# Selecting the Optimal Mine Ventilation System via a Decision Making Framework under Hesitant Linguistic Environment

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## Abstract

**:**

## 1. Introduction

- (1)
- New operational laws and aggregation operators of HFLNs are presented. These new operations can reflect the relationship of the linguistic term and its corresponding membership degrees. Furthermore, a hesitant fuzzy linguistic likelihood is presented to compare two arbitrary HFLNs. It can effectively overcome the limitations of the existing comparison method based on score function and accuracy function.
- (2)
- The concept of HLPRs is proposed to tackle decision making issues under hesitant fuzzy linguistic circumstances. A consistency index using likelihood is defined to check the consistency degree of HLPRs and a consistency-improving model is introduced to get acceptable consistency. Besides, a likelihood-based method is adopted to obtain the final ranking result.
- (3)
- The proposed method is applied in the engineering field of choosing appropriate mine ventilation systems. Thereafter, an in-depth comparison analysis is conducted to demonstrate the validity and merits of the presented method.

## 2. General Concepts

#### 2.1. Linguistic Variables

- (1)
- There is an order: $l{v}_{i}>l{v}_{j}$, when $i>j$;
- (2)
- A negation operator exists: $ne(l{v}_{i})=l{v}_{-i}$.

- (1)
- $l{v}_{i}{\oplus}_{Xu}l{v}_{j}=l{v}_{i+j}$;
- (2)
- $l{v}_{i}{\oplus}_{Xu}l{v}_{j}=l{v}_{j}{\oplus}_{Xu}l{v}_{i}$;
- (3)
- $\rho l{v}_{i}=l{v}_{\rho i}$, $\rho \in [0,1]$.

#### 2.2. Hesitant Fuzzy Sets

**Definition**

**1**

**.**If $X$ is a fixed set, then a hesitant fuzzy set (HFS) on $X$ is in relation to the function, which can go back a set of numbers between zero and one. It is described as the mathematical sign in the following:

**Definition**

**2**

**.**Let $X=\left\{{x}_{1},{x}_{2},\dots ,{x}_{n}\right\}$ be a reference set, then a HFPR $G$ on $X$ is denoted by a matrix $G={({g}_{ij})}_{n\times n}\subset X\times X$, where ${g}_{ij}=\{[{q}_{ij}^{\sigma (l)}|l=1,\dots ,|{l}_{ij}|]\}$ is a HFE expressing whole possible preference degree(s) of the object ${x}_{i}$ over ${x}_{j}$. Furthermore, ${g}_{ij}$ $(i,j=1,2,\dots ,n;\text{}ij)$ should meet the following requirements:

#### 2.3. Hesitant Fuzzy Linguistic Sets

**Definition**

**3**

**.**Let $X=\left\{{x}_{1},{x}_{2},\dots ,{x}_{n}\right\}$ be a fixed set, and $l{v}_{\theta (x)}\in \overline{LV}$. Then, the hesitant fuzzy linguistic set (HFLS) $U$ in $X$ can be described as the subsequent object:

**Definition**

**4**

**.**Given two HFLNs $a=<l{v}_{\theta (a)},{h}_{a}>$ and $b=<l{v}_{\theta (b)},{h}_{b}>$ arbitrarily, and $\lambda \in [0,1]$, then

- (1)
- $a{\oplus}_{Lin}b=<l{v}_{\theta (a)+\theta (b)},{\displaystyle \underset{{r}_{1}\in {h}_{a},{r}_{2}\in {h}_{b}}{\cup}\left\{{r}_{1}+{r}_{2}-{r}_{1}\cdot {r}_{2}\right\}}>$;
- (2)
- $\lambda a=<l{v}_{\lambda \cdot \theta (a)},{\displaystyle \underset{r\in {h}_{a}}{\cup}\{1-{(1-r)}^{\lambda}\}}>$.

**Definition**

**5**

**.**If $a=<l{v}_{\theta (a)},{h}_{a}>$ is a HFLN, then the score function $E(a)$ of $a$ can be described as follows:

**Definition**

**6**

**.**Let $a=<l{v}_{\theta (a)},{h}_{a}>=<l{v}_{\theta (a)},{\cup}_{r\in {h}_{a}}\left\{r\right\}>$ be a HFLN, and the variance function is represented as ${V}^{\ast}({h}_{a})=\frac{1}{\#{h}_{a}}{\displaystyle {\sum}_{r\in {h}_{a}}{[r-s({h}_{a})]}^{2}}$. Hence, the accuracy function $V(a)$ of $a$ can be shown as follows:

**Definition**

**7**

**.**If $a=<l{v}_{\theta (a)},{h}_{a}>$ and $b=<l{v}_{\theta (b)},{h}_{b}>$ are two arbitrary HFLNs, ${r}_{a}^{\sigma (l)}$ and ${r}_{b}^{\sigma (l)}$ are regarded as the $l$th number in ${h}_{a}$ and ${h}_{b}$ respectively, and all membership degrees are arranged in ascending order. Then the comparison method is

- (1)
- If$l{v}_{\theta (a)}\le l{v}_{\theta (b)}$,${r}_{a}^{\sigma (l)}{\le}_{b}^{\sigma (l)}$and${r}_{a}^{\sigma (\#{h}_{a})}\le {r}_{b}^{\sigma (\#{h}_{b})}$, then$a<b$, where at least one of “<” exists,${r}_{a}^{\sigma (l)}\in {h}_{a}$,${r}_{b}^{\sigma (l)}\in {h}_{b}$,$l=1,2,\dots ,\mathrm{min}(\#{h}_{a},\#{h}_{b})$,$\#{h}_{a}$and$\#{h}_{b}$are the numbers of values in${h}_{a}$and${h}_{b}$respectively;
- (2)
- If$E(a)<E(b)$but$a\overline{)<}b$, then$a\prec b$;
- (3)
- If$E(a)=E(b)$and$V(a)<V(b)$, then$a\prec b$;
- (4)
- If$E(a)=E(b)$and$V(a)=V(b)$, then$a=b$.

