Railway Maintenance Reservation Scheduling Considering Train Traffic and Maintenance Demand

Contact information

Author: Bodhi Buurman

Student number: s2265672

Email address: b.buurman@student.utwente.nl bodhi.buurman@prorail.nl

Comissioned by: ProRail - Department ”Capaciteit voor Beheer”

Internship: December 2020 - June 2021 External supervisors: H.C. Zandman

M.J. Brandt

Internal supervisors: Prof. Dr. Ir. E.C. van Berkum Dr. K. Gkiotsalitis

University: University of Twente

Master: Civil Engineering & Management Track: Transport Engineering & Management

Transport & Logistics

Version: Final Version

Amersfoort, June 10, 2021

This thesis is intellectual property of ProRail

Preface

Before you lies the thesis ”Railway maintenance reservation scheduling considering train traffic and maintenance demand”, the basis of which is a mathematical model to solve the maintenance scheduling problem. It has been written to fulfill the graduation requirements of the Civil Engineering and Management masters program at the University of Twente. I was engaged in researching and writing this thesis from December 2020 to June 2021.

The project was undertaken at the request of ProRail, where I undertook an internship. This thesis could not have been written without the support of my supervisors Harmen Zandman and Marco Brandt. I would like to thank both my supervisors for their guidance and support. They provided me with the necessary input, feedback and direction for me to complete the thesis.

Unfortunately, due to the COVID-19 pandemic, I was unable to meet with my supervisors on a regular basis. However, both supervisors invited me to online meetings, which helped me with advancing my thesis and with getting to know some colleagues. I would also like to thank all colleagues from ProRail that helped me with my research.

I would also like to thank Eric van Berkum and Konstantinos Gkiotsalitis for helping me with improving the scientific parts of my thesis by giving comments, feedback and input. I would also like to thank them for being part of my graduation committee.

I would also like to thank all interviewees for providing me with information and insight that helped improve the relevance of this thesis.

A special thanks to my girlfriend, family and friends who helped me find relaxation and keep me socially engaged in this special time for graduating.

After this graduation project, I’ve been offered a job at Dynniq Mobility. I’m very excited to apply the knowledge I obtained during my studies in the field. I’ve enjoyed my internship at ProRail very much and will miss my colleagues.

I hope you enjoy reading my thesis.

Bodhi Buurman

Amersfoort, 10 June 2021

Image front page: ©ProRail

Summary

Railway maintenance works are performed to reduce the probability of the occurrence of a failure on the components of the railway infrastructure. The scheduling of maintenance work is quite important in supporting the normal daily operation of the railway. By proper maintenance scheduling, operational hindrance can be minimized and contractors are given space to execute maintenance activities flexible. This thesis aims to optimize the maintenance schedules for both train operators and maintenance contractors, by considering hindrance and flexibility for both stakeholders respectively.

This thesis achieved this by modelling important factors contributing to both objectives and all relevant constraints in a multi-objective optimization problem. The important factors contributing to the objectives were obtained by interviewing experts in the field and stakeholders.

The most important factors found contributing to hindrance were detouring, rescheduling trains, and relocating parked stock. In order to model detouring of trains, two path finding algorithms were modified to consider physical train travel limitations. The most important factors found contributing to flexibility were the number of maintenance slots and the spread of these slots.

The multi-objective optimization model is solved by two methods. The methods presented for solving the multi-objective model are the ε-constraint method and NSGA-II. Both solution strategies are tested on four toy networks to analyse performance. The comparison between the performance of both strategies is shown in the table below. One can see that the ε-constraint method is faster and as optimal as NSGA-II for small networks (|E| ≤ 15). For larger networks (|E| > 15) NSGA-II is faster and approaches the true optimum, but does not reach true optimal- ity. The ε-constraint method is preferred for small networks and NSGA-II for large networks.

The difference between space coverage with respect to the ε-constraint method increases with problem size.

TN Runtime ε-constraint Runtime NSGA-II % Diff S-metric

1 0.7 s 38.0 s 0

2 6.7 s 43.3 s 0

3 44.0 s 46.5 s 6.4

4 3600 s 75.7 s 19.8

In a case study, the Dutch railway network is assessed and used to create new maintenance schedules based on the new model. The case study did not consider the relocation of parked stock. The solution strategy applied was NSGA-II, since the input pre-processing time for the ε-constraint method is too large. Since the case study networks was very large (|E| = 329), the schedule of current ProRail was given as an initial solution guess. NSGA-II found three new maintenance schedules for the case study. An overview of these solutions in comparison to the current schedule is shown in the table below.

Model 1 Model 2 Model 3

Decrease hindrance total (%) 15.4 -3.4 22.1

Decrease hindrance freight (%) 23.9 3.1 24.4

Decrease hindrance passenger (%) -63.0 -63.0 0

Increase flexibility (%) 85.5 86.6 80.3

The model is able to reduce hindrance by up to 24.4 percent and increase flexibility by up to

86.6 percent. Considering the KPI’s and the main goal of distributing railway capacity between

stakeholders in the Netherlands, solution Model 3 is recommended by the researcher.

Samenvatting

Onderhoud aan het spoor wordt gedaan om de kans op falen van de verschillende infrastruc- turele componenten te verminderen. Het inplannen van onderhoudswerkzaamheden is belangrijk voor het ondersteunen van de dagelijkse operatie. Door het degelijk inplannen van werkzaamhe- den aan het spoor kan de operationele hinder geminimaliseerd worden en krijgen onderhoud- saannemers de ruimte flexibel werkzaamheden uit te voeren. Dit proefschrift probeert onder- houdsroosters te optimaliseren voor zowel vervoerders als aannemers, door rekening te houden met hinder en flexibiliteit respectievelijk.

Dit proefschrift bereikt dit door de belangrijke factoren die bijdragen aan hinder en flexi- biliteit te modelleren in een meerdoelig optimalisatieprobleem. De factoren die bijdragen aan hinder en flexibiliteit zijn gevonden door interviews te houden met experts en belanghebbenden.

De belangrijkste gevonden factoren die bijdragen aan hinder zijn het omrijden, herplannen, en herloceren van treinen. Om het omrijden van treinen te modelleren zijn twee padvind algoritmes aangepast, zodat deze rekening kunnen houden met de fysieke limitaties van treinreizen. De be- langrijkste gevonden factoren die bijdragen aan flexibiliteit zijn het aantal onderhoudsmomenten en de spreiding van deze momenten.

Het meerdoelig optimalisatie model is opgelost met twee methodes. Deze methodes zijn de

”ε-constraint” methode en NSGA-II. Beide strategie¨ en zijn toegepast op fictieve netwerken om de prestatie te analyseren. De vergelijking van de prestaties van beide methodes zijn weergegeven in de tabel hieronder. Er kan gezien worden dat de ”ε-constraint” methode sneller en net zo optimaal is als NSGA-II voor kleine netwerken (|E| ≤ 15). Voor grotere netwerken (|E| > 15) is NSGA-II sneller en benaderd het ware optimum, echter raakt NSGA-II deze niet. ”ε-constraint”

is geprefereerd voor kleine netwerken en NSGA-II voor grote netwerken. De verschil in de S- metric ten opzichte van ”ε-constraint” neemt toe met probleemomvang.

