Three-way decisions based on type-2 fuzzy sets and interval-valued type-2 fuzzy sets

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.

Keywords

Three-way decisions partially ordered sets type-2 fuzzy sets interval-valued type-2 fuzzy sets

1 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.

2 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.

2.1 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 $F (X)$ or Map (X, I) (I = [0, 1]). If R ∈ Map (X × X, I), then R is called a fuzzy relation on X.

Let $A, B \in F (X)$ . The union, intersection and complement are defined pointwise by the following formulae: ∀ x ∈ X, $\begin{matrix} (A \cup B) (x) = A (x) \lor B (x), \\ (A \cap B) (x) = A (x) \land B (x), and \\ (A^{N}) (x) = N (A (x)), esp . (A^{c}) (x) = 1 - A (x) . \end{matrix}$ where N is an involution on I (i.e., N (0) =1, N (1) =0, a ≤ b ⇒ N (b) ≤ N (a) and N (N (a)) = a, ∀ a, b ∈ I). The order relation of fuzzy sets is defined as: for $A, B \in F (X)$ , A ⊆ B iff (if and only if) A (x) ≤ B (x), ∀x ∈ X.

Given α ∈ [0, 1] and $A \in F (X)$ , sets A_α = {x ∈ X : A (x) ≥ α} and $A_{\bar{α}} = {x \in X : A (x) > α}$ 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 $\bar{a}$ is defined by $\bar{a} (u) = {\begin{matrix} 1, u = a, \\ 0, u \neq a . \end{matrix}$

Definition 2.1. [10] Let A, B ∈ Map (J, I), N be an involution over J and * be a binary operation on J. Then the extensions of N and * are defined pointwise as follows: ∀ u ∈ J, $A^{\tilde{N}} (u) = sup_{u = N (v)} A (v) = A (N (u)),$ (1) $(A \tilde{*} B) (u) = sup_{u = u_{1} * u_{2}} A (u_{1}) \land B (u_{2}) .$ (2)

If * =∧ or ∨, then $(A \tilde{\land} B) (u) = sup_{u = u_{1} \land u_{2}} A (u_{1}) \land B (u_{2}),$ (3) $(A \tilde{\lor} B) (u) = sup_{u = u_{1} \lor u_{2}} A (u_{1}) \land B (u_{2}) .$ (4)

Based on formulas (3) and (4), order relations for fuzzy truth values can be defined as follows.

Definition 2.2. [26] [26] For any fuzzy truth values A, B ∈ Map (J, I), it is said that A ⊑ B iff $A \tilde{\land} B = A$ ; ASubsetB iff $A \tilde{\lor} B = B$ ; and A ⪯ B iff A ⊑ B and ASubsetB.

According to [26], relations ⊑, Subset and ⪯ are partial orders.

Definition 2.3. [10] Given two crisp subsets $A, B \subseteq J$ , it is denoted by $A \leq_{\lor} B$ if for each $u_{0} \in A$ there exists a v₀ ∈ B such that u₀ ≤ v₀; similarly, $A \leq_{\land} B$ if for each $v_{0} \in B$ there exists a $u_{0} \in A$ such that u₀ ≤ v₀.

A fuzzy truth value A ∈ Map (J, I) is said to be normal if $A^{sup} = sup_{u \in J} A (u) = 1$ . Let ${Map}_{N} (J, I)$ represent the collection of all normal fuzzy truth values on J.

Remark 2.4. Obviously, the two special fuzzy truth values ${\bar{0}}_{J}, {\bar{1}}_{J}$ are normal, and for all A ∈ Map_N (J, I) it holds that ${\bar{0}}_{J} ⪯ A ⪯ {\bar{1}}_{J}$ . Thus, (Map_N (J, I) , ⪯) is a partially ordered set with an involution $\tilde{N}$ , and ${\bar{0}}_{J}, {\bar{1}}_{J}$ are the minimum and the maximum of Map_N (J, I), respectively.

2.2 Type-2 fuzzy sets and operations

A mapping $\tilde{A} : X \to Map (J, I)$ is called a type-2 fuzzy set (T2 FS, for short) on X, where $\tilde{A} (x) ≜ {\tilde{A}}_{x} \in Map (J, I)$ (x ∈ X), called a fuzzy grade [21], and $\tilde{A} (x) (u) ≜ {\tilde{A}}_{x} (u) ≜ \tilde{A} (x, u)$ (u ∈ J) is the membership grade of $\tilde{A} (x)$ . The family of all T2 FSs on X is denoted by Map (X, Map (J, I)). If $\tilde{R} \in Map (X \times X, Map (J, I))$ , then $\tilde{R}$ 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, $({\tilde{A}}^{\tilde{N}}) (x) = (\tilde{A} (x))^{\tilde{N}},$ (5) $(\tilde{A} ⊓ \tilde{B}) (x) = \tilde{A} (x) \tilde{\land} \tilde{B} (x),$ (6) $(\tilde{A} ⊔ \tilde{B}) (x) = \tilde{A} (x) \tilde{\lor} \tilde{B} (x) .$ (7) where N is an involution on J and $\tilde{A}, \tilde{B} \in Map (X, Map (J, I))$ .

For a T2 FS $\tilde{A} \in Map (X, Map (J, I))$ , if $\tilde{A} (x)$ is normal for each x ∈ X, i.e., $\tilde{A} (x) \in {Map}_{N} (J, I)$ for all x ∈ X, then $\tilde{A}$ is called a normal T2 FS. The family of all normal T2 FSs on X is denoted by Map (X, Map_N (J, I)).

Given two T2 FSs $\tilde{A}, \tilde{B} \in Map (X, Map (J, I))$ , it is defined that $\tilde{A} \subseteq \tilde{B}$ iff $\tilde{A} (x, u) \leq \tilde{B} (x, u)$ , ∀x ∈ X, u ∈ J. According to Remark 2.4 we have another order relation of T2 FSs based on normal fuzzy truth values.

For $\tilde{A}, \tilde{B} \in Map (X, {Map}_{N} (J, I))$ , it is said $\tilde{A} ⊴ \tilde{B}$ iff $\tilde{A} (x) ⪯ \tilde{B} (x)$ , ∀x ∈ X. Obviously, the relation ⊴ on Map (X, Map_N (J, I)) is a partial order.

Remark 2.5. Let $\tilde{\emptyset} (x) = {\bar{0}}_{J}$ and $\tilde{X} (x) = {\bar{1}}_{J}$ for all x ∈ X. Then, it is obvious that $\tilde{\emptyset}, \tilde{X} \in Map (X, {Map}_{N} (J, I))$ and $\tilde{\emptyset} ⊴ \tilde{A} ⊴ \tilde{X}$ , $\forall \tilde{A} \in Map (X, {Map}_{N} (J, I))$ (by Remark 2.4). This means (Map (X, Map_N (J, I)) , ⊴) is a partially ordered set bounded with $\tilde{\emptyset}$ and $\tilde{X}$ .

For more information about T2 FSs we recommend references [10 , 26].

2.3 Interval-valued type-2 fuzzy sets and operations

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⁾).

