Simplified interval-valued intuitionistic fuzzy sets with intuitionistic fuzzy numbers

Abstract

There has been a growing interest and activity in the field of interval-valued intuitionistic fuzzy sets (IVIFSs). It is generally useful to express the variations of the membership function and the non-membership function in fuzzy circumstances. However, the IVIFS is a little bit complicated on expressions and calculations. To overcome this limitation, this paper aims to introduce a succinct way to define IVIFS by two intuitionistic fuzzy numbers (IFNs), and thus, a simplified interval-valued intuitionistic fuzzy number (SIVIFN) is proposed. Afterwards, a series of operational laws and aggregation techniques over the SIVIFNs are developed, and their desirable properties are investigated in detail, which enrich the interval-valued intuitionistic fuzzy set theory. Finally, a numerical example is provided to illustrate the straightforward expressions and calculations of the SIVIFNs.

Keywords

Interval-valued intuitionistic fuzzy set intuitionistic fuzzy numbers operational laws aggregation operators

1 Introduction

Fuzzy set (FS), which was originally introduced by Zadeh [1] in 1965, shows a lot of advantages in dealing with decision making problems. It uses an element to represent the vagueness which we call it the membership degree. However, in some decision making problems, not only the superiority but also the inferiority is a significant factor we cannot ignore. Then, considering this point, Atanassov [2, 3] extended the fuzzy set to the intuitionistic fuzzy set (IFS) by adding a non-membership function. Through this way, we can get the information of superiority, inferiority and hesitation [4] in the IFS. In recent years, the IFS has received great attention since it describes the fuzzy characters of things more detailedly and comprehensively. It has been widely used in decision making [5 –7], clustering analysis [8 –10], distance measures [11, 12], similarity measures [13, 14], correlation measures [15 –17] and pattern recognition [18, 19] and so on.

While the IFS shows its applicability in many aspects, its limitations have been gradually manifested. In some occasions, we cannot indicate the membership degree or the non-membership degree by using a value, but using an interval. Hence, Atanassov and Gargov [20] made great efforts on extending the IFS to the interval-valued intuitionistic fuzzy set (IVIFS), and developed some basic operational laws of IVIFSs. The IVIFS provides an intuitive and solvable method in uncertain and ambiguous problems, and the imprecise and imperfect information can be addressed by the IVIFS. Considering such advantages that the IVIFS has, many scholars have done researches on this area. Xu and Chen [21] managed to replenish another two operations of IVIFSs, and correspondingly, the concept of interval-valued intuitionistic fuzzy number (IVIFN) was put forward [22]. A series of operational laws and aggregation techniques [22] have been proposed to fuse the interval-valued intuitionistic fuzzy information. Afterwards, to distinguish which IVIFN is better, Xu [22] gave the score function, the accuracy function and the ranking principle to compare any two IVIFNs. With these results, the IVIFNs as a useful information representation tool can be widely applied in solving our real-life problems [23 –25]. Unfortunately, the expressions and calculations of IVIFNs are complicated and would cause heavy workloads especially in large decision making problems. In order to avoid this weakness, in this paper, we will greatly simplify the expressions and calculations of IVIFSs and IVIFNs through giving novel versions of interval-valued intuitionistic fuzzy information. By the simplified way, the expressions of the operations and aggregations of interval-valued intuitionistic fuzzy information will be simpler, which are very convenient for actual applications. Furthermore, this work can be more useful in some areas, such as decision making with interval intuitionistic fuzzy sets (IIFSs) [26].

To do so, we organize this paper as follows: Section 2 reviews some basic concepts and operations related to IVIFSs and IVIFNs. Section 3 gives the simplified interval-valued intuitionistic fuzzy set (SIVIFS) and the simplified interval-valued intuitionistic fuzzy number (SIVIFN), and the operations of SIVIFSs. Section 4 puts forward a series of operational laws of SIVIFNs, including addition, subtraction, multiplication and division operations, etc. A family of aggregation operators are introduced in Section 5, and an illustrative example is given in Section 6 to show the facilitation and practicality of SIVIFSs and SIVIFNs in actual applications. Finally, the paper ends up with some conclusions in Section 7.

2 Preliminaries

In the following, let’s review some basic concepts which will be useful in the next sections.

Atanassov [2, 3] extended the fuzzy set [1] to intuitionistic fuzzy set by adding a non-membership function, which is expressed as A = { x, μ_A (x), ν_A (x) |x ∈ X }, where μ_A (x) and ν_A (x) are, respectively, the membership function and the non-membership function, which belong to [0, 1], for every x ∈ X. For convenience, Xu and Yager [27] called α = (μ_α, ν_α) an intuitionistic fuzzy number (IFN) (or an intuitionistic fuzzy value (IFV)), where μ_α ∈ [0, 1], ν_α ∈ [0, 1] and 0 ≤ μ_A (x) + ν_A (x) ≤ 1. In addition, according to some basic operational laws of IFSs [3, 28], Xu and Yager [27, 29] summarized some properties of IFNs as follows:

Theorem 2.1. [27, 29] Let α = (μ_α, ν_α), α₁ = (μ_α₁, ν_α₁) and α₂ = (μ_α₂, ν_α₂) be three IFNs, and λ, λ₁, λ₂ > 0. Then $\begin{matrix} (1) α_{1} \oplus α_{2} = α_{2} \oplus α_{1}; \\ (2) α_{1} \otimes α_{2} = α_{2} \otimes α_{1}; \\ (3) λ (α_{1} \oplus α_{2}) = λ α_{1} \oplus λ α_{2}; \\ (4) {(α_{1} \otimes α_{2})}^{λ} = α_{1}^{λ} \otimes α_{2}^{λ}; \\ (5) λ_{1} α \oplus λ_{2} α = (λ_{1} + λ_{2}) α; \\ (6) α^{λ_{1}} \otimes α^{λ_{2}} = α^{λ_{1} + λ_{2}}; \\ (7) (α_{1} \oplus α_{2}) \oplus α = α_{1} \oplus (α_{2} \oplus α); \\ (8) {(α^{λ_{1}})}^{λ_{2}} = α^{λ_{1} λ_{2}} . \end{matrix}$

Furthermore, Atanassov [30] gave the subtraction and division operations for IFSs. Lei and Xu [31] developed the subtraction and division operations for IFNs, and some properties of the operations [32] were given below:

Theorem 2.2. [32] If α₁and α₂are two IFNs which satisfy α₁ ⊕ ξ = α₂. That is to say, α₁is less than or equal to α₂, then we denote it as $α_{1} {\underline{⊴}}_{\oplus} α_{2}$ . Let α, α₁and α₂be three IFNs, which satisfy $α {\underline{⊴}}_{\oplus} α_{1} {\underline{⊴}}_{\oplus} α_{2}$ , and let λ > 0, 0 ≤ λ₁ ≤ λ₂. Then $\begin{matrix} (1) λ α_{2} ⊖ λ α_{1} = λ (α_{2} ⊖ α_{1}); \\ (2) λ_{2} α ⊖ λ_{1} α = (λ_{2} - λ_{1}) α; \\ (3) (α_{2} \oplus α) ⊖ (α \oplus α_{1}) = α_{2} ⊖ α_{1}; \\ (4) (α_{2} ⊖ α) ⊖ (α_{1} ⊖ α) = α_{2} ⊖ α_{1} \\ (5) (α_{2} ⊖ α_{1}) \oplus (α_{1} ⊖ α) = α_{2} ⊖ α . \end{matrix}$

Theorem 2.3. [32] If α₁ and α₂are two IFNs which satisfy α₁ ⊗ ξ = α₂. In other words, α₁is less than or equal to α₂, then we denote it as $α_{1} {\underline{⊴}}_{\otimes} α_{2}$ . Let α, α₁and α₂be three IFNs, which satisfy $α {\underline{⊴}}_{\otimes} α_{1} {\underline{⊴}}_{\otimes} α_{2}$ , and let λ > 0, 0 ≤ λ₁ ≤ λ₂. Then $\begin{matrix} (1) α_{1}^{λ} ø α_{2}^{λ} = {(α_{1} ø α_{2})}^{λ}; \\ (2) α^{λ_{2}} ø α^{λ_{1}} = α^{(λ_{2} - λ_{1})}; \\ (3) (α ø α_{1}) \otimes (α_{1} ø α_{2}) = α ø α_{2}; \\ (4) (α ø α_{2}) ø (α_{1} ø α_{2}) = α ø α_{1}; \\ (5) (α_{1} \otimes α) ø (α \otimes α_{2}) = α_{1} ø α_{2} . \end{matrix}$

Since the membership function and the non-membership function are intervals in some occasions, Atanassov and Gargov [20] extended the intuitionistic fuzzy set (IFS) to interval-valued intuitionistic fuzzy set (IVIFS). Next, what we will do is to recall some basic concepts and operations of IVIFSs.

