1 Introduction

The fuzzy set was proposed by Zadeh [1] has achieved a great success in various fields, which is considered to be an effective tool to solve the decision making problems, pattern recognition and fuzzy inference [2,3,4]. Due to the fuzziness and uncertainty in the multiple criteria decision making problems, it is difficult to establish the membership degree of fuzzy sets. In order to deal with these situations, Torra [5, 6] introduced the hesitant fuzzy sets (HFSs), which contain several possible values of the membership degree of an element in [0,1], and many researchers have begun to focus on this subject. For example, Xia et al. [7] put forward a series of aggregation operators for hesitant fuzzy information, and Farhadinia [8] developed a series of score functions for HFSs. Recently, Rodr\(\acute{\imath }\)guez et al. [9] reviewed the existing research results of HFSs and discussed their future trends. However, in some situations, it is difficult to use the exact values to present qualitative evaluation because of the complexity of the problems, people often provide their opinions linguistically. In some practical decision making problems, the decision maker regards the linguistic information as the values of linguistic variables, that is to say, the values of the variables are not expressed in terms of numbers but in linguistic terms, such as ”good”, ”better”, ”fair”, ”slightly worse”, ”poor”, etc. Up to now, many people have studied the linguistic multiple criteria decision making problems, Herrera et al. [10] proposed the linguistic assessments in group decision making (GDM) problem in 1993, then Herrera et al. [11] proposed a consensus model for group decision making based on linguistic evaluations information, and Herrera et al. [12] considered several group decision making processes using linguistic ordered weighted averaging (LOWA) operator. Later, Xu [13] proposed a group decision making method based on the uncertain linguistic ordered weighted geometric (LOWG) and the induced uncertain LOWG operators. Furthermore, Xu [14] presented the linguistic hybrid aggregation (LHA) operator and applied it to group decision making. The research on this field has been growing rapidly [15,16,17,18,19,20,21]. However, in some situations, the decision maker cannot express his/her preferences by using just one membership degrees of a linguistic due to the uncertainty of the problem. In order to express the decision makers’ hesitance that exists in the given associated membership degrees of one linguistic term, Rodr\(\acute{\imath }\)guez et al. [22] proposed the hesitant fuzzy linguistic term sets (HFLTSs) based on the HFSs and linguistic term sets (LTSs). The element of the HFLTSs is called hesitant fuzzy linguistic number (HFLN). For example, \(\langle s_{3},(0.2,0.6)\rangle\) is an HFLN, and 0.2, 0.6 are the possible membership degrees of the linguistic term \(s_{3}\). The HFLTSs have made great progress in describing linguistic information and to some extent it can be regarded as an innovative construct.

Distance measure is an important topic in the fuzzy set theory, and it is widely used in some fields, such as decision making, pattern recognition and so on. The concept of distance measures in decision making problems makes it possible to make a decision by comparing different alternatives with the ideal one. The most widely used distance measures are the Hausdorff distance, Hamming distance, Euclidean distance and generalized distance [23,24,25,26,27,28]. Based on these distance measures, people have promoted a lot of extensions. For example, Szmidt [29] gave a distance measure between intuitionistic fuzzy sets based on the Hausdorff metric, Xu et al. [30] introduced a number of distance measures of hesitant fuzzy sets and proposed some hesitant ordered weighted distance operators. Considering the hesitation degree of each element in the fuzzy set, Li et al. [31] proposed a distance measure between hesitant fuzzy sets. Xu et al. [32] extended the distance measure to the linguistic fuzzy sets and developed the linguistic distance operators. Wang et al. [33] introduced the Hausdorff distance based on HFLNs and linguistic scale functions, but the hesitance degree of the hesitant fuzzy elements is not considered. Peng et al. [34] developed an enhanced relative ration method for multi-criteria decision making problem via attitudinal distance measures of interval-valued hesitant fuzzy sets. In fact, the hesitancy degree of the hesitant fuzzy set is the most important character. Based on the literature discussed above, we introduce some new class of distance measures for hesitant fuzzy linguistic sets which include the hesitance degree of hesitant fuzzy element and the linguistic scale function together. The main advantage of the proposed distance measures is that it is not only consider the hesitance of the hesitant fuzzy elements but also deal with linguistic transformation problem under different semantic situations.

The rest of the paper is organized as follows. In Sect. 2, some basic concepts of HFSs, LTSs, HFLSs and linguistic scale functions are briefly reviewed. In Sect. 3, we first introduce the distance measures between HFLNS, and we also define the continuous distance measure between two collections of hesitant fuzzy linguistic term sets (HFLTSs), then discussed some related properties. In Sect. 4, the TOPSIS method for solving multiple criteria decision making (MCDM) problems with HFLSs are developed. In Sect. 5, we give an application of the proposed distance measures between HFLSs, and make comparison analysis with the existing method. The conclusions are given in Sect. 6.

2 Preliminaries

In this section, we will review and discuss some related basic concepts, including HFSs, LTSs, HFLSs and linguistic scale functions.

2.1 HFSs

Definition 1

(Torra [5]) Given a fixed set \(X=\{x_{1},x_{2},\ldots ,x_{n}\}\), then a hesitant fuzzy set(HFS) E on X is often expressed by:

$$\begin{aligned} E=\{<x,h(x)>|x\in X\}, \end{aligned}$$

where h(x) is a set of some values in [0,1]. It denotes the possible membership degree of the element \(x\in X\) to the set E. h(x) is called the hesitant fuzzy element (HFE).

Definition 2

(Li [31]) Let \(h(x_{i})\) be the hesitant fuzzy element on \(X=\{x_{1},x_{2},\ldots ,x_{n}\}\), and for any \(x_{i}\in X\), \(l(h(x_{i})\) represents the number of element in \(h(x_{i})\), then

$$\begin{aligned} u(h(x_{i}))=1-\frac{1}{l(h(x_{i})} \end{aligned}$$

is defined as the hesitance degree of \(h(x_{i})\).

