A Scheduling Approach for Autonomous Vehicle Sequencing Problem at Multi-Intersections

Transcription

1 Internatonal ournal of Operatons Research Internatonal ournal of Operatons Research, Vol 8, No 1, (2011) A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons e Yan 1,, Mahoub Drd 1, and Abdellah El Moudn 1 1 Laboratore Systèmes et Transports, Unversté de Technologe Belfort-Montbelard, Belfort , rance Receved uly 2010; Revsed December 2010, anuary 2011; Accepted anuary 2011 Abstract Ths paper addresses the problem of schedulng autonomous vehcles to traverse several adacent ntersectons wth the consderaton of Vehcle to Infrastructure (V2I) communcaton technology Due to the combnatoral nature of the control strategy, we decentralze the problem nto several solated ntersecton control problems and model each of them as a specal sngle machne schedulng problem wth famly obs that can be processed smultaneously Chan constrants and release dates are also nvolved Two frequently used traffc control measures average queue length and average vehcle watng tme are modeled as the number of late obs and total tardness, respectvely Effcent Branch and Bound algorthms are proposed for two obectves A heurstc that serves as ntal upper bound s presented It can also be used ndependently Computatonal experments and smulatons show the performance gan obtaned by usng the proposed schemes Keywords Intellgent vehcle control, sngle machne schedulng, release date, branch and bound, heurstc 1 INTRODUCTION Transportaton has always been a crucal aspect of human cvlzaton, but t s only snce the second half of the 20th century that the phenomenon of traffc congeston has become predomnant due to the rapd ncrease n the number of vehcles and n the transportaton demand n vrtually all transportaton modes (Hall 2003) Especally over the last decade, the traffc congeston attracted extensve attenton because of the worldwde energy crss and envronmental concerns Snce n modern ctes, the congestons are usually caused by an solated ntersecton wth hgh volume of traffc flow or several adacent ntersectons located n dense street networks, traffc control strateges for solated ntersectons and ntersecton networks are mostly studed n lterature to reduce the congestons In general, the common means of traffc control n modern ctes s the control by traffc sgnals They made t possble to solve conflcts between traffc flows at ntersectons (Gubernc, Senborn et al 2008) The traffc sgnal control for ntersectons usually falls nto two basc categores: pre-tmed control strategy, whch s also called fxed cycle control, and the sem/fully traffc actuated control The frst category ams at calculatng and dvdng a fxed cycle tme for each ntersecton or example, the TRANSYT (Traffc Network Study Tool) system (Robertson 1969) The second strategy, the traffc actuated control, whch s also known as the traffc-responsve strategy, attracts more and more attenton snce the 80s of last century Ths control strategy can change the dvde of cycle tme based on the real-tme measurement lke nductve loops The most famous system of ths knd s SCOOT (Splt Cycle Offset Optmzaton Technque) (Hunt 1982) However, the two strateges are both based on the estmaton of traffc flow rates Snce the flow rate s a contnuous varable that needs a perod of tme to be estmated, there are always bg devatons between the last computed flow rates and actual vehcle arrvals Ths encourages researchers to fnd new approaches for the traffc control at ntersecton Wth the sgnfcant progress acheved n the development of computer and telecommuncaton technologes durng recent decades, more advanced control approaches used n traffc control systems were proposed under the framework of Intellgent Transportaton System (ITS) or example, the development of wreless communcaton technology lke W-, WMax, 3G, and blue tooth enabled the Vehcle to Vehcle communcaton (V2V) and Vehcle Correspondng author s emal: feyan@utbmfr X Copyrght 2011 ORSTW

2 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 58 IOR Vol 8, No 1, (2011) to Infrastructure (V2I) communcaton technologes Meanwhle, mnaturzaton of computng devces and avalablty of Global Postonng System (GPS) made the ntellgent vehcles wth n-vehcle nformaton system (IVIS) become more and more popular Besdes, there are also some researches that focus on predctng vehcle's accurate arrval tme at ntersecton wth both the hstorcal and real-tme GPS vehcle locaton data (see (Chn-Woo, Sungsu et al 2008)) In ths context, the autonomous vehcles start to attract more and more nterests Advanced strateges based on autonomous vehcles have been studed to solve the congeston problem (Wunderlch et al 2008; L et al 2006) In ths paper, we study the autonomous vehcle sequencng problem at an ntersecton network wth the consderaton of V2I communcaton All vehcles are assumed to be equpped wth IVIS that can communcate wth nfrastructure of ntersectons, and they can traverse the ntersecton autonomously when nformed The traffc control