Procedia - Social and Behavioral Sciences 162 ( 2014 ) 51 60

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1 Available online at.sienediret.om SieneDiret Proedia - Soial and Behavioral Sienes 162 ( 2014 ) XVIII Congreso Panameriano de Ingeniería de Tránsito, Transporte y Logístia (PANAM 2014) Train Sheduling in High Speed Railays: Considering Competitive Effets Luis Cadarso a *, Ángel Marín b, José Luís Espinosa-Aranda, Riardo Garía-Ródenas a Rey Juan Carlos University, Departamental III Building, Camino del Molino s/n, Fuenlabrada 28943, Spain b Universidad Politénia de Madrid. Esuela Ténia Superior de IngenierosAeronáutios, Plaza Cardenal Cisneros, 3, Madrid, Spain Universidad de Castilla La Manha. Esuela Superior de Informatia. Paseo de launiversidad, , Ciudad Real, Spain Abstrat The railay planning problem is usually studied from to different points of vie: marosopi and mirosopi. We propose a marosopi approah for the high-speed rail sheduling problem here ompetitive effets are introdued. We study train frequeny planning, timetable planning and rolling stok assignment problems and model the problem as a multi-ommodity netork flo problem onsidering ompetitive transport markets. The aim of the presented model is to maximize the total operator profit. We solve the optimization model using realisti problem instanes obtained from the netork of the Spanish railay operator RENFE, inluding other transport modes in Spain The Authors. Published by Elsevier Ltd. This is an open aess artile under the CC BY-NC-ND liense 2014 The Authors. Published by Elsevier Ltd. ( Seletion and peer-revie under responsibility of PANAM Peer-revie under responsibility of PANAM Keyords: high speed trains; sheduling; rolling stok; ompetitive transport markets 1. Introdution High-Speed Railays (HSR) are urrently regarded as one of the most signifiant tehnologial breakthroughs in passenger transport developed in the seond half of the 20th entury. This type of rail netork is devoted to providing high-speed servies to passengers illing to pay for shorter travel time and a quality improvement in rail transport. Sine the earliest projets started ommerial operation in the 1970s, high-speed rail has been presented as * Corresponding author. Tel.: ; fax: address: luis.adarso@urj.es The Authors. Published by Elsevier Ltd. This is an open aess artile under the CC BY-NC-ND liense ( Peer-revie under responsibility of PANAM doi: /j.sbspro

