New distance measures on hesitant fuzzy sets based on the cardinality theory and their application in pattern recognition

Abstract

As a generalization of fuzzy set, hesitant fuzzy set (HFS) permits the membership of an element to a set having a set of possible values. Distance is one of important tools in measuring the relationship between two HFSs. Based on the cardinality theory, some novel distances which take the cardinal numbers of HFSs into account have been introduced using the concept of “multi-sets.” The main advantage of the distance measures is that they can more objectively and universally measure the relationship between HFSs than the existing methods. Finally, the performance of the proposed distance measures is illustrated through two pattern recognition examples in port enterprise management and transportation infrastructure construction.

1 Introduction

In the real world, it is sometimes very difficult to precisely define the objective membership of an element to a set. To efficiently measure the relationship, “fuzzy set” is proposed by Zadeh (1965) whose characteristic is a membership function with the function value ranging between zero and one. Recently, the theory of fuzzy sets has become a vigorous research area in many applications, for example, management sciences, engineering, decision making and artificial intelligence. Zadel’s fuzzy set has successfully been applied in different branches of science and technology. For example, Tripathy and Baruah (2009) introduced some operators for studying the properties of sequence spaces, and the proposed operators can be applied for studying many other classes of sequences of fuzzy numbers. After the pioneering work by Zadeh (1965), several extensions and generalizations of fuzzy sets have been proposed, for example, intuitionistic fuzzy sets (Atanassov 1986), hesitant fuzzy sets (shorted by HFSs) (Torra and Narukawa 2009; Torra 2010). As an extension of fuzzy set, the HFSs are often used in the situation where there are some difficulties in determining the membership of an element to a set caused by a doubt between a few different values. Since HFS permits the membership having a set of possible values, it has attracted much attentions recently because hesitant situations are very common in different real-world problems.

Distance measure is the fundamental and important definition in the HFS theory. Xu and Xia (2011a) proposed a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. Such measures can be used to alleviate the influence of unduly large (or small) deviations on the aggregation results. Peng et al. (2013) proposed a generalized hesitant fuzzy synergistic weighted distance measure by extending the distance measures in Xu and Xia (2011a, b). Rodrguez et al. (2014) overviewed the advances of the HFSs in details. As discussed in Li et al. (2015a), all the existing distance measures on HFSs only cover the divergence of the values, but not the difference between the cardinal numbers of the hesitant fuzzy elements (shorted by HFEs). Motivated by which, Li et al. (2015a) defined the hesitance degree of HFSs to measure the decision maker’s hesitance. Besides, Li et al. (2015b) also proposed a series of distance measures based on the triangle inequality property.

The existing studies are very wonderful and pioneering. However, there is still a problem that the cardinal numbers of the HFEs are barely used in the computation process. In the distance computation between HFEs with different cardinal numbers, the currently proposed methods are mainly proposed by extending the shorter HFE until both of them have the same cardinal number. For example, by using the existing calculation techniques, the shorter HFE is extended by adding the minimum value, the maximum value, or any value in it. In practice, there are many values in the shorter HFE, the values are empirically selected by the decision makers. Optimists may select the maximum value, while pessimists may select the minimum value. Therefore, the distance between the same pair of HFEs is varying as the decision makers, which weakens the computation objectivity. Especially, corruption is easily exercised in case the distance calculation depends heavily on the decision maker’s personal interests. Besides, when we compare the distances of different pairs of HFEs, all the related HFEs should be unified so that they have the same cardinal number and are in the same algebra space. Thus, it is very necessary to propose some novel distance measures on HFSs which take the cardinal numbers of the HFEs into account scientifically.

In the remainder of this paper, it is organized as follows. In Sect. 2, some classical distance measures on HFSs are reviewed. In Sect. 3, a series of novel distance measures on HFSs are proposed based on the cardinality theory. In Sect.  4, two examples are proposed to demonstrate the performance of the improved distance measures. Finally, some conclusions are proposed in Sect. 5.

2 Preliminaries

In this section, some basic definitions on HFSs are reviewed. For convenience, denote $X=\{x_{1},x_{2},\ldots ,x_{n}\}$ as the discourse set throughout this paper.

