Available online at ScienceDirect. Procedia Engineering 178 (2017 )

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1 Availabl onlin at ScincDirct Procdia Enginring 178 (2017 ) th Confrnc on Rliability and Statistics in Transportation and Communication, RlStat 2016, Octobr, 2016, Riga, Latvia Schduling and Routing Algorithms for Rail Fright Transportation Wojcich Bożjo, Radosław Grymin*, Jarosław Pmpra Wrcocław Univrsity of Scinc and Tchnology, Faculty of Elctronics, Dpartmnt of Control Systms and Mchatronics, Janiszwsigo 11/17, Wrocław, Poland Abstract Th papr dals with schduling and routing rail fright transportation. Thr ar providd mathmatical dscriptions of constraints in th ral rail transportation such as timtabls of passngr trains, safty tim buffrs tc. W dvlopd an algorithm which dtrmins th fastst rout of cargo train in a railway ntwor. It is basd on Dijstra algorithm ida. W xprimntally provd that dtrmination of th fastst rout in vn larg railway ntwor, whr movmnt of a larg numbr of trains was plannd, tas plac in a tim accptabl for dcision mars Th Authors. Publishd by Elsvir by Elsvir Ltd. Ltd. This is an opn accss articl undr th CC BY-NC-ND licns ( Pr-rviw undr rsponsibility of th scintific committ of th 16th Intrnational Confrnc on Rliability and Statistics in Pr-rviw Transportation undr and rsponsibility Communication. of th scintific committ of th Intrnational Confrnc on Rliability and Statistics in Transportation and Communication Kywords: algorithm, schduling, optimization, rail transportation 1. Introduction Fast-growing industry rquirs fficint dlivry of a hug amount of stuff. Transportation tim is oftn vry short and it is not asy to mt a dadlin. Transportation can b accomplishd by th road transportation. Unfortunatly, thr is a high ris of traffic jams that gnrats losss. Howvr, on som roads thr ar wight limits imposd on vhicls so it nforcs slction of longr routs or us of gratr numbr of trucs. All ths factors ma road transportation unconomic and lusiv. Comptitiv approach mploys a rail transport. Propr planning may caus lac of congstion on th railway lin, it allows to carry hug amount of stuff at onc, allows to * Corrsponding author. addrss: radoslaw.grymin@pwr.du.pl Th Authors. Publishd by Elsvir Ltd. This is an opn accss articl undr th CC BY-NC-ND licns ( Pr-rviw undr rsponsibility of th scintific committ of th Intrnational Confrnc on Rliability and Statistics in Transportation and Communication doi: /j.prong

2 Wojcich Bożjo t al. / Procdia Enginring 178 ( 2017 ) shortn dlivry tim and it is mor saf than road transport. Oprational Rsarch among othr things dals with rail transport planning. Rsarch focuss on formulating common problms and dsigning fficint mthods that solv thm. Th fright schduling problm is on of th most difficult problms blonging to th family of transportation problms. Du to its complxity it poss a big challng for modrn Oprational Rsarch studis thus it is in a high dmand. According to Pashchno t al.(2015) it consists of thr sub problms: train schduling problm (choosing propr momnt for train dpartur along its rout), locomotiv assignmnt problm (assigning locomotivs to trains) and locomotiv tam assignmnt problm (assigning tams to locomotivs in an optimal way). Planning train schduls and routs imposs xcution of algorithms on quit big data sts. For instanc currnt polish rail ntwor consists of 999 railway lins (List of railway lins managd by PKP Polsi Lini Koljow S.A., 2016), 588 station buildings (PKP Polsi Kolj Państwow S.A. Our Stations, 2016) and mor than 2500 motor ngins (Camp of PKP Cargo locomotivs, 2016). Howvr, it can b difficult to find a fasibl solution sinc chosn train paths may mutually xclud thmslvs. Trains cannot mov on by on clos to ach othr. A difficulty of finding good solution incrass whn it coms to rail ntwors with a high traffic dnsity. Furthrmor, thr ar many xpctations rlatd to th transportation rgarding safty, spd, capacity and rliability. All mntiond factors forc th us of sophisticatd algorithms to solv schduling and routing problms in a rasonabl tim. Stat of th art According to Cadarso t al. (2014) such planning problms ar usually solvd in two phass. First phas is calld th macroscoping phas. In this phas an xact modl of a rail ntwor is not nown. In this phas th problm is usually formulatd as a multi-commodity ntwor flow problm considring th schduling of th train unit. Rolling stoc assignmnt and train squnc problms ar bing solvd. In th scond phas, calld microscoping phas, all oprations ar considrd in dtails. At this lvl all conflict situations ar dtctd and purgd, compatibility issus ar tan into account and tim allowancs for