Valid inequalities for the synchronization bus timetabling problem

Transcription

1 Vald nequaltes for the synchronzaton bus tmetablng problem P. Foulhoux a, O.J. Ibarra-Rojas b,1, S. Kedad-Sdhoum a, Y.A. Ros-Sols b, a Sorbonne Unverstés, Unversté Perre et Mare Cure, Laboratore LIP6 UMR 7606, 4 place Jusseu Pars, France b Unversdad Autónoma de Nuevo León, Av. Unversdad s/n, San Ncolás de los Garza, 66450, Mexco Abstract Bus transt network plannng s a complex process that s dvded nto several phases such as: lne plannng, tmetable generaton, vehcle schedulng, and crew schedulng. In ths work, we address the tmetable generaton whch conssts n schedulng the departure tmes for all trps of each bus lne. We focus on the Synchronzaton Bus Tmetablng Problem (SBTP) that favors passenger transfers and avods congeston of buses at common stops. These characterstcs and the embedded flexblty of the SBTP are crucal for many transt networks n Latn Amerca. A Mxed Integer lnear Program (MIP) was proposed n the lterature for the SBTP but t fals to solve real bus network nstances. We develop n ths paper four classes of vald nequaltes for ths MIP usng combnatoral propertes of the SBTP on the number of synchronzatons. Expermental results show that the enhanced MIP yelds hgh qualty solutons usng small computatonal tmes. In partcular, nstances based on real transt networks are solved wthn few mnutes wth a relatve devaton from the optmal soluton that s usually less than 2%. Keywords: Integer Programmng, Transportaton, Tmetablng, Vald Correspondng author Emal addresses: perre.foulhoux@lp6.fr (P. Foulhoux), obarrar@uc.cl (O.J. Ibarra-Rojas), safa.kedad-sdhoum@lp6.fr (S. Kedad-Sdhoum), yasmn.rossls@uanl.edu.mx (Y.A. Ros-Sols) 1 Actual afflaton: BRT-Center of Excellence, Department of Transport Engneerng and Logstcs, Pontfca Unversdad Católca de Chle Preprnt submtted to European Journal of Operatons Research June 24, 2014

2 Inequaltes, Urban Bus System. 1. Introducton Plannng and operatng a publc transportaton network n a relatvely bg cty (3 mllons people) s an enormous task f the am s to reduce the costs wthout degradng the qualty servce for the users. In Ceder (2007), the bus transt network problem s presented as a sequence of four man phases. The frst one s the lne plannng phase that defnes the routes, stops, and frequency for each bus lne n a specfc terrtory. Then, the tmetable generaton phase determnes the departure tmes of all the trps of the lnes to acheve a level of qualty servce. The thrd subproblem s the vehcle schedulng phase that assgns vehcles to trps assocated to bus lnes. Fnally, the crew schedulng phase defnes tasks assgned to drvers. The bus transt network problem s commonly tackled by sequental approaches. Therefore, obtanng hgh qualty solutons n short tme for each phase s an mportant ssue snce t s often necessary to terate several tmes to fnd a sutable soluton for the entre plannng process. Partcularly, tmetable generaton s a delcate task snce t has repercussons n the operatonal subproblems whch are vehcle and crew schedulng. In ths artcle, we focus on bus networks havng the followng characterstcs. The passengers of a lne have an estmate of ther watng tme at each stop rather than knowng the whole tmetable of the lne. Therefore, regularly spaced departure tmes for the trps of each lne are requred. Moreover, snce dfferent lnes frequently converge at specfc stops of the network, t s necessary to reduce congeston to mprove the qualty servce. In the sequel, a synchronzaton occurs when the tmetable permts the arrval of two buses at the same stop so that the two buses do not bunch or permts passenger transfers wthout long watng tmes. Therefore, the Synchronzaton Bus Tmetablng Problem (SBTP) conssts n determnng the departure tme of every bus trp n order to maxmze the synchronzaton of dfferent bus lnes of a network. Thus, the SBTP corresponds exactly to the tmetable generaton phase for networks that 2

3 appear n dfferent ctes of Latn Amerca, as for nstance, the network of the cty of Monterrey n Mexco whch has more than 300 bus lnes. Ibarra-Rojas & Ros-Sols (2012) have proved that the SBTP s NP-hard. They have proposed a mult-start terated local search algorthm and a MIP. We denote by SBTP MIP ths formulaton that wll be descrbed n Secton 2. However, the SBTP MIP solved usng a standard lnear solver (CPLEX 12.3) has not succeeded n effcently solvng real sze nstances wthn reasonable CPU tmes. Snce the number of dsruptons such as bus falures or crash accdents, s close to 10% of the vehcles used per day (n Monterrey), bus tmetables must be recomputed several tmes a day. Consequently, heurstc methods are usually used to quckly produce tmetables. However, these solvng methods do not ensure any guarantee on the soluton qualty. In ths artcle, we present an exact approach to solve the SBTP usng nteger programmng technques and show that hgh qualty solutons can be produced for real sze nstances wthn a few mnutes. Partcular cases of the SBTP have been consdered n the lterature. Ceder & Tal (2001) and Ceder (2011) have ntroduced a smpler verson of the SBTP where synchronzatons are defned as smultaneous arrvals at common stops subject to bound constrants on the the separaton tme between consecutve trps of the same lne. Constructve algorthms were developed for ths case. An extenson of Ceder s studes s presented by Erank (2004) where a synchronzaton s redefned as arrvals wthn a tme wndow. Lu et al. (2007) have reformulated the tmetablng problem presented by Ceder et al. (2001) and mplemented a Tabu search to solve the problem. Some closely related problems on transport networks have been studed usng nteger programmng approaches. We can quote the scholar bus schedulng problem ntroduced by Fügenschuh (2009) where startng tmes of schools and startng tmes of scholar buses must be synchronzed to mnmze the number of vehcles to transport all students. The authors have developed dfferent famles of vald nequaltes leadng to a branch-and-cut algorthm. Quadrfoglo et al. (2008) optmzed a weghted objectve functon based on vehcle resources and 3

