1 Introduction
According to the attribute values of alternatives, decision-making theory can be classified into two types. One type is the stochastic decision making, where the attribute values are stochastic variables; the other is the fuzzy decision making, where the attribute values are fuzzy variables. It is worth noting that fuzzy decision-making theory has some advantages to cope with uncertain information. Since Zadeh (
1965) first introduced fuzzy set theory, decision making based on fuzzy sets has been successfully applied in many fields, such as recommender system (Tejeda-Lorente
et al.,
2014; Martínez-Cruz
et al.,
2015; Yager,
2003,
2004), education (Bryson and Mobolurin,
1995), medical care (James and Dolan,
2010), engineering (Chen and Weng,
2006; Lennon
et al.,
2013; Meng
et al.,
2016c), economics (Ölçer
et al.,
2006; Vaidogas and Sakenaite,
2011; Meng
et al.,
2017a,
2016d), reservoir flood control (Fu,
2008), facility location selection (Kahraman
et al.,
2003), new product development (NDP) project screening (Meng and Chen,
2017a), and supplier selection (Meng
et al.,
2017b). With the increasing complexity of the decision-making problems, researchers found that it is insufficient to address decision-making problems by using fuzzy sets, which only permit the decision maker to apply one fuzzy number to denote the uncertainty. Furthermore, fuzzy sets can only express the decision maker’s positive judgment. Thus, several types of generalized fuzzy sets are proposed, such as intuitionistic fuzzy sets (Atanassov,
1986; Atanassov and Gargov,
1989), type-2 fuzzy sets (Zadeh,
1973) and hesitant fuzzy sets (Chen
et al.,
2013a; Torra,
2010).
However, all these types of fuzzy sets can only denote the decision maker’s quantitative cognitions. As Zadeh (
1975) noted, there are many situations, where the decision-making problems are too complex or too ill-defined to use quantitative expressions. To address this issue, Zadeh (
1975) introduced the concept of linguistic variables, which permit the decision maker to use linguistic variables rather than quantitative fuzzy variables to express the judgment. Since then, many studies about decision making based on linguistic variables are developed (Cai
et al.,
2014a,
2014b,
2015; Dong
et al.,
2009,
2016; Gou and Xu,
2016; Herrera and Martínez,
2000; Herrera
et al.,
2000; Ju
et al.,
2016; Li
et al.,
2017; Massanet
et al.,
2014; Morente-Molinera
et al.,
2015; Martínez and Herrera,
2012; Meng
et al.,
2016a; Meng and Chen,
2016b; Pedrycz,
2013; Wei,
2011; Wu and Xu,
2016; Xu,
2004a,
2007; Ye,
2016a). Just as quantitative fuzzy variables, it is still not an easy thing to require a decision maker to apply one linguistic variable to express his/her qualitative judgment. Thus, Xu (
2004b) introduced the concept of uncertain linguistic variables, which permit the decision maker to use an interval linguistic variable rather than one exact linguistic variable to denote information. However, uncertain linguistic variables are inadequate to denote the decision maker’s hesitancy and irresolution. Hesitant fuzzy linguistic term sets (HFLTSs) introduced by Rodríguez
et al. (
2012) can well address this issue, which are composed by several linguistic terms.
All of the above mentioned fuzzy sets can denote either the decision maker’s quantitative or qualitative information. However, none of them can denote these two aspects simultaneously. Following the works of Atanassov (
1986) and Zadeh (
1975), Wang and Li (
2009) presented intuitionistic linguistic sets (ILSs), which are composed by one linguistic variable and an intuitionistic fuzzy variable. Using this type of fuzzy sets, the decision maker can apply one linguistic variable to denote his/her qualitative judgment as well as use an intuitionistic fuzzy variable to show the membership and non-membership degrees about the qualitative judgment. Meng
et al. (
2016e) developed a group decision-making method with intuitionistic linguistic preference relations (ILPRs), where the elements in ILPRs are intuitionistic linguistic fuzzy variables. Later, Liu and Jin (
2012) and Liu (
2013) introduced intuitionistic uncertain linguistic sets (IULSs) and interval-valued intuitionistic uncertain linguistic sets (IVIULSs), respectively. Such generalizations further endow the decision makers with more rights to express their judgments. As researchers (Rodríguez
et al.,
2012; Torra,
2010; Ye,
2015; Meng and An,
2017) noted that the difficulty of expressing the judgments does not arise: there is a margin of error or some possibility distribution on the possibility values but there are several possible values. Recently, Meng
et al. (
2014) presented a new type of fuzzy sets called linguistic hesitant fuzzy sets (LHFSs). This kind of fuzzy sets permits the decision maker to apply several linguistic variables with each having several membership degrees to denote the judgment of one thing. Meanwhile, this type of fuzzy sets can express the qualitative and quantitative cognitions of the decision makers and reflect their hesitancy and inconsistency.
