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1 Available online at ScienceDiect Tanspotation Reseach Pocedia 3 (07) nd Intenational Symposium on Tanspotation and Taffic Theoy Integated Public Tanspot Timetable Synchonization and Vehicle Scheduling with Demand Assignment: A Bi-objective Bi-level Model Using Deficit Function Appoach Tao Liu a, Avishai (Avi) Cede a, b, * a Tanspotation Reseach Cente, Depatment of Civil and Envionmental Engineeing,Univesity of Auckland, Auckland,4 New Zealand b Faculty of Civil and Envionmental Engineeing, Technion-Isael Institute of Technology, Haifa, 3000 Isael Abstact In the opeations planning pocess of public tanspot (PT), timetable synchonization is a useful stategy utilized to educe tansfe waiting time and impove sevice connectivity. Howeve, most of the studies on PT timetable synchonization design have teated the poblem independently of othe opeations planning activities, and have focused only on minimizing tansfe waiting time. In addition, the impact of schedule changes on PT uses oute/tip choice behavio has not been well investigated yet. This wok develops a new bi-objective, bi-level intege pogamming model, taking into account the inteests of PT uses and opeatos in attaining optimization of PT timetable synchonization integated with vehicle scheduling and consideing use demand assignment. Based on the special stuctue chaacteistics of the model, a novel deficit function (DF)-based sequential seach method combined with netwok flow and shifting vehicle depatue time techniques is poposed to achieve a set of Paeto-efficient solutions. The gaphical featues of the DF can facilitate a decision-making pocess fo PT schedules fo finding a desiable solution. Two numeical eamples ae illustated to demonstate the methodology developed. 07 The Authos. Elsevie B.V. All ights eseved. Pee eview unde esponsibility of the scientific committee of the nd Intenational Symposium on Tanspotation and Taffic Theoy. Keywods: public tanspot, timetable synchonization, vehicle scheduling, demand assignment, bi-objective bi-level model, deficit function * Coesponding autho. Tel.: ; fa: addess: a.cede@auckland.ac.nz (A. Cede) The Authos. Elsevie B.V. All ights eseved. Pee eview unde esponsibility of the scientific committee of the nd Intenational Symposium on Tanspotation and Taffic Theoy. 0.06/j.tpo

2 34 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) Intoduction.. Backgound and motivation One of the most challenging poblems in tanspotation planning is how to shift a significant numbe of pivate ca uses to public tanspot (PT) in a sustainable manne. The poblem of impoving PT patonage and its compatibility with use needs is multifaceted. Howeve, one may athe intuitively assume that impoving PT sevice eliability, fom the use pespective, and educing opeations costs, fom the opeato pespective, will lead to an incease in ideship and competiveness compaed with pivate ca use. Fo most PT systems, inte-oute tansfe connections ae impotant factos affecting sevice eliability. Timetable synchonization is a useful stategy used to educe tansfe waiting time, povide a well-connected sevice and impove PT sevice eliability (Cede, 00; Cede, 06). Most pevious studies (e.g., Daganzo, 990; Bookbinde and Desilets, 99; and Cede et al., 00) on the PT timetable synchonization poblem have teated the poblem sepaately, athe than coupled with, o isolated fom, othe opeation planning activities, such as vehicle scheduling, tansit assignment and netwok design. Howeve, as pointed out by Cede (00, 06), it is pefeable that all opeation planning activities can be planned simultaneously in ode to eploit the system s capability to the geatest etent, and theeby, maimize its poductivity and efficiency. Some eseaches (e.g. Cede and Sten, 984; Van den Heuvel et al., 008; Ibaa-Rojas et al., 04; Liu and Cede, 06a; and Liu et al., 06) have focused on integating the PT timetable synchonization poblem (TSP) with the vehicle scheduling poblem (VSP) in ode to analyze the tade-off between level of sevice and opeations costs. To date, thee ae two main decision-making poblems that have not yet been addessed egading an integated appoach to simultaneously analyze timetable synchonization and vehicle scheduling tasks. Fist, almost all pevious studies focused on optimizing opeation paametes only, such as tip offset times, oute headways and vehicle tip chains. Howeve, the effect of the changes of these opeation paametes on PT uses tip choice behavio which have a heavy impact on othe planning activities, is not taken into consideation in the timetable optimization pocess. Second, thee is a lack of effective and efficient solution methods fo finding a set of Paeto-efficient solutions so as to help fo the pupose of assisting in multi-citeia decision analysis fom the pespectives of both PT uses and opeatos. To bidge these gaps, this study addesses the integated PT timetable synchonization and vehicle scheduling poblem with tansit assignment (ITSVS-TA) fo tactical and opeational planning puposes... Liteatue eview The poblem of identifying the optimal synchonized PT timetable is essentially the poblem of deciding on the best dispatching policy fo tansit vehicles on fied outes. This has been dealt with quite etensively in the liteatue. Seveal appoaches and compute-aided softwae packages have been developed. Accoding to the diffeent featues, the appoaches developed can be categoized into fou goups: (i) inteactive gaphical optimization appoach, (ii) analytical modelling appoach, (iii) mathematical pogamming appoach, and (iv) contol theoy appoach. In the fist goup, inteactive gaphical optimization techniques, which is ecognized as the ealiest appoach actually applied in pactice, have been poposed by a few eseaches (e.g., Rapp and Gehne, 976; Désilets and Rousseau, 99; Fleuent et al., 004; and Vuchic, 005). Othe ealie theoetical investigations of the PT timetable synchonization poblem (PT-TSP) ae mainly focused on how to set oute headways and offset times. Salzbon (980) studied a special inte-town oute connected by a sting of feede outes. Daganzo (990) eamined the single tansfe node case, and povided some intuitive ules fo setting the headways of the inbound and outbound outes. The second appoach to solve the PT-TSP employs analytical fomulations fo idealized PT systems. Wiasinghe et al. (977) and Wiasinghe (980) developed appoimate analytical models fo investigating the optimal design paametes of a coodinated ail and bus tansit system atop ectangula gid o ing-adial netwoks. A seies of follow-up studies (e.g., Lee and Schonfeld, 99; Chien and Schonfeld, 998; Chowdhuy and Chien, 00; Ting and Schonfeld, 005; Sivakumaan et al., 0; and Kim and Schonfeld, 04) have been conducted. Knoppes and Mulle (995) investigated the impact of fluctuations in passenge aival times on the possibilities and limitations of synchonized tansfes. As pointed out by Liu and Cede (06b), one limitation of the analytical modelling appoach is that it fails to accuately calculate the measues of the cost components of the objective functions consideed. The thid appoach widely found in the liteatue adopts mathematical pogamming models. Klemt and Stemme