4 qualty servce for a transport network. They defne logc constrants to reduce the feasble space whch allows to reduce up to 90% of the CPU tme when these cuts are added at the begnnng of a branch-and-bound algorthm. In ralway systems, tmetablng problems have been extensvely studed usng nteger programmng technques. In partcular, the perodc tmetablng case has specal structure (not present n the SBTP) that favors the desgn of vald nequaltes commonly mplemented n branch-and-cut algorthms (Cam et al. (2011); Gesemann (2002); Lebchen (2004, 2007); Lebchen & Möhrng (2002, 2007); Schröder & Solchenbach (2006)). Unfortunately, tmetablng problems n urban transport networks do not share the structure of perodc tmetablng. Comprehensve revews have been provded by Guhare & Hao (2008) and Cacchan & Toth (2012). An nequalty s sad to be vald for a MIP f every soluton of the MIP satsfes ths nequalty. Consequently, a vald nequalty can be added to a MIP n order to obtan a stronger formulaton (see Wolsey (1998)) that wll be easer to solve. Our man contrbutons are to defne four famles of vald nequaltes that wll be used for the SBTP MIP. The frst two famles of vald nequaltes bound the number of possble synchronzatons that can occur at a node for a specfc trp. The two other famles are obtaned from the prevous ones usng a generc lftng procedure. Addtonally, some nequaltes of the SBTP MIP are tghtened by lowerng some parameters. The enhanced SBTP MIP s then solved wth a standard lnear solver to obtan optmal solutons for most of the real sze nstances. Moreover, for all the nstances, solutons wth less than 3% of devaton from the optmum are found n less than fve mnutes. The rest of the paper s organzed as follows. In Secton 2, we brefly recall the SBTP MIP that we wll enhance wth our proposed vald nequaltes that are defned n Secton 3. A generc lftng procedure s ntroduced and s appled n order to produce another two famles of vald nequaltes. Secton 4 presents a way to tghten some nequaltes by adjustng some parameters. Expermental results for nstances based on a real transt network are presented n Secton 5 where we show the mpact of some combnaton of the vald nequaltes famles. 4

5 Fnally, conclusons and future research areas are addressed n Secton Problem defnton and mathematcal model In ths secton, we brefly defne the SBTP and present the SBTP MIP proposed by Ibarra-Rojas & Ros-Sols (2012) The synchronzaton bus tmetablng problem The SBTP schedules the departure tmes for all bus lne trps n order to maxmze the synchronzatons between buses. A formal defnton of the problem s gven n the followng. The type of bus network we are nterested n can be represented by a set of lnes denoted as I. As descrbed n Secton 1, the routes and the stops used by each bus lne as well as the frequency of each lne are determned before the tmetable generaton phase. We wll call a synchronzaton node, a specfc stop n the network where two lnes cross each other and where passenger transfers are needed or congeston of buses at common stops can happen. We denote by B the set of all synchronzaton nodes. Let T be the plannng horzon defned n tme unts. For example, T wll be 2 hours n rush perods n the mornng or 3 hours n valley perods n the afternoon. We wll denote by f the frequency of lne I, that s to say the number of trps that have to be scheduled wthn T for lne. We also assume that buses of the same lne wll run at regular speed and we denote by t b, I, the travel tme for a bus from an ntal node, called 110 depot, to a node b B. 115 A headway tme s the separaton tme between consecutve trps of the same lne. For example, f the frst (resp. second) trp of a lne starts at tme unt 3 (resp. 23), there s a headway tme of 20 tme unts between these two trps. Snce regularly spaced departure tmes are requred, each trp of a lne of I wll start every T f tme unts wth a flexblty of δ tme unts. Consequently, h = T f δ (resp. H = T f +δ ) s the mnmum (resp. maxmum) headway tme for a lne I. Moreover, the frst (resp. last) trp of lne must start before H 5

6 (resp. after T H ). These headway tmes guarantee that the entre plannng horzon s covered by the trps whch s one of the man dfferences between the addressed model and the ones studed by Ceder & Tal (2001) and Erank (2004). Consequently, a tmetable gven by the departure tmes X p of each trp T = [8: 00,8: 40] p {1,..., f } of every lne I wll satsfy X 1 H, h X p+1 X p H η = 10 for every {1,..., f 1}, T H X f T. Fgure 1 llustrates regularly spaced trps for a gven lne of frequency = 20% ( h = 2mn) f = 4 wth and wthout flexblty. T T T 1 f 2 f 3 f 4 (a) 0 H T H T (b) H T H T X 1 + h X 1 + H X 2 + h X 2 + H X 3 + h X 3 + H Fgure 1: Two dfferent tmetables for a gven lne : Case (a) wth regularly spaced departure tmes, Case (b) wth flexble headway tmes We recall that a synchronzaton at a stop between two bus trps occurs n two dfferent cases; when the dfference between ther arrval tmes s long enough to reduce congeston of buses belongng to dfferent lnes at common stops n one hand or allow passenger transfers wthout long watng tmes n another hand. A transfer node s a partcular synchronzaton node n the network where passengers may transfer from one lne to some lne j. To allow well tmed passenger transfers at node b, the dfference between two bus arrvals at ths node has to be lower (resp. greater) than a specfc value denoted W b (resp. w b ). A congeston node s a partcular synchronzaton node n the network where two buses (of dfferent lnes) arrvng smultaneously at ths node wll 6

