Journal of Advanced Transportation

Volume 2019, Article ID 2452348, 14 pages

https://doi.org/10.1155/2019/2452348

## Multiobjective Optimal Formulations for Bus Fleet Size of Public Transit under Headway-Based Holding Control

^{1}Institute: Business School, University of Shanghai for Science and Technology, China^{2}Institute: College of Automobile Engineering, Shanghai University of Engineering Science, China

Correspondence should be addressed to Minghui Ma; moc.liamtoh@9891iuhgnimam

Received 9 October 2018; Revised 11 December 2018; Accepted 31 December 2018; Published 10 January 2019

Academic Editor: Dongjoo Park

Copyright © 2019 Shidong Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years, with the development of advanced technologies for data collection, real-time bus control strategies have been implemented to improve the daily operation of transit systems, especially headway-based holding control which is a proven strategy to reduce bus bunching and improve service reliability for high-frequency bus routes, with the concept of regulating headways between successive buses. This hot topic has inspired the reconsideration of the traditional issue of fleet size optimization and the integrated bus holding control strategy. The traditional headway-based control method only focused on the regulation of bus headways, without considering the number of buses on the route. The number of buses is usually assumed as a given in advance and the task of the control method is to regulate the headways between successive buses. They did not consider the bus fleet size problem integrated with headway-based holding control method. Therefore, this work has presented a set of optimal control formulations to minimize the costs for the passengers and the bus company through calculating the optimal number of buses and the dynamic holding time, taking into account the randomness of passenger arrivals. A set of equations were formulated to obtain the operation of the buses with headway-based holding control or the schedule-based control method. The objective was to minimize the total cost for the passengers and the bus company in the system, and a Monte Carlo simulation based solution method was subsequently designed to solve the optimization model. The effects of this optimization method were tested under different operational settings. A comparison of the total costs was conducted between the headway-based holding control and the schedule-based holding control. It was found that the model was capable of reducing the costs of the bus company and passengers through utilizing headway-based bus holding control combined with optimization of the bus fleet size. The proposed optimization model could minimize the number of buses on the route for a guaranteed service level, alleviating the problem of redundant bus fleet sizes caused by bus bunching in the traditional schedule-based control method.

#### 1. Introduction

In a stochastic traffic environment, deviations from the schedules of the bus system are unavoidable. The disturbances in bus travel times and passenger arrivals at the stops can lead to failure in schedule adherence and headway uniformity. A delayed or slowed bus usually encounters a larger number of passengers at each stop than expected, and the bus has to dwell longer, creating further delays. At the same time, the following bus has fewer passengers to board and will move faster. Therefore, large headways tend to become larger and small headways smaller. Eventually, some successive buses have little or even no time between them, and “bus bunching” occurs. Bus bunching results in longer waiting and travel times for the passengers. In addition, the unbalanced load of passengers on successive buses wastes bus capacity, because the leading bus will be quite crowded while the trailing bus will be relatively empty.

Under the schedule-based control method, if the bus cannot travel back to the terminal bus stop when it should depart from the bus stop according to the timetable, it means that the schedule has failed. In general, a bus company will increase the number of buses on the bus route to avoid this issue. However, the redundancy in the bus fleet will bring further chaotic operation and become a burden on urban transportation. Therefore, scholars have focused on real-time headway-based control methods to eliminate bus bunching.

This issue of bus bunching was first presented by Newell and Potts in 1964. Transit agencies have usually handled this issue by introducing slack time into the scheme, giving a delayed bus the ability to recover at control points. Recently, real-time dynamic control strategies have greatly benefited from the advances in technology and the development of monitoring tools such as AVL (Automatic Vehicle Location), APC (Automatic Passenger Counter), and GPS (Global Positioning System). Using these tools, decision makers are able to know the actual behavior of the transit network and implement real-time control strategies. Ibarra-Rojas et al. [1] provided an excellent summary of the previous work on real-time headway control strategies. In terms of spatial configuration, different control strategies can be classified into two categories: station control, including holding strategies [2–4] and stop-skipping strategies [5–7], and interstation control, including bus speed regulation strategies [8] and traffic signal priority strategies [9–11]. In addition, two control methods can be integrated to regulate bus headways or optimize operations of public transport systems [12–16].

