Novel distance and similarity measures on hesitant fuzzy linguistic term sets with application to pattern recognition

Abstract

As two important features of hesitant fuzzy linguistic term sets (HFLTSs), distance and similarity measures have been applied widely in many fields such as pattern recognition, decision making and prediction. Through analyzing the existing distance and similarity measures on HFLTSs, we find that they are not reasonable in some cases. Therefore, we first define the hesitance degree on HFLTSs to reflect the hesitant degree among several linguistic terms. On the basis of hesitance degree on HFLTSs, we develop several novel distance measures and further discuss their properties. Afterwards, several similarity measures based on hesitance degree are proposed and applied to pattern recognition. By comparing our novel proposed distance and similarity measures with the existing methods and giving an example of pattern recognition, we prove that our proposed distance and similarity measures are more reliable than the previous method in some cases.

Keywords

Hesitant fuzzy linguistic term sets distance measure similarity measure hesitance degree pattern recognition

1 Introduction

Distance measure reflects the difference degree between two elements or two sets, while similarity measure reflects the similar degree between two elements or two sets. As two important features of fuzzy set theory, distance and similarity measures have been widely used in pattern recognition [37], decision making [1 , 6– 10], production scheduling [38 , 41– 44], home health care routing [45, 46], etc. Hyung [28] firstly proposed the concept of similarity measures on fuzzy sets (FSs). Then, Liu [39] and Huang [40] defined distance and similarity distance on FSs. Subsequently, distance and similarity measures have been extended to interval-valued fuzzy sets (IVFSs) [29], interval type-2 fuzzy sets (IT2FSs) [30], intuitionistic fuzzy sets (IFSs) [31, 32] and hesitant fuzzy sets (HFSs) [33 –37]. For instance, Zeng and Guo [29] developed normalized distance and similarity measures of IVFSs. Wu and Mendel [30] proposed similarity and uncertainty measures for IT2FSs. Hung and Yang [31] investigated a new Hausdorff distance for IFSs. Ye [32] extended the concept of cosine similarity measure to IFSs. Xia and Xu [33, 34] investigated distance and similarity measures in both discrete and continuous cases for HFSs. Based on the hesitance degree, Li and Zeng [35 –37] investigated several novel distance and similarity measures for HFSs.

Recently, another interesting extension of FSs is hesitant fuzzy linguistic term sets (HFLTSs). In the qualitative decision making process, when evaluating features, attributes or variables, the decision makers might hesitate among several possible linguistic terms. To deal with such issues, Rodriguez et al. [1] proposed the concept of HFLTSs, which permitted that the membership can have a set of several possible linguistic terms. Rodriguez’s study motivated researchers to devote themselves to the topics related to HFLTSs. Majority of the researchers have recently focused on aggregation operator [2 –7], measure [8 –14], preference relationship [15 –19] and multi-criteria decision making (MCDM) [20 –27]. For example, a series of aggregation operators has been extended to HFLTSs, such as weighted averaging (WA) operator and ordered weighted averaging (OWA) operator by Wei et al. [2], WA operator and weighted geometric (WG) operator by Zhang and Qi [3], uncertain hybrid aggregation operator by Zhang et al. [4], weighted arithmetic averaging (WAA) operator and weighted geometric averaging (WGA) operator by Lee and Chen [5], generalized Maclaurin symmetric mean (MSM) operator and generalized geometric MSM operator by Liu and Gao [6], and Hamacher triangle norms based aggregation operator by Zhu and Li [7]. The measure research on HFLTSs included score, distance, similarity and correlation coefficients. Wei et al. [22] investigated score measure on HFLTSs. Liao and Xu [8, 9] investigated some distance measures and similarity measures from algebra forms and geometric forms between two HFLTSs. Hesamian and Shams [11] proposed two similarity measures for HFLTSs. Huang and Yang [12] investigated the pairwise comparison matrix based distance measure method. Meng and Chen [13] investigated generalized distance measures on HFLTSs. Liu et al. [14] combined the Euclidean distance and cosine similarity measure and developed a new similarity measure. Liao et al. [10] developed a series of correlation coefficients and weighted correlation coefficients of HFLTSs. Zhu and Xu [15] extended the concept of preference relations (PRs) to HFLTSs to represent the decision maker’s PRs, and developed the consistency method of hesitant fuzzy linguistic preference relations (HFLPRs). Since then, Zhang and Wu [16] defined the multiplicative consistency of HFLPRs. Wang and Xu [17] investigated PRs with extended HFLTSs. Dong et al. [18] investigated consensus measure and proposed an optimal consensus model. Wu and Xu [19] improved consistency method to handle the problem with HFLPR. So far, some MCDM methods have been applied to hesitant fuzzy linguistic environment, such as HFL-TOPSIS [9 , 21], HFL-TODIM [22, 23], HFL-VIKOR [9 , 25], and HFL-ELECTRE [26, 27].

To apply the HFLTSs into various fields including pattern recognition and decision making more, Liao and Xu [8] investigated a series of distance and similarity measures on HFLTSs for discrete and continuous cases. Since most of the distance measures were proposed from algebra forms, Liao and Xu [9] investigated cosine distance and similarity measures on HFLTSs based on geometric forms.

