^{1}

^{1}

^{1}

^{1}

Traditional transit systems are susceptible to unexpected costs and delays due to unforeseen events, such as vehicle breakdowns. The randomness of these events gives the appearance of an imbalance in the number of operating vehicles and of unreliable transit services. Therefore, this paper proposes the queueing theory as a means to characterize the state of any given transit system considering the risk of vehicle breakdowns. In addition, the proposed method is used to create an optimized model for reserve fleet sizes in transit systems, in order to ensure the reliability of the transit system and minimize the total cost of any transit system exposed to the risks of vehicle breakdowns. The optimization is conducted based on the two main characteristics of all bus systems, namely, operator costs and user costs, in both normal and disruptive situations. In addition, the situations in our optimization are generated in scenarios that have a certain degree of probability of experiencing delays. This paper formulates such an optimization model, presents the formulation solution method, and proves the validity of the proposed method.

Obviously, the traditional transit system fleet size consists of an operating fleet, which is defined as the number of vehicles operating on a line. The fleet size is selected and determined by the desired bus frequency, in order to improve the level of service, as well as a reserve fleet, which serves as backup vehicles when operating vehicles break down or require routine and planned maintenance. In order to ensure the reliability of a transit system, it is necessary to guarantee the size of the reserve fleet under the premise and conditions of capital constraints. For instance, in London Bus, 2.4% of the company’s total scheduled mileage was lost due to serious vehicle breakdowns and traffic or staff problems in 2016/2017. That figure represents 13.2 million km. While minor vehicle failures can be repaired quickly, serious failures require longer repair times. Sometimes, the disabled vehicle will even need to be towed in for lengthy repairs or long-term maintenance. In addition, the original schedule of the broken-down vehicle may deteriorate to the extent where the operating fleet vehicle schedule needs to be adjusted in real time. This has to be done by scheduling backup vehicles from the reserve fleet to cover regular routes, and this in turn depends on the availability of the reserve fleet.

From the perspective of normal public transportation systems, Lee et al. [

From the perspective of disruptive public transportation systems, Li et al. [

In this paper, we construct a new type of optimization model for determining the reserve fleet size in traditional transit systems. This model minimizes the costs to both operators and users in both normal and disruptive situations. The proposed method has been tested on a traditional transit system. The obtained results have shown that the proposed method provides better prediction and achieves better results than the most commonly used method. The paper is organized as follows: In Section

In this section, we explain the reserve fleet scheduling problem in a traditional transit system. The operating fleet in a transit system performs a service trip. This service involves picking up and dropping off passengers at a sequence of bus stops. The reserve fleet performs a backup service trip, acting as a substitute for the main operating fleet when vehicle failures and traffic accidents occur. Traditional transit systems are susceptible to such requirements due to these unforeseen events. In this article, vehicle failures and traffic accidents are generalized as “vehicle breakdowns”. In addition, the scenarios are generated in such a way that they range from scenarios with a very low probability of a delay and a low delay length, to scenarios with a high probability of delay and a high delay length. The reason for generating scenarios in this way was because we wanted to cover very bad days (such as winter days with bad weather conditions or hot summer days with high vehicle failure rates), as well as days where only few disruptions occur. In this paper, uncertain demand and uncertain travel times are not major concerns, so we only focus on the influences of a traditional transit system while considering the risk of vehicle breakdowns.

The optimization of the reserve fleet size is based on some assumptions, which are as follows:

Vehicles breakdowns are mutually independent, and broken-down vehicles can be repaired back to a normal condition.

Daily passenger demand is constant, and passengers normally arrive independent of vehicle arrivals.

The operating fleet and the reserve fleet belong to the same bus line.

Average daily traffic is considered constant, except in cases of vehicle breakdowns.

The operator holds a constant and known preference over the planning horizon.

