New distance and similarity measures for hesitant fuzzy sets and their application in hierarchical clustering

Abstract

The hesitant fuzzy sets (HFSs) are an extension of the classical fuzzy sets. The membership degree of each element in a hesitant fuzzy set can be a set of possible values in the interval [0,1]. On the other hand, distance and similarity measures are important tools in several applications such as pattern recognition, clustering, medical diagnosis, etc. Hence, numerous studies have focused on investigating distance and similarity measures for HFSs. In this paper, some improved distance and similarity measures are introduced for the HFSs, considering the variation range as a hesitance degree for these sets. Comparing the proposed measures to some available distance and similarity measures indicated the better results of the proposed measures. Finally, the application of the proposed measures was investigated in the clustering.

Keywords

hesitant fuzzy set distance measure similarity measure pattern recognition clustering

1 Introduction

After the introduction of fuzzy sets by Zadeh [25], these sets attracted the attention of many researchers, and numerous extensions were introduced. For example, Zadeh [26] introduced the type-2 fuzzy sets and interval-valued fuzzy sets (IVFSs). Dubois and Prade [8] introduced type-n fuzzy sets. Atanassov [3] introduced intuitionistic fuzzy sets (IFSs). Recently, it has also been noted that humans are usually hesitant to make decisions. As an example, suppose two decision-makers who should determine the membership degree of a member like x in a set like A. If one of the decision-makers assigns the value of 0.3 while another assigns the value of 0.5, we are facing a set of membership degrees rather than one membership degree. Torra and Narukawa [18] and Torra [17] introduced the concept of hesitant fuzzy sets (HFSs) to overcome this problem. After the introduction of HFSs, these sets attracted the attention of many researchers. For example, Zhu et al. [30] introduced the dual hesitant fuzzy sets (DHFSs), which in a specific case includes fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets. Chen et al. [4] propose proportional hesitant fuzzy linguistic term sets, which include the proportional information of each generalized linguistic term. Wan et al. [19] develop a new hesitant fuzzy mathematical programming method for hybrid multi-criteria group decision making (MCGDM) with hesitant fuzzy truth degrees and incomplete criteria weight information. Chen et al. [6] develop a novel two-stage aggregation paradigm for hesitant fuzzy linguistic term set possibility distributions. Sajjad Ali Khan et al. [14] proposed a novel approach based on TOPSIS method and the maximizing deviation method for solving multi-attribute decision-making problems where the evaluation information provided by the decision-makers (DMs) is expressed in the form of Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. Sun et al. [16] construct a novel synthetic grey relational degree by considering both the closeness and the variation tendency factors of the data to improve the existing information measures and enhance the grey relational analysis (GRA) theory for the interval-valued hesitant fuzzy sets. Akram et al. [1] introduced a novel hybrid model called hesitant fuzzy N-soft sets.

Distance and similarity measures are important tools in several applications such as pattern recognition, clustering, medical diagnosis, etc. Thus, a large body of research has focused on introducing distance and similarity measures for fuzzy sets and their extension. For example, Wang [20] introduced two similarity measures for fuzzy sets. Hung and Yang [13] introduced similarity measures for type-2 fuzzy sets. Arefi and Taheri [2] proposed a new method to introduce similarity measure between IVFSs. Grzegorzewski [11] suggested new methods based on the Hausdorff metric to measure the distance between IFSs. Xu and Xia [22] introduced different distance and similarity measures for HFSs. Farhadinia [9] investigated the relationship between entropy, similarity, and distance for HFSs and interval-valued hesitant fuzzy sets. Zeng et al. [27] introduced several new distance and similarity measures for HFSs, considering the hesitance degree for these sets. Yang and Hussain [23] proposed the construction of new distance and similarity measures based on the Hausdorff metric for HFSs. Farhadinia and Xu [10] reported the concept of metrical T-norm-based similarity measure for HFSs. They discussed the relationship between the proposed metrical T-norm-based similarity measure and the other type of information measure, known as the metrical T-norm-based entropy measure. Yong et al. [24] proposed a Jaccard similarity measure between cubic hesitant fuzzy sets and investigates their properties. Chen et al. [5] develop a novel hybrid multi-criteria group decision-making model for sustainable building material selection under uncertainty which needs accurate distance measures. Hu et al. [12] propose axiom definitions of the distance measure and the possibility degree of hesitant interval-valued fuzzy sets and generate a series of distance measure models, similarity models, and possibility degree models.

The motivations of this paper: As will be shown in this paper, some of the distance and similarity measures introduced for hesitant fuzzy sets may not be logical in some cases, also is noted that the definition of hesitance degree for hesitant fuzzy sets based solely on the number of membership degrees, may be inadequate. These motivated the paper to introduce improved and new distance and similarity measures for these sets, which will lead to better results for various applications. Therefore, in this paper, by introducing a new hesitance degree for these sets, new distance and similarity measures are proposed for HFSs.

This paper is organized as follows: The required definitions are presented in Section 2. In Section 3, the new distance and similarity measures are proposed for HFSs. In Section 4, the application of the proposed measures in clustering is presented. Finally, Section 5 discusses the conclusion.