Definition 2.6. Let N be an involution on J and [A] , [B] ∈ Map (J, I⁽²⁾). The extension operations for interval-valued fuzzy truth values are defined pointwise as follows: ∀ u ∈ J, $\begin{matrix} [A]^{\tilde{N}} (u) = [(A^{-})^{\tilde{N}} (u), (A^{+})^{\tilde{N}} (u)], \\ ([A] \tilde{\land} [B]) (u) = [(A^{-} \tilde{\land} B^{-}) (u), (A^{+} \tilde{\land} B^{+}) (u)], \\ ([A] \tilde{\lor} [B]) (u) = [(A^{-} \tilde{\lor} B^{-}) (u), (A^{+} \tilde{\lor} B^{+}) (u)] . \end{matrix}$

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⁺ ∈ Map_N (J, I), then [A] is said to be normal. The collection of all normal interval-valued fuzzy truth values on J is denoted by Map_N (J, I⁽²⁾).

Remark 2.7. Let two special interval-valued fuzzy truth values be defined as: $[{\bar{0}}_{J}] = [{\bar{0}}_{J}, {\bar{0}}_{J}]$ and $[{\bar{1}}_{J}] = [{\bar{1}}_{J}, {\bar{1}}_{J}]$ . Then, $[{\bar{0}}_{J}], [{\bar{1}}_{J}] \in {Map}_{N} (J, I^{(2)})$ and $[{\bar{0}}_{J}] ⪯ [A] ⪯ [{\bar{1}}_{J}]$ for any [A] ∈ Map_N (J, I⁽²⁾). Thus, (Map_N (J, I⁽²⁾) , ⪯) is a partially ordered set with an involution $\tilde{N}$ and $[{\bar{0}}_{J}], [{\bar{1}}_{J}]$ are the minimum and the maximum of Map_N (J, I⁽²⁾), respectively.

According to [11] an interval-valued type-2 fuzzy set (IVT2 FS, for short) on X is a mapping $[\tilde{A}] : X \to Map (J, I^{(2)})$ , which is denoted by $[\tilde{A}] = [{\tilde{A}}^{-}, {\tilde{A}}^{+}]$ with ${\tilde{A}}^{-}, {\tilde{A}}^{+} \in Map (X, Map (J, I))$ and ${\tilde{A}}^{-} \subseteq {\tilde{A}}^{+}$ , where ${\tilde{A}}^{-}$ and ${\tilde{A}}^{+}$ are said to be a lower T2 FS and an upper T2 FS of $[\tilde{A}]$ . The family of all IVT2 FSs on X is denoted by Map (X, Map (J, I⁽²⁾)). If $[\tilde{R}] \in Map (X \times X, Map (J, I))$ , then $[\tilde{R}]$ 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, $\begin{matrix} [\tilde{A}]^{\tilde{N}} (x) = [({\tilde{A}}^{-})^{\tilde{N}} (x), ({\tilde{A}}^{+})^{\tilde{N}} (x)] \\ = [({\tilde{A}}^{-} (x))^{\tilde{N}}, ({\tilde{A}}^{+} (x))^{\tilde{N}}], \\ ([\tilde{A}] ⊓ [\tilde{B}]) (x) = [{\tilde{A}}^{-} (x) \tilde{\land} {\tilde{B}}^{-} (x), {\tilde{A}}^{+} (x) \tilde{\land} {\tilde{B}}^{+} (x)], \\ ([\tilde{A}] ⊔ [\tilde{B}]) (x) = [{\tilde{A}}^{-} (x) \tilde{\lor} {\tilde{B}}^{-} (x), {\tilde{A}}^{+} (x) \tilde{\lor} {\tilde{B}}^{+} (x)], \end{matrix}$ for all $[\tilde{A}], [\tilde{B}] \in Map (X, Map (J, I^{(2)}))$ .

In order to be in accordance with aforementioned discussion, only normal IVT2 FSs are taken into consideration, i.e., the IVT2 FSs such that $[\tilde{A}] = [{\tilde{A}}^{-}, {\tilde{A}}^{+}]$ and ${\tilde{A}}^{-}, {\tilde{A}}^{+} \in {Map}_{N} (J, I)$ . The family of all normal IVT2 FSs is denoted by (X, Map_N (J, I⁽²⁾)).

Definition 2.8. Let $[\tilde{A}], [\tilde{B}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ be two normal IVT2 FSs. Then, $[\tilde{A}] ⊴ [\tilde{B}]$ iff $[\tilde{A} (x)] ⪯ [\tilde{B} (x)]$ , ∀x ∈ X iff ${\tilde{A}}^{-} (x) ⪯ {\tilde{B}}^{-} (x)$ and ${\tilde{A}}^{+} (x) ⪯ {\tilde{B}}^{+} (x)$ , ∀x ∈ X.

Obviously, the relation ⊴ is a partial order on Map (X, Map_N (J, I⁽²⁾)). It is said that $[\tilde{B}] ⊵ [\tilde{A}]$ if $[\tilde{A}] ⊴ [\tilde{B}]$ for $[\tilde{A}], [\tilde{B}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ .

Remark 2.9. Let $[\tilde{\emptyset}] (x) = [{\bar{0}}_{J}, {\bar{0}}_{J}]$ and $[\tilde{X}] (x) = [{\bar{1}}_{J}, {\bar{1}}_{J}]$ , ∀x ∈ X. Then, it is obvious that $[\tilde{\emptyset}], [\tilde{X}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ and $[\tilde{\emptyset}] ⊴ [\tilde{A}] ⊴ [\tilde{X}]$ , for all $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ . That is to say, (Map (X, Map_N (J, I⁽²⁾)) , ⊴) is a partially ordered set bounded with $[\tilde{\emptyset}]$ and $[\tilde{X}]$ .

2.4 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 (L_C, ≤ _{L
_C}, N_{L
_C}, 0_{L
_C}, 1_{L
_C}) and (L_D, ≤ _{L
_D}, N_{L
_D}, 0_{L
_D}, 1_{L
_D}) are two partially ordered sets in the following discussion.

Definition 2.10. [9][9] Let U be a decision universe and V be a condition universe. Then a mapping E : Map (V, L_C) → Map (U, L_D) is called a decision evaluation function (short for DEF) of U, if it satisfies the following axioms.

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, L_C), 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, L_C),i.e., N_{L
_D} (E (A)) (x) = E (N_{L
_C} (A)) (x) , ∀ x ∈ U.

Definition 2.11. [9][9] Let U be a decision universe and V a condition universe. Given a DEF E on U, then (U, Map (V, L_C) , L_D, E) is called a three-way decision space (3WDS, for short).

Definition 2.12. [9] Let (U, Map (V, L_C) , L_D, E) be a 3WDS and A ∈ Map (V, L_C). For α, β ∈ L_D and 0_{L
_D} ≤ _{L
_D}β < _{L
_D}α ≤ _{L
_D}1_{L
_D}, the three-way decisions (3WDs, for short) on the basis of A are defined as follows:

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 L_D is a linear order, then UNC_(α,β) (E, A) = {x ∈ U : β < _{L
_D}E (A) (x) < _{L
_D}α}.

3 Three-way decisions based on T2 FSs and T2 FRs

This section establishes 3WDs and 3WDSs based on T2 FSs and T2 FRs.

3.1 Three-way decisions based on T2 FSs

Each T2 FS in Map (X, Map_N (J, I)) can be seen as a DEF of X, which is shown in the following theorem.

