# Ramp Spacing Evaluation of Expressway Based on Entropy-Weighted TOPSIS Estimation Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Evaluation Indicators

#### 2.1. Problem Statement

#### 2.2. Evaluation Indicator System

#### 2.2.1. Traffic Efficiency

#### 2.2.2. Safety

^{−8}∙veh

^{−1})); Let $\sigma =\sqrt{\frac{{\displaystyle {\sum}_{i=1}^{N}{\left({v}_{i}-\overline{v}\right)}^{2}}}{N-1}}$,$\sigma $ denotes the standard deviation of the speed of all vehicles on the expressway section (km/h).

#### 2.2.3. Traffic Accessibility

#### 2.2.4. Economy

## 3. Methodology

#### 3.1. Entropy Weight Method

#### 3.2. TOPSIS Method

## 4. Case Study

#### 4.1. Study Area

#### 4.2. Results and Discussions

#### 4.2.1. Comparison of Ramp Spacing Alternatives

#### 4.2.2. Sensitivity Analysis of TDARs

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Variable | Notation |
---|---|

$i$ | index of towns |

$j$ | index of ramps |

${L}_{i}$ | the comprehensive level of service of roads within the town $i$ |

${l}_{ij}$ | the distance from town $i$ to ramp $j$ |

${M}_{i}$ | the comprehensive aggregation scale of town $i$ |

${\epsilon}_{i}$ | the accessibility of town $i$ when considering distance factor |

${\beta}_{rj}$ | the weight of the ramp $j$ attached to the toll station with location potential level $r$ |

${\lambda}_{i}$ | the accessibility of town $i$ |

${\lambda}_{0}$ | the accessibility of the standard town |

$L{p}_{i}$ | the location potential of town $i$ |

$L{p}_{0}$ | the location potential of the standard town |

$\xi $ | the proportionality coefficient |

$\chi $ | the elastic correction factor for the increase in location potential due to the traffic accessibility |

$\phi $ | the elastic correction factor for the increase in location potential due to the comprehensive aggregation scale |

$coe{f}_{i}$ | the location influence coefficient of town $i$ |

${q}_{i}$ | the traffic demand along the route allocated to town $i$ |

${F}_{j}$ | the cross-sectional flow of the toll station where ramp $j$ is located |

${z}^{t}\left({l}_{ij}\right)$ | the cumulative probability of travel to ramp $j$ from a town which is ${l}_{ij}$ kilometers away from ramp $j$ |

$\zeta $ | parameter of the function related to distance decay theory |

$\psi $ | parameter of the function related to distance decay theory |

${V}_{t}$ | the transit traffic demand of expressway |

${V}_{a}$ | the traffic demand along the route of expressway |

$\mu $ | the proportion of transit traffic demand |

Sets | |

$T$ | the set of towns |

$R$ | the set of ramps set according to spacing alternative |

${R}_{i}$ | the set of ramps that can serve the town $i$, ${R}_{i}\in R$ |

${R}_{0}$ | the set of the original ramps on the expressway |

${R}_{0i}$ | the set of ramps that can serve the town $i$, ${R}_{0i}\in {R}_{0}$ |

${T}_{j}$ | the set of towns served by ramp $j$ |

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**Figure 3.**The distribution of the ramps set according to ${A}_{11}$ within the demand-intensive area: (

**a**) Demand-intensive area I; (

**b**) Demand-intensive area II.

**Figure 4.**The distribution of the ramps set according to ${A}_{14}$ within the demand-intensive area: (

**a**) Demand-intensive area I; (

**b**) Demand-intensive area II.

**Figure 5.**Evaluation of ramp spacing alternatives under different traffic demands along the expressway.

Variable | Notation |
---|---|

Parameters | |

$\overline{v}$ | the average speed |

$\overline{d}$ | the average delay |

${v}_{i}$ | the speed of vehicle $i$ |

${d}_{i}$ | the delay of vehicle $i$ |

$N$ | the sample size of vehicles |

$\u03f5$ | the accident rate of 100 million vehicle- kilometers |

$\sigma $ | the standard deviation of the speed of all vehicles |

${L}_{i}$ | the comprehensive level of service of roads within the town $i$ |

$\alpha $ | the grade of roads |

$le{n}_{\alpha}$ | the length of the road with grade $\alpha $ |

${h}_{\alpha}$ | the evaluation value of the road with class $\alpha $ |

$Le{n}_{i}$ | the total length of the roads passing through the town $i$ |

${a}_{i}$ | the accessibility of town $i$ |

${l}_{ij}$ | the distance from town $i$ to ramp $j$ |

${M}_{i}$ | the comprehensive aggregation scale of town $i$ |

$P$ | the number of evaluation indicators for the comprehensive aggregation scale |

