Abstract
This paper investigates three-way decisions of type-2 fuzzy sets and interval-valued type-2 fuzzy sets based on partially ordered sets. First, the partially ordered sets, constituted by fuzzy truth values and interval-valued fuzzy truth values, are established, respectively. They serve as the basic structures of three-way decision spaces. Second, several kinds of decision evaluation functions for type-2 fuzzy sets are defined. Three-way decision spaces and three-way decisions corresponding to each DEF are provided. In the end, a similar discussion is presented for interval-valued type-2 fuzzy sets. Examples are attached to each section to illustrate our models.
Introduction
Type-2 fuzzy sets were first introduced by Zadeh [39] as an extension of fuzzy sets (or, type-1 fuzzy sets). Type-2 fuzzy sets have grades of membership that are themselves fuzzy. Literatures [10, 26] provide us with definitions and properties of type-2 fuzzy sets and type-2 fuzzy relations. From the application viewpoint [1, 19], type-2 fuzzy sets can help us to characterize an inexact concept more precisely.
The theory of three-way decisions (short for 3WDs) proposed by Yao [28] is an extension of the classical two-way decisions. Its basic idea comes from Pawlak rough sets [22] and probabilistic rough sets [28, 42]. The main purpose is to interpret the positive, negative and boundary regions of rough sets as three kinds of decisions: acceptance, rejection, and uncertainty (or deferment), respectively. Applications of 3WDs can be found in text classification [14, 15], cluster analysis [16, 37], investment decisions [18], multi-classification [16], government decisions [17], recommender systems [40] etc.
Hu [8] studied 3WDs based on fuzzy lattices and established corresponding three-way decision spaces (short for 3WDSs). The main results of his research provide a theoretical foundation for 3WDs. It can be found that existing models of 3WDs are special cases of Hu’s model. However, fuzzy lattices limit the applications of 3WDs spaces, which encourages Hu [9] to research 3WDs based on partially ordered sets. As examples, Hu studied 3WDs for hesitant fuzzy sets [25] and interval-valued hesitant fuzzy sets [7]. Deng and Yao [6] and Sun et al. [23] studied 3WDs for fuzzy sets. Yang et al. [29] studied 3WDs for classical subsets based on fuzzy relations. Zhao and Hu [41, 42] studied 3WDs for fuzzy and interval-valued fuzzy sets based on fuzzy and interval-valued fuzzy relations.
In this age of Artificial Intelligent, the closer the thinking pattern of various models to human beings the more powerful the capability of these models. Perceptions of human beings are at high levels of imprecision for which type-2 fuzzy sets (short for T2 FSs) and interval-valued type-2 fuzzy sets (short for IVT2 FSs) can handle very well. However, 3WDs for T2 FSs and IVT2 FSs have not been studied so far. Besides, existing studies of decision making methods for T2 FSs and IVT2 FSs [3–5, 27] are too complicated for practical problems. Considering the advantages of precisely characterizing property of T2 FSs and IVT2 FSs, it is worthwhile to investigate 3WDs for T2 FSs and IVT2 FSs. On one hand, it completes the theory of 3WDs; on the other, the precisely characterizing property of T2 FSs and IVT2 FSs improves the flexibility of 3WD theory for applications.
The following is divided into three sections. Section 2 recalls definitions and properties of T2 FSs and IVT2 FSs, 3WDs and 3WDSs. Section 3 defines several decision evaluation functions based onT2 FSs and (or) type-2 fuzzy relations and establishes the corresponding 3WDSs. Section 4 discusses 3WDs based on IVT2 FSs and (or) interval-valued type-2 fuzzy relations.
Preliminary
This section mainly recalls some basic concepts, operations as well as properties about T2 FSs and IVT2 FSs, (interval-valued) type-2 fuzzy relations and 3WDs.
Extension operations of fuzzy sets
Given a nonempty universe of discourse X, a mapping A : X → [0, 1] is called a fuzzy set on X. The family of all fuzzy sets on X is denoted by or Map (X, I) (I = [0, 1]). If R ∈ Map (X × X, I), then R is called a fuzzy relation on X.
Let . The union, intersection and complement are defined pointwise by the following formulae: ∀ x ∈ X,
Given α ∈ [0, 1] and , sets A α = {x ∈ X : A (x) ≥ α} and are called α-cut set and strong α-cut set of A, respectively.
Let (J, ≤
J
) be a linearly ordered set. If J is bounded with the minimum 0
J
and the maximum 1
J
and N is an involution on J, then the algebra is denoted as (J, ∨ , ∧ , N, 0
J
, 1
J
). In the following discussion it is always assumed that J is bounded. A fuzzy set A : J → I is called a fuzzy truth value [26] on J. The family of all fuzzy truth values is denoted by Map (J, I). Given a ∈ J, a special fuzzy truth value is defined by
If * =∧ or ∨, then
Based on formulas (3) and (4), order relations for fuzzy truth values can be defined as follows.
According to [26], relations ⊑, Subset and ⪯ are partial orders.
