Sensor Network Design for Multimodal Freight Transportation Systems

Transcription

1 MN WI MI IL IN OH USDOT Regon V Regonal Unversty Transportaton Center Fnal Report NEXTRANS Project No 012IY01 Sensor Network Desgn for Multmodal Freght Transportaton Systems By Xaopeng L Ph.D. student Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn l28@uuc.edu and Eunseok Cho Undergraduate student Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn cho22@uuc.edu and Yanfeng Ouyang Prncpal Investgator Assstant Professor of Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn yfouyang@uuc.edu Report Submsson Date: October 15, 2009

2 DISCLAIMER Fundng for ths research was provded by the NEXTRANS Center, Purdue Unversty under Grant No. DTRT07-G-005 of the U.S. Department of Transportaton, Research and Innovatve Technology Admnstraton (RITA), Unversty Transportaton Centers Program. The contents of ths report reflect the vews of the authors, who are responsble for the facts and the accuracy of the nformaton presented heren. Ths document s dssemnated under the sponsorshp of the Department of Transportaton, Unversty Transportaton Centers Program, n the nterest of nformaton exchange. The U.S. Government assumes no lablty for the contents or use thereof.

3 MN WI MI IL IN OH USDOT Regon V Regonal Unversty Transportaton Center Fnal Report TECHNICAL SUMMARY NEXTRANS Project No 012IY01 Fnal Report, October 2009 Sensor Network Desgn for Multmodal Freght Transportaton Systems Introducton The agrcultural and manufacturng ndustres n the US Mdwest regon rely heavly on the effcency of freght transportaton systems. Whle the growth of freght movement far outpaces that of the transportaton nfrastructure, ensurng the effcency and sustanablty of the transportaton networks becomes a major challenge. The promnent dsbeneft of delay and unrelablty hghlghts the need for an ntegrated, systems-level framework that ncorporates cuttng-edge nformaton technologes and advanced multmodal network modelng technques to montor, manage and plan complex freght transportaton systems. Recent developments n sensng and nformaton technology hold the promse to allow effcent montorng, assessment, and management of complex systems. Fndngs Ths project nvestgated the effect of exstng or off-the-shelf sensors on detectng traffc and nfrastructure condtons for hghway and ral modes. Ths research project developed an analytcal framework to quantfy the benefts and costs of deployng sensors for the major freght transportaton modes. Specfcally, ths project developed a new sensor deployment problem n the context of traffc O- D flow survellance usng vehcle ID nspecton technologes (e.g., RFID). In addton to tradtonal flow coverage benefts based on ndvdual sensors, we nvestgated the path coverage benefts from syntheszng the multple sensors n transportaton networks. We consdered possble sensor dsruptons that are very common for many sensor technologes, yet not well addressed untl very recently. A relable locaton desgn model framework was proposed to optmze sensor deployment beneft. Ths model consdered both flow and path coverage, whle allowng for probablstc sensor falures. A set of effcent algorthms were developed and tested on moderate-sze problem nstances. We found that the LR-based algorthm had the best performance for the tested problems. The greedy-algorthm can yeld good solutons f flow coverage beneft s sgnfcant. Then we appled ths model to the large-scale Chcago ntermodal network, where the hghway network, the ralroad network and ther connectons were consdered n ths study. We extracted detaled nput flow data from lmted data resources. We examned the qualtes of solutons of dfferent algorthms, the best of whch (from the LR-based algorthm) are analyzed to draw out manageral nsghts about sensor deployment. The experments showed that path coverage beneft s more senstve to sensor falures and nstallaton budget. It was also found that path coverage tends to spread out sensor locatons whle hgh falure probabltes tend to cluster sensors together. NEXTRANS Project No 012IY01Techncal Summary - Page 1

4 Recommendatons Future work can be conducted n several drectons. Frst of all, the proposed model addresses probablstc sensor falures but assumes known O-D flow paths. Ths may be reasonable n the freght operaton context but a more comprehensve model that encompasses traffc routng and assgnment wll be desrable. In addton, the current model assumes all sensor falures are ndependent wth equal probablty. Yet more complex sensor falure patterns (e.g, ste-dependent and correlated falures) are not uncommon n the real world. Addtonal work that relaxes these two assumptons s needed. Fnally, t wll be nterestng to explore how alternatve traffc survellance benefts would affect the optmal sensor deployment pattern. Contacts For more nformaton: Yanfeng Ouyang Prncpal Investgator Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn yfouyang@uuc.edu (217) (217) Fax NEXTRANS Center Purdue Unversty - Dscovery Park 2700 Kent B-100 West Lafayette, IN nextrans@purdue.edu (765) (765) Fax NEXTRANS Project No 012IY01Techncal Summary - Page 2

5 MN WI MI IL IN OH USDOT Regon V Regonal Unversty Transportaton Center Fnal Report NEXTRANS Project No 012IY01 Sensor Network Desgn for Multmodal Freght Transportaton Systems By Xaopeng L Ph.D. student Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn l28@uuc.edu and Eunseok Cho Undergraduate student Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn cho22@uuc.edu and Yanfeng Ouyang Prncpal Investgator Assstant Professor of Cvl and Envronmental Engneerng Unversty of Illnos, Urbana-Champagn yfouyang@uuc.edu Report Submsson Date: October 15, 2009

6 ACKNOWLEDGMENTS The data preparaton tasks for the Chcago multmodal network case study was partly supported through the partcpaton of Eunseok Cho n the 2009 NEXTRANS Undergraduate Summer Internshp Program.

