A compromise operator-based approach for multigranulation space 1

Abstract

The notion of compromise operators was systematically explored for multi-attribute decision making problems, which include widely used averaging operators, uninorms and nullnorms as the special cases. In this paper, we employ compromise operators to examine the issue of information fusion in multigranulation spaces. For this purpose, we firstly show that the optimistic multigranuation rough set model can be interpreted from the viewpoint of uninorm operators and the pessimistic multigranulation rough set model can be interpreted from the viewpoint of averaging operators or nullnorm operators. Then, by considering rough membership degrees in each Pawlak space and using the generalized compromise operator, we present a novel approach to information fusion in multigranulation space. Lastly, an illustrative example of information fusion in multigranulation spaces is presented.

Keywords

Multigranulation rough set multigranulation space compromise operator uninorm operator averaging operator

1 Introduction

A multigranulation space [21] refers to a pair (U, E), where U is a universe of objects and E is a family of equivalence relations on U. To date, rough sets in a multigranulation space have been a much studied topic and have received renewed growing interests [1 , 25]. One way to multigranulation spaces is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations [16 , 21]. By using set-theoretic union and intersection, respectively, two types of multigranulation operators, i.e., the so-called optimistic multigranulation rough set model and pessimistic multigranulation rough set model [16, 17], are obtained. These rough set models proposed so far can also be regarded as the results of information fusion [23] in a multigranulation space.

This paper aims to conduct a further study concerning the issue of information fusion in multigranulation spaces. Before proceeding further, let us consider the distinctive feature of information fusion of the proposed multigranulation rough set model. Owing to the fact that a rough set in each Pawlak space divides the universe into three disjoint regions, it is thus safe to say that the existing multigranulation rough set model is the result of fusing different three-valued functions on the universe. Such an information fusion process does not distinguish elements in boundary region in each Pawlak space, or in other words, elements in boundary region are treated equally. It thus lacks sufficient consideration of rough menbership degree [15] of an element relative to the concept. A more reasonable approach should be employed by fusing the rough membership degrees in different Pawlak spaces to obtain a overall membership degree in a multigranulation space.

Cognitive psychology [3, 14] has showed that the human brain is very efficient at processing bipolar (negative and positive) information. Three-way decision theory [22] (3WD for short), proposed by Yao, is an effective approach to deal with such type of information. According to 3WD theory, if an element falls into the acceptance region and the rejection region simultaneously, then it will be classified into the boundary region, that is, non-commitment will be made for these objects. This is indeed a form of trade-off between acceptance and rejection. Similar ideas have also been reflected in some aggregation functions such as uninorm operators [24], averaging operators [2], and nullnorm operators [13]. In [11], by distilling a number of properties that are common to various types of compromise operators (including averaging operators, uninorms and nullnorms), the authors systematically explore the space of possible compromise operators for such multi-attribute decision making problems. The proposed compromise operators prove to be effective to model the various attitudes towards aggregating non-conflicting ratings. In this paper, following the fundamental idea of three-way decision, we aim to employ compromise aggregation operators to carry out the fusion process in multigranulation spaces.

The rest of this paper proceeds as follows. Section 2 briefly recalls some basic notions, such as multigranulation rough set models, t-norm, t-conorm, uninorm and averaging operators. Section 3 provides an interpretation of multigranulation rough set models from the viewpoint of uninorm operators, averaging operators and nullnorm operators, respectively. Then in Section 4, by considering the rough membership degrees in each Pawlak space, we present a novel approach by means of compromise operators. We also employ an illustrative example to show the validity of our approach. Lastly, the present paper is completed with some concluding remarks.

2 Some related notions

In this section, we briefly recall some related notions, such as t-norm, t-conorm, uninorms, nullnorms, averaging operators, compromise operators, etc. All of these notions will be used in the subsequent sections.

2.1 Multigranulation space

Given an equivalence relation E on a universe of objects U, i.e., E is a reflexive, symmetric, and transitive binary relation on U, one can divide U into a family of pairwise disjoint equivalence classes of U, called the partition or the quotient set induced by E. In other words, an equivalence relation gives rise to a granulation of the universe. A multigranulation space [21] refers to a pair (U, E) with U being a nonempty finite set, called the universe of objects, and E being a nonempty collection of equivalence relations on U. In special case of |E|=1, we call the pair (U, E) a single-granulation space, or a Pawlak space [15], which is also denoted by (U, E).

In a Pawlak space (U, E), by considering equivalence classes as the building blocks, one can approximate a subset of U through unions of equivalence classes.

Definition 1. [15] In a Pawlak space (U, E) , a subset X ⊆ U is approximated by the following two sets: $\begin{matrix} {\underline{apr}}_{E} (X) = {x ∣ [x]_{E} \subseteq X}, \\ {\bar{apr}}_{E} (X) = {x ∣ [x]_{E} \cap X \neq \emptyset} . \end{matrix}$

We call ${\underline{apr}}_{E} (X), {\bar{apr}}_{E} (X)$ the lower approximation and upper approximation of X, respectively, with respect to the equivalence relation E, and call ${\underline{apr}}_{E}, {\bar{apr}}_{E} : 2^{U} \to 2^{U}$ rough approximation operators determined by E.

There exists several approaches to rough set approximations in a multigranulation space. One way is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations. By using set intersection and union to combine a family of approximations, respectively, two models can be built as below.

Definition 2. [21] Let (U, E) be a multigranulation space with E = {R₁, R₂}, then for any X ⊆ U, define $\begin{matrix} {\underline{apr}}_{E}^{o} (X) = {\underline{apr}}_{R_{1}} (X) \cup {\underline{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E}^{o} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X), \\ {\underline{apr}}_{E}^{p} (X) = {\underline{apr}}_{R_{1}} (X) \cap {\underline{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E}^{p} (X) = {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) . \end{matrix}$

Then we call the pair $({\underline{apr}}_{E}^{o} (X), {\bar{apr}}_{E}^{o} (X))$ a optimistic multigranulation rough set and call $({\underline{apr}}_{E}^{p} (X), {\bar{apr}}_{E}^{p} (X))$ a pessimistic multigranulation rough set.

In reference [1], by interpreting a multigranulation space as a multiple-source approximation space, Khan and Banerjee introduced the notions of strong and weak lower and upper approximations. Although the interpretations of a multigranulation space in these two studies are different, their results are mathematically equivalent. This new direction of research has attracted much attention. Other types of multigranulation rough set models include fuzzy multigranulation rough sets [19], decision-theoretic multigranulation rough sets [17], etc.

For the sake of our discussion, a multigranulation space containing two equivalence relations is considered below.

2.2 Characteristics of t-norm, t-conorm and uninorm in information fusion

In this subsection, we will briefly review the definitions of t-norm, t-conorm and uninorm, and then analyse the distinct feature of these operators in information fusion.

2.2.1 T-norm and its characteristic in information fusion

A t-norm [6] T is a binary operation on [0, 1] ² satisfying the the following conditions:

∀a, b ∈ [0, 1] , T (a, b) = T (b, a), that is, T is commutative,

∀a, b, c ∈ [0, 1] , T (T (a, b) , c) = T (a, T (b, c)), that is, T is associative,

∀a ∈ [0, 1] , T (a, 1) = a, that is, 1 is the identity element of T,

∀a, b, c, d ∈ [0, 1], if a ≥ b, c ≥ d, then T (a, c) ≥ T (b, d), that is, T is increasing in both arguments.

One basic charasteristic of T is that T (a, b) ≤ min {a, b}, that is, the addition of an argument to a t-norm aggregation never results in an increase.

We now consider the issue of information fusion in multigranulation spaces by using t-norm. Let (U, E) be a multigranulation space and X ⊆ U, a kind of fuzzy set f_{X,R_i} from the set X ⊆ U in Pawlak space (U, E_i) can be derived as follows: $f_{X, R_{i}} (x) = \frac{| [x] \cap X |}{| [x] |}, x \in U .$

As a consequence, for the same subset X, it can induce multiple fuzzy sets over the multigranulation space (U, E). Then, one question arises naturally: what is the aggregated fuzzy set of these multiple fuzzy sets? Indeed, one possible method, among others, to aggregate the collection of fuzzy sets induced from (U, E) can be conducted by using t-norm. That is, $f_{X, E} (x) = T (f_{X, R_{1}} (x), \dots, f_{X, R_{n}} (x)) .$

Clearly, f_X,E (x) =1 if and only if f_X,R₁ (x) = f_X,R₂ (x) = ⋯ = f_{X,R_n} (x) =1. This is similar to the situation of multi-agent decision making. If each agent is completely satisfied with a solution, then the group is completely satisfied with the solution, and vice versa. Moreover, the obtained value is less than or equal to each aggregated one, in particular, if one of the values is equal to 0, then the combined value is also equal to 0, semantically, if one agent rejects one solution, then the group will also reject one solution. Or in other words, the implication of using t-norm is that an agent has an ability to detract from an alternative, in no situation can the agent “support” an alternative. In particular, there exists no possibility for agents to upwardly reinforce each other, only downward reinforcement is possible. For example, for the same concept, if two agents assign different memberships to x with respect to X, e.g., 0.2 and 0.4, and use a product t-norm, then the aggregated membership degree is 0.08 and we see that these two memberships have reinforced each other to even more strongly reject the alternative. On the other hand, if two agents strongly support an alternative, for instance 0.9 and 0.95, then there exists no t-norm that enables these two scores to reinforce each other and more strongly support the alternative.