An IVIFS is $\tilde{A} = {〈 x, {\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x) 〉 | x \in X}$ , where ${\tilde{μ}}_{\tilde{A}} (x) \subset [0, 1]$ and ${\tilde{ν}}_{\tilde{A}} (x) \subset [0, 1]$ , which satisfy $sup {\tilde{μ}}_{A} (x) + sup {\tilde{ν}}_{A} (x) \leq 1$ . Furthermore, Atanassov and Gargov [20] introduced some basic operational laws of IVIFSs:

Definition 2.1. [20] Let $\tilde{A} = {〈 x, {\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x)$ |x∈ X 〉 }, ${\tilde{A}}_{1} = {〈 x, {\tilde{μ}}_{{\tilde{A}}_{1}} (x), {\tilde{ν}}_{{\tilde{A}}_{1}} (x) | x \in X 〉}$ and ${\tilde{A}}_{2} = {〈 x, {\tilde{μ}}_{{\tilde{A}}_{2}} (x), {\tilde{ν}}_{{\tilde{A}}_{2}} (x) | x \in X 〉}$ be three IVIFSs. Then $(1) \tilde{¯} A = {〈 ˜ x, {\tilde{ν}}_{\tilde{A}} (x), {\tilde{μ}}_{\tilde{A}} (x) | x \in X 〉};$ $\begin{matrix} (2) {\tilde{A}}_{1} \cap {\tilde{A}}_{2} \\ = {[x, [min {inf {\tilde{μ}}_{{\tilde{A}}_{1}} (x), inf {\tilde{μ}}_{{\tilde{A}}_{2}} (x)}, \\ min {sup {\tilde{μ}}_{{\tilde{A}}_{1}} (x), sup {\tilde{μ}}_{{\tilde{A}}_{2}} (x)}], \\ [max {inf {\tilde{ν}}_{{\tilde{A}}_{1}} (x), inf {\tilde{ν}}_{{\tilde{A}}_{2}} (x)}, \\ max {sup {\tilde{ν}}_{{\tilde{A}}_{1}} (x), sup {\tilde{ν}}_{{\tilde{A}}_{2}} (x)}] 〉 | x \in X}; \end{matrix}$ $\begin{matrix} (3) {\tilde{A}}_{1} \cup {\tilde{A}}_{2} \\ = {[x, [max {inf {\tilde{μ}}_{{\tilde{A}}_{1}} (x), inf {\tilde{μ}}_{{\tilde{A}}_{2}} (x)}, \\ max {sup {\tilde{μ}}_{{\tilde{A}}_{1}} (x), sup {\tilde{μ}}_{{\tilde{A}}_{2}} (x)}], \\ [min {inf {\tilde{ν}}_{{\tilde{A}}_{1}} (x), inf {\tilde{ν}}_{{\tilde{A}}_{2}} (x)}, \\ min {sup {\tilde{ν}}_{{\tilde{A}}_{1}} (x), sup {\tilde{ν}}_{{\tilde{A}}_{2}} (x)}] 〉 | x \in X}; \end{matrix}$ $\begin{matrix} (4) {\tilde{A}}_{1} + {\tilde{A}}_{2} \\ = {〈 [inf {\tilde{μ}}_{{\tilde{A}}_{1}} (x) + inf {\tilde{μ}}_{{\tilde{A}}_{2}} (x) - inf {\tilde{μ}}_{{\tilde{A}}_{1}} (x) \cdot inf {\tilde{μ}}_{{\tilde{A}}_{2}} (x), \\ sup {\tilde{μ}}_{{\tilde{A}}_{1}} (x) + sup {\tilde{μ}}_{{\tilde{A}}_{2}} (x) - sup {\tilde{μ}}_{{\tilde{A}}_{1}} (x) \cdot sup {\tilde{μ}}_{{\tilde{A}}_{2}} (x)], \\ [inf {\tilde{ν}}_{{\tilde{A}}_{1}} (x) \cdot inf {\tilde{ν}}_{{\tilde{A}}_{2}} (x), sup {\tilde{ν}}_{{\tilde{A}}_{1}} (x) \cdot sup {\tilde{ν}}_{{\tilde{A}}_{2}} (x)] 〉 | x \in X}; \end{matrix}$ $\begin{matrix} (5) {\tilde{A}}_{1} \cdot {\tilde{A}}_{2} \\ = {〈 [inf {\tilde{μ}}_{{\tilde{A}}_{1}} (x) \cdot inf {\tilde{μ}}_{{\tilde{A}}_{2}} (x), sup {\tilde{μ}}_{{\tilde{A}}_{1}} (x) \cdot sup {\tilde{μ}}_{{\tilde{A}}_{2}} (x)], \\ [inf {\tilde{ν}}_{{\tilde{A}}_{1}} (x) + inf {\tilde{ν}}_{{\tilde{A}}_{2}} (x) - inf {\tilde{ν}}_{{\tilde{A}}_{1}} (x) \cdot inf {\tilde{ν}}_{{\tilde{A}}_{2}} (x), \\ sup {\tilde{ν}}_{{\tilde{A}}_{1}} (x) + sup {\tilde{ν}}_{{\tilde{A}}_{2}} (x) - sup {\tilde{ν}}_{{\tilde{A}}_{1}} (x) \cdot sup {\tilde{ν}}_{{\tilde{A}}_{2}} (x)] | x \in X} . \end{matrix}$ As a supplement, Xu and Chen [21] put forward the other two operational laws as below: $\begin{matrix} (6) λ \tilde{A} = {〈 x, [1 - {(1 - inf μ_{A} (x))}^{λ}, 1 - {(1 - inf μ_{A} (x))}^{λ}], \\ [{(inf {\tilde{ν}}_{\tilde{A}} (x))}^{λ}, {(sup {\tilde{ν}}_{\tilde{A}} (x))}^{λ}] 〉 | x \in X}, λ > 0; \end{matrix}$ $\begin{array}{l} \bar{A} = \{〈x, [{(\inf μ_{A} (x))}^{λ}, {(\sup μ_{A} (x))}^{λ}] \\ [1 - {(1 - \inf {\tilde{ν}}_{\tilde{A}} (x))}^{λ}, 1 - {(1 - \sup {\tilde{ν}}_{\tilde{A}} (x))}^{λ}]〉 \\ |x \in X}, λ > 0. \end{array}$

After that, according to Definition 2.1, an IVIFN [22] can be simplified as $\tilde{α} = ([a, b], [c, d])$ , where [a, b] ⊂ [0, 1], [c, d] ⊂ [0, 1] and b + d ≤ 1. Let $\tilde{Θ}$ be the set of all IVIFNs.

Based on the above operations, Xu [22] developed two interval-valued intuitionistic fuzzy aggregation operators:

Definition 2.2. [22] Let ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ be a group of all IVIFNs, and let IIFWA: ${\tilde{Θ}}^{n} \to \tilde{Θ}$ , and IIFWG: ${\tilde{Θ}}^{n} \to \tilde{Θ}$ . If $\begin{matrix} {IIFWA}_{ω} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = ω_{1} {\tilde{α}}_{1} \oplus ω_{2} {\tilde{α}}_{2} \oplus \dots \oplus ω_{n} {\tilde{α}}_{n} \end{matrix}$ (2.1) and $\begin{matrix} {IIFWG}_{ω} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = {\tilde{α}}_{1}^{ω_{1}} \otimes {\tilde{α}}_{2}^{ω_{2}} \otimes \dots \otimes {\tilde{α}}_{n}^{ω_{n}} \end{matrix}$ (2.2) where ω = (ω₁, ω₂, ⋯, ω_n) ^T is the weight vector of ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ , with ω_j ∈ [0, 1], j = 1, 2, ⋯, n, and $\sum_{j = 1}^{n} ω_{j} = 1$ , then the functions IIFWA and IIFWG are, respectively, called an interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operator and an interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator.

