Simultaneous train rerouting and rescheduling on an N-track network: A model reformulation with network-based cumulative flow variables

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1 Simultaneous train rerouting and rescheduling on an N-track network: A model reormulation with network-based cumulative low variables Lingyun Meng Assistant Proessor State Key Laboratory o Rail Traic Control and Saety Beijing Jiaotong University 1201, SiYuan Building, No.3 ShangYuanCun HaiDian District, Beijing , China Tel.: (86) lymeng@bjtu.edu.cn Xuesong Zhou (Corresonding Author) Associate Proessor School o Sustainable Engineering and the Built Environment Arizona State University Teme, AZ 85287, USA Tel.: (1) xzhou74@asu.edu Submitted or ublication in Transortation Research Part B First submission: July 31, 2013 First revised version: Dec 24, 2013 Second revised version: Feb 25, 2014 Abstract Train disatching is critical or the unctuality and reliability o rail oerations, esecially or a comlex rail network. This aer develos innovative integer rogramming models or the roblem o train disatching on an N-track network by means o simultaneously rerouting and rescheduling trains. Based on a time-sace network modeling ramework, we irst adat a commonly used big-m method to reresent comlex i-then conditions or train saety headways in a multi-track context. The track occuancy consideration on tyical single and double tracks is then reormulated using a vector o cumulative low variables. This reormulation technique can rovide an eicient decomosition mechanism through modelling track caacities as side constraints which are urther dualized through a roosed Lagrangian relaxation solution ramework. We urther decomose the original comlex rerouting and rescheduling roblem into a sequence o single train otimization subroblems. For each subroblem, a standard label correcting algorithm is embedded or inding the time deendent least cost ath on a time-sace network. The resulting dual solutions can be transormed to easible solutions through riority rules. We resent a set o numerical exeriments to demonstrate the system-wide erormance beneits o simultaneous train rerouting and rescheduling, comared to commonly-used sequential train rerouting and rescheduling aroaches. Keywords Train disatching, Rail network, Cumulative low variable, Lagrangian relaxation 1

2 1. Introduction Providing unctual and reliable services is a rimary goal o rail industries in order to maintain and urther imrove their cometitive advantages in the raidly changing multimodal transortation market. Train timetabling tyically aims to construct a schedule that seciies a hysical network route and detailed arrival time and dearture times or each train at assing stations, with the objective to minimize e.g. total travel time o all trains. As tactical lans o comlex rail oerations, train schedules are rogrammed and udated every year or every season. During the rocess o executing a lanned train schedule, various sources o erturbations may inluence train running times and dwelling and dearting events, thus causing rimary delays to the lanned train schedule. Due to the high interdeendency between trains, rimary delays could roagate as secondary delays to other trains on a network. The key task o train disatching is to take roer measures that can recover imacted schedules rom stochastic erturbations and urther minimize otential negative consequences. On a high-density rail network with limited caacity, the real-time train disatching task becomes extremely comlicated where ineective schedule adjustment and ad-hoc resonses could signiicantly downgrade the unctuality and reliability o train services and the overall system erormance. The train timetabling and disatching roblems have been well studied in the ast ew decades. Assad (1980) and Cordeau et al. (1998) reviewed many key modeling asects o rail oerations and a recent survey by Hansen (2010), summarized emerging methods and solution techniques or train timetabling and disatching. As real-time train disatching oten relies on lanned schedules rom the stage o train timetabling, we irst start with a brie review on the related train timetabling roblem. Carey and Lockwood (1995) resented a mixed integer rogramming model and solution algorithms or the train timetabling roblem on a double-track rail line with trains oerating at dierent seeds. Carey (1994a) urther develoed an extended model to consider more general and more comlex rail networks with ossible choices o lines and station latorms. A comanion aer by Carey (1994b) roosed an extension rom one-way to two-way rail lines. An alternative discrete event simulation-based modeling ramework was roosed by Dorman and Medanic (2004) to solve train scheduling roblems on a large-scale rail network. Along this line, Li et al. (2008) resented an imroved simulation-based method with a consideration o global train inormation. Recently, how to use theoretically sound ormulations to describe train timetabling on an N-track network has received considerable attention. Carey and Craword (2007) develoed eicient heuristic algorithms or inding and resolving train conlicts in drat schedules in which the choices o lines/tracks between stations were considered. Lee and Chen (2009) roosed an otimization-oriented heuristic to assign routes and tracks and accordingly allocate time slots or trains. Mu and Dessouky (2011) exlored an novel otimization-based rocedure or solving the reight train scheduling roblem on an N-track network in the context o a rail network in United States, which includes (1) a FixedPath model that considers train routing and scheduling sequentially with one exact ath or each train, and (2) a FlexiblePath model that aims to jointly determine train routing and scheduling decisions based on simliying assumtions about the toological structure o a network. We will oer more detailed comarisons with their work at the end o this aer. Yan and Yang (2012) roosed a train routing and scheduling model which alies network low balance constraints and an i-then tye o constraints to model rail traic on a network and saety headways. The i-then tye constraint can be handled through a disjunctive grah modeling aroach or a big-m tye reresentation in an integer 2