2.2.2 T-conorm and its characteristic in information fusion

A t-conorm [6] S is a binary operation on [0, 1] ² satisfying the the following conditions:

∀a, b ∈ [0, 1] , S (a, b) = S (b, a), that is, S is commutative,

∀a, b, c ∈ [0, 1] , S (S (a, b) , c) = S (a, S (b, c)), that is, S is associative,

∀a ∈ [0, 1] , S (a, 0) = a, that is, 0 is the identity element of S,

∀a, b, c, d ∈ [0, 1], if a ≥ b, c ≥ d, then S (a, c) ≥ S (b, d), that is, S is increasing in both arguments.

Or in other words, a t-conorm S is a symmetric, monotonic, associative operator with identity 0.

A trivial result about a t-conorm S is that ∀a ∈ [0, 1] S (1, a) =1 . Moreover, ∀a, b ∈ [0, 1] , S (a, b) ≥ max {a, b}, that is, the addition of an argument to a t-conorm aggregation never results in a decrease.

We now consider the issue of information aggregation in the context of multigranulation space by using t-conorm. One direct method to aggregate the collection of fuzzy sets induced from different Pawlak spaces by using t-conorm is given as follows: $f_{X, E} (x) = S (f_{X, R_{1}} (x), \dots, f_{X, R_{n}} (x)) .$

Clearly, if f_X,R₁ (x) = f_X,R₂ (x) = ⋯ = f_{X,R_n} (x) =0, then f_X,E (x) =0, and vice versa. We can also give the interpretation from the viewpoint of multi-agent decision making: a group is completely dissatisfied with a solution if and only if each member of the group is completely dissatisfied with the solution. Moreover, the obtained result f_X,E (x) is larger than or equal to each aggregated value f_{X,R_i} (x) , i = 1, ⋯ , n, in particular, if one of the aggregated value is equal to 1, then f_X,E (x) is also equal to 1. Semantically, if one agent accepts one solution, then the group will also accept one solution. What is more, the implication of using t-conorm is that an agent has an ability to support an alternative, in no situation can the agent detract from an alternative. In particular, there exists no possibility for agents to downwardly reinforce each other, only upward reinforcement is possible.

2.2.3 Uninorm and its characteristic in information aggregation

The uninorm operators, introduced in [24], provide a unification and generalization of t-norm and t-conorm operators. Up to now, some scholars have studied extensively the properties of these operators.

Definition 3. [24] A uninorm is a mapping U : [0, 1] ² → [0, 1] having the following properties:

∀x, y ∈ [0, 1] , U (x, y) = U (y, x) , that is, U is commutative,

∀x, y, z ∈ [0, 1] , U (U (x, y) , z) = U (x, U (y, z)) , that is, U is associative,

∀x, y, u, v ∈ [0, 1] , x ≥ u, y ≥ v implies that U (x, y) ≥ U (u, v), that is, U is monotonic in both arguments,

There exists some e ∈ [0, 1] such that U (e, x) = x holds for all x ∈ [0, 1] , that is, e is the identity of U .

Observe that uninorm operators have their first three properties in common with the t-norm and t-conorm, but their fourth condition is more general in that it allows for any identity in the unit interval. We note that for a t-norm T, its identity is 1, then condition (iv) is written as T (1, x) = x, while for any t-conorm S, the identity is 0, then condition (iv) can be expressed as S (0, x) = x. That is, both t-norm and t-conorm can be seen as some particular examples of uninorms with the identity 1 and 0, respectively.

Example 1. [5] The following is a uninorm operator with identity $\frac{1}{2} :$ $\begin{matrix} U (x, y) = \frac{xy}{xy + (1 - x) (1 - y)}, \\ U (0, 1) = U (1, 0) \in {0, 1} . \end{matrix}$

Sometimes we use $\frac{xy}{xy + (1 - x) (1 - y)}$ to refer to the underlying uninorm U.

The following proposition provides some important properties of the class of uninorms.

Proposition 1. [5] Let U be a uninorm with neutral element e ∈ (0, 1) . Then there exists a t-norm T and t-conorm S such that U is given by $\begin{matrix} U (x, y) \\ = {\begin{matrix} eT (\frac{x}{e}, \frac{y}{e}), & (x, y) \in [0, e]^{2}, \\ e + (1 - e) S (\frac{x - e}{1 - e}, \frac{y - e}{1 - e}), & (x, y) \in [e, 1]^{2} \end{matrix} \end{matrix}$ and min(x, y) ≤ U (x, y) ≤ max(x, y) , (x, y) ∈ [0, 1] ² \ ([0, e] ² ∪ [e, 1] ²).

2.3 Averaging operator

The fourth type of operators that we consider are averaging operators, which are defined in following manner.

Definition 4. [2, 11] A mapping M : ∪ _n∈N [0, 1] ⁿ → [0, 1] is an averaging operator if it satisfies:

idempotency: ∀a∈ [0, 1] , M (a, ⋯ , a) = a ;

monotonicity: $\forall a_{1}, {a'}_{1}, \dots, a_{n}, {a'}_{n} \in [0, 1], a_{1} \leq {a'}_{1}, \dots, a_{n} \leq {a'}_{n} \Rightarrow M (a_{1}, \dots, a_{n}) \leq M ({a'}_{1}, \dots, {a'}_{n});$ and

symmetry: ∀a₁, ⋯ , a_n ∈ [0, 1] , {p₁, ⋯ , p_n} = {1, ⋯ , n} ⇒ M (a₁, ⋯ , a_n) = M (a_{p
₁}, ⋯ , a_{p
_n}) .

Example 2. [2, 11] The arithmetical mean is a typical averaging operator: $M (a_{1}, \dots, a_{n}) = \frac{a_{1} + a_{2} + \dots + a_{n}}{n} .$

The proposition below gives an important property of averaging operators.

Proposition 2. [2, 11] ∀a₁, ⋯ , a_n ∈ [0, 1] , min {a₁, ⋯ , a_n} ≤ M (a₁, ⋯ , a_n) ≤ max {a₁, ⋯ , a_n} .

2.4 Generalized compromise operators

In [11], the concept of a compromise operator was proposed such that it can cover both uninorm operators and averaging operators as its special cases.

Definition 5. [11] A mapping A : ∪ _n∈N [0, 1] ⁿ → [0, 1] is a general aggregation operator if it satisfies:

idempotency: A (0, 0, ⋯ , 0) =0, A (1, 1, ⋯ , 1) =1 ;

monotonicity: $\forall a_{1}, {a'}_{1}, \dots, a_{n}, {a'}_{n} \in [0, 1], a_{1} \leq {a'}_{1}, \dots, a_{n} \leq {a'}_{n} \Rightarrow A (a_{1}, \dots, a_{n}) \leq A ({a'}_{1}, \dots, {a'}_{n});$ and

symmetry: ∀a₁, ⋯ , a_n ∈ [0, 1] , {p₁, ⋯ , p_n} = {1, ⋯ , n} ⇒ A (a₁, ⋯ , a_n) = A (a_{p
₁}, ⋯ , a_{p
_n}) .

Clearly, uninorm compromise operators and averaging operators are special cases of the general aggregation operator. Of course, t-norms and t-conorms are also special cases since they are special cases of a uninorm.

Based on the concept of general aggregation operator, the notion of generalised compromise operators was defined in the following way.