2 52 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) a suess story in terms of demand and revenues. It has partiularly been vieed in many ountries as a key fator for the revival of railay passenger traffi, a delining business that had lost its momentum due to the fiere ompetition of road and air transport. Hoever, building, maintaining and operating HSR lines is expensive, involves a signifiant amount of resoures and may substantially ompromise both the transport poliy of a ountry and the development of its transport setor for deades. Beause of the upoming deregulation and the advent of high-speed rail netorks, European passenger railroads are battling for ustomers among themselves and ith other means of transportation. To maintain a ompetitive advantage, they rely on the sheduling proess as a key fator in inning market share. Like airlines, rail operators today are looking for advaned deision-support tools in the areas of priing, yield management, shedule planning, and ontrol. Sheduling, ith its produt-positioning omponent, is one of the first steps in the planning proess. The railay operator identifies needs of the ustomer, and one the route struture of the railroad is determined, designs the operating shedule that represents the servies to be offered. Sheduling is therefore a fundamental omponent in maximizing overall profit. Noadays, sheduling has beome more omplex. This omplexity stems from the operators need to build shedules to fit a hanging demand, to meet both onstraint-driven and market-driven riteria, and to allo adjustments. Demand models are used to develop foreasts of passenger demand for eah origin-destination pair as a funtion of attributes suh as average fares, frequenies, market demographis and eonomi onditions. Given these total demand estimates, passenger hoie models are used to estimate for eah ompetitor and eah market, the proportion of demand (or share) it aptures in that market, onsidering market-speifi harateristis, inluding passengers mode preferenes, fares, frequenies and other harateristis (Ben-Akiva and Lerman, 1985). For tratability purposes, shedule design models typially onsider demand for servies to be deterministi and invariant to shedule hanges and ompetition. These assumptions, hoever, have been shon to lead to overestimates of the number of passengers served, the revenue aptured, and shedule profitability (Belobaba, 2009). An effetive shedule planning proess, then, depends ritially on both the aurate estimation of the overall demand for travel; and the aurate understanding of ho passengers ill hoose beteen the ompetitors travel options. The motivation for onsidering multi-modal ompetition stems from the fat that HSRs and airlines are inreasingly ompeting for passengers in many parts of Europe and Asia, espeially in short- to medium-haul markets. The HSR often ompetes by providing similar or even greater servie frequeny and better onnetivity to the ity enters. Moreover, the HSR is often pereived as the safest and most omfortable mode (Jehanno et al., 2011). 2. State of the art Cordeau et al. (1998) present a survey of optimization models for train routing and sheduling problems. Within the sheduling proess, train timetabling (Caprara et al., 2002) and rolling stok assignment (Cadarso and Marín, 2011) may be studied from a marosopi point of vie. Beause detailed infrastruture knoledge is not available (ithin a marosopi approah), high-level planning alone may result in onflits, hih ould hinder the solutions from being operable. Hoever, inorporating detailed infrastruture information ill lead to an extremely huge-sized optimization problem. Therefore, a to-phase approah is usually preferred as a ompromise. The first phase is usually the marosopi one, here the problem is usually modelled as a multi-ommodity netork flo problem onsidering the train unit sheduling (Ghoseiri et al., 2004; Cahiani et al., 2010; Cadarso, 2013; Cadarso and Marín, 2010, 2011, 2012 and 2014). In this phase, the train sequene and rolling stok assignment problems are studied. Real ord onstraints are onsidered jointly ith an objetive funtion hih minimizes the total operating ost based on train servie and empty movement operating osts, routing preferenes and penalties for train and infrastruture shortages and so on. In the seond phase, operational plans are usually studied in detail. Real orld senarios are onsidered at the mirosopi modelling level, i.e. ompatibility issues, time alloanes for oupling/deoupling ativities, and onflits at the system are studied (Kroon et al., 2008). The train sheduling during real-time is studied in order to obtain a onflit-free timetable, onsidering a areful estimation of time separation among trains (D Ariano et al., 2007; Espinosa-Aranda and Garía-Ródenas, 2013).

3 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) Contributions Our ork differs from others in that e develop a ne approah that generates HSR shedules using an integrated shedule optimization model that aounts for passenger demand use apturing the impats of shedule deisions on passenger demand. The main ontributions of this paper are summarized as follos: 1. We develop a marosopi integrated sheduling model that inludes frequeny planning, approximate timetable planning, rolling stok assignment and passenger demand served. 2. We address passenger demand hoie through a logit funtion: the demand split aross modes depends on the servie frequeny (among others), hih is a model variable; 3. We arry out omputational experiments on realisti study ases of the Spanish rail operator RENFE. 2.2 Outline of the paper This paper is organized as follos. Setion 3 introdues our demand modelling approah. Setion 4desribes the problem in detail. Setion 5 is devoted to the mathematial model. In Setion 6 some omputational experiments are presented. Finally, e dra some onlusions. 3. Demand modeling Servie frequeny is the most important attribute on hih transport operators ompete. They an attrat more passengers in a market by inreasing the frequeny on its Origin-Destination (OD) pair. For a given unonstrained total demand, the market share of eah operator depends, among other fators, on its on frequeny and on the frequeny of its ompetitors. Hoever, modelling the market share as simply a funtion of the frequeny share is not enough to model passenger demand behaviour in many markets. This is espeially true in markets here the ompetitor fares are different from eah other. There are other attributes, suh as fares and travel times that an signifiantly affet passengers hoie. Many past studies have modelled market share as a funtion of the attributes (i.e., frequeny, prie and travel time) (Wei and Hansen, 2005; Vaze and Barnhart, 2012a). A standard disrete hoie approah is based on a logit model here the hoie probability is proportional to the exponentials of the systemati utilities of eah operator o for market ( va ( )), vo ( ) e P =, (1) o vo ( ' ) e o' O, here f is the servie frequeny of operator o o in market, p is the prie of operator o o in market, tt is the planned travel time of operator o in market, is the frequeny parameter, is the prie parameter and is the trip time parameter. O is the operator set. The aptured demand by operator o depends on ompetition effets, as explained before. Consequently, unonstrained demand gets split beteen different operators. The aptured demand by operator o in market is Po d, here d is the unonstrained demand in market. As e are studying a tatial problem for hih the timetable is approximate, e model aptured demand as a funtion of the total frequeny (i.e., aross the planning period) offered in the OD pair; i.e., the OD attributes are substituted in the logit model for the frequeny parameter (i.e., od od f is substituted by f being od the OD pair of market : vo ( )= f p tt ). o 4. Problem desription o o o o In this setion, the train sheduling problem in high-speed rail netorks is desribed in detail. First, the railay infrastruture is introdued. Next, e desribe train servies timetable and rolling stok. Finally, e explain ho e treat the passenger demand. o