3 rogramming ramework. Harrod (2011) roosed a directed hyergrah ormulation which is caable to lexibly model both constrained resources and resource transitions by block occuancy side constraints and network transition constraints. By extending otimization models roosed by Zhou and Zhong (2007) and Törnquist and Persson (2007), Castillo et al. (2011) urther incororated user reerence into the train timetabling roblem. We now move to a brie review on the train disatching roblem on a rail network. Jovanovic (1989) introduced a mixed integer rogramming model that minimizes the tardiness cost. Törnquist and Persson (2007) roosed an otimization aroach and various ractically useul solution strategies or managing railway traic on an N-track network. Mannino and Mascis (2009) develoed a real-time automated traic control system to oerate trains on a network o metro stations in Italy. Iqbal et al. (2013) resented and evaluated three dierent search arallelization strategies or designing a greedy train disatching algorithm. By using the multi-strategy based arallel aroach, their greedy algorithm shows its advantages in delivering easible solutions quickly. Meng and Zhou (2011) considered the imact o the stochasticity and dynamicity o erturbations on scheduling robust meet-ass lans by develoing a scenario-based rolling horizon solution aroach. Quaglietta et al. (2013) roosed an innovative simulation-based otimization ramework or analyzing the stability o train disatching lans under a stochastic and dynamic environment. For more inormation about the train disatching roblem, we reer to a recent review by Pacciarelli (2013). Based on a systematic modeling ramework with key comonents such as the alternative grah ormulation, branch-and-bound solution algorithms and heuristic real-time conlict-detection-and-resolving algorithms, D'Ariano et al. (2007a, 2007b, 2008, 2009) and Corman et al. (2009, 2010a, 2010b, 2011) roosed a stream o research that addresses various asects o the train disatching roblem with both small disturbances and heavy disrutions. A laboratory on-line decision suort comuter system, namely ROMA, was designed and imlemented or assisting train disatchers to recover imacted schedules rom erturbations. As indicated by Corman et al. (2010b), comared to conventional solution aroaches assuming ixed train routes, better solution quality could be obtained i train routes are allowed to be otimized in conjunction with the change o train orders. Focusing on the roblem o managing railway traic on a large-scale network, Corman et al. (2012, 2014) roosed otimization models and algorithms or coordinating several disatchers with the objective to drive their behaviors towards globally otimal solutions. Kecman et al. (2013) ut orward a set o macroscoic train disatching models or controlling country-wide railway traic. In this research, we adat the terminology develoed by Hansen (2010) and consider the ollowing our tyical train disatching measures: 1) re-time in terms o changing arrival and dearture times; 2) re-order in terms o changing arrival and/or dearture orders; 3) re-track in terms o using a dierent track; and 4) re-route in terms o using a dierent route on a network. Tables 1 and 2 comare recent studies on the train timetabling and train disatching roblems, resectively, in the key dimensions o scheduling strategy, scheduling measure, mathematical ormulation, roblem size and solution algorithm. Interestingly, we can ind that most existing literature sequentially determines routes and then schedules or trains, and there is a clear trend towards the develoment o detailed mathematical models and eective solution algorithms that can simultaneously (re-)route and (re-)schedule trains. Essentially, there are two categories o modeling aroaches or handling time constraints. The irst category reresents the otimization time horizon in a continuous ashion, and train sequencing constraints are tyically used to ensure saety time headways between a air o train arrival/dearture events occurring at any time o the entire lanning horizon. On the other hand, many 3

4 studies using the Lagrangian relaxation modeling ramework (e.g., Brännlund et al., 1998, Carara et al., 2002) tyically discretize the time eriod o interest to a vector o time intervals (e.g. 1 minute), which are urther considered as discrete resources in the dualized roblems. Table 1 Mathematical ormulations and solution algorithms or train timetabling. Scheduling Scheduling Model Publication Objective strategy measure structure Carara et al. (2002) E T, O IP Maximize the sum o roits o the scheduled trains Problem size U to 500 trains, a rail line o 39 stations Solution algorithm LR, H (Priority rule-based) Dorman and Medanic (2004) E T, O S Minimize the time to clear the line, delay o all trains, and maximum delay or energy cost 36 trains, 31 sections and 31 nodes H (Greedy train advance strategy) Carey and Craword (2007) E T, O, K - Minimize the total cost o scheduled train delays 490 trains, a set o stations with comlex track layout H Cacchiani et al. (2008) Lee and Chen (2009) Cacchiani et al. (2010a) E T, O IP E T, O, K, R MIP E T, O IP Maximize the sum o the roits o the scheduled trains Generate timetables as close to the drat timetable as ossible Maximize the roits o all the selected comatible train aths U to 221 trains, 102 stations U to 128 train services, a single-track rail line o 159.3km and 29 stations, a double-track rail line o 345km and 8 stations U to 210 trains G, H, BCP H (Four ste rocess) D, H (Greedy algorithms) Cacchiani et al. (2010b) Castillo et al. (2011) I T, O, R IP E T, O MIP Introduce as many new reight trains as ossible Minimize the sum o the relative travel times A rail line with u to 69 stations U to 170 trains D, H (Priority rule-based), LR B 4

5 Mu and Dessouky (2011) E, I T, O, K MIP Minimize the total delay o all the trains U to 60 trains on a medium network, 40 trains on a large network H (Look-ahead greedy and global neighborhood search) Table 2 Mathematical ormulations and solution algorithms or train disatching. Scheduling Scheduling Model Publication Objective strategy measure structure Adenso-Diaz et al. (1999) Törnquist and Persson (2007) E R-T, R-O MIP E, I R-T, R-O, R-K, R-R MIP Maximize the number o assenger transorted Minimize the total inal delay o the traic or minimize the total delay costs Problem size 15 train units, a network o km in length 80 trains, a network o 253 segments Solution algorithm H H (Four eicient rescheduling strategies) D'Ariano et al. (2007a) E R-T, R-O MIP Minimize the maximum secondary delay or all trains at all visited stations 54 trains circulating each hour, A network o 20km in length, 86 block sections, 16 latorms B&B, H (FCFS, FLFS) D'Ariano et al. (2008) E R-T, R-O, R-K, R-R MIP Minimize the maximum and average consecutive delays in lexicograhic order 52 trains circulating each hour, A network o 50km in length, 191 block sections, 20 latorms B&B, LS, PR Corman et al. (2009) E R-T, R-O MIP Minimize the (maximum) consecutive delay o all trains and (total) energy consumtion U to 40 trains er hour, a line o 50km in length, 191 block sections, 21 latorms B&B, H (FIFO), PR 5