strategy can be brefly descrbed as follows Suppose there s a small ntersecton network that contans several adacent solated ntersectons A center controller s located n the network and all vehcles equpped wth IVIS are able to communcate wth the controller n some fashon The vtal telemetry data of each vehcle can be obtaned by the controller once the vehcle enters the control range At a basc level, we assume knowledge of each vehcle s accurate arrval tme at the ntersecton whch t s gong to traverse and the tme each vehcle needs to pass through that ntersecton Decson of vehcle passng sequence can be made and broadcasted to all vehcles n the range Vehcles wll pass ther ntersecton accordng to the receved decsons nstead of followng the tradtonal traffc lghts One can note that wth ths control strategy, the traffc control problem changes to a dscrete vehcle sequencng problem Each ntellgent vehcle can be montored and controlled ndvdually wth several parameters such as poston, ntended drecton, arrval tme, etc The controller analyzes all the parameters and decdes a fnal vehcle passng sequence wth a specfc obectve to mprove the traffc stuaton At the same tme, we can observe that the dscreteness brngs us a great number of vehcle passng combnatons Effcent algorthms are needed to fnd the optmal vehcle passng sequence Consderng the dscrete and combnatoral nature of the problem, we decentralze ths problem to several vehcle sequencng problems at solated ntersecton and model each of them as a specal sngle machne schedulng problem Dfferent obectves are studed Effcent Branch and Bound algorthms are proposed to fnd an optmal vehcle passng sequence Heurstc that served as ntal upper bound s also presented It can also be used ndependently to fnd a satsfyng soluton n a very short computaton tme Computatonal experments and smulatons show the mprovement obtaned by usng the proposed schemes The rest of ths paper s structured as follows: a detaled descrpton of the studed problem s provded n Secton 2 We also gve ts schedulng model along wth some useful notatons that wll be used n the sequel Secton 3 descrbes the Branch and Bound algorthms for each obectve Secton 4 presents the results of computatonal experments and smulatons Conclusons are drawn n Secton 5 2 PROBLEM DESCRIPTION AND SCHEDULING MODEL 21 Problem Descrpton The studed ntersecton network conssts of several adacent ntersectons that may have dfferent layout An llustraton s gven n gure 1 There are four solated ntersectons covered by the control range of a center controller The nformaton of each vehcle s gathered by the controller n real-tme and vehcle passng sequences are decded nsde the center controller Snce the layout between two adacent ntersectons are fxed (number of lanes, dstance, etc), we can easly decentralze the mult-ntersecton control problem to several vehcle sequencng problems at solated ntersecton Ths also reduces the complexty of the decson makng process n center controller, whch can ensure the needs of an advanced traffc control system gure 1 Schematcs of the Intersecton Network under Consderaton

3 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 59 IOR Vol 8, No 1, (2011) We now focus on the vehcle sequencng problem at an solated ntersecton At frst, some basc notons should be ntroduced Typcally, an ntersecton conssts of a number of approaches and the crossng area Each approach may be used by several traffc streams or example, the ntersecton I n gure 1, the approach from west to east conssts of three traffc streams (stream 1, 2 and 3) Each stream has ts own lane and an ndependent queue (overtakng s not allowed) The path used by a traffc stream to traverse the ntersecton s called the traectory (wth dashed lnes) A traectory connects an approach on whch vehcles enter the ntersecton to the ntersecton leg on whch these vehcles leave the ntersecton Vehcles belongng to some streams may have more than one traectory whle traversng the ntersecton (eg, stream 3 and stream 10) The obectve of optmal traffc control at ntersecton s to transform nput traffc flows nto output ones whle preventng traffc conflcts and satsfyng a specfc crteron In order to prevent the conflcts of vehcle streams, frequently used traffc conventons provde the noton of compatble streams and ncompatble streams Obvously, when traectores of two traffc streams do not cross, these streams can smultaneously get the rght-of-way We call these two streams compatble streams The lanes on whch the two streams are movng are called compatble lanes or example, stream 1 and stream 7 are compatble streams On the other hand, when traectores of two traffc streams do cross, the streams are n a conflct (eg, stream 2 and 10), and ther smultaneous movement through the ntersecton should not be permtted Some conflct ponts of traffc streams are ndcated n gure 1 When several streams are compatble wth each other, we call the set of these streams a Compatble Stream Group (CSG) In ths example, we can partton the 10 streams nto four compatble