Assumption (i) is very critical to our model, since ignoring the independence of vehicle breakdowns could lead to significant differences in both formulations and optimal solutions. In this paper, we consider the average vehicle breakdown rate in assumption (i); we then extend the discussion to the state of an urban transit system with the queueing theory. We assume passengers arrive independently of the timetable, with no pass-ups in assumption (ii). This is because our paper translates the pass-up passengers cost into the cost of time, rather than considering the behavior of passengers as a means to calculate the fare loss. Besides, we want to determine the reserve fleet size of each line that has the lowest expected total costs in all delay scenarios. If other bus lines exist, it is possible that the backup vehicles of the other bus lines may provide the backup service trip. As a result, we assume that assumption (iii) holds in this paper. We assume the difference in scenarios is only the average vehicle breakdown rate in assumption (iv), because this paper mainly discusses the reserve fleet size considering the risk of vehicle breakdowns. Online requests and uncertain travel times are not major concerns. Additionally, the implementation of station control strategies and interstation control make it possible to keep bus headways evenly distributed. Assumption (v) also has significant importance in this paper, because the issue of uncertain risk preferences leads to another challenging topic in optimization problems. In addition, the implementation of geographical information systems and wireless communication systems in public transit systems makes it possible to handle accidents in a timely fashion when breakdowns occur.

Our idealized model treats the bus route as a dynamic system, with operating vehicles moving at a constant average velocity around a circular route of a given length, with a single depot and bus service workshops. At any point in time, each bus headway is evenly distributed on this circuit as E(H) and related to the frequency, as shown in Figure

Transit system model.

This paper considers daily vehicle breakdowns to be random, accidental events. Breakdown arrival patterns postulate for an ordinary Poisson process, whereby one event at most can occur at any given time. In addition, the broken-down vehicles require fixed repair times for towing the disabled vehicle for repairs and then receiving maintenance from the bus service workshop. The bus operation process assumes that vehicle breakdown arrival patterns postulate for an ordinary Poisson process with parameter

Transit system state flow graph.

The transit system can be one of the possible states

For state 0, there is

For state 1, there is

For state N-1, there is

For state N, there is

For state n-1, there is

For state n, there is

For state N+m-1, there is

The equilibrium probability of each state can be obtained as follows:

However, it is basically impossible for most of the vehicles to fail in the operation process of the real bus line. Therefore, the probability of the state needs to be adjusted according to the historical data of the vehicle breakdowns or the fault tolerance rate

In terms of the number of operating vehicles, when the transit system is in state

In terms of the average headway, when the transit system is in state

Transit system in disruptive situation 1.

When the transit system is in state

Transit system in disruptive situation N+1.

Subsequently, the average headway in different states is defined by

The optimization model for determining the size of the reserve fleet has the objective of minimizing the total daily cost of the transit system, which includes the costs to both the operator and users in both normal and disruptive situations.

Regarding the operator cost, we analyze three main parts of a bus company’s expenses, including the acquisition cost of vehicles, operating and maintenance costs, and the cost of carbon emissions treatment. The vehicle acquisition cost is equivalent to the daily cost of vehicle utilization within the length of the vehicles’ operation life and the transit fleet size. The human cost can be considered as a constant, including employee benefits and bonuses, which are generally neglected in the bus company’s expenses. Operating and maintenance costs also include repair costs and fuel costs related to total vehicle mileage. The cost of carbon emissions treatment is also directly related to vehicle mileage. Vehicle mileage is determined using the frequency under the different states of the transit system defined above.

Subsequently, the daily cost of vehicle utilizations is defined by

Furthermore, the operating and maintenance costs are calculated as

Moreover, the cost of carbon emissions treatment is defined by

The above models can be used to calculate the total operating costs of a transit system, and the function is shown as follows:

Regarding the user costs, we analyze both the waiting time cost and in-vehicle time cost. With regard to the waiting time cost to passengers, since it is assumed that passengers arrive at the stop randomly and uniformly over time, the waiting time may be estimated as half of the headway. As regards the in-vehicle travel time cost to passengers, we translate the pass-up passengers’ cost into the excess of in-vehicle time cost. The unit in-vehicle cost will increase with the decrease in operating fleet size in disruptive situations. To simplify our model, we assume the increase unit of in-vehicle cost is directly proportional to the increase in the in-carriage congestion level.

The waiting time cost to passengers is calculated as

where

The total in-vehicle time cost to passengers can be expressed by

where

The above models can be used to calculate the total user costs of the transit system, and the function is shown as follows:

The above models can be used to calculate the total costs of the transit system in different states. In order to reduce costs and to increase the bus system’s quality, the aim of this paper is to minimize the objective of both the operator cost and the user cost in both normal and disruptive situations, in order to determine the optimum reserve fleet size. The objective function is shown as follows:

where

Equations (

The optimization model in the previous section is nonlinear. A solution algorithm for the optimization model is required, in order to solve the problem and obtain the optimum solution. We used a genetic algorithm to determine the optimum reserve fleet size, and values of reserve fleet size are constantly reset to find the optimal scheme. Figure

Solution algorithm.

