Review of vehicle rostering approaches in railway

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Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Maritime and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

Specialization: Transport Engineering and Logistics Report number: 2014.TEL.7872

Title: Review of vehicle rostering approaches in railway Author: S. Kraijema

Assignment: literature Confidential: no

Initiator (university): Dr.Ir. F. Corman Supervisor: Dr.Ir. F. Corman Date: July 28, 2014

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Delft University of Technology

FACULTY OF MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

Student: S. Kraijema Assignment type: Literature Supervisor (TUD): Dr.Ir. F. Corman Creditpoints (EC): 10

Specialization: TEL

Report number: 2014.TEL.7872 Confidential: No

Subject: Vehicle Rostering

Once a timetable or schedule of operations (i.e. a sequence of transport services to be performed) of a public transport operator or logistic services is defined, an important question is the assignment of vehicles to transport services, in order to match operational requirements (size, capacity), guarantee operations, and reduce fixed costs (for instance, avoid buying an extra vehicle if that is not needed). In general, rostering is the process of matching a transport service to an available vehicle; this problem is solved for shorter time horizons than timetabling but longer than operational control (if timetable considers a year of operations, rosters are generally made weekly).

This literature assignment is to study the problem of vehicle rostering, making an overview of the academic and practical state of the art, including theoretical models, exact and heuristic approaches, and simulation approaches. In particular,

• Identify relevant mathematical optimization models for the rostering problem (e.g. flow-based; constrained paths in networks, ...)

• Make a compendium of the solution approaches found, categorized by their mathematical model, characteristics, performances in quality and computation effort

• Classify the approaches according to their model and solutions, and focus of application (logistic services, bus networks, rail networks, ship networks, other companies)

• Analyze effect of rostering solutions to the overall efficiency of the transport service; and the possibility to give directions of improvements towards the planning stage

This report should be arranged in such a way that all data is structurally presented in graphs, tables, and lists with the belonging descriptions and explanations in the text.

The report should comply with the guidelines of the section. Details can be found on the website. For more information, contact Dr.Ir. F. Corman (B 3 290; f.corman@tudelft.nl).

The professor,

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Summary

While the demand for transportation services is still increasing, the market share of the railway transportation mode is decreasing in favour of air transportation and road transportation. In order to cope with the increasing demand and simultaneously increase the market share, railway companies need to utilise the existing rolling stock fleet more efficiently. The optimization of railway related problems and processes is gaining interest of the operations research community. Significant progress has been made in the last 20 years of research and development of solution approaches to the various optimization problems. The focus of this literature review is the vehicle assignment problem in railway applications. In this field it is mostly referred to as the rolling stock scheduling problem or locomotive scheduling problem depending on the area of application, either freight transportation or passenger railway services. These two main streams in railway transportation services have many differences and require a vastly different approach to solve the vehicle assignment problem. Where freight railway companies focus on providing sufficient locomotive power for transportation of sets of railcars from one yard to another, passenger railway companies focus on providing train units with sufficient seating capacity between one station and another.

The optimization models are mostly based on integer multi commodity flow problems on a graph, where the nodes represent trips and paths represent a set of consecutive trips performed by one or a set of commodities, representing a rolling stock type. A shortest path finding algorithm is applied in order to find the optimal solution.

Due to the increase in effectiveness of recent solving approaches as well as an increase in computation power, the problem formulation and size can be expanded in order to better match the real life situation. More complex models can be solved to satisfying solutions within a reasonable time widow. As a result, real life constraints like maintenance and fuelling requirements are more often included in the recent optimization models. Also the large scale of the real life problems seem to become less of an issue with the more recent approaches.

The solutions that are found by these approaches are beginning to match and even exceed currently used rolling stock schedules that where constructed by experienced practitioners. It is proven that a cost reduction of 10 – 20% can be reached. However, these schedules are still not directly applicable in real life, as some constraints resulting from interdependences with other planning problems were not implemented in the optimization model. In particular local dispatching and shunting possibilities are highly interactive with the rolling stock schedule. These local schedules should be considered in an early stage of rolling stock

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It seems advisable to research the possibility to integrate the next step in the planning process, i.e. the shunting plan, into the optimization model. Such an approach would ensure the feasibility of the compositions used in the rolling stock schedule. The incompatibility between the rolling stock schedule found by the optimization model and the shunting possibilities proves to be one of the main reasons for current solutions not being used directly in real life situations.

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1 Introduction

While the demand for transportation services is still increasing, the market share of the railway transportation mode is decreasing in favour of air transportation and road transportation, see Figure 1. However, one of the European Union's (EU) long term goals is to increase this market share. The target is to shift half of the road transportation usage to railway transportation and inland shipping. The reason behind this goal is the increasing environmental awareness of EU leaders. Biasing the modal split towards railway transportation will decrease the greenhouse gas emissions by approximately 20% in 2020 [1]. The energy consumption of different modes of transportation in the EU is given in Figure 1. To realise the modal shift, railway transportation needs to become more competitive towards road transportation. Greater efficiency of the railway operations is required to accomplish increased flexibility of the services and reduce the kilometre - cost price.

In order to deal with varying demand of freight and passenger transportation, the railway companies need to be able to increase the flexibility and utility rate of its rolling stock fleet. Even though the market share is dropping, the overall demand is growing in recent years. To give an idea of the size, Table 1 gives an overview of railway transportation usage around the world in the last 8 years. The growing demand might force railway companies to expand the rolling stock fleet, but this is very expensive and time consuming. It takes 5 - 7 years from the start of the public tendering procedure to the final delivery of a new batch of train units or locomotives. The average cost of a new locomotive is about 1,5 - 2 million euro and for a high speed train unit about 10 - 15 million euro, both depending on order size and costumer specific systems. It is easy to see that even small improvements in efficiency and effectiveness of the current fleet can result in great savings.

The optimization of railway related problems and processes is gaining interest of the operations research community. Significant progress has been made in the last 20 years of research and development of solution approaches to the various optimization problems. This work gives an literature review on the latest developments in solution approaches for the vehicle assignment problem. Chapter 2 gives background information on the various problems related to the vehicle assignment problem and how the assignment problem applies to railway transportation. Chapter 3 gives an overview of solution approaches to the vehicle assignment problem in railway, found in recent literature. The approaches presented in Chapter 3 are discussed and evaluated in Chapter 4.