A fuzzy truth value A ∈ Map (J, I) is said to be normal if . Let represent the collection of all normal fuzzy truth values on J.
A mapping is called a type-2 fuzzy set (T2 FS, for short) on X, where (x ∈ X), called a fuzzy grade [21], and (u ∈ J) is the membership grade of . The family of all T2 FSs on X is denoted by Map (X, Map (J, I)). If , then is called a type-2 fuzzy relation (T2 FR, for short) on X.
The operations of T2 FSs are defined respectively as follows: ∀x ∈ X,
For a T2 FS , if is normal for each x ∈ X, i.e., for all x ∈ X, then is called a normal T2 FS. The family of all normal T2 FSs on X is denoted by Map (X, MapN (J, I)).
Given two T2 FSs , it is defined that iff , ∀x ∈ X, u ∈ J. According to Remark 2.4 we have another order relation of T2 FSs based on normal fuzzy truth values.
For , it is said iff , ∀x ∈ X. Obviously, the relation ⊴ on Map (X, MapN (J, I)) is a partial order.
For more information about T2 FSs we recommend references [10, 26].
Let I(2) = {[a-, a+] :0 ≤ a- ≤ a+ ≤ 1} be the set of all interval numbers on I. A mapping [A] : J → I(2) is called an interval-valued fuzzy truth value on J, which is denoted by [A] = [A-, A+] with A-, A+ ∈ Map (J, I) being two fuzzy truth values and A- ⊆ A+. The family of all interval-valued fuzzy truth values on J is denoted by Map (J, I(2)).
For two interval-valued fuzzy truth values [A] , [B] ∈ Map (J, I(2)), it is said [A] ⪯ [B] iff A- ⪯ B-, A+ ⪯ B+. To be convenient, it is denoted by [B] ⪰ [A] if [A] ⪯ [B]; [A] ≺ [B] if [A] ⪯ [B] and [A] ≠ [B]. Obviously, relation ⪯ is a partial order on Map (J, I(2)).
Let [A] ∈ Map (J, I(2)) with [A] = [A-, A+]. If A-, A+ ∈ MapN (J, I), then [A] is said to be normal. The collection of all normal interval-valued fuzzy truth values on J is denoted by MapN (J, I(2)).
According to [11] an interval-valued type-2 fuzzy set (IVT2 FS, for short) on X is a mapping , which is denoted by with and , where and are said to be a lower T2 FS and an upper T2 FS of . The family of all IVT2 FSs on X is denoted by Map (X, Map (J, I(2))). If , then is called an interval-valued type-2 fuzzy relation (IVT2 FR, for short) on X.
The operations of IVT2 FS are defined pointwise as follows according to [11]: ∀x ∈ X,
In order to be in accordance with aforementioned discussion, only normal IVT2 FSs are taken into consideration, i.e., the IVT2 FSs such that and . The family of all normal IVT2 FSs is denoted by (X, MapN (J, I(2))).
Obviously, the relation ⊴ is a partial order on Map (X, MapN (J, I(2))). It is said that if for .
Three-way decisions based on partially ordered set
Some underlying concepts about 3WDs and 3WDSs based on partially ordered sets [9] are reviewed in this section. Let U, V be two nonempty universes, where U is to make decisions, called decision universe, and V is to define condition functions, named condition universe. Suppose that (LC, ≤ L C , N L C , 0 L C , 1 L C ) and (LD, ≤ L D , N L D , 0 L D , 1 L D ) are two partially ordered sets in the following discussion.
Minimum element axiom: E (∅) = ∅, i.e., E (∅) (x) =0
L
D
, ∀ x ∈ U; Monotonicity axiom: A ⊆
L
C
B ⇒ E (A) ⊆
L
D
E (B), ∀A, B ∈ Map (V, LC), i.e., A (x) ≤
L
C
B (x) , ∀ x ∈ V ⇒ E (A) (y) ≤
L
D
E (B) (y), ∀y ∈ U; Complement axiom: N
L
D
(E (A)) = E (N
L
C
(A)), ∀A ∈ Map (V, LC),i.e., N
L
D
(E (A)) (x) = E (N
L
C
(A)) (x) , ∀ x ∈ U.
Acceptance region: ACP(α,β) (E, A) = {x ∈ U : E (A) (x) ≥
L
D
α}; Rejection region: REJ(α,β) (E, A) = {x ∈ U : E (A) (x) ≤
L
D
β}; Uncertain region: UNC(α, β) (E, A) = (ACP(α,β) (E, A) ∪ REJ(α,β) (E, A)) c.
If LD is a linear order, then UNC(α,β) (E, A) = {x ∈ U : β <
L
D
E (A) (x) <
L
D
α}.
Three-way decisions based on T2 FSs and T2 FRs
This section establishes 3WDs and 3WDSs based on T2 FSs and T2 FRs.
Three-way decisions based on T2 FSs
Each T2 FS in Map (X, MapN (J, I)) can be seen as a DEF of X, which is shown in the following theorem.