3 Simplified interval-valued intuitionistic fuzzy set (SIVIFS)

As we all know, an IVIFS is a region which is constituted by two closed intervals $[inf {\tilde{μ}}_{\tilde{A}} (x), sup {\tilde{μ}}_{\tilde{A}} (x)]$ and $[inf {\tilde{ν}}_{\tilde{A}} (x), sup {\tilde{ν}}_{\tilde{A}} (x)]$ . It utilizes the length and width of the rectangle (as shown in Fig. 1) to express the IVIFS:

Considering the diagonal line can manifest the character of the rectangle in a coordinate axis plane which would be more simple and convenient, we try to take this advantage into IVIFSs. Based on Fig. 1, we choose the best and the worst values to express IVIFS (as shown in Fig. 2).

We describe these two values as α = (μ₁, ν₁) and β = (μ₂, ν₂), respectively, where they are two ordered IFNs. This property is very meaningful and can be applied to some areas, such as decision making with intuitionistic fuzzy sets (IIFSs). Based on the interval-valued fuzzy set, an IIFS is defined by two ordered IFNs. Thus, the two ordered IFNs can directly turn into the IIFS as B = [〈 x, β (x), α (x) 〉 |x ∈ X].

On the basis of the analysis above, in what follows, we define the concept of simplified IVIFS:

Definition 3.1. Let X be a fixed set. Then $\tilde{A} = {〈 x, α, β 〉 | x \in X}$ (3.1) is called a simplified interval-valued intuitionistic fuzzy set, denoted as SIVIFS, where α and β are two IFNs, which satisfy $α \cup β = α (or α \cap β = β)$ (3.2)

Based on Definition 3.1, we develop some operational laws of SIVIFSs:

Definition 3.2. Let $\tilde{A} = {〈 x, α, β 〉 | x \in X},$ ${\tilde{A}}_{1} = {〈 x, α_{1}, β_{1} 〉 | x \in X}$ . And ${\tilde{A}}_{2} = {〈 x, α_{2}, β_{2} 〉 | x \in X}$ be three SIVIFSs. Then $\begin{matrix} (1) \bar{\tilde{A}} = {〈 x, \bar{β}, \bar{α} 〉 | x \in X}; \\ (2) {\tilde{A}}_{1} \cap {\tilde{A}}_{2} = {〈 x, α_{1} \cap α_{2}, β_{1} \cap β_{2} 〉 | x \in X}; \\ (3) {\tilde{A}}_{1} \cup {\tilde{A}}_{2} = {〈 x, α_{1} \cup α_{2}, β_{1} \cup β_{2} 〉 | x \in X}; \\ (4) {\tilde{A}}_{1} + {\tilde{A}}_{2} = {〈 x, α_{1} \oplus α_{2}, β_{1} \oplus β_{2} 〉 | x \in X}; \\ (5) {\tilde{A}}_{1} \cdot {\tilde{A}}_{2} = {〈 x, α_{1} \otimes α_{2}, β_{1} \otimes β_{2} 〉 | x \in X}; \\ (6) λ \tilde{A} = {〈 x, λ α, λ β 〉 | x \in X}, λ > 0; \\ (7) {\tilde{A}}^{λ} = {〈 x, α^{λ}, β^{λ} 〉 | x \in X}, λ > 0 . \end{matrix}$

Proof. According to Definition 2.1, an IVIFS can be expressed as: $\begin{matrix} \tilde{B} = {〈 x, {\tilde{μ}}_{\tilde{B}} (x), {\tilde{ν}}_{\tilde{B}} (x) 〉 | x \in X} \\ = {〈 x, [inf {\tilde{μ}}_{\tilde{B}} (x), sup {\tilde{μ}}_{\tilde{B}} (x)], \\ [inf {\tilde{ν}}_{\tilde{B}} (x), sup {\tilde{ν}}_{\tilde{B}} (x)] | x \in X} \end{matrix}$ and according to Definition 3.1, a SIVIFS is: $\begin{matrix} \tilde{A} = {〈 x, α, β 〉 | x \in X} \\ = {〈 x, (inf {\tilde{μ}}_{\tilde{B}} (x), sup {\tilde{ν}}_{\tilde{B}} (x)), \\ (sup {\tilde{μ}}_{\tilde{B}} (x), inf {\tilde{ν}}_{\tilde{B}} (x)) 〉 | x \in X} \end{matrix}$

Then we let $\tilde{B} = {〈 x, {\tilde{μ}}_{\tilde{B}} (x), {\tilde{ν}}_{\tilde{B}} (x) 〉 | x \in X}$ , ${\tilde{B}}_{1} = {〈 x, {\tilde{μ}}_{{\tilde{B}}_{1}} (x), {\tilde{ν}}_{{\tilde{B}}_{1}} (x) 〉 | x \in X}$ and be three IVIFSs; $\tilde{A} =$ {〈 x, α, β 〉 |x ∈ X }, ${\tilde{A}}_{1} = {〈 x, α_{1}, β_{1} 〉 | x \in X}$ and ${\tilde{A}}_{2} = {〈 x, α_{2}, β_{2} 〉 | x \in X}$ be SIVIFSs, where $\begin{matrix} α = (inf {\tilde{μ}}_{\tilde{B}} (x), sup {\tilde{ν}}_{\tilde{B}} (x)), β = (sup {\tilde{μ}}_{\tilde{B}} (x), inf {\tilde{ν}}_{\tilde{B}} (x)) \\ α_{1} = (inf {\tilde{μ}}_{{\tilde{B}}_{1}} (x), sup {\tilde{ν}}_{{\tilde{B}}_{1}} (x)), β_{1} = (sup {\tilde{μ}}_{{\tilde{B}}_{1}} (x), inf {\tilde{ν}}_{{\tilde{B}}_{1}} (x)) \\ α_{2} = (inf {\tilde{μ}}_{{\tilde{B}}_{2}} (x), sup {\tilde{ν}}_{{\tilde{B}}_{2}} (x)), β_{2} = (sup {\tilde{μ}}_{{\tilde{B}}_{2}} (x), inf {\tilde{ν}}_{{\tilde{B}}_{2}} (x)) \end{matrix}$ From Definition 2.2, we can get $\begin{matrix} (1) \bar{\tilde{B}} = {〈 x, {\tilde{ν}}_{\tilde{B}} (x), {\tilde{μ}}_{\tilde{B}} (x) 〉 | x \in X} \\ = {〈 x, [inf {\tilde{v}}_{\tilde{B}} (x), sup {\tilde{v}}_{\tilde{B}} (x)], \\ [inf {\tilde{μ}}_{\tilde{B}} (x), sup {\tilde{μ}}_{\tilde{B}} (x)] 〉 | x \in X} \end{matrix}$ and thus, we obtain $\begin{matrix} \bar{\tilde{A}} = {〈 x, (inf {\tilde{ν}}_{\tilde{B}} (x), sup {\tilde{μ}}_{\tilde{B}} (x)), \\ (sup {\tilde{ν}}_{\tilde{B}} (x), inf {\tilde{μ}}_{\tilde{B}} (x)) 〉 | x \in X} \\ = {〈 x, \tilde{β}, \tilde{α} 〉 | x \in X}; \end{matrix}$

