Hesitancy degree-based correlation measures for hesitant fuzzy linguistic term sets and their applications in multiple criteria decision making

Information Sciences 508 (2020) 275–292

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Hesitancy degree-based correlation measures for hesitant fuzzy linguistic term sets and their applications in multiple criteria decision making Huchang Liao a,b, Xunjie Gou a,b, Zeshui Xu a,∗, Xiao-Jun Zeng c, Francisco Herrera b,d a

Business School, Sichuan University, Chengdu 610064, China Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI), University of Granada, Granada 18071, Spain c School of Computer Science, University of Manchester, Manchester M13 9PL, UK d Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia b

a r t i c l e

i n f o

Article history: Received 1 October 2017 Revised 26 August 2019 Accepted 27 August 2019 Available online 28 August 2019 Keywords: Multiple criteria decision making Hesitant fuzzy linguistic term set Correlation measure Correlation coefficient Weight-determining method Qualitative decision making

a b s t r a c t The hesitant fuzzy linguistic term set (HFLTS) turns out to be useful in representing people’s hesitant qualitative information. The aim of this paper is to investigate new correlation measures between HFLTSs and apply them in decision-making process. Firstly, the concepts of mean and hesitancy degree of hesitant fuzzy linguistic elements are introduced. Based on them, we address the drawbacks of the existing correlation measures between HFLTSs. Then, a new correlation coefficient between HFLTSs is established. Additionally, the hesitancy degree of the hesitant fuzzy linguistic correlation coefficient is proposed, which is composed by the upper and lower bounds of the hesitant fuzzy linguistic correlation coefficient. To show the applicability of the proposed correlation measures, a correlation coefficient-based method is developed for multiple criteria decision making in the cases that the weights of criteria are either known or unknown. A practical example concerning the strategic management of Sichuan liquor brands in China is given to validate the proposed method. It is verified that the proposed correlation coefficients between HFLTSs is more convincing than the existing ones and the developed correlation coefficient-based hesitant fuzzy linguistic MCDM with the weights of criteria being either completely known or unknown is applicable. © 2019 Elsevier Inc. All rights reserved.

1. Introduction 1.1. Research background For many decision-making problems, it is easy for decision-makers (DMs) or experts to express their opinions in linguistic terms, such as “low” cost, “high” quality, “good” performance. To formulate linguistic terms, Zadeh [26] proposed the linguistic approach, which uses linguistic variables whose values are not numbers but words or expressions, to represent qualitative opinions. The linguistic variable enhances the feasibility and flexibility of decision-making models, and the progress of analyzing linguistic variables has led to a completely new research area named Computing with Words. ∗

Corresponding author. E-mail address: [email protected] (Z. Xu).

https://doi.org/10.1016/j.ins.2019.08.068 0020-0255/© 2019 Elsevier Inc. All rights reserved.

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The traditional linguistic approach uses single linguistic terms to represent the values of a linguistic variable, and thus cannot represent the hesitant perceptions of DMs. For example, when evaluating a research proposal, an expert may say “it is between good and excellent”. Also, in a group decision-making process, the group’s assessments could be uncertain if there are different opinions among group members. To overcome the inability of the linguistic approach, Rodríguez et al. [18] introduced the concept of hesitant fuzzy linguistic term set (HFLTS). With the use of the HFLTS, experts can provide their uncertain assessments by means of several linguistic terms or comparative linguistic expressions. Therefore, the HFLTS has attracted many scholars’ attention. Please refer to Section 2.1 for details. Surveys on HFLTSs can be found in [14,22]. Correlation measure is an important concept in statistics. It has been extended to different fuzzy environments, yielding fuzzy correlation measures [1], intuitionistic fuzzy correlation measures [5,24], interval-valued intuitionistic fuzzy correlation measures [17], hesitant fuzzy correlation measures [12,19], dual hesitant fuzzy linguistic correlation measures [27], and double hierarchy hesitant fuzzy linguistic correlation measures [2]. Generally, to solve multiple criteria decision-making (MCDM) problems, we need to determine the alternatives and criteria, obtain the weights of criteria and then select the optimal alternatives. It is essential to establish independent criteria and reduce the number of criteria to “seven plus or ˙ [25] conducted a correlation test to obtain independent criteria and minus two” [16]. In this regard, Yurdakul and Tansel IÇ limit the number of criteria to seven plus or minus two. To determine the weights of criteria under the intuitionistic fuzzy environment, Ye [24] proposed the entropy-based weighting models, and then selected the alternatives based on correlation coefficients. Park et al. [17] proposed some optimization models to derive the weights of criteria, and then used the correlation coefficient between each alternative to the ideal one to rank the alternatives. Based on the above analyses, we can find that correlation measures are very important in solving MCDM problems. 1.2. Research challenges and gaps Correlation coefficients are used to represent, in a linear fashion, the closeness degree of two variables. The correlation measures can be classified into two categories, i.e., the statistic-based correlation measures whose values belong to the interval [−1, 1], and the information-energy-based correlation measures, whose values vary within the unit interval [0, 1]. As for the hesitant fuzzy linguistic information, Liao et al. [13] introduced some correlation measures between HFLTSs. However, there are some unsolved issues: (a) The correlation measures in [13] assumed that all HFLEs have an equal length; otherwise, the average linguistic terms are added to the short HFLEs. However, it is common that the HFLEs do not have an equal length in practical applications, considering that different experts may provide different evaluation information. As shown by the examples in Section 3, adding artificial terms to the short HFLEs would change the original information. (b) The existing hesitant fuzzy linguistic correlation coefficients are motivated by the information energy of HFLTSs. Thus, they are restricted within [0, 1]. In other words, these correlation measures cannot distinguish the positive or negative relationships between HFLTSs. (c) Given that the HFLEs are composed of multiple linguistic terms, it would be convincing to take the correlation coefficients between HFLTSs to be hesitant as well. However, all the correlation coefficients calculated by the formulas in [13] are numerical. Thus, they cannot reveal the real situation in hesitant linguistic context. 1.3. Motivation and contributions of this study To solve the above issues, in this study, we investigate the correlation measures between HFLTSs from the statistic points of view. The theoretical contributions of this paper can be summarized as follows: (1) We introduce the concepts of mean and hesitancy degree of the HFLEs. Based on these concepts, we justify that it is unreasonable to add artificial values to the short HFLEs. As we know, many existing methods [6,13] have an underlying assumption that the short HFLEs should be extended by adding artificial values, which will definitely make these methods unreasonable. However, the decision-making method proposed in this paper does not have this problem. This overcomes the first drawback as mentioned above. (2) We establish new correlation coefficients between HFLTSs, whose values lie in the interval [−1, 1]. They can be used to represent the positive or negative relationship between different variables. It overcomes the drawback b) of the existing correlation coefficients proposed in [13] and thus are much more convincing and have wider application potentials. (3) We also introduce the hesitancy degree of the hesitant fuzzy linguistic correlation coefficient, which is composed of the upper and lower bounds of the hesitant fuzzy linguistic correlation coefficient. We further propose the weighted form and ordered weighted form of the hesitant fuzzy linguistic correlation coefficient. In this way, the research gap c) mentioned above can be filled. In addition, as we can see, the MCDM with hesitant fuzzy linguistic information is a hot topic and has been investigated by many scholars. To show the applicability of the proposed correlation measures between HFLEs, we develop a correlation coefficient-based method for hesitant fuzzy linguistic MCDM problems where the weights of criteria are either completely known or unknown. In the case that the weights of criteria are unknown, this paper introduces an entropy-based method

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and a distance-based method to derive the weights of criteria. They are much more convincing than the subjective weightdetermining method that the weights are directly given by experts. To solve hesitant fuzzy linguistic MCDM problems, after establishing the ideal solution, we calculate the correlation coefficient of each alternative to the ideal solution, and then rank these alternatives according to their correlation coefficient values. To show the efficiency of the proposed decision-making method, we apply the method to solve a case study concerning the strategic management of Sichuan liquor brands in China. 1.4. Framework of this study The paper is organized as follows. The next section makes a literature review on HFLTSs and the existing correlation measures between HFLTSs. Section 3 establishes some correlation measures between HFLTSs. Section 4 develops a correlation coefficient-based MCDM method after introducing two weight-determining methods. Section 5 illustrates the proposed method by a case study concerning the strategic management of Sichuan liquor brands. Comparative analyses with the existing hesitant fuzzy linguistic MCDM methods are also given in this section. The paper ends with some concluding remarks in Section 6. 2. Preliminaries This section reviews the HFLTS and the existing correlation measures between HFLTSs. The qualitative MCDM problems are conceptualized at the end of this section. 2.1. Hesitant fuzzy linguistic term set To describe the complex linguistic information, Rodríguez et al. [18] introduced the concept of HFLTS as an ordered finite subset of the consecutive linguistic terms of a given linguistic term set (LTS) S, which can be used to elicit several linguistic terms or linguistic expression for a linguistic variable. Since Rodríguez et al. [18] did not give any mathematical form for HFLTS, Liao et al. [13] extended and formalized the HFLTS as follows: Definition 1 [13]. Let xi ∈ X, i = 1, 2, · · · , N, be fixed and S = {st |t = −τ , · · · , −1, 0, 1, · · · , τ } be an LTS. An HFLTS on X, HS , is in the form of

