means that attributes a_p and a_q have the same ability to describe the membership of object x_i, so let c_μ (a_p, a_q) = 1.

Theorem 1. If $μ_{a}^{p} (x_{i}) \neq μ_{a}^{q} (x_{i})$ , then c_μ (a_p, a_q) = 0, if and only if $μ_{a}^{p} (x_{i}) = 0$ and $μ_{a}^{q} (x_{i}) \neq 0$ (or $μ_{a}^{p} (x_{i}) \neq 0$ , $μ_{a}^{q} (x_{i}) = 0$ ).

Proof In attributes a_p and a_q, if only one attribute has a membership expression of zero, the degree of correlation between attributes is completely determined by the non-membership description and the perturbation coefficient of the attribute. It is easy to know that c_μ (a_p, a_q) =0 by Equation (1).

Some properties of membership correlation degree are given as follows:

Boundedness: 0 ≤ c_μ (a_p, a_q) ≤1;

Local monotonicity: When $μ_{a}^{p} (x_{i})$ is given and $μ_{a}^{p} (x_{i}) \neq μ_{a}^{q} (x_{i})$ , c_μ (a_p, a_q) monotonically increases on interval $(μ_{a}^{p} (x_{i}), 1]$ , and monotonically decreases on interval $[0, μ_{a}^{p} (x_{i}))$ .

Symmetry: c_μ (a_p, a_q) = c_μ (a_q, a_p).

Properties (1) and (3) are easily obtained from Equation (1). The following is only a proof of property (2).

Proof ∀a_p, a_q ∈ A, if $μ_{a}^{q} (x_{i})$ is given, and the interval range of $μ_{a}^{q} (x_{i})$ is $(μ_{a}^{p} (x_{i}), 1]$ . When $μ_{a}^{q} (x_{i})$ increases, $| μ_{a}^{p} (x_{i}) - μ_{a}^{q} (x_{i}) |$ will increase accordingly, and because of $μ_{a}^{p} (x_{i}) \neq μ_{a}^{q} (x_{i})$ , we have $| μ_{a}^{p} (x_{i}) - μ_{a}^{q} (x_{i}) | \leq$ $μ_{a}^{p} (x_{i}) + μ_{a}^{q} (x_{i})$ . Therefore, c_μ (a_p, a_q) is monotonically increasing. Similarly, c_μ (a_p, a_q) monotonically decreases on interval $[0, μ_{a}^{p} (x_{i}))$ holds.

Part 2: Non-membership correlation degree.

The analysis of non-membership correlation is the same as the membership correlation, which can be calculated as follows: $c_{ν} (a_{p}, a_{q}) = - | ν_{a}^{p} (x_{i}) - ν_{a}^{q} (x_{i}) | ln \frac{| ν_{a}^{p} (x_{i}) - ν_{a}^{q} (x_{i}) |}{ν_{a}^{p} (x_{i}) + ν_{a}^{q} (x_{i})}$ (2)

Similarly, when $ν_{a}^{p} (x_{i}) = ν_{a}^{q} (x_{i}) \neq 0$ , let c_ν (a_p, a_q) =1, and let c_ν (a_p, a_q) =0 in the following three cases: $(I) {\begin{matrix} ν_{a}^{p} (x_{i}) = 0 \\ ν_{a}^{q} (x_{i}) \neq 0 \end{matrix} (II) {\begin{matrix} ν_{a}^{p} (x_{i}) \neq 0 \\ ν_{a}^{q} (x_{i}) = 0 \end{matrix} (III) {\begin{matrix} ν_{a}^{p} (x_{i}) = 0 \\ ν_{a}^{q} (x_{i}) = 0 \end{matrix}$

The properties of non-membership correlation degree are as follows:

Boundedness: 0 ≤ c_ν (a_p, a_q) ≤1;

Local monotonicity: When $ν_{a}^{p} (x_{i})$ is given and $ν_{a}^{p} (x_{i}) \neq ν_{a}^{q} (x_{i})$ , c_ν (a_p, a_q) monotonically increases on interval $(ν_{a}^{p} (x_{i}), 1]$ , and monotonically decreases on interval $[0, ν_{a}^{p} (x_{i}))$ .

Symmetry: c_ν (a_p, a_q) = c_ν (a_q, a_p).

Part 3: Disturbance coefficient.

Compared with membership description and non-membership description, hesitation degree is caused by some unknown information. From this perspective, we can think that it is precisely because of a variety of random and uncertain independent factors (or related factors, here we assume that they are independent) collectively cause the existence of hesitation degree. According to the central limit theorem, the influence caused by these factors obeys the Gaussian distribution. For the convenience of analysis, it is assumed to follow the standard Gaussian distribution. Hence, the perturbation coefficients of the attributes a_p and a_q with respect to the object x_i can be expressed as: $ɛ_{η} (a_{p}, a_{q}) = \frac{1}{\sqrt{2 π}} exp (- \frac{(η_{a}^{p} (x_{i}) - η_{a}^{q} (x_{i}))^{2}}{2})$ (3)

Remark 2. When $η_{a}^{p} (x_{i}) = η_{a}^{q} (x_{i}) = 0$ , it means that the intuitionistic fuzzy sets corresponding to attribute a_p and attribute a_q have been degraded to ordinary fuzzy sets, i.e., they no longer have the perturbation coefficient, so let ɛ_η (a_p, a_q) = 0.

In summary, the correlation between the attributes a_p and a_q with respect to the object x_i is calculated as follows: $\begin{matrix} c (a_{p} (x_{i}), a_{q} (x_{i})) = c_{μ} (a_{p} (x_{i}), a_{q} (x_{i})) \\ + c_{ν} (a_{p} (x_{i}), a_{q} (x_{i})) + ɛ_{η} (a_{p} (x_{i}), a_{q} (x_{i})) \end{matrix}$ (4)

Therefore, the correlation degree between any two attributes on domain U can be expressed as: $\begin{matrix} c (a_{p}, a_{q}) = \frac{1}{m} \sum_{i = 1}^{m} \sum_{j = 1}^{n} c_{μ} (a_{j} (x_{i}), a_{k} (x_{i})) + c_{ν} (a_{j} (x_{i}), \\ a_{k} (x_{i})) + ɛ_{η} (a_{j}, (x_{i}) a_{k} (x_{i})), k = 1, 2, \dots, n . \end{matrix}$ (5)

Because Equations (1)–(3) are all symmetrical, the attribute correlation matrix in Definition 11 can be simplified into the form of a lower triangular matrix: $\begin{matrix} \begin{matrix} a_{1} & a_{2} & \dots & a_{n - 1} & a_{n} \end{matrix} \\ ℵ_{mn} = \begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{n - 1} \\ a_{n} \end{matrix} [\begin{matrix} c_{1}^{1} \\ c_{2}^{1} c_{2}^{2} \\ ⋮ ⋮ ⋱ \\ c_{n - 1}^{1} c_{n - 1}^{2} \dots c_{n - 1}^{n - 1} \\ c_{n}^{1} c_{n}^{2} \dots c_{n}^{n - 1} c_{n}^{n} \end{matrix}] \end{matrix}$