Definition 1

(Torra 2010) Let X be a reference set, a HFS E on X is in terms of a function that when applied to X returns a subset of [0, 1], which can be represented as the following mathematical symbol:

$$\begin{aligned} E=\{\langle x,h_{E}(x)\rangle \mid x \in X\}, \end{aligned}$$

(1)

where $h_{E}(x)$ is a set of values in [0, 1], representing the possible membership degrees of the element x to the HFS E. For convenience, $h_{E}(x)$ is called a HFE.

For HFSs, the axiomatic definition of distance measure was addressed by Xu and Xia (2011a) as follows.

Definition 2

(Xu and Xia 2011a) Let M and N be two HFSs on X, then the distance measure on M and N is defined as d(M, N), which satisfies the following properties:

(i) :

$0 \le d(M,N) \le 1;$

(ii) :

$d(M,N) = 0,~if~and~only~ if~M=N;$

(iii) :

$d(M,N)=d(N,M)$.

Since the existing distance measures only depend on the values of HFEs, Li et al. (2015b) called them the value-based distance measures for HFSs. Furthermore, Li et al. (2015b) modified the above definition as follows.

Definition 3

(Li et al. 2015b) Let M, N and O be three HFSs on X, then d is called a value-based distance measure for HFSs if it satisfies the following properties:

(i) :

$0 \le d(M,N) \le 1;$

(ii) :

$d(M,N) = 0,~if~and~only~ if~M=N;$

(iii) :

$d(M,N)=d(N,M);$

(iv) :

$d(M,N)\le d(M,O)+d(O,N);$

(v) :

$if~M \le N \le O,~then,~d(M,N)\le d(M,O)~and~d(N,O)\le d(M,O).$

Drawing on the classical Hamming distance and the Euclidean distance in Euclidean space, Xu and Xia (2011a) defined some important distance measures as follows.

Definition 4

(Xu and Xia 2011a) Let M and N be two HFSs on X. For any $x_{i} \in X$, let $l(h_{M}(x_{i}))$ and $l(h_{N}(x_{i}))$ be the cardinal numbers of $h_{M}(x_{i})$ and $h_{N}(x_{i})$, respectively. Then, the hesitant normalized Hamming distance, Euclidean distance and generalized hesitant normalized distance are defined as follows:

$$\begin{aligned} d_{\mathrm{h}}(M,N)= & {} \dfrac{1}{n}\sum \limits _{i=1}^{n}\left[ \dfrac{1}{l_{x_{i}}}\sum \limits _{j=1}^{l_{x_{i}}}\left| h_{M}^{\sigma (j)}(x_{i})-h_{N}^{\sigma (j)}(x_{i})\right| \right] , \end{aligned}$$

(2)

$$\begin{aligned} d_{\mathrm{e}}(M,N)= & {} \left[ \dfrac{1}{n}\sum \limits _{i=1}^{n}\left( \dfrac{1}{l_{x_{i}}}\sum \limits _{j=1}^{l_{x_{i}}}\left| h_{M}^{\sigma (j)}(x_{i}){-}h_{N}^{\sigma (j)}(x_{i})\right| ^{2}\right) \right] ^{1/2},\nonumber \\ \end{aligned}$$

(3)

$$\begin{aligned} d_{\mathrm{g}}(M,N)= & {} \left[ \dfrac{1}{n}\sum \limits _{i=1}^{n}\left( \dfrac{1}{l_{x_{i}}}\sum \limits _{j=1}^{l_{x_{i}}}\left| h_{M}^{\sigma (j)}(x_{i}){-}h_{N}^{\sigma (j)}(x_{i})\right| ^{\lambda }\right) \right] ^{1/\lambda },\nonumber \\ \end{aligned}$$

(4)

where $l_{x_{i}}= max \left\{ l(h_{M}(x_{i})),l(h_{N}(x_{i})) \right\}$, $\lambda >0,~h_{M}^{\sigma (j)}(x_{i})$ and $h_{N}^{\sigma (j)}(x_{i})$ are the jth largest values in $h_{M}(x_{i})$ and $h_{N}(x_{i})$, respectively.

It is noteworthy that $l(h_{M}(x_{i}))\ne l(h_{N}(x_{i}))$ holds in most cases. To operate correctly, Xu and Xia (2011a) suggested that the shorter one should be extended until the cardinal numbers of $h_{M}(x_{i})$ and $h_{N}(x_{i})$ are the same when we calculate them, and the extension process lies on the decision makers.