6 Corman et al. (2010a) E R-T, R-O, R-K, R-R S Multile criteria such as minimize the total assengers delays 150 trains er hour, a network o 300km in length, more than 1200 block sections AG Corman et al. (2010b) E R-T, R-O, R-K, R-R MIP Minimize the maximum and average consecutive delays in lexicograhic order U to 40 trains, a network o 50km in length, 191 block sections, 21 latorms B&B, H (Tabu search) Corman et al. (2011) E R-T, R-O MIP Minimize train delays along other multile objectives 79 trains er hour, a network o 20 km in length, 600 block sections, 20 latorm tracks B&B, H (Priority rule-based) Meng and Zhou (2013) I R-T, R-O, R-K, R-R IP Minimize the total comletion time o all involved trains U to 40 trains, a network o 287.7km in length, 85 stations and 97 segments LR, H (Priority rule-based) Symbol descritions or Tables 1 and 2: Scheduling strategy: Simultaneous (I): determine train route and arrival/dearture times o trains / Sequential (E): irst seciy train routes and then determine the arrival/dearture times o each train. Scheduling measure: (Re-)time ((R-)T) / (Re-)order ((R-)O) / (Re-)track ((R-)K), tyically known as local re-route which means using a dierent arallel track on the line or within a station / (Re-)route ((R-)R), also known as global re-route which reresents selecting an alternative set o segments and stations rom origin to destination. Model structure: (Mixed) Integer rogramming (M) IP / Simulation-based model (S). Solution algorithm: Branch-and-bound (B&B) / Branch-and-Cut-and-Price (BCP) / Alternative grahs (AG) / Lagrangian relaxation (LR) / Heuristics (H) / Dynamic rogramming (D) / Local search (LS) / Practical rules (PR) / Bisection method (B) / Column generation (G). This aer ocuses on the train disatching roblem on an N-track network, with the main challenge o how to ormulate seciic retiming, reordering, retracking and rerouting otions in combination. We try to oer the ollowing contributions to the growing body o research work on train disatching models: (1) This aer rooses a set o rigorously deined otimization models or the N-track simultaneous train rerouting and rescheduling roblem, in which the routes and assing times at each station along the selected route o each train are jointly otimized. Both roosed big-m-based and 6

7 cumulative low variables-based models allow us to ully exloit available station and track caacity in dierent inrastructure comonents o a rail network. (2) Based on a time-sace reresentation or a multi-track network, we irst introduce dierent methods to reresent the critical saety time headway or trains that might come rom dierent tyes o tracks (single-track vs. double-track) with a large number o ossible routes through the network. By reormulating the inrastructure caacity imlicitly, we then roose a roblem decomosition mechanism where each train-seciic subroblem only has a smaller variable sace comared to the original simultaneous rerouting and rescheduling model involving multile trains. (3) By ully taking advantage o an eicient time-deendent shortest ath algorithm on a time-sace network, we are able to solve the dualized roblem raidly or a medium-size network and urther rovide better lower bound solutions within a Lagrangian relaxation solution ramework. Feasible solutions can be obtained through solution adjustment by riority rules, so the gas between the lower bound and uer bound o generated solutions oer very useul solution quality measures. To evaluate the beneits o the simultaneous train rerouting and retiming aroach, we also develo a sequential train rerouting and rescheduling model which sequentially determines train routes and then the schedules. The simultaneous aroach is roved by exeriments to be able to rovide better disatching solutions comared to the sequential aroach. The remainder o this aer roceeds as ollows. In Section 2, a concetual illustration is resented or the simultaneous train rerouting and rescheduling roblem on an N-track network. Section 3 develos an otimization model through the commonly used big-m method. Section 4 urther constructs a reormulated model based on network cumulative low variables. Based on a time-sace network structure, we use a comutationally eicient Lagrangian relaxation solution ramework to solve large scale roblems in Section 5. The eiciency and eectiveness o roosed models and algorithms, and the beneits o the roosed simultaneous aroach are systematically evaluated in Section 6. Finally, concluding remarks and uture research extensions are discussed in Section 7, ollowed by an aendix that ormulates the sequential train rerouting and rescheduling model based on cumulative low variables. 2. Concetual illustration This section is intended to concetually illustrate the two key roblems o simultaneous and sequential train rerouting and rescheduling, ollowed by a roblem statement and notation used in the roosed models. In our models, we assume that: (1) The length o a train is assumed to be zero (as a virtual dot) or simlicity. That is, our model has a macroscoic reresentation rom train characteristics oint o view. (2) The segment between stations is modelled as a series o block sections or a unidirectional double-track rail line and as one block section or a bidirectional single-track line. A block section is denoted as a cell in our aer (3) I the constraints on siding tracks are not exlicitly considered, a station can be reresented as a node; otherwise a station can be considered as a subnetwork with a number o siding tracks being modelled as a set o cells. (4) Only one train is ermitted on a cell at any given time. (5) The granularity o time is one minute, or a shorter time interval (i required) or some rail systems. 7

