# Evaluating Investment Risks of Metallic Mines Using an Extended TOPSIS Method with Linguistic Neutrosophic Numbers

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## Abstract

**:**

## 1. Introduction

- (1)
- Present a number of distance measures between two LNNs, such as the Hamming distance, the Euclidean distance, and the Hausdorff distance. Equally important, prove relevant properties of these formulas;
- (2)
- Use the thought of maximum deviation for our reference, build a model with respect to linguistic neutrosophic environment to obtain the values of mine risk evaluation criteria weight;
- (3)
- Come up with the extended TOPSIS model with LNNs. Importantly, utilize this method to cope with investment decision-making matter of metallic mine projects;
- (4)
- Compare with other methods, in order to demonstrate the significance and superiority.

## 2. Background

#### 2.1. Risk Factors of Mining Project Investment

#### 2.2. Linguistic Term Sets and Linguistic Scale Function

**Definition 1.**

#### 2.3. Linguistic Neutrosophic Numbers

**Definition 2.**

**Definition 3.**

- (1)
- ${\eta}_{1}\oplus {\eta}_{2}=({s}_{{T}_{1}},{s}_{{I}_{1}},{s}_{{F}_{1}})\oplus ({s}_{{T}_{2}},{s}_{{I}_{2}},{s}_{{F}_{2}})=({s}_{{T}_{1}+{T}_{2}-\frac{{T}_{1}{T}_{2}}{2u}},{s}_{\frac{{I}_{1}{I}_{2}}{2u}},{s}_{\frac{{F}_{1}{F}_{2}}{2u}});$
- (2)
- ${\eta}_{1}\otimes {\eta}_{2}=({s}_{{T}_{1}},{s}_{{I}_{1}},{s}_{{F}_{1}})\oplus ({s}_{{T}_{2}},{s}_{{I}_{2}},{s}_{{F}_{2}})=({s}_{\frac{{T}_{1}{T}_{2}}{2u}},{s}_{{I}_{1}+{I}_{2}-\frac{{I}_{1}{I}_{2}}{2u}},{s}_{{F}_{1}+{F}_{2}-\frac{{F}_{1}{F}_{2}}{2u}});$
- (3)
- $q{\eta}_{1}=q({s}_{{T}_{1}},{s}_{{I}_{1}},{s}_{{F}_{1}})=({s}_{2u-2u{(1-\frac{{T}_{1}}{2u})}^{q}},{s}_{2u{(\frac{{I}_{1}}{2u})}^{q}},{s}_{2u{(\frac{{F}_{1}}{2u})}^{q}}),\text{}q0;$
- (4)
- ${\eta}_{1}{}^{q}={({s}_{{T}_{1}},{s}_{{I}_{1}},{s}_{{F}_{1}})}^{q}=({s}_{2u{(\frac{{T}_{1}}{2u})}^{q}},{s}_{2u-2u{(1-\frac{{I}_{1}}{2u})}^{q}},{s}_{2u-2u{(1-\frac{{F}_{1}}{2u})}^{q}}),\text{}q0.$

**Definition 4.**

**Definition 5.**

- (1)
- ${\eta}_{1}>{\eta}_{2}$ if $SC({\eta}_{1})>SC({\eta}_{2})$;
- (2)
- ${\eta}_{1}>{\eta}_{2}$ if $SC({\eta}_{1})=SC({\eta}_{2})$ and $AC({\eta}_{1})>AC({\eta}_{2})$;
- (3)
- ${\eta}_{1}={\eta}_{2}$ if $SC({\eta}_{1})=SC({\eta}_{2})$ and $AC({\eta}_{1})=AC({\eta}_{2})$.

**Definition 6.**

**Definition 7.**

## 3. Extended TOPSIS Method with Incomplete Weight Information

#### 3.1. Descriptions

#### 3.2. Distance Measures of LNNs

**Definition 8.**

- (1)
- when $\lambda =1$, the Hamming distance$${d}_{Hm}({\eta}_{1},{\eta}_{2})=\frac{1}{3}(|f({s}_{{T}_{1}})-f({s}_{{T}_{2}})|+|f({s}_{2u-{I}_{1}})-f({s}_{2u-{I}_{2}})|+|f({s}_{2u-{F}_{1}})-f({s}_{2u-{F}_{2}})|);$$
- (2)
- when $\lambda =2$, the Euclidean distance$${d}_{Ed}({\eta}_{1},{\eta}_{2})=\sqrt{\frac{1}{3}(|f({s}_{{T}_{1}})-f({s}_{{T}_{2}}){|}^{2}+|f({s}_{2t-{I}_{1}})-f({s}_{2t-{I}_{2}}){|}^{2}+|f({s}_{2t-{F}_{1}})-f({s}_{2t-{F}_{2}}){|}^{2})};$$
- (3)
- the Hausdorff distance$${d}_{Hd}({\eta}_{1},{\eta}_{2})=\mathrm{max}\{|f({s}_{{T}_{1}})-f({s}_{{T}_{2}})|,|f({s}_{2t-{I}_{1}})-f({s}_{2t-{I}_{2}})|,|f({s}_{2t-{F}_{1}})-f({s}_{2t-{F}_{2}})|\}.$$

**Property 1.**

- (1)
- $0\le d({\eta}_{1},{\eta}_{2})\le 1$;
- (2)
- $d({\eta}_{1},{\eta}_{2})=d({\eta}_{2},{\eta}_{1})$;
- (3)
- $d({\eta}_{1},{\eta}_{2})=0$ if ${\eta}_{1}={\eta}_{2}$;
- (4)
- $d({\eta}_{1},{\eta}_{3})\le d({\eta}_{1},{\eta}_{2})+d({\eta}_{2},{\eta}_{3})$.

