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Given the lower efficiency resulting from the overload of bus stops, the capacity and travel time of passengers influenced by skip-stop operation are analyzed under mixed traffic conditions, and the travel time models of buses and cars are developed, respectively. This paper proposes an optimization model for designing skip-stop service that can minimize the total travel time for passengers. Genetic algorithm is adopted for finding the optimal coordination of the stopping stations of overall bus lines in an urban bus corridor. In this paper, Tian-Mu-Shan Road of Hangzhou City is taken as an example. Results show that the total travel time of all travelers becomes 7.03 percent shorter after the implementation of skip-stop operation. The optimization scheme can improve the operating efficiency of the road examined.

In the recent years, with the rapid development of public transport, bus stops face an increasing pressure especially during peak hours, their efficiency decreases continuously, and even serious traffic congestion occurs frequently. The fundamental reason for the above problems is that bus demand exceeds the capacity of bus stops, resulting in some buses waiting in the travel lane until the buses occupying the berth entrance. This not only decreases the level of public transit service but also increases the impact between buses and cars at the location of bus stops on the road without exclusive bus lane. With skip-stop bus services that serve only a subset of stops along certain routes, this problem can be alleviated without technological improvement. For bus passengers, skip-stop services mean improved service levels in the form of lower travel time due to fewer stops and higher between-stop speed. When the skip-stop schedule is adopted by buses, it will reduce the impact of stopping buses on the cars at the location of bus stops, which will increase the car users’ travel speed. In actual practice, skip-stop services in systems such as Transmilenio (Bogota, Colombia) and Metro Rapid (Los Angeles, CA, USA) have been proven to be highly effective [

The literature has a considerable amount of work on the transit operation optimization. Eberlein [

In those studies, the optimization methods are developed inside the transit system, while ignoring the interaction of private and transit vehicle flows. However, there are some roads without exclusive bus lanes in the cities of China. Ignoring this will lead to inaccurate estimates of travel time. Moreover, most of the previous studies were done only for a congested bus line, and they lack the analysis of the overall bus lines that passed by the stops. With these arguments as motivation, this study proposes an optimization method for designing skip-stop services that can minimize the total travel time for passengers under mixed traffic conditions by analyzing the mutual influence between buses and cars, and it uses genetic algorithm to find the optimal coordination of the stop stations of bus lines based on the objective function.

The following notations are used in describing the models in this paper:

is the capacity of link;

is the basic traffic capacity;

is the adjustment factor for bus stops;

is the impacting time by bus stops, which is determined by the number of stopping buses at this stop;

is the probability of no bus being serviced;

is the average dwell time of buses;

is the average bus arrival rate;

is the average arrival rate of buses that stopped at this stop.

is the number of loading areas;

is the number of lanes in one direction,

is the average travel time of vehicle on link

is the free-flow travel time on link

is the volume of traffic on link

are parameters;

is the index of bus line,

is the set of all bus lines in the network;

is the set of all links in the network;

is the set of all stations stopped and skipped by bus line

is the bus volume or car volume on link

is the in-vehicle travel time on link

is the in-vehicle travel time on link

is the average number of passengers on bus line

is the average number of passengers on car;

is the total waiting time for passengers at stop

is the set of all links where bus line

is the number of passengers on buses of line

is the dwell time of buses of line

is the number of passengers boarding buses of line

is the number of passengers alighting buses of line

is the waiting time of passengers boarding buses of line

is the decision variables to indicate stop status of bus line

is the distance between station

is the schedule frequency of bus line

is the time of door opening and closing (s);

is passenger boarding time (s/p);

is passenger alighting time (s/p);

is the transfer penalty;

is the number of buses stopping at stop

is the capacity of stop

Capacity of on-line linear stops (buses/h).

Number of |
Dwell time (s) | ||
---|---|---|---|

15 | 30 | 45 | |

1 | 63 | 43 | 32 |

2 | 117 | 80 | 59 |

3 | 154 | 105 | 78 |

Assuming 15 s clearance time, 25 percent queue probability, 60 percent coefficient of variation of dwell times, and 0.5 g/C.

The travel time depends on the traffic volume and capacity of links, which is influenced by the number of stopping buses at a stop station under mixed traffic conditions. The BPR (Bureau of Public Roads) function, the most classic model, describes the link performance, which states the relationship between resistance and traffic volume [

Curbside bus stops interfere with traffic flows, as the buses stop in the travel lane, resulting in a “bottleneck” (the reduction in the road width) at the location of the stops. Bus bays interfere with passing vehicles primarily, while buses maneuver to pull into and out of the stops. In general, the capacity is calculated by adding adjustment factor for bus stops [

When all passing buses stop at this stop, the impacting time is determined by the numbers of loading areas and passengers getting on and off buses. If the bus stop is considered as a queuing system, the probability of no bus being serviced can be calculated by the following equation [

So the impacting time by bus stops is calculated by

When skip-stop service is adopted, there are two possible scenarios at stops in a skip-stop operation

The probability of the first scenario is determined by the number of buses stopped at this stop, which is computed by the following equation:

The impacting time in a skip-stop operation is approximately calculated using

Therefore, the capacity of links influenced by bus stops is calculated by

The classic BPR function is calibrated using the traffic data of highway. Therefore, the parameters need to be recalibrated when BPR function is used to describe the performance of urban roads. It is difficult to survey traffic operating data under various conditions. So the parameters of BPR function are calibrated using the simulated data in this paper.

