# Offset Optimization Model for Signalized Intersections Considering the Optimal Location Planning of Bus Stops

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## Abstract

**:**

## 1. Introduction

## 2. Optimization Model

#### 2.1. Notations Description

#### 2.2. Decision Variables

#### 2.3. Objective Function

#### 2.4. Problem Constraints

#### 2.4.1. Offset Constraints

#### 2.4.2. Prediction of BRT Delays

- (i)
- The delay characteristics of BRT

- (ii)
- BRT delay at the starting intersection

- (iii)
- BRT delay at the other intersections

#### 2.4.3. Green Wave Bandwidth of Cars

## 3. Solving the Model

#### 3.1. Linearization of the Rounding Function

#### 3.2. Linearization of the Piecewise Functions

## 4. Case Study

#### 4.1. Parameter Input

^{®}Core

^{™}i5-11300H @ 3.10 GHz 2.61 GHz, 16 GB RAM. The average solving time is 8.03 s.

#### 4.2. Comparison and Analysis

#### 4.2.1. Scheme Comparison

#### 4.2.2. Results Analysis

- (i)
- Analysis of BRT delays

- (ii)
- Bandwidth analysis of cars

## 5. Sensitivity Analysis

#### 5.1. Weight Coefficient

#### 5.2. BRT-Vehicle Speed

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Sets | Descriptions |

$I$ | The set of all intersections, $I=\left\{1,2,\cdots ,n\right\},\mathrm{where}$ $n$ denotes the number of intersections on the artery. |

${M}_{1}$ | The set of all BRT vehicles in the outbound, ${M}_{1}=\left\{1,2,\cdots ,{N}_{1}\right\}$, where ${N}_{1}$ denotes the number of BRT vehicles passing the outbound of the artery during the study period. |

${M}_{2}$ | The set of all BRT vehicles in the inbound, ${M}_{2}=\left\{1,2,\cdots ,{N}_{2}\right\}$, where ${N}_{2}$ denotes the number of BRT vehicles passing the inbound of the artery during the study period. |

$M$ | The set of all the two-way BRT vehicles, $M=\left\{{M}_{1}\cup {M}_{2}\right\}$. |

$D$ | The set of the traveling directions, $D=\left\{1,2\right\}$, where $1$ denotes the outbound and $2$ denotes the inbound. |

Parameters | Descriptions |

$C$ | Public cycle length of intersections ($\mathrm{s}$). |

${r}_{id}$ | Red time of intersection $i$ in coordinated direction $d$ ($\mathrm{s}$), $i\in I,d\in D$. |

${L}_{i-1,i,d}$ | Intersection spacing, that is, the distance between intersection $i-1$ and intersection $i$ in direction $d$ ($\mathrm{m}$),$i\in I,$ $d\in D$. |

${v}_{car}$ | Average speed of cars ($\mathrm{m}/\mathrm{s}$). |

${v}_{bus}$ | Average speed of BRT ($\mathrm{m}/\mathrm{s}$). |

${\sigma}_{id}$ | Dwelling time of BRT at intersection $i$ in direction $d$ ($\mathrm{s}$),$i\in I,d\in D$. |

${t}_{i-1,i,d}$ | Travel time of cars from intersection $i-1$ to intersection $i$ in direction $d$ ($\mathrm{s}$), $i\in I,d\in D$. |

$\alpha $ | The weight coefficient in green wave bandwidth of cars, $\alpha \in \left[0,1\right]$. |

${\tau}_{md}$ | The entering time of the BRT vehicle $m$, which denotes the time that the vehicle enters the research area ($\mathrm{s}$), $m\in M,d\in D$. |

$\rho $ | $\rho $ is a weight coefficient in the objective function, $\rho \in \left[0,1\right]$. |

Variables | Descriptions |

${b}_{d}$ | The green wave bandwidth of cars in direction $d$ ($\mathrm{s}$), $d\in D$. |

$B$ | Two-way total green wave bandwidth of cars ($\mathrm{s}$). |

$\Delta {t}_{1d}$ | Total dwelling time at all stops of BRT on the road section from the starting point of the artery to the starting intersection (the first intersection) in direction $d$($\mathrm{s}$), $d\in D$. |

${t}_{i-1,i,d}$ | Total dwelling time at all stops of BRT on the road section from intersection $i-1$ to intersection $i$ in direction $d$($\mathrm{s}$), $i\in I,d\in D$. |

${T}_{i,m,d}$ | Arrival time of BRT $m$ at intersection $i$ in direction $d$ ($\mathrm{s}$), $i\in I,m\in M,d\in D$. |

