ESTIMATING DYNAMIC ORIGIN- DESTINATION DEMANDS FROM LINK AND PROBE COUNTS by Bruce R. Helling A thesis submitted to the Deprtment of Civil Engineering in Conformity with the requirements for the degree of Doctor of Philosophy Queen's University Kingston, Ontrio, Cnd August 1994 Copyright Bruce R. Helling, 1994
Abstrct Mny trffic engineering problems consist of evluting number of lternte schemes in terms of some mesures of performnce, such s totl trvel time, volume to cpcity rtio, verge speed, totl fuel consumption, totl emissions, etc. These evlutions generlly require prior knowledge of the temporl trip mking behviour of drivers by origin nd destintion. This thesis presents the development, ppliction, nd evlution of two models cpble of inferring these temporl origin - destintion (O-D) trffic demnds on the bsis of observed link trffic flows nd ssumptions regrding drivers' route choices. In prticulr, this thesis presents the development nd evlution of Lest Squred Error (LSE) nd Lest Reltive Error (LRE) model, ech of which is cpble of estimting either sttic demnds, time series of sttic demnds, or dynmic demnds. Furthermore, the potentil of using probe dt, from route guidnce system (RGS) equipped vehicles, to enhnce these estimted dynmic O-D demnds is exmined. Both of the LSE nd LRE models' mthemticl formultions re presented. The LSE model formultion prllels tht of lest squred regression s the error function is composed of the sum of the squred bsolute difference between the observed nd estimted link flows. In contrst, the LRE model is formulted on the bsis tht the link flow error, when mesured reltive to the observed flow, is to be minimized insted. Itertive solution lgorithms, tht re modifictions of the Jcobi nd Guss-Seidel techniques, re proposed to solve ech of the model formultions. It is shown, by wy of the ppliction of these itertive lgorithms to severl exmple networks, tht the estimted O-D demnds, which result from these itertive solution techniques, re consistent with the model formultions nd with the nlyticl solutions. Furthermore, it is shown for severl exmples tht, when multiple solutions exist which ech exctly replicte the observed link flows, nd no prior O-D demnd informtion is specified, both the LSE nd LRE models estimte demnds tht closely pproximte the mximum likelihood solution. The proposed itertive solution lgorithms hve been incorported into computer model clled QUEENSOD. This model cn be prcticlly pplied to rel networks using current computer memory constrints. This thesis describes the ppliction of the LSE nd LRE models to 35 km section of multilne urbn freewy in Toronto, Cnd, in which lternte prllel routes exist. Dynmic 15 minute O-D demnds were estimted for the estbound direction for the period from 5 m to 11 m. Despite FTMS detector dt being vilble for only 45% of the network, correltion coefficient of pproximtely 98% ws obtined for both models. This vlue reflects the high liner correltion between estimted nd observed link trffic flow dt for this network. The sttisticl nlysis of the expected qulity of O-D demnds, which re estimted solely on the bsis of RGS probe vehicle dt, indicted tht even for levels of mrket penetrtion of 30-50%, the O-D estimtes re unlikely to be of sufficient qulity to be of prcticl benefit. ii
Acknowledgements Mny people hve contributed to this work in direct nd indirect wys. I m grtefully indebted to my collegues for their helpful discussions nd constructive criticisms; the support stff for cheerfully helping me complete cryptic forms nd innumerble other detils tht thretened to impinge on my snity; fculty members of the Deprtment of Civil Engineering for their interest; Dr. Michel Vn Aerde, my supervisor, for his dvice, direction, optimism, nd his infectious enthusism; my prents nd brothers for cring; nd bove ll, my wife, for her constnt support, encourgement nd ptience. Apprecition is extended to Phil Msters nd Dvid Tsui of the Freewy Trffic Mngement Systems Section of the Ontrio Ministry of Trnsporttion for supporting this reserch nd for mking the FTMS dt vilble. The following orgniztions re grtefully cknowledged for their contributions in funding the reserch described in this thesis: Nturl Sciences nd Engineering Reserch Council of Cnd, Ontrio Ministry of Colleges nd Universities, Trnsporttion Assocition of Cnd, Cndin Trnsporttion Reserch Forum, Ontrio Ministry of Trnsporttion of Ontrio, nd Queen's University. iii
iv to Lorrine
Tble of Contents Abstrct ii Acknowledgements iii List of Figures List of Tbles xviii Nomenclture xx Glossry xxiii CHAPTER 1 INTRODUCTION 1 1.1 Wht re Origin-Destintion Demnds nd Why re They Needed?...1 1.2 How cn O-D Demnds be Obtined?...4 1.3 Wht re Dynmic O-D Demnds?...5 1.4 Wht is the Current Sttus of Synthetic O-D Demnd Estimtion?...6 1.5 Wht re some of the Theoreticl nd Prcticl Problems of O-D Estimtion?...7 1.6 Wht is the Scope nd Approch of this Thesis?...9 1.6.1 Thesis objectives...9 1.6.2 Problems ddressed by this thesis...9 1.6.3 Problems not ddressed by this thesis...10 1.6.4 Similrities nd differences to other pproches...10 1.6.5 Method of pproch...11 1.6.6 Method of vlidtion...11 1.6.7 Method of presenttion...12 CHAPTER 2 ASSESSMENT OF THE CURRENT STATE-OF-THE-ART 14 2.1 Introduction... 14 2.2 Generl Solution Approch Ctegories...15 2.2.1 O-D estimtes from trnsporttion plnning models...15 2.2.2 O-D estimtes from direct smpling...17 2.2.3 O-D estimtes from link counts...18 2.3 Definition of Link-Count Bsed O-D Estimtion Problem...19 2.4 Ctegoriztion of Link-Count Bsed Approches...20 2.4.1 Heuristic vs. mthemticl pproches...21 2.4.2 Intersection vs. network pproches...22 2.4.3 Pre-specified vs. vrible routes...23 xii v
Tble of Contents (continued) 2.4.4 Sttic vs. dynmic pproches...24 2.5 Additionl Issues...26 2.5.1 Under- nd over- specifiction...26 2.5.2 Accurcy of estimted O-D demnds...28 2.5.3 Use of priori informtion...29 2.6 Summry...30 CHAPTER 3 ANALYSIS AND COMPARISON OF TWO MAXIMUM LIKELIHOOD APPROACHES 31 3.1 Introduction... 31 3.1.1 Bckground...31 3.1.2 Structure of chpter...32 3.2 Generlized Model...32 3.2.1 Introduction...32 3.2.2 Mthemticl derivtion...33 3.3 Vehicle Link-Count Bsed Model...38 3.3.1 Model formultion...38 3.3.2 Anlyticl solution for simple network...40 3.3.3 Forml solution...43 3.4 Trip-Count Bsed Model...45 3.4.1 Model formultion...45 3.4.2 Forml solution...47 3.5 Exmintion of Model Assumptions...47 3.5.1 Stirling's pproximtion...48 3.5.2 Totl number of trips...49 3.6 Comprison of Methods...50 3.6.1 Exmple networks...50 3.6.2 Effect of prior informtion...51 3.6.3 Effect of redundnt link flows...53 3.7 Summry...54 CHAPTER 4 DEVELOPMENT OF A LEAST SQUARES O-D ESTIMATION MODEL 56 4.1 Introduction... 56 vi