Definition 12. Let Ω = (U, A, ℵ) be an Association Space, ∀a_r, a_s, a_t ∈ A, the correlation degree between multiple attributes is as follows:

$\begin{matrix} c (a_{r} a_{s} a_{t} | a_{r}) & = min c {a_{{rst} - {r}} | a_{r}} + σ \\ = min {a_{r} a_{s}, a_{r} a_{t}} + σ \end{matrix}$ (6) Where $σ = \sqrt{max c {a_{{rst} - {r}} | a_{r}} - min c {a_{{rst} - {r}} | a_{r}}}$ is the attribute correlation similarity coefficient, and let c (a_s| ∅) = c (∅ | ∅) =0. If Equation (6) is generalized to the degree of joint correlation between any number of attributes, we can get: $\begin{matrix} c (a_{1}, \dots, a_{n} | a_{1}, a_{2}, \dots, a_{j}) = min c {a_{{1 \dots n} - {1 \dots j}} | a_{{1 \dots j}}} + σ \\ = min {a_{j + 1} a_{1}, \dots, a_{n} a_{1}, a_{j + 2} a_{2}, \dots, a_{n} a_{2}, \dots, a_{n} a_{j}} + σ \end{matrix}$ (7)

Theorem 2. ∀a_r, a_s, a_t ∈ A, c₁ (a_ra_sa_t|a_r) = c₂ (a_sa_t|a_r).

Proof According to Definition 12, we can know c₂ (a_sa_t|a_r) = c (a_{rst}-{∅}|a_r) + σ₂ and min {a_ra_r, a_ra_s, a_ra_t} + σ₂ = min {a_ra_s, a_ra_t} + σ₂. Similarly, we can obtain min {a_ra_s, a_ra_t} +σ₁ = c (a_ra_sa_t|a_r). Since the attribute correlation similarity coefficient σ₁ = σ₂, c (a_ra_sa_t|a_r) = c (a_sa_t|a_r) holds.

3.2 Intuitionistic fuzzy three-way concept lattice

3.2.1 Intuitionistic fuzzy concept lattice based on the degree of attribute correlation

Definition 13. Let $\tilde{K} = (U, A, \tilde{I}, ℵ)$ be an intuitionistic fuzzy formal context (IFFC) based on Association Space. Element x_i ∈ U is an object on U, and element a_j ∈ A is an attribute on A. The intuitionistic fuzzy binary relationship between U and A is $\tilde{I} = {< (x, a), μ_{\tilde{I}}^{a} (x), ν_{\tilde{I}}^{a} (x) > | (x, a) \in U \times A}$ , which is abbreviated as $\tilde{I} (x, a) = (μ_{\tilde{I}}^{a} (x), ν_{\tilde{I}}^{a} (x))$ , $μ_{\tilde{I}}^{a} \in [0, 1]$ , $ν_{\tilde{I}}^{a} \in [0, 1]$ , $0 \leq μ_{\tilde{I}}^{a} + ν_{\tilde{I}}^{a}) \leq 1$ .ℵ is the attribute correlation matrix on A. ∀X ∈ (U), $\forall \tilde{B} \subseteq IF (A)$ , $\tilde{B} (x) \in I (x, a)$ , and the definition of operator * is as follows: $\begin{matrix} X^{*} = {a \in A | \forall x \in X, \tilde{I} (x, a) \geq (δ, ρ) \otimes ψ} \\ = {a \in A | \forall x \in X, {(a, μ (x_{i}) \geq δ, ν (x_{i}) \leq ρ)} \otimes ψ} \end{matrix}$ (8)

∀x_i ∈ U, ∀ a_j ∈ A, $ψ ≜ c_{p}^{q} \geq κ (κ \geq 0)$ , δ =∨ {∧ ${{\tilde{I}}_{μ} (x_{i}, a_{j})}}$ , $ρ = \land {\lor {{\tilde{I}}_{ν} (x_{i}, a_{j})}}$ , and the special symbol “ψ” indicates a given condition. For instance, X ⊗ ψ means all elements in X that meet the condition of ψ. $\begin{matrix} {\tilde{B}}^{*} = {x \in U | \forall a \in A, \tilde{I} (x, a) \geq \tilde{B} (δ, ρ) \otimes ψ} \\ = {x \in U | \forall a \in A, {(x_{i}, {\tilde{B}}_{μ} (a) \geq δ, {\tilde{B}}_{ν} (a)) \leq ρ)} \otimes ψ} \end{matrix}$ (9)

In Equation (9), ${\tilde{B}}_{μ} (a)$ and ${\tilde{B}}_{ν} (a)$ respectively represent the membership and non-membership of $\tilde{B} (x)$ with respect to attribute a under the intuitionistic fuzzy relationship $\tilde{I}$ .

Theorem 3. $(X^{*}, {\tilde{B}}^{*})$ is a Galois connection between the partially ordered sets ((U) , ≤) and (IF (A) , ≤).

Proof $\forall {\tilde{B}}_{1} \subseteq {\tilde{B}}_{2} \subseteq \tilde{B} \subseteq IF (A)$ , ∀X₁ ⊆ X₂ ⊆ X ⊆ (U), if $a_{j} \in X_{2}^{*}$ , then for any x_i ∈ X₂, $\tilde{I} (x_{i}, a_{j}) \otimes ψ$ holds. Suppose that x_o ∈ X₂, we can get $\tilde{I} (x_{o}, a_{j}) \otimes ψ$ . And because of X₁ ⊆ X₂, we have $X_{2}^{*} \subseteq X_{1}^{*}$ . Similarly, ${\tilde{B}}_{2}^{*} \subseteq {\tilde{B}}_{1}^{*}$ also holds. Moreover, since $X ({\tilde{B}}^{*})^{*} \Leftrightarrow \tilde{B} \subseteq X ({\tilde{B}}^{*})^{*}$ , we can get $\tilde{B} (X^{*})^{*} \Leftrightarrow X \subseteq \tilde{B} (X^{*})^{*}$ for any a_j ∈ X^*, $x_{i} \in {\tilde{B}}^{*}$ . Hence, $(X^{*}, {\tilde{B}}^{*})$ is a Galois connection between ((U) , ≤) and (IF (A) , ≤).

Definition 14. For any two-tuples $(X, \tilde{B})_{κ}$ , if $X^{*} = \tilde{B}$ and ${\tilde{B}}^{*} = X$ , then $(X, \tilde{B})_{κ}$ is said to be an intuitionistic fuzzy formal concept on $\tilde{K}$ , referred to as κ-concept for short. X and $\tilde{B}$ are κ-extension and κ-intension of $(X, \tilde{B})$ , respectively. The set of all κ-concepts on $\tilde{K}$ is represented by $\underline{B} (\tilde{K})_{κ}$ . It can be seen that $(\underline{B} (\tilde{K})_{κ}, \leq)$ is a concept lattice.