Daganzo [17] proposed a method to achieve the target headways. This method attempts to make the headways presettled static values. Daganzo and Pilachowski [8] proposed a control strategy that continuously adjusts the bus’s cruising speed on the route based on a cooperative two-way system to achieve proper spacing between the buses. As an extension of this idea, Xuan et al. [18] proposed a holding strategy to regularize headways while maximizing the commercial speeds, as well as considering both the forward and the backward headways. Bartholdi and Eisenstein [19] adjusted the trip regularity to achieve a unique and static headway, leading to the satisfactory performance of the system. To integrate the advantages of the “two-way-looking control” with the “self-adaptive equalizing bus headway,” recently, Liang et al. [20] proposed a self-equalizing control strategy based on the two-way-looking control method (the headways between the bus at the control point and both its leading and following buses) with zero slack. More recently, Zhang and Lo [21] proposed two-way-looking self-equalizing headway control that considered multifarious variables, enriching the headway-based bus holding control method system.

In general, the bus fleet size problem is considered a subproblem of Transit Network Timetabling or Vehicle Scheduling Problem instead of independent. Transit Network Timetabling is to define arrival and departure times of buses at all stops along the transit network in order to achieve different goals such as the following: meet a given frequency, satisfy specific demand patterns, maximize the number of well-timed passenger transfers, and minimize waiting times. Vehicle Scheduling Problem is to determine the trips-vehicles assignment to cover all the planned trips such that operational costs based on vehicle usage are minimized.

In terms of Transit Network Timetabling, Mauttone and Urquhart [22] propose a multiobjective optimization approach for minimizing users’ costs (based on in-vehicle time, waiting time, and transfer time) and the fleet size. The authors propose a GRASP algorithm for the multiobjective optimization problem. Medina et al. [23] formulate an optimization problem to simultaneously determine the stop density in a bidirectional corridor and the lines’ frequency for several periods. The objective is to minimize the user costs and operating costs based on waiting time, in-vehicle travel time, fleet operating costs, and stops installation. Later, Nikolić and Teodorović [24] extend their previous approach outlined by Nikolić and Teodorović [19] considering elastic demand and the minimization of the weighted sum of the total number of unsatisfied passengers, the total travel time of all passengers, and the fleet size. Amiripour et al. [25] determine a set of lines to be implemented for an entire year considering seasonal demand patterns with a probability of occurrence. The authors propose a mathematical formulation to minimize the expected value of the weighed sum of passengers’ total waiting time, unused seat capacity, unsatisfied demand (passengers with more than a specific number n of trip legs), and the fleet size.

In terms of Vehicle Scheduling Problem, Baita et al. [26] present several approaches to solve the Vehicle Scheduling Problem considering the following practical elements: it is possible to perform deadheading; it is possible to fuel a vehicle after finishing a trip; and it is possible to hold a bus at the end of the trip to wait for the starting time of the following assigned trip. Moreover, the authors considered criteria such as minimizing the fleet size, minimizing the number of lines that a bus is assigned to, minimizing the deadhead costs, and minimizing the idle times of buses waiting for the next trip. Liu and Shen [27] integrate the Transit Network Timetabling presented by Liu et al. [28] and a multiple deport vehicle scheduling problem minimizing the number of buses and deadhead costs. To solve the problem, the authors develop a Bi-level Nesting Tabu Search which is implemented in a small example. Guihaire and Hao [29] propose a mathematical formulation for the integration of Transit Network Timetabling and Vehicle Scheduling Problem minimizing a weighted objective function based on the following elements: (i) number and quantity of transfers; (ii) headway evenness; (iii) fleet size; and (iv) deadhead costs. The model considers a limited deviation from an initial timetable which allows the authors to define feasible shifting procedures. Kéri and Haase [30] and Kéri and Haase [31] address this problem, minimizing the number of buses and crew costs subject to constraints for task covering, bus-driver coupling, and a flexible timetable. This flexibility is defined based on flexible groups, that is, sets of trips that could be shifted together in order to keep the level of service in terms of waiting times. The proper division of an operational day into smaller planning periods based on demand behavior is an important aspect in the implementation of deterministic optimization approaches. In this matter, Zhoucong et al. [32] propose a clustering-based method to generate short planning periods with low variability of travel times. Ceder [33] addresses an interactive heuristic approach to find timetables with even loads and even headways which take advantage of different vehicle types. The approach is a combination of previously defined tools [34] to compute timetables with even loads, timetables with even headways, and vehicle schedules with minimum operational costs. The weight parameters in weighted objective functions must represent the planner’s preferences for the different objectives. The latter is an issue if two or more objectives are in conflict. Then, Ibarra-Rojas et al. [35] propose a biobjective optimization problem to jointly solve problem considering time windows for departure times and assuming constant demand. The objectives are maximizing the number of passengers benefited by well-timed transfers and minimizing the fleet size. The authors implement a constraint algorithm to obtain Pareto optimal solutions; thus, they are able to measure the “cost” of a vehicle in terms of passengers transfers and vice versa.

