Innovative Applications of O.R.
Scheduling of maintenance work of a large-scale tramway network

https://doi.org/10.1016/j.ejor.2018.04.027Get rights and content

Highlights

  • Generating maintenance plans for long planning horizons requires heuristics.

  • Increasing the yearly maintenance budget reduces the overall cost.

  • The quality of the network benefits particularly from a higher maintenance budget.

  • Most tasks should be performed in night shifts rather than during the day.

  • Performing tasks on neighboring segments at once yields savings.

Abstract

For a public transport network preventive maintenance tasks have to be performed periodically to ensure a functioning system. The tasks include costly ones like replacing rails but also cheaper and more frequent ones like grinding. This paper deals with the strategic scheduling of these tasks for a large-scale network and a planning horizon of up to three decades. This planning also requires the consideration of side-effects like the implementation of replacement services with buses if some segments have to be blocked for the maintenance work, or speed restrictions if the replacement of tracks is overdue. The problem is first formulated as a mixed integer programming model which can be solved via commercial solvers. However, its limitations are revealed when it comes to large real-life problems with a long planning horizon. For this purpose, a metaheuristic based on large neighborhood search is developed. The approach is applied to the Viennese tramway network in a case study.

Introduction

Due to the gradual deterioration of the tracks of an urban public transport network regular maintenance measures are required to prevent a breakdown of parts of the system. Greater renewal works are particularly expensive which makes a thorough planning very important. In addition to the renewal of the infrastructure we consider smaller tasks that can either be scheduled separately or be performed in the course of the renewal works. The problem deals with the scheduling of preventive maintenance for a long planning horizon of three decades to cover the renewal of (almost) the complete network. It is motivated by the practical problem faced by the provider of the Viennese tram network.

Maintenance measures are generally needed in a wide variety of fields. With regard to public transport, infrastructure maintenance problems have mainly been discussed in the context of railways, while literature that focuses particularly on tramway tracks is relatively scarce. Railway infrastructure maintenance clearly shows many similarities with maintaining urban public transport systems. Distinguishing aspects, however, arise from the fact that urban public transport lines are typically characterized by short headways while railway systems rather focus on their schedules. This has distinct implications on the passenger behavior, the rerouting of passengers in case of blockades, and the planning of maintenance during train-free windows. Different designs of train-free windows are analyzed by Lidén and Joborn (2016) for railways. When planning these windows conflicting requests from maintenance and regular train operations have to be taken into account.

The survey by Lidén (2015) gives an overview of the research area of railway infrastructure maintenance. The considered problem would therein fit into the category of possession scheduling that deals with the scheduling of maintenance that requires the serviced track to be closed. The costs that arise from the time a track is blocked due to the maintenance is usually called possession costs. Forsgren, Aronsson, and Gestrelius (2013) consider both train and possession scheduling for a given timetable and a given number of possessions in the same model. Their model seeks to schedule all required maintenance tasks while disturbing the original traffic flow as little as possible.

Budai, Huisman, and Dekker (2006) introduce the preventive maintenance scheduling problem that focuses on grouping maintenance on a segment in order to reduce disturbances in the system. The objective is to minimize possession and maintenance costs while a maximum time gap between two consecutive executions of a routine has to hold. The list of maintenance requests for the planning period is given. In contrast, unknown deterioration processes affect the maintenance tasks in the stochastic tactical railway maintenance problem analyzed by Baldi, Heinicke, Simroth, and Tadei (2016) where the planning is performed for a rolling horizon. Cheung, Chow, Hui, and Yong (1999) deal with the railway track possession assignment problem for the subway in Hong Kong. It aims at generating a weekly assignment plan of railway tracks to scheduled maintenance tasks in a way that the number of assigned jobs is maximized according to their priorities.

Similar problems are analyzed in a more general context in the field of maintenance of multi-unit systems. According to Pargar, Kauppila, and Kujala (2017) the literature in this area is extensive. An overview of advances in this field can be found in the surveys by Dekker, Wildeman, and van der Duyn Schouten (1997) and Van Horenbeek, Pintelon, and Muchiri (2010). Pargar et al. (2017) study the scheduling of preventive maintenance and renewal projects in an integrated way. In particular they analyze the effects of grouping tasks and finding a good balance between maintenance and renewal. In their model grouping is encouraged by reducing the system downtime cost and other synergy effects that can be achieved by scheduling tasks in the same period and on adjacent units. The benefits of grouping tasks are also considered by Caetano and Teixeira (2015) in the context of scheduling renewal operations for the tracks of a railway line. They assume that the longer the jointly renewed segments are, the higher the economies of scale become.