Tble of Contents (continued) 4.2 Development of the LSE Model...57 4.2.1 Mthemticl bsis...57 4.2.2 Exmple with link flow continuity...58 4.2.3 Exmple with link flow discontinuity...59 4.2.4 Exmple with multipth routes...60 4.3 Incorportion of Link Flow Relibility...61 4.3.1 Modified model formultion...61 4.3.2 Exmple considering reltive relibility of link flows...62 4.3.3 Determining pproprite link flow relibility fctors...65 4.4 Chrcteristics of Formultion...69 4.4.1 Negtive demnd estimtes...69 4.4.2 Suitbility of error function...71 4.4.3 Selecting solution when multiple solutions exist...71 4.5 Development of Solution Algorithm...73 4.5.1 Forml solution...73 4.5.2 Limittions of the forml solution...74 4.5.3 Itertive solution lgorithm...75 4.5.4 Algorithm stopping criteri...80 4.5.5 Appliction of itertive lgorithm to simple network hving multiple solutions...82 4.5.6 Appliction of itertive lgorithm to second simple network...85 4.5.7 Appliction of itertive lgorithm to network with non-uniform link relibility fctors...86 4.6 Incorportion of Seed O-D Demnd Relibility...88 4.6.1 Introduction...88 4.6.2 Modifiction to the solution lgorithm...89 4.6.3 Exmple ppliction considering the reltive relibility of seed O-D demnds...90 4.7 Impct of Redundnt Link Flows...93 4.8 Summry...94 CHAPTER 5 DEVELOPMENT OF A LEAST RELATIVE ERROR O-D ESTIMATION MODEL 95 5.1 Introduction... 95 vii
Tble of Contents (continued) 5.2 Development of the LRE Model...95 5.2.1 Mthemticl bsis...96 5.2.2 Exmple with link flow continuity...99 5.2.3 Exmple with link flow discontinuity...101 5.2.4 Exmple with multipth routes...104 5.3 Incorportion of Link Flow Relibility...105 5.4 Chrcteristics of Formultion...105 5.5 Development of Solution Algorithm...106 5.5.1 Forml solution...106 5.5.2 Itertive solution lgorithm...107 5.5.3 Algorithm stopping criteri...108 5.5.4 Appliction of itertive lgorithm to simple network hving multiple solutions...111 5.5.5 Appliction of itertive lgorithm to simple network hving single solution with multiple pths...114 5.6 Incorportion of Seed O-D Demnd Relibility...117 5.6.1 Modifiction to lgorithm...117 5.6.2 Exmple ppliction considering the reltive relibility of seed O-D demnds...118 5.7 Summry...124 CHAPTER 6 DYNAMIC EXTENSION OF STATIC LSE AND LRE MODELS 126 6.1 Introduction... 126 6.2 Estblishing tht Need for Dynmic Demnds Exists...126 6.2.1 Sptil nd temporl vritions evident in field O-D ptterns...127 6.2.2 Effects of violting sttic estimtion ssumptions...130 6.3 Extensions to LSE nd LRE Sttic Models...132 6.3.1 Formultion of dynmic LSE nd LRE models...133 6.3.2 Chrcteristics of dynmic estimtion...134 6.3.3 Generting link use probbilities from known route trees nd link trvel times...135 6.3.4 Identifying extrneous link flows...137 6.4 Appliction of Dynmic Models to Smll Hypotheticl Network...139 6.4.1 Network chrcteristics nd scenrio descriptions...139 viii
Tble of Contents (continued) 6.4.2 Mesures for comprison...140 6.4.3 Anlysis of model results...142 6.5 Limittions of Dynmic Models...143 6.6 Appliction of Dynmic Models to Lrger Hypotheticl Network...144 6.6.1 Network chrcteristics...145 6.6.2 Description of the INTEGRATION model...146 6.6.3 Scenrios being exmined...146 6.6.4 Anlysis of model results...147 6.7 Summry...152 CHAPTER 7 ESTIMATING O-D DEMANDS FOR A FREEWAY CORRIDOR IN TORONTO, CANADA 153 7.1 Introduction... 153 7.1.1 Description of Highwy 401...154 7.1.2 Description of the freewy trffic mngement system...155 7.2 Acquisition of Dt Necessry for O-D Estimtion...155 7.2.1 Network representtion...155 7.2.2 Link flows... 156 7.2.3 Route determintion...157 7.2.4 Link trvel times...158 7.3 Determintion of Time Vrying Demnds...161 7.3.1 Introduction...161 7.3.2 Evlution of estimted demnds with respect to ggregted observed link flows... 162 7.3.3 Evlution of estimted demnds with respect to individul observed link flows... 164 7.3.4 Evlution of estimted demnds with respect to observed origin productions nd destintion ttrctions...167 7.3.5 Further chrcteristics of the estimted O-D demnds...172 7.4 Summry...174 CHAPTER 8 RGS VEHICLE PROBES AS ESTIMATORS OF DYNAMIC O-D DEMANDS 175 8.1 Introduction... 175 8.1.1 Vehicle probe dt...175 ix
Tble of Contents (continued) 8.1.2 Focus of this chpter...176 8.1.3 Nomenclture...177 8.2 Derivtion of Estimte of Popultion O-D Demnds...178 8.2.1 Estimting the popultion O-D demnd D - Approch 1...178 8.2.2 Estimting the popultion O-D demnd D - Approch 2...179 8.2.3 Estimting the popultion probbility P nd totl deprtures D... 179 8.2.4 Estimting the exct level of mrket penetrtion from route flows...180 8.2.5 Estimting the pproximte level of mrket penetrtion from link flows... 180 8.2.6 Summry...181 8.3 Derivtion of Estimte of Relibility of O-D Demnd Estimtes...182 8.3.1 Estimting the relibility of the O-D demnd estimte...182 8.3.2 Estimting the relibility of the popultion probbility...183 8.3.3 Estimting the relibility of the totl number of network deprtures...183 8.3.4 Summry...184 8.4 Prcticl Considertions...184 8.4.1 Popultion size...185 8.4.2 Level of surveillnce...185 8.4.3 Impct on O-D relibility...186 8.4.4 Exmple illustrtion...186 8.5 Evlution of Network Level of Surveillnce...188 8.6 Exmple Network Description nd Results...190 8.6.1 Network structure...190 8.6.2 Results for exmple network...192 8.7 Summry...195 CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS 197 9.1 Overview... 197 9.2 Conclusions...197 9.2.1 Proposed models...198 9.2.2 Appliction of proposed models...198 9.2.3 Potentil of probe dt...199 9.3 Limittions of this Reserch...200 x
Tble of Contents (continued) 9.3.1 Limittions of the LSE nd LRE models...200 9.3.2 Limittions of the sttisticl evlution of probe dt...200 9.4 Significnce of the Thesis Contributions...201 9.4.1 Significnce of the LSE nd LRE models...201 9.4.2 Significnce of the evlution of vehicle probe dt...201 9.5 Recommendtions for Further Work...202 9.5.1 Further development of the LSE nd LRE models...202 9.5.2 Further development of the use of probe dt in estimting dynmic demnds...202 REFERENCES 204 xi