Definition 6. [11] A general aggregation operator is a general compromise operator, denoted as C_τ, with respect to τ ∈ (0, 1) if $\forall τ_{1}^{-}, \dots, τ_{k}^{-} \in [0, τ), τ_{1}^{+}, \dots, τ_{l}^{+} \in (τ, 1] (k, l \geq 1, k + l = n), min {τ_{1}^{-}, \dots, τ_{k}^{-}} \leq C_{τ} (τ_{1}^{-}, \dots, τ_{k}^{-}, τ_{1}^{+}, \dots, τ_{l}^{+})$ $\leq max {τ_{1}^{+}, \dots, τ_{l}^{+}} .$

A general compromise operator C_|Ó is uninorm-like if $τ_{1}^{-}, \dots, τ_{k}^{-} \in [0, τ), τ_{1}^{+}, \dots, τ_{1}^{+} \in (τ, 1] (k, l \geq 1, k + l = n), C_{τ} (τ_{1}^{-}, \dots, τ_{k}^{-}, τ, τ_{1}^{+}, \dots, τ_{1}^{+}) = C_{τ} (τ_{1}^{-}, \dots, τ_{k}^{-}, τ_{1}^{+}, \dots, τ_{1}^{+});$ otherwise it is averaging-like.

2.5 Nullnorms

Definition 7. [4] A binary operator V : [0, 1] × [0, 1] → [0, 1] is a nullnorm if it satisfies:

monotonicity: $\forall a_{1}, {a'}_{1}, a_{2}, {a'}_{2} \in [0, 1], a_{1} \leq {a'}_{1}, a_{2} \leq {a'}_{2} \Rightarrow V (a_{1}, a_{2}) \leq V ({a'}_{1}, {a'}_{2});$

associativity: ∀a₁, a₂, a₃∈ [0, 1] , V (V (a₁, a₂) , a₃) = V (a₁, V (a₂, a₃)) ;

commutativity: ∀a₁, a₂ ∈ [0, 1] , V (a₁, a₂) = V (a₂, a₁); and

annihilator element: ∃τ ∈ [0, 1] , ∀ a ∈ [τ, 1] , V (1, a) = a, ∀ a ∈ [0, τ] , V (0, a) = a .

Proposition 3. [4, 11] Nullnorm is a special case of a general compromise operator.

3 An interpretation of the multigranulation rough set models from the viewpoint of aggregation operators

3.1 Aggregation operation corresponding to optimistic multigranulation rough set model

A Pawlak rough set divides the universe into three disjoint subsets, namely, the positive region (i.e., the lower approximation), the boundary region and the negative region (i.e., the complement of the upper approximation). Semantically, each Pawlak rough set can be represented by a three-valued set on the universe, and combination of approximation results in different Pawlak spaces corresponds to a underlying aggregation function on ${0, \frac{1}{2}, 1}$ .

Let (U, E) be a Pawlak space and X ⊆ U, a three-valued function corresponding to the approximations of X is defined as follows: $\begin{matrix} f_{X, R} (x) = 1, if x \in {\underline{apr}}_{R} (X), \\ f_{X, R} (x) = 0, if x \in ({\bar{apr}}_{R} (X))^{c}, \\ f_{X, R} (x) = \frac{1}{2}, if x \in {\bar{apr}}_{R} (X) \ {\underline{apr}}_{R} (X) . \end{matrix}$

Similarly, given a multigranulation space (U, E) and X ⊆ U, a multigranulation rough set (either optimistic or pessimistic) divides the universe into three disjoint subsets, i.e., the positive region, the negative region and the boundary region. The corresponding three-valued function is denoted by f_X,E, which can be defined in a similar manner as above. Consequently, for (U, E) with E = {R₁, R₂}, we obtain three types of three-valued functions on the universe for X ⊆ U: f_X,E and f_X,R₁, f_X,R₂. The obtained results in [18] explicitly states the underlying aggregation operator.

Considering the fact that in different multigranulation rough set models, there exists different information fusion strategies (or in other words, there exists different relationships between three-valued functions in each Pawlak space and that in concerned multigranulation space), we therefore use ⊗ with different superscripts to denote the corresponding combination operators. Precisely, f_X,E (x) = f_X,R₁ (x) ⊗ f_X,R₂ (x) , x ∈ U .

Proposition 4. Let (U, E) be a multigranulation space with E = {R₁, R₂}. Then the combination operator ⊗^o induced by the optimistic multigranulation rough set model satisfies the following conditions:

$0 \otimes^{o} 0 = 1 \otimes^{o} \frac{1}{2} = \frac{1}{2} \otimes^{o} 0 = 0, \frac{1}{2} \otimes^{o} \frac{1}{2} = \frac{1}{2}, \frac{1}{2} \otimes^{o} 1 = 1 \otimes^{o} \frac{1}{2} = 1 \otimes^{o} 1 = 1 .$

⊗^o is undefined on (0, 1) and (1, 0) .

Proof. To prove (1), it suffices to show that the following conditions are satisfied:

$f_{X, E}^{o} (x) = 1$ if and only if f_X,R₁ (x) =1 or f_X,R₂ (x) =1,

$f_{X, E}^{o} (x) = 0$ if and only if f_X,R₁ (x) =0 or f_X,R₂ (x) =0 .

For (i), we have from $f_{X, E}^{o} (x) = 1$ that $x \in {\underline{apr}}_{E}^{o} (X) = {\underline{apr}}_{R_{1}} (X) \cup {\underline{apr}}_{R_{2}} (X),$ then $x \in {\underline{apr}}_{R_{1}} (X)$ or $x \in {\underline{apr}}_{R_{2}} (X),$ that is, f_X,R₁ (x) =1 or f_X,R₂ (x) =1 . Conversely, if f_X,R₁ (x) =1 or f_X,R₂ (x) =1, then $x \in {\underline{apr}}_{R_{1}} (X)$ or $x \in {\underline{apr}}_{R_{2}} (X) .$ By Definition 2, we have $x \in {\underline{apr}}_{E}^{o} (X) .$ Therefore, $f_{X, E}^{o} (x) = 1$ .

For (ii), we have from $f_{X, E}^{o} (x) = 0$ that $x \notin {\bar{apr}}_{E}^{o} (X)$ . Since $({\bar{apr}}_{E}^{o} (X))^{c} = ({\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X))^{c} = ({\bar{apr}}_{R_{1}} (X))^{c} \cup ({\bar{apr}}_{R_{2}} (X))^{c},$ we conclude immediately that $x \in ({\bar{apr}}_{R_{1}} (X))^{c}$ or $x \in ({\bar{apr}}_{R_{2}} (X))^{c}$ , therefore, f_X,R₁ (x) =0 or f_X,R₂ (x) =0 . The converse direction can be shown in a similar manner.

For (2), it is due to the following fact: f_X,R₁ (x) =1 and f_X,R₂ (x) =0 cannot hold simultaneously (In fact, suppose that f_X,R₁ (x) =1, i.e., x is contained in the lower approximation of X, we then have x ∈ X, and therefore, [x] _{R
₂} ∩ X ≠ ∅ , which means that $f_{X, R_{2}} (x) \in {\frac{1}{2}, 1},$ that is, f_X,R₂ (x) =0 cannot happen). Similarly, f_X,R₁ (x) =0 and f_X,R₂ (x) =1 cannot hold simultaneously.

The aggregation operator ⊗^o is shown in Table 1.

Table 1
Aggregation operator ⊗^o induced by optimistic multigranulation rough set model

⊗^o 0 $\frac{1}{2}$ 1

0 0 0 ×

$\frac{1}{2}$ 0 $\frac{1}{2}$ 1

1 × 1 1

⊗^o	0	$\frac{1}{2}$	1
0	0	0	×
$\frac{1}{2}$	0	$\frac{1}{2}$	1
1	×	1	1

An easy verification shows that the following proposition holds.

Proposition 5. The aggregation operator ⊗^o coincides with the restriction of the uninorm $\frac{xy}{xy + (1 - x) (1 - y)}$ to ${0, \frac{1}{2}, 1} .$

Proof. A trivial verification shows that $\frac{xy}{xy + (1 - x) (1 - y)}$ satisfies both condition (1). Moreover, $\frac{xy}{xy + (1 - x) (1 - y)}$ is undefined on (0,1) and (1,0).

Proposition 5 shows that the optimistic multigranulation rough set model can be obtained by using the uninorm operator $\frac{xy}{xy + (1 - x) (1 - y)}$ on $3 = {0, \frac{1}{2}, 1} .$ Strictly speaking, it is just a partial operation because it is not defined on (0, 1) and (1, 0) .

