PUBLIC-TRANSIT VEHICLE SCHEDULES USING A MINIMUM CREW-COST APPROACH

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1 TOTAL LOGISTIC MANAGEMENT No. PP. Avishi CEDER PUBLIC-TRANSIT VEHICLE SCHEDULES USING A MINIMUM CREW-COST APPROACH Astrt: Commonly, puli trnsit genies, with view towrd effiieny, im t minimizing the numer of vehiles in use to meet pssenger demnd, nd therefore t reduing rew ost. This work ontriutes to hieving these two ojetives y proposing the use of two predominnt hrteristis of puli-trnsit opertions plnning: () different resoure requirements etween pek nd off-pek periods, nd () working during irregulr hours. These hrteristis result in split duties (shifts) with unpid in-etween periods. The outome of this work is n optiml solution for mximizing the unpid shift periods with the ssurne of omplying with the minimum numer of vehiles ttined. The optimiztion prolem utilizes highly informtive grphil tehnique (defiit funtion) for finding the lest numer of vehiles; this enles the onstrution of vehile hins (loks) tht tke into ount mximum unpid shift periods. The ltter onsidertion is intended to help onstrut rew shedules t minimum ost. The methodology developed ws implemented y two lrge us ompnies nd resulted in signifint ost redution. Keywords: puli trnsporttion, vehile sheduling, rew sheduling, defiit funtion.. Introdution Two of the most time-onsuming nd umersome puli trnsit sheduling tsks re reting hins of trips, eh for single dily vehile duty (lled vehile lok), nd ssigning the rew (drivers) to vehile loks. These tsks require the servie of imgintive, experiened shedulers, nd usully it is performed utomtilly. Consequently it is not surprising to lern tht most of the ommerilly ville trnsit-sheduling softwre pkges onentrte primrily on these two tsks, nd espeilly on the rew-sheduling tivities. After ll, from the trnsit geny s perspetive, the lrgest single ost item in the udget is the driver s wge nd fringe enefits. Trnsporttion Reserh Centre, Dept. of Civil nd Environmentl Engineering, Fulty of Engineering, University of Auklnd, Auklnd, New Zelnd, Tel: , Fx: , e-mil:.eder@uklnd..nz

2 A. Ceder This work fouses on the hrteristis of split rew duties (shifts) with unpid in-etween periods; these split duties re the result of different resoure requirements etween pek nd off-pek periods. Usully every trnsit geny desires oth to minimize the numer of vehiles in use tht will omply with pssenger demnd, nd to minimize the rew ost. This work helps in fulfilling these ojetives y providing n optiml solution for mximizing the unpid shift periods, hene reduing the rew osts, with the ssurne of mintining the minimum numer of vehiles required... Bkground Puli trnsit opertions plnning n e thought of s multistep proess. Beuse of the omplexity of this proess eh step is normlly onduted seprtely, nd sequentilly fed into the other. The proess steps re: () designing network of routes; () setting timetles; () sheduling vehiles to trips; nd () ssigning the rew. In order for this proess to e ost-effetive nd effiient, it should emody ompromise etween pssenger omfort nd ost of servie. For exmple, good mth etween vehile supply nd pssenger demnd ours when vehile shedules re onstruted so tht the oserved pssenger demnd is ommodted while the numer of vehiles in use is minimized. The sujet of this work is relted to the third nd fourth steps of the trnsit opertions plnning proess; thus, in this setion, these two steps will e riefly desried elow following literture review. The vehile-sheduling step is imed t reting hins of trips; eh is referred to s vehile shedule ording to given timetles. This hining proess is often lled vehile loking ( lok is sequene of revenue nd non-revenue tivities for n individul vehile). A trnsit trip n e plnned either to trnsport pssengers long its route or to mke dedheding trip in order to onnet two servie trips effiiently. The sheduler s tsk is to list ll dily hins of trips (some dedheding) for eh vehile so s to ensure the fulfillment of oth timetle nd opertor requirements (refueling, mintenne, et.). The mjor ojetive of this step is to minimize the numer of vehiles required. Ceder nd Stern (98) nd Ceder (,, 7, 7) desrie highly informtive grphil tehnique for the prolem of finding the lest numer of vehiles. This tehnique is explited in the following setion onerning kground mterils. It is lso to note tht the min prts of this work pper in the ook y Ceder (7), ut without the tests nd se studies. The gol of the rew sheduling step is to ssign drivers ording to the outome of vehile sheduling. This step is often lled driver-run utting (splitting nd reomining vehile loks into legl driver shifts or runs). This rew-ssignment proess must omply with some onstrints, whih re usully dependent on lor ontrt. Any trnsit geny wishing to utilize its resoures more effiiently hs to del with prolems enountered y the presene of vrious py sles (regulr, overtime, weekends, et.) nd with humn-oriented disstisftion. The purpose of the ssignment funtion is to determine fesile set of driver duties in n optiml mnner. Usully the ojetive is to minimize the ost of duties so tht eh duty piee is inluded in one of the seleted duties.