List of Figures Figure Title Pge 1-1 Simple network used to illustrte the impct of ssuming link flows versus O-D demnds s independent inputs to typicl trffic engineering evlution... 3 2-1 Use of the trip genertion nd trip distribution phses of the trnsporttion plnning process to estimte O-D demnds... 17 2-2 Use of prtil license plte number smpling to estimte O-D demnds... 18 2-3 Use of observed link flows to estimte O-D demnds... 19 2-4 Estimting sttic turning movements for n intersection... 23 2-5 Estimting time vrying turning movements t intersections... 26 2-6 Grphicl representtion of mximum possible reltive O-D error... 29 3-1 Illustrtion of simple exmple hving two possible outcomes... 34 3-2 Comprison of the informtion content contined within possible outcome under the ssumption tht outcomes re eqully likely (ln(g)) nd tht outcomes re weighted by their likelihood of occurring (ln(g'))... 35 3-3 Comprison of the informtion content contined within possible outcomes under the ssumption tht the two outcomes re eqully likely nd under the ssumption tht stte A hs 75% likelihood of occurring... 36 3-4 Simple exmple two-link network... 40 3-5 Informtion objective function s function of estimted demnd when no prior informtion exists... 42 3-6 Informtion objective function mgnitude s function of fesible priori mtrices nd estimted demnd... 42 3-7 Informtion objective function s function of estimted demnd when prior informtion exists but this prior demnd is not fesible solution... 43 3-8 Qulity of two different versions of Stirling's pproximtion... 48 3-9 Comprison of lterntive ssumptions bout the totl number of trips... 50 3-10 Two simple networks used to illustrte severl chrcteristics of the link bsed entropy nd link-count bsed informtion model formultions... 51 4-1 Exmple three-link network... 59 4-2 Exmple four-link network exhibiting multiple pths between n origin nd destintion zone... 61 4-3 Exmple line three-link network used to illustrte the impct of link flow relibility... 63 4-4 Vrition in link flow error s function of link flow relibility... 64 xii
List of Figures (continued) Figure Title 4-5 Effect of link flow relibility on optiml O-D demnd... 64 Pge 4-6 Illustrtion of the impct of vrince on the vlidity of the men... 65 4-7 Vrince of vehicle counts observed on three lne section of Highwy 401 Estbound Collectors t Avenue Rod... 66 4-8 Coefficient of vrition of vehicle counts observed on three lne section of Highwy 401 Estbound Collectors t Avenue Rod... 67 4-9 Exmple three-link network used to illustrte heuristic method of determining link flow relibility... 68 4-10 Impct of trunction on solution optimlity... 70 4-11 Exmple two-link network... 72 4-12 Illustrtion of the fesible O-D solution spce for two-link exmple network... 72 4-13 Effect of the relxtion fctor nd the number of itertions on the squred link flow error (Seed mtrix = (5,5,5))... 83 4-14 Effect of the relxtion fctor nd the number of itertions on selected O-D demnd - T AC (Seed mtrix = (5,5,5))... 84 4-15 Demnd estimtes nd ssocited squred link flow error s function of the itertion for sitution in which the unconstrined optimum results in negtive O-D estimte... 86 4-16 Demnd estimtes nd ssocited squred link flow error s function of the itertion for sitution in which non-uniform link flow relibility fctors exist... 87 4-17 Demnd estimtes nd ssocited squred link flow error s function of the link flow relibility... 88 4-18 Exmple four-link network used to illustrte the impct of the reltive relibility of seed O-D demnds... 90 4-19 Intermedite estimtes of O-D demnd T AD for different levels of seed O- D demnd relibility... 92 4-20 Similrity of estimted demnd (T AD ) to seed demnd (t AD ) s function of seed relibility fctor (R AD )... 92 4-21 Finl estimtes of O-D demnds for different levels of seed O-D demnd relibility (t AD )... 93 5-1 Illustrtion of the functionl form of the LRE model's error function... 97 5-2 Exmple three-link network... 100 xiii
List of Figures (continued) Figure Title Pge 5-3 Grphicl illustrtion of the concvity of the reltive link flow error function in the vicinity of the optiml solution... 103 5-4 Exmple four-link network exhibiting multiple pths between n origin nd destintion zone... 104 5-5 Comprison of ctul verge reltive error nd verge reltive error estimted using Eqution [5-14] over rnge of flow errors... 110 5-6 Comprison of ctul bsolute error nd bsolute error estimted using Eqution [5-16] over rnge of flow errors... 110 5-7 Exmple four-link network in which multiple fesible O-D solutions re possible... 111 5-8 Itertive estimtes of O-D demnds for simple exmple network hving ll-or-nothing routes... 113 5-9 Link flows resulting from estimtes of O-D demnds for simple exmple network hving ll-or-nothing routes... 113 5-10 Rte of convergence of demnds estimted by the LRE model for simple network hving single solution nd multiple pths... 116 5-11 Reltive link flow error resulting from the ppliction of the LRE lgorithm to simple network hving multiple pth routes... 116 5-12 Mrginl improvement in reltive link flow error (E R ) resulting from the ppliction of the LRE lgorithm to simple network hving multiple 117 pths... 5-13 Exmple four-link network used to exmine the reltive relibility of seed O-D demnds... 118 5-14 Similrity of estimted demnd (T AD ) to seed demnd (t AD ) s function of seed relibility fctor (R AD )... 120 5-15 Intermedite estimtes of O-D demnds for Scenrio 2... 121 5-16 Intermedite estimtes of O-D demnds for Scenrio 3... 121 5-17 Reltive error (E R ) of intermedite O-D demnd estimtes for Scenrio 3.. 122 5-18 Intermedite estimtes of O-D demnd T AD for different levels of seed O- D demnd relibility... 123 5-19 Finl estimtes of O-D demnds for different levels of seed O-D demnd relibility (t AD )... 124 6-1 Typicl observed ggregted 15 minute link volumes by time of dy... 128 6-2 Typicl temporl nd sptil vrition in O-D demnds s depicted by totl observed flow from specific origin zones... 129 xiv