**Example**

**1.**

- (1)
- $l{v}_{\theta (b)}=l{v}_{-3}<l{v}_{\theta (a)}=l{v}_{0}$,${r}_{b}^{\sigma (1)}={r}_{a}^{\sigma (1)}=0.1$,${r}_{b}^{\sigma (2)}={r}_{a}^{\sigma (2)}=0.4$, thus$b<a$;
- (2)
- $E(b)=0$,$E(c)=0.125$,i.e.,$E(b)<E(c)$, thus$b\prec c$;
- (3)
- $E(a)=E(c)=0.125$,$V(a)=0.48875$,$V(c)=0.49875$, i.e.,$V(a)<V(c)$, thus$a\prec c$.

## 3. New Operations and Comparison Method

#### 3.1. New Operational Laws and Aggregation Operators

**Definition**

**8.**

- (1)
- $a\oplus b=<l{v}_{\theta (a)+\theta (b)},{\displaystyle \underset{{r}_{1}\in {h}_{a},{r}_{2}\in {h}_{b}}{\cup}\left\{\frac{(\theta (a)+t)\cdot {r}_{1}+(\theta (b)+t)\cdot {r}_{2}}{(\theta (a)+t)+(\theta (b)+t)}\right\}}>;$
- (2)
- $\lambda a=<l{v}_{\lambda \cdot \theta (a)},{h}_{a}>$.

- (1)
- Commutativity: $a\oplus b=b\oplus a$;
- (2)
- Associativity: $(a\oplus b)\oplus c=a\oplus (b\oplus c)$;
- (3)
- Distributivity: $\lambda (a\oplus b)=\lambda a\oplus \lambda b$
**,**$\lambda \in [0,1]$; - (4)
- Distributivity: ${\lambda}_{1}a\oplus {\lambda}_{2}a=({\lambda}_{1}+{\lambda}_{2})a$, ${\lambda}_{1},{\lambda}_{2}\in [0,1]$.

**Definition**

**9.**

**Theorem**

**1.**

**Proof.**

- (1)
- When $n=2$: we have ${\omega}_{1}{a}_{1}=<l{v}_{{C}_{1}},{h}_{{a}_{1}}>$ and ${\omega}_{2}{a}_{2}=<l{v}_{{C}_{2}},{h}_{{a}_{2}}>$, then $HFLWA({a}_{1},{a}_{2})$ = ${\omega}_{1}{a}_{1}\oplus {\omega}_{2}{a}_{2}$ = $<l{v}_{{C}_{1}+{C}_{2}},{\displaystyle \underset{{r}_{1}\in {h}_{{a}_{1}},{r}_{2}\in {h}_{{a}_{2}}}{\cup}\left\{\frac{{D}_{1}\cdot {r}_{1}+{D}_{2}\cdot {r}_{2}}{{D}_{1}+{D}_{2}}\right\}}>$ = $<l{v}_{{\displaystyle \sum _{i=1}^{2}{C}_{i}}},{\displaystyle \underset{{r}_{1}\in {h}_{{a}_{1}},{r}_{2}\in {h}_{{a}_{2}}}{\cup}\left\{\frac{{\displaystyle \sum _{i=1}^{2}{D}_{i}\cdot {r}_{i}}}{{\displaystyle \sum _{i=1}^{2}{D}_{i}}}\right\}}>$.
- (2)
- For $n=k$: If Equation (9) holds, then $HFLA({a}_{1},{a}_{2},\dots ,{a}_{k})=<l{v}_{{\displaystyle \sum _{i=1}^{k}{C}_{i}}},{\displaystyle \underset{{r}_{1}\in {h}_{{a}_{1}},{r}_{2}\in {h}_{{a}_{2}},\dots ,{r}_{k}\in {h}_{ak}}{\cup}\left\{\frac{{\displaystyle \sum _{i=1}^{k}{D}_{i}\cdot {r}_{i}}}{{\displaystyle \sum _{i=1}^{k}{D}_{i}}}\right\}}>$. Hence, for $n=k+1$, from Definition 8, that is $HFLA({a}_{1},{a}_{2},\dots ,{a}_{k+1})$ = $<l{v}_{{\displaystyle \sum _{i=1}^{k}{C}_{i}}},{\displaystyle \underset{{r}_{1}\in {h}_{{a}_{1}},{r}_{2}\in {h}_{{a}_{2}},\dots ,{r}_{k}\in {h}_{ak}}{\cup}\left\{\frac{{\displaystyle \sum _{i=1}^{k}{D}_{i}\cdot {r}_{i}}}{{\displaystyle \sum _{i=1}^{k}{D}_{i}}}\right\}}>\oplus ({\omega}_{k+1}\cdot {a}_{k+1})$, = $<l{v}_{{\displaystyle \sum _{i=1}^{k}{C}_{i}+{C}_{k+1}}},{\displaystyle \underset{{r}_{1}\in {h}_{{a}_{1}},{r}_{2}\in {h}_{{a}_{2}},\dots ,{r}_{k+1}\in {h}_{{a}_{k+1}}}{\cup}\left\{\frac{{\displaystyle \sum _{i=1}^{k}{D}_{i}}\cdot \frac{{\displaystyle \sum _{i=1}^{k}{D}_{i}\cdot {r}_{i}}}{{\displaystyle \sum _{i=1}^{k}{D}_{i}}}+{D}_{k+1}\cdot {r}_{k+1}}{{\displaystyle \sum _{i=1}^{k}{D}_{i}}+{D}_{k+1}}\right\}}>$ = $<l{v}_{{\displaystyle \sum _{i=1}^{k+1}{C}_{i}}},{\displaystyle \underset{{r}_{1}\in {h}_{{a}_{1}},{r}_{2}\in {h}_{{a}_{2}},\dots ,{r}_{k+1}\in {h}_{{a}_{k+1}}}{\cup}\left\{\frac{{\displaystyle \sum _{i=1}^{k+1}{D}_{i}\cdot {r}_{i}}}{{\displaystyle \sum _{i=1}^{k+1}{D}_{i}}}\right\}}>$.

