Lorentzian knowledge measures for atanassov’s intuitionistic fuzzy sets

Abstract

The Lorentzian manifold theory is an important research object in mathematical physics. In this paper, we propose four generalized Lorentzian-like knowledge measures deduced by the Lorentzian inner product for intuitionistic fuzzy sets. Some theorems are given to show the properties of the constructed knowledge measures. Compared with some other knowledge measures, we point out that there exists some counterexamples for some knowledge measures in the frame of the axiom of entropy measure for intuitionistic fuzzy set defined by Szmidt and Kacprzyk. In order to reduce counterexamples, we give some modified orders which extend the definition of the classical order ⪯ in the axiom. And the numerical experiments show that the modified orders are more general than the order ⪯ in a sense, such as the order ≾ can break the constraint that the two intuitionistic fuzzy sets with the order ⪯ must fall in the either side of the line y = x. It is noteworthy that these binary relations are not strict partially ordered relations, and the problem, constructing a universal partial order, is still open. At the end of the paper some numerical experiments show that the proposed knowledge measures work well on different datasets.

Keywords

Intuitionistic fuzzy set Lorentzian knowledge measure order set

1. Introduction

Fuzzy set theory was originally raised by Zadeh [1] in 1965. Considerable attentions has been given to the study of the fuzzy set theory; one can see the monographs [2 –4]. Atanassov established the intuitionistic fuzzy set theory which extended the classical concept of the fuzzy set [5 –7]. Furthermore, some operations and notions were given for intuitionistic fuzzy sets in papers [8 –10].

The theory of entropy and knowledge measure is a very power tool in the study of intuitionistic fuzzy set. One can find an elegant discussion of knowledge measures for Atanassov’s intuitionistic fuzzy sets given by Guo [11]. An axiomatic definition of entropy, distance and similarity measures of fuzzy sets was given by Liu who also systematically considered the basic relations between these measures [12]. Moreover, another axiomatic definition of distance measures was given by Zhang et al., and then they proposed two methods to construct two distances, one based on numerical integration and the another one constructed by using Hausdorff distance [13]. Hung et al. introduced the concept of probability to intuitionistic fuzzy sets and obtained two families of entropy measures of intuitionistic fuzzy sets [14]. Some other knowledge measures were studied and applied to practical problems such as intuitionistic fuzzy evidence theory, group decision making problems and medical diagnosis [15 –18]. In the fuzzy sense, Pham presented the Kolmogorov-Sinai entropy to measure the entropy rate of imprecise systems by using sequence membership grades, in which the underlying deterministic paradigm was replaced with the degree of fuzziness [19]. By using the Shapley function, some Shapley-weighted similarity measures of Atannasov’s interval-valued intuitionistic fuzzy sets were given in [20]. Further, the Shapley-weighted measure was used in pattern recognition, and an optimal fuzzy measure on feature set was raised [21]. Many research papers have been published to consider the knowledge measures of the intuitionistic fuzzy set, and we refer the reader to (see for instance [22 –25]) and references therein for more information on different aspects of this topic.

Moreover, Garg et al. proposed some distance measures for connection number sets based on set pair analysis theory, and investigated its applications to decision-making process [35]. Using single-valued neutrosophic sets theory, some new biparametric distance measures was given, and the comparison between the proposed and the existing measures has been performed in terms of counter-intuitive cases for showing its validity [36]. Under this environment of type-2 intuitionistic fuzzy set, a family of distance measures based on Hamming, Euclidean and Hausdorff metrics were presented [38]. In paper [37] the authors pointed out the weakness of the existing measures and proposed some new similarity and weighted similarity measures between the connection number sets. we refer the reader to (see for instance [39 –41]) and references therein for more information on different aspects of this topic.

Notice that the knowledge and distance measures considered above are all Euclidean-like and Archimedean (the triangle inequality holds). Recently, the non-Euclidean theory and the non-Archimedean theory have been made a breakthrough achievement in the research of Mathematics and Physics, which greatly promote the application and development of non-Euclidean [26 –28]. It is noteworthy that most of non-Euclidean distances cannot be used directly in practical problems such as Lorentzian distance, since the Lorentzian distance function may be complex. By matrix transform method Kerimbekov et al. presented a generalized distance deduced by the Lorentzian norm and used the distance in clustering analysis [29].

It should be clear that the definitions of entropy and knowledge measure all satisfy the following condition: If E ⪯ F, then Knowledge (E) ≥ Knowledge (F) or Entropy (E) ≤ Entropy (F). The notion of the order ⪯ plays an important role in entropy and knowledge measure theory, just as it does in the study of the ordering space theory. However, it is easy to verify that the classical order ⪯ of intuitionistic fuzzy sets is not a strict order, and there exist intuitionistic fuzzy sets E, F such that EnotpreceqF and FnotpreceqE. Thus, we cannot say more about the relationship between Knowledge (E) and Knowledge (F) in this case. Our interest is in pointing out that some orders and knowledge measures can be constructed to reduce these undesired cases by using ordering space theory (See [30, 31]). Then, we will define some orders which extend the classical definition of the order of intuitionistic fuzzy sets. We also modify the definition of the Lorentzian distance and obtain some Lorentzian-like knowledge measures of intuitionistic fuzzy sets. Some numerical experiments are given to show the effectiveness and practicability of the constructed knowledge measures.

This paper is organized as follows. In Section 2, we review relating basic definitions and theoretical backgrounds needed in this paper. In Section 3, we give the definitions of Lorentzian-like knowledge measures and show the principal theorems. Some numerical examples are presented in Section 4. Section 5 ends this paper with a brief conclusion.

2. Mathematical foundation

In this section, we review in some detail notions that we will need later.

2.1. Intuitionistic fuzzy set

Definition 1. [5] An intuitionistic fuzzy set E in $X$ is defined as $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X},$ where $μ_{E} : X \to [0, 1]$ and $ν_{E} : X \to [0, 1]$ represent the degree of membership and the degree of non-membership of the element $x_{0} \in X$ , respectively. Moreover, for every $x_{0} \in X$ we have $0 \leq μ_{E} (x_{0}) + ν_{E} (x_{0}) \leq 1 .$

Definition 2. [5] Let E be an intuitionistic fuzzy set. The intuitionistic fuzzy index (hesitation margin) of the element x₀ ∈ E is defined as $π_{E} (x_{0}) = 1 - μ_{E} (x_{0}) - ν_{E} (x_{0}) .$

Suppose that E = {< x₀, μ_E (x₀), ν_E (x₀), $π_{E} (x_{0}) > | x_{0} \in X}$ is an intuitionistic fuzzy set. The complement E^c is defined as $E^{c} = {< x_{0}, ν_{E} (x_{0}), μ_{E} (x_{0}), π_{E} (x_{0}) > | x_{0} \in X} .$ We will use $𝔽 (X)$ to denote the space of intuitionistic fuzzy sets. Let E, F be intuitionistic fuzzy sets. And we shall give some operations defined by Atanassov [7] as follows: (H1) E ⊂ F if and only if μ_E (x₀) ≤ μ_F (x₀), ν_E (x₀) ≥ ν_F (x₀) for $x_{0} \in X;$ (H2) E = F if and only if E ⊂ F and F ⊂ E; (H3) E ∪ F : = {< x₀, max {μ_E (x₀), μ_F (x₀)}, $min {ν_{E} (x_{0}), ν_{F} (x_{0})} > | x_{0} \in X}$ ; (H4) E ⪯ F; E is less fuzzy that F, i.e. for $x_{0} \in X$ , if μ_F (x₀) ≤ ν_F (x₀) then μ_E (x₀) ≤ μ_F (x₀) and ν_E (x₀) ≥ ν_F (x₀); if μ_F (x₀) ≥ ν_F (x₀) then μ_E (x₀) ≥ μ_F (x₀) and ν_E (x₀) ≤ ν_F (x₀).

With the above notions, Szmidt et. al gave an axiomatic definition of entropy measure for intuitionistic fuzzy sets [32].

Definition 3. [32] Let E, F be intuitionistic fuzzy sets. A function $E : F (X) \to [0, 1]$ is called an intuitionistic fuzzy entropy, if the following conditions hold: (C1) $E (E) = 0$ if and only if E is a crisp set; (C2) $E (E) = 1$ if and only if $μ_{E} (x_{0}) = ν_{E} (x_{0}), x_{0} \in X$ ; (C3) $E (E) = E (E^{)}$ ; (C4) If E ⪯ F then $E (E) \leq E (F)$ .