2 Preliminaries

This section will provide the basic definitions needed for other sections. In these definitions, X = {x₁, x₂, …, x_n} is assumed to be a universal set.

Definition 2.1. [17] The hesitant fuzzy set H on X, is a function that, when applied to X, returns a subset of the interval [0, 1].

In [21] for conveniences, the hesitant fuzzy set H on X is shown as the following mathematical symbol: $H = {< x, h_{H} (x) > ∣ x \in X}$ (1)

Where h_H (x) is a set of possible membership degrees of the element x ∈ X in the set H, and h = h_H (x) is called a hesitant fuzzy element (HFE).

Definition 2.2. [22] Let H₁ and H₂ be two hesitant fuzzy sets on the universal set X, d (H₁, H₂) is a distance measure between H₁ and H₂ if it satisfies the following properties:

0 ≤ d (H₁, H₂) ≤1

d (H₁, H₂) =0 iff H₁ = H₂

d (H₁, H₂) = d (H₂, H₁)

Definition 2.3. [22] Let H₁ and H₂ be two hesitant fuzzy sets on the universal set X, s (H₁, H₂) is a similarity measure between H₁ and H₂ if it satisfies the following properties:

0 ≤ s (H₁, H₂) ≤1

s (H₁, H₂) =1 iff H₁ = H₂

s (H₁, H₂) = s (H₂, H₁)

Remark 2.1. By analyzing definitions 2.2 and 2.3, it is clear that d (H₁, H₂) =1 - s (H₁, H₂).

Definition 2.4. [21] For a hesitant fuzzy element h, $score (h) = \frac{1}{l (h)} \sum_{γ \in h} γ$ is called the score function of h, where l (h) is the number of membership degrees in h. For two fuzzy elements h₁ and h₂, if score (h₁) > score (h₂) then h₁ > h₂; if score (h₁) = score (h₂), then h₁ = h₂.

It should be noted that the number of membership degrees may be different in different hesitant fuzzy elements. As an example, let h₁ and h₂ be two HFEs on the universal set X, it may be l (h₁ (x)) ≠ l (h₂ (x)) for x ∈ X. To have an appropriate performance, the HFE with a fewer number of membership degree is extended until the number of membership degrees of two HFEs becomes equal. Also, the membership degrees of a HFE should be ordered. In this paper, the HFE with a fewer number of membership degree is extended by adding the largest membership degree, and the membership degrees of a HFE are ordered in descending order.

In [28], by defining a hesitancy index (ℏ) for hesitant fuzzy sets, two kinds of new ordering methods for these sets are introduced as follows:

Definition 2.5. [28] Let H₁ and H₂ be two hesitant fuzzy sets on the universal set X, then:

The strict component-wise ordering of HFSs: H₁ ≤ H₂ iff $h_{H_{1}}^{σ (j)} (x_{i}) \leq h_{H_{2}}^{σ (j)} (x_{i}) & ℏ_{H_{2}} (x_{i}) \leq ℏ_{H_{1}} (x_{i}), i = 1, \dots, n, j = 1, \dots, l_{x_{i}}$ ,

The strict total ordering of HFSs: H₁ ⪯ H₂ iff score (H₁) < score (H₂) or score (H₁) = score (H₂) and ℏ (H₂) ≤ ℏ (H₁)

where $h_{H_{1}}^{σ (j)} (x_{i})$ and $h_{H_{2}}^{σ (j)} (x_{i})$ are the j-th largest values in h_{H
₁} (x_i) and h_{H
₂} (x_i), respectively.

assuming that H₁, H₂ and H₃ be three HFSs on X, it is stated in [28] that the distance measure for hesitant fuzzy sets should satisfy the following property:

if H₁ ≤ H₂ ≤ H₃, then d (H₁, H₂) ≤ d (H₁, H₃) and d (H₂, H₃) ≤ d (H₁, H₃)

The above property for the similarity measure between the hesitant fuzzy sets is also stated as follows [28]:

if H₁ ≤ H₂ ≤ H₃, then s (H₁, H₂) ≥ s (H₁, H₃) and s (H₂, H₃) ≥ s (H₁, H₃)

Xu and Xia [22] have introduced various distance measures for hesitant fuzzy sets. Assuming that H₁ and H₂ be two hesitant fuzzy sets on the universal set X, based on the well-known Hamming distance and the Euclidean distance, they proposed the hesitant normalized Hamming distance, the hesitant normalized Euclidean distance and the generalized hesitant normalized distance in the form of Eqs. (2), (3) and (4), respectively.

$\begin{matrix} d_{hnh} (H_{1}, H_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{l_{x_{i}}} \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |] \end{matrix}$ (2)

$\begin{matrix} d_{hne} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}}} \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2})]^{\frac{1}{2}} \end{matrix}$ (3)

$\begin{matrix} d_{ghn} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}}} \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ})]^{\frac{1}{λ}} \end{matrix}$ (4) where $h_{H_{1}}^{σ (j)} (x_{i})$ and $h_{H_{2}}^{σ (j)} (x_{i})$ are the j-th largest values in h_{H
₁} (x_i) and h_{H
₂} (x_i), respectively, l_{x
_i} = max {l (h_{H
₁} (x_i)) , l (h_{H
₂} (x_i))} and λ > 0.