4 54 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) Railay infrastruture The railay netork onsists of traks and stations. We model the infrastruture as a graph ith nodes representing the stations, and ith direted ars (the existing infrastruture linking different stations is represented by ars). Beteen to stations, to different ars exist, one for eah diretion of movement. Therefore, every ar is dened by its departure and arrival station and by its length (e.g., in kilometers).depot stations form a subset of the stations; these are the loations here trains are parked or shunted. The planning time is disretized into time periods. We study a tatial problem here a shedule is to be determined for a planning horizon of, say, one eek. In order to ensure tratability of the problem, e propose an aggregated netork, similar to that presented in Harsha (2008). It allos different levels of time-disretization at eah station. Suh an aggregation sheme redues problem size, ithout ompromising the modelling auray too muh Timetable and rolling stok The train servies are grouped in lines. A line is haraterized by its terminal stations, by a path through the infrastruture beteen the terminals, and by a set of stations along the path. Train servies run up and don beteen the terminals and all at the speied stations underay. We assume that the timetable departure times and frequenies are publily available, so the passengers kno hen the trains depart and plan their traveling aordingly. Finally, e assume that the train shedule ill be periodi, that is, the shedule ill repeat after the planning period ends. To satisfy this, the number of train units of eah type at eah station at the beginning and the end of the planning period must be the same. There are self-propelled train units of different types; they all have a driver seat at both ends. Units of the same type an be attahed to eah other to form trains ompositions. A omposition of train units is a sequene of elements of the same type. Eah line is served ith one train unit type. Eah train unit type has a given apaity; this value inludes seated passengers. The apaity imposes a hard limit of ho many passengers t into the train Passenger demand The demand is haraterized by an origin, a destination and a departure time. This information may be represented by market, defined by a departure station, an arrival station, and the desired departure time. The size of the market is denoted by g. In order to alulate the demand split beteen the different travel options, e use the expression in (1). An important modeling issue is ho to apture the passenger demand satisfation requirements for every market (Vaze and Barnhart, 2012b). In eah market, passengers an hoose any of the orresponding itineraries. Thus the proposed model is itinerary-based. The demand aptured by the train operator ill depend on ompetition effets hih are inorporated in our model through the logit model introdued in Setion High-speed railay sheduling model The aim of the Integrated High-Speed RAilay Sheduling Model (IHSRASM) is to maximise operator s profit, hile passenger demand requirements are satisfied and operator s operational limitations mathed. Within the sheduling proess, train timetabling and rolling stok assignment may be studied from a marosopi point of vie. Hoever, shunting operations based on detailed infrastruture knoledge belong to a loer level planning. Therefore, high-level planning alone may result in onflits suh as rossing, hih ould hinder the solutions from being operable. Hoever, inorporating detailed infrastruture information into our sheduling problem ill lead to an extremely huge-sized optimization problem instane intratable to solve. Therefore, a tophase approah is preferred as a ompromise: a marosopi and a mirosopi approah. The mirosopi approah is presented in Espinosa-Aranda et al. (2014).