However, the existing distance and similarity on HFLTSs has mainly focused on the difference of linguistic terms, instead of considering the hesitance degree on HFLTSs. The distance measures proposed by Liao and Xu [8] are not reasonable in some cases. For instance, let S ={ s₀, s₁, s₂, s₃, s₄, s₅, s₆ } be a set of linguistic terms, h₁ ={ s₆, s₄, s₂ }, h₂ ={ s₄ }, h = {s₅, s₄, s₃} be three HFLTSs in X. Calculated by Liao and Xu’s distance measures [8], the results show that the distance between h₁ and h is equal to the distance between h₂ and h. As such, we cannot know the difference between them and this proves problematic. To overcome the above drawback, the concept of hesitance degree on HFLTSs is firstly introduced to reflect the hesitant degree among several linguistic terms in each HFLTS. Considering hesitance degree on HFLTSs, several distance and similarity measures are developed to measure the hesitant fuzzy linguistic information. The hesitant degree of h₂ is different from h₁ and h, because the decision-makers do not have hesitance in providing the linguistic terms of x to a set h₂, while hesitates between s₆, s₄ and s₂ to h₁ as well as between s₅, s₄ and s₃ to h. Compared with Liao and Xu’s method [8], the novel hesitance degree based distance and similarity measures proposed in our work are better methods in this situation.

Taking the importance of distance and similarity measures in real-world application into account, we also apply the novel distance and similarity measure in pattern recognition. In a nutshell, the main contribution of our work is as follows: (1) the hesitance degree on HFLTSs is defined to reflect the hesitant degree among several linguistic terms in each HFLTS; (2) several distance measures are proposed based on hesitance degree and their properties are discussed; (3) several similarity measures based on hesitance degree are proposed; (4) the novel proposed similarity measures are applied in pattern recognition.

The structure of this paper is organized as follows. In Section 2, we introduce concepts on HFLTSs. Section 3 introduces the concept of hesitance degree of HFLTSs, several distance measures have been proposed based on hesitance degree and their properties have been discussed, and several similarity measures based on hesitance degree have been proposed. Section 4 applies our developed similarity measures on HFLTSs for pattern recognition. Section 5 offers our conclusions.

2 Preliminaries

Definition 1 [10]. Let S ={ s₀, s₁, …, s_g } be a linguistic term set, x_i ∈ X, and i = 1, 2, …, N, then the mathematical form of a HFLTS on X is $H_{S} = {< x_{i}, h_{S} (x_{i}) > | x_{i} \in X}$ (1) where h_S (x_i) : X → S is a function defined on the collection X. For any x_i ∈ X, there is a unique h_S (x_i) corresponding to it. A hesitant fuzzy linguistic element (HFLE) is an ordered finite subset of consecutive linguistic terms of S and is expressed as h_S (x_i) = {s_{δ
_l} (x_i) |s_{δ
_l} (x_i) ∈ S, l = 1, 2, …, L (x_i)}, δ_l ∈ {0, 1, …, g} is the subscript of a linguistic term s_{δ
_l} (x_i), and L (x_i) is the number of linguistic terms. For convenience, a HFLTS H_S is a set of all hesitant fuzzy linguistic elements (HFLEs), and h_S (x_i), s_{δ
_l} (x_i), L (x_i) can be abbreviated as $h_{S}^{i}$ , $s_{δ_{l}}^{i}$ , L_i.

Definition 2 [8]. Let $h_{S}^{1} (x_{i}) = {s_{δ_{j}^{1}} (x_{i}) | s_{δ_{j}^{1}} (x_{i}) \in S}$ and $h_{S}^{2} (x_{i}) = {s_{δ_{j}^{2}} (x_{i}) | s_{δ_{j}^{2}} (x_{i}) \in S}$ be two HFLTSs on the attribute set X = (x₁, x₂, ⋯ , x_n). The distance measure $d (h_{S}^{1}, h_{S}^{2})$ between $h_{S}^{1}$ and $h_{S}^{2}$ satisfies the following properties.

$0 \leq d (h_{S}^{1}, h_{S}^{2}) \leq 1$ ,

$d (h_{S}^{1}, h_{S}^{2}) = 0$ , iff $h_{S}^{1} = h_{S}^{2}$ ,

$d (h_{S}^{1}, h_{S}^{2}) = d (h_{S}^{2}, h_{S}^{1})$ .

Definition 3 [8]. Let $h_{S}^{1} (x_{i}) = {s_{δ_{j}^{1}} (x_{i}) |$ $s_{δ_{j}^{1}} (x_{i}) \in S}$ and $h_{S}^{2} (x_{i}) = {s_{δ_{j}^{2}} (x_{i}) | s_{δ_{j}^{2}} (x_{i}) \in S}$ be two HFLTSs on the attribute set X = (x₁, x₂, ⋯ , x_n). The similarity measure $s (h_{S}^{1}, h_{S}^{2})$ between $h_{S}^{1}$ and $h_{S}^{2}$ satisfies the following properties.

$0 \leq s (h_{S}^{1}, h_{S}^{2}) \leq 1$ ,

$s (h_{S}^{1}, h_{S}^{2}) = 1$ , iff $h_{S}^{1} = h_{S}^{2}$ ,

$s (h_{S}^{1}, h_{S}^{2}) = s (h_{S}^{2}, h_{S}^{1})$ .

