Locomotive Allocation for Toll NZ Sanjay Patel Department of Engineering Science University of Auckland, New Zealand spat075@ec.auckland.ac.nz Abstract A Locomotive is defined as a self-propelled vehicle for pulling freight or passenger cars along railroad tracks. A Locomotive is considered separate from the train itself which comprises carriages or wagons. Locomotives are required to pull trains to their destination. The trains the locomotives are scheduled to take is known as the Locomotive Plan and it ensures that every train has the correct number of locomotives allocated. However, unforeseen events, such as delays or breakdowns, occur frequently which disrupt the Locomotive Plan. The objective of this project was to develop an optimization model for Toll New Zealand (NZ) that re-allocates trains to locomotives due to these daily unforeseen events. The model is to make alterations to the plan ensuring that the correct number of locomotives is assigned to each train and that the alterations are optimal according to preferences indicated by Toll NZ. The optimization model and solution process involved a priori column generation and zero-one integer programming. The model was successfully able to reallocate the locomotives to the trains for a range of possible disruptions at varying times during the week in the Electric Route (coves the Central North Island). Results showed indications towards the model producing a better re-allocation, in terms of the preferred changes, than what is done manually by a Locomotive Controller at Toll NZ. 1 Introduction A locomotive is defined as a self propelled engine that pulls or pushes freight or passenger cars along railway tracks. In this project, a locomotive is considered a separate part from the trains, consisting of wagons or carriages, which the locomotives are required to pull. A Locomotive Plan determines the sequence of trains that a locomotive is to pull. It ensures that every train has the required number of locomotives allocated. This is termed a feasible solution. When there is an unforeseen occurrence, the plan can no longer be adhered to. The aim of this project was to create a model for Toll NZ to identify the optimal reallocation of locomotives to trains based on unforeseen occurrences whenever they arise. Toll NZ have expressed their preferences associated with altering the locomotive plan. At present, these changes are made manually by a Locomotive Controller. Master Coupling Sheets currently exist which can be used to determine the Master Plan. This comprises sequences of trains for each of the locomotives to pull assuming no disruptions and is the preferred Locomotive Plan for the week by Toll NZ.
The Locomotive re-allocation problem requires sequences of train duties to be allocated to the particular locomotives so that all the duties are covered and every locomotive has just one sequence assigned. It can therefore be viewed as a scheduling application and more specifically, the rostering model for crew scheduling. The locomotives correspond to the people and the trains that need to be pulled, correspond to the duties. The solution approach first involves a priori column generation of the possible train sequences each locomotive can now pull given the disruption. The second part of the solution process is the optimisation and involves selecting the best sequence for each locomotive, while ensuring all the train duties are covered. The solution becomes the new locomotive plan. This paper outlines the modelling of the Locomotive re-allocation problem. Section 2 describes the actual problem encountered by Toll NZ. Section 3 gives details on how the data was used to set up the model, the a priori column generation and the ZIP optimisation. Results are presented and discussed for different unforeseen occurrences at different times during the week in Section 5. 1.1 Previous Work The model used in this project is a special case of the set partitioning problem, in particular rostering people. Similar models and solution techniques have been used in many different applications of scheduling including locomotive engineers to trains and airline crew scheduling. 1.2 Background Information Toll NZ operates in four different zones around the country. They hire locomotives from ALSTOM and the fee is based on the total number of kilometres travelled by the locomotives. There are nine different classes of locomotives that are available all with different characteristics. Over the four zones there are approximately nine hundred trains that require a locomotive to be allocated to them throughout the week. This paper looks in detail at one of the four routes - the Electric Route. This runs from Hamilton to Palmerston North. This is shown in Figure 1.1. Only one of the nine types of locomotive is able to travel along this route - the EF Electric Class Locomotive. Figure 1.1: The Electric Route (Runs from Hamilton to Palmerston North) The Master Coupling sheets are supplied by Toll NZ. They indicate the subsequent train for a locomotive to pull given that it comes into a depot pulling a certain train. For example in Table 1.1 the Locomotive arriving pulling train 220 into Hamilton is scheduled to arrive Friday 23:43pm. This locomotive is scheduled to next pull train 221 departing Saturday 02:58am. Two identical rows indicate that two locomotives are