List of Figures (continued) Figure Title Pge 6-3 Typicl temporl nd sptil vrition in O-D demnds s depicted by totl observed flow to specific destintion zones... 129 6-4 Approximting temporl vritions in link flows through the use of time series of sttic O-D demnds... 130 6-5 Exmple liner five-link network used to illustrte the reltive ccurcy of sttic demnd, time-series of sttic demnds, nd dynmic demnd... 130 6-6 O-D estimtion results s function of the estimtion process utilized... 132 6-7 Discretized representtion of demnd propgtion over time nd spce... 136 6-8 Computtion of observed link flows bsed on demnd deprture time nd link trvel times... 136 6-9 Illustrtion of occurrence of flows tht do not result from demnds considered within nlysis period... 138 6-10 Illustrtion of more complex sitution for the determintion of extrneous link flows... 138 6-11 Illustrtion of the occurrence of prtilly extrneous flows... 139 6-12 Exmple five-link network used to illustrte the performnce chrcteristics of the dynmic LSE nd LRE models... 139 6-13 Illustrtion of the sensitivity of the normlized link flow error (E n ) nd the correltion coefficient to errors link flows... 142 6-14 Illustrtion of possible discrepncies between true demnd profile nd observed cpcity constrined flows... 144 6-15 Process used to generte n pproprite time-series of observed link flows nd routes using the INTEGRATION simultion model... 144 6-16 Hypotheticl network used to illustrte the dynmic estimtion bilities of the LSE nd LRE models... 145 6-17 Observed link flows nd flows estimted by the dynmic LRE model from the true dynmic demnd (Scenrio 1)... 150 6-18 Selected O-D demnds estimted by the dynmic LRE model during ech itertion (Scenrio 2b)... 150 6-19 Normlized link flow error resulting from the LRE model s function of the itertion number... 151 6-20 Normlized link flow error resulting from the LSE model s function of the itertion number... 151 7-1 Loction of Highwy 401 study site... 154 xv
List of Figures (continued) Figure Title Pge 7-2 Link nd node representtion of the Highwy 401 estbound network... 156 7-3 Typicl 20 second nd ggregted 15 minute FTMS detector volume dt reflecting conditions on 3 lne section in the estbound collector t Avenue Rod on My 1, 1991... 157 7-4 Normlized link error resulting from the LSE model... 163 7-4b Normlized link error resulting from the LRE model... 163 7-5 Comprison of observed nd estimted flows by time of dy for link 8... 165 7-5b Comprison of observed nd estimted flows by time of dy for link 12... 165 7-6 Correltion between observed link flows nd those estimted by the dynmic LSE model... 166 7-6b Correltion between observed link flows nd those estimted by the dynmic LRE model... 166 7-7 RMS error between estimted nd observed totl origin productions presented s percent of the verge observed totl... 170 7-7b RMS error between estimted nd observed totl destintion ttrctions presented s percent of the verge observed totl... 170 7-8 Correltion between observed totl destintion zone flows nd those estimted by the LSE model... 171 7-8b Correltion between observed totl destintion zone flows nd those estimted by the LRE model... 171 7-9 Averge trip length implied by the time vrying demnd estimted by the LSE nd LRE models s function of itertion number... 173 7-10 Totl number of trips contined within the time vrying demnd estimted by LSE nd LRE models s function of itertion number... 173 8-1 Schemtic representtion of reltionships between severl key vribles which were defined... 178 8-2 Illustrtion of the coefficient of vrition of estimted O-D demnd s function of level of mrket penetrtion ssuming ll links in network re under surveillnce... 187 8-3 Illustrtion of the coefficient of vrition of estimted O-D demnd s function of totl number of trips ssuming ll links in network re under surveillnce... 188 8-4 Configurtion of exmple trffic network... 191 8-5 Process used to produce relistic trffic conditions on exmple network... 191 xvi
List of Figures (continued) Figure Title Pge 8-6 Smple of typicl probe informtion provided by simultion model... 192 8-7 Rndom vrition in the estimted level of mrket penetrtion (m) for globl level of mrket penetrtion (M) of 5% nd 20%... 193 8-8 Anlyticl estimtes nd ctul simultion results for selected O-D demnd over time of dy for n verge level of mrket penetrtion of 20% (O-D pir 1-3)... 194 8-9 Anlyticl estimtes nd simultion results for selected O-D demnd for vrious levels of mrket penetrtion (O-D pir 1-3 for time period from 7:15-7:30 AM)... 194 8-10 Effect of level of mrket penetrtion on the ccurcy of O-D estimtion... 196 xvii
List of Tbles Tble Title Pge 1-1 Chrcteristics of exmple network... 3 1-2 Illustrtion of severl theoreticl difficulties tht cn be encountered when estimting O-D demnds from link flows... 8 3-1 Chrcteristics of exmple networks... 51 3-2 Mximum likelihood O-D estimtes for two-link exmple network... 53 3-3 Mximum likelihood O-D estimtes for four-link network with redundnt link flows... 54 4-1 Squred link flow error for ll potentil O-D solutions in the vicinity of the nlyticl solution (vph 2 )... 60 4-2 Exmple ppliction of heuristic pproch to determining link flow relibility fctors... 69 4-3 Guss-Seidel nd Jcobi itertive method results from simple network hving multiple solutions tht stisfy the system of liner equtions... 77 4-4 Itertive solution lgorithm for estimting O-D demnds tht minimize the squred link flow errors... 80 4-5 Itertive lgorithm results for simple exmple network... 82 4-6 Scenrio configurtions nd estimted demnds used to illustrte the impct of seed O-D demnd relibility fctors... 90 4-7 LSE model O-D estimtes for four-link network with redundnt link 93 flows... 5-1 Left hnd side of non-liner constrints computed for severl O-D demnds (vph)... 101 5-1b Reltive link flow error (E R ) for ll potentil O-D solutions in the vicinity of the nlyticl solution... 101 5-2 Left hnd side of non-liner constrints computed for severl O-D demnds for network exhibiting link flow discontinuity (vph)... 102 5-2b Reltive link flow error (E R ) for ll potentil O-D solutions in the vicinity of the nlyticl solution for network exhibiting link flow discontinuity... 102 5-3 Left hnd side of non-liner constrints computed for severl O-D demnds for network exhibiting multipth routes... 105 5-4 LRE itertive solution lgorithm... 108 5-5 Itertive lgorithm results of LRE model for simple exmple network with multiple solutions nd ll-or-nothing routings... 112 xviii
List of Tbles (continued) 5-6 Link use probbilities ssocited with the four-link exmple network in which multiple pths exist... 114 Tble Title Pge 5-7 Scenrio configurtions nd estimted demnds used to illustrte the impct of seed O-D demnd relibility fctors... 118 6-1 Link counts observed on exmple liner five-link network... 131 6-2 Actul network O-D demnds nd observed link flows... 140 6-3 Description of scenrios evluted to illustrte the dynmic estimtion bilities of the LSE nd the LRE models... 140 6-4 Results from the ppliction of the dynmic LSE nd LRE models to simple five-link network... 143 6-5 Actul time vrying demnd used to determine observed flows... 146 6-6 Description of scenrios evluted to illustrte the dynmic estimtion bilities of LSE nd LRE models... 147 6-7 LSE nd LRE dynmic model results for hypotheticl integrted freewy/rteril network... 149 7-1 Number of links nd nodes used to represent the estbound direction of the Highwy 401 network... 156 7-2 Trunction effect on recorded sttion occupncies... 159 7-3 Mesured trip durtion s function of dy, deprture time, nd route tken, for the estbound direction of Highwy 401... 161 7-4 Definition of zones nd the links tht constitute the zonl flow... 169 xix