Property 1. If $d (h_{S}^{1}, h_{S}^{2})$ and $s (h_{S}^{1}, h_{S}^{2})$ are the distance measure and similarity measure between $h_{S}^{1}$ and $h_{S}^{2}$ , then $d (h_{S}^{1}, h_{S}^{2}) + s (h_{S}^{1}, h_{S}^{2}) = 1$ (2)

Furthermore, the distance measures proposed by Liao and Xu [8] also satisfied the following property.

(D4) For any three HFLTSs $h_{S}^{1}$ , $h_{S}^{2}$ and $h_{S}^{3}$ with the same length L on X = (x₁, x₂, ⋯ , x_n), if $h_{S}^{1, i} (x) \leq h_{S}^{2, i} (x) \leq h_{S}^{3, i} (x)$ , then $d (h_{S}^{1} (x), h_{S}^{2} (x)) \leq d (h_{S}^{1} (x), h_{S}^{3} (x))$ , $d (h_{S}^{2} (x), h_{S}^{3} (x)) \leq d (h_{S}^{1} (x), h_{S}^{3} (x))$ .

The Hamming distance, Euclidean distance, generalized distance on HFLTSs developed by Liao and Xu [8] can be defined as follows. $d_{hh} (h_{S}^{1}, h_{S}^{2}) = \frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |)$ (3)

$d_{he} (h_{S}^{1}, h_{S}^{2}) = {(\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2}))}^{1 / 2}$ (4) $d_{hg} (h_{S}^{1}, h_{S}^{2}) = {(\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ}))}^{1 / λ}$ (5) where λ > 0, g is the subscript of the maximum linguistic term, $δ_{j}^{1} (x_{i})$ and $δ_{j}^{2} (x_{i})$ are the subscripts of j-th linguistic term in i-th attribute on $h_{S}^{1}$ and $h_{S}^{2}$ , L_{1_i} and L_{2_i} are the numbers of i-th attribute on $h_{S}^{1}$ and $h_{S}^{2}$ , L_i = max(L_{1_i}, L_{2_i}).

The numbers of linguistic terms on $h_{S}^{1}$ and $h_{S}^{2}$ are different in most cases and the shorter HFLTS should be extended by adding linguistic term. The method of adding linguistic terms proposed by Zhu and Xu [15] was as follows:

Definition 5 [15]. Let b ={ b_l|l = 1, 2, …, # b } be a HFLTS, where # b is the number of linguistic terms, and the method of adding linguistic term is defined as follows. $\bar{b} = μ b^{+} \oplus (1 - μ) b^{-}$ (6) where b⁺ and b^- are the maximum and minimum linguistic term in b, and μ is an optimized parameter decided by the decision makers’ risk preferences. Optimists prefer to add the maximum linguistic term and μ = 1, while pessimists prefer to add the minimum linguistic term and μ = 0.

Throughout this paper, we assume that the linguistic terms of HFLTSs are in descending order and add the maximum linguistic terms to the shorter HFLTS.

It is noted that the distance and similarity measures proposed by Liao and Xu [8] reflect the difference of linguistic terms. Actually, the distance and similarity measures should not only consider the difference of linguistic terms between $h_{S}^{1}$ and $h_{S}^{2}$ , but also the difference of the number of linguistic terms. Otherwise, the results calculated by Equations (3)– (5) may be unreasonable.

Example 1. Let S ={ s₀, s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈ } be a set of linguistic terms and $h_{S}^{1}$ and $h_{S}^{2}$ be two patterns represented by HFLTSs on X ={ x }, $h_{S}^{1} = {s_{6.5}, s_{4}, s_{2.5}}$ , $h_{S}^{2} = {s_{4}}$ . Now there have a sample h_S = {s₅, s₄, s₃} to be recognized. The sample should be recognized by the following principle of the minimum distance measure between HFLTSs. $d (h_{S}^{i 0}, h_{S}) = min_{1 \leq i \leq 2} {d (h_{S}^{1}, h_{S}), (h_{S}^{2}, h_{S})}$ (7)

It means that the sample h_S belongs to the pattern $h_{S}^{i 0}$ .

Before calculating, $h_{S}^{2}$ should be extended to $h_{S}^{2} = {s_{4}, s_{4}, s_{4}}$ , then the difference of linguistic terms between h_S and $h_{S}^{1}$ is equal to the difference of linguistic terms between h_S and $h_{S}^{2}$ , but hesitance of h_S and $h_{S}^{1}$ are the same and $h_{S}^{2}$ does not have hesitance. From our intuitive analysis, the sample h_S should belong to $h_{S}^{1}$ . The distance measures are calculated by Equations (3) and (4), $d_{hh} (h_{S}, h_{S}^{1}) = 0.0833$ , $d_{hh} (h_{S}, h_{S}^{2}) = 0.0833$ and $d_{he} (h_{S}, h_{S}^{1}) = 0.1141$ , $d_{he} (h_{S}, h_{S}^{2}) = 0.1021$ , $d_{he} (h_{S}, h_{S}^{1}) \geq d_{he} (h_{S}, h_{S}^{2})$ .