Theorem 3.1. For $\tilde{A} \in Map (X, {Map}_{N} (J, I))$ , let $E (\tilde{A}) (x) = \tilde{A} (x), x \in X .$ Then the function E is a DEF of X and the space (X, Map (X, Map_N (J, I)) , Map_N (J, I) , E) is a 3WDS.

Proof. It only needs to prove that function E satisfies axioms (E1)–(E3) in Definition 2.10, which is obvious according to Remark 2.5 and formula (5). □

Given α, β ∈ Map_N (J, I) and ${\bar{0}}_{J} ⪯ α ≺ β ⪯ {\bar{1}}_{J}$ ,the 3WDs for a normal T2 FS $\tilde{A} \in Map (X, {Map}_{N} (J, I))$ are as follows.

Acceptance region: ${ACP}_{(α, β)} (E, \tilde{A}) = {x \in X : E (\tilde{A}) (x) = \tilde{A} (x) ⪰ α}$ ;

Rejection region: ${REJ}_{(α, β)} (E, \tilde{A}) = {x \in X : E (\tilde{A}) (x) = \tilde{A} (x) ⪯ β}$ ;

Uncertain region: ${UNC}_{(α, β)} (E, \tilde{A}) = ({ACP}_{(α, β)} (E, \tilde{A}) \cup {REJ}_{(α, β)} (E, \tilde{A}))^{c} .$

Example 3.2. Let X = {x₁, x₂, …, x₆} be a set of six promising projects. A decision maker wants to invest one or some of these projects. First, he needs to evaluate each of these projects. To be precise, the decision maker, in the first step, grades the investment possibility into six levels: 0.5, 0.6, …, 0.9, 1. (In the opinion of the decision maker if the investment possibility of a project is lower than 0.5, then it is not deserved to be invested.) Then, he estimates the certainty degree of each possibility level for every project based on its development potential. The results are shown in Table 1 based on which a T2 FS $\tilde{A} \in Map (X, Map (J, I))$ is obtained as follows. $\begin{matrix} \tilde{A} (x_{1}) & = & \frac{1}{0.7} + \frac{0.8}{0.8} + \frac{0.7}{0.9}, \\ \tilde{A} (x_{2}) & = & \frac{1}{0.5} + \frac{0.8}{0.7} + \frac{0.6}{0.8} + \frac{0.4}{0.9}, \\ \tilde{A} (x_{3}) & = & \frac{1}{0.5} + \frac{0.8}{0.6} + \frac{0.7}{0.7}, \\ \tilde{A} (x_{4}) & = & \frac{1}{0.7} + \frac{0.8}{0.9}, \\ \tilde{A} (x_{5}) & = & \frac{1}{0.7} + \frac{0.8}{0.8} + \frac{0.8}{0.9} + \frac{0.7}{1}, \\ \tilde{A} (x_{6}) & = & \frac{1}{0.7} + \frac{0.8}{0.8} + \frac{0.7}{0.9}, \end{matrix}$ where J = {0.5, 0.6, …, 1}.

Suppose the evaluation standard is given by the decision maker as: $α = \frac{1}{0.7} + \frac{0.8}{0.8} + \frac{0.7}{0.9}$ and $β = \frac{1}{0.5} + \frac{0.8}{0.6} + \frac{0.7}{0.7}$ . According to Theorem 3.1, 3WDs for $\tilde{A}$ are finally obtained as follows. $\begin{matrix} {ACP}_{(α, β)} (E, \tilde{A}) = {x_{1}, x_{5}, x_{6}}, \\ {REJ}_{(α, β)} (E, \tilde{A}) = {x_{3}}, \\ {UNC}_{(α, β)} (E, \tilde{A}) = {x_{2}, x_{4}} . \end{matrix}$ Results show that projects x₁, x₅, x₆ are going to be invested, projects x₂, x₄ are not for sure and project x₃ will be given up.

Remark 3.3. In the following discussion within this subsection we assume that J = I = [0, 1] and N (u) =1 - u, ∀u ∈ [0, 1].

Let $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ be a normal T2 FS. For a given parameter γ ∈ [0, 1), the fuzzy set ${\tilde{A}}_{γ} (x) = \frac{inf (\tilde{A} (x))_{\bar{γ}} + sup (\tilde{A} (x))_{\bar{γ}}}{2}, x \in X$ (8) is called the γ-fuzzy cut set of $\tilde{A}$ , where $(\tilde{A} (x))_{\bar{γ}} = {u \in I : \tilde{A} (x, u) > γ}$ (x ∈ X).

Theorem 3.4. Let $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ be a normal T2 FS and γ ∈ [0, 1). If it takes $E_{γ}^{1} (\tilde{A}) (x) = {\tilde{A}}_{γ} (x), x \in X,$ then the function $E_{γ}^{1}$ is a DEF on X and the space $(X, Map (X, {Map}_{N} (I, I)), I, E_{γ}^{1})$ is a 3WDS.

Proof. For each γ ∈ [0, 1), let us prove that $E_{γ}^{1}$ satisfies axioms (E1)–(E3) in Definition 2.10 accordingly.

(1) It follows from Remark 2.5 that $(\tilde{\emptyset} (x))_{\bar{γ}} = {0}$ for each x ∈ X, by which it implies that ${\tilde{\emptyset}}_{γ} (x) = 0$ (x ∈ X), i.e., $E_{γ}^{1} (\tilde{\emptyset}) (x) = 0$ , ∀x ∈ X.

(2) Let $\tilde{A}, \tilde{B} \in Map (X, {Map}_{N} (I, I))$ and $\tilde{A} ⊴ \tilde{B}$ , which means $\tilde{A} (x) ⪯ \tilde{B} (x), \forall x \in X$ . It then follows from Definition 2.5 in reference [10] that $sup (\tilde{A} (x))_{\bar{γ}} \leq sup (\tilde{B} (x))_{\bar{γ}}$ and $inf (\tilde{A} (x))_{\bar{γ}} \leq inf (\tilde{B} (x))_{\bar{γ}}$ for all x ∈ X and, consequently, ${\tilde{A}}_{γ} (x) \leq {\tilde{B}}_{γ} (x)$ for each x ∈ X, i.e., $E_{γ}^{1} (\tilde{A}) (x) \leq E_{γ}^{1} (\tilde{B}) (x)$ , ∀x ∈ X.

(3) Let $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ . For each x ∈ X, the following deduction holds. $\begin{matrix} E_{γ}^{1} ({\tilde{A}}^{\tilde{N}}) (x) = ({\tilde{A}}^{\tilde{N}})_{γ} (x) \\ = \frac{inf {u : \tilde{A} (x, N (u)) > γ} + sup {u : \tilde{A} (x, N (u)) > γ}}{2} \\ = \frac{inf {N (u) : \tilde{A} (x, u) > γ} + sup {N (u) : \tilde{A} (x, u) > γ}}{2} \\ = \frac{N (sup {u : \tilde{A} (x, u) > γ}) + N (inf {u : \tilde{A} (x, u) > γ})}{2} \\ = 1 - {\tilde{A}}_{γ} (x) = N (E_{γ}^{1} (\tilde{A}) (x)) = N (E_{γ}^{1} (\tilde{A})) (x) . \end{matrix}$

This completes the proof. □

Theorem 3.5. Let $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ and R be a fuzzy relation on X with ∑_y∈XR (x, y) ≠0 (∀x ∈ U). For any γ ∈ [0, 1), if it takes $E_{γ}^{2} (\tilde{A}) (x) = \frac{\sum_{y \in X} {\tilde{A}}_{γ} (y) R (x, y)}{\sum_{y \in X} R (x, y)}, x \in X,$ then the function $E_{γ}^{2}$ is a DEF on X and the space $(X, Map (X, {Map}_{N} (I, I)), I, E_{γ}^{2})$ is a 3WDS.