Nomenclture Term α β ϕ E n E R n λ C c C d d D d D D 1 D 2 E E R E E bs E n E rel E R n E R G i Description relxtion fctor used in the LSE model's itertive solution lgorithm dimensionless prmeter used in BPR trvel time function dimensionless prmeter used in BPR trvel time function mrginl reduction in verge reltive link flow error (LSE model) mrginl reduction in verge reltive link flow error (LRE model) Lgrnge multiplier for link link number link cpcity (vph) the number of probe deprture clls received for origin i nd destintion j during the given time slice vlue of the constrint eqution for origin i nd destintion j time period when demnd deprts origin i n estimte of D bsed, in prt, on probe nd stndrd loop detector dt the true totl number of trip deprtures initited in the entire network during the given time slice n estimte of the number of trip deprtures between origin i nd destintion j the true number of trip deprtures between origin i nd destintion j in the given time period discontinuity of upstrem node of link discontinuity of downstrem node of link totl squred link flow error (LSE model) totl reltive link flow error (LRE model) verge bsolute link flow error estimted from E true verge bsolute link flow error verge bsolute link flow error estimted s proportion of the observed flow true verge bsolute link flow error computed s proportion of the observed flow verge bsolute link flow error estimted from E R verge reltive link flow error estimted s proportion of the observed flow number of eqully probbly outcomes ssocited with solution origin zone xx
Nomenclture (continued) Term I j K m M M n p o P P t P k p P P d o q r od r v R t T t 0 t T T d l+1 T V V Description quntity of informtion destintion zone constnt used in informtion theory n estimte of M bsed on smple of the RGS equipped nd non-equipped vehicles observed on selected links on the network the popultion network wide level of mrket penetrtion defined s the proportion of ll trips, deprting t some time t, tht re RGS-equipped the popultion level of mrket penetrtion for origin i nd destintion j the totl number of probe vehicles inititing trips in the network during the current period time period when demnd is observed rriving on link mtrix of link use probbilities trnspose of the mtrix P proportion of the flow entering on leg tht exits t leg k n estimte of P mde from some smple of the totl popultion (the probes) the popultion probbility of ny given trip being between origin i nd destintion j proportion of demnd deprting origin i t time d en route to destintion j, tht will rrive on link t time o prior probbility tht counts on link re ssocited with origin i nd destintion j smple correltion coefficient computed between estimted nd true O-D demnds smple correltion coefficient computed between estimted nd observed flows reltive relibility of the flow observed on link totl trvel time (seconds) column vector of unknown O-D trffic demnds free speed trvel time (seconds) prior estimte of the trffic demnd between origin i nd destintion j (vph) trffic demnd deprting from origin i destined for zone j (vph) trffic demnd deprting from origin i t time d destined for zone j (vph) new estimte of the true trffic demnd between origin i nd destintion j (vph) link flow (vph) column vector of observed link flows xxi
Nomenclture (continued) Term V V i V o V o k V' V' o W Description flow estimted to trverse link (vph) observed flow entering the intersection t leg k (vph) flow estimted to rrive on link during time period o (vph) observed flow exiting the intersection t leg k (vph) flow observed to trverse link (vph) flow observed to rrive on link during time period o (vph) mesure of entropy xxii
Glossry Term AADT ATMS BPR CCTV COMPASS continuity CONTRAM COV FHWA FREQ FTMS INTEGRATION ITE IVHS LRE Model LSE Model MOP MTO O-D RGS Seed TMC UTCS Description verge nnul dily trffic dvnced trffic mngement systems Bureu of Public Rods closed circuit television cmers FTMS operting on Highwy 401 in Toronto, Cnd Flow continuity t nodes is used in this thesis to imply tht totl flow into the node is equl to the totl flow out of the node. The term consistency my be substituted for continuity. route choice nd trffic simultion model coefficient of vrition (stndrd devition divided by the men) Federl Highwy Administrtion trffic simultion model bsed on shock wve nlysis freewy trffic mngement system microscopic integrted network trffic simultion model Institute of Trnsporttion Engineers intelligent vehicle highwy systems lest reltive error O-D estimtion model lest squres error O-D estimtion model mesure of performnce Ministry of Trnsporttion of Ontrio origin - destintion route guidnce system seed mtrix is used interchngebly with prior mtrix trffic mngement centre urbn trffic control systems xxiii
CHAPTER 1 INTRODUCTION Potentil solutions to mny trffic engineering problems re typiclly evluted on the bsis of ggregted link-level mesures of performnce such s verge speed, verge dely, nd totl fuel consumption. In mny urbn centres, link-level trffic dt, such s trffic volume, lne occupncy, nd trffic speed, re redily vilble from existing trffic surveillnce systems. Unfortuntely, since link-level dt re often impcted by trffic control systems, such s rmp meters nd trffic signls, the evlution of mny potentil trffic engineering solutions requires tht trip-level dt, such s trip origin, trip destintion, nd trip deprture time, be known. As these trip-level dt re often independent, in the short-term, of trffic control systems, they cn represent the independent input into the trffic engineering evlution process. Unfortuntely, trip-level dt re typiclly not redily vilble nd re usully very costly to obtin through direct mesures. This thesis exmines the problem of estimting generlly unknown trip-level dt (origin - destintion trffic demnds) from redily vilble link-level dt. This chpter introduces the problem of estimting sttic nd dynmic origin - destintion trffic demnds nd briefly discusses pproches to solving this problem. First, four questions re posed nd nswered; "Wht re O-D demnds?", "How cn they be obtined?", "Wht re dynmic O-D demnds?", nd "Wht is the current sttus of O-D demnd estimtion?". Subsequently, severl theoreticl nd prcticl difficulties re identified. Finlly, the scope nd pproch of this thesis re defined nd presented. 1.1 Wht re Origin-Destintion Demnds nd Why re They Needed? Mny trffic engineering problems consist of evluting number of lterntive schemes in terms of some mesures of performnce, such s totl trvel time, volume to cpcity rtio, verge speed, totl fuel consumption, totl emissions, etc. Typiclly, when mking these evlutions, it is ssumed tht trffic flows on links re known nd tht the mgnitudes of 1
