# A New Hesitant Fuzzy Linguistic TOPSIS Method for Group Multi-Criteria Linguistic Decision Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- Lower bound: ${H}_{{S}^{-}}=min\left({s}_{i}\right)={s}_{j}$, ${s}_{i}\in {H}_{S}$ and ${s}_{i}\ge {s}_{j}\forall i$;
- Upper bound: ${H}_{{S}^{+}}=max\left({s}_{i}\right)={s}_{j}$, ${s}_{i}\in {H}_{S}$ and ${s}_{i}\le {s}_{j}\forall i$;
- Complement: ${H}_{S}^{c}=S-{H}_{S}=\{{s}_{i}|{s}_{i}\in S$ and ${s}_{i}\notin {H}_{S}\}$;
- Union: ${H}_{S}^{1}\cup {H}_{S}^{2}=\{{s}_{i}|{s}_{i}\in {H}_{S}^{1}$ or ${s}_{i}\in {H}_{S}^{2}\}$;
- Intersection: ${H}_{S}^{1}\cap {H}_{S}^{2}=\{{s}_{i}|{s}_{i}\in {H}_{S}^{1}$ and ${s}_{i}\in {H}_{S}^{2}\}$;
- Envelope: $env\left({H}_{S}\right)=[{H}_{{S}^{-}},{H}_{{S}^{+}}]$.

- Apply the upper bound ${H}_{{S}^{+}}$ for each HFLTS that is associated with each alternative:$${H}_{{S}^{+}}\left({x}_{i}\right)=\{{H}_{{S}^{+}}^{1}\left({x}_{i}\right),\dots ,{H}_{{S}^{+}}^{m}\left({x}_{i}\right)\},i\in \{1,\dots ,n\}$$
- Obtain the minimum linguistic term for each alternative:$${H}_{{S}_{min}^{+}}\left({x}_{i}\right)=min\{{H}_{{S}^{+}}^{j}\left({x}_{i}\right)|j\in \{1,\dots ,m\}\},i\in \{1,\dots ,n\}$$

- Apply the lower bound ${H}_{{S}^{-}}$ for each HFLTS that is associated with each alternative:$${H}_{{S}^{-}}\left({x}_{i}\right)=\{{H}_{{S}^{-}}^{1}\left({x}_{i}\right),\dots ,{H}_{{S}^{-}}^{m}\left({x}_{i}\right)\},i\in \{1,\dots ,n\}$$
- Obtain the maximum linguistic term for each alternative:$${H}_{{S}_{max}^{-}}({x}_{i})=max\{{H}_{{S}^{-}}^{j}({x}_{i})|j\in \{1,\dots ,m\}\},i\in \{1,\dots ,n\}$$

**Example**

**1.**

## 3. The Proposed TOPSIS for HFLTSs

#### 3.1. A Pseudo-Distance between Two HFLTSs

**Definition**

**2.**

**Proposition**

**1.**

- 1.
- $d({H}_{S}^{1},{H}_{S}^{2})\ge 0$;
- 2.
- $d({H}_{S}^{1},{H}_{S}^{2})=d({H}_{S}^{2},{H}_{S}^{1})$;
- 3.
- $d({H}_{S}^{1},{H}_{S}^{2})\le d({H}_{S}^{1},{H}_{S}^{3})+d({H}_{S}^{3},{H}_{S}^{2})$.

**Proof.**

- The reflexive property: ${H}_{S}^{1}{\u2ab0}_{{H}_{S}}{H}_{S}^{1}$.
- Transitivity: if ${H}_{S}^{1}{\u2ab0}_{{H}_{S}}{H}_{S}^{2}$ and ${H}_{S}^{2}{\u2ab0}_{{H}_{S}}{H}_{S}^{3}$, then ${H}_{S}^{1}{\u2ab0}_{{H}_{S}}{H}_{S}^{3}$.

**Example**

**2.**

#### 3.2. The HFLTS Positive- and Negative-Ideal Solutions

**Definition**

**3.**

**Example**

**3.**

**Definition**

**4.**

#### 3.3. The New Hesitant Fuzzy Linguistic TOPSIS Method

**Step 1**: Let m decision makers $M=\{{d}_{1},\cdots ,{d}_{m}\}$ be asked to assess n alternatives $A=\{{a}_{1},\cdots ,{a}_{n}\}$ with respect to r criteria $C=\{{c}_{1},\cdots ,{c}_{r}\}$ by using HFLTSs on $S=\{{s}_{0},\cdots ,{s}_{g}\}$; decision maker ${d}_{i}(i=1,\cdots ,m)$ with weight ${w}_{i}$ provides the decision matrix ${D}_{i}={({e}_{jk}^{i})}_{n\times r}$ to express his or her assessments, where ${w}_{i}\ge 0$ and ${\sum}_{i=1}^{m}{w}_{i}=1$.