Also, in [22], the hesitant normalized Hamming-Hausdorff distance, normalized Euclidean-Hausdorff distance, hybrid hesitant normalized Hamming distance, and hybrid hesitant normalized Euclidean distance are presented as Eqs. (5), (6), (7), and (8), respectively.

$\begin{matrix} d_{hnhh} (H_{1}, H_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} max_{j} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) | \end{matrix}$ (5)

$\begin{matrix} d_{hneh} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} max_{j} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}]^{\frac{1}{2}} \end{matrix}$ (6)

$\begin{matrix} d_{hhnh} (H_{1}, H_{2}) = \\ \frac{1}{2 n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}}} \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) | \\ + max_{j} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |) \end{matrix}$ (7)

$\begin{matrix} d_{hhne} (H_{1}, H_{2}) = \\ [\frac{1}{2 n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}}} \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2} \\ + max_{j} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2})]^{\frac{1}{2}} \end{matrix}$ (8) It has been noted in [27] that the distance between H₁ and H₂ is not only the difference between the values of their membership degrees and is also related to the difference between the number of membership degrees. Hence the following definition is provided as a definition of the hesitance degree of a hesitant fuzzy element in [27].

Definition 2.6. [27] Let H be a hesitant fuzzy set on the universal set X = {x₁, x₂, …, x_n} and for any x_i ∈ X, l (h_H (x_i)) be the length of h_H (x_i), u (h_H (x_i)) is called the hesitance degree of h_H (x_i) and is defined as the following equation.

$u (h_{H} (x_{i})) = 1 - \frac{1}{l (h_{H} (x_{i}))}$ (9)

Considering the above definition, several distance and similarity measures are introduced in [27].

Definition 2.7. [27] Assuming that H₁ and H₂ are two hesitant fuzzy sets on the universal set X, then the normalized Hamming distance including hesitance degree, the normalized Euclidean distance including hesitance degree, and the normalized generalized distance including hesitance degree between H₁ and H₂ are defined in the form of Eqs. (10), (11) and (12), respectively.

$\begin{matrix} d_{hhd} (H_{1}, H_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{l_{x_{i}} + 1} (| u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) | \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |)] \end{matrix}$ (10)

$\begin{matrix} d_{ehd} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 1} (| u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{2} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}))]^{\frac{1}{2}} \end{matrix}$ (11)

$\begin{matrix} d_{ghd} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 1} (| u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ (12)

3 New distance and similarity measures

While the proposed measures in [27] overcome the problem of ignoring the number of membership degrees in HFEs, it can be demonstrated that considering the hesitance degree only based on the number of membership degrees may be inadequate. The following example shows a condition when the defined measures in [27] do not provide reasonable results.

Example 3.1. Consider the two patterns defined on the universal set X = {x} as hesitant fuzzy sets h₁ = {1.0, 0.95} and h₂ = {1.0, 0.5, 0.45}, also consider a sample to be recognized which is represented by a hesitant fuzzy set h = {0.8, 0.75, 0.7}.

As shown in Table 1, it is not possible to recognize that the sample h belongs to which of the patterns h₁ and h₂ by using the distance measures (2), (3), (4), (5), (6), (7), and (8). Furthermore, the sample h belongs to the pattern h₂ by using the distance measures (10), (11), and (12). However, it is not concluded that the hesitance degree in these sets depends only on the number of their membership degrees. As shown, the membership degrees in the HFS h₁ are close, and it may be concluded that the decision-maker has less hesitation when assigning these membership degrees. On the other hand, the available membership degrees in the HFS h₂ are more diverse with a greater distance, which reflects the greater hesitation. Hence, it is not reasonable to consider only the number of membership degrees as a hesitance degree and, consequently assign the sample h to the pattern h₂.

Table 1
Pattern recognition using distance measures [22] and [27]

Distance Measure d (h, h₁) d (h, h₂) Result

[22] d _hnh 0.2333 0.2333 d (h, h₁) = d (h, h₂)

d _hne 0.2345 0.2345 d (h, h₁) = d (h, h₂)

d_ghn, (λ = 6) 0.2385 0.2385 d (h, h₁) = d (h, h₂)

d_ghn, (λ = 10) 0.2413 0.2413 d (h, h₁) = d (h, h₂)

d _hnhh 0.25 0.25 d (h, h₁) = d (h, h₂)

d _hneh 0.25 0.25 d (h, h₁) = d (h, h₂)

d _hhnh 0.2417 0.2417 d (h, h₁) = d (h, h₂)

d _hhne 0.2424 0.2424 d (h, h₁) = d (h, h₂)

[27] d _hhd 0.2167 0.1750 d (h, h₁) > d (h, h₂)

d _ehd 0.2195 0.2031 d (h, h₁) > d (h, h₂)

d_ghd, (λ = 6) 0.2288 0.2273 d (h, h₁) > d (h, h₂)

d_ghd, (λ = 10) 0.2347 0.2345 d (h, h₁) > d (h, h₂)