coupling and dcoupling oprations ar studid. Planning rail fright transportation tas into account activitis associatd with movmnt of trains and maintnanc tass. Lidén and Joborn (2016) wr daling with dimnsioning maintnanc tim windows. Thy introducd a fright traffic cost modl and a passngr traffic cost modl for valuating ffcts of maintnanc windows on rgional passngr traffic. Thr ar also rsarchs rlatd to ral-tim algorithms intndd for rapid rsponding on unxpctd situations that can disturb th normal cours of daily oprations. Rcnt study of Samà t al. (2016) proposd an approach basd on ant colony huristic whr th ral-tim train slction problm is dscribd as an intgr linar programming formulation. Thr ar rar rsarchs dvotd to intgration of production schduling and rail transportation. In th study of Hajiaghai - Kshtli and Aminnayri (2014) such problm was solvd using th Kshtl algorithm. 2. Problm formulation A fright routing and schduling problm through a rail ntwor can b dfind as follows. Lt us assum that a railway ntwor consisting of of railway junctions from th st {1,, n}is givn. Railway junction may rprsnt a railway station, a cross doc station, an intrmodal trminal, tc. A railway ntwor can b rprsntd as a msh of railway tracs, which is lind in junctions. A rout and a schdul of trains from th stt = {1,, t} ar givn. Th goal is to dtrmin routs and schduls for additional cargo trains. A railway ntwor can b modld using dirctd graphg = (V, E), whr Vis th st of nods ande is th st of arcs. Th nodi V corrsponds to railway junction i, whras arc = (i, j), Ednots unidirctional railway trac from junction i to junction j (bidirctional tracs ar modld as a pair of arcs (i, j)and (j, i)). For ach train Ta squnc of railway junctionsr = (r (1),, r (n )) dfining th rout of th train is nown. Th schdul of train is dtrmind by th dpartur timd (s) from th railway junctionr (s), s = 1,,n 1 and arrival tima (s), s = 2,, n, to junctionr (s).

3 208 Wojcich Bożjo t al. / Procdia Enginring 178 ( 2017 ) Without loss of gnrality, w assum that th spd of all trains is idntical. Th movmnt tim on th trac E is m > 0. Nxt, th safty lag btwn two adjacnt trains moving on th sam trac is constant and it is qual to units of tim. W assum that th capacity of ach junction is unlimitd i.. in vry junction thr is a railway siding with sufficintly high numbr of tracs. W assign a timtablτ to th trac = (i, j)of a railway ntwor. Th timtabl τ contains dpartur tims sortd in chronological ordr of all trains movd from junction i to junctionj, prcisly { d ( s ): r ( s 1) = i,r ( s ) = j,s = {,...,n }, T} τ = 2. (1) Th timtabl τ dtrmin tim windowsw = (w (1),, w (n )),n = W in which it is possibl to start movmnt of an additional train on th trac = (i, j). In th tim windoww (s) = ( (s), l (s)), th train arlist start is ( (s), whras latst start isl (s). W considr som lmntτ (s)of timtablτ. For safty rasons, additional train can start th movmnt onlyl aftr th momntτ (s), for th sam rasons th start of th movmnt cannot start latr than L units of tim bfor th dpartur of th nxt train from timtabl i..τ (s + 1).Th pair (τ (s),τ (s + 1)) gnrats tim window (τ (s) + L,τ (s + 1) L)(obviously ifτ (s + 1) L> τ (s + 1) + L). 3. Dtrmining th fastst rout for singl train In this sction w dscrib an xact algorithm for dtrmining th fastst rout for singl additional train. Th problm is similar to th problm of finding shortst path from fixd start nod to all othr nods in wightd graph, which is solvd in polynomial tim by Dijstra s algorithm. Howvr, du to significant diffrncs btwn problms, w prsnt a modification of Dijstra algorithm adaptd to a problm of finding fastst rout. Lt us assum that a railway ntwor dscribd by dirctd graphg = (V, E) is givn. Th arc Ehas wight qual to movmnt timm and tim windowsw ar nown. Th train is rady to travl in rlas timr. Th goal is to find th rout (and/or schdul) from junction a to junction bwhich minimizs arrival tim to junctionb. Ltε(i) b an stimation of minimal arrival tim to junctioni, i V, Qb a st of unvisitd nods fromv, and Adj(i) b a subst of nods rachabl from th nodi. Th psudocod of algorithm is prsntd on th Figur 1. It is asy to s that th main stps of algorithm not diffr to original Dijstra algorithm. Th fundamntal diffrnc occurs in updating of stimationε(v). Th updating of stimation is ralizd in two stps. In th first stp, for th trac = ( u, v) th tim window (w (s*) = ( (s*), l (s*)such as: l ( s* ) = { l ( s ) : l ( s ) ε( u ),w W } min (2) In th scond stp ε (v) is finally updatd in following way { ( v ), max{ ( u ), ( s*) } } m ε( v ) = min ε + (3) 1. St,,, i a 2. Whil in not mpty do St 5. Forach do updat, Fig. 1. Psudocod of algorithm.

