# Evaluation Method of Highway Plant Slope Based on Rough Set Theory and Analytic Hierarchy Process: A Case Study in Taihang Mountain, Hebei, China

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Site Description

## 3. Construction of Performance Evaluation Index System of the Slope

## 4. Construct the Evaluation Model of Highway Plant Slope

#### 4.1. Calculation Steps of Rough Set Theory

_{p}J as the P positive domain of J, where POS

_{p}J =$\bigcup}_{x\in U/J}PX$, POS

_{p}J denotes those in the argument domain U.

_{1}–U

_{15}.

_{i}in the set of conditional attributes C and the relevance of the conditional attribute D

_{i}in the set of conditional attributes D. It cannot reflect the importance of the conditional attributes in the system itself, and the conditional entropy method can compensate for this drawback, so the problem is solved by defining the conditional entropy, and the conditional entropy of the conditional attributes with respect to the decision attributes is

#### 4.2. Analytic Hierarchy Process Calculation Steps

#### 4.3. Weight Results of First-Level Indicators

_{1}, logistics geotechnical parameter A

_{2}, hydrological condition A

_{3}, and vegetation condition A

_{4}as the first-level target layer. Twelve indicators, such as “soil characteristics” and “slope height,” represented by C, were used as the secondary target layer. By evaluating the relative importance of each indicator using the expert scoring method and generalizing the results by establishing an eigenvalue and eigenvector of each matrix, a consistency test could determine the largest eigenvalue for each level of indicators. The first-level index weight calculation results were obtained, as shown in Table 7. The consistency ratio was C

_{R}< 0.1. The calculation results of each weight were less than 0.1, so it can be determined that the total ranking of the judgment matrix level is consistent, and the judgment matrix does not need to be corrected [50].

#### 4.4. Secondary Index Weight Results

_{1}, S

_{2}, S

_{3}, S

_{4}, S

_{5}, S

_{6}, S

_{7}, S

_{8}, S

_{9}, S

_{10}, S

_{11}, S

_{12}}, and the index weight matrix B obtained by analytic hierarchy process is {B

_{1}, B

_{2}, B

_{3}, B

_{4}, B

_{5}, B

_{6}, B

_{7}, B

_{8}, B

_{9}, B

_{10}, B

_{11}, B

_{12}}. The product of the two matrices and the weight processing can obtain the coupling weight of the rough set theory method and the analytic hierarchy process, as shown in Table 8 [57].

#### 4.5. Establishment of Evaluation Model

_{1}, M

_{2}, M

_{3}, M

_{4}} = {very stable, stable, unstable, to extremely unstable}

## 5. A Case of Evaluation of Highway Plant Slope

_{1}, M

_{2}, M

_{3}, M

_{4}} of the highway plant slope can be obtained as {−0.3564, −0.1529, −0.1797,−0.3308 } by using the extension model theory, and the maximum comprehensive correlation degree is −0.1529. Therefore, the evaluation grade of the slope is secondary. The entropy weight method (EWM) stands out as an excellent and well-studied approach [15]. Combined with the EWM approach, the comprehensive correlation degree {M

_{1}, M

_{2}, M

_{3}, M

_{4}} of the highway plant slope can be obtained as {0.0283, 0.0491, 0.0353, 0.0043} by using the extension model theory, and the maximum comprehensive correlation degree is 0.0491, the evaluation grade of the slope is secondary. The results show that the evaluated results agree with the practical slope, which implies that the proposed method is feasible and reliable.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Region | Side Slope Grade | Slope Defects | Geologic Environment |
---|---|---|---|

Tuling | 1:1.5 | There are valleys on the right side of the slope, the catchment area is large, the valley mouth of the lower river is comprehensive, and there are signs of ancient debris flow | Silty clay soil, stone |

Shangjiaosi | 1:1.25 | Medium soil collapse degree | Loess-like silt, with gravel |

Lingdi | 1:1 | Strongly weathered collapsed rubble visible on both sidesRed clay deposit, poor geology. | Artificial fill, block stone, weathered marl, silt, and other combinations |

Indicators | Notations | Corresponding Grade | Discrete Values |
---|---|---|---|

Drought resistance | C_{1} | Excellent | 1 |

Good | 2 | ||

Average | 3 | ||

Poor | 4 | ||

Cold resistance | C_{2} | Excellent | 1 |

Good | 2 | ||

Average | 3 | ||

Poor | 4 | ||

Salt and alkali resistance | C_{3} | Excellent | 1 |

Good | 2 | ||

Average | 3 | ||

Poor | 4 | ||

Soil characteristics | C_{4} | Excellent | 1 |

Good | 2 | ||

Average | 3 | ||

Poor | 4 | ||

Slope height/m | C_{5} | 0–5 | 1 |

5–10 | 2 | ||

10–15 | 3 | ||

>15 | 4 | ||

Slope gradient/(°) | C_{6} | 0–20 | 1 |

20–40 | 2 | ||

40–60 | 3 | ||

>60 | 4 | ||

Precipitation intensity/mm | C_{7} | 0–20 | 1 |

20–60 | 2 | ||

60–120 | 3 | ||

>120 | 4 | ||

Seepage performance | C_{8} | Excellent | 1 |

Good | 2 | ||

Average | 3 | ||

Poor | 4 | ||

Groundwater level | C_{9} | Dry | 1 |

Wet | 2 | ||

Dripping | 3 | ||

Bubbling | 4 | ||

Plant type | C_{10} | Trees, shrubs and herbs are reasonable | 1 |

Fewer trees, reasonable shrubs and herbs | 2 | ||

Few trees, reasonable shrubs, more reasonable herbs | 3 | ||

No trees, few shrubs, more reasonable herbs | 4 | ||

Purification ability | C_{11} | Excellent | 1 |

Good | 2 | ||

Average | 3 | ||

Poor | 4 | ||

Vegetation cover | C_{12} | 85–100% | 1 |

65–85% | 2 | ||

45–65% | 3 | ||

10–45% | 4 |

Rank | Range of Values |
---|---|

1 | [90, 100] |

2 | [60, 90) |

3 | [40, 60) |

4 | [0, 40) |

Rank | Slope Stability Condition |
---|---|

One | very stable |

Two | stable |

Three | unstable |

Four | severely unstable |

U | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} | C_{11} | C_{12} | D |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