$\forall (X_{1}, B_{1}), (X_{2}, B_{2}) \in \underline{B} (\tilde{K})_{κ}$ , if X₁ ⊆ X₂ (B₁ ⊇ B₂), then (X₁, B₁) is said to be a κ-sub-concept of (X₂, B₂), and (X₂, B₂) is a κ-super-concept of (X₁, B₁), abbreviated as (X₁, B₁) ≤ (X₂, B₂) _κ. If there is not a κ-concept (X₃, B₃) that makes (X₁, B₁) ≤ (X₃, B₃) ≤ (X₂, B₂), then (X₁, B₁) is called the direct sub-concept of (X₂, B₂), and (X₂, B₂) is called the parent concept of (X₁, B₁), which is abbreviated as (X₁, B₁) _κ ≺ (X₂, B₂) _κ.

Example 4. An IFFC $\tilde{K}$ based on Association Space is shown in Table 3. In order to show as much information as possible in $\tilde{K}$ , let κ ≥ 0.

Table 3

An IFFC $\tilde{K}$ based Association Space

I	a	b	c	d
x ₁	(0.7, 0.3)	(0.4, 0.5)	(0.7, 0.2)	(0.8, 0.0)
x ₂	(0.5, 0.3)	(0.6, 0.2)	(0.0, 0.6)	(0.7, 0.2)
x ₃	(0.6, 0.2)	(0.7, 0.1)	(0.1, 0.5)	(0.6, 0.2)
x ₄	(0.7, 0.2)	(0.6, 0.3)	(0.5, 0.2)	(0.5, 0.4)
x ₅	(0.4, 0.3)	(0.4, 0.5)	(0.3, 0.6)	(0.5, 0.3)

According to Equations (5) and (6), we can calculate the degree of attribute correlation in $\tilde{K}$ and obtain the attribute correlation degree matrix ℵ: $\begin{matrix} \begin{matrix} a & b & c & d \end{matrix} \\ ℵ_{4 \times 4} = \begin{matrix} a \\ b \\ c \\ d \end{matrix} [\begin{matrix} 1 \\ 0.169 & 1 \\ 0.163 & 0.134 & 1 \\ 0.217 & 0.160 & 0.157 & 1 \end{matrix}] \end{matrix}$

The corresponding concepts of Table 3 are shown in Fig. 4.

Fig. 4

Intuitionistic fuzzy concept lattice based on Association Space for the formal context $\tilde{K}$ .

It can be found that compared with the classical concept lattice, the intuitionistic fuzzy concept lattice based on Association Space shows the intensions correlation degree between each parent concept and its direct child concept. Therefore, by setting different values of κ, we can show the intension that we are only interested in or the degree of correlation reaches a certain range, so as to achieve the purpose of concept reduction.

3.2.2 IF3WCL based on attribute correlation degree

Definition 15. Let $\tilde{K} = (U, A, \tilde{I}, ℵ)$ be an IFFC based on Association Space. $\forall X_{1}, X_{2}, X \in (U), \forall {\tilde{B}}_{1}, {\tilde{B}}_{2}, \tilde{B} \subseteq IF (A)$ , the definition of operator $\bar{*}$ is as follows: $\begin{matrix} X^{\bar{*}} = {a \in A | \forall x \in X, \tilde{I} (x, a) < (δ, ρ) \otimes ψ} \\ = {a \in A | \forall x \in X, {(a, μ (x_{i}) \geq δ, ν (x_{i}) \leq ρ)} \otimes ψ} \end{matrix}$ (10) $\begin{matrix} {\tilde{B}}^{\bar{*}} = {x \in U | \forall a \in A, \tilde{I} (x, a) < \tilde{B} (δ, ρ) \otimes ψ} \\ = {x \in U | \forall a \in A, {(x_{i}, {\tilde{B}}_{μ} (a) < δ, {\tilde{B}}_{ν} (a)) > ρ)} \otimes ψ} \end{matrix}$ (11)

It can be concluded that under the degree of correlation κ, any object in X “jointly not possessed” the attributes in $X^{\bar{*}}$ , and any attribute in A is “jointly not possessed” by the objects in ${\tilde{B}}^{\bar{*}}$ .

The following is the definition of intuitionistic fuzzy three-way operators $\tilde{⋖}$ : $X^{\tilde{⋖}} = (X^{*}, X^{\bar{*}}), {\tilde{B}}^{\tilde{⋖}} = ({\tilde{B}}^{*}, {\tilde{B}}^{\bar{*}}) .$

The corresponding inverse operator $\tilde{⋗}$ is defined as follows: $\begin{matrix} ({\tilde{B}}_{1}, {\tilde{B}}_{2})^{\tilde{⋗}} = {x \in U | x \in {\tilde{B}}_{1}^{*} \cap x \in {\tilde{B}}_{2}^{\bar{*}}}, \\ (X_{1}, X_{2})^{\tilde{⋗}} = {a \in A | a \in X_{1}^{*} \cap a \in X_{2}^{\bar{*}}} . \end{matrix}$

Proposition 2. Intuitionistic fuzzy three-way operators have the following properties:

$X_{1} \subseteq X_{2} \Rightarrow X_{2}^{\tilde{⋖}} \subseteq X_{1}^{\tilde{⋖}}, {\tilde{B}}_{1} \subseteq {\tilde{B}}_{2} \Rightarrow {\tilde{B}}_{2}^{\tilde{⋖}} \subseteq {\tilde{B}}_{1}^{\tilde{⋖}}$ .

$X \subseteq X^{\tilde{⋖} \tilde{⋗}}, \tilde{B} \subseteq {\tilde{B}}^{\tilde{⋖} \tilde{⋗}}$ .

$X^{\tilde{⋖}} = X^{\tilde{⋖} \tilde{⋗} \tilde{⋖}}, {\tilde{B}}^{\tilde{⋖}} = {\tilde{B}}^{\tilde{⋖} \tilde{⋗} \tilde{⋖}}$ .

$(X_{1} \cup X_{2})^{\tilde{⋖}} = X_{1}^{\tilde{⋖}} \cap X_{2}^{\tilde{⋖}}, ({\tilde{B}}_{1} \cup {\tilde{B}}_{2})^{\tilde{⋖}} = {\tilde{B}}_{1}^{\tilde{⋖}} \cap {\tilde{B}}_{2}^{\tilde{⋖}}$ .

Due to the similarity of the proof method, only the property (1) is proved below.

Proof (1) Let $X_{1}^{\tilde{⋖}} = (X_{1}^{*}, X_{1}^{\bar{*}})$ , if $a_{1} \in X_{1}^{*}$ , according to definitions (12) and (14), we have {(a₁, μ (x_i)≥ δ, ν (x_i) ≤ ρ| ∀ x_i ∈ X₁)}. And $\forall a_{2} \in X_{1}^{\bar{*}}$ , {(a₂, μ (x_i) < δ, ν (x_i) > ρ| ∀ x_i ∈ X₁)} holds. Since X₁ ⊆ X₂, $\forall x_{i}^{•} \in X_{2}$ , we can get ${(a_{1}, μ (x_{i}^{•}) \geq δ, ν (x_{i}^{•}) \leq ρ)}$ and ${(a_{2}, μ (x_{i}^{•}) < δ, ν (x_{i}^{•}) > ρ)}$ . Hence, $a_{1} \in X_{2}^{*}$ , $a_{2} \in X_{2}^{\bar{*}}$ , that is, $X_{2}^{\tilde{⋖}} \subseteq X_{1}^{\tilde{⋖}}$ . ${\tilde{B}}_{1} \subseteq {\tilde{B}}_{2} \Rightarrow {\tilde{B}}_{2}^{\tilde{⋖}} \subseteq {\tilde{B}}_{1}^{\tilde{⋖}}$ obviously holds.