3 Two tiered timing model..... Literture review of vehile sheduling Vehile sheduling refers to the prolem of determining the optiml llotion of vehiles to rry out ll the trips in given trnsit timetle. A hin of trips is ssigned to eh vehile lthough some of them my e dedheding (DH) or empty trips in order to reh optimlity. The numer of fesile solutions to this prolem is extremely high, espeilly in the se in whih the vehiles re sed in multiple depots. Muh of the fous of the literture on sheduling proedures is, therefore, on omputtionl issues. Löel (998, 999) disussed the multiple-depot vehile sheduling prolem nd its relxtion into liner progrmming formultion tht n e tkled using the rnh-nd-ut method. A speil multi-ommodity flow formultion is presented, whih, unlike most other suh formultions, is not r-oriented. A olumn-genertion solution tehnique is developed, lled Lgrngen priing; it is sed on two different Lgrngen relxtions. Heuristis re used within the proedure to determine the upper nd lower ounds of the solution, ut the finl solution is proved to e the rel optimum. Mesquit nd Pixo (999) used tree-serh proedure, sed on multiommodity network flow formultion, to otin n ext solution for the multi-depot vehile sheduling prolem. The methodology employs two different types of deision vriles. The first type desries onnetions etween trips in order to otin the vehile loks, nd the other reltes to the ssignment of trips to depots. The proedure inludes reting more ompt, multi-ommodity network flow formultion tht ontins just one type of vriles nd smller mount of onstrints, whih re then solved using rnh-nd-ound lgorithm. Bnihshemi nd Hghni () nd Hghni nd Bnihshemi () foused on the solvility of rel-world, lrge-sle, multiple-depot vehile sheduling prolems. The se presented inludes dditionl onstrints on route time in order to ount for relisti opertionl restritions suh s fuel onsumption. The uthors proposed formultion of the prolem nd the onstrints, s well s n ext solution lgorithm. In ddition, they desried severl heuristi solution proedures. Among the differenes etween the ext pproh nd the heuristis is the replement of eh inorret lok of trips with legl lok in eh itertion of the heuristis. Applitions of the proedures in lrge ities re shown to require redution in the numer of vriles nd onstrints. Tehniques for reduing the size of the prolem re introdued, using suh modifitions s onverting the prolem into series of single-depot prolems. Freling et l. () nd Huismn et l. (5) presented n integrted pproh for vehile nd rew sheduling for single us route. The two prolems re first defined seprtely; the vehile sheduling prolem is formulted s network-flow prolem, in whih eh pth represents fesile vehile shedule, nd eh node trip. In the omined version, the network prolem is inorported into the sme progrm with set prtitioning formultion of the rew sheduling prolem. Hsse et l. () formulted nother prolem tht inorported oth rew nd vehile sheduling. For vehile sheduling, the se of single depot with homoge-

4 A. Ceder nous fleet is onsidered. The rew sheduling prolem is set prtitioning formultion tht inludes side onstrints for the us itinerries; these onstrints gurntee tht n optiml vehile ssignment n e derived fterwrds. Hghni et l. () ompred three vehile sheduling models: one multipledepot (presented y Hghni nd Bnihshemi, ) nd two single-depot formultions whih re speil ses of the multiple-depot prolem. The nlysis showed tht single-depot vehile sheduling model performed etter under ertin onditions. A sensitivity nlysis with respet to some importnt prmeters is lso performed; the results indited tht the trvel speed in the DH trip ws very influentil prmeter. Huismn et l. () proposed dynmi formultion of the multi-depot vehile sheduling prolem. The trditionl, stti vehile sheduling prolem ssumes tht trvel times re fixed input tht enters the solution proedure only one; the dynmi formultion relxes this ssumption y solving sequene of optimiztion prolems for shorter periods. The dynmi pproh enles n nlysis sed on other ojetives exept for the trditionl minimiztion of the numer of vehiles; tht is, y minimizing the numer of trips strting lte nd minimizing the overll ost of delys. The uthors showed tht solution tht required only slight inrese in the numer of vehiles ould lso stisfy the minimum lte strts nd minimum dely-ost ojetives. To solve the dynmi prolem, luster re-shedule heuristi ws used; it strted with stti prolem in whih trips were ssigned to depots, nd then it solved mny dynmi single-depot prolems. The optimiztion itself ws formulted through stndrd mthemtil progrmming in wy tht ould use stndrd softwre... Literture review of rew sheduling The rew-sheduling prolem hs een disussed undntly oth in nd out the trnsporttion literture; relevnt ppers re found in journls relted to mthemtis, omputing, opertions reserh, nd speilized sheduling resoures. This setion will fous here minly on the ltest developments in this field. The rewsheduling prolem is often formulted s set overing prolem (SCP). For this purpose, lrge set of driver-workdys is defined, nd suset is then hosen tht ttempts to minimize osts, sujet to onstrints tht mke sure tht ll the neessry driving duties re performed. Most rew-sheduling prolem formultions lso verify tht the lor-greement rights of ll drivers re mintined. Pis nd Pixo (99) formulte the rew-sheduling prolem using dynmi progrmming. The serh for solutions employs stte-spe relxtion method using lower-ound solution. Crrresi et l. (995) propose nother olumn-genertion pproh, one tht strts with fesile set of workdys nd itertively reples some workdys to otin etter solution. The pre-onstruted workdys re uilt of duty piees, nd the solution uses Lgrngen-relxtion method. Another, somewht similr olumn-genertion method is proposed y Fores et l. (999). Clement nd Wren (995) introdue solution for the rew-sheduling prolem using geneti lgorithm: group of hromosomes, eh of whih represents fe-