TN Tijd ε-constraint Tijd NSGA-II % Verschil S-metric

1 0.7 s 38.0 s 0

2 6.7 s 43.3 s 0

3 44.0 s 46.5 s 6.4

4 3600 s 75.7 s 19.8

In een case study is gekeken naar het Nederlandse spoor en zijn er onderhoudsroosters gemaakt op basis van het nieuwe model. De case study heeft geen rekening gehouden met hinder ten gevolge van het herloceren van treinen. De gebruikte methode is NSGA-II, gezien het voorberekenen van de input voor de ”ε-constraint” methode te tijdrovend is. Gezien het netwerk groot was (|E| = 329) is het huidige onderhoudsrooster van ProRail gebruikt als initi¨ ele oplossingsgok. NSGA-II heeft drie nieuwe onderhoudsroosters gevonden voor de case study. Een overzicht van deze oplossingingen ten opzichte van het huidige onderhoudsrooster is te zien in de tabel hieronder.

Model 1 Model 2 Model 3

Vermindering hinder totaal (%) 15.4 -3.4 22.1

Vermindering hinder vracht (%) 23.9 3.1 24.4

Vermindering hinder passagier (%) -63.0 -63.0 0

Verhoging flexibliteit (%) 85.5 86.6 80.3

Het model is in staat de hinder te verminderen tot 24.4 procent en de flexibiliteit te verhogen

to 86.6 procent. Rekening houdend met de KPI’s en het hoofd doel in het verdelen van spoor

Contents

1 Introduction 9

2 Literature review 11

2.1 Previous work . . . . 11

2.2 Research gap and contribution . . . . 13

3 Problem description 16 3.1 Interviews . . . . 16

3.1.1 Rail-infra manager . . . . 16

3.1.2 Freight operators . . . . 16

3.1.3 Passenger carriers . . . . 17

3.1.4 Maintenance contractors . . . . 18

3.2 Multi-objective mathematical model . . . . 19

3.2.1 Sets, parameters, and variables . . . . 21

3.2.2 Objective functions . . . . 22

3.2.3 Constraints . . . . 23

4 Methodology 24 4.1 ε-constraint method . . . . 24

4.2 Metaheuristic NSGA-II . . . . 25

5 Numerical experiments 28 5.1 Toy networks . . . . 28

5.1.1 Inputs . . . . 28

5.1.2 Results . . . . 29

5.1.3 Extended testing directions . . . . 31

5.2 Case study: The Dutch railway network . . . . 31

5.2.1 Current Schedule . . . . 31

5.2.2 Inputs . . . . 33

5.2.3 Key performance indicators . . . . 37

5.2.4 Results . . . . 38

6 Conclusion 43

7 Discussion 47

Appendix A Stakeholder analysis 52

Appendix B Method of interviewing 54

Appendix C Modified A* pathfinding algorithm 56

Appendix D Algorithm for fitness functions of NSGA-II 57

Appendix E Modified Depth First Search algorithm 59

Appendix F Detailed methodology for maintenance demand analysis 61

Appendix G Toy network inputs 64

Appendix H Recommended maintenance schedule all nights 75

List of Figures

1 Process of turnround for freight operators . . . . 17

2 Process of turnround for passenger carriers . . . . 18

3 Rail infrastructure contracting areas in the Netherlands . . . . 19

4 Visualisation of toy networks . . . . 28

5 Pareto fronts for both solution strategies for each toy network . . . . 30

6 Changing rail yard vertices to handle hindrance on parked stock . . . . 31

7 Current Schedule for the Monday Tuesday night . . . . 32

8 Dutch railway network . . . . 35

9 Modelling double edges . . . . 35

10 Methodology for maintenance demand analysis . . . . 36

11 Pareto front found for the case study . . . . 39

12 Results of the model versus the current schedule based on KPI’s . . . . 40

13 Resulting maintenance schedules for the Thursday - Friday night . . . . 41

A.14 Power-Interest grid . . . . 53

F.15 Visual representation of the normative period . . . . 63

H.16 Recommended maintenance schedule for the Monday - Tuesday night . . . . 75

H.17 Recommended maintenance schedule for the Tuesday - Wednesday night . . . . . 76

H.18 Recommended maintenance schedule for the Wednesday - Thursday night . . . . 77

H.19 Recommended maintenance schedule for the Thursday - Friday night . . . . 78

H.20 Recommended maintenance schedule for the Friday - Saturday night . . . . 79

H.21 Recommended maintenance schedule for the Saturday - Sunday night . . . . 80

H.22 Recommended maintenance schedule for the Sunday - Monday night . . . . 81

List of Tables 1 Possession time per maintenance activity (Lid´ en, 2014) . . . . 9

1 Literature topics and model objectives. . . . 14

2 Nomenclature multi-objective mathematical model . . . . 21

3 Cardinality of the sets for each toy network . . . . 29

4 NSGA-II specific inputs for the toy networks . . . . 29

5 Computational results for both solutions strategies on the toy networks . . . . . 29

6 Hypervolume for both solutions strategies on the toy networks . . . . 30

7 Data scheme for vertices of the network graph . . . . 34

8 Data scheme for edges of the network graph . . . . 34

9 Constraint violations per constraint without initial solution guessing . . . . 37

10 NSGA-II specific inputs for case study . . . . 38

11 Percentage difference between solutions and current schedule . . . . 45

B.12 Characteristics of all stakeholder interviews . . . . 54

F.13 Validities for every marking in the data . . . . 62

F.14 Prioritization rules for every validity code . . . . 62

G.15 Inputs for toy network 1 . . . . 64

G.16 Inputs for toy network 2 . . . . 66

G.17 Inputs for toy network 3 . . . . 68

G.18 Inputs for toy network 4 . . . . 71

1 Introduction

Rail transport is one of the safest and most environmentally friendly means of conveyance of passengers and goods. By making regions and markets more accessible, it plays a main role in the development of countries due not only to its impact on the economy but also to its social role (Budai et al., 2006). Statistics have shown that the demand for railway transport has increased considerably in the European Union in the last few years (Eurostat, 2020). In the Netherlands, the estimated growth of train traffic demand is 30 percent by 2040 (ProRail, 2021b). In order to satisfy the demand, there is a need for a high quality and modern railway infrastructure, for reliable service, for more trains per hour, for railway safety and improved punctuality. However, increasing the number of trains (and their speed) leads to an increase of deterioration of infrastructure (Budai and Dekker, 2004). Therefore, it is important that enough preventive maintenance of the infrastructure is scheduled.

Preventive railway maintenance works are performed to reduce the probability of the oc- currence of a failure on the components of the railway infrastructure and/or maximize the operational benefit (Kumar et al., 2012). Preventive maintenance on railways can be subdivided into small routine works and projects. The routine (spot) maintenance activities consist of in- spections or small repairs, e.g. inspection of rail, switch, level crossing, overhead wire, signaling system and switch lubrication. These works do not take much time to be performed and are done frequently, few times a year (Esveld, 2001). The projects include renewal works and consists e.g.

of ballast cleaning, rail grinding and tamping. They are carried out once/twice in a few years (Esveld, 2001). Lid´ en (2014) described and categorized the maintenance-oriented activities that take place on railway infrastructure from a planning and scheduling perspective. In table 1, Lid´ en (2014) listed the different activities according to the needed amount of possession time per work shift.