stream groups: CSG 1: stream 1, 7; CSG 2: stream 2, 3 and 8; CSG 3: stream 4 and 9, and CSG 4: stream 5, 6 and 10 One should note that the dvson of these groups s not constant when traffc flow of a specfc stream s much greater durng a peak tme (eg, mornng peak tme or evenng peak tme) However, ths dvson usually remans the same durng a specfc perod Besdes, there s always a lost tme when we swtch the rght-of-way between two vehcles of dfferent CSGs to avod nterference between ncompatble streams Durng the lost tme, no vehcle behnd the stop lne s allowed to pass the ntersecton The traffc control process s to decde a sequence of dstrbutng the rght-of-way to specfc vehcles of each CSG to make these vehcles get through the ntersecton whle satsfyng a specfc obectve Suppose that at a start tme t 0 0, there are n vehcles n the control range approachng the ntersecton from dfferent approaches, and the vtal data of all vehcles n ths range can be receved by control devce mmedately At a basc level, the nformaton from each vehcle contans follow parts: Vehcle dentfcaton (ID): used to dentfy ndvdual vehcle Stream number: whch stream the vehcle belongs to, e, whch lane t s movng on Precse vehcle arrval tme: the precse tme vehcle arrves at the stop lne from t 0 wthout nterference Vehcle passng tme: tme nterval vehcle needs to get through the ntersecton Remark that the vehcle passng tme s actually the tme nterval n whch a vehcle can accelerate from the stop lne untl t reaches a safe dstance wth ts follower on same lane (n same stream) Ths tme nterval depends on the type of vehcle that s gettng through the ntersecton or example, trucks are slower than small vehcles, they need more tme to change the speed and therefore, more tme to accelerate untl t can reach a safe dstance for the followng vehcle to move Typcally, there are three measures for evaluatng the performance of a traffc control approach: the evacuaton tme, average queue sze and average vehcle watng tme The evacuaton tme of an ntersecton can be seemed as an obectve related to the throughput A Dynamc Programmng algorthm and a Branch and Bound algorthm were proposed to fnd an optmal vehcle passng sequence wth mnmzng the evacuaton tme n earler work (Yan, Drd et al 2009; Yan, Drd et al 2010) The average queue sze ndcates the number of vehcles on each lane watng to cross the ntersecton at same tme Average vehcle watng tme measures how long a vehcle has to wat before traversng the ntersecton All the three measures are frequently used to evaluate the performance of a traffc control strategy In ths paper, our obectve s to decde a vehcle passng sequence to mnmze the average queue sze or average vehcle watng tme 22 Schedulng Model or our purpose, we model the studed solated ntersecton as a sngle machne that can process parallel obs Each vehcle s modeled as a ob and ts arrval tme and passng tme can be modeled as the ob release date and processng tme, respectvely The tme each ob should fnsh the traversng procedure wthout any delay can be modeled as the ob due date It equals to the vehcle arrval tme plus ts passng tme obs (vehcles) are parttoned nto dfferent famles accordng to the group of compatble steams All vehcles n same CSG form a ob famly or example, n gure 1, the vehcles approachng ntersecton I can be treated as four famles correspondng to the four CSGs Snce overtakng n same stream s not allowed, vehcles on same lane should traverse ntersecton n rst-in-rst-out way Ths can be modeled as the chan constrants of the sngle machne problem Vehcles n same stream are treated as obs n same chan obs n dfferent chans but same famly can be

4 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 60 IOR Vol 8, No 1, (2011) processed n parallel (vehcles n same compatble group but dfferent streams can use the ntersecton smultaneously) In vehcle passng sequences, the lost tme between two adacent vehcles from dfferent CSGs can be modeled as the famly setup tme whch s only decded by the followng famly Two measures average queue sze and average vehcle watng tme are modeled as number of late obs and total tardness, respectvely Suppose there are n vehcles that are parttoned nto m famles 1, 2,,, m accordng to the compatble stream groups The number of obs n famly s n, where 1 m In each famly, obs are parttoned nto at least one chan or the reason of rgor and mathematcal expresson, we gve the followng notatons: l ( l, ) ( l, ) l, the number of chans n famly th, the l chan n, where 1 l l n, the number of obs n chan l ( l, ) s, the setup tme of (, l, ), the (, l, ) (, l, ) (, l, ) (, l, ) th ob n chan ( l, ) l r, the nteger release date of (, l, ) p, the nteger processng tme of (, l, ) d, the due date of ob (, l, ), e, d(, l, ) r(, l, ) p(, l, ) C, the completon tme of ob (, l, ) s ndexed accordng the release sequence of obs n each chan Note that each ob n famly belongs to one and only one chan, e, { n(, l) } n Only one ob n same chan can be processed on the machne each tme The setup tme s of famly should be