5 Two tiered timing model... 5 sile rew shedule, is sujet to repeted muttions, rossovers, nd other tions, sed on the ide tht the serh for n optiml solution n follow rules similr to geneti survivl mehnism. Severl greedy lgorithms re used for ssigning duties to piees of work. Another solution proedure sed on geneti lgorithm is presented y Kwn et l. (999). The pproh demonstrted y Besley nd Co (996) does not follow SCP onept; only single type of workdy is onsidered rther thn ttempting to serh roder rnge. A Lgrngen-relxtion method provides lower-ound solution, whih is lter improved y using su-grdient optimiztion. Next, tree-serh lgorithm is used to otin the finl optimum. Besley nd Co (998) gin use similr pproh ut, insted of the Lgrngen-relxtion tool, seek the optiml lower ound y using dynmi progrmming lgorithm. Another method tht does not rely on SCP is suggested y Mingozzi et l. (999). The uthors desrie two different duty-sed heuristi solution proedures in whih relxed prolems re formulted; their solutions lso solve the originl CSP. A third proposed solution proedure is sed on set prtitioning prolem (SPP). The dul onept of liner relxtion progrmming is used to otin lower-ound solution. The numer of vriles in this prolem is then redued y using this lower ound; finlly, the redued-size prolem is solved through rnh-nd-ound tehnique. Loureno et l. () ring multi-ojetive rew-sheduling prolem, led y the onept tht in prtie there is need to onsider severl onfliting ojetives when determining the rew shedule. The multi-ojetive prolem is tkled using met-heuristis, Tu-serh tehnique, nd geneti lgorithms. Shen nd Kwn () introdue proess tht involves prtitioning predetermined vehile shedule into set of driver duties. The fous is on refining n existing smll set of workdys; hene, the methodology does not inlude the ommon stge of generting ll fesile solutions. A Tu serh is used to improve the given rew shedule. Tu serh is lss of met-heuristi tht tries to void eing trpped in lol optimum solution y sing the solution hoie in eh itertion on few-itertions-k nlysis; sometimes, this mens tht solution is hosen even if it leds to poorer performne thn the previous itertion. Fores et l. () desrie trditionl integer liner progrmming formultion of the rew-sheduling prolem, with some dded flexiility. The formultion epts different ojetive funtions (minimize the numer of duties, minimize osts, or omintion), different optimiztion tehniques (priml olumn- genertion or dul-steepest edge tehniques), nd different riteri for reduing the numer of fesile workdys. The optimiztion tehnique hosen is used to solve relxed noninteger prolem; rnh-nd-ound proess then finds n integer solution. Finlly, Kroon nd Fishetti () present n rew-sheduling prolem for rilwy rews tht llows some flexiility in speifying penlties for undesirle types of workdys. A dynmi olumn-genertion proedure is used; hene, duties re not generted priori ut in the ourse of the solution proess. Re-genertion nd re-seletion of workdys re rried out in eh itertion. Genertion is preformed in network in whih trips re represented y rs. To solve the SCP, Lgrngen-relxtion method nd su-grdient optimiztion re used insted of the ommon liner progrmming.

6 6 A. Ceder Finlly, more reent ontriutions of integrted multi-depot vehile nd rew sheduling n e found y Mesquit et l. (9), Borndorfer et l. (8) nd Gintner et l. (8) tht use integer mthemtil formultion, relxtion methods nd heuristis to overome the si NP-Hrd prolem. Other relted reent studies serh for relief opportunities in the ttempt to pproh optiml rew sheduling t puli-trnsit stops where the drivers n e swithed. Suh studies re presented y Lplgne et l. (9) nd Kwn nd Kwn (7).. Bkground on the defiit funtion (DF) pproh Following is desription of step funtion pproh desried first y Ceder nd Stern (98) nd lso y Ceder (,, 7, 7), for ssigning the minimum numer of vehiles to llote for given timetle. The step funtion is termed defiit funtion (DF), s it represents the defiit numer of vehiles required t prtiulr terminl in multi-terminl trnsit system. Tht is, DF is step funtion tht inreses y one t the time of eh trip deprture nd dereses y one t the time of eh trip rrivl. To onstrut set of defiit funtions, the only informtion needed is timetle of required trips. The min dvntge of the DF is its visul nture. Let d (k, t, S) denote the DF for the terminl k t the time t for the shedule S. The vlue of d (k, t, S) represents the totl numer of deprtures minus the totl numer of trip rrivls t terminlk, up to nd inluding timet. The mximl vlue of d (k, t, S) over the shedule horizon [T, T ] is designtedd (k, S)... Fixed shedule Let t i s nd t i e denote the strt nd end times of tripi, i S. It is possile to prtition the shedule horizon of d (k, t, S) into sequene of lternting hollow nd mximl intervls. The mximl intervls [ s k i, ] ek i, i =,..., n (k) define the intervl of time over whih d (k, t) tkes on its mximum vlue. Note tht the S will e deleted when it is ler whih underlying shedule is eing onsidered. Index i represents the ith mximl intervls from the left nd n (k) represents the totl numer of mximl intervls ind (k, t). A hollow intervl Hl k, l=,,,...,n(k) is defined s the intervl etween two mximl intervls inluding the first hollow from T to the first mximl intervl, nd the lst hollow-from the lst intervl to T. Hollows my onsist of only one point, nd if this se is not on the shedule horizon oundries(t or T ), the grphil representtion of d (k, t) is emphsized y ler dot. If the set of ll terminls is denoted s T, the sum of D (k) for ll k T is equl to the minimum numer of vehiles required to servie the set T. This is known s the fleet size formul. Mthemtilly, for given fixed shedule S: D (S) = k T D (k) = k T mx d (k, t) () t [T,T ] where D (S) is the minimum numer of uses to servie the set T.