Table 1: Possession time per maintenance activity (Lid´ en, 2014) Possession Time Activity

> 8 hours Catenary wire replacements Track / turnout replacement 4 - 8 hours Tamping of tracks

Grinding

Switch replacement

Catenary inspection and maintenance 1 - 4 hours Tamping of tunrouts

Ultra-sonic testing

Fasteners, joints, rail repair, etc.

0 - 1 hours Inspection

Grinding

Signal repair, vegetation, etc.

Slippery rail, snow removal 1 hour - X days Accidents, urgent repair As train slots Periodic measurement

Fast grinding

Cheung et al. (1999) states it is required to schedule sufficient preventive maintenance work

so that the railway can be operated without any disruption during the operating period. On

being less than five hours a day. To carry out maintenance activities, that particular working zone of the railway system had to be taken out of service (Den Hertog et al., 2005), also known as train-free periods. This is done to ensure the safety of maintenance crews. Moreover, a set of rules and procedures has to be followed in scheduling the maintenance work. Therefore, the scheduling of maintenance work is quite important in supporting the normal daily operation of the railway. The scheduling of maintenance work is to schedule train-free periods.

In the Netherlands, ProRail distributes the capacity of the infrastructure between all its users. This is capacity for traffic and capacity for maintenance. In 2020, 34 passenger and freight carriers asked for capacity on the Dutch railway network for train traffic (ProRail, 2021c). In addition to this, ProRail needs to deal with projects demanding railway network capacity and contractors requiring capacity to perform maintenance activities. In distributing the capacity of the Dutch railway network, the main goal of ProRail is to run more trains over the network, in a safe way and with less hindrance. Working towards the main goal of ProRail, enough capacity for maintenance needs to be reserved, such that contractors can perform all required maintenance activities. The capacity for maintenance activities is thus planned in train-free periods. These train free periods are planned during the night on all sections of the railway infrastructure with a weekly cycle. ProRail always looks to improve the distribution of the capacity between traffic and maintenance. ProRail does this since the capacity reservation for maintenance has implications on the hindrance on train traffic. The larger the maintenance reservation, the more hindrance traffic experiences. This is in conflict with the main goal ProRail has in distributing the capacity of the Dutch railway network. However, a large reservation of capacity for maintenance if favorable for maintenance contractors needing to perform the maintenance. The more train-free periods available, the more flexibility contractors have in planning the required maintenance activities.

Lid´ en (2015) states that the scheduling solution is of critical importance for the safety, re-

liability and efficiency, both for operators and contractors when train-free periods and train

paths are in conflict or influence each other. In this thesis, the problem of scheduling train-free

periods such that train traffic is hindered as little as possible, whilst keeping the flexibility of

maintenance contractors as large as possible. This is modelled by an multi-objective optimiza-

tion model, since the traffic hindrance and maintenance flexibility are experienced by different

stakeholders. The cost of hindrance and the value of flexibility minimised and maximised respec-

tively. This thesis focuses on the medium term planning, on the tactical level, determining how

often (per week) when (which night) en where (what part of the infrastructure) capacity needs

to be reserved for preventive maintenance activities, optimizing the aforementioned hindrance

and flexibility.

2 Literature review

This chapter is used to discuss existing literature on maintenance scheduling within the rail domain (section 2.1). Section 2.2 discusses the research gap identified from the literature. From the identified gap, the research objective, questions and contributions are determined (section 2.2).

2.1 Previous work

Lid´ en (2014) performed a survey on railway maintenance activities and a literature review on mathematical algorithms to solve the planning and scheduling problems. The problems were classified as strategic, tactical, or operational. In the strategic class problems concerning dimensioning, localization, and organization, with time horizons of one to several years are listed.

Tactical problems include scheduling, timetabling and construction of plans covering a medium to long time horizon (weeks to year), often handling resources as categorized, anonymous objects.

In the operational class, problems concerning implementation and effectuation, covering short time horizons (hours to month), usually handling the real resources are described (Lid´ en, 2015).

This research focusses on the tactical level of scheduling and planning, specifically on possession scheduling. Possession here refers to maintenance. Whenever possessions and train paths are in conflict or influence each other, the scheduling solution is of critical importance for the safety, reliability and efficiency, both for operators and contractors (Lid´ en, 2015). Lid´ en (2014) stated that the scheduling of major possessions, coordinated with the train traffic, is perhaps the key planning problem regarding railway infrastructure maintenance. This is because it is conducted all the way from freight corridor to timetable revision planning, has a fundamental impact on the traffic capacity, and frames the work planning and cost conditions.

The definition of possession scheduling is in line with the problem of maintenance scheduling in which both train operators and contractors are involved, and possessions and train paths are influencing each other. Previous research in possession scheduling listed by Lid´ en (2015) are categorized in three main themes:

• The first theme is the construction and scheduling of regular possession patterns that give repeated track access.

• The second theme is the coordination of maintenance tasks, to minimize possession time and maintenance cost.

• The final theme is the adjustment of a given maintenance plan, to maximize transportation throughput.

Higgins (1998) modelled a maintenance schedule aimed at determining the best allocation of maintenance activities and crews to minimize the disruption to and from scheduled trains and to reduce completion time. The solution Higgins (1998) used was a tabu search heuristic for which the neighborhood was defined by swapping the order of jobs, maintenance crews, or both.

The model is subject to constraints such as available budget; maintenance activity precedence;

track availability; and minimum travel time between track links (Higgins, 1998).

Cheung et al. (1999) reports on a case study in Hong Kong on applying constraint-satisfaction

techniques to solve a real-world resource-allocation problem, called the Railway Track Possession

Assignment Problem. The problem is to assign railway tracks to a given set of scheduled

maintenance tasks according to a set of constraints. The primary objective of the Railway

Track Possession Assignment Problem is to produce an assignment plan that maximizes the

requests as possible, before considering any lower priority job requests) and the satisfaction of all rules (Cheung et al., 1999).

Den Hertog et al. (2005) focused on the optimal division of maintenance working zones on the Dutch railway system based on the location of switches. van Zante-de Fokkert et al. (2007) solved the maintenance scheduling problem in two steps. First, they used the working zone division to define single track grids (STGs), which are sets of working zones that can be blocked simultaneously. In the second step, a mixed integer programming (MIP) model was used to assign STGs to nights to create the maintenance schedule. The model of van Zante-de Fokkert et al. (2007) considered the trade-off between the number of nights in the schedule and the contractor workload.

Lake et al. (2000) modelled the short-term scheduling of track maintenance activities, after the train schedule and maintenance activities have been planned. The model schedules the track maintenance activities, including assigning the maintenance crews to the activities, with the objective of minimizing the total maintenance costs. The formulation allows different crews to be assigned to the same activity at different times and incorporates the individual set-up and take-down times required each instance an activity is conducted. This maintenance scheduling model has been solved using a two stage heuristic technique, with the first stage generating a feasible solution and the second stage implementing simulated annealing (Lake et al., 2000).