Based on the paper’s objectives as well as the required details in the analysis, VISSIM (version 4.2) is employed in this paper. The simulation model is developed using VISSIM based on the surveyed data. For the purpose of model validation, the mean speed of values cars and buses observed on the field and simulated by VISSIM are compared. The comparison results of different types of vehicles are shown in Table

Comparison of the investigated values and simulation results.

Statistical parameters | Investigated data | Simulated data |
---|---|---|

Sample size | 390 | 683 |

Mean of speed | 22.3 | 23.1 |

Variance of speed | 21.3 | 34.3 |

A paired

The BPR function is expressed as follows [

In order to improve the reliability of VISSIM model, the simulation runs are made with random number seeds ranging from 41 to 45, and the average of the five values is taken as the final model output. The travel time outputted from the model is as shown in Figure

Relationship between the travel time of car and

In this paper, BPR model is calibrated by the method of Least-squares. First, BPR model should be deformed as a linear function by logarithmic transformation [

Denoting

Second, these above data are transformed to

As indicated in the previous studies, the skip-stop problem can be formulated as a nonlinear 0-1 integer programming problem, with the binary integer variables representing which stops to be skipped by the control vehicles. In this study, for the purpose of solving the exact problem, the stop-skipping problem will be formulated again as a nonlinear 0-1 integer programming problem.

In order to analyze the process, the following assumptions are made in this study.

The total travel demand between any origin and destination node pair (O-D matrix) is fixed and remains the same during the analysis period. It is assumed that the travel demand is not affected by the introduction of skip-stop operation.

There are only two modes of traffic (buses and cars) utilizing the network.

Dwell time at each stop is determined by the number of passengers getting on and off the bus.

Passengers are uniformly distributed throughout the area.

A passenger would not leave the platform, and she or he would not wait more than two times.

Run time models are usually used in understanding the existing service and evaluating several transit planning and operation strategies [

The first term in the objective function is the total travel time by buses, which includes in-vehicle travel time and total waiting time. The next term represents the total travel time by cars. Equation (

The proposed model is a nonlinear programming problem which associates the zero-one variables, and the parameters are tightly related to each other. It is hardly solved with conventional solution methods. The genetic algorithm (GA) is a heuristic search method that imitates the process of natural evolution [

Overall procedure for finding an optimal scenario.

For the genetic algorithm, each gene location in a chromosome represents a possible skip-stop choice, while the vehicles of bus line traverse every station shown in Figure

The coding illustration of a chromosome for the genetic algorithm.

This paper takes Tian-Mu-Shan Road (from Wan-Tang Road to Zhong-Shan Road) as an example, which is a heavy-demand corridor of Hangzhou City. The examined period is the morning peak. The traffic volume and the number of passengers getting on and off at 11 stops are collected, as shown in Tables

Traffic data on Tian-Mu-Shan Road.

No. | Bus stop | Car volume (veh/h) | Distance between adjacent stops (m) |
---|---|---|---|

1 | Gucui Intersection | 930 | — |

2 | Xueyuan Intersection | 975 | 280 |

3 | Qingfeng Village, West | 893 | 522 |

4 | Qingfeng Village, East | 920 | 395 |

5 | Xixi District | 823 | 345 |

6 | Bazi Bridge | 760 | 550 |

7 | Macheng Intersection | 726 | 270 |

8 | City Government | 620 | 510 |

9 | Hushu Intersection | 723 | 280 |

10 | Hangzhou Building | 770 | 420 |

11 | Zhongshan Intersection | 780 | 410 |

Numbers of passengers boarding and alighting at stops (from west to east).