Definition 5

(Xu and Xia 2011a) Let M and N be two HFSs on X, if the weight $w_{i}(i=1,2,\ldots ,n)$ of each element $x_{i} \in X$ is taken into account, the generalized hesitant weighted distance is defined as

$$\begin{aligned}&d_{wg}(M,N)\nonumber \\&\quad {=} \left[ \sum \limits _{i=1}^{n}w_{i}\left( \dfrac{1}{l_{x_{i}}}\sum \limits _{j=1}^{l_{x_{i}}}\left| h_{M}^{\sigma (j)}(x_{i})-h_{N}^{\sigma (j)}(x_{i})\right| ^{\lambda }\right) \right] ^{1/\lambda }{,} \end{aligned}$$

(5)

where $\lambda >0$.

The aforementioned distance measures $d_{\mathrm{h}}(\cdot )$, $d_{\mathrm{e}}(\cdot )$, $d_{\mathrm{g}}(\cdot )$ and $d_{wg}(\cdot )$ are very prospective; many different extensions of them were proposed in the past five years. Though the cardinal numbers of HFSs are dealt with at random in these measures, they are still groundbreaking and pioneering. One flaw cannot obscure the splendor of the jade, and Definition 4 and are also the foundation of our researches.

“multi-sets” is a generalization of crisp sets, where multiple occurrences of an element are permitted. Based on the concept “multi-sets,” a series of novel distance measures on HFSs are proposed in the following section. For more details about “multi-sets,” please see Van and Tauler (2013).

3 Main results

3.1 The hesitant fuzzy multi-sets

Suppose that there are m HFSs on $X=\{x_{1},x_{2},\ldots ,x_{n}\}$, denoted by $E^{*}=\{E_{1},E_{2},\ldots , E_{m}\}$. For any $x_{i} \in X$, the set of membership values of $x_{i}$ to $E_{k}(1\le k \le m)$ is defined by

$$\begin{aligned} h_{E_{k}}(x_{i})= \left\{ h_{E_{k}}^{1}(x_{i}),h_{E_{k}}^{2}(x_{i}),\ldots ,h_{E_{k}}^{l(h_{E_{k}}(x_{i}))}(x_{i}) \right\} , \end{aligned}$$

(6)

where $l(h_{E_{k}}(x_{i}))$ is the cardinal number of $h_{E_{k}}(x_{i})$, $k \in \{1,2,\ldots ,m\}$.

Subsequently, based on the least common multiple [LCM, Reuben et al. (2014)] of $l(h_{E_{1}}(x_{i}))$, $l(h_{E_{2}}(x_{i})),\ldots , l(h_{E_{m}}(x_{i}))$, we define the hesitant fuzzy multi-sets of the HFSs in $E^{*}$.

Firstly, we denote the LCM of $l(h_{E_{1}}(x_{i}))$, $l(h_{E_{2}}(x_{i})),\ldots ,l(h_{E_{m}}(x_{i}))$ as $\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]$.

Secondly, by using $\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]$, the m HFEs $h_{E_{1}}(x_{i}),h_{E_{2}}(x_{i}),\ldots ,h_{E_{m}}(x_{i})$ are extended into m hesitant fuzzy multi-sets with the form of

where the number of occurrences of $h_{E_{k^{*}}}^{j^{*}}(x_{i})(k^{*}=1,2,\ldots ,m;j^{*}=1,2,\ldots ,l(h_{E_{k^{*}}}(x_{i}))$ in $h_{E_{k^{*}}}^{\prime }(x_{i})$ is denoted by

$$\begin{aligned} Cn(h_{E_{k^{*}}}^{j^{*}}(x_{i}))= \dfrac{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}{l(h_{E_{k^{*}}}(x_{i}))}. \end{aligned}$$

(7)

3.2 New distance measures based on cardinality theory

In this subsection, we would propose some distance measures based on the cardinality theory. First of all, for any $k^{*}\in \{1,2,\ldots ,m\}$, all the elements of $h_{E_{k^{*}}}^{\prime }(x_{i})$ are ranked in monotone decreasing order, and the result is denoted as

$$\begin{aligned} \tilde{h}_{E_{k^{*}}}(x_{i}){=}\left\{ \tilde{h}_{E_{k^{*}}}^{\sigma (1)}(x_{i}), \tilde{h}_{E_{k^{*}}}^{\sigma (2)}(x_{i}),\ldots ,\tilde{h}_{E_{k^{*}}}^{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}(x_{i})\right\} . \end{aligned}$$

