# A Bi-Level Programming Model for Optimal Bus Stop Spacing of a Bus Rapid Transit System

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## Abstract

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## 1. Introduction

## 2. Overview of Solutions

## 3. Model Development

#### 3.1. Top-Level Model Establishment

#### 3.1.1. Wait Cost

#### 3.1.2. In-Vehicle Cost

#### 3.1.3. Walk Cost

#### 3.1.4. Operator Cost

#### 3.2. Construction of a Bottom-Level Model

## 4. Case Study

#### 4.1. Model Verification

#### 4.2. Sensitivity Analysis

#### 4.2.1. Coverage Coefficient of Bus Stops

#### 4.2.2. BRT Headway

#### 4.2.3. Normal Speed

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Symbol | Definitions | Unit | Value |
---|---|---|---|

${f}_{1}$ | Equivalent cost (objective value of the top-level model) | ￥ | |

${C}_{u}$ | User cost | ￥ | |

${C}_{o}$ | Operator cost | ￥ | |

${C}_{1}$ | Wait cost | ￥ | |

${C}_{2}$ | In-vehicle cost | ￥ | |

${C}_{3}$ | Walk cost | ￥ | |

${\varphi}_{1}$ | Value of wait time | ￥/s | 0.2 |

${\varphi}_{2}$ | Value of in-vehicle time | ￥/s | 0.1 |

${\varphi}_{3}$ | Value of walk time | ￥/s | 0.2 |

${\varphi}_{4}$ | Average vehicle operating cost | ￥/(vehicle-s) | 0.5 |

${t}_{1}$ | Total wait time for all the passengers | s | |

${t}_{2}$ | Total in-vehicle time | s | |

${t}_{3}$ | Total walk time for all the passengers | s | |

${t}_{4}$ | Round-trip BRT travel time | s | |

t_{ad} | Time for the vehicle acceleration and deceleration | s | |

${t}_{s}$ | Dwell time at the stop station | s | |

${t}_{w}$ | Dwell time caused by traffic signals | s | |

${t}_{r}$ | Unimpeded running time of vehicles | s | |

$n$ | Total number of stops | stops | |

${q}_{i}$ | BRT passenger flow at stop $i$ | persons | |

${t}_{i}$ | Average wait time at stop $i$ | s | |

${q}_{s}$ | BRT demand attraction in relation to stop spacing | persons | |

$l$ | BRT route length | km | 28.3 |

$s$ | BRT stop spacing | m | |

${t}_{i}$ | Average wait time | s | |

$h$ | BRT headway | s | |

${n}_{c}$ | Number of signalized intersections which are located at a considerable distance from stops, and the spacing is at least at a distance of 50 m | intersections | |

$\rho $ | Possibility that vehicles adequately stop for red lights at intersections | 0.5 | |

$T$ | Average cycle time of signalized intersection | s | 80 |

$g$ | Average cycle time of signalized intersection | s | 40 |

${v}_{p}$ | Walk speed of passengers | m/s | 1.2 |

$a$ | Average value of acceleration | m/s^{2} | 0.6 |

$d$ | Average value of deceleration | m/s^{2} | 1 |

$v$ | Normal speed | m/s | |

${v}_{s}$ | Speed of stops | m/s | |

${v}_{\mathrm{max}}$ | Maximum recommended and permissible speed | m/s | 60 |

${x}_{a}$ | Distance travelled during acceleration of the BRT vehicle | m | |

${x}_{d}$ | Distance travelled during deceleration of the BRT vehicle | m | |

$\tau $ | Time required per person to alight or board a bus | s | 3 |

$k$ | Time for the opening and closing of doors | s | 3 |

${n}_{s}$ | Number of signalized intersections | intersections | 23 |

${f}_{2}(s)$ | The coverage coefficient of bus stops caused by stop spacing | ||

${q}_{0}$ | Potential bus traffic | persons | 320,000 |

${d}_{w}$ | Walking distance | m | |

$p({d}_{w})$ | Probability of passengers choosing a BRT | ||

${D}_{1}$ | Area within 300 m from a BRT stop | m^{2} | |

${D}_{2}$ | Area where the distance ranges between 300 and 700 m to a stop | m^{2} | |

$p$ | Probability of passengers in ${D}_{1}$ and ${D}_{2}$ choosing BRT | ||

${C}_{up}$ | Actual service area of the up-run direction of the line | m^{2} | |

${C}_{id}$ | Ideal stop service area | m^{2} | |

${D}_{1\mathrm{max}}$ | Maximum value of ${D}_{1}$ | m^{2} | |

${D}_{2\mathrm{max}}$ | Maximum value of ${D}_{2}$ | m^{2} | |

${a}_{0}$ | Non-repeated service area between adjacent stops | m^{2} | |

${a}_{1}$ | Repeat service areas between adjacent stops in an area where stop spacing is between 0 to 300 m | m^{2} | |

${a}_{2}$ | Repeat service areas between adjacent stops in an area where stop spacing is between 300 to 700 m | m^{2} | |

${s}_{\mathrm{min}}$ | Minimum value of stop spacing | m | |

${s}_{\mathrm{max}}$ | Maximum value of stop spacing | m | |

${s}_{p}$ | BRT stop spacing | m |

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**MDPI and ACS Style**

Cheng, G.; Zhao, S.; Zhang, T.
A Bi-Level Programming Model for Optimal Bus Stop Spacing of a Bus Rapid Transit System. *Mathematics* **2019**, *7*, 625.
https://doi.org/10.3390/math7070625

**AMA Style**

Cheng G, Zhao S, Zhang T.
A Bi-Level Programming Model for Optimal Bus Stop Spacing of a Bus Rapid Transit System. *Mathematics*. 2019; 7(7):625.
https://doi.org/10.3390/math7070625

**Chicago/Turabian Style**

Cheng, Gang, Shuzhi Zhao, and Tao Zhang.
2019. "A Bi-Level Programming Model for Optimal Bus Stop Spacing of a Bus Rapid Transit System" *Mathematics* 7, no. 7: 625.
https://doi.org/10.3390/math7070625