7 share a segment of ther respectve routes. Consequently, to reduce congeston of buses at node b, the length of the tme nterval between two bus arrvals at node b has to be greater than a specfc value denoted w b. In ths case, t s convenent to consder a maxmum separaton tme W b between the bus arrvals at node b such that W b w b < h j. In the bass of the above, a synchronzaton at node b for two bus trps happens f the dfference of ther arrval tmes s n the so-called watng tme wndow [w b, W b ]. For each lne I, let us denote by I() the set of lnes that share a synchronzaton node wth lne. Notce that for, j I, j I() does not mply I(j). Indeed, t could be desrable n some stuatons to enforce the passenger transfers only from lne to lne j. Therefore, the SBTP conssts n determnng the departure tmes Xp of each trp p {1,..., f } of every lne I such that the headway tme bounds are satsfed, X1 H, h Xp+1 Xp H for every I, T H Xf T, wth the objectve of maxmzng the number of pars of synchronzed trps, that s, to maxmze the number of pars of trps p {1,..., f } and q {1,..., f j } such that w b (Xq j + t j b ) (X p + t b ) W b s satsfed, for I, j I(). Fgure 2 llustrates the two types of synchronzaton nodes. Case (1) represents a congeston node b where lnes and j (wth h j = 8) converge and then share a segment of ther routes. The numbers by the sde of node b represent the arrval tmes of a par of trps of these lnes. The mnmum and maxmum separaton tmes to reduce congeston at ths node b are w b = 6 and W b = 12. Case (2) represents a node b where passengers would lke to go from one trp of lne to some trp of lne j. In ths case, 5 tme unts s the maxmum desred passenger watng tme and 2 tme unts s the estmated requred tme for the transfer. Then, the watng tme wndow s gven by [w b, W b ] = [2, 7] Mxed nteger program 165 We consder n the followng an nstance of SBTP and we now recall wth the 0-1 MIP of the problem already descrbed by Ibarra-Rojas & Ros-Sols (2012). For a gven a lne I, we wll denote by (p, ), p { 1,..., f }, the 7

8 8:35 am h j j = 8 b 8:20 am 8:14 am j 8:38 am b (1) [, ] = [ 6,12 ] (2) [, ] = [ 2, 7] w b W b w b W b Fgure 2: Synchronzaton nodes: congeston node (Case 1), transfer (Case 2). 170 p th trp of lne and we assocate to (p, ) the real varable X p that represents ts departure tme. For a par of lnes (, j), j I(), the set B j s the set of all the synchronzaton nodes shared by and j. The bnary decson varable Y j pqb s equal to 1, f and only f, trp (p, ) arrves frst at node b and f trps (p, ) and (q, j) nduce a synchronzaton at node b. In the sequel, we wll denote by (X, Y ) a soluton of an SBTP nstance. Consderng the prevous parameters and decson varables, the SBTP MIP s gven by max I f f j j I() b B j p=1 q=1 Y j pqb s.t. X 1 H I (1) T H X f T I (2) h Xp+1 Xp H I, p = 1,..., f 1(3) ( ) Xq j + t j b ( ( ) Xp + tb) wb + M 1 Y j pqb I, j I(), b B j, ( ) Xq j + t j b ( ( ) Xp + tb) Wb + M 1 Y j pqb p = 1,..., f, q = 1,...,(4) f j I, j I(), b B j, p = 1,..., f, q = 1,...,(5) f j X p R, Y j pqb {0, 1} I, j I(), b Bj, p = 1,..., f, q = 1,...,(6) f j 8

9 175 The objectve functon maxmzes the total number of synchronzatons. Constrants (1) and (2) guarantee that the entre plannng horzon s covered by the trps. Constrants (3) mpose that consecutve trps of lne must happens wth a mnmum (resp. maxmum) headway tme h (resp. H ). Remark that the arrval tme of trp (p, ) at node b s Xp + t b. Hence Constrants (4) and (5) actvate the synchronzaton varables Y j 180 pqb f the dfference between the arrval tmes of (p, ) and (q, j) at node b s wthn [w b, W b ] wth (p, ) arrvng frst at 185 node b. M s a large constant whose value wll be defned n Secton 4. As mentoned by Ibarra-Rojas & Ros-Sols (2012), the SBTP MIP cannot be used to solve real nstances of the SBTP usng a standard lnear programmng solver. The man contrbutons of ths paper s to strengthen SBTP MIP by ntroducng vald nequaltes that allow to effcently solve the real sze nstances of the SBTP. 3. Vald nequaltes To obtan tghter formulatons for the SBTP, we add vald nequaltes to the SBTP MIP before ts resoluton by a lnear programmng solver. In fact, addng vald nequaltes permts to cut fractonal solutons of the lnear relaxatons of nteger programs or to cut non-optmal feasble solutons (Wolsey (1998); Nemhauser & Wolsey (1999)). In the followng, we ntroduce two famles of vald nequaltes for the synchronzaton bus tmetablng problem obtaned from the headway tme parameters and the propagaton of Constrants (1), (2), and (3) Synchronzaton nequaltes 200 We can take advantage of the headway tme parameters to defne a famly of vald nequaltes for each trp to be synchronzed. Let us consder two lnes and j to be synchronzed at a gven node b such that the mnmum headway tme h j of lne j s greater than the length of the watng tme wndow of node b,.e., h j > W b w b. If a trp (p, ) synchronzes wth another trp (q, j), the 9

