Integrated optimization of customized bus routes and timetables with consideration of holding control

Highlights

• Integrates holding control strategy into the customized bus service design.

• Establishes a universal formulation with many realistic concerns and restrictions.

• Combines genetic algorithm and large neighborhood search with embedding branch-and-bound method.

• Proposes an order clustering based divide and conquer approach that achieves a great improvement in solving time.

• Compare between different solution methods.

Abstract

Customized bus (CB) is an innovative urban transport system that plays a significant role in meeting personalized travel needs of passengers. This paper deals with the CB service design problem, which is the problem of driving a fleet of capacitated vehicles to pick up and deliver passengers, with consideration of holding control that keeps vehicles at stops for a period of time. A mixed-integer linear programming model is proposed to jointly optimize a set of vehicle routes and timetables with passenger assignment scheme. Through decomposing the model into a bilevel one, we combine genetic algorithm (GA) and large neighborhood search (LNS) with embedding branch-and-bound (BB) method, and develop a GA-LNS-BB hybrid algorithm. Furthermore, an order clustering based divide and conquer (OC-D&C) approach is provided for the ultra-large-scale problems, in which the order clustering problem is formulated as solving an integer linear programming. Finally, we conduct several numerical experiments on the Sioux Falls network and a real-world city-scale network to evaluate the performance of the proposed model and solution methods. The results indicate that: (1) holding control could enhance the flexibility of CB service, and significantly increase the CB carrier’s profit; (2) the GA-LNS-BB hybrid algorithm and the OC-D&C approach could provide high-quality solutions in short time for large-scale/ultra-large-scale problems compared to other standard algorithms.

Introduction

With the development of the Internet and intelligent transportation facilities, citizens’ travel need is gradually developing from unification to personalization. As one of the most basic transit service modes, bus service that operates on a predetermined route according to a predetermined timetable has been challenging to meet the diverse travel needs of citizens. Under this background, an emerging transit service mode, called customized bus (CB), has been springing up, which arranges vehicle routes and timetables according to passenger needs to provide an advanced, timely, personalized, and flexible transit service. Taking Beijing city as an example, as of September 2022, there are 376 customized bus routes with a daily passenger volume of over 20,000.¹

The CB service aims to offer more personalized and convenient travel experience for groups of passengers with similar travel needs. Compared to traditional bus service, the CB vehicle routes have fewer intermediate stops, and the timetables could be adaptive to the required time windows of passengers (Lyu et al., 2019). The process of CB service is that passengers first submit their orders (including origin/destination stops and expected boarding/alighting time) to the CB carrier through some websites and/or smartphone applications. Based on the submitted orders, the CB carrier would generate vehicle routing schemes. If one passenger’s need is satisfied by some vehicle route, he/she proceeds to reserve a seat and make payment in advance.

With given passenger orders, the CB carrier needs to deal with the service design problem, i.e., the optimization of a set of vehicle routes and timetables with passenger assignment scheme. The goal is to define a set of routes for a planning area, with each route associated with a sequence of stops (Guo et al., 2019, Huang et al., 2020, Ma, Wang, and Xu, 2020), and determine the arriving and departure times at stops of vehicles to meet the passengers’ travel requests (Liu and Ceder, 2015, Ma, Zhao, et al., 2017). Given a set of vehicle routes and timetables, the passenger assignment scheme matches the passengers with the vehicles (Gong et al., 2021, Tong et al., 2017, Wang et al., 2020). Existing studies on CB service design have considered some effective measures to improve the efficiency of CB service, but rarely consider the holding control strategy that keeps vehicles at stops for a period of time. For each vehicle, holding control takes effect by first deciding on some holding stops along the route and then holding vehicle’s departure time within maximum allowed time, which increases the possibility of matching more passenger orders. It is worth noting that although holding control at stops would slightly increase some passengers’ travel time, it may largely reduce the overall operating cost and improve the operating profit of CB service. Thus, this paper aims to fill such a gap by involving holding control in the CB service design.