#### 3.2. Likelihood of Hesitant Fuzzy Linguistic Numbers

**Definition**

**10.**

**Property**

**1.**

- (1)
- $0\le L(a\ge b)\le 1$;
- (2)
- If$l{v}_{\theta (a)}\le l{v}_{\theta (b)},\text{\hspace{0.17em}}{h}_{a}^{+}<{h}_{b}^{-}$, then$L(a\ge b)=0$;
- (3)
- If$l{v}_{\theta (a)}\ge l{v}_{\theta (b)},\text{\hspace{0.17em}}{h}_{a}^{-}>{h}_{b}^{+}$, then$L(a\ge b)=1$;
- (4)
- $L(a\ge b)+L(b\ge a)=1$;
- (5)
- If$L(a\ge b)=L(b\ge a)$, then$L(a\ge b)=L(b\ge a)=0.5$;
- (6)
- If$L(a\ge c)\ge 0.5$, and$L(c\ge b)\ge 0.5$, then$L(a\ge b)\ge 0.5$.

**Proof.**

- (1)
- If $l{v}_{\theta (a)}<l{v}_{\theta (b)},\text{\hspace{0.17em}}{h}_{a}^{+}<{h}_{b}^{-}$ or $l{v}_{\theta (a)}>l{v}_{\theta (b)},\text{\hspace{0.17em}}{h}_{a}^{-}<{h}_{b}^{+}$, according to Definition 10, it is true that $L(a\ge b)+L(b\ge a)=1$.
- (2)
- If $l{v}_{\theta (a)}=l{v}_{\theta (b)}$, the following deduction can be derived: $L(a\ge b)=\frac{1}{\#{h}_{a}\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}\frac{{r}_{a}^{\sigma (i)}}{{r}_{a}^{\sigma (i)}+{r}_{b}^{\sigma (j)}}}}$ and $L(a\le b)=L(b\ge a)=\frac{1}{\#{h}_{a}\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}\frac{{r}_{a}^{\sigma (i)}}{{r}_{a}^{\sigma (i)}+{r}_{b}^{\sigma (j)}}}}$, then $L(a\ge b)+L(a\le b)$=$\frac{1}{\#{h}_{a}\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}\frac{{r}_{a}^{\sigma (i)}}{{r}_{a}^{\sigma (i)}+{r}_{b}^{\sigma (j)}}}}+\frac{1}{\#{h}_{a}\#{h}_{b}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}\frac{{r}_{b}^{\sigma (j)}}{{r}_{a}^{\sigma (i)}+{r}_{b}^{\sigma (j)}}}}$ = $\frac{1}{\#{h}_{a}\#{h}_{b}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}\frac{{r}_{a}^{\sigma (i)}+{r}_{b}^{\sigma (j)}}{{r}_{a}^{\sigma (i)}+{r}_{b}^{\sigma (j)}}}}$ = 1.
- (3)
- If $l{v}_{\theta (a)}\ne l{v}_{\theta (b)}$, similar to proof (2), we can obtain the following: $L(a\le b)=L(b\ge a)=\frac{1}{\#{h}_{a}\#{h}_{\beta}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}\frac{{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}{{f}^{\ast}(l{v}_{\theta (a)})\cdot {r}_{a}^{\sigma (i)}+{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}}}$, $L(a\ge b)+L(a\le b)$ = $\frac{1}{\#{h}_{a}\#{h}_{\beta}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}\frac{{f}^{\ast}(l{v}_{\theta (a)})\cdot {r}_{a}^{\sigma (j)}}{{f}^{\ast}(l{v}_{\theta (a)})\cdot {r}_{a}^{\sigma (i)}+{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}}}+\frac{1}{\#{h}_{a}\#{h}_{\beta}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}\frac{{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}{{f}^{\ast}(l{v}_{\theta (a)})\cdot {r}_{a}^{\sigma (i)}+{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}}}$ = $\frac{1}{\#{h}_{a}\#{h}_{\beta}}{\displaystyle \sum _{j=1}^{\#{h}_{b}}{\displaystyle \sum _{i=1}^{\#{h}_{a}}\frac{{f}^{\ast}(l{v}_{\theta (a)})\cdot {r}_{a}^{\sigma (i)}+{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}{{f}^{\ast}(l{v}_{\theta (a)})\cdot {r}_{a}^{\sigma (i)}+{f}^{\ast}(l{v}_{\theta (b)})\cdot {r}_{b}^{\sigma (j)}}}}$ = 1.