8 2.1 Simultaneous and sequential train routing and scheduling on a rail network Fig.1 deicts a simle rail network with 10 nodes. Nodes 0, 1, 4, 5, 8 and 9 reresent a station without a siding track. The node airs (2,3) and (6,7) resectively corresond to two stations, and in each station there is only one siding track. Each link connects two nodes and it is bidirectional so trains can traverse on both directions. There are our trains to be disatched through this network, trains 1 and 3 traversing rom node 0 to 9, and trains 2 and 4 rom node 9 to 0. These our trains have their earliest dearture times at minute 0, 15, 30 and 45, resectively. As shown in Fig. 2, there are a total o our ossible bidirectional routes or each train, and the corresonding ree-low travel times are 34, 32, 36 and 34 minutes, resectively. A route in this aer can be denoted as a series o nodes. For examle, route 1 corresonds to two directional routes, namely node sequences and Siding Siding 2 3 Fig. 1 A simle rail network with 10 nodes. 6 7 Route 1 Route 2 Route 3 Route Fig. 2 Four bidirectional routes that traverse the rail network. The main decision variables o train disatching are (i) routes o trains and (ii) arrival and dearture times at each node. In ractice, the above otimization model is solved in a sequential ashion. First, we determine the route o each train, or examle, all our trains use the route with the minimum ree-low travel time (i.e. route 2) in Fig. 2. Second, trains orders and assing times at each node are comuted to avoid conlicts. One ossible drawback associated with this sequential routing and scheduling method is that the limited otions given in the irst routing stage could dramatically downgrade the erormance o the second ordering + timing solutions. In this aer, we are mainly interested in constructing joint otimization models or both routing and ordering + timing tasks. This simultaneous train routing and scheduling aroach can serve as a theoretically rigorous benchmark or evaluating the erormance o the above-mentioned ractical sequential scheduling aroach with a redetermined and restrictive route set. From a system-otimal ersective, it is desirable to exloit all ossible routes and ully use caacity on all links, or examle, links 2-3 and 6-7 with residual caacity but on alternative longer routes. 8

9 2.2 Problem statement and notation Given a network o rail stations and segments with dierent numbers o tracks, and a set o trains rom re-seciied origin stations to destination stations, the N-track train disatching roblem needs to determine the routes and arrival/dearture times at each station or a set o trains F, or a given lanning horizon t= 1,, T. T is the length o lanning horizon (e.g., T = 300 or a 5-hour horizon). A iner discretization (e.g. 30 seconds, 15 seconds) is also ossible but could lead to a much larger variable sace (Brännlund et al., 1998, Carara et al., 2002, 2006). In this aer, we reresent an original station-segment rail network as G = (N, E) with a set o nodes N and a set o cells E. The extended rail network structure in terms o nodes and cells can hel us better cature ractical saety oerational rules, through signaling and interlocking methods. Seciically, a hysical double-track segment between stations can be decomosed into a sequence o cells where each cell can corresond to a (directed) hysical track circuit. Interested readers can ind detailed inormation on how track circuits are used in Theeg and Vlasenko (2009). Additionally, a station is reresented as a single or multile siding tracks, and each siding track is modeled as a cell in the roosed model. By doing so, we can ma the station minimal and maximal dwell time as constraints on traveling time o the corresonding cell(s). Accordingly, a route in this aer is deined as a sequence o cells. For each cell, the other given inut data include its ree-low running time or dierent tyes o trains, saety time headways, as well as dwell time requirements. The ollowing arameters or each train are assumed to be given: its origin, destination and earliest dearture time. To describe detailed modeling methods, the general subscrits and inut arameters o the roosed ormulations are irst introduced in Tables 3 and 4. Table 3 General subscrits. Symbol Descrition i, j, k node index, i, j, k N, N is the set o nodes e cell index, denoted by (i, j), e E, E is the set o cells route index, P, P is the set o all routes on a rail network m cell sequence number along a route, m N, N is the number o cells in route t scheduling time index, t= 1,, T, T is the lanning horizon train index, F, F is the set o trains Table 4 Inut arameters. Symbol Descrition E set o sequenced cells o route, E N P set o ossible routes on which train may run, P E set o cells train may use, E E P FT ( i, j ) ree-low running time or train to drive through cell (i, j) EST redetermined earliest starting time o train at its origin node w min ( i, j ) minimum dwell (waiting) time or train on cell (i, j), ( i, j) w max ( i, j ) maximum dwell (waiting) time or train on cell (i, j), ( i, j) Ca( i, j, t ) low caacity on cell (i, j) at time t, Ca( i, j, t )=0 due to maintenance o cell (i, j) at time t, Ca( i, j, t) 1, otherwise 9

10 g h o s saety time interval between occuancy and arrival o trains saety time interval between dearture and release o trains origin nodeo train destination (sink) node o train Ds ( ) reerred arrival time at the destination node o train in the original timetable m, cell maing indicator, reresents the cell that is the m th cell or train along route E OS () i set o cells starting rom or ending at node i E O () i set o cells starting rom node i E S () i set o cells ending at node i set o cells that allow dwell time, reresenting siding tracks in stations 3. Simultaneous train rerouting and rescheduling model Beore detailing the Simultaneous train Rerouting and Rescheduling (SRR) model, we irst introduce the integral decision variables in Table 5. Table 5 Decision variables or the simultaneous train rerouting and rescheduling model. Symbol Descrition 0-1 binary train routing variables, x ( i, j) 1, i train selects cell (i, j) on the x ( i, j ) network, and otherwise x ( i, j) 0 a ( i, j ) arrival time o train at cell (i, j) d ( i, j ) dearture time o train rom cell (i, j) (, ', i, j) 0-1 binary train ordering variables, (, ', i, j) 1 i train ' arrives at cell ( i, j ) ater train, and otherwise (, ', i, j) 0 TT ( i, j ) runnning time or train at cell (i, j) The vector x ( i, j ) catures the routing decisions on a rail network. The vectors a ( i, j ) and d ( i, j ) have been commonly used in revious studies to directly describe the timestams when the arrival and dearture events o train at cell (i, j) occur. The vector (, ', i, j) reresents the train orders, which are tyically used or modelling saety time headways. The objective is to minimize the total deviation time o all involved trains. The total deviation time o train can be described as d ( i, s ) D( s ). Note that this objective unction uses the i same weight on early arrival deviation and late arrival deviation, while it is ossible to have dierent weights on these two schedule deviation comonents (INFORMS RAS, 2012). The SRR model is now ormulated as the ollowing roblem (P1). (P1) Z min d ( i, s ) D( s ) i Subject to Grou I: Flow balance constraints (1) Flow balance constraints at the origin node: 10