Example 1

Let \(h_{1}(x)=\{0.2,0.3,0.4\}\), \(h_{2}(x)=\{0.3,0.5\}\), then \(u(h_{1}(x))=1-\frac{1}{3}=\frac{2}{3}\), \(u(h_{2}(x))=1-\frac{1}{2}=\frac{1}{2}\), the hesitance degree of \(h_{1}(x)\) is greater than that of \(h_{2}(x)\).

Throughout this paper, we assume l(h(x) represents the number of elements in h(x). When we calculate the distance between two HFSs \(h_{1}(x)\) and \(h_{2}(x)\), if the number of elements is not equal, that is \(l(h_{1}(x)\ne l(h_{2}(x)\), we can let \(l=\max \{l(h_{1}(x),l(h_{2}(x)\}\). In order to solve this problem, Xu et al. [30] gave the rules of regulation: the set of fewer number of elements is extended by adding the minimum value, maximum value or any value in it until it has the same number with the more one. When we add the element to the fewer one, the best way is to add the same value in it. In the practical application, the choice of the added element is related to the risk preference of the decision maker. The optimists want to add the maximum value, while the pessimists want to add the minimum value. The decision maker’s risk preferences directly affect the final decisions (Yu et al. [35]). In this paper, we extend the fewer one by adding the minimum value.

2.2 LTSs

The information expressed by the numerical values may be inconvenient. However, when we use a qualitative language, these inconvenience can be avoided. Suppose that \(S=\{s_{\alpha }|\alpha =0,1,\ldots ,\tau \}\) is a finite and totally ordered discrete term set, where \(s_{\alpha }\) represents a possible value for a linguistic variable. For example, a set of seven terms S could be given as follows:

$$\begin{aligned} S=\, & \{s_{0}=very\quad poor, s_{1}=poor, s_{2}=slightly\quad poor, s_{3}=fair, s_{4}=slightly\quad good, s_{5}=good,\\ s_{6}=\,& very\quad good\}. \end{aligned}$$

Xu [36] extended the discrete term set S to the continuous term set \(\overline{S}=\{s_{\alpha }|\alpha \in [0,t]\}\), where \(t(t>\tau )\) is a sufficiently large positive integer. Based on the extended linguistic term set \(\overline{S}\), let \(s_{\alpha },s_{\beta }\in \overline{S}\) and \(\lambda \in [0,1]\), and then the operational laws can be given as follows(Xu [37]):

  1. (1)

    \(s_{\alpha }\oplus s_{\beta }=s_{\alpha +\beta };\)

  2. (2)

    \(\lambda s_{\alpha }=s_{\lambda \alpha };\)

  3. (2)

    \(s_{\alpha }>s_{\beta }\) if \(\alpha >\beta\).

2.3 HFLTSs

Definition 3

(Rodr\(\acute{i}\)guez et al. [22]) Let \(X=\{x_{1},x_{2},\ldots ,x_{n}\}\) be a fixed set and \(s_{\alpha (x)}\in S\), and then an HFLS A in X is defined as:

$$\begin{aligned} A=\{\langle x,s_{A(x)},h_{A}(x)\rangle |x\in X\}, \end{aligned}$$

when \(X=\{x_{1},x_{2},\ldots ,x_{n}\}\) has only one element, the HFLS A is reduced to \(\langle s_{A(x)},h_{A}(x)\rangle\), we denote \(A=\langle s_{A(x)},h_{A}(x)\rangle =\langle s_{A},h_{A}\rangle\). For convenience, A is called as an hesitant fuzzy linguistic number(HFLN). An HFLN is a special case of HFLS, which is flexible for information evaluation.

Example 2

Let \(X=\{x_{1},x_{2},x_{3}\}\) be a universal set. An HFLS \(A=\{\langle x_{1},s_{3},\{0.3,0.4,0.5\}\rangle ,\langle x_{2},s_{4},\{0.3, 0.6,0.8\}\rangle ,\langle x_{3},s_{2},\{0.1,0.4,0.5\}\rangle \}\), it can be divided into three subsets, and each subset contains only one object, then \(\langle s_{3},\{0.3,0.4,0.5\}\rangle ,\langle s_{4},\{0.3,0.6,0.8\}\rangle ,\langle s_{2},\{0.1,0.4,0.5\}\rangle \}\) are HFLNs. For example, for \(\langle x_{1},s_{3},\{0.3, 0.4,0.5\}\rangle ,\) we know that 0.3, 0.4, 0.5 are the possible membership degrees of the element \(x_{1}\) belongs to \(s_{3}\).

2.4 Linguistic Scale Functions

As we all know, operations can’t be carried out straightly when fuzzy numbers are combined directly with the linguistic terms. The linguistic scaling function can assign different semantic values to linguistic terms under different circumstances. In practice, the language scaling functions are very popular because they are very flexible and they can give more deterministic results based on different semantics.

Definition 4

(Wang et al. [33]) Let S be a linguistic term, and \(S=\{s_{i}|i=0,1,\ldots ,2\tau \}.\) If \(\xi _{i}\) is a numeric value between 0 and 1, then the linguistic scale function f can be defined as follows:

$$\begin{aligned} f:s_{i}\rightarrow \xi _{i} (i=0,1,\ldots ,2\tau ), \end{aligned}$$

where \(0\le \xi _{0}<\xi _{1}<\cdots <\xi _{2\tau }\le 1\). The linguistic scale function is strictly monotonously increasing function with respect to \(s_{i}\), in fact, the function value represents the semantics of the linguistic terms.