**Step 2**: For each decision matrix ${D}_{i}={({e}_{jk}^{i})}_{n\times r}$, making use of Equations (11) and (12), we obtain the positive information ${C}_{i}\left({c}_{k}\right)=[{s}_{{p}_{ki}},{s}_{{q}_{ki}}]$ and the negative information ${H}_{i}\left({c}_{k}\right)=[{s}_{{p}_{ki}^{\prime}},{s}_{{q}_{ki}^{\prime}}]$ of ${c}_{k}(k=1,\cdots ,r)$. Then we utilize weight ${w}_{i}(i=1,\cdots ,m)$ and Equations (13) and (14) to calculate the positive and negative information $C\left({c}_{k}\right)=[{s}_{{p}_{k}},{s}_{{q}_{k}}]$ and $H\left({c}_{k}\right)=[{s}_{{p}_{k}^{\prime}},{s}_{{q}_{k}^{\prime}}]$ of each ${c}_{k}(k=1,\cdots ,r)$ provided by m decision makers; we can obtain the HFLTS positive- and negative-ideal solutions as follows:

**Step 3**: We calculate the one decision matrix D by aggregating assessments of decision makers; that is, we use weights $({w}_{1},\cdots ,{w}_{m})$ and the weighted 2-tuple linguistic aggregation operator to aggregate m decision matrices $({D}_{1},\cdots ,{D}_{m})$:

**Step 4**: On the basis of Equation (9) and the HFLTS positive- and negative-ideal solutions of Equation (15), we calculate the positive-ideal separation matrix ${D}^{+}$ and the negative-ideal separation matrix ${D}^{-}$ between assessments of decision makers and the HFLTS positive- and negative-ideal solutions, that is,

**Step 5**: The ranking of alternatives in the original TOPSIS method is based on “the shortest distance from the positive-ideal solution and the farthest from the negative-ideal solution”; formally, this is also fulfilled by the relative closeness degree of each alternative in the existing TOPSIS methods. In the paper, on the basis of ${D}^{+}$ and ${D}^{-}$, we provide the following relative closeness degree $RC\left({a}_{j}\right)$ of each alternative:

**Step 6**: Rank all the alternatives ${a}_{j}(j=1,\cdots ,n)$ according to the relative closeness degree $RC\left({a}_{j}\right)$. The greater the value $RC\left({a}_{j}\right)$, the better the alternative ${a}_{j}$; that is, for any $j,{j}^{\prime}\in \{1,\cdots ,n\}$, ${a}_{j}\u2ab0{a}_{{j}^{\prime}}$ if and only if $RC\left({a}_{j}\right)\ge RC\left({a}_{{j}^{\prime}}\right)$.