Proof. It is only needed to prove that $E_{γ}^{2} ({\tilde{A}}^{\tilde{N}}) = N (E_{γ}^{2} (\tilde{A}))$ for all $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ . This is obvious since $\begin{matrix} E_{γ}^{2} ({\tilde{A}}^{\tilde{N}}) (x) = \frac{\sum_{y \in X} ({\tilde{A}}^{\tilde{N}})_{γ} (y) R (x, y)}{\sum_{y \in X} R (x, y)} \\ = \frac{\sum_{y \in X} (1 - {\tilde{A}}_{γ} (y)) R (x, y)}{\sum_{y \in X} R (x, y)} \\ = 1 - \frac{\sum_{y \in X} {\tilde{A}}_{γ} (y) R (x, y)}{\sum_{y \in X} R (x, y)} = 1 - E_{γ}^{2} (\tilde{A}) (x) . □ \end{matrix}$

Remark 3.6. If R is a classical relation with [x] _R≠ ∅ (x ∈ X), it then follows that $| {\tilde{A}}_{γ} \cap [x]_{R} | = \sum_{y \in X} {\tilde{A}}_{γ} (y) R (x, y)$ (x ∈ X). Thus, function $E_{γ}^{3} (\tilde{A}) (x) = \frac{| {\tilde{A}}_{γ} \cap [x]_{R} |}{| [x]_{R} |}, x \in X,$ is also a DEF on X and $(X, Map (X, {Map}_{N} (I, I)), I, E_{γ}^{3})$ is a 3WDS.

Given α, β ∈ I and 0 ≤ α < β ≤ 1, the 3WDs for a normal T2 FS $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ are given as follows based on DEFs $E_{γ}^{*}$ (*=1, 2, 3).

Acceptance region: ${ACP}_{(α, β)} (E_{γ}^{*}, \tilde{A}) = {x \in X : E_{γ}^{*} (\tilde{A}) (x) \geq α}$ ;

Rejection region: ${REJ}_{(α, β)} (E_{γ}^{*}, \tilde{A}) = {x \in X : E_{γ}^{*} (\tilde{A}) (x) \leq β}$ ;

Uncertain region: ${UNC}_{(α, β)} (E_{γ}^{*}, \tilde{A}) = {x \in X : β < E_{γ}^{*} (\tilde{A}) (x) < α}$ .

Example 3.7. (Continued from Example 3.2). Set γ = 0.7, then by formula (8) the 0.7-fuzzy cut set of $\tilde{A}$ is computed as ${\tilde{A}}_{0.7} = \frac{0.8}{x_{1}} + \frac{0.6}{x_{2}} + \frac{0.6}{x_{3}} + \frac{0.8}{x_{4}} + \frac{0.85}{x_{5}} + \frac{0.8}{x_{6}} .$ By Theorem 3.4 it follows that $E_{0.7}^{1} (\tilde{A}) = {\tilde{A}}_{0.7}$ . Set α = 0.85 and β = 0.75, then 3WDs based on $E_{0.7}^{1}$ for $\tilde{A}$ are $\begin{matrix} {ACP}_{(0.85, 0.75)} (E_{0.7}^{1}, \tilde{A}) = {x_{5}}, \\ {REJ}_{(0.85, 0.75)} (E_{0.7}^{1}, \tilde{A}) = {x_{2}, x_{3}}, \\ {UNC}_{(0.85, 0.75)} (E_{0.7}^{1}, \tilde{A}) = {x_{1}, x_{4}, x_{6}} . \end{matrix}$ That is to say, project x₅ will be invested by the decision maker, projects x₂, x₃ will be rejected and it is not sure for projects x₁, x₄, x₆ with currentinformation.

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 $E_{0.7}^{2} (\tilde{A}) (x_{1}) = 0.7130$ , $E_{0.7}^{2} (\tilde{A}) (x_{4}) = 0.7083$ and $E_{0.7}^{2} (\tilde{A}) (x_{6}) = 0.7563$ . 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.

3.2 Three-way decisions based on T2 FRs

Let $\tilde{R} \in Map (X \times X, {Map}_{N} (J, I))$ be a normal T2 FR and ψ ∈ Map_N (J, I) be a normal fuzzy truth value. The classical subset of X × X, ${\tilde{R}}_{ψ} = {(x, y) \in X \times X : \tilde{R} (x, y) ⪰ ψ},$ is called the ψ-cut set of $\tilde{R}$ . Given x ∈ X, define $[x]_{{\tilde{R}}_{ψ}} = {y \in X : \tilde{R} (x, y) ⪰ ψ} .$

Theorem 3.8. Given a normal T2 FR $\tilde{R} \in Map (X \times X, {Map}_{N} (J, I))$ and a normal fuzzy truth value ψ ∈ Map_N (J, I), suppose that $[x]_{{\tilde{R}}_{ψ}} \neq \emptyset$ for each x ∈ X, then function $E_{γ}^{ψ} (\tilde{A}) (x) = \frac{| {\tilde{A}}_{γ} \cap [x]_{{\tilde{R}}_{ψ}} |}{| [x]_{{\tilde{R}}_{ψ}} |}, x \in X$ is a DEF of X, where $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ is a normal T2 FS and γ ∈ [0, 1) is a given parameter. Moreover, $(X, Map (X, {Map}_{N} (I, I)), I, E_{γ}^{ψ})$ is a 3WDS.

Proof. Suppose $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ is a normal T2 FS and γ ∈ [0, 1). Since $[x]_{{\tilde{R}}_{ψ}}$ is a classical subset of X and not empty for each x ∈ X, it is only needed to verify that equation $E_{γ}^{ψ} ({\tilde{A}}^{\tilde{N}}) (x) = 1 - E_{γ}^{ψ} (\tilde{A}) (x)$ holds for every x ∈ X. In fact, the following deduction $\begin{matrix} | ({\tilde{A}}^{\tilde{N}})_{γ} \cap [x]_{{\tilde{R}}_{ψ}} | = \sum_{y \in [x]_{{\tilde{R}}_{ψ}}} ({\tilde{A}}^{\tilde{N}})_{γ} (y) \\ = \sum_{y \in [x]_{{\tilde{R}}_{ψ}}} (1 - {\tilde{A}}_{γ} (y)) = | [x]_{{\tilde{R}}_{ψ}} | - | {\tilde{A}}_{γ} \cap [x]_{{\tilde{R}}_{ψ}} | \end{matrix}$ holds for every x ∈ X which means $E_{γ}^{ψ} ({\tilde{A}}^{\tilde{N}}) (x) = 1 - E_{γ}^{ψ} (\tilde{A}) (x)$ . □

Remark 3.9. In the following discussion within this subsection it is supposed that J = I and N (u) =1 - u, ∀ u ∈ [0, 1].