Budai et al. (2006) researched the preventive maintenance scheduling problem (PMSP), which can be defined as scheduling a set of routine activities and projects, such that the track possession costs are minimized (Budai et al., 2006). A mathematical formulation was given for a single section, and several greedy heuristics were used to solve the problem fast. The research conducted by Budai et al. (2006) focused on developing operations research tools, which help the maintenance planners to come up with optimal maintenance plans. This research was thus more on the side on the maintenance contractor. The research was extended upon by Budai-Balke (2009), Pouryousef et al. (2010), and Jenema (2011) considering computational time, multiple segments, and/or more detailed costs.

Boland et al. (2013) presented a MIP model for the maintenance scheduling at the Hunter Valley Coal Chain, where the primary objective is to maximize the traffic throughput, and the secondary objective is to minimize deviations from a given initial schedule. Based on a network flow model of the system Boland et al. (2013) proposed a mixed integer programming formulation for this planning task. To deal with the resulting large-scale model which cannot be solved directly by a general-purpose solver, two steps are proposed. The number of binary variables is reduced by choosing a representative subset of the variables of the original problem, and a rolling horizon approach enables the approximation of the long term (i.e. annual) problem by a sequence of shorter problems (for instance monthly). (Boland et al., 2014) solved the problem formulated in 2013 by using four local search-based heuristic methods.

More recent studies into maintenance scheduling focused on several topics. The first topic these studies covered is computational time. Faris et al. (2018) considered the computational time of the railway maintenance scheduling problem. In this study, three distributed optimiza- tion methods were proposed: Parallel Augmented Lagrangian Relaxation (PALR), Alternating Direction Method of Multipliers (ADMM), and Distributed Robust Safe But Knowledgeable (DRSBK). The original distributed algorithms are modified to handle the non-convex nature of the optimization problem and to improve the solution quality. The scalability of these methods was analyzed. The distributed approaches discussed can be seen as heuristic methods to solve the MIP problem because there is no guarantee for convergence toward the global optimum (Boyd et al., 2011).

Another topic covered by research is integrating train timetabling and maintenance schedul-

ing. These activities are mutually exclusive they must be coordinated and should ideally be planned together (Lid´ en and Joborn, 2016). This problem was presented by Ruffing (1993), which included the operational restrictions for trains passing a work site. Albrecht et al. (2013) showed how the Problem Space Search (PSS) meta-heuristic can be used for large scale prob- lems to create timetables in which both train movements and scheduled track maintenance are considered. One key strength of the research is the generality of the Possession Plans On De- mand (PPOD) system, able to model train movements on a variety of rail networks due to the representation of the problem. It is important to note that PPOD is not intended to provide a ready-to-go timetable but a starting platform for refinement by the train controller (Albrecht et al., 2013).

Forsgren et al. (2013) presented a MIP model that optimizes a production plan regarding both trains and preventive maintenance. The goal for the optimization was to find the best possible traffic flow given a fixed set of planned maintenance activities. This study treats the tactical timetable revision planning case, handles a network with both single and multi-track lines, allows trains to be rerouted or cancelled and consider different running times depending on train stops.

Lid´ en and Joborn (2016) presented a mixed integer programming model for solving an inte- grated railway traffic and network maintenance problem. The aim of the model is to find a long term tactical plan that optimally schedules train free windows sufficient for a given volume of regular maintenance together with the wanted train traffic. A spatial and temporal aggregation is used for controlling the available network capacity.

Integrating train timetabling and maintenance scheduling is also considered by Meng et al.

(2018) by modelling maintenance tasks as virtual trains and minimizing the total run time of all trains. Zhang et al. (2019) used linearization techniques to formulate a mixed integer linear programming (MILP) model to identify the operation modes and the timetable of night trains, by integrating the time window selection of regular maintenance on high-speed railways.

Nijland et al. (2021) optimized the maintenance schedule for both train operators and main- tenance contractors, whilst distinguishing maintenance engineering fields to provide a better balance between train traffic and maintenance management. The method presented for design- ing maintenance schedules is a mixed-integer linear programming (MILP) model that considers these aspects. Nijland et al. (2021) assessess the computational costs when using exact (branch- and-bound) or metaheuristic solution methods for solving networks with up to 25 work zones.

Another topic researched is condition-based maintenance planning. One example is the trav- elling maintainer problem (Camci, 2015) for geographically distributed assets using prognostic information obtained from real-time condition-based monitoring. The resulting MINLP problem is solved by heuristics. Su et al. (2017) developed a multi-level approach for both condition- based maintenance planning and the clustering of individual track defects, taking into account the disruption cost to train traffic. Su et al. (2019) developed a multi-level decision making approach for optimal condition-based maintenance planning of a railway network divided into a large number of sections. At higher level, a chance-constrained Model Predictive Control (MPC) controller determines the long-term section-wise maintenance plan, minimizing condi- tion deterioration and maintenance costs for a finite planning horizon, while ensuring that the deterioration level of each section stays below the maintenance threshold. At a lower level, the optimal short-term scheduling of the maintenance interventions is formulated as a capacitated arc routing problem.

2.2 Research gap and contribution

In the past decades, many studies have researched the topic of maintenance (or possession)

example by integrating train timetabling and maintenance scheduling, or by maximizing train throughput. Table 1 shows the topics each paper of the reviewed literature covers with the objectives of the corresponding model.

Table 1: Literature topics and model objectives.

Study Topic Model Objective

Author Timetable Maint. Comp. Hindrance Maint.

Scheduling Scheduling Time on traffic Efficiency

Ruffing (1993) X X

Higgins (1998) X X

Cheung et al. (1999) X X

Lake et al. (2000) X X

Budai et al. (2006) X X

van Zante-de Fokkert et al. (2007) X X

Budai-Balke (2009) X X

Pouryousef et al. (2010) X

Jenema (2011) X X

Albrecht et al. (2013) X X X

Boland et al. (2013) X X X

Forsgren et al. (2013) X X X

Boland et al. (2014) X X X

Camci (2015) X X

Lid´ en and Joborn (2016) X X X X

Su et al. (2017) X X

Faris et al. (2018) X

Meng et al. (2018) X X X

Su et al. (2019) X X

Zhang et al. (2019) X

Nijland et al. (2021) X X X

This study X X X

All studies reviewed focused on maintenance scheduling, whilst some studies also focused on integrating train timetable scheduling. The identified research gaps are:

• All researches that were aimed at minimising traffic hindrance did this based on train travel times and delays. Nijland et al. (2021) is the only study where hindrance on parked stock is included. This is a rising issue due to the increasing amount of rolling stock that needs to be stored over night and the decreasing storage space.

• Studies regarding maintenance efficiency mostly discussed the minimisation of maintenance crew workloads or maintenance costs. However, the value of the maintenance reservation is never considered.