## 4. Numerical Example

**Example**

**4.**

- (1)
- On the basis of Table 3, we can obtain three decision matrices provided by the three decision makers, as follows:$$\begin{array}{ccc}\hfill {D}_{1}& =& \left(\begin{array}{ccc}[{s}_{4},{s}_{6}]& [{s}_{5},{s}_{6}]& [{s}_{4},{s}_{6}]\\ [{s}_{4},{s}_{6}]& [{s}_{4},{s}_{6}]& [{s}_{1},{s}_{3}]\\ [{s}_{5},{s}_{6}]& [{s}_{4},{s}_{6}]& [{s}_{6},{s}_{6}]\end{array}\right),\phantom{\rule{4pt}{0ex}}{D}_{2}=\left(\begin{array}{ccc}[{s}_{4},{s}_{5}]& [{s}_{5},{s}_{6}]& [{s}_{3},{s}_{5}]\\ [{s}_{3},{s}_{5}]& [{s}_{4},{s}_{5}]& [{s}_{2},{s}_{3}]\\ [{s}_{2},{s}_{4}]& [{s}_{3},{s}_{4}]& [{s}_{4},{s}_{5}]\end{array}\right)\hfill \\ \hfill {D}_{3}& =& \left(\begin{array}{ccc}[{s}_{3},{s}_{4}]& [{s}_{4},{s}_{5}]& [{s}_{5},{s}_{6}]\\ [{s}_{5},{s}_{6}]& [{s}_{3},{s}_{4}]& [{s}_{3},{s}_{4}]\\ [{s}_{4},{s}_{5}]& [{s}_{3},{s}_{5}]& [{s}_{5},{s}_{6}]\end{array}\right)\hfill \end{array}$$
- (2)
- On the basis of Equations (11)–(14), we can calculate the positive and negative information of each criterion provided by the three decision makers. For example, for criterion ${c}_{1}$, the positive and negative information provided by decision maker ${d}_{1}$ are ${C}_{1}\left({c}_{1}\right)=[max\{{s}_{4},{s}_{5}\},max\left\{{s}_{6}\right\}]=[{s}_{5},{s}_{6}]$ and ${H}_{1}\left({c}_{1}\right)=[min\{{s}_{4},{s}_{5}\},min\left\{{s}_{6}\right\}]=[{s}_{4},{s}_{6}]$; similarly, ${C}_{2}\left({c}_{1}\right)=[{s}_{4},{s}_{5}]$, ${H}_{2}\left({c}_{1}\right)=[{s}_{2},{s}_{4}]$, ${C}_{3}\left({c}_{1}\right)=[{s}_{5},{s}_{6}]$ and ${H}_{3}\left({c}_{1}\right)=[{s}_{3},{s}_{4}]$. Making use of the weights $(0.3,0.5,0.2)$, we obtain $C\left({c}_{1}\right)=[{s}_{round(0.3\times 5+0.5\times 4+0.2\times 5)},$ ${s}_{round(0.3\times 6+0.5\times 5+0.2\times 6)}]=[{s}_{5},{s}_{6}]$ and $H\left({c}_{1}\right)=[{s}_{round(0.3\times 4+0.5\times 2+0.2\times 3)},{s}_{round(0.3\times 6+0.5\times 4+0.2\times 4)}]=[{s}_{3},{s}_{5}]$; the others are shown in Table 4.
- (3)
- On the basis of the weights $(0.3,0.5,0.2)$, we aggregate ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$ to obtain the one decision matrix D, that is,$$\begin{array}{c}\hfill D=0.3{D}_{1}+0.5{D}_{2}+0.2{D}_{3}=\left(\begin{array}{ccc}[{s}_{4},{s}_{5}]& [{s}_{5},{s}_{6}]& [{s}_{4},{s}_{6}]\\ [{s}_{4},{s}_{6}]& [{s}_{4},{s}_{5}]& [{s}_{2},{s}_{3}]\\ [{s}_{3},{s}_{5}]& [{s}_{3},{s}_{5}]& [{s}_{5},{s}_{6}]\end{array}\right)\end{array}$$
- (4)
- On the basis of the one decision matrix D and the HFLTS positive- and negative-ideal solutions $HPIS=([{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}])$ and $HNIS=([{s}_{3},{s}_{5}],[{s}_{3},{s}_{5}],[{s}_{2},{s}_{3}])$, we use Equations (9), (17) and (18) to calculate the positive- and negative-ideal separation matrices ${D}^{+}$ and ${D}^{-}$, that is,$$\begin{array}{ccc}\hfill {D}^{+}& =& \left(\begin{array}{c}d([{s}_{4},{s}_{5}],[{s}_{5},{s}_{6}])+d([{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{5},{s}_{6}])\\ d([{s}_{4},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{4},{s}_{5}],[{s}_{5},{s}_{6}])+d([{s}_{2},{s}_{3}],[{s}_{5},{s}_{6}])\\ d([{s}_{3},{s}_{5}],[{s}_{5},{s}_{6}])+d([{s}_{3},{s}_{5}],[{s}_{5},{s}_{6}])+d([{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}])\end{array}\right)\hfill \\ & \doteq & \left(\begin{array}{c}0.5+0+0.17\\ 0.17+0.5+0.5\\ 0.5+0.5+0\end{array}\right)=\left(\begin{array}{c}0.67\\ 1.17\\ 1\end{array}\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {D}^{-}& =& \left(\begin{array}{c}d([{s}_{4},{s}_{5}],[{s}_{3},{s}_{5}])+d([{s}_{5},{s}_{6}],[{s}_{3},{s}_{5}])+d([{s}_{4},{s}_{6}],[{s}_{2},{s}_{3}])\\ d([{s}_{4},{s}_{6}],[{s}_{3},{s}_{5}])+d([{s}_{4},{s}_{5}],[{s}_{3},{s}_{5}])+d([{s}_{2},{s}_{3}],[{s}_{2},{s}_{3}])\\ d([{s}_{3},{s}_{5}],[{s}_{3},{s}_{5}])+d([{s}_{3},{s}_{5}],[{s}_{3},{s}_{5}])+d([{s}_{5},{s}_{6}],[{s}_{2},{s}_{3}])\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}0.17+0.5+0.5\\ 0.25+0.17+0\\ 0+0+0.5\end{array}\right)=\left(\begin{array}{c}1.17\\ 0.42\\ 0.5\end{array}\right)\hfill \end{array}$$
- (5)
- On the basis of Equations (21)–(24), we obtain the relative closeness degrees $RC\left({a}_{j}\right)$ of each alternative, which are shown in Table 5.
- (6)
- According to $RC\left({a}_{j}\right)$ of each alternative in Table 5, we obtain that the ranking of alternatives is ${a}_{1}\u2ab0{a}_{3}\u2ab0{a}_{2}$, given that $RC\left({a}_{1}\right)>RC\left({a}_{3}\right)>RC\left({a}_{2}\right)$, and that ${a}_{1}$ is the the most satisfying alternative.