A uninorm is said to be locally internal [12] if for x, y ∈ [0, e) × (e, 1], R (x, y) ∈ {x, y}. It seems reasonable to employ the locally internal uninorms in the subsequent discussion. Suppose that U is a uninorm with identity $0 < e < \frac{1}{2}$ . There are two cases to be considered:

Case 1: U (0, 1) =0: In this case, we have from Definition 3 that $U (0, 0) = U (0, \frac{1}{2}) = U (\frac{1}{2}, 0) = 0,$ $U (\frac{1}{2}, \frac{1}{2}) \in {\frac{1}{2}, 1}$ and $U (0, 1) = U (1, 0) = 0, U (1, \frac{1}{2}) = U (\frac{1}{2}, 1) = U (1, 1) = 1 .$ This produces two types of restrictions of uninorms to ${0, \frac{1}{2}, 1}$ . Precisely, one is the uninorm $\frac{xy}{xy + (1 - x) (1 - y)}$ and the other is given in Table 2.

Table 2

The aggregation operator corresponding to $0 < e < \frac{1}{2}$ and U (0, 1) =0

⊗¹	$\frac{1}{2}$	1
0	0	0
$\frac{1}{2}$	1	1
1	1	1

By using these two types of uninorms, we can obtain two types of multigranulation rough set models correspondingly. One is the optimistic multigranulation rough set model and the other one is given as follows:

$\begin{matrix} {\underline{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X) . \end{matrix}$ (1)

Indeed, we have from the definition of f_X,E that $x \in {\underline{apr}}_{E} (X) \Leftrightarrow f_{X, E} (x) = 1$ . Moreover, weconclude from Table 2 that $f_{X, E} (x) = 1 \Leftrightarrow (f_{X, R_{1}} (x), f_{X, R_{2}} (x) \in {(1, 1), (1, \frac{1}{2}), (\frac{1}{2}, 1), (\frac{1}{2}, \frac{1}{2})} \Leftrightarrow x \in {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X) .$ And hence, ${\underline{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X)$ . Additionally, we observe from Table 2 that the aggregation result of ⊗¹ is either 0 or 1, implying that the boundary region of the corresponding multigranulation rough set model is empty, and therefore, ${\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X) .$ The distincitive feature of multigranulation rough set model defined by (1) is that the boundary region is always empty owing to the fact that for any subset, the lower approximation and the upper approximation of any subset coincide with each other.

Case 2: U (0, 1) =1 . In this case, we have from Definition 3 that $U (0, 0) = 0, U (0, \frac{1}{2}) = U (\frac{1}{2}, 0) \in {0, \frac{1}{2}},$ $U (\frac{1}{2}, \frac{1}{2}) \in {\frac{1}{2}, 1}$ and $U (0, 1) = U (1, 0) = U (1, \frac{1}{2}) = U (\frac{1}{2}, 1) = U (1, 1) = 1 .$ That is, we can obtain four types of operations on ${0, \frac{1}{2}, 1}$ .

Specifically, the first one coincides with the restrictions of the uninorm $\frac{xy}{xy + (1 - x) (1 - y)}$ to ${0, \frac{1}{2}, 1}$ , as shown in Table 1. The second one is provided in Table 2, the corresponding multigranulation rough set model is defined as follows: $\begin{matrix} {\underline{apr}}_{E} (X) = {\underline{apr}}_{R_{1}} (X) \cup {\underline{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) . \end{matrix}$

The third one coincides with ⊗², as shown in Table 3, the corresponding multigranulation rough set model is given as follows: $\begin{matrix} {\underline{apr}}_{E} (X) = {\underline{apr}}_{R_{1}} (X) \cup {\underline{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) . \end{matrix}$

Table 3

The second aggregation operator corresponding to 0 < e < 1 and U (0, 1) =1

⊗²	0	$\frac{1}{2}$	1
0	0	$\frac{1}{2}$	1
$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	1
1	1	1	1

Indeed, we have from the definition of f_X,E that $x \in {\underline{apr}}_{E} (X) \Leftrightarrow f_{X, E} (x) = 1$ . Moreover, weconclude from Table 3 that $f_{X, E} (x) = 1 \Leftrightarrow (f_{X, R_{1}} (x), f_{X, R_{2}} (x) \in {(1, 1), (1, \frac{1}{2}), (\frac{1}{2}, 1))} \Leftrightarrow x \in {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) .$ And hence, ${\underline{apr}}_{E} (X) = {\underline{apr}}_{R_{1}} (X) \cup {\underline{apr}}_{R_{2}} (X), {\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) .$

The fourth one is provided in Table 4, the corresponding multigranulation rough set model is defined as follows: $\begin{matrix} {\underline{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) . \end{matrix}$

Table 4

The fourth aggregation operator corresponding to 0 < e < 1 and U (0, 1) =1

⊗³	0	$\frac{1}{2}$	1
0	0	$\frac{1}{2}$	1
$\frac{1}{2}$	$\frac{1}{2}$	1	1
1	1	1	1

Indeed, we have from the definition of f_X,E that $x \in {\underline{apr}}_{E} (X) \Leftrightarrow f_{X, E} (x) = 1$ . Moreover, we concludefrom Table 4 that $f_{X, E} (x) = 1 \Leftrightarrow (f_{X, R_{1}} (x), f_{X, R_{2}} (x) \in {(1, 1), (1, \frac{1}{2}), (\frac{1}{2}, 1), (\frac{1}{2}, \frac{1}{2})} \Leftrightarrow x \in {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X) .$ Similarly, $x \in ({\bar{apr}}_{E} (X))^{c} \Leftrightarrow f_{X, E} (x) = 0 \Leftrightarrow (f_{X, R_{1}} (x), f_{X, R_{2}} (x) \in {(0, 0)} \Leftrightarrow x \in ({\bar{apr}}_{R_{1}} (X))^{c} \cap ({\bar{apr}}_{R_{2}} (X))^{c} .$ And hence, $({\bar{apr}}_{E} (X))^{c} = ({\bar{apr}}_{R_{1}} (X))^{c} \cap ({\bar{apr}}_{R_{2}} (X))^{c},$ whichimmediately entails ${\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cup {\bar{apr}}_{R_{2}} (X) .$

Suppose that U is a uninorm with identitye = 0. That is, U is a t-conorm. Then we obtainthat $U (0, 0) = 0, U (0, \frac{1}{2}) = U (\frac{1}{2}, 0) = \frac{1}{2}, U (0, 1) = U (1, 0) = 1, U (\frac{1}{2}, \frac{1}{2}) \in {\frac{1}{2}, 1}, U (\frac{1}{2}, 1) = U (1, \frac{1}{2}) = U (1, 1) = 1 .$ That is, two types of operators on ${0, \frac{1}{2}, 1}$ are produced. One coincides with ⊗², as provided in Table 3, and the other one coincides with ⊗³), which is given in Table 4.

Suppose that U is a uninorm with identity e = 1. That is, U is a t-norm. In this case, we have $U (0, 0) = U (0, \frac{1}{2}) = U (\frac{1}{2}, 0) = U (0, 1) = U (1, 0) = 0, U (\frac{1}{2}, \frac{1}{2}) \in {0, \frac{1}{2}}, U (\frac{1}{2}, 1) = U (1, \frac{1}{2}) = \frac{1}{2}, U (1, 1) = 1$ . That is, two types of operators on ${0, \frac{1}{2}, 1}$ can be obtained, as shown in Tables 5 and 6, respectively. The corresponding multigranulation rough set models are given as follows: $\begin{matrix} {\underline{apr}}_{E} (X) = {\underline{apr}}_{R_{1}} (X) \cap {\underline{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E} (X) = {\underline{apr}}_{R_{1}} (X) \cup {\underline{apr}}_{R_{2}} (X), \\ {\underline{apr}}_{E} (X) = {\underline{apr}}_{R_{1}} (X) \cap {\underline{apr}}_{R_{2}} (X), \\ {\bar{apr}}_{E} (X) = {\bar{apr}}_{R_{1}} (X) \cap {\bar{apr}}_{R_{2}} (X) . \end{matrix}$

Table 5

The first aggregation operator corresponding to e = 1

⊗⁴	$\frac{1}{2}$	1
0	0	0
$\frac{1}{2}$	0	$\frac{1}{2}$
1	$\frac{1}{2}$	1

Table 6

The second aggregation operator corresponding to e = 1

⊗⁵	$\frac{1}{2}$	1
0	0	0
$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
1	$\frac{1}{2}$	1

The case concerning $\frac{1}{2} < e < 1$ can be similarly obtained, and so we omit the detailed discussionhere.