10 205 synchronzaton of (p, ) wth trps (q 1, j) or (q +1, j) s mpossble snce there would not be enough tme unts to ensure soluton feasblty. Generalzng the prevous dea, we obtan the followng result. Lemma 1. Let and j be two dstnct lnes wth j I() and b B j, the maxmum number of synchronzatons between one trp of lne and all the Wb w b trps of lne j s 1 +. h j Proof. Let us suppose that there exsts a feasble soluton ( X, Ỹ ) of the SBTP and a trp (p, ) such that there exst r > 1 + W b w b h j trps q1 < q 2 < < q r of lne j that may synchronze wth trp (p, ). We wll show that t s mpossble to schedule these synchronzed trps because of the mposed mnmal headway tmes between trps. The arrval tmes of these trps are wthn the [ feasble synchronzaton tme wndow X p + t b + w b, X p + t b b] + W. Therefore, the dfference between the arrval tmes X j q r +t b and X j q 1 +t b of trps (q 1, j) and (q r, j) must be less than W b w b, consequently we obtan X j q r X j q 1 W b w b. However, snce soluton ( X, Ỹ ) corresponds to a regularly spaced schedule, we have that X q j l+1 X q j l h j. We then have ( X q j r X q j 1 (r 1)h j Wb w b 1 + whch s a contradcton. h j ) h j > ( 1 + ( )) Wb w b h j 1 h j = W b w b Usng Lemma 1, we derve the followng nequaltes that wll be called syn- chronzaton nequaltes. f j q=1 f p=1 Y j pqb 1 + Wb w b h j Y j pqb 1 + Wb w b h I, j I(), b B j, p {1,..., f } (7) I, j I(), b B j, q {1,..., f j }. (8) Let us consder two lnes and j wth j I(), a node b B j, and a trp (p, ). j f Gven a soluton ( X, Ỹ ) of the SBTP, we can remark that s exactly q=1 Ỹ j pqb the number of synchronzaton between trps of lne j and trp (p, ) at node b. Consequently, we have the followng theorem. 10

11 Theorem 1. Synchronzaton nequaltes (7) and (8) are vald for the SBTP MIP Headway nequaltes Usng smlar deas, we can devse the followng class of nequaltes. Y j pqb + j f q =q+1 q 1 Y j pqb + Y j pq b + q =1 f p =p+1 p 1 Y j pq b + p =1 Y j p qb 1 + Wb w b mn(h, h j ) Y j p qb 1 + Wb w b mn(h, h j ) I, j I(), b B j, (9) p {1,..., f }, q {1,..., f j } I, j I(), b B j, (10) p {1,..., f }, q {1,..., f j }. 220 We wll denote these nequaltes as headway nequaltes as they depend on the value of the desred mnmum headway tme between trps. We then have the followng result. Theorem 2. Headway nequaltes (9) and (10) are vald for the SBTP MIP. Proof. We wll consder an nequalty (9) correspondng to the two trps (p, ), (q, j) and a node b B j (the proof for nequaltes (10) s analogous). Let ( X, Ỹ ) be a feasble soluton of the SBTP. We consder the two quanttes c and r defned as follows r = f j q =q+1 Ỹ j pq b and c = If r = 0 then from nequalty (8) we know that f Ỹ j pqb + c p =1 f p =p+1 Ỹ j p qb. Ỹ j p qb 1 + Wb w b Wb w b h 1 + mn(h, h j. ) Smlarly, f c = 0, from nequalty (7), we obtan that j f Ỹ j pqb + r q =1 Ỹ j pq b 1 + Wb w b Wb w b h j 1 + mn(h, h j. ) Let us now suppose that both r 1 and c 1. Snce r 1 trp (p, ) synchronzes wth r trps among trps q + 1,..., f j of lne j. Let q 11

12 be the frst of these trps that s synchronzed wth (p, ). Consequently, arrval tmes of trps q, q + 1,..., q + (r 1) at node b are wthn the tme [ wndow X p + t b + w b, X p + t b b] + W. Snce the mnmal headway tme between two trps of lne j s h j, the arrval tme of trp (q, j) at node b satsfes X j q + tj b X p + t b + W b (r 1)h j. Snce X j q+1 X j q, we fnally obtan X j q+1 + tj b X p + t b + W b (r 1)h j. Smlarly, snce c 1 there are c trps among p + 1,..., f of lne that synchronzes wth (q, j). Let p be the last of these trps that synchronzed wth (q, j), thus we know that X q j + t j b X p + t b + w b. Moreover, snce trps p (c 1),..., p are synchronzed wth (q, j) and because of the mposed mnmum headway tmes of lne, we know that X q j + t j b X p (c 1) + t b + w b + (c 1)h. Snce X p (c 1) X p+1, we obtan X q j + t j b X p+1 + t b + w b + (c 1)h. Because of the mnmum headway tmes between trps of lne j, we then get h j ( X j q+1 + tj b ) ( X j q + t j b ) X p X p+1 + W b w b (r 1)h j (c 1)h. And, because of the mnmum headway tme between trps (p, ) and (p + 1, ), we have X p+1 X p h. Then, we obtan h j h + W b w b (r 1)h j (c 1)h, whch then becomes rh j +ch W b w b. W.l.o.g. we can suppose that h j h. We then obtan Thus we obtan r + c r + c h h j W b w b h j. Ỹ j pqb + r + c 1 + W b w b h j whch proves the valdty of nequalty (9). 12