Now, we introduce three kinds of linguistic scale functions as follows:

$$\begin{aligned} (1)\quad f_{1}(s_{i})=\xi _{i}=\frac{i}{2\tau }(i=0,1,\ldots ,2\tau ). \end{aligned}$$

The evaluation scale of the linguistic information expressed by \(f_{1}(s_{i})\) is divided on average.

$$\begin{aligned} (2)\quad f_{2}(s_{i})=\xi _{i}=\left\{ \begin{array}{llll} \frac{a^{\tau }-a^{\tau -i}}{2a^{\tau }-2},\ \ &{} \quad i=0,1,\ldots ,\tau ,\\ \frac{a^{\tau }+a^{i-\tau }-2}{2a^{\tau }-2},\ \ &{} \quad i=\tau +1,\tau +2,\ldots ,2\tau . \end{array}\right. \end{aligned}$$

For \(f_{2}(s_{i})\), the absolute deviation between adjacent linguistic sets will increase when we extend it from the middle of a given set of languages to both ends.

$$\begin{aligned} (3)\quad f(s_{i})=\xi _{i}=\left\{ \begin{array}{llll} \frac{\tau ^{\alpha ^{'}}-(\tau -i)^{\alpha ^{'}}}{2\tau ^{\alpha ^{'}}},\ \ &{} \quad i=0,1,\ldots ,\tau ,\\ \frac{\tau ^{\beta ^{'}}-(i-\tau )^{\beta ^{'}}}{2\tau ^{\beta ^{'}}},\ \ &{} \quad i=\tau +1,\tau +2,\ldots ,2\tau . \end{array}\right. \end{aligned}$$

For \(f_{3}(s_{i})\), the absolute deviation between adjacent linguistic sets will decrease when we extend it from the middle of a given set of languages to both ends.

The above linguistic scale functions can be developed to \(f^{*}:\overline{S}\rightarrow R^{+}\)(where \(R^{+}\) is a nonnegative real number), which is a continuous and strictly monotonically increasing function.

Definition 5

(Wang et al. [33]) The negation operator of HFLN \(\alpha\) is \(neg(\alpha )=\langle (f^{*})^{-1}(f^{*} (s_{2\tau })-f^{*}(s_{A})),\bigcup _{r\in h_{A}}\{1-r\}\rangle ,\) where \(f^{*}\) is the linguistic scale function.

3 New Distance Measures Between HFLSs

3.1 Distance Measures for Discrete HFLNs

Next, we will present some new class of distance measures between two HFLNs, which includes not only the hesitance degree value of h(x) but also the linguistic scale function \(f^{*}\).

Definition 6

Let \(A=\langle s_{A},h_{A}\rangle\) and \(B=\langle s_{B},h_{B}\rangle\) be any two HFLNs, and let \(f^{*}\) be a linguistic scale function. Then, the hesitant fuzzy linguistic Hamming distance between A and B can be defined as follows:

$$\begin{aligned} d_{hh}(A,B)=|f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|+\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|. \end{aligned}$$
(3.1)

The hesitant fuzzy linguistic Euclidean distance between A and B is defined as:

$$\begin{aligned} d_{he}(A,B)=[|f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|^{2}+\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|^{2}]^{\frac{1}{2}}. \end{aligned}$$
(3.2)

The hesitant fuzzy linguistic generalized distance between A and B is defined as:

$$\begin{aligned} d_{hg}(A,B)=[|f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|^{\lambda }+\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|^{\lambda }]^{\frac{1}{\lambda }}, \end{aligned}$$
(3.3)

where \(\lambda >0\), \(h_{A}^{j}\) and \(h_{B}^{j}\) are the jth values in \(h_{A}\) and \(h_{B}\), respectively, and \(l=\max \{l(h_{A}),l(h_{B})\}\).

Example 3

Let \(A=\langle s_{5},\{0.2,0.4\}\rangle\) and \(B=\langle s_{6},\{0.6,0.7,0.8\}\rangle\) be two HFLNs, and then we know \(u(h_{A})=\frac{1}{2},u(h_{B})=\frac{2}{3}\). If \(f^{*}(s_{i})=\frac{i}{2\tau }\) and \(\tau =3\), firstly, we extend \(h_{A}=\{0.2,0.2,0.4\}\), and then the number of elements in \(h_{A}\) and \(h_{B}\) are same, we have \(d_{hh}(A,B)=0.8084\), \(d_{he}(A,B)=0.5870.\)

If

$$\begin{aligned} f^{*}_{2}(s_{i})=\left\{ \begin{array}{llll} \frac{a^{\tau }-a^{\tau -i}}{2a^{\tau }-2},\ \ &{} \quad i=0,1,\ldots ,\tau ,\\ \frac{a^{\tau }+a^{i-\tau }-2}{2a^{\tau }-2},\ \ &{} \quad i=\tau +1,\tau +2,\ldots ,2\tau . \end{array}\right. \end{aligned}$$

and \(\tau =3,a=1.4,\) then \(d_{hh}(A,B)=0.7723\), \(d_{he}(A,B)=0.5682.\)

If

$$\begin{aligned} f^{*}_{3}(s_{i})=\left\{ \begin{array}{llll} \frac{\tau ^{\alpha ^{'}}-(\tau -i)^{\alpha ^{'}}}{2\tau ^{\alpha ^{'}}},\ \ &{} \quad i=0,1,\ldots ,\tau ,\\ \frac{\tau ^{\beta ^{'}}-(i-\tau )^{\beta ^{'}}}{2\tau ^{\beta ^{'}}},\ \ &{} \quad i=\tau +1,\tau +2,\ldots ,2\tau . \end{array}\right. \end{aligned}$$

and \(\tau =3,\alpha ^{'}=\beta ^{'}=0.8,\) then \(d_{hh}(A,B)=0.5278\), \(d_{he}(A,B)=0.7062.\)

Theorem 1

Let \(A=\langle s_{A},h_{A}\rangle\), \(B=\langle s_{B},h_{B}\rangle\) and \(V=\langle s_{V},h_{V}\rangle\) be any three HFLNs, and let \(f^{*}\) be a linguistic scale function. Then, the hesitant fuzzy linguistic generalized distance \(d_{hg}\) satisfies the following properties:

  1. (1)

    \(d_{hg}(A,B)\ge 0\);

  2. (2)

    \(d_{hg}(A,B)=d_{hg}(B,A);\)

  3. (3)

    \(d_{hg}(A,B)=0\) if and only if \(A=B\);

  4. (4)

    For three HFLNs A, B, V,  if \(s_{A}\le s_{B}\le s_{V}\), \(h_{k}(x)=\{h_{k}^{1}(x),h_{k}^{2}(x),\ldots , h_{k}^{l}(x)\}(k=A,B,V)\) and \(h_{A}^{j}(x)\le h_{B}^{j}(x)\le h_{V}^{j}(x)(j=1,2,\ldots ,l)\), then we have \(d_{hg}(A,B)\le d_{hg}(A,V)\le\) and \(d_{hg}(B,V)\le d_{hg}(A,V).\)

Proof

Properties (1), (2) and (3) are obvious; here, we only present the proof of property (4).