Algorithm 1: The new hesitant fuzzy linguistic TOPSIS method |

Input The decision matrix ${D}_{i}={({e}_{jk}^{i})}_{n\times r}(i=1,\cdots ,m)$ and weights $({w}_{1},\cdots ,{w}_{m})$ of m decision makers. |

Output The ranking of n alternatives $A=\{{a}_{1},\cdots ,{a}_{n}\}$ and the most satisfying alternative A. |

Begin |

for each $i=1,\cdots ,m$ and $k=1,\cdots ,r$ do |

${C}_{i}\left({c}_{k}\right)=[{s}_{{p}_{ki}},{s}_{{q}_{ki}}]$ and ${H}_{i}\left({c}_{k}\right)=[{s}_{{p}_{ki}^{\prime}},{s}_{{q}_{ki}^{\prime}}]$ (in each ${D}_{i}$ by using Equations (11) and (12) to obtain the positive and negative information) |

end |

for $i=1:m$ and each $k=1,\cdots ,r$ do |

$C\left({c}_{k}\right)=[{s}_{{p}_{k}},{s}_{{q}_{k}}]$ and $H\left({c}_{k}\right)=[{s}_{{p}_{k}^{\prime}},{s}_{{q}_{k}^{\prime}}]$ (using weight ${w}_{i}(i=1,\cdots ,m)$, and Equations (13) and (14) to obtain the positive and negative information of each ${c}_{k}$) |

$HPIS=(C\left({c}_{1}\right),\cdots ,C\left({c}_{r}\right))$ and $HNIS=(H\left({c}_{1}\right),\cdots ,H\left({c}_{r}\right))$ (the HFLTS positive- and negative-ideal solutions) |

end |

for $i=1:m$ do |

$D={\sum}_{i=1}^{m}{w}_{i}{D}_{i}$ (using weight ${w}_{i}$ and Equation (16) to obtain the one decision matrix) |

end |

for $k=1:r$ do |

${D}^{+}=d(D,HPIS)$ and ${D}^{-}=d(D,HNIS)$ (using Equations (9), (17) and (18) or (19) and (20) to obtain positive- and negative-ideal separation matrices) |

end |

for $j=1:n$ do |

${D}_{min}^{+}$ and ${D}_{max}^{-}$ (in ${D}^{+}$ and ${D}^{-}$ using Equation (23)) |

$RC\left({a}_{j}\right)=\frac{{D}_{min}^{+}}{{D}_{j}^{+}}+\frac{{D}_{j}^{-}}{{D}_{max}^{-}}$ (using Equation (24) to obtain the relative closeness degree of each alternative) |

end |

Output $A=\{{a}_{j}|\forall {j}^{\prime}\in \{1,\cdots ,n\},RC\left({a}_{j}\right)\ge RC\left({a}_{{j}^{\prime}}\right)\}$ |