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Figure 1: Modal split and energy consumption in passenger transportation [1] Passenger-kilometres (billions) 2006 2007 2008 2009 2010 2011 2012 2013 Europe * 464 469 484 471 473 486 480 476 Russian Federation 178 173 176 154 139 140 145 145 Africa 62 62 62 62 62 49 49 49 America 13 13 14 14 12 21 21 22

Asia Oceania and Middle East 1646 1789 1951 2012 2079 2188 2172 2173

WORLD estimates 2362 2506 2687 2712 2765 2883 2867 2865 Tonne-kilometres (billions) 2006 2007 2008 2009 2010 2011 2012 2013 Europe * 696 723 987 546 701 649 621 604 Russian Federation 1951 2090 2116 1865 1903 2127 2222 2222 Africa 142 139 138 137 139 139 139 139 America 3520 3540 3514 2973 3076 3133 3231 3231

WORLD estimates 9181 9589 10208 8988 9281 9669 9807 9789 Length of lines (kilometres)

2006 2007 2008 2009 2010 2011 2012 2013

Europe * 264205 264630 263806 268466 285408 270342 265116 265231

Russian Federation 85253 84158 85194 85281 85292 85167 84249 84249

Africa 52159 52400 52482 52299 50275 70505 70505 70505

America 385272 389863 386773 383079 375774 369222 369222 381538

WORLD estimates 1008677 1013696 1010082 1013276 1020953 1028806 1021457 1029230

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2 Vehicle assignment problem in railway

The Vehicle Assignment Problem in railway transportation is mostly referred to as the Rolling Stock Scheduling Problem or Locomotive Scheduling Problem depending on the area of application, either freight transportation or passenger railway services. These two main streams in railway transportation services have many differences and require a vastly different approach to solve the vehicle assignment problem. Where freight railway companies focus on transporting sets of railcars from one yard to another, passenger railway companies focus on providing sufficient seating capacity between one station and another. This section describes both applications of the vehicle assignment problem in railway transportation.

2.1 Vehicle assignment problem in freight railway transportation

In freight railway transportation, four main planning and scheduling problems can be indentified in relation to the Vehicle Assignment Problem, or in this case the Locomotive Scheduling Problem. These problems and their interrelations are briefly discussed in this section.

2.1.1 Blocking problem

Freight shipments typically consist of a set of railcars that share the same origin and destination. A typical US railway operator will have to handle about 50,000 shipments in a month [2]. In order to reduce the individual handling cost of these shipments, they are combined into sets sharing the same origin and destination. These sets are commonly known as blocks. The planning problem railway companies have to solve first is how to combine the shipments into blocks in the most cost efficient and effective way, i.e. the Blocking Problem [2]. The bundling, or classification of shipments into blocks can result in longer travel distances and extra shunting operations in case of a block rearrangement, or reclassification during the trip. These are important factors to consider when solving the Blocking Problem. For an introduction on how the Blocking Problem can be solved using a network model see [2]. Figure 2 gives a visualization of such a network representation of the blocking problem. In this network, shipments S1 - S4 need to be transported from their origins, represented by node 1 - 4, to their destinations represented by node 9 - 12. Nodes 5 - 8 represent yards at which shipments can be combined to form blocks. For example, S2 and S3 can form a block at yard 6 and be transported to yard 8 together. The main cost drivers in this problem are travelled distance, shunting operations and locomotive power requirements.

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2.1.2 Train scheduling problem

After the Blocking Problem is solved and a valid blocking plan is established, the railway company has to design the train schedule. This schedule includes information on departure time and location from the origin, routing on the rail infrastructure, intermediate stops for rearrangements, and arrival time and location on the destination of each train. But first of all the trains themselves have to be created, i.e. the composition of the trains have to be determined. Sets of blocks are grouped together to form a train. It is important to note that the composition of a train can change during the trip. Blocks may be added or removed from the train during intermediate stops. The Train Scheduling Problem is a very complicated problem and is therefore most often separated into two subsequent stages [2]. The first stage is to determine which trains are to be routed on the rail infrastructure network. Decisions have to be made on how many trains to run, what time to depart and arrive at origin, intermediate and destination stations, what route to take and what frequency the train should run. The second stage is to determine how the blocks should be routed on this network of trains. For an introduction on how to model the Train Scheduling Problem for freight transportation see [2]. Figure 3 gives a network representation of the Train Scheduling Problem, where the nodes represent stations and the arcs represent rail infrastructure. In this example, block B1 has its origin at station 1 and its destination on station 4. Train B hauls block B1 from station 1 to station 2, where it is rearranged and hauled to its destination station 4 by train A. The main cost drivers of the Train Scheduling Problem are travelled distance, travel time, shunting operations, number of trains, locomotive power requirements.

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2.1.3 Locomotive scheduling problem

The previous two planning problems in freight railway are used as input for the Locomotive Scheduling Problem which is a specific application of the Vehicle Assignment Problem.

The Locomotive Assignment Problem deals with the need to assign locomotives to service the trains that are scheduled according to the solution found from the Train Scheduling Problem. There are three main levels of planning: strategic, tactical, and operational.

Strategic planning is to determine the required number and types of locomotives in the fleet on the long term. This requires estimations of future demand of rail freight transportation and the evolvement of other freight transportation modes and interaction with other modes in terms of intermodal transport. Barge transportation for example, is becoming more and more of a competitor for bulk material handling and container transportation.

Tactical planning is focused on a shorter time horizon and therefore aims more on an operational level. Planning horizon is typically 1 week in advance. At tactical planning level, locomotive types are assigned to trains based on power, pulling and cost requirements. The required power to pull a long heavy train might exceed the power rating of the available locomotives. In that case a set of locomotives can be assigned to the train to form a consist.

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While this is common practice in the US, it is not used very often in Europe where distances are shorter and trains are smaller. Given the fixed fleet size and composition of the railway company, the weekly forecast of demands might show that additional locomotives are needed at some location in the rail network. If this cannot be resolved by repositioning of capacity, railway companies have the opportunity to lease locomotives or use locomotive from other railway companies, also known as foreign power [3].

Locomotives are transferred to other yards or stations in two ways. They either deadhead on a train or they travel light. Deadheading means a locomotive is added to a train but is not part of the active consist. It is pulled like a normal railcar. This way of transferring a locomotive is preferred over light travelling, where the locomotive travels on its own power but without pulling a train of railcars. A big advantage of this way of transfer is the fact that it is independent of the train schedule. It does however take a crew to operate and use energy while not making revenue by hauling shipments for costumers.

