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Pareto-Improving Pricing for Transportation Networks

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Title: Pareto-Improving Pricing for Transportation Networks
Physical Description: 1 online resource (108 p.)
Language: english
Creator: Song, Ziqi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

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Subjects / Keywords: congestion -- pricing -- transportation
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

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Abstract: Since its inception over 90 years ago, congestion pricing has been recognized by many as an efficient method for alleviating traffic congestion. Despite the successes of pricing projects worldwide and growing government support, congestion pricing remains largely unappealing to the general public, and it is this lack of public support that impedes its further development and implementation. This dissertation focuses on a class of congestion pricing strategies that is Pareto-improving (i.e., a pricing scheme that benefits society while ensuring that no one in the system is worse off). It is believed that such pricing strategies should be able to gain more public acceptance. This dissertation provides an in-depth investigation of the-state-of-the-art of Pareto-improving pricing strategies for general transportation networks. First, a systematic study of the existence and properties of Pareto-improving pricing schemes is conducted. Second, an anonymous Pareto-improving pricing scheme in a transportation network with a discrete set of value of times (VOTs) for several distinct user classes is proposed and solution algorithms are developed to solve the proposed model efficiently. Last, a Pareto-improving hybrid policy that combines multiple policy instruments is investigated. The proposed hybrid policy takes advantage of the synergistic effects between congestion pricing and free-travel-right assignment. Numerical results demonstrate that the proposed hybrid policy can achieve substantial improvements in transportation system efficiency while maintaining Pareto-improving. Most importantly, this dissertation tackles a long-standing dilemma for transportation authorities: how to enjoy the efficiency benefits of congestion pricing while keeping the general public happy. The strategies developed demonstrate that Pareto-improving pricing is a viable and promising way to achieve these two seemingly contradictory goals simultaneously. The findings may make congestion pricing no longer a hard sell to decision makers and the general public and lead the nation's transportation system to a more sustainable future.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ziqi Song.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Yin, Yafeng.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

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Permanent Link: http://ufdc.ufl.edu/UFE0043721/00001

Material Information

Title: Pareto-Improving Pricing for Transportation Networks
Physical Description: 1 online resource (108 p.)
Language: english
Creator: Song, Ziqi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: congestion -- pricing -- transportation
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Civil Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Since its inception over 90 years ago, congestion pricing has been recognized by many as an efficient method for alleviating traffic congestion. Despite the successes of pricing projects worldwide and growing government support, congestion pricing remains largely unappealing to the general public, and it is this lack of public support that impedes its further development and implementation. This dissertation focuses on a class of congestion pricing strategies that is Pareto-improving (i.e., a pricing scheme that benefits society while ensuring that no one in the system is worse off). It is believed that such pricing strategies should be able to gain more public acceptance. This dissertation provides an in-depth investigation of the-state-of-the-art of Pareto-improving pricing strategies for general transportation networks. First, a systematic study of the existence and properties of Pareto-improving pricing schemes is conducted. Second, an anonymous Pareto-improving pricing scheme in a transportation network with a discrete set of value of times (VOTs) for several distinct user classes is proposed and solution algorithms are developed to solve the proposed model efficiently. Last, a Pareto-improving hybrid policy that combines multiple policy instruments is investigated. The proposed hybrid policy takes advantage of the synergistic effects between congestion pricing and free-travel-right assignment. Numerical results demonstrate that the proposed hybrid policy can achieve substantial improvements in transportation system efficiency while maintaining Pareto-improving. Most importantly, this dissertation tackles a long-standing dilemma for transportation authorities: how to enjoy the efficiency benefits of congestion pricing while keeping the general public happy. The strategies developed demonstrate that Pareto-improving pricing is a viable and promising way to achieve these two seemingly contradictory goals simultaneously. The findings may make congestion pricing no longer a hard sell to decision makers and the general public and lead the nation's transportation system to a more sustainable future.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ziqi Song.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Yin, Yafeng.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043721:00001


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PARETO-IMPROVINGPRICINGFORTRANSPORTATIONNETWORKSByZIQISONGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011ZiqiSong 2

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Tomyparentsandmywife 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmydeepestgratitudetomyadvisor,Dr.YafengYin.Ihavebeenextremelyfortunatetohaveanadvisorwhoguidedmethroughtheacademicjungle,andprovidedmegreatfreedomtoexploreonmyownatthesametime.Hisguidanceandencouragementhelpedmeovercomenumerousobstaclesincompletingthisdissertation.IamalsodeeplygratefultoDr.SiriphongLawphongpanichforhisprovokingandconstructivecommentsatdifferentstagesofmyresearch.Byintroducingmetoinnovativeoperationsresearchtechniques,hehelpedmeexpandmyresearchinterests.IwouldalsoliketothankDr.LilyElefteriadou,Dr.ScottWashburnandDr.SivaramakrishnanSrinivasanforservingonmydissertationcommittee.Theirvaluableadvicesthroughoutmygraduatestudywerecrucialtomydissertationresearch.Additionally,Iamindebtedtomyfriendsandcolleaguesatthetransportationresearchcenter,whogavemeanopenandstimulatingresearchatmosphere.Particularly,IwouldliketoacknowledgeYingyanLou,LihuiZhangandDiWuformanyfruitfuldiscussionsthathelpedmeunderstandmyresearchbetter.Finally,andmostimportantly,noneofthiswouldhavebeenpossiblewithoutthesupportofmyfamily.ImostwanttothankmybelovedwifeNiChenforherlove,sacriceandpatience,andalsothankmyparentsfortheirfaithandunconditionalloveinme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 1.1Background ................................... 11 1.2Objectives .................................... 14 1.3DissertationOutline .............................. 15 2LITERATUREREVIEW ............................... 16 2.1BackgroundofRoadPricing .......................... 16 2.2FundamentalofCongestionPricing ..................... 17 2.3Network-WideCongestionPricing ...................... 19 2.3.1Fixed-DemandMarginalCostPricing ................. 20 2.3.2Elastic-DemandMarginalCostPricing ................ 22 2.3.3First-BestTollSet ............................ 23 2.3.4Second-BestCongestionPricing ................... 24 2.4CongestionPricingwithHeterogeneousUsers ............... 25 2.5PublicAcceptanceofCongestionPricing .................. 27 2.5.1WeaknessofMarginalCostPricing .................. 27 2.5.2EquityEffectsofCongestionPricing ................. 28 2.6Summary .................................... 29 3PARETO-IMPROVINGCONGESTIONPRICINGAPPROACH .......... 32 3.1TheConceptofPareto-ImprovingPricing .................. 32 3.2DominatingFlowDistribution ......................... 34 3.2.1FeasibleFlowDistribution ....................... 35 3.2.2FindingDominatingFlowDistribution ................. 35 3.2.3DominatingFlowDistributionandNonnegativeTolls ........ 38 3.3NonnegativePareto-ImprovingTolls ..................... 42 3.3.1ExistenceofAnonymousPareto-ImprovingTolls ........... 43 3.3.2ExistenceofDiscriminatoryPareto-ImprovingTolls ......... 47 3.4Pareto-ImprovingTollProblem ........................ 49 3.5Summary .................................... 52 5

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4PARETO-IMPROVINGCONGESTIONPRICINGSCHEMEFORMULTICLASSNETWORK ...................................... 59 4.1MulticlassPareto-ImprovingPricingScheme ................ 59 4.1.1MulticlassNetworkEquilibrium .................... 60 4.1.2DenitionofMulticlassPareto-ImprovingPricingScheme ..... 61 4.1.3FindingMulticlassPareto-ImprovingPricingScheme ........ 63 4.2PropertiesandExistenceofMulticlassPareto-ImprovingPricingScheme 65 4.2.1MulticlassDominatingFlowDistribution ............... 66 4.2.2ExistenceofNonnegativePareto-ImprovingTolls .......... 68 4.3ManifoldSuboptimizationAlgorithm ..................... 70 4.4NumericalExamples .............................. 72 4.5Summary .................................... 74 5PARETO-IMPROVINGHYBRIDPOLICYFORTRANSPORTATIONNETWORKS 81 5.1Pareto-ImprovingPureRoadSpaceRationingPolicy ............ 82 5.1.1ProblemSetting ............................. 83 5.1.2UserEquilibriumunderPureRoadSpaceRationingSchemes ... 84 5.1.3Pareto-ImprovingPureRoadSpaceRationingProblem ....... 86 5.2Pareto-ImprovingHybridPolicy ........................ 87 5.2.1UserEquilibriumunderHybridPolicies ................ 88 5.2.2Pareto-ImprovingHybridPolicyProblem ............... 90 5.3NumericalExamples .............................. 92 5.4Summary .................................... 94 6CONCLUSIONS ................................... 99 6.1SummaryofMajorFindings .......................... 99 6.2FutureResearch ................................ 101 REFERENCES ....................................... 102 BIOGRAPHICALSKETCH ................................ 108 6

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LISTOFTABLES Table page 3-1FlowdistributionsunderUEandMSCpricingfortheve-linknetwork ..... 54 3-2UEanddominatingowdistributions ........................ 55 3-3UEandoptimaldominatingowdistributions ................... 56 3-4Optimaldominatingowdistributionandtolls ................... 56 3-5UEanddominatingdistributions .......................... 57 3-6OD-dependentPareto-improvingtolls ....................... 58 4-1Time-baseddominatingowdistributionandPareto-improvingscheme .... 76 4-2PerformancecomparisonswithUEowdistribution ................ 77 4-3Cost-baseddominatingowdistributionandPareto-improvingscheme ..... 77 4-4PerformancecomparisonswithUEowdistribution ................ 78 4-5Networkattributes .................................. 78 4-6ExactPareto-improvingproblem .......................... 79 4-7ApproximatePareto-improvingproblem ...................... 79 5-1UserequilibriumandPareto-improvingpurerationingproblems ......... 97 5-2Pareto-improvinghybridpolicyproblems ...................... 98 5-3SystemdelayreductionsunderdifferentPareto-improvingstrategies ...... 98 7

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LISTOFFIGURES Figure page 2-1Graphicalrepresentationofmarginalcostpricing ................. 31 3-1Ave-linknetwork .................................. 54 3-2Multi-commodityloopexample ........................... 55 3-3Multi-commodityloopexamplewithloopremoved ................ 55 3-4Ave-linknetworkwithtwoODpairs ........................ 56 3-5Thelongestpathtreeassociatedwithv ...................... 57 4-1Ave-linknetworkwithmulticlassusers ...................... 76 4-2FrequenciesofratiosforSiouxFallsnetwork ................... 80 4-3CumulativedistributionofratiosforSiouxFallsnetwork ............. 80 5-1Amultimodaltransportationnetwork ........................ 96 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyPARETO-IMPROVINGPRICINGFORTRANSPORTATIONNETWORKSByZiqiSongDecember2011Chair:YafengYinMajor:CivilEngineering Sinceitsinceptionover90yearsago,congestionpricinghasbeenrecognizedbymanyasanefcientmethodforalleviatingtrafccongestion.Despitethesuccessesofpricingprojectsworldwideandgrowinggovernmentsupport,congestionpricingremainslargelyunappealingtothegeneralpublic,anditisthislackofpublicsupportthatimpedesitsfurtherdevelopmentandimplementation.ThisdissertationfocusesonaclassofcongestionpricingstrategiesthatisPareto-improving(i.e.,apricingschemethatbenetssocietywhileensuringthatnooneinthesystemisworseoff).Itisbelievedthatsuchpricingstrategiesshouldbeabletogainmorepublicacceptance. Thisdissertationprovidesanin-depthinvestigationofthe-state-of-the-artofPareto-improvingpricingstrategiesforgeneraltransportationnetworks.First,asystematicstudyoftheexistenceandpropertiesofPareto-improvingpricingschemesisconducted.Second,ananonymousPareto-improvingpricingschemeinatransportationnetworkwithadiscretesetofvalueoftimes(VOTs)forseveraldistinctuserclassesisproposedandsolutionalgorithmsaredevelopedtosolvetheproposedmodelefciently.Last,aPareto-improvinghybridpolicythatcombinesmultiplepolicyinstrumentsisinvestigated.Theproposedhybridpolicytakesadvantageofthesynergisticeffectsbetweencongestionpricingandfree-travel-rightassignment.NumericalresultsdemonstratethattheproposedhybridpolicycanachievesubstantialimprovementsintransportationsystemefciencywhilemaintainingPareto-improving. 9

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Mostimportantly,thisdissertationtacklesalong-standingdilemmafortransportationauthorities:howtoenjoytheefciencybenetsofcongestionpricingwhilekeepingthegeneralpublichappy.ThestrategiesdevelopeddemonstratethatPareto-improvingpricingisaviableandpromisingwaytoachievethesetwoseeminglycontradictorygoalssimultaneously.Thendingsmaymakecongestionpricingnolongerahardselltodecisionmakersandthegeneralpublicandleadthenation'stransportationsystemtoamoresustainablefuture. 10

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CHAPTER1INTRODUCTION 1.1Background Trafccongestion,asanalmostinevitableby-productoffasturbanizationaroundtheworld,isundesirablebecauseitcausesdeadweightlosstosocietyasawhole.Therearetwoapproachestohandlingtrafccongestion,namelysupply-sideanddemand-side.Transportationauthoritiesusedtocopewithcongestionproblemsfromthesupplysidebyaddingmorecapacity.However,wheneverandwhereverhighwaycapacityisexpanded,latenttraveldemandsoonllsthecapacitygains.Theproblemisespeciallyacuteformegacitiesindevelopingcountrieswhereroadinfrastructurestrugglestokeeppacewithincreasingvehicleownershipandtraveldemand. Downs ( 1962 2004 )suggestedafundamentallawofhighwaycongestionandconcludedthatoncepeak-hourcongestionhasappearedinaregion;itisimpracticalfortheregiontobuilditswayoutofcongestion. DurantonandTurner ( 2011 )investigatedtherelationshipbetweeninterstatehighwaysandhighwayvehiclemilestraveled(VMT)inUScitiesandfoundthatincreasedprovisionofroadsorpublictransitisunlikelytorelievecongestion.Fromanotherperspective,trafccongestioncanbeviewedasanexampleofthetragedyofthecommons( Garrett 1968 )whichreferstothefactthatpeopletendtooverusefreepublicgoods.Theprimaryreasonforsuchdemand-supplymismatchesisthatroadusersareselshdecisionmakerswhentheyareonlyrequiredtopaytheirprivatecostsinsteadofthetruesocialcosts. Wecontendthatsustainablesolutionstoeliminateoratleastreducetrafccongestionareonthedemandside.Congestionpricingisonetypeoftraveldemandmanagement(TDM)strategiesandhasbeenrecognizedasanefcientmethodforalleviatingtrafccongestion.Thetheoreticalfoundationofcongestionpricingwaslaidoutby Pigou ( 1920 )and Knight ( 1924 ).Manyhavecometorecognizecongestionpricingasaneffectivemarket-basedinstrumenttoallocateroadspaceresourcesto 11

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benetsocietyasawhole,see,e.g.,thereviewarticleof Lindsey ( 2006 ), dePalmaandLindsey ( 2011 )andreferencescitedtherein.Therecentadventofelectronictollingmakescongestionpricingmorepracticalandthereexistseveralsuccessfulimplementationsworldwide(e.g.,SanDiego,Singapore,Oslo,andLondon). Thebasicideaofcongestionpricingistoletroaduserspaythetruesocialcostsofusingtheroad.Ingeneral,roadusersareselshwhentheymaketraveldecisionsandarenotawareofornotwillingtoconsiderthenegativecongestionexternalitytheyimposeonotherusers.Theclassicmarginalsocialcost(MSC)pricingprinciplestatesthatusersshouldbechargedforthedifferencebetweenthemarginalsocialcostsandprivatecosts.Inthisway,roaduserspaythetruesocialcostsandsubsequentlytheirtraveldecisionscomplywiththeinterestsofsociety.AlthoughMSCpricingmakesperfecteconomicsense,itassumesthatthereisnoconstraintonpricing;therefore,itisalsocalledrst-bestpricing.Ontheotherhand,asecond-bestpricingschemeremovesthisstrongassumptionatthepriceofachievinglesssystemefciency.Itisworthmentioningthatallexistingcongestionpricingimplementationsaroundtheworldaresecond-bestpricingschemes. CongestionpricingremainspoliticallydifcultintheUS.Roadusersgenerallyviewcongestionpricingasanothertaxwhentheyareaskedtopayforsomethingtheycurrentlyreceiveforfree.Hence,mostelectedofcialsarereluctanttosupportcongestionpricingknowingroadusersarevoters.Despitestrongtheoreticalargumentsandgrowinggovernmentsupport(e.g.,FHWA'sValuePricingProgram),gettingthepublictoacceptcongestionpricingisstillamajorobstacle.ResidentsinHongKong,Cambridge(England)andEdinburghvotedagainstthecongestionpricingschemesproposedfortheircities.Morerecently,thecongestionpricingplanforNewYorkCitywaskilledbeforeitreachedtheStateAssembly. Hau ( 2005 )concludedthatcongestiontollsleviedoneachroadusertypicallyexceedtraveltime/costsavingstheyyield,which 12