end |

#### 4.1. Comparison with Rodriguez’s and Liao’s Methods

- (1)
- The positive- and negative-ideal solutions: The symbolic aggregation-based method utilizes min _upper and max_lower operators to construct the core information of each alternative. For example, for ${a}_{1}$ of the decision matrix ${D}_{1}$, the min bounds of ${c}_{1}$, ${c}_{2}$ and ${c}_{3}$ are ${s}_{4}$, ${s}_{5}$ and ${s}_{4}$; thus the min_upper of ${a}_{1}$ is ${s}_{5}$. The max bounds of ${c}_{1}$, ${c}_{2}$ and ${c}_{3}$ are ${s}_{6}$, ${s}_{6}$ and ${s}_{6}$; thus the max_lower of ${a}_{1}$ is ${s}_{6}$, and hence the core information of ${a}_{1}$ is $[{s}_{5},{s}_{6}]$. Intuitively, the core information reduces HFLTSs of each alternative with respect to the criteria into a linguistic interval.The HFL-VIKOR method utilizes the score function and the variance function of HFLTSs [45] to rank HFLTSs of all alternatives with respect to each criterion; for example, for ${c}_{3}$ of the decision matrix ${D}_{1}$, according to the score functions and the variance functions of $[{s}_{4},{s}_{6}]$, $[{s}_{1},{s}_{3}]$ and $[{s}_{6},{s}_{6}]$, we obtain $[{s}_{6},{s}_{6}]>[{s}_{4},{s}_{6}]>[{s}_{1},{s}_{3}]$; hence the positive- and negative-ideal solutions of ${c}_{3}$ in the decision matrix ${D}_{1}$ are $[{s}_{6},{s}_{6}]$ and $[{s}_{1},{s}_{3}]$, respectively.The proposed method uses Equations (11) and (12) to obtain the positive and negative information of each criterion. Intuitively, the positive information of each criterion in the decision matrix ${D}_{1}$ is also the optimistic information of all the alternatives provided by decision maker ${d}_{1}$, and the negative information of each criterion is the pessimistic information of all the alternatives, which can be understood as the positive- and negative-ideal solutions provided by decision maker ${d}_{1}$. Table 6 shows the comparison of the three methods.
- (2)
- The ranking of alternatives: In the symbolic aggregation-based method, on the basis of the core information of each alternative, a binary preference relation $p({a}_{j}>{a}_{{j}^{\prime}})$ between two alternatives is calculated on the basis of Equation (7); then the nondominance degree (NDD${}_{j}$) of each alternative is used to obtain the set of nondominated alternatives, which indicates the degree to which alternative ${a}_{j}$ is not dominated by the remaining alternatives.In the HFL-VIKOR method, the hesitant fuzzy linguistic group utility measure HFLGU${}_{j}$ and the hesitant fuzzy individual regret measure HFLIR${}_{j}$ for the alternative ${a}_{j}$ are defined by the hesitant fuzzy linguistic Euclidean ${L}_{p}-$metric; then the hesitant fuzzy linguistic compromise measure HFLC${}_{j}$ is established, that is,$$\begin{array}{ccc}\hfill {HFLC}_{j}& =& \theta \frac{{HFLGU}_{j}-{HFLGU}^{+}}{{HFLGU}^{-}-{HFLGU}^{+}}+(1-\theta )\frac{{HFLIR}_{j}-{HFLIR}^{+}}{{HFLIR}^{-}-{HFLIR}^{+}}\hfill \end{array}$$In the proposed method, positive- and negative-ideal separation matrices ${D}^{+}$ and ${D}^{-}$ are used to obtain the relative closeness degree $RC\left({a}_{j}\right)$ of each alternative, that is,$$\begin{array}{ccc}\hfill {D}^{+}& =& \left(\begin{array}{c}d([{s}_{4},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{6},{s}_{6}])\\ d([{s}_{4},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{1},{s}_{3}],[{s}_{6},{s}_{6}])\\ d([{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{5},{s}_{6}])+d([{s}_{6},{s}_{6}],[{s}_{6},{s}_{6}])\end{array}\right)\doteq \left(\begin{array}{c}0.67\\ 0.84\\ 0.17\end{array}\right)\hfill \\ \hfill {D}^{-}& =& \left(\begin{array}{c}d([{s}_{4},{s}_{6}],[{s}_{4},{s}_{6}])+d([{s}_{5},{s}_{6}],[{s}_{4},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{1},{s}_{3}])\\ d([{s}_{4},{s}_{6}],[{s}_{4},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{4},{s}_{6}])+d([{s}_{1},{s}_{3}],[{s}_{1},{s}_{3}])\\ d([{s}_{5},{s}_{6}],[{s}_{4},{s}_{6}])+d([{s}_{4},{s}_{6}],[{s}_{4},{s}_{6}])+d([{s}_{6},{s}_{6}],[{s}_{1},{s}_{3}])\end{array}\right)\doteq \left(\begin{array}{c}0.67\\ 0\\ 0.67\end{array}\right)\hfill \\ \hfill RC\left({a}_{1}\right)& =& \frac{0.17}{0.67}+\frac{0.67}{0.67}\doteq 1.25,RC\left({a}_{2}\right)=\frac{0.17}{0.84}+0\doteq 0.20,RC\left({a}_{3}\right)=2\hfill \end{array}$$