Contrary to the fact that the optimistic multigranulation rough set model can be interpreted from the viewpoint of uninorm, we will see that the pessimistic multigranulation rough set model cannot be obtained by using any uninorm operator. Indeed, it has been shown in [18] that the aggregation operator corresponding to a pessimistic multigranulation rough set is that in Table 7. ⊗^p is also partially defined owing to the fact that for the same object o and a target concept X ⊆ U, o cannot be a positive element of X under one equivalence relation and a negative element of X under the other equivalence relation simultaneously.

Table 7

Aggregation operator ⊗^p corresponding to pessimistic multigranulation rough set model

⊗^p	0	$\frac{1}{2}$	1
0	0	$\frac{1}{2}$	×
$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
1	×	$\frac{1}{2}$	1

We will show this by considering the following cases:

Suppose that there exists a uninorm U with the identity 0 such that U = ⊗ ^p when restricted to ${0, \frac{1}{2}, 1}$ , that is, U is a t-conorm, then according to the property of U, we have $U (\frac{1}{2}, 1) = 1$ , which, however, contradicts with Table 7.

Suppose that there exists a uninorm U with the identity $\frac{1}{2}$ such that U = ⊗ ^p when restricted to ${0, \frac{1}{2}, 1}$ , we then have $U (\frac{1}{2}, 0) = 0$ , which, however, contradicts with Table 7.

Suppose that there exists a uninorm U with the identity 1 such that U = ⊗ ^p when restricted to ${0, \frac{1}{2}, 1}$ , we then have $U (\frac{1}{2}, 0) = 0$ , which, however, contradicts with Table 7.

Suppose that there exists a uninorm U with the identity between 0 and $\frac{1}{2}$ such that U = ⊗ ^p when restricted on ${0, \frac{1}{2}, 1}$ , then according to the property of a uninorm, we have $U (\frac{1}{2}, 1) = 1$ , which, however, contradicts with Table 7.

Suppose that there exists a uninorm U with the identity between $\frac{1}{2}$ and 1 such that U = ⊗ ^p when restricted on ${0, \frac{1}{2}, 1}$ , then according to the property of a uninorm, we have $U (\frac{1}{2}, 0) = 0$ , which, however, contradicts with Table 7.

However, we have the following proposition stating the relationship between the aggregation operator ⊗^p and the averaging operator in Definition 4.

Proposition 6. The aggregation operator ⊗^p corresponding to the pessimistic multigranulation rough set model is a partial averaging operator on ${0, \frac{1}{2}, 1}^{2} .$

Proof. According to Definition 4, we need show ⊗^p satisfies idempotency, monotonicity and symmetry.

Idempotency: We have from Table 8 that ⊗^p (1, 1) =1, $\otimes^{p} (\frac{1}{2}, \frac{1}{2}) = \frac{1}{2}$ and ⊗^p (0, 0) =0, which shows that idempotency is satisfied.

Monotonicity: It can be trivially checked from Table 8.

Symmetry: We observe from Table 8 that $\otimes^{p} (0, \frac{1}{2}) = \otimes^{p} (\frac{1}{2}, 0) = \otimes^{p} (1, \frac{1}{2}) = \otimes^{p} (\frac{1}{2}, 1)$ ,moreover, ⊗^p is undefined on both (0, 1) and (1, 0), and hence, the symmetry for ⊗^p is satisfied.

It can also be checked that the following propositions hold.

Proposition 7. The aggregation operator ⊗^p corresponding to the pessimistic multigranulation rough set model is a general aggregation operator on ${0, \frac{1}{2}, 1}$ in the sense of Definition 5.

Proof. We have from Definitions 4 and 5 that an averaging operator on ${0, \frac{1}{2}, 1}$ is necessarily a general aggregation operator, then the desired result follows immediately from Proposition 6.

Proposition 8. The aggregation operator ⊗^p corresponding to the pessimistic multigranulation rough set model is a nullnorm operator on ${0, \frac{1}{2}, 1}$ with the annihilator $\frac{1}{2}$ .

Proof. According to Definition 7, we need show that ⊗^p satisfies monotonicity, associativity, commutativity and has an annihilator element.

The proof concerning monotonicity, associativity, commutativity can be given in a similar manner as in Proposition 6 and Proposition 7. In what follows, we will show that $\frac{1}{2}$ is the annihilator element of ⊗^p. Indeed, we have from Table 7 that $\forall a \in [\frac{1}{2}, 1],$ i.e., a = 1 or $a = \frac{1}{2}$ , $\otimes^{p} (1, \frac{1}{2}) = \frac{1}{2}, \otimes^{p} (1, 1) = 1,$ ∀a ∈ [0, τ], a = 0 or $a = \frac{1}{2}$ , $\otimes^{p} (0, \frac{1}{2}) = \frac{1}{2}, \otimes^{p} (0, 0) = 0,$ that is, the condition concerning annihilator element is satisfied.

4 A compromise operator-based approach for multigranulation spaces

Propositions 2–5 in last section show that the existing multigranulation rough set models can be interpreted from the viewpoint of uninorm, averaging operator and nullnorm. Moreover, it is stated in [11] that uninorm operators and averaging operators share one common point, that is, when there are conflicting ratings for an alternative, the overall rating corresponds to some form of trade-off between the best and the worst of the individual ratings. Taking this into consideration, the authors defined the concept of a general compromise operator, which can cover the existing compromise operators: uninorms operators, averaging operators and nullnorms.

Motivated by this fact, in this section, we aim to aggregate rough membership degrees collected from a multigranulation space by using compromise operators.

Definition 8. Let (U, E) be a multigranulation space and X ⊆ U, define M_X,E : U → [0, 1] by $M_{X, E} (x) = C (f_{X, R_{i}} (x), \dots, f_{X, R_{n}} (x)),$ where $f_{X, R_{i}} (x) = \frac{| [x]_{R_{i}} \cap X |}{| [x]_{R_{i}} |}$ , i.e., f_{X,R_i} is a rough membership function for X in the Pawlak space (U, R_i) , we then call M_X,E the rough membership function for X in the multigranulation space(U, E) .

Two important examples are given below.

Example 3. Let (U, E) be a multigranulation space and X ⊆ U, define M_X,E : U → [0, 1] by $M_{X, E} (x) = C_{τ} (f_{X, R_{1}} (x), \dots, f_{X, R_{n}} (x)),$ where $f_{X, R_{i}} (x) = \frac{| [x]_{R_{i}} \cap X |}{| [x]_{R_{i}} |} (i = 1, \dots, n)$ , and C_τ is an uninorm-like compromise operator with the identity element $τ = \frac{1}{2}$ , as defined below: ∀a₁, ⋯ , a_n ∈ [0, 1] ⁿ, $\begin{matrix} C (a_{1}, \dots, a_{n}) \\ = {\begin{matrix} \frac{1}{1 + \prod_{i = 1}^{n} \frac{1 - a_{i}}{a_{i}}}, \forall 1 \leq i \leq n, a_{i} \neq 0, \\ 0, \exists i \in {1, \dots, n}, a_{i} = 0 and a_{j} \neq 1, \\ 0 or 1, otherwise . \end{matrix} \end{matrix}$

Example 4. Let (U, E) be a multigranulation space and X ⊆ U, define M_X,E : U → [0, 1] by $M_{X, E} (x) = \frac{f_{X, R_{1}} (x) + \dots + f_{X, R_{n}} (x)}{n},$ where $f_{X, R_{i}} (x) = \frac{| [x]_{R_{i}} \cap X |}{| [x]_{R_{i}} |} (i = 1, \dots, n)$ .

The rough membership degrees given by the above examples are based on uninorm-like compromise operator and averaging-like compromise operator, respectively.

In [11], the authors defined the various types of compromise operators. In particular, they discussed them in terms of how they reflect the users’ attitudes to risk in aggregation.