3 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) (988) and Domschke (989) povided a quadatic pogamming model of the poblem to minimize passenge tansfe waiting time. Bookbinde and Desilets (99) developed an intege pogamming model and an iteative impovement heuistic pocedue was povided to minimize mean tansfe disutility. Voß (99) poposed a 0- intege pogamming and a tabu seach algoithm to minimize tansfe waiting time. Cede et al. (00) developed a mied intege linea pogamming model and seveal heuistic algoithms to maimize the numbe of simultaneous bus aivals at the tansfe nodes of PT netwoks. Based on this seminal wok, a seies of follow-up studies have been conducted by othe eseaches (e.g., Shafahi and Khani, 00; Ibaa-Rojas and Rios-Solis, 0; Aksu and Akyol, 04; Ibaa-Rojas et al., 05; Fouilhou et al., 06; Wu et al. 06). Wong et al. (008) developed a mied intege pogamming model and an optimization-based heuistic method to minimize the total passenge tansfe waiting time fo the MTR system in Hong Kong. Ibaa-Rojas et al. (04) developed a bi-obejctive, intege pogamming model to maimize the numbe of passenges benefiting fom well-timed tansfes and minimize vehicle opeating costs. In the fouth and last goup, contol theoy based models wee uilized at the opeations contol level to contol the movement of vehicles in ode to impove the sevice eliability, and schedule adheence and headway egulaity of the planned synchonized timetable. This contol appoach is based on feasible opeational contol stategies, such as holding (Dessouky, 003; Hadas and Cede, 00; Liu et al., 04; Daganzo and Andeson, 06), skip-stop (Cede et al., 03), shot-tun (Nesheli et al., 05) and a combination of diffeent selected contol stategies (Liu et al., 05; Nesheli et al., 05; Liu and Cede, 06c). The objective function in most of the afoementioned wok is eithe to minimize the total passenge tansfe waiting time o maimize the numbe of diect tansfes. One limitation of these woks is that they failed to conside othe system pefomance measues. Fo eample, the minimization of passenge tansfe waiting time may lead to impaied pefomance of othe system measues, such as inceased vehicle fleet size, inceased passenge tavel time and moe empty-seat vehicle hous. In addition, PT use oute/tip choice behavio esponding to the changes of the system design paametes, such as oute headways and offset times, has not been taken into consideation in pevious studies. Theefoe, a compehensive, systematic, and multi-citeia decision-making famewok that can take vaious system pefomance measues and uses oute/tip choice behavio into account is needed..3. Objectives and contibutions The pupose of this study is to povide a bi-objective, bi-level decision-making famewok, togethe with a deficit function-based solution scheme, fo the ITSVS poblem consideing passenge demand assignment. The theoetical contibutions of this study to the cuent liteatue ae theefold. Fist, a new bi-objective, bi-level intege pogamming model is developed fo the ITSVS-TA poblem. To the best of ou knowledge, this is the fist time that PT passenge oute choice behavio is incopoated into the ITSVS poblem. Second, in the uppe level ITSVS poblem, compehensive objective function components that take both PT uses and opeatos inteests into account ae fomulated. Thid, an innovative solution method that is based on the deficient function theoy is developed. This wok is compised of si sections including this intoductoy section. Section povides backgound on the deficit function (DF) theoy and the optimization famewok fo the poblem. Section 3 pesents the mathematical fomulations. The DF-based solution method is pesented in Section 4. Section 5 illustates the poposed model and solution method with two detailed eamples. Section 6 concludes ou wok and poposes futue etensions of it.. Deficit function and optimization famewok. Backgound on the deficit function Following is a concise desciption of a step function appoach poposed by Cede and Sten (98) and Cede (007, 06) fo assigning the minimum numbe of vehicles to a given timetable. Linis and Maksim (967) and Getsbach and Guevich (977) have called this step function a DF as its value epesents the deficit numbe of vehicles equied at a paticula teminal. The DF is a step function that inceases by one at the time of each tip depatue and deceases by one at the time of each tip aival. The DF gaphical modelling method has been applied to vaious kinds of PT opeations planning activities (Liu and Cede, 07). Let G g : g,, n denote a set of equied tips.

