Journal of Advanced Transportation

Volume 2019, Article ID 2781590, 13 pages

https://doi.org/10.1155/2019/2781590

## Optimizing Vehicle Scheduling Based on Variable Timetable by Benders-and-Price Approach

Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing, China

Correspondence should be addressed to Shiwei He; nc.ude.utjb@ehwhs

Received 17 October 2018; Revised 18 December 2018; Accepted 28 December 2018; Published 6 January 2019

Academic Editor: Guohui Zhang

Copyright © 2019 Zekang Lan 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 practice, vehicle scheduling is planned on a variable timetable so that the departure times of trips can be shifted in tolerable ranges, rather than on a fixed timetable, to decrease the required fleet size. This paper investigates the vehicle scheduling problem on a variable timetable with the constraint that each vehicle can perform limited trips. Since the connection-based model is difficult to solve by optimization software for a medium-scale or large-scale instance, a designed path-based model is developed. A Benders-and-Price algorithm by combining the Benders decomposition and column generation is proposed to solve the LP-relaxation of the path-based model, and a bespoke Branch-and-Price is used to obtain the integer solution. Numerical experiments indicate that a variable timetable approach can reduce the required fleet size with a tolerable timetable deviation in comparison with a fixed timetable approach. Moreover, the proposed algorithm is greatly superior to GUROBI in terms of computational efficiency and guarantees the quality of the solution.

#### 1. Introduction

In the planning process of public bus transport, timetabling problem, and vehicle scheduling problem (VSP) are dealt with by a sequential approach where the solution of previous subproblem is taken as input of the following subproblem, because of the complexity of solving the integrated model of the two planning processes [1]. Both the timetabling problem and VSP have been well-studied. The former one determines the departure and arrival times at each station for each trip, aiming to offer a high level of service for passengers [2–4]. Based on the timetabled trips, the latter one focuses on minimizing the operational cost by arranging the vehicles to cover each trip exactly once and turning back to their corresponding origin terminals or depots after finishing a defined work period, e.g., one day. The connections of trips depend on the arrival/departure times at the terminals. Hence, the process of vehicle scheduling is highly dependent on the timetable. Thus, in order to lower the number of costly vehicles used in vehicle scheduling, a variable timetable with trips shifting in acceptable tolerances is adopted instead of a fixed timetable. However, the process of shifting is a very time-consuming mission and usually not done in a systematic manner [2]. Ceder [2] and Liu and Ceder [5] have presented deficit-function procedures with and without insertions of deadheading trips to assist vehicle schedulers in achieving the most efficient shifts. However, when taking into account fuel restrictions, maintenance considerations, etc., each vehicle has a limited daily workload that implies that the number of trips taken by one vehicle in one day cannot exceed an upper bound (limited trips, LT). In this case, the deficit-function method is not applicable and a new solution method should be proposed. To the best of our knowledge, the VSP based on a variable timetable with limited trips (VSPVT-LT) has not been studied.

Previous studies typically researched into vehicle scheduling based on a fixed timetable. The usual modeling framework of the VSP is built on a connection-based network, where trips and depots are expressed as vertices, and compatible connections between any two vertices are connected using arcs [6–8]. Considering that the connection-based method could bring an immense number of deadhead connection arcs when the number of trips grows, a space-time network modeling concept has been commonly used [8, 9]. Bertossi et al. [10] have proved that single depot vehicle scheduling problem (SDVSP) can be solved in polynomial time; however multiple depot vehicle scheduling problem (MDVSP) is NP-hard even for two depots. Generally, the connection-based model is hard to solve by optimization software when thousands of trips have to be scheduled. Costa et al. [11] and Hadjar et al. [12] formulated the VSP as a Set Portioning Problem which can be solved successfully by column generation. Bodin et al. [13] and Freling and Paixão [14] discussed the vehicle scheduling problem with time constraint, and they constructed a special connection-based network with inserting additional arcs or alternative graphs with different levels to handle the time constraint. Desaulniers et al. [15] and Hadjar and Soumis [16] researched into the vehicle scheduling problem with time window (VSPTW). However, considering that the time windows of departure times may overlap, the minimum headway requirement, which is usually not considered in the VSPTW, has to be imposed in the VSPVT-LT.

In recent years, integrated optimization models for two consecutive phases of public transit have attracted more and more attention. Ibarra-Rojas et al. [17] proposed a completed biobjective integration model that combines the SDVSP and the timetabling problem, and a -constraint method was implemented to obtain Pareto-optimal solutions. On the other hand, most of the literature addresses a sequential integration of the timetabling problem with the VSP. A sequential integration is an approach that considers the characteristics of one subproblem while another subproblem is being optimized. Van den Heuvel et al. [18] applied an iterated heuristic method between a local search strategy to alter the timetable slightly and a network-flow model for vehicle scheduling. In each iteration, the modification of the timetable with an improvement is always adopted, while with a degradation it is accepted with a probability which decreases in the wake of the algorithm processing. Guihaire and Hao [19] also presented an iterated local search algorithm where each iteration implements trip shifting to obtain a neighboring timetable and then the VSP is optimized by an efficient auction algorithm on the neighboring timetable. Liu et al. [20] provided a new biobjective, bilevel mathematical programming model, and a novel deficit-function-based sequential search approach by combining a network-flow technique and a shifting departure time procedure, presented to solve the problem to achieve a set of Pareto-efficient solutions. Fonseca et al. [21] proposed a metaheuristic that iteratively optimized the mathematical formulation of the integrated timetabling and VSP that permitted a subset of timetabled trips shifting only, whereas solving the full VSP.

The remainder of this paper is organized as follows. In Section 2, the overall problem description and notations are elaborated, and a directed network is built to make provision for the connection-based model in the Appendix and the shortest path algorithm in Section 4.2. The path-based model is put forward in Section 3. A computationally efficient solution method that consists of two phases to solve the path-based model is presented in Section 4. The efficiency of the proposed algorithm and the benefits of the vehicle scheduling in a variable timetable are systematically evaluated with numerical experiments in Section 5. Finally, conclusion and future work extensions are discussed in Section 6.

#### 2. The Problem Description and Notations

This study considers a bus line or a set of bus lines. At the ends of each line are terminals. An initial feasible daily timetable generated by timetabling specifies the origin terminal, destination terminal, departure time at origin terminal, and arrival time at destination terminal of each trip. A timetable is feasible if several constraints such as the minimum headway requirement and the minimum travel time of trip are satisfied, where the headway time is the time interval between the departure times of two adjacent trips with the same origin terminal. Considering that the departure time of each trip can be shifted backward or forward in a given range and respecting the minimum headway requirement of operation, there are a lot of possible feasible timetables. In our case, fuel restrictions and maintenance plans are taken into account; that is, the running mileage or operational time of each vehicle in one day cannot exceed a defined upper bound for the vehicle. Assuming that there are a few differences between the running mileages or travel times of any two trips, the restriction is equivalent to limiting the number of trips carried out by each vehicle. The VSPVT-LT has to find out a feasible timetable with the least fleet size in vehicle scheduling with LT constraints, but also needs to balance the timetable deviation in relation to the initial timetable. We do not consider the inclusion of deadheading trips insertions, and we assume that the travel time of each trip is fixed in timetable modification. For illustrative purpose, some notations for the entire article are firstly introduced in Table 1.