ncurred at start of the sequence and whenever there s a swtch from processng a ob n another famly to a ob n ths famly obs n dfferent chans of same famly can be processed concurrently Besdes, snce our defnton of due date s the sum of ob release date and ts processng tme, a ob can be only on tme or late Accordng to the standard classfcaton scheme for schedulng problems (Graham, Lawler et al 1979), we denote ths problem by 1 p obs, chans, s, r h, where p obs means that obs contaned n same famly can be processed n parallel h represents the U or T whch correspond the number of late obs or total tardness, respectvely 3 PROPOSED ALGORITHMS Over the last several decades, there has been sgnfcant nterest n the sngle machne schedulng problems wth obs can be processed smultaneously These problems always nvolve an element of batchng (see (Monma and Potts 1989)) In 1992, (Potts and Wassenhove 1992) gve a revew that combnes schedulng wth batchng together (Webster and Baker 1995) present an overvew of algorthms and complexty results for schedulng batch processng machnes They call the problem wth processng obs concurrently the burn-n model (or batchng machne model), whch s motvated by the problem of schedulng semconductor burn-n operatons for large scale ntegrated crcut manufacturng (Lee, Uzsoy et al 1992) In ths model, the processng tme of each ob s equal to the maxmum processng tme of any ob assgned to t All obs n same batch start and complete at the same tme There are two varants of the burn-n model: bounded and unbounded, whch depends on whether the sze of batches s bounded by a constant or not (Brucker, Gladky et al 1998) gve a detaled dscusson about the two models In the recent decades, (Potts and Kovalyov 2000) present an updated revew about the usually used algorthms of batchng problem for the case of sngle machne, parallel machne, ob shop and open shop (Cheng and Kovalyov 2001) descrbes the complexty results for varous related problems and obectves, also consderng due dates, but not release dates Some researches wth several constrants lke release dates, famly setup tmes were also studed (see (Lu and Yu 2000; Cheng, Yuan et al 2005) and (Brucker and Kovalyov 1996) for example) However, for our case, the schedulng model has ts dstnct features If we vew ths problem as processng obs n passng groups ( PG ), a passng group PG can be defned as a set of obs from same famly and processed on the machne wthout the nterrupton of obs n any other famles Note that only obs n same famly can be put nto same passng group Snce there may be several chans n one PG, obs n same chan of ths passng group should be processed consecutvely and obs n dfferent chans may be processed smultaneously Thus, the processng tme of ths PG does not equal to the maxmum processng tme of any ob n ths group Meanwhle, famly setup tmes and ob release dates are also nvolved Thus, the frequently used models can not meet the needs of the studed problem 1 l l

5 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 61 IOR Vol 8, No 1, (2011) In fact, wth the defnton of passng group PG, the fnal ob sequence wll have at least m PGs We defne the release date of a passng group r PG as the earlest release date of obs n t, e, rpg mn PG{ r} and the completon tme of the passng group C as the maxmum completon tme of the all obs n ths PG, e, C max { C } PG PG PG Under these consderatons, the optmal soluton of the problem can be vewed as the optmal Passng Group Sequence PGS, e, PGS ( PG1, PG2,, PG b ), where b m These passng groups are separated by setups The decson procedure can be vewed as groupng obs of each famly nto several passng groups and sequencng all passng groups of dfferent famles to get an optmal vehcle sequence 31 Mnmzng Number of Late obs and Total Tardness In lterature, the problem of sngle machne wth mnmzng the number of late obs has already been studed extensvely The problem can be denoted by 1 U It can be solved n O( n log( n )) tme by algorthm proposed n (Moore 1968) However, f obs have release dates (1 r U ), the problem becomes strongly NP-hard (Lenstra, Kan et al 1968) Some researches also concerns about the case wth famly setup tmes or example, (Bruno and Downey 1978) show that for the bnary NP-hard problem 1 s U wth arbtrary number of famles or mnmzng the number of late obs, (Monma and Potts 1989) and (Crauwels, Potts et al 1996) gve some useful propertes than can be used to develop a Branch and Bound algorthm However, wth the ob chan constrants n each famly and the presence that obs n dfferent chans of same famly can be processed smultaneously, those propertes cannot be used any more and the optmal soluton do not even follow the sequence of the form S ( E, L) n lterature, where a partal sequence E of early obs s followed by a partal sequence L of late obs Ths urges us to fnd new propertes of an optmal sequence to reduce the search space or the problem wth mnmzng total tardness on a sngle machne, e, 1 T, ts computatonal complexty remaned open untl ts NP-hardness