**Definition**

**11.**

- (1)
- If$L(a\ge b)>0.5$, then$a$is superior to$b$, expressed by$a>b$;
- (2)
- If$L(a\ge b)<0.5$, then$a$is inferior to$b$, expressed by$a<b$;
- (3)
- If$L(a\ge b)=0.5$, then$a$is indifferent to$b$, expressed by$a=b$.

**Example**

**2.**

- (1)
- $L(a\ge b)=1$,$L(b\ge a)=0$, then$b<a$.
- (2)
- $L(b\ge c)=0$,$L(c\ge b)=1$, then$b<c$.
- (3)
- $L(a\ge c)=0.455$,$L(c\ge a)=0.5446$, then$a<c$.

## 4. Decision Making Framework

#### 4.1. Original Preference Information

**Definition**

**12.**

**Definition**

**13.**

**Example**

**3.**

**Theorem**

**2.**

**Proof.**

**Example**

**4.**

**Theorem**

**3.**

**Proof.**

**Example**

**5.**

#### 4.2. Consistency Checking and Improving Models

**Definition**

**14.**

**Theorem**

**4.**

- (1)
- $0\le L(A\ge B)\le 1$;
- (2)
- $L(A\ge B)+L(B\ge A)=1$;
- (3)
- If$L(A\ge B)=L(B\ge A)$, then$L(A\ge B)=L(B\ge A)=0.5$.

**Definition**

**15.**

**Definition**

**16.**

Algorithm 1. Consistency improving model of HLPRs |

Input: The original HLPR $K={({k}_{ij})}_{n\times n}$, the threshold value $CI=C{I}_{0}$ and the maximum number of iterative times ${s}_{\mathrm{max}}\ge 1$.Output: The adjusted HLPR ${K}_{a}$ and its consistency index $CI({K}_{a})$.Step 1: Let the iterative times $s=0$, and the original HLPR $K={K}^{(0)}={({k}_{ij}^{(0)})}_{n\times n}$. Step 2: According to Equation (14), obtain the corresponding consistent HLPR ${K}^{\ast (s)}={({k}_{ij}^{\ast (s)})}_{n\times n}={(<l{v}_{ij}^{\ast (s)},{r}_{ij}^{\ast (s)}>)}_{n\times n}$ of HLPR ${K}^{(s)}={({k}_{ij}^{(s)})}_{n\times n}$. Step 3: Based on Equation (10), calculate the likelihood $L({k}_{ij}^{(s)}\ge {k}_{ij}^{\ast (s)})$ of the corresponding elements (e.g., ${k}_{ij}^{(s)}$ and ${k}_{ij}^{\ast (s)}$) in the HLPR ${K}^{(s)}={({k}_{ij}^{(s)})}_{n\times n}$ and its consistent HLPR ${K}^{\ast (s)}={({k}_{ij}^{\ast (s)})}_{n\times n}$. Then, construct the likelihood matrix ${L}^{(s)}={({l}_{ij}^{(s)})}_{n\times n}={(L({k}_{ij}^{(s)}\ge {k}_{ij}^{\ast (s)}))}_{n\times n}$ of HLPR ${K}^{(s)}$. Step 4: Calculate the consistency index $CI({K}^{(s)})$ of HLPR ${K}^{(s)}$ by Equation (18). Step 5: If the consistency level of ${K}^{(s)}$ is acceptable, namely $CI({K}^{(s)})<C{I}_{0}$ or the iterative times is maximum, namely $s>{s}_{\mathrm{max}}$, then go to Step 7; or else, go to the next step. Step 6: Find an element ${l}_{ij}^{(s)}$ in the likelihood matrix ${L}^{(s)}={({l}_{ij}^{(s)})}_{n\times n}$, which has the maximum deviation on the diagonal, namely $\mathrm{max}\left\{|{l}_{ij}^{(s)}-\frac{1}{2}|+|{l}_{ji}^{(s)}-\frac{1}{2}|\right\}$. If ${l}_{ij}^{(s)}+{l}_{ij}^{(s)}-1<0$, then the DMs may increase their preference of ${k}_{ij}^{(s)}$; if ${l}_{ij}^{(s)}+{l}_{ij}^{(s)}-1>0$, then the DMs can decrease their values of ${k}_{ij}^{(s)}$. And the modified HLPR is denoted as ${K}^{(s+1)}={({k}_{ij}^{(s+1)})}_{n\times n}={(<l{v}_{ij}^{(s+1)},{r}_{ij}^{(s+1)}>)}_{n\times n}$. Let $s=s+1$, then return to Step 2. Step 7: Let the final adjusted HLPR ${K}^{(s)}={K}_{a}$, Output ${K}_{a}$ and its consistency index $CI({K}_{a})$. |