(8)

Definition 6

Suppose that there are m HFSs on $X=\{x_{1},x_{2},\ldots ,x_{n}\}$ which are denoted as $E^{*}{=}\{E_{1},E_{2},\ldots ,E_{m}\}$. For any $s\in \{1,2,\ldots ,m\},t\in \{1,2,\ldots ,m\}$, the novel hesitant normalized Hamming distance, Euclidean distance and generalized hesitant normalized distance between $E_{s}$ and $E_{t}$ on X are denoted as

$$\begin{aligned} d_{\mathrm{fth}}(E_{s},E_{t})= & {} \dfrac{1}{n}\sum \limits _{i=1}^{n}\left[ \dfrac{1}{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\right. \nonumber \\&{\times }\left. \sum \limits _{j=1}^{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\left| \tilde{h}_{E_{s}}^{\sigma (j)}(x_{i})-\tilde{h}_{E_{t}}^{\sigma (j)}(x_{i})\right| \right] , \end{aligned}$$

(9)

$$\begin{aligned} d_{\mathrm{fte}}(E_{s},E_{t})= & {} \left[ \dfrac{1}{n}\sum \limits _{i=1}^{n}\left( \dfrac{1}{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\right. \right. \nonumber \\&{\times }\left. \left. \sum \limits _{j=1}^{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\left| \tilde{h}_{E_{s}}^{\sigma (j)}(x_{i}){-}\tilde{h}_{E_{t}}^{\sigma (j)}(x_{i})\right| ^{2}\right) \right] ^{1/2},\nonumber \\ \end{aligned}$$

(10)

$$\begin{aligned} d_{\mathrm{ftg}}(E_{s},E_{t})= & {} \left[ \dfrac{1}{n}\sum \limits _{i=1}^{n}\left( \dfrac{1}{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\right. \right. \nonumber \\&\times \left. \left. \sum \limits _{j=1}^{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\left| \tilde{h}_{E_{s}}^{\sigma (j)}(x_{i})-\tilde{h}_{E_{t}}^{\sigma (j)}(x_{i})\right| ^{\lambda }\right) \right] ^{1/\lambda },\nonumber \\ \end{aligned}$$

(11)

where $\lambda > 0$, $\tilde{h}_{E_{s}}^{\sigma (j)}(x_{i})$ and $\tilde{h}_{E_{t}}^{\sigma (j)}(x_{i})$ are the jth ordinal values in $\tilde{h}_{E_{s}}(x_{i})$ and $\tilde{h}_{E_{t}}(x_{i})$, respectively.

Furthermore, if the weight $w_{i}(i=1,2,\ldots ,n)$ of each element $x_{i} \in X$ is taken into account, the novel generalized hesitant weighted distance is defined as

$$\begin{aligned}&d_{ftwg}(E_{s},E_{t})= \left[ \sum \limits _{i=1}^{n}w_{i}\left( \dfrac{1}{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\right. \right. \nonumber \\&\times \left. \left. \sum \limits _{j=1}^{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\left| \tilde{h}_{E_{s}}^{\sigma (j)}(x_{i})-\tilde{h}_{E_{t}}^{\sigma (j)}(x_{i})\right| ^{\lambda }\right) \right] ^{1/\lambda }. \end{aligned}$$

(12)

In addition, one parameter distance measure on HFEs is defined below.

Definition 7

For any $x^{*}\in X$, we take two HFEs $h_{E_{s}}(x^{*})$ and $h_{E_{t}}(x^{*})$ randomly for example. Then, one parameter distance measure on the two HFEs is proposed as

$$\begin{aligned}&d_{ft2}(h_{E_{s}}(x^{*}),h_{E_{t}}(x^{*}))\nonumber \\&\quad = \left( \dfrac{1}{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\sum \limits _{j=1}^{\mathrm{LCM}_{k=1}^{m}[h_{E_{k}}(x_{i})]}\left| \tilde{h}_{E_{s}}^{\sigma (j)}(x^{*})-\tilde{h}_{E_{t}}^{\sigma (j)}(x^{*})\right| ^{\lambda } \right) ^{1/\lambda },\nonumber \\ \end{aligned}$$

(13)

where $\lambda > 0$, $\tilde{h}_{E_{s}}^{\sigma (j)}(x^{*})$ and $\tilde{h}_{E_{t}}^{\sigma (j)}(x^{*})$ are the jth ordinal values in $\tilde{h}_{E_{s}}(x^{*})$ and $\tilde{h}_{E_{t}}(x^{*})$, respectively.