Considering the application of LHFSs, Meng
et al. (
2014) defined several operational laws of LHFSs and then gave a ranking method. After that, the authors developed a method to linguistic hesitant fuzzy multi-attribute decision making with interactive characteristics and incomplete weight information. However, one can see that this method is based on the defined aggregation operators, this makes the process of decision making seem to be complex. Especially, the calculation of the comprehensive attribute values will be very complex with the increase of the number of linguistic hesitant fuzzy sets. Later, Zhou
et al. (
2015) applied a special example to show that the ranking order offered in Meng
et al. (
2014) is unreasonable, and introduced a new ranking method. However, Zhou
et al.’s ranking method is illogical. Furthermore, the Hamming distance on LHFSs offered by Zhou
et al. (
2015) is wrong. Recently, Zhu
et al. (
2016) developed a cloud model method to linguistic hesitant fuzzy multi-attribute decision making and extended the power operators to linguistic hesitant fuzzy environment. However, this method seems also to be complex.
To address the above listed issues in previous researches, the paper continues to research the application of LHFSs. To do this, we first introduce a distance measure between LHFSs that can be seen as an extension of Hamming distance on real numbers. One can check that the new distance measure addresses the issues in Zhou
et al. (
2015). To discriminate the importance of features or attributes, several additive weighted distance measures are defined that are used to calculate the comprehensive ranking values of objects. Meanwhile, a correlation coefficient on LHFSs is provided, and several weighted correlation coefficients are offered. Considering the interactive characteristics and the complexity of fuzzy numbers, Shapley-based distance measures and correlation coefficients with 2-additive measures are provided, which can be seen as extensions of weighted distance measures and correlation coefficients, respectively. Then, an approach to pattern recognition and to multi-attribute decision making with LHFSs is performed, respectively. Meanwhile, associated practical application is offered.
This paper is organized as follows: Section
2 reviews several basic concepts related to LHFSs. Section
3 introduces a distance measure and a correlation coefficient of LHFSs. Section
4 defines two types of hybrid weighted distance measures and correlation coefficients of LHFSs. One is based on additive measures, and the other uses the Shapley function with respect to 2-additive measures. Section
5 develops an approach to pattern recognition and to multi-attribute decision making by using the defined distance measures and correlation coefficients, and then comparison analysis with the existing methods is made. The last section is the conclusion.
2 Some Basic Concepts
Considering the hesitancy and inconsistency of the decision makers, Torra (
2010) defined hesitant fuzzy sets that permit the decision makers to apply several possible values in
$[0,1]$ to denote the membership degree of one thing.
Definition 1 (See Torra, 2010).
Let $X=\{{x_{1}},{x_{2}},\dots ,{x_{n}}\}$ be a finite set. A hesitant fuzzy set (HFS) in X is expressed in terms of a function such that when applied to X it returns a subset of $[0,1]$, denoted by $E=(\langle {x_{i}},{h_{E}}({x_{i}})\rangle |{x_{i}}\in X)$, where ${h_{E}}({x_{i}})$ is a set of some values in $[0,1]$ denoting the possible membership degrees of the element ${x_{i}}\in X$ to the set E.
Sometimes, it is not easy for the decision makers to estimate their information using quantitative values. In this case, linguistic variables are more suitable to only provide the decision makers with the qualitative values. The linguistic reasoning is a technique that represents qualitative aspect using linguistic variables. Let $S=\{{s_{i}}\mid i=1,2,\dots ,t\}$ be a linguistic term set with odd cardinality. Any label ${s_{i}}$ represents a possible value for a linguistic variable, and it should satisfy the following characteristics: (i) The set is ordered: ${s_{i}}>{s_{j}}$, if $i>j$; (ii) Max operator: $\max ({s_{i}},{s_{j}})={s_{i}}$, if ${s_{i}}\geqslant {s_{j}}$; (iii) Min operator: $\min ({s_{i}},{s_{j}})={s_{i}}$, if ${s_{i}}\leqslant {s_{j}}$. For example, a linguistic term set S may be expressed by $S=\{$${s_{1}}$: very poor, ${s_{2}}$: poor, ${s_{3}}$: slightly poor, ${s_{4}}$: fair, ${s_{5}}$: slightly good, ${s_{6}}$: good, ${s_{7}}$: very good}.
Similar to hesitant fuzzy sets, Rodríguez
et al. (
2012) introduced the following concept of hesitant fuzzy linguistic term sets (HFLTSs) that permit a qualitative reference to have several linguistic terms.