7 Two tiered timing model... 7 When dedheding (DH) trips re llowed, the fleet size my e redued elow the level desried in Eqution. Ceder nd Stern (98) desried proedure sed on the onstrution of unit redution DH hin (URDHC), whih, when inserted into the shedule, llows unit redution in the fleet size. The proedure ontinues inserting URDHCs until no more n e inluded or lower oundry on the minimum fleet is rehed. The lower oundry G(S) is determined from the overll defiit funtion defined s g (t, S) = k T d (k, t, S) where G(S) = mx t [T,T ] g (t, S). This funtion represents the numer of trips simultneously in opertion. Initilly, the lower ound ws determined to e the mximum numer of trips in given timetle tht re in simultneous opertion over the shedule horizon. Stern nd Ceder (98) improved this lower ound, to G (S ) > G (S) sed on the onstrution of temporry timetle, S, in whih eh trips is extended to inlude potentil linkges refleted y DH time onsidertion in S. This lower ound ws even further improved y Ceder () y looking into rtifiil extensions of ertin trip-rrivl points without violting the generliztion of requiring ll possile omintions for mintining the fleet size t its lower ound. The lgorithms of the defiit funtion theory re desried in detil y Ceder nd Stern (98) nd Ceder (, 7). However, it is worth mentioning the next terminl (NT) seletion rule nd the URDHC routines. The seletion of the NT in ttempting to redue its mximl defiit funtion my rely on the sis of grge pity violtion, or on terminl whose first hollow is the longest, or on terminl whose overll mximl region (from the strt of the first mximl intervl to the end of the lst one) is the shortest. The rtionle here is to try to open up the gretest opportunity for the insertion of the DH trip. In the URDHC routines there re four rules: R= for inserting the DH trip mnully in onverstionl mode, R= for inserting the ndidte DH trip tht hs the minimum trvel time, R= for inserting ndidte DH trip whose hollow strts frthest to the right, nd R= for inserting ndidte DH trip whose hollow ends frthest to the right. In the utomti mode (R=,, ), if DH trip nnot e inserted nd the ompletion of URDHC is loked, the lgorithm ks up to DH ndidte list nd selets the next DH ndidte on tht list... Construting vehile shedules (hins/loks) nd n exmple At the end of the heuristi lgorithm, ll trips, inluding the DH trips, re hined for onstruting the vehile shedule (loks). Two rules n e pplied for reting the hins: first in-first out (FIFO) nd hin-extrtion proedure desried y Gertsh nd Gurevih (977). The FIFO rule simply links the rrivl time of trip to the nerest deprture time of nother trip (t the sme terminl); it ontinues to rete shedule until no onnetion n e mde. The trips onsidered re deleted, nd the proess ontinues. The hin-extrtion proedure llows n rrivl-deprture onnetion for ny pir within given hollow (on eh DF). The pirs onsidered re deleted, nd the proedure ontinues. Figure illustrtes one DF t k. This d(k,t) hs four hollows,

8 8 A. Ceder Hj k, j=,,,, with Hk hving rrivls of Trips,, nd nd deprtures of Trips, 5, nd 6. Below Figure re the FIFO onnetions (within this hollow) s well s other lterntives; in ll, the minimum fleet size t k, D(k), is mintined. d(k,t) Mx d(k,t) = D(k) 8 6 H H k k 6 5 H k H k T T Time FIFO Set of onnetions: [( - ), ( - 5), ( 6)] Other sets of onnetions: [( - ), ( - 6), ( 5)], [( - 5), ( - 6), ( )], [( - 6), ( - ), ( 5)], [( - 5), ( - ), ( 6)], [( - 6), ( - 5), ( )] Fig.. Exmple of reting trips onnetions within one hollow, H k, using the FIFO rule nd ll other possiilities while mintining the minimum fleet size ttined A nine-trip exmple with four terminls (,,, nd d) is presented in Tle nd Figure ; Tle shows the dt required for this simple exmple. Tle. Input dt for the prolem illustrted in Figure Trip No. Deprture Terminl Deprture Time Arrivl Terminl Arrivl Time Dedheding (DH) Trips Between Terminls DH Time (sme for oth diretions) 6: 6: min 6: 6:5 6: 7: min 7: 7: 5 7: 7: d 6 min 6 7: 8: 7 d 7:5 d 8: min 8 d 8: 8: 9 8: d 9: d min - d min

9 Two tiered timing model... 9 Four DFs re onstruted long with the overll DF. Aording to the NT proedure, terminl d (whose first hollow is the longest) is seleted for possile redution in D(d). The DH-insertion proess ontinues using the riterion R=. The first UR- DHC is DH +DH, nd the seond DH. The result is tht D() nd D(d) re redued from to nd from to, respetively; hene, N = D(S) = 5, nd G is inresed from to using three inserted DH trips. The five FIFO-sed loks re s follows: [-5-DH -9], [-DH -7], [-DH -6], [], [8]. Time 6: 6: 6: 7: 7: 7: 8: 8: 8: 9: Fixed Shedule d(,t) 5 d 6 d 8 9 d 7 d D()= d(,t) DH D()= d(,t) - DH DH D()= d(d,t) - D(d)= g(t) G= 6: 6: 6: 7: 7: 7: 8: 8: 8: 9: Time Fig.. Nine-trip exmple with DH trip insertions for reduing fleet size.

10 A. Ceder.. Vrile sheduling A smll mount of shifting in sheduled deprture times eomes lmost ommon in prtie when ttempting to minimize fleet size or the numer of vehiles required. However, the trnsit sheduler who employs shifting in trip-deprture times is not lwys wre of the onsequenes tht ould rise from these shifts. Ceder (, 7) presented methods, mostly ording to the DF, to relize vrile trip shedule in n effiient mnner. Let [t i - i( ), t i + i(+) ] e the tolerne time intervl of the deprture time of trip i, in whih: i( ) = mximum dvne of the trip s sheduled deprture time (the se of n erly deprture), nd i(+) = mximum dely from the sheduled deprture time (the se of lte deprture). Time 6: 6: 6: 7: 7: 7: 8: 8: 8: 9: Fixed Shedule d(,t) d 9 d 7 d d D()= d(,t) DH D()= d(,t) - DH DH D()= d(d,t) - D(d)= g(t) G= 6: 6: 6: 7: 7: 7: 8: 8: 8: 9: Time Fig.. The nine-trip exmple (of Figure ), first with shifting nd seond with DH trip insertion, for reduing fleet size

