Bus Rescheduling for Long-Term Benefits: An Integrated Model Focusing on Service Capability and Regularity

Abstract

Unplanned disruptions, such as vehicle breakdowns, in a public transportation system can lead to severe delays and even service interruptions, preventing the successful implementation of subsequent plans and the overall stability of transit services. A common solution to address such issues is implementing a bus bridging service using an experience-based response strategy, involving the deployment of spare buses to continue affected services. However, with this approach, it becomes impractical and challenging to generate a feasible and rational rescheduling scheme for the remaining transit services when spare buses are insufficient or widespread disruptions occur. In response to this challenge, we propose an innovative model that integrates service capability and regularity, aiming to minimize rescheduling costs through timetable adjustments and scheduling reassignments. We apply dynamic programming to comprehensively consider the hysteresis effects of disruptions and achieve a long-term optimal rescheduling scheme. To efficiently solve the proposed model, the large neighborhood search algorithm is improved by incorporating operational rules. Finally, several experiments are conducted under an actual transit operation scenario in Shenzhen. The results demonstrate that our method significantly reduces trip cancellations and, simultaneously, diminishes the increase in the departure interval resulting from the adjusted schedule by 23.27%.

1. Introduction

A stable and reliable bus service system is crucial for urban mobility, providing convenient travel services for residents. However, buses are susceptible to disruptions due to the heterogeneous traffic conditions. Unforeseen events such as vehicle breakdowns frequently disrupt bus schedules, resulting in severe delays and even service interruptions. These disruptions prevent transit services from successful implementation as planned, and running a bus bridging service using an experience-based response strategy is a common solution. However, delaying or canceling services is often necessary when there are insufficient spare buses or widespread disruptions occur. The delays and cancellations of services increase passenger waiting times, making services irregular and negatively impacting both bus service quality and passenger satisfaction. Although several predictive-based offline or scroll generation methods have been developed to enhance schedule robustness [1], it is important to note that the real-world events leading to delays and cancellations mentioned above have not been properly considered due to their inherently random nature of occurrence. Therefore, bus rescheduling, which adjusts initial fleet plans such as timetables and bus schedules in response to the current conditions of a public transportation system, is imperative to restore transit functionality and ensure stable operation for the remainder of the day.

Bus rescheduling aims to refine the original schedule to minimize the impacts of disruptions on both service capability and regularity [2,3]. A small case is illustrated in Figure 1, where Figure 1a shows a schematic diagram of the actual travel processes of two bus lines, and Figure 1b–f shows the corresponding operational schedule. $tri{p}_2$, $tri{p}_4$, $tri{p}_6$, and $tri{p}_7$ are trips along line 1, whereas $tri{p}_1$, $tri{p}_3$, and $tri{p}_5$ are trips along line 2. The departure time is indicated above the top-left corner of each trip node, whereas the arrival time is indicated above the top-right corner. Each bus serves multiple trips; for example, bus 1 operates $tri{p}_1\to tri{p}_4$, bus 2 manages $tri{p}_3\to tri{p}_6$, and bus 3 is assigned $tri{p}_2\to tri{p}_5\to tri{p}_7$. And, a deadhead is the movement of an empty bus to a destination without any passengers, which helps relocate bus resources [4]. In this case, bus 2 experiences a breakdown while starting $tri{p}_3$ at 7:50, impacting the remaining trips (i.e., $tri{p}_3$ and $tri{p}_6$). At that moment, bus 1 is preparing to start $tri{p}_4$, while bus 3 is in the process of $tri{p}_2$. In scenario 1, no additional operations are applied, leading to the cancellation of both $tri{p}_3$ and $tri{p}_6$ and an increase in the maximum departure interval of line 1 by 35 min (100% of the original). Scenario 2 employs an intuitive rescheduling method, which completes $tri{p}_6$ by assigning it to an available vehicle (bus 1 in this case) without affecting other trips. However, the maximum departure interval still increases by 35 min (100% of the original). With overall consideration of the entire schedule and vehicle resources, all subsequent trips are completed following a regular departure interval in scenario 3, with only a 10-minute increase in the maximum departure interval (28.57% of the original), resulting in minor impacts on the original schedule. From this small case, two interesting conclusions can be drawn: (1) bus services can be recovered through rescheduling when disruptions occur, and (2) the comprehensive consideration of both subsequent schedules and vehicle resources is essential for ensuring both service capability and regularity.

However, the rescheduling process involves the comprehensive consideration of various complex factors, including trip departure times, the number of trips, and bus assignments. Numerous studies have demonstrated the NP-hard nature of the bus rescheduling problem, which involves timetabling and vehicle scheduling, in different scenarios, such as single-line rescheduling [3,5,6], single-depot rescheduling [4], and multi-depot rescheduling [7]. Some researchers [2,8] have formulated the vehicle rescheduling problem (VRSP) based on information on the remaining trips and operating buses. The VRSP is a combinatorial optimization problem that focuses on minimizing trip cancellations and deviations from the original bus schedules when recovering operations. With consideration of trip delays, studies have derived practical schemes through cancellations and reassignments to guarantee the successful operation of as many remaining trips as possible. However, trip delays are predefined based on the idle time of a single bus and remain unmodified. Individually adjusting each trip’s departure time would overlook the overall trip departure distribution, potentially resulting in irregular departure intervals and destabilizing the bus service. Therefore, it is necessary to modify departure times with consideration of neighboring trips.

The hysteresis effects of disruptions also make it difficult to determine a long-term viable solution for the remaining transit operations. When disruptions occur, their impacts propagate backward along schedules, turning rescheduling into a long-term decision-making challenge for the entire day. Manual rescheduling is the most common method during actual operation. According to controllers’ experience, the impacts of disruptions can be mitigated through local schedule adjustments. However, this approach only provides limited optimization for trips departing within a short time frame, thereby giving rise to subsequent operational challenges. For instance, inadequate consideration of the overall schedule may lead to numerous cancellations due to the inappropriate allocation of bus resources, resulting in significantly divergent plans from the original schedule. Moreover, the rescheduling problem becomes more intricate and difficult to solve manually as the number of buses and trips increases. And, the NP-hard nature of the problem also necessitates the introduction of more effective solution approaches.

To address the above challenges in the VRSP problem, we propose an integrated service capability and regularity (ISCR) model, and an improved large neighborhood search (LNS) algorithm based on the dynamic programming (DP) framework, named DP-LNS, is introduced to solve the problem effectively. Our main contributions are as follows:

1.

Service regularity is innovatively combined with service capability in constructing the ISCR model. Rather than separate adjustments in the timetable before rescheduling, we directly introduce the overall departure regularity into the formulation of the objective function, which simultaneously achieves more sufficient and regular services.

2.

The rescheduling problem is formulated with the dynamic programming framework, and an improved large neighborhood search (LNS) algorithm is applied based on it. To achieve long-term optimization for the subsequent operations of the day, dynamic programming decomposes the problem into a series of stages over time and considers the aftereffects of previous periods. In addition, by incorporating operational rules, we improve the LNS algorithm with scheduling-based heuristic operators to solve the problem effectively.

3.

Several experiments are conducted under a real public transit operation scenario in Shenzhen, China. The method offers an effective and robust rescheduling scheme for bus services, facilitating the recovery of operations and ensuring stable service provision.