n ordnary sense was establshed by 1987 (Du and Leung 1990) Researches wth ths obectve always concern about some specal cases such as the release date (Baptste, Carler et al 2004), setup tmes (Luo and Chu 2006), etc rom these researches, we can easly deduce that our problem wth mnmzng the number of late obs and total tardness are at least bnary NP-hard for arbtrary number of famles However, accordng to the solated ntersecton confguraton, famly number s usually constant durng a specfc tme, we ntend to use Branch and Bound algorthm to fnd an optmal soluton for both the problem wth U and T Suppose PG x s a passng group n an optmal sequence and obs n PG x are from famly The completon tme of the last passng group before PG s C We can have the followng property about the frst UN-grouped ob x PG r n each chan of same famly after the completon tme C PG r of a partal sequence Property 1 or problem1 p obs, chans, s, r h, there exsts an optmal passng group sequence, n whch any passng group PG x contans the frst UN-grouped ob wth the mnmum completon tme of all chans Proof By contradcton The proof s presented for the problem1 p obs, chans, s, r U, and the problem wth f total tardness s smlar Suppose there s a partal sequence, n whch some passng groups have been formed, but no decson has been taken yet on groupng the remanng UN-grouped obs The completon tme of the last passng group n the partal sequence s C Assume that there are only two chans n, e, l (,1) and l (,2) Let the frst PG r UN-grouped ob after C n these two chans be (,1, y) and (,2, y'), respectvely See gure 2 as an example In PG r the example, the completon tme of (,1, y) s the smallest gure 2 Example of Property 1

6 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 62 IOR Vol 8, No 1, (2011) Suppose that there exsts an optmal sequence PGS n whch PG contans only one ob x (,2, y'), we can obtan a new passng group sequence PGS ' by nsertng ob (,1, y) nto PG x Obvously, ths change does not ncrease the number of late obs after C, and then PGS ' s optmal too Contnue ths procedure; we wll get an optmal passng PG r group sequence as the property above One can clearly observe that ths property stands even have the followng corollary has more than two chans By Property 1, we can easly Corollary 2 or problem 1 p obs, chans, s, r h, f there are several obs of same famly that have the same startng tme and processng tme, there exsts an optmal sequence that these obs are contaned n same passng group Note that for the studed problem, the obs of same famly wth same startng tme or release dates must be located ndfferent chans The proof s smlar as that n Property 1 32 Branch and Bound Algorthm and Heurstc Based on the results gven above, we propose a Branch and Bound algorthm for the problem 1 p obs, chans, s, r h as follows 321 Branchng Scheme In ths Branch and Bound scheme, each node denotes a partal group sequence and t s parttoned nto k branches: one branch ndcates the last group ust formed should be nserted more obs of same famly f there are stll UN-grouped obs n t; other k 1 branches ndcate that we gve the authorzaton of usng machne to obs n other k 1 famles that stll have obs watng for process, where k m Each tme consderng obs of famly, 1 m, the new node s gong to contan the ob/obs that satsfy wth Property 1 or Corollary 2 The search tree s constructed n a depth-frst fashon 322 athomng and Backtrackng In the proposed Branch and Bound algorthm, a node s fathomed f: It s a leaf node, e, a complete soluton The lower bound exceeds or equals the ncumbent upper bound In searchng process, f a complete soluton has smaller obectve value (e, smaller number of late obs or total tardness) than the current upper bound s found, ths value should be regarded as a new upper bound athomng ntates backtrackng to the frst node that stll not fathomed If no such node s found, the search termnates 323 Lower Bound and Intal Upper Bound or a partal group sequence wth the last passng group PG r s PG p CPG r and C PG p s the completon tme of PG r and PG from famly, suppose the group adacently before r p PG, respectvely Clearly, PG s not from The proposed lower bound of ths partal sequence conssts of two parts: LBS (or LBS '): the number of late obs (or the total tardness) n the partal sequence that already scheduled, whch can be easly counted by checkng the startng tme and release tme of each ob n the partal sequence Note that snce the last passng group n partal sequence PG may contan more obs from, we only count LBS (or LBS ') from the begnnng of the partal sequence to the last ob n PG p LBR (or LBR ' ): the lower bound of late obs (or the total tardness) for the rest UN-grouped obs, Ths can be computed by comparng the release date wth the mnmum startng tme of each UN-grouped ob If the mnmum startng tme s bgger than ts release date, we count the ob as a ob that wll be late (or count ts mnmum tardness) Ths process s gven n gure 3 We can observe that the lower bound LBR / LBR ' conssts of two parts: the number of late obs (sum of tardness) whch are gong to have n famly, and that n other famles The fnal lower bound LB / LB' can be computed by the sum of LBS / LBS ' and LBR / LBR ', e, LB LBS LBR, LB' LBS ' LBR ' (1) r p