**Proof.**

- (1)
- Because $f({s}_{i})=\frac{i}{2u}\in [0,1]$ $\Rightarrow $ $|f({s}_{{T}_{1}})-f({s}_{{T}_{2}})|\in [0,1]$, $|f({s}_{2u-{I}_{1}})-f({s}_{2u-{I}_{2}})|$ and $|f({s}_{2u-{F}_{1}})-f({s}_{2u-{F}_{2}})|$, as $\lambda >0$, then $0\le d({\eta}_{1},{\eta}_{2})\le 1$.
- (2)
- This proof is obvious.
- (3)
- $\begin{array}{cc}\hfill \mathrm{Since}& \text{}{\eta}_{1}={\eta}_{2},\text{}\mathrm{then}\text{}SC({\eta}_{1})=SC({\eta}_{2})\text{}\mathrm{and}\text{}AC({\eta}_{1})=AC({\eta}_{2})\hfill \\ & \Rightarrow \text{}(4u+{T}_{1}-{I}_{1}-{F}_{1})/(6u)=(4u+{T}_{2}-{I}_{2}-{F}_{2})/(6u)\text{}\mathrm{and}\text{}({T}_{1}-{F}_{1})/(2u)=({T}_{2}-{F}_{2})/(2u)\hfill \\ & \Rightarrow \text{}{T}_{1}-{I}_{1}-{F}_{1}={T}_{2}-{I}_{2}-{F}_{2}\text{}\mathrm{and}\text{}{T}_{1}-{F}_{1}={T}_{2}-{F}_{2}\text{}\Rightarrow \text{}{I}_{1}={I}_{2}\text{}\mathrm{and}\text{}{T}_{1}-{F}_{1}={T}_{2}-{F}_{2}.\hfill \end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{cc}\hfill \mathrm{Thus},\text{}d({\eta}_{1},{\eta}_{2})& ={(\frac{1}{3}(|f({s}_{{T}_{1}})-f({s}_{{T}_{2}}){|}^{\lambda}+|f({s}_{2u-{I}_{1}})-f({s}_{2u-{I}_{2}}){|}^{\lambda}+|f({s}_{2u-{F}_{1}})-f({s}_{2u-{F}_{2}}){|}^{\lambda}))}^{\frac{1}{\lambda}}\hfill \\ & ={(\frac{1}{3}(|\frac{{T}_{1}-{T}_{2}}{2u}{|}^{\lambda}+|\frac{{I}_{2}-{I}_{1}}{2u}{|}^{\lambda}+|\frac{{F}_{2}-{F}_{1}}{2u}{|}^{\lambda}))}^{\frac{1}{\lambda}}\hfill \\ & ={(\frac{1}{3}(|\frac{{T}_{1}-{F}_{1}+{F}_{1}-{T}_{2}}{2u}{|}^{\lambda}+|\frac{{I}_{2}-{I}_{1}}{2u}{|}^{\lambda}+|\frac{{F}_{2}-{T}_{2}+{T}_{2}-{F}_{1}}{2u}{|}^{\lambda}))}^{\frac{1}{\lambda}}\hfill \\ & ={(\frac{1}{3}(|\frac{{T}_{2}-{F}_{2}+{F}_{1}-{T}_{2}}{2u}{|}^{\lambda}+|\frac{{I}_{2}-{I}_{1}}{2u}{|}^{\lambda}+|\frac{{F}_{1}-{T}_{1}+{T}_{2}-{F}_{1}}{2u}{|}^{\lambda}))}^{\frac{1}{\lambda}}\hfill \\ & ={(\frac{1}{3}(|\frac{{F}_{1}-{F}_{2}}{2u}{|}^{\lambda}+|\frac{{I}_{1}-{I}_{2}}{2u}{|}^{\lambda}+|\frac{{T}_{2}-{T}_{1}}{2u}{|}^{\lambda}))}^{\frac{1}{\lambda}}\hfill \\ & ={(\frac{1}{3}(|f({s}_{{T}_{2}})-f({s}_{{T}_{1}}){|}^{\lambda}+|f({s}_{2u-{I}_{2}})-f({s}_{2u-{I}_{1}}){|}^{\lambda}+|f({s}_{2u-{F}_{2}})-f({s}_{2u-{F}_{1}}){|}^{\lambda}))}^{\frac{1}{\lambda}}\hfill \\ & =d({\eta}_{2},{\eta}_{1})\hfill \end{array}$
- (4)
- $\begin{array}{cc}\mathrm{As}\text{}\hfill & |f({s}_{{T}_{1}})-f({s}_{{T}_{2}})|\text{}=\text{}|f({s}_{{T}_{1}})-f({s}_{{T}_{2}})+f({s}_{{T}_{2}})-f({s}_{{T}_{3}})|\hfill \\ \le & |f({s}_{{T}_{1}})-f({s}_{{T}_{2}})|+|f({s}_{{T}_{2}})-f({s}_{{T}_{3}})|,\hfill \\ & |f({s}_{2u-{I}_{1}})-f({s}_{2u-{I}_{2}})|=|f({s}_{2u-{I}_{1}})-f({s}_{2u-{I}_{2}})+f({s}_{2u-{I}_{2}})-f({s}_{2u-{I}_{3}})|\hfill \\ \le & |f({s}_{2u-{I}_{1}})-f({s}_{2u-{I}_{2}})|+|f({s}_{2u-{I}_{2}})-f({s}_{2u-{I}_{3}})|,\hfill \\ \mathrm{and}\text{}\hfill & |f({s}_{2u-{F}_{1}})-f({s}_{2u-{F}_{2}})|=|f({s}_{2u-{F}_{1}})-f({s}_{2u-{F}_{2}})+f({s}_{2u-{F}_{2}})-f({s}_{2u-{F}_{3}})|\hfill \\ \le & |f({s}_{2u-{F}_{1}})-f({s}_{2u-{F}_{2}})|+|f({s}_{2u-{F}_{2}})-f({s}_{2u-{F}_{3}})|,\hfill \\ \mathrm{hence},\text{}\hfill & d({\eta}_{1},{\eta}_{3})\le d({\eta}_{1},{\eta}_{2})+d({\eta}_{2},{\eta}_{3}).\hfill \end{array}$