Definition 16. Let $\tilde{K} = (U, A, \tilde{I}, ℵ)$ be an IFFC based on Association Space. ∀X, Y ∈ (U), $\forall \tilde{B}, \tilde{C} \subseteq IF (A)$ , if and only if $X_{κ}^{\tilde{⋖}} = (\tilde{B}, \tilde{C})$ and $(\tilde{B}, \tilde{C})_{κ}^{\tilde{⋗}} = X$ , then $(X, (\tilde{B}, \tilde{C}))_{κ}$ is said to be the object-induced intuitionistic fuzzy three-way concepts (OE_κ concept for short). X and $(\tilde{B}, \tilde{C})$ are κ-extension and κ-intension of $(X, (\tilde{B}, \tilde{C}))_{κ}$ , respectively.

∀X, Y ∈ (U), $\forall \tilde{B}, \tilde{C}, \tilde{E}, \tilde{F} \subseteq IF (U)$ , $(X, (\tilde{B}, \tilde{C}))_{κ}$ and $(Y, (\tilde{E}, \tilde{F}))_{κ}$ are both OE_κ concepts. If $(X, (\tilde{B}, \tilde{C})) \leq (Y, (\tilde{E}, \tilde{F})) \Leftrightarrow X \subseteq Y ((\tilde{B}, \tilde{C}) \supseteq (\tilde{E}, \tilde{F}))$ , then $(X, (\tilde{B}, \tilde{C}))_{κ}$ is a sub-concept of $(Y, (\tilde{E}, \tilde{F}))_{κ}$ . The set of all OE_κ concepts on $\tilde{K}$ is recorded as ${OE}_{κ} (\tilde{K})$ .

Example 5. In the IFFC $\tilde{K}$ corresponding to Table 3, let κ = 0.2. $({x_{2}, x_{3}})^{\tilde{⋖}} = ({abd}, {c})_{0.302}$ , $({abd}, {c})_{0.302}^{\tilde{⋗}} = {x_{2}, x_{3}}$ , $({x_{1}, x_{5}})^{\tilde{⋖}} = ({d},$ {b}) _0.16, $({d}, {b})_{0.16}^{\tilde{⋗}} = {x_{1}, x_{5}}$ . Due toκ = 0.2 > 0.16, ({x₁, x₃} , ({abd} , {c}) _0.302) is the _0.2 concept.

Theorem 4. $OEB (\tilde{K})_{κ}$ is a complete lattice, and the supremum and infimum of $(X, (\tilde{B}, \tilde{C}))$ and $(Y, (\tilde{E}, \tilde{F}))$ are respectively: $\begin{matrix} (X, (\tilde{B}, \tilde{C}))_{κ} \lor (Y, (\tilde{E}, \tilde{F}))_{κ} = (X \cup Y)_{κ}^{\tilde{⋖} \tilde{⋗}}, ((\tilde{B}, \tilde{C})_{κ} \cap (\tilde{D}, \tilde{E})_{κ}), \\ (X, (\tilde{B}, \tilde{C}))_{κ} \land (Y, (\tilde{E}, \tilde{F}))_{κ} = ((X \cap Y)_{κ}, (\tilde{B}, \tilde{C})_{κ} \cup (\tilde{D}, \tilde{E}))_{κ}^{\tilde{⋖} \tilde{⋗}}) . \end{matrix}$

Proof Since $(X \cup Y)_{κ} = (\tilde{B}, \tilde{C})_{κ}^{\tilde{⋗}} \cup (\tilde{E}, \tilde{F})_{κ}^{\tilde{⋗}} = ((\tilde{B}, \tilde{C}) \cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗}}$ , we have $(X, (\tilde{B}, \tilde{C}))_{κ} \lor (Y, (\tilde{E}, \tilde{F}))_{κ} = (((\tilde{B}, \tilde{C}) \cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗}},$ $((\tilde{B}, \tilde{C}) \cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗} \tilde{⋖}})$ , so we only need to prove $(((\tilde{B}, \tilde{C})$ $\cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗}}, ((\tilde{B}, \tilde{C}) \cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗} \tilde{⋖}}))$ is a OE_κ concept. Because of $((\tilde{B}, \tilde{C}), (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗} \tilde{⋖} \tilde{⋗}} = ((\tilde{B}, \tilde{C})^{*} \cap (\tilde{E}, \tilde{F})^{\bar{*}})_{κ}^{\tilde{⋖} \tilde{⋗}} = (((\tilde{B}, \tilde{C})^{*}$ $\cap (\tilde{E}, \tilde{F})^{\bar{*}})_{κ}^{*} \supseteq (\tilde{B}, \tilde{C})_{κ}^{*} \cap (\tilde{E}, \tilde{F})_{κ}^{\bar{*}}$ , we can get $((\tilde{B}, \tilde{C}), (\tilde{E}, \tilde{F}))_{κ}$ $\subseteq ((\tilde{B}, \tilde{C}), (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗} \tilde{⋖}}$ and $((\tilde{B}, \tilde{C}), (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗} \tilde{⋖} \tilde{⋗}} \subseteq ((\tilde{B}, \tilde{C}), (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗}}$ .

Thus, $(((\tilde{B}, \tilde{C}) \cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗}}, ((\tilde{B}, \tilde{C}) \cap (\tilde{E}, \tilde{F}))_{κ}^{\tilde{⋗} \tilde{⋖}}))$ is a OE_κ concept lattice, and the infimum is similarly provable.

Example 6. The corresponding ₀ concepts of Table 3 are shown in Fig. 5.

Fig. 5

OE₀ concept lattice for the formal context $\tilde{K}$ .

Similarly, the following is the definition of the IF3WCL of attribute-induced.

Definition 17. Let $\tilde{K} = (U, A, \tilde{I}, ℵ)$ be an IFFC based on Association Space. ∀X, Y ∈ (U), $\forall \tilde{B} \subseteq IF (A)$ , if and only if $X, Y)^{\tilde{⋗}} = \tilde{B}$ and ${\tilde{B}}^{\tilde{⋖}} = (X, Y)$ , then $((X, Y), \tilde{B})_{κ}$ is said to be the attribute-induced intuitionistic fuzzy three-way concepts (AE_κ concept for short). (X, Y) and $\tilde{B}$ are κ-extension and κ-intension of $((X, Y), \tilde{B})_{κ}$ , respectively.

∀X, Y, Z, W ∈ P (U), $\forall \tilde{B}, \tilde{C} \subseteq IF (A)$ , $((X, Y), \tilde{B})$ and $((Z, W), \tilde{C})$ are both AE_κ concepts. If $((X, Y), \tilde{B}) \leq ((Z, W),$ $\tilde{C}) \Leftrightarrow (X, Y) \subseteq (Z, W) (\Leftrightarrow \tilde{B} \supseteq \tilde{C})$ , then $((X, Y), \tilde{B})_{κ}$ is a sub-concept of $((Z, W), \tilde{C})_{κ}$ , and $((Z, W), \tilde{C})_{κ}$ is the parent concept of $((X, Y), \tilde{B})_{κ}$ . The set of all AE_κ concepts on $\tilde{K}$ is recorded as ${AE}_{κ} (\tilde{K})$ .