13 Lftng procedure In ths secton, we present a generc lftng method that permts to compute new vald nequaltes from the prevous ones. 230 Let us frst ntroduce a useful notaton. Let ξ be the set of all 5-tuples (, j, p, q, b) wth I, j I(), p {1,..., f }, q {1,..., f j } and b B j. Notce that set ξ s n a one-to-one correspondence wth the set of Y s varables, that s to say, each 5-tuple (, j, p, q, b) ξ corresponds to a potental synchronzaton between trps (p, ) and (q, j) at node b so that trp (p, ) arrves frst at node b. Let us consder the followng nequalty: (,j,p,q,b) E Y j pqb γ (11) where E ξ and γ s an upper bound over the number of synchronzatons n E. The nequaltes of type (11) are clearly vald. We note that synchronzaton and headway nequaltes belong to ths class. Gven ( 0, j 0, p 0, q 0, b 0 ) ξ, we defne E 0j0 p 0q 0b 0 as a set of 5-tuples (, j, p, q, b) of E so that trps (p, ) and (q, j) cannot be synchronzed at node b f trps (p 0, 0 ) and (q 0, j 0 ) are synchronzed at node b 0 and trp (p 0, 0 ) arrves frst at node b 0, that s to say that for a gven soluton ( mples Ỹ j pqb = 0 for all (, j, p, q, b) E0j0 p 0q 0b 0. X, Ỹ ) of the SBTP, Ỹ 0j0 p 0q 0b 0 = 1 Then nequalty (11) can be lfted to create the followng nequalty j Y (,j,p,q,b) E E 0 j 0 p 0 q 0 b 0 ( pqb γ 1 Y 0j0 p 0q 0b 0 ) (12) whch s clearly vald. In the followng, we present a generc lftng procedure to obtan nequaltes of type (12). Ths procedure wll be appled to both synchronzaton and headway nequaltes n order to obtan lfted synchronzaton nequaltes and lfted headway nequaltes. Remark that the set of varables Y 245 of such a lfted nequalty s related to dfferent lnes whle the set of varables of a headway or a synchronzaton nequalty s related to only two lnes. To compute the lfted nequaltes, gven a 5-tuple (, j, p, q, b), we wll need to fnd a lst of 5-tuples that can not be synchronzed f (, j, p, q, b) s synchronzed. 13

14 To acheve ths, we wll compare the feasble departure tme wndow of the potentally synchronzed trps, that s to say the tme wndow durng whch a trp has to start. Some of the results ntroduced by Ibarra-Rojas & Ros-Sols (2012) for a preprocessng stage, can be used to devse these tme wndows. Gven trp (p, ) wth I and p { 1,..., f } [ ], we denote by Dp = D p, D p a feasble departure tme wndow of trp (p, ). Ibarra-Rojas & Ros-Sols (2012) have been proved that [ ] [ ] Dp = D p, D p d p, d d p = max { (p 1) h, T ( f (p 1) ) H } p wth d p = mn { ph, T ( f p ) h }. (13) [ ] The tme wndow d p, d p consttutes a generc feasble departure tme wndow. As t wll turn out, when two trps synchronze, t wll be possble to tghten ths wndow. Ths can be done usng logcal nferences that are gven by the followng theorem whch gathers several results gven by Ibarra-Rojas & Ros- Sols (2012). Theorem 3. Let (, j, p, q, b) ξ and D p = [ ] [ ] D p, D p (resp. Dq j = D j q, Dj q ) be a feasble departure tme wndow of trp (p, ) (resp. (q, j)). By settng [ ] [ ] [α, β] = D p + t b + w b, D p + t b + W b D j q + t j b, Dj q + t j b, we obtan that ) (q, j) s synchronzed wth trp (p, ) arrvng frst at b f and only f [α, β], ) and, n that case, [α t j b, β tj b ] s a tghter feasble departure tme wndow for trp (q, j) and Dp [α W b t j b, β w b t j b ] s a tghter feasble departure tme wndow for trp (p, ). An example of how to use the prevous theorem s the followng. Consder [ ] [ ] feasble departure tmes D p, D p = [15, 20] and D p, D p = [18, 25] for trps (p, ) and (q, j), respectvely. Assume we have the parameters values t b = 10, t j b = 22, and [w b, W b ] = [3, 8]. Then, the arrval of trp (q, j) at node b s wthn the nterval [ , ] = [40, 47] and the arrval of trp (p, ) at node b s wthn [ , ] = [25, 30]. If t s possble to synchronze trp (p, ) wth trp (q, j), arrval tme of trp (q, j) at node b must be wthn 14