When a train arrives at its final destination, the locomotive or consist is either assigned to the next outbound train on its schedule or it goes to the depot to wait for the next scheduled event. This event can either be the assignment to the next train, a regular safety inspection or a maintenance task or repair in the workshop. When the train used a consist and no direct train to train connection is made because one or more locomotives are assigned to another train or consist, regrouping takes place. This regrouping is known as consist busting. Since many different railway companies operate from the same stations and yards, these standby pools contain locomotives from different owners. As mentioned earlier, railway companies sometimes use locomotives owned by other companies to fill in on short term shortages during high power demand peaks. This is called using foreign power.

Operational planning consists of making decisions on a short time horizon of typically 12 hours to one day. Whereas tactical planning schedules locomotive types, operational planning schedules locomotives on actual identification number level. Additional operational requirements and constraints are to be taken into account when planning at this level, such as fuelling and maintenance requirements.

In Europe most main routes of the rail infrastructure are covered with overhead power lines so rail transportation is mostly electrically driven. In this case the fuelling issue is not applicable. However, it introduces another issue to take into account in real life, and that is the fact that various areas in Europe use different power supplies to feed the overhead lines. The supply voltage may vary but also the current type may be alternating or direct. This adds an additional requirement to the locomotives when scheduling trains. In the US most freight railway transportation is performed by diesel or diesel-electric powered locomotives. Also in the US interest is growing to electrify more of the rail infrastructure in order to reduce emissions and fossil energy consumption.

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Maintenance requirements are to be obeyed according to railway safety legislation. Periodical safety checks and inspections are to be carried out by capable mechanics. When a locomotive is almost due for maintenance it cannot and may not be scheduled to service a train on a trip on which it might exceed the maintenance interval. Or when the trip will take the locomotive to far from the work shop to make it back in time. The locomotive would then have to deadheaded to the nearest workshop. This is an expensive move and the operating railway company will risk losing its licence to operate. Besides the need to obey the maintenance schedule, also the resulting technical failures and malfunctions of a locomotive are putting extra strain on the availability of the fleet for operational and maybe even tactical planning. Small failures with a short downtime will only effect the operational planning, but some failures will take several days, weeks, or even months to repair and can have big consequences on tactical planning level. This downtime for preventive and corrective maintenance should be accounted for on strategic level when determining the fleet size. The extra capacity is called the technical reserve of the fleet and is normally around 5% of the total fleet size. This number is based on experience from the past and or maintenance schedules and failure rates given by the manufacturer of the locomotive. As the behaviour is very difficult to predict and failures may come all at once, the technical reserve is no guarantee. On a higher level, service demands might grow and a decision has to made on whether or not to buy new inventory for the fleet or to increase the operational availability of the fleet by cutting down the technical reserve. Most often the last option is chosen, being unaware of the consequences for planning reliability.

In literature, the Locomotive Scheduling Problem is mostly represented as a network model. This is covered extensively in Chapter 3, but for now a brief description is given by means of Figure 4. Each locomotive follows a path starting at origin node O and ending at destination node D. The intermediate nodes 1 - 6 represent the trains to be serviced during a specific time window. In this example, trip 1 is serviced by consist of locomotive A and B.

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The main cost drivers for the Locomotive Scheduling Problem are energy consumption, deadheading time, light travelling time and distance, amount of consist busting, crew requirements, own capacity shortage , and overdue time for maintenance.

2.1.4 Train dispatching problem

After the train schedule is completed, it gives an master schedule of train arrival and departure times on main stations and contains the master plan for routing. But more detail on train movements is needed to prevent trains from conflicting each other somewhere along the track, for example on main crossings, single line track sections, and in and around stations and yards. This detailed level of traffic control is determined in the Train Dispatching Problem. An example of a freight railway transportation specific constraint to consider is the fact that it is prohibited to stop a freight train transporting toxic or highly flammable products near a populated area. Another example is the length of a passing section, called siding, on a single line track might be too short for the train to use. A distinction can be made in local and global dispatching. Local dispatching determines detailed train movements in the near neighbourhood of a single station or yard at operational level. This is very sensitive for delays and other disturbances so real time adjustments are constantly needed. On a tactical level, the planning focuses on constructing a valid master timetable schedule on a weekly periodic basis that defines arrival and departure times on a detailed yet global level. On a strategic level, the rail infrastructure is to be evaluated and possible investment decisions must be made to modify bottlenecks in the network for improved traffic flow and therefore greater capacity handling.

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A visualization of the Train Dispatching Problem is given in Figure 5. The problem is represented by a time - space graph. In this example, at least station C and station E have the possibility for two trains to pass each other when train 2 waits on a side track.

Train dispatching is a much researched problem and additional information on the real life problem and the current state of models and solving algorithms can easily be found, see for example [4] and [5] for recent survey's of solving approaches.

2.2 Vehicle assignment problem in passenger railway transportation

In passenger railway transportation, three main planning and scheduling problems can be indentified in relation to the Vehicle Assignment Problem, or in this case the Rolling Stock Scheduling Problem. These problems and their interrelations are briefly discussed in this section. Figure 6 gives an overview of these problems at all planning levels.

Figure 6: Overview of planning problems in passenger railway transportation

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2.2.1 Time tabling problem

The main goal in passenger railway is to provide enough capacity to transport passengers from their origin to their destinations. In order to do this, passenger transportation by rail is based on a periodically repeated time table. The time table defines arrival and departure times for each train on each station in the network. The interval at which trains travel between stations is based on the capacity needed to meet the demands to a high level of customer satisfaction. Data on travel demands of passengers can be collected by conducting surveys on trajectories of interest. It is important to collect this data and analyze trends in order to be able to predict future demands. Data like this can also be used to make a passenger behaviour prediction model. Such a model could also support decision making in the case of an emergency that requires a deviation from the normal train schedule. See [6] for an application of such a model. The Time Tabling Problem also involves platform and route planning. This means that for a time table to be applicable in a real life situation, platform availability and the actual route of each train must be determined as well. This is a very complicated problem to solve and up till now no solving approach have been developed to give a useful solution within an acceptable time frame [7].

To make a cross reference to freight railway transportation, the Time Tabling Problem deals with an equivalent of the Blocking Problem, the Train Scheduling Problem and the global Dispatching Problem all related in one problem. Where the varying passenger flow demand per time unit between stations can be seen as individual shipments needed to be classified into blocks. Just considering the total passenger flow is not sufficient, while this could lead to a time table that may provide enough capacity between all stations but, at the same time, would be very inconvenient for the customers to use. It would not take costumer preferred routes, i.e. shortest routes into consideration. Customers therefore have to be handled as packages, or groups of individuals sharing the same origin and destination, just like individual shipments in the Blocking Problem. The train route planning, usually referred to as line planning in passenger railway, is a very similar problem to the Train Scheduling Problem as described for freight transport.