	Distance Measure	d (h, h₁)	d (h, h₂)	Result
[22]	d _hnh	0.2333	0.2333	d (h, h₁) = d (h, h₂)
	d _hne	0.2345	0.2345	d (h, h₁) = d (h, h₂)
	d_ghn, (λ = 6)	0.2385	0.2385	d (h, h₁) = d (h, h₂)
	d_ghn, (λ = 10)	0.2413	0.2413	d (h, h₁) = d (h, h₂)
	d _hnhh	0.25	0.25	d (h, h₁) = d (h, h₂)
	d _hneh	0.25	0.25	d (h, h₁) = d (h, h₂)
	d _hhnh	0.2417	0.2417	d (h, h₁) = d (h, h₂)
	d _hhne	0.2424	0.2424	d (h, h₁) = d (h, h₂)
[27]	d _hhd	0.2167	0.1750	d (h, h₁) > d (h, h₂)
	d _ehd	0.2195	0.2031	d (h, h₁) > d (h, h₂)
	d_ghd, (λ = 6)	0.2288	0.2273	d (h, h₁) > d (h, h₂)
	d_ghd, (λ = 10)	0.2347	0.2345	d (h, h₁) > d (h, h₂)

To overcome the above problem, this paper defines a new hesitance degree for hesitant fuzzy sets.

Definition 3.1. Assuming that H is a hesitant fuzzy set on the universal set X, for x_i ∈ X, r (h_H (x_i)) is called the range of h_H (x_i) and defined as the following equation:

$r (h_{H} (x_{i})) = h_{H}^{+} (x_{i}) - h_{H}^{-} (x_{i})$ (13) where $h_{H}^{+} (x_{i}) = max h_{H} (x_{i})$ and $h_{H}^{-} (x_{i}) = min h_{H} (x_{i})$ . The range of hesitant fuzzy set H is also defined as follows:

$r (H) = \frac{1}{n} \sum_{i = 1}^{n} r (h_{H} (x_{i}))$ (14)

As indicated by definition mentioned above, if the decision-maker assigns membership degrees with great distances to an element like x_i, r will have a greater value which seems reasonable since the greater distance among the membership degrees indicates the greater hesitation of the decision-maker. Similar to Definition 2, the following definition is considered for ordering hesitant fuzzy sets in this paper.

Definition 3.2. Let H₁ and H₂ be two hesitant fuzzy sets on the universal set X, then:

The strict component-wise ordering of HFSs: H₁ ≤ H₂ iff $h_{H_{1}}^{σ (j)} (x_{i}) \leq h_{H_{2}}^{σ (j)} (x_{i})$ & r (h_H
₂ (x_i)) ≤ r (h_H
₁ (x_i)) , i = 1, …, n, j = 1, …, l_{x
_i},

The strict total ordering of HFSs: H₁ ⪯ H₂ iff score (H₁) < score (H₂) or score (H₁) = score (H₂) and r (H₂) ≤ r (H₁)

Hence, considering r as a new hesitance degree for HFSs, this paper proposes the following new distance and similarity measures for these sets.

Definition 3.3 Assuming that H₁ and H₂ are two hesitant fuzzy sets on the universal set X, then the new hesitant normalized Hamming distance, the new hesitant normalized Euclidean distance, and the new generalized hesitant normalized distance between H₁ and H₂ are defined in the form of Eqs. (15), (16) and (17), respectively.

$\begin{matrix} d_{nhnh} (H_{1}, H_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) | \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) | \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |)] \end{matrix}$ (15)

$\begin{matrix} d_{nhne} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{2} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{2} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}))]^{\frac{1}{2}} \end{matrix}$ (16)

$\begin{matrix} d_{nghn} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ (17) where λ > 0.

Example 3.2. Taking into account the patterns and sample presented in Example 3.1, as can be seen from Table 2, if the new proposed distance measures are used, the result obtained in Example 3.1 is changed, and the sample h is assigned to the Pattern h₁. Since the hesitation in sample h is closer to the hesitation in pattern h₁, the results from the new measures appear to be more rational.

Table 2

Pattern recognition using distance measures [22], [27] and proposed distance measures

	Distance Measure	d (h, h₁)	d (h, h₂)	Result
[22]	d _hnh	0.2333	0.2333	d (h, h₁) = d (h, h₂)
	d _hne	0.2345	0.2345	d (h, h₁) = d (h, h₂)
	d_ghn, (λ = 6)	0.2385	0.2385	d (h, h₁) = d (h, h₂)
	d_ghn, (λ = 10)	0.2413	0.2413	d (h, h₁) = d (h, h₂)
	d _hnhh	0.25	0.25	d (h, h₁) = d (h, h₂)
	d _hneh	0.25	0.25	d (h, h₁) = d (h, h₂)
	d _hhnh	0.2417	0.2417	d (h, h₁) = d (h, h₂)
	d _hhne	0.2424	0.2424	d (h, h₁) = d (h, h₂)
[27]	d _hhd	0.2167	0.1750	d (h, h₁) > d (h, h₂)
	d _ehd	0.2195	0.2031	d (h, h₁) > d (h, h₂)
	d_ghd, (λ = 6)	0.2288	0.2273	d (h, h₁) > d (h, h₂)
	d_ghd, (λ = 10)	0.2347	0.2345	d (h, h₁) > d (h, h₂)
(Proposed)	d _nhhd	0.1833	0.2300	d (h, h₁) < d (h, h₂)
	d _nehd	0.1976	0.2711	d (h, h₁) < d (h, h₂)
	d_nghd, (λ = 6)	0.2204	0.3478	d (h, h₁) < d (h, h₂)
	d_nghd, (λ = 10)	0.2295	0.3833	d (h, h₁) < d (h, h₂)