Since \(s_{A}\le s_{B}\le s_{V}\), \(0\le h_{A}^{j}(x)\le h_{B}^{j}(x)\le h_{V}^{j}(x)(j=1,2,\ldots ,l)\), and \(f^{*}\) is a continuous and strictly monotonically increasing function, we have \(f^{*}(s_{A})\le f^{*}(s_{B})\le f^{*}(s_{V})\), then

$$\begin{aligned}&f^{*}(s_{A})h_{A}^{j}(x)\le f^{*}(s_{B})h_{B}^{j}(x)\le f^{*}(s_{V})h_{V}^{j}(x),\\&|f^{*}(s_{A})h_{A}^{j}(x)-f^{*}(s_{B})h_{B}^{j}(x)|^{\lambda }\le |f^{*}(s_{A})h_{A}^{j}(x)- f^{*}(s_{V})h_{V}^{j}(x)|^{\lambda }. \end{aligned}$$

Therefore,

$$\begin{aligned}&\left[ |f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|^{\lambda }+\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|^{\lambda }\right] ^{\frac{1}{\lambda }}\\&\quad \le \left[ |f^{*}(s_{A})u(h_{A})-f^{*}(s_{V})u(h_{V})|^{\lambda }+\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{V})h_{V}^{j}|^{\lambda }\right] ^{\frac{1}{\lambda }}. \end{aligned}$$

\(d_{hg}(B,V)\le d_{hg}(A,V)\) can be proven in a similar way. Here, we complete the proof of property (4). \(\square\)

Remark

When \(\lambda =1\) or \(\lambda =2\), \(d_{hg}\) is reduced to \(d_{hh}\) or \(d_{he}\), respectively.

If we have different preferences for the membership values and the hesitance degree, we can present the distance measures with preference as follows:

Definition 7

Let \(A=\langle s_{A},h_{A}\rangle\) and \(B=\langle s_{B},h_{B}\rangle\) be any two HFLNs, and let \(f^{*}\) be a linguistic scale function. Then, the hesitant fuzzy linguistic Hamming distance between A and B with different preferences for the membership values and the hesitance degrees can be defined as follows:

$$\begin{aligned} d_{phh}(A,B)=p|f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|+(1-p)\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|. \end{aligned}$$

The hesitant fuzzy linguistic Euclidean distance between A and B with different preferences for the membership values and the hesitance degree is defined as follows:

$$\begin{aligned} d_{phe}(A,B)=\left[ p|f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|^{2}+(1-p)\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|^{2}\right] ^{\frac{1}{2}}. \end{aligned}$$

The hesitant fuzzy linguistic generalized distance between A and B with different preferences for the membership values and the hesitance degree can be defined as follows:

$$\begin{aligned} d_{phg}(A,B)=\left[ p|f^{*}(s_{A})u(h_{A})-f^{*}(s_{B})u(h_{B})|^{\lambda }+(1-p)\frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{A})h_{A}^{j}-f^{*}(s_{B})h_{B}^{j}|^{\lambda }\right] ^{\frac{1}{\lambda }}, \end{aligned}$$
(3.4)

where \(\lambda >0\), \(0\le p\le 1\), \(h_{A}^{j}\) and \(h_{B}^{j}\) are the jth values in \(h_{A}\) and \(h_{B}\), respectively, and \(l=\max \{l(h_{A}),l(h_{B}\}\).

Remark

If \(p=0\), which means we ignore the influence of the hesitance degree of hesitant fuzzy element, the distance measure reduces to the metric distance measure that we often consider.

Theorem 2

The distance measures \(d_{phh}(A,B),d_{phe}(A,B)\) and \(d_{phg}(A,B)\) satisfy the properties (1)–(4) in Theorem 1.

The proof is obvious.

3.2 Distance Measures for Continuous HFLSs

The distance measures defined above are discrete, and we only consider the distance measures of hesitant fuzzy linguistic term set over one linguistic variable. But in many practical situations, we should consider different criteria in multiple criteria decision making. Sometimes, we also need to consider the weight of different criteria. When the universe of discourse X and the weights of elements are continuous, and the evaluation information is represented by collections of hesitant fuzzy linguistic term set, we can introduce the distance measures between hesitant fuzzy linguistic term sets from continuous case, which are defined as follows:

Definition 8

Let \(X=[c,d]\) be a continuous universe of discourse, \({\bar{A}}=\langle s_{{\bar{A}}(x)},h_{{\bar{A}}}(x)\rangle\) and \({\bar{B}}=\langle s_{{\bar{B}}(x)},h_{{\bar{B}}}(x)\rangle\) be any two collections of hesitant fuzzy linguistic term sets over the element \(x(x\in X)\). The weight of x is \(\omega (x)\), satisfying \(\omega (x)\in [0,1]\) and \(\int _{c}^{d}\omega (x)dx=1\), and let \(f^{*}\) be a linguistic scale function. Then, based on (3.1) (3.2) and (3.3), the continuous weighted hesitant fuzzy linguistic Hamming distance between \({\bar{A}}\) and \({\bar{B}}\) can be defined as follows:

$$\begin{aligned} d_{chh}({\bar{A}},{\bar{B}})= & {} \int _{c}^{d}\omega (x)|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|dx\nonumber \\&+\,\int _{c}^{d}\omega (x) \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|\mathrm{d}x. \end{aligned}$$
(3.5)