As rescheduling is carried out through bus dispatch and timetable adjustments [9], we focus on a single bidirectional line, including one depot and two routes, to maintain a sufficiently large yet manageable experimental scenario. The remainder of this paper is organized as follows. Section 2 reviews previous related studies. Section 3 establishes the proposed optimization model, and the solution algorithm is described in Section 4. Section 5 presents the computational results. Finally, the conclusions are discussed in Section 6.

2. Related Works

Rescheduling is an essential aspect of transportation systems, including buses, metros, railways, and aircraft [10,11]. Many studies have been carried out on formulations, model establishment, and solution approaches.

To better formulate the problem and design a mathematical model, researchers [6,8] often construct an underlying network based on trip information and bus status. Time-space networks (TSNs) and connection networks (CN)s are the most popular basic structures. A TSN offers exact time and space information of buses, making it easy to describe the actual state of a public transportation system [12]. Van den Heuvel et al. [13] developed routines based on TSNs to reduce the wastage of bus resources. And, some studies [14,15] have combined variable speed and TSNs to achieve a more practical rescheduled timetable. However, TSNs focus more on the details of the travel process, which makes rescheduling complicated. Compared with a TSN, a CN is constructed based on relationships between trips and vehicles and has fewer elements, offering a more intuitive and brief computational representation [16,17]. It has been proven that CNs are more conducive to the development of effective methods when travel information is given [4,12]. However, this approach limits the solution space to specific trip departure times. Recently, researchers [9,18] have found that extending the connection network with time-delay windows can further improve the quality of solutions. In this paper, our main focus is on bus assignments and the precise modification of departure times, rather than the intricacies during a single trip. Therefore, we adopted the concept of a connection network with delayed trip copies to formulate the rescheduling problem in a clear and effective manner.

The purpose of bus rescheduling is to minimize the impact of disruptions on the service capability and regularity of public transportation systems. Early works mainly considered recovering operations by canceling trips [2,8], which significantly reduced bus service reliability. Thus, Li et al. [4] reconsidered real-world situations and expanded previous models with reassignments to ensure the executability of trips. However, trips are still prone to failure due to insufficient vehicles. Some studies [9,18,19] have introduced time-delay windows into models and obtained higher-quality rescheduled plans through reasonable time-sharing of vehicle resources. Wang [20] additionally calculated the delays of each train to reduce the aftereffects of rescheduling. Based on the above aspects, researchers have constructed models with objectives concerning trip delays, trip cancellations, and deviations from bus schedules and made some achievements in guaranteeing service capability. Nevertheless, few studies have paid attention to service regularity after rescheduling, which is significant for public transit passengers. Tessitore [21] perceived the regularities of train arrivals and departures as essential to measuring the performance of a transit system. Gkiotsalitis [3,22] raised the idea that service regularity is directly related to the excessive waiting times of passengers. Accordingly, researchers [23,24] have built models considering passenger delays and waiting times. Moreover, Long [25] took the maximum train-loading rate into account. Researchers [26,27,28,29] have also added deviations from departure intervals to models to satisfy service requirements. Service regularity has gradually become increasingly crucial during rescheduling, which can be represented by departure intervals. However, the cancellation and retiming of individual trips within transportation services can lead to significant service gaps and disrupt service regularity. There has been a noticeable lack of emphasis on addressing the issue of maintaining service regularity both before and after rescheduling, as well as its integration with service capability. Consequently, we propose a time-based service regularity model and integrate it with service capability to further enhance the rescheduling model.

The rescheduling problem is a long-term decision-making problem. An intuitive idea to achieve comprehensive optimization is to perform rescheduling on the whole schedule directly, but this requires significant computational support. By decomposing complicated problems into several sub-stages and constructing correlations among them, dynamic programming can help efficiently obtain long-term beneficial solutions [30]. Moreover, the scheduling model based on dynamic programming can adeptly take into account time-based variations, which help improve resource utilization efficiency and achieve robust schemes [31]. Therefore, dynamic programming has attracted considerable interest in operational optimization and may be more practical for performing rescheduling. Considering service capability, the rescheduling problem is described as a binary linear programming model. Many deterministic algorithms can solve this problem and obtain analytical solutions, such as column generation [14], and branch-and-bound [6,32] algorithms. However, the problem becomes nonlinear when service regularity is integrated because trips can be canceled or retimed, which complicates the solution. As the problem has been proven to be NP-hard [3,4,5,6,7], several approximation algorithms have been applied to effectively address the problem. In [8], the auction algorithm was used to solve the model pseudo-polynomially. Lagrangian relaxation [4,9] can help reduce the complexity by decomposing the problem into multiple sub-problems. Nevertheless, the computational cost still explodes with the number of remaining trips. Thus, researchers have designed many heuristic-based algorithms to reduce the solution space. For example, the greedy heuristic algorithm [11], insertion heuristic algorithm [4], variable neighborhood search heuristic algorithm [33], tabu search algorithm [34], neighborhood search algorithm [35], simulated-annealing algorithm [28], and genetic algorithm [36] have been applied to gradually modify initial solutions by performing local adjustments. Heuristic algorithms are highly effective for solving combinatorial optimization problems and obtaining approximate optimal solutions.

Local search (LS) is a prevalent heuristic algorithm employed for combinatorial optimization problems. Typically, the algorithm preserves a feasible solution and iteratively replaces the solution with its neighbor of higher quality until a locally optimal solution is achieved [37]. The large neighborhood search (LNS) algorithm was developed based on the principle of local search, where the neighborhood is defined by recombination methods (often heuristics) that are used to destroy and repair a current solution [38]. Through the use of heuristics, LNS is capable of exploring complex neighborhoods efficiently and finding better candidate solutions in each iteration, making it one of the most successful paradigms for solving various transportation and scheduling problems [37]. The recombination methods function by strategically destroying the solution at specific positions and subsequently repairing it using predefined rules. Various destroy methods are utilized in bus operation research, including random removal [39], worst removal [40], and neighborhood-based removal [41]. Common repair methods involve additions and shifts, such as stop insertion for bus routes [42], trip insertion for bus schedules [40], or timetable shifts [43]. The flexible nature of these configurations makes LNS a powerful tool for solving bus operation problems. Through the improvement of LNS with appropriately designed recombination rules, the aim of ensuring service capability and regularity can be effectively achieved in the bus rescheduling problem.

Therefore, to comprehensively meet the dual demands of service capability and regularity within the context of rescheduling, we innovatively propose the integrated ISCR model, leveraging the existing connection network. Additionally, we develop the DP-LNS algorithm as a powerful solution to address this challenge. ISCR uniquely incorporates departure time as another crucial variable alongside trip numbers, allowing for a more thoughtful consideration of service capability and regularity requirements. To mitigate the enduring effects of disruptions, the dynamic programming algorithm decomposes the problem, helping achieve long-term optimized solutions. Simultaneously, LNS is improved based on existing research [37,44,45] to efficiently tackle the ISCR model, making use of domain-specific knowledge. Finally, we empirically validate the method’s effectiveness and robustness through experiments conducted in real operational scenarios in Shenzhen, China.

3. Mathematical Formulation

In this section, we commence with a summary of the notations and definitions utilized in this paper. Following that, we provide a brief overview of the construction of the connection network on which the ISCR model is based. Subsequently, we elaborate on the approach to integrating service capability and regularity within the ISCR model, taking into account operational rules through the formulation of corresponding constraints.