HS = {< xi , hS (xi ) > |xi ∈ X }

(1)

where hS (xi ) is a set of some continuous values in S, denoting the possible degrees of the linguistic variable xi to S, and

hS (xi ) = {sϕl (xi )|sϕl (xi ) ∈ S; l = 1, 2, · · · , L(xi )}

(2)

with L(xi ) being the number of linguistic terms in hS (xi ). For convenience, hS (xi ) is called a hesitant fuzzy linguistic element (HFLE). Remark 1. In Definition 1, the linguistic terms are chosen in discrete form from S and the subscript ϕl belongs to {−τ , · · · , −1, 0, 1, · · · , τ }. We can extend it to a continuous form, i.e., ϕl ∈ [−τ , τ ], which is much more general and flexible. As the HFLTS is not similar to the human way of thinking and reasoning, the context-free grammar [18] was proposed to generate simple but elaborated linguistic expressions that align human expressions. With the transformation function EGH [18], it is easy to transform the initial linguistic expression l l to an HFLE hS . There are many results related to HFLTSs. Different kinds of distance measures, similarity measures, correlation coefficients and entropy measures for HFLTSs can be found in [3,9,10,13,20]. Different MCDM methods with HFLTSs have been proposed [7,8,11,15]. 2.2. The existing correlation measures between HFLTSs From the information energy point of view, Liao et al. [13] proposed some correlation measures between HFLTSs. The information energy of an HFLTS HS was defined as:

E (HS ) =

n 



i=1

Li 1 Li l=1



ϕl ( xi ) 2τ

2 

(3)

For two HFLTSs HS1 = {< xi , h1S (xi ) > |xi ∈ X } and HS2 = {< xi , h2S (xi ) > |xi ∈ X } with hkS (xi ) = {sϕ k (xi )|sϕ k (xi ) ∈ S, l = 1, 2,

· · · , Li }, k = 1, 2, the correlation between HS1 and HS2 was defined as [13]:

C0 (HS1 , HS2 ) =

n  i=1



Li 1 Li l=1



|ϕl1 (xi )| |ϕl2 (xi )| · 2τ 2τ

l

l



(4)

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The correlation coefficient between HS1 and HS2 was defined as [13]:

ζ0 (HS1 , HS2 ) = 

C( E(

HS1

HS1 , HS2

) · E(

n

)

HS2

i=1

1 |ϕl ( xi )| Li 1 Li

2τ

l=1

·

|ϕl2 (xi )| 2τ



1 / 2 =  

1 2   2 2 1/2 n ) ϕl ( xi ) ϕl ( xi ) Li n 1 L i 1 i=1

Li

 1 L i 

l=1

2τ

·

i=1

 n |ϕl1 (xi )| · |ϕl2 (xi )| i=1 Li l=1 =

 2  n 1 Li  2 2 1/2 n Li 1 · i=1 Li l=1 ϕl (xi ) ϕl1 (xi ) i=1 Li l=1

Li

l=1

2τ

(5)

where Li is the maximum number of linguistic terms in h1S (xi ) and h2S (xi ) (the shorter one should be extended till equal length), and n is the cardinality of X. Liao et al. [13] proved that for any two HFLTSs HS1 and HS2 , 0 ≤ ζ0 (HS1 , HS2 ) ≤ 1 holds. That is to say, this correlation coefficient between HFLTSs lies in the unit interval [0, 1]. 2.3. Description of the qualitative MCDM problems To show the applicability of the correlation measures, this paper dedicates to developing a method to solve qualitative MCDM problems with hesitant fuzzy linguistic information. Let A = {A1 , A2 , . . . , Am } be a set of alternatives, and C = {C1 , C2 , . . . , Cn } be a set of criteria whose weight vector is ω = (ω1 , ω2 , . . . , ωn )T with ω j ∈ [0, 1], j = 1, 2, . . . , n, and n j=1 ω j = 1. The evaluation value of alternative Ai with respect to criterion C j is represented by a linguistic expression l li j . Based on the transformation function EGH , we can transfer each linguistic expression l li j to the HFLE hSi j , where hS = ij

{si(jl ) |si(jl ) ∈ S j ; l = 1, 2, . . . , Li j ; i = 1, 2, . . . , m; j = 1, 2, . . . , n}. Then, a hesitant fuzzy linguistic decision matrix H = (hSi j )m×n is established as:

⎡

hS11 ⎢ hS21 H=⎢ . ⎣ .. hSm 1

hS12 hS22 .. . hSm 2

··· ··· .. . ···

⎤

hS1n hS2n ⎥ .. ⎥ ⎦ . hSmn

(6)

In hesitant fuzzy linguistic MCDM problems, basically, there are three types of criteria: benefit type, cost type, and target type. For the target criterion, we have a certain ideal value that is neither the upper bound as the benefit criterion nor the lower bound like the cost criterion. Generally, the optimal value hoj of the target criterion C j is determined in advance by

T the DM. According to different types of criteria, it is easy to furnish the ideal solution H + = (h+ , h+ , . . . , h+ n ) to the hesitant 1 2 fuzzy linguistic MCDM problem where

h+j

=

⎧ max hS ⎪ ⎨i=1,2,...,m i j min hSi j

⎪ ⎩i=1o ,2,...,m hj

for the benefit criterion C j for the cost criterion C j

, j = 1, 2, . . . , n,

(7)

for the target criterion C j

The ideal solution sometimes does not exist because the criteria may conflict with each other. It is obvious that the best alternative should be the one that has the highest correlation coefficient to the ideal solution H + . To solve hesitant fuzzy linguistic MCDM problems, after establishing the ideal solution, we need to calculate the correlation coefficient of each alternative to the ideal solution, and then rank these alternatives according to their correlation coefficient values. In this sense, the correlation coefficient is significant in aiding qualitative decision making. 3. The mean and hesitancy degree of HFLEs To show the drawbacks of the existing correlation measure between HFLTSs, this section introduces the mean and hesitancy degree of HFLEs. It is observed that the correlation measures proposed by Liao et al. [13] were based on the assumption that each HFLE has an equal length; otherwise, some linguistic terms are added to the shorter HFLEs. However, it is common that the HFLEs do not have an equal length in practical applications considering that different experts may provide different evaluation information. In addition, arbitrarily adding artificial terms to an HFLE would change the original information of the HFLE. This problem can be clarified by the mean and hesitancy degree of HFLEs. Definition 2. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS and hiS = {sϕ i |sϕ i ∈ S, l = 1, 2, . . . , Li } be an HFLE with Li being l l the number of linguistic terms in hiS . The mean of hiS is defined as: Li 1 ϕli h¯ iS = sϕ¯ i , where ϕ¯ i = Li l=1

(8)

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279

The hesitancy degree of hiS is defined as:



δ( ) = hiS

 

Li Li  i 2  1  1 ϕl − ϕ¯ i =  Li Li l=1





ϕ − i l

l=1

Li 1 ϕli Li

2 (9)

l=1

It is noted that the mean of the HFLE hiS , h¯ iS , is a linguistic term, while the hesitancy degree of the HFLE hiS , δ (hiS ), is a crisp number. Example 1. Consider three HFLEs h1S = {s2 }, h2S = {s2 , s3 } and h3S = {s0 , s1 , s2 , s3 }. According to Definition 2, the means of these three HFLEs are h¯ 1S = s2 , h¯ 2S = s2.5 and h¯ 3S = s1.5 , respectively. The hesitancy degrees of these three HFLEs are δ (h1S ) = 0, δ (h2S ) = 0.5, and δ (h3S ) = 1.118, respectively. Example 2. Consider Example 3 in [13]. For two HFLTSs HS1 = {{s−3 , s−2 , s−1 }, {s0 , s1 , s2 }, {s2 , s3 }} and HS2 = {{s2 , s3 }, {s−1 , s0 }, {s−3 }}, to calculate the correlation degree between HS1 and HS2 , the authors extended HS2 to   H 2 = {{s2 , s2.5 , s3 }, {s−1 , s−0.5 , s0 }, {s−3 , s−3 }}. Based on Definition 2, we have h¯ 21 = s2.5 , h¯ 22 = s−0.5 , h¯ 23 = s−3 , and h¯ 21 = s2.5 , C

S

S

S

S

 ¯ 23 h¯ 22 S = s−0.5 , hS = s−3 . We can see that the mean of each HFLE is unchanged. This is because they extended the shorter HFLEs by adding the average linguistic terms. If we add other artificial linguistic terms, the mean values should be changed. In terms of the hesitancy degree, based on Definition 2, we have δ (h21 ) = 0.5, δ (h22 ) = 0.5, δ (h23 ) = 0, and S S S    21 22 23 ¯ ¯ ¯ δ (h ) = 0.4082, δ (h ) = 0.4082 and δ (h ) = 0. S