• Regarding constraints in literature, the working capacity of maintenance crews is con-

sidered by van Zante-de Fokkert et al. (2007) and Nijland et al. (2021). Maintenance

thresholds of deteriorating infrastructure were considered by Su et al. (2019), and Jenema

(2011) took into account the length of maintenance activities. These types of constraints

are used to realistically model the maintenance crews performing the maintenance. How-

ever, little to no constraints considered the routes of train traffic.

• Maintenance scheduling is always formulated as a single objective optimization problem (SOOP) considering either hindrance on traffic or maintenance efficiency, or considering both in a single objective function. However, hindrance on traffic is experienced by a differ- ent stakeholder than the stakeholder experiencing the maintenance efficiency. By consid- ering the multi objective optimization problem (MOOP), multiple optimal distributions of the maintenance schedule can be determined, which opens room for decision-making between stakeholders.

This research aims to find multiple optimal maintenance schedules that balance the require- ments of train traffic and maintenance. The thesis tries to achieve this by creating an MOOP, with the objectives of minimizing hindrance for train operators and maximizing contractor flex- ibility. Train operator hindrance is similar to hindrance on traffic and contractor flexibility is similar to maintenance efficiency in this case. The aim of the thesis is translated into the following research question:

Main question: How can optimal maintenance schedules be created when minimiz- ing hindrance for train operators and maximizing flexibility for contractors?

Two sub-questions are formulated to support answering the main question. The first sub- question is formulated to understand train operator hindrance and contractor flexibility. A stakeholder analysis is performed and key stakeholders will be interviewed to uncover the aspects of hindrance and flexibility for operators and contractors respectively. The second sub-question is formulated to support help guide the research in solving the MOOP. Two solutions strategies are researched and implemented which will generate Pareto optimal solutions.

Sub-question 1: What are the factors contributing to train operator hindrance and contractor flexibility, and how can these be modelled?

Sub-question 2: How can the multi-objective optimization problem be solved in order to find Pareto optimal maintenance schedules?

The contribution of the thesis is thus twofold. First the objectives are modelled in more detail and are evaluated separately in a MOOP, since the costs/benefits are for different stakeholders.

Second, the model is solved by using meta-heuristics, such that it is solvable for large railway networks, improving the practical applicability of the thesis. The result is an overarching model that generates maintenance schedules that are applicable for railway networks, reflecting the requirements of train operators and maintenance contractors more realistically than the current state-of-the-art.

Finally, in this research two path finding algorithms (A* pathfinding and Depth First Search)

were altered to include the limitations of train travel.

3 Problem description

The problem investigated in this thesis is the design of a weekly repeating maintenance sched- ule for preventive maintenance activities that can be performed during nights to prevent large disturbances for train traffic. The maintenance schedule consists of distributed maintenance slots, in which preventive maintenance activities can be planned by contractors. Maintenance slots have a spatial scope (they cover a certain part of the railway infrastructure) and a temporal scope (they are planned in a certain night of the week). Creating a weekly maintenance sched- ule thus consists of distributing maintenance slots over time and space, such that every part of the railway infrastructure can be maintained. Section 3.1 discusses the, according to stake- holders, important aspects contributing to hindrance and flexibility. Section 3.2 captures the maintenance scheduling problem with all stakeholder aspects in an multi-objective optimization model.

3.1 Interviews

In order to understand the requirements for railway maintenance scheduling, experts in the field are interviewed. Interviews are performed with four stakeholder groups, namely the rail-infra manager, freight operators, passenger carriers, and maintenance contractors. These interviews gave insight into what each party requires with respect to the maintenance schedule.

This section is used to discuss the main findings of the interviews and how these findings are used in the proposed scheduling model. For the stakeholder analysis see Appendix A and for the method of interviewing see Appendix B.

3.1.1 Rail-infra manager

The rail-infra manager owns and maintains the maintenance schedule. The most important aspect for the rail-infra manager is that the maintenance schedule takes into account both the requirements from train operators and maintenance contractors. This means that the mainte- nance schedule should at least meet the traffic and maintenance demand. One exception on the maintenance demand is made. The infra manager accepts some maintenance slots to be planned during the realisation phase of the maintenance schedule, called incidental slots. This means the maintenance schedule does not have to meet the maintenance demand on all edges. For example, if for a certain edge the maintenance demand is 26 slots in eight weeks. Then, if the maintenance schedule provides 3 slots per week on this edge, the schedule will provide 24 slots in eight weeks. This would not be sufficient and the schedule is forced to have 4 slots per week on the edge. Now, if we accept 2 incidental slots on this edge, 3 slots per week would become sufficient (3 * 8 + 2 = 26). This could be helpful in reducing hindrance for train operators, because if less slots are planned, more rail-infrastructure is available for traffic. The number of maintenance slots planned during realisation accepted by the rail-infra manager is captured by parameter ψ ^slot .

Planning slots during the realisation phase also brings downsides. If such a slot is planned on a traffic path, traffic needs to be rescheduled and other regular maintenance slots could be invalidated. This is also a form of hindrance, which is captured in objective function (1) and constraint (4).

3.1.2 Freight operators

For freight train operators it is important that the cost of the route they take is as low as possible and that trains can always run. The cost of a route is mainly dependent on operational costs. These operational costs consist of two parts. First, the cost of utilizing the infrastructure.

Railway operators are charged by the railway manager for using the infrastructure. The more

infrastructure is used by an operator, the more this operator needs to pay. Second, the cost of

operating trains. This consists of the cost of personnel, gas, wear to material. The operational costs a freight operator has are defined in this research per kilometer by the value Λ ^detour _{f reight} .

The cost of a route is proportional to the time the trip takes. The longer a trip takes, the more expensive it is, since trains are used longer, personnel works longer, and the infrastructure is used more. Detouring thus is the main hindrance freight operators experience. Furthermore, freight train operators are hindered by ”kop maken”. “Kop maken” is the action of turning the driving direction of the train as illustrated in figure 1”. The english translation for ”kop maken”

is turnround (Gutter, 2016). This process takes roughly 30 minutes for a freight operator to complete.

Figure 1: Process of turnround for freight operators

This thesis assumes that on all freight corridors, the same speed is driven, set at 100 km/h.

This means the optimal route for an freight operator is the shortest route in distance. With this assumed driving speed, turnround is equal to 50 kilometers distance (30 minutes multiplied by 100 kilometers per hour). Every time a train needs to perform turnround, 50 kilometers is added to the distance of the route. To calculate the cost per route for a freight train operator, Λ ^detour _{f reight} is multiplied by the distance of the route, including the distance penalty each time of turnround.

This is modelled in objective function (1) in section 3.2.2. Turnround is also included in two shortest path algorithms as described in Appendix C and Appendix E.

For a freight operator, if a train cannot make its trip, all revenue is lost. For this reason, all the traffic demand a freight train operator has, must be able to make the trip. This is modelled by equations (6), (7), and (10) in section 3.2.3.