Chpter 1: Assessment of the Current Stte-of-the-Art 2 these trffic flows re not ffected by the performnce of the scheme being evluted. For exmple, the nlysis of isolted trffic signls ssumes tht the trffic flows on ech pproch link re known priori. It is further ssumed tht these pproch flows remin unffected by the performnce of the trffic signl regrdless of how well or how poorly the trffic signl opertes. This ssumption is not limited to the nlysis of isolted intersections. The TRANSYT model (Robertson, 1969; FHWA, 1984), which is cpble of optimizing coordinted signl networks, lso ssumes tht the mgnitude of ll pproch flows re known nd remin unchnged by the performnce of the signls. For these evlution pproches, link flows re considered to be independent input dt. However, trffic networks re dynmic environments in which multiple feedbck loops exist. Individul drivers my lter their routing ptterns in response to the trffic conditions tht they experience, nticipte, or re informed of. Thus, the mgnitude of the link trffic flows re ctully outcomes of process tht my be dependent on the network trffic conditions tht re experienced. The obvious question then, is "Wht re the independent inputs?". In the context of trffic routing nlyses, which re conducted for the purpose of evluting opertionl control strtegies, it cn often be ssumed tht trip origin nd trip destintion loctions re fixed nd re therefore the independent inputs. Consider n utomobile trip mde for the purpose of commuting to work. The origin of the trip is the driver's residence, while the destintion is the driver's plce of employment. In the short term, both of these loctions cn often be ssumed to be fixed nd thus they re not ffected by the existing trffic conditions. One my now consider ll of the drivers within n urbn re, ech deprting from their own residence (origin) t some distinct time, en route to their own plce of employment (destintion). This is the most disggregte level t which the system cn be represented. By nture, origin-destintion (O-D) demnds re discrete rther thn continuous quntities s vehicles re discrete entities, for which frctionl vlues hve no physicl mening. Of course, even s discrete quntities, it is not often prcticl, for even smll networks, to nlyze the trip mking problem t this micro level. Rther, it is usully more convenient to ggregte both time nd spce in some mnner. First, the network is divided into severl geogrphic zones. Any trip beginning from loction within the boundries of tht zone, is considered to originte from tht zone. Similrly, trip's destintion zone is tht zone whose boundries encompss the specific loction of the trip's destintion. Trip deprture time is usully lso ggregted into number of periods of finite durtion. All trips tht hve deprted fter the strt of period nd prior to the end of tht sme period, re often ggregted s if they hd deprted t uniform rte throughout tht time period. The ggregted trip mking behviour, of ll of the individul drivers, is referred to s the origin-destintion demnd for prticulr network within prticulr time frme. For ech origin-destintion pir, there will exist some number of vehicles tht will, on verge, mke trip from the origin zone to the destintion zone. Usully this demnd is represented s the number of vehicles tht wish to mke trip per unit of time (i.e. vehicles per hour). It is useful to illustrte the potentil impct of considering link flows s the independent inputs versus considering O-D demnds s the independent inputs. Consider the simple network illustrted in Figure 1-1, hving two origins, one destintion, nd six links. A bridge, hving fixed cpcity of 4000 vph (vehicles per hour) exists on link 5. The existing O-D demnds nd resulting link flows re provided in Tble 1-1. It is desired tht the impct on
Chpter 1: Assessment of the Current Stte-of-the-Art 3 the system of incresing the bridge cpcity to 6000 vph be estimted. It is ssumed tht link trvel time increses linerly with flow, so tht t cpcity, trvel times re twice s long s the free flow trvel time. It is further ssumed tht ll drivers choose their routes so s to minimize their own trvel time. If it is ssumed tht link flows re independent inputs, then link flows re fixed nd the only impct of incresing the cpcity of the bridge is decrese in trvel time cross the bridge s the volume to cpcity rtio is reduced from 1.0 to 0.66. If, however, link flows re ssumed to be vrible nd O-D demnds re considered to be the independent inputs, then in response to the incresed cpcity on the bridge, drivers could lter their routes to minimize their own trvel times nd the flows on the links could chnge. As indicted, the mgnitude of the expected impct on system dely, of incresing the cpcity of the bridge, cn be significntly different depending on whether link flows or O-D demnds re ssumed s the independent input. A B 4 3 1 2 5 6 C Demnds: A-C = 4000 vph B-C = 2000 vph Figure 1-1: Simple network used to illustrte the impct of ssuming link flows versus O-D demnds s independent inputs to typicl trffic engineering evlution Tble 1-1: Chrcteristics of exmple network Link Cpcity Free Flow Existing Flow Independent O-D Independent (vph) Time (min.) Flows (vph) Link Flows Link Trvel Time Link Flows Link Trvel Time 1 4000 10 4000 4000 20.0 4000 20.0 2 4000 20 2500 2500 32.5 2182 30.9 3 4000 10 1500 1500 13.8 1818 14.5 4 4000 10 2000 2000 15.0 2000 15.0 5 4000 10 3500 3500 18.8 3818 16.4 6 6000 10 6000 6000 20.0 6000 20.0 System Dely 6625 hours 6440 hours The ssumption tht trip origins nd destintions re fixed is not strictly true in the context of regionl plnning nlysis, in which long term interctions between lnd use development, employment opportunities, residentil loctions, nd trnsporttion infrstructure, re ll inter-relted. For exmple, the cretion of lower cost suburbn housing permits people to chnge the loction of their residence. Alterntively, the supply of new trnsporttion infrstructure my stimulte the development of new businesses, which in turn provide new trip destintions t these new employment opportunities. However, these chnges generlly occur slowly, such tht O-D demnds cn be considered to remin constnt within the much shorter time spn tht is considered in the evlution of most trffic control strtegies. For O-D demnds to be considered the independent inputs, it must lso be ssumed tht lterntive control strtegies do not significntly lter the demnds. Depending on the control strtegy being investigted, O-D demnds my chnge in severl wys. For exmple, the