According to Remark 2.1, the following new similarity measures for hesitant fuzzy sets H₁ and H₂ on the universal set X can be defined:

$\begin{matrix} s_{nhnh} (H_{1}, H_{2}) = 1 - d_{nhnh} (H_{1}, H_{2}) = \\ 1 - \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) | \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) | \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |)] \end{matrix}$ (18)

$\begin{matrix} s_{nhne} (H_{1}, H_{2}) = 1 - d_{nhne} (H_{1}, H_{2}) = \\ 1 - [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{2} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{2} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}))]^{\frac{1}{2}} \end{matrix}$ (19)

$\begin{matrix} s_{nghn} (H_{1}, H_{2}) = 1 - d_{nghn} (H_{1}, H_{2}) = \\ 1 - [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ (20) where λ > 0.

Theorem 3.1. Let H₁ and H₂ be two HFSs on the universal set X; the new distance measures satisfy the properties (D1)-(D3), and the new similarity measures satisfy the properties (S1)-(S3). Also considering hesitant fuzzy sets H₁, H₂ and H₃ which have the same length (l (h_{H
₁} (x_i)) = l (h_{H
₂} (x_i)) = l (h_{H
₃} (x_i)) , i = 1, 2, …, n), such that H₁ ≤ H₂ ≤ H₃, the new distance measures and new similarity measures satisfy the properties (D4) and (S4), respectively.

Proof 1. Considering the three HFSs H₁, H₂, and H₃ on the universal set X, the proof for the distance measure d_nghn is performed as follows. Clearly, by putting λ = 1 and λ = 2, the proof for distance measures d_nhnh and d_nhne is also clear.

(D1) For x_i ∈ X, i = 1, 2, …, n, j = 1, 2, …, l_{x
_i}:

0 ≤ |r (h_{H
₁} (x_i)) - r (h_{H
₂} (x_i)) |^λ ≤ 1, 0 ≤ |u (h_{H
₁} (x_i)) - u (h_{H
₂} (x_i)) |^λ ≤ 1, $0 \leq \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ} \leq l_{x_{i}}$ , $\Rightarrow 0 \leq (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ})) \leq 1$ $\Rightarrow 0 \leq [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \leq 1$ ⇒0 ≤ d_nghn (H₁, H₂) ≤1

(D2) if d_nghn (H₁, H₂) =0 then for x_i ∈ X, i = 1, 2, …, n, j = 1, 2, …, l_{x
_i}: $\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}) = 0$ $\Rightarrow h_{H_{1}}^{σ (j)} (x_{i}) = h_{H_{2}}^{σ (j)} (x_{i}) \Rightarrow H_{1} = H_{2}$ if H₁ = H₂ then for x_i ∈ X, i = 1, 2, …, n: $h_{H_{1}}^{σ (j)} (x_{i}) = h_{H_{2}}^{σ (j)} (x_{i}) \Rightarrow d_{nghn} (H_{1}, H_{2}) = 0$

(D3) $\begin{matrix} d_{nghn} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ $\begin{matrix} = & [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{2}} (x_{i})) - r (h_{H_{1}} (x_{i})) |^{λ} \\ + | u (h_{H_{2}} (x_{i})) - u (h_{H_{1}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{2}}^{σ (j)} (x_{i}) - h_{H_{1}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ $\begin{matrix} = d_{nghn} (H_{2}, H_{1}) \end{matrix}$

(D4) if H₁ ≤ H₂ ≤ H₃ with the same length, then for x_i ∈ X, i = 1, 2, …, n, j = 1, 2, …, l_{x
_i} according to Definition 3.2 that $h_{H_{1}}^{σ (j)} (x_{i}) \leq h_{H_{2}}^{σ (j)} (x_{i}) \leq h_{H_{3}}^{σ (j)} (x_{i})$ and r (h_{H
₁} (x_i)) ≥ r (h_{H
₂} (x_i)) ≥ r (h_{H
₃} (x_i)): $| h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) | \leq | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{3}}^{σ (j)} (x_{i}) |$ , $| h_{H_{2}}^{σ (j)} (x_{i}) - h_{H_{3}}^{σ (j)} (x_{i}) | \leq | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{3}}^{σ (j)} (x_{i}) |$ , |r (h_{H
₁} (x_i)) - r (h_{H
₂} (x_i)) | ≤ |r (h_{H
₁} (x_i)) - r (h_{H
₃} (x_i)) |, |r (h_{H
₂} (x_i)) - r (h_{H
₃} (x_i)) | ≤ |r (h_{H
₁} (x_i)) - r (h_{H
₃} (x_i)) |, |u (h_{H
₁} (x_i)) - u (h_{H
₂} (x_i)) | = |u (h_{H
₁} (x_i)) - u (h_{H
₃} (x_i)) |, |u (h_{H
₂} (x_i)) - u (h_{H
₃} (x_i)) | = |u (h_{H
₁} (x_i)) - u (h_{H
₃} (x_i)) |, $\begin{matrix} \Rightarrow [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \\ \leq \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{3}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{3}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{3}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}}, \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{2}} (x_{i})) - r (h_{H_{3}} (x_{i})) |^{λ} \\ + | u (h_{H_{2}} (x_{i})) - u (h_{H_{3}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{2}}^{σ (j)} (x_{i}) - h_{H_{3}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \\ \leq \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{3}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{3}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{3}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}}, \end{matrix}$ ⇒d_nghn (H₁, H₂) ≤ d_nghn (H₁, H₃) and d_nghn (H₂, H₃) ≤ d_nghn (H₁, H₃) For the similarity measures, proof can be done similarly. □