Definition 2 (See Rodríguez et al., 2012).
An HFLTS, ${H_{S}}$, is an ordered finite subset of consecutive linguistic terms of S, where $S=\{{s_{1}},\dots ,{s_{t}}\}$ is a linguistic term set.
For example, let S be a linguistic term set as shown above, and let Q be a qualitative reference. An HFLTS could be ${H_{S}}(Q)=\{{s_{2}},{s_{3}},{s_{4}}\}$.
As pointed out in introduction, HFLTSs only denote the hesitancy and inconsistency of the decision makers’ qualitative references, and it is based on the assumption that the decision makers have the same cognition degrees of the given linguistic terms in an HFLTS. However, this might be not true. For instance, to evaluate the quietness of the refrigerator, the decision maker might hesitate to give the value 15% or 20% for slightly good, the value 30%, 40% or 50% for good, and the value 15% for very good. To address this situation, HFLTSs and HFSs seem to be insufficient. Linguistic hesitant fuzzy sets (LHFSs) introduced in Meng
et al. (
2014) can well address this problem.
Definition 3 (See Meng et al., 2014).
Let $S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set. A LHFS in S is a set that when applied to the linguistic terms of S it returns a subset of S with several values in $[0,1]$, denoted by $\mathit{LH}=\{({s_{\theta (i)}},lh({s_{\theta (i)}})\mid {s_{\theta (i)}}\in S)\}$, where $lh({s_{\theta (i)}})=\{{r_{1}},{r_{2}},\dots ,{r_{{m_{i}}}}\}$ is a set with ${m_{i}}$ values in $[0,1]$ denoting the possible membership degrees of the element ${s_{\theta (i)}}\in S$ to the set $\mathit{LH}$.
In the example of evaluating the quietness of the refrigerator, the decision maker’s judgment can be expressed by a LHFS $\mathit{LH}=\{({s_{5}},0.15,0.2),({s_{6}},0.3,0.4,$ $0.5),({s_{7}},0.15)\}$. To compare LHFSs, Meng et al. (2014) introduced the following method:
Definition 4 (See Meng et al., 2014).
Let ${\mathit{LH}_{1}}$ be a LHFS for the predefined linguistic term set $S=\{{s_{1}},\dots ,{s_{t}}\}$. Suppose that $l{h_{i}}=({s_{\theta (i)}},lh({s_{\theta (i)}}))\in {\mathit{LH}_{1}}$ with $lh({s_{\theta (i)}})=\{{r_{{i_{1}}}},{r_{{i_{2}}}},\dots ,{r_{{i_{m}}}}\}$, then the expectation value of $l{h_{i}}$ is defined by $E(l{h_{i}})=\frac{\theta (i){\textstyle\sum _{k=1}^{m}}{r_{{i_{k}}}}}{m}$, and its variance is given as $V(l{h_{i}})=\frac{{\textstyle\sum _{k=1}^{m}}{(\theta (i){r_{{i_{k}}}}-E(l{h_{i}}))^{2}}}{m}$.
Let ${\mathit{LH}_{1}}$ be a LHFS for the predefined linguistic term set $S=\{{s_{1}},\dots ,{s_{t}}\}$. Suppose that $l{h_{i}}=({s_{\theta (i)}},lh({s_{\theta (i)}}))\in {\mathit{LH}_{1}}$ with $lh({s_{\theta (i)}})=\{{r_{{i_{1}}}},{r_{{i_{2}}}},\dots ,{r_{{i_{m}}}}\}$ and $l{h_{j}}=({s_{\theta (j)}},lh({s_{\theta (j)}}))\in {\mathit{LH}_{1}}$ with $lh({s_{\theta (j)}})=\{{r_{{j_{1}}}},{r_{{j_{2}}}},\dots ,{r_{{j_{n}}}}\}$. Then, their order relationship is defined as follows:
If $E(l{h_{i}})\leqslant E(l{h_{j}})$, then $l{h_{i}}\leqslant l{h_{j}}$;
If $E(l{h_{i}})=E(l{h_{i}})$, then $\left\{\begin{array}{l@{\hskip4.0pt}l}V(l{h_{i}})>V(l{h_{j}}),\hspace{1em}& l{h_{i}}<l{h_{j}},\\ {} V(l{h_{i}})<V(l{h_{j}}),\hspace{1em}& l{h_{i}}>l{h_{j}},\\ {} V(l{h_{i}})=V(l{h_{j}}),\hspace{1em}& l{h_{i}}=l{h_{j}}.\end{array}\right.$
3 Distance Measure and Correlation Coefficient of LHFSs
Distance measure and correlation coefficient are two useful tools to decision making, by which we can obtain the best choice or rank the objects. This section introduces a distance measure and a correlation coefficient of LHFSs.