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impliesthatroadusersareusuallymadeworseoffunlesstollrevenuesareredistributedtothemincertainforms. Itwouldpresumablybeeasytogainsupportforacongestionpricingschemefromthegeneralpublicifimplementingtheschemeimprovesnetsocialbenetswithoutmakinganystakeholder(roadusers,societyandtransportationauthorities)worseoff. LawphongpanichandYin ( 2010 )recentlyintroducedanewclassofpricingschemescalledPareto-improvingcongestionpricing,whichimprovessystemefciencywhileensuringthat,intermsoftraveltime,nouserismadeworseoffandsomeusersarebetteroffwhencomparedwiththesituationwithoutanypricingintervention,evenbeforetollrevenueredistribution. Inrecentyears,theconceptofPareto-improvingpricinghasbeenpickedupbymanyresearchers.Theyproposedtocompensatetheunhappyroaduserswhoaremadeworseoffbycongestionpricingusingthetollrevenueraised,usuallythrougharevenuerefundingscheme. Liuetal. ( 2009 )proposedaPareto-improvingschemethatchargespositivetollsonroadusersandrefundsalltollrevenuecollectedtotransituserswithuniformlydistributedusers'valueoftime(VOT)functions. NieandLiu ( 2010 )examinedhowusers'VOTdistributionsmayaffecttheexistenceofaPareto-improvingpricingandrefundingscheme. GuoandYang ( 2010 )investigatedPareto-improvingpricingandrefundingschemesingeneraltransportationnetworks.TheyconcludedthatacongestionpricingschemeisPareto-improvingifitreducessystemtraveltimeandalltollrevenueisrefundedtousers.Admittedlytollrevenuerefundingisaneffectivewaytomakeacongestionpricingschememoreappealingtothegeneralpublic.However,ifusersanticipatetollrefundinginsomeformwhentheypay,thebehavioralimplicationsaremuchmorecomplicatedthanwhatareassumedinmodelsintheliterature,whichmayleadtounexpectedoutcomes.Therefore,inthisdissertation,wefocusprimarilyonnonnegativePareto-improvingcongestionpricingschemes. 13

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1.2Objectives Themainobjectiveofthisdissertationistoprovideanin-depthinvestigationofthe-state-of-the-artofPareto-improvingpricingstrategiesforgeneraltransportationnetworks.Morespecically,threemajorcomponentsdiscussedinthisdissertationareasfollows: First,forthePareto-improvingpricingproblemformulatedin LawphongpanichandYin ( 2010 ),theexistenceofthepricingschemeisnotguaranteed.Whyandwhensuchpricingschemesexistarestillopenquestions.ThisdissertationprovidesasystematicstudyoftheexistenceandpropertiesofPareto-improvingschemesingeneraltransportationnetworks,whichisalmostanunexploredareaintheliterature. Second,onecomplicationthatmakespracticalimplementationsofcongestionpricingchallengingisthatroadusersareheterogeneousinmanyaspects.Manydebatesoncongestionpricingrootinuserheterogeneity.Forinstance,roaduserswithdifferentVOTsmayhavedifferentperceptionsofananonymous(uniform)tollandreactdifferentlytothetollingintervention.ThePareto-improvingpricingproblemformulatedby LawphongpanichandYin ( 2010 )onlyconsideredroaduserswithhomogeneousVOT.ThisdissertationexplorestheproblemofdevelopingPareto-improvingpricingschemesfornetworkswithheterogeneoususers,whichmayhaveimportantpracticalpolicyimplicationsandisurgentlyneeded. Last,apartfrommarket-basedcongestionpricingstrategies,regulatorydemandmanagementpolicies,suchasroadspacerationing,canalsoeffectivelymitigatetrafccongestion(see,e.g., Wangetal. 2010 and Hanetal. 2010 ).However,intransportationliterature,regulatoryandmarket-baseddemandmanagementpoliciesareusuallyevaluatedseparately.Thepotentialofbundlingmultiplepolicyinstrumentsaregenerallyoverlooked. Downs ( 2004 )concludedthatthemosteffectiveoverallstrategyforreducingcongestionprobablyshouldconsistofbothmarket-basedandregulatoryelements.Thisdissertationllsthisneededgapintheliteraturebyinvestigatinghowtodesignhybrid 14

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policiesthattakeadvantageofthesynergisticeffectsbetweencongestionpricingandfree-travel-rightassignment. 1.3DissertationOutline Theremainderofthisdissertationisorganizedasfollows.Chapter2reviewsthebackgroundofcongestionpricingandhighlightstwoaspectsofcongestionpricing:userheterogeneityandequityissues.Chapter3systematicallyinvestigatesthePareto-improvingcongestionpricingapproach.Chapter4presentsthePareto-improvingcongestionpricingproblemfornetworkwithmulticlassusers.Chapter5proposesaPareto-improvinghybridpolicythatcombinesmultiplepolicyinstruments.Chapter6discussesfutureresearchdirectionsandconcludesthedissertation. 15

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CHAPTER2LITERATUREREVIEW 2.1BackgroundofRoadPricing Broadlyspeaking,roadpricingreferstoallchargesthatuserspayforusingtheroadsystem,whichincludesfueltaxes,vehicleregistrationfees,parkingfees,roadtollsandcongestionpricingcharges,etc.Peoplemaythinkroadsareprovidedbythegovernmentforfree,however,theyarenotawareofthefactthattheyareactuallypayingfortheirtravelthroughvarioustaxesandfees. Roadpricingisnotanewconcept,tollroadswerecommoninBritainandtheUnitedStatesduringmostofthe19thcentury( Lindsey 2006 ).TheHighwayTrustFund(HTF)wasfoundedin1956toensuredependablenancingfortheinterstatehighwaysystem.TheHTFcandatebacktothe1920swhenOregonrstadoptedamotorfueltaxthatwasearmarkedforroadconstructionandmaintenance.Currentlyitisderivedfromtwomainsources:federalexcisetaxesonmotorfuels(gasoline,diesel,andspecialfuels)andtruck-relatedtaxes(truckandtrailersales,trucktires,andheavy-vehicleuse).TheHTFisfacingshortfallbecauseofdecliningfueltaxincomeduetomorefuel-efcientvehiclesontheroadandrisingcostsoftransportationprojects.Despitean$8billioninfusionfromtheGeneralFundoftheTreasuryinSeptember2008toreplenishtheaccount,HTFneedsanadditionalinfusionoffunds,about$15billion,toremainsolventthroughtheendofscalyear2010( GAO 2009 ).Anurgentreformoftheexistinghighwaynancesystemisneeded.Interestingly,whenitwasrstproposedin1920s,fueltaxeswereconsciouslyadoptedasimperfectsubstitutesbystatelegislatures.Theybelievedthatdirecttollingatthetimeandplaceofuseismoreappropriate,buttollswereexpensiveandawkwardtocollect( Wachs 2003 ).Astheadventofelectronictolling,weshouldthinkofshiftingfromrelianceonfueltaxtodirectusage-basedcharges. Amongvariousformsofdirectusage-basedcharges,inthischapter,wewillfocusoncongestionpricing,whichparticularlyaimsatmitigatingcongestionusingpricing 16

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instruments.Forliteraturereviewongeneralroadpricing,pleasereferto Smalletal. ( 1989 )and Gomez-Ibanezetal. ( 1999 ). 2.2FundamentalofCongestionPricing Theprincipleofcongestionpricingwasrstproposedby Pigou ( 1920 )and Knight ( 1924 )over80yearsago.Theworkhasbeenelaboratedandextendedbymanyresearchers,see,e.g., Walters ( 1961 ), MohringandHarwitz ( 1962 ), Beckmannetal. ( 1956 )and Vickrey ( 1969 ).Thebasicconceptofcongestionpricingistochargeroaduserstheirnegativeexternalimpactsonothers.Considersocietyofroadusers,morespecically,theexternalityismainlythecongestionimpactthatanindividualuserimposesonallotherroadusers. Wardrop ( 1952 )statedtheuserequilibrium(UE)conditionsanddescribedroadusers'selshroutingbehavior.Undersuchbehaviorassumption,whenusersenterroadnetwork,theyeitherarenotawareofornotwillingtoconsiderthecongestiontheyimposeonothers.Theytendtoonlyconsidertheirownprivatecost(marginalprivatecost).However,thecost(marginalsocialcost)bornebysocietyismorethantheprivatecost.Thediscrepancyisthefundamentalcauseofinefciencyinroadresourceallocation.Therefore,toachievethemaximumefciencytosocietyeachroaduserisrequiredtopayauserchargeequaltothedifferencebetweenthemarginalsocialcostandprivatecost,whichensuresthattheindividualuser'sdecisionalsoreectstheinterestsofsociety.Suchcharge,knownasmarginalcostpricingorrst-bestcongestionpricingtoll,maximizesthenetbenetofsocietyandalsoinducesaPareto-optimalsituation,thatis,noonecanbemadebetteroffwithoutmakingsomeoneelseworseoff. TheunderlyingprincipleofmarginalcostpricingcanbeexplainedgraphicallyusingFigure 2-1 forasingleroadsegment(bottleneck)withhomogenousroadusers.TheMCcurverepresentsthemarginalsocialcostofanadditionalusertotrafcowforthenewtrip-makerandallexistingroadusers,whiletheACcurverepresentsthemarginalprivatecostbornebythenewtrip-makeralone.Notethatwhenallroadusers 17

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areassumedtobeprice-takers,perceivedmarginalprivatecostequalstotheaverageprivatecost.Withoutanytollingintervention,theequilibriumowisVa,whereACandinversedemandcurveintersect.Fromsociety'spointofview,Vaisnotoptimalbecausethelastuserthatenterstheroadnetworkenjoysabenetof abwhileimposingamarginalcostof actosociety.Bychargingusersthemarginalcosttoll,theequilibriumowshiftsfromVatoVo.Withthistollscheme,users'interestscoincidewithsociety'sinterests,thereby,theowVoisoptimaltosociety.Thetotalsocialwelfare,giveninthearea ghqm,isalsomaximized.TheadditionaltrafcowbeyondVogeneratesacostofthearea acheandonlyenjoysabenetofthearea abhe.Awelfaregainofthearea bch(theshadowarea)isapparentduetothemarginalcostpricingtollscheme. Mathematically,letc(v),vdenoteroadusers'averagetravelcostfunctionortheACcurveandtrafcowontheroadrespectively.Thenthemarginalsocialcost,MC,isdenedasMC(v)=@c(v)v @v=c(v)+v@c(v) @v Themarginalcostpricingtoll,bydenition,isthedifferencebetweenMCandACcurve.Therefore,themarginalcostpricingtoll,(v)is(v)=v@c(v) @v Theaboveresultsalsocanbederivedbysolvingasocialwelfaremaximizationproblem.Socialwelfare(SW)isdenedastotalsocialbenetminustotalsocialcost.SW(v)=Zv0D)]TJ /F9 7.97 Tf 6.58 0 Td[(1(!)d!)]TJ /F4 11.955 Tf 11.95 0 Td[(c(v)v whereD)]TJ /F9 7.97 Tf 6.59 0 Td[(1(v)istheinversedemandfunctionorbenetfunction.MaximizingSW(v)withrespecttovyieldsthenecessaryrst-ordercondition:D)]TJ /F9 7.97 Tf 6.59 0 Td[(1(v))]TJ /F4 11.955 Tf 11.96 0 Td[(c(v))]TJ /F4 11.955 Tf 11.96 0 Td[(v@c(v) @v=0 18

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Bydenitionofmarginalcostpricingtoll,theoptimalequilibriumow,vo,happenswhenMCcurveandinversedemandcurveintersect.D)]TJ /F9 7.97 Tf 6.59 0 Td[(1(v)=c(v)+(v) Thus,theoptimaltollis(v)=v@c(v) @v Graphically,thesocialwelfarecanbeseenasthesumofconsumersurplusofthearea hqpandthegovernmentsurplusofthearea ghpm. 2.3Network-WideCongestionPricing Inthissection,wewillreviewcongestionpricingproblemsingeneralnetworks. Wardrop ( 1952 )introducedtwoimportantprinciplesintransportationnetworkmodeling,namely,Wardrop'srstprincipleandsecondprinciple.Therstprincipledepictsusers'selshroutingbehavior,thatis,allusersinthenetworktrytominimizetheirownprivatetravelcosts.TheresultingequilibriumstateiscalledUEstate,wheretravelcostsonallutilizedpathsareequalorlessthanthoseonanyunusedpathsforaspecicODpair.WhenthenetworktrafcowdistributionisinUEstate,nouserscanfurtherreducetheirtravelcostsbyunilaterallychangingtheirpaths.Thesecondprincipleassumesthereisacentralcontrollertocoordinatethetrafcowsothatusersworkcollaborativelytoachievethebestsystemperformance.Therefore,suchaowdistributionisalsocalledsystemoptimum(SO)state.NotethataSOstateisnotsustainablebecausesomeusersmaybeabletolowertheirtravelcostsbyswitchingtootherpaths.TheSOstategivestheupperboundofsystemperformance,thusitisanimportantbenchmarkandalsothetargetgoalforplanningandoperations.Ingeneralnetworks,aUEowdistributiongenerallyisdifferentfromaSOowdistributionduetocongestionexternality.Wewillshowinthissectionthatthemarginalcostpricingprincipleisstillvalidinthenetworkcontext,whichcandriveaUEowdistributiontoaSOone. 19

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Notationsthatwillbeusedinthissectionaregivenasfollows.LetNandLdenotesthesetofnodesandlinksinaroadnetwork,andWbethesetofODpairs.Alink(i,j)2Lrepresentsaroadsegment.ForODpairw,xwijdenotestheowonlink(i,j)ofthatODpair.ThevectorxistheowvectorforODpairwwithxwijasitselements.Theaggregateowonlink(i,j)isexpressedbyvij=Pwxwij.Duetocongestioneffects,traveltimefunctiontij(vij)isassumedtobeacontinuousandmonotonicallyincreasingfunctionoftheaggregatelinkowvij.LetAbethenode-arcincidencematrixforthenetworkandEwdenoteavectorinRm,wheremisthenumberofnodes.Specically,Ewisaninput-outputvectorandhasexactlytwonon-zerocomponents:onehasavalue1correspondingtheoriginnodeofODpairwandtheotheronehasavalue-1inthecomponentforthedestination. 2.3.1Fixed-DemandMarginalCostPricing ConsiderthecasewheretraveldemandforODpairw,dw,isgivenandhomogeneous,totalsystemtraveltimecanbesetasthesystemperformancemeasure.ThusSOowdistributionminimizesthetotalsystemtraveltime.Mathematically,theSOowdistributioncanbeobtainedbysolvingthefollowingmathematicalprogram:min(v,x)X(i,j)2Ltij(vij)vij (2)s.t.Axw=Ewdw8w2W (2)v=Xw2Wxw (2)x0. (2) Beckmannetal. ( 1956 )formulatedtrafcassignmentproblemasamathematicalprogramandprovedtheequivalencebetweentheoptimalityconditionsoftheproposedmodelandtheWardrop'srstprinciple.Thecorrespondingmathematicalprogramisas 20

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follows:min(v,x)X(i,j)2LZvij0tij(!)d! (2)s.t.Axw=Ewdw8w2W (2)v=Xw2Wxw (2)x0. (2) ComparingtheoptimalityconditionsoftheaboveSOandUEformulations,wenotethatSOproblemisequivalenttoaUEproblemwiththefollowingmodiedtravelcostfunction ~tij(vij)=tij(vij)+vijdtij(vij) dvij.(2) Thersttermofthemodiedtravelcostfunctionistheuser'sprivatetraveltimefunctionandthesecondtermistheadditionaltraveltimeimposedonallexistingusersbyanadditionaluser.Ifallusersadoptthemodiedtravelcostfunctiontomaketheirroutechoicedecisions,theresultingowdistributionwillbeautomaticallyaSOowdistribution.However,inrealityusersnormallyonlyconsidertheirownprivatecost,tij(vij).Tomakeusers'decisioncomplywithsociety'sinterests,thesecondtermofthemodiedtravelcostfunctionshouldbechargedasacongestiontoll.Themarginalcostpricingtollforeachlinkcanbeexpressedasfollows: ij=vijdtij(vij) dvijvij=vSOij(i,j)2L(2) Notethat~tij(vij)=dvijtij(vij) dvij,themodiedtravelcostfunctionisexactlythemarginalcostofaddinganadditionalusertolink(i,j).Furthermore,tij(vij)isassumedtobeseparable,thereforewehave~tij(vij)=dP(i,j)2Lvijtij(vij) dvij,whichisalsothemarginalsocialcostofhavinganadditionalusertothesystem.Hence,wecanexpressWardrop's 21