7 TABLE OF CONTENTS Page LIST OF FIGURES... CHAPTER 1. INTRODUCTION Background and motvaton Lterature revew and study objectves Organzaton of the research...5 CHAPTER 2. MODELING AND SOLUTION TECHNIQUES Model ntroducton Soluton approach Greedy algorthm LR-based Algorthm Algorthm test...17 CHAPTER 3. CASE STUDY: THE INDOT-MAINTAINABLE NETWORK Data preparaton Results...27 CHAPTER 4. CONCLUSIONS Summary Future research drectons...34 REFERENCES... 35

8 LIST OF FIGURES Fgure Page Fgure 2.1. The Soux-Falls test network. (Source: 17 * Fgure 2.2. Relatonshp between N, q and z for the Soux-Falls network Fgure 2.3. Optmal deployment of N=3 nstallatons n the Soux-Falls network Fgure 3.1. Network representaton of an ntersecton Fgure 3.2. Network dagram of Chcago Fgure 3.3. Network and freght movement dagram Fgure 3.4. Sensor deployment wth falure probablty of 0% and 20% Fgure 3.5. Sensor deployment wth flow coverage and path coverage Fgure 3.6. Sensor deployment wth 10 sensors and 20 sensors Fgure 3.8. Number of nstallatons vs. net beneft Fgure 3.7. Falure probablty vs. net beneft... 31

9 1 CHAPTER 1. INTRODUCTION 1.1 Background and motvaton The US Mdwest regon generates about 20 percent of the naton's overall gross domestc product (GDP). The backbone ndustres (such as agrculture and manufacturng) and regonal economy are heavly dependent of the effcency of regonal freght transportaton systems. Whle the transportaton nfrastructure development has reached a plateau n recent years (approxmately one percent ncrease of lane mles per year) 1, the volume of freght movement has been growng, and wll contnue to grow, dramatcally. It s estmated that the number of freght trucks n the Mdwest states wll ncrease by more than 60 percent by 2020 (Meller 2007). As the demand for freght transportaton nfrastructure contnues to grow at an ncreasng pace, ensurng the effcency and sustanablty of the transportaton networks for current and future generatons s a major challenge. Natonwde, after 40 years of steady declnaton, freght logstcs costs have been rsng agan n both absolute quantty and percentage GDP. A sgnfcant porton of ths cost ncrease has been attrbuted to the deteroraton of delay and unrelablty n our hghly congested freght transportaton systems across modes (hghway, ral, ocean waterway). Ths poses promnent dsbenefts to the socety and hghlghts the need for an ntegrated, systems-level framework that ncorporates cuttngedge nformaton technologes and advanced multmodal network modelng technques to montor and manage the complex freght transportaton systems. Such an ntegrated framework wll enable decson makers to (1) understand condtons of multple system components (traffc, nfrastructure, etc.) at varous spatal and temporal scales, and (2)

10 2 dentfy effectve plannng and management solutons to acheve desrable operatng condtons and ensure effcent operatons of freght transportaton across modes. Recent developments n sensng and nformaton technology and ts applcatons to the feld of transportaton hold the promse to allow effcent montorng, assessment, management, and plannng of complex networks wth system-wde nteractons among multple system components. Ths s possble by combnng nformaton from parallel sensng systems wth ntegrated mult-scale modelng and decson support. State transportaton agences have made sgnfcant nvestment n deployng varous types of sensors (e.g., loop detectors, RFID transponders) on state and local hghways, whch enable a wde range of fundamental applcatons n traffc management systems (TMS) such as traffc condton survellance (e.g., ncdent detecton), predcton (e.g., travel tme estmaton), and control (e.g., freeway ramp meterng and traffc dverson). Pror success on traffc database development and travel tme predcton have demonstrated the beneft of collectng traffc and speed data from loop detectors, radar sensor statons, and toll collecton transponders. The ralroads have also nvested mllons of dollars n advanced sensng technology (e.g., track-sde machne vson devces) to montor ralcar traffc. 1.2 Lterature revew and study objectves Sensor technologes (e.g., loop detectors, survellance cameras, rado frequency dentfcatons/rfid) have been wdely used on transportaton networks. Real-tme traffc nformaton s sampled by these sensors to montor traffc status and to develop control strateges. The effectveness of a traffc survellance system depends on not only the accuracy of the sampled nformaton but also the coverage over the transportaton network. However, mplementng these new technologes usually requres large nvestment. Accuracy and coverage are often two conflctng objectves due to lmted resources: collectng hgh-qualty nformaton usually reles on sophstcated and expensve technologes and thus lmted budget would restrct the number of nstallatons; on the other hand, due to the lmted effectve range of most sensors, complete coverage

11 3 over a network usually requres dense nstallatons. To balance ths trade-off, ntensve studes have been conducted to determne effcent and relable deployment of survellance systems. Yang et al. (1991) conducted a robust analyss on the utlty of traffc countng ponts for traffc O-D flow estmaton. Yang and Zhou (1998) proposed a sensor deployment framework to maxmze such utlty. Ths framework has been extended to accommodate turnng traffc nformaton (Banco et al., 2001), exstng nstallatons and O-D nformaton content (Ehlert et al., 2006), screen lne problem (Yang et al., 2006), tme-varyng network flows (Fe et al., 2007; Fe and Mahmassan, 2008) and ralcar nspecton under potental sensor falures (Ouyang et al., 2009). Despte numerous studes on O-D flow coverage, research on the usage of sensors for network O-D travel tme estmaton has been relatvely scarce. To the best of our knowledge, only Banet al. (2009) developed sensor deployment algorthm for travel tme estmaton n a sngle freeway corrdor---lttle research has addressed the problem n general networks. Accurate travel tme estmaton provdes mportant nformaton for decson support n both prvate sectors (e.g., tackng fleets for truckng companes, traveler nformaton provson) and publc agences (e.g., congeston mtgaton, accdent management). For a transportaton network, we may want to know as much as possble the real-tme travel tme between all possble O-D pars. However, tradtonal survellance technologes (e.g., loop detectors) would encounter sgnfcant challenges due to ther nablty to accurately capture O-D flows (Kerner and Rehborn, 1996; L et al., 2009). New sensor technologes, on the other hand, are able to dentfy vehcle IDs and therefore hold the promse to overcome these challenges by syntheszng vehcle ID nformaton from dfferent sensors. For example, the consecutve tme stamps of a vehcle at two sensor locatons would provde an accurate estmate of travel tme. Despte numerous studes on O-D flow coverage, research on the usage of sensors for network O-D travel tme estmaton has been relatvely scarce. To the best of our knowledge, only Ban et al. (2009) developed sensor deployment algorthm for travel tme estmaton n a sngle freeway corrdor---lttle research has addressed the problem n general networks. Accurate travel tme estmaton provdes mportant nformaton for decson support n both prvate sectors (e.g., tackng fleets for truckng companes,