${\gamma}_{k}$ | the weight of the $k-\mathrm{th}$ evaluation indicator |

${u}_{ik}$ | the value of the $k-\mathrm{th}$ evaluation indicator of town $i$ |

$\varphi $ | the accessibility of the study district |

$\mathsf{\Omega}$ | the project cost of constructing all ramps along the expressway section |

$\rho $ | the density of ramps along the expressway section |

${l}_{c}$ | the relevant parameter of the project cost |

${\delta}_{c}$ | |

${\beta}_{c}$ | |

Sets | |

$R$ | the set of ramps set according to spacing alternative |

${R}_{i}$ | the set of ramps that can serve the town $i$, $R\in {R}_{i}$ |

$T$ | the set of towns |

Indicator (Criterion) | Dimension | Indicator Source |
---|---|---|

Average speed ($\overline{v}$) | positive | VISSIM simulation |

Average delay ($\overline{d}$) | negative | VISSIM simulation |

Accident rate ($\u03f5$) | negative | VISSIM simulation and calculation |

Traffic accessibility ($\varphi $) | positive | Calculation based on data |

Project cost ($\mathsf{\Omega}$) | negative | Evaluation based on ramp data and terrain conditions |

Indicator (Criterion) | Average Speed | Average Delay | Accident Rate | Traffic Accessibility | Project Cost |
---|---|---|---|---|---|

Weight (${w}_{j}$) | 0.20211 | 0.25285 | 0.16918 | 0.13340 | 0.24246 |

Spacing Alternative | Decision Matrix | ||||
---|---|---|---|---|---|

Average Speed | Average Delay | Accident Rate | Traffic Accessibility | Project Cost | |

${A}_{1}$ | 0.47 | 0.66 | 0.54 | 1.00 | 0.00 |

${A}_{2}$ | 0.00 | 0.29 | 0.66 | 0.98 | 0.05 |

${A}_{3}$ | 0.65 | 0.72 | 0.92 | 0.99 | 0.10 |

${A}_{4}$ | 1.00 | 0.80 | 0.67 | 0.96 | 0.15 |

${A}_{5}$ | 0.43 | 0.60 | 0.87 | 0.94 | 0.20 |

${A}_{6}$ | 0.81 | 0.74 | 0.80 | 0.92 | 0.25 |

${A}_{7}$ | 0.55 | 0.70 | 1.00 | 0.91 | 0.30 |

${A}_{8}$ | 0.96 | 1.00 | 0.41 | 0.86 | 0.35 |

${A}_{9}$ | 0.58 | 0.92 | 0.67 | 0.83 | 0.40 |

${A}_{10}$ | 0.44 | 0.46 | 0.29 | 0.85 | 0.45 |

${A}_{11}$ | 0.89 | 0.68 | 0.64 | 0.71 | 0.50 |

${A}_{12}$ | 0.04 | 0.49 | 0.00 | 0.71 | 0.55 |

${A}_{13}$ | 0.70 | 0.64 | 0.42 | 0.60 | 0.60 |

${A}_{14}$ | 0.19 | 0.08 | 0.19 | 0.56 | 0.65 |

${A}_{15}$ | 0.79 | 0.35 | 0.59 | 0.55 | 0.70 |

${A}_{16}$ | 0.65 | 0.43 | 0.21 | 0.51 | 0.75 |

${A}_{17}$ | 0.35 | 0.00 | 0.42 | 0.44 | 0.80 |

${A}_{18}$ | 0.52 | 0.36 | 0.32 | 0.37 | 0.85 |

${A}_{19}$ | 0.13 | 0.12 | 0.24 | 0.24 | 0.90 |

${A}_{20}$ | 0.28 | 0.05 | 0.32 | 0.20 | 0.95 |

${A}_{21}$ | 0.45 | 0.07 | 0.34 | 0.00 | 1.00 |

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## Share and Cite

**MDPI and ACS Style**

Ma, J.; Zeng, Y.; Chen, D.
Ramp Spacing Evaluation of Expressway Based on Entropy-Weighted TOPSIS Estimation Method. *Systems* **2023**, *11*, 139.
https://doi.org/10.3390/systems11030139

**AMA Style**

Ma J, Zeng Y, Chen D.
Ramp Spacing Evaluation of Expressway Based on Entropy-Weighted TOPSIS Estimation Method. *Systems*. 2023; 11(3):139.
https://doi.org/10.3390/systems11030139

**Chicago/Turabian Style**

Ma, Jie, Yilei Zeng, and Dawei Chen.
2023. "Ramp Spacing Evaluation of Expressway Based on Entropy-Weighted TOPSIS Estimation Method" *Systems* 11, no. 3: 139.
https://doi.org/10.3390/systems11030139