In a similar way, we can prove the following operations: $\begin{matrix} (2) {\tilde{A}}_{1} \cap {\tilde{A}}_{2} \\ = {〈 x, (min {inf {\tilde{μ}}_{{\tilde{B}}_{1}} (x), inf {\tilde{μ}}_{{\tilde{B}}_{2}} (x)}, \\ max {sup {\tilde{ν}}_{{\tilde{B}}_{1}} (x), sup {\tilde{ν}}_{{\tilde{B}}_{2}} (x)}), \\ (min {sup {\tilde{μ}}_{{\tilde{B}}_{1}} (x), sup {\tilde{μ}}_{{\tilde{B}}_{2}} (x)}, \\ max {inf {\tilde{ν}}_{{\tilde{B}}_{1}} (x), inf {\tilde{ν}}_{{\tilde{B}}_{2}} (x)}) 〉 | x \in X} \\ = {{x, α_{1} \cap α_{2}, β_{1} \cap β_{2}} | x \in X}; \end{matrix}$ $\begin{matrix} (3) {\tilde{A}}_{1} \cup {\tilde{A}}_{2} \\ = {〈 x, (max {inf {\tilde{μ}}_{{\tilde{B}}_{1}} (x), inf {\tilde{μ}}_{{\tilde{B}}_{2}} (x)}, \\ min {sup {\tilde{ν}}_{{\tilde{B}}_{1}} (x), sup {\tilde{ν}}_{{\tilde{B}}_{2}} (x)}), \\ (max {sup {\tilde{μ}}_{{\tilde{B}}_{1}} (x), sup {\tilde{μ}}_{{\tilde{B}}_{2}} (x)} \\ min {inf {\tilde{ν}}_{{\tilde{B}}_{1}} (x), inf {\tilde{ν}}_{{\tilde{B}}_{2}} (x)}) 〉 | x \in X} \\ = {{x, α_{1} \cup α_{2}, β_{1} \cup β_{2}} | x \in X}; \end{matrix}$ $\begin{matrix} (4) {\tilde{A}}_{1} + {\tilde{A}}_{2} \\ = {〈 x, (inf {\tilde{μ}}_{{\tilde{B}}_{1}} (x) + inf {\tilde{μ}}_{{\tilde{B}}_{2}} (x) \\ - inf {\tilde{μ}}_{{\tilde{B}}_{1}} (x) \cdot inf {\tilde{μ}}_{{\tilde{B}}_{2}} (x), sup {\tilde{ν}}_{{\tilde{B}}_{1}} (x) \cdot sup {\tilde{ν}}_{{\tilde{B}}_{2}} (x)), \\ (sup {\tilde{μ}}_{{\tilde{B}}_{1}} (x) + sup {\tilde{μ}}_{{\tilde{B}}_{2}} (x) - sup {\tilde{μ}}_{{\tilde{B}}_{1}} (x) \cdot sup {\tilde{μ}}_{{\tilde{B}}_{2}} (x), \\ inf {\tilde{ν}}_{{\tilde{B}}_{1}} (x) \cdot inf {\tilde{ν}}_{{\tilde{B}}_{2}} (x)) 〉 | x \in X} \\ = {{x, α_{1} \oplus α_{2}, β_{1} \oplus β_{2}} | x \in X}; \end{matrix}$ $\begin{matrix} (5) {\tilde{A}}_{1} \cdot {\tilde{A}}_{2} \\ = {〈 x, (inf {\tilde{μ}}_{{\tilde{B}}_{1}} (x) \cdot inf {\tilde{μ}}_{{\tilde{B}}_{2}} (x), sup {\tilde{ν}}_{{\tilde{B}}_{1}} (x) + sup {\tilde{ν}}_{{\tilde{B}}_{2}} (x) \\ - sup {\tilde{ν}}_{{\tilde{B}}_{1}} (x) \cdot sup {\tilde{ν}}_{{\tilde{B}}_{2}} (x)), (sup {\tilde{μ}}_{{\tilde{B}}_{1}} (x) \cdot sup {\tilde{μ}}_{{\tilde{B}}_{2}} (x), \\ inf {\tilde{ν}}_{{\tilde{B}}_{1}} (x) + inf {\tilde{ν}}_{{\tilde{B}}_{2}} (x) - inf {\tilde{ν}}_{{\tilde{B}}_{1}} (x) \cdot inf {\tilde{ν}}_{{\tilde{B}}_{2}} (x) 〉 | x \in X} \\ = {{x, α_{1} \otimes α_{2}, β_{1} \otimes β_{2}} | x \in X}; \end{matrix}$ $\begin{matrix} (6) λ \tilde{A} = {〈 x, (1 - {(1 - inf {\tilde{μ}}_{\tilde{B}} (x))}^{λ}, {(sup {\tilde{ν}}_{\tilde{B}} (x))}^{λ}) \\ (1 - {(1 - sup {\tilde{μ}}_{\tilde{B}} (x))}^{λ}, {(inf {\tilde{ν}}_{\tilde{B}} (x))}^{λ}) com>rangle \in X}, λ > 0 \\ = {〈 x, λ α, λ β 〉 | x \in X}, λ > 0; \end{matrix}$ $\begin{matrix} (7) {\tilde{A}}^{λ} = {〈 x, ({(inf {\tilde{μ}}_{\tilde{B}} (x))}^{λ}, 1 - {(1 - sup {\tilde{ν}}_{\tilde{B}} (x))}^{λ}), \\ ({(sup {\tilde{μ}}_{\tilde{B}} (x))}^{λ}, 1 - {(1 - inf {\tilde{ν}}_{\tilde{B}} (x))}^{λ}) 〉 | x \in X}, λ > 0 \\ = {〈 x, α^{λ}, β^{λ} 〉 | x \in X}, λ > 0 . \end{matrix}$

In order to facilitate the actual applications, we further develop another two operations of SIVIFSs:

Definition 3.3. Let ${\tilde{A}}_{1} = {〈 x, α_{1}, β_{1} 〉 | x \in X}$ and ${\tilde{A}}_{2} = {〈 x, α_{2}, β_{2} 〉 | x \in X}$ be any two SIVIFSs. Then $\begin{matrix} (1) If α_{2} {\underline{⊴}}_{\oplus} α_{1} and β_{2} {\underline{⊴}}_{\oplus} β_{1}, then \\ {\tilde{A}}_{1} - {\tilde{A}}_{2} = {〈 x, α_{1} ⊖ α_{2}, β_{1} ⊖ β_{2} 〉 | x \in X}; \end{matrix}$ $\begin{matrix} (2) If α_{1} {\underline{⊴}}_{\otimes} α_{2} and β_{1} {\underline{⊴}}_{\otimes} β_{2} then \\ {\tilde{A}}_{1} / {\tilde{A}}_{2} = {〈 x, α_{1} ø α_{2}, β_{1} ø β_{2} 〉 | x \in X} . \end{matrix}$

It follows from Definition 3.1 that the basic component of an SIVIFS is an ordered pair which is constituted by two IFNs. This ordered pair is a simplified interval-valued intuitionistic fuzzy number (SIVIFN). For convenience, a SIVIFN can be denoted as $\tilde{a} = 〈 α, β 〉$ , and let ${\tilde{Θ}}^{'}$ be the set of all SIVIFNs.

Motivated by all above operations of SIVIFSs, we can drive some basic operational laws of SIVIFNs:

Definition 3.4. Let $\tilde{a} = 〈 α, β 〉, {\tilde{a}}_{1} = 〈 α_{1}, β_{1} 〉$ and ${\tilde{a}}_{2} = 〈 α_{2}, β_{2} 〉$ be three SIVIFNs. Then $\begin{matrix} (1) \bar{\tilde{a}} = 〈 \bar{β}, \bar{α} 〉; \\ (2) {\tilde{a}}_{1} \cap {\tilde{a}}_{2} = 〈 α_{1} \cap α_{2}, β_{1} \cap β_{2} 〉; \\ (3) {\tilde{a}}_{1} \cup {\tilde{a}}_{2} = 〈 α_{1} \cup α_{2}, β_{1} \cup β_{2} 〉; \\ (4) {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} = 〈 α_{1} \oplus α_{2}, β_{1} \oplus β_{2} 〉; \\ (5) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = 〈 α_{1} \otimes α_{2}, β_{1} \otimes β_{2} 〉; \\ (6) λ \tilde{a} = 〈 λ α, λ β 〉; \\ (7) {\tilde{a}}^{λ} = 〈 α^{λ}, β^{λ} 〉 . \end{matrix}$ Before we introduce the fundamental operations of SIVIFNs about subtraction and division, we define two orders of SIVIFNs:

Definition 3.5. Let ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ be two SIVIFNs, and $\exists \tilde{χ}$ is an SIVIFN which satisfies ${\tilde{a}}_{1} \oplus \tilde{χ} = {\tilde{a}}_{2}$ , then we get that ${\tilde{a}}_{1}$ is less than or equal to ${\tilde{a}}_{2}$ , denoted as ${\tilde{a}}_{1} \underline{⊴} \oplus_{} {\tilde{a}}_{2}$ . If $\tilde{χ} \neq ([0, 0], [1, 1])$ , then ${\tilde{a}}_{1}$ is less than ${\tilde{a}}_{2}$ , denoted as ${\tilde{a}}_{1} ⊴_{\oplus} {\tilde{a}}_{2}$ .

Definition 3.6. Let ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ be two SIVIFNs, and there exists $\tilde{χ}$ an SIVIFN which satisfies ${\tilde{a}}_{1} \otimes \tilde{χ} = {\tilde{a}}_{2}$ , then we get that ${\tilde{a}}_{2}$ is less than or equal to ${\tilde{a}}_{1}$ , denoted as ${\tilde{a}}_{2} \underline{⊴} \otimes {\tilde{a}}_{1}$ . If $\tilde{χ} \neq ([1, 1], [0, 0])$ , then ${\tilde{a}}_{2}$ is less than ${\tilde{a}}_{1}$ , denoted as ${\tilde{a}}_{2} ⊴_{\otimes} {\tilde{a}}_{1}$ .