5 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) In our model, the relationships beteen the data and variables are onsidered ithin a direted spae time graph, G(S,A),here S is the set of stations and A is the set of ars. The purpose of the onstraints is summarized as follos: 1. As for the shedule, servie frequeny is determined, approximate departure times are deided and headay times are enfored. 2. The passenger demand, hih depends on the variable of servie frequeny, is linked to the apaity of the alloated train units. 3. As for the rolling stok, the amount of used rolling stok is limited; eah trip gets a omposition assigned; the storage of the stations is ontrolled Sets In order to be able to formulate the IHSRASM, e need to dene the folloing sets: S : set of nodes indexed by s; eah node represents a station. DS : set of depot stations indexed by s. OD : set of origin-destination pairs indexed by od. W : set of markets indexed by ; eah element in this set is defined by an origin-destination pair and a departure time period. A : set of traks indexed by a; eah element is defined by (onseutive) origin and destination stations. C : set of train ompositions indexed by. L : set of train servies indexed by l; eah element is defined by origin and destination stations, a departure time period, an arrival time period and a route. : set of lines indexed by ; eah element is defined by a sorted set of traks. I : set of itineraries indexed by i; eah element is defined by origin and destination stations, the traks hih onnet them, and the departure time. I : subset of itineraries serving market. I l : subset of itineraries using train servie l. I a :subset of itineraries using ar a. L od : subset of servies attending the pair of stations od. L a :subset of train servie using ar a. L :subset of train servie using line. PAS : subset of train servies oming through station s at period t. s, t 5.2. Parameters p i : average tiket prie for a passenger in market travelling in itinerary i. l : unitary operating ost for train servie l ith omposition. g : size of market. a lst,,: timetable of train servie l; it takes value 1(-1) if servie l arrives at (departs from) station s during time period t. q : seating apaity in omposition. n : fleet size for omposition. P HSR : probability of passengers from market seleting alternative HSR among all the alternatives. Note that this depends on the frequeny deision variable. : running time for train servie l. l

6 56 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) b : average available running time aross the planning period per omposition. h : average headay apaity for station s during time period t. s, t tu : number of train units in omposition. ap : train unit apaity for eah station s. s 5.3. Variables The most entral deision variables are number for a servie l L f od, i s k l ith omposition C, dened for, : servie frequeny aross the planning period in od pair. ll C. Their values indiate the frequeny. The model ontains the folloing additional variables: g : number of passengers from market that travel in itinerary i ithin line. y : train inventory of omposition in station s at the beginning of the planning period Objetive funtion The objetive funtion of the model reads as follos: (2) max z p g k, HSR i i l l Wi I llc The objetive terms, in the given order, model the folloing quantities: revenue given by served passengers and operating osts of sheduled train servies Sheduling onstraints The rst set of sheduling onstraints enfores the headay requirements. kl hs, t ss, tt (3) lpasst, C fod kl odod (4) llod C k = kl s DS, C (5) l tt lla : lst,, 1 tt lla : lst,, 1 Constraints (3) are headay requirements in term of station and time period. Constraints (4) make sure that the servie frequeny in eah od pair is not greater than the number of operated train servies. Constraints (5) ensure that the operated shedule is symmetri, i.e., the number of departures and arrivals at eah station aross the planning period is the same Passenger demand onstraints, gi PHSRg, W (6) ii, gi qk l aa,, lla L (7) iilia W C Constraints (6) ensure that the number of passengers served in eah market does not exeed those aptured in that market. Note that P HSR depends on the deision variables f od. The onstraints (7) say that for eah ar a A, line and train servie l L, the ombined apaity of the trains on the ar during the time interval is enough to aommodate the passenger demand.

7 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) Rolling stok onstraints ys n C (8) ss lklbn C (9) ll ys kl kl sds, tt, C (10) t' T: t' tlla : lst,, ' 1 lla : lst,, ' 1 tuys aps s S (11) C The onstraints (8) are fleet size onstraints. Fleet utilization onstraints (9) ensure that the utilization of eah omposition is not greater than the available total blok hours during the planning period. The onstraints (10) are flo balane onstraints at the depot stations. Constraints (11) are the depot train apaity at station s Variable definition kl ll, C (12) fod od OD (13) ys s S, C (14), g W, i I, (15) i The onstraints (12)-(15) are variable domain onstraints. 6. Computational experiments Our experiments are based on realisti ases dran from RENFE s Madrid-Seville high speed rail orridor in Spain (Figure 1). This orridor onsists of three types of lines. Lines 2 and 3 are operated ith the same rolling stok, hih differs from the one used in line 1 (see Table 1). Figure 1 shos in detail the stations, railay segments and the definition of the 3 lines. Table 2 shos operating osts for the train servies; note that every train servie ithin a line operates all the line; onsequently, there are four different operating osts (reall there are to types of rolling stok). Table 3 shos the size of eah market ithin a day. Tables 4 and 5 sho the average tiket pries for line 1 and lines 2 and 3, respetively. A eekly planning is onsidered, ith a daily time interval beteen [4; 24] hours. Figure 1: Madrid-Seville orridor