15 270 [25 + 3, ] = [28, 38] whch s mpossble snce [28, 38] [40, 47] =. [ ] Knowng a tghter feasble departure tme wndow Dp = D p, D p for a gven trp (p, ), a tghter departure tme wndow D p, for all trps p p, can be nferred usng the headway tmes between trps of lne. Ths can be done usng the procedure P ropagate(d p). Algorthm 1 P ropagate(d p) 1: for p = 1 to p 1 do 2: D p := D p [ D p + (p p)h, D p + (p p)h ] 3: end for 4: for p = p + 1 to f do 5: D p := D p [ D p (p p )H, D p (p p )h ] 6: end for Algorthm 2, called Generc Lftng, shows the steps to generate lfted nequaltes. The general dea to obtan these nequaltes s to consder a 5-tuple ( 0, j 0, p 0, q 0, b 0 ) ξ and see what mplcatons arse f the correspondng synchronzaton s set, that s to say when Y 0j0 p 0q 0b 0 = 1. Step 2 conssts n computng the feasble departure tme wndows of every trp from Formula (13). Then, assumng that ( 0, j 0, p 0, q 0, b 0 ) s synchronzed at node b 0, we compute tghter feasble departure tme wndows D 0 p 0 and D j0 q 0 (Steps 3 5) usng Theorem 3). Usng procedure P ropagate, we then update the departure tme wndows for the rest of the trps of lnes 0 and j 0 (Step 6). We then apply a specfc procedure for every nequaltes (,j,p,q,b) E Y j pqb γ of type (11): we determne a set E of 5-tuples (, j, p, q, b) of E that cannot be synchronzed and we then can create the correspondng lfted nequaltes. Unfortunately, Steps 7-9 must be dedcated to each of the two consdered type of nequaltes n order to consder sets E wth known upper bounds. In fact, lnes 7-10 have to be replaced by Algorthm 4 for synchronzaton nequaltes and Algorthm 5 for headway nequaltes. In order to fnd whch trps cannot be synchronzed, we ntroduce another procedure (Algorthm 3), called Test synchronzaton. Ths procedure deter- 15

16 Algorthm 2 Generc Lftng 1: for each ( 0, j 0, p 0, q 0, b 0 ) ξ do [ ] 2: Dp := d p, d p for every trp (p, ) 3: [α, β] := [ D 0 p 0 + t 0 b 0 + w b0, D 0 p 0 + t 0 b 0 + W b0 ] 4: Dq j0 0 := [α t j0 b 0, β t j0 b 0 ] 5: Dp 0 0 := D 0 p 0b 0 [α W b0 t j0 b 0, β w b0 t j0 b 0 ] 6: P ropagate(d 0 p 0 ) and P ropagate(d j0 q 0 ) [ ] D j0 q 0 + t j0 b 0, D j0 q 0 + t j0 b 0 7: for each nequalty of type (11) (,j,p,q,b) E Y j pqb γ do 8: Fnd a set E of 5-tuples (, j, p, q, b) of E that cannot be synchronzed 9: Create nequalty ( ) (,j,p,q,b) E Y j pqb γ 1 Y 0j0 p 0q 0b 0 10: end for 11: end for 290 mnes f trp (p, ) cannot be synchronzed wth trp (q, j) due to the fact that ( 0, j 0, p 0, q 0, b 0 ) s synchronzed. Usng Theorem 3), lnes 2-5 tests f (, j, p, q, b) s a potental synchronzaton and then, f ths synchronzaton becomes mpossble after the update of the departure tme wndows. Algorthm 3 Test synchronzaton(, j, p, q, b) [ ] [ ] 1: [α, β] := D p + t b + w b, D p + t b + W b D j q + t j b, Dj q + t j b 2: f ([α, β] ) then 3: D j q := [α t j b, β tj b ] 4: Dp := Dp [α W b t j b, β w b t j b [ ] [ ] ] 5: f D p + t b + w b, D p + t b + W b D j q + t j b, Dj q + t j b 6: Return False 7: end f 8: end f 9: Return True = then 295 For creatng lfted synchronzaton nequaltes, Algorthm 4 enumerates every synchronzaton nequalty of type (7) whch can be affected by the mod- fcaton of lnes 0 and j 0 (Steps 1-4). Each potental trp q s then tested by 16

17 300 Algorthm 3 to know f (, j, p, q, b) becomes mpossble and we then create a lfted synchronzaton nequalty from the set E whch represents the set of mpossble synchronzatons. The procedure that generates the lfted synchronzaton nequaltes from the synchronzaton nequaltes (8) s analogous to Algorthm 4. Algorthm 4 Create lfted synchronzaton nequaltes( 0, j 0, p 0, q 0, b 0 ) 1: f ({, j} { 0, j 0 } and (, j, b) ( 0, j 0, b 0 )) then 2: for ( p = 1 to f ) do 3: E := 4: for ( q = 1 to f j) do 5: f T est synchronzaton(, j, p, q, b)= False then 6: E := E {(, j, p, q, b)} 7: end f 8: end for 9: Create Y j pqb ( 1 + W b w b ) ( ) h 1 Y 0j0 j p 0jq 0b 0 (,j,p,q,b) E 10: end for 11: end f Usng smlar deas, Algorthm 5 computes lfted headway nequaltes from the headway nequaltes of type (9). A smlar algorthm can be devsed for the headway nequaltes of type (10) Tghtenng Constrants (4) and (5) An mportant aspect n nteger programmng s to compute tght parameters to reduce the computatonal tme of solvng the lnear relaxaton of nteger programs. In a smlar way that we use feasble departure, arrval, and synchronzaton tme wndows to defne the lftng nequaltes, we can use them to bound bg M parameters for Constrants (4) and (5) of the SBTP MIP. We can recall that the earlest arrval tme of trp (q, j) at node b s d j q + t j b and the latest arrval tme of trp (p, ) at node b s d p + t b. Therefore, the 17

