# Cosine Distance Measure between Neutrosophic Hesitant Fuzzy Linguistic Sets and Its Application in Multiple Criteria Decision Making

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## Abstract

**:**

## 1. Introduction

- (1)
- A neutrosophic hesitant fuzzy linguistic set (NHFLS) is defined based on NHFS and linguistic term set, its operations and comparison rules are given based on linguistic scale function, which can describe some decision information more accurately.
- (2)
- A new method is proposed to construct a similarity measure of NHFLTS based on the generalized neutrosophic hesitant fuzzy linguistic distance measure and the cosine similarity measure, which satisfies with the axiom of similarity measure. Then the corresponding cosine distance measure between NHFLTSs is obtained according to the relationship between the similarity measure and the distance measure.
- (3)
- The proposed cosine distance measure with TOPSIS method is developed to solve multiple criteria decision making with neutrosophic hesitant fuzzy linguistic environment, which can deal with the related decision information not only from the point of view of algebra but also from the point of view of geometry.

## 2. Preliminaries

#### 2.1. NS

**Definition**

**1**

#### 2.2. LTS

- (1)
- ${s}_{k}\oplus {s}_{l}={s}_{k+l};$
- (2)
- $\eta {s}_{k}={s}_{\eta k};$
- (3)
- ${s}_{k}>{s}_{l}$ if $k>l$.

#### 2.3. HFS

**Definition**

**2**

#### 2.4. NHFS

**Definition**

**3**

#### 2.5. Linguistic Scale Functions

**Definition**

**4**

**Definition**

**5**

## 3. Neutrosophic Hesitant Fuzzy Linguistic Term Set

**Definition**

**6.**

**Example**

**1.**

**Definition**

**7.**

- (1)
- $neg(N)=<{\phi}^{*-1}({\phi}^{*}({s}_{2\tau})-{\phi}^{*}({s}_{N})),{\bigcup}_{{f}_{N}\in {F}_{N},{i}_{N}\in {I}_{N},{t}_{N}\in {T}_{N}}({f}_{N},1-{i}_{N},{t}_{N})>;$
- (2)
- ${N}_{1}\oplus {N}_{2}=<{\phi}^{*-1}({\phi}^{*}({s}_{{N}_{1}})+{\phi}^{*}({s}_{{N}_{2}})),{\bigcup}_{{t}_{{N}_{j}}\in {T}_{{N}_{j}},{i}_{{N}_{j}}\in {I}_{{N}_{j}},{f}_{{N}_{j}}\in {F}_{{N}_{j}}}({t}_{{N}_{1}}+{t}_{{N}_{2}}-{t}_{{N}_{1}}{t}_{{N}_{2}},{i}_{{N}_{1}}{i}_{{N}_{2}},{f}_{{N}_{1}}{f}_{{N}_{2}})>;$$(j=1,2)$
- (3)
- ${N}_{1}\otimes {N}_{2}=<{\phi}^{*-1}({\phi}^{*}({s}_{{N}_{1}}){\phi}^{*}({s}_{{N}_{2}})),{\bigcup}_{{t}_{{N}_{j}}\in {T}_{{N}_{j}},{i}_{{N}_{j}}\in {I}_{{N}_{j}},{f}_{{N}_{j}}\in {F}_{{N}_{j}}}({t}_{{N}_{1}}{t}_{{N}_{2}},{i}_{{N}_{1}}+{i}_{{N}_{2}}-{i}_{{N}_{1}}{i}_{{N}_{2}},{f}_{{N}_{1}}+{f}_{{N}_{2}}-{f}_{{N}_{1}}{f}_{{N}_{2}})>;$$(i=1,2)$
- (4)
- $\lambda N=<{\phi}^{*-1}(\lambda {\phi}^{*}({s}_{N})),{\bigcup}_{{t}_{N}\in {T}_{N},{i}_{N}\in {I}_{N},{f}_{N}\in {F}_{N}}(1-{(1-{t}_{N})}^{\lambda},{i}_{N}^{\lambda},{f}_{N}^{\lambda})>;$
- (5)
- ${N}^{\lambda}=<{\phi}^{*-1}({({\phi}^{*}({s}_{N}))}^{\lambda}),{\bigcup}_{{t}_{N}\in {T}_{N},{i}_{N}\in {I}_{N},{f}_{N}\in {F}_{N}}({t}_{N}^{\lambda},1-{(1-{i}_{N})}^{\lambda},1-{(1-{f}_{N})}^{\lambda})>;$