Example 7. In the IFFC $\tilde{K}$ corresponding to Table 3, let κ = 0.2, ${b, c}_{0.134}^{\tilde{⋖}} = ({x_{4}}, {x_{5}})$ , $({x_{4}}, {x_{5}})^{\tilde{⋗}} = {b, c}_{0.134}$ , ${a, d}_{0.217}^{\tilde{⋖}} = ({x_{1}, x_{2},$ x₃} , ∅), $({x_{1}, x_{2}, x_{3}}, \emptyset)^{\tilde{⋗}} = {a, d}_{0.217}$ . Since κ = 0.2 > 0.134, (({x₁, x₂, x₃} , ∅) , {a, d} _0.217) is the _0.2 concept.

Theorem 5. $\forall ((X, Y), \tilde{B}), ((Z, W), \tilde{C}) \in_{κ} (\tilde{K})$ , then their supremum and infimum are as follows: $\begin{matrix} ((X, Y), \tilde{B})_{κ} \lor ((Z, W), \tilde{C})_{κ} = (((X, Y) \cup (Z, W))_{κ}^{\tilde{⋗} \tilde{⋖}}, (\tilde{B} \cap \tilde{C})_{κ}), \\ ((X, Y), \tilde{B})_{κ} \land ((Z, W), \tilde{C})_{κ} = (((X, Y)_{κ} \cap (Z, W)_{κ}), (\tilde{B} \cup \tilde{C})_{κ}^{\tilde{⋖} \tilde{⋗}}) . \end{matrix}$

Example 8. The corresponding ₀ concepts of Table 3 are shown in Fig. 6.

Fig. 6

AE₀ concept lattice for the formal context $\tilde{K}$ .

4 The relationship between IF3WCL and 3WD

The main idea of 3WD is to divide the domain into three disjoint parts, namely the positive domain POS, the negative domain NEG and the boundary domain BND. Furthermore, we adopt corresponding strategies for different domains, and finally achieve the purpose of optimal decision-making. In IF3WCL, the intension of each concept OE_κ is essentially a set of intuitionistic fuzzy sets that satisfy the (δ, ρ) condition, expressed as follows: $\begin{matrix} \tilde{B} = \underset{a \in A}{\cup} {\tilde{I} (x, a) \geq (δ, ρ) | (a, μ (x_{i}) \geq δ, ν (x_{i}) \leq ρ, π (x_{i})} \otimes ψ, \\ \tilde{C} = \underset{a \in A}{\cup} {\tilde{I} (x, a) < (δ, ρ) | (a, μ (x_{i}) < δ, ν (x_{i}) > ρ, π (x_{i})} \otimes ψ . \end{matrix}$

To make this more precise, the intension of each OE_κ concept is composed of a membership degree and a non-membership degree that simultaneously satisfies the above-mentioned conditional attributes. For any OE_κ concept, its three-way division rules about intension can be expressed as: $\begin{matrix} POS (X) : {a \in A | \tilde{B} : \underset{a \in A}{\cup} \tilde{I} (x, a) \geq (δ, ρ)_{κ}}, \\ NEG (X) : {a \in A | \tilde{C} : \underset{a \in A}{\cup} \tilde{I} (x, a) < (δ, ρ)_{κ}}, \\ BND (X) : {a \in A | (\tilde{B} \cup \tilde{C})^{C} : (POS (X) \cup NEG (X))_{κ}^{C}} . \end{matrix}$

Example 9. ({x₁, x₅} , ({d} , {b}) _0.16) is a _0.1 concept in $\tilde{K}$ according Table 3, and its 3WD division is as follows: $\begin{matrix} {\tilde{B}}_{1} = {a \in A | (d, 0.8 (x_{1}), 0 (x_{1})), (d, 0.5 (x_{5}), 0.3 (x_{5}))}, \\ {\tilde{C}}_{1} = {a \in A | (b, 0.4 (x_{1}), 0.5 (x_{1})), (b, 0.4 (x_{5}), 0.5 (x_{5}))}, \\ ({\tilde{B}}_{1} \cup {\tilde{C}}_{1})^{C} = {a \in A | {(a, 0.7 (x_{1}), 0.3 (x_{1})), (a, 0.4 (x_{5}), \\ 0.3 (x_{5}))}, {(c, 0.7 (x_{1}), 0.2 (x_{1})), (c, 0.3 (x_{5}), 0.6 (x_{5}))}} \end{matrix}$

Therefore, the positive domain is POS (A) = {d}, the negative domain is NEG (A) = {b}, and the boundary domain is BND (A) = {a, c} of concept ({x₁, x₅} , ({d} , {b}) _0.16).

Similarly, the three-way division rules for the extension of the AE_κ concept are as follows: $\begin{matrix} POS (X) : {x \in U | X : \underset{a \in A}{\cup} \tilde{I} (x, a) \geq \tilde{B} (δ, ρ)_{κ}}, \\ NEG (X) : {x \in U | Y : \underset{a \in A}{\cup} \tilde{I} (x, a) < \tilde{B} (δ, ρ)_{κ}}, \\ BND (X) : {x \in U | (X \cup Y)^{C} : (POS (X) \cup NEG (X))_{κ}^{C}} . \end{matrix},$

Example 10. ({x₄} , {x₅}) , (bc) _0.134) is a _0.1 concept in $\tilde{K}$ according Table 3, and its 3WD division is as follows: $\begin{matrix} X_{1} = {x \in U | (x_{4}, {\tilde{X}}_{1} = {0.6 (b), 0.3 (b), (0.5 (c), 0.2 (c)}}, \\ Y_{1} = {x \in U | (x_{5}, {\tilde{X}}_{2} = {0.4 (b), 0.5 (b), (0.3 (c), 0.6 (c)}}, \\ (X_{1} \cup Y_{1})^{C} = {x \in X | {{\tilde{X}}_{3} = {x_{1}, {(0.4 (b), 0.5 (b)), \\ (0.7 (c), 0.2 (c))}, {(x_{2}, {(0.6 (b), 0.2 (b)), (0 (c), 0.6 (c))}, \\ {(x_{3}, {(0.7 (b), 0.1 (b)), (0.1 (c), 0.5 (c))}} . \end{matrix}$

Therefore, the positive domain is POS (X) = {x₄}, the negative domain isNEG (X) = {x₅}, and the boundary domain is BND (X) = {x₁, x₂, x₃} of concept ({x₄} , {x₅}) , (bc) _0.134).

4.1 The secondary decision of boundary intension object in the OE_κ concept

In the case of insufficient information, the boundary domain provides a good buffer space for the optimal decision, but the boundary object cannot always be in a delayed state, and it will eventually be transformed into two-way decisions, i.e., the positive or negative domains. To scientifically and objectively solve the problem of reclassification of boundary domain objects in the OE_κ concept, a secondary decision strategy is given below.