The passenger train route planning part of the Timetabling Problem determines which lines are to be serviced by trains and at which frequencies they should run. These lines should be created in such a way that the minimum required capacity is provided in the most efficient and cost effective way, while retaining a high service level for the customers. This means least possible transfers and the shortest possible travel times for the majority of the passengers. The process of defining lines to be used as input for the timetable is often referred to as the Line Planning Problem [8]. The Line Planning Problem is solved on a strategic planning level and uses forecasted passenger flow demands and available rail infrastructure as direct input. A completed line plan contains not only information on the origin and destination of each line and the route that connect those two but also the line type that gives details on which intermediate stops to take for each line. In passenger

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railway the cycle time of a line plan in usually one hour. During quiet hours in the early morning or night, line frequencies may be reduced.

On a tactical planning level railway companies use the line plan with desired frequencies to generate the timetable. The time table contains arrival and departure times at all stations for each line in the line plan. The generated times must be compatible with the infrastructure capacity constraints such as track width, i.e. the number of parallel tracks and therefore the ability to pass other trains, or the number of platforms at a station. For safety reasons, trains are not allowed to be in the same track section within a certain period of time. Trains have to have a full stopping distance between them, called headway. The headway is depending on the actual stopping distance of a specific train type with some safety distance added on top of that. The time table must also comply with some customer service related constraints such as train connections on key points in the network system. It is obviously not possible to create direct connections between every station in the network, so passengers must be given the opportunity to switch trains on some main stations. This introduces a new contradicting constraint because normally dwell times at stations in to be minimized in order maximize the stations train handling capacity. On the other hand, longer dwell times are favourable to relax the required precision of arrival times to ensure proper train connections. To come back to the cross reference with freight transportation, the difference in solution approaches between the Train Scheduling Problem for freight transportation and the Timetabling Problem for passenger railway, is mainly caused by the fact that passenger railway requires more precision. It has to deal with many more restrictions [8]. For that same reason and added to the fact that freight trains predominantly use the same rail infrastructure as passenger trains, passenger train schedules often form the basis around which the freight train schedule is constructed. The main drivers to observe when creating a time table for passenger railway are to minimize operational cost, to maximize passenger satisfaction and to maximize robustness. These objectives are contradicting while high service levels mean high operating costs and vice versa. This is because high service levels are achieved by reduced waiting time by increased line frequencies, many direct connections between stations by running more separate lines, and the ability for each customer to attain a seat by offering a surplus of seating capacity. Robustness is the ability of the timetable to remain operational in case of disruptions of disturbances, which is later described in more detail. In short maximizing robustness consists of finding a time table that is insensitive to the delay of a single train. Meaning that this delay will not have an effect on the scheduled arrival and departure times of other trains. Many research publications focus on increasing the robustness of timetables. For more information on robustness of timetables see [9] and [10] for recent publications.

While routing on dense rail infrastructural networks as the ones found in most European countries, as well as areas around the major cities in the US offer many possibilities, it is practically not possible to consider all possible routings within stations and junctions at once. The Timetabling Problem is often split into different levels of detail. In this case the time

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tabling problem plans on a higher aggregation level with a lower level of detail. The high level of detail on the infrastructural network of stations, yards, and junctions is ignored in this approach and dealt with on in later stage. The timetable is designed on this adjusted network. The resulting flows on these detailed points in the network are then used as input for the local detailed routing plan. If the result of the high level time table does not allow for feasible detailed routing, it has to be adjusted. This is a repeating process until a satisfying solution for Timetabling Problem is found. This second level of routing is called the Train Routing Problem or Train Scheduling Problem. In case the detailed section consists of a station the problem is called the Train Platforming Problem or the Train Dispatching Problem which is described later on in this chapter. To read more information on the interaction between these problems, see [11].

2.2.2 Rolling stock scheduling problem

Once a satisfying timetable is found that identifies the trips needed to fulfil both passenger and railway operator demands, the next step is to schedule appropriate rolling stock to service these trips, i.e. the Rolling Stock Scheduling Problem. This is exactly the same problem as described earlier for freight transportation, where it was called the Locomotive Assignment Problem. Both problems are specific applications of the Vehicle Assignment Problem. In literature different names are given to the same problem, but Locomotive Assignment Problem is mostly used in freight applications and Rolling Stock Scheduling Problem is most often used for train unit assignment in passenger railway. For that reason this text uses the same notation. Energy consumption, maintenance cost and the required shunting activities are the main cost drivers for the Rolling Stock Scheduling Problem.

The rolling stock used in passenger railway is in many ways different to that used for freight transportation. A freight train consists of a sets of railcars that are pulled by a locomotive or a consist. Some passenger trains are operated much alike freight trains when they consists of a locomotive pulling trailing passenger carriages. However, a typical passenger train consists of one or more train units like the VIRM used by Dutch operator NS. These train units are actually a set of powered passenger carriages that can be combined together in order to provide more capacity. The majority of the locomotives and driven train units are electrically powered, while most of the European rail infrastructure is electrified by means of powered overhead lines.

Train units have the advantage of reduced need for shunting activities because of their fixed capacity and ability to run both ways. Most train units are so called two-directional vehicles. Shunting is the rearrangement or relocation of rolling stock, whether it be locomotives, carriages, or train units, between the active track and the shunting yard. The routing and timing of these relocations are defined by a shunting plan. This movements may not interfere with the operational timetable activities of schedule trains. Also the placement of parked rolling stock must be planned with care, because it may not form an obstruction for

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The Rolling Stock Scheduling Problem can be separated into three planning levels: strategic planning, tactical planning, and operational planning.

Strategic rolling stock planning is about supporting decision making for long term fleet management. At this level, the size and composition of the fleet is evaluated. It has to be decided whether the current configuration of the fleet is sufficient for the next 10 to 20 years of operation. Rolling stock has a expected life time of about 30 years. After that period, a decisions have to be made whether to do a complete overhaul and modernize some of the systems to stretch the expected life time another 10 to 20 years or to scrap or sell the current rolling stock and replace it with new units. However, it may be decided that the fleet has to down size because of decreasing demand. The ability to satisfy customer demands in the near and far future are highly dependent on the composition and size of the fleet of a railway operator. A great amount of money is involved to increase fleet size. The need for research on this topic and development of decision making support tools is just recently recognized and started about 10 years ago, see [14] for an approach to find fleet requirements based on a timetable [15]. Demand forecasting is of great importance to make valid long term decisions. These demand forecasts are typically based on scenario analysis. The most important topics to gain knowledge on with this kind of analysis is the future passenger capacity demand and how this is spread across the network and the demand variation during the day.