Let $\tilde{R} \in Map (X \times X, {Map}_{N} (I, I))$ be a normal T2 FR and γ ∈ [0, 1). The γ-fuzzy cut set of $\tilde{R}$ is defined as follows: ∀ x, y ∈ X, ${\tilde{R}}_{γ} (x, y) = \frac{inf (\tilde{R} (x, y))_{\bar{γ}} + sup (\tilde{R} (x, y))_{\bar{γ}}}{2},$ (9) where $(\tilde{R} (x, y))_{\bar{γ}} = {u \in I : \tilde{R} (x, y) (u) > γ}$ .

Theorem 3.10. Given a normal T2 FR $\tilde{R} \in Map (X \times X, {Map}_{N} (J, I))$ and a parameter γ ∈ [0, 1), suppose that $\sum_{y \in X} {\tilde{R}}_{γ} (x, y) \neq 0$ for each x ∈ X. For any normal T2 FS $\tilde{A} \in Map (X, {Map}_{N} (I, I))$ , let $E_{γ}^{4} (\tilde{A}) (x) = \frac{\sum_{y \in X} {\tilde{A}}_{γ} (y) {\tilde{R}}_{γ} (x, y)}{\sum_{y \in X} {\tilde{R}}_{γ} (x, y)}, x \in X .$ Then,the function $E_{γ}^{4}$ is a DEF of X and the space $(X, Map (X, {Map}_{N} (I, I)), I, E_{γ}^{4})$ is a 3WDS.

Proof. By formula (9), relation ${\tilde{R}}_{γ}$ is a fuzzy relation on X for any given normal T2 FR $\tilde{R}$ and any given parameter γ ∈ [0, 1). Since $\sum_{y \in X} {\tilde{R}}_{γ} (x, y) \neq 0$ for each x ∈ X, then function $E_{γ}^{4}$ is a DEF of X according to Theorem 3.5. □

Let $\tilde{R} \in Map (X \times X, {Map}_{N} (I, I))$ be a normal T2 FR. Define $\begin{matrix} {\tilde{R}}_{γ}^{(λ)} (x, y) & = & (1 - λ) inf (\tilde{R} (x, y))_{\bar{γ}} \\ + λ sup (\tilde{R} (x, y))_{\bar{γ}}, \\ | ({\tilde{R}}_{x})_{γ}^{(λ)} | & = & \sum_{y \in X} {\tilde{R}}_{γ}^{(λ)} (x, y), x, y \in X, \end{matrix}$ in which γ ∈ [0, 1) and λ ∈ [0, 1]. Especially, $\begin{matrix} | ({\tilde{R}}_{x})_{γ}^{(0)} | & = & \sum_{y \in X} inf (\tilde{R} (x, y))_{\bar{γ}} and \\ | ({\tilde{R}}_{x})_{γ}^{(1)} | & = & \sum_{y \in X} sup (\tilde{R} (x, y))_{\bar{γ}} . \end{matrix}$

Theorem 3.11. Let $\tilde{A}$ and $\tilde{R}$ be a normal T2 FS and a normal T2 FR on X, respectively, and parameters γ ∈ [0, 1), λ ∈ [0, 1]. Suppose $| ({\tilde{R}}_{x})_{γ}^{(0)} | \neq 0$ for all x ∈ X, then function $E_{γ}^{(λ)} (\tilde{A}) (x) = \frac{\sum_{y \in X} {\tilde{A}}_{γ} (y) {\tilde{R}}_{γ}^{(λ)} (x, y)}{| ({\tilde{R}}_{x})_{γ}^{(λ)} |}, x \in X$ is a DEF of X and $(X, Map (X, {Map}_{N} (I, I)), I, E_{γ}^{(λ)})$ is a 3WDS.

Proof. This is obvious by Theorem 3.5 since ${\tilde{R}}_{γ}^{(λ)}$ is a fuzzy relation on X and $| ({\tilde{R}}_{x})_{γ}^{(λ)} | \neq 0, \forall x \in X$ . □

Remark 3.12. Note that if it takes λ = 0.5, then $E_{γ}^{(0.5)} (\tilde{A}) (x) = E_{γ}^{4} (\tilde{A}) (x)$ , ∀x ∈ X.

Given α, β ∈ I and 0 ≤ α < β ≤ 1, the 3WDs for a normal T2 FS $\tilde{A}$ based on DEFs $E_{γ}^{*}$ (* = ψ, 4, (λ)) are given as follows.

Acceptance region: ${ACP}_{(α, β)} (E_{γ}^{*}, \tilde{A}) = {x \in X : E_{γ}^{*} (\tilde{A}) (x) \geq α}$ ,

Rejection region: ${REJ}_{(α, β)} (E_{γ}^{*}, \tilde{A}) = {x \in X : E_{γ}^{*} (\tilde{A}) (x) \leq β}$ ,

Uncertain region: ${UNC}_{(α, β)} (E_{γ}^{*}, \tilde{A}) = {x \in X : β < E_{γ}^{*} (\tilde{A}) (x) < α}$ .

4 Three-way decisions based on IVT2 FSs and IVT2 FRs

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.

4.1 Three-way decisions based on IVT2 FSs

Theorem 4.1. Let $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ and $E ([\tilde{A}]) (x) = [\tilde{A}] (x), x \in X .$ Then, the function E is a DEF of X and the space (X, Map (X, Map_N (J, I⁽²⁾)) , Map_N (J, I⁽²⁾) , E) is a 3WDS.

Proof. Since $[\tilde{A}] (x) = [{\tilde{A}}^{-} (x), {\tilde{A}}^{+} (x)]$ for every x ∈ X, the proof is obvious according to Theorem 3.1. □

Given α, β ∈ Map_N (J, I⁽²⁾) and $[{\bar{0}}_{J}] ⪯ α ≺ β ⪯ [{\bar{1}}_{J}]$ , the 3WDs for $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ based on DEF E are produced as follows.

Acceptance region: ${ACP}_{(α, β)} (E, [\tilde{A}]) = {x \in X : E ([\tilde{A}]) (x) = [\tilde{A}] (x) ⪰ α}$ ;

Rejection region: ${REJ}_{(α, β)} (E, [\tilde{A}]) = {x \in X : E ([\tilde{A}]) (x) = [\tilde{A}] (x) ⪯ β}$ ;

Uncertain region: ${UNC}_{(α, β)} (E, [\tilde{A}]) = ({ACP}_{(α, β)} (E, [\tilde{A}]) \cup {REJ}_{(α, β)} (E, [\tilde{A}]))^{c}$ .

Remark 4.2. In the following discussion within this subsection it is assumed that J = I = [0, 1] and N (u) =1 - u, ∀u ∈ [0, 1].