In the initial population generation logical block, we use the binary code of reserve fleet size to reduce the efficiency of the searching procedure. In the fitness computation logical block, the total cost of the transit system in different states is used as the fitness. We convert binary code to decimal integer code that can uniformly generate a random number of reserve fleet size from the feasible integer set. This process can not only satisfy constraints (

In this section, we present several numerical examples to illustrate the performance of our models and algorithm. We begin by testing the models on a common bus line, and we compare the optimum reserve fleet size with the minimum reserve fleet size. All related information pertaining to the bus line is presented in Table

Related bus line information.

Index | Bus Line |
---|---|

L [km] | 20 |

m [vehicle] | 20 |

n | 10 |

H [hours/day] | 12 |

| 5% |

t [minute] | 2 |

Q | 10000 |

d [km] | 2 |

| 0.15 |

Average vehicle speed [km/h] | 20 |

E [g/km] | 57 |

| 500000 |

| 2 |

| 0.00005 |

| 15 |

| 10 |

We applied the mathematical software MATLAB to determine the total cost of the transit system and the optimal reserve fleet size, as obtained by the genetic algorithm. The comparison of the minimum reserve fleet size obtained via the experiential method (whereby the reserve fleet size is determined by the minimum reserve fleet rate) and the optimal reserve fleet size as obtained by the proposed model is presented in Table

Comparison of optimal and minimum reserve fleet size as obtained by the proposed model.

Parameter | Experiential method | Proposed model | Value added |
---|---|---|---|

N | 1 | 2 | 1 |

| 2876.71 | 3013.70 | 136.99 |

| 9048.48 | 9287.16 | 238.68 |

| 12.89 | 13.23 | 0.34 |

| 4572.44 | 4441.02 | -131.42 |

| 10763.27 | 10393.57 | -369.7 |

G [¥] | 13636.90 | 13574.34 | -62.56 |

Based on the results presented in Table

Transit system daily total cost with different

Figure

In addition, we tested the models on different discrete values of average accident rate

Transit system daily total cost under different

As shown in Figure

In this paper, we implement an optimization model for finding reserve fleet sizes in traditional transit systems that are exposed to the risks of vehicle breakdowns. The scenarios are generated in ways where they range from scenarios with a very low probability of a delay and a low delay length, to scenarios with a high probability and a high delay length. Using the M/M/n/m+N/m queueing model, we characterize the state of the transit system considering the risk of vehicle breakdowns, and we analyze the influence on both the operator cost and the user cost in disruptive situations. Furthermore, we minimize the objective of the operator cost and the user cost in both normal and disruptive situations, in order to determine the optimum reserve fleet size. From our tests, several conclusions can be found, as follows.

The comparison of our proposed model and the experiential method was conducted according to both the total operator cost and user cost. The proposed model can effectively reduce the total user costs of the transit system and improve the quality of the bus service, but it will also increase the total operating costs of the transit system.

Through an examination to determine the optimum reserve fleet size on the different weight coefficients of the operator cost, we can see that a change of the weight coefficient will have an influence on the objective formulation. If the weight coefficient of the operator cost is too small, the objective of balancing the operator cost and the user cost cannot be achieved.

Compared with different average accident rates on our proposed model, we observe that, whatever the reserve fleet size is, the total cost with different average accident rates changes only slightly when the reserve fleet size is more than 5. Therefore, if the accident rate of the traditional transit systems is uncertain within limits, there will be a reserve fleet size that is sufficient to cover the broken-down vehicles.

Rules above would be beneficial for the determination of reserve fleet sizes. We will incorporate capacity constraints, uncertain demand, and uncertain correlated disruptions in our future work. In addition, we will consider multiple bus lines belong to the bus company, and we will optimize the total reserve fleet size serving different bus lines.

The 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.

The authors acknowledge Jilin Tong Tai Traffic Science and Technology Consulting Co., Ltd., for bus line data. The work was supported by the National Natural Science Foundation of China (Grant no. 51608224 and Grant no. 71871103).