${U}_{i-1,i,d}$ | Travel time of BRT from intersection $i-1$ to intersection $i$ in direction $d$ ($\mathrm{s}$), $i\in I,d\in D$. |

${w}_{id}$ | ${w}_{i1}$ denotes the time interval from the left side of the green wave bandwidth of the car to the red light end time at the intersection $i$ in the outbound, and ${w}_{i2}$ denotes the time interval from the right side of the green wave bandwidth of the car to the red light start time at the intersection $i$ in the inbound ($s$), $i\in I,d\in D$. |

${\phi}_{i-1,i,d}$ | ${\phi}_{i-1,i,1}$ is the time interval from the intersection $i-1$ on the left side of the green wave bandwidth of cars to the red light start time at the intersection $i$ in the outbound ($\mathrm{s}$). ${\phi}_{i-1,i,2}$ is the time interval from the intersection $i-1$ on the right side of the green wave bandwidth of cars to the red light end time at the intersection $i$ in the inbound ($\mathrm{s}$), $i\in I,d\in D$. |

${O}_{i-1,i,d}$ | Relative offset of intersection $i$ relative to intersection $i-1$ in direction $d$ ($\mathrm{s}$), $i\in I,d\in D$. |

${\theta}_{id}$ | Absolute offset of the intersection $i$ in direction $d$($\mathrm{s}$), selecting the first intersection in direction $d$ as the reference intersection, $i\in I,d\in D$. |

${\delta}_{id}$ | Location planning of BRT stops at the intersection $i$ in direction $d.$ ${\delta}_{id}$ is the binary variable, the value of “1” indicates that stop is arranged upstream of the intersection, and the value of “0” indicates that stop is arranged downstream of the intersection, $i\in I,d\in D$. |

${D}_{i,m,d}$ | Signal delay of BRT $m$ at intersection $i$ in direction $d$ ($\mathrm{s}$), $i\in I,m\in M,d\in D$. |

${D}_{a}$ | Average delay of BRT on the artery during the study period ($\mathrm{s}$). |

Entering Order | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Entering time | 7:12 | 7:24 | 7:36 | 7:48 | 8:00 |

Basic Parameters | |||||

$\mathrm{BRT}$ speed ${v}_{bus}$($\mathrm{m}/\mathrm{s}$) | Car speed ${v}_{car}$($\mathrm{m}/\mathrm{s}$) | Cycle $C\left(\mathrm{s}\right)$ | Dwell time of BRT at stops ${\sigma}_{id}\left(\mathrm{s}\right)$ | Weight coefficient in objective function $\rho $ | Weight Coefficient of Bandwidth $\alpha $ |

11 | 15 | 150 | 26 | 0.5 | 0.45 |

Intersection spacing and signal timing parameters | |||||

Intersections | Outbound | Inbound | |||

Intersection spacing $\left(\mathrm{m}\right)$ | Red time $\left(\mathrm{s}\right)$ | Intersections | Intersection spacing $\left(\mathrm{m}\right)$ | Red time $\left(\mathrm{s}\right)$ | |

Beiyuan Street | 220 | 95 | Jiefang Road | 220 | 90 |

Huangtai Road | 671 | 75 | South Shanda Road | 698 | 91 |

Huayuan Road | 354 | 103 | Lilongzhuang Road | 376 | 76 |

Lilongzhuang Road | 698 | 76 | Huayuan Road | 698 | 103 |

South Shanda Road | 376 | 91 | Huangtai Road | 354 | 75 |

Jiefang Road | 698 | 90 | Beiyuan Street | 671 | 95 |

Schemes | Direction | Location Planning of BRT Stops | |||||
---|---|---|---|---|---|---|---|

Beiyuan Street | Huangtai Road | Huayuan Road | Lilongzhuang Road | South Shanda Road | Jiefang Road | ||

