Dynamc contrant generaton n HASTUS-CrewOpt, a column generaton approach for trant crew chedulng By Alan Dallare, Charle Fleurent, and Jean-Marc Roueau Introducton Trant crew chedulng a challengng practcal optmzaton problem that ha tmulated a lot of reearch actvty over the pat three decade. Although everal algorthmc approache to th problem have been tred over the year (ee [7]), column generaton generally condered the mot powerful method currently avalable (ee [1, 3]). Frt ued commercally n Publc Tranport by GIRO (ee [2, 3, 6]), column generaton wa ntally retrcted to problem of mall to moderate ze. Snce then, gnfcant development have enabled t to olve problem of all ze effcently, leadng GIRO to elect CrewOpt a the tandard crew chedulng algorthm for all our clent (ee [3]). Now that problem ze not a crtcal a t once wa, GIRO ha devoted a great deal of reearch actvty to addng new capablte to the algorthm. For ntance, although CrewOpt ha been able to obtan le cotly crew oluton by changng trp ln from ource vehcle chedule for many year, t can now do o more effcently. Th approach n eepng wth a recent reearch trend amed at degnng optmzaton method that ntegrate vehcle and crew chedulng (ee [4, 5]). Another recent feature of CrewOpt t ablty to generate coteffectve oluton that repect the charactertc of an extng duty et a much a poble, o change to the chedulng envronment can be accommodated wth mnmal drupton to prevou crew chedule. Thee new feature are well receved by cheduler becaue they mae t poble to produce oluton that are both le cotly and more relable from an operatonal perpectve. However, the new capablte alo nduce larger optmzaton problem that mae them more dffcult to olve effcently. For ntance, when modfcaton to ln between trp are permtted, a large number of lnear contrant mut be ncluded n the mathematcal model to lmt ncreae to the number of vehcle that operate durng pea hour and to enure the new trp ln are feable for all dute. If many trp, depot, and vehcle group are nvolved, the number of contrant can reach thouand. Lewe, when mlar oluton are dered, lnear contrant mut be added to prevent creatng dute that are mlar to more than one duty from the orgnal crew chedule. In the next ecton, we explan how both goal can be ntegrated nto a et-parttonng model wth addtonal contrant that can be handled by column generaton. Secton 3 focue on the practcal mplementaton of th model when olvng large-cale problem, ncludng the dynamc generaton of contrant and tratege ued to accelerate the proce. In Secton 4, numercal reult are provded for ome real applcaton, whle ecton 5 conclude th paper. 75, rue de Port-Royal Et 1 514 383 0404 Bureau 500 1 514 383 4971 Montréal (Québec) www.gro.ca Canada H3L 3T1 nfo@gro.ca
Some advanced feature of CrewOpt Bac model The trant crew chedulng problem cont of dentfyng an optmal et of dute that completely cover a gven vehcle chedule, whle repectng everal contrant. Although a lot of data needed to completely decrbe a practcal problem (relef opportunte, walng tme for drver, collectve agreement, etc.), t can bacally be formulated a a et-parttonng problem (ee [2] for ntance). Out of the et all poble dute, the lowet-cot ubet of dute that cover each drvng ta exactly once (.e. ndvble porton of drvng between conecutve relef pont) mut be dentfed. In practce, everal addtonal contrant mut alo be taen nto account to control charactertc of ndvdual dute and of the oluton a a whole. A general model can be ummarzed a follow. Mn c x f f.t. L L a x = 1, j x { 0,1} b x = d, 0, 0, L L j J L Where: I the et of all poble dute; J the et of all drvng ta; L the et of addtonal contrant; c the cot of duty I ; ( P) 1 f duty cover drvng ta j, I, j J a j = 0 otherwe; b the coeffcent of varable n f addtonal contrant L ; the cot of the lac varable of L ; addtonal contrant f the cot of the urplu varable of addtonal contrant L ; the lac varable of addtonal contrant L ; the urplu varable of addtonal contrant L ; d the rght-hand de value of addtonal contrant L ; 1 f duty elected n the oluton, x = 0 otherwe. In the objectve functon, the frt term repreent the um of the cot of elected dute, whle the econd and thrd term correpond to the um of cot of lac and urplu varable from the addtonal contrant. The frt et of contrant pecfe that each drvng ta mut be covered by