3.1 A Distance Measure of LHFSs
Distance measure is an effective tool to measure the deviations of different arguments, which is applied in many fields, such as ranking fuzzy numbers (Tran and Duckstein,
2002), determining the weights (Yue,
2011), decision making (Cabrerizo
et al.,
2015; Gong
et al.,
2016; Peng
et al.,
2013; Xu,
2010b), economics (Merigó and Casanovas,
2011), pattern recognition (Hung and Yang,
2004; Zeng
et al.,
2016), and cluster analysis (Yang and Lin,
2009). Similar to the OWA operator (Yager,
1988), Xu and Chen (
2008) defined the ordered weighted distance measure (OWDM), which can decrease the influence of extreme values. Later, Zeng and Su (
2011) introduced an OWDM on intuitionistic fuzzy sets. However, the OWDM only considers the importance of the ordered positions, but it does not give the importance of the elements. It is worth noting that a distance measure corresponds to a similarity measure (Hung and Yang,
2004; Xu and Xia,
2011; Yang and Lin,
2009). This subsection gives a distance measure of LHFSs.
Definition 5.
Let
$S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set, and let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
S. Without loss of generality, suppose that
$l{h_{i}}=({s_{\theta (i)}},lh({s_{\theta (i)}}))\in {\mathit{LH}_{1}}$ with
$lh({s_{\theta (i)}})=\{{r_{{i_{1}}}},{r_{{i_{2}}}},\dots ,{r_{{i_{m}}}}\}$ and
$l{h_{j}}=({s_{\theta (j)}},lh({s_{\theta (j)}}))\in {\mathit{LH}_{2}}$ with
$lh({s_{\theta (j)}})=\{{r_{{j_{1}}}},{r_{{j_{2}}}},\dots ,{r_{{j_{n}}}}\}$. The distance measure from
$l{h_{i}}$ to
$l{h_{j}}$ is defined as follows:
and the distance measure from
$lh\textit{j}$ to
$lh\textit{i}$ is defined as follows:
Furthermore, the distance measure between
$l{h_{i}}$ and
$l{h_{j}}$ is defined as follows:
Property 1.
Let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
$S=\{{s_{1}},\dots ,{s_{t}}\}$. Suppose that
$l{h_{i}}=({s_{\theta (i)}},lh({s_{\theta (i)}}))\in {\mathit{LH}_{1}}$ and
$l{h_{j}}=({s_{\theta (j)}},lh({s_{\theta (j)}}))\in {\mathit{LH}_{2}}$ are given as shown in Definition
5. Then, we have:
-
(i) $d(l{h_{i}},l{h_{j}})=0$ if and only if there is ${r_{{j_{p}}}}\in lh({s_{\theta (j)}})$ such that $\theta (j){r_{{j_{p}}}}=\theta (i){r_{{i_{k}}}}$ for all $k=1,2,\dots ,m$, and there is ${r_{{i_{k}}}}\in lh({s_{\theta (i)}})$ such that $\theta (i){r_{{i_{k}}}}=\theta (j){r_{{j_{p}}}}$ for all $p=1,2,\dots ,n$;
-
(ii) $0\leqslant d(l{h_{i}},l{h_{j}})\leqslant 1$;
-
(iii) $d(l{h_{i}},l{h_{j}})=d(l{h_{j}},l{h_{i}})$;
-
(iv) Let
${\mathit{LH}_{3}}$ be another LHFS with
$lhg=({s_{\theta (g)}},lh({s_{\theta (g)}}))\in {\mathit{LH}_{3}}$. If we have
for all
${r_{{j_{p}}}}\in lh({s_{\theta (j)}})$,
${r_{{i_{k}}}}\in lh({s_{\theta (i)}})$ and
${r_{{i_{z}}}}\in lh({s_{\theta (g)}})$, then
$d(l{h_{i}},l{h_{g}})\leqslant d(l{h_{i}},l{h_{j}})$.
Proof.