11 Two tiered timing model... The nine-trip exmple illustrted in Figure is used for possile shifting deprture times in Figure, whih employs the DF disply. The tolernes of this exmple re i(+) = i( ) = 5 minutes for ll trips in the shedule. Strting with shifting Trip kwrd nd Trip forwrd y 5 minutes results in reduing D() from to. This my e ontinued with shifting Trips 7 nd 8 to redue D(d) from to. Beuse no further shifting in deprture times is fesile for the given tolernes, the proess eomes one of serhing for URDHC using DH trip insertion. This yields three DH trips resulting in Min N = G (S new ) = metrionverterprodutid, in, in whih S new is the new shedule. The three loks re determined y FIFO: [-5-DH - 9], [-DH -7-8], nd [--DH -6]. In se DH trip insertion is not llowed, the shifting proess will end with Min N = 5 nd the FIFO-sed loks: [-5], [-9], [-], [7-8], [9].. Vehile-hin onstrution using rew-ost pproh There re two predominnt hrteristis in trnsit-opertions plnning: () different resoure requirements etween pek nd off-pek periods, nd () working during irregulr hours. These hrteristis result in split duties (shifts) with unpid inetween periods. Often it lled swing time. The inonveniene ompnying split duties led driver (rew) unions to negotite for n extension of the mximum llowed driver s idle time for whih the driver n still get pid. It is ommon, therefore, to hve onstrint in lor union greement speifying this mximum pid idle time (swing time), to e termed T mx. The rew-sheduling prolem from the geny s perspetive is known to e the minimum rew-ost prolem. With this minimum-ost orienttion in mind, the DF (defiit funtion) properties n e used to onstrut vehile hins (loks) tht tke into ount T mx.in other words, to mximize idle times (swing times) tht re lrger thn T mx, nd hene to redue rew osts... Arrivl-deprture joinings within hollows The following desription uses the nottion nd definitions ssoited with the DFs. Eh hollow of DF, d(k,t) t terminl k, ontins the sme numer of deprtures nd rrivls, exept for the first nd lst hollow t the eginning nd end of the shedule horizon. This is due to the ft tht eh rrivl redues d(k,t) y one nd eh deprture inreses it y one, so tht the hollow strts nd ends t D(k). For given hollow, Hm, k let Im k e the set of ll rrivl epohs t i e in Hm, k nd let Jm k e the set of ll deprture epohs t j s in Hm. k The differene in time etween deprture nd rrivl is defined s ij = t j s t i e for t j s > t i e in Hm. k The joining (onnetion) etween t i e nd t j s in vehile lok is effetively the idle time etween trips; hene, ij my represent this idle time. In ddition lol pek uv is defined within hollow Hm k s d(k,t uv ) etween t u s nd t v e in whih e k m < t u s t uv t v e < s k m+, where Hm k strts nd ends t e k m nd s k m+, respetively. Note tht if the strt nd/or end

12 A. Ceder of lol pek, uv, hs more thn one deprture or rrivl, then it suffies to refer to only one of them (s u or v). Let d k,m uv e the numer of deprtures in Hm k efore nd inluding t u s, nd k,m uv e the numer of rrivls in Hm k efore t u s. Lemm. The numer of rrivl-deprture joinings in hollow H k m efore t u s must e d k,m uv. Proof. If some deprture epohs efore lol pek, uv, re left without joining, it will e impossile to onnet them with rrivl epohs fter t v e. Tht is, eh deprture epoh efore nd inluding t u s must hve joining to n erlier rrivl time within H k m. This n e seen in Figure (). Lemm (. The numer ) of rrivl-deprture joinings tht n e onstruted fter t v e within Hm k is k,m uv d k,m uv. Proof. Given hollow Hm k nd lol pek uv, then sed on Lemm nd the hrteristis of lol peks in hollows, d k,m uv, deprture epohs must nd n e joined to erlier rrivl ( epohs in ) Hm; k hene, the numer of rrivl epohs left over without joinings is k,m uv d k,m uv for ll lol peks. Figure () displys this explntion. Lemm. The sum of ll idle times within ny hollow is fixed numer nd independent of ny proedure imed t joining rrivl nd deprture epohs; tht is i,j ij = onstnt. Proof. Let H k m hve n rrivls nd n deprtures. It is noted previously tht the numer of deprtures nd rrivls re the sme within eh middle hollow (i.e., exluding the first nd lst hollows). Let two different n-joining rrngements with idle times ij nd ij for ll joinings etween i Ik m nd j J k m e expressed s follows: ij = i,j i,j ( ) t j s ti e = j t j s i t i e; nd similrly ij = i,j j t j s i t i e

13 Two tiered timing model... d(k,t) () H k m d(k,t uv ) k m e u t s v t s s k m d(k,t) () 6 5 k, m u v k, m k, m u v - d uv k, m k, m u v - d uv d(k,t) () Time Fig.. Prt () desries exmples of rrivl-deprture joining to support Lemm ; prt () interprets Lemm ; prt () shows two -joining exmples of Lemm It is known tht the sum of ll deprture or rrivl times in hollow is fixed numer; hene i,j ij = i,j ij = onstnt. Figure () further lrifies this rgument.

14 A. Ceder.. Ojetive funtion nd formultion In onstruting the loks t eh DF, the im is to mximize the numer of times in whih ij T mx ; in other words, to redue rew ost. At the sme time, however, for ses in whih ij < T mx, from rew s firness perspetive, it will e resonle to ttempt to hve equitle pid idle times. This ws for simple reson, to eliminte sitution in whih some drivers will hve long nd some short pid idle times. In wht follows is formultion of the min ojetive nd then seondry ojetive. For given hollow H k m t terminl k, let x ij e vrile ssoited with trip-joining etween the rrivl of the i th trip to H k m nd the deprture from H k m of the j th. The prolem of finding the mximum numer of idle times greter thn or equl to T mx in hollow H k m is s follows: Prolem P. Mx Z = x ij () Sujet to: i I k m j J k m x ij, i I k m () j J k m x ij, j Jm k () i I k m The inry deision vriles re determined y:, t j s t i e T mx x ij =, otherwise x ij = {, }, i I k m, j J k m (5) A solution with x ij = indites tht joining trips i (rrivl epoh) nd j (deprture epoh) results in n idle time lrger thn or equl T mx. Constrints () nd () insure tht eh trip in H k m my e joined with, t most, one suessor trip, nd one predeessor trip, respetively. Trips tht were not joined in the solution of P re sujet to seondry ojetive: equitle pid idle times. It is shown elow tht joinings with this seondry ojetive re sed on the FIFO rule. Blning ij for ij < T mx is the sme s minimizing the differene etween eh ij nd its verge ij either y solute differene or y lest-squre differene. The FIFO rule used for this lning is stted in the following theorem. Theorem. Minimizing the lest-squre differenes etween ij nd eh ij for ll i I k m nd j J k min H k u is omplished y onstruting joinings using the FIFO rule. Proof. It is suffiient to prove Theorem on simple, ut generlized exmple, s illustrted in Figure 5, with hollow ontining two rrivls nd two deprtures.