The method that integrated “two-way-looking control” with “self-adaptive equalizing bus headway” has been proven to perform well in reducing bus bunching and improving the service level for passengers. However, this kind of headway-based control method only focused on the regulation of bus headways without considering the number of buses on the route. The number of buses is usually assumed in advance and the task of the control method is to regulate the headways between successive buses. The control method may fail to regulate the bus headways due to quite enough fleet size on the route, even though the control method itself is effective. In addition, the benefits brought about from the advanced control method cannot be achieved directly by the bus company, such as reducing the cost of buying buses or operating costs. Therefore, the primary objective in this paper has been to optimize the bus system using the bus holding control method considering the bus fleet size. Furthermore, the fleet size problem was usually considered a subproblem of schedule-based bus operation process. However, with the development of real-time headway-based holding control, the headway between two successive buses is paid more attention, instead of the time the bus arrived at the control point. However, if the headway-based holding control replaces the schedule-based method, the bus fleet size should be considered under the novel operation process, which has not been addressed.

Therefore, the main contribution of this paper has been to integrate the self-adaptive control method with the optimal bus fleet size in order to optimize the bus system, which can increase the performance of the bus holding control method and bring benefits created by the advanced holding control method to the bus company.

The remainder of this paper has been organized as follows. In Section 2, the problem description and notations have been given first. The buses’ operating process has been described by formulas under the scheme-based bus holding control in Section 3, followed by the basic control scheme that has been used in this paper. The enhanced self-adaptive control method has also been proposed, which can dynamically decide which bus should dwell at the control point and its holding time. In Section 4, a cost objective function has been formulated including buying buses, operation, and cost for passengers waiting at bus stops. Finally, a set of tests have been conducted in Section 5, using the proposed scheme-based control method and the enhanced self-adaptive control method, respectively.

#### 2. Problems Statement and Notations

In practice holding control is usually applied to regulate the bus headways on the bus route and resist bus bunching. Compared with the no control scene or schedule-based control method, the headway-based control method can make the bus operation process, maintaining stronger stability. There are three major advantages for the public transit system.

First, the more balanced distribution of bus headways can obviously reduce the waiting time for passengers at the bus stops, because the large bus headways are vanished. Therefore, with the same bus fleet size, the headway-based control method can provide better service level for the passengers. In other words, in order to provide the same service level for passengers, the bus headway-based control method may use less number of buses.

Second, the unbalanced load of passengers on successive buses wastes bus capacity. This is because the leading bus is quite crowded, while the trailing bus is relatively empty. Therefore, the chaos bus operation process cannot make maximum utilization of bus fleet size. If the full bus capacity of each bus can be used properly, the bus fleet size, to some extent, can be cut down.

Third, as shown in the previous research works (e.g., [17, 20]), the proper headway-based control method can improve the cruising speed of buses on the bus route. Because the holding time or slack time can be greatly cut down, the bus can run back to the terminal bus stop with less time with the headway-based control method. If the length of bus route and the bus headway are fixed, the bus fleet size can be saved (the number of buses is assumed to be obtained by the bus travel time of one cycle divide the bus headway).

Therefore, although the bus headway-based control method was used to regulate bus headways and resist bus bunching, it can further cut down the redundancy buses on the bus route to save cost for bus company, instead of saving the time cost for passengers only. The redundancy buses will waste the money for company of buying buses, bus operation, and pay for drivers, for example, and the redundancy buses will increase burden for urban traffic. This is the insight for the relationship between the headway-based control method and the bus fleet size, and the motivation of optimizing the bus fleet size under holding control.

In this paper, the fleet size optimization problem is integrated with the headway-based control method. In fact, the headway-based control method is an optimization problem as well to regulate the bus headways and maintain the stable bus operation. If the bilevel optimization model, including fleet size optimization problem and headway-based optimization control method, is formulated to obtain the optimal fleet size, the model will be complex to be solved. Therefore, we select an analytical version of headway-based control method proposed by Liang et al. [20] and Zhang and Lo [21], which can obtain the optimal real-time holding time by calculation model according to the location between the bus at control point and its leading and trailing bus instead of solving the complex objective function. Therefore, the bilevel optimization model can become down to one single objective function of cost, including the cost of passengers’ total travel time and the cost of company for buying buses and operation. The “lower level optimization” can be replaced by the analytical headway-based control.

The fleet size optimization is a static problem, because for a long time (several years) the total fleet size cannot be changed. However, the headway-based control method is a real-time problem; the dynamic holding time is determined by the current state of bus system. Therefore, the problem should be solved integrated by the Monte Carlo simulation based method. According to enough simulation tests, we can obtain a relatively reliably optimal fleet size which can make the minimum cost of bus system. The theoretical optimal fleet size can be obtained.