required to pull the train. The Master Plan can be determined by linking the trains to form sequences (as shown in bold in Table 1.1) from the Master Couplings for each of the locomotives. Loco's Location Arriving Service Arrival Day Arrival Time Depart Service Depart Day Depart Time Layover Ef Hamilton 220 Fri Fri 23:43 221 Sat Sat 02:58 3:15 Ef Hamilton 220 Fri Fri 23:43 221 Sat Sat 02:58 3:15 Ef Hamilton 222 Sat Sat 01:58 215 Sat Sat 05:21 3:23 Ef : : : : : : : : Ef : : : : : : : : Ef Palm Nth 229 Fri Fri 23:37 230 Sat Sat 03:30 3:53 Ef Palm Nth 231 Sat Sat 02:02 312 Sat Sat 05:50 3:48 Ef Palm Nth 211 Sat Sat 04:28 234 Sat Sat 08:48 4:20 Ef : : : : : : : : Ef Palm Nth 221 Sat Sat 11:24 242 Sat Sat 18:05 6:41 Ef Palm Nth 221 Sat Sat 11:24 242 Sat Sat 18:05 6:41 Ef : : : : : : : : Table 1.1 Excerpts from the Master Couplings 1.3 Sequence Representation This project deals with the allocation of locomotives to trains. A concise way of representing the sequences for each of the locomotives has been chosen. Each locomotive has a particular identification number and the trains have characteristic parameters (such as departure and arrival times and towns). To simplify these representations, numbers are assigned to each locomotive to distinguish them from one another and similarly to each train. A sequence for a locomotive is represented as shown: e.g. Loco 7 1 5 9 15 If locomotive 7 was assigned this sequence, the trains for the locomotive to pull are 1, followed by 5, 9 then 15. 2 The Problem A plan is feasible and can be adhered to as long as there are no unforeseen disruptions. Each day however there are events which disrupt the locomotive plan. The model is to identify the optimal way of reallocating the trains to the EF electric class locomotives on the Electric Route to obtain feasibility again. The possible disruptions that need to be modelled and the measures which determine an optimal change are outlined in this section. 2.1 Possible Disruptions The possible disruptions that make the plan infeasible are: A delay in the arrival time of a train. A locomotive breakdown (Tags) Tags are a term used by Toll NZ that indicate that a particular locomotive is partly or fully damaged and cannot work at full capacity. Different letters denote different tags. An extra freight train needing to be added to the schedule. Cancelling a freight train from the schedule. Locomotive scheduled maintenance. Altering the number of locomotives required for each of the trains.
2.2 Quality of the Alterations as denoted by Toll NZ In altering the Locomotive Plan, the following measures of quality in priority order are: Ensuring all the trains depart on time all the trains are to have the correct number of locomotives assigned to pull them to their destinations. Minimising the number of kilometres travelled The trains that are required to run are fixed therefore this objective requires minimising the number of kilometres each locomotive travels by itself to reposition. Trying to incorporate locomotive maintenance service checks into the schedule Maximising the compliance with the Master Couplings. Maximising the utilisation of the available locomotives this provides robustness in terms of the system recovery from a disruption. 3 The Model The requirement of the model was to determine the optimal way to reallocate locomotives to trains. It was to provide solutions to the possible different circumstances such as the various disruptions and the times at which these occur throughout the week in the Electric Route. The aim is to produce better quality changes to the plan than those that are made manually by the Loco Controller. Certain assumptions were made to allow the optimisation model to be formulated: There is always a locomotive engineer available to drive a locomotive Locomotives are always able to pass each other along the track Linking up of trains and locomotives at the boundaries of the four routes is ignored. Changing the departure times of trains is not considered directly. The model aims to find the optimal reallocation to cover only the trains within a set planning horizon. This is typically one to two days. The process to determine the optimal reallocation initially involved transforming the necessary inputs into a workable form. Two parts to the model then followed. The first is the a priori generation which creates all the feasible sequences of trains that each locomotive can now take. The second is the optimisation which finds the optimal sequence for each locomotive so that all the trains have the required number of locomotives allocated. The process is outlined in Figure 3.1 and explained in more detail in the following sections. Disruption encountered Provide the necessary input data Generate all the feasible sequences for each locomotive due to the disruption A Priori Generation Optimisation New Sequence of trains to pull for each Locomotive found Figure 3.1: Outline of the model to get to the solution
