Offset Optimization Model for Signalized Intersections Considering the Optimal Location Planning of Bus Stops
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
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 |
The set of all intersections, denotes the number of intersections on the artery. | |
The set of all BRT vehicles in the outbound, , where denotes the number of BRT vehicles passing the outbound of the artery during the study period. | |
The set of all BRT vehicles in the inbound, , where denotes the number of BRT vehicles passing the inbound of the artery during the study period. | |
The set of all the two-way BRT vehicles, . | |
The set of the traveling directions, , where denotes the outbound and denotes the inbound. | |
Parameters | Descriptions |
Public cycle length of intersections (). | |
Red time of intersection in coordinated direction (), . | |
Intersection spacing, that is, the distance between intersection and intersection in direction (), . | |
Average speed of cars (). | |
Average speed of BRT (). | |
Dwelling time of BRT at intersection in direction (),. | |
Travel time of cars from intersection to intersection in direction (), . | |
The weight coefficient in green wave bandwidth of cars, . | |
The entering time of the BRT vehicle , which denotes the time that the vehicle enters the research area (), . | |
is a weight coefficient in the objective function, . | |
Variables | Descriptions |
The green wave bandwidth of cars in direction (), . | |
Two-way total green wave bandwidth of cars (). | |
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 (), . | |
Total dwelling time at all stops of BRT on the road section from intersection to intersection in direction (), . | |
Arrival time of BRT at intersection in direction (), . | |
Travel time of BRT from intersection to intersection in direction (), . | |
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 in the outbound, and 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 in the inbound (), . | |
is the time interval from the intersection on the left side of the green wave bandwidth of cars to the red light start time at the intersection in the outbound (). is the time interval from the intersection on the right side of the green wave bandwidth of cars to the red light end time at the intersection in the inbound (), . | |
Relative offset of intersection relative to intersection in direction (), . | |
Absolute offset of the intersection in direction (), selecting the first intersection in direction as the reference intersection, . | |
Location planning of BRT stops at the intersection in direction 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, . | |
Signal delay of BRT at intersection in direction (), . | |
Average delay of BRT on the artery during the study period (). |
Entering Order | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Entering time | 7:12 | 7:24 | 7:36 | 7:48 | 8:00 |
Basic Parameters | |||||
speed () | Car speed () | Cycle | Dwell time of BRT at stops | Weight coefficient in objective function | Weight Coefficient of Bandwidth |
11 | 15 | 150 | 26 | 0.5 | 0.45 |
Intersection spacing and signal timing parameters | |||||
Intersections | Outbound | Inbound | |||
Intersection spacing | Red time | Intersections | Intersection spacing | Red time | |
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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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
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 StyleWu, 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