Therefore, the sample h_S cannot be recognized by Hamming distance. By Euclidean distance, the sample h_S belongs to the pattern $h_{S}^{2}$ , which is contrary to our intuitive analysis. Thus, a more reasonable distance measure method between HFLTSs needs to be reconsidered. In such a case, the hesitance degree based distance and similarity measures may overcome the above shortcomings that are contrary to intuitive analysis.

3 Hesitance degree based distance and similarity measures

First, we introduce the concept of hesitance degree on HFLTSs.

Definition 6. Let h_S be a HFLTS on X = (x₁, x₂, ⋯ , x_n), for any x_i ∈ X, l (h_S (x_i)) be the length of h_S (x_i). The hesitance degree of h_S (x_i) is defined as follows: $u (h_{S} (x_{i})) = 1 - \frac{1}{l (h_{S} (x_{i}))}$ (8)

The value of u (h_S (x_i)) expresses the hesitant degree when decision makers evaluating an alternative or indicator hesitating among several linguistic terms, 0 ≤ u (h_S (x_i)) < 1. The larger the value of u (h_S (x_i)), the more hesitant the decision makers. When u (h_S (x_i)) = 0, it means that the decision maker has no hesitance in determining the values of linguistic terms. When u (h_S (x_i)) → 1, it means that the decision maker hardly decides the values of linguistic terms and is completely hesitant.

Example 2. Let S ={ s₀, s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈ } be a linguistic term set, $h_{S}^{1} (x_{i}) = {s_{2}, s_{5}}$ and $h_{S}^{2} (x_{i}) = {s_{1}, s_{4}, s_{6}}$ be two HFLTSs. Thus, the hesitance degree $u (h_{S}^{1} (x_{i})) = 1 - \frac{1}{2} = \frac{1}{2}$ and $u (h_{S}^{2} (x_{i})) = 1 - \frac{1}{3} = \frac{2}{3}$ . The hesitance degree of $h_{S}^{2} (x_{i})$ is greater than that of $h_{S}^{1} (x_{i})$ .

Based on the above hesitance degree of HFLTSs, some novel hesitance degree based distance and similarity measures are developed as follows.

Definition 7. Let $h_{S}^{1}$ and $h_{S}^{2}$ be two HFLTSs on X = (x₁, x₂, ⋯ , x_n), the hesitance degree based normalized Hamming distance, Euclidean distance and generalized distance between $h_{S}^{1} (x_{i})$ and $h_{S}^{2} (x_{i})$ are defined as follows. $\begin{matrix} d_{h} (h_{S}^{1}, h_{S}^{2}) = \frac{1}{2 n} \sum_{i = 1}^{n} (| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) | \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |) \end{matrix}$ (9) $\begin{matrix} d_{e} (h_{S}^{1}, h_{S}^{2}) = (\frac{1}{2 n} \sum_{i = 1}^{n} ({| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{2} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} {+ \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2}))}^{1 / 2} \end{matrix}$ (10) $\begin{matrix} d_{g} (h_{S}^{1}, h_{S}^{2}) = (\frac{1}{2 n} \sum_{i = 1}^{n} ({| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} \\ \begin{matrix} \end{matrix} {\begin{matrix} \end{matrix} + \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ}))}^{1 / λ} \end{matrix}$ (11) where λ > 0, $δ_{j}^{1} (x_{i})$ and $δ_{j}^{2} (x_{i})$ are the subscripts of j-th linguistic term in i-th attribute on $h_{S}^{1}$ and $h_{S}^{2}$ , and $L_{i} = max {l (h_{S}^{1} (x_{i})), l (h_{S}^{2} (x_{i}))}$ .

Example 3. (Continued with Example 1). According to Equation (9), the Hamming distance measures considering hesitance degree are $d_{h} (h_{S}, h_{S}^{1}) = 0.0417$ and $d_{h} (h_{S}, h_{S}^{2}) = 0.3750$ , then $d_{h} (h_{S}, h_{S}^{1}) \leq d_{h} (h_{S}, h_{S}^{2})$ .

According to Equation (10), the Euclidean distance measures considering hesitance degree are $d_{e} (h_{S}, h_{S}^{1}) = 0.0807$ , $d_{e} (h_{S}, h_{S}^{2}) = 0.4769$ , $d_{e} (h_{S}, h_{S}^{1}) \leq d_{e} (h_{S}, h_{S}^{2})$ .

The comparison of our proposed distance measures with distance measures by Liao and Xu [8] is shown in Table 1. Know from Table 1, h_S and $h_{S}^{1}$ have two linguistic terms, while $h_{S}^{2}$ has only one linguistic term and does not have hesitance. From our intuitive analysis, the sample h_S should belong to $h_{S}^{1}$ . However, by distance measures proposed from Liao and Xu [8], the sample h_S cannot be recognized by Hamming distance, and belongs to pattern $h_{S}^{2}$ by Euclidean distance, which is contrary to our intuitive analysis. However, by our proposed distance measures, the sample h_S belongs to $h_{S}^{1}$ from both Hamming distance and Euclidean distance. Our proposed distance measures based on hesitance degree are closer to intuitive analysis and therefore are more reasonable.