4 Wojcich Bożjo t al. / Procdia Enginring 178 ( 2017 ) Cas study It is wll-nown that railway transport is th most nvironmnt-frindly typ of land transport. An amount of nrgy ndd to carry popl and goods pr ilomtr is much smallr than in cas whn a comptitiv car transport is usd. Morovr rail transport rlis on lctrical nrgy that can b producd from rnwabl sourcs. Low nrgy consumption rsults also in lowr costs of transportation. Unfortunatly, dspit mntiond advantags, rail transport is not attractiv for transport companis by virtu of low fficincy that vntuats, intr alia, from charactr of managing accss to a rail infrastructur (railway lin). Mart rquirmnts forc transport companis to ma quic dcisions which mods of transport should b usd to raliz an ordr and fulfill xpctations of customrs. In th currnt sction w will prov that th algorithm introducd in th prvious sction can in a timly fashion dtrmin a rout and a schdul ordrd by a transportr. Th objct of study is a sgmnt of th Polish rail ntwor consisting of 7 citis: Gdańs, Kraów, Lublin, Toruń, Warszawa, Wrocław and Poznań. Figur 2 prsnts this rail ntwor. A lin conncting rail nods dnots a ral rail connction that dos not pass any othr city. Tabl 1 shows th ral passngr train timtabl. On th othr hand, Tabl 2 prsnts stimatd tim of travl for cargo trains on ntwor sgmnts. Gdańs Toruń Poznań Warszawa Wrocław Lublin Kraów Fig. 2. Map of Poland with mard considrd citis and dirct connctions. A principal commissiond transit of crtain numbr of wagons with commodity from Port of Gdańs to Lublin. All shipping and logistic activitis will b finishd at 8:20 am. Th aim is to dtrmin rout and schdul of transportation counting plannd passngr traffic and a thirty minut safty buffr. At th bginning w dtrmin th shortst tim of transportation from Port of Gdańs to all citis, spcially to th city pointd by th principal, that is Lublin. For this purpos w us Dijstra algorithm. Th rsults of th algorithm on data prsntd in Tabl 2 wr collctd in Tabl 3. In th scond column w prsnt total transportation tim consists of transit tims on all sctions of th fastst rout from fixd start nod (Gdańs) to th nod mntiond in first column. In th third column w prsnt th station which prsds th targt station in fasts rout. Th shortst tim of transportation from th Port of Gdańs to Lublin is 11 h 43 min. Th shortst rout lads through rail nod locatd in Warszawa.