7 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 63 IOR Vol 8, No 1, (2011) The algorthm for the lower bound LBR / LBR ' s gven as follows: Algorthm 1 Lower bound LBR / LBR ' Begn /* LBR 0, LBR ' 0, start 0 */ or f 1 to m wth f or l 1 to l f PGr start C s f f ; otherwse start C s ; f PGp or y to n ( f, l), /* ( f, l, y) s the frst UN-grouped ob n l ( fl ) */ If start r( f, l, ) LBR LBR 1 (for the problem U ); LBR ' LBR ' ( start r ) T ); ( f, l, ) (for the problem f End start start p ; ( f, l, ) gure 3 Algorthm of the Lower Bound LBR / LBR ' or the proposed Branch and Bound algorthm, we gve the followng algorthm to fnd an ntal group sequence The ntal upper bound can be obtaned from ths ntal sequence to ncrease the rate wth whch nodes are fathomed Each tme consderng obs n a famly, we group the obs that satsfy Property 1 or Corollary 2 to a temporary group Then add the temporary group that has the earlest release date (ERD) of all famles Let PGS be the partal sequence that already grouped; ths procedure can be descrbed n gure 4 Algorthm 2 Intal Sequence for U and T Begn /* CPG r 0, PGS */ whle there stll are un-grouped obs, do ; for famly, [1, m] End Add the ob/obs n whch can satsfy the Property 1 or Corollary 2 to ; Add the ob wth earlest release date n to PGS ; C PGr C ; PGS gure 4 Algorthm for ndng an Intal Sequence The ntal upper bound of number of late obs and total tardness can be easly obtaned from ths ntal sequence One should note that we can also use ths procedure as an ndependent heurstc 324 A numercal example In order to llustrate the search procedure, a numercal example of problem 1 p obs, chans, s, r U s present as follows Suppose there are 15 vehcles at tme t 0 0 approachng the ntersecton and vehcles are parttoned nto

8 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 64 IOR Vol 8, No 1, (2011) three CSGs The lost tme of each CSG s s1 1, s s2 2, s s3 3s The arrval tme and passng tme of vehcles are gven n Table 1 Table 1 Arrval tme and passng tme of vehcles n example (1,1,1) (1,1,2) (1,1,3) (1,2,1) (1,2,2) (1,3,1) (2,1,1) (2,1,2) (2,2,1) (2,2,2) (3,1,1) (3,1,2) (3,1,3) (3,2,1) (3,2,2) r p By followng the ntal upper bound of the gven algorthm, we can have an ntal sequence wth 13 late vehcles The branchng scheme wll then start from root node wth three new branches; each branch ndcates the vehcle sequence wll start from a CSG The lower bound of the three new branches s 4, 5 and 7 late vehcles, respectvely Then the branchng scheme wll contnue from the frst branch (the branch wth only one passng group from the frst CSG) snce t has the smallest number of late obs nally, we can have the optmal sequence wth 8 late vehcles The fnal sequence s gven n gure 5, n whch the late vehcles (obs) are (2,1,1), (2,2,1), (1,2,2), (3,1,1), (3,1,2), (3,1,3), (3,2,1), (3,2,2) gure 5 nal optmal sequence of example The search procedure s smlar wth the problem1 p obs, chans, s, r T 33 Applcaton of Algorthms and Coordnaton of Intersectons Snce at the real-world ntersectons, vehcles keep enterng the control range of the center controller; the algorthm should be executed separately to each solated ntersecton whenever there are new vehcles approachng ths ntersecton However, f a group of vehcles have the rght-of-way to pass an ntersecton, the recalculaton process should be postponed untl all vehcles n that group have passed through the ntersecton Snce mentoned n (Hall 2003), for an advanced traffc control system, the decson makng process should be executed for about each 2 seconds (for the effcency of traffc control and the safety of drvers), the recalculaton procedure should follow the rules below: If there are not any vehcles detected n 2 seconds, the recalculaton wll be executed when vehcle comes If there are one or more vehcles detected wthn 2 seconds, the recalculaton wll be executed only once at end of the 2 seconds wth the consderaton of all new vehcles Besdes, n the shared space between two neghbor ntersectons, for example, consderng the lanes (from west to east) between ntersecton I and ntersecton II n gure 1, the output traffc streams of ntersecton I s also the nput traffc stream of ntersecton II The vehcles already traversed Intersecton I (from west to east) can be seemed as the new comng vehcles for ntersecton II, and vce versa The average queue length and average vehcle watng tme can be obtaned by the average of all vehcles n ths ntersecton network Moreover, snce the layout between two adacent ntersectons are fxed n both number of lanes and dstance The tme used by vehcles from upstream ntersecton to downstream ntersecton can be seemed as the same Thus, when some vehcles are grouped together to pass the upstream ntersecton accordng to ther arrval tmes and passng tme, they can stll be scheduled n same group f ther ntentons keep the same wth each other (gure 6) The computatonal tme wll also be reduced ths way