For (i): when we have
$d(l{h_{i}},l{h_{j}})=0$, by the equation (
3) we have
$\overrightarrow{d(l{h_{i}},l{h_{j}})}=\overrightarrow{d(l{h_{j}},l{h_{i}})}=0$. According to the equations (
1) and (
2), we get
${\min _{{r_{{j_{p}}}}\in lh({s_{\theta (j)}})}}|\theta (i){r_{{i_{k}}}}-\theta (j){r_{{j_{p}}}}|={\min _{{r_{{i_{p}}}}\in lh({s_{\theta (i)}})}}|\theta (j){r_{{j_{k}}}}-\theta (i){r_{{i_{p}}}}|=0$. Thus, there is
${r_{{j_{p}}}}\in lh({s_{\theta (j)}})$ such that
$\theta (j){r_{{j_{p}}}}=\theta (i){r_{{i_{k}}}}$ for all
$k=1,2,\dots ,m$, and there is
${r_{{i_{k}}}}\in lh({s_{\theta (i)}})$ such that
$\theta (i){r_{{i_{k}}}}=\theta (j){r_{{j_{p}}}}$ for all
$p=1,2,\dots ,n$. On the other hand, one can easily derive that
$d(l{h_{i}},l{h_{j}})=0$ from the equations (
1)–(
3) according to the listed conditions in (i).
For (ii): from the equations (
1) and (
3), we have
$0\leqslant \overrightarrow{d(l{h_{i}},l{h_{j}})},\overrightarrow{d(l{h_{j}},l{h_{i}})}\leqslant 1$ by
$0\leqslant {\min _{{r_{{i_{k}}}}\in lh({s_{\theta (i)}})}}|\theta (j){r_{{j_{k}}}}-\theta (i){r_{{i_{p}}}}|$,
${\min _{{r_{{j_{p}}}}\in lh({s_{\theta (j)}})}}|\theta (i){r_{{i_{k}}}}-\theta (j){r_{{j_{p}}}}|\leqslant t$ for each
$k=1,2,\dots ,m$ and each
$p=1,2,\dots ,n$. According to the equation (
3), we get
$0\leqslant d(l{h_{i}},l{h_{j}})\leqslant 1$.
For (iii): from the equation (
3), we obtain
$d(l{h_{i}},l{h_{j}})=\frac{\overrightarrow{d(l{h_{i}},l{h_{j}})}+\overrightarrow{d(l{h_{j}},l{h_{i}})}}{2}=\frac{\overrightarrow{d(l{h_{j}},l{h_{i}})}+\overrightarrow{d(l{h_{i}},l{h_{j}})}}{2}=d(l{h_{j}},l{h_{i}})$.
For (iv): from
$\left\{\begin{array}{l}{\min _{{r_{{i_{z}}}}\in lh({s_{\theta (g)}})}}\big|\theta (i){r_{{i_{k}}}}-\theta (g){r_{{i_{z}}}}\big|\leqslant {\min _{{r_{{j_{p}}}}\in lh({s_{\theta (j)}})}}\big|\theta (i){r_{{i_{k}}}}-\theta (j){r_{{j_{p}}}}\big|\\ {} {\min _{{r_{{i_{z}}}}\in lh({s_{\theta (g)}})}}\big|\theta (g){r_{{i_{z}}}}-\theta (i){r_{{i_{k}}}}\big|\leqslant {\min _{{r_{{i_{k}}}}\in lh({s_{\theta (i)}})}}\big|\theta (j){r_{{j_{p}}}}-\theta (i){r_{{i_{k}}}}\big|\end{array}\right.$ for all
${r_{{j_{p}}}}\in lh({s_{\theta (j)}})$,
${r_{{i_{k}}}}\in lh({s_{\theta (i)}})$ and
${r_{{i_{z}}}}\in lh({s_{\theta (g)}})$, we have
From the equation (
3), we derive
$d(l{h_{i}},l{h_{g}})\leqslant d(l{h_{i}},l{h_{j}})$. □
Definition 6.
Let
$S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set, and let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
S. Without loss of generality, suppose that
$l{h_{i}}=({s_{\theta (i)}},lh({s_{\theta (i)}}))\in {\mathit{LH}_{1}}$ and
$l{h_{j}}=({s_{\theta (j)}},lh({s_{\theta (j)}}))\in {\mathit{LH}_{2}}$. Then, the distance measure from
$l{h_{i}}$ to
${\mathit{LH}_{2}}$ is defined as follows:
and the distance measure from
$l{h_{j}}$ to
${\mathit{LH}_{1}}$ is defined as follows:
Remark 1.
Let $S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set, and let ${\mathit{LH}_{1}}$ and ${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set S. Let $l{h_{i}}=({s_{\theta (i)}},lh({s_{\theta (i)}}))\in {\mathit{LH}_{1}}$, if we have $\overrightarrow{d(l{h_{i}},{\mathit{LH}_{2}})}=d(l{h_{i}},l{h_{j}})$ with $l{h_{j}}\in {\mathit{LH}_{2}}$, then we denote $l{h_{j}}$ as $l{h_{j}^{i}}=({s_{{\theta ^{i}}(j)}},lh({s_{{\theta ^{i}}(j)}}))$.
Definition 7.