Table 1

The results of different distance measures

$h_{S}^{1} = {s_{6.5}, s_{4}, s_{2.5}}$	$h_{S}^{2} = {s_{4}}$
h_S = {s₅, s₄, s₃}
d _hh	0.0833	0.0833	$d_{hh} (h_{S}^{1}, h_{S}) = d_{hh} (h_{S}^{2}, h_{S})$
d _h	0.1141	0.1021	$d_{he} (h_{S}^{1}, h_{S}) \geq d_{he} (h_{S}^{2}, h_{S})$
d _he	0.0417	0.3750	$d_{h} (h_{S}^{1}, h_{S}) \leq d_{h} (h_{S}^{2}, h_{S})$
d _e	0.0807	0.4769	$d_{e} (h_{S}^{1}, h_{S}) \leq d_{e} (h_{S}^{2}, h_{S})$

Furthermore, if differences of linguistic terms and hesitance degrees have different weights, the distance measures with preference on HFLTSs are defined as follows: $\begin{matrix} d_{ph} (h_{S}^{1}, h_{S}^{2}) = \frac{1}{n} \sum_{i = 1}^{n} (α | u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) | \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |) \end{matrix}$ (12) $\begin{matrix} d_{pe} (h_{S}^{1}, h_{S}^{2}) = (\frac{1}{n} \sum_{i = 1}^{n} (α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{2} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2}))}^{1 / 2} \end{matrix}$ (13) $\begin{matrix} d_{pg} (h_{S}^{1}, h_{S}^{2}) = (\frac{1}{n} \sum_{i = 1}^{n} (α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ}))}^{1 / λ} \end{matrix}$ (14) where 0 ≤ α, β ≤ 1 and α + β = 1.

If α = 0, d_ph, d_pe and d_pg are the same as d_hh, d_he and d_hg. Namely, the distance measures d_hh, d_he and d_hg are the special cases of d_ph, d_pe and d_pg.

In some cases, the distance measures of HFLTSs should consider the weight of the element x ∈ X. Here, the weighted Hamming distance, Euclidean distance and generalized distance for HFLTSs are defined as follows: $\begin{matrix} d_{wh} (h_{S}^{1}, h_{S}^{2}) = \frac{1}{2} \sum_{i = 1}^{n} ω_{i} (| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) | \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} | \end{matrix}$ (15) $\begin{matrix} d_{we} (h_{S}^{1}, h_{S}^{2}) = (\frac{1}{2} \sum_{i = 1}^{n} ω_{i} ({| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{2} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} {+ \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2}))}^{1 / 2} \end{matrix}$ (16) $\begin{matrix} d_{wg} (h_{S}^{1}, h_{S}^{2}) = (\frac{1}{2} \sum_{i = 1}^{n} ω_{i} ({| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} \\ \begin{matrix} \end{matrix} {\begin{matrix} \end{matrix} + \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ}))}^{1 / λ} \end{matrix}$ (17) where ω_i is the weight of the element x_i, and satisfy 0 ≤ ω_i ≤ 1, $\sum_{i = 1}^{n} ω_{i} = 1$ , λ > 0.

The weighted distance measures considering the preference between hesitance degree and the values of linguistic terms are defined as follows: $\begin{matrix} d_{ω ph} (h_{S}^{1}, h_{S}^{2}) = \sum_{i = 1}^{n} ω_{i} (α | u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) | \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |) \end{matrix}$ (18) $\begin{matrix} d_{ω pe} (h_{S}^{1}, h_{S}^{2}) = (\sum_{i = 1}^{n} ω_{i} (α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{2} \\ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2}))}^{1 / 2} \end{matrix}$ (19) $\begin{matrix} d_{ω pg} (h_{S}^{1}, h_{S}^{2}) = (\sum_{i = 1}^{n} ω_{i} (α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} \\ {\begin{matrix} \end{matrix} \begin{matrix} \end{matrix} + β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ}))}^{1 / λ} \end{matrix}$ (20) where 0 ≤ ω_i ≤ 1, $\sum_{i = 1}^{n} ω_{i} = 1$ , λ > 0, and 0 ≤ α, β ≤ 1, α + β = 1.

If $ω = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , the distance measures d_wph, d_wpe and d_wpg are equal to the distance measures d_ph, d_pe and d_pg. If α = 0 and $ω = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , the distance measures d_wph, d_wpe and d_wpg are the same as the distance measures d_hh, d_he and d_hg proposed by Liao and Xu [8].

Definition 8. Let $h_{S}^{1}$ and $h_{S}^{2}$ be two HFLTSs on X = (x₁, x₂, ⋯ , x_n), $h_{S}^{j -} (x) = min_{i} h_{S}^{ij} (x)$ , $h_{S}^{j +} (x) = max_{i} h_{S}^{ij} (x)$ . The inclusion operator of the two HFLTSs $h_{S}^{1}$ and $h_{S}^{2}$ is defined as $h_{S}^{1} \subseteq h_{S}^{2}$ , iff for any x ∈ X, $h_{S}^{1 +} (x) \leq h_{S}^{2 -} (x)$ .