**Example 1.**

#### 3.3. Weight Model Based on Maximum Deviation

- (1)
- If there is a tiny difference of evaluation values ${\eta}_{ij}$ among all objects under criteria ${a}_{j}$$(j=1,2,\dots ,m)$, it indicates that the criteria ${a}_{j}$ has little effect on the sorting results. Accordingly, it is appropriate to allocate a small value of the related weight ${\omega}_{j}$.
- (2)
- Conversely, if there is a significant variance of assessment information ${\eta}_{ij}$ among all alternatives under criteria ${a}_{j}$$(j=1,2,\dots ,m)$, then the criteria ${a}_{j}$ may be very important to the ranking orders. In this case, giving a large weight value ${\omega}_{j}$ is reasonable.
- (3)
- Notably, if ${\eta}_{ij}$ are the same values among all options under criteria ${a}_{j}$ ($(j=1,2,\dots ,m)$), it means that the criteria ${a}_{j}$ doesn’t affect the ranking results. Therefore, we can make the corresponding weight ${\omega}_{j}=0$.

#### 3.4. The Extended TOPSIS Method with LNNs

**Step 1:**Obtain the normalized decision-making matrix ${N}^{\u2022(l)}={({\eta}_{ij}^{\u2022(l)})}_{n\times m}={({s}_{{T}_{ij}}^{\u2022(l)},{s}_{{I}_{ij}}^{\u2022(l)},{s}_{{F}_{ij}}^{\u2022(l)})}_{n\times m}$. If the criteria belong to cost type, let ${s}_{{T}_{ij}}^{\u2022(l)}={s}_{2u-{T}_{ij}}^{(l)}$, ${s}_{{I}_{ij}}^{\u2022(l)}={s}_{2u-{I}_{ij}}^{(l)}$ and ${s}_{{F}_{ij}}^{\u2022(l)}={s}_{2u-{F}_{ij}}^{(l)}$. If the criteria belong to benefit type, then the matrix remains, that is to say ${s}_{{T}_{ij}}^{\u2022(l)}={s}_{{T}_{ij}}^{(l)}$, ${s}_{{I}_{ij}}^{\u2022(l)}={s}_{{I}_{ij}}^{(l)}$ and ${s}_{{F}_{ij}}^{\u2022(l)}={s}_{{F}_{ij}}^{(l)}$.

**Step 2:**Get the comprehensive decision-making matrix ${N}^{\u2022}={({\eta}_{ij}^{\u2022})}_{n\times m}={({s}_{{T}_{ij}}^{\u2022},{s}_{{I}_{ij}}^{\u2022},{s}_{{F}_{ij}}^{\u2022})}_{n\times m}$ using the LNWAM operator or LNWGM operator on the basis of Formula (3) or Formula (4).

**Step 3:**Use the weight model to calculate the weight values ${\omega}_{j}$ $(j=1,2,\dots ,m)$ based on Formula (9), and then normalize the weight information in line with Formula (10), denoted as ${\omega}_{j}^{\u2022}$ $(j=1,2,\dots ,m)$.

**Step 4:**Establish the weight standardized decision-making matrix ${N}^{*}={({\eta}_{ij}^{*})}_{n\times m}={({s}_{{T}_{ij}}^{*},{s}_{{I}_{ij}}^{*},{s}_{{F}_{ij}}^{*})}_{n\times m}$ through multiplying the normalized matrix with weight vector, where ${s}_{{T}_{ij}}^{*}={\omega}_{j}{s}_{{T}_{ij}}^{\u2022}$, ${s}_{{I}_{ij}}^{*}={\omega}_{j}{s}_{{I}_{ij}}^{\u2022}$ and ${s}_{{F}_{ij}}^{*}={\omega}_{j}{s}_{{F}_{ij}}^{\u2022}$.

**Step 5:**Distinguish the positive ideal solution ${\eta}^{+}$ and the negative ideal solution ${\eta}^{-}$, respectively, then:

**Step 6:**Based on Formula (5), calculate the distance measures of the positive ideal solution to all options, and the distance measures of the negative ideal solution to all options in proper sequence. The computation formulas are:

**Step 7:**For each option ${x}_{i}$ $(i=1,2,\dots ,n)$, compute the values of correlation coefficient ${D}_{i}$ with the following equation:

**Step 8:**Achieve the ranking orders according to the values of ${D}_{i}$ $(i=1,2,\dots ,n)$. The bigger the value of ${D}_{i}$, the better the alternative ${x}_{i}$ is.