12 4 traveler nformaton provson) and publc agences (e.g., congeston mtgaton, accdent management). For a transportaton network, we may want to know as much as possble the real-tme travel tme between all possble O-D pars. However, tradtonal survellance technologes (e.g., loop detectors) would encounter sgnfcant challenges due to ther nablty to accurately capture O-D flows (Kerner and Rehborn, 1996; L et al., 2009). New sensor technologes, on the other hand, are able to dentfy vehcle IDs and therefore hold the promse to overcome these challenges by syntheszng vehcle ID nformaton from dfferent sensors. For example, the consecutve tme stamps of a vehcle at two sensor locatons would provde an accurate estmate of travel tme. Lke many other IT technologes, most exstng sensors are subject to performance dsruptons due to system errors, adverse weather condtons, or ntentonal sabotages (Rajagopal and Varaya, 2007; Carbunar et al., 2005). Intutvely, such falures may substantally mpar the survellance effectveness. Potental dsruptons need to be addressed n a relable desgn so that the sensor system not only has a good performance n the normal scenaro but also s reslent aganst possble loss n falure scenaros. In recent years, relable faclty locaton problems have been studed n the supply chan desgn (Daskn, 1983; Snyder and Daskn, 2005; Cu et al., 2009; L and Ouyang, 2009) and ralroad defect detecton sensor desgn contexts (Ouyang et al., 2009). However, despte these recent efforts, few studes n the network traffc survellance context have addressed the possblty of sensor falures. Ths research ams to fll these gaps. It bulds on the relable faclty locaton lterature and develops a lnear nteger model to determne optmal locatons for vehcle ID nspecton sensors for travel tme estmaton as well as traffc O-D flow count. The model allows probablstc sensor falures n general transportaton networks. The formulated problem s complex by nature, and the real-world nstances are generally of large scale. Ths mposes prohbtve computatonal burden f we solve ths model wth standard solvers. We therefore propose customzed algorthms to solve the problem effcently. Case studes are conducted to test the algorthms and to draw nsghts.

13 5 1.3 Organzaton of the research The remander of the research s organzed as follows. Chapter 2 develops the mathematcal model and proposes dfferent algorthms to solve ths problem. The performance of these algorthms s tested wth a moderate-sze example. We found the Lagrangan-relaxaton-based algorthm outperforms the others n general. Chapter 3 apples ths model to a large-scale real problem, the Chcago ntermodal network. A set of data-processng procedures ncludng heurstcs have been taken to extract detaled nput from lmted macroscopc data. Manageral nsghts are drawn from result analyss. Chapter 5 summarzes the research and provdes future research drectons.

14 6 CHAPTER 2. MODELING AND SOLUTION TECHNIQUES Ths chapter ntroduces the model formulaton and soluton algorthms. Secton 2.1 formulates the problem nto a mathematcal model. Secton 2.2 ntroduces soluton algorthms and analyzes ther propertes. Secton 2.3 tests these algorthms wth a moderate-scale example and dscusses ther performances. 2.1 Model ntroducton We select sensor locatons n a transportaton network to maxmze the expected beneft from both O-D volume estmaton and travel tme measurement. For any O-D flow, the total traffc volume can be nspected by a sngle sensor f and only f the flow passes the sensor (Yang and Zhou, 1998). In ths case, we say that the flow s covered by the sensor n the sense of flow coverage (see fgure 2). Such ndvdual sensor nformaton can also be used to nfer travel tme based on speed measurements (Ban et al., 2009). However, sensors Fg. 2 Flow and Path Coverage (partcularly those wth vehcle-id capabltes; e.g., RFID) can work n pars to provde an accurate measurement of travel tme between ther nstallaton locatons. Assume that the traffc state along the traffc paths remans

15 7 relatvely stable durng the nomnal travel tme. 1 Intutvely, accurate travel tme estmaton for an O-D path benefts all traffc on ths path, whle the accuracy depends on the span of sensors---the wder a par of sensors span over an O-D path, the larger porton of the path s measured and the better t helps to estmate travel tme of that O-D path. Thus the travel tme survellance beneft, whch we denote by path coverage (see fgure 2), depends on not only the nspected traffc volume but also the lengths of covered O-D paths by sensor pars. We assume for smplcty that path coverage for an O-D path s proportonal to both ts traffc volume and covered length. Let I be the set of O-D paths on the network. Each path I s specfed by ts traffc volume set of canddate locatons, f, whch s assumed to be determnstc and known. Each path locaton j on path has a correspondng mleage, drecton of J passes a, where sensors can be potentally nstalled. Each canddate m j, ncreasng along the traffc f. The collecton of all canddate locatons over the network s J : = J. For convenence of notaton, let Note that I I. j j = I j denote the set of paths that pass the same locaton j. Due to lmted budget, no more than N sensors can be bult on the network. For I, f s nspected f an operatonal sensor s located at j. Smlar to the tradtonal j maxmal coverng models (Yang and Zhou, 1998), f f s nspected by at least one sensor, the beneft of flow coverage s bf, where b s a nonnegatve coeffcent. If c c f passes at least two sensors, we can record ts travel tme between the frst functonng (head) sensor t passes, at locaton j h, and the last functonng (rear) sensor t passes, at locaton e j. The beneft of path coverage can be expressed as bf m t ( e h j j m ), where bt s also a nonnegatve coeffcent. 1 Wthout losng generalty a path can be dvded nto multple short segments to make ths assumpton reasonable.