3.1 Adjusting the Input Parameters The required attributes for the trains that were stored were: Identification number Departure and arrival times Departure and arrival days Departure and arrival towns Number of locomotives required to pull the train Delay in arrival time Indications as to the type of train (e.g. freight, passenger, extra heavy etc) The following characteristics for the locomotives were required: Tags assigned In-repair maintenance check details Maintenance check requirement details Breakdown details 3.2 A Priori Column Generation A Priori column generation is basically an enumeration process which involves generating possible sequences of trains that a locomotive can now take based on the disruption and its current placement. Sequences of trains that cannot be pulled by a locomotive are not generated. All the sequences are created before entering them into an optimisation package. Costs are then assigned to each of the sequences. If a locomotive arrives pulling a certain train, there are several following trains that can then be allocated to the locomotive dependent on the disruption. These following trains form a list known as the subsequence list. A subsequence list can therefore be created for every train. The list sizes were limited by applying practicality rules. This decreased the time taken to generate all the possible sequences. From the subsequence lists, the possible sequences of trains for each of the locomotives can be generated given the disruption. Each disruption has a different effect on the system and therefore needs to be dealt with in a different way. The algorithm for generating all the possible sequences is shown in Figure 3.2. Each of the sequences of trains that were generated was assigned a total cost. An individual cost was calculated for each of the objectives explained in section 2.2 and each was weighted in terms of its importance to give the total cost. The total cost corresponded to how favourable or unfavourable the sequence was in terms of the measures of quality outlined by Toll NZ (section 2.2). 3.3 Optimisation The objective of the optimisation process is to determine the optimal sequences of trains such that each train has the required number of locomotives and each locomotive is given exactly one sequence of train duties. This problem was modelled as a zero-one integer programming model which can be solved with the ZIP 4.0 optimisation package (Ryan 1981). Allocating trains to locomotives can be accomplished by modelling the problem as a generalised set partitioning problem. This is a special form of a zero-one integer programming problem. The problem involves allocating a set of trains to each of the locomotives. Each sequence of trains that is generated in the a priori generation comprises a subset of all the trains that require pulling within the planning horizon. The optimal feasible solution selects a set of sequences so that each train is covered by the
required number of locomotives and each locomotive is assigned just one sequence of trains at minimum total cost. START Generate for next locomotive until finished Start the sequence with the locomotive s initial / current train(s) Limited subsequence lists already generated for Locomotive no Remove the last train added yes More than one train in the outputted sequence? Output the sequence no Check the last train in the sequences subsequence list for any more trains to add? yes Update the sequence yes no Feasible to add? Check whether the train to add to the sequence departs within the time window and the locomotive is able to take the train Figure 3.2 Algorithm for generating the sequences for each of the locomotives The mathematical model used to solve the locomotive allocation problem is acquired from the set partitioning model. The set partitioning model is as follows: Minimise: Subject to: Z = C T x Ax = B x j {0,1} Where: -x j is the variable corresponding to sequence j -C j is the cost of sequence j as described in section 3.2 -A ij = 1 if sequence j includes pulling train i, 0 otherwise or 1 if sequence j is for locomotive i, 0 otherwise The columns of the A matrix correspond to each of the possible sequences created in the a priori generation. Each sequence generated is transformed into a column which completely represents the information shown by the sequence. The variable x j corresponds to sequence j. If sequence j is included in the feasible solution, x j =1 otherwise x j =0. Each row in the A matrix represents a separate constraint. Two forms of constraints are required for the locomotive allocation problem being locomotive constraints and train constraints. C j is the associated cost of sequence j and the objective is to choose the set of sequences to minimise the total cost while ensuring the constraints are satisfied. The locomotive constraints ensure that exactly one sequence is chosen for each locomotive:
n j= 1 a x ij j = 1 for i = 1,2,,L (L= number of locomotives), n = # variables The train constraints ensure the correct number of locomotives is allocated to each train: n a ij x j = j=1 for i = 1,2,,z; z = # of trains within the planning horizon, n = # variables t i represents the number of locomotives required for train i ZIP 4.0 solves the Linear Program (LP) relaxation (ignores the integer constraints) and then applies Branch and Bound to achieve integer solutions. Due to the characteristics of the set partitioning problem, the solution to the LP relaxation produced near-integer solutions so minimal work was required in the Branch and Bound component. The output from the model is a new Locomotive Plan that is feasible taking into account the disruption. 4 Results The model is able to provide a solution for all known disruptions at any point throughout the week. Two scenarios are used to show the scope of the solutions generated. Also a comparison is made between the quality of the model solution and the solution created using the current process (performed manually by a Loco Controller) at Toll NZ. Each of the schedules produced are analysed in terms of quality. The objectives used to optimise the quality of the schedules, in order of priority, were to: Maximise the number of trains that depart on time (i.e. % of trains that have the correct number of assigned locomotives) Minimise the number of times a locomotive moves on its own to a different depot Maximise the number of locomotives that are available for their maintenance check Maximise the couplings in the schedule produced that comply with the Master Couplings Maximise the utilisation of the locomotives that are available 4.1 Example of Solutions to Different Hypothetical Scenarios A particular example of a train delay is as follows: Currently the time is 13:20pm on Thursday and it is found out that train 84 has been delayed and will be arriving 8 hours 20 minutes behind its scheduled arrival (i.e. 06:46am on Friday) Locomotives 16 and 17 are currently in scheduled maintenance. The locomotives are currently following the schedule determined by the master couplings as this has been feasible up till now. The train delay makes the original plan infeasible. i.e. The next train that was supposed to be pulled by the delayed locomotive does not have a locomotive to pull it. To regain feasibility, a proposed solution is found by the model and is shown in Figure 4.1. t i
Trains 74 75 76 77 78 79 80 Locomotives 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 1 X X X 2 X X X 3 X - - - - - - - - - - X - - - - - 4 X X X 5 X X 6 X X X 7 X X 8 X X X 9 X X 10 X X 11 X X 12 X X X 13 X X 14 X X X 15 X X X 16 IN SCHEDULED MAINTENANCE X 17 IN SCHEDULED MAINTENANCE - Represents a train that is to be pulled by the locomotive X Original Plan (based on Master Couplings) Represents the delayed train - - - - - - - - Locomotive travels on its own to reposition in between pulling the two scheduled trains Figure 4.1: Train Delay Example The delay disrupts the schedule as locomotive 10 can no longer pull train 95. The only way to resolve this issue was to send locomotive 3 from Palmerston North to Hamilton without pulling a train. Statistics of the solution to the train delay example are shown in Figure 4.1. The model is able to find a feasible re-allocation to ensure all the trains have the required number of locomotives assigned which is very favourable (shown by 100% of the trains that will depart on time). As a high priority, the model tries to minimise the number of times a locomotive repositions. A locomotive is required to travel on its own to account for the delay in train 84. A large proportion of the schedule complies with the Master Couplings which is very attractive. Planning Horizon 1 Day % of the Scheduled Trains that have an assigned 100% locomotive(s) that will depart on time Number of times a Locomotive travels on its own 1 to reposition without pulling a train % of Locomotives that are available for their No Maintenance Checks scheduled maintenance check % of train changeovers that comply with the 87.50% Master Couplings % utilisation of the available locomotives 94% Table 4.1: Statistics from the Train Delay Example Another example is a locomotive breakdown: On Tuesday 24 th August at 06:40 am it is found out that Locomotive 11 has fully broken down 59km from Palmerston North pulling train number 40 (Service 221 on Tuesday). It cannot pull any more trains until it is repaired. Train Service 221 required two locomotives to pull it and therefore the other attached locomotive, number 6 is stranded with the train and broken locomotive 11. It cannot do anything until another locomotive is sent to replace the pulling power