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secondprincipleinanotherway:allutilizedpathshavethesamemarginalcostforaspecicODpair. 2.3.2Elastic-DemandMarginalCostPricing Intheelasticdemandcase,traveldemandforaspecicODpairw,dwfollowsademandfunctionDw(w),wherewisthetravelcostbetweenthatODpair.LetD)]TJ /F9 7.97 Tf 6.59 0 Td[(1w(dw)denotetheinversedemandfunctionforODpairw.Totalsystemtraveltimeisnotappropriatetobeusedasthesystemperformancemeasureanymorebecauseofdemandelasticity.Fortheelasticdemandcase,SOproblemcanbedenedintermsofmaximizationofsocialwelfare.ThemathematicalprogramthatcorrespondingtotheSOproblemisasfollows:max(v,x,d)Xw2WZdw0D)]TJ /F9 7.97 Tf 6.59 0 Td[(1w(!)d!)]TJ /F8 11.955 Tf 16.03 11.35 Td[(X(i,j)2Ltij(vij)vij (2)s.t.Axw=Ewdw8w2W (2)v=Xw2Wxw (2)x0. (2) TheUEproblemwithelasticdemandisformulatedasfollows:min(v,x,d)X(i,j)2LZvij0tij(!)d!)]TJ /F8 11.955 Tf 14.28 11.36 Td[(Xw2WZdw0D)]TJ /F9 7.97 Tf 6.58 0 Td[(1w(!)d! (2)s.t.Axw=Ewdw8w2W (2)v=Xw2Wxw (2)x0. (2) Usingtheexactsameprocedureasthelastsection,wecanobtainthemarginalcostpricingtollforeachlinkasfollows: ij=vijdtij(vij) dvijvij=vSOij(i,j)2L(2) 22

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NotethatalthoughSOstatedenitionsaredifferentforxedandelasticdemandcase,themarginalcostpricingtollvectors,( 2 )and( 2 )areidentical. 2.3.3First-BestTollSet Inmostliterature,themarginalcostpricingtollvector( 2 and 2 )derivedinprevioustwosectionshasbeentheonlyrst-bestpricingscheme.However, HearnandRamana ( 1998 )foundthatthelink-basedmarginalcostpricingtollschemedoesnotnecessarilyhavetobeuniqueandproposedarst-besttollset,whichcontainsalltollvectorsthatcandriveaUEstatetoaSOone.Thetraditionalmarginalcostpricingtollvectorisjustoneelementofthistollset.Considerthefollowingsystemofequations:xwij(tij(vij)+ij)]TJ /F7 11.955 Tf 11.95 0 Td[(wi+wj)=08(i,j)2L,w2W (2)tij(vij)+ijwi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj8(i,j)2L,w2W (2)Axw=Ewdw8w2W (2)v=Xw2Wxw (2)x0 (2) wherewirepresentsthenodepotential Ahujaetal. ( 1993 )ofnodeiforODpairw.TheseconditionsessentiallyaretheKarush-Kuhn-Tucker(KKT)conditionsforthetolledUEproblem.Recallthattheobjectiveofrst-besttollistodrivethesolutionofthetolledUEproblemtocoincidewiththeun-tolledSOproblem.Therefore,ifandonlyifthereexistsavectorsuchthatthecombinedvector(,)satisesthefollowingconditions,thevectorisavalidrst-besttollvector( HearnandRamana 1998 ).xwij(tij(vSOij)+ij)]TJ /F7 11.955 Tf 11.95 0 Td[(wi+wj)=08(i,j)2L,w2W (2)tij(vSOij)+ijwi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj8(i,j)2L,w2W. (2) 23

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Moreconcisely,therst-besttollsetcanbeexpressedinvectorfromasfollows:(t(vSO)+)TvSO=Xw2W(Ew)Tdww (2)t(vSO)+ATw8w2W. (2) Sincetherst-besttollsetgenerallyhasmultipleelements,includingthetraditionalmarginalcostpricingtoll,itgivesmoreexibilityinsettinglinktollstomeetsecondaryobjectives,suchasminimumnumberoftollbooth,maximumgovernmentrevenue,etc. YinandLawphongpanich ( 2009 )furtherinvestigatedtherst-besttollsetandconcludedthatallelementsofthetollsetareactuallystillinlinewiththemarginalcostpricingprincipleinpathlevelalthoughtheymaybedifferentfromthetraditionallink-basedmarginalcostpricingtollvector. 2.3.4Second-BestCongestionPricing Marginalcostpricingorrst-bestcongestionpricingtollschemesassumethatthereisnoconstraintonpricingandreducecongestiontoitsminimumlevelwhentraveldemandisxedorincreasethesocialwelfaretoitsmaximumlevelwhentraveldemandiselastic.Ontheotherhand,forpoliticalreasonsorotherwise,someroadsarenottollableandotherrestrictionsmayalsoapply.Becauseoftheserestrictions,generallythistypeoftollschemecannotachievethemaximumsystemefciencyandiscalledsecond-bestcongestionpricingscheme.Intheliterature,manyformulatedtheproblemasabi-leveloptimizationproblem(see,e.g., Bard 1998 ; Shimizuetal. 1997 )oramathematicalprogramwithequilibriumconstraints(see,e.g., Luoetal. 1996 ).Generallyspeaking,theupper-levelproblemsortheobjectivesofthesecond-besttollschemesareoptimizingcertainsystemperformancemeasuresandthelower-levelproblemsorcomplimentaryconstraintsareusedtodescriberoadusers'routechoicebehaviors,i.e.,tolledUEconditions.Othertypesofsecond-bestpricingschemesincludearea-based(see,e.g., MaruyamaandHarata 2006 ; MaruyamaandSumalee 2007 ) 24

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andcordonpricing(see,e.g., Akiyamaetal. 2004 ; Sumalee 2004 ; ZhangandYang 2004 ). 2.4CongestionPricingwithHeterogeneousUsers Itiswellknownthatroadusersconsistsofheterogeneoususers.Userheterogeneitycanbecategorizedintotwotypes,namelyobservableandunobservableheterogeneity.Observableuserheterogeneityreferstousers'differentvehicletypes,e.g.,truck,bus,car,etc.Unobservableuserheterogeneityassumesthatallusershavethesametypeofvehicles,whiletheirsocioeconomiccharacteristicsvaryacrossdifferentusergroups,suchasvalueoftimes(VOT). Fortheobservableuserheterogeneity,duetothedifferencesoftheirvehicletypes,differentusergroupsmayhavedifferentcongestionexternalitiesimposingonothers.Sincetheirphysicaldifferenceisobservable,itispossibletochargedifferentiatedtollsforeachvehicletypebasedonthemarginalcostpricingprinciple.Moredetaileddiscussionsofcongestionpricingonthistypeofuserheterogeneitycanbefoundin Patriksson ( 1994 )and Nagurney ( 1999 ). Forunobservableuserheterogeneity,allusersdifferfromoneanotherinunobservableways,itisverydifcultifnotimpossibletointroducedifferentiatedornon-anonymoustolls( ArnottandKraus 1998 ).SinceusershavedifferentVOTs,theymayreactdifferentlytoananonymous(uniform)tollscheme.Itistheninterestingtoinvestigatewhethermarginalcostpricingprincipleisstillvalidinthiscase.Intransportationliterature,thistypeofuserheterogeneityiscapturedbyassumingeitherusers'VOTsfollowacontinuousdistributionoradiscretesetofVOTsforseveraldistinctuserclasses. YinandYang ( 2004 )concludedthatananonymouslinktollcanstillbederivedbasedonthemarginalcostpricingprincipletoachievethecost-basedsystemoptimum(totalvalueofsystemtraveltimeminimization).Userexternalityoftraveltimeisthesameforallusergroupsbecausetheyusethesametypeofvehicle.Thus,theoptimalanonymoustollisequaltotheuserexternalityoftraveltimemultipliedbythearithmetic 25

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meanofVOTsofalluserstraversingthatlink,whichisexactlytheadditionaltravelcostthatamarginaluserimposesonothersalreadytravelingonthatlink.Ontheotherhand,thesystemmanagermaywanttoachieveatime-basedsystemoptimum(totalsystemtraveltimeminimization).Inthiscase,althoughtheuserexternalityoftraveltimeisstillthesameacrossdifferentusergroups,noanonymoustollcanbederivedbasedonthemarginalcostpricingprinciple.Thereasoniscongestiontollscanonlybechargedinmonetaryunit.Tointernalizetheexternality,acongestiontollequalstotheuserexternalityoftraveltimemultipletheusergroup'sVOTshouldbechargedtoeachusergroup,whichisnolongeranonymous. YangandHuang ( 2004 )furtherprovedthatananonymoustollschemethatsupportsamulticlassuserequilibriumowdistributionasatime-basedsystemoptimumexistsandcanbeobtainedbysolvingaproposedduallinearprogramming(LP)problem.However,theanonymoustollderiveddoesnotreecttheuserexternality. YinandYang ( 2004 )extendedtheworkandproposedageneraltollsetcontainingallfeasibleanonymoustollpatternsthatsupportamulticlassuserequilibriumowdistributionasatime-basedsystemoptimum.Thetollsetcanbeexpressedasanonemptypolyhedroninalinearinequalityandequalitysystem.Inasimilarway,thegeneraltollsetthatsupportsacost-basedsystemoptimumcanalsobederived.And,themarginalcostlinktollpatternisanelementoftheset. EngelsonandLindberg ( 2006 )pointedoutthatfromapureeconomicperspectivethatcost-basedsystemoptimumprovidesthemaximumoveralleconomicefciencytosocietywithheterogeneoususes. EngelsonandLindberg ( 2006 )investigatedthemarginalcostpricingprincipleinacost-basedframeworkandconcludedthattheequilibriumowdistributionisnotuniqueinacost-basedsystemoptimumsupportedbythemarginalcostlinktollwhiletheaggregatelinkowisindeedunique.Therefore,implementingthemarginalcostlinktoll,theresultingtollequilibriumneedsnotcoincidewiththecost-basedsystemoptimumowdistribution,whichmaycausemarginalcostpricingprincipletofailtowork 26

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inreal-worldimplementation.However,theyfurtherprovedthattheobjectivefunctionvalueofthecost-basedsystemoptimum,i.e.,thetotalvalueofsystemtraveltimeisuniqueforanyresultingtolledequilibriumdistributionbyimplementingthemarginalcostlinktoll,eventhoughtheobjectivefunctioningeneralisnon-convex. 2.5PublicAcceptanceofCongestionPricing Economistshavelongrecognizedthatcongestionpricingisaneffectivetooltomanageandreducetrafccongestion.However,gettingthepublictoacceptcongestionpricingisstillanobstacle.Inthissection,wewillbrieydiscusswhythemarginalcostpricingtollschemeisnotappealingtothegeneralpublic.Equityissues,themajorconcernpreventingcongestionpricinggainingpublicacceptance,willalsobeinvestigated. 2.5.1WeaknessofMarginalCostPricing Marginalcostpricingcanincreasesocialwelfaretothemaximumlevel,whichisbenecialtosocietyasawhole.However,itturnsoutthatallroadusersinthesystemareworseoffcomparedtotheun-tolledscenario.Theonlystakeholderthatgainssurplusisthegovernmentwhocollectstollrevenues. Hau ( 2005 )pointedoutthatmarginalcostpricingismostlikelydoomedtobepoliticalfailuresbecauseuserswillndthemselvesworseoff.UseFigure 2-1 toillustrate,excessowbeyondtolledequilibriumowVoispricedofftoothertimesorpathesbecausetheirwillingnesstopayisnothighenough.Theseuserssufferalossinconsumersurplusofthearea bhk.Forthoseusersremainingontheroad,theirtraveltimeisreducedatthecostofpayingcongestiontolls.Theirtraveltimesaving(area mnkg)isalwayslessthanthetollstheypay(area mphg).Adecitinconsumersurplusofthearea nphkisexpectedwhenmarginalcostpricingtollisintroduced.Ifthegovernmentdoesnotcompensateusersdirectlybyrevenuerefundingorindirectlybyimprovementsininfrastructureandpublictransit,roadusersaregenerallymadeworseoff. 27

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2.5.2EquityEffectsofCongestionPricing Equityeffectswerelargelyignoredbyearlyresearchersoncongestionpricing. Richardson ( 1974 )believedthattheequityargumentsaremurkyanddecisionshouldbemadeinlightofefciency. Giuliano ( 1994 )claimedthatcongestionpricingisnomoreregressivethanotherexistingroadrelatedcharges,suchasfueltax.Indeed,equityeffectscouldbehighlysubjectiveandeconomistshavenotreachedaconsensusonthedenitionofequityyet.Ontheotherhand,equityissuesplayanimportantroleinimplementinganypublicpolicies.Thereisnoexceptionforcongestionpricing.Withtheadventofelectronictolling,congestionpricingbecomesmorepracticalandmanycitiesaroundtheworldareconsideringimplementingit.Therefore,inrecentyears,researchershaveshownanincreasinginterestinequityeffectsofcongestionpricing.Duetothecomplexityofequityissues,threedimensionsofequityconcernsarediscussedinthissection. Assumingroadusersareheterogeneousintheirincomes,social(vertical)equityissuesarisewhenananonymoustollischargedbecausetheuniformchargetendstotakeupahigherpercentageofthebudgetofapersonorfamilywithalowerincome. Foster ( 1974 )wasthersttoarguethatcongestionpricingisregressiveanddiscriminatesagainstthepoor.Sincethen,manystudieshavebeendoneonwelfare/revenueredistributiontomakecongestionpricingschemeprogressiveorlessregressive. Eliasson ( 2001 )proposedatollschemethatreducesaggregatetraveltimeandmakeseveryonebetteroffbyredistributingtollrevenuesequallytoallusers. Yangetal. ( 2004 )developedanintegratedpricingschemeforabimodaltransportationnetworkwithtransitsubsidy. EliassonandMattsson ( 2006 )claimedthatwiththetollrevenuespentonpublictransportation,whichbenetswomenandthosewithlowerincomes,theproposedStockholmroadpricingschemeisprogressive. Spatial(horizontal)equityemphasizesthatthebenetsandcostsofcongestionpricingshouldbedistributedequallyoverspace( Viegas 2001 ).Congestionpricingmay 28

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havesignicantlydifferentimpactsonthegeneralizedtravelcostsbetweendifferentODpairs. YangandZhang ( 2002 )designedasecond-besttollschemewithexplicitconstraintsonsocialandspatialequity. YinandYang ( 2004 )triedtondthemostequitableanonymouslinktollpatternthatcansupportthemulticlass,bi-criterionuserequilibriumowpatternasasystemoptimum.Thetotaltransformedtraveldisutilitywasusedastheinequalitymeasurement.Socialandspatialinequalitieswerecalculatedforthepurposeofcomparison. Anotherdimensionofequityissuewhichisoftenoverlookedisintergenerationalortemporalequity.Let'sconsiderabroaderroadpricingconcept,tollrevenuesareusuallyusedtonancenewroadsandcapacityexpansionsundercurrentpracticesoftransportationinfrastructurenancing.Towhatextentthegainsandlossesaredistributedbetweenthepresentandfuturegenerationsisthekeyissueofintergenerationalequity. SzetoandLo ( 2006 )denedagapfunctiontocapturetheintergenerationalequityandformulatedatime-dependentnetworkdesignproblemtoobtainthemostdesirabletollpatternsforeachplanningperiod(generation). Sincethereisnoabsoluteequity,itismoremeaningfultodiscussthedegreeofequityusingsomeequitymeasures. Ramjerdi ( 2006 )examinedvariousequitymeasuresforaproposedpricingschemeinOslousingtwocasestudiesandconcludedthatitisnotappropriatetomakejudgmentabouttheequityimplicationofapolicyonthebasisofasingleequitymeasure,therefore,multipleequitymeasuresshouldbeemployed. 2.6Summary Thischapterbrieyreviewedtheclassicmarginalcostpricingprincipleintheframeworkofnetworkmodeling.Recentdevelopmentsinrst-bestandsecond-bestpricingwerealsocovered.Twoaspectsofcongestionpricingwerehighlighted:userheterogeneityandpublicacceptance.Generalspeaking,unobservableuserheterogeneityismoredifculttodealwithbecauseonlyanonymoustollisfeasibletochargeinthiscaseandsomeequityconcernsmayariseduetotheregressiveness 29

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ofanonymoustolling.Publicacceptanceisthekeyissueofimplementingcongestionpricingschemes.Weexplainedwhymarginalcostpricingschemeisnotappealingtothegeneralpublicandhowthreedimensionsofequityconcernsplayimportantrolesindesigningcongestionpricingtollschemes. Toaddressthepublicacceptanceissues,ideallyweshoulddirectlytackleequityconcernswhendevelopingacongestionpricingscheme.However,theequityissueisnottodeterminewhetherapricingschemeisequalornot,butitisaproblemaboutthedegreeofequity,whichreliesonsomeequitymeasures.Unfortunately,notallequitymeasuresarewelldenedandsuitabletobeincorporatedinoptimizationmodels.Therefore,insteadofdirectlyutilizingvariousequitymeasures,aPareto-improvingapproachisadoptedinthisdissertation,thatis,developingcongestionpricingschemesthatimprovesystemefciencywhileensuringthateveryoneisbetteroffandnooneisworseoffthantheun-tolledsituation,whichwillbediscussedinmoredetailsinthenextchapter.AlthoughthePareto-improvingapproachdoesnotdirectlyaddressequityconcerns,itiseasyforthegeneralpublictounderstand.Sincenooneisworseoffinthetollscheme,itwouldpresumablybeeasiertogainpublicacceptanceandservesasasteppingstonetowardsmorecomprehensiveroadpricingschemes. 30

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Figure2-1. Graphicalrepresentationofmarginalcostpricing 31

