The logarithmic operational laws of intuitionistic fuzzy sets and intuitionistic fuzzy numbers

Abstract

Intuitionistic fuzzy set (IFS) is an extension of fuzzy set. The basic element of an IFS is the ordered pair called intuitionistic fuzzy number (IFN). So far, some basic operational laws of IFNs are defined, but not including the logarithmic operation. In this paper, a logarithmic operational law about IFNs is defined, in which the base λ is a real number in (0, 1). Then some properties of the operational law are investigated, and four intuitionistic fuzzy information aggregation methods are proposed. Finally, the influences of logarithmic operation for IFNs and the selection of the logarithmic base λ in practice are discussed, and an example is illustrated to show the applications of the operational law and the aggregation methods.

Keywords

Intuitionistic fuzzy numbers logarithmic operational law information aggregation parameter selection

1 Introduction

Since Atanassov [1] proposed the definition of the intuitionistic fuzzy set (IFS) in 1983, IFS has receiv- ed great attention and has been applied to various fields [2, 3]. Inspired by the concept of IFS, Xu and Yager [4] defined the intuitionistic fuzzy number (IFN) as an ordered pair which consists of a membership degree and a non-membership degree. Lots of work has been done on various problems with IFNs, such as decision making [5 –7], pattern recognition [8, 9], medical diagnosis [10, 11], clustering [12], and game theory [13] etc.

As an important research topic to the theory and application of IFSs and IFNs, over the last decades, lots of work has been done on the operations of IFSs and IFNs [14–16 , 29–31]. Atanassov [1, 2] and De et al. [16] introduced several basic operations on IFSs, such as ‘supplement’, ‘union’, ‘intersection’, ‘power’ and so on. Afterwards, Xu and Yager [4], Xu [17] defined some basic operational laws for IFNs, including ‘addition’, ‘multiplication’ and ‘scalar multiplication’ etc. Inspired by Atanassov et al. [18], Lei and Xu [19] developed the subtraction and division operations for IFNs. In addition, Xu et al. [4 , 20–22], Wei et al. [23 –25], Zhou et al. [26, 27], and Yu et al. [28] have focused on the subject of aggregation techniques for intuitionistic fuzzy information. They have defined operational laws of IFNs, and introduced a series of operators for aggregating intuitionistic fuzzy information, including the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, intuitionistic fuzzy hybrid averaging operator, intuitionistic fuzzy geometric operator, intuitionistic fuzzy weighted geometric operator, intuitionistic fuzzy ordered weighted geometric operator, intuitionistic fuzzy hybrid geometric operator, intuitionistic fuzzy bonferroni means, generalized intuitionistic fuzzy aggregation operators, intuitionistic fuzzy aggregation operators based on Choquet integral, induced generalized intuitionistic fuzzy aggregation operators, dynamic intuitionistic fuzzy weighted geometric operator and uncertain dynamic intuitionistic fuzzy weighted geometric operator etc. They have also applied these operators to the field of multi-attribute decision making. Moreover, other extended aggregation operators also are proposed. Recently, Yu [31] presented a scientometrics review on the development of aggregation operator; Xu and Gou [22] gave an overview about the intuitionistic fuzzy and interval intuitionistic information aggregations.

As the study of the IFSs and IFNs theory extends in both depth and scope, new operational laws and aggregation methods were proposed increasingly. Recently, Gou et al. [32] introduced the exponential operational law as a supplement of operational laws of IFSs. But as a kind of important mathematical operation, the logarithmic operational law of IFSs and IFNs is necessary to be developed. So, in this paper, we define a logarithmic operational law of IFSs and IFNs, in which the base is a positive real number in open interval (0, 1) and the values are IFNs. Base on this, many properties of the operational law will be discussed. Furthermore, in the process of designing aggregation operators, the most important basic operators for aggregating arguments are the following four operators: the weighted averaging (WA) operator, weighted geometric (WG) operator, ordered weighted averaging (OWA) operator and ordered weighted geometric (OWG) operator. Motivated by the methods from Xu [3], we also propose four intuitionistic fuzzy information aggregation methods based on the logarithmic operational law. At last, the influences of logarithmic operation for IFNs and the selection of the logarithmic base λ in practice are discussed, and a simple example is given to show the applications of the operational law and the aggregation methods.

2 Preliminaries

Let’s start with some basic concepts related to IFSs and IFNs.

Definition 2.1. [1] Let X be a fixed set. Then the following set $A = {〈 x, μ_{A} (x), v_{A} (x) 〉 | x \in X}$ can be defined as an intuitionistic fuzzy set, each element of which is characterized by a membership function μ_A (x) and a non-membership function v_A (x), with the conditions 0 ≤ μ_A (x), v_A (x) ≤1 and 0 ≤ μ_A (x) + v_A (x) ≤1, for any x ∈ X. Furthermore, π_A (x) =1 - μ_A (x) - v_A (x) is called a hesitancy function. When π_A (x) =0 (∀ x ∈ X), the IFS reduces to the traditional fuzzy set.

Based on the concept of IFS, Xu and Yager [4] defined the order pair (μ, v) as an IFN, which satisfies the conditions: 0 ≤ μ, v ≤ 1 and 0 ≤ μ + v ≤ 1.

Additionally, for any IFN α = (μ, v), the score of α can be evaluated by the score function s [33] as shown s (α) = μ - v, where s (α) ∈ [-1, 1]. In order to distinguish any two IFNs, Hong and Chori [34] defined an accuracy function of α, which is denoted by h (α) = μ + v. Based on the two functions s (α) and h (α), Xu and Yager [4] developed a method for comparison and ranking for two IFNs as below:

Definition 2.2. [4] Let α₁ = (μ₁, v₁) and α₂ = (μ₂, v₂) be two IFNs, s (α₁) = μ₁ - v₁ and s (α₂) = μ₂ - v₂ the scores of the IFNs α₁ and α₂ respectively, and h (α₁) = μ₁ + v₁ and h (α₂) = μ₂ + v₂ the accuracy degrees of the IFNs α₁ and α₂ respectively. Then

If s (α₁) < s (α₂), then the IFN α₁ is smaller than the IFN α₂, denoted by α₁< α₂ ;

If s (α₁) = s (α₂), then

If h (α₁) = h (α₂), the IFNs α₁ and α₂ are equal, i.e., μ₁ = μ₂ and v₁ = v₂, denoted by α₁ = α₂ ;

If h (α₁) < h (α₂), the IFN α₁ is smaller than the IFN α₂, denoted by α₁< α₂ ;

If h (α₁) > h (α₂), the IFN α₁ is larger than the IFN α₂, denoted by α₁ > α₂ .