U_{1} | 2 | 3 | 1 | 2 | 3 | 4 | 3 | 2 | 4 | 3 | 3 | 3 | Four |

U_{2} | 2 | 3 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 4 | 4 | 3 | Three |

U_{3} | 3 | 4 | 4 | 2 | 4 | 2 | 3 | 1 | 3 | 4 | 3 | 4 | Four |

U_{4} | 2 | 2 | 3 | 2 | 3 | 4 | 1 | 3 | 2 | 3 | 3 | 2 | Two |

U_{5} | 2 | 3 | 4 | 2 | 2 | 3 | 3 | 2 | 1 | 1 | 2 | 3 | Two |

U_{6} | 4 | 3 | 1 | 2 | 3 | 4 | 3 | 2 | 4 | 3 | 3 | 3 | One |

U_{7} | 2 | 2 | 3 | 2 | 3 | 1 | 3 | 2 | 3 | 4 | 4 | 3 | One |

U_{8} | 1 | 4 | 2 | 1 | 2 | 2 | 3 | 2 | 2 | 3 | 2 | 1 | Two |

U_{9} | 2 | 3 | 4 | 2 | 2 | 3 | 3 | 2 | 1 | 2 | 2 | 3 | Four |

U_{10} | 2 | 3 | 4 | 2 | 2 | 3 | 3 | 2 | 3 | 1 | 2 | 3 | Three |

U_{11} | 2 | 2 | 4 | 2 | 3 | 4 | 1 | 3 | 2 | 3 | 3 | 2 | Two |

U_{12} | 3 | 4 | 4 | 2 | 4 | 2 | 3 | 1 | 3 | 2 | 3 | 4 | Three |

U_{13} | 2 | 3 | 3 | 2 | 2 | 1 | 4 | 1 | 2 | 3 | 2 | 3 | Two |

U_{14} | 2 | 2 | 3 | 2 | 3 | 4 | 1 | 3 | 4 | 3 | 3 | 2 | Four |

U_{15} | 3 | 4 | 2 | 2 | 4 | 2 | 3 | 1 | 3 | 4 | 3 | 4 | Two |

Canonical Scale | Definition | Explanation |
---|---|---|

1 | Equally important | One factor is as important as the other |

3 | A little important | One factor is slightly more important than the other |

5 | Clearly important | The importance of one factor outweighs the other |

7 | Strongly important | One factor is significantly more important than the other |

9 | Absolutely important | One factor is more essential than the other |

A_{0} | A_{1} | A_{2} | A_{3} | A_{4} | Weight | Other Values |
---|---|---|---|---|---|---|

A_{1} | 1 | 2 | 4 | 3 | 0.4427 | ${\lambda}_{\mathrm{max}}=4.2295$ C _{R} = 0.0859 |

A_{2} | 1/2 | 1 | 5 | 4 | 0.3545 | |

A_{3} | 1/4 | 1/5 | 1 | 2 | 0.1123 | |

A_{4} | 1/3 | 1/4 | 1/2 | 1 | 0.0905 |

Indicator | Analytic Hierarchy Process Weight | Rough Set Weight | Coupling Weight |
---|---|---|---|

W_{1} | 0.3025 | 0.0503 | 0.2086 |

W_{2} | 0.0885 | 0.0527 | 0.0639 |

W_{3} | 0.0517 | 0.0959 | 0.0680 |

W_{4} | 0.0921 | 0.0743 | 0.0938 |

W_{5} | 0.1463 | 0.0767 | 0.1538 |

W_{6} | 0.1161 | 0.0983 | 0.1564 |

W_{7} | 0.0780 | 0.0792 | 0.0847 |

W_{8} | 0.0107 | 0.0815 | 0.0120 |

W_{9} | 0.0236 | 0.1007 | 0.0326 |

W_{10} | 0.0516 | 0.1031 | 0.0729 |

W_{11} | 0.0088 | 0.0818 | 0.0098 |

W_{12} | 0.0301 | 0.1055 | 0.0435 |

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**MDPI and ACS Style**

Liu, L.; Dou, Y.; Qiao, J.
Evaluation Method of Highway Plant Slope Based on Rough Set Theory and Analytic Hierarchy Process: A Case Study in Taihang Mountain, Hebei, China. *Mathematics* **2022**, *10*, 1264.
https://doi.org/10.3390/math10081264

**AMA Style**

Liu L, Dou Y, Qiao J.
Evaluation Method of Highway Plant Slope Based on Rough Set Theory and Analytic Hierarchy Process: A Case Study in Taihang Mountain, Hebei, China. *Mathematics*. 2022; 10(8):1264.
https://doi.org/10.3390/math10081264

**Chicago/Turabian Style**

Liu, Luliang, Yuanming Dou, and Jiangang Qiao.
2022. "Evaluation Method of Highway Plant Slope Based on Rough Set Theory and Analytic Hierarchy Process: A Case Study in Taihang Mountain, Hebei, China" *Mathematics* 10, no. 8: 1264.
https://doi.org/10.3390/math10081264