4 344 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) The tips ae conducted between a set of teminals U u : u,, q, each tip is seviced by a single vehicle, and g g g g each vehicle is able to sevice any tip. Each tip g can be epesented as a 4-tuple p, ts, q, t e, in which the odeed elements denote depatue teminal, depatue (stat) time, aival teminal, and aival (end) time. It is assumed that each tip g g g lies within a schedule hoizon T, T, i.e., T ts te T. The set of all tips g g g g g g S p, t, q, t : p, q U, g G constitutes the timetable. Two tips, g, g may be seviced sequentially s e g g (feasibly joined) by the same vehicle if and only if (a) t e t g g s and (b) q p. Let d u, t, S denote the DF fo teminal u at time t fo schedule S. The value of d u, t, S epesents the total numbe of depatues minus the total numbe of tip aivals at teminal u, up to and including time t. The maimum value of d u, t, S ove the schedule hoizont, T, designated D u, S, depicts the deficit numbe of vehicles equied at u. g g Let t and s t denote the stat and end times of tip g, g G. It is possible to patition the schedule hoizon of e d u, t, S into a sequence of altenating hollow and maimal intevals. The maimal intevals u u si, e i, i,..., nu define the inteval of time ove which dut, takes on its maimum value. Note that the S will be deleted when it is clea which undelying schedule is being consideed. Inde i epesents the i th maimal u intevals fom the left and nu epesents the total numbe of maimal intevals in dut,. A hollow inteval H, l l 0,,,..., nu is defined as the inteval between two maimal intevals including the fist hollow fom T to the fist maimal inteval, and the last hollow fom the last inteval to T. Hollows may consist of only one point, and if this case is not on the schedule hoizon boundaies T o T, the gaphical epesentation of dut, is emphasized by a clea dot. The sum of all DFs ove u is defined as the oveall DF, g() t dut (,). The maimum value of g(t), G(S) is eploited fo a detemination of the initial lowe bound on the fleet size. Theoem (The deficit function fleet size theoem). If, fo a set of teminals U and a fied set of equied tips G, all tips stat and end within the schedule hoizon [T, T ] and no deadheading (DH) insetions ae allowed, then the minimum numbe of vehicles equied to sevice all tips in G is equal to the sum of all the deficits. Min N S D( u,s) ma d(u, t, S) DF () uu uu t[ T, T] Poof. A fomal poof of this theoem can be found in Cede (06). When deadheading (DH), o empty, vehicle tip insetion and shifting depatue times (SDT) ae allowed fo, the fleet size may be futhe educed below the level descibed in Eq. ().. Outline of the optimization famewok The ITSVS poblem can be descibed as a Stackelbeg o leade-followe game, whee the leade is a PT opeato who ceates timetables and vehicle schedules (S) to optimize system-wide pefomance measues, and the followes ae the PT uses who choose thei tavel paths in a use optimal manne esponding to the opeato s decision. The opeato s decision-making on S can influence, but cannot contol PT uses tavel choice behavio, which will esult in passenge loads ( L ) on vehicles. The deivation of passenge loads L is actually a non-linea and non-continuous mapping of S. The passenge loads defined by the lowe level TA seve as the input of the uppe level ITSVS optimization poblem. This kind of poblem can be mathematically fomulated as a bi-level optimization pogamming model (Yang and Bell, 998; Yin, 00). The bi-objective, bi-level optimization famewok of the poblem is systematically outlined in Fig.. The uppe level model fom the pespective of PT opeatos aims to: (i) minimize total opeation costs, which ae elated to fleet size, and (ii) minimize total passenge-hou cost, which include the total passenge in-vehicle tavel time, total passenge initial waiting time, total passenge tansfe waiting time, and total passenge load discepancy o ovecowding hous. The lowe level poblem fom the pespective of PT passenges is a standad schedule-based tansit assignment poblem with capacity constaints. uu

5 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) Min F( S, L(S) ) [ F( S, L(S) ), F ( S, L(S) )] S Min fsl (, ) L 3. Model fomulation 3. Nomenclatue Fig.. Schematic epesentation of the bi-objective, bi-level optimization famewok Conside a connected netwok composed of a diected gaph G N, A with a finite numbe of nodes N connected by A acs. The notations used thoughout the pape ae listed as follows unless othewise specified: Route pogessive path initiated at a given PT teminal and teminated at a cetain node while tavesing given acs in sequence Tansfe path pogessive path that uses moe than one oute R set of PT outes S s set of PT oute segments on which the epected load is moe than the desied occupancy M m set of numbe of depatues TR t set of all tansfe paths N set of nodes located on oute N set of nodes located on tansfe path t t d passenge demand between i and j, i, j, N, iding on oute ij t d passenge demand between i and j along the tansfe path ij t d initial aggegated passenge O-D demand between i and j ij vehicle fequency associated with oute F F minimum fequency (ecipocal of policy headway) equied min t aveage tavel time between i and j on oute ij t t ij aveage tavel time between i and j on tansfe path t (do not include tansfe penalties) s t aveage tavel time of oute segment s of oute on which the epected load is moe than the desied occupancy t p w oveall tavel time on oute between its stat and end maimum passenge load on oute passenge initial waiting time on oute

6 346 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) L i initial aggegated passenge load at i C v vehicle capacity d desied occupancy on each vehicle (load standad) o oveload facto, tansfe t moves thought oute t a 0, othewise m, if m even-headway depatues ae selected fo oute 0, othewise 3. The uppe level integated timetable synchonization and vehicle scheduling (ITSVS) poblem 3.. Two pincipal objective functions The uppe level ITSVS poblem is based on two pincipal objective functions, minimum Z and minimum Z, acoss the diffeent sets of PT outes: Z PH i, j IWT i, j3 TWT i, j4 LD () i, jn i, jn i, jn R Z FS (3) whee P H i, j = passenge hous between nodes i and j, i, j, N (defined as passenge iding time in a vehicle on an houly basis; it measues the time spent by passenges in vehicles between the two nodes). IWT i, j = initial waiting time between nodes i and j, i, j, N (defined as the amount of time passenges spend at the boading stops between the two nodes). TWT i, j = tansfe waiting time between nodes i and j, i, j, N (defined as the amount of time passenges spend at the tansfe stops between the two nodes). LD = passenge load discepancy on oute (defined as the diffeence between the epected load and the desied occupancy; passenge load discepancy measues the ovecowding seat use on vehicles). FS = fleet size (defined as the numbe of vehicles needed to povide all tips along a chosen set of outes). k = monetay o othe weights, k,,3,4. Fo given weights of o without units, Eq. () esults in units of passenge hous (pass-h). Eq. (3) is simply the minimum fleet size equied. 3.. Objective function components Eqs. () and (3) essentially combine five objective function components. The fist objective component is to minimize the total passenge in-vehicle tavel hous in the system. This is stictly fom the pespective of PT uses. The fomulation of this objective component takes the following fom: Min PH i, j (4) i, jn whee is the monetay value of h in-vehicle tavel time. Specifically, its fomulation is: PH i, j d t t t d t (5) ij ij ij ij i, jn R i, jn ttr i, jnt The second objective component is to minimize total initial passenge waiting time at boading stops. This is stictly fom the pespective of the PT uses. The following is the fomulation of this objective component: Min IWT i, j (6) i, jn