Usually, in real applications, different members x_i, (i = 1, 2, …, n) of the universal set have different important degrees. Therefore, weighted distance measures for these sets should also be defined. Assuming w_i, (i = 1, 2, …, n) is the weight of x_i ∈ X where 0 ≤ w_i ≤ 1 and $\sum_{i = 1}^{n} w_{i} = 1$ , the weighted new distance measures are defined as follows:

$\begin{matrix} d_{wnhnh} (H_{1}, H_{2}) = \\ \sum_{i = 1}^{n} w_{i} [\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) | \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) | \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |)] \end{matrix}$ (21)

$\begin{matrix} d_{wnhne} (H_{1}, H_{2}) = \\ [\sum_{i = 1}^{n} w_{i} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{2} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{2} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}))]^{\frac{1}{2}} \end{matrix}$ (22)

$\begin{matrix} d_{wnghn} (H_{1}, H_{2}) = \\ [\sum_{i = 1}^{n} w_{i} (\frac{1}{l_{x_{i}} + 2} (| r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ (23)

If there are different preferences between the hesitance degrees u and r and the difference of membership degrees, the new distance measures are defined as follows.

$\begin{matrix} d_{nphnh} (H_{1}, H_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{l_{x_{i}} + 2} (α | r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) | \\ + β | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) | \\ + γ \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |)] \end{matrix}$ (24)

$\begin{matrix} d_{nphne} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (α | r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{2} \\ + β | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{2} \\ + γ \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}))]^{\frac{1}{2}} \end{matrix}$ (25)

$\begin{matrix} d_{npghn} (H_{1}, H_{2}) = \\ [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{l_{x_{i}} + 2} (α | r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + β | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + γ \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ (26) where 0 ≤ α ≤ 1, 0 ≤ β ≤ 1, 0 ≤ γ ≤ 1 and α + β + γ = 1.

Fig. 1

The clustering dendrograms.

Also, considering the different weights for the members of the universal set as well as the different preferences for the hesitance degrees and the difference of membership degrees, the new distance measures can be considered as follows.

$\begin{matrix} d_{wnphnh} (H_{1}, H_{2}) = \\ \sum_{i = 1}^{n} w_{i} [\frac{1}{l_{x_{i}} + 2} (α | r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) | \\ + β | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) | \\ + γ \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |)] \end{matrix}$ (27)

$\begin{matrix} d_{wnphne} (H_{1}, H_{2}) = \\ [\sum_{i = 1}^{n} w_{i} (\frac{1}{l_{x_{i}} + 2} (α | r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{2} \\ + β | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{2} \\ + γ \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{2}))]^{\frac{1}{2}} \end{matrix}$ (28)

$\begin{matrix} d_{wnpghn} (H_{1}, H_{2}) = \\ [\sum_{i = 1}^{n} w_{i} (\frac{1}{l_{x_{i}} + 2} (α | r (h_{H_{1}} (x_{i})) - r (h_{H_{2}} (x_{i})) |^{λ} \\ + β | u (h_{H_{1}} (x_{i})) - u (h_{H_{2}} (x_{i})) |^{λ} \\ + γ \sum_{j = 1}^{l_{x_{i}}} | h_{H_{1}}^{σ (j)} (x_{i}) - h_{H_{2}}^{σ (j)} (x_{i}) |^{λ}))]^{\frac{1}{λ}} \end{matrix}$ (29)

4 Application

Data clustering is considered as one of the important methods of data analysis. Hence, recently, uncertain data clustering, such as hesitant fuzzy data, has attracted a lot of researchers’ attention [7 , 29]. Since distance and similarity measures have significant effects on data clustering, in this paper, the proposed measures are used for the clustering of several energy projects to show their application. The idea of considering these projects for clustering was obtained from [22].

The single linkage hierarchical clustering algorithm [15] is used to cluster the data. Supposing that HFSs H₁, H₂, … H_m over the universal set X are hesitant fuzzy data to be clustered, each set is considered as a cluster firstly. Next, by calculating the distance matrix D = [d_ij] _m×m whose element d_ij represents the distance between the clusters C_i and C_j, two closest clusters are merged in each step. This process continues until all data is placed in one cluster. It should be noted that the distance between the clusters C_i and C_j is calculated using Equation (30).