11 x ( i, j) 1, (2) o i, j:( i, j) E ( o ) E Flow balance constraints at intermediate nodes: x ( i, j) x ( j, k),, j N o s (3) s o i:( i, j) E ( j) E k:( j, k ) E ( j) E Flow balance constraints at the destination node: x ( i, j) 1, (4) s i, j:( i, j) E ( s ) E Grou II: Time-sace network constraints Starting time constraints at the origin node: a ( o, j) EST, (5) j:( o, j) E Cell-to-cell transition constraints: d ( i, j) a ( j, k),, j N s e (6) i, j:( i, j) E j, k:( j, k ) E Maing constraints between time-sace network and hysical network x ( i, j) 1 a ( i, j) x ( i, j) M,,( i, j) E (7) x ( i, j) 1 d ( i, j) x ( i, j) M,,( i, j) E (8) Grou III: Running time and dwell time constraints Running time constraints: TT ( i, j) d ( i, j) a ( i, j),,( i, j) E (9) Minimum running time constraints: TT ( i, j) (1 x ( i, j)) M FT ( i, j),,( i, j) E (10) Minimum and maximum dwell time constraints: TT i j x i j M w i j FT i j i j E (11) min (, ) (1 (, )) (, ) (, ),,(, ) TT i j x i j M w i j FT i j i j E (12) max (, ) ( (, ) 1) (, ) (, ),,(, ) Grou IV: Maing constraints between train orders and cell usage on the same track x ( i, j) x ( i, j) 1 (, ', i, j) ( ',, i, j) 3 x ( i, j) x ( i, j),, ', ',( i, j) E (13) ' ' (, ', i, j) x ( i, j),, ', ',( i, j) E (14) ' (, ', i, j) x ( i, j),, ', ',( i, j) E (15) Grou V: Caacity constraints on the same track a ( i, j) (3 x ( i, j) x ( i, j) (, ', i, j)) M d ( i, j) g h,, ', ',( i, j) E (16) ' ' a ( i, j) (3 x ( i, j) x ( i, j) ( ',, i, j)) M d ( i, j) g h,, ', ',( i, j) E (17) ' ' In Grou I, constraints (2), (3) and (4) ensure low balance on the network at the origin node o train, intermediate nodes, and the destination node resectively. In Grou II, constraints (5) make sure that trains do not deart earlier than the redetermined earliest starting time at their origin nodes. Constraints (6) aim to guarantee a ( j, k) d ( i, j) i the adjacent cells ( i, j ) and ( jk, ) are both used by train. Constraints (7) and (8) are imosed to ma 11

12 the variables a ( i, j ) and d ( i, j ) in time-sace network to the variables x ( i, j ) in hysical network, so as to describe whether cell ( i, j ) is selected by or traversing the network rom its origin to destination. In Grou III, constraints (10) enorce the required minimum running time and constraints (11, 12) guarantee minimum and maximum station dwell times by the variables TT ( i, j ). TT ( i, j ) means the running time or train on cell ( i, j ), which can be comuted by Eq. (9). It should be noted that the inut arameter w max ( i, j ) is normally set to be equal to the value used in train timetabling stage. For a congested art o a network, it should be a suiciently large value (e.g. 1 hour) to avoid model ineasibility (Carey and Lockwood, 1995). In Grou IV, constraints (13) link train orders variables (, ', i, j) and cell usage variables x ( i, j ). More seciically, constraints (13) make sure that, i and only i both trains and ' traverse on cell ( i, j ), i.e., x ( i, j) x ( i, j) 1, then both inequalities reduce to (, ', i, j) ( ',, i, j) 1. ' This equality (with two 0-1 binary variables o (, ', i, j) and ( ',, i, j) ) urther indicates that, either train ' arrives at cell ( i, j ) ater train or train arrives at cell ( i, j ) ater train '. I x ( i, j) 0, x '( i, j) 1 or x ( i, j) 1, x '( i, j) 0 or x ( i, j) x '( i, j) 0, then constraints (13) reduces to non-active inequalities, since (, ', i, j) ( ',, i, j) is always between 0 and 2. Constraints (14) and (15) urther ensure that any (, ', i, j) and ( ',, i, j) are always less than x ( i, j ) and x (, ) ' i j. M is a suiciently large ositive number. In Grou V, constraints (16) and (17) exlicitly ensure the cell caacity requirement by setting a saety headway which is comuted by g+h between the dearture o a receding train and the arrival o a ollowing train i those two trains are running on the same cell. Seciically, or train and ' traversing on cell ( i, j ), i.e., x ( i, j) x ( i, j) 1, constrains (16, ' 17) can be reduced to common i-then conditions as discussed below. (1) I train ' arrives at cell ( i, j ) ater train (i.e., (, ', i, j)=1), then there should be a saety time headway g+h between the arrival time o train ' and the dearture time o train on cell ( i, j ), lease see constraints (16); (2) I train arrives at cell ( i, j ) ater train ' (i.e., ( ',, i, j)=1), then there should be a saety time headway g+h between the arrival time o train and the dearture time o train ' on cell ( i, j ), lease see constraints (17). Moreover, Table 6 enumerates dierent statuses o constraints (16) and (17) through combined binary indicators ' CBI ' 3 x ( i, j) x ( i, j) ( ',, i, j) in more detail. ' CBI 3 x ( i, j) x ( i, j) (, ', i, j) and Table 6 Statuses o constraints (16) and (17). Combined variables x ( i, j) x ( i, j) 1 ' (, ', i, j) 0, (, ', i, j) 1, ( ',, i, j) 1 ( ',, i, j) 0 x ( i, j) 0, x ( i, j) 1 ' or x ( i, j) x ( i, j) 0 ' 12

