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Although transit stop location problem has been extensively studied, the two main categories of modeling methodologies, i.e., discrete models and continuum approximation (CA) ones, seem have little intersection. Both have strengths and weaknesses, respectively. This study intends to integrate them by taking the advantage of CA models’ parsimonious property and discrete models’ fine consideration of practical conditions. In doing so, we first employ the state-of-the-art CA models to yield the optimal design, which serves as the input to the next discrete model. Then, the stop location problem is formulated into a multivariable nonlinear minimization problem with a given number of stop location variables and location constraint. The interior-point algorithm is presented to find the optimal design that is ready for implementation. In numerical studies, the proposed model is applied to a variety of scenarios with respect to demand levels, spatial heterogeneity, and route length. The results demonstrate the consistent advantage of the proposed model in all scenarios as against its counterparts, i.e., two existing recipes that convert CA model-based solution into real design of stop locations. Lastly, a case study is presented using real data and practical constraints for the adjustment of a bus route in Chengdu (China). System cost saving of 15.79% is observed by before-and-after comparison.

Transit route design problem can be divided into two categories: transit network design and single transit route design [

The transit stop location problem has been extensively studied in the literature. Methodologically, two categories can be identified: discrete models and continuum approximation (CA) models. A majority of studies belong to the discrete-method category. For instance, Vuchic and Newell and Vuchic [

In the second category, CA models had been developed as an alternative option in locating transit stops. Instead of based on dozens of location variables, these models were built upon a single stop density/spacing variable or function. This parsimonious property endows the CA model with the high-efficient finding of the global optimum solution, or sometimes the closed-form solution. The first endeavor in this vein was made by Newell [

The CA models, however, have been criticized being too idealized with unrealistic assumptions, such as a continuous space for locating stops at anywhere along the route. Thus, it is recognized that the designs offered by CA models are not ready for implementation. Endeavors had been made to enhance the applicability of CA models. In Wirasinghe and Ghoneim and Medina et al. [

This paper intends to fill the gap. We propose an optimization framework that integrates CA models with discrete ones for locating bus stops with respect to location constraint. The idealized design of the CA model serves as input to the discrete model, which accordingly defines a given number of stop location variables and formulates the location constraint. The corresponding problem is a nonlinear multivariate optimization problem. A heuristic solution algorithm is presented to find the optimal solution. To the best of our knowledge, this is the first work connecting CA and discrete models so as to furnish implementation-ready transit route designs.

The remainder of the paper is organized as follows. The next section introduces the existing CA and discretization models. After that, a novel optimization model is proposed for locating bus stops. In Section

Section

Notation.

Variables | Unit | Descriptions |

Number of operation time periods | ||

km | Bus route length | |

Number of stops | ||

RMB/day | Total generalized system cost | |

RMB | Patrons’ access/egress time cost at | |

RMB | Patrons’ waiting time cost at | |

RMB | Patrons’ in-vehicle travel time cost at | |

RMB | Agency’s distance-based cost during period | |

RMB | Agency’s time-based cost at location | |

RMB | Agency’s amortized infrastructure cost at | |

RMB | Patrons’ access/egress time cost during period | |

RMB | Patrons’ in-vehicle travel time cost during period | |

RMB | Agency’s distance-based cost during period | |

RMB | Agency’s time-based cost during period | |

RMB | Agency’s amortized infrastructure cost during period | |

h | Duration of period | |

pax/km/h | Boarding and alighting demand density at | |

pax | Onboard flow passing stop | |

km | Location of stop | |

The restricted locations to be avoided from being stop locations | ||

km | Left and right coverage boundaries of stop | |

pax/h | Boarding and alighting volumes at stop | |

h | Headway during period | |

pax/h | Onboard flow at | |

Stop/km | Stop density at location | |

Subscript representing time periods | ||

Parameters | Unit | Descriptions |

m/s^{2} | Vehicle acceleration and deceleration | |

Pax/vehicle | Vehicle capacity | |

pax/stop | Stop capacity | |

km | Minimum distance between bus stops and restricted locations | |

RMB/h | Value of access time | |

RMB/h | Value of in-vehicle travel time | |

RMB/km | Unit cost of distance-based operation cost, e.g., vehicle fuel consumption cost | |

RMB/h | Unit cost of the time-based cost, e.g., drivers’ wage and amortized vehicle purchase cost | |

RMB/h | Unit amortized costs of stop construction | |

RMB/h | Unit cost of stop maintenance | |

km/h | Patrons’ average walking speed | |

km/h | Vehicles’ cruising speed during period | |

h | Time delay due to bus deceleration and acceleration at stops | |

h | Average boarding and alighting delays per passenger | |

h | Average delay caused by opening door and closing door |

Consider a linear bus route with length

From the first-order conditions of (

Based on the above analytical results, the efficient algorithm can be readily developed using the iteration method to find the optimal solution (see again in Su et al. and Medina et al. [

Example of the optimal stop density function

In the literature of CA transit route design models, we found two discretization recipes for translating

Illustration of two discretization recipes.

The two methods are formulated as follows. First, define

The endpoint method further includes a default stop at

Consequently, discrete system metrics can be computed: e.g., boarding and alighting volumes at each stop by

It is worth noting that although (

Other than arbitrarily determining stop locations, we propose a multivariate optimization model to do so and admit constraint of stop locations. Given the knowledge of the total number of stops obtained from the CA model, we accordingly define

Computations of

Note in (

Problem (

Admittedly, the above solution method does not guarantee a globally-optimal solution due to the nonconvex nature of (

To demonstrate the effectiveness of the proposed model, Section

Following Vaughan and Cousins [

Values of parameters.