Definition 2.3. [4, 32] Let α = (μ, v), α₁ = (μ₁, v₁) and α₂ = (μ₂, v₂) be three IFNs. Then

$\bar{α} = (v, μ);$

α₁∩ α₂ = (min {μ₁, μ₂}, max {v₁, v₂}) ;

α₁∪ α₂ = (max {μ₁, μ₂}, min {v₁, v₂}) ;

α₁⊕ α₂ = (1 - (1 - μ₁) (1 - μ₂), v₁v₂) ;

α₁⊗ α₂ = (μ₁μ₂, 1 - (1 - v₁) (1 - v₂)) ;

λα = (1 - (1 - μ) ^λ, v^λ), λ> 0 ;

α^λ = (μ^λ, 1 - (1 - v) ^λ), λ> 0 ;

$α_{1} ⊖ α_{2} = {\begin{matrix} (\frac{μ_{1} - μ_{2}}{1 - μ_{2}}, \frac{v_{1}}{v_{2}}), & if μ_{1} \geq μ_{2}, v_{1} \leq v_{2}, \\ v_{2} > 0 and v_{1} π_{2} \leq \\ v_{2} π_{1}; \\ 0, & otherwise; \end{matrix}$

$α_{1} ø α_{2} = {\begin{matrix} (\frac{μ_{1}}{μ_{2}}, \frac{v_{1} - v_{2}}{1 - v_{2}}), & if μ_{1} \leq μ_{2}, v_{1} \geq v_{2}, \\ μ_{2} > 0 and μ_{1} π_{2} \leq \\ μ_{2} π_{1}; \\ (0, 1), & otherwise; \end{matrix}$

$λ^{α} = {\begin{matrix} (λ^{1 - μ}, 1 - λ^{v}), & λ \in (0, 1); \\ ((1 / λ)^{1 - μ}, 1 - (1 / λ)^{v}), & λ \geq 1 . \end{matrix}$

Based on the basic operational laws in Definition 2.3, Xu and Cai [3] introduced an intuitionistic fuzzy weighted averaging operator and an intuitionistic fuzzy weighted geometric operator:

Definition 2.4. [3] Let α_i = (μ_i, v_i) (i = 1, 2, ⋯, n) be a collection of IFNs, and let IFWA: Θⁿ → Θ. If IFWA_ω (α₁, α₂, ⋯, α_n) = ω₁α₁ ⊕ ω₂α₂ ⊕ ⋯ ⊕ ω_nα_n, then the function IFWA is called an intuitionistic fuzzy weighted averaging (IFWA) operator, where ω = (ω₁, ω₁, ⋯, ω_n) ^T is the weight vector of α_i, with ω_i ∈ [0, 1], (i = 1, 2, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ . The aggregated value by using the IFWA operator is also an IFN, and ${IFWA}_{ω} (α_{1}, α_{2}, \dots, α_{n}) = (1 - \prod_{i = 1}^{n} (1 - μ_{i})^{ω_{i}}, \prod_{i = 1}^{n} v_{i}^{ω_{i}}) .$

Definition 2.5. [3] Let α_i = (μ_i, v_i) (i = 1, 2, ⋯, n) be a collection of IFNs, and let IFWG: Θⁿ → Θ. If ${IFWG}_{ω} (α_{1}, α_{2}, \dots, α_{n}) = α_{1}^{ω_{1}} \otimes α_{2}^{ω_{2}} \otimes \dots \otimes α_{n}^{ω_{n}},$ then the function IFWG is called an intuitionistic fuzzy weighted geometric (IFWG) operator, where ω = (ω₁, ω₁, ⋯, ω_n) ^T is the exponential weight vector of α_i, with ω_i ∈ [0, 1], (i = 1, 2, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ . The aggregated value by using the IFWG operator is also an IFN, and ${IFWG}_{ω} (α_{1}, α_{2}, \dots, α_{n}) = (\prod_{i = 1}^{n} μ_{i}^{ω_{i}}, 1 - \prod_{i = 1}^{n} (1 - v_{i})^{ω_{i}}) .$

3 The logarithmic operational laws of IFSs and IFNs

In Section 2, some basic operational laws of IFNs are introduced in Definition 2.3. It is well known that the logarithm is a kind of important mathematical operation in real numbers. Motivated by the definition of exponential operational law of IFSs and IFNs, it is natural for us to define the new logarithmic operational law about IFSs and IFNs. In this section, the definition of logarithmic operational law about IFSs and IFNs will be proposed, and many properties of the new operational law will be investigated. At last, we will introduce four intuitionistic fuzzy information aggregation methods for logarithmic IFNs.

3.1 The definitions of logarithmic operational laws of IFSs and IFNs

In the following, the logarithmic operational law about IFSs and IFNs will be defined respectively. Let O = (0, 1) and E = (1, 0). Since log _λ0 is meaningless and log ₁x is not defined in real number field, we suppose that α ≠ O and λ ≠ 1 in this paper, where α is an IFN and λ is a real number.

Definition 3.1. Let X be a fixed set, and A ={ 〈 x, μ_A (x), v_A (x) 〉 |x ∈ X } be an IFS, then we can define a logarithmic operational law of the IFS A as follows: $\begin{matrix} {log}_{λ} A \\ = {〈 x, 1 - {log}_{λ} μ_{A} (x), {log}_{λ} (1 - v_{A} (x)) 〉 | x \in X}, \\ 0 < λ \leq μ \leq 1, λ \neq 1 . \end{matrix}$

It can be proved that log _λA is also an IFS. In fact, by the definition of IFS, the membership function and the non-membership function of A satisfy: $\begin{matrix} μ_{A} : X \to [0, 1], \forall x \in X \to μ_{A} (x) \in [0, 1]; \\ v_{A} : X \to [0, 1], \forall x \in X \to v_{A} (x) \in [0, 1]; \end{matrix}$ and 0 ≤ μ_A (x) + v_A (x) ≤1, x ∈ X, so μ_A (x) ≤1 - v_A (x) and 0 ≤ 1 - v_A (x) ≤1. If 0 < λ ≤ μ ≤ 1 and λ ≠ 1, then the membership function: $\begin{matrix} 1 - {log}_{λ} μ_{A} : X \to [0, 1], \\ \forall x \in X \to 1 - {log}_{λ} μ_{A} (x) \in [0, 1], \end{matrix}$ and the non-membership function: $\begin{matrix} {log}_{λ} (1 - v_{A}) : X \to [0, 1], \\ \forall x \in X \to {log}_{λ} (1 - v_{A} (x)) \in [0, 1], \end{matrix}$ and the indeterminacy function: $\begin{matrix} 0 & \leq & 1 - (1 - {log}_{λ} μ_{A} (x) + log (1 - v_{A} (x))) \\ \leq & 1, x \in X . \end{matrix}$

So, $\begin{matrix} {log}_{λ} A \\ = {〈 x, 1 - {log}_{λ} μ_{A} (x), {log}_{λ} (1 - v_{A} (x)) 〉 | x \in X}, \end{matrix}$ (0 < λ ≤ μ ≤ 1, λ ≠ 1) is an IFS.

Definition 3.2. Let α = (μ, v) be an IFN. If ${log}_{λ} α = (1 - {log}_{λ} μ, {log}_{λ} (1 - v)),$ where 0 < λ ≤ μ ≤ 1, λ ≠ 1, then the function log _λα is called a logarithmic operator, and the value log _λα is called a logarithmic IFN (Log-IFN).

It can be proved that log _λα is also an IFN. Let 0 < λ ≤ μ ≤ 1, λ ≠ 1, by the definition of IFN, we can get that α satisfies 0 < μ ≤ 1, 0 ≤ v < 1 and 0 < μ + v ≤ 1. They can be changed by 0 < μ ≤ 1 - v, then 0 ≤ 1 - log _λμ ≤1, 0 ≤ log _λ (1 - v) ≤1 and 0 ≤ 1 - log _λμ + log _λ (1 - v) ≤1.