2.2. Lorentzian Distance

Given a Lorentzian manifold (M, g_L) and a timelike unitary vector field $E \in X (M)$ we can construct the Riemannian metric $g_{R} = g_{L} + 2 ω \otimes ω,$ (1)ω being the g_L-metrically equivalent one from to E. This construction is frequently used to exploit the positive definitions of g_R, which provides some conclusions about g_L. A similar construction has been used to induce a Riemannian metric on a lightlike hypersurface of a Lorentzian manifold. The dual construction of (end-add-1), i.e., given a Riemannian manifold (M, g_R) defines the Lorentzian metric $g_{L} = g_{R} - 2 ω \otimes ω .$ (2) Then, the metric matrix on the Lorentzian manifold $L_{1}^{n}$ can be given as $G = [\begin{matrix} Λ_{(n - 1) \times (n - 1)} & 0 \\ 0 & - λ \end{matrix}],$ (3) where Λ_(n-1)×(n-1) is diagonal and its diagonal entries and λ are all positive. Let x = (x₀, x₁, …, x_n), y = (y₀, y₁, …, y_n). The Lorentzian inner product is defined as $< x, y > = - x_{0} y_{0} + x_{1} y_{1} + . . . + x_{n} y_{n} .$

The Lorentzian norm deduced by Lorentzian inner product can be defined as $∥ x ∥ = {((\sum_{i = 1}^{n} x_{i}^{2}) - x_{0}^{2})}^{\frac{1}{2}},$ (4) where x = (x₀, x₁, …, x_n).

Definition 4. The Lorentzian distance deduced by the Lorentzian inner product can be defined as $D (x, y) = {((\sum_{i = 1}^{n} | x_{i} - y_{i} |^{2}) - | x_{0} - y_{0} |^{2})}^{\frac{1}{2}},$ (5) where x = (x₀, x₁, …, x_n) and y = (y₀, y₁, …, y_n).

Notice that the Lorentzian distance is a non-Euclidean linear distance. The Lorentzian distance may be complex, so we need to modify the definition when we apply it in the space of intuitionistic fuzzy sets. By using the absolute value function, we obtain a natural generalization of Lorentzian distance as follows:

Definition 5. The modified Lorentzian distance is defined as $D_{0} (x, y) = {| (\sum_{i = 1}^{n} | x_{i} - y_{i} |^{2}) - | x_{0} - y_{0} |^{2} |}^{\frac{1}{2}},$ (6) where x = (x₀, x₁, …, x_n) and y = (y₀, y₁, …, y_n).

Remark 1. In fact, $x \in ℝ^{n + 1}$ is space-like, light-like or time-like, if its Lorentzian norm is positive, zero or pure imaginary, respectively. If we replace |x₀ - y₀|² by |x_i - y_i|² in (add-1), then we will obtain n + 1 different definitions of modified Lorentzian distances, and denote it by $D_{i}, i \in {0, 1, 2, … n} .$ .

In order to avoid to use the absolute value function, we then give the following definition.

Definition 6. The generalized Lorentzian distance is defined as $\begin{matrix} D (x, y) = \\ {max_{k \in Γ} {(\sum_{i \neq k, i \in Γ} | x_{i} - y_{i} |^{2}) - | x_{k} - y_{k} |^{2}}}^{\frac{1}{2}}, \end{matrix}$ (7) where x = (x₀, x₁, …, x_n), y = (y₀, y₁, …, y_n), and Γ = {0, 1, 2, …, n}.

3. Lorentzian Knowledge Measure

In this section, we propose the notions of Lorentzian knowledge measure and prove the main results of this paper.

3.1. Some Orders for Intuitionistic Fuzzy Sets

Szmidt and Nguyen gave some comparative analysis for the different distances and similarity measures, and showed that most of them could not avoid the counter-intuitive cases [10, 25]. The main reason is that in condition (H4) the order ⪯ is not a strict partial order, since it does not satisfy antisymmetry.

Definition 7. Let $E, F \in 𝔽 (X)$ be two intuitionistic fuzzy sets. We call E ≾ ₀F, if and only if for all $a \in X$ , $μ_{E} (a) ν_{E} (a) \geq μ_{F} (a) ν_{F} (a) .$ (8)

Remark 2. The order ≾₀ is not a strict partial order, because it does not satisfy antisymmetry. In fact, it is hard to construct a universal partial order on $𝔽 (X)$ satisfying the strict definition of the partial order, and we can only find some result on the well-defined orders on some strong constructions, such as lattice and algebra. See that the order ⪯ in (H4) is also not a partial order. (II) E ≾ ₀F and F ≾ ₀E don’t imply that E = F. It is easy to give the following counterexamples: E ≾ ₀F and F ≾ ₀E for E = {< x₀, 1, 0, 0 >}, F = {< x₀, 0, 1, 0 >} and $X = {x_{0}}$ . (III) For $E, F \in 𝔽 (X)$ , we have E ≾ ₀F or F ≾ ₀E. And in the case of ⪯, this result doesn’t hold. (IV) For $E, F \in 𝔽 (X)$ if E ≾ ₀F, then E^c ≾ ₀F, E ≾ ₀F^c and E^c ≾ ₀F^c.

Definition 8. Let $E, F \in 𝔽 (X)$ be two intuitionistic fuzzy sets. We call E ≾ ₁F, if and only if for all $a \in X$ , $ν_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \geq ν_{F} (a)^{2} + μ_{F} (a) ν_{F} (a) .$ (9)

Remark 3. The order ≾₁ is not a strict partial order, because it does not satisfy antisymmetry. (II) E ≾ ₁F and F ≾ ₁E don’t imply that E = F. Assume that ν_E (x₀) ² + μ_E (x₀) ν_E (x₀) = ν_F (x₀) ² + μ_F (x₀) ν_F (x₀). One can verify that the set of solutions of the above equation is not empty. Assume that (μ_E, ν_E, μ_F, ν_F) is a solution of the above equation. We then have E ≾ ₁F and F ≾ ₁E for E = {< x₀, μ_E (x₀), ν_E (x₀), π_E (x₀) >}, F = {< x₀, μ_F (x₀), ν_F (x₀), π_F (x₀) >} and $X = {x_{0}}$ . There exist some unreasonable ordering pairs of intuitionistic fuzzy sets such as the above [E, F]. (III) For $E, F \in 𝔽 (X)$ , we have E ≾ ₁F or F ≾ ₁E. (IV) For $E, F \in 𝔽 (X)$ if E ≾ ₁F, F^c can not compare with E^c.

Definition 9. Let $E, F \in 𝔽 (X)$ be two intuitionistic fuzzy sets. We call E ≾ ₂F, if and only if for all $a \in X$ , $μ_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \geq μ_{F} (a)^{2} + μ_{F} (a) ν_{F} (a) .$ (10)

Remark 4. The order ≾₂ is not a strict partial order, because it does not satisfy antisymmetry. (II) E ≾ ₂F and F ≾ ₂E don’t imply that E = F. Assume that μ_E (x₀) ² + μ_E (x₀) ν_E (x₀) = μ_F (x₀) ² + μ_F (x₀) ν_F (x₀). One can verify that the set of solutions of the above equation is not empty. We then have E ≾ ₂F and F ≾ ₂E for E = {< x₀, μ_E (x₀), ν_E (x₀), π_E (x₀) >} and F = {< x₀, μ_F (x₀), ν_F (x₀), π_F (x₀) >} and $X = {x_{0}}$ . There exist some unreasonable ordering pairs of intuitionistic fuzzy sets such as above [E, F]. (III) For $E, F \in 𝔽 (X)$ , we have E ≾ ₂F or F ≾ ₂E. (IV) For $E, F \in 𝔽 (X)$ if E ≾ ₂F, we can not compare the two intuitionistic fuzzy sets, F^c and E^c.