Let $[\tilde{A}] = [{\tilde{A}}^{-}, {\tilde{A}}^{+}] \in Map (X, {Map}_{N} (I, I^{(2)}))$ with $[\tilde{A}] = [{\tilde{A}}^{-}, {\tilde{A}}^{+}]$ and ${\tilde{A}}^{-}, {\tilde{A}}^{+} \in {Map}_{N} (I, I)$ . For parameter [γ^-, γ⁺] ∈ I⁽²⁾ and x ∈ X, let $[\tilde{A}]_{[γ^{-}, γ^{+}]} (x) = \frac{({\tilde{A}}^{-})_{γ^{-}} (x) + ({\tilde{A}}^{+})_{γ^{+}} (x)}{2},$ (10) called the [γ^-, γ⁺]-fuzzy cut set of $[\tilde{A}]$ , where $\begin{matrix} ({\tilde{A}}^{-})_{γ^{-}} (x) & = & \frac{inf ({\tilde{A}}^{-} (x))_{\bar{γ^{-}}} + sup ({\tilde{A}}^{-} (x))_{\bar{γ^{-}}}}{2}, \\ ({\tilde{A}}^{+})_{γ^{+}} (x) & = & \frac{inf ({\tilde{A}}^{+} (x))_{\bar{γ^{+}}} + sup ({\tilde{A}}^{+} (x))_{\bar{γ^{+}}}}{2}, \\ ({\tilde{A}}^{-} (x))_{\bar{γ^{-}}} & = & {u \in I : {\tilde{A}}^{-} (x, u) > γ^{-}}, \\ ({\tilde{A}}^{+} (x))_{\bar{γ^{+}}} & = & {u \in I : {\tilde{A}}^{+} (x, u) > γ^{+}} . \end{matrix}$

With the concept of [γ^-, γ⁺]-fuzzy cut set, more decision evaluation functions can be defined as follows.

Theorem 4.3. Let [γ^-, γ⁺] ∈ I⁽²⁾ be a given parameter. For $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ , if it takes $E_{[γ^{-}, γ^{+}]}^{1} ([\tilde{A}]) (x) = [\tilde{A}]_{[γ^{-}, γ^{+}]} (x), x \in X,$ then the function $E_{[γ^{-}, γ^{+}]}^{1}$ is a DEF of X and the space $(X, Map (X, {Map}_{N} (J, I^{(2)})), I, E_{[γ^{-}, γ^{+}]}^{1})$ is a 3WDS.

Proof. Let $[\tilde{A}], [\tilde{B}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ and [γ^-, γ⁺] ∈ I⁽²⁾.

(1) By Remark 2.9 and the proof of Theorem 3.4, it follows that $[\tilde{\emptyset}]_{[γ^{-}, γ^{+}]} (x) = 0$ for all x ∈ X.

(2) Suppose that $[\tilde{A}] ⊴ [\tilde{B}]$ which means ${\tilde{A}}^{-} (x) ⪯ {\tilde{B}}^{-} (x)$ and ${\tilde{A}}^{+} (x) ⪯ {\tilde{B}}^{+} (x)$ , ∀x ∈ X. According to the proof of Theorem 3.4 it follows that $({\tilde{A}}^{-})_{γ^{-}} (x) \leq ({\tilde{B}}^{-})_{γ^{-}} (x)$ and $({\tilde{A}}^{+})_{γ^{+}} (x) \leq ({\tilde{B}}^{+})_{γ^{+}} (x)$ for all x ∈ X, from which it concludes that $[\tilde{A}]_{[γ^{-}, γ^{+}]} (x) \leq [\tilde{B}]_{[γ^{-}, γ^{+}]} (x)$ for all x ∈ X, i.e., $E_{[γ^{-}, γ^{+}]}^{1} ([\tilde{A}]) (x) \leq E_{[γ^{-}, γ^{+}]}^{1} ([\tilde{B}]) (x)$ , ∀x ∈ X.

(3) According to the proof of Theorem 3.4, for each x ∈ X the following holds.

$\begin{matrix} [{\tilde{A}}^{\tilde{N}}]_{[γ^{-}, γ^{+}]} (x) \\ = \frac{(({\tilde{A}}^{-})^{\tilde{N}})_{γ^{-}} (x) + (({\tilde{A}}^{+})^{\tilde{N}})_{γ^{+}} (x)}{2} \\ = \frac{1 - ({\tilde{A}}^{-})_{γ^{-}} (x) + 1 - ({\tilde{A}}^{+})_{γ^{+}} (x)}{2} \\ = 1 - [\tilde{A}]_{[γ^{-}, γ^{+}]} (x) . □ \end{matrix}$

Theorem 4.4. Let R be a fuzzy relation on X with ∑_y∈XR (x, y) ≠0 (x ∈ X) and parameter [γ^-, γ⁺] ∈ I⁽²⁾. For $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ , if it takes $\begin{matrix} E_{[γ^{-}, γ^{+}]}^{2} ([\tilde{A}]) (x) \\ = \frac{\sum_{y \in X} [\tilde{A}]_{[γ^{-}, γ^{+}]} (y) R (x, y)}{\sum_{y \in X} R (x, y)}, x \in X, \end{matrix}$ then the function $E_{[γ^{-}, γ^{+}]}^{2}$ is a DEF of X and the space $(X, Map (X, {Map}_{N} (J, I^{(2)})), I, E_{[γ^{-}, γ^{+}]}^{2})$ is a 3WDS.

Proof. It is obvious that $E_{[γ^{-}, γ^{+}]}^{2}$ satisfies axioms (E1) and (E2) in Definition 2.10 for any [γ^-, γ⁺] ∈ I⁽²⁾. It is only needed to prove that $E_{[γ^{-}, γ^{+}]}^{2} ([\tilde{A}]^{\tilde{N}}) = N (E_{[γ^{-}, γ^{+}]}^{2} ([\tilde{A}]))$ for each $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ . In fact, for every x ∈ X,

$\begin{matrix} E_{[γ^{-}, γ^{+}]}^{2} ([\tilde{A}]^{\tilde{N}}) (x) \\ = \frac{\sum_{y \in X} ([\tilde{A}]^{\tilde{N}})_{[γ^{-}, γ^{+}]} (y) R (x, y)}{\sum_{y \in X} R (x, y)} \\ = \frac{\sum_{y \in X} (1 - [\tilde{A}]_{[γ^{-}, γ^{+}]} (y)) R (x, y)}{\sum_{y \in X} R (x, y)} \\ = 1 - \frac{\sum_{y \in X} [\tilde{A}]_{[γ^{-}, γ^{+}]} (y) R (x, y)}{\sum_{y \in X} R (x, y)} \\ = 1 - E_{[γ^{-}, γ^{+}]}^{2} ([\tilde{A}]) (x) . □ \end{matrix}$

Remark 4.5. If R is a classical relation with [x] _R≠ ∅ (x ∈ X), then function $E_{[γ^{-}, γ^{+}]}^{3} ([\tilde{A}]) (x) = \frac{| [\tilde{A}]_{[γ^{-}, γ^{+}]} \cap [x]_{R} |}{| [x]_{R} |}, x \in X,$ is also a DEF.

Given 0 ≤ α < β ≤ 1, 3WDs based on DEFs $E_{[γ^{-}, γ^{+}]}^{*}$ (*=1, 2, 3) for $[\tilde{A}] \in Map (X, {Map}_{N} (J, I^{(2)}))$ are given as follows.