**Example**

**2.**

- (1)
- $neg({N}_{1})=\langle {s}_{5},\{0.7,0.8\},\left\{0.5\right\},\{0.3,0.4\}\rangle $;
- (2)
- ${N}_{1}\oplus {N}_{2}=\langle {s}_{5.3216},\{0.56,0.64,0.72,0.76\},\{0.1,0.15\},\{0.56,0.64\}\rangle $;
- (3)
- ${N}_{1}\otimes {N}_{2}=\langle {s}_{0.5740},\{0.12,0.16,0.18,0.24\},\{0.6,0.65\},\{0.94,0.96\}\rangle $;
- (4)
- $3{N}_{1}=\langle {s}_{4.4117},\{0.657,0.784\},\left\{0.125\right\},\{0.343,0.512\}\rangle $;
- (5)
- ${N}_{1}^{3}=\langle {s}_{0.0432},\{0.027,0.064\},\left\{0.875\right\},\{0.657,0.488\}\rangle $.

**Theorem**

**1.**

- (1)
- ${N}_{1}\oplus {N}_{2}={N}_{2}\oplus {N}_{1}$;
- (2)
- ${N}_{1}\otimes {N}_{2}={N}_{2}\otimes {N}_{1}$;
- (3)
- $\lambda ({N}_{1}\oplus {N}_{2})=\lambda {N}_{1}\oplus \lambda {N}_{2}$;
- (4)
- $({\lambda}_{1}+{\lambda}_{2}){N}_{1}={\lambda}_{1}{N}_{1}+{\lambda}_{2}{N}_{1}$;
- (5)
- ${({N}_{1}\otimes {N}_{2})}^{\lambda}={N}_{1}^{\lambda}\otimes {N}_{2}^{\lambda}$;
- (6)
- ${N}_{1}^{{\lambda}_{1}+{\lambda}_{2}}={N}_{1}^{{\lambda}_{1}}\otimes {N}_{2}^{{\lambda}_{2}}$;

**Proof.**

**Definition**

**8.**

- (1)
- If $S({N}_{1})>S({N}_{2}),$ then ${N}_{1}\succ {N}_{2}$;
- (2)
- If $S({N}_{1})<S({N}_{2}),$ then ${N}_{1}\prec {N}_{2}$;
- (3)
- If $S({N}_{1})=S({N}_{2}),$ then
- if $H({N}_{1})>H({N}_{2})$, ${N}_{1}\succ {N}_{2}$;
- if $H({N}_{1})=H({N}_{2})$, ${N}_{1}\sim {N}_{2}$.

**Example**

**3.**

## 4. Cosine Distance and Similarity Measures between NHFLSs

- (p1)
- $0\le \rho (\alpha ,\beta )\le 1$;
- (p2)
- $\rho (\alpha ,\beta )=1$ if and only if $\alpha =\beta $;
- (p3)
- $\rho (\alpha ,\beta )=\rho (\beta ,\alpha )$,

**Example**

**4.**

#### 4.1. Distance Measures between NHFLSs

**Definition**

**9.**

**Example**

**5.**

**Remark**

**1.**

**Theorem**

**2.**

- (1)
- $0\le {d}_{gnhlw}({N}_{1},{N}_{2})\le 1$;
- (2)
- ${d}_{gnhlw}({N}_{1},{N}_{2})=0$ if and only if ${N}_{1}={N}_{2}$;
- (3)
- ${d}_{gnhlw}({N}_{1},{N}_{2})={d}_{g\omega nhl}({N}_{2},{N}_{1})$;