Definition 10. Let $E, F \in 𝔽 (X)$ be two intuitionistic fuzzy sets. We call E ≾ F, if and only if for all $a \in X$ , $\begin{matrix} max {μ_{E} (a), ν_{E} (a)}^{2} + μ_{E} (a) ν_{E} (a) \\ \geq max {μ_{F} (a), ν_{F} (a)}^{2} + μ_{F} (a) ν_{F} (a) . \end{matrix}$ (11)

Remark 5. (I) By Definition E ≾ F is equivalent to the following four cases:

If μ_E (a) ≥ ν_E (a) and μ_F (a) ≥ ν_F (a), then E ≾ ₂F and $μ_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \geq μ_{F} (a)^{2} + μ_{F} (a) ν_{F} (a);$

If ν_E (a) ≥ μ_E (a) and μ_F (a) ≥ ν_F (a), then $ν_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \geq μ_{F} (a)^{2} + μ_{F} (a) ν_{F} (a);$

If μ_E (a) ≥ ν_E (a) and ν_F (a) ≥ μ_F (a), then $μ_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \geq ν_{F} (a)^{2} + μ_{F} (a) ν_{F} (a);$

If ν_E (a) ≥ μ_E (a) and ν_F (a) ≥ μ_F (a), then E ≾ ₁F and $ν_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \geq ν_{F} (a)^{2} + μ_{F} (a) ν_{F} (a) .$

(II) It is easy to see that the order ≾ satisfies reflexivity and transitivity. It is not a partial order since it does not satisfy antisymmetry. In the next section, we will detailedly illustrate the difference between ≾ and ⪯. (III) For

E, F \in 𝔽 (X)

, we have E ≾ F or F ≾ E. (IV) For

E, F \in 𝔽 (X)

if E ≾ F, then E^c ≾ F, E ≾ F^c and E^c ≾ F^c.

Example 1. From the geometric representation of the intuitionistic fuzzy set, the intuitionistic fuzzy sets all fall in the triangle area surrounded by the point (1, 0, 0), the point (0, 1, 0) and the point (0, 0, 1). The two intuitionistic fuzzy sets with the order ⪯ just fall in the either side of the line segment defined by the point (0, 0, 1) and the point (0.5, 0.5, 0). But there are no restrictions for the intuitionistic fuzzy sets with the order ≾.

3.2. Lorentzian-like Knowledge Measures

We will define knowledge measures based on generalized Lorentzian distances.

Definition 11. Let $X = {x_{1}, x_{2}, . . ., x n}$ and $E = {< μ_{E} (a), ν_{E} (a), π_{E} (a) > | a \in X}$ be an intuitionistic fuzzy set. The Lorentzian knowledge measure $K_{0} (E)$ based on generalized Lorentzian distance is defined as $\begin{matrix} K_{0} (E) & = \frac{\sqrt{2}}{n} \sum_{i = 1}^{n} {| μ_{1} (x_{i})^{2} + μ_{2} (x_{i})^{2} - μ_{3} (x_{i})^{2} |}^{\frac{1}{2}} \\ = \frac{2}{n} \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{i}) ν_{E} (x_{i})}, \end{matrix}$ (12) where μ₁ (x_i) = μ_E (x_i), μ₂ (x_i) = ν_E (x_i), μ₃ (x_i) =1 - π_E (x_i) = μ_E (x_i) + ν_E (x_i).

Definition 12. Let $X = {x_{1}, x_{2}, . . ., x_{n}}$ and $E = {< μ_{E} (a), ν_{E} (a), π_{E} (a) > | a \in X}$ be an intuitionistic fuzzy set. The Lorentzian knowledge measure $K_{1} (E)$ based on generalized Lorentzian distance is defined as $\begin{matrix} K_{1} (E) & = \frac{1}{\sqrt{2} n} \sum_{i = 1}^{n} {| μ_{2} (x_{i})^{2} + μ_{3} (x_{i})^{2} - μ_{1} (x_{i})^{2} |}^{\frac{1}{2}} \\ = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i})}, \end{matrix}$ (13) where μ₁ (x_i) = μ_E (x_i), μ₂ (x_i) = ν_E (x_i), μ₃ (x_i) =1 - π_E (x_i) = μ_E (x_i) + ν_E (x_i).

Definition 13. Let $X = {x_{1}, x_{2}, . . ., x_{n}}$ and $E = {< μ_{E} (a), ν_{E} (a), π_{E} (a) > | a \in X}$ be an intuitionistic fuzzy set. The Lorentzian knowledge measure $K_{2} (E)$ based on generalized Lorentzian distance is defined as $\begin{matrix} K_{2} (E) & = \frac{1}{\sqrt{2} n} \sum_{i = 1}^{n} {| μ_{1} (x_{i})^{2} + μ_{3} (x_{i})^{2} - μ_{2} (x_{i})^{2} |}^{\frac{1}{2}} \\ = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{0})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i})}, \end{matrix}$ (14) where μ₁ (x_i) = μ_E (x_i), μ₂ (x_i) = ν_E (x_i), μ₃ (x_i) =1 - π_E (x_i) = μ_E (x_i) + ν_E (x_i).

Definition 14. Let $X = {x_{1}, x_{2}, . . ., x_{n}}$ and $E = {< μ_{E} (a), ν_{E} (a), π_{E} (a) > | a \in X}$ be an intuitionistic fuzzy set. The Lorentzian knowledge measure $K (E)$ based on generalized Lorentzian distance is defined as $\begin{matrix} K (E) = \frac{1}{\sqrt{2} n} \\ \sum_{i = 1}^{n} {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ = \frac{1}{n} \sum_{i = 1}^{n} {max {μ_{E} (x_{i})^{2}, ν_{E} (x_{i})^{2}} + μ_{E} (x_{i}) ν_{E} (x_{i})}^{\frac{1}{2}}, \end{matrix}$ (15) where Λ = {1, 2, 3}, μ₁ (x_i) = μ_E (x_i), μ₂ (x_i) = ν_E (x_i), μ₃ (x_i) =1 - π_E (x_i) = μ_E (x_i) + ν_E (x_i).

Remark 6. We can obtain that $K (E) \geq max {\frac{1}{2} K_{0} (E), K_{1} (E), K_{2} (E)}$ and $K_{1} (E) \geq \frac{1}{2} K_{0} (E), K_{2} (E) \geq \frac{1}{2} K_{0} (E) .$

The dual non-probabilistic entropy measure can be defined as $E (E) = 1 - K (E) .$ (16) We can obtain the following theorems.

Theorem 1. Let $E = {< μ_{E} (x_{i}), ν_{E} (x_{i}), π_{E} (x_{i}) > | x_{i} \in X}$ and $F = {< μ_{F} (x_{i}), ν_{F} (x_{i}), π_{F} (x_{i}) > | x_{i} \in X}$ be intuitionistic fuzzy sets. We have (R1) $K_{0} (E) = 1$ if and only if $μ_{E} (x_{i}) = ν_{E} (x_{i}) = \frac{1}{2}$ for $x_{i} \in X$ ; (R2) $K 0 (E) = 0$ if and only if E is a crisp set or π_E (x_i) =1; (R3) $0 \leq K_{0} (E) \leq 1$ ; (R4) $K_{0} (E) = K_{0} (E)$ ; (R5) If E ≾ ₀F, i.e. E is less fuzzy than F, then $K_{0} (E) \geq K_{0} (F)$ .

Proof. (R1) Assume that $K_{0} (E) = 1$ . We have $K_{0} (E) = \frac{2}{n} \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{i}) ν_{E} (x_{i})} = 1 .$ (17) By Definition for $x_{i} \in X$ we get 0 ≤ μ_E (x_i) ≤1, 0 ≤ ν_E (x_i) ≤1 and μ_E (x_i) + ν_E (x_i) ≤1. By basic inequality, we have $\begin{matrix} 0 \leq K_{0}^{i} (E) : = 2 \sqrt{μ_{E} (x_{i}) ν_{E} (x_{i})} \\ \leq μ_{E} (x_{i}) + ν_{E} (x_{i}) = 1, \end{matrix}$ (18)

and we get maximum value $K 0 (E) = 1$ if and only if $μ_{E} (x_{i}) = ν_{E} (x_{i}) = \frac{1}{2} .$ $K_{o} (E) = \frac{1}{n} \sum_{i = 0}^{n} k_{0}^{i} (E) = 1$ and $0 \leq K_{0}^{i} (E) \leq 1$ imply that $K 0 (E) = 1$ for i = {1, 2, 3, …, n}. Then, we have $μ_{E} (x_{i}) = ν_{E} (x_{i}) = \frac{1}{2}$ for $x_{i} \in X$ .

Suppose that $μ_{E} (x_{i}) = ν_{E} (x_{i}) = \frac{1}{2}$ for $x_{i} \in X$ . It is easy to verify that $K_{0} (E) = 1 .$

(R2) Setting $K_{0} (E) = 0$ , one can obtain μ_E (x_i) =0 or ν_E (x_i) =0 or π_E (x_i) =1. Thus, E is a crisp set or π_E (x_i) =1 for $x_{i} \in X$ .

It is obviously that $K_{0} (E) = 0$ , when E is a crisp set or π_E (x_i) =1.

(R3) By (add-equ-5), (R1) and (R2), it is easy to obtain this result.