## 4. Case Study

**Step 1:**Obtain the normalized decision matrix. As all the criteria are risk element, regarded as a part of cost, then normalizing evaluation values with function ${s}_{{T}_{ij}}^{\u2022(l)}={s}_{2u-{T}_{ij}}^{(l)}$, ${s}_{{I}_{ij}}^{\u2022(l)}={s}_{2u-{I}_{ij}}^{(l)}$ and ${s}_{{F}_{ij}}^{\u2022(l)}={s}_{2u-{F}_{ij}}^{(l)}$. The followings (Table 7, Table 8 and Table 9) are the normalized decision-making matrix of each expert.

**Step 2:**Using the LNWAM operator in line with Formula (3) to get the comprehensive decision matrix as Table 10:

**Step 3:**Calculate the values of the criteria weight ${\omega}_{j}$ (suppose $\lambda =1$) on the basis of Formula (9) as follows: ${\omega}_{1}\approx 0.17$, ${\omega}_{2}\approx 0.42$, ${\omega}_{3}\approx 0.31$, ${\omega}_{4}\approx 0.55$ and ${\omega}_{1}\approx 0.63$. Normalize them based on Formula (10): ${\omega}_{1}^{\u2022}=\frac{{\omega}_{1}}{{\omega}_{1}+{\omega}_{2}+{\omega}_{3}+{\omega}_{4}+{\omega}_{5}}\approx 0.08$, ${\omega}_{2}^{\u2022}\approx 0.20$, ${\omega}_{3}^{\u2022}\approx 0.15$, ${\omega}_{4}^{\u2022}\approx 0.27$ and ${\omega}_{5}^{\u2022}\approx 0.30$.

**Step 4:**Establish the weight standardized decision-making matrix as Table 11.

**Step 5:**Identify the positive ideal solution and the negative ideal solution, respectively. See Table 12.

**Step 6:**In line with Formula (5), the distances are measured as follows (assume $\lambda =1$): ${d}_{1}^{+}\approx 9.88$, ${d}_{2}^{+}\approx 5.06$, ${d}_{3}^{+}=7.13$, ${d}_{4}^{+}=5.01$, ${d}_{1}^{-}\approx 1.22$, ${d}_{2}^{-}\approx 5.50$, ${d}_{3}^{-}\approx 3.68$ and ${d}_{4}^{-}\approx 6.04$.

**Step 7:**Compute the values of correlation coefficient: ${D}_{1}\approx 0.11$, ${D}_{2}\approx 0.52$, ${D}_{3}\approx 0.34$ and ${D}_{4}\approx 0.55$.

**Step 8:**Since ${D}_{4}>{D}_{2}>{D}_{3}>{D}_{1}$, then the ranking order is ${x}_{4}>{x}_{2}>{x}_{3}>{x}_{1}$, and the best metal mine is ${x}_{4}$.

## 5. Comparison Analysis

- (1)
- Evaluating the risk degree of mining projects under qualitative criteria by means of LNNs is a good choice. As all the consistent, hesitant, and inconsistent linguistic information are taken into account.
- (2)
- The flexibility has increased because various distance measures, aggregation operators, and linguistic scale functions can be chosen according to the savants’ experience or reality.
- (3)
- A common situation, in which the criteria weight information is unknown, is under consideration. There are many complex risk factors in the process of metallic mining investment. Thus, it is difficult or unrealistic for decision makers to give the weight vector directly. The weight model based on the thought of maximum deviation may be a simple and suitable way to resolve this problem.
- (4)
- Instead of using absolute ideal points, the extended TOPSIS method defined the relative ideal solutions. The strength of it is that different ideal solutions are calculated corresponding with the different original information of different mining projects. This may be more in line with reality.

## 6. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Risk Factors | Explanations |
---|---|

Financial risk | Caused by the unexpected changes in the mine’s balance of payments. It largely consists of financial balance, exchange rate, interest rate, and other factors. |

Production risk | Caused by accident, which makes it impossible to produce the production plan according to the predetermined cost. Mainly including production cost, technical conditions, selection scheme, and so on. |

Market risk | Caused by the unexpected changes in the market, which makes the mine unable to sell its products according to the original plan. It chiefly contains demand forecasting, substitution products, peer competition, and other factors. |

Personnel risk | Caused by accident or change of the important personnel in the mine, which causes a significant impact on the production and operation of the mine. The main factors include accidental casualties, confidential leaks, and personnel changes. |

Environmental risk | Caused by the changes of the external environment of the mining industry, which primarily comprises the national policies, geological conditions, and pollution control. |

Assessment Indicators | |
---|---|

Primary indicators | Secondary indicators |

Production risk | Mining type, production equipment level, and mining technology |

Geological risk | Geological grade, mine reserves, hydrogeology, and surrounding rock conditions |

Social environment | Marco economy, national industrial policy, and international environment |

Market risk | Marketing ability, product market price, and potential competition |

Management risk | Rationality of enterprise organization, scientific decision, and management personnel |

Grade | 0~19 | 20~29 | 30~39 | 40~49 | 50~59 | 60~69 | 70~79 | 80~89 | 90~100 |
---|---|---|---|---|---|---|---|---|---|

Evaluation | exceedingly low | pretty low | low | slightly low | medium | slightly high | high | pretty high | exceedingly high |