Proposed scheme | Outbound | 0 | 0 | 1 | 0 | 1 | 0 |

Inbound | 0 | 0 | 1 | 0 | 0 | 0 | |

Scheme 1 | Outbound | 1 | 1 | 0 | 1 | 1 | 0 |

Inbound | 0 | 1 | 0 | 1 | 0 | 0 | |

Scheme 2 | Outbound | 1 | 1 | 0 | 1 | 1 | 1 |

Inbound | 1 | 0 | 0 | 0 | 0 | 1 | |

Scheme 3 | Outbound | 1 | 1 | 0 | 1 | 1 | 0 |

Inbound | 0 | 0 | 1 | 0 | 1 | 0 | |

Scheme 4 | Outbound | 1 | 1 | 1 | 1 | 1 | 1 |

Inbound | 1 | 1 | 1 | 1 | 1 | 1 | |

Scheme 5 | Outbound | 1 | 1 | 1 | 1 | 1 | 1 |

Inbound | 1 | 1 | 1 | 1 | 1 | 1 | |

Scheme 6 | Outbound | 0 | 0 | 0 | 0 | 0 | 0 |

Inbound | 0 | 0 | 0 | 0 | 0 | 0 | |

Scheme 7 | Outbound | 0 | 0 | 0 | 0 | 0 | 0 |

Inbound | 0 | 0 | 0 | 0 | 0 | 0 | |

Schemes | Direction | Offset of intersections (s) | |||||

Beiyuan Street | Huangtai Road | Huayuan Road | Lilongzhuang Road | South Shanda Road | Jiefang Road | ||

Proposed scheme | Outbound | 0.00 | 97.54 | 58.18 | 145.73 | 10.82 | 74.27 |

Inbound | 75.73 | 23.27 | 133.91 | 71.45 | 86.55 | 0.00 | |

Scheme 1 | Outbound | 0 | 44 | 66 | 78 | 14 | 114 |

Inbound | 36 | 80 | 102 | 114 | 50 | 0 | |

Scheme 2 | Outbound | 0 | 44 | 66 | 78 | 14 | 114 |

Inbound | 36 | 80 | 102 | 114 | 50 | 0 | |

Scheme 3 | Outbound | 0 | 75.2 | 75.2 | 136.8 | 15.6 | 90.1 |

Inbound | 59.9 | 135.1 | 135.1 | 46.7 | 75.5 | 0 | |

Scheme 4 | Outbound | 0 | 44 | 66 | 78 | 14 | 114 |

Inbound | 36 | 80 | 102 | 114 | 50 | 0 | |

Scheme 5 | Outbound | 0 | 91.9 | 51.8 | 139.4 | 14.1 | 68.7 |

Inbound | 81.3 | 23.3 | 133.1 | 70.6 | 95.4 | 0.0 | |

Scheme 6 | Outbound | 0 | 44 | 66 | 78 | 14 | 114 |

Inbound | 36 | 80 | 102 | 114 | 50 | 0 | |

Scheme 7 | Outbound | 0.0 | 91.0 | 50.9 | 139.4 | 6.2 | 67.8 |

Inbound | 82.2 | 23.3 | 133.1 | 71.6 | 88.5 | 0.0 |

Schemes | Entering Time | Direction | Delay of BRT at the Intersections (s) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Beiyuan Street | Huangtai Road | Huayuan Road | Lilongzhuang Road | South Shanda Road | Jiefang Road | Total One-Way Delay | Total Two-Way Delay | |||

Proposed scheme | 7:12 | Outbound | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Inbound | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |||

7:24 | Outbound | 0.0 | 0.0 | 30.0 | 0.0 | 0.0 | 0.0 | 30.0 | 41.8 | |

Inbound | 0.0 | 0.0 | 0.0 | 11.8 | 0.0 | 0.0 | 11.8 | |||

7:36 | Outbound | 15.0 | 0.0 | 45.0 | 0.0 | 0.0 | 0.0 | 60.0 | 101.8 | |

Inbound | 0.0 | 0.0 | 0.0 | 30.5 | 1.3 | 10.0 | 41.8 | |||

7:48 | Outbound | 45.0 | 0.0 | 45.0 | 0.0 | 0.0 | 0.0 | 90.0 | 161.8 | |

Inbound | 0.0 | 0.0 | 0.0 | 30.5 | 1.3 | 40.0 | 71.8 | |||

8:00 | Outbound | 75.0 | 0.0 | 45.0 | 0.0 | 0.0 | 0.0 | 120.0 | 221.8 | |

Inbound | 0.0 | 0.0 | 0.0 | 30.5 | 1.3 | 70.0 | 101.8 | |||

Scheme 1 | 7:12 | Outbound | 79.0 | 0.0 | 0.0 | 0.0 | 15.2 | 35.5 | 129.7 | 336.5 |