3.1. Preliminaries

Table 1. Notations.

Indices	Description
$i,j,u$	Trip index, $i,j=1,2,\dots ,R_n$; u refers to the remainder of the trip currently being performed by the breakdown bus.
b	Bus index, $b=1,2,\dots ,B_n$.
l	Line index, $l=1,2,\dots ,L_n$.
Parameters
$t_i$	Travel time of trip i.
$S_i$	$S_i\in \left\{l\right|l=1,2,\dots ,L_n\}$. The line that trip i belongs.
$O_l$	Departure stop of line l.
$D_l$	Arrival stop of line l.
$p_j^b$	$p_j^b\in \{0,1\}$; 1 if trip j is covered by bus b in the original schedule, 0 otherwise.
$t_{min}^{idle}$	The minimal bus idle time.
$d_{max}$	The maximum trip delay time.
$C^Q$	$C^Q\in {\mathbb{R}}^{+}$. The cost of a single trip cancellation.
$C_{i,j}^D$	$C_{i,j}^D\in {\mathbb{R}}^{+}$. The cost of deadheading from trip i to trip j.
$C^P$	$C^P\in {\mathbb{R}}^{+}$. The reassignment penalty resulting from changes in vehicle departure times and sequences.
$C^H$	$C^H\in {\mathbb{R}}^{+}$. The cost of minute changes in trip departure intervals.
${\overline{h}}^s$	${\overline{h}}^s\in {\mathbb{N}}^{+}$. The original average departure intervals of the trips departing from stop s.
$h_{max}$	$h_{max}\in {\mathbb{N}}^{+}$. The maximum departure interval.
$\epsilon$	$\epsilon \in {\mathbb{N}}^{+}$. The adjustment parameter for updating $h_{max}$.
$T_{st}^b$	Start time of bus b’s daily working schedule.
$T_{ed}^b$	End time of bus b’s daily working schedule.
Decision Variables
$e_{i,j}^b$	$e_{i,j}^b\in \{0,1\}$; 1 if bus b operates from trip i to trip j sequentially, 0 otherwise.
$T_i$	Departure time of trip i.

provides an overview of the primary notations used in this paper. Within a transit schedule, there are $L_n$ bus lines, $R_n$ trips on different bus lines, and $B_n$ buses serving all trips. A trip i is a service designed on line $S_i=l$, starting from departure stop $O_{S_i}$ at time $T_i$, following the entire sequence of stops along line $S_i$, and arriving at the destination, $D_{S_i}$, with a total travel time of $t_i$. And, the timetable is formed by assembling all trips and providing comprehensive information on both the timing and route details of transit services. According to the timetable, each bus is scheduled to serve multiple trips sequentially per day; for example, $(i\to j\to \dots )$ for bus b. After completing the prior trip i, bus b engages in deadheading, which is the movement of an empty bus to help manage vehicle resources, to reach the subsequent trip service j. Therefore, two consecutive trips within any bus schedule adhere to trip compatibility, as outlined below:

Trip compatibility: Trips $i,j$ are compatible if the following conditions are satisfied:

• The arrival time of trip i is earlier than the departure time of trip j;
• There exists adequate time for a bus to deadhead from trip i to trip j;
• The idle time of the bus is guaranteed.

Let $\phi (i,j)=1$ denote that the trip pair $(i,j)$ is compatible, and 0 otherwise. $\mathbf{tra}(D_{S_i},O_{S_j})$ calculates the travel time from $D_{S_i}$ to $O_{S_j}$. In summary, trip compatibility can be described as follows:

$$\phi (i,j)=1\iff T_i+t_i+\mathbf{tra}(D_{S_i},O_{S_j})+t_{min}^{idle}\le T_j$$

(1)

Sometimes, bus b experiences a breakdown during trip i, impacting the subsequent schedule $(j\to \dots )$. To mitigate the impacts of disruptions, rescheduling is applied to all trips and buses, ensuring both service capability and regularity.

3.2. Generation of the Connection Network

Drawing from existing methods [4,9], we construct the connection network with time-delay windows vehicle by vehicle. Figure 2 serves as an illustrative example of a single bus’s connection network, corresponding to bus 3 in the breakdown scenario depicted in Figure 1. In this graphical representation, the rectangular nodes correspond to individual trips, whereas the edges symbolize specific buses. The different colors of the rectangular nodes indicate the corresponding bus lines that trips travel along. When breakdowns occur, buses may be either in transit or parked, and the manager needs to determine how to reschedule the buses to minimize the impact of breakdowns. For example, combined with the information from Figure 1, it can be seen that bus 1 is parked at the arrival stop of $tri{p}_1$, and bus 3 is serving $tri{p}_2$ when bus 2 breaks down. Therefore, we introduce ${sd}^b$ to represent the starting point of a specific bus b, indicating the specific availability time and location from which b can be deployed. For example, ${sd}^3$ means that bus 3 is available for rescheduling starting from the arrival stop of $tri{p}_2$ at 7:55. Additionally, ${td}^b$ is used to denote that b has completed its service and returned to the depot. A complete schedule of bus b after disruptions is depicted by a unique link from ${sd}^b$ to ${td}^b$.

We commence by constructing the connection network with all compatible trip pairs based on the remaining trips. Let ${\mathbf{V}}_t$ denote the nodes corresponding to the trips that depart after disruptions. ${\mathbf{V}}_t^b\subseteq {\mathbf{V}}_t$ is the set of trip nodes that can be performed by b. According to Formula (1), we establish connections between the nodes using directed edges and derive a set of edges ${\mathbf{E}}_t^b=\left\{(i,j)\right|\phi (i,j)=1,i,j\in {\mathbf{V}}_t^b\}$ pertaining to deadheads between $(i,j)$, as indicated by the solid edges in Figure 2. Subsequently, the connection network of bus b without considering delays can be represented by $G^b=({\mathbf{V}}^b,{\mathbf{E}}^b)$, where ${\mathbf{V}}^b={\mathbf{V}}_t^b\cup \{{sd}^b,{td}^b\}$ and ${\mathbf{E}}^b={\mathbf{E}}_t^b\cup ({sd}^b\times {\mathbf{V}}_t^b)\cup ({\mathbf{V}}_t^b\times {td}^b)\cup \left\{({sd}^b,{td}^b)\right\}$.