Linguistic term | ${s}_{0}$ | ${s}_{1}$ | ${s}_{2}$ | ${s}_{3}$ | ${s}_{4}$ | ${s}_{5}$ | ${s}_{6}$ | ${s}_{7}$ | ${s}_{8}$ |

${\mathit{N}}^{(\mathbf{1})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{1},{s}_{2},{s}_{1})$ | $({s}_{2},{s}_{3},{s}_{2})$ | $({s}_{4},{s}_{4},{s}_{3})$ | $({s}_{1},{s}_{5},{s}_{1})$ | $({s}_{3},{s}_{3},{s}_{2})$ |

${x}_{2}$ | $({s}_{2},{s}_{6},{s}_{2})$ | $({s}_{3},{s}_{8},{s}_{2})$ | $({s}_{2},{s}_{4},{s}_{1})$ | $({s}_{3},{s}_{1},{s}_{2})$ | $({s}_{1},{s}_{2},{s}_{1})$ |

${x}_{3}$ | $({s}_{2},{s}_{3},{s}_{1})$ | $({s}_{3},{s}_{2},{s}_{3})$ | $({s}_{1},{s}_{4},{s}_{1})$ | $({s}_{3},{s}_{5},{s}_{1})$ | $({s}_{5},{s}_{2},{s}_{4})$ |

${x}_{4}$ | $({s}_{3},{s}_{1},{s}_{2})$ | $({s}_{1},{s}_{7},{s}_{1})$ | $({s}_{4},{s}_{6},{s}_{3})$ | $({s}_{2},{s}_{5},{s}_{1})$ | $({s}_{4},{s}_{6},{s}_{4})$ |

${\mathit{N}}^{(\mathbf{2})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

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

${x}_{2}$ | $({s}_{1},{s}_{4},{s}_{1})$ | $({s}_{3},{s}_{2},{s}_{1})$ | $({s}_{2},{s}_{3},{s}_{4})$ | $({s}_{4},{s}_{0},{s}_{5})$ | $({s}_{2},{s}_{6},{s}_{4})$ |

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

${x}_{4}$ | $({s}_{2},{s}_{7},{s}_{2})$ | $({s}_{4},{s}_{6},{s}_{1})$ | $({s}_{3},{s}_{7},{s}_{2})$ | $({s}_{4},{s}_{4},{s}_{2})$ | $({s}_{3},{s}_{8},{s}_{4})$ |

${\mathit{N}}^{(\mathbf{3})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

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

${x}_{2}$ | $({s}_{1},{s}_{2},{s}_{1})$ | $({s}_{2},{s}_{4},{s}_{2})$ | $({s}_{1},{s}_{5},{s}_{3})$ | $({s}_{4},{s}_{2},{s}_{0})$ | $({s}_{0},{s}_{5},{s}_{6})$ |

${x}_{3}$ | $({s}_{2},{s}_{3},{s}_{3})$ | $({s}_{1},{s}_{5},{s}_{2})$ | $({s}_{2},{s}_{4},{s}_{5})$ | $({s}_{0},{s}_{4},{s}_{6})$ | $({s}_{3},{s}_{2},{s}_{4})$ |

${x}_{4}$ | $({s}_{2},{s}_{3},{s}_{2})$ | $({s}_{4},{s}_{2},{s}_{1})$ | $({s}_{1},{s}_{4},{s}_{3})$ | $({s}_{3},{s}_{4},{s}_{5})$ | $({s}_{0},{s}_{4},{s}_{5})$ |

${\mathit{N}}^{\u2022(\mathbf{1})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7},{s}_{6},{s}_{7})$ | $({s}_{6},{s}_{5},{s}_{6})$ | $({s}_{4},{s}_{4},{s}_{5})$ | $({s}_{7},{s}_{3},{s}_{7})$ | $({s}_{5},{s}_{5},{s}_{6})$ |

${x}_{2}$ | $({s}_{6},{s}_{2},{s}_{6})$ | $({s}_{5},{s}_{0},{s}_{6})$ | $({s}_{6},{s}_{4},{s}_{7})$ | $({s}_{6},{s}_{7},{s}_{6})$ | $({s}_{7},{s}_{6},{s}_{7})$ |

${x}_{3}$ | $({s}_{6},{s}_{5},{s}_{7})$ | $({s}_{5},{s}_{6},{s}_{5})$ | $({s}_{7},{s}_{4},{s}_{7})$ | $({s}_{5},{s}_{3},{s}_{7})$ | $({s}_{3},{s}_{6},{s}_{4})$ |

${x}_{4}$ | $({s}_{5},{s}_{7},{s}_{6})$ | $({s}_{7},{s}_{1},{s}_{7})$ | $({s}_{4},{s}_{2},{s}_{5})$ | $({s}_{6},{s}_{3},{s}_{7})$ | $({s}_{4},{s}_{2},{s}_{4})$ |

${\mathit{N}}^{\u2022(\mathbf{2})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7},{s}_{2},{s}_{7})$ | $({s}_{4},{s}_{5},{s}_{4})$ | $({s}_{6},{s}_{2},{s}_{6})$ | $({s}_{5},{s}_{3},{s}_{6})$ | $({s}_{3},{s}_{6},{s}_{4})$ |

${x}_{2}$ | $({s}_{7},{s}_{4},{s}_{7})$ | $({s}_{5},{s}_{6},{s}_{7})$ | $({s}_{6},{s}_{5},{s}_{4})$ | $({s}_{4},{s}_{8},{s}_{3})$ | $({s}_{6},{s}_{2},{s}_{4})$ |