**Theorem**

**5.**

**Example**

**6.**

#### 4.3. Likelihood-Based Ranking Method

Algorithm 2. Likelihood-based ranking method |

Input: The initial HLPR $K={({k}_{ij})}_{n\times n}$.Output: The optimal alternative ${x}^{\ast}$.Step 1: Obtain the acceptable HLPR ${K}_{a}$ by Algorithm 1. Step 2: Utilize the HFLA operator based on Equation (8) to aggregate each row of the HLPR ${K}_{a}$, then determine the overall preference degree ${p}_{i}$ of each alternative ${x}_{i}$ ($i=1,2,\dots ,n$). Step 3: According to Equation (10), calculate the likelihood ${l}_{ij}=L({p}_{i}\ge {p}_{j})$ between ${p}_{i}$ and ${p}_{j}$ ($i=1,2,\dots ,n$, $j=1,2,\dots ,n$), then construct a likelihood matrix $L={({l}_{ij})}_{n\times n}$. Step 4: Calculate the dominance degree $\phi ({x}_{i})=\frac{1}{n}{\displaystyle {\sum}_{j=1}^{n}{l}_{ij}}$ of alternative ${x}_{i}$$(i=1,2,\dots ,n)$, where $\phi ({x}_{i})$ represents the degree of ${x}_{i}$ preferred to other alternatives. Obviously, the greater the value of $\phi ({x}_{i})$, the better the alternative ${x}_{i}$. Step 5: Rank all the alternatives on the basis of the dominance degree $\phi ({x}_{i})$ of each alternative ${x}_{i}$($i=1,2,\dots ,n$). Then obtain the ranking results and the optimal alternative(s) is denoted as ${x}^{\ast}$. |