9 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 65 IOR Vol 8, No 1, (2011) gure 6 Illustraton of same vehcle passng group from upstream to downstream ntersecton 4 COMPUTATIONAL EXPERIMENTS AND SIMULATIONS 41 Computatonal experments A traffc control system should satsfy the need of a real-tme system; for example, the decson makng process should be completed wthn 2 seconds Thus, the runnng tme of the proposed control algorthms should be tested In ths secton, experments are carred out to evaluate the computatonal performance of the proposed Branch and Bound algorthms and the heurstcs Accuracy of the heurstc s also calculated The computatonal experments are performed at an solated ntersecton wth four approaches and four compatble stream groups Wthout loss of the generalty, we assume that there are 2 lanes, 3 lanes and 4 lanes for ncomng vehcles n each of the four compatble stream groups Table 2 shows the computaton tme for dfferent obectves wth the proposed Branch and Bound algorthms and heurstc We assume that the number of vehcles approachng the ntersecton s vared from 10 to 50; the loss tme (setup tme) for each compatble group s randomly generated ntegers vared from 3 to 8 seconds and passng tme of vehcles (processng tme) are vared from 2 to 8 seconds Vehcles n each compatble group are equally dstrbuted among the streams To llustrate the performance of the heurstc n terms of accuracy, experments are also carred out to test the maxmum devaton between heurstc and Branch and Bound algorthm for each computaton All approaches are coded n C++ and run on a desktop computer wth Lnux system (kernel 2632) The results llustrate that for most cases, the proposed Branch and Bound algorthms can handle nearly 50 vehcles n ts control range n a very short tme The CPU tme augments wth the number of lanes contaned by each compatble stream group from 0018s to 0622s for mnmzng U ; and from 0065s to 1047s for mnmzng the total tardness The effcent computaton tme makes t possble to apply the proposed Branch and Bound algorthms to a real-tme control system Heurstc performs much better than the Branch and Bound algorthm n runnng tme wth the maxmum devaton of 982% n each computaton All the computatonal experments are performed for an upstream ntersecton, the Branch and Bound algorthms for downstream ntersecton wll be more effcent snce some vehcle groups from upstream ntersecton can be used drectly l * Table 2 Computaton Tme for Dfferent Obectves wth B&B and Heurstc Mnmzng U (n seconds) Mnmzng T (n seconds) Heurstc N B&B B&B B&B B&B B&B B&B CPU average mn max average mn max tme Devaton % % % % % % % % % l represents the number of lanes contaned n each compatble stream group (famly) *

10 Average Queue Length (veh) Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 66 IOR Vol 8, No 1, (2011) 42 Smulaton Experments In ths subsecton, we analyze the two measures wth dfferent relatve traffc loads Comparsons are gven among the Branch and Bound algorthm, the heurstc, an optmzed fxed cycle and an actuated traffc lght control scheme The smulaton s mplemented at an ntersecton network wth four ntersectons (see gure 1 for example) The dstance between any two adacent ntersectons s the same and all vehcles need 15s to travel from an upstream ntersecton to a downstream ntersecton Each ntersecton has four approaches and each approach has two lanes for ncomng vehcles Vehcles approachng each ntersecton are parttoned nto four CSGs Accordng to statstcs, we consder that the maxmum traffc load for each of the four approaches s 1800 vehcles/h (one vehcle every 2 seconds) and the traffc load for each ncomng lane s quarter of the maxmum load of one approach Other confguratons lke vehcle passng tme and lost tme are the same as n the computatonal experments Each data pont s obtaned by takng the average over several separate smulatons Each smulaton runs 10 mnutes of traffc flow By applyng the Branch and Bound algorthm and heurstc wth mnmzng the number of late obs, the smulaton result of average queue sze s presented n gure 7 We can fnd that the new control strategy reduces the average queue length for more than 60% at certan traffc flow rates Branch & Bound x cycle tme Heurstc Actuated Relatve Traffc Load (vehcle/s) gure 7 Average Queue Length for 10 mnutes Traffc low Meanwhle, the average vehcle watng tme of all vehcles n ths network durng 10 mnutes by applyng algorthms wth mnmzng total tardness are shown n gure 8 The results show that even for hgh traffc load (05 vehcle/s), the average vehcle watng tme before traversng the ntersecton network decreased from 671s to 204s Ths means drvers of ntellgent vehcles wll spend about 70% less tme n ths area 5 CONCLUSIONS Ths paper proposed a new approach to sequence ntellgent vehcles to pass an ntersecton network va V2I communcatons The consdered mult-ntersectons were decentralzed to several solated ntersectons At each solated ntersecton, vehcles were treated as dscrete ndvduals n the control strategy and our obectve was to mnmze the average queue length or average vehcle watng tme We modeled ths problem as a specal sngle machne schedulng problem that obs n same famly can be processed n parallel, setup tmes and chan constrants were also consdered Two obectves were consdered Branch and Bound algorthms and a heurstc were developed based on the analyss of structural propertes of the problem Results showed that ther average runnng tme can satsfy the need of a real-tme control system Smulatons showed the sgnfcant mprovement by applyng the proposed algorthms