Passenger railway transportation defers from freight in many obvious ways. One important aspect to consider in rolling stock planning is that passengers can chose between two different service levels aboard the same train. On a tactical level this means care should be taken when assigning train units to trips, while two different demands should be satisfied with the same units. Train units have a fixed share of first and second class departments, while a train consisting of a locomotive and separate carriages can be adjusted to the desired balance for a trip. This does require expensive and time consuming shunting operations. The balance between first and second class demands is also something to be considered on strategic level when investment decisions are made for new rolling stock. As described earlier a railway operator can decide to lease or rent train units, should a short term capacity shortage occur.

Another passenger railway specific characteristic is that the demand is of non-deterministic nature, while the majority of lines is operated without a seat reservation system. Some exceptions can be found on special hi-speed, long distance lines like the Fyra, TGV, or ICE. This is one of the main reasons why the robustness of the time table and rolling stock schedule is of such great importance. To achieve a robust schedule, the operator must avoid coupling and uncoupling activities as much as possible as these are prone to lead to delays. According to [15], rolling stock scheduling is normally performed on line basis to prevent delays from propagating to other lines.

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Tactical level planning of rolling stock consists of assigning train units to the trips defined by the timetable. To organise this in an robust, efficient and cost effective manner, while retaining a high level of customer service and robustness is the main objective of the rolling stock schedule. The time horizon for tactical planning of rolling stock is typically 1 year. The planning focuses on the assignment of train unit types or combinations between them to the different lines according to the capacity demands and timetable. Train unit types can be combined in various ways to increase seating capacity. Two or more units of the same type or subtype may be coupled together. An example of different sub types are the 4 car and 6 car VIRM vehicles as used by NS. They share the same traction units and have compatible driving characteristics so they can be operated as a coupled pair. The capacity demand during the day is not constant and therefore it would be sufficient to use a smaller capacity train unit combination than it would during rush hours. This would require a switch between one combination of train units and another or maybe even a reconfiguration of the current unit. Both require costly shunting activity and increase the risk of delays. Besides capacity requirements, there are also operational constraints such as the maximum length of the train unit combination in order to match the size of the platforms along the line. Another operational constraint is defined by the specific driving characteristics desired for the line. For example on long distance trains with few intermediate stops between the origin and destination, the comfort and high speed stability offered by the train unit type is more important than for regional short distance trains. On lines with countless intermediate stops, the acceleration and deceleration ability is more important. During peak demand hours, passenger carrying and boarding flow capacity will be more important than comfort, even for longer distances.

On an operational planning level, the actual rolling stock has to be assigned to the yearly plan resulting from the tactical planning level on a daily basis. The time horizon is typically 1 day to a week and assignment is performed on a line to line basis for improved robustness as described earlier. Train units are now assigned on actual identification number level. At this level more detailed constraints have to be taken into account. The most important being maintenance requirements of each specific train unit.

Train units need preventive maintenance checks on a regular basis. Usually the intervals are somewhere around 25.000km for basic inspections. Intervals are longer for the more extensive maintenance tasks, such as oil changes, bearing condition checks and overhauls. All maintenance tasks are performed at specific locations, not necessarily at or near a regular station. This means they are not easily combined with the regular routes that were planned to service lines. The rolling stock schedule that was constructed on a tactical level does not take this maintenance schedule into account. It only schedules train unit types or combinations of train unit types to trips. Special care should be taken when planning train units that are nearly due for maintenance tasks, meaning within the planning horizon. This is usually realized by modifying the particular part of the tactical schedule in such a way that

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according to the timetable. See [15] for a detailed description and [16] for a proposal for a decision support system of this particular scheduling problem.

The main cost drivers for the Rolling Stock Scheduling Problem are energy consumption, shunting activity, crew requirements, own capacity shortage , and overdue time for maintenance.

2.2.3 Local train dispatching problem

The Train Dispatching Problem deals with local routing of trains on a complex part of the rail infrastructure, for example near a station. This problem is also referred to as the Platforming Problem in some literature. The main goal is to route trains in such a way that both safety requirements and timetable requirements are met. Safety requirements are for example a safe head distance between two trains on the same track. The timetable is split into two levels of detail. The local dispatch schedule has to fit both the global and surrounding local timetables. It also has to be compatible with the local shunting plans. Train dispatching is a complicated and much researched problem and additional information on the real life problem and the current state of models and solving algorithms can easily be found, see for example [4] and [5] for recent survey's of solving approaches.

2.3 Difficulties of the vehicle assignment problem in real life

The problems as described in the previous parts of this paper, only deal with static situations. Their aim is to create a schedule for a predefined time horizon, based on known and fixed constraints. However, in real life there will be all kinds of disturbances effecting the viability of the schedule. Even the most robust schedule will need to be adjusted in case of a seriously conflicting event. In literature a conflicting event is often called a disruption, disturbance or emergency. In this paper the term disruption will be used for events with minor to major impact on the current timetable and rolling stock schedule.

Most timetables are constructed in such a way that passenger convenience heavily depends on connections between lines. For passenger convenience sake, the next train in the connection should wait for the delayed train. This on the other hand increases waiting time for passenger wanting to board that particular train on the next stations. It also effects the next incoming train scheduled to stop on the occupied platform. The delay of one single train can quickly propagate throughout the schedule. When a disruption is caused by a obstruction somewhere in the infrastructure, this has a major effect on all trains originally routed through the blocked section. Trains will have to be rerouted to avoid the obstruction or part of the line will be cancelled and replaced with shuttle busses.

In passenger railway transportation it is common practice to match train unit compositions of incoming and outgoing trains. This means that an arrived train that just finished its trip, is assigned to a next trip with the same composition requirements. This is called the turning pattern of rolling stock and is illustrated in Figure . Due to an obstruction on the route, it

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may not be possible to maintain the original turning pattern and it has to be modified. As explained earlier one leg of the turning pattern may be split into two because it is blocked by the obstruction. Two additional turning points are added just before and after the obstruction until the infrastructure is cleared. The adjustment of the turning pattern is illustrated in Figure 7.