(R4) $\begin{matrix} K_{0} (E) = \frac{2}{n} \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{i}) ν_{E} (x_{i})} \\ = \frac{2}{n} \sum_{i = 1}^{n} \sqrt{ν_{E} (x_{i}) μ_{E} (x_{i})} = K_{0} (E^{c}) . \end{matrix}$ (19)

(R5) If E ≾ ₀F, for $x_{i} \in X$ we get $μ_{E} (x_{i}) ν_{E} (x_{i}) \geq μ_{F} (x_{i}) ν_{F} (x_{i}) .$ (20) By Definition 11, one can obtain $\begin{matrix} K_{0} (E) = \frac{2}{n} \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{i}) ν_{E} (x_{i})} \\ \geq \frac{2}{n} \sum_{i = 1}^{n} \sqrt{μ_{F} (x_{i}) ν_{F} (x_{i})} \\ = K_{0} (F) . □ \end{matrix}$ (21)

Remark 7. (I) Compared with Definition 3, (R1) and (R2) don’t obey the duality given by (add-equ-6) with (C1) and (C2), respectively. (II) (R4) and (R5) are identical with (C3) and (C4) in the sense of dual non-probabilistic entropy measure, respectively.

Theorem 2. Let $E = {< μ_{E} (x_{i}), ν_{E} (x_{i}), π_{E} (x_{i}) > | x_{i} \in X}$ and $F = {< μ_{F} (x_{i}), ν_{F} (x_{i}), π_{F} (x_{i}) > | x_{i} \in X}$ be intuitionistic fuzzy sets. We have (R1) $K_{1} (E) = 1$ if and only if ν_E (x_i) =1 for $x_{i} \in X$ ; (R2) $K_{1} (E) = 0$ if and only if ν_E (x_i) =0 or π_E (x_i) =0 for $x_{i} \in X$ ; (R3) $0 \leq K_{1} (E) \leq 1$ ; (R4) If E ≾ ₁F, i.e. E is less fuzzy than F, then $K_{1} (E) \geq K_{1} (F)$ .

Proof. (R1) Assume that $K_{1} (E) = 1$ . We have $\begin{matrix} K_{1} (E) & = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i})} = 1 . \end{matrix}$ (22) One can show that for $x_{i} \in X$ $0 \leq ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i}) \leq (μ_{E} (x_{i}) + ν_{E} (x_{i}))^{2} \leq 1 .$ (23) By (7) and (8), we have for $x_{i} \in X$ $ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i}) = ν_{E} (x_{i}) (ν_{E} (x_{i}) + μ_{E} (x_{i})) = 1 .$ (24) Again by 0 ≤ ν_E (x_i) ≤1 and 0 ≤ (μ_E (x_i) + ν_E (x_i)) ≤1, the unique solution of (9) is ν_E (x_i) =1 for $x_{i} \in X$ .

Suppose that ν_E (x_i) =1 for $x_{i} \in X$ . By Definition 8, it is easy to compute that $K_{1} (E) = 1$ .

(R2) By the definition of the intuitionistic fuzzy set, we know that 0 ≤ μ_E (x_i) ≤1 and 0 ≤ ν_E (x_i) ≤1. Then, it is easy to prove this conclusion. Here, we omit it.

(R3) By (8), we have $0 \leq K_{1} (E) = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i})} \leq 1 .$ (25) (R3) follows from (R1), (R2) and (10).

(R4) If E ≾ ₁F, then we have for $x_{i} \in X$ $ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i}) \geq ν_{F} (x_{i})^{2} + μ_{F} (x_{i}) ν_{F} (x_{i}) .$ (26) Then, $\begin{matrix} K_{1} (E) & = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{ν_{E} (x_{i})^{2} + μ_{E} (x_{i}) ν_{E} (x_{i})} \\ \geq \frac{1}{n} \sum_{i = 1}^{n} \sqrt{ν_{F} (x_{i})^{2} + μ_{F} (x_{i}) ν_{F} (x_{i})} \\ = K_{1} (F) . \end{matrix}$ (27)

□

Theorem 3. Let E = {< μ_E (x_i), ν_E (x_i), π_E (x_i) > $| x_{i} \in X}$ and F = {< μ_F (x_i), ν_F (x_i), π_F (x_i) > $| x_{i} \in X}$ be intuitionistic fuzzy sets. We have (R1) $K_{2} (E) = 1$ if and only if μ_E (x_i) =1 for $x_{i} \in X$ ; (R2) $K_{2} (E) = 0$ if and only if μ_E (x_i) =0 or π_E (x_i) =0 for $x_{i} \in X$ ; (R3) $0 \leq K_{2} (E) \leq 1$ ; (R4) If E ≾ ₂F, i.e. E is less fuzzy than F, then $K_{2} (E) \geq K_{2} (F)$ .

Proof. The proof of this theorem is similar to the proof of Theorem 2. □

Remark 8. $K_{0}, K_{1}$ and $K_{2}$ corresponded to the orders ≾₀, ≾₁ and ≾₂ are called Lorentzian-like knowledge measures.

Theorem 4. Let $E = {< μ_{E} (x_{i}), ν_{E} (x_{i}), π_{E} (x_{i}) > | x_{i} \in X}$ and $F = {< μ_{F} (x_{i}), ν_{F} (x_{i}), π_{F} (x_{i}) > | x_{i} \in X}$ be intuitionistic fuzzy sets. We have (R1) $K (E) = 1$ if and only if E is a crisp set; (R2) π_E (x_i) =1 if and only if $K (E) = 0$ ; (R3) $0 \leq K (E) \leq 1$ ; (R4) $K (E) = K (E)$ ; (R5) If E ≾ F, i.e. E is less fuzzy than F, then $K (E) \geq K (F)$ .

Proof. (R1) Assume that $K (E) = 1$ . We have $\begin{matrix} \frac{1}{\sqrt{2} n} \sum_{i = 1}^{n} {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ = 1, \end{matrix}$ (28) where Λ = {1, 2, 3}. Then, $\begin{matrix} \sum_{i = 1}^{n} {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ = \sqrt{2} n . \end{matrix}$ (29) It is easy to see that $\begin{matrix} 0 \leq {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ \leq \sqrt{2} . \end{matrix}$ (30) By (3) and (4), we obtain $\begin{matrix} {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ = \sqrt{2} . \end{matrix}$ (31) By (5) and 0 ≤ μ_i (x_i) ² ≤ 1, i = {1, 2, 3}, we get μ₁ (x_i) =1, μ₂ (x_i) =0, μ₃ (x_i) =1 or μ₁ (x_i) =0, μ₂ (x_i) =1, μ₃ (x_i) =1.

And suppose that E is a crisp set. We have μ_E (x_i) =1 or ν_E (x_i) =1 for $x_{i} \in X$ . It is clear that $K (E) = 1 .$

(R2) Assume that $π_{E} (x_{i}) = 1, x_{i} \in X$ . By Definition 14, one obtains $K (E) = 0$ .

If $K (E) = 0$ , then we have for $x_{i} \in X$ $max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}} = 0 .$ (32) (8) can be transformed into $max {2 μ_{E} (x_{i})^{2}, 2 ν_{E} (x_{i})^{2}} + 2 μ_{E} (x_{i}) ν_{E} (x_{i}) = 0 .$ (33) 0 ≤ μ_E (x_i) ≤1 and 0 ≤ ν_E (x_i) ≤1 imply that the unique solution of (9) is $μ_{E} (x_{i}) = μ_{E} (x_{i}) = 0 .$ Thus, π_E (x_i) =1.

(R3) Let $K_{i} (E) = max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}},$ (34) where Λ = {1, 2, 3}, μ₁ (x_i) = μ_E (x_i), μ₂ (x_i) = ν_E (x_i), μ₃ (x_i) =1 - π_E (x_i).

By Definition 1 and Definition 2, we have 0 ≤ μ_i ≤ 1, i ∈ {1, 2, 3}. We can obtain $0 \leq K_{i} (E) \leq 2$ . By (1), we have $0 \leq K (E) \leq 1 .$

(R4) It is easy to prove the result. Here, we omit it.