3.1.3 Passenger carriers

Passenger carriers experience hindrance on parked stock at rail yards when maintenance

requires the rail tracks to be free of material or the catenary system to be voltage-free. This is

a problem since the storage space is limited and the amount of material passenger carriers have

increases. If one rail yard is closed for maintenance during the night and no material can be

stored there, nearby rail yards need to remain open. Modelling the hindrance on parked stock

for passenger carriers is described in objective function (1) in section 3.2.2 and by constraint (9)

in section 3.2.3.

Another form of hindrance for passenger carriers is the detouring of the nightly train traffic.

This hindrance is modelled the same as hindrance for freight operators with a different opera- tional cost per kilometer (see equation (1) in section 3.2.2). The operational cost per kilometer for passenger carriers is equal to Λ ^detour _passenger . Furthermore, the process of turnround is not a problem for passenger operators, as illustrated in figure 2. The distance penalty for turnround is thus not included in the route lengths for passenger carriers.

Figure 2: Process of turnround for passenger carriers

3.1.4 Maintenance contractors

The most important aspect for contractors is that they can perform all required maintenance activities. This is to ensure the quality of the railway infrastructure is high. To achieve this, contractors want to be able to work as flexible as possible, meaning that contractors want as much capacity reserved for maintenance as possible. This is modelled in equation (2) in section 3.2.2. By doing this, contractors can decide when and where they employ maintenance crews to perform the required maintenance activities. Contractors thus want the maximum number of reservations possible per maintenance zone per week, or as much maintenance slots as possible.

Contractors are also limited by the amount of staff they have. The maintenance for the

Dutch railway system is split into nine zones (see figure 3). For each zone, a singular maintenance

contractor is tendered. The contractor is responsible for all maintenance activities in the zone(s)

the contractor is tendered for. Due to the contractor being limited by its staff, the spread of

maintenance slots in a week is important. This thesis assumes that one maintenance crew

maintains one zone per night. For example, if a contractor has two maintenance crews, then

the contractor needs to be able to go to at least two maintenance zones each night such that

the contractor can deploy all available maintenance crews. In other words, the maintenance

contractor needs enough work available each night, such that the contractor can employ all its

personnel. This means that, in this example, the contractor needs two maintenance slots per

night in all zones the contractor is responsible for as a minimum. This is modelled by equation

(5) in section 3.2.3.

Esri Nederland, Community Maps Contributors

Figure 3: Rail infrastructure contracting areas in the Netherlands

Finally, maintenance slots cannot be planned in all nights. For example, the Friday to Saturday night, almost no maintenance work is performed. This is based on conditions such as maintenance crew availability, type of day (weekday or weekend), or time of maintenance (nightly). To ensure the model only plans maintenance is allowed nights, constraint (8) is added.

3.2 Multi-objective mathematical model

The goal of the this research is to assign edges of a train network to certain nights, thus creating maintenance slots on these edges. In order to do this, a multi-objective mathematical model is formulated. The model takes into account the following aspects from the interviews:

• Detouring of freight and passenger trains (including turnround). The detouring of a train is equal to the distance (in kilometers) that train has to travel more than the shortest route between its origin and destination.

• Relocating parked stock when closing rail yards. The hindrance on parked stock for closing

rail yard X is equal to the number of trains that can be stored at rail yard X multiplied

by the average route length to the other nearby rail yards Y . This is multiplied by the

cost of driving parked stock one kilometer.

• The scheduling of slots during the realisation phase of the schedule, or incidental slots. This means that the maintenance schedule can provide less slots per edge than the maintenance demand of the contractor. Further, hindrance on trains due to incidental slots is included.

• Ensuring contractors can perform all required maintenance activities. The number of scheduled slots per maintenance zone should equal the maintenance demand of a contrac- tor. By spreading maintenance slots, the limitations of maintenance staff are included.

Further, maintenance slots cannot be planned in all nights.

Data is required to include the aspects from the interviews into the model. For example, to model detouring, the train demand, the possible routes the train demand can travel over the network, and the excessive length of all these routes is required. The data required for the model are:

• A set of vertices and edges that make up a railway network graph.

• The pairs of edges that cannot be scheduled for maintenance for the same night.

• The nights certain edges cannot be scheduled for maintenance.

• The contracting areas and the corresponding contracting area for all edges.

• A set of origin-destination pairs and the corresponding traffic demand.

• The maintenance demand per normative period (a period in which the maintenance de- mand peaks, see also ”normative period” in Appendix F) per edge of the network graph.

• A set of considered routes, with for each origin-destination pair the edges present in the routes and the excessive length of each of these routes (extra length on top of the shortest route) including turnround.

• The hindrance on parked stock when scheduling an edge for maintenance.

• The monetary cost of detouring trains, relocating parked stock, and rescheduling trains.

The monetary value of maintenance slots.

The results of the model are:

• Pareto optimal maintenance schedules, with a distribution of maintenance slots for each night in the maintenance schedule.

• The hindrance for train operators based on detouring, rescheduling and relocating for each Pareto optimal solution.

• The flexibility for contractors based on the number of slots planned above the maintenance demand for each Pareto optimal solution.

The rest of this section is used to discuss the sets, parameters, variables, objective functions,

and constraints of the multi-objective mathematical model.

3.2.1 Sets, parameters, and variables

The sets, parameters, and variables are shown in table 2.

Table 2: Nomenclature multi-objective mathematical model Sets

V vertices of the network

E edges of the network

N nights of the maintenance schedule

C contracting areas

O operators

D ⊂ V × V origin-destination pairs, i.e. if vertices v

and v

form an origin-destination pair and v

6= v

, then v

, v

∈ D

S ⊂ E × E set of not allowed edge combinations, i.e. if edges e

and e

cannot be planned for maintenance in the same night and e

6= e

, then e

, e

∈ S

R = {Z

| ≤ min(K)} considered routes, where K = {k

, k

, ..., k

} in which k

is the maximum number of routes possible between OD-pair d, d ∈ D

Parameters

δ

maintenance demand for edge e in the normative period, e ∈ E

δ

train traffic demand for operator o for OD-pair d in night n, o ∈ O, d ∈ D, n ∈ N ω

binary parameter that indicates whether the train demand for OD-pair d in night n is

larger than 0 (ω

= 1), or not (ω

= 0), d ∈ D, n ∈ N

α

binary parameter that indicates whether edge e belongs to contracting area c (α

= 1), or not (α

= 0), e ∈ E, c ∈ C

β

binary parameter that indicates whether edge e can be assigned to night n (β

= 1), or not (β

= 0), e ∈ E, n ∈ N

ζ

minimum number of required maintenance slots for contracting area c in night n, c ∈ C, n ∈ N

ρ

binary parameter that indicates whether edge e is used in route r for OD-pair d, d ∈ D, r ∈ R, e ∈ E

ν

excessive length of route r for OD-pair d compared to the length of the shortest route for OD-pair d, d ∈ D, r ∈ R

Λ

monetary value of a single maintenance slot

Λ

monetary cost for detouring one kilometer for train operator o, o ∈ O

Λ

monetary cost of relocating parked stock due to closing edge e. If the edge e is a rail yard then Λ

> 0, otherwise Λ

= 0.