16 8 In the long run, sensors may be dsrupted or malfunctonal from tme to tme. When sensors fal, the flow coverage and path coverage patterns n the network also change. Hence we consder the expected survellance beneft across all sensor falure scenaros n addton to the deal non-falure scenaro. The head (or rear) sensor for each may vary over dfferent falure scenaros. In other words, dfferent head (or rear) sensors are assgned to accordng to falure scenaros. Sensors on can be ranked nto dfferent prorty levels accordng to such head (or rear) assgnment such that n any scenaro the sensor wth the lowest level among all functonng ones, f avalable, s the head (or rear) sensor. In the normal scenaro (wthout any falure), the most upstream sensor on serves as the head sensor, and thus t s the level-zero head sensor for. If ths sensor fals, ts mmedately downstream sensor takes over to serve, and thus ths second sensor s the level-one head sensor for. Ths process can be repeated to label every nstalled sensor on wth a unque head sensor assgnment level. Smlarly, each sensor on can be labeled wth a unque rear sensor assgnment level that starts from zero for the most downstream sensor and ncreases upstream. Supposng that there are sensors nstalled on path, we see that once the locatons wth nstallatons on are gven (.e., S { j, j,.., j } 0 1 S 1 ordered from upstream to downstream), ther head and rear assgnment levels are determned by the followng smple rule Defnton 1. (vald assgnment rule) A sensor at locaton j s s the level-s head sensor and the level- ( S 1 s) rear sensor for traffc path. Snce each sensor nstalled on receves a unque head (or rear) assgnment level to, there are at most R : = mn( J, N), levels of possble head (or rear) assgnment. Let r = 0,1,, R 1 denote a possble head (or rear) assgnment level for a sensor on. x j The prmal decson varables x : = { } determne where to nstall sensors, where 1, f a sensor s nstalled at locaton j; x j = 0, otherwse.

17 9 Gven h jr x, the auxlary varables h = { } and e = { } decde how sensors are h jr assgned to paths accordng to the vald assgnment rule;.e., h jr and e jr 1, f a sensor s nstalled at j and t s assgned to as a level- r head sensor; = 0, otherwse, 1, f a sensor s nstalled at j and t s assgned to as a level- r rear sensor; = 0, otherwse. Assume that each sensor fals ndependently wth an dentcal probablty 0 q < 1. The objectve of ths two-sensor-coverng problem (TSC) s to maxmze the expected total beneft of flow coverage and path coverage for all O-D paths. h, e R 1 r j t j jr t j c jr r= 0 (TSC) max z( x) : = max I J q (1 q) f [ bm h + ( bm + b ) e ], (1) x subject to J x N, (2) j j R 1 hjr = xj, I, j J, r= 0 (3) R 1 ejr = xj, I, j J, (4) r= 0 j J hjr 1, I, r = 0; (5a),, 1,, j J hjr j J hj( r 1) I r = R 1, (5b) j J ejr j J hjr, I, r = 0,1,, R 1, (6) x, h, e {0,1}, I, j J, r = 0,1,, R 1. (7) j jr jr Constrant (2) enforces the budget lmt, whle constrants (3) - (7) postulate the vald assgnment rule. Constrants (3) (or (4)) ensure that each nstalled sensor s assgned to each of ts correspondng paths at one and only one head (or rear) assgnment level. Constrants (5) and (6) ndcate that no more than one head or rear sensor s assgned to

18 10 each path at each level, and each rear assgnment must be accompaned by a head assgnment. Constrants (6) also mply that for each path, all the mplemented head j J jr = assgnment levels, { r h 1}, start from 0 and form a consecutve sequence. Constrants (7) postulate all decson varables to be bnary. The followng proposton reveals the relatonshp between the above formulaton and the vald assgnment rule. Proposton 1: The optmal soluton to the TSC problem (1)-(7) satsfes the vald assgnment. * * * Proof. Let x, h, e denote the optmal soluton to TSC. Agan locatons wth nstalled sensors on each path are ndexed wth { j, j,.., j } from upstream to downstream. 0 1 S 1 h Let denote the set of all mplemented head assgnment levels to ;.e., R h e R : = { r J = 1}. Smlarly, let R : = { r = 1}. For the case of q = 0, j h jr j J e jr there s no falure and only the level-0 assgnment affects the objectve value. It s obvous that the optmal soluton enforces all non-trval assgnments (at level-0) to be consstent wth the vald assgnment rule. Now we consder the case wth $q>0$. Snce each nstalled sensor on $$ corresponds to only one mplemented head (or rear) assgnment level (from (3) and (4)) and dfferent sensor cannot have the same head (or rear) assgnment level (from (5) h e and (6)), t s obvous that R = R =. S For the head assgnment, due to constrants (5), h R contans a sequence of levels e from 0 to S 1. Due to constrants (6), R R h h e. Thus R = R = {0,1,, S 1}, and we denote them by (or rear) assgnment level n R. Therefore on path, each sensor R j s s labeled wth a unque head. At optmalty, a more upstream sensor shall have a lower head assgnment level and a hgher rear assgnment level. Thus j s corresponds to

19 11 the level-s head assgnment and the level- ( S 1 s) rear assgnment to, whch s the vald assgnment rule. It shall be noted that the TSC modal can be easly adapted for cases where exstng nstallatons are already present (Ehlert et al., 2006). We smply enforce x = 1 f a sensor s already nstalled at locaton j; the model stll has the same structure and complexty. j 2.2 Soluton approach We have bult a lnear nteger mathematcal program that determnes sensor deployment to optmze traffc survellance benefts from both ndvdual sensor flow coverage (e.g., for traffc volume statstcs) and syntheszed sensors (e.g., for travel tme estmaton). Ths model bulds on the relable faclty locaton lterature and allows sensors to be subject to probablstc falures (e.g., due to techncal flaws or envronmental hazards). The formulated problem s complex by nature, whch mposes a prohbtve computatonal burden on solvng ths problem wth commercal solvers (e.g., CPLEX). We therefore propose customzed greedy and Lagrangan relaxaton algorthms to solve the problem effcently. These proposed algorthms are appled to ths project to yeld very good results, whle CPLEX often has dffculty n solvng most of tested nstances. Those who are nterested n detals are referred to L and Ouyang (2009). TSC s NP-hard because the maxmal coverng problem s a specal case of TSC (wth b = 0 and q = 0 ). As we wll show n Secton 4, commercal optmzaton software t (e.g., CPLEX) would work well only for small-scale nstances but t usually runs nto dffculty when problem sze ncreases. We hence propose customzed algorthms to obtan near-optmal solutons for large-scale problems. The frst algorthm s based on a smple greedy heurstc, whch can yeld good solutons for many realstc applcatons. But t does not provde nformaton on how close these solutons are from the true optma. Hence we propose a second algorthm based on Lagrangan relaxaton (LR), whch provdes not only good feasble solutons but also optmalty gaps.