In addition, in this paper, the schedule-based bus holding control means the fixed departure interval of buses from the control point. The bus control model assumes that the positions of all the buses are known, as well as the number of passengers in each bus and the number of passengers that are waiting at each stop.

The bus system can be completely defined by the following state variables:(i): number of bus stops on the bus route(ii): number of buses on the bus route(iii): index of bus stops on the bus route ()(iv): index of buses on the bus route ()(v): bus n’s arrival time at bus stop s(vi): bus n’s departure time at bus stop s(vii): dwell time required for bus n at stop s to provide service to passengers(viii): bus travel time between two successive bus stops, s-1 and s(ix): target bus fixed departure interval for schedule-based bus holding(x): capacity of buses on the bus route(xi): average arrival rate of passengers at the bus stop s(xii): passengers that have arrived in the period , (xiii): passenger boarding time (min/passenger)(xiv): passenger alighting time (min/passenger)(xv): number of passengers waiting at stop s(xvi): number of passengers that board bus n at stop s(xvii): number of passengers that alight bus n at stop s(xviii): number of the rest passengers at stop s because of bus n capacity limitation(xix): number of passengers on bus n when it departs bus stop s(xx): length of the bus route(xxi)SC: the control point.

#### 3. Bus Operation Process Description

In this section, the bus operation processes have been described under the schedule-based control method and the headways-based control method, respectively. The schedule-based control method means the bus leaves the control point with a fixed headway. The headway-based control method refers to the enhanced driven self-adaptive control method described by Liang et al. [20]

##### 3.1. Bus Operation Process with Schedule-Based Bus Holding

Each bus run can be regarded as a series of ‘events’ of arrivals and departures which specify the arrival and departure times of each bus n at each stop s along the service route. The evolution of is subject to the boundary condition at the stop s, shown in

The evolution of bus n from bus stop s-1 to s can be written as Eq. (2). Bus travel time between two successive bus stops s-1 and s can be obtained or estimated from the detectors [36, 37]:

In Eq. (1) and Eq. (2), the dwell time of bus n at stop s is an unknown variable. In the traditional method, the dwell time can be determined by the maximum service time for passengers to board and alight, which can be written as

We assume that the passengers’ arrival process obeys the Poisson distribution. Therefore, in a short period , passengers have arrived at the bus stop. The passengers that arrived during can be expressed by .

With respect to the boarding passengers, the passengers waiting at the bus stop s can be expressed as . Some of these passengers may not board because of the bus’s capacity limitation, and the number of passengers waiting to board should be discussed. If the bus’s capacity is less than the total number of passengers on the bus and also waiting at the bus stop, some passengers are not allowed to board because of capacity limitation. Therefore, the number of boarding passenger can be expressed by

When the bus n departs from bus stop s, the rest of the passengers at bus stop s because of capacity limitation can be expressed by

The unknown variable can be regarded as the function of the number of passengers at the upstream bus stop in bus n, and the number of alighting and boarding passengers at bus stop s, which can be written as

As mentioned in the beginning of this section, under the schedule-based bus operation, the buses are asked to depart the first bus stop with a fixed headway for its leading bus. If the headway between the bus at bus stop SC (control point) and its leading bus is less than H, the bus at bus stop SC should be delayed for a certain time. However, in this research, the bus fleet may not be large enough and, when the bus arrives at bus stop SC, if the headway with its leading bus is larger than H, the bus that just arrived at bus stop SC should depart at once after picking up the waiting passengers. The departure time from control point can be written as

##### 3.2. Bus Operation Process with Headway-Based Bus Holding

In this section, the headway-based control method described by Liang et al. [20] has been presented first. In the original control model, the random variables were not considered. Therefore, in order to fix these gaps, the enhanced control method that has been proposed in this section makes the self-adaptive control method more flexible in handling the variables in the public transit system, such as the stochastic arrival process of the passengers.

###### 3.2.1. Enhanced Version of Headway-Based Control Procedure

As shown in the Appendix, the bus headways will iterate to convergence by continually equalizing the headways between the bus at the control point with its forward and backward buses (). In practice, the headway is defined as the time between two successive buses leaving the same bus stop. Therefore, only three buses have been focused on at once, which are the bus at the control point and its following and leading buses. To obtain the bus headways among the three buses, it is necessary to predict the time of the last bus among the three buses leaving the control point, and the related research is relatively mature [38, 39]. Therefore, according to this control concept, a procedure has been designed to select the bus that should be controlled and find its corresponding holding time. The advantage of this method is that extending the self-equalizing bus headway control method makes it more flexible in handling the stochastic variables. The procedure of the proposed control method can be described in the flowchart shown in Figure 1.