The continuous weighted hesitant fuzzy linguistic Euclidean distance between \({\bar{A}}\) and \({\bar{B}}\) can be defined as:

$$\begin{aligned} d_{che}({\bar{A}},{\bar{B}})= & {} \left[ \int _{c}^{d}\omega (x)|f^{*}(s_{{\bar{A}}(x)})u (h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{2}dx\right. \nonumber \\&\left. +\,\int _{c}^{d}\omega (x) \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*} (s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|^{2}dx\right] ^{\frac{1}{2}}. \end{aligned}$$
(3.6)

The continuous weighted hesitant fuzzy linguistic generalized distance between \({\bar{A}}\) and \({\bar{B}}\) is defined as follows:

$$\begin{aligned} d_{chg}({\bar{A}},{\bar{B}})= & {} \left[ \int _{c}^{d}\omega (x)|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{\lambda }\mathrm{d}x\right. \nonumber \\&\left. +\,\int _{c}^{d}\omega (x) \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}} (x)})h_{{\bar{B}}}^{j}(x)|^{\lambda }\mathrm{d}x\right] ^{\frac{1}{\lambda }}, \end{aligned}$$
(3.7)

where \(\lambda >0,\) for \(x\in X\), \(h_{{\bar{A}}}^{j}(x)\) and \(h_{{\bar{B}}}^{j}(x)\) are the jth values in \(h_{{\bar{A}}}(x)\) and \(h_{{\bar{B}}}(x)\), respectively, and \(l=\max \{l(h_{{\bar{A}}}(x)),l(h_{{\bar{B}}}(x))\}\).

Remark

For \(x\in [c,d],\) if \(\omega (x)=\frac{1}{d-c}\), then (3.5)–(3.7) can be reduced to the continuous normalized hesitant fuzzy linguistic Hamming distance measure \(d_{cnhh}\), the continuous normalized hesitant fuzzy linguistic Euclidean distance measure \(d_{cnhe}\) and the continuous normalized hesitant fuzzy linguistic generalized distance measure \(d_{cnhg}\) between collections of hesitant fuzzy linguistic term sets \({\bar{A}}\) and \({\bar{B}},\) respectively, which can be shown as:

$$\begin{aligned} d_{cnhh}({\bar{A}},{\bar{B}})= & {} \frac{1}{d-c}|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|\\&+\frac{1}{d-c}\int _{c}^{d} \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|\mathrm{d}x.\\ d_{cnhe}({\bar{A}},{\bar{B}})= & {} \left[ \frac{1}{d-c}|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{2}\right. \\&\left. +\frac{1}{d-c}\int _{c}^{d} \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*} (s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|^{2}\mathrm{d}x\right] ^{\frac{1}{2}}.\\ d_{cnhg}({\bar{A}},{\bar{B}})= & {} \left[ \frac{1}{d-c}|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{\lambda }\mathrm{d}x\right. \\&\left. +\frac{1}{d-c}\int _{c}^{d} \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)}) h_{{\bar{B}}}^{j}(x)|^{\lambda }\mathrm{d}x\right] ^{\frac{1}{\lambda }}, \end{aligned}$$

where \(\lambda >0.\)

Next, we will consider different preferences for the membership values and the hesitance degree in the continuous hesitant fuzzy linguistic distance measures, and we can give the definition of the continuous hesitant fuzzy linguistic distance measures with preference as follows:

Definition 9

Let \(X=[c,d]\) be a continuous universe of discourse, \({\bar{A}}=\langle s_{{\bar{A}}(x)},h_{{\bar{A}}}(x)\rangle\) and \({\bar{B}}=\langle s_{{\bar{B}}(x)},h_{{\bar{B}}}(x)\rangle\) be any two collections of hesitant fuzzy linguistic term sets over the element \(x(x\in X)\). The weight of x is \(\omega (x)\), satisfying \(\omega (x)\in [0,1]\) and \(\int _{c}^{d}\omega (x)dx=1\), and let \(f^{*}\) be a linguistic scale function. Then, the continuous weighted hesitant fuzzy linguistic Hamming distance between \({\bar{A}}\) and \({\bar{B}}\) with different preferences for the membership values and the hesitance degree can be defined as follows:

$$\begin{aligned} d_{pchh}({\bar{A}},{\bar{B}})= & {} p\int _{c}^{d}\omega (x)|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|\mathrm{d}x\nonumber \\&+(1-p)\int _{c}^{d}\omega (x) \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|\mathrm{d}x. \end{aligned}$$
(3.8)

The continuous weighted hesitant fuzzy linguistic Euclidean distance between \({\bar{A}}\) and \({\bar{B}}\) with different preferences for the membership values and the hesitance degree is defined as:

$$\begin{aligned} d_{pche}({\bar{A}},{\bar{B}})= & {} \left[ p\int _{c}^{d}\omega (x)|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{2}\mathrm{d}x\right. \nonumber \\&\left. +(1-p)\int _{c}^{d}\omega (x) \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}} (x)})h_{{\bar{B}}}^{j}(x)|^{2}\mathrm{d}x\right] ^{\frac{1}{2}}. \end{aligned}$$
(3.9)

The continuous weighted hesitant fuzzy linguistic generalized distance between \({\bar{A}}\) and \({\bar{B}}\) with different preferences for the membership values and the hesitance degree can be defined as follows:

$$\begin{aligned} d_{pchg}({\bar{A}},{\bar{B}})= & {} \left[ p\int _{c}^{d}\omega (x)|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{\lambda }dx\right. \nonumber \\&\left. +(1-p)\int _{c}^{d}\omega (x) \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|^{\lambda }dx \right] ^{\frac{1}{\lambda }}, \end{aligned}$$
(3.10)

where \(\lambda >0,0\le p\le 1,\) for \(x\in X\), \(h_{{\bar{A}}}^{j}(x)\) and \(h_{{\bar{B}}}^{j}(x)\) are the jth values in \(h_{{\bar{A}}}(x)\) and \(h_{{\bar{B}}}(x)\), respectively, and \(l=\max \{l(h_{{\bar{A}}}(x)),l(h_{{\bar{B}}}(x))\}\).