Definition 18. Let $\tilde{K} = (U, A, \tilde{I}, ℵ)$ be an IFFC based on Association Space. The membership and non-membership weights of attribute a_j are as follows: $ω_{μ} = {- \frac{1}{m} \sum_{i = 1}^{m} μ_{\tilde{I}}^{a} (x_{i}) ln \frac{μ_{\tilde{I}}^{a} (x_{i}) + 1}{μ_{\tilde{I}}^{⌢} a (x) + | M |}}$ (12) $ω_{ν} = {\frac{1}{m} \sum_{i = 1}^{m} ν_{\tilde{I}}^{a} (x_{i}) ln \frac{ν_{\tilde{I}}^{a} (x_{i}) + 1}{ν_{\tilde{I}}^{⌢} a (x) + | M |}}$ (13)

$μ_{\tilde{I}}^{⌢} a (x)$ and $ν_{\tilde{I}}^{⌢} a (x)$ are the membership and non-membership means of the attribute a_j, respectively. |M| represents the cardinality of the set U. It is easy to find that the greater the certainty of the attribute a_j, the greater its weight.

Theorem 6. (1) $\forall a_{j} \in A, μ_{\tilde{I}}^{a} (x_{1}) \leq μ_{\tilde{I}}^{a} (x_{2}), ω_{μ} (x_{1})$ ≤ω_μ (x₂) .

(2) ∀a_j ∈ A, $ν_{\tilde{I}}^{a} (x_{1}) \geq ν_{\tilde{I}}^{a} (x_{2})$ , ω_ν (x₁) ≥ ω_ν (x₂).

Proof (1) Let $μ_{1}^{a} = \frac{μ_{\tilde{I}}^{a} (x_{1}) + 1}{μ_{\tilde{I}}^{⌢} a (x) + | M |}$ , $μ_{2}^{a} = \frac{μ_{\tilde{I}}^{a} (x_{2}) + 1}{μ_{\tilde{I}}^{⌢} a (x) + | M |}$ .

Since $0 \leq μ_{\tilde{I}}^{a} (x_{1}) \leq 1$ , $μ_{\tilde{I}}^{a} (x_{1}) \leq μ_{\tilde{I}}^{a} (x_{2})$ , we can get $0 \leq \ln μ_{1}^{a} \leq \ln μ_{2}^{a}$ . Thus, we need to proof $μ_{\tilde{I}}^{a} (x_{1}) \cdot \ln μ_{1}^{a} \leq μ_{\tilde{I}}^{a} (x_{2}) \cdot \ln μ_{2}^{a}$ . Let $f (i) = μ_{\tilde{I}}^{a} \cdot$ $ln μ_{i}^{a}$ , then $f^{'} (i) = 1 + ln μ_{i}^{a}$ , we have $f^{'} (i) = 1 + ln μ_{i}^{a} \geq (ln 1) + 1 > 0$ .

So f (i) is an increasing function on [0,1]. Since $μ_{\tilde{I}}^{a} (x_{1}) \leq μ_{\tilde{I}}^{a} (x_{2})$ , f (x₁) ≤ f (x₂), it is obvious that $μ_{\tilde{I}}^{a} (x_{1}) \cdot \ln μ_{1}^{a} \leq μ_{\tilde{I}}^{a} (x_{2}) \cdot \ln μ_{2}^{a}$ holds.

For the secondary decision problem of boundary objects in the OE_κ concept, we mainly need to consider two factors. In the first place, the 3WD division of IF3WCL is based on the degree of attribute correlation. In the second place, the objects in the boundary domain are closely related to the (ρ, δ) condition. So, considering the combination of these two factors and adding the influence of attribute weights, the calculation method of attribute central degree is given as follows: $ϖ_{μ} = \sum_{j = 1}^{n} \sum_{i = 1}^{m} ω_{μ}^{j} \cdot c_{ij}, ϖ_{ν} = \sum_{j = 1}^{n} \sum_{i = 1}^{m} ω_{ν}^{j} \cdot c_{ij}$ (14)

Definition 19. Let $(X, (\tilde{B}, \tilde{C}))$ be an OE_κ concept in $\tilde{K}$ , and a_j be an object in the boundary domain. χ = (ϖ_μ, ϖ_ν) and ℏ (a_j) represent the attribute central degree of the attributes of A and a, respectively. Under the extension X, the relationship between χ and ℏ (a_j) has the following three cases:

Remark 3. A solid line indicates that a boundary value is included, and a dotted line indicates that a boundary value is not included. If the central degree of the boundary object is completely within the range of χ, it will appear as POS-internal absorption, otherwise, it will show NEG-intension separation. The formal expression is as follows: $\forall a_{j} \in A, {(ϖ_{μ} \geq ℏ (a_{j})) \land (ϖ_{ν} \leq ℏ (a_{j}))} \in POS (X)$ (15) $\forall a_{j} \in A, {(ϖ_{μ} < ℏ (a_{j})) \land (ϖ_{ν} > ℏ (a_{j}))} \in NEG (X)$ (16)

For the situation of intension separation, we can use the Definition 9 to calculate and compare the Euclidean distance of χ and (ℏ _μ (a_j) , ℏ _ν (a_j)) for final division.

Proposition 3. For any OE_κ concept, as the $\tilde{I} (x, a)$ value of the boundary object changes, the variation ranges of intension absorption and intension separation are approximately consistent with the Gaussian distribution.

Proof Let $(X, (\tilde{B}, \tilde{C}))$ be an OE_κ concept, $a_{j} \in \tilde{B}$ , $a_{o} \in \tilde{C}$ . On the one hand, ∀x_i ∈ X, if $\tilde{I} (x_{i}, a_{j}) < \tilde{I} (x_{i}, a_{o})$ , the increase of $μ_{a}^{j} (x_{i})$ causes the membership correlation degree between a_j and a_o to increase, and reaches the peak when $μ_{a}^{j} (x_{i}) = μ_{a}^{o} (x_{i})$ . Correspondingly, c_μ (a_j (x_i) , a_o (x_i)) becomes smaller as $μ_{a}^{j} (x_{i})$ continues to increase, and it reaches a minimum when $μ_{a}^{j} (x_{i}) = 1$ . Therefore, the change in value approximates a Gaussian distribution. On the other hand, it can be known from Theorem 6 that both the membership weight ω_μ (a_j) and the non-membership weight ω_ν (a_j) are determined by their certainty degree, so as the value of $\tilde{I} (x_{i}, a_{j})$ changes, the central degree (ϖ_μ, ϖ_ν) of a_j also closely follows the Gaussian distribution. We know that the intension absorption and intension separation are positively related to the attribute central degree, so the proposition 3 holds.

Based on the above ideas and strategies, we give the secondary decision algorithm of object-induced IF3WCL.