5 210 Wojcich Bożjo t al. / Procdia Enginring 178 ( 2017 ) Tabl 1. Tim tabl. Dirction Dpartur Gdańs-Kraów 23:32 Gdańs-Toruń 5:33, 9:33, 13:34, 17:40, 21:50, 23:45 Gdańs-Warszawa 7:46, 8:51, 9:50, 10:52, 11:14, 11:49, 12:52, 13:44, 13:52, 14:52, 15:50, 16:58, 17:07, 17:52, 17:59, 18:46, 18:52, 19:52 Gdańs-Poznań 6:21, 7:32, 8:22, 12:21, 14:18, 16:23, 18:21, 21:57 Kraów-Gdańs 21:25 Kraów-Lublin 6:00, 18:08 Kraów-Toruń 22:16 Kraów-Warszawa 7:44, 8:22, 9:46, 10:13, 11:46, 13:41, 14:14, 14:31, 15:16, 15:38, 16:44, 17:19,17:46, 19:07, 20:00, 20:07, 20:33 Kraów-Wrocław 6:18, 10:43, 11:15, 12:22, 14:54, 16:24, 18:11, 18:45, 22:25 Kraów-Poznań 7:03, 11:03, 13:13, 15:00, 21:10 Lublin-Gdańs 5:38, 8:20, 12:22, 16:20, 20:23 Lublin-Kraów 6:39, 18:30, 00:28 Lublin- Warszawa 7:52, 9:44, 12:15, 13:52, 16:05, 18:07, 19:37, 20:23, 6:11, 09:12, 11:35, 13:13, 14:50, 17:23, 19:20,20:22 Toruń-Wrocław 12:25 Toruń-Poznań 7:32, 8:59, 11:38, 12:52, 15:28, 17:00, 19:00, 19:40 Warszawa-Gdańs 7:16, 8:05, 08:20, 09:20, 09:30, 10:20, 11:05, 11:20, 12:20, 13:00, 13:20, 14:20,15:20, 16:20, 17:20, 18:00, 19:20, 20:20, 22:10 Warszawa-Kraów 7:50, 8:15, 9:03, 9:55, 10:30, 10:55, 11:55, 12:15, 13:00, 13:45, 14:50, 15:50, 16:17,16:55, 17:50, 18:45, 18:50, 19:50, 20:25, 21:10 Warszawa-Lublin 7:50, 9:50, 12:10, 13:35, 15:50, 11:30, 17:50, 19:50, 20:50 Warszawa-Toruń 6:30, 08:30, 10:25, 12:25, 14:15, 16:25, 18:27, 20:30, 22:00 Warszawa-Wrocław 7:45, 08:07, 09:35, 11:45, 12:05, 13:50, 16:13, 17:30, 19:25 Warszawa-Poznań 7:25, 10:00, 11:30, 13:05, 14:00, 14:35, 14:39, 15:00, 16:01, 17:00, 18:00, 19:00,22:35, 23:23 Wrocław-Kraów 6:13, 8:11, 10:50, 13:09, 14:53, 16:00, 17:27, 18:32 Wrocław-Lublin 9:23, 12:12 Wrocław-Warszawa 6:50, 8:25, 10:30, 10:45, 13:03, 14:47, 15:00, 17:20, 18:27 Wrocław-Poznań 7:29, 8:17, 08:55, 09:35, 10:21, 10:49, 11:05, 12:24, 13:27, 14:32, 15:54, 16:23, 16:42, 17:37, 18:47, 20:05, 21:31, 23:35, 23:55 Poznań-Gdańs 6:34, 8:47, 10:50, 12:45, 14:41, 17:30, 18:47 Poznań-Kraów 7:38, 9:43, 11:40, 15:35, 17:36 Poznań-Toruń 6:46, 9:51, 10:55, 14:46, 14:51 Poznań-Warszawa 7:44, 8:41, 9:33, 9:40, 10:40, 10:45, 12:40, 14:29, 15:40, 17:40, 18:42, 19:30, 20:40 Poznań-Wrocław 5:53, 10:33, 14:32, 22:21 Tabl 2. Transport tims. Gdańs Kraów Lublin Toruń Warszawa Wrocław Poznań Gdańs 8:52 2:42 4:03 3:43 Kraów 9:00 4:13 9:09 4:21 5:05 7:02 Lublin 4:09 2:56 7:27 Toruń 2:59 6:53 2:53 2:23 Warszawa 5:55 5:05 7:40 2:56 5:21 3:41 Wrocław 4:58 7:27 5:30 3:30 Poznań 3:42 6:52 2:25 3:55 3:20

6 Wojcich Bożjo t al. / Procdia Enginring 178 ( 2017 ) Tabl 3. Minimal transportation tim. Targt station Total transportation tim Prvious station Kraów 08:52 Gdańs Lublin 11:43 Warszawa Toruń 02:42 Gdańs Warszawa 04:03 Gdańs Wrocław 07:03 Poznań Poznań 03:43 Gdańs Proposd in prvious sction algorithm dtrmins a rout and schdul for cargo train from th chosn initial nod to all othr nods. W will dmonstrat xcution of algorithm for two tims in which a fright train will b rady to transportation:r = 8 andr = 8:30. Th rsults of th algorithm wr collctd in two tabls 4 and 5. For ach targt station w show arrival tim, station which prsids in fastst rout, th dpartur tim from this station and total transportation tim. Th total transportation tim is th diffrnc btwn th tim of arrival to targt station and th tim of cargo train rady to go. It consists with transits tims and downtims in rail nods. It is a masur of occupancy of th railway infrastructur. Tabl 4. Train rout and schdul forr = 8:30. Targt station Arrival tim Total tim Prvious station Dpartur tim Kraów 17:12 08:52 Gdańs 08:20 Lublin 20:20 12:00 Warszawa 12:40 Toruń 11:02 02:42 Gdańs 08:20 Warszawa 12:23 04:03 Gdańs 