Chpter 1: Assessment of the Current Stte-of-the-Art 4 introduction of flexible work hours will ffect trip deprture time. Similrly, ny ltertions to the utility, or reltive ttrctiveness, of one or more of the vilble trnsporttion modes my ffect trip mode choice between the use of privte utomobiles versus public trnsit. Finlly, the reloction of lrge employment centers, such s government offices, will ffect the loction of some work trip destintions. Any of these chnges my lso cuse some trip mkers to decide not to mke the trip, or cuse dditionl trips to be mde. All of these scenrios reflect significnt chnges to lnd use or trnsporttion infrstructure, nd s such re usully not nlyzed using trffic engineering techniques. These situtions re therefore not exmined in this thesis. 1.2 How cn O-D Demnds be Obtined? O-D demnds cn be observed directly only through some form of survey, in which the ctul origin nd destintion of vehicle or trip mker re obtined. Typiclly, one of three methods is used to crry out n O-D survey. Within the first method, subset of households re identified in such wy s to cpture representtive smple of the popultion. Ech household is then telephoned nd the respondent is sked to indicte the origin nd destintion of their typicl home-to-work nd other types of trips. A second pproch is to stop some portion of en route vehicles for rodside survey, in which the driver of ech vehicle is sked to indicte the origin nd destintion of her/his current trip. A third pproch relies on recording vehicle license plte numbers t severl strtegic sites within study re. The license numbers recorded t one site re subsequently mtched ginst observtions t other sites to determine the origin nd destintion zone of ech vehicle. Ech of these pproches provides direct observtion of the O-D demnds for smple of the popultion. However, these pproches re lso expensive, nd re susceptible to vrious smpling errors s well s systemtic errors. Furthermore, it is rrely possible to extrct dynmic demnds tht re pplicble for short periods, of sy 5 to 15 minutes, from direct survey dt. In the bsence of direct observtion, O-D demnds re sometimes lso estimted bsed on lnd use dt. Typiclly, within the trditionl four stge trnsporttion plnning process (trip genertion, mode split, trip distribution, nd ssignment), O-D demnds re estimted bsed on vrious zonl lnd use dt, such s popultion, employment opportunities, nd mesures of ffluence. Zonl sttistics re obtined from lnd use plns nd census dt, nd trip productions nd ttrctions re obtined through household surveys. Reltionships between the number of trips produced or ttrcted re then typiclly determined through the clibrtion of multiple liner regression models. These totl zonl ttrctions nd productions re subsequently distributed into individul O-D demnds, typiclly through the use of grvity type of model. Conversely, O-D demnds my be estimted through the use of direct demnd plnning model, in which ll four elements of the trnsporttion process re integrted into single model. Unfortuntely, both pproches rely on ggregte zonl chrcteristics to determine O-D demnds. This level of nlysis my be well suited to ggregte plnning nlyses, but it is not pproprite for the estimtion of O-D demnds for trffic control strtegy evlutions. The reltionships between socio-economic fctors nd trip genertion rtes re insufficiently strong to permit the estimtion of the time of trip deprture dt tht re necessry for the estimtion of dynmic demnds. Furthermore, the level of detil considered in most trffic
Chpter 1: Assessment of the Current Stte-of-the-Art 5 nlyses, precludes the formtion of lrge zones, for which trip genertion rtes could be dequtely determined bsed only on the ggregte zonl chrcteristics. A third pproch ttempts to infer the unknown O-D demnd from observed link trffic flow dt. This method circumvents the difficulty of dequtely estimting zonl trip productions nd ttrctions from ggregte zonl chrcteristics, s the trffic flows cn be observed directly on the links within the network using vehicle loop detectors. Typiclly, the objective of the O-D estimtion technique is to estimte n O-D demnd tht replictes the observed flows s closely s possible. Within these techniques, there is no need to clibrte reltionships linking trip mking behviour to socio-economic fctors, but unfortuntely, there re severl other difficulties tht must be overcome, which will be discussed in detil lter. Recently, demonstrtion projects hve commenced using systems tht ttempt to improve the efficiency of trffic networks by trnsmitting ner rel-time trffic dt (i.e. link trvel times) to suitbly equipped vehicles. Computers on-bord these vehicles cn then utilize these dt to pln route towrds the driver's stted destintion. In ddition to receiving informtion, such equipped vehicles lso trnsmit dt, including the vehicle's origin nd destintion, bck to the trffic control centre. This two-wy communiction provides n opportunity to directly collect, in rel-time, time-vrying demnds for smple of the totl vehicle popultion. The sttisticl relibility of such dt s estimtors of popultion chrcteristics, is susceptible to some of the sme smpling errors s re other forms of direct survey, however, probe bsed O-D estimtes my be obtined economiclly if two-wy communiction link exists, nd these estimtes my be dynmic. These lst two pproches re expected to hold the most promise for the relible estimtion of dynmic O-D demnds tht re suitble for trffic nlyses. As such, this thesis focuses on exmining the current stte-of-the-rt of these pproches, identifying remining shortcomings, nd developing nd evluting two estimtion methods tht fll within these pproch ctegories. 1.3 Wht re Dynmic O-D Demnds? Erlier sections of this thesis hve determined wht O-D demnds re, nd why they re needed. In ddition, four potentil methods of obtining O-D demnds hve been identified. At this point it is necessry to more clerly differentite between wht re considered to be sttic nd dynmic O-D demnds. Dynmic demnds re considered to differ from sttic demnds in tht sttic demnds ssume tht every trip is, or cn be pproximted to be, completed within single nlysis time slice such tht ny temporl nd sptil interctions between consecutive time slices cn be ignored. In order to void violting this ssumption, sttic demnds re often developed to be pplicble for reltively long period, sy from 30 minutes up to severl hours. Within this time period, vehicles re further ssumed to deprt their origins t constnt rte. However, s will be illustrted lter in this thesis, field dt show tht O-D demnds cn vry significntly with time of dy. Sttic demnds, with ppliction periods of s much s severl hours, re therefore often not dequte to reflect these temporl vritions. It is possible, however, to develop time series of sttic demnds, which reflect to some extent the temporl vritions observed in the ctul field demnds. In creting this time series of