13 x ( i, j) 1, x ( i, j) 0 ' CBI 1 Constraints (16) non-active 0 Constraints (16) active 2 Constraints (16) non-active 3 Constraints (16) non-active CBI' 0 Constraints (17) active 1 Constraints (17) non-active 2 Constraints (17) non-active 3 Constraints (17) non-active 4. Model reormulation based on network cumulative low variables Considering the above big-m tye reresentation or the either-or constraints in the train disatching roblem, many studies are devoted to eicient decomosition mechanisms or reducing the model comlexity, and heuristic algorithms or obtaining easible solutions within reasonable comutational time (e.g., train-based decomosition by Carey and Lockwood, 1995, Lee and Chen, 2009). In the SRR model, the couled caacity constraints and cell usage constraints (13-17) are extremely diicult to decomose into dierent solution branches, as there are a large number o ossible routes leading to very dierent cell usage combinations. Carara et al. (2002, 2006) roosed a method to model track caacity constraints in an imlicit ashion through a set o clique inequality constraints, which are urther dualized in a Lagrangian relaxation solution ramework. Their models are constructed in a single line context while the number o clique constraints can grow exonentially as the number o trains increases. As a result, they adoted a relax-and-cut method to consider a limited set o active constraints. In a rail network, modelling track caacities and saety headway requirements through clique inequality constraints could become even more diicult esecially at nodes with double tracks merging into a single track or a single track diverging to double tracks. Based on network cumulative low variables, this section aims to roose a REFormulated Simultaneous train Rerouting and Rescheduling model (REF-SRR). In this reormulated model, we can easily model temoral and satial occuancy o trains on tracks and saety time headway under a multi-track context. Moreover, through the time-sace discretization o track occuancy eatures, the track caacities in a network can be modelled as discretized resources and REF-SRR rovides a tractable oundation or eicient train-based decomosition. We irst introduce the decision variables and the key modelling eatures o satial occuancy and saety time headway through network cumulative low variables, and then describe REF-SRR model in detail. Recall that x ( i, j ) is used to cature the routing decisions on a rail network. Two more sets o variables are introduced in Table 7 to describe the comlex relationshi between routes and train (time-sace) aths. Seciically, vector y ( i, j, t ) describes a detailed train ath through the extended time-sace network, and cumulative low variables a ( i, j, t ) and d ( i, j, t ) are used to reresent both temoral and satial resource consumtion o trains. 13

14 Table 7 Decision variables or the reormulated simultaneous train rerouting and rescheduling model. Symbol Descrition 0-1 binary time-sace occuancy variables or time-sace network, y a d ( i, j, t ) ( i, j, t ) ( i, j, t ) y ( i, j, t) 1, i train occuies cell (, ) y ( i, j, t) 0 i j at time t, and otherwise 0-1 binary cumulative arrival low variables, a ( i, j, t) 1 i train has already arrived at cell (, ) i j by time t, and otherwise a ( i, j, t) binary cumulative dearture low variables, d ( i, j, t) 1, i train has already dearted rom cell (, ) i j by time t, and otherwise d ( i, j, t) Modeling satial occuancy and saety headway through network cumulative low variables This section aims to give a detailed discussion on network cumulative low variable-based modeling techniques or handling satial occuancy and saety headway constraints. On an N-track network, a challenging issue is how to consider temoral and satial caacity constraints or both single-track and double-track lines. Without loss o generality, the lanning horizon is discretized and denoted as integers rom time index 1 to T. As illustrated in Fig. 3(a), train arrives at cell ( i, j ) at time 8 and dearts at time 10. It can be urther reormulated through cumulative low variables a ( i, j, t ) and d ( i, j, t ). As shown in Fig. 3(b), there is a change o a ( i, j, t ) at time 8 and a change o d ( i, j, t ) at time 10, rom 0 to 1. The transormation rom a ( i, j ) to a ( i, j, t ) can be ormally reresented by T (18) a ( i, j) t [ a ( i, j, t) a ( i, j, t 1)] t 1 Similarly, we have the ollowing reresentation or dearture variables d ( i, j ) : T (19) d ( i, j) t [ d ( i, j, t) d ( i, j, t 1)] t 1 Time horizon Time horizon (a) train a (, i j)=8 d ( i, j)=10 d a ( i, j, t) ( i, j, t) train (b) j i Cell (i, j) Fig. 3 Reormulation o arrival and dearture time variables through cumulative low variables We now need to consider how to model satial occuancy and saety headways. A set o shited a i j t g and d ( i, j, t h) is introduced in this aer to reresent cumulative low variables (,, ) whether train starts or ends occuying cell e by time t, by considering minimum saety time headway g and h. Finally, we can reresent the satial occuancy o a train through a simle equation 14

15 y ( i, j, t) a ( i, j, t g) d ( i, j, t h). Let s assume g = h =1, the gray rectangle block in Fig. 4 corresonds to y ( i, j, t) 1 or t , y ( i, j, t) 0 otherwise. Time horizon d ( i, j, t) shited d ( i, j, t h) shited a ( i, j, t g) a ( i, j, t) train h = y (, i j, t) a (, i j, t g) d (, i j, t h) g = 1 j i Cell (i, j) Fig. 4 Satial occuancy o cell ( i, j ) by train between time 7 min and 11 min. We now move to a double-track segment case with our sequencing cells e 1, e 2, e 3 and e 4. As shown in Fig. 4, the stacks o gray rectangles reresent the detailed occuancy by trains and ' in a time-sace network. On a double-track segment, it is ossible that two trains are running at the same time (e.g. at minute 14 in Fig. 5), while there is only one train being allowed on each cell (i.e. track circuit). The roosed reormulation using cumulative low variables can nicely cature the above two requirements through y ( i, j, t) y ( i, j, t) 1 '. More seciically, at minute 14 o Fig. 5, both trains and ' are running at the segment rom station A to B, but trains and ' do not occuy any cell in the segment at the same time. For examle, trains occuies cell e 2 (i, j) in the time san between t =7 and t = 10, and train ' occuies e 2 (i, j) between t =13 and t = 15. Time horizon Distance Outbound Station B j i Station A h Cell e 4 Cell e 3 Cell e 2 Cell e 1 train g train ' Segment l Fig. 5 Cell decomosition o a double-track segment l between stations A and B. A single-track case is illustrated in Fig. 6. We need to introduce directed cell e rom station i to j and cell e' rom station j to i, in order to allow trains running on oosite directions. Let us consider train using e and train ' using e'. Since cells e and e' corresond to the same segment, we can use a constraint o y ( i, j, t) y ( j, i, t) 1 ' to model the saety headway requirement. More seciically, 15