of locomotive 11 and help move the entire train to Palmerston North (repair location). Locomotive 2 has a P-Tag - restricted in the trains it can pull Locomotives 14 and 15 are required for scheduled maintenance checks. Locomotive 16 and 17 are currently in scheduled maintenance. The sequences determined by the master couplings have been feasible up to this point in time. This was the plan in place. Based on the breakdown, a new plan must be generated by the model to regain feasibility. This requires rescuing the broken down locomotive and re-allocating the locomotives to the trains so that each train has the required number of locomotives assigned to it. The proposed solution is shown in Figure 4.2. Trains 30 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Locomotives 1 X X 2 X X X 3 X X 4 X X X 5 X X 6 X X X 7 X X X 8 X X X 9 X X X 10 X X X 11 X BROKEN 12 X X X 13 X X X 14 X X CHECK 15 X X REQ D FOR CHECK 16 IN SCHEDULED MAINTENANCE 17 IN SCHEDULED MAINTENANCE X X - Represents a train that is to be pulled by the locomotive X Original Plan (based on Master Couplings) Represents the locomotive that rescues the broken down locomotive Figure 4.2: Locomotive Breakdown Example Points to note concerning the solution: Locomotive 13 rescues the breakdown (3rd locomotive to train number 40). Locomotive 6 is back at Palmerston North in time to pull train 51 Locomotive 11 is damaged and cannot pull any more trains. Statistics were formed from the optimal solution in Figure 4.2 to determine the quality of the schedule produced. Table 4.2 shows that the solution generated by the model is of high quality. The re-allocation allocates the locomotives so that all the trains have the required number assigned. All of the locomotives are available for their scheduled maintenance checks when required. The master couplings do not incorporate maintenance checks. Therefore when the plan must be altered to include maintenance checks, compliance with the master couplings is degraded. However 70.21% compliance with the master couplings is of high standard.
Time Window 2 Days % of the Scheduled Trains that have an assigned locomotive(s) that will depart on time 100% Number of times a Locomotive travels on its own to reposition without pulling a train 0 % of Locomotives that are available for their scheduled maintenance check 100% % of train changeovers that comply with the Master Couplings 70.21% % utilisation of the available locomotives 100% Table 4.2: Statistics from the locomotive breakdown example 4.2 Comparison with Toll NZ s Schedule Table 4.3 gives a comparison of the details of the solution (schedule) produced by model and that created manually by the Locomotive Controller at Toll NZ. The reallocation proposed by the model is of higher quality than Toll NZ s solution in terms of a much higher percentage of the couplings complying with the master couplings in the model solution. Both the figures for the compliance with the master couplings are low due to the cancellations in certain trains. Both solutions effectively allocate the locomotives to ensure that all the scheduled trains have the required number assigned. Toll NZ s Solution Model Solution Time Window 1 ½ Days % of the Scheduled Trains that have assigned locomotive(s) that will depart on time 100% 100% Number of times a Locomotive travels on its own to reposition without pulling a train 0 0 % of Locomotives that are available for their scheduled maintenance check 100% 100% % of train changeovers that comply with the Master Couplings 8.00% 56.52% % utilisation of the available locomotives 87.5% 93.75% Table 4.3: Comparison of the Statistics from the solutions from 11:40 August 7 2004 5 Conclusions The model successfully produced results that were in compliance with the objectives set by Toll NZ. Assigning component costs and weightings to the sequences produced from the a priori generation allowed the objectives to be taken into account simultaneously and forced the schedule to prioritise certain characteristics. The statistics from the hypothetical examples showed that the new schedules produced to account for the disruption, achieve feasibility and are of high quality. Only very weak conclusions can be made on the model producing better quality results than with the current situation at Toll NZ despite the favourable statistics. The model only takes into account one zone and one type of locomotive. Currently at Toll NZ, all four zones are considered together and demands for different types of locomotives and changeovers of trains at the borders of the zones are incorporated. Possible future work is to extend the model to include the four different zones so more valid comparisons can be made.