11 Average Watng Tme (s) Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 67 IOR Vol 8, No 1, (2011) Branch & Bound x cycle tme Heurstc Actuated Relatve Traffc Load (vehcle/s) gure 8 Average Watng Tme for 10 mnutes Traffc low 6 ACKNOWLEDGEMENT The authors are grateful for the constructve comments of the referees on an earler verson of ths paper REERENCES 1 Baptste, P, Carler, and ouglet, A (2004) A Branch-and-Bound procedure to mnmze total tardness on one machne wth arbtrary release dates, European ournal of Operatonal Research, 158(3): Brucker, P, Gladky, A, Hoogeveen, H, Kovalyov, M, Potts, CN, Tautenhahn, T and Velde, SVD (1998) Schedulng a batchng machne, ournal of Schedulng, 1: Brucker, P and Kovalyov, M Y (1996) Sngle machne batch schedulng to mnmze the weghted number of late obs, Mathematcal Methods of Operatons Research, 43: Bruno, and Downey P (1978) Complexty of task sequencng wth deadlnes, set-up tmes and changeover costs, SIAM ournal on Computng, 7: Cheng, T C E and Kovalyov, M Y (2001) Sngle machne batch schedulng wth sequental ob processng, IIE Transactons, 33(5): Cheng, T C E, Yuan, and Yang, A (2005) Schedulng a batch-processng machne subect to precedence constrants, release dates and dentcal processng tmes, Computers and Operatons Research, 32(4): Chn-Woo, T, Sungsu, P, Lu, H, Qng, X and Lau, P (2008) Predcton of Transt Vehcle Arrval Tme for Sgnal Prorty Control: Algorthm and Performance, IEEE Transactons on Intellgent Transportaton Systems, 9(4): Crauwels, H A, Potts, C N and Van Wassenhove, LN (1996) Local search heurstcs for sngle-machne schedulng wth batchng to mnmze the number of late obs, European ournal of Operatonal Research, 90(2): Du, and Leung, YT (1990) Mnmzng total tardness on one machne s NP-hard, Mathematcs of Operatons Research, 15: Graham, R, Lawler, E, Lenstra, and Rnnooy Kan, A (1979) "Optmzaton and approxmaton n determnstc machne schedulng: A survey, Annuals of Dscrete Mathematcs, 5: Gubernc, S, Senborn, G and Lazc, E (2008) Optmal Traffc Control: Urban Intersectons, CRC Press, Taylor & rancs Group 12 Hall, R W (2003) Handbook of Transportaton Scence, Sprnger 13 Hunt, P B (1982) The SCOOT on-lne traffc sgnal optmsaton technque, Traffc Engneerng & Control, 23: Lee, CY, Uzsoy, R and Martn-Vega, L A (1992) Effcent algorthms for schedulng semconductor burn-n operatons, Operatons Research, 40: Lenstra, K, Rnnoy Kan, A H G and Brucker, P (1968) Complexty of machne schedulng problems, Annuals of Dscrete Mathematcs, 1: L, L and Wang, Y (2006) Cooperatve drvng at blnd crossngs usng ntervehcle communcaton, Vehcular Technology, IEEE Transactons on Vehcular Technology, 55(6):

12 Yan, Drd, and El Moudn: A Schedulng Approach for Autonomous Vehcle Sequencng Problem at Mult-Intersectons 68 IOR Vol 8, No 1, (2011) 17 Lu, Z and Yu, W (2000) Schedulng one batch processor subect to ob release dates, Dscrete Appled Mathematcs, 105(1-3): Luo, X and Chu, (2006) A branch and bound algorthm of the sngle machne schedule wth sequence dependent setup tmes for mnmzng total tardness, Appled Mathematcs and Computaton, 183(1): Monma, C L and Potts, C N (1989) On the complexty of schedulng wth batch setup tmes, Operatons Research, 37: Moore, M (1968) An n ob one machne algorthm for mnmzng the number of late obs, Management Scence, 15: Potts, C N and Kovalyov, M Y (2000) Schedulng wth batchng: A revew, European ournal of Operatonal Research, 120(2): Potts, C N and Wassenhove, L N V (1992) Integratng schedulng wth batchng and lotszng: A revew of algorthms and complexty, ournal of Operatonal Research Socety, 43: Robertson, D I (1969) TRANSYT: A traffc network study tool, Crowthorne, England, Mnstry of Transport 24 Webster, S and Baker, K R (1995) Schedulng groups of obs on a sngle machne, Operatons Research, 43: Wunderlch, R, Lu, C, Elhanany, I and Urbank, T (2008) A novel sgnal-schedulng algorthm wth qualty-of-servce provsonng for an solated Intersecton, IEEE Transactons on Intellgent Transportaton Systems, 9(3): Yan,, Drd, M and EI Moudn, A (2009) Autonomous vehcle sequencng algorthm at solated ntersectons, 12th Internatonal IEEE Conference on Intellgent Transportaton Systems, ITSC '09, Yan,, M Drd and EI Moudn, A (2010) New vehcle sequencng algorthms wth vehcular nfrastructure ntegraton for an solated ntersecton, Telecommuncaton Systems, In press