8 58 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) Table 1. Capaities for the different rolling stok types Composition Line 1 Line 2 Line Table 2. Operating osts Composition Line 1 Line 2 Line Table 3. Daily market sizes OD Table 4. Average tiket pries in line 1 OD Table 5. Average tiket pries in lines 2 and 3 OD The IHSRASM is a non-linear mixed integer programming model. The non-linearity is in onstraints (6). The aptured demand is a non-linear funtion of the train operator s frequeny values ( f od ). In order to solve the model, e linearize this expression using pieeise funtions. We approximate the relationship beteen the fration of passengers seleting the HSR and the frequeny values by a pieeise linear funtion. In order to linearize the nonlinear relationship e use speial ordered set variables, hih is an ordered set of non-negative variables, of hih at most to an be non-zero, and if to are non-zero these must be onseutive in their ordering. Speial ordered

9 Luis Cadarso et al. / Proedia - Soial and Behavioral Sienes 162 ( 2014 ) sets are typially used to approximately inorporate non-linear funtions of a variable into a linear model. When embedded in a Branh and Bound ode these variables enable truly global optima to be found, and not just loal optima (Beale and Tomlin, 1970). We used for our tests a personal omputer ith an Intel Core i7 at 2.8 GHz and 8 GB of RAM, running under Windos 7 64-Bit, and e implemented the models in GAMS/Cplex We solved all models to a maximum 1% optimality gap. Our ase study fouses on studying the sensibility of the solutions to the ompetitors frequeny. Table 6 shos these results. The first olumn (COM) shos the ompetitors frequeny value. The seond olumn (FREQ) shos the frequeny value as predited by the IHSRASM. The third olumn (%MS) displays the average market share the train operator ill be able to serve ith the predited shedule. The fourth olumn (RSD) shos the rolling stok distribution by depot station at the beginning of the planning period; this olumn is subdivided into three different sub olumns, one per depot station (i.e., Madrid (MAD), Puertollano (PU) and Seville (SEV)), and again subdivided into to different sub olumns, one per available train omposition (i.e., 1, 2).The fifth olumn shos the objetive funtion value (OF) for eah study ase. Finally, the last to olumns sho the number of iterations (#ITER) and omputational time (TIME) needed to reah the solution. When the frequeny of the ompetitors inreases, the market share obtained by the train operator strongly dereases. Hoever, the frequeny value as predited by the IHSRASM slightly dereases beause the train operator tries to maintain its ompetitive presene in the markets. It is also useful to kno the optimal initial distribution of the rolling stok, hih varies depending on the shedule to be implemented. Obviously, hen the market share drops, the number of served passengers also drops, and therefore the profit, hih is given by the objetive funtion, strongly drops. Table 6. IHSRASM solutions RSD MAD PU SEV COM FREQ %MS OF( x 10 ) #ITER TIME(s) , , , , , , , , * 317 0, , , , ,85 Conlusions *this ase study as solved to a 3.5% optimality gap We have developed an integrated sheduling and rolling stok model in order to study the high-speed rail tatial planning. The model inludes frequeny planning, approximate timetable, rolling stok assignment and passenger demand hoie. We formulate the model using mixed integer non-linear programming. The novel aspets of this formulation are the modelling of the modal and operator hoie by the passengers, that is, the unonstrained demand is split beteen the different operators. The aptured demand is modeled as a funtion of the total eekly frequeny offered in the OD pairs. The model is solved using real data instanes obtained from RENFE, the Spanish highspeed rail operator. Aknoledgements This researh as supported by projet grant TRA C03-01 and TRA C03-02 by the Spanish Ministerio de Eonomía y Competitividad.

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