So log _λα = (1 - log _λμ, log _λ (1 - v)) is an IFN.

3.2 The properties of the logarithmic operational law of IFNs

In what follows, the properties of logarithmic operational law of IFNs will be discussed. At first, we give two results about the relationship between logarithmic operational law and exponential operational law.

Theorem 3.1.Letα = (μ, v) be an IFN. If 0 < λ ≤ μ ≤ 1, λ ≠ 1, thenλ^log_λα = α.

Proof. According to Definitions 2.3 and 3.2, $\begin{matrix} λ^{{log}_{λ} α} & = & (λ^{1 - (1 - {log}_{λ} μ)}, 1 - λ^{{log}_{λ} (1 - v)}) \\ = & (λ^{{log}_{λ} μ}, 1 - (1 - v)) \\ = & (μ, v) \\ = & α . \end{matrix}$

The proof is completed. Theorem 3.1 shows that the logarithm is the inverse operation to exponentiation in intuitionistic fuzzy environment.

Theorem 3.2.Letα = (μ, v) be an IFN. If 0 < λ ≤ μ ≤ 1, λ ≠ 1, then log _λλ^α = α.

Proof. From Definitions 2.3 and 3.2, it follows that $\begin{matrix} {log}_{λ} λ^{α} & = & {log}_{λ} (λ^{1 - μ}, 1 - λ^{v}) \\ = & (1 - {log}_{λ} λ^{1 - μ}, {log}_{λ} (1 - (1 - λ^{v})) \\ = & (μ, v) \\ = & α . \end{matrix}$

Similar to the properties of exponential operational law due to Gou et al. [32], we can get a series of results in logarithmic operational law. So we just list them without proofs as follows.

Theorem 3.3.Letα₁ = (μ₁, v₁) and α₂ = (μ₂, v₂) be two IFNs, 0 < λ ≤ min {μ₁, μ₂} andλ ≠ 1. Then

log _λα₁⊕ log _λα₂ = log _λα₂ ⊕ log _λα₁ ;

log _λα₁ ⊗ log _λα₂ = log _λα₂ ⊗ log _λα₁ .

Theorem 3. 4.Letα_i = (μ_i, v_i) (i = 1, 2, 3) be three IFNs, 0 < λ ≤ min {μ₁, μ₂, μ₃} andλ ≠ 1. Then $\begin{matrix} (1) & ({log}_{λ} α_{1} \oplus {log}_{λ} α_{2}) \oplus {log}_{λ} α_{3} \\ = {log}_{λ} α_{1} \oplus ({log}_{λ} α_{2} \oplus {log}_{λ} α_{3}); \\ (2) & ({log}_{λ} α_{1} \otimes {log}_{λ} α_{2}) \otimes {log}_{λ} α_{3} \\ = {log}_{λ} α_{1} \otimes ({log}_{λ} α_{2} \otimes {log}_{λ} α_{3}) . \end{matrix}$

Theorem 3.5.Letα₁ = (μ₁, v₁) andα₂ = (μ₂, v₂) be two IFNs, 0 < λ ≤ min {μ₁, μ₂}, λ ≠ 1, and k > 0. Then

k (log _λα₁⊕ log _λα₂) = k log _λα₁ ⊕ k log _λα₂ ;

(log _λα₁ ⊗ log _λα₂) ^k = (log _λα₁) ^k ⊗ (log _λα₂) ^k .

Actually, we can easily extend Theorem 3.5 to the general forms with a collection of n Log-IFNs log _λα₁, log _λα₂, ⋯, log _λα_n, i.e., $\begin{matrix} k ({log}_{λ} α_{1} \oplus {log}_{λ} α_{2} \oplus \dots \oplus {log}_{λ} α_{n}) \\ = k ({log}_{λ} α_{1} \oplus {log}_{λ} α_{2} \oplus \dots \oplus {log}_{λ} α_{n}); \\ ({log}_{λ} α_{1} \otimes {log}_{λ} α_{2} \otimes \dots \otimes {log}_{λ} α_{n})^{k} \\ = ({log}_{λ} α_{1})^{k} \otimes ({log}_{λ} α_{2})^{k} \otimes \dots \otimes ({log}_{λ} α_{n})^{k} . \end{matrix}$

Theorem 3.6.Letα = (μ, v) be an IFN, 0 < λ ≤ μ ≤ 1, λ ≠ 1, andk₁, k₂ > 0. Then

k₁ log _λα⊕ k₂ log _λα = (k₁ + k₂) log _λα ;

(log _λα) ^{k
₁}⊗ (log _λα) ^{k
₂} = (log _λα) ^k₁+k₂ ;

((log _λα) ^{k
₁}) ^{k
₂} = (log _λα) ^{k
₁
k
₂} .

Similar as Theorem 3.5, we can also extend Theorem 3.6 to the following general forms: $\begin{matrix} k_{1} {log}_{λ} α \oplus k_{2} {log}_{λ} α \oplus \dots \oplus k_{n} {log}_{λ} α \\ = (k_{1} + k_{2} + \dots + k_{n}) {log}_{λ} α; \\ ({log}_{λ} α)^{k_{1}} \otimes ({log}_{λ} α)^{k_{2}} \otimes \dots \otimes ({log}_{λ} α)^{k_{n}} \\ = ({log}_{λ} α)^{k_{1} + k_{2} + \dots + k_{n}} . \end{matrix}$

Theorem 3.7.Letα₁ = (μ₁, v₁) andα₂ = (μ₂, v₂) be two IFNs, 0 < λ ≤ min {μ₁, μ₂}, and λ ≠ 1. Then

Ifμ₁ ≥ μ₂, v₁ ≤ v₂, v₂ ≥ 0 andv₁π₂ ≤ v₂π₁, then (log _λα₁⊖ log _λα₂) ⊕ log _λα₂ = log _λα₁ ;

Ifμ₁ ≤ μ₂, v₁ ≥ v₂, μ₂ ≥ 0 andμ₁π₂ ≤ μ₂π₁, then (log _λα₁ ø log _λα₂) ⊗ log _λα₂ = log _λα₁ .

Theorem 3.8.Letα_i = (μ_i, v_i) (i = 1, 2, 3) be three IFNs, 0 < λ ≤ min {μ₁, μ₂, μ₃}, and let log _λα_i = (μ_{log_λα_i}, v_{log_λα_i}) (i = 1, 2, 3), with $\begin{matrix} {log}_{λ} α_{1} ⊖ {log}_{λ} α_{2} \\ = (μ_{{log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}}, v_{{log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}}), \\ {log}_{λ} α_{1} ø {log}_{λ} α_{2} \\ = (μ_{{log}_{λ} α_{1} ø {log}_{λ} α_{2}}, v_{{log}_{λ} α_{1} ø {log}_{λ} α_{2}}) . \end{matrix}$