exactly one duty, whle the econd et nclude lnear contrant that decrbe requrement for a pecfc applcaton context. Although the problem (P) correctly decrbe trant crew chedulng problem, t cannot be olved drectly n practce, a the cardnalty of et I can ealy reach mllon for problem of typcal ze. In our column generaton approach, a mater lnear program mlar to (P) repeatedly olved for ubet I I n whch I are much maller et. Thee ubet of dute are generated by heurtcally olvng contraned hortet path problem (auxlary problem) n everal networ, where cot on arc come from duty cot and dual value obtaned from the reoluton of the prevou mater problem. Contrant that apply to ndvdual dute are handled when the auxlary problem are olved, o only vald dute are generated. Retrcton that apply to the global oluton are formulated a de contrant n the mater problem. The column generaton method now well nown and ha been wdely tuded and ued n academa. It, however, relatvely complex and n practcal context the effcency of t mplementaton crtcal. CrewOpt ha the dtnct advantage of ung the GENCOL pacage a the core of t the column generaton code. GENCOL the product of more than 20 year of ongong reearch on thee technque at GERAD (Groupe d Étude et de Recherche en Analye de Décon) and nclude everal mechanm that contrbute to t performance. Integrated bu and duty chedulng It clear that multaneouly chedulng vehcle and dute provde flexblty that can lead to gnfcant avng n overall oluton cot. For th reaon, GIRO crew chedulng algorthm have alway offered the ablty to reve the ource vehcle chedule whle dute are optmzed. Our former heurtc algorthm SuperMcro (ee [7]), whch wa ued by many trant compane durng the eghte and early nnete, could change extng ln n the ource vehcle chedule when uch revon led to le cotly oluton. Snce there generally are fewer contrant on vehcle than on dute, the reultng vehcle chedule often remaned acceptable, even though the only meaure ued to protect them were penalte on change to trp ln durng the pea hour (when uch modfcaton are more lely to ncreae the requred number of vehcle). -2-
When CrewOpt wa frt ntroduced, GIRO made the adaptaton neceary to permt revon to the ource vehcle chedule. Thee adaptaton eentally conted of modfyng the tructure of the networ ued by the auxlary problem o the dute generated can mplctly nclude change to trp ln, and addng lnear contrant to the mater problem o decon affectng trp ln and drver relef are compatble. However, we oberved that for CrewOpt, penalzng modfcaton to trp ln durng pea hour wa not alway an effectve way of lmtng ncreae n vehcle uage. In ome ntance, becaue of t accuracy, the column generaton method produced oluton that avoded ung penalzed relef pont yet tll ncreaed the number of vehcle to reduce the overall cot of dute. We thu decded to nclude a et of addtonal lnear contrant n our model to forbd ncreae n the number of vehcle durng ome perod of the day. Informaton requred to generate thee contrant n the mater problem can be extracted from the dute that are condered at a partcular teraton. More recently, we ntroduced the opton of addng even more contrant to prevent ncreae to the number of vehcle by depot and by vehcle group n multple-depot chedulng context. Thee contrant can be drectly added to the mater problem a long a ome retrcton are mpoed on the tructure of dute generated when olvng the auxlary problem. Snce we uually tart from a prevouly optmzed ource vehcle chedule, we mply forbd the ntroducton of dute f they nclude new trp ln that would lead to a change of depot or vehcle group n the ource vehcle oluton. In the centfc lterature, there are two man approache to model and olve ntegrated vehcle and crew chedulng problem. The frt approach (ee [4]) ue a mxed model that nclude networ flow varable normally aocated wth ngle-depot vehcle chedulng problem, bnary varable aocated to a et parttonng component for the electon of dute, and addtonal contrant that bnd both et of varable nto a compatble ntegrated oluton. Although th model ha ome theoretcal nteret, t ha not yet led to oluton method that can handle large-cale practcal problem. The econd approach (ee [5]) rele on a et-parttonng model that eentally mlar to what CrewOpt ue. Although CrewOpt can multaneouly generate an ntegrated vehcle/crew chedule from cratch, n practce