#### 4.2. Comparison with Beg and Rashid’s Method

- (1)
- On the basis of Equations (1) and (2), we aggregate ${D}_{1}$, ${D}_{2}$ and ${D}_{3}$ to obtain the one decision matrix D:$$\begin{array}{ccc}\hfill D& =& {([min\{ma{x}_{i=1}^{3}(min{e}_{jk}^{i}),mi{n}_{i=1}^{3}(max{e}_{jk}^{i})\},max\{ma{x}_{i=1}^{3}(min{e}_{jk}^{i}),mi{n}_{i=1}^{3}(max{e}_{jk}^{i})\}])}_{3\times 3}\hfill \\ & =& \left(\begin{array}{ccc}[{s}_{4},{s}_{4}]& [{s}_{5},{s}_{5}]& [{s}_{5},{s}_{5}]\\ [{s}_{5},{s}_{5}]& [{s}_{4},{s}_{4}]& [{s}_{3},{s}_{3}]\\ [{s}_{4},{s}_{5}]& [{s}_{4},{s}_{4}]& [{s}_{5},{s}_{6}]\end{array}\right)\hfill \end{array}$$
- (2)
- On the basis of Equations (3) and (4), we calculate the HFLTS positive- and negative-ideal solutions ${A}^{+}$ and ${A}^{-}$; here, we suppose that the criteria are beneficial, that is,$$\begin{array}{ccc}\hfill {A}^{+}& =& \left(\right[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}],[{s}_{6},{s}_{6}\left]\right),{A}^{-}=\left(\right[{s}_{2},{s}_{4}],[{s}_{3},{s}_{4}],[{s}_{1},{s}_{3}\left]\right)\hfill \end{array}$$
- (3)
- On the basis of the distance $d({H}_{S}^{1},{H}_{S}^{2})=|{q}^{\prime}-q|+|{p}^{\prime}-p|$ between ${H}_{S}^{1}$ and ${H}_{S}^{2}$, we obtain the positive (negative)-ideal matrices ${D}^{+}$ (${D}^{-}$) between D and ${A}^{+}$ (${A}^{-}$), that is,$$\begin{array}{ccc}\hfill {D}^{+}& =& \left(\begin{array}{c}d\left(\right[{s}_{4},{s}_{4}],[{s}_{5},{s}_{6}\left]\right)+d\left(\right[{s}_{5},{s}_{5}],[{s}_{5},{s}_{6}\left]\right)+d\left(\right[{s}_{5},{s}_{5}],[{s}_{6},{s}_{6}\left]\right)\\ d\left(\right[{s}_{5},{s}_{5}],[{s}_{5},{s}_{6}\left]\right)+d\left(\right[{s}_{4},{s}_{4}],[{s}_{5},{s}_{6}\left]\right)+d\left(\right[{s}_{3},{s}_{3}],[{s}_{6},{s}_{6}\left]\right)\\ d\left(\right[{s}_{4},{s}_{5}],[{s}_{5},{s}_{6}\left]\right)+d\left(\right[{s}_{4},{s}_{4}],[{s}_{5},{s}_{6}\left]\right)+d\left(\right[{s}_{5},{s}_{6}],[{s}_{6},{s}_{6}\left]\right)\end{array}\right)=\left(\begin{array}{c}6\\ 10\\ 6\end{array}\right)\hfill \\ \hfill {D}^{-}& =& \left(\begin{array}{c}d\left(\right[{s}_{4},{s}_{4}],[{s}_{2},{s}_{4}\left]\right)+d\left(\right[{s}_{5},{s}_{5}],[{s}_{3},{s}_{4}\left]\right)+d\left(\right[{s}_{5},{s}_{5}],[{s}_{1},{s}_{3}\left]\right)\\ d\left(\right[{s}_{5},{s}_{5}],[{s}_{2},{s}_{4}\left]\right)+d\left(\right[{s}_{4},{s}_{4}],[{s}_{3},{s}_{4}\left]\right)+d\left(\right[{s}_{3},{s}_{3}],[{s}_{1},{s}_{3}\left]\right)\\ d\left(\right[{s}_{4},{s}_{5}],[{s}_{2},{s}_{4}\left]\right)+d\left(\right[{s}_{4},{s}_{4}],[{s}_{3},{s}_{4}\left]\right)+d\left(\right[{s}_{5},{s}_{6}],[{s}_{1},{s}_{3}\left]\right)\end{array}\right)=\left(\begin{array}{c}11\\ 7\\ 9\end{array}\right)\hfill \end{array}$$