Inbound | 39.0 | 67.8 | 49.5 | 14.8 | 35.5 | 0.0 | 206.7 | |||

7:24 | Outbound | 0.0 | 46.0 | 17.8 | 19.5 | 40.8 | 35.5 | 159.7 | 396.5 | |

Inbound | 39.0 | 67.8 | 49.5 | 14.8 | 65.5 | 0.0 | 236.7 | |||

7:36 | Outbound | 0.0 | 0.0 | 93.8 | 19.5 | 40.8 | 35.5 | 189.7 | 456.5 | |

Inbound | 39.0 | 67.8 | 49.5 | 14.8 | 85.5 | 10.0 | 266.7 | |||

7:48 | Outbound | 19.0 | 0.0 | 0.0 | 0.0 | 15.2 | 35.5 | 69.7 | 366.5 | |

Inbound | 39.0 | 67.8 | 49.5 | 14.8 | 85.5 | 40.0 | 296.7 | |||

8:00 | Outbound | 49.0 | 0.0 | 0.0 | 0.0 | 15.2 | 35.5 | 99.7 | 426.5 | |

Inbound | 39.0 | 67.8 | 49.5 | 14.8 | 85.5 | 70.0 | 326.7 | |||

Scheme 2 | 7:12 | Outbound | 79.0 | 0.0 | 0.0 | 0.0 | 15.2 | 9.5 | 103.7 | 284.4 |

Inbound | 13.0 | 41.8 | 51.9 | 0.0 | 0.0 | 74.0 | 180.7 | |||

7:24 | Outbound | 0.0 | 46.0 | 17.8 | 19.5 | 40.8 | 9.5 | 133.6 | 194.3 | |

Inbound | 13.0 | 41.8 | 5.9 | 0.0 | 0.0 | 0.0 | 60.7 | |||

7:36 | Outbound | 0.0 | 0.0 | 93.8 | 19.5 | 40.8 | 9.5 | 163.6 | 254.3 | |

Inbound | 13.0 | 41.8 | 35.9 | 0.0 | 0.0 | 0.0 | 90.7 | |||

7:48 | Outbound | 19.0 | 0.0 | 0.0 | 0.0 | 15.2 | 9.5 | 43.7 | 164.4 | |

Inbound | 13.0 | 41.8 | 51.9 | 0.0 | 0.0 | 14.0 | 120.7 | |||

8:00 | Outbound | 49 | 0.0 | 0.0 | 0.0 | 15.2 | 9.5 | 73.7 | 224.5 | |

Inbound | 13.0 | 41.8 | 51.9 | 0.0 | 0.0 | 44.0 | 150.7 | |||

Scheme 3 | 7:12 | Outbound | 0.0 | 0.0 | 18.7 | 0.0 | 0.0 | 64.8 | 83.5 | 91.5 |

Inbound | 0.0 | 0.0 | 0.0 | 0.0 | 8.0 | 0.0 | 8.0 | |||

7:24 | Outbound | 0.0 | 0.0 | 48.7 | 0.0 | 0.0 | 64.8 | 113.5 | 151.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 38.0 | 0.0 | 0.0 | 38.0 | |||

7:36 | Outbound | 15.0 | 0.0 | 63.7 | 0.0 | 0.0 | 64.8 | 143.5 | 211.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 58.0 | 0.0 | 10.0 | 68.0 | |||

7:48 | Outbound | 45.0 | 0.0 | 63.7 | 0.0 | 0.0 | 64.8 | 173.5 | 271.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 58.0 | 0.0 | 40.0 | 98 | |||

8:00 | Outbound | 75.0 | 0.0 | 63.7 | 0.0 | 0.0 | 64.8 | 203.5 | 331.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 58.0 | 0.0 | 70.0 | 128 | |||

Scheme 4 | 7:12 | Outbound | 79.0 | 0.0 | 78.8 | 45.5 | 40.8 | 9.5 | 253.7 | 434.5 |

Inbound | 39.0 | 41.8 | 25.9 | 0.0 | 0.0 | 74.0 | 180.7 | |||

7:24 | Outbound | 0.0 | 46.0 | 0.0 | 37.4 | 40.8 | 9.5 | 133.7 | 344.5 | |

Inbound | 39.0 | 41.8 | 64.4 | 0.0 | 65.5 | 0.0 | 210.7 | |||

7:36 | Outbound | 0.0 | 0.0 | 67.8 | 45.5 | 40.8 | 9.5 | 163.7 | 254.5 | |

Inbound | 39.0 | 41.8 | 9.9 | 0.0 | 0.0 | 0.0 | 90.7 | |||

7:48 | Outbound | 19.0 | 0.0 | 78.8 | 45.5 | 40.8 | 9.5 | 193.7 | 314.5 | |

Inbound | 39.0 | 41.8 | 25.9 | 0.0 | 0.0 | 14.0 | 120.7 | |||

8:00 | Outbound | 49.0 | 0.0 | 78.8 | 45.5 | 40.8 | 9.5 | 223.7 | 374.5 | |

Inbound | 39.0 | 41.8 | 25.9 | 0.0 | 0.0 | 44.0 | 150.7 | |||

Scheme 5 | 7:12 | Outbound | 79.0 | 0.0 | 64.6 | 0.0 | 0.0 | 64.8 | 207.5 | 339.5 |