we fnd t much better to frt generate an optmzed vehcle chedule to ue a a tartng pont for a ubequent phae. The advantage of uch an approach are gnfcant when pecalzed algorthm wth more feature are ued to generate the ntal vehcle oluton. For example, GIRO Mnbu algorthm nclude many advanced opton that permt multple depot/vehcle-group chedulng, handlng of complex contrant, ue of mdday parng, and controlled devaton to mnmum layover and deadhead duraton. It can alo permt trp hftng (mnor hft to trp tme), whch ue an ntegrated tmetablng/vehcle chedulng approach that can lead to ubtantal avng n the fnal oluton. Thu a long a an effcent, tate-of-the-art vehcle chedulng algorthm avalable, there nothng to loe, but rather much to gan, by generatng the bet poble vehcle chedule a a tartng pont for crew chedulng. Generatng mlar crew chedulng oluton An algorthm that can generate a cot-effectve crew oluton that mlar to an ntal crew chedule can alo have mportant practcal beneft. Among other thng, t can lead to mproved operatonal relablty and reduced tranng need f mlar dute are contently propoed to drver. One way of achevng th goal by defnng a dtance functon to evaluate quanttatvely how mlar two dute are, and reducng the cot of ome generated dute when they are condered mlar to one n the tartng oluton. Several factor can be condered n the evaluaton of the dtance of two dute, ncludng ther crew bae and duty type, number of brea, percentage of common drvng tme, etc. When the algorthm generate a duty that condered mlar, t alo mportant to aocate t to a counterpart n the tartng oluton. Th mae t poble to eep the ame dentfer o thee dute can be recognzed and agned to the ame drver when poble. Addtonal lnear contrant mut alo be added to the mater problem (P), to enure that no more than one duty aocated to each duty n the tartng oluton. -3-
Handlng large number of contrant In many of the older heurtc approache to crew chedulng (ee [7]), contrant volaton had to be penalzed n the objectve functon, and penalty value had to be adjuted wth Lagrangean method that were not alway robut. An mportant advantage of the column generaton method rede n t ablty to drectly handle lnear contrant that can be added to a mater problem, whch olved by lnear programmng. However, when revon to the vehcle chedule are permtted or a mlar oluton ought, the number of contrant can grow and become a crtcal ue n the reoluton proce. We fnd that drectly addng a large number of lnear contrant to the mater problem generally not worthwhle becaue t mae the optmzaton proce le effcent. Frt, the lnear program are larger and tae more tme to olve. Second, we oberve that better oluton are often found when contrant are added progrevely. Although we cannot yet provde a defntve explanaton for th behavor, our tet ndcate that th trategy hould be preferred n mot cae. In our mplementaton, ome addtonal contrant are ntally gnored and progrevely added a volaton are detected. For ntance, vehcle count contrant can be dregarded n the early tage, o the algorthm can frt concentrate on generatng effcent dute that cover all drvng ta at low cot. Once a good oluton to the contnuou relaxaton of the mater problem found, vehcle can be counted, and the approprate contrant are ncluded n the ubequent mater problem to be olved. To avod fxng fractonal varable to nteger value when a contrant volaton occur, addng contrant ha prorty over any other branchng decon. In addton, everal tratege are ued to accelerate the reoluton proce. Thee nclude a varety of branchng cheme to control the exploraton of the branch and bound tree. It poble, for example, to branch on the decon to ln ndvdual drvng ta nto a oluton or the decon to nclude a full duty. Many parameter alo control threhold value for whch ome varable wll be fxed, a well a the number and charactertc of varable that wll be fxed at each node of the branch and bound tree. Mot of thee tratege are part of the GENCOL pacage and have been fne-tuned by GIRO over the year to yeld the bet reult for trant chedulng problem. Reult CrewOpt now GIRO tandard crew chedulng optmzaton algorthm and repeatedly ued by the majorty of our clent (Montréal, New Yor, Chcago, Barcelona, Lyon, Turn, Rotterdam, Geneva, Venna, Hamburg, Canberra, to name a few). Th wde acceptance enable u to tet our development on a varety