Acceptance region: ${ACP}_{(α, β)} (E_{[γ^{-}, γ^{+}]}^{*}, [\tilde{A}]) = {x \in X : E_{[γ^{-}, γ^{+}]}^{*} ([\tilde{A}]) (x) \geq α}$ ;

Rejection region: ${REJ}_{(α, β)} (E_{[γ^{-}, γ^{+}]}^{*}, [\tilde{A}]) = {x \in X : E_{[γ^{-}, γ^{+}]}^{*} ([\tilde{A}]) (x) \leq β}$ ;

Uncertain region: ${UNC}_{(α, β)} (E_{[γ^{-}, γ^{+}]}^{*}, [\tilde{A}]) = {x \in X : β < E_{[γ^{-}, γ^{+}]}^{*} ([\tilde{A}]) (x) < α}$ .

Example 4.6. Nine scenic spots x₁, x₂, …, x₉ are going to be evaluated as the best scenic area. Their membership of being the best were given by experts through an IVT2 FS $[\tilde{A}] \in Map (X, {Map}_{N} (I, I^{(2)}))$ , where X = {x₁, x₂, …, x₉}. For x_i ∈ X the interval-valued fuzzy grades of $[\tilde{A}]$ is ${\tilde{A}}^{-} (x_{i}) = \int_{0 . i - 0.1}^{0 . i} \frac{10 u + (1 - i)}{u} + \int_{0 . i}^{0 . i + 0.1} \frac{- 10 u + i + 1}{u}$ (i = 1, 2, …, 9), ${\tilde{A}}^{+} (x_{i}) = \int_{0 . i - 0.1}^{0 . i} \frac{10 u + (1 - i)}{u} + \int_{0 . i}^{0 . i + 0.1} \frac{1}{u} + \int_{0 . i + 1}^{0 . i + 0.2} \frac{- 10 u + i + 2}{u}$ (i = 1, 2, …, 8) and ${\tilde{A}}^{+} (x_{9}) = \int_{0.8}^{0.9} \frac{10 u - 8}{u} + \int_{0.9}^{1} \frac{1}{u}$ .

Given [γ^-, γ⁺] ∈ I⁽²⁾, it follows that $({\tilde{A}}^{-} (x_{i}))_{\bar{γ^{-}}} = (\frac{i + γ^{-} - 1}{10}, \frac{i - γ^{-} + 1}{10})$ (i = 1, 2, …, 9), $({\tilde{A}}^{+} (x_{i}))_{\bar{γ^{+}}} = (\frac{i + γ^{+} - 1}{10}, \frac{i - γ^{+} + 2}{10})$ (i = 1, 2, …, 8)and $({\tilde{A}}^{+} (x_{9}))_{\bar{γ^{+}}} = (\frac{γ^{+} + 8}{10}, 1]$ . Thus, we have $({\tilde{A}}^{-})_{\bar{γ^{-}}} (x_{i}) = \frac{i}{10}$ (i = 1, 2, …, 9) and $({\tilde{A}}^{+})_{\bar{γ^{+}}} (x_{i}) = \frac{2 i + 1}{20}$ (i = 1, …, 8) and $({\tilde{A}}^{+})_{\bar{γ^{+}}} (x_{9}) = 0.9 + \frac{γ^{+}}{20}$ . According to (10), it follows that $[\tilde{A}]_{[γ^{-}, γ^{+}]} (x_{i}) = \frac{4 i + 1}{40}$ (i = 1, …, 8) and $[\tilde{A}]_{[γ^{-}, γ^{+}]} (x_{9}) = 0.9 + \frac{γ^{+}}{40}$ .

Set [γ^-, γ⁺] = [0.5, 0.6], α = 0.65 and β = 0.45, then the 3WDs based on $E_{[0.5, 0.6]}^{1}$ are finally obtained as follows. $\begin{matrix} {ACP}_{(0.65, 0.45)} (E_{[0.5, 0.6]}^{1}, [\tilde{A}]) = {x_{7}, x_{8}, x_{9}}, \\ {REJ}_{(0.65, 0.45)} (E_{[0.5, 0.6]}^{1}, [\tilde{A}]) = {x_{1}, x_{2}, x_{3}, x_{4}}, \\ {UNC}_{(0.65, 0.45)} (E_{[0.5, 0.6]}^{1}, [\tilde{A}]) = {x_{5}, x_{6}} . \end{matrix}$

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 $E_{(0.6, 0.5)}^{2} ([\tilde{A}]) (x_{5}) = 0.457$ and $E_{(0.6, 0.5)}^{2} ([\tilde{A}]) (x_{6}) = 0.602$ . 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.

4.2 Three-way decisions based on IVT2 FRs

Let $[\tilde{R}] \in Map (X \times X, {Map}_{N} (J, I^{(2)}))$ and [ψ] ∈ Map_N (J, I⁽²⁾). The classical subset of X × X, $[\tilde{R}]_{[ψ]} = {(x, y) \in X \times X : [\tilde{R}] (x, y) ⪰ [ψ]},$ is called the [ψ]-cut set of $[\tilde{R}]$ . Given x ∈ X, define $[x]_{[\tilde{R}]_{[ψ]}} = {y \in X : [\tilde{R}] (x, y) ⪰ [ψ]} .$

Theorem 4.7. Let $[\tilde{R}] \in Map (X \times X, {Map}_{N} (J, I^{(2)}))$ , [ψ] ∈ Map_N (J, I⁽²⁾) and parameter [γ^-, γ⁺] ∈ I⁽²⁾. Suppose that $[x]_{[\tilde{R}]_{[ψ]}} \neq \emptyset$ for each x ∈ X. For $[\tilde{A}] \in Map (X, {Map}_{N} (I, I^{(2)}))$ , if it takes $E_{[γ^{-}, γ^{+}]}^{[ψ]} ([\tilde{A}]) (x) = \frac{| [\tilde{A}]_{[γ^{-}, γ^{+}]} \cap [x]_{[\tilde{R}]_{[ψ]}} |}{| [x]_{[\tilde{R}]_{[ψ]}} |}, x \in X,$ then the function $E_{[γ^{-}, γ^{+}]}^{[ψ]}$ is a DEF of X and the space $(X, Map (X, {Map}_{N} (I, I^{(2)})), I, E_{[γ^{-}, γ^{+}]}^{[ψ]})$ is a 3WDS.

Proof. It only needs to prove that $E_{[γ^{-}, γ^{+}]}^{[ψ]} ([\tilde{A}]^{\tilde{N}}) (x) = 1 - E_{[γ^{-}, γ^{+}]}^{[ψ]} ([\tilde{A}]) (x)$ , ∀x ∈ X. This is obvious since $\begin{matrix} | ([\tilde{A}]^{\tilde{N}})_{[γ^{-}, γ^{+}]} \cap [x]_{[\tilde{R}]_{[ψ]}} | \\ = \sum_{y \in [x]_{[\tilde{R}]_{[ψ]}}} ([\tilde{A}]^{\tilde{N}})_{[γ^{-}, γ^{+}]} (y) \\ = \sum_{y \in [x]_{[\tilde{R}]_{[ψ]}}} (1 - [\tilde{A}]_{[γ^{-}, γ^{+}]} (y)) \\ = | [x]_{[\tilde{R}]_{[ψ]}} | - | [\tilde{A}]_{[γ^{-}, γ^{+}]} \cap [x]_{[\tilde{R}]_{[ψ]}} | . □ \end{matrix}$

Remark 4.8. In the following discussion within this subsection it is assumed that J = I and N (u) =1 - u, ∀u ∈ I.