**Proof.**

#### 4.2. Similarity Measures between NHFLSs

**Definition**

**10.**

- $Cos({N}_{1},{N}_{2})=\frac{{\sum}_{i=1}^{m}{\omega}_{i}{\phi}^{*}({s}_{{N}_{1}}({x}_{i})){\phi}^{*}({s}_{{N}_{2}}({x}_{i}))\frac{{\sum}_{j=1}^{{l}_{1}}{\alpha}_{1j}({x}_{i}){\alpha}_{2j}({x}_{i})+{\sum}_{j=1}^{{l}_{2}}{\beta}_{1j}({x}_{i}){\beta}_{2j}({x}_{i})+{\sum}_{j=1}^{{l}_{3}}{\gamma}_{1j}({x}_{i}){\gamma}_{2j}({x}_{i})}{(2\tau +1)(2\tau +1)}}{{H}_{1}\u2022{H}_{2}}$,
- ${H}_{1}={({\sum}_{i=1}^{m}{\omega}_{i}{({\phi}^{*}({s}_{{N}_{1}}({x}_{i})))}^{2}*\frac{{\sum}_{j=1}^{{l}_{1}}{({\alpha}_{1j}({x}_{i}))}^{2}+{\sum}_{j=1}^{{l}_{2}}{({\beta}_{1j}({x}_{i}))}^{2}+{\sum}_{j=1}^{{l}_{3}}{({\gamma}_{1j}({x}_{i}))}^{2}}{{(2\tau +1)}^{2}})}^{\frac{1}{2}}$,
- ${H}_{2}={({\sum}_{i=1}^{m}{\omega}_{i}{({\phi}^{*}({s}_{{N}_{2}}({x}_{i})))}^{2}*\frac{{\sum}_{j=1}^{{l}_{1}}{({\alpha}_{2j}({x}_{i}))}^{2}+{\sum}_{j=1}^{{l}_{2}}{({\beta}_{2j}({x}_{i}))}^{2}+{\sum}_{j=1}^{{l}_{3}}{({\gamma}_{2j}({x}_{i}))}^{2}}{{(2\tau +1)}^{2}})}^{\frac{1}{2}}$;

**Example**

**6.**

**Theorem**

**3.**

- (1)
- $0\le {S}_{nhlw}({N}_{1},{N}_{2})\le 1$;
- (2)
- ${S}_{nhlw}({N}_{1},{N}_{2})=1$ if and only if ${N}_{1}={N}_{2}$;
- (3)
- ${S}_{nhlw}({N}_{1},{N}_{2})={S}_{nhlw}({N}_{2},{N}_{1})$.

**Proof.**

- (1)
- $Cos({N}_{1},{N}_{2})$ is the cosine value, so $0\le Cos({N}_{1},{N}_{2})\le 1$. According to Theorem 2, one easily sees $0\le 1-{d}_{gnhlw}({N}_{1},{N}_{2})\le 1$, then we have $0\le {S}_{nhlw}({N}_{1},{N}_{2})\le 1.$
- (2)
- If ${N}_{1}={N}_{2}$, ${d}_{gnhlw}({N}_{1},{N}_{2})=0$ and $Cos({N}_{1},{N}_{2})=1$ are exist, we can get ${S}_{nhlw}({N}_{1},{N}_{2})=\frac{1+1-0}{2}=1$. On the other hand, since $0\le Cos({N}_{1},{N}_{2})\le 1$ and $0\le 1-{d}_{gnhlw}({N}_{1},{N}_{2})\le 1$, if ${S}_{nhlw}({N}_{1},{N}_{2})=1$, $Cos({N}_{1},{N}_{2})=1$ and ${d}_{gnhlw}({N}_{1},{N}_{2})=0$ must hold simultaneously. By Theorem 2, we can obtain ${N}_{1}={N}_{2}$.
- (3)
- According to Definition 10, ${S}_{nhlw}({N}_{1},{N}_{2})={S}_{nhlw}({N}_{2},{N}_{1})$ can be obtained directly.