of envronment and dentfy robut mplementaton. For ntance, we found that the dynamc contrant generaton mechanm decrbed n the prevou ecton contently mprove reult, ether by acceleratng the reoluton proce or by generatng oluton of better qualty. We beleve that the nferor reult obtaned when all contrant are preent from the tart can be explaned by the fact that dual nformaton from a larger number of contrant ha to be paed from the mater to the auxlary problem, whch can lead the heurtc mechanm to overloo crtcal dute. Although ome mprovement and parameter ettng could allevate th dffculty, dynamc contrant generaton a very effectve and relable technque n practce. To llutrate thee fndng, we provde typcal reult oberved for real example from two major European trant compane (tet were conducted on a 3 GHz Xeon proceor wth 1 GB of memory). In the frt example, we conder a problem nvolvng 900 drvng ta. Wth th problem, more than 2000 addtonal contrant theoretcally have to be added to control vehcle count ncreae. However, only about 50 of thee are requred when they are dynamcally generated a volaton are detected. In th cae, a better oluton found and the proce gnfcantly fater wth dynamc contrant generaton (ee Table 1). All contrant alway preent Soluton cot 53,461 # of addtonal contrant Executon tme (econd) 2,037 11,803 3,815 Dynamc contrant generaton 52749 (1.3 % mprovement) 52 (2.5 % of poble contrant) Table 1 Reult for the frt tet problem. -4-
In our econd example (ee Table 2), the problem ntance nvolve 706 drvng ta and 795 potental addtonal contrant. In th cae, dynamcally generatng contrant tae more tme but produce a much better oluton. The ncreae n executon tme can be explaned by the fact that a larger percentage of addtonal contrant had to be generated and that the algorthm pent more tme generatng new column and optmzng. However, th turned out to be benefcal by provdng more accurate dual nformaton early n the proce, and preventng heurtc mechanm to overloo worthy dute. All contrant alway preent Soluton cot 48,412 # of addtonal contrant Executon tme (econd) 795 Dynamc contrant Generaton 47473 (1.9 % mprovement) 239 (30 % of poble contrant) 6,683 13,063 Table 2 Reult for the econd tet problem. Concluon Column generaton currently recognzed a the bet optmzaton approach for trant crew chedulng. A algorthmc advance have enabled u to tacle problem of large ze, development can now focu on addng new capablte to the extng method. Controlled revon to vehcle chedule and the ablty to generate crew oluton that are mlar to a reference crew chedule are two example of nteretng feature that are now avalable to cheduler. Thee opton put ome preure on the reoluton proce a everal addtonal contrant are added to the theoretcal model. Fortunately, careful mplementaton can allevate ome of that preure, and lead to better chedule that are currently put on the treet every day. Reference [1] Borndörfer, R., Grötchel, M., Löbel, A. (2001). Schedulng Dute by Adaptve Column Generaton. ZIB-Report 01-01, Konrad-Zue- Zentrum für Informatontchn, Berln.[2] Derocher, M., Soum, F. (1989). A Column Generaton Approach to the Urban Trant Crew Schedulng Problem. Tranportaton Scence 23: 1-13. [3] Fleurent, C., Ségun, L. (2002). CrewOpt: A Column Generaton Method for Trant Crew Schedulng. Preented at the Internatonal Conference on Operaton Reearch SOR 2002, Unverty of Klagenfurt, Autra, September 2002. [4] Frelng, R., Wagelman, APM., Paxao, JMP. (1999). An Overvew of Model and Technque for Integratng Vehcle and Crew Schedulng. In: Wlon NHM (ed), Computer-Aded Trant Schedulng, Lecture Note n Economc and Mathematcal Sytem 471, Sprnger, Berln, pp 441-460. [5] Haae, K., Deaulner, G., Deroer, J. (2001). Smultaneou Vehcle and Crew Schedulng n Urban Ma Trant Sytem. Tranportaton Scence 35: 286-303. [6] Roueau, J.-M., Deroer, J. (1995). Reult Obtaned wth CrewOpt, a Column Generaton Method for Trant Crew Schedulng. In: Daduna JR, Branco I, Paxao JMP (ed), Computer-Aded Schedulng of Publc Tranport, Lecture Note n Economc and Mathematcal Sytem 430, Sprnger, Berln, pp 349-358. [7] Roueau, J.-M., Wren, A. (1995). Bu Drver Schedulng An Overvew. In: Daduna JR, Branco I, Paxao JMP (ed), Computer-Aded Schedulng of Publc Tranport, Lecture Note n Economc and Mathematcal Sytem 430, Sprnger, Berln, pp 173-187. -5-