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CHAPTER3PARETO-IMPROVINGCONGESTIONPRICINGAPPROACH 3.1TheConceptofPareto-ImprovingPricing Paretoefciency,orParetooptimality,isacentralconceptinwelfareeconomics(see,e.g., Ng 1979 ).AchangefromoneeconomicarrangementtoanotherthatcanmakeatleastonepersonbetteroffwithoutmakinganyotherpersonworseoffiscalledaParetoimprovement.AsituationissaidtobePareto-efcientorPareto-optimalwhenthereisnofurtherParetoimprovementcanbemade.ItisnamedafterVilfredoPareto(1848-1923),anItalianeconomistwhousedtheconceptinhisstudiesofeconomicefciencyandincomedistribution.AlmosteveryonewouldagreethatsocietyshouldavoidsituationsthatarenotPareto-optimal.SinceaParetoimprovementdoesnothurtanyone,aPareto-improvingarrangementwouldpresumablyhavelessobstaclesofgainingpublicacceptance.Itiswellknownthatifthereisperfectcompetitionandnoexternality,Paretooptimalitywouldhold.However,thestatementingeneralisnottruewhenexternalityexists. Intransportationliterature,usersareusuallyassumedtomaketheirroutechoicesselshlytominimizetheirowntravelcosts.TheowdistributionthatfollowsthisroutechoiceprincipleiscalledaWardrop'suserequilibrium(UE)distribution.Becauseoftheexistenceofcongestionexternality,ParetooptimalitygenerallydoesnotholdinaUEstate.Fromthesystem'sperspective,systemefciencyisnotoptimizedunderaUEowdistribution.Systemoptimum(SO),ontheotherhand,makesanassumptionthatallusersworkcooperativelytoimprovesystemefciencyand,furthermore,aSOowdistributionisPareto-optimal.Ingeneral,aPareto-optimalowdistributionneedsnottooptimizesystemefciency,althoughaSOowdistributionisalwaysPareto-optimal.ASOowdistributionisidealtosociety;however,itisunstableinthesensethatsometravelerscouldunilaterallychangeroutestoreducetheirowntravelcosts.Intheliterature,along-establishedmarket-basedapproachtoinduceaUEowdistributionto 32

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aSOowdistributionisthemarginalsocialcost(MSC)congestionpricingstrategy(see,e.g., Pigou 1920 ; Beckmannetal. 1956 ; Button 1993 and ArnottandSmall 1994 ). AlthoughaSOowdistributionandMSCcongestionpricingimprovesystemefciencytoitsoptimallevel,theyarenotappealingtothegeneralpublicwithgoodreason( Hau 2005 ).SomeusersmayndthemselvessufferinglongertraveltimeinaSOowdistributionevenbeforetolling.Moreover,afterintroducingtheMSCtolls,someusersmayexperiencehighertravelcosts(timepluscongestiontolls)thanthoseundertheoriginalUEcondition.ToillustratetheconceptofMSCcongestionpricing,considertheexampleinFigure 3-1 from HagstromandAbrams ( 2001 )inwhichthereisonlyoneODpair(1,4)withademandof3.6. Table 3-1 displaysthelinkandpathowsunderUEandMSCpricingalongwiththeassociatedcosts.RecallthattollsunderMSCpricingareoftheformdtij(vij)=dvijandvijistheowonlink(i,j).Whenimplemented,MSCpricingforces1.54userstousepath1-2-4withatotalcostof101.70,ofwhich75.85isthetraveltimeandtherest(25.85)isfortolls.Thus,these1.54userssuffertwice,onceforhavingtousealongerroute(75.85insteadof71.06)andtheotherforhavingtopaytolls.Overall,thetotalcosttothe3.6usersunderMSCpricingis366.13whichismorethanthetotalcost(225.80)underUE,acostconsistingentirelyoftimeordelay.However,MSCpricingyieldslesstotaldelay(227.11)andgeneratestollrevenue(139.02)forthetransportationauthority. Inthischapter,insteadofconsideringMSCorsimilarpricing,weintroduceaclassoftollingorpricingschemesthatimprovesnetsocialbenetswithoutmakinganystakeholder(roadusers,societyorthetransportationauthority)worseoff.WetermthisclassofpricingschemesPareto-improvingcongestionpricing. Theoutlineofthischapterisasfollows:Section2discussesaclassofnon-equilibriumowdistributioncalleddominatingowdistributionthatimprovessystemefciencywithoutmakinganyoneworseoff.Mathematicalformulationstondsuchaowdistributionaregiven.Theconditionswhenadominatingowdistributioncanbe 33

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supportedbyanonymousnonnegativetollsareexplored.Section3investigatespropertiesandtheexistenceofanonymousanddiscriminatoryPareto-improvingtolls.Section4providesmathematicalformulationstoobtaindominatingowdistributionsandequilibrium-inducingPareto-improvingtollvectorssimultaneously.Section5isthesummaryofthischapter. 3.2DominatingFlowDistribution EventhoughaSOowdistributioncanreducetotalsystemdelaytoitsminimumlevel,someusersmaysufferlongertraveltime,andthusareworseoff.Suchaowdistributionmayencountererceoppositionfromthegeneralpublic,andsubsequentlysomeelectedofcialsarereluctanttoadvocatesuchasolution.Intuitively,ifthereexistsaowdistributionthatimprovessystemefciencywithoutmakinganyuserworseoff,theowdistributionwouldbemucheasiertogainpublicsupport.Inthisdissertation,suchaowdistributionisreferredtoasadominatingowdistribution.AowdistributiondominatestheUEdistributionifitstrictlyimprovesameasureofsystemefciencyandallowsalluserstouseroutesthatarenolongerthanthoseunderUE.Forbrevity,wealsosaythataowdistributionisdominatingifitdominatestheUEdistribution.WhencomparedtotheUEdistribution,weshowlaterthatourdenitionofdominationisconsistentwiththetraditionalone,i.e.,nouserisworseoffandsomearebetteroffunderadominatingowdistribution.Consequently,wesaythatapricingortollingschemeisPareto-improvingifitinducesaowdistributionthatdominatestheUEdistributionandgeneratesageneralizedtravelcost(i.e.,timepluscongestiontolls)foreveryuserthatisnolargerthanhisorhertraveltimeunderUE. In HagstromandAbrams ( 2001 )and AbramsandHagstrom ( 2006 ),theauthorsrefertoadominatingowdistributionasaGeneralizedBraessParadox.Theyfoundnon-equilibriumowdistributionsmayachievelowertraveltimeforeveryuserthantheUEowdistributiondoesandalsosubsequentlyreducetotalsystemdelay.TheseeminglyparadoxcanbeexplainedbythefactthattheUEowdistributionmaynotbe 34

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Pareto-optimal,whichimpliesitispossibletondbetteralternativesforatleastoneuserwithoutmakinganyotheruserworseoff. 3.2.1FeasibleFlowDistribution Whendemandsarexed,VFdenotesasetofallfeasibleowdistributions,eachofwhichisrepresentedasv.Thenumberofnodesandlinksinthetransportationnetworkaremandnrespectively.Inparticular,v2RnandvF2Rn.ThesetvFcanbedescribedusingeitherpathorlinkowvariables.Usingtheformer,letfwranddwdenotetheamountofowsonpath(orroute)randthegivendemandforODpairw,respectively.Then,VF=fv:vij=XwXr2Pwijrfwr;Xr2Pwfwr=dw,8w;fwr0,8w,rg, wherePwisthesetofpathsforODpairwandijr(equals0or1)indicateswhetherarc(i,j)isonpathr.Alternatively,letAbethenode-arcincidencematrixanditisofsizemn.WeusewtorepresenttheindexforODpairwanddwdenotesthedemandbetweenODpairw.Ew2Rmisaninput-outputvectorandhasexactlytwononzerocomponents,theonecorrespondingtotheoriginofODpairwhasavalue1andtheonecorrespondingtothedestinationhasavalueof-1.LetLbethesetoflinksinthenetworkanditselement(i,j)representsalinkfromnodeitonodej.Traveltimefunctiontij(vij)isassumedtobeseparable,continuous,strictlymonotoneandtij(vij)>0forallv2VF.ThefeasibleowdistributionVFcanbewrittenasfollows:VF=fv:v=Xwxw,Axw=Ewdw,8wg, wherexw2RnisavectorwhosecomponentsarelinkowsofODpairw. 3.2.2FindingDominatingFlowDistribution Todescribeadominatingowdistributionmathematically,letfwrbethepathowonpathrofODpairwandPwbethesetofpathsbetweenODpairw.Aowdistributionv2VFdominatesagivenUEowdistribution,orisdominatingifvanditspathowfwr 35

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satisfytheowingconditions:X(i,j)2wrtij(vij)cUEw8w,r2Pw+ (3)t(v)Tv0gisasetofutilizedpathsassociatedwiththeowdistributionv.vUEistheUEowdistribution,i.e.,vUEsatisest(vUE)T(v)]TJ /F4 11.955 Tf 12.42 0 Td[(vUE)0,8v2VF.Inparticular,condition( 3 )ensuresthatallusersarenotworseoffandcondition( 3 )guaranteesthatsomearebetteroff.Ifvisdominating,thencondition( 3 )impliesthattheremustexistatleastoneutilizedpathforsomeODpairw,suchthatP(i,j)2w~rtij(vij)
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problem.Theoptimizationproblemisequivalenttotheformulationin AbramsandHagstrom ( 2006 )ofndingaGeneralizedBraessParadox.P1:minvt(v)Tv (3)s.t.v2VF (3)fwr(X(i,j)2wrtij(vij))]TJ /F4 11.955 Tf 11.95 0 Td[(cUEw)08w,r2Pw (3) Intheabove,constraint( 3 )requiresvtobeafeasibleowdistribution.Constraint( 3 )ensuresthatwhenpathrisutilized,itstraveltimeshouldbelessthanorequaltothatundertheUEowdistribution.Theobjectivefunctionistominimizetotalsystemdelay.Thus,iftheoptimalsolutiontoP1isvandt(v)v0,P(i,j)2wrtij(vij))]TJ /F4 11.955 Tf 11.95 0 Td[(cUEw0andhence,condition( 3 )alsoholds.Furthermore,ifthereexistsanotherdominatingowdistributionotherthanvthathaslesstotalsystemdelay,vcannotbetheoptimalsolutiontoproblemP1.Thus,ifdominatingowdistributionsexist,vmustbetheoptimaldominatingowdistribution.Whent(v)Tv=t(vUE)TvUE,thereisnov2VFthatsatisesboth( 3 )and( 3 ),i.e.,nodominatingdistributionexists.Notethatfortheve-linknetwork,DF-F-3inTable2solvestheP1problemandtheothertwodonot. Tomaketheproblemmorecomputationallytractable,weformulatethefollowinglink-basedoptimizationproblem:P2:min(v,)t(v)Tv (3)s.t.v2VF (3)xwij)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(wi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj)]TJ /F4 11.955 Tf 11.95 0 Td[(tij(vij)08w,(i,j)2L (3)wo(w))]TJ /F7 11.955 Tf 11.95 0 Td[(wd(w)cUEw8w (3) Whenalink(i,j)isutilized,i.e.,xwij>0,constraint( 3 )reducesto)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(wi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj)]TJ /F4 11.955 Tf 11.96 0 Td[(tij(vij)0.AddingthisexpressionupforeachlinkalongautilizedpathrofODpairwyieldsthe 37

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following:X(i,j)2wrtij(vij)X(i,j)2wr(wi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj)=wo(w))]TJ /F7 11.955 Tf 11.95 0 Td[(wd(w), where,forODpairw,wrisasetcontaininglinksonpathrando(w)andd(w)representtheoriginanddestinationnodes.Thus,alongwithconstraint( 3 ),itimpliesthatcondition( 3 )issatisedforeveryutilizedpathandODpair.WithasimilarargumenttoproblemP1,wecanconcludethattheoptimalsolutiontoP2istheoptimaldominatingowdistributionifthereexistsadominatingowdistribution. Asformulatedabove,mathematicalprogramP1andP2areoptimizationproblemswithcomplementarityconstraints(MPCC),adifcultclassofproblemtosolveoptimally(see,e.g., ScheelandScholtes 2000 ),and AbramsandHagstrom ( 2006 )proposeapracticalmethodologyforndingalocaloptimalsolutiontoproblemP1byallowingonlyroutesutilizedintheUEdistributiontohavepositiveows. 3.2.3DominatingFlowDistributionandNonnegativeTolls Althoughadominatingowdistributioncanimprovetrafcconditionwithoutmakinganyoneworseoff,theowdistributionisnotinequilibriumstateandhenceisunstable.Inreal-worldimplementation,thedominatingowdistributionhastoberealizedthrougheitherregulatoryormarket-basedapproach.Inthischapter,roadcongestionpricingisusedasaninstrumenttoinducesuchadominatingowdistribution.Inthissection,wewouldliketoexploretherelationshipbetweenadominatingowdistributionandnonnegativetolls.Morespecically,wewilldiscusswhenadominatingowdistributioncanbesupportedbyananonymousnonnegativetollvectorasaWardropequilibriumowdistribution. Beforewestartansweringthisquestion,theconceptofGallagerloop,orloopinGallager'ssense( Gallager 1977 )needstobeintroducedrst. DenitionofGallagerloop:Anaggregatefeasibleowvthatsatisesdemanddcontainsloopsifitcanbeexpressedasv=v+~v,wherevisafeasibleowdistributionfordand~v>0isacirculationow,suchthatA~v=0. 38

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Inotherwords,foraowdistributionwithGallagerloop,ifthecirculationow~visremoved,visstillafeasibleowdistributionfordemandd.HereafterwewillusethetermcommoditytorefertousersbetweenaparticularODpair.BythedenitionofGallagerloop,aowdistributionisloop-freeifthereexistsnodecompositionintocommoditylinkowsthatusealllinksofadirectedcycle. Gallager ( 1977 )alsoprovidedthefollowingnecessaryandsufcientconditionforaowdistributiontocontainaGallagerloop. Lemma1:Anaggregatefeasibleowvcontainsloopsifandonlyifvisanotherfeasibleaggregateowforthesamedemandwithv
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demand,wehaveanotheraggregateowvandv
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Lemma2:Theabsenceofloops(single-andmulti-commodity)isnecessaryandsufcientfortheexistenceofanonymousnonnegativetollstosupportagivenfeasibleowdistributionasaWardropequilibrium. Proof:Weprovethenecessarypartrst.Ifafeasibleow,v,containsnoloop,accordingtoTheorem1,itisrealizablewithpositivecostc(v).Thecostfunctioncanbewrittenasfollows:cij(vij)=tij(vij)+ij.Thetollforlink(i,j)canbeexpressedasij=cij(vij))]TJ /F4 11.955 Tf 11.95 0 Td[(tij(vij).InatolledUE,thefollowingsystemofequationsholds:X(i,j)2wrcij(vij)=X(i,j)2wr(tij(vij)+ij)=w,8wandr2Pw+ (3)X(i,j)2wrcij(vij)=X(i,j)2wr(tij(vij)+ij)w,8wandr2Pw0 (3) wherePw+=fr:r2Pwandfwr>0gandPw0=fr:r2Pwandfwr=0g.WefurtherknowthattheUEowdistributiondoesnotchangewhenthetravelcostfunctionofeachlinkismultipliedbyapositivescale,i.e.,iflinktravelcostbecomescij(vij)foreverylink(i,j)2L,theUEowdistributionstillholds.Therefore,weknowthatij=cij(vij))]TJ /F4 11.955 Tf 12.07 0 Td[(tij(vij)isalsoavalidtolltosupporttheUEow.Bychangingthescale,theexistenceofanonnegativetollvectorisguaranteed. Toprovethesufcientpart,generallyeverylinkisassumedtohaveapositivefreeowtraveltimeinatransportationnetwork,whichimpliesthattij(vij)>0.Therefore,whenthereisanonnegativetollvectortosupportafeasibleowdistributionv,wehavecij(vij)=tij(vij)+ij>0.ByTheorem1,weknowthatthefeasibleowdistributionvdoesnotcontainanyloop(single-andmulti-commodity). Corollary1:Theabsenceofloops(single-andmulti-commodity)isnecessaryandsufcientfortheexistenceofanonymousnonnegativetollstosupportadominatingowdistributionasaWardropequilibrium. Essentially,adominatingowdistributionisjustaspecialclassoffeasibleowdistribution.Asaresult,byLemma2,wehavetheabovecorollary. 41