## 5. Multiple Criteria Decision Making With Distance Measure Based on TOPSIS Approach

**Step****1**- When the number of elements of NHFLSs about the truth-membership degree, the indeterminacy-membership degree and the falsity-membership degree are different, we can extend the fewer one by adding the minimum value respectively until they have the same number as the greater one; the new decision matrix ${D}^{{}^{\prime}}$ is obtained.
**Step****2**- Normalize the decision matrix ${D}^{{}^{\prime}}$.For the benefit-type criterion, we do not do anything. For the cost-type criterion, we should use the negation operator in Definition 7 to make NHFLNs normalized.
**Step****3**- For each alternative under different criteria, using the comparison rule defined in Definition 8, the positive ideal solution ${\alpha}^{+}$ and the negative ideal solution ${\alpha}^{-}$ can be obtained as follows:$$\begin{array}{c}\hfill {\alpha}^{+}=\{{N}_{1}^{+},{N}_{2}^{+},\cdots ,{N}_{m}^{+}\},{N}_{i}^{+}=max\{S({N}_{1i}),S({N}_{2i}),\cdots ,S({N}_{ki})\},\phantom{\rule{1.em}{0ex}}(i=1,2,\cdots ,m).\end{array}$$$$\begin{array}{c}\hfill {\alpha}^{-}=\{{N}_{1}^{-},{N}_{2}^{-},\cdots ,{N}_{m}^{-}\},{N}_{i}^{-}=min\{S({N}_{1i}),S({N}_{2i}),\cdots ,S({N}_{ki})\},\phantom{\rule{1.em}{0ex}}(i=1,2,\cdots ,m),\end{array}$$
**Step****4**- Utilizing the proposed distance measure ${d}_{nhlw}^{*}$ to calculate the separation of each alternative between the positive ideal solution and the negative ideal solution. Here the separation measure between ${A}_{j}(j=1,2,\cdots ,k)$ and ${\alpha}^{+}$ is: ${d}_{j}({A}_{j},{\alpha}^{+})={\sum}_{i=1}^{m}{\omega}_{i}{d}_{nhlw}^{*}({N}_{ji},{\alpha}^{+})$; The separation measure between ${A}_{j}(j=1,2,\cdots ,k)$ and ${\alpha}^{-}$ is: ${d}_{j}({A}_{j},{\alpha}^{-})={\sum}_{i=1}^{m}{\omega}_{i}{d}_{nhlw}^{*}({N}_{ji},{\alpha}^{-})$. Then we calculate the closeness coefficient of alternative ${R}_{j}$, that is$$\begin{array}{c}\hfill {R}_{j}=\frac{{d}_{j}({A}_{j},{\alpha}^{+})}{{d}_{j}({A}_{j},{\alpha}^{-})+{d}_{j}({A}_{j},{\alpha}^{+})},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}(j=1,2,\cdots ,k).\end{array}$$
**Step****5**- Rank all the alternatives according to the closeness coefficient ${R}_{j}$, the smaller closeness coefficient ${R}_{j}(j=1,2,\cdots ,k)$, the better ${A}_{j}(j=1,2,\cdots ,k)$ will be.