S

S

From Example 2, we can find that by adding the average linguistic terms to the shorter HFLEs to make them have an equal length to the longer one would change their original information (the hesitancy degrees are changed). It is easy to check that if we add the maximum or minimum linguistic terms to the shorter HFLEs, both the means and hesitancy degrees of HFLEs would be changed. This is the main shortcoming of the existing correlation measures between HFLTSs. Based on the mean and hesitancy degree of HFLEs, we can develop a scheme to rank HFLEs. Scheme 1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For two HFLEs hiS = {sϕ i |sϕ i ∈ S, l = 1, 2, . . . , Li} (i = 1, 2), if l

l

h¯ 1S > h¯ 2S , then h1S > h2S ; if h¯ 1S = h¯ 2S , then h1S > h2S if δ (h1S ) < δ (h2S ) and h1S = h2S if δ (h1S ) = δ (h2S ). 4. Correlation measures for hesitant fuzzy linguistic information from a statistic point of view As we can see from Example 2, it is not reasonable to add artificial linguistic terms to the shorter HFLEs. However, this is just the precondition of the existing correlation measure between HFLTSs as given in Eq. (5). In this sense, the existing correlation measure between HFLTSs is not adequate for practical application. In addition, as we justified in the introduction, the existing correlation measures between HFLTSs are restricted within [0, 1], which cannot reflect the positive or negative relationships between two HFLTSs. What’s more, it cannot depict the hesitant linguistic information since the correlation coefficient is a crisp value. To avoid these shortcomings, this section introduces new correlation measures from the perspective of classical statistics. To broaden their applicability, we also propose the weighted and ordered weighted forms of the correlation measures between HFLTSs. 4.1. Correlation measures for hesitant fuzzy linguistic information from a statistic point of view Firstly, we propose some definitions related to HFLTSs. Definition 3. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For an HFLTS HS = {< xi , hS (xi ) > |xi ∈ X } with hS (xi ) = {sϕl (xi )|sϕl (xi ) ∈ S, l = 1, 2, · · · , Li}, the mean of the HFLTS HS is defined as: n n 1 i 1 ϕ¯ = H¯ S = sϕ¯ , where ϕ¯ = n n i=1



i=1

Li 1 ϕli Li



(10)

l=1

The variance of the HFLTS HS is defined as: n n 2 1  1  i V ar (HS ) = ϕ¯ − ϕ¯ = n n i=1



i=1

Li n 1 1 ϕli − Li n l=1

i=1



Li 1 ϕli Li

2 (11)

l=1

Definition 4. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For two HFLTSs HS1 = {< xi , h1S (xi ) > |xi ∈ X } and HS2 = {< xi , h2S (xi ) > |xi ∈ X } with hkS (xi ) = {sϕ k (xi )|sϕ k (xi ) ∈ S, l = 1, 2, · · · , Lki } and Lki being the number of linguistic terms in hkS (xi ), l

l

k = 1, 2, the correlation between HS1 and HS2 is defined as:

C ( HS1 , HS2 ) =

n    1   i1 ϕ¯ − ϕ¯ 1 · ϕ¯ i2 − ϕ¯ 2 n i=1

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n 1 = n



i=1

1

Li n 1 1  i ϕ − l n L1i i=1 l=1



1

Li 1  ϕli L1i

 

2

Li n 1 1  · 2 ϕli − n Li i=1

l=1



l=1

2

Li 1  ϕli L2i

 (12)

l=1

Note 1. The cardinalities of different HFLEs may be different and we do not need to require that the HFLEs hkS (xi ) (i = 1, 2, . . . , n, k = 1, 2) have the same length. It is quite different from Eq. (5). In addition, the correlation given in Definition 4 could be either positive or negative. It is also different from Eq. (5). Proposition 1. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For the HFLTS HS = {< xi , hS (xi ) > |xi ∈ X } with hS (xi ) = {sϕl (xi )|sϕl (xi ) ∈ S, l = 1, 2, . . . , Li}, we have

C (HS , HS ) = V ar (HS )

(13)

In analogous to traditional statistics, we can define the correlation coefficient between HFLTSs. Definition 5. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For two HFLTSs HS1 = {< xi , h1S (xi ) > |xi ∈ X } and HS2 = {< xi , h2S (xi ) > |xi ∈ X } with hkS (xi ) = {sϕ k (xi )|sϕ k (xi ) ∈ S, l = 1, 2, . . . , Lki } and Lki being the number of linguistic terms in hkS (xi ), l

l

k = 1, 2, the correlation coefficient between HS1 and HS2 is defined as:

ζ(

HS1 , HS2

   ϕ¯ i1 − ϕ¯ 1 · ϕ¯ i2 − ϕ¯ 2 )=    1 / 2 =  1 n ¯ i1 − ϕ¯ 1 |2 · 1n ni=1 |ϕ¯ i2 − ϕ¯ 2 |2 V ar (HS1 ) · V ar (HS2 ) i=1 |ϕ n  1

1   2

2  Li Li Li Li 1 n 1 ϕli − 1n ni=1 L11 l=1 ϕli · L12 l=1 ϕli − 1n ni=1 L12 l=1 ϕli i=1 L1 n l=1 i i i i = ! !

1

1 2

2

2 2 Li Li Li Li 1 n 1 1 n 1 1 n 1 1 n 1 i i i ϕ − ϕ · ϕ − ϕi 1 1 2 i =1 i =1 i =1 i =1 n n n n l=1 l l=1 l l=1 l l=1 l L L L L2 C (HS1 , HS2 )

1 n

i

n



i=1

i

i

(14)

i

Based on Proposition 1, we can get the following proposition immediately. Proposition 2. The correlation coefficient ζ (HS1 , HS2 ) between the HFLTSs HS1 and HS2 satisfies: (1) ζ (HS1 , HS2 ) = ζ (HS2 , HS1 ); (2) ζ (HS1 , HS1 ) = 1. As we indicated in Note 1, the correlation could be either positive or negative, which is quite different from the existing correlation measures for HFLTSs. Actually, we can prove that the correlation coefficient proposed in Definition 5 lies in the interval [−1, 1], which is just the same as that in the traditional statistics. Theorem 1. The correlation coefficient ζ (HS1 , HS2 ) between the HFLTSs HS1 and HS2 satisfies −1 ≤ ζ (HS1 , HS2 ) ≤ 1. Proof. According to the Cauchy-Schwarz inequality (x1 y1 + x2 y2 + · · · + xn yn )2 ≤ (x21 + x22 + · · · + x2n ) · (y21 + y22 + · · · + y2n ), where xi , yi ∈ R, i = 1, 2, . . . , n we have

|C (

HS1 , HS2

" " "  n    i2 " "1 i1 1 2 " )| = " ϕ¯ − ϕ¯ · ϕ¯ − ϕ¯ " " " n i=1   ≤

n 1  i1 |ϕ¯ − ϕ¯ 1 |2 · n i=1

n  1 / 2 1  i2 |ϕ¯ − ϕ¯ 2 |2 = V ar (HS1 ) · V ar (HS2 ) n

(15)

i=1

Thus, we have |ζ (HS1 , HS2 )| ≤ 1, i.e., −1 ≤ ζ (HS1 , HS2 ) ≤ 1. This completes the proof of Theorem 1.



Definition 6. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For an HFLTS HS = {< xi , hS (xi ) > |xi ∈ X } with hS (xi ) = {sϕl (xi )|sϕl (xi ) ∈ S, l = 1, 2, · · · , Li}, the negative set of HS is defined as: HS− = {< xi , h−S (xi ) > |xi ∈ X } with h−S (xi ) = {s−ϕl (xi )|s−ϕl (xi ) ∈ S, l = 1, 2, . . . , Li}. Proposition 3. The correlation coefficient ζ (HS , HS− ) = −1. Propositions 2 and 3 indicate that the correlation coefficient between an HFLTS and itself attains the maximal value, while the correlation coefficient between an HFLTS and its corresponding negative set attains the minimal correlation degree. However, for any two distinct HFLTSs HS1 and HS2 , it is not the common case to get the maximum value or the minimum value. We can obtain a crisp value of the correlation coefficient between HFLTSs according to Eq. (14). However, we should not forget that the essential feature of the HFLTS is hesitation. Thus, only using a crisp value to represent their relationship may not make sense or lose useful information related to uncertainty and hesitation. It is natural to suppose that the correlation coefficient has a hesitancy degree. From this point of view, we shall determine some indicators to further represent the correlation degree between HFLTSs. The simplest way to measure the hesitancy is the upper and low bounds of the correlation coefficient.

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According to Eq. (14), the correlation coefficient between HFLTSs HS1 and HS2 can be simplified as:



   ϕ¯ i1 − ϕ¯ 1 · ϕ¯ i2 − ϕ¯ 2 ) =#  2 # n  2 n  i 1 ¯ − ϕ¯ 1 · ¯ i2 − ϕ¯ 2 i=1 ϕ i=1 ϕ n

ζ(

HS1 , HS2

i=1

(16)

We define the upper and low bounds of the correlation coefficient ζ (HS1 , HS2 ) as



n

ζ ( U

HS1 , HS2

)=

#

max

pi ∈{ϕli1 },qi ∈{ϕli2 } i=1,2,··· ,n

i=1



n

i=1

pi − ϕ¯ 1



n

ζ ( L

HS1 , HS2

)=

#

min

pi ∈{ϕli1 },qi ∈{ϕli2 } i=1,2,...,n



i=1

 2 # n  ·

pi − ϕ¯

i=1

n

 

pi − ϕ¯ 1 · qi − ϕ¯ 2

pi − ϕ¯ 1

i=1

· qi − ϕ¯ 2

·

i=1

2

(17)