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ForatransportationnetworkwithasingleODpair,therelationshipbetweentheoptimaldominatingowdistributionandnonnegativetollscanbefurtherstrengthened. Theorem2:WhenatransportationnetworkhasasingleODpair,therealwaysexistsanonnegativetollvectortosupporttheoptimaldominatingowdistribution. Proof:Toobtainacontradiction,let(v,)bethesolutiontotheoptimizationproblemP2.IfvcontainsaGallager'sloop,itcanbeexpressedasv=v+~v,wherevisanotherfeasibleow,v2VF,andv
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followingdiscussionswillprovideananswertothisquestion.Inthissection,weassumethatadominatingdistributionisgiven. 3.3.1ExistenceofAnonymousPareto-ImprovingTolls Letvdenoteagivendominatingdistribution.WithrespecttothetravelcostsunderUE,anonnegativetollvector,,isPareto-improvingifitsatisesthefollowingconditions:X(i,j)2wr(tij(vij)+ij)=w8w,r2Pw+ (3)X(i,j)2wr(tij(vij)+ij)w8w,r2Pw0 (3)>0 (3)wcwUE8w (3) Similartobefore,Pw+=fr:r2Pwandfwr>0gandPw0=fr:r2Pwandfwr=0g.ForeachODpair,conditions( 3 )and( 3 )ensurethatisavalidtollvector(see,e.g., HearnandRamana 1998 ),inthattheyforcetheutilizedpathsassociatedwithvtohavethesamegeneralizedtravelcostthatisnogreaterthanthoseforthenon-utilizedpaths.Condition( 3 )ensuresthatthegeneralizedcostforeachODpairisnogreaterthantheuserequilibriumtraveltime.Withoutthenonnegativityrequirementin( 3 ),setting=)]TJ /F4 11.955 Tf 9.3 0 Td[(t(v)triviallysatisesconditions( 3 )( 3 )andtheresultinggeneralizedtravelcostiszeroforallpaths.Thus,=)]TJ /F4 11.955 Tf 9.3 0 Td[(t(v)isParetoimproving.Whenmustbenonnegative,aPareto-improvingtollvectormaynotexist.Asdiscussedintheprevioussection,ifadominatingowdistributioncontainsasingle-ormulti-commodityloop,nononnegativetollscanbefoundtosupportthedominatingowdistribution.However,evenifadominatingowdistributioncanbesupportedbyanonnegativetollvector,doesthatindicatethatwecanalwaysndanonnegativePareto-improvingtollvectortoinducetheowdistribution?Wewillillustratetheproblemusingthefollowingnumerical 43

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example.Theve-linknetworkinFigure 3-4 hastwoODpairs(1,4)and(3,2)withdemandsof3.6and1.0respectively. ToobtainadominatingowdistributionoftheexampleinFigure 3-4 ,wecansolvetheoptimizationproblemP2.TheUEowdistributionandoptimaldominatingowdistributionaregiveninTable 3-3 WecanobservefromTable 3-3 thatusersbetweenODpair(3,2)arebetteroffintermsoflowertraveltime,andtotalsystemdelayisalsoreduced.Letafeasibleowdistributionvbeadominatingowdistribution.ThefollowinglinearprogramisconstructedtoexaminetheexistenceofanonnegativePareto-improvingtollvectortosupportthedominatingowdistributionv.P4:min(y,,)y (3)s.t.xwij)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(tij(vij)+ij)]TJ /F7 11.955 Tf 11.95 0 Td[(wi+wj=08w,(i,j)2L (3)tij(vij)+ijwi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj8w,(i,j)2L (3)wo(w))]TJ /F7 11.955 Tf 11.96 0 Td[(wd(w))]TJ /F4 11.955 Tf 11.96 0 Td[(cUEwy8w (3),y0 (3) Constraints( 3 )and( 3 )ensurethatthetollvectorsupportstheowdistributionvasauserequilibrium,i.e.,(t(v)+)T(v)]TJ /F3 11.955 Tf 12.81 0 Td[(v)0,8v2VF.Whenthedominatingowdistributioncontainsnoloop,itcanbesupportedbyanonnegativetollvector.Bysettingyequalstoalargenumber,constraint( 3 )isnotbindinganymore.Theproblemreducestosolvingasystemofequationstogetanonnegativetollvectorthatsupportsthedominatingowdistribution.Sincethenon-tolledUEowdistributionisalwaysfeasibletothesystemofequations,theoptimizationproblemalwayshasasolution.Iftheoptimalobjectivefunctionvalueislessthanorequalto0and>0,wendaPareto-improvingtollvectortosupportthedominatingowdistribution.Iftheoptimalobjectionvalueisstrictlygreaterthan0,wecanconcludethatthereisno 44

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Pareto-improvingtollvectortosupportthedominatingowdistribution.Inthemeantime,theprogramalsogeneratesanonnegativetollvectorthatcaninducethedominatingowdistributiontoauserequilibriumowdistribution. ThefollowingexampledemonstratesthatanonnegativePareto-improvingtollvectormaynotexisttosupportadominatingowdistributioneventhoughtheowdistributioncanbesupportedbynonnegativetolls. TheUEanddominatingowdistributionsarethesameasthepreviousexampleinFigure 3-2 .BysolvingproblemP4,theoptimalobjectivefunctionvalueis12.16,whichindicatesthatthereisnoPareto-improvingtollvectortosupportthegivendominatingowdistribution.ThesolutionstoproblemP4alsodemonstratethatthegivendominatingowdistributionactuallycanbesupportedbyanonnegativetollvector.ThetollvectorislistedinthelastcolumnofTable 3-4 .TheequilibriumtravelcostofODpair(1,4)increasesfrom71.25to82.69afterimplementingthenonnegativetolls,andhencetheequilibrium-inducingtollvectorisnotPareto-improving. Generally,theremaynotexistasetofPareto-improvingtollsthatinducesagivendominatingowdistribution.Thetheorembelowprovidesanecessaryandsufcientconditionunderwhichsuchtollsexist. Theorem3:Letvbeadominatingowdistributionsuchthatv=Pwxw,whereAxw=Ewdwandxw0forallw.Then,thereexistsaPareto-improvingtollvectorthatinducesvifandonlyifthevector(xw,0)solvesthefollowinglinearprogram:min(y,z)t(v)T(Xwyw)+XwcUEwzw (3)s.t.Ayw+Ewzw=Ewdw8w (3)Xwywijvij8(i,j)2L (3)yw,zw08w (3) Proof:BecausetheKarush-Kuhn-Tucker(KKT)conditionsarebothnecessaryandsufcientforlinearprograms,(xw,0)solvestheabovelinearprogramifandonlyifthere 45

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existsmultipliersw,,wandwsuchthatthefollowinghold:tij(v)+ij)]TJ /F3 11.955 Tf 11.96 0 Td[((wi)]TJ /F7 11.955 Tf 11.95 0 Td[(wj))]TJ /F7 11.955 Tf 11.95 0 Td[(wij=0,8w,(i,j)2L (3)cUEw)]TJ /F4 11.955 Tf 11.96 0 Td[(ETww)]TJ /F7 11.955 Tf 11.96 0 Td[(w=0,8w (3)(xw)Tw=0,8w (3),w,w0 (3) Becausexw>0whenlink(i,j)isonautilizedpath,( 3 )impliesthatwij=0forall(i,j)onautilizedpath.Byaddingequationsin( 3 )associatedwitharcsonthesamepathtogetherandusingthatthefactthatwij=0forlinksonutilizedpath,thefollowingmustholdforeveryODpairw=(o,d):X(i,j)2wr(tij(v)+ij)=(wo)]TJ /F7 11.955 Tf 11.95 0 Td[(wd),8r2Pw0 (3)X(i,j)2wr(tij(v)+ij)(wo)]TJ /F7 11.955 Tf 11.95 0 Td[(wd),8r2Pw0 (3) where,similartobefore,Pw+=fr:r2Pwandfwr>0gisthesetofutilizedpathsassociatedwithvandPw0=fr:r2Pwandfwr=0gisthesetofunutilizedpaths.Next,observethat( 3 )impliesthatcUEw)]TJ /F4 11.955 Tf 12.05 0 Td[(ETww=w0orcUEwwo)]TJ /F7 11.955 Tf 12.05 0 Td[(wdforallw.Thus,lettingw=wo)]TJ /F7 11.955 Tf 12.18 0 Td[(wdyieldsatriplet(v,,)thatsatises( 3 )( 3 ).Becausevisdominating,itfollowsthatt(v)Tv0alsocontainsnodirectedcycle. 46

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UsingthealgorithminDial(1999)fortheoneoriginproblem,itispossibletoconstructatollvector0suchthat,foreachODpair,thegeneralizedcostsofallutilizedpathsarethesameandatleastoneofwhichistoll-free.Thelatterimpliesthat,foreveryODpairw,P(i,j)2w~rij=0forsome~r2Pw+anditfollowsfrom( 3 )that,foreveryr2Pw+,w=X(i,j)2wr(tij(v)+ij)=X(i,j)2w~rtij(v)cUEw,8r2Pw+ Fromtheabove,satises( 3 )( 3 ),andvisdominating.Thus,isPareto-improving. Toillustrate,considerthevelinkexampleinFigure 3-1 .Table 3-5 providestheUEandadominatingdistribution.ThelattersolvesproblemP2andislabeledas`DF-F-3'inTable 3-5 ThealgorithminDial(1999)constructsthelongestpathtreeinFigure 3-5 usingthelinkcosttij(v)inTable 3-5 .Basedonthislongestpathtree,Dial'salgorithmwouldsetthetollsonlink(3,2)and(3,4)tobe32=(51.36-22.4)-10.61=18.35and34=(71.06-22.4)-42.74=5.91.Doingsoproducesthetollvector1=[0.0,0.0,18.35,5.91,0.0]Tandmakesthegeneralizedcost(timeplustolls)ofeverypathequal71.06.However,otherPareto-improvingtollvectorsexist.Forexample,2=[5.91,0.0,12.44,0.0,0.0]Tand3=[2.26,0.0,16.09,3.65,0.0]TarebothPareto-improving.Whenconsideringasecondaryobjective,1and2maybemoreattractivebecause,e.g.,theyrequireasmallernumberoftollcollectionfacilitiesthan3.Withallthreetollvectors,w=cwUEbecausethelongestpathsundervandUEhavethesamelength. 3.3.2ExistenceofDiscriminatoryPareto-ImprovingTolls Sofarinthischapter,allpricingschemeswehavediscussedareanonymouspricingschemes,i.e.,roadusersarechargedthesameamountofmoneyonlinksthataresubjecttocongestionpricingregardlessoftheirindividualdifferences.However,asthetechnologyofvehiclepositioningsystemandelectrictollingadvances,discriminatorycongestionpricingschemesarebecomingmorereadilyavailabletoreal-world 47

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implementations.Oneformofdiscriminatorypricingistochargeusersdifferentlybasedontheiroriginsanddestinations,whichisreferredtoascommodity-dependent(OD-dependent)pricing.Inthissection,wewillinvestigatetherelationshipbetweenOD-dependenttollsanddominatingowdistributions. Letvdenoteagivendominatingdistribution.WithrespecttothetravelcostsunderUE,aOD-dependentnonnegativetollvector,,isPareto-improvingifitsatisesthefollowingconditions:X(i,j)2wr(tij(vij)+wij)=w8w,r2Pw+ (3)X(i,j)2wr(tij(vij)+wij)w8w,r2Pw0 (3)0 (3)wcwUE (3) wherePw+=fr:r2Pwandfwr>0gandPw0=fr:r2Pwandfwr=0g.Similartoitsanonymouspricingcounterpart,amathematicalprogramisconstructedtoexaminetheexistenceofaPareto-improvingtollvectortosupportthedominatingowdistributionv.P5:min(y,,)y (3)s.t.xwij)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(tij(vij)+wij)]TJ /F7 11.955 Tf 11.96 0 Td[(wi+wj=08w,(i,j)2L (3)tij(vij)+wijwi)]TJ /F7 11.955 Tf 11.96 0 Td[(wj8w,(i,j)2L (3)wo(w))]TJ /F7 11.955 Tf 11.96 0 Td[(wd(w))]TJ /F4 11.955 Tf 11.95 0 Td[(cUEwy8w (3),y0 (3) UsingthesamedemandandnetworksettingsasthepreviousexampleinFigure 3-4 ,wesolveproblemP5andpresenttheresultsinTable 3-5 OD-dependenttollsarelistedinthelasttwocolumnsofTable 3-6 .WecanobservethatroadusersbetweenODpair(3,2)arebetteroffandtotalsystemdelayisalso 48

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reduced,therefore,theresultingOD-dependenttollingschemeisPareto-improving.TheabovenumericalexamplesinTable 3-4 and 3-6 demonstratethatadominatingowdistributionthatcannotbesupportedbyanonymousPareto-improvingtollsmaybesupportedasatolledUEbyanOD-dependentPareto-improvingtollvector. HagstromandAbrams ( 2009 )concludedthatanyfeasibleowdistributionwithnosingle-commodityloopcanberealizedasaWardropequilibriumusingcommodity-dependentnonnegativetollseveniftheowcontainsamulti-commodityloop.ThefollowingLemmaisadirectresultoftheirconclusion. Lemma3:Ifadominatingowdistributiondoesnotcontainasingle-commodityloop,thentherealwaysexistsanonnegativeOD-dependenttollvectortosupportit. ToexploretherelationshipbetweenagivendominatingowdistributionandOD-dependentPareto-improvingtollvectorsingeneral,wehavethefollowingTheorem. Theorem5:Givenadominatingowdistributionwithoutasingle-commodityloop,wecanalwaysndanonnegativeOD-dependentPareto-improvingtollvectortosupportitasaWardropequilibrium. Proof:Thesystemofequations( 3 )to( 3 )canbedecomposedintoseparatesub-problemsforeachODpair.Eachsub-problemisequivalenttondingananonymousPareto-improvingnonnegativetollvectorforagivendominatingowdistributionwithsingleODpair.UsingthealgorithminDial(1999),itispossibletoconstructatollvectorw0forthesub-problemofODpairw,suchthatallutilizedpathsofODpairwhavethesamecostandatleastoneofthemistoll-free.ByrunningthealgorithmforeachODpair,wecanconstructanonnegativeOD-dependentPareto-improvingtollvectortoinduceanydominatingowdistributionwithoutasingle-commodityloop. 3.4Pareto-ImprovingTollProblem TheprevioustwosectionsinvestigatedpropertiesandexistenceofdominatingowdistributionsandPareto-improvingtolls.TondaPareto-improvingtollvector, 49

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atwo-stepprocedurecanbefollowed.Foragivennetworkconguration,ndadominatingowdistributionrst(problemP1orP2),andifitexiststhentrytondcorrespondingPareto-improvingtollvectors(problemP4orP5)toinducethedominatingowdistributiontoaWardropequilibriumowdistribution.Nevertheless,thetwo-stepproceduremaynotbeparticularlyefcientinndingaPareto-improvingtollvector.Inthissection,weintroduceanewmathematicalproblemcalledPareto-improvingtollproblemtondadominatingowdistributionandequilibrium-inducingPareto-improvingtollssimultaneously.Ingeneral,Pareto-improvingpricingschemesmaybeclassiedassecond-bestpricingschemes,whichdonotnecessarilyreducecongestiontotheminimumlevelpossible. TondananonymousPareto-improvingtollvector,thefollowingmathematicalprogramisformulated:P6:min(v,x,,)t(v)Tv (3)s.t.v2VF (3)xwij)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(tij(vij)+ij)]TJ /F7 11.955 Tf 11.95 0 Td[(wi+wj=08w,(i,j)2L (3)tij(vij)+ijwi)]TJ /F7 11.955 Tf 11.95 0 Td[(wj8w,(i,j)2L (3)wo(w))]TJ /F7 11.955 Tf 11.96 0 Td[(wd(w)cUEw8w (3)0 (3) Theobjectivefunctionoftheaboveproblemistominimizethetotaltraveltimeordelayinthesystem.Whencombinedtogether,constraints( 3 )and( 3 )aretheKKTconditionsassociatedwith(t(v)+)T(u)]TJ /F4 11.955 Tf 12.26 0 Td[(v)0,8u2VF.Inotherwords,thesetwoconstraintsensureowdistributionvisatolledequilibriumowdistribution.Whenlink(i,j)isutilized,i.e.,xwij>0,constraint( 3 )forcestheequationtij(vij)+ij=wi)]TJ /F7 11.955 Tf 11.28 0 Td[(wjtohold.Combingtogetherthisequationforeachlinkonautilizedpathyieldsthefollowing:X(i,j)2wr(tij(vij)+ij)=X(i,j)2wr(wi)]TJ /F7 11.955 Tf 11.95 0 Td[(wj)=wo(w))]TJ /F7 11.955 Tf 11.95 0 Td[(wd(w) 50

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where,forODpairw,wrisasetcontaininglinksonpathrando(w)andd(w)representtheoriginanddestinationnodes.Thus,constraint( 3 )impliesthatthegeneralizedcost(timeplustolls)ofeveryutilizedpathequals(wo(w))]TJ /F7 11.955 Tf 12.14 0 Td[(wd(w))forODpairw.Consequently,constraint( 3 )guaranteesthatnoutilizedpathcostsmorethancUEw,thecostunderUE,anddoingsomakesnooneworseoff.Finally,constraint( 3 )requiresthelinktollstobenonnegative. Asformulated,problemP6isalwaysfeasible.Inparticular,letvUEdenoteaUEdistribution,i.e.,vUEsatisest(vUE)T(u)]TJ /F4 11.955 Tf 13.12 0 Td[(vUE)0,8u2VF.Then,(v,,)=(vUE,UE,0)isfeasibletoproblemP6,whereUEistheKKTmultipliersassociatedwiththeprecedingvariationalinequality.Ontheotherhand,adominatingowdistributionorPareto-improvingtollsmaynotexist.Let(v,,)denoteanoptimalsolutiontoproblemP6.IfP(i,j)2Ltij(vij)vij
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distribution.Constraint( 3 )connesequilibriumtravelcostofeachODpairtonomorethanthatunderoriginalUEcondition.Asformulated,bysetting=0,vUEisalwaysafeasiblesolutiontoproblemP7.However,anOD-dependentPareto-improvingtollvectoronlyexistsiftheoptimalvalueoftheobjectivefunctionofproblemP7isstrictlylessthant(vUE)TvUE. TheabovePareto-improvingtollproblemsareformulatedasMPCC,aclassofoptimizationproblemsdifculttosolve(see,e.g., ScheelandScholtes 2000 ).StandardstationarityconditionssuchastheKKTconditionsdonotholdforMPCCandmany(see,e.g., Luoetal. 1996 andreferencescitedtherein)haveproposedspecialalgorithmstosolvethem.Themanifoldsuboptimizationalgorithmdevelopedby LawphongpanichandYin ( 2010 )isworthparticularmentioningbecauseitcansolvelarge-scaleMPCCproblemseffectively.Theiralgorithmfocusesonndingastronglystationarysolutionbysolvingasequenceofrelaxedproblems.Moredetailsaboutthealgorithmcanbefoundin LawphongpanichandYin ( 2010 ). 3.5Summary ThischaptersystematicallyinvestigatedPareto-improvingcongestionpricingtolls.Inthebeginningofthechapter,wediscussedwhyaSOowdistributionisnotappealingtothegeneralpublic.ThefundamentalreasonthataPareto-improvingcongestionpricingschemeisachievableisthataUEowdistributionmaynotbePareto-optimal.AParetoimprovementovertheUEowdistributiontermedasadominatingowdistributionwasintroducedinSection2.Mathematicalformulationstonddominatingowdistributionswerepresented.Sinceadominatingowdistributionisnotinanequilibriumstate,howtoinduceadominatingowdistributionusingnonnegativetollswasdiscussed.Theabsenceofloops(single-andmulti-commodity)wasidentiedasthenecessaryandsufcientconditionfortheexistenceofanonymousnonnegativetollstosupportagivendominatingowdistribution.ForatransportationnetworkwithoneODpair,wefurtherconcludedthattherealwaysexistsanonnegativetollvector 52