18 Algorthm 5 Create lfted headway nequaltes( 0, j 0, p 0, q 0, b 0 ) 1: f ({, j} { 0, j 0 } and (, j, b) ( 0, j 0, b 0 )) then 2: for ( p = 1 to f and q = 1 to f j) do 3: E := 4: for ( q q to f j) do 5: f T est synchronzaton(, j, p, q, b)= False then 6: E := E {(, j, p, q, b)} 7: end f 8: end for 9: for ( p > p to f ) do 10: f T est synchronzaton(, j, p, q, b)= False then 11: E := E {(, j, p, q, b)} 12: end f 13: end for 14: Create 15: end for 16: end f (,j,p,q,b) E Y j pqb (1 + ) ( ) Wb w b mn(h,h j ) 1 Y 0j0 p 0q 0b 0 18

19 mnmum dfference of arrval tmes between trps (q, j) and (p, ) at node b s d p + t b dj q tj b. Consequently, gven a soluton ( X, Ỹ ) of the SBTP, we know that ( ) ( ) Xj qb + tj b X pb + t b d p + t b d j q t j b. Smlarly, we can remark that the maxmum dfference of arrval tmes between trps (q, j) and (p, ) at node b satsfes the followng nequalty ( ) ( ) Xj qb + tj b X pb + t b d j q + t j b d p tj b. 315 In the bass of the above, for Constrants (4) and (5) correspondng to (, j, p, q, b) ξ, we can replace M by m j pqb = d p + t b dj q t j b d j q + t j b d p tj b respectvely. and M j pqb = 5. Expermental results 320 In ths secton, some computatonal results are presented usng the nteger lnear solver CPLEX 12.3 on a Mac OS X wth an Intel Core 2 Duo 3.06 GHz processor and 4 GB RAM. The computatonal effort to generate the vald nequaltes for the SBTP MIP s neglgble (less than one second) for the nstances we consder. We compare several combnatons of the proposed vald nequaltes wthn the SBTP MIP n order to solve effcently real SBTP nstances Instances We use the nstance generaton scheme proposed by Ibarra-Rojas & Ros- Sols (2012) as well as some nformaton provded by a company of Monterrey s transt network. All the nstances have the followng common characterstcs: a plannng perod of T = 240 mnutes; the frequency f for each lne s randomly generated n [13,18]; the travel tme t b from the depot to synchronzaton node b for each lne s randomly generated n [20,60]; the mnmum (resp. maxmum) watng tme for each synchronzaton node b s randomly generated n [3,5] (resp. [9,12]); 19

20 fnally, the number of dfferent pars of lnes to synchronze at each node b s randomly generated between 1 and 7. The derved nstances are grouped nto 9 types dependng on the followng parameters: the number I of lnes, the number B of synchronzaton nodes, and the flexblty parameter δ. Ths latter parameter s randomly generated so that the rato between δ and T f belongs to a flexblty nterval denoted by [F mn, F max ]. As t s hghlghted by Ibarra-Rojas & Ros-Sols (2012), for a lnear solver, the larger F mn s, the more ntractable s the related nstance usng the basc SBTP MIP. Accordng to the planners of Monterrey s transt network, t s reasonable to consder ether [10,20] or [25,35] for [F mn, F max ]; and a number of lnes 5 tmes the number of synchronzaton nodes. We randomly generate 10 nstances for each of the 9 nstance types (a total of 90 nstances) to analyze the algorthm performance. The name of the nstance types and ther parameters are summarzed n Table 1. Notce that nstances of type A9 have a very large number of synchronzaton nodes. In fact, we wll use these non-realstc nstances to show the lmts of our soluton approach. Inst. I B [F mn, F max ] n % A [10,20] A [25,35] A [10,20] A [25,35] A [10,20] A [25,35] A [10,20] A [25,35] A [25,35] Table 1: Instance characterstcs 20

21 5.2. Solvng to optmalty Table 2 summarzes the expermental results obtaned for dfferent combnatons of vald nequaltes wthn a tme lmt of one hour. Prelmnary experments were conducted to dentfy the promsng combnatons of vald nequaltes. The followng 4 dfferent combnatons have been kept, that we wll denote (1)-(6), (1)-(10), (1)-(8)+lft and (1)-(8)+lft 20. Formulaton (1)-(6) corresponds to the basc SBTP MIP; (1)-(10) to the SBTP MIP enhanced wth synchronzaton and headway nequaltes; (1)-(8)+lf t to the SBTP MIP wth synchronzaton nequaltes and all the lfted headway nequaltes obtaned usng Algorthm 5. Fnally (1)-(8)+lft 20 s smlar to the latter formulaton where only 20% of the generated lfted headway nequaltes are ncluded. Indeed, the nequaltes mantaned n the formulaton correspond to those havng the hghest number of synchronzaton varables n ther left hand sdes. Ths rato have been set after prelmnary experments: ths value corresponds to the best compromse between the qualty of the soluton and the computaton tme. For each of the 4 combnatons, Table 2 provdes the followng entres: #Nodes : the average number of nodes n the branchng tree, Rel : the average value of the objectve functon at the root node, Gap : the average relatve error between the best obtaned value and the best obtaned lower bound, CPU : the average computatonal tme n seconds In the frst block of rows of Table 2, we can see that the orgnal formulaton of SBTP s ntractable snce we obtan large gaps n one hour of computaton tme. Notce that no feasble soluton can be found wthn ths tme lmt for nstances A9. The second block of rows of Table 2 corresponds to the case where all the vald nequaltes (1)-(10) have been added to the orgnal formulaton. We can see that addng these nequaltes to the orgnal formulaton leads to very good solutons wthn reasonable CPU tmes, the gaps are ndeed sgnfcantly reduced 21