Let
$S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set, and let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
S. Then, the distance measure between
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ is defined as follows:
where
$\overrightarrow{d({\mathit{LH}_{1}},{\mathit{LH}_{2}})}=\frac{1}{|{\mathit{LH}_{1}}|}{\textstyle\sum _{i=1}^{|{\mathit{LH}_{1}}|}}\overrightarrow{d(l{h_{i}},{\mathit{LH}_{2}})}$ and
$\overrightarrow{d({\mathit{LH}_{2}},{\mathit{LH}_{1}})}=\frac{1}{|{\mathit{LH}_{2}}|}{\textstyle\sum _{j=1}^{|{\mathit{LH}_{2}}|}}$ $\overrightarrow{d(l{h_{j}},{\mathit{LH}_{1}})}$ with
$l{h_{i}}\in {\mathit{LH}_{1}}$ and
$l{h_{j}}\in {\mathit{LH}_{2}}$,
$|{\mathit{LH}_{1}}|$ and
$|{\mathit{LH}_{2}}|$ denote the cardinalities of the linguistic variables in
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$, respectively.
Corollary 1.
Let ${\mathit{LH}_{1}}$ and ${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set $S=\{{s_{1}},\dots ,{s_{t}}\}$. Then, the distance measure $d({\mathit{LH}_{1}},{\mathit{LH}_{2}})$ between ${\mathit{LH}_{1}}$ and ${\mathit{LH}_{2}}$ satisfies:
-
(i) $d({\mathit{LH}_{1}},{\mathit{LH}_{2}})=0$ if and only if there is $l{h_{j}}\in {\mathit{LH}_{2}}$ such that $\overrightarrow{d(l{h_{i}},l{h_{j}})}=0$ for any $l{h_{i}}\in {\mathit{LH}_{1}}$, and there is $l{h_{i}}\in {\mathit{LH}_{1}}$ such that $\overrightarrow{d(l{h_{j}},l{h_{i}})}=0$ for any $l{h_{j}}\in {\mathit{LH}_{2}}$;
-
(ii) $0\leqslant d({\mathit{LH}_{1}},{\mathit{LH}_{2}})\leqslant 1$;
-
(iii) $d({\mathit{LH}_{1}},{\mathit{LH}_{2}})=d({\mathit{LH}_{2}},{\mathit{LH}_{1}})$;
-
(iv) Let ${\mathit{LH}_{3}}$ be another LHFS with $l{h_{g}}=({s_{\theta (g)}},lh({s_{\theta (g)}}))\in {\mathit{LH}_{3}}$. If we have $\left\{\begin{array}{l}{\min _{l{h_{g}}\in {\mathit{LH}_{3}}}}\overrightarrow{d(l{h_{i}},l{h_{g}})}\leqslant {\min _{l{h_{j}}\in {\mathit{LH}_{2}}}}\overrightarrow{d(l{h_{i}},l{h_{j}})}\\ {} {\min _{l{h_{i}}\in {\mathit{LH}_{}}}}\overrightarrow{d(l{h_{g}},l{h_{i}})}\leqslant {\min _{l{h_{i}}\in {\mathit{LH}_{1}}}}\overrightarrow{d(l{h_{j}},l{h_{i}})}\end{array}\right.$ for all $l{h_{i}}\in {\mathit{LH}_{1}}$, $l{h_{j}}\in {\mathit{LH}_{2}}$ and $l{h_{g}}\in {\mathit{LH}_{3}}$ , then $d({\mathit{LH}_{1}},{\mathit{LH}_{3}})\leqslant d({\mathit{LH}_{1}},{\mathit{LH}_{2}})$.
Proof.
Similar to Property
1, one can easily derive the conclusions. □
For example, let
${\mathit{LH}_{1}}=({s_{2}},0.2,0.3),({s_{3}},0.3,0.5,0.7),({s_{4}},0.1)$ and
${\mathit{LH}_{2}}=({s_{4}},0.5,0.6),({s_{5}},0.1,0.2)$ be two LHFSs for the predefined linguistic term set
$S=\{{s_{1}},{s_{2}},{s_{3}},{s_{4}},{s_{5}},{s_{6}},{s_{7}}\}$. Then, the distance measure between
$l{h_{1}^{1}}=({s_{2}},0.2,0.3)$ and
$l{h_{1}^{2}}=({s_{4}},0.5,0.6)$ is
and the distance between
$l{h_{1}^{1}}=({s_{2}},0.2,0.3)$ and
${\mathit{LH}_{2}}$ is
Furthermore, the distance between
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ is
$d({\mathit{LH}_{1}},{\mathit{LH}_{2}})=0.034$.
Remark 2.