Definition 9. [11] For a general compromise operator C_τ,

it is τ-high-optimistic if $\forall τ_{1}^{+}, \dots, τ_{n}^{+} \in (τ, 1], C_{τ} (τ_{1}^{+}, \dots, τ_{n}^{+}) \geq max {τ_{1}^{+}, \dots, τ_{n}^{+}},$

it is τ-low-optimistic if $\forall τ_{1}^{-}, \dots, τ_{n}^{-} \in [0, τ), τ \geq C_{τ} (τ_{1}^{-}, \dots, τ_{n}^{-}) \geq max {τ_{1}^{-}, \dots, τ_{n}^{-}},$

it is τ-high-pessimistic if $\forall τ_{1}^{+}, \dots, τ_{n}^{+} \in (τ, 1], τ \leq C_{τ} (τ_{1}^{+}, \dots, τ_{n}^{+}) \leq min {τ_{1}^{+}, \dots, τ_{n}^{+}},$

it is τ-low-pessimistic if $\forall τ_{1}^{-}, \dots, τ_{n}^{-} \in [0, τ), C_{τ} (τ_{1}^{-}, \dots, τ_{n}^{-}) \leq min {τ_{1}^{-}, \dots, τ_{n}^{-}},$

it is τ-high-neutral if $\forall τ_{1}^{+}, \dots, τ_{n}^{+} \in (τ, 1], min {τ_{1}^{+}, \dots, τ_{n}^{+}} \leq C_{τ} (τ_{1}^{+}, \dots, τ_{n}^{+}) \leq max {τ_{1}^{+}, \dots, τ_{n}^{+}},$

it is τ-low-neutral if $\forall τ_{1}^{-}, \dots, τ_{n}^{-} \in [0, τ), min {τ_{1}^{-}, \dots, τ_{n}^{-}} \leq C_{τ} (τ_{1}^{-}, \dots, τ_{n}^{-}) \leq max {τ_{1}^{-}, \dots, τ_{n}^{-}} .$

In what follows, a general compromise operator is said to be optimistic, if it is τ-low-optimistic and τ-high optimistic; A general compromise operator is said to be pessimistic, if it is π-low-pessimistic and π-high-pessimistic; A general compromise operator is optimistic-pessimistic, if it is π-low-optimistic and π-high-pessimistic; a general compromise operator is pessimistic-optimistic, if it is π-low-pessimistic and π-high-optimistic; For a general compromise operator, it is optimistic-neutral, if it is π-low-optimistic and π-high-neutral; it is pessimistic-neutral, if it is π-low-pessimistic and π-high-neutral; it is neutral-optimistic, if it is π-low-neutral and π -high-optimistic; it is neutral-pessimistic, if it is π-low-neutral and π-high-pessimistic.

Various types of compromise operators in Definition 9 reflect the users’ attitudes to risk in aggregation. More precisely, it is stated in reference [11] that Definition 9(1) means that positive ratings are aggregated in a risk-seeking way; Definition 9(2) means that negative ratings are aggregated in a risk-seeking way; Definition 9(3) means that the positive ratings are aggregated in a risk-averse way; Definition 9(4) means that the negative ratings are aggregated in a risk-averse way; Definition 9(5) means that the positive ratings are aggregated in a risk-neutral way; and Definition 9(6) means that the negative ratings are aggregated in a risk-neutral way.

On the basis of rough membership function M_X,E, we can perform three-way decision by means of multigranulation rough set model. Formally, for thresholds α, β with α < β, if M_X,E (x) > β, then classify x into the positive region (denoted by POS_E (X)) related to X, if M_X,E (x) < α, then classify x into the negative region (denoted by NEG_E (X)) related to X and if α ≤ M_X,E (x) ≤ β, then classify x into the boundary region (denoted by BND_E (X)) related to X. Since τ is the threshold between positive information and negative information, it seems reasonable to assume that β > τ > α . One can also conduct three-way decision in each Pawlak space (U, R_i) based on the thresholds α, β, the corresponding three regions are denoted by POS_{R
_i} (X) , NEG_{R
_i} (X) and BND_{R
_i} (X), respectively.

The following proposition establishes the relationship between three-way decision in each Pawlak space and that in a multigranulation space.

Proposition 9. (i) If C_τ is τ-high-optimistic, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X),$

(ii) If C_τ is τ-low-pessimistic, then $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X) .$

Proof. (i) Choose arbitrarily $x \in \cap_{i = 1}^{n} {POS}_{R_{i}} (X),$ then we have that ∀i ∈ {1, 2, ⋯ , n} , x ∈ POS_{R
_i} (X) , which means that ∀i, f_{X,R_i} (x) > β . Considering the preliminary condition that β > τ, we then have from Definitions 8 and 9(i) that M_X,E = C (f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)) ≥ max {f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)} > β, and hence, x ∈ POS_E (X) .

(ii) Choose arbitrarily $x \in \cap_{i = 1}^{n} {NEG}_{R_{i}} (X),$ then we have that ∀i ∈ {1, 2, ⋯ , n} , x ∈ NEG_{R
_i} (X) , which means that ∀i, f_{X,R_i} (x) < α . Considering the preliminary condition that β > τ, we then have from Definition 8 and Definition 9(i) that M_X,E = C (f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)) ≤ min {f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)} < α, and hence, x ∈ NEG_E (X) .

A general compromise operator is optimistic, if it is τ-low-optimistic and τ-high optimistic; A general compromise operator is pessimistic, if it is τ-low-pessimistic and τ-high-pessimistic; A general compromise operator is optimistic-pessimistic, if it is τ-low-optimistic and τ-high-pessimistic; a general compromise operator is pessimistic-optimistic, if it is τ-low-pessimistic and τ-high-optimistic; For a general compromise operator, it is optimistic-neutral, if it is τ-low-optimistic and τ-high-neutral; it is pessimistic-neutral, if it is τ-low-pessimistic and τ-high-neutral; it is neutral-optimistic, if it is τ-low-neutral and τ -high-optimistic; it is neutral-pessimistic, if it is τ-low-neutral and τ-high-pessimistic.

For the thresholds α, β satisfying α < τ < β and X ⊆ U, we have the following result.

Proposition 10. (i) If C_τ is optimistic, then we have $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {BND}_{E} \cup {NEG}_{E} (X)$ ,

(ii) If C_τ is pessimistic, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X) \cup {BND}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X)$ ,

(iii) If C_τ is optimistic-pessimistic, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X) \cup {BND}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X) \cup {BND}_{E} (X)$ ,

(iv) If C_τ is pessimistic-optimistic, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X)$ hold,

(v) If C_τ is optimistic-neutral, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X) \cup \cup {BND}_{E} (X)$ ,

(vi) If C_τ is pessimistic-neutral, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X)$ ,

(vii) If C_τ is neutral-optimistic, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X)$ ,

(viii) If C_τ is neutral-pessimistic, then $\cap_{i = 1}^{n} {POS}_{R_{i}} (X) \subseteq {POS}_{E} (X) \cup {BND}_{E} (X)$ and $\cap_{i = 1}^{n} {NEG}_{R_{i}} (X) \subseteq {NEG}_{E} (X)$ .

Proof. We will give the proof of (v), and the others can be shown in a similar manner.

(v) Choose arbitrarily $x \in \cap_{i = 1}^{n} {POS}_{R_{i}} (X),$ then we have that ∀i ∈ {1, 2, ⋯ , n} , x ∈ POS_{R
_i} (X) , which means that ∀i, f_{X,R_i} (x) > β . Considering the preliminary condition that β > τ and C_τ is optimistic-neutral, we then have from Definitions 8 and 9 that M_X,E = C (f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)) ≥ min {f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)} > β, and hence, x ∈ POS_E (X) .

Similarly, choose arbitrarily $x \in \cap_{i = 1}^{n} {NEG}_{R_{i}} (X),$ then we have that ∀i ∈ {1, 2, ⋯ , n} , x ∈ NEG_{R
_i} (X) , which means that ∀i, f_{X,R_i} (x) < α . Since C_τ is optimistic-neutral, we then have from Definitions 8 and 9 that τ ≥ M_X,E = C (f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)) ≥ max {f_{X,R_i} (x) , ⋯ , f_{X,R_n} (x)} . And hence, x ∈ NEG_E (X) ∪ BND_E (X) .

Proposition 10 is obtained under the assumption that α < τ < β. Other cases concerning different relationships of α, β and τ can be obtained in a similar way.

5 An example

In this section, we employ an example to illustrate the application of compromise operator in aggregating the information from a complete information system.

Before proceeding further, we recall the following result concerning the construction of compromise operators.

In [11], the authors discussed the issue of how such operators can actually be constructed. More specifically, we give a number of ways to construct these operators from the existing t-norms, t-conorms and various compromise operators. For a given τ ∈ [0, 1] , a non-decreasing 1-1 mapping h : [0, τ] → [0, 1] is a τ^--constructor if h (0) =0 and h (τ) =1; and a non-decreasing 1-1 mapping h : [τ, 1] → [0, 1] is a τ⁺-constructor if h (τ) =0 and h (1) =1 .