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tosupporttheoptimaldominatingowdistribution.InSection3,weinvestigatedtheexistenceofnonnegativePareto-improvingtollvectorsthatcansupportagivendominatingowdistributionasanequilibriumowdistributioninbothanonymousanddiscriminatorycongestionpricingsettings.TwoexampleswereusedtoshowthatanOD-dependentPareto-improvingtollvectormayexisttosupportadominatingowdistributionthatcannotbesupportedbyananonymousPareto-improvingtollvector.Theseexampleswereparticularlyenlighteningbecausetheydemonstratedthepotentialbenetsofintroducingdiscriminatorypricingschemes.WethenprovedthatanonnegativeOD-dependentPareto-improvingtollvectorcanalwaysbefoundtoinduceanydominatingowdistributionwithoutasingle-commodityloopasaWardropequilibriumowdistribution.InSection4,insteadoffollowingatwo-stepproceduretondadominatingowdistributionanditscorrespondingPareto-improvingtollvector,weformulatedaMPCCthatcansolvebothproblemssimultaneously. 53

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Figure3-1. Ave-linknetwork Table3-1. FlowdistributionsunderUEandMSCpricingfortheve-linknetwork UE MSCpricing Link Flow Time Flow(SO) Time Toll Gen.cost (1,3) 3.60 36.00 2.06 20.64 20.64 41.28 (1,2) 0.00 50.00 1.54 51.54 1.54 53.07 (3,2) 2.28 12.28 0.90 10.90 0.90 11.79 (3,4) 1.32 35.06 1.17 31.21 29.21 60.43 (2,4) 2.28 22.78 2.43 24.32 24.32 48.63 Path 1-3-4 1.32 71.06 1.17 51.85 49.85 101.70 1-3-2-4 2.28 71.06 0.90 55.85 45.85 101.70 1-2-4 0.00 72.78 1.54 75.85 25.85 101.70 Coststousers 255.80 227.11 139.02 366.13 54

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Table3-2. UEanddominatingowdistributions Linkows Link UE DF-F-1 DF-F-2 DF-F-3 (1,3) 3.60 1.90 2.24 2.24 (1,2) 0.00 1.70 1.36 1.36 (3,2) 2.28 0.00 0.45 0.61 (3,4) 1.32 1.90 1.79 1.63 (2,5) 2.28 1.70 1.81 1.97 Costofthelongestutilizedpath 71.06 68.70 69.40 71.06 Systemcostortotaldelay 255.78 247.10 241.17 234.99 Figure3-2. Multi-commodityloopexample Figure3-3. Multi-commodityloopexamplewithloopremoved 55

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Figure3-4. Ave-linknetworkwithtwoODpairs Table3-3. UEandoptimaldominatingowdistributions Link UE Optimaldominatingow OD1-4ow OD3-2ow OD1-4ow OD3-2ow (1,3) 3.60 0.00 2.13 0.00 (1,2) 0.00 0.00 1.47 0.00 (3,2) 2.25 1.00 0.55 1.00 (3,4) 1.35 0.00 1.57 0.00 (2,4) 2.25 0.00 2.03 0.00 Longestutilizedpath 71.75 13.25 71.75 11.55 Totaldelay 271.55 245.09 Table3-4. Optimaldominatingowdistributionandtolls UE Dominatingowandtoll Link OD1-4 OD3-2 Time OD1-4 OD3-2 Time Toll ow ow ow ow (1,3) 3.60 0.00 36.00 2.13 0.00 21.27 18.65 (1,2) 0.00 0.00 50.00 1.47 0.00 51.47 0.00 (3,2) 2.25 1.00 13.25 0.55 1.00 11.55 0.00 (3,4) 1.35 0.00 35.75 1.57 0.00 41.31 1.46 (2,4) 2.25 0.00 22.50 2.03 0.00 20.28 10.94 Equilibriumcost 71.75 82.69 ofOD1-4 Equilibriumcost 13.25 11.55 ofOD3-2 Totaldelay 271.55 245.09 56

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Table3-5. UEanddominatingdistributions Linkow vUE vorDF-F-3tij(v) (1,3) 3.60 2.24 22.40 (1,2) 0.00 1.36 51.36 (3,2) 2.28 0.61 10.61 (3,4) 1.32 1.63 42.75 (2,5) 2.28 1.97 19.70 Costofthelongestutilizedpath 71.06 71.06 Systemcostortotaldelay 255.78 234.99 Figure3-5. Thelongestpathtreeassociatedwithv 57

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Table3-6. OD-dependentPareto-improvingtolls Link UE Pareto-improvingtoll OD1-4 OD3-2 Time OD1-4 OD3-2 Time OD1-4 OD3-2 ow ow ow ow toll toll (1,3) 3.60 0.00 36.00 2.13 0.00 21.27 6.14 0.00 (1,2) 0.00 0.00 50.00 1.47 0.00 51.47 0.00 0.00 (3,2) 2.25 1.00 13.25 0.55 1.00 11.55 12.51 1.44 (3,4) 1.35 0.00 35.75 1.57 0.00 41.31 3.04 0.00 (2,4) 2.25 0.00 22.50 2.03 0.00 20.28 0.00 0.00 Equilibrium 71.75 71.75 costOD1-4 Equilibrium 13.25 12.99 costOD3-2 Totaldelay 271.55 245.09 58

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CHAPTER4PARETO-IMPROVINGCONGESTIONPRICINGSCHEMEFORMULTICLASSNETWORK LawphongpanichandYin ( 2010 )introducedandproposedanalgorithmforndingPareto-improvingtollsongeneralnetworksassumingthatusersarehomogenous.Becausethereissubstantialheterogeneityinvalueoftraveltime(VOT)acrossthedrivingpopulation(e.g.,between$20and$40perhourobservedby BrownstoneandSmall 2005 )andtheheterogeneitymayhaveimportantimplicationsindesigningcongestionpricingschemes( Smalletal. 2005 ; SmallandYan 2001 ),thischapterextendstheworkof LawphongpanichandYin ( 2010 )tothecasewithuserheterogeneity.Inparticular,thischapterpresentsoptimizationmodelsthatdetermineananonymousPareto-improvingtollvectororindicatethatonedoesnotexist.Intheliterature,tollsareanonymousoruniformiftheyarethesameforalluserclasses.SuchtollsareofinterestbecauseitisgenerallydifcultorimpracticaltodeterminetheVOTofauserarrivingatatollfacility(e.g., YangandHuang 2004 ). Fortheremainderofthischapter,Section2introducesthedenitionofMulticlassPareto-improvingcongestionpricingschemeandpresentstwomathematicalprogrammingmodelsforndingaMulticlassPareto-improvingscheme.Section3analyzestheproblemandestablishestheexistenceconditionsofMulticlassPareto-improvingschemesthroughatwo-stepprocedure.Section4proposesanalgorithmthatcansolvetheproblemusingmanifoldsuboptimizationprocedure.Section5presentsnumericalexamples,followedbyconcludingremarksinSection6. 4.1MulticlassPareto-ImprovingPricingScheme ThissectiondenesaMulticlassPareto-improvingpricingschememathematicallyandformulatestheproblemofndingsuchaschemeasamathematicalprogramwithcomplementarityconstraints(MPCC).Tohighlightkeyideas,weassumethatthetraveldemandforeveryODpairisxed. 59

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4.1.1MulticlassNetworkEquilibrium Similartomanypreviousstudiesintheliterature(e.g., Dafermos 1972 ; EngelsonandLindberg 2006 ; YangandHuang 2004 ),werepresentuserheterogeneitybyadiscretesetofVOTs,andclassifyaccordinglytheusersintomultipleclasses.Foreachuserclassk,letdw,kandkdenotethetraveldemandbetweenODpairwandthecorrespondingVOT,respectively. Todescribefeasibleowdistributions,letnbethenumberoflinksinthenetworkandK,thenumberofuserclasses.Then,afeasiblemulticlassowdistributionisrepresentedasaK-tupleofvectorsinRn,i.e.(v1,...,vK),wherevi2Rn,i=1,...,K,isavectoroflinkowsforclasskwithvkaasitselement.Mathematically,thesetofallfeasiblemulticlassowdistributions,denotedasvF,canbedescribedusingeitherpathorlinkowvariables.Usingtheformer,letfw,krbetheowsofuserclasskonpathrforODpairw.Then, VF=(v1,vK):vka=XwXr2Pwarfw,kr;Xr2Pwfw,kr=dw,k,8w,k;fw,kr0,8w,k,r(4) wherePwisthesetofpathsforODpairwandar=1iflinkaisonpathr,0otherwise.Todescribethesetintermsoflinkows,letAbethenode-arcincidencematrixforthenetworkandEwdenoteavectorinRm,wheremisthenumberofnodes.Specically,Ewisaninput-outputvectorandhasexactlytwonon-zerocomponents:onehasavalue1correspondingtheoriginnodeofODpairandtheotheronehasavalue-1inthecomponentforthedestination.Then,VFcanbewrittenas VF=(v1,,vK):vk=Pwxw,k,Axw,k=Ewdw,k,xw,k0,8w,k. Intheabove,xw,k2RnisavectorwhosecomponentsarelinkowsofuserclasskforODpairw. Asanalternativeto(v1,,vK),wealsolet(u1,,uK)denoteanelementofVF.Inaddition,thelettersuandvwithoutanysuperorsubscriptdenotetheaggregation 60

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ofclass-speciclinkowvectors,i.e.,u=Pkukandv=Pkvk.Generally,uUEanduSOdenotetheaggregatemulticlassuserequilibrium(UE)andsystemoptimum(SO)distribution. Foreachlinka,ta(ua)denotesthetraveltimefunctionthatisconvexandmonotonicallyincreasingwithrespecttotheaggregateowua.ThemulticlassUEowdistributionuUEcanbeobtainedbysolvingthefollowingvariationalinequality(e.g.,EngelsonandLindberg,2006): Pkt(uUE)T(vk)]TJ /F4 11.955 Tf 11.96 0 Td[(uk,UE)0,8v2VF Undertheaboveassumptionsconcerningthetraveltimefunctions,theaggregatevectoruUEisunique.However,theclass-speciclinkowsuk,UEmaynotbeunique.TheremaybeanotherK-tuple(v1,,vK)suchthatuUE=PKk=1vk. 4.1.2DenitionofMulticlassPareto-ImprovingPricingScheme Consideringanetworkwithananonymouspricingscheme2Rn,weassumethatusersperceivetheirtravelcosts(disutility)asasumoftravel-timecostandtollfare,althoughothercriteriamayapply(e.g., Nagurney 2000 ; YinandYang 2004 ).Wefurtherassumethatthetravelcostsareadditiveandthusageneralizedroutetravelcostisthesumofgeneralizedcostsassociatedwithlinksthattheroutecomprises.Consequently,thetolledmulticlassUEdistribution,~v,canbeobtainedbysolvingthefollowingvariationalinequality: Pk)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(kt(~u)+T(vk)]TJ /F3 11.955 Tf 12.19 0 Td[(~uk)0,8v2VF Anonnegativetollvector,,isPareto-improvingifthefollowingconditionsaresatised:Xaar(kta(~ua)+a)=w,k8w,k,r2~Pw,k+ (4)Xaar(kta(~ua)+a)w,k8w,k,r2~Pw,k0 (4)0 (4)w,kcUEw,k8w,k (4) 61

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t(~u)T~u0g,~Pw,k0=fr:r2Pwand~fw,kr=0g,andcUEw,kdenotesthetravelcostofuserclasskforODpairwundertheUEconditionwithouttolling,i.e.,cUEw,k=kPaarta(uUEa)forallpathsrutilizedbyusersinclasskunderUE. ForeachODpairanduserclass,thersttwoconditionsensurethat,withatollpattern,thetolledUEholdsandtheequilibriumcostisw,k.Condition( 4 )requiresalltollstobenonnegativebecauseitismoresensibletopenalizeortaxnegativeexternalitiessuchastrafccongestion,insteadofsubsidizingorrewarding.Moreover,pricingschemeswithsubsidiesaregenerallymorecomplextoimplementinpractice.Condition( 4 )requiresthatthegeneralizedcostforeachODpairanduserclasswithtollsisnogreaterthanthesituationwithout,i.e.,nouserismadeworseoff.Condition( 4 )or( 4 )guaranteesastrictimprovementinsystemperformance.Theformerensuresthatthetotalsystemtraveltimewillbestrictlyreducedwhilethelatterisconcernedwiththetotalsystemtravelcost.Thesetwoobjectiveshavedifferentpolicyimplications.Intheformer,traveltimesamongallusersareweightedequallywhilethelatterweighstraveltimesbyusers'VOT.BecauseVOTispositivelycorrelatedwithincome,theformerseemsmoreprogressive( MayetandHansen 2000 ).Todifferentiate,theformeriscalledastime-basedPareto-improvingschemewhilethelattercost-based. Asstatedabove,condition( 4 )appliestoallODpairsanduserclasses.However,thisisunnecessarybecauseitissufcienttorequire( 4 )toholdonlyfortheuserclasswiththelowestVOT,i.e.,( 4 )canbereplacedby w,LcUEw,L,8k(4) 62

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whereLisaclassindexsuchthatLk,8k.Assumethat(~u,,)satises( 4 )( 4 ),( 4 ),( 4 ),and( 4 ).Letr2~Pw,L+beapathutilizedbyusersinclassLunder~u.Then,thefollowingmustholdfromthefactthatLk,8k: Paar(ta(~ua)+a k)Paar(ta(~ua)+a L)=w,L L=cUEw,L L=tUEw Whenmultipliedbyk,theaboveimpliesthatPaar(kta(~ua)+a)ktUEw=cUEw,k.Thus,( 4 )implies( 4 )forallr2~Pw,L+.ForpathsnotutilizedbyusersinclassLunder~u,i.e.,r02~Pw,L0,assumethereexistsaclassksuchthatr0isutilizedunder~u,i.e.,r02~Pw,k+,andPaar0(kta(~ua)+a)=w,k>cUEw,k.However,thelatterimpliesthatpathr0costsmorethanr2~Pw,L+whichcontradictsthefactthatr0isutilizedbyusersinclasskand~uisinuserequilibrium. 4.1.3FindingMulticlassPareto-ImprovingPricingScheme Conditions( 4 )( 4 )or( 4 )deneaPareto-improvingpricingschemethatmakesnouserworseoffwhileimprovingsystemperformanceandgeneratingtollrevenue.Theschememaynotalwaysexistinageneralroadnetwork.ThesectionpresentsmathematicalformulationsforndingsuchaPareto-improvingpricingscheme,ifitexists. Amulticlasstime-basedPareto-improving(MTPI)schemecanbeobtainedbysolvingthefollowingmathematicalprogram:MTPI-P:mint(u)Tu (4)s.t.u2VF (4)fw,krXaar)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(kta(ua)+a)]TJ /F7 11.955 Tf 11.96 0 Td[(w,k=08w,k,r2Pwk (4)Xaar)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(kta(ua)+aw,k8w,k,r2Pwk (4)w,kcUEw,k8w,k (4)0 (4) 63