(R5) By Definition 14, (1) can be rewritten as $\begin{matrix} K (E) \\ = \frac{1}{n} \sum_{i = 1}^{n} {max {μ_{E} (x_{i}), ν_{E} (x_{i})}^{2} + μ_{E} (x_{i}) ν_{E} (x_{i})}^{\frac{1}{2}} . \end{matrix}$ (35) By Definition 10 and (7), for $x_{i} \in X$ we have $\begin{matrix} E ≾ F \Leftrightarrow \\ max {μ_{E} (x_{i}), ν_{E} (x_{i})}^{2} + μ_{E} (x_{i}) ν_{E} (x_{i}) \\ \geq max {μ_{F} (x_{i}), ν_{F} (x_{i})}^{2} + μ_{F} (x_{i}) ν_{F} (x_{i}) . \end{matrix}$ (36) Then, $\begin{matrix} K (E) \\ = \frac{1}{\sqrt{2} n} \sum_{i = 1}^{n} {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} μ_{j} (x_{i})^{2}) - μ_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ = \frac{1}{n} \sum_{i = 1}^{n} {max {μ_{E} (x_{i})^{2}, ν_{E} (x_{i})^{2}} + μ_{E} (x_{i}) ν_{E} (x_{i})}^{\frac{1}{2}} \\ \geq \frac{1}{n} \sum_{i = 1}^{n} {max {μ_{F} (x_{i})^{2}, ν_{F} (x_{i})^{2}} + μ_{F} (x_{i}) ν_{F} (x_{i})}^{\frac{1}{2}} \\ = \frac{1}{\sqrt{2} n} \sum_{i = 1}^{n} {max_{k \in Λ} {(\sum_{j \neq k, j \in Λ} ν_{j} (x_{i})^{2}) - ν_{k} (x_{i})^{2}}}^{\frac{1}{2}} \\ = K (F), \end{matrix}$ (37) where Λ = {1, 2, 3}, μ₁ (x_i) = μ_E (x_i), μ₂ (x_i) = ν_E (x_i), μ₃ (x_i) =1 - π_E (x_i) = μ_E (x_i) + ν_E (x_i) and ν₁ (x_i) = μ_F (x_i), ν₂ (x_i) = ν_F (x_i), ν₃ (x_i) = μ_F (x_i) + ν_F (x_i). □

Example 2. If we replace ≾ by ⪯, (R5) is not true. In fact, there exist some different knowledge measures based on ⪯ such as the knowledge measures in [25] and [11]. We see that Guo [11] gave a new definition of knowledge measure by modifying the condition (KP_AIFS3). There has a conclusion in [11], which shows that if E ⪯ F, one can obtain K (E) ≥ K (F). Reversely, if K (E) ≥ K (F), E ⪯ F is not true. Then, the condition KP_AIFS3 is a sufficient condition in the sense of the order ⪯. Nguyen in [25] presented a novel knowledge measure based normalized Euclidean distance, and got a conclusion in [25] about a sufficient and necessary condition (A1.5) of K (E) ≥ K (F). But A1.5 is not well-defined. There has a counterexample. Let E ⪯ F for two intuitionistic fuzzy sets E, F. By Definition 1 in [25] we have $\begin{matrix} K (E) = \frac{1}{n \sqrt{2}} \\ \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{i})^{2} + ν_{E} (x_{i})^{2} + (μ_{E} (x_{i}) + ν_{E} (x_{i}))^{2}}, \end{matrix}$ (38) and $\begin{matrix} K (F) = \frac{1}{n \sqrt{2}} \\ \sum_{i = 1}^{n} \sqrt{μ_{F} (x_{i})^{2} + ν_{F} (x_{i})^{2} + (μ_{F} (x_{i}) + ν_{F} (x_{i}))^{2}} . \end{matrix}$ (39) By E ⪯ F, for $x_{i} \in X$ we can obtain the following results:

If μ_F (x_i) ≤ ν_F (x_i) then μ_E (x_i) ≤ μ_F (x_i) and ν_E (x_i) ≥ ν_F (x_i);

If μ_F (x_i) ≥ ν_F (x_i) then μ_E (x_i) ≥ μ_F (x_i) and ν_E (x_i) ≤ ν_F (x_i).

For case (i), we cannot obtain the result K (E) ≥ K (F), because μ_E (x_i) ² + ν_E (x_i) ² + (μ_E (x_i) + ν_E (x_i)) ² is not always larger than μ_F (x_i) ² + ν_F (x_i) ² + (μ_F (x_i) + ν_F (x_i)) ². It is easy to construct a counterexample such as for $X = {x_{0}}, E = < x_{0}, 0.2, 0.65, 0.15 >,$ F = < x₀, 0.3, 0.6, 0.1>. By the definition of the order ⪯, one can verify that E ⪯ F. By (10) and (11), we get K (E) =0.770 < K (F) =0.794. One can construct some similar counterexamples for case (ii). In fact, if the result A1.5 is true, some additional conditions should be given. And we give the following order.

Definition 15. Let $E, F \in 𝔽 (X)$ be two intuitionistic fuzzy sets. We call E ≾ _eF, if and only if for $a \in X$ , $\begin{matrix} μ_{E} (a)^{2} + ν_{E} (a)^{2} + μ_{E} (a) ν_{E} (a) \\ \geq μ_{F} (a)^{2} + ν_{F} (a)^{2} + μ_{F} (a) ν_{F} (a) . \end{matrix}$ (40) Then, A1.5 will turn into the following statement: A1.5 If E ≾ _eF, i.e. E is less fuzzy than F, then $K (E) \geq K (F)$ , where

\begin{matrix} K (E) = \frac{1}{n \sqrt{2}} \\ \sum_{i = 1}^{n} \sqrt{μ_{E} (x_{i})^{2} + ν_{E} (x_{i})^{2} + (μ_{E} (x_{i}) + ν_{E} (x_{i}))^{2}} . \end{matrix}

(41)

For the above example, we see that F ≾ _eE and K (E) < K (F) satisfying the dual entropy measure.

Remark 9. Compared with other knowledges, we notice that almost all of them are in the frame of the axiomatic of entropy measure defined by Szmidt and Kacprzyk. It is easy to see that the notion of order ⪯ can only represent the intuitionistic fuzzy sets all fall in the triangle area surrounded by the point (1, 0, 0), the point (0, 1, 0) and the point (0, 0, 1), and the knowledge measurement only work in this case. But the constructed Lorentzian-like knowledge measures can be used any two intuitionistic fuzzy sets with the order ≾. It is also a significative extension in theory that is given a flexible way to define a knowledge measure in a new axiomatic frame with out the order ⪯. Then, one can choose the proper order and knowledge measure for the special dataset or application.

4. Numerical Examples

Example 3. Let $X = {x_{n}, i \in {1, 2, . . ., 5}}$ and $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X}$ be an intuitionistic fuzzy set. Assume that <x₁, 0.1, 0.8>, <x₂, 0.3, 0.5>, <x₃, 0.6, 0.2>, <x₄, 0.9, 0.0> and <x₅, 1.0, 0.0>.

Using the method proposed by Kumar et al. [33], one can construct the following intuitionistic fuzzy sets: For $n \in ℝ^{+}$ , $E^{n} = {< x_{0}, μ_{E} (x_{0})^{n}, 1 - (1 - ν_{E} (x_{0}))^{n} > | x_{0} \in X} .$ E^0.5 means that if E is LARGE, then E^0.5 is MORE or LESS LARGE. It is clear that for n > m one can get Eⁿ ⪯ E^m. By the theory proposed by Hung and Yang [14], one can get the following the result: For n > m, n, we have $Entropy (E^{n}) < Entropy (E^{m}) .$ (42) Let n ∈ {0.5, 1, 2, 3, 4}. We will verify the equation (12) with the constructed knowledge measures. The results are shown in Table 1. As can be seen form Table 1, we have $K_{0} (E^{0.5}) < K_{0} (E) < K_{0} (E^{2}) < K_{0} (E^{3}) < K_{0} (E^{4})$ and $K_{2} (E^{0.5}) < K_{2} (E) < K_{2} (E^{2}) < K_{2} (E^{3}) < K_{2} (E^{4}) .$ They both satisfies the equation (12) in the sense of dual entropy. It is completely consistent with the Theorem 1. From Table 1, one can see that knowledge measure $K_{gu}$ , $K_{1}$ and Entropy E_bb obtain the opposite results. Entropy E_sz, E_li, E_mc and E_ye satisfies the equation (12), and others do not. In a sense, it implies that the definitions of entropy and knowledge are dependent on the definition of the order. And this example shows the actions of the different knowledge measures on the dataset with ⪯.