Then, we expand the connection network to further explore which trip could be performed by bus b with a permissible delay. Let the delay time for trips be denoted as d minutes. Consider $W^b$ as a copy of ${\mathbf{V}}_t$, where the nodes are arranged in order of departure time to ensure that all nodes will be visited. The expansion starts by visiting the first node in $W^b$, assuming it to be ${sd}^3$, and iterating over all nodes $\left\{j\right|\phi ({sd}^3,j)\ne 1,j\in {\mathbf{V}}_t\}$ that cannot be operated from ${sd}^3$ without delay. If there exists $d\le d_{max}$ such that trip j can be operated from ${sd}^3$ after a d-minute delay in departure time, a copy of trip j, denoted by $j^{\left(d\right)}$ and departing at $T_{j^{\left(d\right)}}=T_j+d$, could be reasonably performed following the completion of ${sd}^3$. Next, we add the edge $({sd}^3,j^{\left(d\right)})$ and the edges connecting $j^{\left(d\right)}$ to all its compatible trips to the graph, such as the trip pairs $({sd}^3,3^{\left(10\right)})$ and $(3^{\left(10\right)},6^{\left(10\right)})$ shown in Figure 2. Following this, the currently explored node ${sd}^3$ is removed, the derived nodes are added to $W^b$, and $W^b$ is reordered. Let ${\widehat{\mathbf{V}}}_t^b=\left\{j^{\left(d\right)}\right\}$ denote the trip copies and ${\widehat{\mathbf{E}}}_t^b=\left\{({sd}^3,j^{\left(d\right)})\right\}\cup \left\{(j^{\left(d\right)},i)\right|\phi (j^{\left(d\right)},i)=1,i\in {\mathbf{V}}^b\}$ indicate the edges connecting the trip copies, as shown by the dashed nodes and edges in Figure 2, respectively. By repeating the above process until list $W^b$ is empty, the complete connection network of bus b is updated with ${\widehat{G}}^b=({\widehat{\mathbf{V}}}^b,{\widehat{\mathbf{E}}}^b)$, where ${\widehat{\mathbf{V}}}^b={\mathbf{V}}^b\cup {\widehat{\mathbf{V}}}_t^b$, ${\widehat{\mathbf{E}}}^b={\mathbf{E}}^b\cup {\widehat{\mathbf{E}}}_t^b$.

While exploring, only the compatible copy with minimal delays is considered to avoid producing additional delay times. Then, the connection network for each bus can be constructed in a consistent manner, resulting in many identical trip nodes being shared among them. To facilitate better bus coordination, these networks are merged into a single unified network $G=(\mathbf{V},\mathbf{E})={\bigcup }_b{\widehat{G}}^b$, where $\mathbf{V}={\bigcup }_b{\widehat{\mathbf{V}}}^b$ and $\mathbf{E}={\bigcup }_b{\widehat{\mathbf{E}}}^b$.

3.3. Construction of the Integrated Model

Based on the connection network, we utilize the decision variables $e_{i,j}^b$ and $T_i$ to construct our ISCR model. $e_{i,j}^b=1$ if edge $(i,j)\in {\widehat{\mathbf{E}}}^b$ is selected. ${\widehat{\mathbf{E}}}^b\left(i\right)$ contains the edges in ${\widehat{G}}^b$ pointing to trip node i. Then, we propose the objective function of the ISCR model, considering both service capability and regularity.

Service capability: Service capability focuses on the performance and successful operation of the transit schedule. The number of actually dispatched trips is a key indicator for measuring the capability of public transit services. During disruptions, it becomes crucial to ensure the completion of as many trips as possible, meaning that a sufficient number of nodes should be connected by edges within the connection network. Part of the objective considering service capability is given in Formula (2), where $R_n^l$ is the number of trips in $\left\{i\right|S_i=l,i\in {\mathbf{V}}_t\}$ and $R_n={\sum }_l{R}_n^l$.

$$z_Q=C^Q\sum _l(R_n^l-\sum _b\sum _{(i,j)\in \{(i,j)|(i,j)\in {\widehat{\mathbf{E}}}^b,S_j=l\}}{e}_{i,j}^b)$$

(2)

Furthermore, the rescheduling scheme should be easy to follow to mitigate additional risks. This entails minimizing the dispatch of vehicles and reducing changes in the assignment of vehicles to trips. The dispatch of empty vehicles, referred to as deadheading, introduces more instability into the schedule implementation and results in additional operational costs, such as idle time and fuel consumption. At the same time, extensive reassignments hinder the swift implementation of new schedules. To address these concerns, we include the cost of deadheads and reassignments in the service capability. The remaining objectives, which take service capability into account, are related to the deadheading time and the number of reassignments, as shown below:

$$z_C=\sum _b\sum _{(i,j)\in {\widehat{\mathbf{E}}}^b}{C}_{i,j}^D{e}_{i,j}^b$$

(3)

$$z_P=C^P\sum _b\sum _j\left[\sum _{i:(i,j)\in {\widehat{\mathbf{E}}}^b\left(j\right)}{e}_{i,j}^b(e_{i,j}^b-p_j^b)\right]$$

(4)

Service regularity: Service regularity strives to provide a comparable level of transit service after rescheduling. Service frequency reflects the maximum waiting time of passengers, which is essential to evaluate the service. We introduce the changes in trip departure intervals to represent the variation in service frequency. The service regularity objective $z_H$ is formulated as follows:

$$z_H=C^H\sum _l\sum _{(i,j)\in {\mathbf{E}}_{sel}^l}\mid T_j-T_i-{\overline{h}}^l\mid$$

(5)

$${\mathbf{E}}_{sel}^l=\{(i,j)|j=i+1,i\in {\mathbf{V}}_{sel}^l\}$$

(6)

$${\mathbf{V}}_{sel}^l=\{i|\sum _b\sum _{j:(j,i)\in {\widehat{\mathbf{E}}}^b\left(i\right)}{e}_{j,i}^b=1,T_{i+1}>T_i,S_i=l,i\in \mathbf{V}\}$$

(7)

where ${\mathbf{V}}_{sel}^l$ represents the trip sequence in the solution that travels along line l, and ${\overline{h}}^l$ denotes the original average departure intervals of the corresponding trips. ${\mathbf{E}}_{sel}^l$ contains consecutive trip pairs in ${\mathbf{V}}_{sel}^l$. It is evident that $z_H$ acts as the focal point for feedback regulation within the framework of a rescheduling scheme, rendering the model challenging to solve. We address the solution to this issue in Section 4.

Finally, constraints are formulated for operational rules, except for the minimum bus idle time, which has already been satisfied during the construction of the connection network. The mathematical model is as follows:

$$minz=z_Q+z_C+z_P+z_H$$

(8)

s.t.

$$\sum _b\sum _{(i,j)\in \{(i,j)|(i,j)\in {\widehat{\mathbf{E}}}^b,S_j=l\}}{e}_{i,j}^b\le R_n^l\forall l$$

(9)

$$\sum _{j\in {\widehat{\mathbf{V}}}^b}{e}_{{sd}^b,j}^b=1\forall b$$

(10)

$$\sum _{j\in {\widehat{\mathbf{V}}}^b}{e}_{j,{td}^b}^b=1\forall b$$

(11)

$$\sum _b\sum _{i:(i,u)\in {\widehat{\mathbf{E}}}^b\left(u\right)}{e}_{i,u}^b=1$$

(12)

$$\sum _b\sum _{i:(i,j)\in {\widehat{\mathbf{E}}}^b\left(j\right)}{e}_{i,j}^b\le 1\forall j\in \mathbf{V}$$

(13)

$$\sum _{j:(i,j)\in {\widehat{\mathbf{E}}}^b}{e}_{i,j}^b=\sum _{j:(j,i)\in {\widehat{\mathbf{E}}}^b}{e}_{j,i}^b\forall i\in {\widehat{\mathbf{V}}}^b-\{{sd}^b,{td}^b\},\forall b$$

(14)

$$T_j-T_i\le h_{max}+\epsilon \forall (i,j)\in {\mathbf{E}}_{sel}^l,\forall l$$

(15)

Operational rules: Constraint (9) means that the total number of trips will not increase after rescheduling. Constraints (10) and (11) provide only one trip schedule for each bus. Constraint (12) guarantees the retrieval of stranded passengers on trip u, whereas constraint (13) places a restriction on the in-degree of nodes in the connection network, ensuring that each trip can be performed once at most. Constraint (14) guarantees flow conservation, making paths unique and feasible. Finally, constraint (15) ensures that trip departure intervals do not exceed a maximum. It should be noted that trip departure intervals are determined by travel demand situations, e.g., the departure intervals at night may be longer due to fewer passengers. Therefore, we construct a soft constraint (15) with an adjustment parameter $\epsilon$. This parameter can be employed when the solution is unstable in multiple iterations to prevent the generation of low-quality solutions and the resulting increase in total costs.