15 Two tiered timing model... 5 d(k,t) D(k) Non FIFO FIFO t Δ Δ Δ Δ Fig. 5. Exmple of omprison of joinings sed on FIFO nd other rules It my e shown tht ( ) + ( ) < ( ) + ( ) (6) where is the verge rrivl-deprture joining length in the exmple. Using n lgeri expression, then (6) eomes + < ( + ) (7) Lemm sttes tht + = +, nd hene the right-hnd side of expression (7) is zero. From Lemm one n further otin ( + ) = ( + ) or + =. The ltter is inserted into (7) to yield < (8) Bsed, gin, on Lemm, let = = B or = B nd = B; these lst two equtions re inserted into (8) to otin ( B) < ( B), whih yields >. The lst result must e orret from Figure 5, nd therefore it grees with expression (6).. Mximum unpid idle times The mthemtil progrmming formultion in Equtions () (5) is imed t mximizing the numer of idle times tht re longer thn or equl to T mx. However, this formultion my involve very lrge numer of omputtions (NP-Complete), hene entiling the use of nother (more simplified) proedure. Suh proedure is desried in flow digrm in Figure 6 nd ontins oth T mx nd FIFO rule onsidertions; the ltter is for joining rrivls nd deprtures with pid idle times. Let us ll this proedure lgorithm T m F.

16 6 A. Ceder Initiliztion: rrivl nd deprture times t k; T mx Construt d(k,t) Find D(k) List D(k) rrivls t T eh with trip numer Step move on d(k,t) Yes d(k,t)=d(k,t )? No Yes Anymore disjoined deprture rry? No STOP An rrivl exist, too? Yes Yes A deprture exit? No No Yes Ersing rrivl rry Adding unpid joining rry Δ ij T mx? No Adding deprture rry in pid joining (list of deprtures) Adding disjoined rrivl rry Construting FIFO joining rrys Adding joining rrys Yes Being t D(k), or is Lemm pplied? No STOP Construting FIFO joining rrys Adding joining rrys Fig. 6. Flow digrm of lgorithm T mf

17 Two tiered timing model... 7 The input for lgorithm T m F for eh terminl k onsists of two rrys, the rrivl nd deprture rrys, nd given T mx. This input enles onstruting DF t k nd otining D(k) following the insertion of DH trips nd the shifting of deprture times for minimizing fleet size (see Setion ). Beuse D(k) vehiles re required t k, we ssume their rrivls there to e t (or efore) T ( the strt of the shedule horizon). Algorithm T m F moves y steps on d(k,t), in whih eh step refers to hnge in d(k,t) or the detetion of dot on d(k,t); the dot mens tht rrivl nd deprture epohs t k overlpped t t. Algorithm T m F ontinues with hek of the end of the shedule horizon nd detets the nture of the hnge (or dot) in d(k,t). For eh deprture epoh, ij is exmined to determine whether it is greter thn or equl to T mx ; if greter, then n unpid joining rry is dded, otherwise disjoined deprture time rry is dded. Eh rrivl epoh (deteted in step move in Figure 6) is dded s disjoined rrivl rry. If deprture epoh is identified in step move, the lgorithm looks for possile dot on d(k,t), dding its rrivl epoh to the list of disjoined rrivl rrys. At the end of the proess, the lgorithm onstruts joining rrys from the disjoined rrivl nd deprture rrys, using the FIFO rule. The omplete proess is shown in Figure Rel-life exmples An exmple of onstruting vehile hins (loks), inluding the employment of lgorithm T m F, is shown in Figures 7 9. This exmple is sed on rel-life sheduling dt from EGGED the lrge ntionl us rrier of Isrel. The exmple, onsisting of three terminls nd -trip shedule, is exhiited in Figure 7, inluding DH trvel time mtrix, shifting tolerne, T mx, nd shedule horizon. It should e noted, though, tht DH trvel time etween terminls nd is onsidered in oth diretions lthough there is only servie route etween nd. The fleet-redution proedure, involving the shifting of deprture times nd DH trip insertions, is shown in Figure ; here it is pplied to the exmple of Figure 7 in Figure 8. Two DH trips nd two shifts re introdued into the proess to redue D() nd D() from four to three, resulting in fleet size of eleven vehiles. The shifts re shown in Figure 8 y their shifting length nd trip numer. It n e seen from this Figure tht the only middle hollow ontining more thn single deprture is the seond hollow of d(,t); hene, only this hollow is sujet to the proess of lgorithm T m F. Figure 9() desries the solution for lgorithm T m F in omprison with solution sed only on the FIFO rule in Figure 9(). The trip numers of the exmple, ppering in Figure 7 re dded to Figure 9. AlgorithmT m F results in two unpid joinings etween the rrivls of trips nd nd the deprtures of trips 5 nd 7, respetively. In oth ses, ij > T mx = minutes. The remining joinings in Figure 9() re sed on the FIFO rule. The use of only the FIFO rule for the entire proess results in only one unpid joining (tht etween trips nd 5) s is shown in Figure 9().