Real time disruption detection and management is needed in order to reduce both delay propagation and passenger inconvenience. Disruption management can be divided into three main steps: adjustment of the timetable, reschedule rolling stock and crew according to the new timetable.

Adjustment of the timetable in order to continue to service the lines as efficiently and effectively as possible given the current circumstances is called the Timetable Rescheduling Problem. The main objectives for the timetable rescheduling problem are to minimize delay propagation and / or passenger inconvenience. In literature lots of studies and different approaches can be found. Some focus more on minimizing delay propagation and others focus more on minimizing passenger inconvenience. See for example [17], where the authors present an algorithm to reschedule the time table in such a way that the inconvenience for the passengers is minimized. In [18] the authors present a real time disruption management system called ROMA to minimize the delay propagation. In [19] a bi-objective approach is presented to find a solution that minimizes both delay propagation and missed connections by passengers.

Once an acceptable solution for the timetable adjustment is found, the next step is to adjust the rolling stock schedule accordingly. This is often referred to as the Rolling Stock Rescheduling Problem. The new timetable may introduce problems for planned composition changes. As lines may be split due to an obstruction, it is possible that the required train unit type is not available to form the specified composition. The amount and composition of the available rolling stock at a station is called the inventory of that particular station. The difference between the scheduled inventory at the end of the day and the actual inventory is referred to as the end of day off-balance of a station or depot. The goal for most railway operators is to return to the original schedule as soon as possible. That is why the end of day off-balance is important because it directly effects the available rolling stock for the next day. If the end of day off-balance leaves an undesirable starting situation for the next day, it may be necessary to reposition rolling stock over night. The end of day off-balance should be minimized or even better avoided.

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Figure 8: Standard turning pattern between two stations

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2.3.1 Rescheduling decisions

According to [20] rolling stock rescheduling decisions can be divided into three main groups: changing shunting plan, changing turning patterns, and repositioning rolling stock.

Changes in the shunting plan may be necessary due to changes in the composition of train units assigned to a specific trip. This may cause problems for the local dispatchers as they may not have sufficient capacity to realize this additional shunting operations. There may be a capacity shortage of personnel, equipment, storage or shunting space. Changes in the shunting plan should therefore always be communicated and negotiated with local dispatchers. Care should be taken when the originally planned shunting operation involves coupling of train units because the total length of the new composition may exceed the shortest platform length on the route. In case an uncoupling shunting action was planned originally and the available composition consists of smaller train units then planned, the capacity of the new compositions may be insufficient. Changes in train unit types or changes in the number of train unit types to be coupled or uncoupled do not interfere with the original shunting plan as it requires the same movements.

In passenger railway transportation it is common practice to match train unit compositions of incoming and outgoing trains. This greatly reduces the need for shunting operations, while no coupling or uncoupling actions are required. Often the timetable does not even offer the time to make changes to the composition. This travelling back and forth of train compositions is called a turning pattern. By changing the connections in the turning pattern between incoming and outgoing trains, the dispatcher can change the composition assigned to a trip to better match the required capacity. Or provide more time to perform coupling or uncoupling activities. The adjustment of the turning pattern to change composition assignment is illustrated in Figure 9. Changing the turning pattern effects both global and local dispatching plans.

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Another possibility for adjusting the rolling stock schedule to the new temporary timetable is to reposition rolling stock. This may be necessary if the inventory at a particular station or shunting yard is insufficient to provide the required capacity. Rolling stock has to be repositioned from one stations inventory to another. This can be done at night when line frequencies are low and track capacity allows additional movement of trains. A better solution is to reschedule train destinations in such a way that the repositioning is performed while servicing a trip.

The main objectives of the Rolling Stock Rescheduling Problem are to minimize cancellation of trains, to minimize rolling stock end of day off-balance, and to minimize changes in shunting plans, while the provided capacity still meets the passenger demand for the lines that are not cancelled.

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3 Solution approaches to the vehicle assignment problem in railway

There is a great amount of literature available for all stages of planning problems in both passenger and freight railway transportation. This section gives an overview of the most recent solving approaches for the Rolling Stock Scheduling Problem and the Rolling Stock Rescheduling Problem. It compares and reviews the different assumptions and solving approaches and assesses their ability to be used as a decision support system in real life situations.

First a brief overview of the evolution of solving approaches to rolling stock scheduling problems is given. Then a basic generic mathematical model is described, covering the basic objective function and constraints to be used when solving the rolling stock scheduling problem. After this basic model representation is explained, an overview of the most recent solution approaches found in literature is given. Starting with the rolling stock scheduling solution approaches aimed for application in both freight and passenger transportation. Followed by the rolling stock rescheduling approaches for both freight and passenger applications.

3.1 Evolution of solving approaches

For an overview of the very early beginning of combinatorial optimization solving methods for fleet management problems starting in the 1950's, see [21]. A very early publication on the locomotive assignment problem was presented in 1972, see [22]. It formulates a solution approach for a mathematical representation of the locomotive scheduling problem. It aims to find the minimum cost solution for providing sufficient engine power to trains by finding the most cost effective composition of consists. The problem was greatly simplified and focuses only on power requirements and a cost function. The cost function consists of depreciation and maintenance costs. It presents a method to solve the linear program in such a way that it is able to find satisfying solutions for medium scaled problems. This method was used by the Canadian National Railway Company to support future fleet size decisions over a 15 year time horizon. Another early approach for solving the rolling stock scheduling problem was presented in 1980, see [23]. The author presents a heuristic method to find integer solutions to a linear programming model. The objective is to find a minimum cost solution for locomotive scheduling to trains with a fixed starting time. The algorithm was able to find solutions for up to 50 scheduled trains in about 3 to 10 minutes, depending heavily on the variation in starting times and the number of available locomotive types. The lack of computing power and the limitations of the available integer linear programming (ILP) relaxation methods, forced these early solving approaches to simplify their models to great extents. Even with these simplifications these methods were not able to really support large scale and / or short term decision making.