Theorem 1. Let $h_{S}^{1}$ and $h_{S}^{2}$ be two HFLTSs on X = (x₁, x₂, ⋯ , x_n), $d_{h} (h_{S}^{1}, h_{S}^{2})$ , $d_{e} (h_{S}^{1}, h_{S}^{2})$ , $d_{g} (h_{S}^{1}, h_{S}^{2})$ , $d_{ph} (h_{S}^{1}, h_{S}^{2})$ , $d_{pe} (h_{S}^{1}, h_{S}^{2})$ , $d_{pg} (h_{S}^{1}, h_{S}^{2})$ , $d_{wh} (h_{S}^{1}, h_{S}^{2})$ , $d_{we} (h_{S}^{1}, h_{S}^{2})$ , $d_{wg} (h_{S}^{1}, h_{S}^{2})$ and $d_{wph} (h_{S}^{1}, h_{S}^{2})$ , $d_{wpe} (h_{S}^{1}, h_{S}^{2})$ , $d_{wpg} (h_{S}^{1}, h_{S}^{2})$ satisfy the properties (D1)–(D4).

Proof.

It is straightforward.

If $d (h_{S}^{1}, h_{S}^{2}) = 0$ , namely, $\sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ} = 0$ , ${| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} = 0$ , then $δ_{j}^{1} (x_{i}) = δ_{j}^{2} (x_{i})$ , thus $h_{S}^{1} = h_{S}^{2}$ . If $h_{S}^{1} = h_{S}^{2}$ , then $δ_{j}^{1} (x_{i}) = δ_{j}^{2} (x_{i})$ , thus $d (h_{S}^{1}, h_{S}^{2}) = 0$ . From the above analysis, $d (h_{S}^{1}, h_{S}^{2}) = 0$ , iff $h_{S}^{1} = h_{S}^{2}$ .

It is easily noted that: $\begin{matrix} d (h_{S}^{1}, h_{S}^{2}) \\ = (\frac{1}{n} \sum_{i = 1}^{n} ((α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} \\ {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ})))}^{1 / λ} \\ = (\frac{1}{n} \sum_{i = 1}^{n} ((α {| u (h_{S}^{2} (x_{i})) - u (h_{S}^{1} (x_{i})) |}^{λ} \\ {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{2} (x_{i}) - δ_{j}^{1} (x_{i})}{g} |}^{λ})))}^{1 / λ} \\ = d (h_{S}^{2}, h_{S}^{1}) . \end{matrix}$

Thus, $d (h_{S}^{1}, h_{S}^{2}) = d (h_{S}^{2}, h_{S}^{1})$ .

If $h_{S}^{1, i} (x) \leq h_{S}^{2, i} (x) \leq h_{S}^{3, i} (x)$ , i = 1, 2, …, L,

thus

u (h_{S}^{1} (x_{i})) = u (h_{S}^{2} (x_{i})) = u (h_{S}^{3} (x_{i})),

| h_{S}^{1} (x) - h_{S}^{2} (x) | \leq | h_{S}^{1} (x) - h_{S}^{3} (x) |,

| h_{S}^{2} (x) - h_{S}^{3} (x) | \leq | h_{S}^{1} (x) - h_{S}^{3} (x) | .

Because λ > 0, then $\begin{matrix} (\frac{1}{n} \sum_{i = 1}^{n} ((α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{λ} \\ {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{λ})))}^{1 / λ} \\ \leq (\frac{1}{n} \sum_{i = 1}^{n} ((α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{3} (x_{i})) |}^{λ} \\ {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{3} (x_{i})}{g} |}^{λ})))}^{1 / λ}, \end{matrix}$ $\begin{matrix} (\frac{1}{n} \sum_{i = 1}^{n} ((α {| u (h_{S}^{2} (x_{i})) - u (h_{S}^{3} (x_{i})) |}^{λ} \\ {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{2} (x_{i}) - δ_{j}^{3} (x_{i})}{g} |}^{λ})))}^{1 / λ} \\ \leq (\frac{1}{n} \sum_{i = 1}^{n} ((α {| u (h_{S}^{1} (x_{i})) - u (h_{S}^{3} (x_{i})) |}^{λ} \\ {+ β \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{3} (x_{i})}{g} |}^{λ})))}^{1 / λ} \end{matrix} .$

Namely, $d (h_{S}^{1} (x), h_{S}^{2} (x)) \leq d (h_{S}^{1} (x), h_{S}^{3} (x))$ and $d (h_{S}^{2} (x), h_{S}^{3} (x)) \leq d (h_{S}^{1} (x), h_{S}^{3} (x))$ .□

Theorem 2. Let $h_{S}^{1}$ and $h_{S}^{2}$ be two HFLTSs on X = (x₁, x₂, ⋯ , x_n). If $h_{S}^{1}$ and $h_{S}^{2}$ have the same hesitance degree, $u (h_{S}^{1} (x_{i})) = u (h_{S}^{2} (x_{i}))$ , then the distance d_ph (or d_pe, d_pg) is linear with d_hh (or d_he, d_hg).

From theorem 2, we can know that, the distance measures d_hh, d_he and d_hg are special cases of d_ph, d_pe and d_pg.

Proposition 1. Let $h_{S}^{1}$ , $h_{S}^{2}$ and $h_{S}^{3}$ be three HFLTSs which have the same length L on X = (x₁, x₂, ⋯ , x_n), if $h_{S}^{1} \subseteq h_{S}^{2} \subseteq h_{S}^{3}$ , then $d_{*} (h_{S}^{1}, h_{S}^{3}) \geq max {d_{*} (h_{S}^{1}, h_{S}^{2}), d_{*} (h_{S}^{2}, h_{S}^{3})}$ , where d_* represents d_h, d_e, d_g, d_ph, d_pe or d_pg.