${x}_{3}$ | $({s}_{5},{s}_{3},{s}_{6})$ | $({s}_{6},{s}_{4},{s}_{5})$ | $({s}_{7},{s}_{2},{s}_{3})$ | $({s}_{5},{s}_{3},{s}_{5})$ | $({s}_{6},{s}_{2},{s}_{7})$ |

${x}_{4}$ | $({s}_{6},{s}_{1},{s}_{6})$ | $({s}_{4},{s}_{3},{s}_{7})$ | $({s}_{5},{s}_{1},{s}_{6})$ | $({s}_{4},{s}_{4},{s}_{6})$ | $({s}_{5},{s}_{0},{s}_{4})$ |

${\mathit{N}}^{\u2022(\mathbf{3})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{6},{s}_{4},{s}_{7})$ | $({s}_{5},{s}_{3},{s}_{6})$ | $({s}_{3},{s}_{7},{s}_{4})$ | $({s}_{6},{s}_{2},{s}_{7})$ | $({s}_{5},{s}_{5},{s}_{6})$ |

${x}_{2}$ | $({s}_{7},{s}_{6},{s}_{7})$ | $({s}_{6},{s}_{4},{s}_{6})$ | $({s}_{7},{s}_{3},{s}_{5})$ | $({s}_{4},{s}_{6},{s}_{8})$ | $({s}_{8},{s}_{3},{s}_{2})$ |

${x}_{3}$ | $({s}_{6},{s}_{5},{s}_{5})$ | $({s}_{7},{s}_{3},{s}_{6})$ | $({s}_{6},{s}_{4},{s}_{3})$ | $({s}_{8},{s}_{4},{s}_{2})$ | $({s}_{5},{s}_{6},{s}_{4})$ |

${x}_{4}$ | $({s}_{6},{s}_{5},{s}_{6})$ | $({s}_{4},{s}_{6},{s}_{7})$ | $({s}_{7},{s}_{4},{s}_{5})$ | $({s}_{5},{s}_{4},{s}_{3})$ | $({s}_{8},{s}_{4},{s}_{3})$ |

${\mathit{N}}^{\u2022}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{6.74},{s}_{3.63},{s}_{7})$ | $({s}_{5.12},{s}_{4.22},{s}_{5.24})$ | $({s}_{4.58},{s}_{3.83},{s}_{4.93})$ | $({s}_{6.18},{s}_{2.62},{s}_{6.65})$ | $({s}_{4.44},{s}_{5.31},{s}_{5.24})$ |

${x}_{2}$ | $({s}_{6.74},{s}_{3.63},{s}_{6.65})$ | $({s}_{5.38},{s}_{0},{s}_{6.32})$ | $({s}_{6.18},{s}_{3.91},{s}_{5.19})$ | $({s}_{4.83},{s}_{6.95},{s}_{5.24})$ | $({s}_{8},{s}_{3.3},{s}_{3.83})$ |

${x}_{3}$ | $({s}_{5.71},{s}_{4.22},{s}_{5.94})$ | $({s}_{6.18},{s}_{4.16},{s}_{5.31})$ | $({s}_{6.74},{s}_{3.17},{s}_{3.98})$ | $({s}_{8},{s}_{3.3},{s}_{4.12})$ | $({s}_{4.89},{s}_{4.16},{s}_{4.82})$ |

${x}_{4}$ | $({s}_{5.71},{s}_{3.27},{s}_{6})$ | $({s}_{5.48},{s}_{2.62},{s}_{7})$ | $({s}_{5.71},{s}_{2},{s}_{5.31})$ | $({s}_{5.11},{s}_{3.63},{s}_{5.01})$ | $({s}_{8},{s}_{0},{s}_{3.63})$ |

${\mathit{N}}^{\mathit{w}}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{1.1},{s}_{7.51},{s}_{7.91})$ | $({s}_{1.48},{s}_{7.04},{s}_{7.35})$ | $({s}_{0.96},{s}_{7.16},{s}_{7.44})$ | $({s}_{2.64},{s}_{5.92},{s}_{7.61})$ | $({s}_{1.73},{s}_{7.07},{s}_{7.05})$ |

${x}_{2}$ | $({s}_{1.1},{s}_{7.51},{s}_{7.88})$ | $({s}_{1.6},{s}_{0},{s}_{7.63})$ | $({s}_{1.59},{s}_{7.19},{s}_{7.5})$ | $({s}_{1.77},{s}_{7.7},{s}_{7.14})$ | $({s}_{8},{s}_{6.13},{s}_{6.41})$ |

${x}_{3}$ | $({s}_{0.76},{s}_{7.6},{s}_{7.81})$ | $({s}_{2.05},{s}_{7.02},{s}_{7.37})$ | $({s}_{1.94},{s}_{6.96},{s}_{7.2})$ | $({s}_{8},{s}_{6.3},{s}_{6.69})$ | $({s}_{1.97},{s}_{6.57},{s}_{6.87})$ |

${x}_{4}$ | $({s}_{0.76},{s}_{7.45},{s}_{7.82})$ | $({s}_{1.65},{s}_{6.4},{s}_{7.79})$ | $({s}_{1.37},{s}_{6.5},{s}_{7.53})$ | $({s}_{1.92},{s}_{6.46},{s}_{7.05})$ | $({s}_{8},{s}_{0},{s}_{6.31})$ |