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | |
---|---|---|---|

${x}_{1}$ | $\{{s}_{1},{s}_{2},{s}_{3}\}$ | $\{{s}_{4},{s}_{5}\}$ | $\left\{{s}_{4}\right\}$ |

${x}_{2}$ | $\{{s}_{2},{s}_{3}\}$ | $\left\{{s}_{3}\right\}$ | $\{{s}_{0},{s}_{1},{s}_{2}\}$ |

${x}_{3}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | $\{{s}_{1},{s}_{2}\}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ||
---|---|---|---|---|

${a}_{1}$ | $\{{s}_{1},{s}_{2},{s}_{3}\}$ | $\{{s}_{4},{s}_{5}\}$ | $\{{s}_{3},{s}_{4}\}$ | |

${d}_{i}$ | ${a}_{2}$ | $\{{s}_{2},{s}_{3}\}$ | $\{{s}_{3},{s}_{4}\}$ | $\{{s}_{0},{s}_{1},{s}_{2}\}$ |

${a}_{3}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | $\{{s}_{1},{s}_{2}\}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ||
---|---|---|---|---|

${a}_{1}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | $\{{s}_{5},{s}_{6}\}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | |

${d}_{1}\left(0.3\right)$ | ${a}_{2}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | $\{{s}_{1},{s}_{2},{s}_{3}\}$ |

${a}_{3}$ | $\{{s}_{5},{s}_{6}\}$ | $\{{s}_{4},{s}_{5},{s}_{6}\}$ | $\left\{{s}_{6}\right\}$ | |

${a}_{1}$ | $\{{s}_{4},{s}_{5}\}$ | $\{{s}_{5},{s}_{6}\}$ | $\{{s}_{3},{s}_{4},{s}_{5}\}$ | |

${d}_{2}\left(0.5\right)$ | ${a}_{2}$ | $\{{s}_{3},{s}_{4},{s}_{5}\}$ | $\{{s}_{4},{s}_{5}\}$ | $\{{s}_{2},{s}_{3}\}$ |

${a}_{3}$ | $\{{s}_{2},{s}_{3},{s}_{4}\}$ | $\{{s}_{3},{s}_{4}\}$ | $\{{s}_{4},{s}_{5}\}$ | |

${a}_{1}$ | $\{{s}_{3},{s}_{4}\}$ | $\{{s}_{4},{s}_{5}\}$ | $\{{s}_{5},{s}_{6}\}$ | |

${d}_{3}\left(0.2\right)$ | ${a}_{2}$ | $\{{s}_{5},{s}_{6}\}$ | $\{{s}_{3},{s}_{4}\}$ | $\{{s}_{3},{s}_{4}\}$ |

${a}_{3}$ | $\{{s}_{4},{s}_{5}\}$ | $\{{s}_{3},{s}_{4},{s}_{5}\}$ | $\{{s}_{5},{s}_{6}\}$ |

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ||
---|---|---|---|---|

${d}_{1}\left(0.3\right)$ | ${C}_{1}$ | $[{s}_{5},{s}_{6}]$ | $[{s}_{5},{s}_{6}]$ | $[{s}_{6},{s}_{6}]$ |

${H}_{1}$ | $[{s}_{4},{s}_{6}]$ | $[{s}_{4},{s}_{6}]$ | $[{s}_{1},{s}_{3}]$ | |

${d}_{2}\left(0.5\right)$ | ${C}_{2}$ | $[{s}_{4},{s}_{5}]$ | $[{s}_{5},{s}_{6}]$ | $[{s}_{4},{s}_{5}]$ |

${H}_{2}$ | $[{s}_{2},{s}_{4}]$ | $[{s}_{3},{s}_{4}]$ | $[{s}_{2},{s}_{3}]$ | |

${d}_{3}\left(0.2\right)$ | ${C}_{3}$ | $[{s}_{5},{s}_{6}]$ | $[{s}_{4},{s}_{5}]$ | $[{s}_{5},{s}_{6}]$ |