Λ

monetary cost for rescheduling trains when incidental slots are planned during the reali- sation phase of the maintenance schedule

ψ

maximum number of incidental slots per edge the schedule can be less than the main- tenance demand per normative period (allows for ψ

incidental slots per normative period per edge)

ψ

number of days in the normative period

ψ

ψ

= b

c, conversion parameter from the planning window to a normative period planning

M a big positive number

Variables

x

binary variable that indicates whether edge e is closed for maintenance in night n (x

= 1), or not (x

= 0), e ∈ E, n ∈ N

y

binary variable that indicates whether route r is used for OD-pair d in night n (y

= 1), or not (y

= 0), d ∈ D, r ∈ R, n ∈ N

z

positive integer variable that indicates the number of incidental slots to be planned during the realisation phase of the schedule per normative period for edge e, e ∈ E

Note that the consideration of the set R is not valid if any of the OD-pairs in set D have an adjacent origin and destination, and either the origin or destination vertex has degree one (the vertex only has one adjacent vertex).

3.2.2 Objective functions

Using the nomenclature from table 2, the first objective function is shown in equation (1).

The first objective function minimizes the cost caused by hindrance over all train operators.

Minimisef

(x) = X

o∈O

(Λ

X

d∈D

X

r∈R

X

n∈N

(δ

× ν

× y

))

| {z }

Total cost of detouring

+ Λ

ψ

X

e∈E

(z

)

| {z }

Total cost of rescheduling

+ X

e∈E

(Λ

X

n∈N

(x

))

| {z }

Total cost of relocating

(1)

In equation (1), the part δ _o,d,n ^train ×ν _d,r ×y _d,r,n is equal to the excessive length of the chosen route r for OD-pair d in night n multiplied by the train demand for operator o for that same OD-pair d and the same night n. The excessive length multiplied by the demand is summed over all nights, OD-pairs, and routes. The summation, which equals the total amount of detoured kilometers for operator o, is multiplied with the cost of detouring one kilometer for the corresponding operator o. This results in the total cost of detouring. The sum over all edges of z _e is equal to the number of slots to be planned during realisation per normative period for the entire network. This is multiplied by the train rescheduling cost Λ ^reschedule and divided by the conversion parameter ψ ^window to get the rescheduling hindrance per planning period. Finally, the sum of x _e,n over all nights is equal to the number of times a maintenance slot is planned on edge e. Multiplying this with Λ ^relocate _e results in the hindrance on parked stock per edge e. By summing this over all edges the total hindrance on parked stock is calculated.

The second objective function maximizes the amount of scheduled slots in the maintenance schedule, by maximizing the value of the maintenance slots above the maintenance demand.

Using the nomenclature from table 2, the second objective function is shown in equation (2).

Maximise f 2 (x) = Λ ^slot ψ ^window

X

z∈Z

(ψ ^window X

n∈N

(x e,n ) − δ ^maint _e ) (2)

In equation (2), the part P

n∈N (x _e,n ) calculates how many times an maintenance slot is planned on edge e over all nights. This is multiplied by the conversion factor to get from the planning window to the normative period. For example, if the planning window is 1 week and the normative period is 8 weeks, ψ ^window is equal to 8. Then, from the planned slots per edge over all nights, the maintenance demand per normative period δ _e ^maint for that edge is subtracted.

This results in the difference in planning and demand per edge, i.e. how many more slots are

scheduled than demanded per edge per normative period. These differences are summed over

all edges. The total difference is multiplied by the value of a maintenance slot Λ ^slot divided by

the conversion factor ψ ^window . The result is the value of the reserved slots per week above the maintenance demand. This is equal to the flexibility of a contractor.

3.2.3 Constraints

Using the nomenclature from table 2, the constraints are shown by equations (3) to (13).

ψ ^window × X

n∈N

(x _e,n ) ≥ δ ^maint _e − ψ ^slot ∀ e ∈ E (3)

ψ ^window × X

n∈N

(x e,n ) + z e ≥ δ ^maint _e ∀ e ∈ E (4)

X

e∈E

(x _e,n × α _e,c ) ≥ ζ _c,n ∀ c ∈ C, n ∈ N (5)

− M × (1 − y _d,r,n ) ≤ X

e∈E

(ρ _d,r,e × x _e,n ) ∀ d ∈ D, r ∈ R, n ∈ N (6) X

e∈E

(ρ _d,r,e × x _e,n ) ≤ M × (1 − y _d,r,n ) ∀ d ∈ D, r ∈ R, n ∈ N (7)

x _e,n ≤ β _e,n ∀ e ∈ E, n ∈ N (8)

x _e

_,n + x _e

_,n ≤ 1 ∀ (e ₁ , e ₂ ) ∈ S, n ∈ N (9)

X

r∈R

(y _d,r,n ) = ω _d,n ∀ d ∈ D, n ∈ N (10)

x e,n ∈ {0, 1} ∀ e ∈ E, n ∈ N (11)

y d,r,n ∈ {0, 1} ∀ d ∈ D, r ∈ R, n ∈ N (12)

z e ∈ Z ≥0 ∀ e ∈ E (13)

Constraint (3) ensures that the number of slots planned per edge meet the demand for that edge minus ψ ^slot slots. The ψ ^slot remaining slots can be planned during the realisation phase of the schedule. (4) is used to set z _e to the difference between the demand and planning, i.e. the number of slots to be planned during realisation. Constraint (5) ensures that each contracting area has enough slots available each night. Constraints (6) and (7) ensure that always one route is available for each OD-pair in each night if necessary. These constraints flow from the if-then condition: if a route is available and necessary in a certain night for an OD-pair (y _d,r,n = 1), then all the edges in this route for that OD-pair should not be planned for maintenance in this night P

e∈E (ρ _d,r,e × x _e,n ) = 0. Constraint (8) ensures that maintenance slots are only planned in the nights they are allowed. Constraint (9) ensures that edges that cannot be planned for maintenance in the same night are not planned for maintenance in the same night. Constraint (10) ensures that only one route is used for each OD-pair in each night if that OD-pair has train demand in that night, or no route is used for each OD-pair in each night if that OD-pair has no train demand in that night. Constraints (11) and (12) ensure that both variables x e,n and y d,r,n

are binary. Finally, constraint (13) ensures that z e is an integer larger than or equal to 0.

4 Methodology

The proposed model is a multi-objective optimization model with two objectives. A multi- objective optimization problem involves several conflicting objectives and has a set of Pareto optimal solutions (Zhou et al., 2011). The Pareto optimal solutions (or non-dominated solutions) form a Pareto front. The solution approach will result in multiple Pareto optimal solutions, which balance hindrance and flexibility. This chapter discusses the solution strategy employed to solve the multi-objective mathematical model. Two strategies are used. The first strategy is the ε-constraint method as described in section 4.1. The second strategy is a metaheuristic called NSGA-II, which is used to solve scalability problems of the ε-constraint method. The metaheuristic NSGA-II is described in section 4.2.

4.1 ε-constraint method

The first method used to solve the MOOP is the ε-constraint method adapted from (Haimes et al., 1971). The ε-constraint method requires one of objective functions (1) and (2) to be set as an constraint. The objective that is set equal to a constraint than needs to be ≤ ε if the objective was to minimise or ≥ ε if the objective was to maximise. Here, ε is a parameter with a single value. Objective function (2) is made into an constraint in equation (14).