In Sect. 4, two pattern recognition examples are proposed to illustrate the efficiency of the new proposed distance measures. The first example comes from the project of the National Social Science Foundation of China (No. 15CJY057) entitled “Research on the motivation, effect and mechanism of the third-party logistics embedded in the global supply chain.” The second example comes from the project of the National Natural Science Foundation of China (No. 51508319) entitled “Study on travel choice behavior uncertainty decision mechanism of the drivers of rigid travel demand in front of the diversions isolation facilities in the main road under the condition of emergency incident in the city.”

4 Illustrative examples

4.1 Example 1

In Zhabei district, Shanghai city, P. R. China, there is a tugboat company who has just bought two harbor operational tugs. At present, there are three fleets of tugs in Shanghai city who are recruiting tugs and many tugboats are recruited to provide the tug services for port operation in 2016. Since the “performance assessments” and the “all-business environments” of the three fleets are almost the same, the mentioned company wants to join one fleet that is most similarly managed as itself. More details about the concept “management philosophy” can be seen in Vanelslander and Sys (2014). Thereafter, this company hires a committee of experts to evaluate the management philosophy of the three ports and himself. The evaluation criteria include “the efficiency in decision-making,” “the sense of urgency” and “the quality of service.” For convenience, we denote the three fleets by $E_{1},E_{2} and E_{3}$, denote the tugboat company by $E_{0}$, and denote the above three criteria by $P=\{P_{1},P_{2} and P_{3}\}$. Moreover, the weights of each criterion $P_{i}(i=1,2,3)$ are evaluated, and the result is $W=(0.35,0.35,0.30).$

According to the three criteria $P_{1},P_{2} \ { and }\, P_{3}$, the three fleets $E_{1},E_{2} \; { and } \ E_{3}$ and the company $E_{0}$ are evaluated.

Because it is difficult to define the state of each $E_{j}(j=0,1,2,3)$ under each criterion in precisely mathematical language, the fuzzy set theory is used to describe them in this study. By aggregating decision information, the evaluation results have the following matrix

Obviously, this is actually a pattern recognition problem in hesitant fuzzy setting. For more details about pattern recognition, please see Nguyen (2015). Therefore, the distance measures proposed in Sect. 3 are used to solve this problem.

Firstly, for the three criteria $P_{1},P_{2} and P_{3}$, we calculate the LCM of the cardinal numbers of their corresponding HFEs, and the results are

$$\begin{aligned}&\mathrm{LCM}_{k=0}^{3}[h_{E_{k}}(P_{1})]=\mathrm{LCM}_{k=0}^{3}[h_{E_{k}}(P_{2})]\\&\quad =\mathrm{LCM}_{k=0}^{3}[h_{E_{k}}(P_{3})]=12. \end{aligned}$$

Subsequently, by Eq. (7), the four HFSs $E_{k}(k=0,1,2,3)$ are extended into four multi-sets, where