Line | Stop | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |

1 | 19/14 | 14/45 | 24/41 | 27/63 | 42/56 | 36/33 | 16/22 | 49/65 | 21/34 | 28/58 | 16/22 |

2 | — | — | — | — | — | — | — | — | 18/57 | 18/49 | 21/34 |

3 | 26/18 | 6/23 | 14/26 | 44/58 | 40/48 | 26/57 | 44/36 | — | — | — | — |

4 | 13/8 | 22/18 | 28/33 | 26/63 | 28/23 | 14/41 | 20/34 | 54/38 | 24/33 | 17/55 | 19/27 |

5 | 18/20 | 10/28 | 8/28 | 25/87 | 26/34 | 37/38 | 32/43 | — | — | — | — |

6 | — | — | — | — | — | — | — | — | 17/51 | 20/77 | 36/45 |

7 | 17/37 | 8/32 | 6/46 | 14/60 | 34/50 | 23/45 | 10/22 | 58/68 | 31/36 | 7/40 | 24/36 |

8 | 22/12 | 12/16 | 3/34 | 26/47 | 29/64 | — | — | — | — | — | — |

9 | 29/13 | 6/22 | 5/52 | 17/56 | 38/35 | — | — | — | — | — | — |

10 | 12/21 | 12/34 | 26/26 | 40/51 | 22/52 | 49/44 | — | — | — | — | — |

11 | — | — | — | — | — | — | — | — | 29/39 | 16/62 | 35/24 |

12 | 21/22 | 13/25 | 21/76 | 40/62 | 57/38 | 27/67 | 68/47 | — | — | — | — |

13 | 34/14 | 8/27 | 8/38 | 32/40 | 18/41 | 50/76 | — | — | — | — | — |

14 | 17/18 | 10/16 | 12/14 | 14/58 | — | — | — | — | — | — | — |

15 | 24/26 | 8/28 | 6/36 | 11/44 | — | — | — | — | — | — | — |

16 | 16/19 | 6/46 | 19/40 | 44/61 | 64/69 | 44/57 | 30/26 | 34/56 | 10/28 | 25/57 | 12/34 |

17 | — | — | — | — | — | — | — | — | 13/55 | 10/65 | 35/62 |

18 | — | — | — | — | — | — | — | — | — | 52/82 | 46/40 |

19 | — | — | — | — | — | — | — | — | — | 37/75 | 36/66 |

Parameters values of the model and the algorithm are shown as follows: the population of chromosomes (individuals) for each generation ^{2}, and

The optimal scenario of skip-stop operation.

Line | Stop | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |

1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |

2 | — | — | — | — | — | — | — | — | 1 | 1 | 0 |

3 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | — | — | — | — |

4 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |

5 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | — | — | — | — |

6 | — | — | — | — | — | — | — | — | 1 | 1 | 1 |

7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |

8 | 1 | 0 | 1 | 0 | 1 | — | — | — | — | — | — |

9 | 0 | 1 | 1 | 1 | 1 | — | — | — | — | — | — |

10 | 1 | 1 | 0 | 1 | 1 | 1 | — | — | — | — | — |

11 | — | — | — | — | — | — | — | — | 1 | 1 | 0 |

12 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | — | — | — | — |

13 | 0 | 1 | 1 | 0 | 1 | 1 | — | — | — | — | — |

14 | 1 | 0 | 1 | 1 | — | — | — | — | — | — | — |

15 | 1 | 1 | 0 | 1 | — | — | — | — | — | — | — |

16 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |

17 | — | — | — | — | — | — | — | — | 1 | 1 | 1 |

18 | — | — | — | — | — | — | — | — | — | 1 | 1 |

19 | — | — | — | — | — | — | — | — | — | 1 | 1 |

Note that “1” represents stopping, “0” represents skipping, and “—” represents not passing by.

The diagram of skip-stop operation scenario. Note that “

The travel times of all travelers before and after the optimization are compared, as shown in Table

The comparison between all-stop and skip-stop operations.

Scenario | Car travelers | Bus travelers | Total travel time (min) | ||
---|---|---|---|---|---|

Travel time (min) | Out-off-vehicle time (min) | In-vehicle time (min) | Travel time (min) | ||

Original scenario | 34,966 | 8,505 | 296,553 | 305,058 | 340,024 |

Optimal scenario | 30,471 | 12,228 | 273,427 | 285,655 | 316,126 |

Change | −12.86% | 43.77% | −7.80% | −6.36% | −7.03% |

The flow capacity is adjusted at the section of stop zone based on the analysis of the mutual influence between buses and cars at bus stops. BPR function including the modified capacity is calibrated using traffic simulation. Then, this paper proposes an optimization model for designing skip-stop service that can minimize the total travel time of passengers under mixed traffic conditions. The genetic algorithm is used to find the optimal coordination of the stopping stations of overall bus lines in an urban bus corridor. The validation of the model and the algorithm have been proved with the help of a real-world case. Results show that the operational efficiency of buses and cars is improved with skip-stop operation under mixed traffic conditions. Although skip-stop operation is complicated for the operators and it can confuse passengers at the beginning of the service, it can certainly reduce passengers’ total travel time. It should be noted that the dwell time of a bus is assumed to be based on the number of passengers boarding and alighting, as a matter of fact, which is also influenced by the number of stopping buses at the stop. Therefore, an interesting project for future research would be to propose an optimization method for designing skip-stop services with the consideration of the bus operating features at stops.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (no. 51278454).