7 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) whee is the monetay value of h initial waiting time. Diffeent fomulations of the epected initial waiting time fo PT passenges can be found in Maguie and Cede (984). Fo andomly aiving passenges, the epected initial waiting time on oute can be calculated as: E( H ) Va( H ) w, fo all R E ( H) (7) whee E( H ) and Va( H ) ae, espectively, the mean and vaiance of headway time H between vehicles. Assuming that PT passenges ae aiving andomly at the stops and that the vehicle headways ae elatively shot and distibuted in a deteministic and egula manne, this epected initial waiting time on oute is the half headway: w, fo all R F (8) whee F can be calculated using the maimum load point method [See Cede (06) fo details.]: L F ma, Fmin d o (9) If F F, then the load pofile will have no influence on the fequency detemination. Theefoe, the total min initial passenge waiting time can be calculated by: t t IWT i, j dij dij a i, j N R F i, jn i, jnt (0) The thid objective component is to minimize the total passenge tansfe waiting time at tansfe stops. This is stictly fom the pespective of the PT uses. The fomulation of this objective component takes the following fom: Min 3 TWT i, j () i, jn whee 3 is the monetay value of h tansfe waiting time. n Conside a goup of passenges p kiqj tansfeing fom the k-th i tip of the oute that node i is in to the q j -th tip of the oute that node j is in at tansfe stopn. Let epesent the set of feasible tansfe connecting q j tip pattens defined by: q j D A W 0, k i K i, q j Q j, n N i, j () t q j q j k i k i q j n whee A is the aival time of the k k i -th i tip of the oute that node i is in at tansfe stop n; D is the depatue q j time of the q j -th tip of the oute that node j is in at tansfe stop n; W is the tansfe walking time needed kiq jn fom the aival place of the k-th i tip of the oute that node i is in to the depatue place of the q j -th tip of the oute that node j is in at tansfe stop n; is the constant time needed fo alighting and boading. It is assumed that the same goup of passenges has the same tansfe walking time and constant time needed fo alighting and boading, and passenges will boad the fist feasible tansfe connecting tip, then the tansfe waiting time fo the passenge n goup p kiqj can be epesented as follows: T D A W kiki n N ij (3) n,, t, kiqj q j ki kiq jn whee q j, the fist feasible tansfe connecting tip, is epesented by: q j ag min q j (4) qj qj Thus, the total passenge tansfe waiting time can be epesented by: n TWT i, j p D kiq j q A W j ki kiq jn (5) ij, N nnt ij, ki Ki, qjqj The fouth objective component is to minimize the total passenge load discepancy. The fomulation is: Min 4 LD (6) R

8 348 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) whee is the monetay value of h in-vehicle cowding tavel time. 4 Define p as the total numbe of passenges obseved at the maimum load point of oute. p is a common sevice measue used by PT agencies woldwide, as descibed by Cede (06) and Liu and Cede (06a). The m passenge load discepancy cost of oute associated with m s LD ma p m do,0t is defined as: (7) ss Thus, LD LD (8) m m R RmM The fifth objective component is to minimize the numbe of vehicles equied. This is stictly fom the pespective of the PT opeatos. This objective component takes the fom: Min FS (9) Remak : Estimation of the minimum fleet size can utilize the DF theoy descibed in the pevious section. Note that it may be sufficient to use the DF modelling technique to detemine the stonge FS lowe and uppe bounds fo this estimation. Pactically speaking, the ITSVS poblem usually involves a vast quantity of computations of oute sets. Thus, the lowe and uppe bounds-based FS calculations can ease the computation effot fo each oute consideed. Objective function components (4), (6), (), and (6) ae all in tems of passenge-hou cost. Theefoe, fo the sake of simplicity, they can be summed up to Min Z as shown in Eq. (). The objective function component (8) stands alone to some etent and is temed Min Z in Eq. (3) Constaints The mathematical mode includes fou goups of constaints. The fist goup of constaints ae the bundle depatue constaints: U m, R (0) m L whee m is an inde of the numbe of depatues unning fom L to U, i.e., m L, L, L, U, U ; L and U ae espectively the lowe and uppe bounds of the numbe of vehicle depatues fo a given oute. The second goup of constaints is the DF bounds constaints: du, t Du, t T, T, u U () whee dut, is the net numbe of depatues less aivals that occu befoe o at time t at teminal u as detemined m by the value of m. Thus, the left hand side of Eq. () can be epesented as a linea function of the vaiables. The value of this function fo a given solution is that of an associated DF fo teminal u at time t. This goup of constaints ensue that the numbe of vehicles used at a given teminal u befoe and up to time t does not eceed the numbe of vehicles D u assigned to teminal u. The second goup of constaint is the fleet size constaint: D u N0 FS () uu 0 D( u), uu (3) whee N is the total fleet size. This constaint indicates that the sum of vehicles assigned to all teminals should not 0 be moe than the equied minimum fleet size FS= N 0. The fouth goup of constaints is the decision vaiable constaints: m, if m even headway depatues ae selected fo oute, mm, R (4) 0, othewise.