Then $\begin{matrix} (1) & {log}_{λ} α_{1} ⊖ {log}_{λ} α_{2} ⊖ {log}_{λ} α_{3} \\ = {log}_{λ} α_{1} ⊖ ({log}_{λ} α_{2} \oplus {log}_{λ} α_{3}), \end{matrix}$ where $\begin{matrix} μ_{{log}_{λ} α_{1}} \geq μ_{{log}_{λ} α_{2}}, v_{{log}_{λ} α_{1}} \leq v_{{log}_{λ} α_{2}}, v_{{log}_{λ} α_{2}} \geq 0, \\ v_{{log}_{λ} α_{1}} π_{{log}_{λ} α_{2}} \leq v_{{log}_{λ} α_{2}} π_{{log}_{λ} α_{1}}, μ_{{log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}} \geq \\ μ_{{log}_{λ} α_{3}}, v_{{log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}} \leq v_{{log}_{λ} α_{3}}, v_{{log}_{λ} α_{3}} \geq 0 \end{matrix}$ and v_{log_λα₁⊖log_λα₂}π_{log_λα₃} ≤ v_{log_λα₃}π_{log_λα₁⊖log_λα₂}. $\begin{matrix} (2) & {log}_{λ} α_{1} ø {log}_{λ} α_{2} ø {log}_{λ} α_{3} \\ = {log}_{λ} α_{1} ø ({log}_{λ} α_{2} \otimes {log}_{λ} α_{3}), \end{matrix}$ where $\begin{matrix} μ_{{log}_{λ} α_{1}} \leq μ_{{log}_{λ} α_{2}}, v_{{log}_{λ} α_{1}} \geq v_{{log}_{λ} α_{2}}, μ_{{log}_{λ} α_{2}} \geq 0, \\ μ_{{log}_{λ} α_{1}} π_{{log}_{λ} α_{2}} \leq μ_{{log}_{λ} α_{2}} π_{{log}_{λ} α_{1}}, μ_{{log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}} \leq \\ μ_{{log}_{λ} α_{3}}, v_{{log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}} \geq v_{{log}_{λ} α_{3}}, μ_{{log}_{λ} α_{3}} \geq 0 \end{matrix}$ and μ_{log_λα₁⊖log_λα₂}π_{log_λα₃} ≤ μ_{log_λα₃}π_{log_λα₁⊖log_λα₂}.

Theorem 3.9.Letα₁ = (μ₁, v₁) andα₂ = (μ₂, v₂) be two IFNs, 0 < λ ≤ min {μ₁, μ₂}, k > 0, andλ ≠ 1. Then $(1) k {log}_{λ} α_{1} ⊖ k {log}_{λ} α_{2} = k ({log}_{λ} α_{1} ⊖ {log}_{λ} α_{2}),$ where μ₁ ≥ μ₂, v₁ ≤ v₂, v₂ ≥ 0 and v₁π₂ ≤ v₂π₁; $(2) ({log}_{λ} α_{1})^{k} ø ({log}_{λ} α_{2})^{k} = ({log}_{λ} α_{1} ø {log}_{λ} α_{2})^{k},$ where μ₁ ≤ μ₂, v₁ ≥ v₂, μ₂ ≥ 0 and μ₁π₂ ≤ μ₂π₁.

In general, if we have n Log-IFNs which satisfy the requirements of the subtraction and division of Log-IFNs, we can extend Theorem 3.9 to the following forms: $\begin{matrix} k {log}_{λ} α_{1} ⊖ k {log}_{λ} α_{2} ⊖ \dots ⊖ k {log}_{λ} α_{n} \\ = k ({log}_{λ} α_{1} ⊖ {log}_{λ} α_{2} ⊖ \dots ⊖ {log}_{λ} α_{n}); \\ ({log}_{λ} α_{1})^{k} ø ({log}_{λ} α_{2})^{k} ø \dots ø ({log}_{λ} α_{n})^{k} \\ = ({log}_{λ} α_{1} ø {log}_{λ} α_{2} ø \dots ø {log}_{λ} α_{n})^{k} . \end{matrix}$

Theorem 3.10.Letα = (μ, v) be an IFN, k₁ ≥ k₂ > 0, 0 < λ ≤ μ ≤ 1 andλ ≠ 1. Then

k₁ log _λα⊖ k₂ log _λα = (k₁ - k₂) log _λα ;

(log _λα) ^{k
₂} ø (log _λα) ^{k
₁} = (log _λα) ^(k₂-k₁) .

Except for the above properties, there are some special properties of logarithmic operational laws of IFNs.

Theorem 3.11.Letα = (μ, v) be an IFN. If 0 < λ₁ ≤ λ₂ ≤ u ≤ 1, and λ₁, λ₂ ≠ 1, then log _{λ
₁}α ≥ log _{λ
₂}α.

Proof. By Definition 3.2, it follows that $\begin{matrix} {log}_{λ_{1}} α = (1 - {log}_{λ_{1}} μ, {log}_{λ_{1}} (1 - v)), \\ {log}_{λ_{2}} α = (1 - {log}_{λ_{2}} μ, {log}_{λ_{2}} (1 - v)) . \end{matrix}$

If 0 < λ₁ ≤ λ₂ ≤ u ≤ 1, and λ₁, λ₂ ≠ 1, then $\begin{matrix} 1 - {log}_{λ_{1}} μ \geq 1 - {log}_{λ_{2}} μ, \\ {log}_{λ_{1}} (1 - v) \leq {log}_{λ_{2}} (1 - v) . \end{matrix}$

Theorem 3.11 shows that log _{λ
₁}α ≥ log _{λ
₂}α when λ₁ ≤ λ₂. Similar with Theorem 3.11, we can deduce that

Theorem 3.12.Letα₁ = (μ₁, v₁) andα₂ = (μ₂, v₂) be two IFNs. Ifμ₁ ≤ μ₂, v₁ ≥ v₂, i . e ., α₁ ≤ α₂, 0 < λ ≤ min {μ₁, μ₂} ≤1, and λ ≠ 1, then log _λα₁ ≤ log _λα₂.

3.3 The aggregation technique of Log-IFNs

Unlike the traditional IFWA and IFWG operator, in what follows, we utilize the Log-IFN to develop a Log-IFN weighted average operator and a Log-IFN weighted geometric (IFWG) operator.

Definition 3.3. Let α_i = (μ_i, v_i) (i = 1, ⋯, n) be a collection of IFNs, 0 < λ_i ≤ μ_i ≤ 1, λ_i ≠ 1, (i = 1, ⋯, n) and let Log-IFWA: Θⁿ → Θ. If $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ = ω_{1} {log}_{λ_{1}} α_{1} \oplus ω_{2} {log}_{λ_{2}} α_{2} \oplus \dots \oplus ω_{n} {log}_{λ_{n}} α_{n}, \end{matrix}$ then the function Log - IFWA is called a logarithmic intuitionistic fuzzy weighted averaging (Log-IFWA) operator, where ω = (ω₁, ω₂, ⋯, ω_n) ^T is the weighting vectors of log _λα_i (i = 1, ⋯, n), with ω_i ∈ [0, 1], (i = 1, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ .

Remark. If $ω_{i} = λ_{i} / \sum_{j = 1}^{n} λ_{j}, (i = 1, \dots, n)$ , then ω_i ∈ (0, 1), (i = 1, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ . This is one of objective weighting methods in practice.