Theorem 1. [11] For a general compromise operator C_τ,

it is τ-high-optimistic if $\forall τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+} \in (τ, 1], C_{τ} (τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+}) = h^{- 1} (S (h (τ_{1}^{+}), \dots, h (τ_{1}^{+}))),$ where h is a τ⁺-constructor,

it is τ-low-optimistic if $\forall τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{n}^{-} \in [0, τ), C_{τ} (τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{n}^{-}) = h^{- 1} (S (h (τ_{1}^{-}), \dots, h (τ_{1}^{-}))),$ where h is a τ^--constructor,

it is τ-high-pessimistic if $\forall τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+} \in (τ, 1], C_{τ} (τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+}) = h^{- 1} (T (h (τ_{1}^{+}), \dots, h (τ_{1}^{+}))),$ where h is a τ⁺-constructor,

it is τ-low-pessimistic if $\forall τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{n}^{-} \in [0, τ), C_{τ} (τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{n}^{-}) = h^{- 1} (T (h (τ_{1}^{-}), \dots, h (τ_{1}^{-}))),$ where h is a τ^--constructor.

Example 5. Some examples of t-norms andt-conorms are listed as below.

Idempotent operators: T_I (a₁, ⋯ , a_n) = min {a₁, ⋯ , a_n} , S_I (a₁, ⋯ , a_n) = max {a₁, ⋯ , a_n} ,

Probability operators: $T_{P} (a_{1}, \dots, a_{n}) = \prod_{i = 1}^{n} a_{i},$ $S_{P} (a_{1}, \dots, a_{n}) = 1 - \prod_{i = 1}^{n} (1 - a_{i}),$

Dombi operators: $T_{D} (a_{1}, \dots, a_{n}) = \frac{1}{\sum_{i = 1}^{n} \frac{1}{a_{i}} - (n - 1)},$ $S_{D} (a_{1}, \dots, a_{n}) = 1 - \frac{1}{\sum_{i = 1}^{n} \frac{1}{1 - a_{i}} - (n - 1)} .$

For a given τ ∈ [0, 1], a non-decreasing 1-1 mapping h : [0, τ] → [0, τ] is a τ^--generator if h (0) =0 and h (τ) = τ; and a non-decreasing 1-1 mapping h : [τ, 1] → [τ, 1] is a τ⁺-generator if h (τ) = τ and h (1) =1.

Theorem 2. [11] For a general compromise operator C_τ,

it is τ-high-neutral if $\forall τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+} \in (τ, 1], C_{τ} (τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+}) = h^{- 1} (M (h (τ_{1}^{+}), \dots, h (τ_{1}^{+}))),$ where h is a τ⁺-genearator,

it is τ-low-neutral if $\forall τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{n}^{-} \in [0, τ), C_{τ} (τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{n}^{-}) = h^{- 1} (M (h (τ_{1}^{-}), \dots, h (τ_{1}^{-}))),$ where h is a τ^--generator.

For a general compromise operator C,

it is uninorm-like if $\forall τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{m}^{-} \in [0, τ], τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+} \in (τ, 1]$ ,

$C_{τ} (τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{m}^{-}, τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+}) = h^{- 1} (U_{τ} (h (τ_{1}^{-}), h (τ_{2}^{-}), \dots, h (τ_{m}^{-}), h (τ_{1}^{+}), h (τ_{2}^{+}), \dots, h (τ_{n}^{+})))$ where h is a τ-generator; and

it is averaging-like if $\forall τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{m}^{-} \in [0, τ], τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+} \in (τ, 1]$ ,

$C_{τ} (τ_{1}^{-}, τ_{2}^{-}, \dots, τ_{m}^{-}, τ_{1}^{+}, τ_{2}^{+}, \dots, τ_{n}^{+}) = h^{- 1} (M (h (τ_{1}^{-}), h (τ_{2}^{-}), \dots, h (τ_{m}^{-}), h (τ_{1}^{+}), h (τ_{2}^{+}), \dots, h (τ_{n}^{+})))$ where h is a [0,1]-generator.

Table 8 depicts a complete information system containing some information about an emporium investment project. Locus, Investment and Population density are the conditional attributes of the system, whereas Decision is the decision attribute. The attribute domains are as follows: V_L = {good ; common ; bad}; V_I = {high ; low}; V_P = {big ; small ; medium} and V_D = {Yes ; No} .

Table 8
A complete target information system about emporium investment project

Project Locus Investment Popularity Decision

density

a ₁ Common High Big Yes

a ₂ Bad High Big Yes

a ₃ Bad Low Small No

a ₄ Bad Low Small No

a ₅ Bad Low Small No

a ₆ Bad High Medium Yes

a ₇ Common High Medium No

a ₈ Good High Medium Yes

Project	Locus	Investment	Popularity	Decision
a ₁	Common	High	Big	Yes
a ₂	Bad	High	Big	Yes
a ₃	Bad	Low	Small	No
a ₄	Bad	Low	Small	No
a ₅	Bad	Low	Small	No
a ₆	Bad	High	Medium	Yes
a ₇	Common	High	Medium	No
a ₈	Good	High	Medium	Yes

Let A₁ = {Locus}, then the partition induced by A₁ is {{a₁, a₇} , {a₂, a₃, a₄, a₅, a₆} , {a₈}}, the corresponding equivalence relation is denoted by R₁.

Let A₂ = {Investment}, then the partition induced by A₂ is {{a₁, a₂, a₆, a₇, a₈} , {a₃, a₄, a₅}}, the corresponding equivalence relation is denoted by R₂.

Let A₃ = {Popularity density}, then the partition induced by A₃ is {{a₁, a₂} , {a₃, a₄, a₅} , {a₆, a₇, a₈}}, the corresponding equivalence relation is denoted by R₃.

Let X = {a₃, a₅, a₇}, we have from $f_{X, R_{i}} (x) = \frac{| [x]_{R_{i}} \cap X |}{| [x]_{R_{i}} |} (i = 1, 2, 3)$ that $\begin{matrix} f_{X, R_{1}} & = & \frac{\frac{1}{2}}{a_{1}} + \frac{\frac{2}{5}}{a_{2}} + \frac{\frac{2}{5}}{a_{3}} + \frac{\frac{2}{5}}{a_{4}} + \frac{\frac{2}{5}}{a_{5}} \\ + \frac{\frac{2}{5}}{a_{6}} + \frac{\frac{1}{2}}{a_{7}} + \frac{0}{a_{8}}, \\ f_{X, R_{2}} & = & \frac{\frac{1}{5}}{a_{1}} + \frac{\frac{1}{5}}{a_{2}} + \frac{\frac{2}{3}}{a_{3}} + \frac{\frac{2}{3}}{a_{4}} + \frac{\frac{2}{3}}{a_{5}} \\ + \frac{\frac{1}{5}}{a_{6}} + \frac{\frac{1}{5}}{a_{7}} + \frac{\frac{1}{5}}{a_{8}}, \\ f_{X, R_{3}} & = & \frac{0}{a_{1}} + \frac{0}{a_{2}} + \frac{\frac{2}{3}}{a_{3}} + \frac{\frac{2}{3}}{a_{4}} + \frac{\frac{2}{3}}{a_{5}} \\ + \frac{\frac{1}{3}}{a_{6}} + \frac{\frac{1}{3}}{a_{7}} + \frac{\frac{1}{3}}{a_{8}} . \end{matrix}$

The results of these aggregations, with respect to different risk attitudes, are as shown in Table 9 (we assume the threshold (neutral element) is 0.5), where O means optimistic, P means pessimistic and N means neutral. In this table, we use (A^-, A⁺, A^±) to denote the compromise operator in the form of: $\begin{matrix} C_{0.5} (τ_{1}, τ_{2}, τ_{3}) \\ = {\begin{matrix} A^{-} (τ_{1}, τ_{2}, τ_{3}), & if τ_{1}, τ_{2}, τ_{3} \in [0, 0.5], \\ A^{+} (τ_{1}, τ_{2}, τ_{3}), & if τ_{1}, τ_{2}, τ_{3} \in (0.5, 1], \\ A^{\pm} (τ_{1}, τ_{2}, τ_{3}), & otherwise . \end{matrix} \end{matrix}$ (2)