9 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) The optimal solution set m m indicates the optimal numbe of depatues. Fom, an optimal timetable can be constucted with pope oute offset times. Remak : Given the geneated timetable, a vehicle schedule can be easily deived using the fist-in-fist-out (FIFO) ule [see Cede and Sten, (98) and Cede, (06) fo details] o the chain etaction (CE) pocedue [see Getsbach and Guevich, (977) fo details]. 3.3 The lowe level tansit assignment (TA) poblem The lowe level of the bi-level model epesents the PT use oute and vehicle un choice behavio esponding to the uppe level timetables and vehicle schedules. The accuate calculation of the numbe of passenges assigned to each vehicle un needs a sound tansit assignment model. In the liteatue, thee ae vaious kinds of tansit assignment models which have been developed. Geneally speaking, they can be classified into fequency-based models and schedule-based models. The fequency-based models utilize aggegated epesentation of PT sevice and ae usually used fo stategic o long ange planning poblems. The schedule-based models ae based on the actual vehicle un tajectoies. They can detemine the passenge loads on each vehicle un. Thus, they ae moe suitable fo opeation design poblems. Fo a detailed desciption of the ecent developments of the fequency-based and schedule-based models, the eades may efe to Gentile et al. (06). Since we ae doing a timetable design poblem that aims at optimizing passenge tansfes in a netwok, it is clea that schedule-based tansit assignment models should be adopted. Accoding to Gentile et al. (06), thee ae thee main appoaches used fo developing schedule-based models: (i) the diachonic gaph-based appoach, (ii) the dual gaph-based appoach, and (iii) the passenge choice set based appoach. The diachonic gaph-based appoach is adopted in this study fo its advantages in modelling un based assignment that is closely elated to DF based modelling fo vehicle scheduling. In a diachonic gaph, also called a space-time netwok, a node is modelled though a specific sub-gaph whose nodes have space and time coodinates accoding to the timetable (see Nuzzolo et al., 00; Nuzzolo, 00; and Gentile et al., 06 fo details). Fig. shows an eample of how to tansfom a thee node physical netwok to a geneic diachonic gaph in which black acs epesent waiting activities. The diachonic gaph-based appoach povides a moe natual way to model the schedule-based tansit assignment poblem. In a diachonic gaph, time elated elements ae built into its topological stuctue and thus it can clealy descibe each vehicle un in a chonological ode. Fig.. An illustation of tansition fom a physical netwok to a diachonic gaph The lowe level assignment model utilizes a utility function to descibe uses peceived tavel costs, i.e., genealized tavel cost, tavelling fom oigin i to destination j (Tong and Wong, 999; Pabo, 04). The fomation of this genealized tavel cost function, which is epessed in minutes, is given as follows: tij t t 3t3 4t4 5t5 6n (5) whee t is the total walking time needed fo getting fom the oigin place to the fist stop and fom the last stop to the destination place, t is the waiting time at the fist stop, t 3 is the in-vehicle tavel time, t 4 is the tansfe walking

10 350 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) time, t 5 is the tansfe waiting time, n is the time cost of the numbe of tansfes involved in using path, i i,,, 6 ae the coesponding weighting factos. Remak 3: The genealized tavel cost function can be etended by: (i) consideing a time-dependent vesion of it, and (ii) incopoating the values of times (VoTs) to diffeent goups of uses. Based on the calculation of genealized tavel costs, a shotest (minimum cost) tee can be easily constucted on a diachonic gaph by pocessing all nodes and using the Pallottino algoithm. [See Pallottino and Scutellà, 998; and Gentile et al., 06 fo details.] With the help of this shotest tee, a netwok loading pocedue is pefomed to conduct the demand assignment. The netwok loading pocedue is done by pefoming a capacity-estained incemental assignment, which is outlined as follows, by taking vehicle capacity constaints into account. Algoithm Capacity estained incemental assignment Step 0 (Peliminaies): Calculate initial passenge O-D demand d ij and numbe of passenges L i on vehicle at stop i. Set n:. Step (Incemental loading): If d ijn C v L, then the packet of in d passenge demand is loaded onto ijn the n -shotest path, i.e., L : L d C L demand is Step in in, and stop; othewise, the packet of ijn v in loaded onto the n -shotest path, i.e., Lin : Lin Cv Lin. (Update): Set n: n, dijn dij( n) Cv Li ( n), and go to Step. Remak 4: It should be pointed out that the poposed capacity-estained incemental assignment pocedue assumes that PT passenges have complete knowledge of the entie netwok and timetables, and that vehicles always keep to thei schedules, i.e., without schedule delays. It will be inteesting to etend it to a stochastic vesion to captue the peception and measue eos and heteogeneity of pefeences of PT passenges. 3.4 Model integation By combining the uppe level ITSVS model and the lowe level TA model, the esulting integated timetable synchonization and vehicle scheduling poblem with passenge assignment can be epesented as follows: Min Z( SLS, ( )) PH i, j IWT i, j 3 TWT i, j 4 LD (6) S Min Z (, ( )) S i, jn i, jn i, jn R SLS (7) s.t. Timetable synchonization and vehicle scheduling constaints: (0)-(4) (8) whee the passenge load on vehicles LS ( ) is obtained by solving the following TA poblem: Min fsl (, ) (9) L s.t. vehicle capacity constaint: Li Cv, i N (30) passenge vehicle un choice and flow popagation constaints defined in Algoithm (3) The natue of the uppe level ITSVS model is non-linea, bi-objective intege pogamming with linea constaints, which is a special case of the intege quadatic pogamming poblem that is known to be a NP-had poblem. In addition, solving the uppe level ITSVS poblem equies solving the lowe level TA poblem that is in effect a nonlinea constaint of the uppe level model, which makes the whole poblem non-conve. Due to its intinsic, non-linea and non-conve compleity, the bi-objective, bi-level ITSVS-TA poblem is etemely difficult, to solve mathematically, especially fo lage scale netwoks, fo an optimum global solution. In the net section, we popose a DF-based heuistic solution method fo this poblem by eploing its special stuctue popeties. 4. Solution method A new DF-based solution method has been developed to handle this poblem fo pactical implementations. Befoe descibing the DF-based solution method, a ma flow technique fo fied schedule vehicle scheduling is intoduced. A possible shifting tip depatue time pocedue is also intoduced to futhe optimize the esults.