Given α, β ∈ MapN (J, I) and ,the 3WDs for a normal T2 FS are as follows. Acceptance region: ; Rejection region: ; Uncertain region:
Suppose the evaluation standard is given by the decision maker as: and . According to Theorem 3.1, 3WDs for are finally obtained as follows.
Let be a normal T2 FS. For a given parameter γ ∈ [0, 1), the fuzzy set
(1) It follows from Remark 2.5 that for each x ∈ X, by which it implies that (x ∈ X), i.e., , ∀x ∈ X.
(2) Let and , which means . It then follows from Definition 2.5 in reference [10] that and for all x ∈ X and, consequently, for each x ∈ X, i.e., , ∀x ∈ X.
(3) Let . For each x ∈ X, the following deduction holds.
This completes the proof. □
Given α, β ∈ I and 0 ≤ α < β ≤ 1, the 3WDs for a normal T2 FS are given as follows based on DEFs (*=1, 2, 3). Acceptance region: ; Rejection region: ; Uncertain region: .
Furthermore, suppose relationship between these projects is represented by a reflexive and symmetric relation R (shown in Table 2, only the lower triangle is presented because of the reflexivity and symmetry of R). By taking this relation into consideration, we adopt the DEF in Theorem 3.5. Thus, it is computed that , and . By setting α = 0.75 and β = 0.72 we can put project x6 into the acception region and projects x1, x4 into the rejection region.
Let be a normal T2 FR and ψ ∈ MapN (J, I) be a normal fuzzy truth value. The classical subset of X × X,
Let be a normal T2 FR and γ ∈ [0, 1). The γ-fuzzy cut set of is defined as follows: ∀ x, y ∈ X,
Let be a normal T2 FR. Define
Given α, β ∈ I and 0 ≤ α < β ≤ 1, the 3WDs for a normal T2 FS based on DEFs (* = ψ, 4, (λ)) are given as follows. Acceptance region: , Rejection region: , Uncertain region: .
When the lower and upper T2 FSs of an IVT2 FS are equal, the IVT2 FS degenerates into a T2 FS. That is to say, T2 FSs are special cases of IVT2 FSs which is one of the motivations for studying 3WDs and 3WDSs based on IVT2 FSs and IVT2 FRs in this section. The latter allows more flexibility for representing human’s perception.
Three-way decisions based on IVT2 FSs
Given α, β ∈ MapN (J, I(2)) and , the 3WDs for based on DEF E are produced as follows. Acceptance region: ; Rejection region: ; Uncertain region: .
Let with and . For parameter [γ-, γ+] ∈ I(2) and x ∈ X, let
With the concept of [γ-, γ+]-fuzzy cut set, more decision evaluation functions can be defined as follows.
(1) By Remark 2.9 and the proof of Theorem 3.4, it follows that for all x ∈ X.
(2) Suppose that which means and , ∀x ∈ X. According to the proof of Theorem 3.4 it follows that and for all x ∈ X, from which it concludes that for all x ∈ X, i.e., , ∀x ∈ X.
(3) According to the proof of Theorem 3.4, for each x ∈ X the following holds.
Given 0 ≤ α < β ≤ 1, 3WDs based on DEFs (*=1, 2, 3) for are given as follows. Acceptance region: ; Rejection region: ; Uncertain region: .
Given [γ-, γ+] ∈ I(2), it follows that (i = 1, 2, …, 9), (i = 1, 2, …, 8)and . Thus, we have (i = 1, 2, …, 9) and (i = 1, …, 8) and . According to (10), it follows that (i = 1, …, 8) and .
Set [γ-, γ+] = [0.5, 0.6], α = 0.65 and β = 0.45, then the 3WDs based on are finally obtained as follows.
Results show that scenic spots x7, x8, x9 are selected as the best, spots x1, x2, x3, x4 are rejected to be the best, current information is not enough to make certain decisions on spots x5, x6.
Furthermore, suppose the relationship between these nine scenic spots is depicted by a fuzzy relation (shown in Table 3) and set α = 0.6, β = 0.5. By Theorem 4.4 it obtains and . Thus, based on this new information we can say that spot x5 belongs to the rejection decision region and spot x6 to the acception region.
Let and [ψ] ∈ MapN (J, I(2)). The classical subset of X × X,
Let [γ-, γ+] ∈ I(2). The [γ-, γ+]-fuzzy cut set of is defined as
Given [γ-, γ+] ∈ I(2) and λ ∈ [0, 1], for a normal IVT2 FR , let
Given 0 ≤ α < β ≤ 1, we have the following 3WDs based on DEFs (* = [ψ] , 4, (λ)) on the basis of a normal IVT2 FS . Acceptance region: ; Rejection region: ; Uncertain region: .
Footnotes
Acknowledgments
The work described in this paper is supported by the National Nature Science Foundation of China (Grant Nos. 11571010, 61179038) and the Fundamental Research Funds for the South-Central University for Nationalities (Grant No. CZQ16013).