Let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
$S=\{{s_{1}},\dots ,{s_{t}}\}$. Let
Then,
$S({\mathit{LH}_{1}},{\mathit{LH}_{2}})$ is a similarity measure between
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$, which satisfies: (i)
$S({\mathit{LH}_{1}},{\mathit{LH}_{2}})=1$ if and only if
$d({\mathit{LH}_{1}},{\mathit{LH}_{2}})=0$; (ii)
$0\leqslant S({\mathit{LH}_{1}},{\mathit{LH}_{2}})\leqslant 1$; (iii)
$S({\mathit{LH}_{1}},{\mathit{LH}_{2}})=S({\mathit{LH}_{2}},{\mathit{LH}_{1}})$; (iv) Let
${\mathit{LH}_{3}}$ be another LHFS. If
$d({\mathit{LH}_{1}},{\mathit{LH}_{2}})\leqslant d({\mathit{LH}_{1}},{\mathit{LH}_{3}})$, then
$S({\mathit{LH}_{1}},{\mathit{LH}_{2}})\geqslant S({\mathit{LH}_{1}},{\mathit{LH}_{3}})$.
3.2 A Correlation Coefficient of LHFSs
Correlation coefficient is a powerful tool to measure the linear relation between stochastic variables. Recently, researchers applied the correlation coefficient to measure the similarity between fuzzy variables and discussed their application in digital image processing (Van der Weken
et al.,
2004), clustering analysis (Chen
et al.,
2013b; Xu
et al.,
2008), pattern recognition (Liang and Shi,
2003), artificial intelligence (Zhang
et al.,
2013,
2014), and multi-attribute decision making (Park
et al.,
2009; Wei
et al.,
2011; Ye,
2016b; Tong and Yu,
2016). Recently, Meng and Chen (
2015) noted the issues of the correlation coefficient of hesitant fuzzy sets in Chen
et al. (
2013b) and defined several new ones, which need not consider the lengths of HFEs and the arrangement of the possible values. Furthermore, Meng
et al. (
2016b) defined several correlation coefficients of interval-valued hesitant fuzzy sets in a similar way to Meng and Chen (
2015). Following the work of Meng and Chen (
2015), this section introduces a correlation coefficient of LHFSs.
Definition 8.
Let
$S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set, and let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
S. The correlation coefficient between
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ is defined as follows:
where
$\overrightarrow{C({\mathit{LH}_{1}},{\mathit{LH}_{2}})}$ is the correlation of
${\mathit{LH}_{1}}$ with respect to
${\mathit{LH}_{2}}$ defined as follows:
with
$|\theta (i){r_{{i_{k}}}}-{\theta ^{i}}(j){r_{{j_{p}}}^{{i_{k}}}}|={\min _{{r_{{j_{p}}}^{i}}\in lh({s_{{\theta ^{i}}(j)}})}}|\theta (i){r_{{i_{k}}}}-{\theta ^{i}}(j){r_{{j_{p}}}^{i}}|$ and
$|lh({s_{\theta (i)}})|$ being the cardinality of
$lh({s_{\theta (i)}})$.
$\overrightarrow{C({\mathit{LH}_{2}},{\mathit{LH}_{1}})}$ is the correlation of
${\mathit{LH}_{2}}$ with respect to
${\mathit{LH}_{1}}$ defined as follows:
with
$|\theta (j){r_{{j_{p}}}}-{\theta ^{j}}(i){r_{{i_{k}}}^{{j_{p}}}}|={\min _{{r_{{i_{k}}}^{j}}\in lh({s_{{\theta ^{j}}(i)}})}}|\theta (j){r_{{j_{p}}}}-{\theta ^{j}}(i){r_{{i_{k}}}^{j}}|$ and
$|lh({s_{\theta (j)}})|$ being the cardinality of
$lh({s_{\theta (j)}})$.
Property 2.
Let
$S=\{{s_{1}},\dots ,{s_{t}}\}$ be a linguistic term set, and let
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ be any two LHFSs for the predefined linguistic term set
S. The correlation coefficient between
${\mathit{LH}_{1}}$ and
${\mathit{LH}_{2}}$ satisfies:
-
(i) $CC({\mathit{LH}_{1}},{\mathit{LH}_{1}})=1$;
-
(ii) $CC({\mathit{LH}_{1}},{\mathit{LH}_{2}})=CC({\mathit{LH}_{2}},{\mathit{LH}_{1}})$;
-
(iii) $0\leqslant CC({\mathit{LH}_{1}},{\mathit{LH}_{2}})\leqslant 1$.
Proof.