18 8 A. Ceder Routes DH trvel times (minutes) Shifting tolerne = ± minutes T mx = minutes [T,T ] = [6:, 9:5] Trip numer Deprture terminl Deprture time 6: 6:5 6: 6: 7: 7: 7:5 8: 8: 6: 6: 6:5 6:5 6: 7: 7:5 7:5 8:5 6: 7: 7:5 7:5 7: 7:8 Arrivl terminl Arrivl time 6: 7: 7:5 7:8 8: 8:5 8:5 8:5 8:5 6: 6: 7: 7: 7: 8: 8: 8:5 9: 7:5 7: 7:5 7:5 8:5 9: Fig. 7. Exmple onsisting of trips nd terminls for onstruting vehile hins with the T mx onstrint

19 Two tiered timing model... 9 Time 6: 6:5 6: 6:5 7: 7:5 7: 7:5 8: 8:5 8: 8:5 9: d(,t) (6:) D() = d(,t) DH D() = 5 d(,t) DH 6: 6:5 6: 6:5 7: 7:5 7: 7:5 8: 8:5 8: 8:5 9: Time Trip numer D() = Fig. 8. The -trip exmple (depited y three DFs) undergoing DH trip-insertion proedure, omined with shifting deprture times d(,t) DH () Joining sed on T m F : 6: 7: 7: 8: 8: 9: 7 D(k) = Trip numer 8 9 d(,t) DH () Joining sed on FIFO : 6: 7: 7: 8: 8: 9: Time 7 D(k) = Fig. 9. Arrivl-deprture joinings for onstruting vehile hins (loks) in the middle hollow of terminl, utilizing in () the TmF lgorithm, nd in () the FIFO rule 7 8 9

20 A. Ceder The finl phse of the rrivl-deprture joining proess is to onstrut vehile loks. This will ontin the joinings reted nd other FIFO-sed joinings in order to mke omplete set of loks. The eleven loks of the -trip-shedule exmple, sed on lgorithm T m F (t terminl ), re given y their numers in the following list: [-DH --7], [--], [-5], [-], [-7], [-5], [-DH -9], [-6], [9-6], [-8], [-8]. The proess sed only on the FIFO rule results in the sme loks, exept for the 5 th nd 9 th loks, whih eome [-6] nd [9-7], respetively. Finlly it is worth noting tht the ide presented turned to e very useful in prtie. It llows the sheduler to do things oth mnully nd utomtilly. In ddition of using this ide in the EGGED us rrier, with out uses, it ws implemented y the lrge KMB us ompny in Hong Kong with out uses. In oth ses this implementtion resulted in signifint ost redution. 6. Conluding remrk The riteri for trnsit rew sheduling re sed on n effiient use of mnpower resoures while mintining the integrity of ny work-rule greements. The onstrution of the seleted rew shedule is usully result of the following su-funtions: (i) duty piee nlysis; (ii) work-rules oordintion; (iii) fesile duty onstrution; nd (iv) duty seletion. The duty-piee nlysis prtitions eh vehile lok t seleted relief points into set of duty piees. These duty piees re ssemled in fesile duty-onstrution funtion. Other required informtion: trvel times etween relief points nd list of relief points designted s required duty stops nd strt lotions. The fous of this work is relted indiretly to the ssemling of duty piees using prtil minimum-ost pproh in the proess of onstruting vehile shedules. From the trnsit geny s perspetive, the lrgest single ost item in the udget is the driver s wge nd fringe enefits. Beuse of the importnt implitions of rew sheduling for providing good trnsit servie, prtitioners ought to omprehend the root of the prolem, nd e equipped with si tools to e le to rrive t solution. This work provides useful tool using the hrteristis of split rew duties (shifts) with unpid in-etween periods; these split duties re the result of different resoure requirements etween pek nd off-pek periods. The nlysis presented provides n optiml solution for mximizing the unpid shift periods, hene reduing the rew osts, with the ssurne of mintining the minimum numer of vehiles required. Referenes [] Bnihshemi, M. nd Hghni, A., (). Optimiztion model for lrge-sle us trnsit sheduling prolems. Trnsporttion Reserh Reord, 7, pp. [] Besley, J. E. nd Co, E. B., (996). A tree serh lgorithm for the rew sheduling prolem. Europen Journl of Opertionl Reserh, 9, pp [] Besley, J. E. nd Co, E. B., (998). A dynmi progrmming sed lgorithm for the rew sheduling prolem. Computers & Opertions Reserh, 5, pp