${\eta}_{1}^{+}$ | ${\eta}_{2}^{+}$ | ${\eta}_{3}^{+}$ | ${\eta}_{4}^{+}$ | ${\eta}_{5}^{+}$ |

$({s}_{1.1},{s}_{7.51},{s}_{7.88})$ | $({s}_{1.6},{s}_{0},{s}_{7.63})$ | $({s}_{1.94},{s}_{6.96},{s}_{7.2})$ | $({s}_{8},{s}_{6.3},{s}_{6.69})$ | $({s}_{8},{s}_{0},{s}_{6.31})$ |

${\eta}_{1}^{-}$ | ${\eta}_{2}^{-}$ | ${\eta}_{3}^{-}$ | ${\eta}_{4}^{-}$ | ${\eta}_{5}^{-}$ |

$({s}_{0.76},{s}_{7.6},{s}_{7.81})$ | $({s}_{1.48},{s}_{7.04},{s}_{7.35})$ | $({s}_{0.96},{s}_{7.16},{s}_{7.44})$ | $({s}_{1.77},{s}_{7.7},{s}_{7.14})$ | $({s}_{1.73},{s}_{7.07},{s}_{7.05})$ |

Approaches | Ranking Orders | Optimal Alternatives | Worst Alternatives |
---|---|---|---|

Approach with the LNWAM operator [54] | ${x}_{4}>{x}_{2}>{x}_{3}>{x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

Approach with the LNWGM operator [54] | ${x}_{4}>{x}_{3}>{x}_{2}>{x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

Approach with ${u}_{ij}=\frac{1}{3}{s}_{{T}_{ij}}$ [50] | ${x}_{4}>{x}_{3}>{x}_{2}>{x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

Approach with ${u}_{ij}=\frac{1}{3}{s}_{{T}_{ij}}+\frac{1}{6}{s}_{{I}_{ij}}$ [50] | ${x}_{1}>{x}_{3}>{x}_{2}>{x}_{4}$ | ${x}_{1}$ | ${x}_{4}$ |

Approach with ${u}_{ij}=\frac{1}{3}{s}_{{T}_{ij}}+\frac{1}{3}{s}_{{F}_{ij}}$ [50] | ${x}_{2}>{x}_{1}>{x}_{3}>{x}_{4}$ | ${x}_{2}$ | ${x}_{4}$ |

Approach with SVNLN-TOPSIS [42] | ${x}_{4}>{x}_{2}>{x}_{3}>{x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

The presented approach | ${x}_{4}>{x}_{2}>{x}_{3}>{x}_{1}$ | ${x}_{4}$ | ${x}_{1}$ |

${\mathit{N}}^{\mathit{c}\mathit{o}(\mathbf{1})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7/3},{s}_{13/3})$ | $({s}_{2},{s}_{11/3})$ | $({s}_{4/3},{s}_{3})$ | $({s}_{7/3},{s}_{10/3})$ | $({s}_{5/3},{s}_{11/3})$ |

${x}_{2}$ | $({s}_{2},{s}_{8/3})$ | $({s}_{5/3},{s}_{2})$ | $({s}_{2},{s}_{11/3})$ | $({s}_{2},{s}_{13/3})$ | $({s}_{7/3},{s}_{13/3})$ |

${x}_{3}$ | $({s}_{2},{s}_{4})$ | $({s}_{5/3},{s}_{11/3})$ | $({s}_{7/3},{s}_{11/3})$ | $({s}_{5/3},{s}_{10/3})$ | $({s}_{1},{s}_{10/3})$ |

${x}_{4}$ | $({s}_{5/3},{s}_{13/3})$ | $({s}_{7/3},{s}_{8/3})$ | $({s}_{4/3},{s}_{7/3})$ | $({s}_{2},{s}_{10/3})$ | $({s}_{4/3},{s}_{2})$ |

${\mathit{N}}^{\mathit{c}\mathit{o}(\mathbf{2})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7/3},{s}_{3})$ | $({s}_{4/3},{s}_{3})$ | $({s}_{2},{s}_{8/3})$ | $({s}_{5/3},{s}_{3})$ | $({s}_{1},{s}_{10/3})$ |

${x}_{2}$ | $({s}_{7/3},{s}_{11/3})$ | $({s}_{5/3},{s}_{13/3})$ | $({s}_{2},{s}_{3})$ | $({s}_{4/3},{s}_{11/3})$ | $({s}_{2},{s}_{2})$ |

${x}_{3}$ | $({s}_{5/3},{s}_{3})$ | $({s}_{2},{s}_{3})$ | $({s}_{7/3},{s}_{5/3})$ | $({s}_{5/3},{s}_{8/3})$ | $({s}_{2},{s}_{3})$ |

${x}_{4}$ | $({s}_{2},{s}_{7/3})$ | $({s}_{4/3},{s}_{10/3})$ | $({s}_{5/3},{s}_{7/3})$ | $({s}_{4/3},{s}_{10/3})$ | $({s}_{5/3},{s}_{4/3})$ |

${\mathit{N}}^{\mathit{c}\mathit{o}(\mathbf{3})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{2},{s}_{11/3})$ | $({s}_{5/3},{s}_{3})$ | $({s}_{1},{s}_{11/3})$ | $({s}_{2},{s}_{3})$ | $({s}_{5/3},{s}_{11/3})$ |

${x}_{2}$ | $({s}_{7/3},{s}_{13/3})$ | $({s}_{2},{s}_{10/3})$ | $({s}_{7/3},{s}_{8/3})$ | $({s}_{4/3},{s}_{14/3})$ | $({s}_{8/3},{s}_{5/3})$ |