Table 1

The comparison results for different entropy or knowledge measures

$𝔽 (X)$	E_mc [43]	E_li[42]	E_ye[16]	E_sz[32]	E_bb [34]	K_ng [25]	$K_{gu}$ [11]	$K$	$K_{0}$	$K_{1}$	$K_{2}$
E ^0.5	0.417	0.310	0.501	0.319	0.460	0.857	0.785	0.830	0.442	0.299	0.796
E	0.375	0.294	0.494	0.307	0.600	0.835	0.788	0.815	0.407	0.376	0.677
E ²	0.315	0.293	0.395	0.301	0.660	0.840	0.805	0.816	0.287	0.454	0.539
E ³	0.196	0.189	0.333	0.212	0.672	0.847	0.854	0.839	0.204	0.493	0.461
E ⁴	0.121	0.154	0.294	0.176	0.680	0.852	0.877	0.850	0.145	0.518	0.410

Example 4. Let $X = {x_{n}, i \in {1, 2, . . ., 5}}$ and $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X}$ be an intuitionistic fuzzy set. Assume that <x₁, 0.1, 0.8>, <x₂, 0.3, 0.5>, <x₃, 0.5, 0.4>, <x₄, 0.9, 0.0> and <x₅, 1.0, 0.0>. Form Table 2, we have

Table 2

The comparison results for different entropy or knowledge measures

$𝔽 (X)$	E_mc [43]	E_li[42]	E_ye[16]	E_sz[32]	E_bb [34]	K_ng [25]	$K_{gu}$ [11]	$K$	$K_{0}$	$K_{1}$	$K_{2}$
E ^0.5	0.455	0.341	0.545	0.344	0.409	0.859	0.767	0.827	0.487	0.330	0.793
E	0.444	0.351	0.525	0.374	0.500	0.847	0.761	0.810	0.447	0.416	0.672
E ²	0.259	0.192	0.363	0.197	0.490	0.874	0.865	0.865	0.303	0.503	0.531
E ³	0.133	0.124	0.259	0.131	0.467	0.893	0.911	0.891	0.128	0.545	0.450
E ⁴	0.066	0.098	0.205	0.109	0.468	0.900	0.926	0.899	0.145	0.568	0.397

$K_{0} (E^{0.5}) > K_{0} (E) > K_{0} (E^{2}) > K_{0} (E^{3}) < K_{0} (E^{4}),$

$K_{1} (E^{0.5}) < K_{1} (E) < K_{1} (E^{2}) < K_{1} (E^{3}) < K_{1} (E^{4}),$

$K_{2} (E^{0.5}) > K_{2} (E) > K_{2} (E^{2}) > K_{2} (E^{3}) > K_{2} (E^{4})$ and $K (E) < K (E^{0.5}) < K (E^{2}) < K (E^{3}) < K (E^{4}) .$ Knowledge measure $K_{0}$ , E_mc, E_ye and $K_{2}$ obtain the right results in the sense of dual entropy, and others do not. It is easy to see that not all the knowledge and entropy work well with the order ⪯.

Example 5. Let $X = {x_{n}, i \in {1, 2, . . ., 5}}$ and $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X}$ be an intuitionistic fuzzy set. Assume that $\begin{matrix} E_{1} = & < x_{1}, 0.2955, 0.1169 >, < x_{2}, 0.8432, 0.1428 >, \\ < x_{3}, 0.7986, 0.1601 >, < x_{4}, 0.4155, 0.1474 >, \\ < x_{5}, 0.8052, 0.0876 >, \\ E_{2} = & < x_{1}, 0.3287, 0.0053 >, < x_{2}, 0.4609, 0.0859 >, \\ < x_{3}, 0.7738, 0.0275 >, < x_{4}, 0.8690, 0.0587 >, \\ < x_{5}, 0.5148, 0.0014 >, \\ E_{3} = & < x_{1}, 0.5881, 0.0003 >, < x_{2}, 0.2266, 0.1477 >, \\ < x_{3}, 0.7519, 0.0168 >, < x_{4}, 0.7918, 0.0112 >, \\ < x_{5}, 0.5343, 0.0009 >, \\ E_{4} = & < x_{1}, 0.1119, 0.0016 >, < x_{2}, 0.6879, 0.0420 >, \\ < x_{3}, 0.1949, 0.0327 >, < x_{4}, 0.1514, 0.0297 >, \\ < x_{5}, 0.8508, 0.0005 >, \\ E_{5} = & < x_{1}, 0.9296, 0.0001 >, < x_{2}, 0.5864, 0.0150 >, \\ < x_{3}, 0.8792, 0.0013 >, < x_{4}, 0.0006, 0.0870 >, \\ < x_{5}, 0.6126, 0.0001 > . \end{matrix}$

It is easy to verify that $E_{1} ⪯_{0} E_{2} ⪯_{0} E_{3} ⪯_{0} E_{4} ⪯_{0} E_{5} .$ (43)

Table 3

The comparison results for different entropy or knowledge measures

$𝔽 (X)$	E_mc [43]	E_li[42]	E_ye[16]	E_sz[32]	E_bb [34]	K_ng [25]	$K_{gu}$ [11]	$K$	$K_{0}$	$K_{1}$	$K_{2}$
E ₁	0.448	0.409	0.688	0.424	1.187	0.708	0.665	0.694	0.561	0.311	0.694
E ₂	0.258	0.362	0.646	0.425	1.874	0.609	0.674	0.607	0.256	0.134	0.607
E ₃	0.249	0.378	0.633	0.450	1.930	0.602	0.665	0.594	0.170	0.096	0.594
E ₄	0.111	0.522	0.756	0.614	2.897	0.411	0.462	0.410	0.141	0.074	0.410
E ₅	0.027	0.319	0.523	0.382	1.888	0.621	0.690	0.621	0.061	0.047	0.605

As shown in Table 5, one can get $K_{0} (E_{1}) > K_{0} (E_{2}) > K_{0} (E_{3}) > K_{0} (E_{4}) > K_{0} (E_{5}),$ $K_{1} (E_{1}) > K_{1} (E_{2}) > K_{1} (E_{3}) > K_{1} (E_{4}) > K_{1} (E_{5}) .$ Thus, $K_{0}$ satisfies Theorem 1 and the definition of the knowledge measure in the sense of ≾₀. We also notice that $K_{1}$ and E_mc works well on this intuitionistic fuzzy sets, since the knowledge measure $K_{1}$ is mainly affected by the degree of membership μ_E on these intuitionistic fuzzy sets. Moreover, the other knowledge measures and entropy cannot obtain the effective results for this dataset in the sense of ≾₀.

Example 6. Let $X = {x_{n}, i \in {1, 2, . . ., 5}}$ and $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X}$ be an intuitionistic fuzzy set. Assume that $\begin{matrix} E_{1} = & < x_{1}, 0.3442, 0.2669 >, < x_{2}, 0.2242, 0.3060 >, \\ < x_{3}, 0.4822, 0.1012 >, < x_{4}, 0.2777, 0.4279 >, \\ < x_{5}, 0.7064, 0.0354 >, \\ E_{2} = & < x_{1}, 0.7351, 0.0760 >, < x_{2}, 0.6075, 0.0057 >, \\ < x_{3}, 0.8696, 0.0229 >, < x_{4}, 0.7908, 0.0223 >, \\ < x_{5}, 0.5279, 0.0321 >, \\ E_{3} = & < x_{1}, 0.4734, 0.0353 >, < x_{2}, 0.6412, 0.0046 >, \\ < x_{3}, 0.9025, 0.0097 >, < x_{4}, 0.5073, 0.0057 >, \\ < x_{5}, 0.8188, 0.0054 >, \\ E_{4} = & < x_{1}, 0.5325, 0.0259 >, < x_{2}, 0.8103, 0.0005 >, \\ < x_{3}, 0.8561, 0.0062 >, < x_{4}, 0.7125, 0.0017 >, \\ < x_{5}, 0.8773, 0.0039 >, \\ E_{5} = & < x_{1}, 0.8631, 0.0023 >, < x_{2}, 0.5862, 0.0005 >, \\ < x_{3}, 0.7779, 0.0003 >, < x_{4}, 0.5724, 0.0010 >, \\ < x_{5}, 0.5247, 0.0019 > . \end{matrix}$

By Definition 8, it is clear that $E_{1} ⪯_{1} E_{2} ⪯_{1} E_{3} ⪯_{1} E_{4} ⪯_{1} E_{5} .$

Table 4

The comparison results for different entropy or knowledge measures

$𝔽 (X)$	E_mc [43]	E_li[42]	E_ye[16]	E_sz[32]	E_bb [34]	K_ng [25]	$K_{gu}$ [11]	$K$	$K_{0}$	$K_{1}$	$K_{2}$
E ₁	0.562	0.614	0.870	0.692	1.828	0.574	0.498	0.533	0.516	0.352	0.500
E ₂	0.231	0.262	0.518	0.303	1.310	0.723	0.787	0.722	0.280	0.144	0.722
E ₃	0.069	0.277	0.528	0.337	1.596	0.675	0.759	0.675	0.159	0.080	0.675
E ₄	0.042	0.201	0.409	0.245	1.173	0.762	0.838	0.762	0.122	0.062	0.762
E ₅	0.011	0.270	0.531	0.336	1.670	0.666	0.767	0.666	0.053	0.027	0.666

As shown in Table 6, one can get $K_{0} (E_{1}) > K_{0} (E_{2}) > K_{0} (E_{3}) > K_{0} (E_{4}) > K_{0} (E_{5}),$ $K_{1} (E_{1}) > K_{1} (E_{2}) > K_{1} (E_{3}) > K_{1} (E_{4}) > K_{1} (E_{5}) .$ Thus, $K_{1}$ satisfies Theorem 2 and the definition of the knowledge measure in the sense of ≾₁. We also notice that $K_{0}$ and E_mv both work well on this intuitionistic fuzzy sets. Moreover, the other knowledge measures and entropy cannot obtain the effective results in the sense of ≾₁.