Remark

Naturally, for \(x\in [c,d],\) if \(\omega (x)=\frac{1}{d-c}\), then (3.8)–(3.10) are reduced to the continuous normalized hesitant fuzzy linguistic Hamming distance measure \(d_{pcnhh}\), the continuous normalized hesitant fuzzy linguistic Euclidean distance measure \(d_{pcnhe}\) and the continuous normalized hesitant fuzzy linguistic generalized distance measure \(d_{pcnhg}\) between hesitant fuzzy linguistic term sets \({\bar{A}}\) and \({\bar{B}}\) with different preferences for the membership values and the hesitance degree, respectively, which can be shown as:

$$\begin{aligned} d_{pcnhh}({\bar{A}},{\bar{B}})= & {} \frac{p}{d-c}|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|\\&+\frac{1-p}{d-c}\int _{c}^{d} \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|\mathrm{d}x.\\ d_{pcnhe}({\bar{A}},{\bar{B}})= & {} \left[ \frac{p}{d-c}|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{2}\right. \\&\left. +\frac{1-p}{d-c}\int _{c}^{d} \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|^{2}\mathrm{d}x\right] ^{\frac{1}{2}}.\\ d_{pcnhg}({\bar{A}},{\bar{B}})= & {} \left[ \frac{p}{d-c}|f^{*}(s_{{\bar{A}}(x)})u(h_{{\bar{A}}}(x))-f^{*}(s_{{\bar{B}}(x)}) u(h_{{\bar{B}}}(x))|^{\lambda }dx\right. \\&\left. +\frac{1-p}{d-c}\int _{c}^{d} \frac{1}{l}\sum _{j=1}^{l} |f^{*}(s_{{\bar{A}}(x)})h_{{\bar{A}}}^{j}(x)-f^{*}(s_{{\bar{B}}(x)})h_{{\bar{B}}}^{j}(x)|^{\lambda }\mathrm{d}x\right] ^{\frac{1}{\lambda }}, \end{aligned}$$

where \(\lambda >0,0\le p\le 1.\)

Remark

We can easy know the continuous hesitant fuzzy linguistic distance measures satisfy the properties (1)-(4) in Theorem 1.

4 Multiple Criteria Decision Making with Distance Measures Based on TOPSIS Approach

In this section, we apply the proposed distance measures for HFLSs to develop a multiple criteria decision making(MCDM) method. Suppose the decision makers decide to choose the best one from m alternatives according to n criteria. Let \(X=\{X_{1},X_{2},\ldots ,X_{m}\}\) be a set of alternatives, and \(C=\{C_{1},C_{2},\ldots ,C_{n}\}\) be a set of criteria. A number of decision makers provide their evaluation values of HFLSs \(X_{ij}=\langle s_{\theta _{ij}},h_{ij}\rangle (i=1,2,\ldots ,m;\;j=1,2,\ldots ,n)\) for alternative \(X_{i}\) with respect to \(C_{j}\). The weight of criteria \(C_{j}\) is \(\omega _{j}\), satisfying \(\omega _{j}\ge 0\;(j=1,2,\ldots ,n)\) and \(\sum _{j=1}^{n}\omega _{j}=1\), and the decision matrix D with HFLSs information is formed as follows:

$$\begin{aligned} D=\left( \begin{array}{ccccc} X_{11} &{}X_{12} &{} X_{13} &{}\cdots &{}X_{1n} \\ X_{21} &{}X_{22} &{} X_{23} &{}\cdots &{} X{2n} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ X_{m1} &{}X_{m2} &{} X_{m3} &{} \cdots &{} X_{mn} \\ \end{array} \right) , \end{aligned}$$

where \(X_{ij}(i=1,2,\ldots ,m;\;j=1,2,\ldots ,n)\) are HFLSs.

In the following, we will give a multiple attribute group decision making method. We will take advantage of the new distance measures between HFLSs and extend the TOPSIS method to obtain the ranking of the alternatives, the step of the approach for MCDM involves the following steps:

  • Step 1 Calculate the hesitance degree of \(h_{ij}\), then we extend the fewer one by adding the minimum value until it has the same number with the more one.

  • Step 2 Normalize the decision matrix D.

    For the benefit-type criteria, we do not do anything; but for the cost-type criteria, we should use the negation operator in Definition 5 to make HFLNs normalized.

  • Step 3 Determine the positive ideal solution \(X^{+}\) and the negative ideal solution \(X^{-}\).

    $$\begin{aligned} X^{+}=\langle (s_{\max _{\theta _{ij}}})_{_{i=1,2,\ldots ,m;j=1,2,\ldots ,n.}},\{X_{1}^{+},X_{2}^{+},\ldots ,X_{n}^{+}\}\rangle , \end{aligned}$$

    where \(X_{j}^{+}=\bigcup _{q_{1j}\in h_{1j},\ldots ,q_{mj}\in h_{mj}}\max \{q_{1j},q_{2j},\ldots ,q_{mj}\}\) and \(j=1,2,\ldots ,n;\)

    $$\begin{aligned} X^{-}=\langle (s_{\min _{\theta _{ij}}})_{_{i=1,2,\ldots ,m;j=1,2,\ldots ,n.}},\{X_{1}^{-},X_{2}^{-},\ldots ,X_{n}^{-}\}\rangle , \end{aligned}$$

    where \(A_{j}^{-}=\bigcup _{q_{1j}\in h_{1j},\ldots ,q_{mj}\in h_{mj}}\max \{q_{1j},q_{2j},\ldots ,q_{mj}\}\) and \(j=1,2,\ldots ,n.\)

    We use the approach in Wang et al. [28], if there are the same elements in \(\{X_{1}^{+},X_{2}^{+},\ldots ,X_{n}^{+}\}\) or \(\{X_{1}^{-},X_{2}^{-},\ldots ,X_{n}^{-}\}\), and we should delete the repeated elements and ensure each element appears in a set only once.