4. Solution Approach

In this section, we first formulate the rescheduling problem as a dynamic programming problem to comprehensively consider the hysteresis effects of disruptions. Moreover, the VRSP problem is proven to be NP-hard, and our proposed ISCR model needs feedback regulation, which poses a challenge in obtaining an exact solution within polynomial time. To address this, we improve the large neighborhood search algorithm (LNS) based on existing research, which combines domain knowledge to efficiently generate a reasonable and feasible optimization scheme.

4.1. Dynamic Programming

The public transit rescheduling problem exhibits several significant characteristics: (a) the initial schedule is phased; (b) disruptions have a backward propagating impact along schedules, which represents a long-term decision challenge; and (c) the computational complexity increases with the number of buses and trips. Therefore, the application of staged decision making over time becomes practical. Drawing inspiration from this concept, we introduce dynamic programming to break down the problem and facilitate state transitions across stages. This approach allows us to derive a suitable response to disruptions and address a smaller-scale problem with LNS in each stage, consequently reducing the solution space in a rational manner and efficiently obtaining a long-term optimal solution.

According to the typical scheduling phase, we divide the rescheduling problem into hourly stages. Each sub-stage, denoted as ${stage}_k$, is defined by a start time $T_{min}^k$ and an end time $T_{max}^k$, spanning one hour. These stages are guided by the objective presented in Formula (8), and the allocation of buses ${\mathbf{B}}_k$ and trips ${\mathbf{R}}_k$ is determined as follows:

$${\mathbf{B}}_k=\left\{b\right|T_{st}^b\le T_{max}^k\&T_{min}^k<T_{ed}^b\}$$

(16)

$${\mathbf{R}}_k=\left\{i\right|T_{min}^k\le T_i<T_{max}^k,i\in \mathbf{V}\}$$

(17)

where $T_{st}^b$ and $T_{ed}^b$ represent the start and end times of bus b’s daily working schedule.

In each stage, we input the initial spatiotemporal information of the arranged buses and scheduled trips for that period and employ LNS to generate the schedule for that specific time frame. A bus b becomes available for dispatch after completing its previous trip. Thus, the input state for sub-stage ${stage}_k$ can be represented as ${\mathbf{ST}}_{al}^k=\left\{(s_{al}^{b,k},T_{al}^{b,k})\right|b\in {\mathbf{B}}_k\}$, where $T_{al}^{b,k}$ signifies the time when the bus completes its last trip in the previous stage, and $s_{al}^{b,k}$ corresponds to the arrival stop. With specific values of $({\mathbf{B}}_k,{\mathbf{ST}}_{al}^k,{\mathbf{R}}_k)$, we construct the connection network and ultimately produce the solution for ${stage}_k$, denoted as $f_k=\left\{(e_{i,j}^b,T_j)\right|b\in {\mathbf{B}}_k,i,j\in {\mathbf{R}}_k\}$. The corresponding cost $z_k$ can be calculated using (8).

Between stages, the states of buses undergo transitions, allowing us to consider the propagation of disruptions. Buses are scheduled to perform trips in ${stage}_{k+1}$ upon the completion of the ${stage}_k$ schedule. Consequently, the input state for ${stage}_{k+1}$ is exclusively determined by the input state and decisions made in ${stage}_k$. We denote $g_k$ as the process governing the state’s evolution from ${stage}_k$ to ${stage}_{k+1}$, and the transition function is defined in Formula (18).

$${\mathbf{ST}}_{al}^{k+1}=g_k({\mathbf{ST}}_{al}^k,f_k)$$

(18)

With the state representation of each stage and the transition function, we can formalize the problem’s optimization. Given the initial state ${\mathbf{ST}}_{al}^1$ when disruptions occur and K sub-stage solutions $\{f_1,f_2,\dots ,f_K\}$, the total cost of the optimization scheme is represented by Formula (19). Finally, to minimize the total cost, the optimal value of an initial state is given by function (20) [30].

$$J({\mathbf{ST}}_{al}^1;f_1,f_2,\dots ,f_K)=\sum _{k=1}^K{z}_k$$

(19)

$$J^{*}\left({\mathbf{ST}}_{al}^1\right)=minJ({\mathbf{ST}}_{al}^1;f_1,f_2,\dots ,f_K)$$

(20)

Let $J_k^{*}\left({\mathbf{ST}}_{al}^k\right)$, as shown in (21), denote the optimal value of the tail sub-problem that starts from ${stage}_k$, i.e., the minimal rescheduling cost from ${stage}_k$ to ${stage}_K$. Then, dynamic programming solves Function (21) by considering all possible states ${\mathbf{ST}}_{al}^k$ and then proceeds with Formula (22).

$$J_k^{*}\left({\mathbf{ST}}_{al}^k\right)=minJ({\mathbf{ST}}_{al}^k;f_k,f_{k+1},\dots ,f_K)$$

(21)

$$J_k^{*}\left({\mathbf{ST}}_{al}^k\right)=min\left[z_k+J_{k+1}^{*}\left({\mathbf{ST}}_{al}^{k+1}\right)\right]$$

(22)

Following the principles of dynamic programming, we divide the original problem into multiple sub-stages, where the input state of each sub-stage is influenced by the state and decisions from the preceding one. Within each sub-stage, the small-scale problem can be efficiently addressed with LNS, considering dependencies from the previous time period. This approach allows us to preserve a set of sub-schemes $\{f_k^1,f_k^2,\dots \}$ representing unique schedules in ${stage}_k$. Finally, we combine these sub-schemes to obtain a complete solution. As the impact of disruptions may decrease over time, they can often be resolved during earlier sub-stages. To achieve this, we utilize the Dijkstra algorithm [46] to identify an optimal solution and create a comprehensive rescheduling plan. Algorithm 1 outlines the details of the algorithm, and we have established the following two termination conditions for it:

(a) The last stage has been reached.

(b) The solution for the new stage does not incur additional costs, i.e., it matches the original schedule.