22 (1)-(6) (1)-(10) (1)-(8)+lft (1)-(8)+lft20 #Node Rel Gap CPU #Node Rel Gap CPU #Node Rel Gap CPU #Node Rel Gap CPU A % % % % 45.4 A % % % % A % % % % A % % % % 22 A % % % % 3254 A % % % % A % % % % A % % % % A % Table 2: Results for the nstance types A1-A9 usng CPLEX 12.3 and dfferent combnatons of vald nequaltes. 22

23 n comparson wth the ones obtaned for the orgnal formulaton. Instances A1- A8 are almost all solved to optmalty n less than 48 mnutes. More precsely, all nstances A1-A2, A4 and A6-A8 have been solved to optmalty n less than 30 mnutes and only few A3 and A5 nstances have not been solved to optmalty wthn one hour. The average number of nodes n the tree search decreases sgnfcantly as well as the average value of the objectve functon at the root node. We can notce that for the A9 nstances, even f the gap s mportant, at least a feasble soluton s found. If we look at the results provded n Table 2 for the thrd block correspondng to the case where all the lfted headway nequaltes are added to (1)-(8) formulaton, we can see that the average values of the objectve functon at the root node are mproved regardng the orgnal formulaton but are comparable to (1)-(10) wth a greater computatonal effort. If we compare the results of the two last blocks of Table 2 correspondng to the formulaton ncludng lfted headway nequaltes, we can notce that keepng 20% of the generated lfted nequaltes leads to very good results both n terms of gaps and CPU tmes regardng (1)-(8)+lf t formulaton. Partcularly, the gaps obtaned for A7-A8 nstances are better wth lower computatonal tmes. We can see that a greater number of nodes are explored wth no ncrease of CPU tmes leadng to good solutons. To sum up ths frst set of experments, we can conclude that the vald nequaltes proposed for the SBTP MIP are useful, they sgnfcantly mprove the gaps and the values of the objectve functon at the root node. However, a huge computatonal effort s requred. The lfted nequaltes can also be useful to reach the same effcency. In that case however, t can be trcky to fnd the relevant and adequate set of nequaltes to mantan. To enhance the effcency, a perspectve of ths work s to develop a Branch-and-Cut solvng method for the SBTP. 23

24 6. Solvng up to 3% Table 3 provdes a comparson between the best soluton found usng the (1)- (10) formulaton wth a stoppng crteron of 3% of relatve gap and a Mult-start Iterated Local Search (MILS) proposed by Ibarra-Rojas & Ros-Sols (2012) for the same problem. The dea of ths MILS s to use constrant propagaton shown n Algorthm 1 to defne and explore the search space. In partcular, constrant propagaton s used to desgn randomzed constructve algorthms and mplement lne departure tmes shftng wthn the feasble space to nduce more synchronzatons. (1)-(10) MILS Gap Tme Gap Tme A1 1.9% % 0.4 A2 1.3% % 0.5 A3 2.6% % 1.4 A4 0.8% % 1.8 A5 2.1% % 7.0 A6 1.2% % 8.6 A7 1.7% % 26.7 A8 1.7% % 29.4 Table 3: Results of solvng nstances A1-A8 mplementng nequaltes (1)-(10) wth a stoppng crteron of 3% of relatve gap versus Mult-start Iterated Local Search (MILS) proposed by Ibarra-Rojas & Ros-Sols (2012). 415 To get the gap for the MILS mplementaton, we compute the devaton of the feasble soluton obtaned by MILS and the best upper bound found by CPLEX The exact approach based on (1)-(10) formulaton leads to the best results for all nstance types n terms of relatve gaps. Although, MILS s more effcent consderng the executon tme, our exact approach reaches the stoppng crteron n less than 2 mnutes for most of the nstances. 24

25 Conclusons We defne an exact soluton approach for the NP-hard synchronzaton bus tmetablng problem. Ths problem determnes regular spaced departure tme for all the trps of each lne to allow well tmed passenger transfers and reduce congeston of buses belongng to dfferent lnes at common stops. The flexblty n the SBTP gven by headway tme bounds (nstead of a fxed headway tme) allows us to defne dfferent famles of vald nequaltes to tghten the SBTP MIP. Our soluton approach s to strengthen the SBTP MIP usng our proposed vald nequaltes and mplement an nteger lnear solver. Numercal results show that hgh qualty solutons (optmal for most cases) can be found for large nstances of SBTP n a short tme. Moreover, there s a fast convergence of our approach to solutons wth less than 3% of relatve devaton from the optmal soluton n seconds. These results are very nterestng f we want an ntegrated approach wth the vehcle schedulng snce we have a stronger formulaton that can be used n an sequental approach or n a ntegraton wth other subproblems of the entre transt network plannng problem. Although, we obtan hgh qualty solutons n a short tme, there are nterestng research lnes such as determnng the dmenson of the vald nequaltes proposed n ths study. A natural mprovement for ths work s to develop a polyhedral study along wth a branch-and-cut approach to handle unsolvable nstances. Indeed, the lftng procedure that does not show sgnfcant emprcal results n ths work may be of partcular nterest n a branch-and-cut. Integraton of SBTP wth other subproblems of transt network plannng such as vehcle and crew schedulng s a challengng research area. Moreover, the generalzaton of SBTP to cover the entre day nstead of short plannng perods s needed to defne accurate ntegrated approaches. 25

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