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and penalizes these violations to prevent them from entering the solution. So it estimates a solution for the LP relaxation problem. A solution for relaxed LP problem is an estimation for the solution of the original ILP problem. This LP relaxation was then solved for a suboptimal solution by means of a heuristic algorithm, of which diving heuristic searches where considered to be the most successful. Lagrangian relaxation algorithms take a large number of iterations before converging to a solution, but each iteration takes little time to calculate. The ease of implementation made Lagrangian relaxation the preferred method to some more complicated methods like surrogate relaxation. The performance characteristics of Lagrangian relaxation in comparison to other methods is discussed extensively in literature. Solving the LP relaxation of large arc ILP models still requires a lot of calculation time and capacity today. Lagrangian relaxation is still a very viable method to estimate the LP relaxation. But with the use of column generation it is possible to find the LP relaxation of large path ILP models in short calculation times [24]. Column generation considers only a subset of the constraint variables and adds contributing constraint variables at each iteration in order to find the LP relaxation efficiently. This approach is not as effective when an ILP model contains a lot of constraint variables. There are some other variations on the column generation method like the separation method. The separation method solves for a subset of the constrain variables and then checks to see if the other constraints are violated and adds them when this is the case. Diving heuristics are still one the most used heuristic methods to find a near optimal solution for the LP relaxation [24].

Even though solving methods have greatly improved in the last years, still the majority of solutions found are only matching the performance of handmade schedules at best [24]. However, the solving time is greatly reduced when a modern solving approach is used and therefore more possibilities for experimentation with different scenarios is enabled.

3.2 Generic mathematical model

Most approaches model the Rolling Stock Scheduling Problem as an Integer Multi-Commodity Flow (IMCF) problem on a graph with arcs and nodes [24]. The IMCF problem defines a set of commodities C and a directed multigraph G = (V, A) with a set of nodes, or vertices V and a set of edges, or arcs A. The set of vertices V has a source node s and a target node t. The set of arcs A contains a subset Ac for each commodity c ϵ C. Each arc (u , v) ϵ A

has a starting node u and an end node v. Every node has a demand dv. Every arc has a flow

capacity ac for each commodity assigned to that arc and an associated cost per flow unit

used in the solution. The flow of commodity c between u and v is represented by xc, (u, v). The

cost associated with the use of an arc is directly coupled to the flow along that arc. The objective of the problem is to find a path P ⊆ Ac from s to t for every c in such a way that

demands are satisfied, along with the additional constraints at minimum costs. See Figure 10 for a visualization of an IMCF problem on graph G.

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This standard form translates to rolling stock scheduling as follows. Rolling stock types are assigned to trips following from the timetable. Trips are represented as nodes in V in the graph G. Rolling stock types are each represented by a commodity c. Each arc (u, v) ϵ Ac

represent a possible connection between trips u and v for commodity c. A connection is only possible if the departure time of trip v is sufficiently spaced with arrival time of trip u. If the arrival station of trip u is not the same as the departure station of trip v, addition costs for the transfer has to be accounted to arc (u, v). Path P contains all subsequent trips to be serviced by the same composition of rolling stock types in the calculated time window. Each trip represented by node v has a capacity demand dv which has to be satisfied by the

capacity ac of commodity c assigned to node v by arc (u, v). This means that the total

provided capacity of paths visiting node v must meet the required demand dv. The capacity

can either be locomotive power or seating capacity, depending on the application. Whether or not arc (u, v) ϵ Ac is used in path P, is indicated by decision variable xc, (u, v) = 0 or 1 if an arc

ILP model representation is used. In case a path ILP model is used to represent the problem, the use of path P for commodity c can be indicated on path level by decision variable xc, P = 0

or 1. In this case Pc represents a set of paths P ϵ Pc that are assigned to commodity c. In a

path ILP formulation, the feasibility of subsequent assignment of trips u and v to commodity

c is represented by the existence of arc (u, v) in G(V, Ac). When an arc formulation is used,

the feasibility of subsequent assignment of trips u and v to commodity c is implemented in the cost associated with the arc. The cost is increased significantly if commodity c is not able to service trip u and trip v in one day. The cost is directly connected to the time between arrival of trip u and departure of trip v.

In most recent approaches a path ILP representation is used for solving rolling stock

scheduling problems [24]. Now, an example of such a ILP representation is given. Figure 10: Integer Multi Commodity Flow Problem on graph G

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When variable xc, P = 1, cost fc, P is added to the total cost. The total cost is defined by:

� � 𝑓𝑐,𝑃 𝑥𝑐,𝑃 𝑃ϵ𝑃𝑐

𝑐ϵ𝐶 (1)

In order to ensure that only one path is assigned to each commodity, the following constraint is defined:

� 𝑥𝑐,𝑃 𝑃ϵ𝑃𝑐

= 1, 𝑐 ϵ 𝐶 ₍₂₎

The following constraint is defined in order to ensure that sufficient capacity ac is provided

by commodities c ϵ C at each node with demand dv , where Pc, v is defined as the set of paths P in the solution, that are assigned to commodity c and routed though node v:

� � 𝑎𝑐 𝑥𝑐,𝑃 𝑃ϵ𝑃𝑐,𝑣

𝑐ϵ𝐶

≥ 𝑑𝑣 , 𝑣 ϵ 𝑉 ₍₃₎

According to [24] this formulation of the constraint would lead to a weak LP relaxation and the resulting loose lower bound would lead to sub optimal solutions. The author of [24] describes two approaches to increase the strength of constraint (3). The first method reformulates this constraint by redefining the demand coefficients in such a way that they are minimal.

The first step is to redefine the demand dv of node v in such a way that it can be matched

exactly by a subset S of commodities C: min_{𝑆⊆ 𝐶}�� 𝑎𝑐

𝑐∈ 𝑆

∶ � 𝑎𝑐 𝑐∈ 𝑆

≥ 𝑑𝑣� (4)

The second step is to iteratively redefine ac,v for every commodity c ϵ C, starting with ac, v := ac and continue according to (5) until the demand ac,v does not change anymore:

max 𝑆⊆ 𝐶\ {𝑐}�𝑑𝑣 − � 𝑎𝑑,𝑣 𝑑∈ 𝑆 ∶ 𝑎𝑐,𝑣 + � 𝑎𝑑,𝑣 𝑑∈ 𝑆 ≥ 𝑑𝑣� (5)

These two steps have to be repeated until the demands are redefined for all nodes in V. Constraint (3) can now be replaced by the stronger constraint:

� � 𝑎𝑐,𝑣 𝑥𝑐,𝑃 𝑃ϵ𝑃𝑐,𝑣

𝑐ϵ𝐶

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According to [24] having this minimal coefficients is still not sufficient for a strong LP relaxation and it is advised to create m constraints for each node:

� � 𝑏𝑐,𝑣𝑖 𝑥𝑐,𝑃 𝑃ϵ𝑃𝑐,𝑣

𝑐ϵ𝐶

≥ 𝑑𝑣 , 𝑣 ϵ 𝑉, 𝑖 = 1, . . . , 𝑚 ₍₇₎

The second method is to simply replace the demands for each node by a constraint that ensures that every node has at least one commodity assigned to it. In this approach constraint (3) is replaced with:

� � 𝑥𝑐,𝑃 𝑃ϵ𝑃𝑐,𝑣 𝑐ϵ𝐶

≥ 1 , 𝑣 ϵ 𝑉 ₍₈₎

The solution found by using this constraint may not provide sufficient capacity to service the trip represented by the nodes in V. This is not acceptable for rolling stock scheduling.