Theorem 3.13.Letα_i = (μ_i, v_i) (i = 1, ⋯, n) be a collection of IFNs, 0 < λ_i ≤ μ_i ≤ 1, λ_i ≠ 1, (i = 1, ⋯, n) . Then, the aggregated value by using the Log-IFWA operator is also an IFN, where $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}}) . \end{matrix}$

In particular,

If λ₁ = ⋯ = λ_n, 0 < λ ≤ min {μ₁, ⋯, μ_n} ≤1, then Log-IFWA reduces to the following: $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{i})}^{\frac{1}{n}}, \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{i}))}^{\frac{1}{n}}) . \end{matrix}$

If 1 - μ_i = v_i (i = 1, 2, ⋯, n), then Log-IFWA reduces to the following: $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}}) . \end{matrix}$

Proof. It is easy to get the results by employing mathematical induction on n, so the proofs are omitted here.

Definition 3.4. Let α_i = (μ_i, v_i) (i = 1, ⋯, n) be a collection of IFNs, 0 < λ_i ≤ μ_i ≤ 1, λ_i ≠ 1, (i = 1, ⋯, n) and let Log-IFWG: Θⁿ → Θ. If $\begin{matrix} Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) \\ = ({log}_{λ_{1}} α_{1})^{ω_{1}} \otimes ({log}_{λ_{2}} α_{2})^{ω_{2}} \\ \otimes \dots \otimes ({log}_{λ_{n}} α_{n})^{ω_{n}}, \end{matrix}$ then the function Log - IFWG is called a logarithmic intuitionistic fuzzy weighted geometric (Log-IFWA) operator, where w = (w₁, w₂, ⋯ w_n) ^T is the weighting vectors of log _λα_i (i = 1, ⋯, n), with ω_i ∈ [0, 1] (i = 1, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ .

Similar with Theorem 3.13, by employing mathematical induction on n, we can easily prove the following:

Theorem 3.14.Letα_i = (μ_i, v_i) (i = 1, ⋯, n) be a collection of IFNs, 0 < λ_i ≤ μ_i ≤ 1, λ_i ≠ 1, (i = 1, ⋯, n) . Then, the aggregated value by using the Log-IFWG operator is also an IFN, where $\begin{matrix} Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) \\ = (\prod_{i = 1}^{n} {(1 - {log}_{λ_{i}} μ_{i})}^{ω_{i}}, \prod_{i = 1}^{n} (1 - (1 - {log}_{λ_{i}} (1 - v_{i}))^{ω_{i}})) . \end{matrix}$

In particular,

If λ₁ = λ₂ = ⋯ = λ_n, 0 < λ ≤ min {μ₁, μ₂, ⋯, μ_n} ≤1, then Log-IFWA reduces to the following:

$\begin{matrix} Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) \\ = (\prod_{i = 1}^{n} {(1 - {log}_{λ} μ_{i})}^{\frac{1}{n}}, \prod_{i = 1}^{n} (1 - (1 - {log}_{λ} (1 - v_{i}))^{\frac{1}{n}})) . \end{matrix}$

If 1 - μ_i = v_i (i = 1, 2, ⋯, n), then Log-IFWG reduces to the following:

$\begin{matrix} Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) \\ = (\prod_{i = 1}^{n} {(1 - {log}_{λ_{i}} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} (1 - (1 - {log}_{λ_{i}} μ_{i})^{w_{i}})) . \end{matrix}$

Similar to Definitions 3.3 and 3.4, we have

Definition 3.5. A logarithmic intuitionistic fuzzy ordered weighted averaging (Log-IFOWA) operator is a mapping Log-IFOWA: Θⁿ → Θ, such that ω = (ω₁, ω₂, ⋯, ω_n) ^T, with ω_i ∈ [0, 1], (i = 1, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ , and $\begin{matrix} Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) \\ = ω_{1} {log}_{λ_{1}} α_{(1)} \oplus ω_{2} {log}_{λ_{2}} α_{(2)} \\ \oplus \dots \oplus ω_{n} {log}_{λ_{n}} α_{(n)}, \end{matrix}$ where ((1), (2), ⋯, (n)) is a permutation of (1, 2, ⋯, n), such that α_(i-1) ≥ α_(i), for any i.

Definition 3.6. A logarithmic intuitionistic fuzzy ordered weighted geometric (Log-IFOWG) operator is a mapping Log-IFOWG: Θⁿ → Θ, such that ω = (ω₁, ω₂, ⋯, ω_n) ^T, with ω_i ∈ [0, 1], (i = 1, ⋯, n) and $\sum_{i = 1}^{n} ω_{i} = 1$ , and $\begin{matrix} Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) \\ = ({log}_{λ_{1}} α_{(1)})^{ω_{1}} \otimes ({log}_{λ_{2}} α_{(2)})^{ω_{2}} \\ \otimes \dots \otimes ({log}_{λ_{n}} α_{(n)})^{ω_{n}}, \end{matrix}$ where ((1), (2), ⋯, (n)) is a permutation of (1, 2, ⋯, n), such that α_(i-1) ≥ α_(i), for any i.

Similar as Theorems 3.13 and 3.14, it follows that

Theorem 3.15. The aggregated value by using the Log-IFOWA operator and the Log-IFOWG operator are two IFNs, where $\begin{matrix} Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{(i)})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{(i)}))}^{w_{i}}), \\ Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) \\ = (\prod_{i = 1}^{n} {(1 - {log}_{λ_{i}} μ_{(i)})}^{ω_{i}}, \prod_{i = 1}^{n} (1 - (1 - {log}_{λ_{i}} (1 - v_{(i)}))^{ω_{i}})) . \end{matrix}$

In particular,

If λ₁ = ⋯ = λ_n, 0 < λ ≤ min {μ₁, ⋯, μ_n} ≤1, then Log-IFOWA and Log-IFOWG reduce to the following: $\begin{matrix} Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{(i)})}^{\frac{1}{n}}, \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{(i)}))}^{\frac{1}{n}}), \\ Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) \\ = (\prod_{i = 1}^{n} {(1 - {log}_{λ} μ_{(i)})}^{\frac{1}{n}}, \prod_{i = 1}^{n} (1 - (1 - {log}_{λ} (1 - v_{(i)}))^{\frac{1}{n}})) . \end{matrix}$

If 1 - μ_i = v_i (i = 1, 2, ⋯, n), then Log-IFOWA and Log-IFOWG reduce to the following: $\begin{matrix} Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{(i)})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{(i)})}^{w_{i}}), \\ Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) \\ = (\prod_{i = 1}^{n} {(1 - {log}_{λ_{i}} μ_{(i)})}^{w_{i}}, \prod_{i = 1}^{n} (1 - (1 - {log}_{λ_{i}} μ_{(i)})^{w_{i}})) . \end{matrix}$

Next we give more properties to the Log-IFWA operators.

Theorem 3.16. (Idempotency)If all IFNsα_i = (μ_i, v_i) (i = 1, ⋯, n) are equal, i.e.,α_i = α, i = 1, ⋯, n, then $Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) = {log}_{λ} α,$ where λ₁ = λ₂ = ⋯ = λ_n = λ.