Algorithm 1: DP-LNS

Initialize the state ${\mathbf{ST}}_{al}^1$, the list of stages Initialize $C^Q$, $C_{i,j}^D$, $C^P$, $C^H$, $d_{max}$, $t_{min}^{idle}$, $h_{max}$, $\epsilon$, $\gamma$ Create a list, ${\mathbf{L}}_s$, to store all possible input states Begin with ${\mathbf{L}}_s$ containing the initial state, ${\mathbf{ST}}_{al}^1$ $\{f_1^{*},f_2^{*},\dots ,f_k^{*},\dots \}=$ the optimal integrated rescheduling scheme Until termination condition is met    Calculate the cost of operation from the initial state to each state in ${\mathbf{L}}_s$ and sort ${\mathbf{L}}_s$ based on these costs    Select the first element of ${\mathbf{L}}_s$ as the current state, denoted as ${\mathbf{ST}}_{al}^k$, and construct the connection network based on it    Generate multiple sub-schemes $\{f_k^1,f_k^2,\dots \}$ for the current state using Algorithm 2. Calculate new states $\{{\mathbf{ST}}_{al}^{k+1,1},{\mathbf{ST}}_{al}^{k+1,2},\dots \}$ according to the transition function provided in Formula (18)    for each new state:       Update the new state’s corresponding integrated rescheduling schemes $\{f_1^{*},f_2^{*},\dots ,f_k^{*}\}$    end for    Add the newly generated states to ${\mathbf{L}}_s$ and remove the current state from ${\mathbf{L}}_s$ return  $\{f_1^{*},f_2^{*},\dots ,f_k^{*},\dots \}$ correlated to the final state

4.2. Large Neighborhood Search Algorithm

The NP-hard characteristics of the VRSP problem and the nonlinear nature of the proposed ISCR model require a trade-off between solution quality and computational efficiency. Large neighborhood search (LNS) is an effective approach for solving bus operation problems, capable of rapidly converging toward an approximate optimal solution. The LNS algorithm commences with an initial solution and iteratively refines the current solution through a cycle of destroy and repair rules until a locally optimal solution is reached [37]. The primary content of LNS is illustrated in Algorithm 2. With flexible configurations of the initial solutions and destroy and repair rules, our study improves the LNS by incorporating bus operational rules relevant to service capability and regularity to effectively address the bus rescheduling problem.

Algorithm 2: LNS

Initialize parameters Initial solution: Generate an initial solution f Until the costs of results remain stable within a small range    Calculate the selection probability of each position in solution f based on heuristic operators    Destroy: Apply destroy for the solution f based on the probability    Choose an allowed neighborhood to be repaired    Repair: Apply repair for the solution f based on the rule    Update the solution f with any improvements return f

According to the above algorithm, we mainly focus on the design of the initial solution and the destroy and repair rules. The input parameters contain the connection network G, $C^Q$, $C_{i,j}^D$, $C^P$, $C^H$, $d_{max}$, $t_{min}^{idle}$, $h_{max}$, $\epsilon$, and $\gamma$.

Initial solution: In this study, an initial solution f is generated based on the connection network G. The structure of the proposed model reveals the inherent nonlinearity in both the objective $z_H$ and constraint (15), significantly increasing the complexity of the problem. In particular, $z_H$ serves as an objective necessitating feedback regulation, which can be effectively managed through localized adjustments. Additionally, constraint (15) offers a higher level of flexibility. Taking these aspects into account, we opt to relax these elements, thus obtaining a feasible initial solution $f_k$ by addressing the following problem:

$$min{z}^{\prime }=z_Q+z_C+z_P$$

(23)

$$\mathrm{s.t.} \left(9\right)\sim \left(14\right)$$

Problem (23) primarily focuses on ensuring service capability, which constitutes a 0–1 linear programming problem that can be efficiently resolved using existing tools. In this paper, we utilize the Gurobi solver to address problem (23) and procure an initial solution for the subsequent recombination processes.

Destroy rules: Service regularity is a primary challenge to be addressed during the recombination process, and it is closely associated with service frequency. Therefore, we devise a heuristic operator that considers service frequency and makes adjustments to departure times to assist in formulating recombination rules. The destroy operation is intended to identify a trip pair $(i,j)$ for adjustment.

The heuristic operator is designed as the difference between the rescheduled service frequency and the baseline, as shown in Equation (24). Service frequency is characterized by trip departure intervals, with the original average value during the corresponding time periods, denoted as ${\overline{h}}^l$, serving as the baseline. The probability of selecting a trip pair $(i,j)$ is determined by Equation (25). When a departure interval is less than the baseline, it indicates an excess of service supply, potentially leading to resource wastage, and thus necessitates an enlargement of the interval. Conversely, if the departure interval exceeds the baseline, it results in longer passenger waiting times, requiring a reduction. Similar to the expansion of the connection network, we only consider the approach of delaying trips. Therefore, ${cp}_{(i,j)}$ plus or minus indicates which trip needs to be adjusted.

$${cp}_{(i,j)}=(T_j-T_i)-{\overline{h}}^l\forall (i,j)\in {\mathbf{E}}_{sel}^l,\forall l$$

(24)

$$p_{(i,j)}=\frac{\mid {cp}_{(i,j)}\mid }{{\sum }_l{\sum }_{(m,n)\in {\mathbf{E}}_{sel}^l}\mid {cp}_{(m,n)}\mid }$$

(25)

Repair rules: The repair operation is realized by adjusting the departure time of the destroyed trip. Let $\gamma$ denote the adjustment rate. Then, the updated departure time for the selected trip i is calculated using (26).

$$T_i\Leftarrow T_i+\mid {cp}_{(i,j)}\mid \gamma$$

(26)

To ensure the executability of trips, it is also essential to ensure compatibility between the adjusted trip and the neighboring trips performed by the same bus. Furthermore, we limit the departure time adjustments to a range that preserves the order of all trips, thus ensuring timetable stability. Therefore, Formula (26) is modified to create Formula (27).

$$\begin{array}{cc}\hfill T_i\Leftarrow min& [T_i+\mid {cp}_{(i,j)}\mid \gamma ,\hfill \\ \hfill & T_j-1,\hfill \\ \hfill & \sum _b\sum _{j^{\prime }:(i,j^{\prime })\in {\widehat{\mathbf{E}}}^b}{e}_{i,j^{\prime }}^b(T_{j^{\prime }}-t_{min}^{idle}-\mathbf{tra}(D_{S_i},O_{S_{j^{\prime }}})-t_i)]\hfill \end{array}$$

(27)

Finally, we iteratively refine several initial solutions through recombination until convergence is achieved. These outputs constitute the sub-schemes for the stage of dynamic programming, ultimately contributing to the development of the integrated optimal rescheduling scheme.

5. Case Study and Discussion

This section considers the public transit operation of a bidirectional line, the Shenzhen No.10 bus line, as the scenario. There is one depot and two different routes in this case. We perform a series of experiments that simulate breakdown disruptions to assess the effectiveness of our proposed model in ensuring the smooth operation and service regularity of this transit system.

The Shenzhen No.10 bus line boasts comprehensive operational data, allowing us to establish operation-related parameters based on actual data. This particular line covers a total distance of 13.25 km, with 22 stops in the upward direction (departing from stop $s=1$) and 21 stops in the downward direction (departing from stop $s=2$), as illustrated in Figure 3. The data reveal that the line operates approximately 285 trips per day from 06:00 to 23:00, with a maximum departure interval of about 20 min and an average interval of about 8 min. The average travel times for trips in the upward and downward directions are roughly 50 min and 46 min, respectively. In conjunction with navigation information, we established the travel times for buses to deadhead in the upward and downward directions at 35 min and 30 min, respectively. Finally, the experiments are conducted based on operational data for the No.10 bus line over the course of a single day in September 2019.

5.1. Scenario Description

In the breakdown scenario, bus_No_19 experiences a breakdown at 15:00, with one active trip and four remaining trips scheduled. This situation results in a decrease in the number of available vehicles, leaving a total of 115 trips remaining to be completed.