The problem considered in the paper at hand builds upon some aspects discussed by Budai et al. (2006). These aspects include the scheduling of recurrent tasks with known maintenance intervals. In contrast to Budai et al. (2006), maintenance can be performed in different shift types. Performing a task during the day time necessitates the blockade of some segments. In general, multiple segments are affected by a blockade, while Budai et al. (2006) consider only the blockade of the segment under maintenance. Moreover, grouping effects are specified in a different way. In our model grouping similar tasks on neighboring segments yields savings, while Budai et al. (2006) identify a set of tasks that can be performed in parallel and thereby blocking the segment only once. Furthermore, we consider costs for necessary speed restrictions if a segment has not been maintained for a long time.

Particularly when rails reach the end of their lifetime the question arises whether it is economically preferable to perform the maintenance when it is due or to prepone the renewal work. This aspect of the problem is similar to the discussion about balancing maintenance and renewal by Pargar et al. (2017). Another similarity between our model and the one by Pargar et al. (2017) refers to the consideration of grouping affects. With respect to the system downtime cost our model differs from theirs as a scheduled task typically does not affect the whole system. Similarly to Caetano and Teixeira (2015), the expenses for the maintenance in each period are constrained by a certain budget.

In the literature the closely related problems are typically approached by representing them as mixed integer programming MIP models. This includes the work by Budai et al. (2006), Caetano and Teixeira (2015), and Pargar et al. (2017). Caetano and Teixeira (2015) use the proposed MIP model together with a commercial solver as their sole solution approach. In addition to their MIP approach, Budai et al. (2006) present two heuristics that iteratively schedule routine works first and improve the schedule afterwards. Another two heuristics are presented for a restricted version of the problem. Pargar et al. (2017) suggest a decomposition heuristic that generates plans for each component individually. The schedule is then used to enhance their MIP approach in two different ways.

The contribution of the paper is twofold. First, a comprehensive MIP model for scheduling maintenance measures is presented. In combination with a powerful solver the MIP approach is applicable to medium- to large-sized urban transportation networks with a limited planning horizon. The second contribution refers to a fast metaheuristic based on large neighborhood search (LNS) by Shaw (1998) particularly dedicated to long planning horizons. The paper is structured as follows. Section 2 gives a detailed description of the problem. The corresponding MIP model is presented in Section 3. In Section 4, the basic LNS approach is explained first, followed by a description of a set covering (SC) phase that extends the algorithm. The benefits of the different features of the proposed approach are analyzed in Section 5. A case study about the Viennese tramway network is also presented in Section 5. Finally, Section 6 concludes.

Section snippets

Problem description

The network is represented by a graph that consists of nodes that are connected with edges, while only those edges are incorporated that are actually traversed by lines. The maintenance has to be planned for the edges, or synonymously segments. If a maintenance task is scheduled on a segment, this task is performed on the whole segment, not just parts of it. In general, the network can be partitioned in an arbitrary way. However, nodes may be defined naturally by using the stations of the

Mathematical model

In this section the MIP model that represents the problem is described. The notation is listed in Table 1, where the first part refers to the index sets and the second part to the model parameters. The model extends the one by Budai et al. (2006) as discussed in the introduction.

Algorithm

In this section the developed metaheuristic based on LNS by Shaw (1998) is presented. Its basic idea is to repetitively destroy and subsequently repair parts of the incumbent solution. The results of the metaheuristic can be improved by recombining the partial schedules generated by LNS in a SC phase. These partial schedules refer to maintenance plans of single segments. The combination of metaheuristics and exact methods, often denoted as matheuristics, has been successfully applied by several

Numerical tests

In this section, the MIP and the LNS approach are tested on instances based on the the Viennese tram network. The results are generated by two Intel Xeon E5-2650v2 processors with eight cores each running at 2.6 gigahertz. The LNS approach is restricted to a single core, though. The results of LNS are generally based on 10 runs, each with a runtime of 3 minutes for small instances, 15 minutes for intermediate instances, and 60 minutes for the large instance plus a maximum of 5 minutes for the

Conclusion

The paper presented a comprehensive strategic model for scheduling maintenance measures for the infrastructure of a public transport system. The model incorporates various features including different shift types for performing the work, speed restrictions if segments are not maintained regularly, costs for replacement services if the activities are scheduled during the daytime, and synergy effects whenever similar tasks are grouped on neighboring segments. In order to achieve these synergy

Acknowledgments

The computational results presented have been achieved (in part) using the Vienna Scientific Cluster (VSC). The financial support by the Austrian Federal Ministry of Science, Research and Economy and the National Foundation for Research, Technology and Development is gratefully acknowledged.

References (19)

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