Below we define another two operations of SIVIFNs:

Definition 3.7. Let ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ be two SIVIFNs, then $\begin{matrix} (1) If {\tilde{a}}_{2} {\underline{⊴}}_{\oplus} {\tilde{a}}_{1}, then {\tilde{a}}_{1} ⊖ {\tilde{a}}_{2} = 〈 α_{1} ⊖ α_{2}, β_{1} ⊖ β_{2} 〉; \\ (2) If {\tilde{a}}_{1} {\underline{⊴}}_{\otimes} {\tilde{a}}_{2}, then {\tilde{a}}_{1} ø {\tilde{a}}_{2} = 〈 α_{1} ø α_{2}, β_{1} ø β_{2} 〉 . \end{matrix}$

Motivated by the score function and the accuracy function of IVIFN [22], we introduce the two functions of SIVIFN as follows:

Definition 3.8. Let $\tilde{a} = 〈 α, β 〉$ be a SIVIFN. Then we call $s (\tilde{a}) = \frac{1}{2} [s (α) + s (β)]$ (3.3) the score of $\tilde{a}$ , where s is the score function of $\tilde{a}$ , $s (\tilde{a}) \in [- 1, 1]$ .

Example 3.1. Suppose $\tilde{a} = 〈 α, β 〉$ is a SIVIFN, where α = (0.1, 0.3) and β = (0.5, 0.2). Then we have $s (\tilde{a}) = \frac{1}{2} [(0.1 - 0.3) + (0.5 - 0.2)] = 0.05$

Obviously, the greater the value of $s (\tilde{a})$ , the larger the $\tilde{a}$ . If $s (\tilde{a}) = 1$ , then $\tilde{a}$ is the largest SIVIFN: 〈 (1, 0), (0, 1)〉; On the contrary, If $s (\tilde{a}) = - 1$ , then $\tilde{a}$ is the smallest SIVIFN: 〈 (0, 1), (1, 0)〉.

Actually, we sometimes will come across the situations where two SIVIFNs get the same score. For example, if ${\tilde{a}}_{1} = 〈 (0.1, 0.5), (0.2, 0.3) 〉$ and ${\tilde{a}}_{2} = 〈 (0.1, 0.3), (0.2, 0.5) 〉$ , then $s ({\tilde{a}}_{1}) = s ({\tilde{a}}_{2}) = - 0.25$ . In this case, we cannot point out which one is better, but we know that the two SIVIFNs must be different. For the purpose of solving this issue, in the following, we introduce an accuracy function:

Definition 3.9. Let $\tilde{a} = 〈 α, β 〉$ be an SIVIFN. Then an accuracy function of $\tilde{a}$ is: $h (\tilde{a}) = \frac{1}{2} [h (α) + h (β)]$ (3.4) where $h (\tilde{a}) \in [0, 1]$ .

Based on the score function and accuracy function, we put forward a comparison method for two SIVIFNs:

Definition 3.10. Let ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ be any two SIVIFNs. Then

If $s ({\tilde{a}}_{1}) < s ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} ≺ {\tilde{a}}_{2}$ ;

If $s ({\tilde{a}}_{1}) = s ({\tilde{a}}_{2})$ , then

√ If

h ({\tilde{a}}_{1}) = h ({\tilde{a}}_{2})

, then

{\tilde{a}}_{1} \sim {\tilde{a}}_{2}

;

√ If $h ({\tilde{a}}_{1}) < h ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} ≺ {\tilde{a}}_{2}$ .

Example 3.2. Assume that ${\tilde{a}}_{1} = 〈 (0.3, 0.5),$ (0.6, 0.2)〉, ${\tilde{a}}_{2} = 〈 (0.2, 0.4), (0.6, 0.3) 〉$ , ${\tilde{a}}_{3} = 〈 (0.5, 0.5), (0.6, 0.4) 〉$ are three SIVIFNs. Then we first calculate their scores $s ({\tilde{a}}_{1}) = 0.1$ , $s ({\tilde{a}}_{2}) = 0.05$ , $s ({\tilde{a}}_{3}) = 0.1$ . Therefore $s ({\tilde{a}}_{1}) = s ({\tilde{a}}_{3}) > s ({\tilde{a}}_{2})$ . Since the scores of ${\tilde{a}}_{1}$ and ${\tilde{a}}_{3}$ are equivalent, then we utilize the accuracy function to compare them: $h ({\tilde{a}}_{1}) = 0.8$ , $h ({\tilde{a}}_{2}) = 0.7$ . Thus, we obtain ${\tilde{a}}_{1} ≻ {\tilde{a}}_{3} ≻ {\tilde{a}}_{2}$ .

From the above example, we can know that the score function and the accuracy function of SIVIFNs are different from those of IVIFNs, but they remain the same essence. Moreover, we can understand them easily and clearly that these two functions are decided by synthesizing the score function and the accuracy function of two IFNs which are the elements of SIVIFNs.

4 Some properties of SIVIFNs

To comprehend the operations of SIVIFNs deeply, in what follows, we will investigate some desirable properties of SIVIFNs:

Theorem 4.1.Let λ, λ₁, λ₂ > 0, and $\tilde{a}, {\tilde{a}}_{1}, {\tilde{a}}_{2}$ be three SIVIFNs. Then $\begin{matrix} (1) {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} = {\tilde{a}}_{2} \oplus {\tilde{a}}_{1}; \\ (2) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = {\tilde{a}}_{2} \otimes {\tilde{a}}_{1}; \\ (3) λ ({\tilde{a}}_{1} \oplus {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} \oplus λ {\tilde{a}}_{2}; \\ (4) {({\tilde{a}}_{1} \otimes {\tilde{a}}_{2})}^{λ} = {\tilde{a}}_{1}^{λ} \otimes {\tilde{a}}_{2}^{λ}; \\ (5) λ_{1} \tilde{a} \oplus λ_{2} \tilde{a} = (λ_{1} + λ_{2}) \tilde{a}; \\ (6) {\tilde{a}}^{λ_{1}} \otimes {\tilde{a}}^{λ_{2}} = {\tilde{a}}^{(λ_{1} + λ_{2})}; \\ (7) {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} \oplus \tilde{a} = {\tilde{a}}_{1} \oplus ({\tilde{a}}_{2} \oplus \tilde{a}); \\ (8) {({\tilde{a}}^{λ_{1}})}^{λ_{2}} = {(\tilde{a})}^{{λ_{1}}^{λ_{2}}} . \end{matrix}$ Proof.

(1) From Theorem 2.1, we know that

α₁ ⊕ α₂ = α₂ ⊕ α₁ and β₁ ⊕ β₂ = β₂ ⊕ β₁, thus ${\tilde{a}}_{1} \oplus {\tilde{a}}_{2} = 〈 α_{1} \oplus α_{2}, β_{1} \oplus β_{2} 〉 = {\tilde{a}}_{2} \oplus {\tilde{a}}_{1}$ ;

(2) It follows from Theorem 2.1 that α₁ ⊗ α₂ = α₂ ⊗ α₁ and β₁ ⊗ β₂ = β₂ ⊗ β₁, thus

${\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = 〈 α_{1} \otimes α_{2}, β_{1} \otimes β_{2} 〉 = {\tilde{a}}_{2} \otimes {\tilde{a}}_{1}$ ;

(3) Since λ (α₁ ⊕ α₂) = λα₁ ⊕ λα₂ and λ (β₁ ⊕ β₂) =λβ₁ ⊕ λβ₂, then $\begin{matrix} λ ({\tilde{a}}_{1} \oplus {\tilde{a}}_{2}) = λ 〈 α_{1} \oplus α_{2}, β_{1} \oplus β_{2} 〉 \\ = 〈 λ (α_{1} \oplus α_{2}), λ (β_{1} \oplus β_{2}) 〉 \end{matrix}$ and $\begin{matrix} λ {\tilde{a}}_{1} \oplus λ {\tilde{a}}_{2} = 〈 λ α_{1}, λ β_{1} 〉 \oplus 〈 λ α_{2}, λ β_{2} 〉 \\ = 〈 λ α_{1} \oplus λ α_{2}, λ β_{1} \oplus λ β_{2} 〉 \end{matrix}$ thus $λ ({\tilde{a}}_{1} \oplus {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} \oplus λ {\tilde{a}}_{2}$ ;