2

(18)

qi − ϕ¯ 2

  1

2 # n 





qi − ϕ¯ 2

The following theorem verifies the upper and lower bounds of the correlation coefficient between any HFLTSs. Theorem 2. Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be an LTS. For two HFLTSs HS1 = {< xi , h1S (xi ) > |xi ∈ X } and HS2 = {< xi , h2S (xi ) > |xi ∈ X } with hkS (xi ) = {sϕ k (xi )|sϕ k (xi ) ∈ S, l = 1, 2, . . . , Lki } and Lki being the number of linguistic terms in hkS (xi ), l

l

k = 1, 2. Let pUi = maxl=1,2,··· ,L1 {ϕli1 }, pLi = maxl=1,2,··· ,L1 {ϕli1 }, qUi = maxl=1,2,...,L2 {ϕli2 } and qLi = maxl=1,2,...,L2 {ϕli2 }. Then, it i

follows

i

i

i

ζ L (HS1 , HS2 ) ≤ ζ (HS1 , HS2 ) ≤ ζ U (HS1 , HS2 ) where



n

ζ U (HS1 , HS2 ) = #

i=1

n



i=1

(19)

 

pUi − ϕ¯ 1 · qUi − ϕ¯ 2

pUi − ϕ¯

2 # n  1 ·

i=1



n

qUi − ϕ¯ 2

2

ζ L (HS1 , HS2 ) = #

,



i=1



n

i=1

 

pLi − ϕ¯ 1 · qLi − ϕ¯ 2

pLi − ϕ¯

 2 # n  1 ·

i=1



qLi − ϕ¯ 2

2

(20)

Proof. We only prove the right side of the inequality concerning ζ U (HS1 , HS2 ) . The other side can be proven analogously. Let pi ∈ {ϕ1i1 , ϕ2i1 , . . . , ϕ i11 }, pUi = max{ϕ1i1 , ϕ2i1 , . . . , ϕ i11 }, αi = pUi − ϕ¯ 1 , and βi = pi − ϕ¯ 1 . Then, we have αi ≥ βi . Let qi ∈ Li

Li

{ϕ1i2 , ϕ2i2 , . . . , ϕLi22 }, qUi = max{ϕ1i2 , ϕ2i2 , . . . , ϕLi22 }, γi = qUi − ϕ¯ 2 , and λi = qi − ϕ¯ 2 . Then, we have γi ≥ λi . i

i

2 2 Let fi ( p1 , p2 , . . . , pi−1 , pi+1 , . . . , pn ) = nj=1, j=i ( p j − ϕ¯ 1 ) = fi and gi (q1 , q2 , . . . , qi−1 , qi+1 , . . . , qn ) = nj=1, j=i (q j − ϕ¯ 2 ) = gi . Then it is evident that fi > 0 and gi > 0. With the above representation, we have



n

#



n

j=1, j=i

p j − ϕ¯ 1

2

i=1



+ pUi − ϕ¯

2 # n 1 ·

According to Lemma 1 in [12], f (x ) = then, it follows



n

i=1

αi · γi 

f i + αi 2 ·

gi + γi 2

=

n i=1

= #



f i + αi 2



n

i=1

n  i=1



q j − ϕ¯ 2

·

γi

≥

gi + γi 2 n 

2

i=1



i=1

+ pi − ϕ¯

 

 2 # n  1 ·

n

pi − ϕ¯

pi − ϕ¯ 1 · qi − ϕ¯ 2

pi − ϕ¯

2



+ qUi − ϕ¯ 2

2

=



n

i=1

αi · γi 

fi + αi · 2

i=1



  1

βi fi + βi

· q − ϕ¯

i 2 # n 1



·

qi − ϕ¯ 2

2

·

 2

j =1, j =i



λi g i + λi

q j − ϕ¯ 2

2

2

= 

n

i=1 2

#

n

j=1, j=i



p j − ϕ¯ 1

2

i=1





 

pUi − ϕ¯ 1 · qUi − ϕ¯ 2

+ pUi − ϕ¯ 1

2 # n ·



+ qi − ϕ¯ 2

j=1, j=i

g i + λi

2

2 (22)

2





βi · λi 

fi + βi ·

Combining Eqs. (21) and (22), for all pi ∈ {ϕli1 }, qi ∈ {ϕli2 }, i = 1, 2, . . . , n, we obtain

n

(21)

gi + γi 2

is a monotonically increasing function when a > 0. Since αi ≥ βi and γi ≥ λi ,

p j − ϕ¯ 1

n 



j=1, j=i

√x x2 +a

αi

j =1, j =i

= #

 

pUi − ϕ¯ 1 · qUi − ϕ¯ 2

q j − ϕ¯ 2

n

2



+ qUi − ϕ¯ 2

i=1

 2 ≥ # n  i=1



 

pi − ϕ¯ 1 · qi − ϕ¯ 2

pi − ϕ¯ 1

 2 # n  ·

i=1



qi − ϕ¯ 2

2 (23)

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H. Liao, X. Gou and Z. Xu et al. / Information Sciences 508 (2020) 275–292

In the denominator of the left side of Eq. (23), we can let pi = pUi , qi = qUi . Thus, it follows



n

#



n

≥

i=1

2

 

pUi − ϕ¯ 1 · qUi − ϕ¯ 2

2 # n



¯ 1 + pUi − ϕ¯ 1 · j=1, j=i p j − ϕ    n  ¯ 1 · qi − ϕ¯ 2 i=1 pi − ϕ

#

n



i=1

pi − ϕ

¯1

2 # n  ·

i=1

n

ζ U (HS1 , HS2 ) = #



i=1

n



i=1

pUi



j=1, j=i

qi − ϕ

¯2

Combing Eqs. (24) and (17), it yields

−ϕ

 

2 # n  ·

n

q j − ϕ¯ 2

2



i=1



+ qUi − ϕ¯ 2

 2 = # n  i=1

 

pUi − ϕ¯ 1 · qUi − ϕ¯ 2

pUi − ϕ¯ 1

 2 # n  ·

i=1

i1 i2 2 , for all pi ∈ {ϕl }, qi ∈ {ϕl }, i = 1, 2, . . . , n

pUi − ϕ¯ 1 · qUi − ϕ¯ 2 ¯1



i=1

qUi



qUi − ϕ¯ 2

2

(24)

 −ϕ

¯2

(25)

2

In addition, on the right side of Eq. (24), let pi = ϕ¯ i1 and qi = ϕ¯ i2 . Then, we have

   ϕ¯ i1 − ϕ¯ 1 · ϕ¯ i2 − ϕ¯ 2 #   2 # n  U 2 ≥ # n  2 # n  2 n U 1 2 i1 − ϕ 1 p − ϕ ¯ · q − ϕ ¯ ϕ ¯ ¯ · ¯ i2 − ϕ¯ 2 i i=1 i=1 i=1 ϕ i i n

i=1



 

pUi − ϕ¯ 1 · qUi − ϕ¯ 2



n



i=1

(26)

i.e., ζ U (HS1 , HS2 ) ≥ ζ (HS1 , HS2 ). This completes the proof of Theorem 2. Based on the upper and lower bounds of the correlation coefficient, it is natural to introduce the hesitancy degree of the correlation coefficient. Definition 7. The hesitancy degree of the correlation coefficient ζ (HS1 , HS2 ) defined as Eq. (12) is calculated as

δ (HS1 , HS2 ) = ζ U (HS1 , HS2 ) − ζ L (HS1 , HS2 )

(27)

where ζ U (HS1 , HS2 ) and ζ L (HS1 , HS2 ) are the upper and lower bounds of ζ (HS1 , HS2 ), respectively. 4.2. Weighted correlation measures for hesitant fuzzy linguistic information It is observed that sometimes the arguments corresponding to different objects may be associated with different weights. Thus, when calculating the correlation coefficient between HFLTSs, we shall consider the influence resulted from the weights. In the following, we propose different weighted correlation measures between HFLTSs. For two HFLTSs HS1 = {< xi , h1S (xi ) > |xi ∈ X } and HS2 = {< xi , h2S (xi ) > |xi ∈ X } with hkS (xi ) = {sϕ k (xi )|sϕ k (xi ) ∈ S, l = l

l

1, 2, . . . , Lki }, k = 1, 2, suppose that the weight associated with each objective xi is ωi , which satisfies ωi ∈ [0, 1] (i = 1, 2, . . . , n) and ni=1 ωi = 1. (1) The weighted mean of the HFLTS HS is n n 1 1 ωi ϕ¯ i = H¯ Sω = sϕ¯ ω , where ϕ¯ ω = n n i=1



i=1

Li ωi 

Li

 ϕ

i l

(28)

l=1

(2) The weighted variance of the HFLTS HS is n n 2 1  1  V ar (HSω ) = ωi ϕ¯ i − ϕ¯ ω = n n i=1



Li ωi 

Li

i=1

ϕ

i l

n 1 − n i=1

l=1



Li ωi 

Li

2 ϕ

i l

(29)

l=1

(3) The weighted correlation between HS1 and HS2 is

Cω ( HS1 , HS2 ) =

n    1  ωi ϕ¯ i1 − ϕ¯ ω1 · ωi ϕ¯ i2 − ϕ¯ ω2 n i=1

n 1 = n i=1



1

Li ωi 

L1i

l=1

n 1 ϕli − n i=1



 

1

Li ωi 

L1i

l=1

ϕli

·

2

Li ωi 

L2i

l=1

n 1 ϕli − n i=1





2

Li ωi 

L2i

l=1

ϕli

(30)