The above proposition can be easily proved by Definition 8 and Theorem 1.

Moreover, a numerical example is given to compare similarity measures based on the above distance measures with the existing similarity measures.

Example 4. Let S ={ s₀, s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈ }, X ={ x } and $h_{S}^{1}$ and $h_{S}^{2}$ be two HFLTSs. The calculated similarity measures between $h_{S}^{1}$ and $h_{S}^{2}$ are shown in Table 2.

Table 2

The results of different similarity measures

	1	2	3
$h_{S}^{1}$	{s₃}	{s₃, s₅}	{s₃, s₄, s₅}
$h_{S}^{2}$	{s₄}	{s₄}	{s₄}
$s_{hh} (h_{S}^{1}, h_{S}^{2})$	0.8750	0.8750	0.9167
$s_{he} (h_{S}^{1}, h_{S}^{2})$	0.8750	0.8750	0.8979
$s_{h} (h_{S}^{1}, h_{S}^{2})$	0.9375	0.7083	0.6250
$s_{e} (h_{S}^{1}, h_{S}^{2})$	0.9116	0.6392	0.5231

The similarity measures of Table 2 are defined as follows: $\begin{matrix} s_{hh} (h_{S}^{1}, h_{S}^{2}) = 1 - d_{hh} (h_{S}^{1}, h_{S}^{2}) \\ \begin{matrix} \end{matrix} = 1 - \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |] \end{matrix}$ (21) $\begin{matrix} s_{he} (h_{S}^{1}, h_{S}^{2}) = 1 - d_{he} (h_{S}^{1}, h_{S}^{2}) \\ \begin{matrix} \end{matrix} = 1 - {[\frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2}]]}^{1 / 2} \end{matrix}$ (22) $\begin{matrix} s_{h} (h_{S}^{1}, h_{S}^{2}) = 1 - d_{h} (h_{S}^{1}, h_{S}^{2}) \\ \begin{matrix} \end{matrix} = 1 - \frac{1}{n} \sum_{i = 1}^{n} ((| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) | \\ \begin{matrix} \end{matrix} + \frac{1}{L_{i}} | \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |)) \end{matrix}$ (23) $\begin{matrix} s_{e} (h_{S}^{1}, h_{S}^{2}) = 1 - d_{e} (h_{S}^{1}, h_{S}^{2}) \\ \begin{matrix} \end{matrix} = 1 - (\frac{1}{n} \sum_{i = 1}^{n} (({| u (h_{S}^{1} (x_{i})) - u (h_{S}^{2} (x_{i})) |}^{2} \\ {\begin{matrix} \end{matrix} + \frac{1}{L_{i}} \sum_{j = 1}^{L_{i}} {| \frac{δ_{j}^{1} (x_{i}) - δ_{j}^{2} (x_{i})}{g} |}^{2})))}^{1 / 2} \end{matrix}$ (24)

From Table 2, the similarity measures s_hh and s_he only cover the differences of linguistic terms and have no consideration of the difference on the numbers of linguistic terms. s_hh and s_he cannot be distinguished in some cases. Our proposed similarity measures s_h and s_e are more reasonable.

4 Numerical example

Assume that there have m patterns represented by HFLTSs $h_{S}^{i} = {< x_{j}, h_{S}^{i} (x_{j}) > | x_{j} \in X}$ (i = 1, 2, …, m) on X ={ x₁, x₂, …, x_n }. Now the sample h_S ={ < x_j, h_S (x_j) > |x_j ∈ X } needs to be recognized. The sample should be recognized as the pattern which has the maximum similarity measure, the principle is as follows. $s (h_{S}^{i 0}, h_{S}) = max_{1 \leq i \leq m} {s (h_{S}^{i}, h_{S})}$ (25)

It means that the sample h_S belongs to the pattern $h_{S}^{i 0}$ .

Example 5. Let S ={ s₀, s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈ } be a set of linguistic terms, $h_{S}^{1}$ , $h_{S}^{2}$ and $h_{S}^{3}$ be three patterns represented by HFLTSs on X ={ x₁, x₂ }. $h_{S}^{1} = (< x_{1}, {s_{5}, s_{4}} >, < x_{2}, {s_{6}, s_{5}} >)$ $h_{S}^{2} = (< x_{1}, {s_{5}, s_{4}, s_{3}} >, < x_{2}, {s_{5}} >)$ $h_{S}^{3} = (< x_{1}, {s_{8}, s_{5}, s_{3}} >, < x_{2}, {s_{5}, s_{4}, s_{3}} >)$

Table 3

The comparison between Liao and Xu’s method and our proposed method

Liao and Xu’s method			Our proposed method
$s_{h} (h_{S}, h_{S}^{1})$	0.8047	h_S belongs to $h_{S}^{3}$	$s_{hh} (h_{S}, h_{S}^{1})$	0.8177	h_S belongs to $h_{S}^{2}$
$s_{h} (h_{S}, h_{S}^{2})$	0.7656		$s_{hh} (h_{S}, h_{S}^{2})$	0.9063
$s_{h} (h_{S}, h_{S}^{3})$	0.9141		$s_{hh} (h_{S}, h_{S}^{3})$	0.8906
$s_{e} (h_{S}, h_{S}^{1})$	0.7817	h_S belongs to $h_{S}^{3}$	$s_{he} (h_{S}, h_{S}^{1})$	0.7761	h_S belongs to $h_{S}^{2}$
$s_{e} (h_{S}, h_{S}^{2})$	0.6072		$s_{he} (h_{S}, h_{S}^{2})$	0.8346
$s_{e} (h_{S}, h_{S}^{3})$	0.8670		$s_{he} (h_{S}, h_{S}^{3})$	0.8288