Table 9

Aggregation outcomes of compromise operators with different risk attitudes

Objects	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈
Rough membership degree under L	$\frac{1}{2}$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{1}{2}$	0
Rough membership degree under I	$\frac{1}{5}$	$\frac{1}{5}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{5}$	$\frac{1}{5}$	$\frac{1}{5}$
Rough membership degree under P	0	0	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
Compromise aggregation operators
O $(A_{o}^{-}, A_{o}^{+}, A^{\pm})$	0.5	0.41	0.72	0.72	0.72	0.48	0.5	0.36
P $(A_{p}^{-}, A_{p}^{+}, A^{\pm})$	0	0	0.72	0.72	0.72	0.12	0.17	0
N $(A_{n}^{-}, A_{n}^{+}, A^{\pm})$	0.18	0.17	0.72	0.72	0.72	0.30	0.31	0.16
P-O $(A_{p}^{-}, A_{o}^{+}, A^{\pm})$	0	0	0.72	0.72	0.72	0.12	0.17	0
O-P $(A_{o}^{-}, A_{p}^{+}, A^{\pm})$	0.5	0.41	0.72	0.72	0.72	0.48	0.5	0.36
N-O $(A_{n}^{-}, A_{o}^{+}, A^{\pm})$	0.18	0.17	0.72	0.72	0.72	0.30	0.31	0.16
N-Pc $(A_{n}^{-}, A_{p}^{+}, A^{\pm})$	0.18	0.24	0.72	0.72	0.72	0.30	0.31	0.16
O-N $(A_{o}^{-}, A_{n}^{+}, A^{\pm})$	0.5	0.41	0.72	0.72	0.72	0.48	0.5	0.36
P-N $(A_{p}^{-}, A_{n}^{+}, A^{\pm})$	0	0	0.72	0.72	0.72	0.12	0.17	0

$A^{-} \in {A_{0}^{-}, A_{p}^{-}, A_{n}^{-}}$ and $A^{+} \in {A_{0}^{+}, A_{p}^{+}, A_{n}^{+}}$ , which are given in Table 10.

Table 10

Various aggregation components corresponding to different risk attitudes

Risk attitude	x₁, x₂, x₃ ∈ [0, 0.5]	x₁, x₂, x₃ ∈ (0.5, 1]
Optimistic	$A_{0}^{-} (x_{1}, x_{2}, x_{3}) = 0.5 - \frac{0.5}{\sum_{i = 1}^{3} \frac{1}{1 - 2 x_{i}} - 2}$	$A_{0}^{+} (x_{1}, x_{2}, x_{3}) = 1 - \frac{1}{\sum_{i = 1}^{3} \frac{1}{1 - x_{i}} - 4}$
Pessimistic	$A_{p}^{-} (x_{1}, x_{2}, x_{3}) = \frac{0.5}{\sum_{i = 1}^{3} \frac{1}{2 x_{i}} - 2}$	$A_{p}^{+} (x_{1}, x_{2}, x_{3}) = \frac{0.5}{\sum_{i = 1}^{3} \frac{1}{2 x_{i} - 1} - 2} + 0.5$
Neutral	$A_{n}^{-} (x_{1}, x_{2}, x_{3}) = \frac{1}{π} arcsin (\frac{sin (π x_{1}) + sin (π x_{2}) + sin (π x_{3})}{3})$	$A_{n}^{+} (x_{1}, x_{2}, x_{3}) = \frac{3}{2} - \frac{3}{\sum_{i = 1}^{3} \frac{2}{3 - 2 x_{i}}}$

In the case where a₁, a₂, a₃ ∈ [0, 0.5], row 0.5-low optimistic branch $A_{o}^{-}$ is obtained, by Theorem 1, from 0.5-constructor h (x) =2x and Dombi t-conorm in Example 5; row 0.5-low pessimistic branch $A_{p}^{-}$ is obtained, by Theorem 1, from 0.5-constructor h (x) =2x and Dombi t-norm in Example 5; row 0.5-low neutral branch $A_{n}^{-}$ is obtained, by Theorem 2, from 0.5^--generator $h (x) = \frac{1}{2} sin (π x)$ and the averaging compromise operator in Example 2.

In the case where a₁, a₂ ∈ (0.5, 1], row 0.5-high optimistic branch $A_{0}^{+}$ is obtained from 0.5⁺-constructor h (x) =2x - 1 and Dombi t-conorm; row 0.5-high pessimistic branch $A_{p}^{+}$ is obtained from 0.5⁺-constructor h (x) =2x - 1 and Dombi t-norm; row 0.5-high neutral branch $A_{n}^{+}$ is obtained from 0.5⁺-generator $h (x) = \frac{1}{3 - 2 x}$ and the averaging compromise operator in Example 2. Moreover, let $h (x) = {\begin{matrix} \frac{2 x}{1 + 2 x}, & x \in [0, 0.5], \\ \frac{1}{3 - 2 x}, & x \in [0.5, 1] . \end{matrix}$ (3) and $U (x_{1}, x_{2}) = \frac{x_{1} x_{2}}{x_{1} x_{2} + (1 - x_{1}) (1 - x_{2})},$ then according to Theorem 2, A^± can be given by: A^± (x₁, x₂, x₃) = h^-1 (U (h (x₁) , h (x₂) , h (x₃))) .

Accordingly, with respect to the different compromise operators, the various orderings for all the objects are shown in Table 11, where RAANR and RAAPR denote risk attitude in aggregating negative ratings and risk attitude in aggregating positive ratings, respectively. R-S, R-A and R-N mean risk-seeking, risk-averse, and risk-neutral, respectively. It can be seen clearly that as the type of compromise operator we adopt changes, the corresponding orderings of the overall ratings are different. Moreover, since both uninorm operators and averaging operators are special cases of compromise operators, we can thus conclude that the existing multigranulation rough set models can be put into the framework of our approach.

Table 11

The different aggregation’s overall rating orderings on the objects

RAANR	RAAPR	Compromise	Ordering on the objects
		operators
R-S	R-S	O	{a₃, a₄, a₅} , {a₁, a₇} , {a₆} , {a₂} , {a₈}
R-A	R-A	P	{a₃, a₄, a₅} , {a₇} , {a₆} , {a₁, a₂, a₈}
R-N	R-N	N	{a₃, a₄, a₅},{a₇},{a₆} , {a₁} , {a₂},{a₈}
R-A	R-S	P-O	{a₃, a₄, a₅} , {a₇} , {a₆} , {a₁, a₂, a₈}
R-S	R-A	O-P	{a₃, a₄, a₅}, {a₁, a₇}, {a₆},{a₂} , {a₈}
R-N	R-S	N-O	{a₃, a₄, a₅},{a₇},{a₆} , {a₁} , {a₂},{a₈}
R-N	R-A	N-P	{a₃, a₄, a₅},{a₇},{a₆} , {a₂} , {a₁},{a₈}
R-S	R-N	O-N	{a₃, a₄, a₅}, {a₁, a₇},{a₆},{a₂},{a₈}
R-A	R-N	P-N	{a₃, a₄, a₅}, {a₇}, {a₆},{a₁, a₂, a₈}

6 Conclusion

Information fusion has been extensively studied in recent years. In this paper, we have studied this topic in the context of multigranulation spaces. For this purpose, we firstly show that the optimistic multigranuation rough set model can be interpreted from the viewpoint of uninorm while the pessimistic multigranulation rough set model can be interpreted from the viewpoint of averaging operator or nullnorm operator. Then, by considering rough membership degrees in each Pawlak space and using the generalized compromise operator, we present a novel approach to information fusion in multigranulation space. Lastly, an illustrative example is employed to show the validity of our approach.

Our approach is clearly related to three-way decision theory [22] since for two conflicting membership degrees, their combined value is always in-between. The present study is restricted to classical multigranulation spaces, that is, the binary relation is an equivalence relation. A natural way to extend the present study is to consider other types of binary relations. Furthermore, information fusion in fuzzy multigranulation spaces is an interesting topic, and we will report the results in our forthcoming papers.

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A compromise operator-based approach for multigranulation space 1

Abstract

Keywords

1 Introduction

2 Some related notions

2.1 Multigranulation space

2.2 Characteristics of t-norm, t-conorm and uninorm in information fusion

2.2.1 T-norm and its characteristic in information fusion

2.2.2 T-conorm and its characteristic in information fusion

2.2.3 Uninorm and its characteristic in information aggregation

2.3 Averaging operator

2.4 Generalized compromise operators

2.5 Nullnorms

3 An interpretation of the multigranulation rough set models from the viewpoint of aggregation operators

3.1 Aggregation operation corresponding to optimistic multigranulation rough set model

Table 1 Aggregation operator ⊗ o induced by optimistic multigranulation rough set model ⊗ o 0 1 2 1 0 0 0 × 1 2 0 1 2 1 1 × 1 1

5 An example

References

Table 1
Aggregation operator ⊗^o induced by optimistic multigranulation rough set model

⊗^o 0 $\frac{1}{2}$ 1

0 0 0 ×

$\frac{1}{2}$ 0 $\frac{1}{2}$ 1

1 × 1 1