Similarly, we have

(4) Since ${(α_{1} \otimes α_{2})}^{λ} = α_{1}^{λ} \otimes α_{2}^{λ}$ and ${(β_{1} \otimes β_{2})}^{λ} = β_{1}^{λ} \otimes β_{2}^{λ}$ , then $\begin{matrix} {({\tilde{a}}_{1} \otimes {\tilde{a}}_{2})}^{λ} = {〈 α_{1} \otimes α_{2}, β_{1} \otimes β_{2} 〉}^{λ} \\ = 〈 {(α_{1} \otimes α_{2})}^{λ}, {(β_{1} \otimes β_{2})}^{λ} 〉 \\ {\tilde{a}}_{1}^{λ} \otimes {\tilde{a}}_{2}^{λ} = 〈 α_{1}^{λ}, β_{1}^{λ} 〉 \otimes 〈 α_{2}^{λ}, β_{2}^{λ} 〉 \\ = 〈 α_{1}^{λ} \otimes α_{2}^{λ}, β_{1}^{λ} \otimes β_{2}^{λ} 〉 \end{matrix}$

hence ${({\tilde{a}}_{1} \otimes {\tilde{a}}_{2})}^{λ} = {\tilde{a}}_{1}^{λ} \otimes {\tilde{a}}_{2}^{λ}$ ;

(5) Since λ₁ α ⊕ λ₂ α = (λ₁ + λ₂) α and λ₁β ⊕ λ₂β = (λ₁ + λ₂) β, then $\begin{matrix} λ_{1} \tilde{a} \oplus λ_{2} \tilde{a} = 〈 λ_{1} α, λ_{1} β 〉 \oplus 〈 λ_{2} α, λ_{2} β 〉 \\ = 〈 λ_{1} α \oplus λ_{2} α, λ_{1} β \oplus λ_{2} β 〉 \\ (λ_{1} + λ_{2}) \tilde{a} = 〈 (λ_{1} + λ_{2}) α, (λ_{1} + λ_{2}) β 〉 \end{matrix}$ therefore $λ_{1} \tilde{a} \oplus λ_{2} \tilde{a} = (λ_{1} + λ_{2}) \tilde{a}$ ;

(6) Since α^λ₁ ⊗ α^λ₂ = α^{(λ₁ +λ₂)} and β^λ₁ ⊗ β^λ₂ =β^{(λ₁ +λ₂)}, then $\begin{matrix} {\tilde{a}}^{λ_{1}} \otimes {\tilde{a}}^{λ_{2}} = 〈 α^{λ_{1}}, β^{λ_{1}} 〉 \otimes 〈 α^{λ_{2}}, β^{λ_{2}} 〉 \\ = 〈 α^{λ_{1}} \otimes α^{λ_{2}}, β^{λ_{1}} \otimes β^{λ_{2}} 〉 \\ {\tilde{a}}^{(λ_{1} + λ_{2})} = 〈 α^{(λ_{1} + λ_{2})}, β^{(λ_{1} + λ_{2})} 〉; \end{matrix}$

i.e., ${\tilde{a}}^{λ_{1}} \otimes {\tilde{a}}^{λ_{2}} = {\tilde{a}}^{(λ_{1} + λ_{2})}$ ; $\begin{matrix} (7) {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} \oplus \tilde{a} = 〈 α_{1} \oplus α_{2} \oplus α, β_{1} \oplus β_{2} \oplus β 〉 \\ = 〈 α_{1} \oplus (α_{2} \oplus α), β_{1} \oplus (β_{2} \oplus β) 〉 \\ = 〈 α_{1}, β_{1} 〉 \oplus 〈 α_{2} \oplus α, β_{2} \oplus β 〉 \\ = {\tilde{a}}_{1} \oplus ({\tilde{a}}_{2} \oplus \tilde{a}); \end{matrix}$ $\begin{matrix} (8) {({\tilde{a}}^{λ_{1}})}^{λ_{2}} & = {〈 α^{λ_{1}}, β^{λ_{1}} 〉}^{λ_{2}} = 〈 α^{λ_{1} λ_{2}}, β^{λ_{1} λ_{2}} 〉 \\ = {〈 α, β 〉}^{λ_{1} λ_{2}} = {(\tilde{a})}^{λ_{1} λ_{2}} . \end{matrix}$

Some properties of SVIFNs about subtraction and division operations can be shown as:

Theorem 4.2.Let λ > 0, 0 ≤ λ₁ ≤ λ₂ and $\tilde{a}, {\tilde{a}}_{1}, {\tilde{a}}_{2}$ be any three SVIFNs. ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ satisfy ${\tilde{a}}_{2} {\underline{⊴}}_{\oplus} {\tilde{a}}_{1}$ . Then $\begin{matrix} (1) λ ({\tilde{a}}_{1} ⊖ {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} ⊖ λ {\tilde{a}}_{2}; \\ (2) λ_{2} \tilde{a} ⊖ λ_{1} \tilde{a} = (λ_{2} - λ_{1}) \tilde{a} . \end{matrix}$

Theorem 4.3.Let λ > 0, 0 ≤ λ₁ ≤ λ₂ and $\tilde{a}, {\tilde{a}}_{1}, {\tilde{a}}_{2}$ be any three SVIFNs. ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ satisfy ${\tilde{a}}_{1} {\underline{⊴}}_{\otimes} {\tilde{a}}_{2}$ . Then $\begin{matrix} (1) {({\tilde{a}}_{1} ø {\tilde{a}}_{2})}^{λ} = {\tilde{a}}_{1}^{λ} ø {\tilde{a}}_{2}^{λ}; \\ (2) {\tilde{a}}^{λ_{2}} ø {\tilde{a}}^{λ_{1}} = {\tilde{a}}^{(λ_{2} - λ_{1})} . \end{matrix}$

Besides, we can put forward some mixed operations of SIVIFNs:

Theorem 4.4.Let $\tilde{a} = 〈 α, β 〉, {\tilde{a}}_{1} = 〈 α_{1}, β_{1} 〉$ and ${\tilde{a}}_{2} = 〈 α_{2}, β_{2} 〉$ satisfy $\tilde{a} {\underline{⊴}}_{\oplus} {\tilde{a}}_{1} {\underline{⊴}}_{\oplus} {\tilde{a}}_{2}$ . Then $\begin{matrix} (1) ({\tilde{a}}_{2} \oplus \tilde{a}) ⊖ (\tilde{a} \oplus {\tilde{a}}_{1}) = {\tilde{a}}_{2} ⊖ {\tilde{a}}_{1}; \\ (2) ({\tilde{a}}_{2} ⊖ \tilde{a}) ⊖ ({\tilde{a}}_{1} ⊖ \tilde{a}) = {\tilde{a}}_{2} ⊖ {\tilde{a}}_{1}; \\ (3) ({\tilde{a}}_{2} ⊖ {\tilde{a}}_{1}) \oplus ({\tilde{a}}_{1} ⊖ \tilde{a}) = {\tilde{a}}_{2} ⊖ \tilde{a} . \end{matrix}$ Theorem 4.5.Let $\tilde{a} = 〈 α, β 〉, {\tilde{a}}_{1} = 〈 α_{1}, β_{1} 〉$ and ${\tilde{a}}_{2} = 〈 α_{2}, β_{2} 〉$ satisfy $\tilde{a} {\underline{⊴}}_{\otimes} {\tilde{a}}_{1} {\underline{⊴}}_{\otimes} {\tilde{a}}_{2}$ . Then $\begin{matrix} (1) (\tilde{a} ø {\tilde{a}}_{1}) \otimes ({\tilde{a}}_{1} ø {\tilde{a}}_{2}) = \tilde{a} ø {\tilde{a}}_{2}; \\ (2) (\tilde{a} ø {\tilde{a}}_{2}) ø ({\tilde{a}}_{1} ø {\tilde{a}}_{2}) = \tilde{a} ø {\tilde{a}}_{1}; \\ (3) ({\tilde{a}}_{1} \otimes \tilde{a}) ø (\tilde{a} \otimes {\tilde{a}}_{2}) = {\tilde{a}}_{1} ø {\tilde{a}}_{2} . \end{matrix}$

Here, we omit the proofs of Theorem 4.2 to Theorem 4.5.

5 Simplified interval-valued intuitionistic fuzzy aggregation operators

In this section, we will introduce some aggregation operators for SIVIFNs.

Definition 5.1. Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ be a collection of SIVIFNs, and let SIIFWA: ${({\tilde{Θ}}^{'})}^{n} \to {\tilde{Θ}}^{'}$ . If $\begin{matrix} {SIIFWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = ω_{1} {\tilde{a}}_{1} \oplus ω_{2} {\tilde{a}}_{2} \oplus \dots \oplus ω_{n} {\tilde{a}}_{n} \end{matrix}$ (5.1) where ω = (ω₁, ω₂, ⋯, ω_n) ^T is the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , with ω_j ∈ [0, 1] (j = 1, 2, ⋯, n) and $\sum_{j = 1}^{n} ω_{j} = 1$ , then the function SIIFWA is called a simplified interval-valued intuitionistic fuzzy weighted averaging (SIIFWA) operator.