## 6. Illustrative Example

#### 6.1. Background

#### 6.2. An Illustration of the Proposed Method

**Step****1**- We extend the fewer elements of NHFLSs about the truth-membership degree, the indeterminacy-membership degree and the falsity-membership degree by adding the minimum value until they have the same number with the more one, respectively. For example, we can extend $\langle {s}_{4},\{0.4,0.5\},\left\{0.2\right\},\left\{0.3\right\}\rangle $ to $\langle {s}_{4},\{0.4,0.5\},\{0.2,0.2\},\{0.3,0.3\}\rangle $ by adding the minimum value 0.2, 0.3 respectively. Then Table 2 is transformed as follows:
**Step****2**- Normalize the transformation of neutrosophic hesitant fuzzy linguistic decision matrix.Because the criterion ${C}_{3}$ is the cost-type criterion, we use the negation operator in Definition 7 to make the transformation of neutrosophic hesitant fuzzy linguistic decision matrix normalized, which is shown in Table 3:
**Step****3**- Determine the positive ideal solution ${\alpha}^{+}$ and negative ideal solution ${\alpha}^{-}$ by Definition 8, we have:$$\begin{array}{c}\hfill {\alpha}^{+}=\{\langle {s}_{5},\{0.7,0.7\},\{0.1,0.2\},\{0.1,0.1\}\rangle ,\langle {s}_{5},\{0.6,0.6\},\{0.1,0.1\},\{0.2,0.2\}\rangle ,\langle {s}_{3},\{0.2,0.2\},\{0.2,0.2\},\{0.5,0.5\}\rangle \},\end{array}$$$$\begin{array}{c}\hfill {\alpha}^{-}=\{\langle {s}_{4},\{0.3,0.4\},\{0.2,0.2\},\{0.3,0.3\}\rangle ,\langle {s}_{4},\{0.6,0.6\},\{0.1,0.1\},\{0.2,0.2\}\rangle ,\langle {s}_{5},\{0.5,0.5\},\{0.2,0.2\},\{0.1,0.2\}\rangle \}.\end{array}$$
**Step****4**- It is already known the weight vector of ${C}_{i}(i=1,2,3)$ is $\omega =\{{\omega}_{1},{\omega}_{2},{\omega}_{3}\}=(0.35,0.25,0.4).$ For different $\lambda $, we utilize the cosine distance measure ${d}_{nhlw}^{*}$ to calculate the separation distance measure between ${A}_{i}$ and ${\alpha}^{+}$ $(i=1,2,3,4)$, and calculate the closeness coefficient of alternative ${A}_{j}$. Then the closeness coefficient ${R}_{j}$ is given in Table 4:
**Step****5**- Rank the alternatives according to ${R}_{j}(j=1,2,\cdots ,4)$.For different $\lambda $, we can see that the ranking order of alternatives is ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$, and the best alternative is ${A}_{2}$.To illustrate the influence of the linguistic scale function ${\phi}^{*}$ on the decision making method, we utilize different linguistic scale functions in the distance measure ${d}_{nhlw}^{*}$.Let$$\begin{array}{c}\hfill \begin{array}{c}\phantom{\rule{1.em}{0ex}}{\phi}^{*}={\phi}_{2}({s}_{i})=\left\{\begin{array}{cc}\frac{{a}^{\tau}-{a}^{\tau -i}}{2{a}^{\tau}-2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}i=0,1,\cdots ,t,\hfill \\ \frac{{a}^{\tau}+{a}^{i-\tau}-2}{2{a}^{\tau}-2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}i=\tau +1,\tau +2,\cdots ,2\tau ,\hfill \end{array}\right.\hfill \end{array}\end{array}$$Let$$\begin{array}{c}\hfill \begin{array}{c}{\phi}^{*}={\phi}_{3}({s}_{i})=\left\{\begin{array}{cc}\frac{{\tau}^{{\alpha}^{{}^{\prime}}}-{(\tau -i)}^{{\alpha}^{{}^{\prime}}}}{2{\tau}^{{\alpha}^{{}^{\prime}}}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}i=0,1,\cdots ,\tau ,\hfill \\ \frac{{\tau}^{{\beta}^{{}^{\prime}}}-{(i-\tau )}^{{\beta}^{{}^{\prime}}}}{2{\tau}^{{\beta}^{{}^{\prime}}}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}i=\tau +1,\tau +2,\cdots ,2\tau ,\hfill \end{array}\right.\hfill \end{array}\end{array}$$

#### 6.3. Comparison Analysis with Existing Method

- (1)
- It not only considers the relationship of different criteria, but also provides a new way for processing the inconsistent and indeterminate linguistic decision information.
- (2)
- The proposed cosine distance measure between NHFLSs is defined based on linguistic scale function ${\phi}^{*}$, the decision makers can flexibly select the linguistic scale function ${\phi}^{*}$ on the basis of their preferences. This avoids the information distort in the decision making process.
- (3)
- The proposed cosine distance measure between NHFLSs is constructed based on the generalized distance measure and cosine similarity measure, which can solve the decision information with NHFLSs not only from the point view of algebra but also from the point view of geometry.