20 Greedy algorthm The greedy algorthm for TSC smply selects sensor locatons sequentally based on the best margnal ncrease of objectve (1), untl all N nstallaton locatons have been selected. The exact steps are as follows. Step 0: Intalzaton. Let the set of selected locaton ndces Q := and the teraton ndex n : = 1. Set x = 0, j J ; Step 1: Search for the of objectve th n th n j locaton that wll brng the maxmum margnal mprovement ;.e., select = arg max { ( x ) : = 1, ff Q { }}. The * j k J \ Q z xj j k correspondng margnal objectve mprovement s denoted by ρ n : = z( x ) z( x ), where x j ff j j * = 1, Q { }. Let x * 1 j * = and Q = Q { j }. Setp 2: If n=n, stop and return x and the correspondng objectve value otherwse, n= n+ 1, and go to step 1. N ρn ; n= 1 Greedy heurstc s wdely appled to many practcal problems not only because of ts smplcty but also due to ts reasonable practcal performance. For example, n case of the classc maxmal coverng problem (a specal case of TSC where q=0 and b t =0), Fege (1998) proved that the objectve value of any greedy soluton s no smaller than (1 1 / e) of the true optmum;.e., the approxmaton rato s e/( e 1). More mportantly, no known polynomal-tme algorthm can beat the greedy algorthm n terms of ths approxmaton rato bound (Fege, 1998). We can obtan a smlar approxmaton rato for the maxmal coverng problem wth probablstc faclty falures (a specal case of TSC where b = 0 and q > 0 ). t For general TSC, however, the approxmaton rato of the proposed greedy algorthm s not bounded. Ths can be seen from the followng smple example. Suppose a network has three nodes J = {1, 2, 3}, two lnks {( 1,2),(2,3)}, and two consecutve O-D flow paths,.e., I = { ab, } wth f a = 0, f b = 1, and J = a {1, 2} and J = {2,3} b. If b 0 c =,

21 13 b > 0 and N = 2, a possble soluton from the greedy algorthm s Q = {1,2}, whch t yelds z( x ) = 0. Yet the optmal soluton s obvously Q = {2,3}, whch gves a postve objectve value. Hence, the proposed greedy algorthm for TSC does not have a performance bound, and we propose an LR-based algorthm n the next secton LR-based Algorthm Relaxed subproblems and bounds We relax constrants (5) and (6), and add them to the objectve (1) wth nonnegatve Lagrangan multplers λ = { λ r } and γ = { γ r } TSC (RTSC) becomes:, respectvely. The relaxed (RTSC) R 1 mn λγ, 0 zr ( λ, γ) : = max xhe,, I j J ( H h E ) jr jr + jrejr + I λ 0 (8) r= 0 s.t. (2)-(4) and (7), where the beneft of an nstallaton at locaton j as a level-r head sensor for any I j s H jr r q (1 q) fbm t j λr + λ ( r+ 1) + γr, r = 0,1,, R 2; = r q (1 q) fbm t j λr + γr, r = R 1, (9) and the beneft of ths nstallaton as a level-r rear sensor s E jr r = q (1 q) f ( bm + b ) γ. (10) t j c r For any gven λ and γ, the exact value of z R ( λ, γ ) provdes an upper bound of (1), and t can be obtaned from the followng decomposton scheme. When (5) and (6) are relaxed, assgnments are no longer dependent across j. Constrants (3) requre that the hear assgnment of each j wth sensor nstalled s conducted at exactly one level for each

22 14 I j. Thus to acheve the optmal beneft, j s assgned to as a head sensor at the level correspondng to the maxmum H jr across all r. Smlarly, the correspondng rear assgnment level s chosen to maxmze E jr across all r. Therefore, n RTSC, the contrbuton of nstallng a sensor at j, n terms of objectve (8), s B = [max ( H ) + ma x ( E )] (11) j I r jr j r jr Obvously, the optmal soluton to (8) s to set x j = 1 for the N locatons wth the largest B j values, and accordngly, set h jr = 1 (or e jr = 1) f x j = 1 and r maxmzes H jr (or E jr ) across all r.2 Then the optmal objectve value of RTSC s λ γ = J I λ (12) zr(, ) j Bjxj + 0 Snce the soluton obtaned from the above procedure s probably not feasble to the orgnal TSC problem, heurstc methods are used to construct a feasble soluton. Although such constructve heurstcs do not guarantee the exact optmal soluton, prevous experments (Cornuejols et al., 1977; Caprara et al., 1999) yeld very good feasble, often exactly optmal, solutons (and tght lower bounds of the orgnal objectves) f the Lagrangan multplers are near convergence. One smple heurstc s that we nstall all facltes that are obtaned from RTSC, and then apply the vald assgnment rule to determne the feasble h and e accordngly. If the lower bound equals the upper bound at any teraton, then the optmal soluton s found. Otherwse, the dfference between these bounds provdes an optmalty gap - the dfference between the true optmum and the feasble soluton s sure to be no larger than ths gap. For the classc maxmal coverng problem (q=0and b t =0), Cornuejols et al. (1977) proved that the relatve gap between the optmal LR soluton and the optmal TSC 2 Tes can be broken arbtrarly