Next the sample h_S needs to be recognized.

h_{S} = (< x_{1}, {s_{5}, s_{4}, s_{3}} >, < x_{2}, {s_{5}, s_{4}, s_{3}, s_{2}} >)

The results of the similarity measures are calculated according to Property 1, Equations (9) and (10), and shown as follows. $\begin{matrix} s_{h} (h_{S}, h_{S}^{1}) & = 1 - d_{h} (h_{S}, h_{S}^{1}) = 0.8047, s_{h} (h_{S}, h_{S}^{2}) \\ = 1 - d_{h} (h_{S}, h_{S}^{2}) = 0.7656, s_{h} (h_{S}, h_{S}^{3}) \\ = 1 - d_{h} (h_{S}, h_{S}^{3}) = 0.9141, \end{matrix}$ and $\begin{matrix} s_{e} (h_{S}, h_{S}^{1}) = 0.7817, \\ s_{e} (h_{S}, h_{S}^{2}) = 0.6072, \\ s_{e} (h_{S}, h_{S}^{3}) = 0.8670 . \end{matrix}$

Assume that the weights are ω = (0.4, 0.6) ^T, then the results of the similarity measures are calculated by Property 1, Equations (15) and (16), and shown as follows: $\begin{matrix} s_{wh} (h_{S}, h_{S}^{1}) = 0.7906, s_{wh} (h_{S}, h_{S}^{2}) = 0.7188, \\ s_{wh} (h_{S}, h_{S}^{3}) = 0.9135, \end{matrix}$ and $\begin{matrix} s_{we} (h_{S}, h_{S}^{1}) = 0.7690, s_{we} (h_{S}, h_{S}^{2}) = 0.5697, \\ s_{we} (h_{S}, h_{S}^{3}) = 0.8734 . \end{matrix}$

Therefore, known by the results of similarity measures s_wh and s_we, the sample h_S belongs to the pattern $h_{S}^{3}$ .

If we use the similarity measures proposed by Liao and Xu [8], we obtain that $s_{hh} (h_{S}, h_{S}^{1}) = 0.8177$ , $s_{hh} (h_{S}, h_{S}^{2}) = 0.9063$ , $s_{hh} (h_{S}, h_{S}^{3}) = 0.8906$ , s_he $(h_{S}, h_{S}^{1}) = 0.7761$ , $s_{he} (h_{S}, h_{S}^{2}) = 0.8346$ , s_he (h_S, $h_{S}^{3}) = 0.8288$ , shown in Table 3. According to Equation (25), the sample h_S belongs to the pattern $h_{S}^{2}$ . The number of linguistic terms of x₂ between $h_{S}^{2}$ and h_S have a large difference. Before calculating, $h_{S}^{2}$ should be extend to $h_{S}^{2} = (< x_{1}, {s_{5}, s_{4}, s_{3}} >, < x_{2}, {s_{5}, s_{5}, s_{5}, s_{5}} >)$ . Synthetically, we should consider both differences of linguistic terms and the numbers of linguistic terms. However, the sample h_S belongs to the pattern $h_{S}^{3}$ by Equations (9) and (10). The results between Liao and Xu’s method and our proposed method are different, because the hesitance degree of x₂ on $h_{S}^{2}$ is 0, $h_{S}^{3}$ is 0.6667 and h_S is 0.75. Thus, h_S is closer to $h_{S}^{3}$ than to $h_{S}^{2}$ . The hesitance degree based similarity measures s_h and s_e are more reasonable than s_hh and s_he.

5 Concluding remarks

As a fundamental feature of HFLTSs, hesitance degree expresses the hesitant degree when decision makers evaluate an alternative or indicator hesitating among several linguistic terms. Based on hesitance degree of HFLTSs, several distance and similarity measures are proposed and applied to pattern recognition in our work. The main contribution of our work is as follows: (1) the hesitance degree on HFLTSs is defined to reflect the hesitant degree among several linguistic terms in each HFLTS; (2) several distance measures are proposed based on hesitance degree and their properties are discussed, and by comparing with the distance measures proposed by Ref. [8], we find that the proposed distance measures on HFLTSs are closer to our intuitive analysis and more reasonable; (3) several similarity measures based on hesitance degree are proposed; (4) the proposed similarity measures are applied in pattern recognition, the results indicate that the proposed similarity measures produce better results than the previous method in some cases.

In the further work, we will further apply our proposed distance and similarity measures in some fields such as medical diagnosis, performance evaluation, and decision making and compare them with the previous method in real-word application.

Footnotes

Acknowledgments

This work is supported by National Natural Science Foundation of China (No.71672128), the Fundamental Research Funds for the Central Universities, Tongji University (No. 1200219368), National Key Research and Development Program of China (2018YFC0830400).

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