Because the Rolling Stock Scheduling Problem consists of different rolling stock types and many trips to service with many constraints on arcs and paths, it is a very large scale ILP problem that cannot be solved for an optimal solution. Therefore LP relaxation is necessary in order to find a lower bound for the optimal solution of the ILP. As stated earlier, column generation is very well suited to find the LP relaxation of large path ILP models.

In column generation the original problem gets split into two problems. One is the original problem with only a subset of constraint variables considered, called the main problem. The other problem, or sub problem is to identify a new constraint variable to include in the subset of constraint variables considered in the main problem. The objective function of the sub problem looks for negative reduced cost variables with respect to the current dual variables, also referred to as shadow prices of the master problem. When negative reduced cost variables are found, they are added to the main problem. This cycle is repeated until no negative reduced cost variables are found in the sub problem and the solution to the master problem is optimal.

In the path ILP model (1)(2)(7), the reduced cost of variable xc,P is calculated as follows.

Define αc as the dual variable of (2) and define βvi as the dual variable of (7), then the

reduced cost of xc,P is:

𝑓𝑐,𝑃− 𝛼𝑐− � � 𝑏𝑐,𝑣𝑖 𝛽𝑣𝑖 𝑚

𝑖=1 𝑣ϵ𝑉𝑃

, 𝑖 = 1, . . . , 𝑚 (9)

It should be noted that this generic model only covers the very basics of the rolling stock scheduling problem. A complete representation contains at least additional constraints on the availability of commodities, the size and composition of commodity subsets assigned to

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3.3 Solution approaches to the rolling stock scheduling problem

In this section various solution approaches to the Rolling Stock Scheduling Problem as presented in recent literature are described and evaluated. An overview of the reviewed literature is given in Table 2. For each approach the area of application and the planning level is given. Then the basic model structure on with the problem representation is based and the solution method are described. Finally whether or not the results of the solution approach are tested to be satisfactory in real life situations.

Reference Application Planning

level Model structure Solution method Verification [25] Passenger

Train Unit Assignment

Operational Integer Multi Commodity Flow Problem Heuristic Compared with real world schedule [26] Passenger Train Unit Assignment

Operational Integer Multi Commodity Flow Problem Heuristic Compared with other models [27] Passenger Train Unit Assignment

Operational Integer Multi Commodity Flow Problem Heuristic Compared with other models [28] Locomotive

Assignment Operational Vehicle Routing Problem with Time

Window

Genetic

heuristic Compared with exact solution [3] Locomotive

Assignment Strategic, Tactical, Operational

Task Graph Approximate Dynamic Programming Tested on real world case [29] Passenger Train Unit Assignment Operational Asymmetric Travelling Salesman Problem Ant Colony

Optimization Tested on real world case Table 2: Overview Rolling Stock Scheduling Problem solution approaches

Integer Multi Commodity Flow Model, Heuristic (Cacchiani, Caprara, Toth, 2010)

In [25], the authors present a solution approach for the assignment of available train units to trips scheduled according to a predefined timetable. The problem is presented as an Integer Multi Commodity Flow Problem, where trips are represented by nodes in a directed multigraph. In addition to these nodes there is a source node and an target node defined. Paths consisting of a set of arcs are defined between the source node and the target node. These paths only connect trips that can be serviced by the same train unit on the same day. The LP relaxation of the path ILP formulation is solved using column generation based procedure in conjunction with a three step heuristic method.

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First a diving rule is applied to fix the value of some variables of the current optimal LP solution. After this the LP is re-optimized with the new fixing constraints. Then a constructive heuristic algorithm is applied at each iteration. This algorithm defines paths to each train unit type at a time. It starts with the train unit type with the lowest cost / capacity ratio and moves up when all train units of that type are fully utilized. For each train unit type it starts assigning the paths that were already fixed in the previous step and finds the remaining paths with a column generation based algorithm. This algorithm basically follows the dual profits of the uncovered trips, while trying to match assigned capacity to required capacity at equality and trying to service uncovered trips by over-covering trips that are already covered. Then a refinement algorithm is applied to each solution of the constructive heuristic algorithm in order to improve the solution. The refinement algorithm is based on an arc ILP formulation with covering constraints of the same problem, but only considering the number of times that a trip is assigned to each train unit type. It is not considering the paths in the solution of the arc ILP formulation. The number of times that a trip is assigned to a specific train unit type in the solution of the arc ILP formulation is used as a requirement for the solution of the constructive algorithm.

Maintenance appointments are be added to this model by introducing a subset of arcs corresponding to sequences of two trips that allow enough time for maintenance in between for a particular train unit type. An additional constraint will be needed to ensure that enough paths for a particular train unit type contain an arc in the maintenance subset. The results of this approach are compared to the rolling stock schedules made by practitioners. The authors claim to have improved the schedule by 10-20%. Even though some of the real life constraints were not taken into account by this solution approach, the authors claim that these additional constraints do not have a big impact on the solution quality.

Integer Multi Commodity Flow Model, Heuristic (Cacchiani, et al., 2012)

In [26], the authors present a solution approach for the assignment of available train units to trips scheduled according to a predefined timetable. The problem is presented as an Integer Multi Commodity Flow Problem, where trips are represented by nodes in a directed acyclic graph. The authors present a heuristic solution approach that is based on the calculation of a lower bound obtained by solving an ILP representation of the problem.

First a lower bound is found by solving an ILP representation of the problem for a set of incompatible trips, i.e. trips that cannot be performed by the same train unit on the same day, during the peak period of the day. The objective is to find the number of train units of each type that need to be assigned to each trip in the set of incompatible trips to cover the capacity requirements of these trips during the peak period. The value of the solution to this ILP representation is a valid lower bound for the original problem. The number of train units

Review of vehicle rostering approaches in railway

Summary

Contents

1 Introduction

2 Vehicle assignment problem in railway

3 Solution approaches to the vehicle assignment problem in railway