23 15 soluton s bounded by 1/e. Ths bound holds for more general problems wth postve falure probablty q>0. It should be noted that the computatonal tme for solvng the RTSC problem (8) and for obtanng an feasble soluton are bounded by ON ( I + J I R) and ON ( I ), respectvely Multpler updatng Functon z ( λ, γ ) s known to be convex over λ and γ. RTSC can be solved wth R an teratve subgradent search. We update λ and γ teratvely to fnd the tghtest upper bound mn λγ, 0 zr ( λ, γ ). We add subscrpt k to dstngush varables n teraton k. The ntal values of the multplers are obtaned wth heurstcs (e.g., the dual soluton to the lnear relaxaton of the orgnal problem). At the end of each teraton, multplers are updated as follows. ( ) 1 λ k+ = max 0, λ k + t k Δλ k, I, r = 0,1,, R 1, r r r ( ) 1 γ k+ = max 0, γ k + t k Δγ k, I, r = 0,1,, R 1, r r r where the subgradents are k Δ λr : = j J h jr j J 1, r = 0 h j( r 1), otherwse, r j J jr jr k and Δ γ : = ( e h ). Step sze t s usually set to k t k = μ ( z ( λ, γ ) z ) k k k LB R R 1 k 2 k 2 I ( Δ λr) + ( Δγr) r= 0,

24 16 where k μ s a control scaler, and LB z s the objectve value of the best-known feasble k 0 soluton. Tradtonally, control scaler μ s determned by settng μ = 2 k μ f R ( and halvng k k z λ, γ ) s not mproved after a fxed number of teratons (Fsher, 1981). Ths approach s modfed by (Fsher, 1981) Caprara et al. (1999) for faster convergence. The dea s to set 0 μ k = 0.1, and compare the best and worst values of (, k z λ γ ) n every certan number (e.g., 20) of teratons: decrease μ k f the dfference s greater than a larger threshold (e.g., 1% ) and ncrease μ k f the dfference s less than a smaller threshold (e.g., 0.1%). In our case study, we use the tradtonal approach when multplers are far from ther optmal values and then swtch to the second approach near convergence. In prncple, the LR algorthm s termnated f one of the followng condtons s satsfed: () the lower bound equals the upper bound, () the optmalty gap stops reducng, and () the soluton tme exceeds a reasonable lmt. Our experence shows that condton () termnates the algorthm most of the tme. In case that happens, we may use the followng branch and bound procedure to further reduce the optmalty gap. R Branch and bound If the aforementoned LR algorthm ends up havng a non-zero optmalty gap, we mplement the LR algorthm nto a branch and bound framework. We branch on varables x n a depth-frst manner, and use a greedy heurstc to choose the next varable x j for branchng: nstallaton at j shall brng n the greatest ncrease of the objectve value (1) gven the varables that have already been branched. We branch each varable frst to 1 (enforcng nstallaton) and then to 0 (forbddng nstallaton). At each node, we run the LR algorthm to determne the lower and upper bounds, whle extra constrants for already-branched varables are exerted. If the upper bound s lower than the best feasble soluton so far, the node no longer has potental and s trmmed. If the current node has already had N enforced or J N forbdden nstallatons, only one non-trval feasble

25 17 soluton exsts and s returned as both the lower and the upper bounds. At each branchng, the multplers of a parent node are passed down to ts chld nodes as ther ntal multplers. 2.3 Algorthm test The Soux-Falls network has 24 vertces and 76 lnks, as shown n Fgure 2.1. Assume that all the vertces are canddate locatons,.e., J = 24. There are 528 traffc O-D pars. For smplcty, we assume that each O-D par only has one flow path that s determned by the shortest path algorthm5, and hence I = 528. Assume too that the sensor at a node can detect all traffc passng that node from dfferent drectons Fgure 2.1. The Soux-Falls test network. (Source:

26 18 The experments are mplemented on a PC wth 2.0 GHz CPU and 2 GB memory. We set a soluton tme lmt of 1800 seconds, and run a seres of nstances b t = 1, b {0,1,10}, N {3,5,7}, and q {0,0.05,0.2,0.5}. The results are summarzed n c Table 1. In ths table, denote the optmal objectve value for the LR-based algorthm by * z, the soluton tme by T, and the optmalty gap by ε. The objectve value found by the greedy algorthm by G z. For comparson, we solve the same nstances wth commercal software CPLEX, and let z C, C C T and ε be the objectve value, the soluton tme and the resdual optmalty gap, respectvely. Let α : = b / ( b + ) be an ndcator of the relatve mportance of path coverage beneft.3 As we can see, the LR-based algorthm found optmal solutons for almost all the nstances ( ε = 0% t t b c ). CPLEX has a comparable performance only when α s small (.e., flow coverage domnates). Otherwse, the performance of CPLEX s sgnfcantly worse than the LR-based algorthm: CPLEX cannot fnd the optmal soluton wthn 1800 seconds for many nstances, and sometmes C t even cannot fnd a meanngful feasble soluton (where z = 0 or ε C = INF% ). The greedy algorthm fnds a good feasble soluton (.e., G z * z ) when α s small. For most nstances wth α = 1, however, the results from the greedy algorthm are qute far from the optma. Ths mples that the greedy algorthm does not work as well when path coverage s the domnatng objectve. Ths s probably because a sensor's contrbuton to path coverage hghly depends on other sensors' locatons. 3 Note that once α s fxed, scalng the value of b (or b ) does not affect the optmal sensor deployment. t c

27 19 Fgure 2.2. Relatonshp between N, q and * z for the Soux-Falls network. * In Table 1, z ncreases wth N and decreases wth q, as expected. Fgure 2.2 further reveals ther relatonshps by plottng values. In * z over N and q for dfferent parameter Fgure 2.2a, curves 1 and 2 are for path coverage only ( α = 1) and curves 3 and 4 are for flow coverage only ( α = 0 ). We see that curves 3 and 4 quckly flatten out whle curves 1 and 2 almost lnearly ncrease untl N s close to J. Ths suggests that path coverage beneft s more senstve to value of N. Ths s probably because the margnal path coverage beneft depends not only on the addtonal nstallaton tself, but also on other nstallatons that form pars wth the addtonal one. The dfferences between curves 1 and 2, and that between 3 and 4 represent the expected coverage loss due to probablstc sensor falures. Although such loss s small for flow coverage, t s sgnfcant for path coverage. Ths s further confrmed by Fgure 2.2b whch shows how * z vares wth q. Curves 5 and 6 are for path coverage whle curves 7 and 8 are for flow coverage. We see that when q s not too large (e.g., q<0.5, whch s true for most realworld cases), curves 5 and 6 drop much faster than curves 7 and 8. Ths confrms the observaton that the beneft loss due to falures s more sgnfcant for path coverage. In ths case, sensor falures should be addressed carefully. It s also nterestng to notce that curves 5 and 6 are rather convex whle 7 and 8 are rather concave, ndcatng opposte senstvty behavors n dfferent q value ranges.