Let [γ^-, γ⁺] ∈ I⁽²⁾. The [γ^-, γ⁺]-fuzzy cut set of $[\tilde{R}] \in Map (X \times X, {Map}_{N} (I, I^{(2)}))$ is defined as $[\tilde{R}]_{[γ^{-}, γ^{+}]} (x, y) = \frac{({\tilde{R}}^{-})_{γ^{-}} (x, y) + ({\tilde{R}}^{+})_{γ^{+}} (x, y)}{2},$ (11)x, y ∈ X, where $\begin{matrix} ({\tilde{R}}^{-})_{γ^{-}} (x, y) & = & \frac{inf ({\tilde{R}}^{-} (x, y))_{\bar{γ^{-}}} + sup ({\tilde{R}}^{-} (x, y))_{\bar{γ^{-}}}}{2}, \\ ({\tilde{R}}^{+})_{γ^{+}} (x, y) & = & \frac{inf ({\tilde{R}}^{+} (x, y))_{\bar{γ^{+}}} + sup ({\tilde{R}}^{+} (x, y))_{\bar{γ^{+}}}}{2}, \end{matrix}$ and $\begin{matrix} ({\tilde{R}}^{-} (x, y))_{\bar{γ^{-}}} & = & {u \in I : {\tilde{R}}^{-} (x, y) (u) > γ^{-}}, \\ ({\tilde{R}}^{+} (x, y))_{\bar{γ^{+}}} & = & {u \in I : {\tilde{R}}^{+} (x, y) (u) > γ^{+}} . \end{matrix}$

Theorem 4.9. Let $[\tilde{R}] \in Map (X \times X, {Map}_{N} (I, I^{(2)}))$ and [γ^-, γ⁺] ∈ I⁽²⁾. Suppose $\sum_{y \in X} [\tilde{R}]_{[γ^{-}, γ^{+}]} (x, y) \neq 0$ for all x ∈ X. For $[\tilde{A}] \in Map (X, {Map}_{N} (I, I^{(2)}))$ , let $\begin{matrix} E_{[γ^{-}, γ^{+}]}^{4} ([\tilde{A}]) (x) \\ = \frac{\sum_{y \in X} [\tilde{A}]_{[γ^{-}, γ^{+}]} (y) [\tilde{R}]_{[γ^{-}, γ^{+}]} (x, y)}{\sum_{y \in X} [\tilde{R}]_{[γ^{-}, γ^{+}]} (x, y)}, \end{matrix}$ x ∈ X. Then, the function $E_{[γ^{-}, γ^{+}]}^{4}$ is a DEF of X and the space $(X, Map (X, {Map}_{N} (I, I^{(2)})), I, E_{[γ^{-}, γ^{+}]}^{4})$ is a 3WDS.

Proof. The proof is obvious according to Theorem 4.4 since $[\tilde{R}]_{[γ^{-}, γ^{+}]} \in Map (X \times X, I)$ is a fuzzy relation and $\sum_{y \in X} [\tilde{R}]_{[γ^{-}, γ^{+}]} (x, y) \neq 0, \forall x \in X$ . □

Given [γ^-, γ⁺] ∈ I⁽²⁾ and λ ∈ [0, 1], for a normal IVT2 FR $[\tilde{R}] \in Map (X \times X, {Map}_{N} (I, I^{(2)}))$ , let $\begin{matrix} [\tilde{R}]_{[γ^{-}, γ^{+}]}^{(λ)} (x, y) \\ = (1 - λ) ({\tilde{R}}^{-})_{γ^{-}} (x, y) + λ ({\tilde{R}}^{+})_{γ^{+}} (x, y), \end{matrix}$ (12) $| ([\tilde{R}]_{x})_{[γ^{-}, γ^{+}]}^{(λ)} | = \sum_{y \in X} [\tilde{R}]_{[γ^{-}, γ^{+}]}^{(λ)} (x, y) .$ Especially, $\begin{matrix} | ([\tilde{R}]_{x})_{[γ^{-}, γ^{+}]}^{(0)} | & = & \sum_{y \in X} inf ({\tilde{R}}^{-})_{γ^{-}} (x, y) and \\ | ([\tilde{R}]_{x})_{[γ^{-}, γ^{+}]}^{(1)} | & = & \sum_{y \in X} sup ({\tilde{R}}^{+})_{γ^{+}} (x, y) . \end{matrix}$

Theorem 4.10. Let $[\tilde{R}] \in Map (X \times X, {Map}_{N} (I, I^{(2)}))$ and [γ^-, γ⁺] ∈ I⁽²⁾, λ ∈ [0, 1]. Suppose $| ([\tilde{R}]_{x})_{[γ^{-}, γ^{+}]}^{(0)} | \neq 0$ for all x ∈ X. Given $[\tilde{A}] \in Map (X, {Map}_{N} (I, I^{(2)}))$ , if it takes $\begin{matrix} E_{[γ^{-}, γ^{+}]}^{(λ)} ([\tilde{A}]) (x) \\ = \frac{\sum_{y \in X} [\tilde{A}]_{[γ^{-}, γ^{+}]} (y) [\tilde{R}]_{[γ^{-}, γ^{+}]}^{(λ)} (x, y)}{| ([\tilde{R}]_{x})_{[γ^{-}, γ^{+}]}^{(λ)} |}, \end{matrix}$ x ∈ X, then $E_{[γ^{-}, γ^{+}]}^{(λ)}$ is a DEF of X and $(X, Map (X, {Map}_{N} (I, I^{(2)})), I, E_{[γ^{-}, γ^{+}]}^{(λ)})$ is a 3WDS.

Proof. The proof is obvious according to Theorem 4.4 since $[\tilde{R}]_{[γ^{-}, γ^{+}]}^{(λ)} \in Map (X \times X, I)$ is a fuzzy relation and $| ([\tilde{R}]_{x})_{[γ^{-}, γ^{+}]}^{(λ)} | \neq 0, \forall x \in X$ . □

Given 0 ≤ α < β ≤ 1, we have the following 3WDs based on DEFs $E_{[γ^{-}, γ^{+}]}^{*}$ (* = [ψ] , 4, (λ)) on the basis of a normal IVT2 FS $[\tilde{A}] \in Map (X, {Map}_{N} (I, I^{(2)}))$ .

Acceptance region: ${ACP}_{(α, β)} (E_{[γ^{-}, γ^{+}]}^{*}, [\tilde{A}]) = {x \in X : E_{[γ^{-}, γ^{+}]}^{*} ([\tilde{A}]) (x) \geq α}$ ;

Rejection region: ${REJ}_{(α, β)} (E_{[γ^{-}, γ^{+}]}^{*}, [\tilde{A}]) = {x \in X : E_{[γ^{-}, γ^{+}]}^{*} ([\tilde{A}]) (x) \leq β}$ ;

Uncertain region: ${UNC}_{(α, β)} (E_{[γ^{-}, γ^{+}]}^{*}, [\tilde{A}]) = {x \in X : β < E_{[γ^{-}, γ^{+}]}^{*} ([\tilde{A}]) (x) < α}$ .

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).

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