Λ ^slot ψ ^window

X

z∈Z

(ψ ^window X

n∈N

(x e,n ) − δ ^maint _e ) ≥ ε (14)

By doing this the multi-objective mathematical program is transformed in an integer linear program (ILP), and can be formally written as:

Minimise f 1 (x) (15)

s.t. Eqs.(3) − (14) (16)

ε is given different values to find the Pareto optimal solutions. The pseudo code of the ε-constraint algorithm for solving the multi-objective optimization model is given in algorithm 1. This algorithm starts with an initial value of ε, and solves the program stated by equations (15) and (16). This results in solution ˆ x, which is used to calculate the flexibility (f ₂ (ˆ x)). The new value of ε is then set to the f 2 (ˆ x) + ∆, where ∆ is a small number (∆ ≤ 0.01). This process continues until the stated problem becomes infeasible. The initial value of ε is set equal to the minimal possible flexibility. The minimal possible occurs when no maintenance slots are planned and is equal to _ψ

^Λ

× − P

e∈E (δ _e ^maint ). Finally, the dominated solutions are filtered from all found solutions, such that only Pareto optimal solutions remain.

Algorithm 1: ε-constraint method for solving the MOOP Result: set sols containing Pareto optimal solutions sols ← ∅;

ε ← − _ψ

^Λ

× P

e∈E (δ _e ^maint );

while min{f 1 (x)| eqs. (3) − (14)} is feasible do ˆ

x ← arg min{f ₁ (x)| eqs. (3) − (14)};

sols ← sols ∪ ˆ x;

ε ← f ₂ (ˆ x) + ∆;

end

Filter dominated solutions in sols

4.2 Metaheuristic NSGA-II

In order to solve the scalability problem of the ε-constraint method, the meta-heuristic NSGA-II (Non-dominated Sorting Genetic Algorithm II) is used as a second solution method.

NSGA-II is an evolutionary algorithm that has the following three features: It uses an elitist principle, i.e. the elites of a population are given the opportunity to be carried to the next gen- eration. It uses an explicit diversity preserving mechanism (crowding distance). It emphasizes the non-dominated solutions. The pseudo-code of the NSGA-II main loop is shown in algorithm 2.

The pseudocode of NSGA-II shown in algorithm 2 calls several functions. These functions make the NSGA-II problem specific. The MOOP has two objectives, namely hindrance and flexibility. These objectives are calculated in the hindrance and flexibility functions of the NSGA-II algorithm. All called functions are:

• hindrance: This function calculates the hindrance an individual solution yields. The hindrance is calculated according to function f 1 (x). If any of the constraints (6), (7), or (10) are violated, the hindrance is penalized by M . Together with the flexibility, the hindrance determines the solution fitness.

• flexibility: This function calculates flexibility an individual solution yields. The flexibility is calculated according to function f ₂ (x). If any of constraints (3), (4), (5), (8), or (9) are violated, the flexibility is penalized by −M . Together with the hindrance, the flexibility determines the solution fitness.

• non dominated sort: This function ranks the individual solutions based on the Pareto- front they are in, based on domination. Individual A is said to dominate another individual B, if and only if there is no objective of A worse than that objective of B and there is at least one objective of A better than that of B. In the case of the MOOP, solution A dominates solution B if and only if solution A has at least equal hindrance and flexibility as solution B and either the hindrance of solution A is less than that of B or the flexibility of A is more than that of B. The algorithm of non-dominated sorting is based on the fast non-dominated sorting as described by (Deb et al., 2000).

• crowding distance: This function calculates the crowding distance of each individual solution. The crowding distanceof a solution is the normalized cuboid distance between its left and right nearest neighboring solutions. For solutions with only one neighbor (either left or right), the crowding distance is set equal to ∞.

• binary tournament: This function returns two parent solutions based on binary tour- nament selection. From all solutions, two random solutions are picked. These solutions are compared based on rank first and crowding distance second. Then the best solution is picked as an parent solution. The lower the rank, and the higher the crowding distance, the better the solution. This process is repeated twice.

• crossover: This function performs single point crossover on two parenting solutions. The solutions are two dimensional (x e,n ∈ {0, 1} ^E×N ). The crossover is performed vertically on the solutions. The crossover point ˆ n is randomly chosen where ˆ n ∈ {2, 3, ..., |N | − 1}

is chosen. Suppose the two parenting solutions are x ¹ _e,n and x ² _e,n , then the solutions after

vertical crossover are: ˆ x ¹ _{e,{1,...,ˆ n}} = x ¹ _{e,{1,...,ˆ n}} and ˆ x ¹ _e,{ˆ n+1,...,|N |} = x ² _e,{ˆ n+1,...,|N |} , ˆ x ² _{e,{1,...,ˆ n}} =

x ² _{e,{1,...,ˆ n}} and ˆ x ² _e,{ˆ n+1,...,|N |} = x ¹ _e,{ˆ n+1,...,|N |} .

• mutate: This function performs mutation on an individual solution. The mutation is performed horizontally. The mutation point is randomly chosen where ˆ e is randomly chosen where ˆ e ∈ {1, 2, ..., |E|}. The solution is horizontally mutated at the mutate point by randomly generating a new binary array of N elements. Suppose the individual solution is x e,n , then the mutated solution is: ˆ x e,n = x e,n ∀ e ∈ E − {ˆ e} and ˆ x _{ˆ e,n} = random binary array of size N

Algorithm 2: Main loop of the NSGA-II algorithm

Data: sets, parameters, η (number of individuals), γ (number of generations), σ (crossover probability), µ (mutate probability)

Result: X ^γ,parent (parent solution population) initialize random X ^1,parent ∈ {0, 1} ^η,Z,N

f ¹ , f ² ← {0} ∗ η for i ∈ {1, 2, ..., η} do

f _i ¹ ← hindrance(X _i ^i,parent ) f _i ² ← flexibility(X _i ^i,parent ) end

rank ← non dominated sort(X ^1,parent ) crowding ← crowding distance(X ^1,parent ) for i ∈ {1, 2, ..., γ} do

X ^i,child ← ∅

for j ∈ {1, 2, ..., η} do

children ← binary tournament(X ^i,parent ) if random x ∈ U v [0, 1] ≤ σ then

children ← crossover(children) for child ∈ children do

if random x ∈ U v [0, 1] ≤ µ then child ← mutate(child)

add child to X ^i,child end

end

X ^i,total ← X ^i,parent ∪ X ^i,child f ¹ , f ² ← {0} × (2 × η) for i ∈ {1, 2, ..., (2 × η)} do

f _i ¹ ← hindrance(X _i ^i,total ) f _i ² ← flexibility(X _i ^i,total ) end

rank ← non dominated sort(X ^i,total ) crowding ← crowding distance(X ^i,total )

sort X ^i,total , f ¹ , f ² , rank, crowding on rank and crowding X ^i+1,parent ← {X _k ^i,total |1 ≤ k ≤ 100}

end