Proof. Let α = (μ, v), then

$\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{i}))}^{w_{i}}) \\ = (1 - \prod_{i = 1}^{n} {({log}_{λ} μ)}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ} (1 - v))}^{w_{i}}) \\ = (1 - {({log}_{λ} μ)}^{\sum_{i = 1}^{n} w_{i}}, {({log}_{λ} (1 - v))}^{\sum_{i = 1}^{n} w_{i}}) \\ = (1 - {log}_{λ} μ, {log}_{λ} (1 - v)) \\ = {log}_{λ} α . \end{matrix}$

Theorem 3.17. (Boundedness)Ifα_i = (μ_i, v_i), (i = 1, ⋯, n) with $\begin{matrix} α^{-} & = & (min_{i} {μ_{i}}, max_{i} {v_{i}}), \\ α^{+} & = & (max_{i} {μ_{i}}, min_{i} {v_{i}}), \end{matrix}$ then $\begin{matrix} {log}_{λ} α^{-} & \leq & Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ \leq & {log}_{λ} α^{+}, (*) \end{matrix}$ where λ₁ = λ₂ = ⋯ = λ_n = λ.

Proof. For any i, since $\begin{matrix} min_{i} {μ_{i}} \leq μ_{i} \leq max_{i} {μ_{i}}, \\ min_{i} {v_{i}} \leq v_{i} \leq max_{i} {v_{i}}, \end{matrix}$ it follows that $\begin{matrix} 1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{i})}^{w_{i}} \geq 1 - \prod_{i = 1}^{n} {({log}_{λ} (min_{i} {μ_{i}}))}^{w_{i}} \\ = 1 - {({log}_{λ} (min_{i} {μ_{i}}))}^{\sum_{i = 1}^{n} w_{i}} = 1 - {log}_{λ} (min_{i} {μ_{i}}), \\ \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{i}))}^{w_{i}} \geq \prod_{i = 1}^{n} {({log}_{λ} (1 - min_{i} {v_{i}}))}^{w_{i}} \\ = {({log}_{λ} (1 - min_{i} {v_{i}}))}^{\sum_{i = 1}^{n} w_{i}} = {log}_{λ} (1 - min_{i} {v_{i}}), \\ 1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{i})}^{w_{i}} \leq 1 - \prod_{i = 1}^{n} {({log}_{λ} (max_{i} {μ_{i}}))}^{w_{i}} \\ = 1 - {({log}_{λ} (max_{i} {μ_{i}}))}^{\sum_{i = 1}^{n} w_{i}} = 1 - {log}_{λ} (max_{i} {μ_{i}}), \\ \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{i}))}^{w_{i}} \leq \prod_{i = 1}^{n} {({log}_{λ} (1 - max_{i} {v_{i}}))}^{w_{i}} \\ = {({log}_{λ} (1 - max_{i} {v_{i}}))}^{\sum_{i = 1}^{n} w_{i}} = {log}_{λ} (1 - max_{i} {v_{i}}) . \end{matrix}$

Let $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) = {log}_{λ} α = (μ_{α}, v_{α}), \\ {log}_{λ} α^{-} = (μ_{α^{-}}, v_{α^{-}}), {log}_{λ} α^{+} = (μ_{α^{+}}, v_{α^{+}}) . \end{matrix}$

Then, based on the definitions of score function and monotonicity of logarithmic function, we have $\begin{matrix} s ({log}_{λ} α) & = & 1 - {log}_{λ} μ - {log}_{λ} (1 - v) \\ \leq & 1 - {log}_{λ} (max_{i} {μ_{i}}) - {log}_{λ} (1 - min_{i} {v_{i}}) \\ = & μ_{α^{+}} - v_{α^{+}} \\ = & s ({log}_{λ} α^{+}), \\ s ({log}_{λ} α) & = & 1 - {log}_{λ} μ - {log}_{λ} (1 - v) \\ \geq & 1 - {log}_{λ} (min_{i} {μ_{i}}) - {log}_{λ} (1 - max_{i} {v_{i}}) \\ = & μ_{α^{-}} - v_{α^{-}} \\ = & s ({log}_{λ} α^{-}) . \end{matrix}$

In what follows, we discuss three cases:

If s (log _λα^-) < s (log _λα) < s (log _λα⁺), then Equation (*) holds by Definition 2.2.

If s (log _λα) = s (log _λα⁺), then μ_α - v_α = μ_{α
⁺} - v_{α
⁺}, μ_α = μ_{α
⁺}, v_α = v_{α
⁺}, and hence $\begin{matrix} h ({log}_{λ} α) & = & μ_{α} + v_{α} = μ_{α^{+}} + v_{α^{+}} \\ = & h ({log}_{λ} α^{+}) . \end{matrix}$

In this case, according to Definition 2.2, we have Log - IFWA (α₁, ⋯, α_n) = log _λα⁺.

If s (log _λα) = s (log _λα^-), then μ_α - v_α = μ_{α
^-} - v_{α
^-}, μ_α = μ_{α
^-}, v_α = v_{α
^-}, which implies that $h ({log}_{λ} α) = μ_{α} + v_{α} = μ_{α^{-}} + v_{α}^{-} = h ({log}_{λ} α^{-}) .$

So we have Log - IFWA_ω (α₁, ⋯, α_n) = log _λα^- .

Now, Theorem 3.16 holds from (1)–(3). This completes the proof.

Theorem 3.18. (Monotonicity)Letα_i = (μ_i, v_i), (i = 1, ⋯, n) and $α_{i}^{*} = (μ_{i}^{*}, v_{i}^{*}) (i = 1, \dots, n)$ be two collections of IFNs. If $μ_{i} \leq μ_{i}^{*}, v_{i} \geq v_{i}^{*}$ , and $0 < λ_{i} \leq min {μ_{i}, μ_{i}^{*}} \leq 1$ holds for any i, then $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ \leq Log - {IFWA}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}) . \end{matrix}$

Proof. Let $\begin{matrix} α & = & Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ = & (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}}) \end{matrix}$ and $\begin{matrix} α^{*} & = & Log - {IFWA}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}) \\ = & (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i}^{*})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}^{*}))}^{w_{i}}) \end{matrix}$

For any i, since $μ_{i} \leq μ_{i}^{*}$ and $v_{i} \geq v_{i}^{*}$ , it follows that $\begin{matrix} 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}} \leq 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i}^{*})}^{w_{i}}, \\ \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}} \geq \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}^{*}))}^{w_{i}} . \end{matrix}$

Hence, $\begin{matrix} 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}} - \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}} \\ \leq 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i}^{*})}^{w_{i}} - \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}^{*}))}^{w_{i}} . \end{matrix}$

So, by the score function, s (α) ≤ s (α^*) holds.

If s (α) ≤ s (α^*), then $\begin{matrix} Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) \\ \leq Log - {IFWA}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}) . \end{matrix}$

If s (α) < s (α^*), then $\begin{matrix} 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}} - \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}} \\ = 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i}^{*})}^{w_{i}} - \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}^{*}))}^{w_{i}} . \end{matrix}$

Therefore, by the condition $μ_{i} \leq μ_{i}^{*}, v_{i} \geq v_{i}^{*}$ , and $0 < λ_{i} \leq min {μ_{i}, μ_{i}^{*}} \leq 1$ , for any i, $\begin{matrix} 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}} = 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i}^{*})}^{w_{i}}, \\ \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}} = \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}^{*}))}^{w_{i}} . \end{matrix}$

Thus, $\begin{matrix} h (α) & = & 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}} + \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}} \\ = & 1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i}^{*})}^{w_{i}} + \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}^{*}))}^{w_{i}} \\ = & h (α^{*}) . \end{matrix}$

So, $Log - {IFWA}_{ω} (α_{1}, \dots, α_{n}) = Log - {IFWA}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}) .$

Now, we can conclude that Theorem 3.18 holds.