Example 11. ({x₁, x₅} , ({d} , {b}) _0.16) is a OE_0.1 concept in $\tilde{K}$ according Table 3, and its boundary domain is BND = {a, c}. We can get ϖ_μ = (0.42, 0.38, 0.28, 0.43), ϖ_ν = (0.34, 0.37, 0.46, 0.32) and ℏ (a) = (0.51, 0.27), ℏ (c) = (0.48, 0.35), Since ℏ_μ (a) >0.42, ℏ_ν (a) <0.34, ℏ_μ (c) >0.28, ℏ_ν (c) <0.46.

Therefore, the secondary decision result is that both attributes a and c belong to the positive domain.

4.2 The secondary decision of boundary extension object in the AE_κ concept

Similarly, each AE_κ concept has a definite intension, and its extension is an intuitive fuzzy representation of intension. Therefore, the problem of re-decision on the boundary objects of the AE_κ concept is finally transformed into an analysis and processing problem for the intuitionistic fuzzy set in intension.

Definition 20. Let R = (ρ, δ) be the relationship on ${AE}_{κ} (\tilde{K})$ . P = (POS (U) , R) and N = (NEG (U) , R) respectively represent the positive domain space and negative domain space on U. Then the P-membership coverage degree and N-non-membership coverage degree of the extension object are defined as follows:

$\begin{matrix} {Cov}_{P} = (((X, Y), \tilde{B}) \in {AE}_{κ} B (\tilde{K}), \\ μ_{R} (x_{i}) \in P | {min}_{i = 1}^{m} {\frac{| μ_{R} (X_{i}) |}{| A |}}) \end{matrix}$ (17)

$\begin{matrix} {Cov}_{N} = (((X, Y), \tilde{B}) \in {AE}_{κ} B (\tilde{K}), \\ ν_{R} (x_{i}) \in N | m {ax}_{i = 1}^{m} {\frac{| ν_{R} (Y_{i}) |}{| A |}}) \end{matrix}$ (18)

In the Equations (17) and (18), | · | represents the cardinality of the set. In order to avoid full coverage (or full separation) of the boundary objects due to too low (or too high) coverage, the values 0(or 1) should be eliminated during the calculation.

Theorem 7. When|X| = |U|(or |X| =∅), Cov_P = 1(or Cov_P = 0).

Lemma 1. For any AE_κ concept, the higher Cov_P value, the richer extension X “jointly possessed", and the lower Cov_N value, the rarer extension Y “jointly not possessed".

Proof For any AE_κ concept, it is easy to know from Definition 20 and Theorem 7 that the more objects that meet the μ_R (X_i) condition, the larger its cardinality, and the higher the Cov_P value, that is to say, the more objects in the set X. In contrast, the fewer objects that meet the ν_R (Y_i)condition, the lower the Cov_N value, namely, the fewer objects in the set Y.

Definition 21. Let $((X, Y), \tilde{B})$ be an AE_κ concept in $\tilde{K}$ , and (Cov_P, Cov_N)be the P-membership coverage degree and N-non-membership coverage degree of U. Under the intension $\tilde{B}$ , there are two cases between extension (X, Y) and (Cov_P, Cov_N), as shown below:

If the boundary object x_i is completely covered by (Cov_P, Cov_N), it will appear as POS-extension coverage, otherwise, it will show NEG-extension separation, which can be expressed as: $\begin{matrix} \forall x_{i} \in U, {(X, Y) \subseteq ({Cov}_{P}, {Cov}_{N})} \in POS (\tilde{B}) \\ \Leftrightarrow {(x_{i}, μ_{R} (X_{i}) \geq {Cov}_{P}, ν_{R} (Y_{i}) \leq {Cov}_{N})} \in POS (\tilde{B}) \end{matrix}$ (19) $\begin{matrix} \forall x_{i} \in U, {(X, Y) ⊄ ({Cov}_{P}, {Cov}_{N})} \in NEG (\tilde{B}) \\ \Leftrightarrow {(x_{i}, μ_{R} (X_{i}) \geq {Cov}_{P}, ν_{R} (Y_{i}) \leq {Cov}_{N})}^{C} \in NEG (\tilde{B}) \end{matrix}$ (20)

According to the Definition 21 and Theorem 7, we give the secondary decision algorithm of attribute-induced IF3WCL as follows.

Example 12. ({x₄} , {x₅}) , (bc) _0.134) is a _0.1 concept in $\tilde{K}$ according Table 3, and its boundary domain is BND = {x₁, x₂, x₃}. According to Equations (19) and (20), we can obtain membership coverage and non-membership coverage as Cov_P = 0.25, Cov_N = 0.5. We can get, x₁ (((0.4 (b) , 0.7 (c)) , ((0.5 (b) , 0.2 (c))) ⊆ (Cov_P, Cov_N) , x₂ (((0.6 (b) , 0 (c)) , (0.2 (b) , 0.6 (c)))⊄ (Cov_P, Cov_N) , x₃ (((0.7 (b) , (0.1 (c)) , (0.1 (b) , 0.5 (c))) ⊄ (Cov_P, Cov_N) .

Therefore, the final results of the secondary decision are x₁ ∈ POS and x₂, x₃ ∈ NEG.

5 Numerical example

To illustrate the application of the proposed model, this section will consider an example of a multinational company, which wants to achieve the group’s strategic development goals through the best capital investment (adapted from [45]). After preliminary screening, there are three (U = {x₁, x₂, x₃}) investment plans: (1) x₁, car company; (2) x₂, food company; and (3) x₃, computer company. In addition, the evaluation indicators can be divided into five (G = {G₁, G₂G₃, G₄, G₅}) categories: (1) G₁, the impact; (4) G₄, the environmental impact; and (5) G₅, the development of the society, as shown in the following matrix: $\begin{matrix} \begin{matrix} G_{1} & G_{2} & G_{3} & G_{4} & G_{5} \end{matrix} \\ D = \begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix} [\begin{matrix} (0.2, 0.5) (0.4, 0.2) (0.5, 0.4) (0.3, 0.3) (0.7, 0.1) \\ (0.2, 0.7) (0.6, 0.3) (0.4, 0.3) (0.4, 0.4) (0.6, 0.1) \\ (0.2, 0.7) (0.5, 0.3) (0.4, 0.5) (0.3, 0.4) (0.6, 0.2) \end{matrix}] \end{matrix}$

In the following, we will make a decision on the above problems through the method proposed in this paper. The method involves the following steps:

Step 1. Utilize Equation (6) to calculate the correlation between each attribute. The obtained attribute correlation matrix is as follows: $\begin{matrix} \begin{matrix} G_{1} & G_{2} & G_{3} & G_{4} & G_{5} \end{matrix} \\ ℵ_{5 \times 5} = \begin{matrix} G_{1} \\ G_{2} \\ G_{3} \\ G_{4} \\ G_{5} \end{matrix} [\begin{matrix} 1 \\ 0.099 1 \\ 0.097 0.118 1 \\ 0.096 0.087 0.112 1 \\ 0.092 0.110 0.099 0.093 1 \end{matrix}] \end{matrix}$

Step 2. Utilize the attribute correlation information given in matrix ℵ and Equations (10) and (11), we can obtain the AE₀ (κ = 0) concepts and the corresponding decisions, as shown in Table 4.