A visual representation of this scenario is depicted in Figure 4. The x-axis represents the departure time, whereas the y-axis represents the identity of the bus. Each rectangle in the figure represents a trip departing at a specific time, with different colors denoting different directions, and the length of each rectangle represents the travel time. Thus, each row in the figure corresponds to a bus schedule. The remaining trips of the disabled bus are highlighted by black boxes.

In this paper, we analyze the impact of different methods on the breakdown scenario using various performance indicators. In terms of the timetable, we consider metrics such as the number of canceled trips (CT), the associated cancellation cost ($z_Q$), and the cost of modifying service frequency ($z_H$). Regarding bus schedules, we calculate the ratio of reassigned trips (RR), the corresponding reassignment cost ($z_P$), and the deadhead cost ($z_C$). Specifically, RR represents the proportion of trips for which the execution relationships change in the optimized output scheme.

5.2. Parameter Configuration

The parameters are established with consideration of the actual operating costs, local economic factors, and results of a sensitivity analysis.

In alignment with the average load rate of the No. 10 bus line, the cancellation cost ($C^Q$) is set at 2000, whereas the deadhead cost ($C_{i,j}^D$) is determined by multiplying the deadheading time by a factor of 10.

Subsequently, a sensitivity analysis encompassing the number of backup vehicles (N), reassignment penalty ($C^P$), and minute change cost of intervals ($C^H$) is conducted. It is anticipated that sustaining such a scheme continuously could be challenging if a specific aspect is significantly compromised. Based on the findings detailed in

Table 2. Results of sensitivity experiments.

N	${\mathit{C}}^{\mathit{P}}$	${\mathit{C}}^{\mathit{H}}$	Timetable			Bus Schedule
			CT	$z_Q$	$z_H$	RR	$z_P$	$z_C$
0	50	10	0	0	2778.26	18.26%	1050	0
		20	0	0	5441.77	17.39%	1000	0
	500	10	0	0	2368.42	12.17%	7000	950
		20	0	0	4601.84	13.91%	8000	950
1	50	10	0	0	2698.59	21.74%	1250	0
		20	0	0	4833.06	13.91%	1050	0
	500	10	0	0	2553.23	15.65%	9000	1250
		20	0	0	4381.10	15.65%	9000	1250

, we perform the following parameter adjustments: the number of backup vehicles (N) is set at 0, the reassignment penalty ($C^P$) is fixed at 500, and the minute change cost for the departure interval ($C^H$) is established at 10.

Taking into account the actual operational context, we configure the parameters as follows: the maximum delay ($d_{max}$) is limited to 10 min, the minimum rest time ($t_{min}^{idle}$) is set at 3 min, and the maximum departure interval ($h_{max}$) is established as 20 min. The adjustment parameter ($\epsilon$) for the soft constraint is defined as 5 min, and the learning rate ($\gamma$) for the recombination rules is set to 0.05. With these parameter settings, the computational cost of the sub-stages is approximately 15 s.

5.3. Comparison Method

The coordination of vehicles requires a series of calculations and feedback, along with consideration of subsequent schedules, which makes manual rescheduling a highly challenging task. Therefore, the most common approach to rescheduling during actual operations is rule-based decentralized control [4]. Consequently, in the absence of backup vehicles, the most common practice is to cancel the remaining trip schedules of the disabled buses when a breakdown occurs. This method does not require reassignments and can be carried out manually with ease.

In this section, our experimental analysis focuses on comparing the effectiveness of different modeling approaches and algorithms. The central consideration in constructing the ISCR model is whether delays are introduced for trips. We further examine the following two scenarios:

(a) With time delay (WT): Time delay is introduced during the construction of the connection network to expand the solution space, and the ISCR model is developed based on this connection network with delayed trips.

(b) Without time delay (WOT): Only trips from the original timetable are taken into consideration, and the ISCR model is constructed based on the connection network without delayed trips.

Furthermore, the following optimization algorithms are compared. Algorithms II-IV solve the problem after dividing it into periods, but they differ in their sub-stage solution approaches and how they create comprehensive scheduling schemes.

(I) Manual approach (MA): Recovers public transit operations using the above manual optimization methods.

(II) Greedy algorithm (Greedy): According to problem (23), the solution of each sub-stage is obtained directly using the Gurobi solver, and the final optimal solution is output by the greedy algorithm.

(III) Greedy with LNS (Greedy-LNS): According to problem (8), the solution of each sub-stage is found by the LNS algorithm, and the final optimal solution is output by the greedy algorithm.

(IV) DP with LNS (DP-LNS): According to problem (8), the solution of each sub-stage is found by the LNS algorithm, and the final optimal solution is output by the Dijkstra algorithm.

5.4. Analysis of Results

Table 3. Computational results in the breakdown scenario.

		Timetable			Bus Schedule			Total Cost	Compared with MA
		CT	$z_Q$	$z_H$	RR	$z_P$	$z_C$	$z$
	MA	5	10,000	2559.19	0.00%	0	1950	14,509.19	-
WOT	Greedy	2	4000	2422.57	12.39%	7000	1950	15,372.57	+5.95%
	Greedy-LNS	2	4000	1681.30	8.85%	5000	1950	12,631.30	−12.94%
	DP-LNS	2	4000	1807.62	6.19%	3500	1950	11,257.62	−22.41%
WT	Greedy	0	0	3285.93	13.04%	7500	1650	12,435.93	−14.29%
	Greedy-LNS	0	0	2512.72	13.91%	8000	950	11,462.72	−21.00%
	DP-LNS	0	0	2368.42	12.17%	7000	950	10,318.42	−28.88%

shows the computational results in terms of the timetables and bus schedules.

Compared to the MA, the rescheduling model (ISCR) and its variants, solved by various intelligent algorithms, consistently demonstrate a notable enhancement in solution quality, achieving at least a 12.94% cost reduction. Through the coordination of multiple buses, the number of CTs decreases from 5 to about 2. This indicates that even an experienced controller using the MA would struggle to accurately analyze problems and thus effectively coordinate buses. As the severity of disruptions increases, the application of the rescheduling model for comprehensive rescheduling becomes necessary. Notably, solving the WOT model using the greedy approach increases the total cost by 5.95%. This increment in cost occurs because the solution reduces the number of CTs but requires more reassignments compared to the MA. Controllers can evaluate the pros and cons of different schemes based on specific situational requirements.

Comparing the results of the different models indicates that considering delay times allows trip cancellations to be significantly reduced. Thus, introducing delay times allows trips to be performed more flexibly, making it easier to obtain an optimal solution containing sufficient trips. However, we also find that modeling using delays changes the service frequency, increasing the corresponding cost $z_H$ by about 35.64%. This additional cost is unavoidable and requires LNS algorithms for further optimization.

The LNS algorithm is implemented based on the heuristic operator designed in this study, which is more effective in identifying irregular departure intervals for optimization. Thus, we can alleviate the imbalanced service frequency caused by introducing delayed trips. The results of WT modeling show that the cost $z_H$ decreases by 23.53% using the proposed LNS algorithm. Moreover, dynamic programming can help obtain a beneficial long-term rescheduling scheme. The available information on trips and buses needs to be determined based on the scheme in the previous period. This means that the state of sub-stages will change according to the different schemes of the previous sub-stage, resulting in a distinct connection network. Dynamic programming can comprehensively evaluate the hysteresis effects of the sub-schemes and, therefore, obtain a comprehensively optimal solution. Significantly, the results also show that at most 13.91% of trips, including those with practical conflicts, need to be reassigned to mitigate the impact of disruptions in the current scenario, which provides a preliminary impact range.