H. Liao, X. Gou and Z. Xu et al. / Information Sciences 508 (2020) 275–292

283

(4) The weighted correlation coefficient between HS1 and HS2 is

ζ (

1 2 ω HS , HS

 i1    ωi ϕ¯ − ϕ¯ ω1 · ωi ϕ¯ i2 − ϕ¯ ω2 )=   1/2 =  1 n ¯ i1 − ϕ¯ ω1 |2 · 1n ni=1 |ωi ϕ¯ i2 − ϕ¯ ω2 |2 Var (HS1ω ) · Var (HS2ω ) i=1 |ωi ϕ n  1

1   2

2  Li Li Li Li ωi 1 n ϕ i − 1n ni=1 ωL1i l=1 ϕli · ωL2i l=1 ϕli − 1n ni=1 ωL2i l=1 ϕli i=1 L1 n l=1 l i i i i = ! !

1

1 2

2

2 2 Li Li Li Li 1 n 1 ϕ i − 1n ni=1 L11 l=1 ϕli · 1n ni=1 L12 l=1 ϕli − 1n ni=1 L12 l=1 ϕli i=1 L1 n l=1 l Cω (HS1 , HS2 )

1 n

i

n

i=1

i

i

(31)

i

It is easy to verify that ζω (HS1 , HS2 ) satisfies: (i) ζω (HS1 , HS2 ) = ζω (HS2 , HS1 ); (ii) ζω (HS1 , HS1 ) = 1; (iii) ζω (HS , HS− ) = −1; (iv) −1 ≤ ζω (HS1 , HS2 ) ≤ 1. (5) The upper and lower bounds of the weighted correlation coefficient ζω (HS1 , HS2 ) are defined as:



max

   ωi pi − ϕ¯ ω1 · ωi qi − ϕ¯ ω2 # 2 # n  2 n  ¯ ω1 · ¯ ω2 i=1 ωi pi − ϕ i=1 ωi qi − ϕ

min

   ωi pi − ϕ¯ ω1 · ωi qi − ϕ¯ ω2 # 2 # n  2 n  ¯ ω1 · ¯ ω2 i=1 ωi pi − ϕ i=1 ωi qi − ϕ

n

ζ ( U

1 2 ω HS , HS

)=

i=1

pi ∈{ϕli1 },qi ∈{ϕli2 } i=1,2,...,n

n

ζ ( L

1 2 ω HS , HS

)=

(32)



i=1

pi ∈{ϕli1 },qi ∈{ϕli2 } i=1,2,...,n

(33)

Thus, we can obtain ζωL (HS1 , HS2 ) ≤ ζω (HS1 , HS2 ) ≤ ζωU (HS1 , HS2 ) and the hesitancy degree of ζω (HS1 , HS2 ) , i.e., δω (HS1 , HS2 ) = − ζωL (HS1 , HS2 ), where

ζωU (HS1 , HS2 )

   ωi pUi − ϕ¯ ω1 · ωi qUi − ϕ¯ ω2 # 2 n  U 2 n  U ¯ ω1 · ¯ ω2 i=1 ωi pi − ϕ i=1 ωi qi − ϕ n

ζωU (HS1 , HS2 ) = #

(34)

   ωi pLi − ϕ¯ ω1 · ωi qLi − ϕ¯ ω2 # 2 n  L 2 n  L ¯ ω1 · ¯ ω2 i=1 ωi pi − ϕ i=1 ωi qi − ϕ n

ζωL (HS1 , HS2 ) = #



i=1



i=1

(35)

with pUi = maxl=1,2,...,L1 {ϕli1 }, pLi = maxl=1,2,...,L1 {ϕli1 }, qUi = maxl=1,2,...,L2 {ϕli2 } and qLi = maxl=1,2,...,L2 {ϕli2 }. i

i

i

i

It is noted that when ω = (1/n, 1/n, . . . , 1/n )T , all the above-weighted concepts reduce to the normal cases as those given in Section 4.1, respectively. Example 3. Suppose that we have three HFLTSs on S, HS = {{s2 , s3 }, {s3 }, {s3 }, {s2 , s3 }, {s2 , s3 }}, HS1 = {{s1 , s2 }, {s0 , s1 }, {s0 , s1 }, {s1 , s2 }, {s2 }}, HS2 = {{s−2 }, {s−1 , s−2 }, {s−2 }, {s−2 }, {s−2 , s−3 }}. The associated weight vector is ω = (0.3, 0.2, 0.2, 0.1, 0.2)T . According to Eq. (28), we have ϕ¯ ω = 0.54; ϕ¯ ω1 = 0.24; ϕ¯ ω2 = −0.4. According to Eq. (29), we obtain V ar (HSω ) = 0.0274, V ar (HS1ω ) = 0.0234, V ar (HS2ω ) = 0.0200. According to Eq. (30), we can calculate the weighted correlations Cω ( HS , HS1 ) = 0.0094, Cω ( HS , HS2 ) = −0.0180. Based on Eq. (31), we have ζω (HS , HS1 ) =

Cω (HS ,HS1 )

1/2

[V ar (HSω )·V ar (HS1ω )]

= 0.3712,

ζω (HS , HS2 ) = −0.7689. Since ζω (HS , HS1 ) > 0 while ζω (HS , HS2 ) < 0, we can say that HS1 has a weak positive correlation with HS but HS2 has a strong negative correlation with HS .

4.3. Ordered weighted correlation measures for hesitant fuzzy linguistic information The above-weighted correlation measure weights each assessment value. In many decision-making problems, we need to first rank the alternatives/objectives and then weight them according to their ranking positions. This issue has been discussed widely [10]. In the following, we investigate the ordered weighted correlation measure between HFLTSs. To calculate the ordered weighted correlation degree, we first rank the HFLEs in each HFLTS according to their means and hesitancy degrees and get the ranking of them. Suppose that the weight associated with each position is λi and satisfies λi ∈ [0, 1] (i = 1, 2, . . . , n) and ni=1 λi = 1. Then, the following concepts related to the ordered weighted correlation measure can be introduced: (1) The ordered weighted mean of the HFLTS HS is n n 1 1 λi ϕ¯ (i) = H¯ Sλ = sϕ¯ λ , where ϕ¯ λ = n n i=1

i=1



L (i ) λi 

L (i )

l=1

 (i )

ϕl

(36)

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H. Liao, X. Gou and Z. Xu et al. / Information Sciences 508 (2020) 275–292

(2) The ordered weighted variance of the HFLTS HS is n n 2 1  1  V ar (HSλ ) = λi ϕ¯ (i) − ϕ¯ ω = n n i=1



L (i ) λi 

L (i )

i=1

(i )

ϕl

n 1 − n



i=1

l=1

L (i ) λi 

Li

2 (i )

ϕl

(37)

l=1

(3) The ordered weighted correlation between HS1 and HS2 is

Cλ ( HS1 , HS2 ) =

n    1  λi ϕ¯ (i)1 − ϕ¯ λ1 · λi ϕ¯ (i)2 − ϕ¯ λ2 n i=1

⎡

⎛

L1

⎞⎤ ⎡

L1

⎛

L2

⎞⎤

L2

(i ) (i ) (i ) (i ) n n n   1 1  λi  (i ) 1  λ  ⎣ ⎝ λi ⎝ λi = ϕl − ϕl(i) ⎠⎦·⎣ 2i ϕl(i) − ϕl(i) ⎠⎦ 1 1 2 n n n L (i ) L (i ) L (i ) L (i )

i=1

i=1

l=1

l=1

i=1

l=1

(38)

l=1

(4) The ordered weighted correlation coefficient between the HFLTSs HS1 and HS2 is

ζ (

1 2 λ HS , HS

Cλ (HS1 , HS2 )

)= 

1 / 2 = 

Var (HS1λ ) · Var (HS2λ ) 1 n

= ! 1 n

It is easy to verify that −1 ≤ ζλ (HS1 , HS2 ) ≤ 1.

n

i=1

n

i=1



1 λi L(i)

L1(i )

L1 (i ) 1 L1i

ζλ (HS1 , HS2 )

l=1

l=1

ϕl(i) −

ϕl(i) −

1 n

n

satisfies: (i)

 ( i )1    λ ϕ¯ − ϕ¯ λ1 · λi ϕ¯ (i)2 − ϕ¯ λ2  |λ ϕ − ϕ¯ λ1 |2 · n1 ni=1 |λi ϕ¯ (i)2 − ϕ¯ λ2 |2

1 n i i=1 n n 1 ( i )1 i ¯ i=1 n

1 n

i=1

n

  L2

L2  (i ) (i ) ϕl(i) · Lλ2i l=1 ϕl(i) − 1n ni=1 Lλ2i l=1 ϕl(i) (i ) (i )  ! L2

L2 2 (39) L1(i) (i ) 2 ( i ) (i ) n n (i ) (i ) 1 1 1 1 ϕ · ϕ − ϕ i=1 L2 i=1 L2 n n l=1 l l=1 l l=1 l

i=1



1 L1(i )



1 λi L(i)

L1(i )

ζλ (HS1 , HS2 )

l=1

(i )

=

ζλ (HS2 , HS1 );

(ii)

ζλ (HS1 , HS1 )

(i )

= 1; (iii) ζλ (HS , HS− ) = −1; (iv)

(5) The upper and lower bounds of the ordered weighted correlation coefficient ζλ (HS1 , HS2 ) are defined as:

   λi pi − ϕ¯ λ1 · λi qi − ϕ¯ λ2 ζλ ( )= max # 2 # n  2 n  pi ∈{ϕli1 },qi ∈{ϕli2 } 1 p − ϕ ¯ ¯ λ2 λ i i i=1,2,...,n i=1 i=1 λi qi − ϕ λ ·    n  ¯ λ1 · λi qi − ϕ¯ λ2 i=1 λi pi − ϕ L 1 2 ζλ (HS , HS ) = min # 2 # n  2 n  pi ∈{ϕli1 },qi ∈{ϕli2 } 1 p − ϕ ¯ ¯ λ2 λ i i i=1,2,...,n i=1 i=1 λi qi − ϕ λ · n

U



i=1

HS1 , HS2

(40)

(41)

Thus, we can obtain ζλL (HS1 , HS2 ) ≤ ζλ (HS1 , HS2 ) ≤ ζλU (HS1 , HS2 ) and the hesitancy degree of ζλ (HS1 , HS2 ) is δλ (HS1 , HS2 ) = − ζλL (HS1 , HS2 ), where

ζλU (HS1 , HS2 )

   λi pUi − ϕ¯ λ1 · λi qUi − ϕ¯ λ2 # 2 n  U 2 n  U ¯ λ1 · ¯ λ2 i=1 λi pi − ϕ i=1 λi qi − ϕ    n  L ¯ λ1 · λi qLi − ϕ¯ λ2 i=1 λi pi − ϕ ζλL (HS1 , HS2 ) = # # 2 n  L 2 n  L ¯ λ1 · ¯ λ2 i=1 λi pi − ϕ i=1 λi qi − ϕ n

ζλU (HS1 , HS2 ) = #



i=1

(42)

(43)

with pUi = maxl=1,2,...,L1 {ϕli1 }, pLi = maxl=1,2,...,L1 {ϕli1 }, qUi = maxl=1,2,...,L2 {ϕli2 } and qLi = maxl=1,2,...,L2 {ϕli2 }. i

i

i

i

It is noted that, when λ = (1/n, 1/n, . . . , 1/n )T , all the above ordered weighted concepts reduce to the normal cases as those given in Section 4.2, respectively. 5. An approach to decision making based on the correlation measures between HFLTSs Based on the proposed correlation measures between HFLTSs, in this section, we propose a method to solve the MCDM problems with HFLEs as described in Section 2.3. 5.1. Methods to determine the weights of criteria For a given MCDM problem, sometimes the weight information is completely given by DMs but sometimes not. In the case that the weight vector of criteria is unknown, we should first determine the weight of each criterion. If there is no evidence to show the difference between criteria, we can suppose that all criteria are considered equally; otherwise, we should find methods to determine the weights of criteria. This section dedicates to solving this problem.

H. Liao, X. Gou and Z. Xu et al. / Information Sciences 508 (2020) 275–292

285

5.1.1. Information entropy-based weight-determining method According to the information entropy theory, it is convincing to assign the higher weight to the criterion with smaller entropy, because the DMs provide unanimous and useful information on that criterion. Otherwise, the criterion will be deemed as unimportant by most DMs. Hence, the entropy measure can be used to determine the weights of criteria. Gou et al. [3] introduced different entropy and cross-entropy measures for HFLTSs. They also proved that the similarity between HFLE and its complementary set can be taken as an entropy of that HFLE. Note that Liao et al. [10,11] introduced a series of distance and similarity measures for HFLTSs. All these similarity measures can be transferred to the entropy measures of HFLTSs. For this reason, here we do not further discuss the details about the entropy measures of HFLTSs, but focus on how to use the information entropy of HFLTSs given as Eq. (3) as a representation form to derive the weights of criteria. Based on Eq. (3), the entropy of criterion C j can be calculated as

E (C j ) =

m 



i=1

Li j 1  Li j



l=1

ϕl ( hi j ) 2τ

2 

(44)

Then, the entropy-based weight of criterion C j can be represented as

ω j = n

1 − hj

j=1

(1 − h j )

=

where m 1 1  h j = E (C j ) = m m i=1

1 − hj n − nj=1 h j



Li j 1  Li j



l=1

(45)

ϕl ( hi j ) 2τ

2  (46)

It is obvious that ω j ∈ [0, 1], j = 1, 2, . . . , n, and

n j=1

ω j = 1.

5.1.2. Distance-based weight-determining method Additionally, we can utilize the generalized distance measure proposed by Liao et al. [10] to develop a weightdetermining method. As we know, if the performance of each alternative has little difference under a criterion, then it implies that this criterion plays a less important role in the priority procedure and therefore should be assigned a small weight. For criterion C j , we can define the average distance measure of an alternative Ai to all other alternatives:

d¯i j =

m m     1 1 d hi j , hι j = m−1 m−1

ι=1,ι=i



ι=1,i=ι

L 1 L l=1



ϕl ( hi j ) − ϕl ( hι j ) 2τ

λ 1/λ

(47)

Then, the overall distance measure of C j , i.e., the divergence degree of all alternatives corresponding to criterion C j , can be expressed as

⎛

dj =

m 

d¯i j =

i=1

m  i=1

m  ⎝ 1 m−1

ι=1,i=ι



L 1 L l=1



ϕl ( hi j ) − ϕl ( hι j ) 2τ

 λ  1 /λ

⎞

⎠, λ ≥ 1

(48)

Thus, the weights can be established as:

dj w j = n j=1

dj

, j = 1, 2, . . . , n

(49)

5.2. A correlation coefficient-based approach to deal with hesitant fuzzy linguistic MCDM problems After determining the positive ideal solution and the weight vector of criteria for the hesitant fuzzy linguistic MCDM problem, we can calculate the weighted correlation coefficient as well as its hesitancy degree between each alternative and the ideal solution. The larger the weighted correlation coefficient is and the smaller the hesitancy degree is, the better the alternative should be. Based on the above analysis, we can develop a correlation coefficient-based approach to solve the hesitant fuzzy linguistic MCDM problems. The approach is given in stepwise for the facility of application. Step 1. For an MCDM problem, the linguistic evaluation values of alternatives with respect to criteria are represented by linguistic expressions l li j (i = 1, 2, . . . , m, j = 1, 2, . . . , n). Then, according to the transformation function EGH , we convert all linguistic expressions to HFLEs and construct the hesitant fuzzy linguistic decision matrix H = (hSi j )m×n . Go to the next step.

T Step 2. Find the ideal solution H + = (h+ , h+ , . . . , h+ n ) for the hesitant fuzzy linguistic MCDM problem, where the ideal 1 2 criterion value h+j is given by Eq. (7). Go to the next step.

Step 3. Check whether the weight vector ω = (ω1 , ω2 , . . . , ωn )T of criteria is given by the DMs or not. If yes, go to Step 5. If not, go to the next step.

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Step 4. Establish the weight vector by Eqs. (45) or (49). Go to the next step. Step 5. Calculate the weighted correlation coefficient between each alternative Ai and the ideal solution H + by Eqs. (31) or (39). Then, we calculate the hesitancy degree between each alternative and the ideal solution H + by Eqs. (32) and (33), or Eqs. (40) and (41). Go to the next step. Step 6. Rank the alternatives according to the correlation coefficient and the hesitancy degree of each alternative. The bigger the correlation coefficient is and the smaller the hesitancy degree is, the better the alternative should be. 6. Case study concerning the strategic management of Sichuan liquor brands In the following, a case study concerning the strategic management of Sichuan liquor brands in China is employed to illustrate the usefulness and efficiency of the correlation coefficient-based qualitative MCDM method. 6.1. Case description of the evaluation of Sichuan liquor brands Sichuan is the biggest province located in the Southwest of China. Besides the Panda and Sichuan cuisine, Sichuan is also famous for its different types of liquors. Basically, there are three different categories of alcoholic beverage in terms of different main features of technology, including the distilled liquor (such as brandy, whisky, gin, vodka, rum, Chinese liquor), fermented wine (such as beer, fruit wine, yellow rice wine), and compound wine (such as vermouth, sweet-scented osmanthus) [29]. There is a very long history of Chinese liquor which has existed since 50 0 0 years ago. Generally, Chinese liquor can be further classified into different varieties according to their fragrance, such as the “sauce” fragrance, strong fragrance, light fragrance, rice fragrance, phoenix fragrance, and mixed fragrance. As we know, any liquor is unique owning to the unique natural environmental conditions such as the climate, terrain and source of water. Researches have also confirmed that the quality of function bacteria affects the quality and flavor of the Sichuan liquor through the pit mud [21]. Feng et al. [1] investigated the microbial community of the water in Maotai town to reveal the deep relationship between the water and the most famous Chinese liquor, Maotai liquor. It is observed that many fragrance categories are only an advertising hype to distinguish a particular brand from other brands [4]. There are many different brands of Sichuan liquor. The most famous brands include Wuliangye, Luzhou Lao Jiao, Jiannanchun, Tuopai Liquor, Quanxing Daqu, and Langjiu. The title of these six kinds of liquor wine competitions is known as the “Six Golden Flowers”. In the above six brands, only Langjiu belongs to the “sauce” fragrance (also known as Maotai flavor liquor), while the remaining five species belong to the strong fragrance liquor (also known as Luzhou flavor liquor). With the fantastic spur liquor industry in Sichuan, people are likely to choose good products from different kinds of Sichuan liquor. However, there are many problems in the development of Sichuan liquor, especially in the brand management. For example, some firms give an exaggerated account of some properties of a wine product, which results in a bad influence on their brands. People may have a bad impression on a brand because of purchasing fake and shoddy win products. For the better development of Sichuan liquor, it is important to evaluate the six golden flowers of Sichuan liquor to find out the best brand, and then figure out the feasible reasons for its blossom and call for the other firms to learn from it. To evaluate the six golden flowers of Sichuan liquor, several experts as a committee were invited to give their subjective preferences according to the following criteria: brand productivity (C1 ), brand communication (C2 ), brand innovation (C3 ), brand defense (C4 ), and brand growth (C5 ). As we can see, all the five criteria are qualitative variables and thus it is natural and sensible to use linguistic expressions to represent the preferences of the experts over the Sichuan liquor brands. Suppose that the LTSs over the five criteria are the same and determined as S = {s−3 = very low, s−2 = low, s−1 = a lit t le low, s0 = normal, s1 = a lit t le high, s2 = high, s3 = very high}. The linguistic evaluation information given by the expert committee is shown in Table 1. The linguistic expressions shown in Table 1 are similar to the human way of thinking. According to the transformation function, we can construct a hesitant fuzzy linguistic decision matrix (see Table 2).