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WhentheoptimalobjectivefunctionvalueofMTPI-Pisstrictlylessthant(uUE)TuUE,thesolutionisaPareto-improvingpricingscheme.Asformulated,MTPI-PisanMPCC,aclassofoptimizationproblemsdifculttosolve.StandardstationarityconditionssuchastheKarush-Kuhn-TuckerconditionsdonotholdforMPCCandmanyhaveproposedspecialalgorithmstosolvethem(e.g., ScheelandScholtes 2000 ). LawphongpanichandYin ( 2010 )appliedconceptsfrommanifoldsuboptimizationandproposedanewalgorithmthatconvergestoastronglystationarysolutioninanitenumberofiterations.ThealgorithmwillbeappliedtosolvetheaboveMTPI-Pproblemforstronglystationarysolutionslaterinthischapter. Althoughintuitive,theaboveformulationisnotcomputationallyconvenientbecauseitrequiresknowingorgeneratingallpathsbetweeneachODpair.Analternativeformulationusinglinkowvariablesandnodepotentials(see Ahujaetal. 1993 )canbeformulatedbelow:MTPI-L:mint(u)Tu (4)s.t.u2VF (4)xw,ka)]TJ /F7 11.955 Tf 5.48 -9.69 Td[(kta(ua)+a)]TJ /F7 11.955 Tf 11.96 0 Td[(w,ki+w,kj=08a,w,k,a=(i,j) (4)kta(ua)+aw,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj8a,w,k,a=(i,j) (4)w,ko(w))]TJ /F7 11.955 Tf 11.95 0 Td[(w,kd(w)cUEw,k8w,k (4)0 (4) Intheabove,w,karenodepotentials.Moreover,nodesiandjarethestartingandendingnodesoflinka,ando(w)andd(w)representtheoriginanddestinationnodesofODpairw.Asstatedabove,w,kareunrestricted.However,itispossibletosetw,kd(w)=0anddoingsoforcesw,kitobenonnegativeforalli=1,...,m. SincePareto-improvingtollschememaynotalwaysexist,whenitdoesnotexist,wecontendthatacongestionpricingschemewouldstillbeappealingifusersarenotworse 64

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offbyasignicantamount.ThisleadstotheconceptofapproximatePareto-improvingtollschemes,whichcanbedeterminedbysolvingthefollowingMPCC:MTPI-L:mint(u)Tu (4)s.t.u2VF (4)xw,ka)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(kta(ua)+a)]TJ /F7 11.955 Tf 11.96 0 Td[(w,ki+w,kj=08a,w,k,a=(i,j) (4)kta(ua)+aw,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj8a,w,k,a=(i,j) (4)w,ko(w))]TJ /F7 11.955 Tf 11.95 0 Td[(w,kd(w)(1+)cUEw,k8w,k (4)0 (4) Whereiscalledrelaxationfactor,whichindicatesusersareallowedtobeworseoffbyatmostcomparingwiththeirtravelcostsunderUEcondition.Theaboveformulationisconceptuallysimilarasthebi-levelprogrammingformulationproposedby YangandZhang ( 2002 ). Jahnetal. ( 2005 )proposeasimilarformulationthatusesroutinginsteadoftollingtoachieveapproximateParetoimprovement. Similarly,apathorlink-basedMPCCcanbeformulatedforndingacost-basedPareto-improvingschemebyreplacingtheobjectivefunctionoftheaboveformulationswithminPkkt(u)Tvk. 4.2PropertiesandExistenceofMulticlassPareto-ImprovingPricingScheme ThesectiondiscussesthepropertiesofaPareto-improvingschemeandestablishestheexistenceconditions. ThefundamentalreasonfortheexistenceofaPareto-improvingpricingschemeisthattheoriginalWardropianuserequilibriummaynotbeParetooptimal.Ingametheory( Fisk 1984 ),aWardropianuserequilibriumcorrespondstoaNashequilibrium,whereaplayercorrespondstoatraveleroruser.Astheprisoner'sdilemma( Poundstone 1992 )oftenusedtodemonstrate,aNashequilibriumneedsnotbestronglyParetooptimal.Theremaybeanothersituationthatisasgoodforeveryoneandstrictlypreferred 65

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bysome.Innetworkowmodeling,suchapropertyischaracterizedby( HagstromandAbrams 2001 )asgeneralizedBraess'sparadox.GivenaUEowdistributioninageneralroadnetwork,ageneralizedBraess'sparadoxoccursifthereexistssomeotherfeasibleowdistributionunderwhichthetotalsystemtraveltimeissmallerandallutilizedpathsarenolongerthanthoseunderUE.SuchaowdistributionmaybemoredesirablesinceitdominatestheUEdistribution( LawphongpanichandYin 2010 ).Becausethedominatingowdistributionisnotinequilibriumandthusnon-sustainable,weusetollsasamechanismtoevolvethetrafcowtothedominatingdistribution.Inotherwords,weseekatollingschemesuchthatthetolledUEowdistributionisthedominatingowdistribution.Ifafterpayingthetolls,nousershavelargertravelcoststhanbefore,thetollingschemeisParetoimproving. Therefore,ndingPareto-improvingcongestionpricingschemescanbedecomposedintotwosteps:therststepistodetectwhetherageneralizedBraessparadoxoccurs,i.e.,ndingadominatingow.Givenadominatingdistribution,wetheninvestigatewhetherthereexistsanonnegativetollingschemethatinducesthedominatingowdistributionwithoutmakinganyuserworseoff. 4.2.1MulticlassDominatingFlowDistribution Aowdistributionv2VFdominatesagivenmulticlassUEdistributionorisdominatingifitstrictlyreducesthetotalsystemtraveltimeortravelcostandallowsalluserstouseroutesthatarenomoreexpensivethanthoseunderUE.Mathematically,vanditspathowsfw,krsatisfythefollowingconditions:Xaarkta(ua)cUEw,k,8w,k,r2Pw,k+ (4)t(u)Tu
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wherePw,k+=fr:r2Pwandfw,kr>0g.Itfollowsimmediatelyfromcondition( 4 )thatadominatingowdistributionmakesnouserexperiencelongertraveltimewhencomparedtotheUEdistribution.Condition( 4 )or( 4 )ensuresthatthesystemperformanceisstrictlyimprovedunderadominatingowdistribution,whichimpliesthatsomeuserswillbebetteroff. Tondatime-basedmulticlassdominatingowdistribution,weformulatethefollowingpath-basedoptimizationproblem:TBDF-P:mint(u)Tu (4)s.t.v2VF (4)fw,kr(Xaarkta(ua))]TJ /F4 11.955 Tf 11.95 0 Td[(cUEw,k)08w,k,r2Pw (4) Intheabove,therstconstraintrequiresvtobeafeasibleowdistributionandthesecondensuresthat,ifpathrisutilized,itstravelcostforuserk,Paarkta(ua),mustbenogreaterthanthecorrespondingUEtravelcost.Theobjectiveoftheproblemistominimizetotalsystemtraveltime.Thus,ifvDFsolvestheTBDFPproblemandt(uDF)TuDF
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Intheabove,whenlinkaisutilizedbyuserclassk,i.e.,xw,ka>0,thesecondconstraintreducestow,ki)]TJ /F7 11.955 Tf 12.57 0 Td[(w,kj)]TJ /F7 11.955 Tf 12.57 0 Td[(kta(ua)0.Collectively,thisexpressionforeachlinkonautilizedpathyieldsthefollowing Pa2wrkta(ua)Pa2wr,a=(i,j)(w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj)=w,ko(w))]TJ /F7 11.955 Tf 11.95 0 Td[(w,kd(w), wherewrisasetcontaininglinksonpathrbetweenODpairw.Thus,thesecondconstraintimplicitlyensuresthatthelengthofeveryutilizedpathforuserclasskandODpairwisnogreaterthan(w,ko(w))]TJ /F7 11.955 Tf 12.44 0 Td[(w,kd(w)).Consequently,thelastconstraintguaranteesthatnoutilizedpathhasmoreexpensivetravelcostthantheUEcost.NotethatTBDF-PandTBDF-LaresimilarinstructuretoMTPI-PandMTPI-LandcanbetransformedintoMPCCbyintroducingauxiliaryvariables.AndifwereplacetheobjectivefunctionswithminPkkt(u)Tvk,theproblemofndingthecost-baseddominatingowdistributionscanbeformulated. 4.2.2ExistenceofNonnegativePareto-ImprovingTolls Foragivendominatingowdistribution,thissectioninvestigateswhetherthereexistsaPareto-improvingtollvectorthatcaninducesuchaowdistribution. LetuDFdenotesagivenaggregatedominatingowdistribution.ThesetofnonnegativePareto-improvingtollpatternsconsistsofthecomponentofthesolution(,,^v)tothefollowingsystemofequationsandinequalities:XkkXa(ta(uDFa)+a)^vka=XkXww,kdw,k (4)Xaar(kta(uDFa)+a)w,k8w,k,r2Pwk (4)uDF=Xk^vk (4)^v2VF (4)w,kcUEw,k8w,k (4) Intheabove,foreachODpairandeachuserclass,therstfourconditionsensurethatafteraddingtoll,thetolledmulticlassUEholdsandtheequilibriumcostforuserclasskandODpairwisw,k.Notethat^visnotnecessarilyvDF,whichistheclassspecic 68

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owdistributionassociatedwithuDFobtainedbysolvingTBDF-L.Condition( 4 )ensuresthatthegeneralizedcostforeachODpairandeachuserclassunderthetolledUEisnogreaterthantheoriginalUEtravelcost. Theabovetollsetcouldbeempty.Toestablishanexistencecondition,considerthefollowingproblemforagivendominatingowuDF:MPIT:mint(uDF)T(XwXkkyw,k)+XwXkcUEw,kzw,k (4)s.t.Ayw,k+Ewzw,k=Ewdw,k8w,k (4)XwXkyw,kauDFa8a (4)yw,k,zw,k08w,k (4) WerefertotheaboveproblemasthemulticlassPareto-improvingtoll(MPIT)problembecausethetheorembelowshowsthatthedualvariablesormultipliersassociatedwiththesecondconstraintbecomesPareto-improvingtolls.ObservethattheMPITproblemislinearwithrespecttoitsdecisionvariables,yw,kaandzw,k,becauseuDFandcUEw,karegiven.ThefollowingtheoremgivesanecessaryandsufcientconditionfortheexistenceofaPareto-improvingtollvector. Theorem:LetuDFbeamulticlassaggregatedominatingowdistribution.APareto-improvingtollvectorexistsifandonlyif(^xw,k,0)solvestheMPITproblemandtheinequalityconstraintsarebinding,i.e.,PwPk^xw,k=uDF. Proof:SinceKKTconditionsarebothnecessaryandsufcientforlinearprograms,(^xw,k,0)solvestheMPITproblemifandonlyifthereexistsmultipliersw,ki,a,w,kandw,kasuchthatthefollowinghold: kta(uDFa))]TJ /F3 11.955 Tf 11.95 0 Td[((w,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj)+a)]TJ /F7 11.955 Tf 11.95 0 Td[(w,ka=08w,kanda=(i,j) (4)cUEw,k)]TJ /F3 11.955 Tf 11.96 0 Td[((Ew)Tw,k)]TJ /F7 11.955 Tf 11.95 0 Td[(w,k=08w,k (4)(^xw,k)Tw,k=08w,k (4) 69

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,,0 (4)A^xw,k=Ewdw,k (4)XwXk^xw,k=uDF (4) Because^xw,ka>0whenlinkaisonautilizedpath,thecomplementaryslacknessconditionimpliesthatw,ka=0forallaonautilizedpath.Byaddingtherstequationassociatedwithlinksonthesamepathtogetherandusingthefactthatw,ka=0forlinksonutilizedpaths,thefollowingmustholdforeveryuserclasskandODpairw=(o,d):Xaar(kta(uDFa)+a)=w,ko)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kd8w,kandr2^Pw,k+ (4)Xaar(kta(uDFa)+a)w,ko)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kd8w,kandr2^Pw,k0 (4) where^Pw,k+=fr:r2Pwand^fw,kr>0g,and^Pw,k0=fr:r2Pwand^fw,kr=0g.Letw,ko)]TJ /F7 11.955 Tf 12.44 0 Td[(w,kd=w,k.Togetherwith( 4 )and^vk=Pw^xw,k,theabovetwoconditionsimplythatconditions( 4 )and( 4 )hold(HeranandRamana,1998). Fromequation( 4 ),wehavecUEw,k)]TJ /F3 11.955 Tf 11.62 0 Td[((Ew)Tw,kk=w,k0,i.e.,cUEw,kw,ko)]TJ /F7 11.955 Tf 9.3 0 Td[(w,kd=w,kforallw,k.Thus,theaboveequations( 4 )( 4 )yieldasolution(,,^v)satisfyingconditions( 4 )( 4 ),andisPareto-improving. Noticethatimposingtheabovetollvectorwillnotnecessarilyreplicate^vk,becausethemulticlasstolledUEproblemmayadmitmultiplenetworkequilibrium.However,thecorrespondingaggregatelinkowpattern,thepathtraveltimesandthegeneralizedpathtravelcostsarethesameundertheseequilibrium(Proposition2,EngelsonandLindberg,2006).Inotherwords,theresultingaggregatelinkowwillbeuDFaforeachlinkandthegeneralizedtravelcostw,kcUEw,kforeachODpairanduserclass,therebyensuringthatisParetoimproving. 4.3ManifoldSuboptimizationAlgorithm TheproblemsweformulatedinSections2and3,namelyMTPI-P,MTPI-L,TBDF-P,andTBDF-L,areMPCCproblem,whichisaclassofoptimizationproblems 70

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difculttosolve.Thereasonsaretwofolds.OneisthefactthatMPCCviolatestheMagasarian-Fromovitzconstraintqualication(MFCQ)andtheotheristhatitsfeasibleregionisnon-convex(see,e.g., ChenandFlorian 1995 ; ScheelandScholtes 2000 ).StandardalgorithmsfornonlinearprogramsbecomeineffectiveforMPCC. LawphongpanichandYin ( 2010 )developedaManifoldSuboptimizationalgorithmforndingastronglystationarysolutiontoPareto-improvingproblembysolvingasequenceofrelaxedproblems.Moredetailsaboutthealgorithmcanbefoundin LawphongpanichandYin ( 2010 ). BelowisaversionofthemanifoldsuboptimizationalgorithmforsolvingMTPI-Lproblem. Manifold Suboptimization Algorithm Step0 LetvbeaUEdistributionandxw,kijdenotesthecorrespondinglinkowforODpairwofuserclassk.Setn=1,1x,w,k=(i,j):xw,kij=0,and1z,w,k=(i,j):xw,kij>0. Step1 Let(vn,n,n,xn,zn)solvethefollowingproblem:R-MTPI:mint(u)Tv (4)s.t.u=XwXkxw,k (4)Axw,k=Ew,kdw,k8w,k (4)zw,kij=ktij(uij)+ij)]TJ /F7 11.955 Tf 11.96 0 Td[(w,ki+w,kj8w,kand(i,j)2L (4)w,ko(w))]TJ /F7 11.955 Tf 9.3 0 Td[(w,kd(w)cUEw,k8w,k (4)xw,kij=08w,kand(i,j)2nx,w,k (4)zw,kij=08w,kand(i,j)2nz,w,k (4)xw,kij08w,kand(i,j)=2nx,w,k (4)zw,kij08w,kand(i,j)=2nz,w,k (4)ij08(i,j)2L (4) 71

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Step2 Let)]TJ /F6 7.97 Tf 6.78 4.34 Td[(nw,k=f(i,j)2nx,w,k:nij,w,k<0andznij,w,k=0g,wherenij,w,kisthemultiplierassociatedwiththeconstraintxw,kij=0.If)]TJ /F6 7.97 Tf 6.78 4.34 Td[(nw,kisemptyforallw,k,stopand(vn,n,n,xn,zn)isstronglystationary.Otherwise,dothefollowingandgotoStep1: a)Setn+1x,w,k=nx,w,k)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 7.97 Tf 6.78 4.34 Td[(nw,k, b)Setn+1z,w,k=f(i,j):znij,w,k=0g, c)Setn=n+1. Oneoftheadvantagesofthisalgorithmisitsfastcomputationtime,whichisverydesirableinlarge-scaleapplications.WewillusethisalgorithmtosolveMTPIproblemsinthreetestnetworksinthenextsection. 4.4NumericalExamples ConsiderthenetworkinFigure 4-1 inwhichthereisonlyoneODpair(1,4)andtwouserclasses,whoseVOTsare0.6and1.4respectively.Thetotaldemandis3.6andeachuserclasshasademandof1.8. InsteadofdirectlysolvingthePareto-improvingtollproblemsproposedinSection2,weadoptthetwo-stepproceduretogainmoreinsights.WerstsolvetheTBDF-Fproblemforatime-baseddominatingowandthenobtainaPareto-improvingschemebysolvingMPITandexaminingthemultipliers.Table 4-1 presentstheresultsandcomparisonswiththemulticlassUEowdistributionarereportedinTable 4-2 ForthecasewithPareto-improvingtolls,theresultsinTable 4-2 indicatethatthetravelcostforuserclass1isthesameasbeforewhilethecostforclass2decreasesfrom99.48to94.76.Inthepresenceofanonymoustolls,theuserswithlowerVOTsaremorelikelytobemadeworseoffthanthosewithhigherVOTs( YinandYang 2004 ).IftheuserswiththelowestVOTarenoworseoff,thenalltheotheruserswillbebetteroff.Atthesametime,thePareto-improvingtollsreducethetotalsystemtraveltime(andcost),andgenerateanamountoftollrevenueof18.57.Therefore,thepricingschemeisbenecialtosociety,thegovernmentandusersofclass2withouthurting 72