$\begin{matrix} d_{ij} & = distance (C_{i}, C_{j}) \\ = min {distance (a, b) | a \in C_{i}, b \in C_{j}} \end{matrix}$ (30)

Example 4.1. This example deals with the clustering of the performance of five energy projects evaluated by several evaluators. Assume that A_i, (i = 1, …, 5) are five energy projects; each energy project consists of four features including P₁: technological, P₂: environmental, P₃: socio-political and P₄: economic. Several evaluators were asked to comment on the performance of these projects, and their comments for each project were collected as a hesitant fuzzy set. Some values in these comments may be repeated more than once, which do not indicate their importance relative to the values with fewer repetitions. For example, the value which existed once, may be presented by an evaluator who is elite in the field, and the value repeated more than once may be presented by evaluators which are not sufficiently familiar with that field. The collected comments are shown in Table 3.

Table 3

Collected comments for five energy projects

	P ₁	P ₂	P ₃	P ₄
A ₁	{0.9, 0.85, 0.7}	{0.8, 0.75, 0.7}	{0.7, 0.6, 0.5}	{1.0, 0.8}
A ₂	{0.9, 0.35, 0.3}	{0.8, 0.15, 0.1}	{0.9, 0.6, 0.5}	{0.6, 0.2}
A ₃	{0.8, 0.7}	{0.7, 0.65, 0.6}	{0.6, 0.5, 0.4}	{1.0, 0.8}
A ₄	{0.6, 0.5}	{0.5, 0.45, 0.4}	{0.8, 0.7}	{0.8, 0.5}
A ₅	{0.8, 0.3}	{0.7, 0.25, 0.2}	{1, 0.5, 0.4}	{0.6, 0.2}

To compare the performance of the aforementioned distance measures, clustering has been done using distance measures d_hnh, d_hhd, and d_nhnh, and the results are shown in Tables 4, 5, and 6, respectively. As can be seen from the results, the distance measures d_hnh and d_hhd are not able to provide an accurate clustering, given the number of clusters equal to 2. But if the proposed distance measure d_nhnh is used, the projects are precisely clustered, taking into account the number of clusters equal to 2. Figure 1 shows the dendrograms corresponding to the clustering results.

Table 4

Clustering results using d_hnh

Number of clusters	Clustering result
5	{A₁} , {A₂} , {A₃} , {A₄} , {A₅}
4	{A₁, A₃} , {A₂} , {A₄} , {A₅}
3	{A₁, A₃} , {A₂, A₅} , {A₄}
2	{A₁, A₃, A₄} , {A₂, A₅} or {A₁, A₃} , {A₂, A₄, A₅}
1	{A₁, A₂, A₃, A₄, A₅}

Table 5

Clustering results using d_hhd

Number of clusters	Clustering result
5	{A₁} , {A₂} , {A₃} , {A₄} , {A₅}
4	{A₁, A₃} , {A₂} , {A₄} , {A₅}
3	{A₁, A₃} , {A₂, A₅} , {A₄}
2	{A₁, A₃, A₄} , {A₂, A₅} or {A₁, A₃} , {A₂, A₄, A₅}
1	{A₁, A₂, A₃, A₄, A₅}

Table 6

Clustering results using d_nhnh

Number of clusters	Clustering result
5	{A₁} , {A₂} , {A₃} , {A₄} , {A₅}
4	{A₁, A₃} , {A₂} , {A₄} , {A₅}
3	{A₁, A₃} , {A₂, A₅} , {A₄}
2	{A₁, A₃, A₄} , {A₂, A₅}
1	{A₁, A₂, A₃, A₄, A₅}

In order to make more comparisons, the clustering algorithm introduced in [28] has also been used to clustering the energy projects using similarity measures s_hnh = 1 - d_hnh, s_hhd = 1 - d_hhd, and s_nhnh = 1 - d_nhnh. The clustering results are presented in Tables 7, 8, and 9, respectively. As can be seen, the clustering method was not able to produce two clusters using the s_hnh and s_hhd similarity measures but, by using the s_nhnh similarity measure, clustering with two clusters will be obtained accurately.

Table 7

Clustering results using the clustering algorithm introduced in [28] and s_hnh

Number of clusters	Clustering result
5	{A₁} , {A₂} , {A₃} , {A₄} , {A₅}
4	{A₁, A₃} , {A₂} , {A₄} , {A₅}
3	{A₁, A₃} , {A₂, A₅} , {A₄}
2	-
1	{A₁, A₂, A₃, A₄, A₅}

Table 8

Clustering results using the clustering algorithm introduced in [28] and s_hhd

Number of clusters	Clustering result
5	{A₁} , {A₂} , {A₃} , {A₄} , {A₅}
4	{A₁, A₃} , {A₂} , {A₄} , {A₅}
3	{A₁, A₃} , {A₂, A₅} , {A₄}
2	-
1	{A₁, A₂, A₃, A₄, A₅}

Table 9

Clustering results using the clustering algorithm introduced in [28] and s_nhnh

Number of clusters	Clustering result
5	{A₁} , {A₂} , {A₃} , {A₄} , {A₅}
4	{A₁, A₃} , {A₂} , {A₄} , {A₅}
3	{A₁, A₃} , {A₂, A₅} , {A₄}
2	{A₁, A₃, A₄} , {A₂, A₅}
1	{A₁, A₂, A₃, A₄, A₅}

5 Conclusion

In this paper, new and various distance and similarity measures were proposed for HFSs by introducing a new hesitance degree for these sets. It was shown that the proposed measures could provide more reasonable results in pattern recognition than some available distance measures. Furthermore, the proposed measures were used for the clustering of several energy projects to show their application. The comparison of clustering results from the proposed measures with those of using some available measures indicated that the use of the proposed measures in clustering leads to more accurate clustering. Researchers can focus on adjusting preference parameters for the hesitance degrees and distance between membership degrees, in addition to adjusting weight values in the proposed distance and similarity measures for future works.