Example 7. Let $X = {x_{n}, i \in {1, 2, . . ., 5}}$ and $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X}$ be an intuitionistic fuzzy set. Assume that $\begin{matrix} E_{1} = & < x_{1}, 0.1788, 0.3784 >, < x_{2}, 0.2622, 0.3889 >, \\ < x_{3}, 0.0112, 0.5288 >, < x_{4}, 0.3245, 0.6101 >, \\ < x_{5}, 0.0616, 0.5409 >, \\ E_{2} = & < x_{1}, 0.0368, 0.7694 >, < x_{2}, 0.0369, 0.7333 >, \\ < x_{3}, 0.0032, 0.7423 >, < x_{4}, 0.0322, 0.9404 >, \\ < x_{5}, 0.0486, 0.6506 >, \\ E_{3} = & < x_{1}, 0.0306, 0.6392 >, < x_{2}, 0.0289, 0.5293 >, \\ < x_{3}, 0.0015, 0.9242 >, < x_{4}, 0.0503, 0.4510 >, \\ < x_{5}, 0.0291, 0.6602 >, \\ E_{4} = & < x_{1}, 0.0213, 0.6764 >, < x_{2}, 0.0198, 0.6106 >, \\ < x_{3}, 0.0002, 0.7913 >, < x_{4}, 0.0240, 0.6118 >, \\ < x_{5}, 0.0126, 0.5279 >, \\ E_{5} = & < x_{1}, 0.0178, 0.5399 >, < x_{2}, 0.0151, 0.5249 >, \\ < x_{3}, 0.0001, 0.7935 >, < x_{4}, 0.0072, 0.9613 >, \\ < x_{5}, 0.0065, 0.7492 > . \end{matrix}$

By Definition 9, it is easy to verify that

E_{1} ⪯_{2} E_{2} ⪯_{2} E_{3} ⪯_{2} E_{4} ⪯_{2} E_{5} .

Table 5

The comparison results for different entropy or knowledge measures

$𝔽 (X)$	E_mc [43]	E_li[42]	E_ye[16]	E_sz[32]	E_bb [34]	K_ng [25]	$K_{gu}$ [11]	$K$	$K_{0}$	$K_{1}$	$K_{2}$
E ₁	0.609	0.560	0.868	0.626	1.715	0.598	0.547	0.565	0.514	0.565	0.310
E ₂	0.195	0.212	0.437	0.241	1.006	0.784	0.837	0.783	0.293	0.783	0.150
E ₃	0.175	0.312	0.584	0.372	1.656	0.656	0.729	0.655	0.236	0.655	0.122
E ₄	0.129	0.299	0.585	0.363	1.704	0.652	0.747	0.651	0.178	0.651	0.090
E ₅	0.059	0.238	0.465	0.290	1.385	0.719	0.798	0.718	0.139	0.718	0.070

As shown in Table 7, one can obtain

\begin{matrix} K_{0} (E_{1}) > K_{0} (E_{2}) > K_{0} (E_{3}) > K_{0} (E_{4}) > K_{0} (E_{5}), \\ K_{2} (E_{1}) > K_{2} (E_{2}) > K_{2} (E_{3}) > K_{2} (E_{4}) > K_{2} (E_{5}) . \end{matrix}

Thus, the knowledge measure

K_{2}

satisfies Theorem 3 and the definition of the knowledge measure in the sense of ≾₂. We also notice that

K_{0}

and E_mc both work well on these intuitionistic fuzzy sets. Moreover, the other knowledge measures and entropy cannot obtain the effective results in the sense of ≾₂.

Example 8. Let $X = {x_{n}, i \in {1, 2, . . ., 5}}$ and $E = {< x_{0}, μ_{E} (x_{0}), ν_{E} (x_{0}) > | x_{0} \in X}$ be an intuitionistic fuzzy set. Assume that $\begin{matrix} E_{1} = & < x_{1}, 0.3412, 0.5346 >, < x_{2}, 0.4530, 0.3062 >, \\ < x_{3}, 0.4716, 0.0949 >, < x_{4}, 0.3668, 0.2406 >, \\ < x_{5}, 0.1992, 0.4050 >, \\ E_{2} = & < x_{1}, 0.6023, 0.1438 >, < x_{2}, 0.4546, 0.0742 >, \\ < x_{3}, 0.2155, 0.3547 >, < x_{4}, 0.1813, 0.3357 >, \\ < x_{5}, 0.2772, 0.2948 >, \\ E_{3} = & < x_{1}, 0.4729, 0.3949 >, < x_{2}, 0.2426, 0.2584 >, \\ < x_{3}, 0.2848, 0.2655 >, < x_{4}, 0.2094, 0.3048 >, \\ < x_{5}, 0.2971, 0.2360 >, \\ E_{4} = & < x_{1}, 0.2247, 0.4054 >, < x_{2}, 0.2510, 0.2494 >, \\ < x_{3}, 0.2607, 0.2437 >, < x_{4}, 0.2852, 0.2378 >, \\ < x_{5}, 0.2895, 0.2146 >, \\ E_{5} = & < x_{1}, 0.4272, 0.1088 >, < x_{2}, 0.2510, 0.2493 >, \\ < x_{3}, 0.2567, 0.2455 >, < x_{4}, 0.2703, 0.2457 >, \\ < x_{5}, 0.2634, 0.2651 > . \end{matrix}$

By Definition 10, it is easy to verify that $E_{1} ⪯ E_{2} ⪯ E_{3} ⪯ E_{4} ⪯ E_{5} .$ (44)

Table 6

The comparison results for different entropy or knowledge measures

$𝔽 (X)$	E_mc [43]	E_li[42]	E_ye[16]	E_sz[32]	E_bb [34]	K_ng [25]	$K_{gu}$ [11]	$K$	$K_{0}$	$K_{1}$	$K_{2}$
E ₁	0.745	0.659	0.946	0.731	1.587	0.603	0.482	0.551	0.637	0.455	0.494
E ₂	0.752	0.655	0.917	0.733	2.066	0.526	0.452	0.488	0.515	0.361	0.443
E ₃	0.818	0.838	0.996	0.924	2.034	0.515	0.334	0.438	0.590	0.416	0.422
E ₄	0.802	0.837	0.991	0.917	2.338	0.463	0.312	0.398	0.526	0.378	0.372
E ₅	0.808	0.855	0.979	0.918	2.417	0.453	0.311	0.388	0.496	0.335	0.388

As shown in Table 8, we have $K_{ng} (E_{1}) > K_{ng} (E_{2}) > K_{ng} (E_{3}) > K_{ng} (E_{4}) > K_{ng} (E_{5}),$ $K_{gu} (E_{1}) > K_{gu} (E_{2}) > K_{gu} (E_{3}) > K_{gu} (E_{4}) > K_{gu} (E_{5}),$ and $K (E_{1}) > K (E_{2}) > K (E_{3}) > K (E_{4}) > K (E_{5}) .$ Thus, the knowledge measure $K$ satisfies Theorem 1 and the definition of the knowledge measure in the sense of ≾. We also notice that $K_{ng}$ and $K_{gu}$ work well on this intuitionistic fuzzy sets. Moreover, the other knowledge measures and entropy cannot obtain the effective results in the sense of ≾.