11 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) Netwok flow technique fo vehicle scheduling with fied schedule A netwok flow technique is employed to estimate the minimum fleet size of a given schedules. A tip joining aay fo S may be constucted by associating the g th ow with the aival event of the g th tip, and the g th column with the depatue event of the g th tip. Cell ( g, g ) will be admissible if g and g can be joined feasibly. Othewise, ( g, g ) will be an inadmissible cell. Let gg be a 0- vaiable associated with cell ( g, g ) and G be the set of equied tips. Then, conside the following poblem: Ma N (3) gg gggg s.t. gg, g G (33) gg gg, g G (34) gg gg 0,, all ( gg, ) admissible (35) gg 0, all ( gg, ) inadmissible A solution with gg = indicates that tips g and g ae joined. The objective function maimizes the numbe of such joinings. Constaint (33) insues that each tip may be joined with no moe than one successo tip. Similaly, constaint (34) indicates that each tip may be joined with no moe than one pedecesso tip. This poblem is equivalent to a special aangement of the maimum flow (ma-flow) poblem. The ma-flow algoithm that solves the vehicle scheduling poblem with DH tips is called an augmenting path algoithm. It is addessed at length in the classic book by Fod and Fulkeson (96). A complete desciption of the augmenting path algoithm can be found in Cede (06). The vehicle scheduling poblem can be tansfomed to a unit capacity bipatite netwok in which the / solution time has the compleity of O( n m ) with n nodes (depatue times) and m acs. The following theoem states that maimizing N is tantamount to minimizing the numbe of chains fo a tip schedule of size n. Theoem (The ma-flow fleet size theoem). Let N MF S and n denote the numbe of chains and tips of schedule S, espectively. Then, M in N MF S n Ma N (36) Poof. Given a set of G g : g,, n equied tips. Assigning each tip sepaately to an individual vehicle esults in a fleet size of n vehicles. If gg, then tip g can be pefomed afte tip g by the same vehicle v. Thus, g the vehicle v g assigned to tip g can be saved. The equied fleet size thus can be educed fom n to n. Similaly, the value of ma-flow Ma N means Ma N vehicles can be saved by linking tips togethe. Thus, the minimum numbe of vehicles equied to pefom all tips in G is n Ma N. This completes the poof. Table. Tip schedule S fo the eample poblem Table. Aveage DH tavel time (minutes) mati fo the eample in Table Tip numbe g Depatue teminal g p Depatue time t g s Aival teminal g q Aival time t b 6:00 b 6:30 a 7:05 c 8:05 3 c 7:0 a 8:00 4 b 8:30 a 9:0 5 a 9:00 b 9:45 g e Depatue teminal Aival teminal a b c a b c Eample: Conside the thee teminal poblem defined by the data in Tables and and Fig. 3(a). The data in Table and Fig. 3(a) ae tansfomed into the geneic diachonic gaph in Fig. 3(c) and netwok-flow epesentation in Fig. 3(d), which has two dummy nodes: a souce node s and a sink node t. The nodes, being connected fom s, ae the aival times of the eample, with an indication, in paentheses, of the aival teminal. The nodes connected to t ae

12 35 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) the depatue times, with an indication of the depatue teminal. Feasible connections between the aival and depatue times, utilizing the DH data in Table, establish the acs between the left and ight nodes, based on Eq. (3). Each ac capacity epesents the numbe of connections that can flow though the ac. In ou case, thee is only a unit capacity assigned to each ac, because only one connection (if any) between a given aival time and teminal and a given depatue time and teminal is possible. (a) A thee-teminal PT netwok (b) The DF solution (c) The coesponding geneic diachonic gaph (d) The ma-flow solution Fig. 3. An eample vehicle scheduling poblem using ma-flow technique and its equivalent DF solution The moe flow ceated, the fewe chains will be equied as stated by Theoem. The objective function N equals the flow to be ceated at s and absobed at t. Since ma-flow = minimum s-t cut in the oiginal netwok flow. This minimum s-t cut is shown in Fig. 3(d). The esult of the eample is ma-flow = Ma N = 3, and, following Theoem, M in N MF S n Ma N = 5-3 = chains. Restated, the timetable in Table can be caied out by a minimum of two vehicles having the connections shown in Fig. 3(d). The equivalent DF solution, shown in Fig. 3(b), also esults in two vehicles: M in N DF S D( a ) D(b ) D( c ) 0. Eplicitly the two blocks, by thei tip numbe, ae [-DH --DH 3-5] and [3-DH -4]. These two blocks have thee DH tips fo connecting aival and depatue teminals: DH (b-a) and DH 3 (c-a) (in the fist block) and DH (a-b) (in the second block), with the total of =0 minutes DH time. The M in N MF S can be used as an initial lowe bound of the equied fleet size when using a DF-based sequential seach method fo solving the poblem, which is descibed in the following section. 4. Deficit function based sequential seach method Based on the special stuctue of the model, we developed a novel DF-based sequential seach (DF-SS) method to geneate Paeto-efficient solutions. The key point behind the DF-SS method is to decompose the oiginal bi-objective pogamming model to a seies of one-objective pogamming models. The decomposition is achieved by estimating