#### 6.4. Managerial Implications

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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

${A}_{1}$ | $\langle {s}_{4},\{0.4,0.5\},\left\{0.2\right\},\left\{0.3\right\}\rangle $ | $\langle {s}_{5},\left\{0.4\right\},\{0.2,0.3\},\left\{0.3\right\}\rangle $ | $\langle {s}_{3},\left\{0.2\right\},\left\{0.2\right\},\left\{0.5\right\}\rangle $ |

${A}_{2}$ | $\langle {s}_{5},\left\{0.6\right\},\{0.1,0.2\},\left\{0.2\right\}\rangle $ | $\langle {s}_{5},\left\{0.6\right\},\left\{0.1\right\},\left\{0.2\right\}\rangle $ | $\langle {s}_{5},\left\{0.5\right\},\left\{0.2\right\},\{0.1,0.2\}\rangle $ |

${A}_{3}$ | $\langle {s}_{4},\{0.3,0.4\},\left\{0.2\right\},\left\{0.3\right\}\rangle $ | $\langle {s}_{5},\left\{0.5\right\},\left\{0.2\right\},\left\{0.3\right\}\rangle $ | $\langle {s}_{4},\left\{0.5\right\},\{0.2,0.3\},\left\{0.2\right\}\rangle $ |

${A}_{4}$ | $\langle {s}_{5},\left\{0.7\right\},\{0.1,0.2\},\left\{0.1\right\}\rangle $ | $\langle {s}_{4},\left\{0.6\right\},\left\{0.1\right\},\left\{0.2\right\}\rangle $ | $\langle {s}_{4},\left\{0.6\right\},\left\{0.3\right\},\left\{0.2\right\}\rangle $ |

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

${A}_{1}$ | $\langle {s}_{4},\{0.4,0.5\},\{0.2,0.2\},\{0.3,0.3\}\rangle $ | $\langle {s}_{5},\{0.4,0.4\},\{0.2,0.3\},$$\{0.3,0.3\}\rangle $ | $\langle {s}_{3},\{0.2,0.2\},\{0.2,0.2\},$$\{0.5,0.5\}\rangle $ |

${A}_{2}$ | $\langle {s}_{5},\{0.6,0.6\},\{0.1,0.2\},\{0.2,0.2\}\rangle $ | $\langle {s}_{5},\{0.6,0.6\},\{0.1,0.1\},$$\{0.2,0.2\}\rangle $ | $\langle {s}_{1},\{0.5,0.5\},\{0.2,0.2\},$$\{0.1,0.2\}\rangle $ |

${A}_{3}$ | $\langle {s}_{4},\{0.3,0.4\},\{0.2,0.2\},\{0.3,0.3\}\rangle $ | $\langle {s}_{5},\{0.5,0.5\},\{0.2,0.2\},$$\{0.3,0.3\}\rangle $ | $\langle {s}_{2},\{0.5,0.5\},\{0.2,0.3\},$$\{0.2,0.2\}\rangle $ |

${A}_{4}$ | $\langle {s}_{5},\{0.7,0.7\},\{0.1,0.2\},\{0.1,0.1\}\rangle $ | $\langle {s}_{2},\{0.6,0.6\},\{0.1,0.1\},$$\{0.2,0.2\}\rangle $ | $\langle {s}_{4},\{0.6,0.6\},\{0.3,0.3\},$$\{0.2,0.2\}\rangle $ |

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

${A}_{1}$ | $\langle {s}_{4},\{0.4,0.5\},\{0.2,0.2\},\{0.3,0.3\}\rangle $ | $\langle {s}_{5},\{0.4,0.4\},\{0.2,0.3\},$$\{0.3,0.3\}\rangle $ | $\langle {s}_{3},\{0.5,0.5\},\{0.8,0.8\},$$\{0.2,0.2\}\rangle $ |