Based on the above analysis, the proposed DP-LNS algorithm can maintain sufficient service capability during different disruptions. However, it is difficult to ensure service regularity because of the variations in timetables caused by delays and cancellations. From the perspective of timetables, departure times decide when passengers can use the transit service, whereas departure intervals denote the maximum waiting time for passengers. Therefore, to further analyze the performance of different algorithms in solving WT models, we calculate the average departure time bias (DB) and the average interval bias (IB) between the original and the optimized schedules. The computations are described by Formulas (28) and (29), where $T_i^{op}$ and $T_i^{or}$ are the departure times of the i-th trip in the optimized and original schedules, respectively.

$$DB=\frac{1}{\parallel {\mathbf{V}}_{sel}^l\parallel }\sum _{i\in {\mathbf{V}}_{sel}^l}\mid T_i^{op}-T_i^{or}\mid \forall l$$

(28)

$$IB=\frac{1}{\parallel {\mathbf{E}}_{sel}^l\parallel }\sum _{(i,j)\in {\mathbf{E}}_{sel}^l}\mid T_j-T_i-{\overline{h}}^l\mid \forall l$$

(29)

Table 4. Analysis of the regularity of transit services.

	Upward				Downward
	DB (min)		IB (min)		DB (min)		IB (min)
	Average	Total	Average	Total	Average	Total	Average	Total
Greedy	6.73	188.33	3.04	173.08	7.49	200.52	2.73	155.52
Greedy-LNS	5.79	127.43	2.19	124.78	6.31	179.0	2.22	126.50
DP-LNS	5.25	110.23	2.10	119.64	6.32	185.52	2.06	117.20

shows the computational results of different algorithms, where a larger value means a greater difference between the optimized and original schedules. From the results, we can conclude that the DP-LNS algorithm not only effectively ensures overall service regularity but also controls the range of the variation, with about a 23.27% bias reduction. Moreover, the service frequency provided by the DP-LNS algorithm is more similar to the original schedule compared to the schedules generated by the other algorithms. This saves passengers from disruptions caused by excessive waiting times while also guaranteeing the reliability of public transit services.

Figure 5 illustrates the advantages of the DP-LNS algorithm, where the x-axis indicates the departure time and the y-axis indicates the cumulative number of departed trips. The curves in the figure represent the implementation of different departure schedules. Each point (x, y) signifies the departure time of the y-th trip at time x under the respective scheme. Therefore, when two curves have points closer together at the same y-coordinate, it signifies a closer alignment of the departure times for the two schemes. As observed in Figure 5, the red curve aligns more closely with the blue curve, indicating that the rescheduling scheme derived from the DP-LNS algorithm exhibits higher consistency with the original schedule. This significantly contributes to the maintenance of bus service stability.

In addition to the solution, the optimization process can provide guidance for the actual operation of a public transportation system. Figure 6 shows the variation curve of the departure intervals, where the x-axis is the sequential number of intervals and the y-axis is the value of the corresponding interval. The greedy algorithm solves problem (23) without considering service frequency. Thus, it may select some trips within the time-delay windows to guarantee sufficient trips, generating disproportionate intervals. For instance, as highlighted by the arrow in Figure 6b, an interval of over 30 min occurs for downward trips, significantly reducing the reliability of the public transit service and passenger satisfaction. However, this also means that there is a shortage of vehicle resources during the corresponding period, suggesting that the use of additional vehicles could improve the quality of the transit service.

We use the local schedule around the interval highlighted by the arrow in Figure 6b as an example to compare the effects of different algorithms, as shown in Figure 7. The local schedule covers part of the optimized schemes during both periods: 18:00–19:00 and 19:00–20:00. Because of the breakdown of bus_No_19 at 15:00, several trips need to be rearranged or canceled, such as trip 8 in Figure 7. Without backup buses, MA controllers cancel the above trips, as shown in Figure 7a, which reduces the capability of bus services. Therefore, the ISCR model adds trips to other buses to generate a certain number of trips. For example, trips are inserted behind trips 1, 4, and 6 to fill the gap in trip numbers during the periods 18:00–19:00 and 19:00–20:00. However, the greedy algorithm ignores the hysteresis effects of the solutions of previous stages, leading to additional rescheduling (e.g., the cancellation of trip 7 in Figure 7b). Thus, the interval exceeding 30 min occurs due to cancellations and reassignments following common methods, leading to fluctuating service frequencies. Therefore, we introduce the LNS algorithm and combine it with dynamic programming to design the DP-LNS algorithm. The heuristic operator, designed by combining domain knowledge, allows the improved LNS to identify the maximum departure interval bias and repair the irregular departure intervals by delaying trip 5 within an allowable time window, as shown in Figure 7c. This adjustment aims to offer a more consistent service for passengers. Compared to the greedy algorithm, dynamic programming comprehensively combines the sub-schemes of different stages to obtain a long-term optimal solution, preventing poor decisions in the early stages that could compromise subsequent plans. Therefore, the requirement of trip numbers in the period 18:00–19:00 can be satisfied by more reassignments, and bus_No_15 can perform trip 7 with a 9 min delay to avoid extra operational costs, as shown in Figure 7d. Finally, dynamic programming produces an optimized scheme that is more similar to the original schedule, making it easier to ensure successful operation and minimize variation in service frequency.

Finally, A visual representation of the optimization scheme for the DP-LNS algorithm is shown in Figure 8.

6. Conclusions

Bus rescheduling is a critical approach to ensuring the operation of a public transit system. When disruptions occur, the most useful methods for restoring operation schedules are rearranging, canceling, or reassigning trips. However, combinations of the above methods are various and intricate. In this paper, we construct a transit rescheduling optimization model based on the connection network. Studies have focused on the successful operation of subsequent trips. However, changes in service frequency caused by delays or cancellations also need to be considered, as these alterations can detrimentally affect passengers’ travel experiences. Therefore, we introduce the deviation from departure intervals and construct the ISCR model to ensure both service capability and regularity, which makes the optimal scheme more achievable and reliable.

The challenges lie in comprehensively considering the hysteresis effects of disruptions and efficiently solving the NP-hard problem and the proposed nonlinear model. Therefore, we propose combining domain knowledge by designing the DP-LNS algorithm. Dynamic programming decomposes the problem and then makes decisions for sub-stages by incorporating the effects of previous stages, which helps overcome long-term issues. Moreover, the LNS algorithm is improved to solve the problem and guarantee the sufficient capability and regularity of the transit service. Finally, we conduct several computational experiments based on an actual public transit scenario in Shenzhen, China. The results reveal that the DP-LNS algorithm effectively reduces the number of canceled trips to facilitate the recovery of transit operations. Moreover, DP-LNS also ensures service regularity through the modification of departure intervals. In conclusion, our method achieves the best rescheduling performance when compared to other approaches.

However, this paper only analyzes rescheduling on a single line. The coordination of multiple lines would further expand the solution space because of the increased availability of resources, which could enable higher-quality solutions to be achieved. In addition, adding a feedback mechanism could improve our algorithm, and considering a multi-layer model will be a topic of our future research.