16 y ( i, j, t) y ( j, i, t) 1 ' or t between 3 and 9, 11 and 16, ' y ( i, j, t) y ( j, i, t) 0 or t between 0 and 2, 9 and 11 (2 time units buer time), 16 and 25. In a rail network context in which there are both double-track segments and single-track segments, one needs to aly the above two methods to reresent the double/single track segments by cells. Moreover, as ointed by Harrod (2011), on a rail network, deadlock(s) may occur i block/cell occuancy are simly imlemented resulting in inlated objective values. Thanks to the introduction o saety headways g / h, which can revent more than one train arriving at a node at the same time, the roosed cell occuancy reresentation is caable o avoiding the occurrence o deadlock(s). Time horizon Station j train ' Cell e Cell e' Station i train Segment l Fig. 6 Two cells corresonding to a single-track segment l rom station i to j. Cumulative low counts-based methods have been widely used in highway traic engineering. Seciically, cumulative low counts are tyically treated as continuous variables to describe macroscoic characteristics and relationshis between dierent traic stream measures and vehicular travel times (Hall, 2003; Cassidy, 2003). In comarison, the cumulative low variables used in this aer are a secial vector o 0-1 binary variables that allows us to clearly deine (microscoic and train-seciic) temoral and satial caacity constraints in a time-sace network. In the area o air traic low management, Bertsimas and Stock (1998) introduced 0-1 variables as the summation o low variables by time t to construct a series o time-deendent multi-commodity network low models. Although they have not seciically used the terminology o cumulative low variables, such ormulation acilitates them to model connectivity between sectors along a redetermined light route. In the train timetabling/disatching ield, to the best o our knowledge, this aer is the irst study using cumulative low variables-based modeling methods that allow simultaneous rerouting and rescheduling decisions. In our aer, the roosed network cumulative low variables-based reresentation enables many unique modeling eatures. First, it can easily cature comlex saety headway constraints in a network with both single and double tracks, with/without a redetermined route, through reormulating the temoral and satial resource occuancy o trains. Second, it enables a very eicient roblem decomosition mechanism by trains, while each subroblem is relatively simle to solve in its extended time-sace network. 4.2 Simultaneous train rerouting and rescheduling model based on network cumulative low variables The reormulated simultaneous train rerouting and rescheduling model based on network cumulative low variables is now ormulated as the ollowing roblem (P2). 16

17 (P2) Z min t d ( i, s, t) d ( i, s, t 1) D( s ) t s i:( i, s ) E ( s ) E (20) Subject to Grou I: Flow balance constraints Flow balance constraints at the origin node: x ( i, j) 1, (21) o i, j:( i, j) E ( o ) E Flow balance constraints at intermediate nodes: x ( i, j) x ( j, k),, j N o s (22) s o i:( i, j) E ( j) E k:( j, k ) E ( j) E Flow balance constraints at the destination node: x ( i, j) 1, (23) s i, j:( i, j) E ( s ) E Grou II: Time-sace network constraints Starting time constraints at the origin node: a ( o, j, t) 0,, t EST (24) j:( o, j) E d ( o, j, t) 0,, t EST (25) j:( o, j) E Within cell transition constraints: d ( i, j, t FT ( i, j)) a ( i, j, t),,( i, j) E, t (26) Cell-to-cell transition constraints: d ( i, j, t) a ( j, k, t),, j N s e, t (27) i, j:( i, j) E j, k:( j, k ) E Maing constraints between time-sace network and hysical network x ( i, j) a ( i, j, T),,( i, j) E (28) Grou III: Running time and dwell time constraints Running time constraints: TT ( i, j) { t [ d ( i, j, t) d ( i, j, t 1)]} { t [ a ( i, j, t) a ( i, j, t 1)]},,( i, j) E (29) t t Minimum running time constraints: TT ( i, j) FT ( i, j),,( i, j) E (30) Minimum and maximum dwell time constraints: w ( i, j) FT ( i, j) TT ( i, j) w ( i, j) FT ( i, j),,( i, j) E (31) min max Grou IV: Caacity constraints Cell occuancy indication constraints: y ( i, j, t) a ( i, j, t g) d ( i, j, t h),,( i, j) E, t (32) Cell caacity constraints: [ y ( i, j, t)] [ y '( j, i, t)] Ca( i, j, t), i, j, t (33) :( i, j) E ':( j, i) E ' Grou V: Time-connectivity constraints or cumulative low variables 17