Table 1 Linguistic evaluation expressions on the six golden flowers of Sichuan liquor. C1

C2

C3

C4

C5

Wuliangye Luzhou Lao Jiao

Greater than high At least high

Very high Higher than normal

At least high High

High At least high

Jiannanchun

High

High

At least high

Tuopai Liquor

High

Quanxing Daqu

Between a little high and high Between a little high and high

Between a little high and high Between a little high and high At least normal

Very high Between a little high and high High Between normal and a little high Between a little high and high Between normal and a little high

Between a little high and high Between a little high and high Between a little high and high

Between a little high and high Between a little high and high High

Langjiu

Between normal and a little high

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Table 2 Hesitant fuzzy linguistic decision matrix on the six golden flowers of Sichuan liquor.

Wuliangye Luzhou Lao Jiao Jiannanchun Tuopai Liquor Quanxing Daqu Langjiu

C1

C2

C3

C4

C5

{s2 , s3 } {s2 , s3 } {s2 } {s2 } {s1 , s2 } {s1 , s2 }

{s3 } {s1 , s2 , s3 } {s1 , s2 } {s1 , s2 } {s1 , s2 , s3 } {s0 , s1 }

{s3 } {s1 , s2 } {s2 } {s0 , s1 } {s1 , s2 } {s0 , s1 }

{s2 , s3 } {s2 } {s2 } {s1 , s2 } {s1 , s2 } {s1 , s2 }

{s2 } {s2 , s3 } {s2 , s3 } {s1 , s2 } {s1 , s2 } {s2 }

Table 3 Calculation results with respect to the given weights.

H1 H2 H3 H4 H5 H6 H+

ωi ϕ¯ i1

ωi ϕ¯ i2

ωi ϕ¯ i3

ωi ϕ¯ i4

ωi ϕ¯ i5

ϕ¯ ωi

Var (HSi ω )

Cω ( HSi , HS+ )

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

0.75 0.75 0.6 0.6 0.45 0.45 0.75

0.6 0.4 0.3 0.3 0.4 0.1 0.6

0.6 0.3 0.4 0.1 0.3 0.1 0.6

0.25 0.2 0.2 0.15 0.15 0.15 0.25

0.4 0.5 0.5 0.3 0.3 0.4 0.5

0.52 0.43 0.4 0.29 0.32 0.24 0.54

0.0306 0.0356 0.02 0.0304 0.0106 0.0234 0.0274

0.0282 0.0243 0.0180 0.0189 0.0162 0.0094 –

0.9739 0.7780 0.7689 0.6549 0.9506 0.3712 –

0.5913 0.1014 0.1685 0.0594 −0.0387 −0.3352 –

0.9963 0.9471 0.9783 0.7941 0.9915 0.8949 –

0.4050 0.8457 0.8098 0.7347 1.0302 1.2301 –

Fig. 1. Values of weighted correlation coefficients and hesitancy degrees with known weights.

6.2. Solve the problem in which the weights of criteria are known Suppose that the weight vector of criteria is given by the experts as ω = (0.3, 0.2, 0.2, 0.1, 0.2)T . Below we use the proposed correlation coefficient-based approach to solve this problem. Step 1. The linguistic evaluations are given and listed in Table 1, and the hesitant fuzzy linguistic decision matrix is constructed as Table 2. Step 2. It is observed that all the five criteria are benefit criteria. According to Eq. (7), it is easy to find the positive ideal solution H + for this hesitant fuzzy linguistic MCDM problem, which is H + = ({s2 , s3 }, {s3 }, {s3 }, {s2 , s3 }, {s2 , s3 } )T . We can see that there is no alternative which meets this ideal solution. Step 3. The weight vector is given as ω = (0.3, 0.2, 0.2, 0.1, 0.2)T . Thus we go to Step 5 directly. Step 5. Calculate the weighted correlation coefficients between each alternative and the positive ideal solution H + and their corresponding hesitancy degrees. The calculation results are set out in Table 3 and Fig. 1. The calculation processes are similar to those in Example 4, and thus here we do not illustrate them.

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Table 4 Calculation results with respect to the entropy weights.

1

H H2 H3 H4 H5 H6 H+

ωi ϕ¯ i1

ωi ϕ¯ i2

ωi ϕ¯ i3

ωi ϕ¯ i4

ωi ϕ¯ i5

ϕ¯ ωi

Var (HSi ω )

Cω ( HSi , HS+ )

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

0.4933 0.4933 0.3946 0.3946 0.2960 0.2960 0.4933

0.5985 0.3990 0.2993 0.2993 0.3990 0.0998 0.5985

0.6135 0.3068 0.4090 0.1023 0.3068 0.1023 0.6135

0.5035 0.4028 0.4028 0.3021 0.3021 0.3021 0.5035

0.3946 0.4933 0.4933 0.2960 0.2960 0.3946 0.4933

0.5207 0.4190 0.3998 0.2788 0.3200 0.2389 0.5404

0.0063 0.0049 0.0038 0.0092 0.0016 0.0140 0.0029

0.0038 −0.0032 −0.0019 −0.0038 0.0013 −0.0061 –

0.8949 −0.8396 −0.5688 −0.7304 0.6116 −0.9577 –

−0.3640 −0.9218 −0.8540 −0.8605 −0.9882 −0.9830 –

0.9879 0.6475 0.1033 0.3121 0.9873 0.5770 –

1.3519 1.5693 0.9573 1.1726 1.9755 1.5600 –

Step 6. According to the values of correlation coefficients in Table 3, we can find that the first brand has the highest correlation to the ideal solution. The ranking of these brands is Wuliangye (H 1 ) Quanxing Daqu(H 5 ) Luzhou Lao Jiao (H 2 ) Jiannanchun (H 3 ) Tuopai Liquor (H 4 ) Langjiu (H 6 ). Thus, Wuliangye is the best brand of Sichuan liquor. In this example, we can see that the mean of Quanxing Daqu (H 5 ) is much lower than that of Luzhou Lao Jiao (H 2 ). However, the variance of Quanxing Daqu (H 5 ) is also much lower than that of Luzhou Lao Jiao (H 2 ). This is the main reason why Quanxing Daqu (H 5 ) has a much higher correlation to the ideal solution than that of Luzhou Lao Jiao (H 2 ). Besides, we also calculate the upper and lower bounds of the weighted correlation coefficients. Then, the hesitancy degree, δω (HSi , HS+ ), of each alternative is obtained. From Fig. 1, we can clearly find that the weighted correlation coefficient of each alternative is between its upper bound and lower bound. Additionally, the ranking of hesitancy degrees of all alternatives is Wuliangye (H 1 ) Tuopai Liquor (H 4 ) Jiannanchun (H 3 ) Luzhou Lao Jiao (H 2 ) Quanxing Daqu (H 5 ) Langjiu (H 6 ). Obviously, even though the Quanxing Daqu (H 5 ) has a high value of correlation coefficient, its hesitancy degree is very high. 6.3. Solve the problem in which the weights of criteria are unknown It is observed that in the above example, the weight vector is given in advance. However, in some cases, it is probable that the weight information is unknown. If this is the case, we can use the entropy-based method or the distance-based method to derive the weights of criteria. 6.3.1. Determine the weights of criteria by the entropy-based method According to Eq. (44), we can calculate the entropy of each criterion as E (C1 ) = 0.7222, E (C2 ) = 0.6620, E (C3 ) = 0.5278, E (C4 ) = 0.6111, E (C5 ) = 0.7222. Thus, according to Eq. (46), we have h1 = 0.1204, h2 = 0.1103, h3 = 0.0880, 0.1019, h5 = 0.1204. Based on Eq. (45), the weight vector of the criteria is calculated as ω = (0.1973, 0.1995, 0.2045, 0.2014, 0.1973 )T . It seems that the weights of criteria are close to each other. This is because the differences between the criteria are not very large. As we can see from Table 2, the first and fifth columns are the same if we do not consider the permutation of the HFLEs. Thus, according to the principle of the entropy-based weight-determining method, the weights of the first and fifth criteria should be the same. This is confirmed in the weight vector. If we use the derived weight vector to calculate the weighted correlation coefficients and their hesitancy degrees, the results can be obtained, which are shown in Table 4 and Fig. 2. According to the values of correlation coefficients in Table 4, the ranking of these brands is Wuliangye (H 1 ) Quanxing Daqu (H 5 ) Jiannanchun (H 3 ) Tuopai Liquor (H 4 ) Luzhou Lao Jiao (H 2 ) Langjiu (H 6 ). Thus, Wuliangye is also the best brand of Sichuan liquor. Additionally, from Fig. 2, we can also find that the weighted correlation coefficient of each alternative is between its upper bound and lower bound of the weighted correlation coefficient. Based on the hesitancy degree of each alternative, the ranking of these brands is Jiannanchun (H 3 ) Tuopai Liquor (H 4 ) Wuliangye (H 1 ) Luzhou Lao Jiao (H 2 ) Langjiu (H 6 ) Quanxing Daqu (H 5 ). 6.3.2. Determine the weights of criteria by the distance-based method According to Eqs. (47–49), we can calculate five weight vectors of the criteria with respect to different values of λ (λ = 1, 2, 5, 8, 10), which are shown in Table 5. Then the weighted correlation coefficients, the upper and lower bounds, and hesitancy degrees of these weighted correlation coefficients, and the final rankings of alternatives can be obtained, which are shown in Tables 6–10 and Figs. 3 and 4. From Tables 6–10, we can find that all the rankings are Wuliangye (H 1 ) Quanxing Daqu (H 5 ) Jiannanchun (H 3 ) Luzhou Lao Jiao (H 2 ) Tuopai Liquor (H 4 ) Langjiu (H 6 ). All the weighted correlation coefficients of the alternatives are between their upper and lower bounds of the weighted correlation coefficients. In addition, with the increase of the values of λ, the change of the weighted correlation coefficients of each alternative is very small (see Fig. 3). Furthermore, from Fig. 4, the hesitancy degrees of all alternatives are not similar to the increase of λ. When λ varies from 1 to 5, the changes