Parameters | Values |
---|---|

1 m/s | |

20 km/h, 30 km/h | |

6.6 RMB/h | |

9.9 RMB h | |

3.3 RMB/h | |

3 s | |

5.1 s/stop, 7.64 s/stop | |

1 m/s^{2}, 1.2 m/s^{2} | |

1.55 s | |

0.99 s | |

37 RMB/h | |

2.68 RMB/km | |

1.67 RMB/h | |

0.6 RMB/h | |

80 pax/vehicle | |

120 pax/stop |

Experiments are conducted under a variety of demand scenarios with respect to

Comparisons between the proposed method and existing approaches.

Parameters | Cost items | Cost saving against, % | |||||
---|---|---|---|---|---|---|---|

Midpoint approach | Endpoint approach | ||||||

Max | Min | Avg. | Max | Min | Avg. | ||

Spatial variation, | System cost | 0.91 | 0.12 | 0.50 | 2.36 | 0.88 | 1.68 |

Passenger cost | 3.36 | −1.43 | 0.27 | −0.76 | −2.68 | −1.82 | |

Operator cost | 2.18 | −0.28 | 0.61 | 5.39 | 2.56 | 3.41 | |

Corridor length, | System cost | 0.86 | 0.13 | 0.36 | 2.39 | 0.44 | 1.12 |

Passenger cost | 2.61 | −0.93 | 0.37 | −0.62 | −5.34 | −1.84 | |

Operator cost | 1.40 | −0.18 | 0.50 | 6.00 | 2.21 | 3.36 | |

Demand level, | System cost | 0.58 | 0.12 | 0.28 | 2.95 | 0.46 | 1.11 |

Passenger cost | 1.03 | −2.51 | −0.12 | −0.68 | −3.94 | −1.68 | |

Operator cost | 2.23 | −0.14 | 0.51 | 6.44 | 1.82 | 3.11 |

Note that the values in Table

We apply the proposed model to bus route no. 3 in Chengdu (China), as depicted in Figure

The existing bus stop locations of bus route no. 3 in Chengdu (China).

Intersections along bus route no. 3.

Intersection label | Location (km) | Intersection label | Location (km) |
---|---|---|---|

1 | 0.17 | 23 | 11.14 |

2 | 0.5 | 24 | 11.59 |

3 | 0.78 | 25 | 12.1 |

4 | 1.08 | 26 | 12.2 |

5 | 1.71 | 27 | 12.6 |

6 | 2.42 | 28 | 13.4 |

7 | 3.01 | 29 | 13.86 |

8 | 3.58 | 30 | 14.24 |

9 | 4 | 31 | 14.7 |

10 | 4.74 | 32 | 14.93 |

11 | 5 | 33 | 15.12 |

12 | 5.3 | 34 | 16.04 |

13 | 6.1 | 35 | 16.29 |

14 | 6.4 | 36 | 16.59 |

15 | 6.79 | 37 | 17.28 |

16 | 7.35 | 38 | 17.61 |

17 | 7.7 | 39 | 17.86 |

18 | 8.6 | 40 | 18 |

19 | 9 | 41 | 18.4 |

20 | 9.72 | 42 | 18.64 |

21 | 9.8 | 43 | 18.71 |

22 | 10.61 |

In preparation for bus route adjustment, boarding and alighting demand at 35 stops were surveyed on April 10th, 2018. Correspondingly, the density functions of boarding and alighting demand are fitted using spline interpolation, as shown in Figure

Boarding and alighting demand along the bus route (the peak period, for example).

Based on the CA model and proposed discretization recipe, we redesign the current transit service. Figure

Stop locations for the optimized transit service. (a) Stop locations with and without the location constraint. (b) Locations of 10th and 23th stops for different results.

Table

Comparison of cost items among different scenarios.

Scenarios | Average passenger cost (RMB) | Average agency cost (RMB) | Average system cost (RMB) | Location constraint |
---|---|---|---|---|

Current service | 5.44 | 5.39 | 10.83 | No |

Idealized design | 5.68 | 3.42 | 9.10 | Violated |

Optimal design with location constraint | 5.66 | 3.46 | 9.12 | Satisfied |

−0.35% | 1.65% | 0.22% | — |

This paper proposes a modeling framework that connects continuum approximation methods and discrete ones in optimizing bus stop locations. To our best knowledge, this is the first work in the transit route design literature. Our model is no longer limited by the given set of candidate stop locations as the conventional discrete models. Meanwhile, our design outreaches the idealized design of CA models and explicitly addresses practical stop locating restrictions. The proposed hybrid model not only bears the solution efficiency of CA models due to the parsimonious property but also produces implementation-ready designs as do by discrete models. Numerical studies of various scenarios demonstrate the effectiveness of the proposed model. A case study in Chengdu (China) illustrates how the model is applied to bus line redesign/adjustment in reality.

Of note, the present study still has several limitations. For instance, more realistic concerns (e.g., socioeconomic and political ones) are involved in locating bus stops, which may require further fine tuning. The local conditions (e.g., design and safety) of streets may also influence the decision of bus stop location [

The boarding and alighting data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This study was funded by the National Nature Science Foundation of China (NSFC 51608455), Sichuan Provincial Science & Technology Innovation Cooperation Funds (2020YFH0038), Doctoral Innovation Fund Program of Southwest Jiaotong University (DCX201826), Chongqing Municipal Transportation Engineering Key Laboratory Open Project (2018TE04), and National Key R&D Program of China (2018YFB1601100).