Similarly, the Log-IFWG, Log-IFOWA and Log-IFOWG operators also have the same properties.

Theorem 3.19.

(Idempotency) If all IFNsα_i (i = 1, ⋯, n) are equal, i.e., α_i = α, i = 1, ⋯, n, then $\begin{matrix} Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) = {log}_{λ} α, \\ Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) = {log}_{λ} α, \\ Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) = {log}_{λ} α, \end{matrix}$ whereλ₁ = λ₂ = ⋯ = λ_n = λ.

(Boundedness)For anyω, $\begin{matrix} {log}_{λ} α^{-} & \leq & Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) \\ \leq & {log}_{λ} α^{+}, \\ {log}_{λ} α^{-} & \leq & Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) \\ \leq & {log}_{λ} α^{+}, \\ {log}_{λ} α^{-} & \leq & Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) \\ \leq & {log}_{λ} α^{+}, \end{matrix}$ whereλ₁ = λ₂ = ⋯ = λ_n = λ, and $\begin{matrix} α^{-} = (min_{i} {μ_{i}}, max_{i} {v_{i}}), \\ α^{+} = (max_{i} {μ_{i}}, min_{i} {v_{i}}) . \end{matrix}$

(Monotonicity)Let $α_{i}^{*} = (μ_{i}^{*}, v_{i}^{*}) (i = 1, \dots, n)$ be a collections of IFNs. If $μ_{i} \leq μ_{i}^{*}, v_{i} \geq v_{i}^{*}$ , and $0 < λ_{i} \leq min {μ_{i}, μ_{i}^{*}} \leq 1$ holds for anyi. Then $\begin{matrix} Log - {IFWG}_{ω} (α_{1}, \dots, α_{n}) \\ \leq Log - {IFWG}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}), \\ Log - {IFOWA}_{ω} (α_{1}, \dots, α_{n}) \\ \leq Log - {IFOWA}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}), \\ Log - {IFOWG}_{ω} (α_{1}, \dots, α_{n}) \\ \leq Log - {IFOWGA}_{ω} (α_{1}^{*}, \dots, α_{n}^{*}) . \end{matrix}$

4 The influences of logarithmic operation for IFNs and λ selection in practice

In this section, the influences of logarithmic operator for IFNs and the selection of the logarithmic base λ in practice will be discussed. It is well known that the essence of logarithmic operator is a kind of transformation for IFNs, i.e., for a given α an IFN, then log _λα is another IFN, where log _λα = α cannot hold usually. The differences of log _λα and α is varied with the logarithmic base λ. It is very important to understand the specified influence of logarithmic transformation for IFNs, so we will study the selection of λ.

Theorem 4.1.Letα = (μ, v) be an IFN, log _λα = (1 - log _λμ, log _λ (1 - v)) and 0 < λ ≤ μ < 1. Then

There exists a real number $λ_{1} = μ^{\frac{1}{1 - μ}}$ such that

Ifλ = λ₁, then 1 - log _λμ = μ;

Ifλ > λ₁, then 1 - log _λμ < μ;

Ifλ < λ₁, then 1 - log _λμ > μ.

There exists a real number $λ_{2} = (1 - v)^{\frac{1}{v}}$ , such that

Ifλ = λ₂, then log _λ (1 - v) = v;

Ifλ > λ₂, then log _λ (1 - v) > v;

Ifλ < λ₂, then log _λ (1 - v) < v.

If $λ_{1} = μ^{\frac{1}{1 - μ}} \leq λ_{2} = (1 - v)^{\frac{1}{v}}$ andμ + v = 1, then $λ_{1} = μ^{\frac{1}{1 - μ}} = λ_{2} = (1 - v)^{\frac{1}{v}} .$

Ifλ < λ₁ < λ₂, then log _λα > α, i.e., the value ofαwill be increase after a logarithmic operator.

Ifλ₁ < λ < λ₂, then 1 - log _λμ < μ and log _λ (1 - v) < v, i.e., the membership degree and the non-membership degree ofαall decreased after a logarithmic operator.

Ifλ > λ₂ > λ₁, then log _λα < α, i.e., the value ofαwill be decrease after a logarithmic operator.

Before presenting the proof of Theorem 4.1, we need the following lemma.

Lemma 4.1. Let $f (x) = (1 - x)^{\frac{1}{x}}, 0 < x < 1$ . Then f (x) is a monotonic decreasing function.

Proof. It is well known that ln f (x) and f (x) have the same monotonicity. Thus, it suffices to prove that ln f (x) is a monotonic decreasing function, when 0 < x < 1.

Obviously, $ln f (x) = \frac{1}{x} ln (1 - x)$ and $\frac{d (ln f (x))}{dx} = - \frac{1}{x^{2}} (ln (1 - x) + \frac{x}{1 - x}) .$

If 0 < x < 1, then $ln (1 - x) + \frac{x}{1 - x} > 0$ , and hence, $\frac{d (ln f (x))}{dx} = - \frac{1}{x^{2}} (ln (1 - x) + \frac{x}{1 - x}) < 0 .$

This completes the proof.

Proof of Theorem 4.1. Firstly, when 0 < λ < 1, log _λx increases as λ increases, however, the larger the value of x, the smaller the value of log _λx is, where x is a real number variable. From the properties of the logarithmic function of real numbers, the results (1) and (2) holds immediately.

Secondly, let α = (μ, v) an IFN, 0 < μ + v ≤ 1, i.e., 0 < μ ≤ 1 - v < 1 holds. By Lemma 4.1, the result (3) is proved.

Finally, combining the results (1), (2) and (3), we can see that (4)–(6) also holds naturally. This completes the proof.

Theorem 4.1 shows that: giving an IFN α = (μ, v), if you select a relatively small number λ such that λ < λ₁ and λ < μ, then the logarithmic transformation will enlarge the value of IFNs; if you select a relatively large number λ with λ > λ₂ and λ < μ, then the logarithmic transformation will reduce the value of IFNs. In practice, we can select a desirable λ based on Theorem 4.1 and actual requirements. For example, in comprehensive evaluation, if we want to enlarge an IFN α = (μ, v) which is undervalued for poor information or evaluator’s preferences, then we can use a logarithmic operator with a relatively small number λ, and vice versa.

In what follows, a simple example in comprehensive evaluation will be illustrated to show the applications of logarithmic operator.