## 5. Selection of Mine Ventilation Systems

#### 5.1. Illustrative Example

#### 5.2. Comparative Analysis

- (1)
- The HFLNs can closely depict the experts’ preferences as the membership degrees of a certain linguistic value are given. And they can reserve the completeness of initial information in some extents, which is the guarantee for obtaining ideal results.
- (2)
- Only one element which greatly affects the consistency needs to be adjusted by professionals. The revised alternatives may be diverse according to the reality. Specialists make a decision in the light of a recommended direction as they are acquainted with their current positions.
- (3)
- The experts may change the linguistic scale function under different semantics on the basis of their preferences and reality. Then different ranking results may be achieved if another linguistic scale function is applied. The flexibility and practicability of the method can be reflected.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Euler, D.S. Application of ventilation management programs for improved mine safety. Int. J. Min. Sci. Technol.
**2017**, 27, 647–650. [Google Scholar] - Geng, F.; Luo, G.; Zhou, F.B.; Zhao, P.T.; Ma, L.; Chai, H.L.; Zhang, T.T. Numerical investigation of dust dispersion in a coal roadway with hybrid ventilation system. Powder Technol.
**2017**, 313, 260–271. [Google Scholar] [CrossRef] - Xia, M.M.; Xu, Z.S.; Liao, H.C. Preference relations based on intuitionistic multiplicative information. IEEE Trans. Fuzzy Syst.
**2013**, 21, 113–133. [Google Scholar] - Luo, S.Z.; Cheng, P.F.; Wang, J.Q.; Huang, Y.J. Selecting project delivery systems based on simplified neutrosophic linguistic preference relations. Symmetry
**2017**, 9, 151. [Google Scholar] [CrossRef] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci.
**1974**, 8, 199–249. [Google Scholar] [CrossRef] - Liang, W.Z.; Zhao, G.Y.; Hong, C.S. Selecting the optimal mining method with extended multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA) approach. Neural Comput. Appl.
**2018**, 1–16. [Google Scholar] [CrossRef] - Cabrerizo, F.J.; Morente-Molinera, J.A.; Pedrycz, W.; Taghavi, A.; Herrera-Viedma, E. Granulating linguistic information in decision making under consensus and consistency. Expert Syst. Appl.
**2018**, 99, 83–92. [Google Scholar] [CrossRef] - Wang, H.; Xu, Z.S. Interactive algorithms for improving incomplete linguistic preference relations based on consistency measures. Appl. Soft Comput.
**2016**, 42, 66–79. [Google Scholar] [CrossRef] - Massanet, S.; Riera, J.V.; Torrens, J.; Herrera-Viedma, E. A model based on subjective linguistic preference relations for group decision making problems. Inf. Sci.
**2016**, 355, 249–264. [Google Scholar] [CrossRef] - Xu, Y.J.; Wang, H. A group consensus decision support model for hesitant 2-tuple fuzzy linguistic preference relations with additive consistency. J. Intell. Fuzzy Syst.
**2017**, 33, 41–54. [Google Scholar] [CrossRef] - Rodriguez, R.M.; Martinez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Zhang, Z.M.; Wu, C. Hesitant fuzzy linguistic aggregation operators and their applications to multiple attribute group decision making. J. Intell. Fuzzy Syst.
**2014**, 26, 2185–2202. [Google Scholar] - Lee, L.W.; Chen, S.M. Fuzzy decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets and hesitant fuzzy linguistic operators. Inf. Sci.
**2015**, 294, 513–529. [Google Scholar] [CrossRef] - Liu, P.D.; Teng, F. Some Interval neutrosophic hesitant uncertain linguistic Bonferroni aggregation operators for multiple attribute decision-making. Int. J. Uncertain. Quantif.
**2017**, 7, 525–572. [Google Scholar] [CrossRef] - Liao, H.C.; Xu, Z.S.; Zeng, X.J. Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inf. Sci.
**2014**, 271, 125–142. [Google Scholar] [CrossRef] - Lee, L.W.; Chen, S.M. Fuzzy decision making and fuzzy group decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets 1. J. Intell. Fuzzy Syst.
**2015**, 29, 1119–1137. [Google Scholar] [CrossRef] - Chen, S.M.; Hong, J.A. Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation of fuzzy sets. Inf. Sci.
**2014**, 286, 63–74. [Google Scholar] [CrossRef] - Adar, T.; Delice, E.K. Evaluating mental work load using multi-criteria hesitant fuzzy linguistic term set (HFLTS). Int. J. Fuzzy Syst.
**2017**, 8, 90–101. [Google Scholar] - Joshi, D.K.; Kumar, S. Trapezium cloud TOPSIS method with interval-valued intuitionistic hesitant fuzzy linguistic information. Granul. Comput.
**2018**, 3, 139–152. [Google Scholar] [CrossRef] - Liu, D.H.; Chen, X.H.; Peng, D. Distance measures for hesitant fuzzy linguistic sets and their applications in multiple criteria decision making. Int. J. Fuzzy Syst.
**2018**. [Google Scholar] [CrossRef] - Adem, A.; Çolak, A.; Dağdeviren, M. An integrated model using SWOT analysis and hesitant fuzzy linguistic term set for evaluation occupational safety risks in life cycle of wind turbine. Saf. Sci.
**2018**, 106, 184–190. [Google Scholar] [CrossRef] - Zhu, B.; Xu, Z.S. Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Trans. Fuzzy Syst.
**2014**, 22, 35–45. [Google Scholar] [CrossRef] - Zhang, Z.M.; Wu, C. On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations. Knowl.-Based Syst.
**2014**, 72, 13–27. [Google Scholar] [CrossRef] - Wang, H.; Xu, Z.S. Some consistency measures of extended hesitant fuzzy linguistic preference relations. Inf. Sci.
**2015**, 297, 316–331. [Google Scholar] [CrossRef] - Wu, Z.B.; Xu, J.P. Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations. Omega
**2016**, 65, 28–40. [Google Scholar] [CrossRef] - Gou, X.J.; Xu, Z.S.; Liao, H.C. Group decision making with compatibility measures of hesitant fuzzy linguistic preference relations. Soft Comput.
**2017**, 1–17. [Google Scholar] [CrossRef] - Li, C.C.; Rodríguez, R.M.; Martínez, L.; Dong, Y.C.; Herrera, F. Consistency of hesitant fuzzy linguistic preference relations: An interval consistency index. Inf. Sci.
**2018**, 432, 347–361. [Google Scholar] [CrossRef] - Xu, Y.J.; Wen, X.; Sun, H.; Wang, H.M. Consistency and consensus models with local adjustment strategy for hesitant fuzzy linguistic preference relations. Int. J. Fuzzy Syst.
**2018**, 1–18. [Google Scholar] [CrossRef] - Wang, J.Q.; Wu, J.T.; Wang, J.; Zhang, H.Y.; Chen, X.H. Multi-criteria decision-making methods based on the Hausdorff distance of hesitant fuzzy linguistic numbers. Soft Comput.
**2016**, 20, 1621–1633. [Google Scholar] [CrossRef] - Wang, J.Q.; Wu, J.T.; Wang, J.; Zhang, H.Y.; Chen, X.H. Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf. Sci.
**2014**, 288, 55–72. [Google Scholar] [CrossRef] - Lin, R.; Zhao, X.F.; Wei, G.W. Models for selecting an ERP system with hesitant fuzzy linguistic information. J. Intell. Fuzzy Syst.
**2014**, 26, 2155–2165. [Google Scholar] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Xu, Z.S. Deviation measures of linguistic preference relations in group decision making. Omega
**2005**, 33, 249–254. [Google Scholar] [CrossRef] - Liang, W.Z.; Zhao, G.Y.; Wu, H. Evaluating investment risks of metallic mines using an extended TOPSIS method with linguistic neutrosophic numbers. Symmetry
**2017**, 9, 149. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Wang, H.Y.; Chen, S.M. Evaluating students’ answerscripts using fuzzy numbers associated with degrees of confidence. IEEE Trans. Fuzzy Syst.
**2008**, 16, 403–415. [Google Scholar] [CrossRef] - Chen, S.M.; Wang, N.Y.; Pan, J.S. Forecasting enrollments using automatic clustering techniques and fuzzy logical relationships. Expert Syst. Appl.
**2009**, 36, 11070–11076. [Google Scholar] [CrossRef] - Chen, S.M.; Chen, C.D. Handling forecasting problems based on high-order fuzzy logical relationships. Expert Syst. Appl.
**2011**, 38, 3857–3864. [Google Scholar] [CrossRef] - Chen, S.M.; Tanuwijaya, K. Fuzzy forecasting based on high-order fuzzy logical relationships and automatic clustering techniques. Expert Syst. Appl.
**2011**, 38, 15425–15437. [Google Scholar] [CrossRef] - Chen, S.M.; Munif, A.; Chen, G.S.; Liu, H.C.; Kuo, B.C. Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights. Expert Syst. Appl.
**2012**, 39, 6320–6334. [Google Scholar] [CrossRef] - Liang, W.Z.; Zhao, G.Y.; Hong, C.S. Performance assessment of circular economy for phosphorus chemical firms based on VIKOR-QUALIFLEX method. J. Clean. Prod.
**2018**, 196, 1365–1378. [Google Scholar] [CrossRef] - Liang, W.Z.; Zhao, G.Y.; Wu, H.; Chen, Y. Assessing the risk degree of goafs by employing hybrid TODIM method under uncertainty. Bull. Eng. Geol. Environ.
**2018**. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Bedregal, B.; Bustince, H.; Dong, Y.C.; Farhadinia, B.; Kahraman, C.; Martinez, L.; Torra, V.; Xu, Y.J.; Xu, Z.S.; et al. A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making. Towards high quality progress. Inf. Fusion
**2016**, 29, 89–97. [Google Scholar] [CrossRef] - Liu, N.; Meng, S.S. Approaches to the selection of cold chain logistics enterprises under hesitant fuzzy environment based on decision distance measures. Granul. Comput.
**2018**, 3, 27–38. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; Torra, V. Decomposition theorems and extension principles for hesitant fuzzy sets. Inf. Fusion
**2018**, 41, 48–56. [Google Scholar] [CrossRef] - Xia, M.M.; Xu, Z.S. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason.
**2011**, 52, 395–407. [Google Scholar] [CrossRef] [Green Version] - Zhu, B. Studies on consistency measure of hesitant fuzzy preference relations. Proc. Comput. Sci.
**2013**, 17, 457–464. [Google Scholar] [CrossRef] - Tian, Z.P.; Wang, J.; Wang, J.Q.; Zhang, H.Y. A likelihood-based qualitative flexible approach with hesitant fuzzy linguistic information. Cogn. Comput.
**2016**, 8, 670–683. [Google Scholar] [CrossRef] - Zhang, Z.M. Multi-criteria decision-making using interval-valued hesitant fuzzy QUALIFLEX methods based on a likelihood-based comparison approach. Neural Comput. Appl.
**2017**, 28, 1835–1854. [Google Scholar] [CrossRef] - Yang, Y.; Hu, J.H.; An, Q.X.; Chen, X.H. Group decision making with multiplicative triangular hesitant fuzzy preference relations and cooperative games method. Int. J. Uncertain. Quantif.
**2017**, 7, 271–284. [Google Scholar] [CrossRef] - Dong, Y.C.; Li, C.C.; Chiclana, F.; Herrera-Viedma, E. Average-case consistency measurement and analysis of interval-valued reciprocal preference relations. Knowl.-Based Syst.
**2016**, 114, 108–117. [Google Scholar] [CrossRef] - Liu, F.; Pedrycz, W.; Wang, Z.X.; Zhang, W.G. An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations. Fuzzy Set. Syst.
**2017**, 322, 1–18. [Google Scholar] [CrossRef] - Li, X.B.; Li, D.Y.; Liu, Z.X.; Zhao, G.Y.; Wang, W.H. Determination of the minimum thickness of crown pillar for safe exploitation of a subsea gold mine based on numerical modelling. Int. J. Rock Mech. Min.
**2013**, 57, 42–56. [Google Scholar] [CrossRef]