${H}_{3}$ | $[{s}_{3},{s}_{4}]$ | $[{s}_{3},{s}_{4}]$ | $[{s}_{3},{s}_{4}]$ | |

C | $[{s}_{5},{s}_{6}]$ | $[{s}_{5},{s}_{6}]$ | $[{s}_{5},{s}_{6}]$ | |

H | $[{s}_{3},{s}_{5}]$ | $[{s}_{3},{s}_{5}]$ | $[{s}_{2},{s}_{3}]$ |

${\mathit{D}}_{\mathit{j}}^{+}$ | ${\mathit{D}}_{\mathit{j}}^{-}$ | $\mathit{RC}\left({\mathit{a}}_{\mathit{j}}\right)$ | |
---|---|---|---|

${a}_{1}$ | $0.67$ | $1.17$ | $\frac{0.67}{0.67}+\frac{1.17}{1.17}=2$ |

${a}_{2}$ | $1.17$ | $0.42$ | $\frac{0.67}{1.17}+\frac{0.42}{1.17}\doteq 0.92$ |

${a}_{3}$ | 1 | $0.5$ | $\frac{0.67}{1}+\frac{0.5}{1.17}\doteq 1.1$ |

${D}_{min}^{+}=0.67$ | ${D}_{max}^{-}=1.17$ |

The Positive-Ideal Solution | The Negative-Ideal Solution | The Core Information | |
---|---|---|---|

The method [24] | − | − | $\left(\right[{s}_{5},{s}_{6}],[{s}_{3},{s}_{4}],[{s}_{6},{s}_{6}\left]\right)$ |

The method [45] | $\left(\right[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}],[{s}_{6},{s}_{6}\left]\right)$ | $\left(\right[{s}_{4},{s}_{6}],[{s}_{6},{s}_{6}],[{s}_{1},{s}_{3}\left]\right)$ | − |

The proposed method | $\left(\right[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}],[{s}_{6},{s}_{6}\left]\right)$ | $\left(\right[{s}_{4},{s}_{6}],[{s}_{4},{s}_{6}],[{s}_{1},{s}_{3}\left]\right)$ | − |

NDD${}_{\mathit{j}}$, HFLC${}_{\mathit{j}}$ or $\mathit{RC}\left({\mathit{a}}_{\mathit{j}}\right)$ | The Ranking | The Best | |
---|---|---|---|

The method [24] | $(0,0.5,1)$ | ${a}_{1}\prec {a}_{2}\prec {a}_{3}$ | ${a}_{3}$ |

The method [45] | $({0}^{*},{1}^{-},0.6074)$ | ${a}_{2}\prec {a}_{3}\prec {a}_{1}$ | ${a}_{1}$ |

The proposed method | $(1.25,0.20,2)$ | ${a}_{2}\prec {a}_{1}\prec {a}_{3}$ | ${a}_{3}$ |

Weights | HPIS and HNIS | $\mathit{RC}\left({\mathit{a}}_{\mathit{j}}\right)$ | The Ranking | The Best | |
---|---|---|---|---|---|

The method [63] | − | $\left(\right[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}],[{s}_{6},{s}_{6}\left]\right)$ | $(0.65,0.41,0.6)$ | ${a}_{2}\prec {a}_{3}\prec {a}_{1}$ | ${a}_{1}$ |

$\left(\right[{s}_{2},{s}_{4}],[{s}_{3},{s}_{4}],[{s}_{1},{s}_{3}\left]\right)$ | |||||

The proposed method | √ | $\left(\right[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}],[{s}_{5},{s}_{6}\left]\right)$ | $(2,0.92,1.1)$ | ${a}_{2}\prec {a}_{3}\prec {a}_{1}$ | ${a}_{1}$ |

$\left(\right[{s}_{3},{s}_{5}],[{s}_{3},{s}_{5}],[{s}_{2},{s}_{3}\left]\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Ren, F.; Kong, M.; Pei, Z.
A New Hesitant Fuzzy Linguistic TOPSIS Method for Group Multi-Criteria Linguistic Decision Making. *Symmetry* **2017**, *9*, 289.
https://doi.org/10.3390/sym9120289

**AMA Style**

Ren F, Kong M, Pei Z.
A New Hesitant Fuzzy Linguistic TOPSIS Method for Group Multi-Criteria Linguistic Decision Making. *Symmetry*. 2017; 9(12):289.
https://doi.org/10.3390/sym9120289

**Chicago/Turabian Style**

Ren, Fangling, Mingming Kong, and Zheng Pei.
2017. "A New Hesitant Fuzzy Linguistic TOPSIS Method for Group Multi-Criteria Linguistic Decision Making" *Symmetry* 9, no. 12: 289.
https://doi.org/10.3390/sym9120289