28 20 Table 1 Results for Soux-Falls test network Fgure 2.3 shows the mpact of α and q on the optmal sensor deployment. The lnk wdth llustrates flow volumes. The dark nodes are the optmal nstallaton locatons, whch are generally at places wth heavy traffc. The optmal deployment for path coverage ( α = 1) s more spread-out than that for flow coverage ( α = 0 ). Ths s ntutve

29 21 because more scattered sensor pars can cover longer paths. On the other hand, hgher falure probablty generally leads to a hgher degree of sensor clusterng. Fgure 2.3. Optmal deployment of N=3 nstallatons n the Soux-Falls network.

30 22 CHAPTER 3. CASE STUDY: THE INDOT-MAINTAINABLE NETWORK Chapter 3 dscusses the applcaton of the proposed model to a large-scale real problem, the Chcago ntermodal network. Secton 3.1 ntroduces how to obtan nput data by ntegratng multple data sources. Heurstcs are taken to extract detaled nformaton from lmted data wth coarse granularty. Secton 3.2 analyzes solutons wth dfferent algorthms for a set of nstances. Manageral nsghts are drawn on how coverage measure and falure probabltes nfluence optmal sensor deployment. 3.1 Data preparaton Collectng accurate and qualty data s very mportant to llustrate that the proposed model and soluton technques can be appled to realstc scenaros. Wthout data or wth nvald data, the results of the experments could be msleadng. The data preparaton efforts consst of network porton and freght movement porton. Network We frst had to fgure out the network system n a smple and realstc way, and to transform t n a mathematcal representaton. In Chcago, ntermodal traffc s transported through two networks: hghway and ral. Unlke the hghway network system whch s easly recognzable, ral network system s more complcated and harder to access. Besdes, there were smply not enough data avalable for ral traffc to fgure out complete pcture of ntermodal traffc movement. Our team decded to focus our network on hghway network and ral termnals where hghway and ral nterchange occurs. Here are some of concepts and terms we defned regardng the network.

31 23 1. Access Pont An access pont s an entrance pont that connects the hghway network n Chcago and the outsde world. There are 8 access ponts n the Chcago network, and these ponts bascally defne the network boundary. 2. Conjuncton A conjuncton s an nterchange of two hghways. Conjunctons play an mportant role n ths network as they represent the local populaton. To represent Chcago populaton, we assumed that conjunctons are domnant over local hghway networks meanng conjunctons absorb all local populaton. Bascally, conjunctons functon as local exts as we do not consder any local exts. o Sensors wll be deployed at conjunctons: Logcally, t makes sense to nstall sensors at conjunctons rather than n the mddle of hghway snce traffc only can change ther route at conjunctons. 3. Termnals A termnal s bascally a ral yard where freght transton occurs between ral and truck. There are 17 termnals n Chcago land area, and ther locatons are represented at the nearest hghway ext for the purpose of not dealng wth local road networks. 4. Network Representaton (Sheff, 1985) Recallng the lmted effectveness range of RFID sensors (~31ft), one nstallaton of RFID detecton structure can only cover one travel bound on a hghway. In other words, to cover the entre conjuncton wth RFID, all four drectons of the conjuncton needs to be nstalled wth RFID detecton separately. To break down a conjuncton, we used the network representaton approach developed by Yosef Sheff n Urban Transportaton Networks n Fgure 3.1 explans how a conjuncton breaks down nto multple nodes and drectons.

32 24 Fgure 3.1. Network representaton of an ntersecton When a conjuncton s broken, t creates 4 nodes, each representng a traffc drecton. Usng ths concept, we broke down all 21 conjunctons to obtan 72 equvalent network nodes. Fgure 3.2. Network dagram of Chcago

33 25 Fgure 3.2 s the network dagram of Chcago hghway and termnals. It shows all the locatons of conjunctons, termnals, and access ponts. After the codng of the network, we obtaned 89 total nodes, 353 total lnks, and 1046 O-D flows. Freght Movement Understandng the movement of ntermodal freght traffc s a crucal part of the data preparaton efforts. The prmary data source we used was from the webste of Bureau of Transportaton Statstcs. 4 Gven data that descrbes only a small porton of the entre traffc, we had to utlze the data as much as possble to draw the most realstc pcture as possble. The data summary s shown n Table 2. Table 2. Chcago freght volume table from Bureau of Transportaton Statstcs (unt: thousand tons) Inbound Outbound All Modes 384, ,993 Sngle Mode 371, ,750 Truck 312, ,611 Truck: Outer States 117,289 87,778 Ral 34,343 43,957 Mult Modes 5,926 9, Sngle Mode Sngle mode traffc descrbes the freght traffc that s transported by truck only and has Chcago as destnaton or orgn. Inbound sngle mode 4 Commodty Flow Survey: Metropoltan Areas (2002) Retreved June 20, 2009 from Bureau of Transportaton Statstcs Webste: gan_cty_l_n_w_csa_l_part/ndex.html

34 26 traffc orgnates from other states and crosses an access pont to enter the network and ends up at local Chcago exts represented by conjunctons. Outbound traffc starts at conjunctons and ends up n outer states. These traffc volumes are obtaned at the webste of Bureau of Transportaton Statstcs under Commodty Flow Survey The route that sngle mode traffc chooses s based on the shortest path between Chcago and the metropoltan cty of the state of orgn, and we assgn that traffc volume to the access pont they cross. After assgnng all the volume to the access ponts, then we use the gravty model method to dstrbute the volume to 21 conjunctons accordng to nearby populaton weghts. 2. Intermodal Intermodal traffc here descrbes freght traffc that travels on both ral and hghway. Inbound traffc travels by ral and gets dropped off at termnals wthn the network. Then they travel from termnals to conjunctons by truck, and that s the only part we are concerned about. 3. There are several assumptons we made to smplfy the dffculty assocated wth data scarcty. Frst, we gnored the traffc that goes through Chcago. Gven lmted data, t was almost mpossble to dstngush through traffc from others. Moreover, as we suspected that most of through traffc would take more of perpheral routes around metropoltan area rather than gong through them, we decded to take them out of the pcture. Another area of uncertanty was the ral termnal transacton freght volumes. Ral freght tonnage numbers are gven, but there s smply no data avalable for us to track down how the freght s dstrbuted or whch route they take. Thus, we assumed that all ral freght volume that s transported to the termnal s destned to conjunctons and vce versa.