References

Akram

, Adeel

and Alcantud

J.C.R

, Hesitant fuzzy N-soft sets: A new model with applications in decision-making, Journal of Intelligent & Fuzzy Systems 36(6) (2019), 6113–6127.

Arefi

and Taheri

S.M.

, Weighted similarity measure on interval-valued fuzzy sets and its application to pattern recognition, Iranian Journal of Fuzzy Systems 11(5) (2014), 67–79.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

Chen

Z.S.

, Chin

K.S.

, Li

Y.L.

and Yang

, Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making, Information Sciences 357 (2016), 61–87.

Chen

Z.S.

, Martínez

, Chang

J.P.

, Wang

X.J.

, Xionge

S.H.

and Chin

K.S.

, Sustainable building material selection: A QFD-and ELECTRE III-embedded hybrid MCGDM approach with consensus building, Engineering Applications of Artificial Intelligence 85 (2019), 783–807.

Chen

Z.S.

, Martí nez

, Chin

K.S.

and Tsui

K.L.

, Two-stage aggregation paradigm for HFLTS possibility distributions: A hierarchical clustering perspective, Expert Systems with applications 104 (2018), 43–66.

Chen

, Xu

and Xia

, Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modelling 37(4) (2013), 2197–2211.

Dubois

and Prade

H.M.

, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.

Farhadinia

, Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets, Information Sciences 240 (2013), 129–144.

10.

Farhadinia

and Xu

, Hesitant Fuzzy Information Measures Derived From t-norms and s-norms, Iranian Journal of Fuzzy Systems 15(5) (2018), 157–175.

11.

Grzegorzewski

, Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric, Fuzzy sets and systems 148(2) (2004), 319–328.

12.

, Lan

and Wang

, A distance measure, similarity measure and possibility degree for hesitant interval-valued fuzzy sets, Computers & Industrial Engineering 137 (2019), 106088.

13.

Hung

W.L.

and Yang

M.S.

, Similarity measures between type-2 fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12(6) (2004), 827–841.

14.

Sajjad Ali Khan

, Ali

, Abdullah

, Amin

and Hussain

, New extension of TOPSIS method based on Pythagorean hesitant fuzzy sets with incomplete weight information, Journal of Intelligent & Fuzzy Systems 35(5) (2018), 5435–5448.

15.

Sneath

P.H.A.

and Sokal

, Numerical taxonomy. The principles and practice of numerical classification, 1973.

16.

Sun

, Guan

, Yi

and Zhou

, Multi-Attribute Decision Making with Interval-Valued Hesitant Fuzzy Information, a Novel Synthetic Grey Relational Degree Method, Informatica 29(3) (2019), 517–537.

17.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25(6) (2010), 529–539.

18.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, 2009 IEEE International Conference on Fuzzy Systems (2009), 1378–1382.

19.

Wan

S.P.

, Qin

Y.L.

and Dong

J.Y.

, A hesitant fuzzy mathematical programming method for hybrid multi-criteria group decision making with hesitant fuzzy truth degrees, Knowledge-Based Systems 138 (2017), 232–248.

20.

Wang

W.J.

, New similarity measures on fuzzy sets and on elements, Fuzzy sets and systems 85(3) (1997), 305–309.

21.

Xia

and Xu

, Hesitant fuzzy information aggregation in decision making, International journal of approximate reasoning 52(3) (2011), 395–407.

22.

and Xia

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(11) (2011), 2128–2138.

23.

Yang

M.S.

and Hussain

, Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering, Soft Computing 23(14) (2019), 5835–5848.

24.

Yong

, Zhu

and Ye

, Multiple attribute decision method using similarity measure of cubic hesitant fuzzy sets, Journal of Intelligent & Fuzzy Systems 37(1) (2019), 1075–1083.

25.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

26.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences 8(3) (1975), 199–249.

27.

Zeng

, Li

and Yin

, Distance and similarity measures between hesitant fuzzy sets and their application in pattern recognition, Pattern Recognition Letters 84 (2016), 267–271.

28.

Zhang

and Xu

, Novel distance and similarity measures on hesitant fuzzy sets with applications to clustering analysis, Journal of Intelligent & Fuzzy Systems 28(5) (2015), 2279–2296.

29.

Zhang

and Xu

, Hesitant fuzzy agglomerative hierarchical clustering algorithms, International Journal of Systems Science 46(3) (2015), 562–576.

30.

Zhu

, Xu

and Xia

, Dual hesitant fuzzy sets,13 pages, Journal of Applied Mathematics 2012, Article ID 879629 (2012).