5. Conclusion

The order plays an important role in the definition of the entropy and knowledge measure of intuitionistic fuzzy sets in the frame of the axiom of entropy measure for intuitionistic fuzzy set defined by Szmidt and Kacprzyk. It is easy to see that there exists some counterexamples for some knowledge measures. In order to reduce counterexamples, we give some modified orders which extend the definition of the classical order ⪯. And the numerical experiments show that the new orders are more general than the order ⪯ in a sense, such as the order ≾ can break the constraint that the two intuitionistic fuzzy sets with the order ⪯ must fall in the either side of the line y = x. In the remarks, we show that these binary relations are not strict partially ordered relations. And the problem, constructing a universal partial order, is still open.

With the above notations, we then present four linear Lorentzian-like knowledge measures of intuitionistic fuzzy sets. The constructed knowledge measures are different from the Euclidean-like knowledge measures based on normalized Euclidean distance or normalized Hamming distance, in which the triangle inequality of Lorentzian distance does not hold. Some numerical experiments results show that the new knowledge measures work well on different datasets, and they have the ability to recognize the different patterns.

At the end of the paper, we will point out also some shortcomings on these constructed Lorentzian knowledge measures, like: why the hesitation of intuitionistic fuzzy sets becomes now the counter-part of "time dimension" in physics? This may, in a sense, be seen as Lorentzian knowledge measure for hesitation margin of an intuitionistic fuzzy set. Then, $K_{1}$ and $K_{2}$ can be seen as Lorentzian knowledge measures for the degree of membership and the degree of non-membership. Each of them can be used to measure the biases on the degree of membership, the degree of non-membership and hesitation margin in several scenarios, such as purchase intention. Moreover, we may define a total Lorentzian knowledge measure as $K_{T} = W K^{T},$ where W = [w₁, w₂, w₃] is the weight of the degree of membership, the degree of non-membership and hesitation margin and $K = [K_{0}, K_{1}, K_{2}]$ .

Competing interests

The authors declare that they have no competing interests.

Footnotes

Acknowledgment

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References

L.A.

Zadeh , Fuzzy sets, Information and Control 8 (1965), 338–358.

J.J.

Buckley , Fuzzy complex numbers, Fuzzy Sets and Systems 33 (1989), 333–345.

Bede and

S.G.

Gal , Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005), 581–599.

Ramot ,

Milo ,

Friedman and

Kandel , Complex fuzzy sets, IEEE Transactions on Fuzzy Systems 10 (2002), 171–186.

Atanassov , Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov , More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989), 37–46.

Atanassov , New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems 61 (1994), 137–142.

Atanassov , The most general form of one type of intu-itionistic fuzzy modal operators, Notes on Intuitionistic Fuzzy Sets 12 (2006), 36–38.

Atanassov , Norms and metrics over intuitionistic fuzzy sets, Busefal 55 (1993), 11–20.

10.

Szmidt , Distances and Similarities in Intuitionistic Fuzzy Sets, Springer International Publishing, 2014.

11.

H.H.

Guo , Knowledge measure for Atanassov's intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systems 24 (2016), 1072–1078.

12.

X.C.

Liu , Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems 52 (1992), 305–318.

13.

H.M.

Zhang and

L.Y.

Yu , New distance measures between intuitionistic fuzzy sets and interval-valued fuzzy sets, Information Sciences 245 (2013), 181–196.

14.

W.L.

Hung and

M.S.

Yang , Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent Systems 21 (2006), 443–451.

15.

Y.F.

Song ,

X.D.

Wang and

H.L.

Zhang , A distance measure between intuitionistic fuzzy belief functions, Knowledge-Based Systems 86 (2015), 288–298.

16.

Ye , Two effective measures of intuitionistic fuzzy entropy, Computing 87 (2010), 55–62.

17.

Dügenci , A new distance measure for interval valued intuitionistic fuzzy sets and its application to group decision making problems with incomplete weights information, Applied Soft Computing 41 (2016), 120–134.

18.

Muthukumar and

G.S.

Krishnan , A similarity measure of intuitionistic fuzzy soft sets and its application in medical diagnosis, Applied Soft Computing 41 (2016), 148–156.

19.

T.D.

Pham , The Kolmogorov-Sinai entropy in the setting of fuzzy sets for image texture analysis and classification, Pattern Recognition 53 (2016), 229–237.

20.

F.Y.

Meng and

X.H.

Chen , Entropy and similarity measure for Atannasov's interval-valued intuitionistic fuzzy sets and their application, Fuzzy Optimization and Decision Making 15 (2016), 75–101.

21.

F.Y.

Meng and

X.H.

Chen , Entropy and similarity measure of Atannasov's intuitionistic fuzzy sets and their application to pattern recognition based on fuzzy measures, Pattern Analysis and Applications 19 (2016), 11–20.

22.

G.A.

Papakostas ,

A.G.

Hatzimichailidis and

V.G.

Kaburla-sos , Distance and similarity measures between intuitionistic fuzzy sets, A comparative analysis form a pattern recognition point of view, Pattern Recognition Letters 34 (2013), 1609–1622.

23.

Quiros ,

Alonso ,

Bustince ,

Diaz and

Montes , An entropy measure definition for finite interval-valued hesitant fuzzy sets, Knowledge-Based Systems 84 (2015), 121–133.

24.

Han ,

Z.P.

Yang ,

Sun and

G.L.

Xu , Chordal distance and non-Archimedean chordal distance between Atanassov's intuitionistic fuzzy set, Journal of Intelligent & Fuzzy Systems 33 (2017), 3889–3894.

25.

Nguyen , A novel similarity/dissimilarity measure for intuitionistic fuzzy sets and its application in pattern recognition, Expert Systems With Applications 45 (2016), 97–107.

26.

T.Z.

Xu ,

Z.P.

Yang and

J.M.

Rassias , Direct and fixed point approaches to the stability of an AQ-functional equation in non-Archimedean normed spaces, Journal of Computational Analysis and Applications 17 (2014), 697–706.

27.

Szmidt and

Kacprzyk , Distance between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000), 505–518.

28.

Ichiki and

Nishimura , Recognizable classification of Lorentzian distance-squared mappings, Journal of Geometry and Physics 81 (2014), 62–71.

29.

Kerimbekov ,

H.S.

Bilge and

H.H.

Ugurlu , The use of Lorentzian distance metric in classification problems, Pattern Recognition Letters 84 (2016), 170–176.

30.

T.Z.

Xu and

Z.P.

Yang , A fixed point approach to the stability of functional equations on noncommutative spaces, Results in Mathematics 72 (2017), 1639–1651.

31.

L.G.

Huang and

Zhang , Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathe-matical Analysis and Applications 33 (2007), 1468–1476.

32.

Szmidt and

Kacprzyk , Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems 118 (2001), 467–477.

33.

Kumar ,

Biswas and

A.R.

Roy , Some operations on intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000), 477–484.

34.

Burillo and

Bustince , Entropy on intuitionistic fuzzy sets, Fuzzy Sets and Systems 78 (1996), 305–316.

35.

Garg and

Kumar , Distance measures for connection number sets based on set pair analysis and its applications to decision-making process, Applied Intelligence. 10.1007/s10489-018-1152-z

36.

Garg

Nancy , Some new biparametric distance measures on single-valued neutrosophic sets with applications to pattern pecognition and medical diagnosis, Information 8 (2017), 162. doi: 10.3390/info8040162

37.

Singh and

Garg , Distance measures between type-2 intuitionistic fuzzy sets and their application to multicrite-ria decisionmaking process, Applied Intelligence 46 (2017), 788–799.

38.

Garg and

Kumar , An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making, Soft Computing 22 (2018), 4959–4970.

39.

Rani and

Garg , Distance measures between the complex intuitionistic fuzzy sets and its applications to the decision-making process, International Journal for Uncertainty Quantification 7 (2017), 423–439.

40.

Garg and

Arora , Distance and similarity measures for dual hesitant fuzzy soft sets and their applications in multi-criteria decision making problem, International Journal for Uncertainty Quantification 7 (2017), 229–248.

41.

Garg , Distance and similarity measures for intuitionistic multiplicative preference relation and its applications, International Journal for Uncertainty Quantification 7 (2017), 117–133.

42.

J.Q.

Li ,

G.N.

Deng ,

H.X.

Li and

W.Y.

Zeng , The relationship between similarity measure and entropy of intuitionistic fuzzy sets, Information Sciences 188 (2012), 314–321.

43.

X.S.

Fan ,

C.H.

Li and

Wang , Strict intuitionistic fuzzy entropy and application in network vulnerability evaluation, Soft Computing. 10.1007/s00500-018-3474-5