${x}_{3}$ | $({s}_{2},{s}_{10/3})$ | $({s}_{7/3},{s}_{3})$ | $({s}_{2},{s}_{7/3})$ | $({s}_{8/3},{s}_{2})$ | $({s}_{5/3},{s}_{10/3})$ |

${x}_{4}$ | $({s}_{2},{s}_{11/3})$ | $({s}_{4/3},{s}_{13/3})$ | $({s}_{7/3},{s}_{3})$ | $({s}_{5/3},{s}_{7/3})$ | $({s}_{8/3},{s}_{7/3})$ |

${\mathit{N}}^{\mathit{t}\mathit{r}(\mathbf{1})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7},<1,6/7,1>)$ | $({s}_{6},<1,5/6,1>)$ | $({s}_{5},<4/5,4/5,1>)$ | $({s}_{7},<1,3/7,1>)$ | $({s}_{6},<5/6,5/6,1>)$ |

${x}_{2}$ | $({s}_{6},<1,1/3,1>)$ | $({s}_{6},<5/6,0,1>)$ | $({s}_{7},<6/7,4/7,1>)$ | $({s}_{7},<6/7,1,6/7>)$ | $({s}_{7},<1,6/7,1>)$ |

${x}_{3}$ | $({s}_{7},<6/7,5/7,1>)$ | $({s}_{6},<5/6,1,5/6>)$ | $({s}_{7},<1,4/7,1>)$ | $({s}_{7},<5/7,3/7,1>)$ | $({s}_{6},<1/2,1,2/3>)$ |

${x}_{4}$ | $({s}_{7},<5/7,1,6/7>)$ | $({s}_{7},<1,1/7,1>)$ | $({s}_{5},<4/5,2/5,1>)$ | $({s}_{7},<6/7,3/7,1>)$ | $({s}_{4},<1,1/2,1>)$ |

${\mathit{N}}^{\mathit{t}\mathit{r}(\mathbf{2})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7},<1,2/7,1>)$ | $({s}_{5},<4/5,1,4/5>)$ | $({s}_{6},<1,1/3,1>)$ | $({s}_{6},<5/6,1/2,1>)$ | $({s}_{6},<1/2,1,2/3>)$ |

${x}_{2}$ | $({s}_{7},<1,4/7,1>)$ | $({s}_{7},<5/7,6/7,1>)$ | $({s}_{6},<1,5/6,2/3>)$ | $({s}_{8},<1/2,1,3/8>)$ | $({s}_{6},<1,1/3,2/3>)$ |

${x}_{3}$ | $({s}_{6},<5/6,1/2,1>)$ | $({s}_{6},<1,2/3,5/6>)$ | $({s}_{7},<1,2/7,3/7>)$ | $({s}_{5},<1,3/5,1>)$ | $({s}_{7},<6/7,2/7,1>)$ |

${x}_{4}$ | $({s}_{6},<1,1/6,1>)$ | $({s}_{7},<4/7,3/7,1>)$ | $({s}_{6},<5/6,1/6,1>)$ | $({s}_{6},<2/3,2/3,1>)$ | $({s}_{5},<1,0,4/5>)$ |

${\mathit{N}}^{\mathit{t}\mathit{r}(\mathbf{3})}$ | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{3}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

${x}_{1}$ | $({s}_{7},<6/7,4/7,1>)$ | $({s}_{6},<5/6,1/2,1>)$ | $({s}_{7},<3/7,1,4/7>)$ | $({s}_{7},<6/7,2/7,1>)$ | $({s}_{6},<5/6,5/6,1>)$ |

${x}_{2}$ | $({s}_{7},<1,6/7,1>)$ | $({s}_{6},<1,2/3,1>)$ | $({s}_{7},<1,3/7,5/7>)$ | $({s}_{8},<1/2,3/4,1>)$ | $({s}_{8},<1,3/8,1/4>)$ |

${x}_{3}$ | $({s}_{6},<1,5/6,5/6>$ | $({s}_{7},<1,3/7,6/7>)$ | $({s}_{6},<1,2/3,1/2>)$ | $({s}_{8},<1,1/2,1/4>)$ | $({s}_{6},<5/6,1,2/3>)$ |

${x}_{4}$ | $({s}_{6},<1,5/6,1>)$ | $({s}_{7},<4/7,6/7,1>)$ | $({s}_{7},<1,4/7,5/7>)$ | $({s}_{5},<1,4/5,3/5>)$ | $({s}_{8},<1,1/2,3/8>)$ |

© 2017 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.; Wu, H.
Evaluating Investment Risks of Metallic Mines Using an Extended TOPSIS Method with Linguistic Neutrosophic Numbers. *Symmetry* **2017**, *9*, 149.
https://doi.org/10.3390/sym9080149

**AMA Style**

Liang W, Zhao G, Wu H.
Evaluating Investment Risks of Metallic Mines Using an Extended TOPSIS Method with Linguistic Neutrosophic Numbers. *Symmetry*. 2017; 9(8):149.
https://doi.org/10.3390/sym9080149

**Chicago/Turabian Style**

Liang, Weizhang, Guoyan Zhao, and Hao Wu.
2017. "Evaluating Investment Risks of Metallic Mines Using an Extended TOPSIS Method with Linguistic Neutrosophic Numbers" *Symmetry* 9, no. 8: 149.
https://doi.org/10.3390/sym9080149