European Journal of Operational Research 256 (2017) 44–54 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Discrete Optimization Optimal allocation of buffer times to increase train schedule robustness Predrag Jovanovi ´ c a,* , Pavle Kecman b , Nebojša Bojovi ´ c a , Dragomir Mandi ´ c a a Faculty of Transport and Traffic Engineering, University of Belgrade, Serbia b Department of Science and Technology, Linköping University, Sweden a r t i c l e i n f o Article history: Received 5 July 2015 Accepted 7 May 2016 Available online 12 May 2016 Keywords: Buffer times Capacity Knapsack problem Timetable robustness a b s t r a c t Reliability and punctuality of railway traffic are among the key performance indicators, which have a significant impact on user satisfaction. A way to improve the reliability and on-time performance in the timetable design stage is by improving the timetable robustness. In order to increase the robustness, most railway companies in Europe insert a fixed amount of buffer time between possibly conflicting events in order to reduce or prevent delay propagation if the first event occurs with a delay. However, this often causes an increase of capacity consumption which is a problem for heavily utilised lines. A sufficient amount of buffer time can therefore not be added between every two conflicting events. Thus, buffer times need to be allocated carefully to protect events with the highest priority. In this paper we consider the problem of increasing the robustness of a timetable by finding an optimal allocation of buffer times on a railway corridor. We model this resource allocation problem as a knapsack problem, where each can- didate buffer time is treated as an object with the value (priority for buffer time assignment) determined according to the commercial and operational criteria, and size equal to its time duration. The validity of the presented approach is demonstrated on a case study from a busy mixed-traffic line in Sweden. © 2016 Elsevier B.V. All rights reserved. 1. Introduction Reliability and punctuality of railway traffic are among the key performance indicators, which have a significant impact on user satisfaction (Hansen & Pachl, 2014). Trains typically run accord- ing to a timetable that contains the scheduled departure and ar- rival times for all trains. A train path in a timetable is represented as a sequence of running times over railway line sections and dwell times in stations. An important constraint of railway traf- fic scheduling is that the trains which run over the same infras- tructure elements (block sections between two signals or station tracks and routes) need to be separated in order to prevent colli- sions. In real time operations, this task is performed by the safety and signalling systems. However, in order to prevent conflicts, that often result in unnecessary braking and re-accelerating of hindered trains, the conflicting train paths in a timetable need to be sep- arated at least by minimum headway time. Minimum headway times are computed with respect to microscopic train routes using blocking time theory and added between train paths in stations (Hansen & Pachl, 2014). * Corresponding author. Tel.: +381 113091239. E-mail addresses: p.jovanovic@sf.bg.ac.rs (P. Jovanovi ´ c), pavle.kecman@liu.se (P. Kecman), nb.bojovic@sf.bg.ac.rs (N. Bojovi ´ c), drama@sf.bg.ac.rs (D. Mandi ´ c). The inevitable variability of running and dwell times, may cause delays and render a timetable infeasible. Moreover, in busy and heavily utilised networks, a delay of a single train can easily prop- agate to other trains that share the same infrastructure and/or have a planned passenger transfer, rolling-stock or crew connec- tion (Goverde, 2010; Kecman, Corman, D’Ariano, & Goverde, 2013). In order to increase the robustness of a timetable to delay prop- agation resulting from the deviations in process times, minimum headway times are often extended with buffer times (Goverde & Hansen, 2013). Fig. 1 shows how buffer times are inserted be- tween train paths. The purpose of a buffer time is to (partially) absorb the deviation of the first train from the scheduled trajec- tory and prevent delay propagation to the second train (Kroon, Maróti, Helmrich, Vromans, & Dekker, 2008). Therefore, in case of an initial delay of a train, buffer times are essential for keeping the planned schedule of other trains feasible. However, this approach may cause an increase of travel times and infrastructure capacity consumption. Moreover, the unused time reserves represent a di- rect loss of available capacity (Mattsson, 2007; Vromans, Dekker, & Kroon, 2006). For that reason, finding a well-balanced allocation of buffer times is an important problem in the stage of tactical plan- ning and timetable design. In this paper we present a knapsack problem based approach for buffering a timetable in a heavily utilised railway network. For http://dx.doi.org/10.1016/j.ejor.2016.05.013 0377-2217/© 2016 Elsevier B.V. All rights reserved.