13 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) the lowe bound and uppe bound of objective function Z, i.e., the fleet size, which can be accomplished by using the netwok flow and DF techniques. The poposed DF-SS method is outlined below in a step by step manne. Algoithm Deficit Function-based Sequential Seach (DF-SS) method Step 0 (Lowe bound and uppe bound calculation): Apply the netwok flow technique to obtain an initial lowe bound of the equied fleet size N. Constuct the DF fo each oute L with m L, R, and Step calculate the lowe bound of the fleet size as with m U (Initialization): Set v,, R, and calculate the uppe bound of the fleet size as 0 Z, NUNL N L min N L, D u; constuct the DF fo each oute uu X 0. Z, NU NLR NU min n, D u uu. v v Step ( Z Calculation): Decompose the oiginal bi-objective model to a single objective model with Z v known. Solve the esulting single objective intege pogamming model. This yields the value of Z and a solution set of,,, mv, mv, mv, R v -th ow of X with mv, mv, mv,,,,. R v v Step 3 (Move): Set Z Z. v v. Replace the v -th ow of Z with Z, Z ; eplace the Step 4 v (Stopping ule): If Z N U, go to Step 6; othewise, set v: v and go to Step 3. Step 5 (Solution output): Geneate Paeto-efficient solutions fom mati X with associated objective function values Z Z geneated by mati Z., In Step, some available intege pogamming solves, such as CPLEX and GAMS, can be utilized to efficiently solve the esulting single objective intege pogamming model. In the DF-SS method, the fistly identified lowe and uppe bounds of the equied vehicle fleet size can significantly educe the compleity of the poblem. 4.3 A possible shifting depatue time (SDT) pocedue It is cetain that the esults may be impoved by allowing the timetable obtained by the DF-SS method to be shifted within given shifting toleances. The implications ae that fo each tansfe stop and fo each two time points: t, t (t < t ) at which thee ae vehicle aivals at the tansfe stop, thee is an attempt to shift depatue times (offset times) of all vehicles aiving at the node at t so that they will aive at time t. If this succeeds, the timetable is changed accodingly, and the passenge-hou cost may be educed. In addition, afte pefoming SDT, the vehicle fleet size may be also futhe educed (Cede 00). Howeve, it should be noted that due to the impact of the changed schedule on passenges oute/tip choice behavio, the vehicle passenge loads L may also be changed afte the SDT pocedue, which may lead to unbalanced passenge loads, i.e., ovecowding o empty seat hous, and bus bunching poblems. Theefoe, the SDT pocedue must be pefomed vey caefully by epeienced schedules with the help of DF gaphical displays of the optimized timetables. See Getsbakh and Sten (978) and Cede (00) fo details. 4.4 Oveall solution pocedue An oveview of the solution pocedue is outlined as follows: Step 0 Estimate possible oute fequencies based on O-D demand, vehicle capacity, policy fequency equiement and othe paamete values. Step Geneate Paeto-efficient solutions using Algoithm and Algoithm.

14 354 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) Step Step 3 Modify oute offset times using the SDT pocedue, based on schedules DF obsevations, pesonal pefeences o othe pactical consideations, so as to geneate moe possible Paeto-efficient solutions. Display the combined Paeto-efficient solutions obtained in Step and Step in a fleet cost twodimensional (D) space. 5. Numeical Studies Two numeical eamples ae pesented fo compehending the application of the poposed methodology. The fist is a small-sized netwok. The second is a medium-sized netwok adapted fom Spiess and Floian (989). 5. A small illustative eample A small eample is povided to illustate the poposed model and solution method. The eample PT netwok, which is adapted fom Liu and Cede (06b) and shown in Fig 4(a), has two temini ( a andb), two outes ( a b and b a ) and one tansfe stop (node 3). The input data duing the given time peiod (7:00-8:00) consist of aveage tavel times (in minutes) and an estimated O-D demand mati in pat b of Fig. 4. Based on the input data, the passenge load pofiles pesented in Fig. 4(c) and Fig. 4(d) ae constucted fo outes a b and b, espectively. Fo oute a a, the load on oute segment 3- is the maimum load (360 passenges) of the O-D demand: -, -5, 3-, 3-5, 5- b (see Fig. (b)). Similaly to oute b a, the load on oute segment 3-4 is the maimum load (340 passenges) of the O-D demand: 5-4, 5-, 3-4, 3-, -4. Othe input data fo this eample netwok ae pesented in Tables 3-5. The aveage oute tavel times fo oute a b and oute b ae set as 30 minutes and 0 minutes, espectively. The a desied occupancy fo both of the two outes is set up to 70 passenges. In addition, vehicle dwell times at tansfe stop 3 of vehicles of the two outes ae minute. The tansfe walking time W needed is set up to 0.5 minutes. kiq jn The constant time needed fo alighting and boading is set as 0.5 minutes. The fou weights k, k,,3,4 take the value of. Note that in this eample netwok, thee is only one feasible path between each O-D pai. Thus, the stop/link passenge flows esulted fom the lowe level tansit assignment will emain fied. a b b a a b b a Fig. 4. Eample netwok, its demand data, and the constuction of load pofiles The input data esult in thee sets of numbe of depatues ( q 4, q 5 and q 6) fo oute a, and two sets of b numbe of depatues ( q 4 and q 5) fo oute b. Thus, thee ae five associated vaiables, namely a,, 3

15 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) , and 4, as shown in Table 3. The LD cost is calculated by using Eq. (7) based on the maimum load data, 5 oute segment tavel time, desied occupancy and numbe of depatues. Fo eample, fo decision vaiable, its associated LD cost is ma , 0 0 / pass-h. The LD costs fo othe decision vaiables, as displayed in Table 3, ae calculated in this way. The initial passenge waiting time is calculated by using Eq. (0). Fo this eample poblem, fo passenges taking oute a b with q 4 depatues, the initial waiting time is IWT a b, / pass-h. Fo othe decision vaiables, the initial passenge waiting time can also be calculated in this way. Thus, we have: IWT a b, pass-h, IWT a b, pass-h, IWT b a, 4 =65.00 pass-h and IWT b a, pass-h. The total passenge tansfe waiting time cost is calculated by using Eq. (5). The total passenge tansfe waiting time cost can be calculated by using the aveage tansfe waiting time of diffeent goups of tansfeing passenges. That is TWTi, j The in-vehicle passenge iding 3 5 i, jn time is calculated by Eq. (5). Thus, Eq. (4) yields PH i, j/ i, jn pass-h. Adding the fou passenge hou cost components yields the objective function: LD IWT i, j R i, jn Z TWT i, j PH i, j i, jn i, jn The eample poblem can now be fomulated as the following bi-objective nonlinea intege pogamming model: Min Z (37) Min Z FS N 0 (38) s.t. 3 (39) 4 5 (40) 33 Da (4) D a (4) Da (43) Da (44) 4 35 D b (45) Db (46) D b (47) Db (48) D a D b N 0 (49) 0 D(a), Db ( ) (50) i 0 o ; i,,3,4,5 (5) whee Eqs. (37) and (38) ae the two pincipal objective functions; Eqs. (39)-(40) ae the bundle depatue constaints; Eqs. (4)-(48) ae the DF bound constaints, which ae constucted by using the timetable infomation given in Table 4 and Table 5. Eqs. (4)-(48) ae constucted in the same way. Eqs. (49)-(50) ae the fleet size constaints. Eq. (5)