$\mathit{V}\mathit{S}$ | $\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{4}}$ |
---|---|---|---|---|

$\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{3},\{0.2,0.3,0.6\}>$ | $<l{v}_{1},\{0.4,0.6,0.8\}>$ | $<l{v}_{2},\{0.3,0.4,0.8\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $<l{v}_{-3},\{0.2,0.3,0.6\}>$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{-2},\{0.3,0.4,0.7\}>$ | $<l{v}_{3},\{0.2,0.5,0.6\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $<l{v}_{-1},\{0.4,0.6,0.8\}>$ | $<l{v}_{2},\{0.3,0.4,0.7\}>$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{-1},\{0.4,0.5,0.9\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{4}}$ | $<l{v}_{-2},\{0.3,0.4,0.8\}>$ | $<l{v}_{-3},\{0.2,0.5,0.6\}>$ | $<l{v}_{1},\{0.4,0.5,0.9\}>$ | $<l{v}_{0},\left\{1\right\}>$ |

$\mathit{V}{\mathit{S}}^{\mathbf{\ast}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{4}}$ |
---|---|---|---|---|

$\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{2.2},\{\frac{1}{4},\frac{25}{88},\frac{7}{11},\frac{7}{11}\}>$ | $<l{v}_{2.5},\{\frac{9}{50},\frac{13}{50},\frac{9}{20}\}>$ | $<l{v}_{3.6},\{\frac{1}{6},\frac{41}{159},\frac{67}{159}\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $<l{v}_{-2.2},\{\frac{1}{4},\frac{25}{88},\frac{7}{11}\}>$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{-1.4},\{\frac{3}{14},\frac{9}{56},\frac{29}{56}\}>$ | $<l{v}_{1.8},\{\frac{1}{36},\frac{2}{9},\frac{11}{72}\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $<l{v}_{-2.5},\{\frac{9}{50},\frac{13}{50},\frac{9}{20}\}>$ | $<l{v}_{1.4},\{\frac{3}{14},\frac{9}{56},\frac{29}{56}\}>$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{3.2},\{\frac{3}{64},\frac{15}{128},\frac{11}{64}\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{4}}$ | $<l{v}_{-3.6},\{\frac{1}{6},\frac{41}{159},\frac{67}{159}\}>$ | $<l{v}_{-1.8},\{\frac{1}{36},\frac{2}{9},\frac{11}{72}\}>$ | $<l{v}_{-3.2},\{\frac{3}{64},\frac{15}{128},\frac{11}{64}\}>$ | $<l{v}_{0},\left\{1\right\}>$ |