The CB service design problem with holding control is a variant of the vehicle routing problem with pickup and delivery time windows (VRPPDTW). To tackle with VRPPDTW, previous studies mainly focused on exact algorithms and heuristic algorithms. The exact algorithms include branch-and-price algorithm (Dou, Meng, & Liu, 2021), Lagrangian relaxation (Mahmoudi & Zhou, 2016), branch-and-cut (Cordeau, 2006), and so on, which usually have limited solving capability and are only suitable for small-scale problems (up to 20 nodes). For solving large-scale VRPPDTW problems, various heuristic algorithms were designed to derive a good enough solution in a short time, including genetic algorithm (GA) (Arbex and da Cunha, 2015, Xu et al., 2021), ant colony optimization (ACO) (Asghari, Al-e, & Rekik, 2022), particle swarm optimization (PSO) (Kechagiopoulos & Beligiannis, 2014), simulated annealing (SA) (Ma, Guan, & Wang, 2020), large neighborhood search (LNS) (Montenegro, Sörensen, & Vansteenwegen, 2021), adaptive LNS (Xue, Song, He, An, & Wang, 2022), and so on. In these heuristic algorithms, GA is the most widely used due to its ease of implementation and extensibility. However, GA has poor local search ability, making it easy to fall into local optima prematurely. A general optimization technique is to combine GA with other methods, such as the combinations of GA and SA (Sun, Chien, Hu, Chen, & Jiang, 2020), GA and branch-and-cut algorithm (Guo, Guan, & Zhang, 2018), GA and variable neighborhood search (Chen, Wang, & Ma, 2021), etc. It is confirmed that the combination may use the advantages of both algorithms and overcome GA’s deficiency. Along this direction, this study is devoted to implementing a method combining GA with large neighborhood search (LNS) (Shaw, 1998). LNS method could explore a complex neighborhood through the use of a two-phase procedure which partially relaxes a given solution and repairs it. In addition, cluster-first route-second methods are also a kind of commonly used heuristic techniques in solving vehicle routing related issues, which determine clusters of nodes to be visited and solve a traveling salesman problem for each cluster (Prins, Lacomme, & Prodhon, 2014). After reviewing the existing literature, we find that cluster-first route-second methods have rarely been adopted in the field of CB service design. Since the CB service design problem has more realistic concerns and restrictions, how to use cluster-first route-second methods to route the CB vehicles is worthy of study.

This paper incorporates the issues of vehicle route design, timetable development and passenger assignment into the CB service design problem. The objective is to maximize the profit of the CB carrier. In order to construct a universal formulation, the flow balance constraints, travel time constraints, holding time constraints, order assignment constraints, time window constraints, vehicle capacity constraints, and route length constraints are all involved. A comprehensive solution framework is proposed to solve problems of different scales. The branch-and-bound (BB) method is taken to solve the small-scale problems. For solving the large-scale problems, we combine GA and LNS with embedding BB method, and develop a GA-LNS-BB hybrid algorithm. Finally, an order clustering based divide and conquer (OC-D&C) approach is provided to solve the ultra-large-scale problems, which performs a spatial–temporal clustering of the order set and then determines the vehicle routes on each cluster.

The main contributions can be summarized as follows:

(1) This paper establishes a mixed-integer programming (MIP) model to formulate the CB service design problem, where decision-making considerations in vehicle routing, vehicle timetabling with holding control, and passenger assignment are concurrently integrated. The consideration of holding control improves the flexibility of vehicle scheduling in CB service and increases the possibility of matching more passenger orders.

(2) Through decomposing the proposed MIP model into a bilevel one, a GA-LNS-BB hybrid algorithm is proposed to solve the large-scale problems. The combination of GA and LNS is responsible for determining the outer-level vehicle routing decision, while the BB method is adopted to optimize the inner-level vehicle timetables to match the passengers’ time window requirements under a given set of vehicle routes, which makes this heuristics get more accurate and stable results.

(3) Based on the spatial–temporal characteristics of CB orders, an OC-D&C approach is developed for the ultra-large-scale problems, in which the order clustering problem is formulated as solving an integer linear programming.

The rest of this paper is organized as follows. Section 2 reviews the related literature. Section 3 introduces the problem description. Section 4 discusses the model formulation. Section 5 provides the solution methods for the proposed model with details. Section 6 conducts numerical experiments to evaluate the performance of the model and solution methods. Finally, we conclude this paper and discuss future work in Section 7.