$$\begin{aligned} h_{E_{1}}^{\prime }(P_{1})= & {} \left\{ 0.80,0.80,0.80,0.80,0.85,0.85,0.85,0.85,\right. \\&\left. 0.90, 0.90, 0.90, 0.90 \right\} ,\\ h_{E_{2}}^{\prime }(P_{1})= & {} \left\{ 0.86,0.86,0.86,0.86,0.86,0.86,0.88,\right. \\&\left. 0.88,0.88, 0.88, 0.88, 0.88 \right\} ,\\ h_{E_{3}}^{\prime }(P_{1})= & {} \left\{ 0.68,0.68,0.68, 0.72,0.72,0.72, 0.74,0.74,\right. \\&\left. 0.74,0.75,0.75,0.75 \right\} ,\\ h_{E_{0}}^{\prime }(P_{1})= & {} \left\{ 0.80,0.80,0.80,0.80,0.80,0.80,0.85,\right. \\&\left. 0.85,0.85,0.85,0.85,0.85 \right\} ,\\ h_{E_{1}}^{\prime }(P_{2})= & {} \left\{ 0.72,0.72,0.72,0.72,0.75,0.75,0.75,0.75,\right. \\&\left. 0.78,0.78,0.78,0.78 \right\} ,\\ h_{E_{2}}^{\prime }(P_{2})= & {} \left\{ 0.82,0.82,0.82,0.82,0.82,0.82,0.87,0.87,\right. \\&\left. 0.87,0.87,0.87,0.87 \right\} ,\\ h_{E_{3}}^{\prime }(P_{2})= & {} \left\{ 0.78,0.78,0.78,0.80,0.80,0.80,0.85,0.85,\right. \\&\left. 0.85,0.88,0.88,0.88 \right\} ,\\ h_{E_{0}}^{\prime }(P_{2})= & {} \left\{ 0.82,0.82,0.82,0.82,0.82,0.82,0.88,0.88,\right. \\&\left. 0.88,0.88,0.88,0.88 \right\} ,\\ h_{E_{1}}^{\prime }(P_{3})= & {} \left\{ 0.85,0.85,0.85,0.85,0.88,0.88,0.88,\right. \\&\left. 0.88,0.90,0.90,0.90,0.90 \right\} ,\\ h_{E_{2}}^{\prime }(P_{3})= & {} \left\{ 0.70,0.70,0.70,0.70,0.70,0.70,0.72,\right. \\&\left. 0.72,0.72,0.72,0.72,0.72 \right\} ,\\ h_{E_{3}}^{\prime }(P_{3})= & {} \left\{ 0.82,0.82,0.82,0.85,0.85,0.85,0.87,0.87,\right. \\&\left. 0.87,0.90,0.90,0.90 \right\} ,\\ h_{E_{0}}^{\prime }(P_{3})= & {} \left\{ 0.85,0.85,0.85,0.85,0.85,0.85,0.90,\right. \\&\left. 0.90,0.90,0.90,0.90,0.90 \right\} . \end{aligned}$$

Then, for any $k=0,1,2,3$ and $i=1,2,3$, all the elements of each $h_{E_{k}}^{\prime }(P_{i})$ are ranked with monotone decreasing order.

With the weights of $P_{1},P_{2}$ and $P_{3}$, we calculate the distances between $E_{0}$ and $E_{k}(k=1,2,3)$ by Eq. (12) respectively, and the results are listed in Table 1.

Table 1 Results obtained by distance measure $d_{ftwg}$

In Table  1, we see that the pattern recognition results are influenced by the parameter $\lambda$. Specifically, when $\lambda \le 6$, the company should join the fleet $E_{1}$; when $\lambda \ge 7$, the company should join the fleet $E_{3}$. In real life, when $\lambda \le 6$, the overall evaluation information of the entire committee of experts is appreciated, and there is relatively little influence of unduly large or small evaluation values; meanwhile, when $\lambda \ge 7$, the evaluation information of the individual influence is appreciated, and there is relatively big influence of unduly large or small evaluation values. Furthermore, the larger the parameter $\lambda$ is, the bigger the influence of the unduly large or small evaluation values is. Therefore, to solve a specific pattern recognition problem, researchers should firstly choose a suitable parameter $\lambda$ according to the specific pattern recognition circumstance.

4.2 Example 2

On urban streets, different types of bus stops could have distinct impact on the operation of various road users. Then, it is very important to choose a suitable stop type while a new bus stop needs to be built. In most cases in China, the bus stops are usually located near the bike lanes or bike paths on the right side of urban streets. In this example, four types of bus stops are considered: (1) Type 1. Near this kind of bus stop, the bike lane is not physically separated from the vehicle lanes. When a bus arrives, bicyclists would go through from either the right or left side of the bus. (2) Type 2. Near this kind of bus stop, the bike lane is physically separated from the vehicle lanes. At the bus stop, the physical structure is as the same as Type 1. (3) Type 3. The bicycle lane is totally physically separated from the vehicle lanes, and the bus stop is a curbside design. (4) Type 4. The bicycle lane is totally physically separated from the vehicle lanes, and the bus stop is designed with a bus bay. For convenience, the above four kinds of bus stops are denoted by $A_{1},~A_{2},~A_{3}$ and $A_{4}$.