${\mathit{L}}^{(\mathbf{0})}$ | $\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{4}}$ |
---|---|---|---|---|

$\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | 0.5000 | 0.5076 | 0.6078 | 0.5693 |

$\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | 0.3656 | 0.5000 | 0.5642 | 0.7844 |

$\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | 0.8069 | 0.6378 | 0.5000 | 0.6862 |

$\mathit{v}{\mathit{s}}_{\mathbf{4}}$ | 0.9287 | 0.6195 | 1.0000 | 0.5000 |

$\mathit{V}{\mathit{S}}^{(\mathbf{1})}$ | $\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $\mathit{v}{\mathit{s}}_{\mathbf{4}}$ |
---|---|---|---|---|

$\mathit{v}{\mathit{s}}_{\mathbf{1}}$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{3},\{0.2,0.3,0.6\}>$ | $<l{v}_{1},\{0.4,0.6,0.8\}>$ | $<l{v}_{2},\{0.3,0.4,0.8\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{2}}$ | $<l{v}_{-3},\{0.2,0.3,0.6\}>$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{-2},\{0.3,0.4,0.7\}>$ | $<l{v}_{3},\{0.2,0.5,0.6\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{3}}$ | $<l{v}_{-1},\{0.4,0.6,0.8\}>$ | $<l{v}_{2},\{0.3,0.4,0.7\}>$ | $<l{v}_{0},\left\{1\right\}>$ | $<l{v}_{-1},\{0.1,0.2,0.3\}>$ |

$\mathit{v}{\mathit{s}}_{\mathbf{4}}$ | $<l{v}_{-2},\{0.3,0.4,0.8\}>$ | $<l{v}_{-3},\{0.2,0.5,0.6\}>$ | $<l{v}_{1},\{0.1,0.2,0.3\}>$ | $<l{v}_{0},\left\{1\right\}>$ |

$\mathit{L}$ | ${\mathit{p}}_{\mathbf{1}}$ | ${\mathit{p}}_{\mathbf{2}}$ | ${\mathit{p}}_{\mathbf{3}}$ | ${\mathit{p}}_{\mathbf{4}}$ |
---|---|---|---|---|

${\mathit{p}}_{\mathbf{1}}$ | 0.5000 | 0.6001 | 0.5756 | 0.6586 |

${\mathit{p}}_{\mathbf{2}}$ | 0.3999 | 0.5000 | 0.4745 | 0.5620 |

${\mathit{p}}_{\mathbf{3}}$ | 0.4244 | 0.5255 | 0.5000 | 0.5872 |

${\mathit{p}}_{\mathbf{4}}$ | 0.3414 | 0.4380 | 0.4128 | 0.5000 |

Methods | Consistency Checking | Consistency Improving | Ranking Approaches | Ranking Results |
---|---|---|---|---|

Zhang and Wu [23] | Distance measure | Iterative algorithm | Score functions | $v{s}_{1}\succ v{s}_{3}\succ v{s}_{4}\succ v{s}_{2}$ |

Wang and Xu [24] | Graph theory | Not given | Not given | Unavailable |

Wu and Xu [25] | Distance measure | Feedback mechanism | Score functions | Uncertain |

Gou et al. [26] | Compatibility measure | Not given | Complementary matrix | $v{s}_{1}\succ v{s}_{2}\succ v{s}_{3}\succ v{s}_{4}$ |

Li et al. [27] | Linear programing model | Not given | Not given | Unavailable |

Xu et al. [28] | Distance measure | Iterative algorithm | Score functions | $v{s}_{1}\succ v{s}_{3}\succ v{s}_{2}\succ v{s}_{4}$ |

The proposed method | likelihood | Feedback mechanism | Likelihood matrix | $v{s}_{1}\succ v{s}_{3}\succ v{s}_{2}\succ v{s}_{4}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liang, W.; Zhao, G.; Luo, S.
Selecting the Optimal Mine Ventilation System via a Decision Making Framework under Hesitant Linguistic Environment. *Symmetry* **2018**, *10*, 283.
https://doi.org/10.3390/sym10070283

**AMA Style**

Liang W, Zhao G, Luo S.
Selecting the Optimal Mine Ventilation System via a Decision Making Framework under Hesitant Linguistic Environment. *Symmetry*. 2018; 10(7):283.
https://doi.org/10.3390/sym10070283

**Chicago/Turabian Style**

Liang, Weizhang, Guoyan Zhao, and Suizhi Luo.
2018. "Selecting the Optimal Mine Ventilation System via a Decision Making Framework under Hesitant Linguistic Environment" *Symmetry* 10, no. 7: 283.
https://doi.org/10.3390/sym10070283