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Table 5 The weight vectors of criteria with respect to different values of λ. The weight vectors

λ=1 λ=2 λ=5 λ=8 λ = 10

(0.1455,0.2636,0.3182,0.1273,0.1455)T (0.1552,0.2569,0.2937,0.1391,0.1552)T (0.1579,0.2593,0.2814,0.1435,0.1579)T (0.1572,0.2627,0.2796,0.1433,0.1572)T (0.1568,0.2641,0.2791,0.1431,0.1568)T

Table 6 The weighted correlation coefficients, hesitancy degrees and final rankings when λ = 1.

H1 H2 H3 H4 H5 H6

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

The final ranking

0.9951 0.8687 0.8870 0.0918 0.9529 −0.7115

0.9152 −0.4303 0.4906 −0.6669 0.0606 −0.9582

0.9988 0.9868 0.9947 0.3513 0.9952 0.7448

0.0836 1.4171 0.5041 1.0182 0.9346 1.7030

H1 H5 H3 H2 H4 H6

Table 7 The weighted correlation coefficients, hesitancy degrees and final rankings when λ = 2.

H1 H2 H3 H4 H5 H6

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

The final ranking

0.9926 0.7753 0.8304 −0.0332 0.9424 −0.8019

0.8692 −0.6312 0.2845 −0.7496 −0.3944 −0.9671

0.9983 0.9851 0.9882 0.3373 0.9955 0.6654

0.1291 1.6163 0.7037 1.0869 1.3899 1.6325

H1 H5 H3 H2 H4 H6

Table 8 The weighted correlation coefficients, hesitancy degrees and final rankings when λ = 5.

1

H H2 H3 H4 H5 H6

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

The final ranking

0.9915 0.7267 0.8005 −0.0220 0.9387 −0.8380

0.8481 −0.6952 0.1604 −0.7395 −0.5868 −0.9761

0.9981 0.9866 0.9855 0.3688 0.9958 0.6325

0.1500 1.6818 0.8251 1.1083 1.5853 1.6086

H1 H5 H3 H2 H4 H6

Table 9 The weighted correlation coefficients, hesitancy degrees and final rankings when λ = 8.

1

H H2 H3 H4 H5 H6

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

The final ranking

0.9917 0.7328 0.8032 0.0119 0.9398 −0.8428

0.8506 −0.6941 0.1484 −0.7222 −0.5826 −0.9801

0.9981 0.9872 0.9863 0.3886 0.9960 0.6360

0.1475 1.6813 0.8379 1.1108 1.5786 1.6161

H1 H5 H3 H2 H4 H6

Table 10 The weighted correlation coefficients, hesitancy degrees and final rankings when λ = 10.

1

H H2 H3 H4 H5 H6

ζω (HSi , HS+ )

ζωL (HSi , HS+ )

ζωU (HSi , HS+ )

δω (HSi , HS+ )

The final ranking

0.9918 0.7369 0.8055 0.0272 0.9404 −0.8441

0.8524 −0.6915 0.1468 −0.7140 −0.5744 −0.9817

0.9981 0.9873 0.9870 0.3969 0.9960 0.6382

0.1457 1.6788 0.8402 1.1109 1.5704 1.6199

H1 H5 H3 H2 H4 H6

289

290

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Fig. 2. Values of weighted correlation coefficients and hesitancy degrees with unknown weights.

Fig. 3. The weighted correlation coefficients with different values of λ.

of the hesitancy degrees of the alternatives H 1 , H 4 and H 6 are very small, while the hesitancy degrees of the alternatives H 2 , H 3 and H 5 are increasing. When λ varies from 5 to 10, the hesitancy degrees of all alternatives are steady with the increase of λ.

6.4. Comparisons with the existing hesitant fuzzy linguistic MCDM methods We can make some comparisons between the proposed approach and other existing hesitant fuzzy linguistic MCDM methods [9–11,28] and summarize the advantages of the proposed method.

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291

Fig. 4. The hesitant degrees of the weighted correlation coefficients with different values of λ.

(1) The concepts of the mean and hesitancy degree of HFLEs introduced in this paper ensure the integrity of the hesitant fuzzy linguistic decision-making theory. Additionally, they can also be regarded as a kind of normalization method. As we know, the cosine distance-based HFL-TOPSIS method and HFL-VIKOR method [9], the satisfaction degrees-based method [10] and the HFL-VIKOR method [11] are based on the distance or similarity measures. However, the normalization methods given in [9–11] did not consider the hesitant degrees of HFLEs. They need to utilize the normalization method to make all HFLEs have the same length by adding some linguistic terms. But, as we indicated in Section 3, adding artificial linguistic terms may change the original linguistic information. Therefore, the proposed method is more accurate than the existing MCDM methods. (2) In the proposed correlation coefficient-based approach, the ranking of alternatives is obtained according to the correlation coefficients and their hesitant degrees. We do not need to use aggregation operators to calculate the collective values, which saves time and makes the calculations simple. However, the decision-making method proposed by Zhang and Wu [28] was based on aggregation operators. There is a significant shortcoming: when we aggregate HFLEs into a collective one, the number of linguistic terms will be very large and thus the computational complexity is high. Suppose that the numbers of linguistic terms in the HFLEs are n j ( j = 1, 2, . . . , m), respectively. Then, the ( number of linguistic terms in the aggregated result would be m j=1 n j . In other words, the obtained HFLE would be very complicated and it is very hard to justify the meaning of the obtained results. (3) How to determinate the weights of criteria is taken into consideration in the proposed correlation coefficient-based approach, and the information entropy-based weight-determining method and the distance-based weight-determining method are introduced. However, lots of MCDM methods [9–11] did not consider this process. 7. Conclusions In this paper, we introduced new correlation measures for HFLTSs. We firstly defined the concepts of mean and hesitancy degree of HFLEs, and then proposed a new correlation coefficient. Compared with the existing correlation measures, the most important property of the proposed new correlation measure is that it is capable to distinguish the positive and negative correlations. We further introduced the hesitancy degree of the hesitant fuzzy linguistic correlation coefficient, which is composed of the upper and lower bounds of the hesitant fuzzy linguistic correlation coefficient. We also gave the weighted form and the ordered weighted form of the hesitant fuzzy linguistic correlation coefficients. To show the applicability of the proposed correlation measures, we developed a correlation-based approach for hesitant fuzzy linguistic MCDM problems in the case that the weights are either known or unknown. For the case where the weight information is unknown, an entropy-based method and a distance-based method were proposed to derive the weights of criteria. We applied the developed hesitant fuzzy linguistic MCDM method to evaluate the brands of Sichuan liquor in China as a case study. In the future, we will further investigate the applicability of the proposed correlation measures in clustering analysis and medical diagnosis. Using the correlation measures in the consensus reaching process and group decision making with hesi-

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tant fuzzy linguistic information are also good research topics. The correlation coefficient for complex linguistic information [23] is another research issue for future study. Declaration of Competing Interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgements The work was supported by the National Natural Science Foundation of China (71771156, 71971145), and the 2016 Key Project of the Key Research Institute of Humanities and Social Sciences in Sichuan Province (CJZ16-01, CJCB2016-02, Xq16B04). References [1] Q.Q. Feng, L. Han, X. Tan, Y.L. Zhang, T.Y. Meng, J. Lu, J. Lv, Bacterial and archaeal diversities in Maotai section of the Chishui river, China, Curr. Microbiol. 73 (2016) 924–929. [2] X.J. Gou, H.C. Liao, X.X. Wang, Z.S. Xu, F. 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