In Pudong district, Shanghai city, P. R. China, a new bus stop is designed to be built on a given road. Since that the road is an important new sub-arterial one in the five-year plan of the local government, all of the four kinds of bus stops meet the requirements of the related national codes (Ministry of Housing and Urban-Rural Development of the People’s Republic of China 2010; Ministry of Construction of the People’s Republic of China 1995; Ministry of Housing and Urban-Rural Development of the People’s Republic of China 2006). Then, a committee of experts is organized to choose the best stop type.

According to the field investigation, the local administrators mainly assess the effect of bus stop on the traffic flows. So the space cost is not included in the factors of bus stop construction. In this study, the committee considers six attributes, i.e., “saturation status of bus arrival flow at rush hour,” “saturation status of vehicle flow at rush hour,” “saturation status of bicycle flow at rush hour,” “saturation status of bus arrival flow at non-rush hour,” “saturation status of vehicle flow at non-rush hour” and “saturation status of bicycle flow at non-rush hour” on the studied road (Zhang 2016). For the convenience, the six attributes are denoted by $f_{1},~f_{2},~f_{3},~f_{4},~f_{5}$ and $f_{6}$.

To choose the suitable stop type, 12 typical roads in Yangzi delta area (around Shanghai city) are investigated in that they have the similar traffic lane conditions as the studied one and different stop types. Thereafter, the traffic flow conditions of the studied road and the chosen 12 roads are assessed through our proposed methods. Considering that even on the same road, the above six attribute values are changing at any time in a day, and the fuzzy set theory rather than precise crisp set theory is incorporated into the study. The evaluated membership degree of the traffic flow conditions of the chosen 12 roads to each attribute $f_{i}(i=1,2,\ldots ,6)$ is listed in Table  2. Denote the studied road by $A_{0}$, the membership degree of the traffic flow condition of $A_{0}$ to each attribute $f_{i}(i=1,2,\ldots ,6)$ in five years can be predicted as in Table  3.

Table 2 Traffic flow conditions of the chosen 12 roads

Table 3 Forecast traffic flow conditions of the studied road

After intensive communication with the committee, a weight vector for $f_{1},f_{2},\ldots ,f_{6}$ is obtained as $W=(0.25,0.25,0.2,0.1,0.1,0.1)$. Then, the new proposed distance measures on HFSs are utilized to choose the suitable stop type.

By Eq. (1), the attribute values of Tables  2 and  3 are transferred as the matrix

Obviously, the stop type selection is transferred into a pattern recognition problem in hesitant fuzzy setting. In what follows, the distance measures proposed in Sect. 3 are used to solve this problem.

Firstly, for the six criteria $f_{1},f_{2},\ldots ,f_{6}$, we calculate the LCM of the cardinal numbers of all $h_{A_{k}}(f_{i})(i=1,2,\ldots ,6;k=0,1,2,3,4)$, and the results are

$$\begin{aligned} \mathrm{LCM}_{k=0}^{4}[h_{A_{k}}(f_{i})]=4 \end{aligned}$$

for any $i=0,1,2,\ldots ,6$.

Table 4 Results obtained by distance measure $d_{ftwg}$

Subsequently, by Eq. (7), the HFS $A_{0}$ is extended into a multi-set, where

$$\begin{aligned} h_{A_{0}}^{\prime }(f_{1})= & {} \left\{ 0.75,0.75,0.80,0.80 \right\} ,\nonumber \\ ~~ h_{A_{0}}^{\prime }(f_{2})= & {} \left\{ 0.80,0.80,0.85,0.85 \right\} ,\\ h_{A_{0}}^{\prime }(f_{3})= & {} \left\{ 0.74,0.74,0.76,0.76 \right\} ,\\ h_{A_{0}}^{\prime }(f_{4})= & {} \left\{ 0.55,0.55,0.65,0.65\right\} ,\\ h_{A_{0}}^{\prime }(f_{5})= & {} \left\{ 0.59,0.59,0.61,0.61 \right\} ,\\ h_{A_{0}}^{\prime }(f_{6})= & {} \left\{ 0.50,0.50,0.54,0.54\right\} , \end{aligned}$$

and the four HFSs $A_{k}(k=1,2,3,4)$ keep the same as their original values, i.e., $h_{A_{k}}^{\prime }(f_{i})$ is the same as which of the matrix $H_{2}$ for any $i=1,2,3,4,5,6$, $k=1,2,3,4$. Obviously, all $h_{A_{k}}^{\prime }(f_{i})(i=1,2,3,4,5,6;j=0,1,2,3,4)$ are ranked with monotonically decreasing order.