35 27 Fgure 3.3. Network and freght movement dagram Fgure 3.3 llustrates the Chcago network and the all the freght movements around the network. 3.2 Results We conducted a set of experments on a PC wth 2.0 GHz CPU and 2 GB memory for the greedy algorthm, the LR-based algorthm, and CPLEX. The soluton tme lmt s set to be 1200 seconds. We have run a seres of nstances for b {0,1}, b {0,1,4}, N {10, 20,30}, and q {0,0.2,0.5}. The results are summarzed n Table 3. Due to the ncreased problem sze, CPLEX cannot even get a meanngful feasble soluton for most nstances. The LR-based algorthm always yelds a near-optmum soluton wth a reasonable resdual gap ( 15% ). From our experments, the dfference between the nearoptmal soluton and the optmum s often much smaller than the resdual gap. Thus these solutons are sutable for engneerng practce. t c

36 28 Table 3. Results for Chcago ntermodal network The followng comparsons helped us draw manageral nsghts nto the sensor deployment problem. a. Varyng sensor falure probabltes at flow coverage wth 10 sensors (Fgure 3.4)

37 29 Fgure 3.4. Sensor deployment wth falure probablty of 0% and 20% When the sensors are 100% relable (.e. 0% falure probablty), 10 sensors wll have 96.8% coverage rate; under 20% falure rate, the coverage rate drops to 89.4%. From the fgures, we also see that the sensor deployment at 20% falure probablty shows more scatterng pattern of sensors than 0% falure probablty. Ths s ntutve snce wth hgher falure probablty, sensors need to be densely deployed n the central area to back each other up n case one fals. b. Flow coverage and path coverage (Fgure 3.5) Fgure 3.5. Sensor deployment wth flow coverage and path coverage

38 30 Then we compared flow and path coverage at 0% falure probablty. Path coverage showed a coverage rate of 67.8% whch s a lot less than flow coverage. If we compare the two bg red crcles n the dagram above, we see sensors n the path coverage scenaro are more spread apart than ones n flow coverage. Because the nature of path coverage s to cover as much traffc as possble, path coverage scenaro tends to prortze quantty of nformaton over the qualty of nformaton. c. Number of sensors: 10 vs. 20 (Fgure 3.6) Fgure 3.6. Sensor deployment wth 10 sensors and 20 sensors Fnally, we compared 10 sensors versus 20 sensors on path coverage. Blue dots on rght dagram shows 10 new added sensors onto the orgnal network. The coverage rate s 67.8% for 10 sensors and 92.3% for 20 sensors, and t s obvous that more sensors mprove the coverage rate of the network.

39 31 Fgure 3.7. Falure probablty vs. net beneft Fgure 3.8. Number of nstallatons vs. net beneft Fgures 3.7, 3.8 summarze the comparson of flow and path coverage. We graphed the relatonshps between net beneft and falure probablty and one between net beneft and number of sensor nstallatons. The general observaton s that net beneft ncreases at lower sensor probablty and hgher number of nstallatons. The notable nsght we gan from these graphs s that path coverage beneft s much more senstve

40 32 toward any changes n the network. From the graphs, we fnd that path coverage lnes are steeper than flow coverage lnes, and that shows the senstve nature of path coverage.

41 33 CHAPTER 4. CONCLUSIONS Ths chapter summarzes the research, hghlghts ts contrbutons, and proposes drectons for future research. 4.1 Summary Ths research addresses a new sensor deployment problem n the context of traffc O-D flow survellance usng vehcle ID nspecton technologes (e.g., RFID). In addton to tradtonal flow coverage benefts based on ndvdual sensors, we nvestgated the path coverage benefts from syntheszng the multple sensors n transportaton networks. We consder possble sensor dsruptons, that are very common for many sensor technologes, yet not well addressed untl very recently. A relable locaton desgn model framework s proposed to optmze sensor deployment beneft. Ths model consders both flow and path coverage, whle allowng for probablstc sensor falures. A set of effcent algorthms are developed, and tested on a moderate-sze problem. We fnd that the LR-based algorthm has good performance for the tested problems. The greedy-algorthm can yeld good solutons f flow coverage beneft s sgnfcant. Then we apples ths model to the large-scale Chcago ntermodal network. We study both hghway and ralroad networks and ther connectons. We descrbe our efforts to extract detaled nput flow data from lmted data resources. We examne the qualtes of solutons of dfferent algorthms, the best of whch (from the LR-based algorthm) are analyzed to draw out manageral nsghts about sensor deployment. The experments show that path coverage beneft s more senstve to sensor falures and nstallaton

42 34 budget. It s also found that path coverage tends to spread out sensor locatons whle hgh falure probabltes tend to cluster sensors together. 4.2 Future research drectons Future work can be conducted n several drectons. Frst of all, the proposed model addresses probablstc sensor falures but assumes known O-D flow paths. Ths may be reasonable n the freght operaton context but a more comprehensve model that encompasses traffc routng and assgnment wll be desrable. In addton, the current model assumes all sensor falures are ndependent wth equal probablty. Yet more complex sensor falure patterns (e.g, ste-dependent and correlated falures) are not uncommon n the real world. Addtonal work that relaxes these two assumptons s needed. Fnally, t wll be nterestng to explore how alternatve traffc survellance benefts would affect the optmal sensor deployment pattern.