# Optimization of Ramp Locations along Freeways: A Dynamic Programming Approach

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## Abstract

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## 1. Introduction

## 2. Problem Statement and Model Development

#### 2.1. Total Travel Cost

#### 2.1.1. Access Cost

#### 2.1.2. Freeway Travel Cost

#### 2.2. Construction Cost

#### 2.3. Safety Issue

#### 2.4. Model Development

## 3. Dynamic Programming Approach

## 4. Case Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Ramp placement on the Beijing–Hong Kong–Macao Expressway. (

**a**) Existing placement. (

**b**) Optimized placement.

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

Indices | |

i | Index of traffic regions |

j | Index of existing ramps and candidate ramps |

g | Index of traffic regions along the freeway section between ramp j and ramp k $\left(j=1,2,\dots ,N-1\right)$ |

Parameters | |

${C}_{a}$ | Cost of access time |

${C}_{f}$ | Cost of freeway travel time |

${C}_{s}$ | Construction cost |

${d}_{gj}$ | Distance between traffic region g and ramp j |

${p}_{gj}$ | Distance decay probability between traffic region g and ramp j |

${Q}_{g}$ | Travel demand of travel region g |

${q}_{gj}$ | Traffic demand between traffic region g and ramp j |

${\phi}_{a}$ | Unit value of access cost |

${\overline{v}}_{a}$ | Average access speed |

${V}_{jk}$ | Traffic volume between ramp j and ramp k$\left(j=1,2,\dots ,N-1\right)$ |

${\mathrm{t}}_{f}^{jk}$ | Travel time between ramp j and ramp k$\left(j=1,2,\dots ,N-1\right)$ |

${\overline{v}}_{f}$ | Average free-flow travel speed on the freeway |

c | Freeway capacity |

${t}_{d}^{jk}$ | Travel delay existing in the location between ramp j and ramp k$\left(j=1,2,\dots ,N-1\right)$ |

$\mathrm{e}$ | Critical gap acceptance for vehicles entering or exiting the ramps |

${\mathrm{m}}_{jk}$ | Average waiting interval between ramp j and ramp k$\left(j=1,2,\dots ,N-1\right)$ |

${\mathrm{T}}_{jk}$ | Average non-gap duration between ramp j and ramp k$\left(j=1,2,\dots ,N-1\right)$ |

${\lambda}_{j}$ | Average vehicle arrival rate of ramp j |

${\phi}_{f}$ | Unit value of freeway travel cost |

${C}_{j}$ | Construction cost of ramp j |

${\gamma}_{s}$ | Service life of freeway |

${H}_{jk}$ | Number of lane-changing behaviors between ramp j and ramp k |

${H}_{max}$ | Maximum number of lane changes |

${L}_{jk}$ | Spacing between ramp j and ramp k (j = 1, 2, …, N − 1) |

${L}_{min}$ | Lower limit of the spacing |

Sets | |

${G}_{g}$ | Set of available ramp locations of traffic region g |

S | Set of existing ramps |

$\overline{S}$ | Set of existing ramps and selected candidate ramps |

Variables | |

${x}_{j}$ | Binary variable that equals 1 if ramp j ($j$∉$\overline{S}$) is selected to be newly constructed ramps and 0 if otherwise |

j | Index of existing ramps and candidate ramps |

k | Downstream ramp of ramp j |

Ramp Number (j) | Service Area | ||
---|---|---|---|

Traffic Region Number (i) | ${\mathit{d}}_{\mathit{i}\mathit{j}}\left(\mathbf{km}\right)$ | ${\mathit{Q}}_{\mathit{i}}(\mathbf{veh}/\mathbf{h})$ | |

1 | 1 | 8 | 103 |

2 | 2 | 84 | |

3 | 5.5 | 107 | |

4 | 5.4 | 232 |

Symbol | Description | Value |
---|---|---|

${\phi}_{a}$ | Unit value of access cost | 10 |

${\overline{v}}_{a}$ | Average access speed | 50 km/h |

c | Freeway capacity | 3600 pcu/h |

$\mathsf{\epsilon}$ | Critical gap acceptance for vehicles entering or exiting the ramps | 4 s |

${\phi}_{f}$ | Value of freeway travel cost | 5 |

${C}_{j}$ | Construction cost of ramp j | $590,000 |

${\gamma}_{s}$ | Service life of the freeway | 15 years |

Existing Locations | Optimized Locations | |
---|---|---|

Total number of ramps | 38 | 46 |

Average access time (h per vehicle) | 0.169 | 0.157 (7% reduction) |

Average freeway travel time (h per vehicle) | 1.667 | 1.207 (27.6% reduction) |

Total travel cost (USD) | 3603 | 3052 (15.3% reduction) |

Construction cost (USD) | 2472 | 22,252 |

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**MDPI and ACS Style**

Chen, D.; Mo, F.; Chen, Y.; Zhang, J.; You, X.
Optimization of Ramp Locations along Freeways: A Dynamic Programming Approach. *Sustainability* **2022**, *14*, 9718.
https://doi.org/10.3390/su14159718

**AMA Style**

Chen D, Mo F, Chen Y, Zhang J, You X.
Optimization of Ramp Locations along Freeways: A Dynamic Programming Approach. *Sustainability*. 2022; 14(15):9718.
https://doi.org/10.3390/su14159718

**Chicago/Turabian Style**

Chen, Dawei, Fangxu Mo, Ye Chen, Jun Zhang, and Xinyu You.
2022. "Optimization of Ramp Locations along Freeways: A Dynamic Programming Approach" *Sustainability* 14, no. 15: 9718.
https://doi.org/10.3390/su14159718