08:20 Wrocław 15:55 07:35 Poznań 12:35 Poznań 12:35 04:15 Gdańs 08:52 Tabl 5. Train rout and schdul forr = 8:30. Targt station Arrival tim Total tim Prvious station Dpartur tim Kraów 17:22 08:52 Gdańs 08:30 Lublin 21:35 13:05 Kraów 17:22 Toruń 11:12 02:42 Gdańs 08:30 Warszawa 14:23 05:53 Gdańs 10:20 Wrocław 15:55 07:25 Poznań 12:35 Poznań 12:35 04:05 Gdańs 08:52 It can b asily noticd that in cas of citis: Kraów, Toruń and Warszawa total tim of transportation (s 3rd column) is minimal. It mans that transport to ths citis is currid out on th shortst way without any dlay. In cas of Poznań dpartur from Gdańs was dlayd until 8:52 am bcaus of conflict with passngr train dparting from Gdańs toward Poznań. Dlayd arrival to Wrocław is causd by dlayd arrival to Poznań. Finally, train riding from Lublin must wait in Warsaw from 12:23 am until 12:40 am bcaus of conflict with passngr train dpartur at 12:10 am in th sam dirction. ForR = 8:30 algorithm dtrmind a nw rout to Lublin through Kraów. Th nw rout guarants arlir arrival to a targt station thn through Warszawa, bcaus rout Gdańs-Warszawa du to safty tim buffrs is not availabl sinc 8:21 am (dpartur 8:51) until 10:20 am (dpartur 9:50 am).

7 212 Wojcich Bożjo t al. / Procdia Enginring 178 ( 2017 ) Conclusion W dvlopd an algorithm dtrmining th fastst rout for singl additional train assuming that two trains cannot mov from th sam station in th sam dirction on by on spcial safty tim buffr must b assumd. In our xprimnts w obtaind valuabl solutions in a snsibl tim. Usually th problm is to find svral fast routs for trains. Each rout is dscribd by initial and targt nod. Our algorithm can b usd to dvlop an algorithm solving such problm. In such algorithm w add pairs (initial and targt nod) to th priority quu. In ach stp w ta off th first lmnt from th quu and dtrmin th fastst rout for on additional train using our algorithm. W updat th rail ntwor and continu with taing off lmnts, dtrmining th fastst rout and updating th rail ntwor till th priority quu is mpty. Solution consists of such problm consists of routs found by our algorithm and can b assssd by a fitnss function that dtrmins quality of th solution. Obtaind solution (and its quality) dpnds on ordr of lmnts in th priority quu. W can chc all possibl solutions and provid an xhaustiv sarch or involv mtahuristics, s Bożjo t al.(2016), such as tabu sarch or simulatd annaling, also in paralll vrsions. Rfrncs Bożjo W., Uchrońsi M. and Wodci M. (2016) Paralll mtahuristics for th cyclic flow shop schduling problm. Computrs & Industrial Enginring 95, Cadarso L., Marín Á., Espinosa-Aranda J. L. and García-Ródnas R. (2014) Train schduling in high spd railways: Considring comptitiv ffcts. Procdia-Social and Bhavioral Scincs 162, Camp of PKP Cargo locomotivs. URL Hajiaghai-Kshtli M. and Aminnayri M. (2014) Solving th intgratd schduling of production and rail transportation problm by Kshtl algorithm, Applid Soft Computing 25, Lidén T. and Joborn M. (2016) Dimnsioning windows for railway infrastructur maintnanc: Cost fficincy vrsus traffic impact. Journal of Rail Transport Planning & Managmnt, 6(1), Pashchno F., Kuzntsov N., Ryabyh N., Minashina I., Zaharova E. and Tsvtova O. (2015) Implmntation of train schduling systm in rail transport using assignmnt problm solution. Procdia Computr Scinc, 63, pp Samà M., Pllgrini P., D'Ariano A., Rodriguz J. and Pacciarlli D. (2016) Ant colony optimization for th ral-tim train routing slction problm. Transportation Rsarch Part B: Mthodological, 85, PKP Polsi Kolj Państwow S.A., Our stations, Wb pag, accss URL List of railway lins managd by PKP Polsi Lini Koljow S.A., Wb pag, accss URL