  • Step 4 Utilize the new distance measures to calculate the separation of each alternative between the positive ideal solution and the negative ideal solution.

    The separation measure between \(X_{i}(i=1,2,\ldots ,m)\) and \(X^{+}\): \(d_{i}(X_{i},X^{+})=\sum _{j=1}^{n}\omega _{j}d(X_{ij},X^{+})\); the separation measure between \(X_{i}(i=1,2,\ldots ,m)\) and \(X^{-}\): \(d_{i}(X_{i},X^{-})=\sum _{j=1}^{n}\omega _{j}d(X_{ij},X^{-})\), where \(d(X_{ij},X^{+})\) or \((d (X_{ij},X^{-}))\) is the distance measure between \(X_{ij}\) and \(X^{+}\) or \((X^{-})\), and \(\omega _{j}\) is the corresponding weight of \(C_{j}\).

  • Step 5 Calculate the closeness coefficient of alternative \(X_{i}\).

    $$\begin{aligned} R_{i}=\frac{d_{i}(X_{i},X^{+})}{d_{i}(X_{i},X^{-})+d_{i}(X_{i},X^{+})}. \end{aligned}$$

  • Step 6 Rank all alternatives according to the closeness coefficient \(R_{i}\), the smaller closeness coefficient \(R_{i}\;(i=1,2,\ldots ,m)\) is, the better \(X_{i}(i=1,2,\ldots ,m)\) will be.

5 Illustrative Example

5.1 Background

The following background is adapted from Wang et al. [33].

ABC Nonferrous Metals Co, ltd., is a company whose business is producing and selling nonferrous metals. In order to expand its business, the company is ready to invest overseas. The overseas investment department is going to choose some alternatives from foreign countries based on preliminary investigation. After a careful investigation, five alternatives \(\{X_{1},X_{2},X_{3},X_{4},X_{5}\}\) are considered. The panel evaluates the alternatives from the following four factors, \(C_{1}:\) resources, \(C_{2}:\) policy, \(C_{3}:\) profitability, \(C_{4}:\) infrastructure. Suppose that the weight vector of \(C_{j}(j=1,2,3,4)\) is \(\omega =\{\omega _{1},\omega _{2},\omega _{3},\omega _{4}\}=(0.2,0.4,0.3,0.1).\) The panel makes evaluation using the linguistic term set \(S=\{s_{0}=very\quad poor, s_{1}=poor, s_{2}=slightly\quad poor, s_{3}=fair, s_{4}=slightly\quad good, s_{5}=good, s_{6}=very\quad good\}\). The evaluation information \(X_{ij}\) is represented by the hesitant fuzzy linguistic numbers(HFLNs) listed in Table 1.

Table 1 The hesitant fuzzy linguistic decision matrix
Full size table

5.2 An Illustration of the Proposed Method

In order to get the best alternative, at first, we let the linguistic scale function \(f^{*}=f_{1}(s_{i})=\frac{i}{2\tau }(\tau =3),\) and the method proposed in Sect. 4 will be used to identify the best alternative (Table 2).

  • Step 1 At first, we calculate the hesitance degree of each \(X_{ij}\), for example, the hesitance degree of \(\langle s_{4},\{0.3,0.4\}\) is \(\frac{1}{2}\), then we can extend \(\{0.3,0.4\}\) to \(\{0.3,0.3,0.4\}\) by adding the minimum value 0.3. The transformation of Table 1 is obtained as follows:

Table 2 The transformation of hesitant fuzzy linguistic decision matrix
Full size table

  • Step 2 Normalize the transformation of hesitant fuzzy linguistic decision matrix.

    We do not need to normalize the transformation of hesitant fuzzy linguistic decision matrix because all the criteria are of the maximizing type .

  • Step 3 Determine the positive ideal solution \(X^{+}\) and the negative ideal solution \(X^{-}\). We know \(X^{+}=\langle s_{6},\{0.6,0.7,0.8\}\rangle\), \(X^{-}=\langle s_{2},\{0.2,0.3,0.4\}\rangle\).

  • Step 4 Calculate the separation distance measure \(d_{hh}\) for each alternative.

    The separation measure \(d_{hh}\) between \(X_{i}\) and \(X^{+}\):

    $$\begin{aligned}&d_{hh}(X_{1},X^{+})=0.7667, d_{hh}(X_{2},X^{+})=0.8084, d_{hh}(X_{3},X^{+})=0.7756, d_{hh}(X_{4},X^{+})=1.0489, \\&d_{hh}(X_{5},X^{+})=0.6228; \end{aligned}$$

    The separation measure \(d_{hh}\) between \(X_{i}\) and \(X^{-}\):

    $$\begin{aligned}&d_{hh}(X_{1},X^{-})=0.3222, d_{hh}(X_{2},X^{-})=0.3028, d_{hh}(X_{3},X^{-})=0.4933, d_{hh}(X_{4},X^{-})=0.2511, \\&d_{hh}(X_{5},X^{-})=0.4383; \end{aligned}$$

  • Step 5 Calculate the closeness coefficient of alternative \(X_{i}\). We can obtain the closeness coefficient \(R_{i}\) as follows:

    $$\begin{aligned} R_{1}=0.7041, R_{2}=0.7275,R_{3}=0.6112,R_{4}=0.8068,R_{5}=0.5869. \end{aligned}$$

  • Step 6 Rank the alternatives according to \(R_{i}(i=1,2,\ldots ,5)\).

We can find that the ranking order of five alternative is \(X_{5}\succ X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\), and the best alternative is \(X_{5}\).