For (i): From the equation (
11), we have the following:
On the other hand, from
$\overrightarrow{d(l{h_{i}},{\mathit{LH}_{1}})}=d(l{h_{i}},l{h_{i}})=0$, we derive the following:
namely,
$|\theta (i){r_{{i_{k}}}}-{\theta ^{i}}(j){r_{{j_{p}}}^{{i_{k}}}}|={\min _{{r_{{j_{p}}}^{i}}\in lh({s_{{\theta ^{i}}(j)}})}}|\theta (i){r_{{i_{k}}}}-{\theta ^{i}}(j){r_{{j_{p}}}^{i}}|={\min _{{r_{{i_{l}}}}\in lh({s_{\theta (i)}})}}|\theta (i){r_{{i_{k}}}}-\theta (i){r_{{i_{l}}}}|=|\theta (i){r_{{i_{k}}}}-\theta (i){r_{{i_{k}}}}|=0$. Thus,
${\theta ^{i}}(j){r_{{j_{p}}}^{{i_{k}}}}=\theta (i){r_{{i_{k}}}}$.
According to the equation (
8), we obtain the following:
Thus,
$CC({\mathit{LH}_{1}},{\mathit{LH}_{1}})=\frac{D({\mathit{LH}_{1}})}{D({\mathit{LH}_{1}})}=1$.
For (ii): From the equation (
8), one can easily check that this conclusion holds.
For (iii): From
$CC({\mathit{LH}_{1}},{\mathit{LH}_{2}})\geqslant 0$, we only need to show
$CC({\mathit{LH}_{1}},{\mathit{LH}_{2}})\leqslant 1$. By the Cauchy–Schwarz inequality, we have
From
$\sqrt{D({\mathit{LH}_{1}})D({\mathit{LH}_{2}^{{\mathit{LH}_{1}}}})}\leqslant \frac{D({\mathit{LH}_{1}})+D({\mathit{LH}_{2}^{{\mathit{LH}_{1}}}})}{2}\leqslant \max \{D({\mathit{LH}_{1}}),D({\mathit{LH}_{2}^{{\mathit{LH}_{1}}}})\}$, we derive
Similarly, we have
$\overrightarrow{C({\mathit{LH}_{2}},{\mathit{LH}_{1}})}\leqslant \max \{D({\mathit{LH}_{2}}),D({\mathit{LH}_{1}^{{\mathit{LH}_{2}}}})\}$. Thus,
□
Remark 3.
Similar to the correlation coefficient defined in the equation (
7), we can define other correlation coefficients of LHFSs in a similar way to Meng and Chen (
2015). When the membership degree of each linguistic term in LHFSs is one, then LHFSs degenerate to HFLTSs. In this situation, the correlation coefficient (
7) reduces to the correlation coefficient for HFLTSs. Furthermore, when all LHFSs only have the same one linguistic term, then the correlation coefficient (
7) reduces to the correlation coefficient for HFSs given by Meng and Chen (
2015).
For example, let ${\mathit{LH}_{1}}$ and ${\mathit{LH}_{2}}$ be two LHFSs for the predefined linguistic term set $S=\{{s_{1}},{s_{2}},{s_{3}},{s_{4}},{s_{5}},{s_{6}},{s_{7}}\}$, where ${\mathit{LH}_{1}}$ and ${\mathit{LH}_{2}}$ as shown above, namely, ${\mathit{LH}_{1}}=\{({s_{2}},0.2,0.3),({s_{3}},0.3,0.5,0.7),({s_{4}},0.1)\}$ and ${\mathit{LH}_{2}}=\{({s_{4}},0.5,0.6),({s_{5}},0.1,0.2)\}$. Then, the correlation coefficient between ${\mathit{LH}_{1}}$ and ${\mathit{LH}_{2}}$ is $CC({\mathit{LH}_{1}},{\mathit{LH}_{2}})=0.8747$.
6 Conclusions
As we know, distance measure and correlation coefficient are two important tools to decision making. Considering the application of linguistic hesitant fuzzy sets, this paper defines a distance measure, and then introduces a correlation coefficient. After that, we develop two types of the hybrid weighted distance measures and the hybrid weighted correlation coefficients for linguistic hesitant fuzzy sets, by which we can derive the comprehensive evaluated values of the objects. Furthermore, we study their applications to pattern recognition and to multi-attribute decision making.
Comparing with the previous researches about decision making with LHFSs, there are several contributions of our method:
However, we only present one distance measure and one correlation coefficient, and it will be interesting to study other distance measures and correlation coefficients for linguistic hesitant fuzzy sets. Furthermore, we shall discuss their application in other fields, such as expert systems, digital image processing, and clustering analysis. Moreover, we can extend the developed theoretical results to other types of fuzzy sets, such as hesitant interval neutrosophic linguistic sets (Ye,
2013).