21 Two tiered timing model... [] Borndorfer, R., Loel, A., nd Weider, S., (8). A Bundle Method for Integrted Multi-Depot Vehile nd Duty Sheduling in Puli Trnsit. Computer-Aided Systems in Puli Trnsport (M. Hikmn, P. Mirhndni, S. Voss, eds). Leture notes in eonomis nd mthemtil systems, Vol. 6, Springer, pp. [5] Crrresi, P., Nonto, M., nd Girrd, L., (995). Network models, lgrngen relxtion nd sugrdients undle pproh in rew sheduling prolems. In Computer-ided Trnsit Sheduling. Leture Notes in Eonomis nd Mthemtil Systems, (J. R. Ddun, I. Brno, nd J. M. P. Pixo, eds.), pp. 88, Springer-Verlg [6] Ceder, A., () A step funtion for improving trnsit opertions plnning using fixed nd vrile sheduling, in Trnsporttion nd Trffi Theory, M.A.P. Tylor, (ed), pp., Elsevier Siene [7] Ceder, A., (). Puli trnsport timetling nd vehile sheduling. Chpter in Advned Modeling for Trnsit Opertions nd Servie Plnning (W. Lm nd M. Bell, eds.), pp 57, Pergmon Imprint, Elsevier Siene [8] Ceder, A., (7). Optiml Single-Route Trnsit Sheduling. Trnsporttion & Trffi Theory (R. Allsop, M. Bell, B. Heydeker, eds), ISTTT-7, Elsevier Siene & Pergmon Pu., pp [9] Ceder, A., (7). Puli Trnsit Plnning nd Opertion: Theory, Modeling nd Prtie, Elsevier, Butterworth-Heinemnn, Oxford, UK, 6 p. [] Ceder, A. nd Stern, H.I., (98). Defiit funtion us sheduling with dedheding trip insertion for fleet size redution. Trnsporttion Siene, 5 (), pp. 8 6 [] Clement, R. nd Wren, A., (995). Greedy geneti lgorithms, optimizing muttions nd us driver sheduling. In Computer-Aided Trnsit Sheduling. Leture Notes in Eonomis nd Mthemtil Systems, (J. R. Ddun, I. Brno, nd J. M. P. Pixo, eds.), pp. 5, Springer-Verlg [] Fores, S., Proll, L., nd Wren, A., (999). An improved ILP system for driver sheduling. In Computer-Aided Trnsit Sheduling. Leture Notes in Eonomis nd Mthemtil Systems, 7 (N. H. M. Wilson, ed.), pp. 6, Springer-Verlg [] Fores, S., Proll, L., nd Wren, A., (). Experienes with flexile driver sheduler. In Computer-Aided Sheduling of Puli Trnsport. Leture Notes in Eonomis nd Mthemtil Systems, 55 (S. Voss nd J. R. Ddun, eds.), pp. 7 5, Springer-Verlg [] Freling, R., Huismn, D., nd Wgelmns, A. P. M., (). Applying n integrted pproh to vehile nd rew sheduling in prtie. In Computer-Aided Sheduling of Puli Trnsport. Leture Notes in Eonomis nd Mthemtil Systems, 55 (S. Voss nd J. R. Ddun, eds.), pp. 7 9, Springer-Verlg. [5] Gertsh, I. nd Gurevih, Y., (977). Construting n optiml fleet for trnsporttion shedule. Trnsporttion Siene,, pp. 6 [6] Gintner, V., Kliewer, N. nd Suhl, L., (8). A Crew Sheduling Approh for Puli Trnsit Enhned with Aspets from Vehile Sheduling. Computer-Aided Systems in Puli Trnsport (M. Hikmn, P. Mirhndni, S. Voss, eds). Leture notes in eonomis nd mthemtil systems, Vol. 6, Springer, pp. 5 [7] Hse, K., Desulniers, G., nd Desrosiers, J., (). Simultneous vehile nd rew sheduling in urn mss trnsit systems. Trnsporttion Siene, 5(), pp. 86

22 A. Ceder [8] Hghni, A. nd Bnihshemo, M., (). Heuristi pprohes for solving lrge-sle us trnsit vehile sheduling prolem with route time onstrints. Trnsporttion Reserh, 6A, pp. 9 [9] Hghni, A., Bnihshemi, M., nd Ching, K. H., (). A omprtive nlysis of us trnsit vehile sheduling models. Trnsporttion Reserh, 7B, pp. [] Huismn, D., Freling, R., nd Wgelmns, A.O.M., (). A roust solution pproh to the dynmi vehile sheduling prolem. Trnsporttion Siene, 8 (), pp [] Huismn, D., Freling, R., nd Wgelmns, A. P. M., (5). Models nd lgorithms for integrtion of vehile nd rew sheduling. Trnsporttion Siene, 9, pp. 9 5 [] Kroon, L. nd Fishetti M., (). Crew sheduling for Netherlnds rilwys destintion: ustomer. In Computer-Aided Sheduling of Puli Trnsport. Leture Notes in Eonomis nd Mthemtil Systems, 55 (S. Voss nd J. R. Ddun, eds.), pp. 8, Springer- Verlg [] Kwn, A. S. K., Kwn R. S. K., nd Wren, A., (999). Driver sheduling using geneti lgorithms with emedded omintoril trits. In Computer-Aided Trnsit Sheduling. Leture Notes in Eonomis nd Mthemtil Systems, 7 (N. H. M. Wilson, ed.), pp. 8, Springer-Verlg [] Kwn R.S.K nd Kwn A.S.K., (7). Effetive Serh Spe Control for Lrge nd/or Complex Driver Sheduling Prolems. Annls of Opertions Reserh 55, pp. 7 5 [5] Lplgne, I., Kwn R.S.K., nd Kwn A.S.K., (9). Critil Time Window Trin Driver Relief Opportunities. Puli Trnsport plnning nd Opertions (), pp [6] Löel, A., (998), Vehile sheduling in puli trnsit nd lgrngen priing. Mngement Siene, (), pp [7] Löel, A., (999). Solving lrge-sle multiple-depot vehile sheduling prolems. In Computer- Aided Trnsit Sheduling. Leture Notes in Eonomis nd Mthemtil Systems, 7 (N. H. M. Wilson, ed.), pp. 9, Springer-Verlg [8] Loureno, H. R., Pixo, J. P., nd Portugl, R., (). Multiojetive metheuristis for the us-driver sheduling prolem, Trnsporttion Siene, 5(), pp. [9] Mesquit, M. nd Pixo, J.M.P., (999). Ext lgorithms for the multi-depot vehile sheduling prolem sed on multiommodity network flow type formultions. In Computer-Aided Trnsit Sheduling. Leture Notes in Eonomis nd Mthemtil Systems, 7 (N. H. M. Wilson, ed.), pp., Springer-Verlg [] Mesquit, M., Pis, A., nd Respiio, A., (9). Brnhing Approhes for Integrted Vehile nd Crew Sheduling. Puli Trnsport Plnning nd Opertions (), pp. 7 [] Mingozzi, A., Boshetti, M. A., Riirdelli, S, nd Bino, L., (999). A set prtitioning pproh to the rew sheduling prolem. Opertions Reserh, 7, pp [] Pis, A. nd Pixo, J.M.P., (99). Stte spe relxtion for set-overing prolems relted to us driver sheduling. Europen Journl of Opertionl Reserh, 7, pp. 6 [] Shen, Y. nd R. S. K. Kwn., (). Tu serh for driver sheduling. In Computer-Aided Sheduling of Puli Trnsport. Leture Notes in Eonomis nd Mthemtil Systems, 55 (S. Voss nd J. R. Ddun, eds.), pp. 5, Springer-Verlg [] Stern, H.I. nd Ceder, A., (98). An improved lower ound to the minimum fleet size prolem. Trnsporttion Siene, 7 (), pp. 7 77