Furthermore, we use the hesitant fuzzy linguistic Euclidean distance \(d_{he}\)and the hesitant fuzzy linguistic generalized distance \(d_{hg}(\lambda =6,10,15,20)\) to calculate the closeness coefficient of each alternative, and then we get the rankings of the alternative, which are listed in Table 3.

Table 3 Results obtained by distance measure \(d_{hg}\)
Full size table

We can see that the best alternative is still \(X_{5}\).

To illustrate the influence of the linguistic scale function \(f^{*}\) on decision making based on the proposed method, we utilize different linguistic scale functions in the distance measures.

Let

$$\begin{aligned} f^{*}=f_{2}(s_{i})=\left\{ \begin{array}{llll} \frac{a^{\tau }-a^{\tau -i}}{2a^{\tau }-2},\ \ &{} \quad i=0,1,\ldots ,t,\\ \frac{a^{\tau }+a^{i-\tau }-2}{2a^{\tau }-2},\ \ &{} \quad i=\tau +1,\tau +2,\ldots ,2\tau \end{array},\right. \end{aligned}$$

and \(a=1.4,\tau =3,\) then the ranking results with \(d_{hh}\) and \(d_{he}\) are listed in Table 4.

Let

$$\begin{aligned} f^{*}=f_{3}(s_{i})=\left\{ \begin{array}{llll} \frac{\tau ^{\alpha ^{'}}-(\tau -i)^{\alpha ^{'}}}{2\tau ^{\alpha ^{'}}},\ \ &{} \quad i=0,1,\ldots ,\tau ,\\ \frac{\tau ^{\beta ^{'}}-(i-\tau )^{\beta ^{'}}}{2\tau ^{\beta ^{'}}},\ \ &{} \quad i=\tau +1,\tau +2,\ldots ,2\tau \end{array}\right. , \end{aligned}$$

and \(\alpha ^{'}=\beta ^{'}=0.8,\tau =3,\) and then the ranking results with \(d_{hh}\) and \(d_{he}\) are listed in Table 4.

Table 4 Results obtained with different linguistic scale functions
Full size table

As we can see from Table 3, it is obvious that the ranking results are always \(X_{5}\succ X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\). Based on Table 4, the alternative \(X_{5}\) is still the best choice in most cases, the rankings of the alternatives may be a little different when the linguistic scale function \(f^{*}=f_{2}(s_{i})\)(the linguistic scale function which can be considered as the actual semantic situation), and then the decision makers can select the appropriate linguistic scale function \(f^{*}\) according to their interests.

Next, we will give different preferences to the membership value and the hesitance degree, i.e., we will apply the distance measures of \(d_{phh}(\lambda =1),d_{phe}(\lambda =2)\) and \(d_{phg}\) to determine the best alternative. To investigate it in detail, here we only consider linguistic scale function \(f^{*}=f_{1}(s_{i})=\frac{i}{2t}(\tau =3).\) In order to understand the effect of preference parameter p, we adopt the parameter \(p(p=0.1,0.2,0.4,0.6,0.8),\) respectively. The results are listed in Tables 5, 6, 7, 8 and 9.

Table 5 Results obtained by distance measure \(d_{phg}\) with \(p=0.1\)
Full size table
Table 6 Results obtained by distance measure \(d_{phg}\) with \(p=0.2\)
Full size table
Table 7 Results obtained by distance measure \(d_{phg}\) with \(p=0.4\)
Full size table
Table 8 Results obtained by distance measure \(d_{phg}\) with \(p=0.6\)
Full size table
Table 9 Results obtained by distance measure \(d_{phg}\) with \(p=0.8\)
Full size table

As can be seen from Tables 5, 6, 7, 8 and 9, the most desirable alternative is \(X_{5}\). The ranking of the alternatives may be a little different when the parameter p (which can be considered as the influence of the hesitance degree, \(1-p\) can be considered as the influence of the membership values) changes; consequently, the decision makers have more choices when the different values of the parameter are given in the proposed distance measures according to their preference.

5.3 Comparison Analysis with Existing Distance Measure

To illustrate the validity and advantages of the proposed distance measures for MCDM problems, we now use the existing Hausdorff distance of HFLNs (Wang et al. [33]) for comparison analysis. If \(f^{*}(s_{i})=\frac{i}{2\tau }(\tau =3)\), the Hausdorff distance of HFLNs is used to solve the same illustrative example, the ranking order is \(X_{5}\succ X_{3}\succ X_{1}\succ X_{2}\succ X_{4}\), and the most desirable alternative is \(X_{5}\). It is the same ranking as that obtained by the proposed new distance measure when \(f^{*}(s_{i})=f_{1}(s_{i})=\frac{i}{2t}(\tau =3)\), this shows the effectiveness of the proposed method. The ranking result obtained by Wang et al. [33] is not exactly the same as the proposed method derived by \(d_{phg}\), the reason is that the hesitance degree of hesitant fuzzy element has different influence on the calculation.

According to the comparison analyses, the method for MCDM problems in this paper has the following advantages. First, it is reasonable that the distance measures include the hesitance degree which influences the decision results. Second, the proposed distance for HFLSs is defined based on linguistic scale function \(f^{*}\), and the decision makers can flexibly select the linguistic scale function \(f^{*}\) depending on their preferences and the actual semantic situations.

6 Conclusions

In this paper, we propose some new class of distance measures for HFLSs which include the hesitance degree of hesitant fuzzy element and the linguistic scale function together. Furthermore, the proposed distance measures based on TOPSIS method for hesitant fuzzy linguistic multiple criteria decision making problem are developed, and the effectiveness and advantages of the proposed methods are demonstrated by an illustrative example and comparison analyses. The main characteristics of the proposed distance measures are that it not only considers the hesitance of the hesitant fuzzy elements but also deals with linguistic transformation problem under different semantic situations, which efficiently avoid information loss and distortion. In future research, the developed distance measures will be extended to the hesitant intuitionistic fuzzy linguistic set and it can be applied in supply chain selection, project evaluation and other related decision making.