Chpter 1: Assessment of the Current Stte-of-the-Art 6 demnds, s the durtion over which ech individul sttic demnd is ctive becomes shorter, the ssumption, tht ll trips cn be completed within the deprture time slice, becomes incresingly more violted. Dynmic O-D demnd estimtes, which do not require trips to be completed within their deprture time slices, my hve deprture time slices of ny durtion without violting these ssumptions. The sttic pproch of inferring O-D demnds from link flows is esily dpted to the estimtion of dynmic O-D demnds. Link flows re usully obtined through the use of induction loop detectors nd re typiclly vilble on 20 second to one minute bsis from urbn trffic control systems (UTCS) nd freewy trffic mngement systems (FTMS). Vehicle probe dt lso provide n opportunity to obtin efficiently, in ner rel-time, O-D dt for smple of the vehicle popultion. These O-D estimtion pproches, therefore, lend themselves rther well to the tsk of estimting dynmic O-D demnds. 1.4 Wht is the Current Sttus of Synthetic O-D Demnd Estimtion? Over the lst four decdes, much work hs been crried out to develop systemtic pproches for relibly estimting O-D demnds. Initilly, these efforts were limited to developing methods pproprite for longer term trnsporttion plnning purposes s the highwy network went through rpid expnsion phse. More recently, due in lrge prt to the need to mnge existing utomobile trnsporttion infrstructure more efficiently, there hs been greter emphsis on estimting O-D demnds tht my be used for more short term trffic nlyses. In discussing the current sttus of O-D estimtion pproches, it is pproprite to introduce system for clssifying existing nd newly proposed pproches. This system is intended only to provide structure within which n ssessment of different techniques cn be mde, nd is not intended to be definitive ctegoriztion of ll proposed methods. In fct, some pproches cn rgubly be ssigned to more thn one ctegory. However, the chosen clssifiction permits severl importnt fundmentl distinctions between different pproches to be mde more esily nd concisely. In ddition to the four clssifictions tht re defined, severl specific problems, tht must be ddressed in estimting O-D demnds, re identified nd discussed. The first distinction tht cn be mde is between heuristic nd mthemticl pproches. Severl heuristic pproches hve been proposed, however, since they often lck sound mthemticl foundtion, less confidence cn be plced in their estimtion bilities. In generl the preferred pproch hs mthemticl bsis tht hs more clerly defined theoreticl properties nd my lso be prcticlly pplied. A second distinction is mde bsed on the scope of the pproch's pplicbility. One cn generlize this distinction to clssify pproches s being either pplicble to turning movements t isolted intersections only, or to more generl networks. As the estimtion of intersection turning movements is simpler thn the estimtion of O-D demnds for generl networks, much of the erlier reserch reported on in the literture hs exmined only intersections, but is not necessrily vlid for the purposes of this thesis.
Chpter 1: Assessment of the Current Stte-of-the-Art 7 The third distinction, tht is mde between different techniques, is bsed on the pproches' ssumption regrding the types of routes tht drivers utilize. Routes must usully be known in order to convert estimtes of O-D demnds into estimtes of link flows. The vst mjority of pproches therefore ssume tht routes re known priori nd do not vry with levels of O-D demnd. A few methods hve explicitly recognized tht the routes tht drivers utilize re dependent on the s yet unknown O-D demnds. These pproches must ttempt to estimte simultneously the O-D demnds nd the routes. Severl such mthemticl models hve been formulted, however, dequte solution lgorithms for these models hve not yet been fully developed. The lst distinction, tht is mde between the cpbilities of different pproches, is to distinguish if the techniques estimte sttic nd/or dynmic O-D demnds. It should be noted tht the problem of estimting dynmic demnds is considerbly more demnding thn tht of estimting sttic demnds. In ddition, even though sttic estimtion is considerbly less chllenging, no single sttic estimtion method hs been shown to be both theoreticlly nd prcticlly superior to ll others. Perhps the gretest impct on the current stte of sttic estimtion hs been mde by Vn Zuylen nd Willumsen (1980; Vn Zuylen, 1981; Willumsen, 1992), with their development of comptible yet contrsting trip-count bsed entropy mximizing nd link-count bsed informtion minimizing models. In generl, much less hs been ccomplished in developing dynmic O-D demnd estimtion methods. Significnt contributions hve been mde by Cremer nd Keller (1987; 1984; 1981; Cremer 1991) nd others (Kessci, et l., 1989) to estimting dynmic intersection turning movements, but to-dte it hs either not been possible to extend the formultion of these methods of ppliction to generl networks, or their performnce upon ppliction to generl networks hs proved unstisfctory. 1.5 Wht re some of the Theoreticl nd Prcticl Problems of O-D Estimtion? Lstly, severl theoreticl nd prcticl problems, tht re encountered when estimting network demnds, re used to provide nother mens of compring different pproch strtegies. These difficulties, illustrted in Tble 1-2, include the theoreticl possibility tht multiple O-D demnds my exist tht exctly replicte the observed link flows. In prctice, link flow continuity t nodes rrely exists within set of observed dt, with the result tht often no O-D demnd exists tht exctly replictes the observed flows. However, multiple solutions, tht hve equl levels of link flow error ssocited with them, my still exist. It is usully not known priori into which of the four ctegories prticulr estimtion problem cn be clssified.