Inbound | 0.0 | 0.0 | 0.0 | 50.1 | 6.9 | 74.0 | 132.0 | |||

7:24 | Outbound | 0.0 | 0.0 | 23.6 | 0.0 | 0.0 | 64.8 | 87.5 | 99.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 12.0 | 0.0 | 0.0 | 12.0 | |||

7:36 | Outbound | 0.0 | 0.0 | 53.6 | 0.0 | 0.0 | 64.8 | 118.4 | 159.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 41.1 | 0.0 | 0.0 | 41.1 | |||

7:48 | Outbound | 19.0 | 0.0 | 64.6 | 0.0 | 0.0 | 64.8 | 148.4 | 219.4 | |

Inbound | 0.0 | 0.0 | 0.0 | 50.1 | 6.9 | 14.0 | 71 | |||

8:00 | Outbound | 49.0 | 0.0 | 64.6 | 0.0 | 0.0 | 64.8 | 178.4 | 279.4 | |

Inbound | 0.0 | 0.0 | 0.0 | 50.1 | 6.9 | 44.0 | 101.0 | |||

Scheme 6 | 7:12 | Outbound | 0.0 | 42.0 | 0.0 | 37.4 | 40.8 | 9.5 | 129.7 | 336.5 |

Inbound | 39.0 | 41.8 | 64.4 | 0.0 | 61.5 | 0.0 | 206.7 | |||

7:24 | Outbound | 0.0 | 72.0 | 0.0 | 37.4 | 40.8 | 9.5 | 159.7 | 246.5 | |

Inbound | 39.0 | 41.8 | 0.0 | 0.0 | 5.9 | 0.0 | 86.7 | |||

7:36 | Outbound | 15.0 | 0.0 | 78.8 | 45.5 | 40.8 | 9.5 | 189.7 | 306.5 | |

Inbound | 39.0 | 41.8 | 25.9 | 0.0 | 0.0 | 10.0 | 116.7 | |||

7:48 | Outbound | 45.0 | 0.0 | 78.8 | 45.5 | 40.8 | 9.5 | 219.7 | 366.5 | |

Inbound | 39.0 | 41.8 | 25.9 | 0.0 | 0.0 | 40.0 | 146.7 | |||

8:00 | Outbound | 75.0 | 0.0 | 78.8 | 45.5 | 40.8 | 9.5 | 249.7 | 426.5 | |

Inbound | 39.0 | 41.8 | 25.9 | 0.0 | 0.0 | 70.0 | 176.7 | |||

Scheme 7 | 7:12 | Outbound | 0.0 | 0.0 | 18.7 | 0.0 | 0.0 | 64.8 | 83.5 | 91.5 |

Inbound | 0.0 | 0.0 | 0.0 | 8.0 | 0.0 | 0.0 | 8.0 | |||

7:24 | Outbound | 0.0 | 0.0 | 48.7 | 0.0 | 0.0 | 64.8 | 113.5 | 151.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 38.0 | 0.0 | 0.0 | 38.0 | |||

7:36 | Outbound | 15.0 | 0.0 | 63.7 | 0.0 | 0.0 | 64.8 | 143.5 | 211.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 58.0 | 0.0 | 10.0 | 68.0 | |||

7:48 | Outbound | 45.0 | 0.0 | 63.7 | 0.0 | 0.0 | 64.8 | 173.5 | 271.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 58.0 | 0.0 | 40.0 | 98.0 | |||

8:00 | Outbound | 75.0 | 0.0 | 63.7 | 0.0 | 0.0 | 64.8 | 203.5 | 331.5 | |

Inbound | 0.0 | 0.0 | 0.0 | 58.0 | 0.0 | 70.0 | 128 |

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## Share and Cite

**MDPI and ACS Style**

Wu, W.; Luo, X.; Shi, B.
Offset Optimization Model for Signalized Intersections Considering the Optimal Location Planning of Bus Stops. *Systems* **2023**, *11*, 366.
https://doi.org/10.3390/systems11070366

**AMA Style**

Wu W, Luo X, Shi B.
Offset Optimization Model for Signalized Intersections Considering the Optimal Location Planning of Bus Stops. *Systems*. 2023; 11(7):366.
https://doi.org/10.3390/systems11070366

**Chicago/Turabian Style**

Wu, Wei, Xiaoyu Luo, and Baiying Shi.
2023. "Offset Optimization Model for Signalized Intersections Considering the Optimal Location Planning of Bus Stops" *Systems* 11, no. 7: 366.
https://doi.org/10.3390/systems11070366