Especially if ω=(1/n,1/n,...,1/n)^T, then the SIIFWA operator reduces to a simplified interval-valued intuitionistic fuzzy averaging (SIIFA) operator, which can be expressed as: $\begin{matrix} SIIFA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{n} ({\tilde{a}}_{1} \oplus {\tilde{a}}_{2} \oplus \dots \oplus {\tilde{a}}_{n}) \end{matrix}$ (5.2)

Definition 5.2. Let SIIFWG: ${({\tilde{Θ}}^{'})}^{n} \to {\tilde{Θ}}^{'}$ . If $\begin{matrix} {SIIFWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {\tilde{a}}_{1}^{ω_{1}} \otimes {\tilde{a}}_{2}^{ω_{2}} \otimes \dots \otimes {\tilde{a}}_{n}^{ω_{n}} \end{matrix}$ (5.3) where ω = (ω₁, ω₂, ⋯, ω_n) ^T is the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , with ω_j ∈ [0, 1] (j = 1, 2, ⋯, n) and $\sum_{j = 1}^{n} ω_{j} = 1$ , then the function SIIFWG is called a simplified interval-valued intuitionistic fuzzy weighted geometric (SIIFWG) operator.

Particularly, if ω=(1/n,1/n,...,1/n)^T, then the SIIFWG operator reduces to a simplified interval-valued intuitionistic fuzzy geometric (SIIFG) operator, which can be expressed as: $\begin{matrix} SIIFG ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {({\tilde{a}}_{1} \otimes {\tilde{a}}_{2} \otimes \dots \otimes {\tilde{a}}_{n})}^{\frac{1}{n}} \end{matrix}$ (5.4)

With the above aggregation operators, we can drive the following results:

Theorem 5.1.Let ${\tilde{a}}_{j} = 〈 α_{j}, β_{j} 〉$ , where α_j = (μ_{α_j}, ν_{α_j}) and β_j = (μ_{β
_j}, ν_{β
_j}) (j = 1, 2, ⋯, n) are two collections of SIVIFNs. Then the aggregated result of Equation (5.1) is a SIVIFN, and $\begin{matrix} {SIIFWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = 〈 {IFWA}_{ω} (α_{1}, α_{2}, \dots, α_{n}), {IFWA}_{ω} (β_{1}, β_{2}, \dots, β_{n}) 〉 \end{matrix}$ (5.5)where ω = (ω₁, ω₂, ⋯, ω_n) ^Tis the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , with ω_j ∈ [0, 1] (j = 1, 2, ⋯, n) and $\sum_{j = 1}^{n} ω_{j} = 1$ .

Proof. Through mathematical induction, we can prove Theorem 5.1 as below: (1) When n = 2, $\begin{matrix} IIFWA *_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}) \\ = ω_{1} {\tilde{a}}_{1} \oplus ω_{2} {\tilde{a}}_{2} \\ = 〈 ω_{1} α_{1}, ω_{1} β_{1} 〉 \oplus 〈 ω_{2} α_{2}, ω_{2} β_{2} 〉 \\ = 〈 ω_{1} α_{1} \oplus ω_{2} α_{2}, ω_{1} β_{1} \oplus ω_{2} β_{2} 〉 \\ = 〈 {IIFWA}_{ω} (α_{1}, α_{2}), {IIFWA}_{ω} (β_{1}, β_{2}) 〉 \end{matrix}$

(2) Suppose that when n = k, Equation (5.1) holds, i.e., $\begin{matrix} IIFWA *_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{k}) \\ = ω_{1} {\tilde{a}}_{1} \oplus ω_{2} {\tilde{a}}_{2} \oplus \dots \oplus ω_{k} {\tilde{a}}_{k} \\ = {IIFWA}_{ω} 〈 (α_{1}, α_{2}, \dots, α_{k}), (β_{1}, β_{2}, \dots, β_{k}) 〉 \end{matrix}$ then when n = k + 1, we have $\begin{array}{l} I I F W A^{*} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{k + 1}) \\ = ω_{1} {\tilde{a}}_{1} \oplus ω_{2} {\tilde{a}}_{2} \oplus \dots \oplus ω_{k + 1} {\tilde{a}}_{k + 1} \\ = I I F W A_{ω} (α_{1}, α_{2}, \dots, α_{k}), I I F W A_{ω} (β_{1}, β_{2}, \dots, β_{k})〉 \\ \oplus ω I I F_{k + 1} {\tilde{a}}_{k + 1} \\ = 〈I I F W A_{ω} (α_{1}, α_{2}, \dots, α_{k}), I I F W A_{ω} (β_{1}, β_{2}, \dots, β_{k})〉 \\ \oplus 〈ω_{k + 1} α_{k + 1,} ω_{k + 1}, β_{k + 1},〉 \\ = 〈I I F W A_{ω} (α_{1}, α_{2}, \dots, α_{k}), \oplus ω_{k + 1} α_{k + 1}, \\ I I F W A_{ω} (β_{1}, β_{2}, \dots, β_{k}) \oplus ω_{k + 1} β_{k + 1}〉 \\ = 〈ω_{1} β_{1} \oplus ω_{2} β_{2} \oplus \dots \oplus ω_{k + 1} α_{k + 1}, \\ ω_{1} β_{1} \oplus ω_{2} β_{2} \oplus \dots \oplus ω_{k + 1} β_{k + 1}〉 \\ = 〈I I F W A_{ω} (α_{1}, α_{2}, \dots, α_{k + 1}) \\ I I F W A_{ω} (β_{1}, β_{2}, \dots, β_{k + 1})〉 \end{array}$

Therefore, combining the results in (1) and (2), we can know that Equation (5.1) holds for any n, which completes the proof of Theorem 5.1.

Particularly, if α_j = β_j for all j = 1, 2, ⋯, n, then the SIVIFNs ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ reduce to IFNs, and correspondingly, the SIIFWA operator reduces to an IFWA operator.

Similarly, we can prove the following theorem easily:

Theorem 5.2. The aggregated result of Equation (5.3) is also a SIVIFN, and we have $\begin{matrix} {SIIFWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = 〈 {IFWG}_{ω} (α_{1}, α_{2}, \dots, α_{n}), {IFWG}_{ω} (β_{1}, β_{2}, \dots, β_{n}) 〉 \end{matrix}$ (5.6)

Particularly, if α_j = β_j for all j = 1, 2, ⋯, n, then the collection of SIVIFNs ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ reduce to IFNs, and therefore, the SIIFWG operator reduces to the IFWG operator.

In the following, we give an example to illustrate the aggregation process of the above two operators:

Example 5.1. Let ${\tilde{a}}_{1} = 〈 (0.3, 0.5), (0.5, 0.3) 〉$ , ${\tilde{a}}_{2} = 〈 (0.2, 0.4), (0.6, 0.1) 〉$ , ${\tilde{a}}_{3} = 〈 (0.3, 0.6), (0.8, 0.3) 〉$ , ${\tilde{a}}_{4} = 〈 (0.1, 0.7), (0.6, 0.3) 〉$ and ${\tilde{a}}_{5} = 〈 (0.2, 0.5),$ (0.7, 0.4)〉 be five SIVIFNs, and let ω = (0.15, 0.3, 0.1, 0.2, 0.25) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ . Then $\begin{matrix} {SIIFWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3}, {\tilde{a}}_{4}, {\tilde{a}}_{5}) \\ = 〈 (0.1435, 0.5094), (0.6409, 0.2319) 〉 \end{matrix}$ and $\begin{matrix} {SIIFWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, {\tilde{a}}_{3}, {\tilde{a}}_{4}, {\tilde{a}}_{5}) \\ = 〈 (0.1289, 0.5337), (0.6245, 0.2737) 〉 \end{matrix}$

However, in some situations, for example, in gymnastics competition, we need to consider the importance of ordered positions of the scores of athletes, therefore, some operators should be given to aggregate the data by weighting their ordered positions. Motivated by the ordered idea [36], we develop a simplified interval-valued intuitionistic fuzzy ordered weighted averaging (SIIFOWA) operator and a simplified interval-valued intuitionistic fuzzy ordered weighted geometric (SIIFOGA) operator, respectively:

Theorem 5.3.Let SIIFOWA: ${({\tilde{Θ}}^{'})}^{n} \to {\tilde{Θ}}^{'}$ . If $\begin{matrix} {SIIFOWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = ω_{1} {\tilde{a}}_{σ (1)} \oplus ω_{2} {\tilde{a}}_{σ (2)} \oplus \dots \oplus ω_{n} {\tilde{a}}_{σ (n)} \end{matrix}$ (5.7)where ω = (ω₁, ω₂, ⋯, ω_n) ^Tis the weighting vector which is associated with the function SIIFOWA with ω_j ∈ [0, 1] (j = 1, 2, ⋯, n) and $\sum_{j = 1}^{n} ω_{j} = 1$ , (σ (1), σ (2), ⋯, σ (n)) is any permutation of (1, 2, ⋯, n), such that ${\tilde{a}}_{σ (j - 1)} > {\tilde{a}}_{σ (j)}$ , for anyj, then the function SIIFOWA is called a simplified interval-valued intuitionistic fuzzy ordered weighted averaging (SIIFOWA) operator. Particularly, if ω=(1/n,1/n,...,1/n), then the function SIIFOWA reduces to the SIIFA operator.