18 a ( i, j, t) a ( i, j, t 1),,( i, j) E, t (34) d ( i, j, t) d ( i, j, t 1),,( i, j) E, t (35) In Grou I, similar to constraints (2), (3), (4), constraints (21), (22) and (23) ensure low balance on the network at the origin node o train, intermediate nodes, and the destination node resectively. In Grou II, constraints (24) and (25) make sure that trains do not deart earlier than redetermined earliest starting time at their origin nodes. Constraints (26) ensure the transition constraints within cells. Constraints (27) aim to guarantee a ( j, k, t) d ( i, j, t) i the adjacent cells ( i, j ) and (, ) jk are both used by train. Constraints (28) are imosed to ma the variables a ( i, j, t ) in time-sace network to the variables x ( i, j ) in hysical network, so as to describe whether cell ( i, j ) is selected by or traversing the network rom its origin to destination. In Grou III, constraints (30) and (31) enorce the required minimum running time as well as minimum and maximum station dwell times by the variables TT ( i, j ). TT ( i, j ) means the running time or train on cell ( i, j ), which can be comuted by Eq. (29). In Grou IV, constraints (32) link y ( i, j, t ) with a ( i, j, t g) and d ( i, j, t h) 18. Note that the irst term will be 1 i train has started occuying cell ( i, j ) by time t and the second term will be 1 i train has ended occuying cell ( i, j ) by time t. Thereore, the only trains that contribute a value o 1 to the dierence a ( i, j, t g) d ( i, j, t h) reresent the trains that are occuying cell ( i, j ) at time t, i.e., y ( i, j, t) 1. Recall that in Fig. 4, y ( i, j, t) 1 or t and y ( i, j, t) 0 otherwise. Furthermore, constraints (33) make sure the number o trains that are occuying cell ( i, j ) is less that the caacity o cell ( i, j ), which imlicitly ensures saety time headways between trains. The additive structure o caacity usage (let ortion o constraints (33)) is a key technique o the reormulated model as it can decoule the original roblem into many train-seciic subroblems. The mechanism is later used by a Lagrangian relaxation solution ramework in section 5. It should be noted that the occuancy o cell rom j to i should also be counted into the occuancy o cell rom i to j by train, and vice versa, as cell ( i, j ) and ( ji, ) essentially reer to one hysical track circuit. In Grou V, constraints (34) and (35) reresent connectivity in time. Thus, i train has arrived at or dearted rom cell (, ) later time eriods, t' i j by time t, then a ( i, j, t ) or d ( i, j, t ) has to have a value o 1 or all t. Thanks to the roosed modeling aroach based on network-wide cumulative low variables, the above model can easily model satial occuancy and saety headways and work on a comlex network with both unidirectional double tracks and bidirectional single tracks. This relaxes the assumtion that each track can only be traversed in one direction by a train whose traveling direction is given in Mu and Dessouky (2011). We will discuss how the REF-SRR model, roosed in this aer, is related to the revious work o Mu and Dessouky (2011) in detail in Section 7. In order to evaluate the beneits o the roosed simultaneous train rerouting and rescheduling aroach, comared to ractically-used sequential rerouting/rescheduling aroaches, we also develo a SEQuential train rerouting and rescheduling model (SEQ) based on cumulative low variables as a benchmark, which is detailed as model (P3) in the aendix. 5. Lagrangian relaxation based solution rocedure Although considerable rogress has been made on both linear/integer rogramming ormulations and solution algorithms or train rerouting and rescheduling roblems, the existing methods either rely

19 on commercial otimization solvers to solve the ormulated model, which could take a signiicant amount o running time and memory sace to solve a real-world roblem to otimality, or many heuristic rules that could eiciently ind ractically-satisactory solutions but still with a lack o solution quality assessment. Brännlund et al. (1998) irst introduced a Lagrangian relaxation aroach to determine a roit maximizing schedule in which track resources are modeled by additive constraints and then relaxed in a quadratic otimization subroblem. Their resource constraints are relatively simle in a sense that there can be only one train on each block at each time eriod. Based on a grah-theoretic ormulation or the eriodic-timetabling roblem, Carara et al. (2002) modeled limited resources by incomatible arcs (i.e. conlicting oerations) and orbid the simultaneous selection o such arcs through a novel concet o clique constraints. They resented a Lagrangian relaxation solution method, and many additional ractical constraints are incororated seciically in the study by Carara et al. (2006). In a single-track corridor, Zhou and Zhong (2007) modeled limited track resources at both segments and stations to consider headway constraints, which are urther relaxed through the Lagrangian relaxation technique to comute a lower bound. It should be remarked that, the above Lagrangian relaxation techniques are alied to solve the train routing and scheduling roblem in a single or double track rail line with a route being seciied a riori or each train. In comarison, our roosed Lagrangian relaxation solution ramework urther allows trains to select an otimal route (among dierent easible routes) and the corresonding schedule that can minimize the deviation time in a comlex time-sace network. This section aims to solve the roosed model (P2) which simultaneously reroute and reschedule trains in an N-track network with both unidirectional and bidirectional tracks. We irst dualize the comlex constraints in (P2) and solve the relaxed roblem using a comutationally eicient shortest ath algorithm in a careully constructed time-sace network reresentation. The Lagrangian relaxation solution ramework used in this research can hel to construct a tight lower bound and then rovide a good base solution or generating easible solutions with valid uer bounds. We next detail the overall Lagrangian relaxation solution ramework and the underlying label-correcting algorithm or solving the time-deendent least cost ath roblem, as well as the riority rule-based method or transorming dual solutions to easible solutions. 5.1 Lagrangian relaxation solution ramework The key modeling asect need to be discussed in the roosed Lagrangian relaxation ramework is which constraints should be relaxed. The constraints in (P2) can be classiied into two categories. The irst category includes Grous I, II, III and V, and they are all directly related to individual trains. The second category contains cell caacity constraints o Grou IV, which is in general diicult to solve as it involves all trains on the same cell. Note that the diiculties o dierent constraints grous are examined in section 6. We introduce a set o nonnegative Lagrangian multiliers to dualize the additive cell i, j, t caacity constraints (33) as a enalty term in the ollowing relaxed model (P4): min Z t d ( i, s, t) d ( i, s, t 1) D( s ) t s i:( i, s ) ( ) (P4) E s E i, j, t [ y ( i, j, t)] [ y '( j, i, t)] Ca( i, j, t) i, j t :( i, j) E ':( j, i) E ' 19 (36)