16 356 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) epesents the decision vaiable constaints. By applying the fleet size estimation method in Algoithm, the fleet size lowe bound and uppe bound ae obtained by the DFs shown in Fig. 5(a) and 5(b). Fig. 5(a) is based on the minimum numbe of depatues fo tip bundles, i.e.,, 4, which yields the fleet size lowe bound N L D a D b 4. Fig. 5(b) is based on the maimum numbe of depatues fo tip bundles, i.e., 3, 5, which yields the fleet size uppe bound NU Da Db Based on the calculated fleet size lowe bound and uppe bound, the oiginal biobjective nonlinea intege pogamming poblem is decomposed to thee one-objective intege pogamming poblems with known objective function Z FS N0 4,5,6. The esulting thee one-objective intege pogamming poblems can be easily solved, yielding the following thee sets of Paeto-efficient solutions: Z Set, 4 Z 4 ; Z 3.34 Set, 5 Z 5 ; Z Set 3 3, 5 Z 6. Table 3. The basic input data fo the numeical eample Route Aveage Time No. of Desied tavel span pa occ. d o time P (min) LD cost (pass.h) 30 7:00- a b 8:00 0 7:00- b a 8:00 Vaiable LD cost (pass.h) Numbe of depatues (Fequencies) q 3 q 4 q 5 q 6 Vaiable LD cost (pass.h) Vaiable LD cost (pass.h) 0 Vaiable 3 5 Table 4. Depatue and aival times at teminal nodes and node 3, and passenge eady fo depatue times at node 3 (oute a ) b Depatue teminal a Aival teminal b Vaiable 3 Depatue time 7:5 7:30 7:45 8:00 7: 7:4 7:36 7:48 8:00 7:0 7:0 7:30 7:40 7:50 8:00 Aival time 7:45 8:00 8:5 8:30 7:4 7:54 8:06 8:8 8:30 7:40 7:50 8:00 8:0 8:0 8:30 Aive time at node 7:5 7:40 7:55 8:0 7: 7:34 7:46 7:58 8:0 7:0 7:30 7:40 7:50 8:00 8:0 3 Depatue time 7:6 7:4 7:56 8: 7:3 7:35 7:47 7:59 8: 7: 7:3 7:4 7:5 8:0 8: Fom node 3 Passenge eady fo depatue time at node 3 7:6 7:4 7:56 8: 7:3 7:35 7:47 7:59 8: 7: 7:3 7:4 7:5 8:0 8: Table 5. Depatue and aival times at teminal nodes and node 3, and passenge eady fo depatue times at node 3 (oute b a) Depatue teminal b Aival teminal a Vaiable 4 5 Depatue time 7:5 7:30 7:45 8:00 7: 7:4 7:36 7:48 8:00 Aival time 7:35 7:50 8:05 8:0 7:3 7:44 7:56 8:08 8:0 Aive time at node 3 7: 7:37 7:5 8:07 7:9 7:3 7:43 7:55 8:07 Depatue time fom node 3 7:3 7:38 7:53 8:08 7:0 7:3 7:44 7:56 8:08 Passenge eady fo depatue time at node 3 7:3 7:38 7:53 8:08 7:0 7:3 7:44 7:56 8:08 In addition, by applying the SDT pocedue to modify the offset times of the timetable geneated by solution set, as shown in Fig. 6, the fleet size can be educed fom 4 to 3, but which esults in inceased passenge-hou costs. This Z poduces anothe Paeto-efficient solution: Set 4 Z 3. The fou Paeto-efficient solutions ae depicted in Fig. 7. PT schedules need to make a tadeoff between passengehou cost Z and vehicle fleet size Z displayed in a fleet-cost D space. With this gaphical infomation in hand,

17 Tao Liu et al. / Tanspotation Reseach Pocedia 3 (07) the PT schedules ae able to choose a desied solution o a desied set of solutions based on thei pefeences and pactical consideations, by taking account of both PT use and opeato inteests. Fig. 5. Deficit function fo estimating fleet size lowe bound and uppe bound of the eample netwok timetable Z 4 3 Min Z Min Z Z Fig. 6. Using SDT pocedue to educe the fleet size fom 4 to 3 Fig. 7. Paeto-efficient solutions: tadeoff between passengehou cost Z and vehicle fleet size Z of the eample poblem in 5. Application to the Spiess-Floian netwok The model and solution method descibed in pevious sections wee applied to the netwok used by Spiess and Floian (989), which is a classic netwok used in tansit assignment poblems. The netwok, depicted in Fig. 8, compises fou tansit outes and fou tansfe stops. Thee ae two tansfe stops, stop b and c, in this netwok. The ed oute opeates in a sepaated coido and the othe thee outes, outes, 3 and 4, opeate along the same coido. It is assumed that stop d is within a cental business distict (CBD), and anothe thee stops ae diected to stop d. In this netwok, tavels have the choice between a numbe of diffeent paths to each thei final destination. Fo eample, fo tips depating fom stop a and aiving at stop d, thee ae five diffeent paths fo eaching stop d, namely -a-d, -a-c-3-d, -a-c-4-d, -a-b-3-d, -a-b-3-c-4-d. The input data, which includes oute data shown in Table 6, oute segment data shown in Table 7 and O-D demand data shown in Table 8, ae taken fom Noekel et al. (06). The schedule hoizon is [7:00, 8:00] epesenting the moning peak hou. The weighting factos in objective function Z ae set as,.5, 3 and 4.. The weighting factos in the genealized tavel cost function, Eq. (5) ae set as 0,.5, 3, 4.5, 5 and 6 5 minutes pe tansfe stop. The policy fequencies (the minimum equied fequencies) fo all fou outes ae set as veh/h. The tansfe walking time fo all the tansfes made by taveles in the netwok ae set as minute. The desied occupancy fo all fou outes ae set as the capacity of the vehicles, i.e., do Cv 80. The vehicle oveload facto is set as.8. It is assumed that passenge O-D tavel demands ae evenly distibuted in the schedule hoizon. It is also assumed that all passenges elate to the VoT to the same degee.