${A}_{2}$ | $\langle {s}_{5},\{0.6,0.6\},\{0.1,0.2\},\{0.2,0.2\}\rangle $ | $\langle {s}_{5},\{0.6,0.6\},\{0.1,0.1\},$$\{0.2,0.2\}\rangle $ | $\langle {s}_{1},\{0.1,0.2\},\{0.8,0.8\},$$\{0.5,0.5\}\rangle $ |

${A}_{3}$ | $\langle {s}_{4},\{0.3,0.4\},\{0.2,0.2\},\{0.3,0.3\}\rangle $ | $\langle {s}_{5},\{0.5,0.5\},\{0.2,0.2\},$$\{0.3,0.3\}\rangle $ | $\langle {s}_{2},\{0.2,0.2\},\{0.7,0.8\},$$\{0.5,0.5\}\rangle $ |

${A}_{4}$ | $\langle {s}_{5},\{0.7,0.7\},\{0.1,0.2\},\{0.1,0.1\}\rangle $ | $\langle {s}_{4},\{0.6,0.6\},\{0.1,0.1\},$$\{0.2,0.2\}\rangle $ | $\langle {s}_{2},\{0.2,0.2\},\{0.7,0.7\},$$\{0.6,0.6\}\rangle $ |

$\mathit{\lambda}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | Ranking |
---|---|---|---|---|---|

1 | $0.6492$ | 0.2028 | $0.5017$ | $0.3694$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

2 | $0.6182$ | 0.2523 | $0.4945$ | $0.3134$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

3 | $0.6043$ | 0.2615 | $0.4896$ | $0.2958$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

5 | $0.5941$ | 0.2698 | $0.4920$ | $0.2864$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

7 | $0.5902$ | 0.2747 | $0.4959$ | $0.2842$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

9 | $0.5879$ | 0.2775 | $0.4989$ | $0.2836$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

$\mathit{\lambda}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | Ranking |
---|---|---|---|---|---|

1 | $0.6553$ | 0.2146 | $0.5029$ | $0.3824$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

2 | $0.6361$ | 0.2182 | $0.4925$ | $0.3809$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

3 | $0.6264$ | 0.2162 | $0.4920$ | $0.3857$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

5 | $0.6163$ | 0.2168 | $0.4941$ | $0.3815$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

7 | $0.6113$ | 0.2193 | $0.4965$ | $0.3814$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

9 | $0.6082$ | 0.2214 | $0.4986$ | $0.3813$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

$\mathit{\lambda}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | Ranking |
---|---|---|---|---|---|

1 | $0.6467$ | 0.2014 | $0.5001$ | $0.3647$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

2 | $0.6290$ | 0.2157 | $0.4764$ | $0.3423$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

3 | $0.6222$ | 0.2206 | $0.4709$ | $0.3308$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

5 | $0.6169$ | 0.2277 | $0.4732$ | $0.3213$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

7 | $0.6148$ | 0.2324 | $0.4766$ | $0.3177$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

9 | $0.6136$ | 0.2353 | $0.4792$ | $0.3160$ | ${A}_{2}\succ {A}_{4}\succ {A}_{3}\succ {A}_{1}$ |

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## Share and Cite

**MDPI and ACS Style**

Liu, D.; Chen, X.; Peng, D.
Cosine Distance Measure between Neutrosophic Hesitant Fuzzy Linguistic Sets and Its Application in Multiple Criteria Decision Making. *Symmetry* **2018**, *10*, 602.
https://doi.org/10.3390/sym10110602

**AMA Style**

Liu D, Chen X, Peng D.
Cosine Distance Measure between Neutrosophic Hesitant Fuzzy Linguistic Sets and Its Application in Multiple Criteria Decision Making. *Symmetry*. 2018; 10(11):602.
https://doi.org/10.3390/sym10110602

**Chicago/Turabian Style**

Liu, Donghai, Xiaohong Chen, and Dan Peng.
2018. "Cosine Distance Measure between Neutrosophic Hesitant Fuzzy Linguistic Sets and Its Application in Multiple Criteria Decision Making" *Symmetry* 10, no. 11: 602.
https://doi.org/10.3390/sym10110602