Example. Let α₁ = (0 . 3, 0 . 5), α₂ = (0 . 2, 0 . 7), α₃ = (0 . 8, 0 . 1), α₄ = (0 . 7, 0 . 3) and α₅ = (0 . 5, 0 . 4) be five IFNs, which is a student’s achievement coming from five professor’s independent evaluations in a graduate admission interview. α₁ = (0 . 3, 0 . 5) implies that the ‘acceptance degree’ is 30 percent, ‘objection degree’ 50 percent and ‘hesitancy degree’ 20 percent according to the first professor’s evaluation. The others four IFNs have similar meaning. In the following, we show four kinds of comprehensive evaluation results as following:

Using the traditional IFWA operator with the same weight to get the integrate score M₁ of the student. Assume that ω₁ = ω₂ = ω₃ = ω₄ = ω₅ = 0.2, i.e., the five professors have the same position, then $\begin{matrix} M_{1} & = & {IFWA}_{ω} (α_{1}, α_{2}, \dots, α_{5}) \\ = & (1 - \prod_{i = 1}^{5} (1 - μ_{i})^{ω_{i}}, \prod_{i = 1}^{5} v_{i}^{ω_{i}}) \\ = & (1 - (1 - 0.3)^{0.2} \times (1 - 0.2)^{0.2} \\ \times (1 - 0.8)^{0.2} \times (1 - 0.7)^{0.2} \\ \times (1 - 0.5)^{0.2}, 0 . 5^{0.2} \times 0 . 7^{0.2} \\ \times 0 . 1^{0.2} \times 0 . 3^{0.2} \times 0 . 4^{0.2}) \\ = & (0.558, 0.335) . \end{matrix}$

Using Log-IFWA operator with the same weight to get the integrate score M₂ of the student. Suppose five professors have the same position like (1), but the values of α_i (i = 1, 2, ⋯, 5) are all undervalued for the student’s poor speech skills. In order to get a more reasonable integrate score of the student, examination committee agrees to adjust the score using logarithmic operator. According to Theorem 4.1, we can get $\begin{matrix} (λ_{11}, λ_{21}, λ_{31}, λ_{41}, λ_{51}) \\ = (0.179, 0.134, 0.328, 0.305, 0.250), \\ (λ_{12}, λ_{22}, λ_{32}, λ_{42}, λ_{52}) \\ = (0.250, 0.179, 0.349, 0.305, 0.279), \end{matrix}$ where λ_i1 and λ_i2 are the threshold values of α_i (i = 1, 2, ⋯, 5) in the results (1) and (2) of Theorem 4.1. In this case, suppose that λ₁ = λ₂ = λ₃ = λ₄ = λ₅ = 0.1, then $\begin{matrix} M_{2} & = & Log - {IFWA}_{ω} (α_{1}, α_{2}, \dots, α_{5}) \\ = & (1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{i}))}^{w_{i}}) \\ = & (1 - ({log}_{0.1} 0.3)^{0.2} \times ({log}_{0.1} 0.2)^{0.2} \times ({log}_{0.1} 0.8)^{0.2} \\ \times ({log}_{0.1} 0.7)^{0.2} \times ({log}_{0.1} 0.5)^{0.2}, ({log}_{0.1} (1 - 0.5))^{0.2} \\ \times ({log}_{0.1} (1 - 0.7))^{0.2} \times ({log}_{0.1} (1 - 0.1))^{0.2} \\ \times ({log}_{0.1} (1 - 0.3))^{0.2} \times ({log}_{0.1} (1 - 0.4))^{0.2}) \\ = & (0.722, 0.190) . \end{matrix}$

Using Log-IFWA operator with different weights to get the integrate score M₃ of the student. Suppose that the five professors have different position for their academic research field. For example, selecting a statistical professional student, a statistical professor’s evaluation may be more reasonable than others. So, we suppose λ₁ = λ₂ = λ₃ = λ₄ = λ₅ = 0.1 like (2) and ω₁ = ω₂ = 0.125, ω₃ = ω₄ = ω₅ = 0.25 for some acceptable reasons, then

$\begin{matrix} M_{3} & = & Log - {IFWA}_{ω} (α_{1}, α_{2}, \dots, α_{5}) \\ = & (1 - \prod_{i = 1}^{n} {({log}_{λ} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ} (1 - v_{i}))}^{w_{i}}) \\ = & (1 - ({log}_{0.1} 0.3)^{0.125} \times ({log}_{0.1} 0.2)^{0.125} \\ \times ({log}_{0.1} 0.8)^{0.25} \times ({log}_{0.1} 0.7)^{0.25} \times ({log}_{0.1} 0.5)^{0.25}, \\ ({log}_{0.1} (1 - 0.5))^{0.25} \times ({log}_{0.1} (1 - 0.7))^{0.125} \\ \times ({log}_{0.1} (1 - 0.1))^{0.25} \times ({log}_{0.1} (1 - 0.3))^{0.25} \\ \times ({log}_{0.1} (1 - 0.4))^{0.25}) \\ = & (0.771, 0.158) . \end{matrix}$

Using Log-IFWA operator with different weights to get the integrate score M₄ of the student, where the values of α_i (i = 1, 2, ⋯, 5) are all overvalued for evaluator’s preferences. In order to induce the values of α_i (i = 1, 2, ⋯, 5) reasonable and get an objective weights, we suppose that λ₁ = 0.28, λ₂ = 0.18, λ₃ = λ₄ = 0.5, λ₅ = 0.3 according to the threshold values λ_i1 and λ_i2 of α_i (i = 1, 2, ⋯, 5), and $w_{i} = \frac{λ_{i}}{\sum_{j = 1}^{n} λ_{j}} (i = 1, \dots, n), w_{i} \in (0, 1)$ , (i = 1, ⋯, n) and $\sum_{i = 1}^{n} w_{i} = 1$ . It is easy to get that w₁ = 0.159, w₂ = 0.102, w₃ = 0.284, w₄ = 0.284 and w₅ = 0.170. Then, $\begin{matrix} M_{4} & = & Log - {IFWA}_{ω} (α_{1}, α_{2}, \dots, α_{5}) \\ = & (1 - \prod_{i = 1}^{n} {({log}_{λ_{i}} μ_{i})}^{w_{i}}, \prod_{i = 1}^{n} {({log}_{λ_{i}} (1 - v_{i}))}^{w_{i}}) \\ = & (1 - ({log}_{0.28} 0.3)^{0.159} \times ({log}_{0.18} 0.2)^{0.102} \\ \times ({log}_{0.5} 0.8)^{0.284} \times ({log}_{0.5} 0.7)^{0.284} \\ \times ({log}_{0.3} 0.5)^{0.170}, ({log}_{0.28} (1 - 0.5))^{0.159} \\ \times ({log}_{0.18} (1 - 0.7))^{0.102} \times ({log}_{0.5} (1 - 0.1))^{0.284} \\ \times ({log}_{0.5} (1 - 0.3))^{0.284} \times ({log}_{0.3} (1 - 0.4))^{0.170}) \\ = & (0.462, 0.367) . \end{matrix}$

Obviously, M₃ > M₂ > M₁ > M₄. The results are consistent with our expectation.

5 Conclusions

In this paper, the logarithmic operational law of IFSs and IFNs is given to be a useful supplement of the existing intuitionistic fuzzy aggregation techniques. Base on the logarithmic operational laws, we have investigated a series of properties of these Log-IFNs, and also proposed four aggregation methods Log-IFWA, Log-IFWG, Log-IFOWA and Log-IFOWG. Finally, the influences of logarithmic operation for IFNs and the selection of logarithmic base λ in practice are discussed, and a simple example is illustrated to show the applications of the operational law and the aggregation methods. In the future, the derived results can be applied to some other fields, such as decision making, medical diagnosis, and macro-economic analysis etc.

Footnotes

Acknowledgments

This research was funded in part by Guangdong Science and Technology Plan Projects (N0. 2016A 020210101) and the earmarked fund for modern agro-industry technology research system of China (CARS-01-33).

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