Note that the weights of $f_{1},f_{2},\ldots ,f_{6}$ are known, we calculate the distances between $A_{0}$ and $A_{k}(k=1,2,3,4)$ by Eq. (12), respectively, and the results are listed in Table 4.

In Table  4, we see that the suitable bus stop type is influenced by the parameter $\lambda$. Specifically, when $\lambda \le 3$, the studied road should be build with the stop as $A_{2}$; when $4 \le \lambda \le 15$, the studied road should be build with the stop as $A_{1}$; when $16 \le \lambda$, the studied road should be build with the stop as $A_{2}$. The explanation of the above results are as follows. (i) The condition that $\lambda \le 3$ means that the overall evaluation information of the entire committee of experts is appreciated, and there is relatively little influence of unduly large or small evaluation values. (ii) The condition that $4 \le \lambda \le 15$ means that the evaluation information of the individual influence is appreciated, and there is influence of unduly large or small evaluation values. (iii) The condition that $16 \le \lambda$ means that the evaluation information of the individuals influence is appreciated, and there is relatively much larger influence of the relatively more fewer unduly large or small evaluation values than the condition $4 \le \lambda \ge 15$. Obviously, the larger the parameter $\lambda$ is, the bigger the influence of the unduly large or small evaluation values is. Therefore, to solve this specific pattern recognition problem, researchers should firstly choose a suitable parameter $\lambda$ according to the specific pattern recognition circumstance.

5 Conclusion

In this paper, the HFSs are studied from the view of the cardinality theory. And then, a series of improved distance measures on HFSs, which take the cardinal numbers of the HFSs into account, are proposed by virtue of the concept “multi-sets.” According to the two illustrative examples, some conclusions are obtained as follows.

(1) When calculating the distance of two HFEs with different cardinal numbers, the past distance measures on HFSs usually extend the shorter HFE until they have the same cardinal number. Since there are many values in the shorter HFE which could be empirically selected by the decision makers, the extending process is usually dealt with optionally which weakens the objectivity of the calculating process. By contrast, the novel distance measures $d_{fth}(\cdot )$, $d_{fte}(\cdot )$, $d_{ftg}(\cdot )$ and $d_{ftwg}(\cdot )$ proposed in this paper overcome the above shortcoming by using the concept “hesitant fuzzy multi-sets.” which ensures that the calculating process is objective.

(2) In the novel distance measures $d_{fth}(\cdot )$, $d_{fte}(\cdot )$, $d_{ftg}(\cdot )$ and $d_{ftwg}(\cdot )$ proposed in this paper, it is noteworthy that the calculation results of the distance of any two HFSs are influenced by the parameter $\lambda$. The smaller the parameter $\lambda$ is, the more the group’s influence is appreciated, whereas the bigger the parameter $\lambda$ is, the more the individual’s influence is appreciated. The underlying cause of this phenomenon is the concept “a large number annihilating a small number in addition operation of two numbers (Goldberg 1991).” In this paper, by using the parameter $\lambda$, the novel distance measures could combine the subjective and objective decision-making information well.

(3) The novel proposed distance measures $d_{fth}(\cdot )$, $d_{fte}(\cdot )$, $d_{ftg}(\cdot )$ and $d_{ftwg}(\cdot )$ on HFSs all satisfy Definition 3 in their calculation process, i.e., they are valued-based distance measures which satisfy the triangle inequality property.

(4) Similar to the novel distance measures proposed in this paper, one could get the improved distance measures on dual HFSs (Singh 2014) and on interval-valued HFSs (Farhadinia 2013). Since the principal of them is the same, they are not explored here. Besides, to concentrate our energies on explaining the new proposed distance measures clearly, the concept “hesitance degree” is also not mentioned (Li et al. 2015a).