Table 4
The extension and intension of AE₀ concept induced by attributes

Extension Intension POS NEG BND

¹ AE ₀ {U, U} {ϕ} – – –

² AE ₀ {{x₁, x₂, x₃} , ϕ} {G₅} {x₁, x₂, x₃} {ϕ} {ϕ}

³ AE ₀ {ϕ, {x₁, x₂, x₃}} {G₁, G₃, G₄} {ϕ} {x₁, x₂, x₃} {ϕ}

⁴ AE ₀ {ϕ, {x₃}}

{G₁, G₂, G₃, G₄} {ϕ} {x₃} {x₁, x₂}

⁵ AE ₀ {ϕ, ϕ} {G} {ϕ} {ϕ} {x₁, x₂, x₃}

	Extension	Intension	POS	NEG	BND
¹ AE ₀	{U, U}	{ϕ}	–	–	–
² AE ₀	{{x₁, x₂, x₃} , ϕ}	{G₅}	{x₁, x₂, x₃}	{ϕ}	{ϕ}
³ AE ₀	{ϕ, {x₁, x₂, x₃}}	{G₁, G₃, G₄}	{ϕ}	{x₁, x₂, x₃}	{ϕ}
⁴ AE ₀	{ϕ, {x₃}}
	{G₁, G₂, G₃, G₄}	{ϕ}	{x₃}	{x₁, x₂}
⁵ AE ₀	{ϕ, ϕ}	{G}	{ϕ}	{ϕ}	{x₁, x₂, x₃}

From Table 4, we can observe that a total of five AE₀ concepts are obtained. Since the intension of concept ¹AE₀ is empty set, we will not consider it in our analysis. In the other four cases, as the concept intension changes, the corresponding decision-making is also different. Taking ⁴AE₀ as an example, when the intension is {G₁, G₂, G₃, G₄}, the corresponding positive domain is {ϕ}, the negative domain is {x₃}, and the boundary domain is {x₁, x₂}. In other words, scheme x₃ cannot be adopted, and schemes x₁ and x₂ need be determined after further investigation. The interpretation of ²AE₀, ³AE₀, and ⁵AE₀ is similarly to ⁴AE₀.

Step 3. Calculate membership coverage and non-membership coverage, and make secondary decisions for boundary domain objects.

Table 4 shows that ⁴AE₀ and ⁵AE₀ need to make a secondary decision. For ⁴AE₀, using Equations (17) and (18), we get Cov_P = 0.2, Cov_N = 0.4. $\begin{matrix} x_{1} ((0.2 (G_{1}), 0.4 (G_{2}), 0.5 (G_{3}), 0.3 (G_{4})), \\ (0.5 (G_{1}), 0.2 (G_{2}), 0.4 (G_{3}), 0.3 (G_{4})) ⊄ ({Cov}_{P}, {Cov}_{N}), \\ x_{2} ((0.2 (G_{1}), 0.6 (G_{2}), 0.4 (G_{3}), 0.4 (G_{4})), \\ (0.7 (G_{1}), 0.3 (G_{2}), 0.3 (G_{3}), 0.4 (G_{4})) ⊄ ({Cov}_{P}, {Cov}_{N}) . \end{matrix}$ Similarly, for $_{0}^{5}$ , we get Cov_P = 0.2, Cov_N = 0.6. $\begin{matrix} x_{1} ((0.2 (G_{1}), 0.4 (G_{2}), 0.5 (G_{3}), 0.3 (G_{4}), 0.7 (G_{5})), (0.5 (G_{1}), \\ 0.2 (G_{2}), 0.4 (G_{3}), 0.3 (G_{4}), 0.1 (G_{5})) \subseteq ({Cov}_{P}, {Cov}_{N}), \\ x_{2} ((0.2 (G_{1}), 0.6 (G_{2}), 0.4 (G_{3}), 0.4 (G_{4}), 0.6 (G_{5})), (0.7 (G_{1}), \\ 0.3 (G_{2}), 0.3 (G_{3}), 0.4 (G_{4}), 0.1 (G_{5})) ⊄ ({Cov}_{P}, {Cov}_{N}) . \\ x_{3} ((0.2 (G_{1}), 0.5 (G_{2}), 0.4 (G_{3}), 0.3 (G_{4}), 0.6 (G_{5})), (0.7 (G_{1}), \\ 0.3 (G_{2}), 0.5 (G_{3}), 0.4 (G_{4}), 0.2 (G_{5})) ⊄ ({Cov}_{P}, {Cov}_{N}) . \end{matrix}$

Therefore, the secondary decision result is as follows: $^{4} {AE}_{0} : x_{1}, x_{2} \in NEG,$ $^{5} {AE}_{0} : x_{1} \in POS, x_{2}, x_{3} \in NEG$

Further analysis shows that different κ values will lead to changes in the number of AE concepts, as shown in Table 5.

Table 5

AE concept generated under different κ values

Value	Number	AE_κ concept
κ = 0	5	{¹AE, ²AE, ³AE, AE, ⁵AE}
κ = 0.15	4	{²AE, ³AE, ⁴AE, ⁵AE}
κ = 0.25	3	{²AE, ³AE, ⁵AE}
κ = 0.3	1	{⁵AE}

To sum up, when considering the influence of all factors on the investment plan, x₁ is the most desirable alternative, while x₂ and x₃ are not suitable for investment.

6 Conclusions

At present, uncertainty analysis and decision-making has become more and more worthy of attention. FCA can objectively describe the uncertainty problem from two dimensions of attribute and object.

This paper mainly combines the intuitionistic fuzzy set with the 3WFCA model, and uses the attribute correlation degree to analyze the relationship between each concept. Meanwhile, we give the secondary decision strategy and algorithm of boundary object in AE-concept and OE-concept. The proposed models, OI-IF3WCL and AI-IF3WCL, can not only satisfy the quantitative analysis of uncertainty in intuitionistic fuzzy environment, but also enhance the information expression ability and application scope of the traditional 3WFCA model. In future work, we consider extending the existing model to the Pythagorean fuzzy environment [46, 47]. In addition, the combination of 3WFCA and the support intuitionistic fuzzy set [48] is also a subject worth of research.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 61877004, and the Beijing Normal University Interdisciplinary Research Foundation for the First-Year Doctoral Candidates No. BNUXKJX1925.

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Intuitionistic fuzzy three-way formal concept analysis based attribute correlation degree

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Three-way decision

Table 1 Cost function matrix Action Cost function X ¬X a P λ PP λ PN a B λ BP λ BN a N λ NP λ NN

Table 2 A formal context K I a b c d x 1 × × × x 2 × × x 3 × × × x 4 × × × x 5 ×

3 Intuitionistic fuzzy 3WFCA

3.1 Correlation degree of intuitionistic fuzzy attributes

3.2.1 Intuitionistic fuzzy concept lattice based on the degree of attribute correlation

4.1 The secondary decision of boundary intension object in the OE κ concept

Footnotes

Acknowledgments

References

Table 1
Cost function matrix

Action Cost function

X ¬X

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

Table 2
A formal context K

I a b c d

x ₁ × × ×

x ₂ × ×

x ₃ × × ×

x ₄ × × ×

x ₅ ×

4.1 The secondary decision of boundary intension object in the OE_κ concept