In a similar way, we can propose another function:

Theorem 5.4.Let SIIFOWG: ${({\tilde{Θ}}^{'})}^{n} \to {\tilde{Θ}}^{'}$ . If $\begin{matrix} {SIIFOWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = {({\tilde{a}}_{σ (1)})}^{ω_{1}} \otimes {({\tilde{a}}_{σ (2)})}^{ω_{2}} \otimes \dots \otimes {({\tilde{a}}_{σ (n)})}^{ω_{n}} \end{matrix}$ (5.8) where ω = (ω₁, ω₂, ⋯, ω_n) ^T is the weighting vector which is associated with the function SIIFOWG with ω_j ∈ [0, 1] (j = 1, 2, ⋯, n) and $\sum_{j = 1}^{n} ω_{j} = 1$ , (σ (1), σ (2), ⋯, σ (n)) is any permutation of (1, 2, ⋯, n), such that ${\tilde{a}}_{σ (j - 1)} > {\tilde{a}}_{σ (j)}$ , for any j, then the function SIIFOWG is called a simplified interval-valued intuitionistic fuzzy ordered weighted geometric (SIIFOWG) operator. Particularly, if ω=(1/n,1/n,...,1/n), then the function SIIFOWG reduces to the SIIFG operator.

6 Illustrative example

Now we utilize a practical example (adapted from [37]) to demonstrate the aggregation operations of SIVIFNs developed in the previous sections.

Example 6.1. In a manufacturing company, searching for the best global supplier is very critical at its assembling process. They have four potential global suppliers to be chosen. We denote them as S_i (i = 1, 2, 3, 4). We have to consider total cost of the product A₁, quality of the produce A₂, service performance of the supplier A₃, the supplier’s profile A₄ and the risk factor A₅ these five attributes in selection process. An expert group is formed which consists four experts E_k (k = 1, 2, 3, 4) whose weight vector is ξ = (0.2, 0.3, 0.2, 0.3) ^T. The SIVIFNs $a_{ij}^{(k)} (i, j = 1, 2, 3, 4, 5)$ are attribute values provided by the four experts E_k (k = 1, 2, 3, 4), which represent the characteristics of the potential suppliers S_i (i = 1, 2, 3, 4) with respect to the attributes A_i (i = 1, 2, 3, 4, 5) (whose weight vector is ω = (0.2, 0.15, 0.3, 0.25, 0.1) ^T. All $a_{ij}^{(k)} (i, j = 1, 2, 3, 4, 5)$ are respectively contained in the simplified interval-valued intuitionistic fuzzy decision matrices $R^{(k)} = (r_{ij}^{(k)})_{4 \times 5} (k = 1, 2, 3, 4)$ , as shown in Tables 1–4, where the former IFN is $α_{ij}^{(k)}$ of $a_{ij}^{(k)}$ , and the latter IFN is $β_{ij}^{(k)}$ of $a_{ij}^{(k)}$ .

First, according to the experts’ weight vector ξ = (0.2, 0.3, 0.2, 0.3) ^T, we can aggregate the attribute values given by all experts: $\begin{matrix} a_{ij} = 0.2 a_{ij}^{(1)} \oplus 0.3 a_{ij}^{(2)} \oplus 0.2 a_{ij}^{(3)} \oplus 0.2 a_{ij}^{(4)} \\ = 〈 \begin{matrix} 0.2 α_{ij}^{(1)} \oplus 0.3 α_{ij}^{(2)} \oplus 0.2 α_{ij}^{(3)} \oplus 0.3 α_{ij}^{(4)}, \\ 0.2 β_{ij}^{(1)} \oplus 0.3 β_{ij}^{(2)} \oplus 0.2 β_{ij}^{(3)} \oplus 0.3 β_{ij}^{(4)} \end{matrix} 〉 \end{matrix}$

Thus, we get the collective interval-valued intuitionistic fuzzy decision matrix $R = (r_{ij})_{4 \times 5}$ :

Then, we take the SIIFWA operators for example to aggregate the collective attribute values r_ij (j = 1, 2, 3, 4, 5). By Equations (5.1) and (5.5), we obtain: $\begin{matrix} {\tilde{a}}_{1} = 〈 (0 . 2057, 0 . 4717), (0 . 5275, 0 . 1999) 〉 \\ {\tilde{a}}_{2} = 〈 (0 . 1853, 0 . 4094), (0 . 4940, 0 . 1974) 〉 \\ {\tilde{a}}_{3} = 〈 (0 . 1852, 0 . 4613), (0 . 5518, 0 . 1340) 〉 \\ {\tilde{a}}_{4} = 〈 (0 . 1761, 0 . 4902), (0 . 6031, 0 . 1833) 〉 \end{matrix}$

Calculating the scores $s ({\tilde{a}}_{i}) (i = 1, 2, 3, 4)$ of ${\tilde{a}}_{i} (i = 1, 2, 3, 4)$ by Equation (3.3), we can obtain: $s ({\tilde{a}}_{1}) = 0.0308$ , $s ({\tilde{a}}_{2}) = 0.0363$ , $s ({\tilde{a}}_{3}) = 0.0709$ , $s ({\tilde{a}}_{4}) = 0.0 529$ . Rank the four potential global suppliers S_i (i = 1, 2, 3, 4) as: S₃ ≻ S₄ ≻ S₂ ≻ S₁. Therefore, the best choice is S₃.

From this example we can know the good manifestations of the depictions and operations of SIVIFNs: (1) the expression of each SIVIFN using two IFNs is straightforward and much simpler than the traditional IVIFN; (2) the operation processes related to SIVIFNs are simplified by using a series of succinct aggregation operators (such as the SIIFWA and SIIFWG operators). These two advantages not only make the decision making processes briefer, but also save us a lot of energy and time when using these succinct operations and aggregations of SIVIFNs.

7 Conclusions

Interval-valued intuitionistic fuzzy information is a significant and useful preference representation tool used by the experts in decision making processes when the experts cannot express their preferences over the objects (alternatives and attributes) in exact numerical values. However, the traditional depictions of interval-valued intuitionistic fuzzy set (IVIFS) and interval-valued intuitionistic fuzzy number (IVIFN) seem too complicated. To overcome this drawback, in this paper, we have proposed a simplified interval-valued intuitionistic fuzzy set (SIVIFS) and the corresponding simplified interval-valued intuitionistic fuzzy number (SIVIFN) which are characterized by two IFNs. Then, we have given the operational laws and investigated their desirable properties. After that, we have developed a series of aggregation techniques for SIVIFNs, such as the SIIFWA, SIIFWG, SIIFOWA, and SIIFOWG operators. An illustrative example has been presented to show the succinctness and practicality of the operations related to SIVIFNs. By introducing the SIVIFS, the expressions of the operations and aggregations of the intuitionistic-valued fuzzy information get simpler, which brings the benefits in our practical applications when we handle imprecise and imperfect information. Additionally, the SIVIFS can be applied into some areas, like decision making with interval intuitionistic fuzzy sets (IIFSs), while the IVIFS cannot. Therefore, the SIVIFS is of much practicability.

Last but not least, it should be stated that our work of this paper is just in the initial stage. Applications in other practical fields, such as clustering analysis, data mining, information retrieval distance measures, machine learning, and pattern recognition, etc., are interesting future researches.

Footnotes

Acknowledgments

The authors would like to thank the anonymous referees for their insightful and constructive comments and suggestions. The work was supported in part by the National Natural Science Foundation of China (No. 61273209).

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