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usersofclass1.Thetotalsystemtraveltimeisreducedfrom255.80to234.99,whichisveryclosetothetotalsystemtraveltimeunderSOcondition,227.11.Thereductionisapproximately72.53%ofthemaximumpossiblereduction.Itisalsointerestingtonotethatthetime-basedclass-specicowdistributionfromsolvingtheTBDF-Lproblem,vDF,1=f0.99,0.81,0.30,0.69,1.11gTandvDF,2=f1.25,0.55,0.30,0.94,0.86gT,doesnotsolvetheMPITproblemwhiletheonereportedinTable1does.However,sincetheircorrespondingaggregateowsarethesame(bothareuDFreportedinthetable),theresultingsystemtraveltimeisthesame. Wealsosolveforacost-baseddominatingowandtheresultingowdistributionalongwiththetollsareshowninTable 4-3 .Similarly,Table 4-4 comparestheresultingperformancemeasureswiththemulticlassUE.Bymakingsimilarcomparisons,theresultsinTable 4-4 showthatthePareto-improvingschemebenetseveryotherstakeholderwithoutmakingusersofclass1worseoff.Asexpected,thecost-basedPareto-improvingschemewillleadtosmallertotalcostwhilethetime-basedschemewillresultinasmallertotalsystemtime. ToexploretheexistencesandpropertiesofmulticlassPareto-improvingtollschemesingeneralnetworks,wefurthersolvethreenetworksintheliterature:nine-node(see,e.g., HearnandRamana 1998 ),SiouxFalls(see,e.g., Baietal. 2004 ),andHull(see,e.g., Florianetal. 1987 ).Wekeeptheoriginalnetwork-relatedsettingsbutintroducetwoclassesofusers,whosevalueoftimeis0.6and1.4respectively.Wefurtherassumeauniform6040%splitbetweenthelowandhighVOTusersforeachODpair.SomegeneralattributesofthesethreenetworksarelistedinTable 4-5 .Table 4-6 showsthetotalsystemtraveltimeundermulticlasssystemoptimum(MSO),multiclassuserequilibrium(MUE)andmulticlasstime-basedPareto-improvingtollscheme(MTPI).ThelastcolumnofthetablereportsthereductionintraveltimefromMUEdelaytoMTPIdelayasafractionofthedifferencebetweenMUEdelayandSOdelay,whichisthemaximumpossiblereduction.ObservefromTable 4-6 thatno 73

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Pareto-improvingtollschemeexistsfornine-nodeandSiouxFallsnetworkandtheresultingMTPIdelayofHullnetworkreducesthetraveldelaybynomorethan1%ofthemaximumpossibleamount,i.e.,Pareto-improvingtollschemesdonotleadtoasignicantimprovementintraveldelay.Table 4-7 reportstheresultsfromsolvingtheapproximatePareto-improvingtollproblemorAMITPIwith=0.05,0.1,0.15and0.20.Theseresultsillustratethatallowinguserstobeslightlyworseoffmayleadtosubstantiallyimprovementinthesystemperformance. AlthoughmarginalcostpricingcanachievetheSOowdistributionandreducethetotalsystemtraveltimetoitsminimumlevel,itmaycauseinequalityamongusers,suchasspatialinequality(e.g., YangandZhang 2002 ).Figure 4-2 and 4-3 showsthefrequenciesandcumulativedistributionsoftworatios,CSOw=CUEwandCAMTPIw=CUEw,whereCSOwisthetravelcost(timeplustoll)betweenODpairwunderthemarginalcostpricing;CUEwandCAMTPIwaretheequilibriumtravelcostsunderMUEandanapproximateMTPIschemewith=15%respectively.ThedataforthegurearefromSiouxFalls.Figure 4-2 showsthatmarginalcostpricingmakesallusersworstoffthantheMUEcondition.Evenwithoutincludingthetolls,around12%oftheODpairsunderMSOwillhavemorethan15%highertraveltimesthanMUE.FromFigure 4-3 ,asexpected,equilibriumtravelcostsofallODpairsarenomorethan15%ofthoseunderMUE.TheGinicoefcientsassociatedwiththeratioofCSOw=CUEwandCAMTPIw=CUEware0.122and0.016respectively,suggestingthatcomparedwithmarginalcostpricing,Pareto-improvingschememayinducelessspatialinequalitywithrespecttothestatusquo.ItisalsoclearfromFigure 4-3 thatthehighVOTuserstendtobebetteroffthanthelowVOTusers.Unfortunately,suchsocialinequalitymaybeinevitablewithanonymoustollingandcanonlybeaddressedbyothersubsidyschemes(e.g., YinandYang 2004 ). 4.5Summary ThischapterdiscussesPareto-improvingpricingschemesforageneralnetworkwhenusersbelongtoadiscretesetofclasses,eachonewithadifferentVOT.We 74

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determinewhetherananonymousPareto-improvingpricingschemeexistsbysolvinganoptimizationandobservingitsoptimalobjectivevalue.WeprovideformulationsforndingaPareto-improvingschemeaswellasadominatingowdistribution.Givenadominatingowdistribution,weprovidenecessaryandsufcientconditionsfortheexistenceofanonnegativePareto-improvingtollvector.ThenumericalresultsfromthreenetworksillustratethatmulticlassPareto-improvingpricingschemesarelessprevalentwhencomparedtoitssingle-classcounterpart.TheexistenceandeffectivenessPareto-improvingpricingschemesnotonlydependonthenetworkcongurationsbutalsoondemographicfeatures(e.g.,VOTvalues)ofthepopulation. Tofacilitatethepresentationofthekeyideas,thischapterassumesthattraveldemandisdeterministic.Thetheoryforelasticdemandcanbesimilarlyestablished,e.g.,see LawphongpanichandYin ( 2010 )forthecasewithoneuserclass. 75

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Figure4-1. Ave-linknetworkwithmulticlassusers Table4-1. Time-baseddominatingowdistributionandPareto-improvingscheme UE Pareto-improving LinkuUEt(uUE)uDF^v1^v2t(uDF) (1,3) 3.60 36.00 2.24 0.44 1.80 22.40 0.00 (1,2) 0.00 50.00 1.36 1.36 0.00 51.36 0.00 (3,2) 2.28 12.28 0.61 0.00 0.61 10.61 20.96 (3,4) 1.32 35.06 1.63 0.44 1.19 42.75 3.55 (2,4) 2.28 22.78 1.97 1.36 0.61 19.70 0.00 76

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Table4-2. PerformancecomparisonswithUEowdistribution Userequilibrium Pareto-improving Class1 Class2 Class1 Class2 Pathtraveltime 1-3-4 71.06 71.06 65.15 65.15 1-3-2-4 71.06 71.06 52.71 52.71 1-2-4 72.78 72.78 71.06 71.06 Longestutilizedpath 71.06 71.06 71.06 65.15 Pathtravelcost 1-3-4 42.64 99.48 42.64 94.76 1-3-2-4 42.64 99.48 52.59 94.76 1-2-4 43.67 101.89 42.64 99.48 Equilibriumcost 42.64 99.48 42.64 94.76 Totalsystemtraveltime 255.80 234.99 Totalsystemtravelcost 255.80 228.74 Tollrevenue 0.00 18.57 Table4-3. Cost-baseddominatingowdistributionandPareto-improvingscheme UE Pareto-improving LinkuUEt(uUE)uDF^v1^v2t(uDF) (1,3) 3.60 36.00 2.15 0.35 1.80 21.49 0.00 (1,2) 0.00 50.00 1.45 1.45 0.00 51.45 0.00 (3,2) 2.28 12.28 0.51 0.00 0.51 10.51 21.96 (3,4) 1.32 35.06 1.64 0.35 1.29 42.98 3.96 (2,4) 2.28 22.78 1.96 1.45 0.51 19.61 0.00 77

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Table4-4. PerformancecomparisonswithUEowdistribution Userequilibrium Pareto-improving Class1 Class2 Class1 Class2 Pathtraveltime 1-3-4 71.06 71.06 64.47 64.47 1-3-2-4 71.06 71.06 51.61 51.61 1-2-4 72.78 72.78 71.06 71.06 Longestutilizedpath 71.06 71.06 71.06 64.47 Pathtravelcost 1-3-4 42.64 99.48 42.64 94.21 1-3-2-4 42.64 99.48 52.92 94.21 1-2-4 43.67 101.89 42.64 99.49 Equilibriumcost 42.64 99.48 42.64 94.21 Totalsystemtraveltime 255.80 235.09 Totalsystemtravelcost 255.80 228.64 Tollrevenue 0 17.68 Table4-5. Networkattributes Network Links Nodes ODpairs Nine-node 9 18 4 SiouxFalls 76 24 528 Hull 798 501 158 78

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Table4-6. ExactPareto-improvingproblem Network MSOdelay MUEdelay MTPIdelay Reduction (%ofMax) Nine-node 2253.92 2455.87 2455.87 0.00% SiouxFalls 3514.39 3654.46 3654.46 0.00% Hull 50542.64 51350.74 51347.14 0.44% Table4-7. ApproximatePareto-improvingproblem Nine-node SiouxFalls Hull Relaxation AMTPI Reduction AMTPI Reduction AMTPI Reduction factor() delay (%ofMax) delay (%ofMax) delay (%ofMax) 5% 2455.87 0% 3650.73 2.67% 51279.49 8.82% 10% 2392.27 31.50% 3649.88 3.27% 51274.31 9.46% 15% 2385.81 34.70% 3614.85 28.28% 51274.31 9.46% 20% 2385.81 34.70% 3607.94 33.21% 51208.48 17.60% 79

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Figure4-2. FrequenciesofratiosforSiouxFallsnetwork Figure4-3. CumulativedistributionofratiosforSiouxFallsnetwork 80

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CHAPTER5PARETO-IMPROVINGHYBRIDPOLICYFORTRANSPORTATIONNETWORKS LawphongpanichandYin ( 2010 )proposedaPareto-improvingcongestionpricingschemethatleadsatransportationsystemtoaParetoimprovementoverstatusquoevenbeforetollrevenueredistribution.ThefundamentalreasonforanonnegativePareto-improvingtollvectortoexististheuserequilibrium(UE)owdistributionmaynotbePareto-optimal,whichindicatestheremayexistanotherowdistributionthatmakesatleastoneuserbetteroffwithoutmakinganyotheruserworseoff. Songetal. ( 2009 )and Wuetal. ( 2011 )extendedthePareto-improvingcongestionpricingmodeltonetworkswithmulticlassandmodes.AlthoughitisrelativelyeasytoimplementanonnegativePareto-improvingpricingscheme,theexistenceofthepricingschemeisnotguaranteed.Evenitdoesexist;wefoundthereductionintotalsystemdelaymaynotbesubstantial. ThischapterexploresthepotentialofcombingmultiplepolicyinstrumentstoachieveParetoimprovements.Weconsiderahybridpolicythatcombinestravel-rightassignmentandcongestionpricing.Inthepresenceofcongestiontolling,thepolicyallocatesroaduserstherightsoftollexemptionondesignateddaysinsteadofpayingtollsoneverysingleday.Ondaysthatusersareexempted(freedays),theyareallowedtonavigatethroughtheroadnetworkwithoutbeingchargedwhiletheothersaresubjecttocongestiontolls.Usersmayexperiencedifferenttraveltimesfromdaytodaydependingonwhethertheyareexemptedfromcongestiontollsonaparticularday.Tomakemeaningfulcomparisonswiththestatusquo,averagetravelcostisdenedastheweightedaverageoftravelcostsonfreedaysandrestricteddays.Subsequently,usersareconsideredbetteroffiftheiraveragetravelcostsarelessthanthosebeforetheimplementationofthehybridpolicy. Fromanotherperspective,theallocationofrightsfortravelingforfreecanbeviewedasaformofroadspacerationing.Asaregulatorytraveldemandmanagement 81

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strategy,roadspacerationinghasbeenusedtomitigatetrafccongestionfordecades.CitiessuchasAthens,Santiago,MexicoCity,BeijingandGuangzhouhaveresortedtousingplate-number-basedroadspacerationingtoreducecongestionandairpollution(see,e.g., Wangetal. 2010 and Hanetal. 2010 ). Downs ( 2004 )concludedthatthemosteffectiveoverallstrategyforreducingtrafccongestionprobablyshouldconsistofbothmarket-basedandregulatoryelements. Daganzo ( 1995 )proposedaschemethatcanbeviewedashybridbetweenpricingandrationingtocontrolowthroughabottleneck.Usersareallowedtochoosetheformofthepenaltytheymustpayforusingthebottleneckbasedontheirpersonalneedsanddifferences.ThehybridstrategycouldachieveaParetoimprovementintheory. Inasimilarspirittotheoneby Daganzo ( 1995 ),thehybridpolicyproposedinthischapterattemptstomakeuseofthesynergisticeffectsofmultiplepolicyinstrumentstoachieveParetoimprovementsinageneralnetwork.Theaimofthischapteristoestablishamathematicalframeworkofdesigninganoptimalroadspacerationingpolicyandanoptimalhybridpolicyforageneralnetworkandcomparestheperformanceofthesetwopolicies.Theremainderofthischapterisorganizedasfollows.Section2discussesroadspacerationingschemesandformulatesamathematicalprogramtodesignoptimalPareto-improvingroadspacerationingscheme.Section3formulatesamathematicalprogramtodesignanoptimalPareto-improvinghybridpolicy.Section4comparesthesetwopoliciesonatoynetwork.Thelastsectionoffersconcludingremarks. 5.1Pareto-ImprovingPureRoadSpaceRationingPolicy Roadspacerationinghasbeenusedbymanycitiesforyearstomitigatetrafccongestionand/orairpollution.However Wangetal. ( 2010 )isthersttodemonstratethatroadspacerationingschemescouldleadtoParetoimprovements.Furthermore,inpractice,rationingratio,whichindicatesthepercentageofusersthatareprohibitedfromusingtheroadnetwork,isdeterminedprimarilybasedonengineeringjudgment. 82

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ThissectionfollowsanoptimizationapproachandprovidestheformulationofaPareto-improvingpureroadspacerationingproblem,whichwebelievemayoffermoreinsightsintothepureroadspacerationingproblem. 5.1.1ProblemSetting LetG=(N,L)beadirectedtransportationnetwork,whereNisthesetofnodesandListhesetofdirectedlinks.Aisthenode-arcincidencematrixofamulti-modalnetwork,whichconsistsoftwosub-networks,namelyroadnetworkandtransitnetwork.Aisthenode-arcincidencematrixofthetransitnetworkonly.AnelementofLisdenotedas(i,j)andtij(vij)representsthetraveltimefunctionforlink(i,j),whichisassumedtobeseparable,continuousandmonotonicallyincreasingrespecttotheaggregatelinkowvij.LetLnandLtdenotethesetoflinksintheroadandtransitnetwork,respectively.Itisassumedthateachorigin-destination(OD)pairisconnectedbyatransitlinewhosetraveltimeisconstant.Toberealistic,thetransittraveltimeisassumedtobelongerthanthetraveltimeonroadforthesameODpair.LetWbethesetofODpairsanddwbethedeterministicnumberoftripsmadebetweenODpairw2W.Userscanbedividedintotwogroups(k=1,22K),namely,generalusersandrestrictedusers.ForODpairw,xw,kijrepresentsthelinkowofusergroupkonlink(i,j)andvij=PwPk=1,2xw,kij. Inapureroadspacerationingscheme,generallyrationingisenforcedonallroadlinks,Ln.Thepercentageofrestrictedroaduserswhoareforcedtousealternativemodes,e.g.,transitservice,istermedasrationingratio.Eachdayafractionofroadusers(usergroupk=1),100(1)]TJ /F7 11.955 Tf 12.35 0 Td[()%ofthetotaldemand,areallowedtoaccesstheentiretransportationnetworkfreely.Therest(usergroupk=2),100%ofthetotaldemand,areprohibitedfromusingtheroadnetwork. 83

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5.1.2UserEquilibriumunderPureRoadSpaceRationingSchemes Theuserequilibriumproblemunderapureroadspacerationingpolicycanbeformulatedasfollows:P1:min(x,v)X(i,j)2LnZvij0tij(!)d!+X(i,j)2Lttijvij (5)s.t.Axw,1=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()Ewdw8w (5)Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5) whereconstraints( 5 )and( 5 )describeowbalanceconstraintsforgeneralandrestrictedusersrespectively.Ewisaninput-outputvector,i.e.,avectorwithexactlytwonon-zerocomponents,tospecifytheoriginanddestinationofODpairw.ThecomponentofEwcorrespondingtheoriginhasavalue1andtheonecorrespondingthedestinationhasavalue-1.Constraints( 5 )to( 5 )essentiallyarethefeasibleowregionofamulticlassmultimodalnetwork,whichisconvex.Theobjectivefunctionisstrictlyconvexwithrespecttotheaggregatelinkowvij.Therefore,problemP1isastrictlyconvexoptimizationproblemandthereisauniquesolutionoftheuserequilibriumlinkowdistribution.Forastrictlyconvexmathematicalprogram,theKarushKuhnTucker(KKT)conditionsarebothnecessaryandsufcientforoptimality.Toshowtheabovemathematicalprogramisequivalenttotheuserequilibriumunderapureroadspacerationingpolicy,theKKTconditionsoftheaboveoptimizationproblemcanbestatedasfollows,tij(vij)+w,1i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,1j08(i,j)2Ln,w (5)xw,1ijtij(vij)+w,1i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,1j=08(i,j)2Ln,w (5)tij+w,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj08(i,j)2Lt,w,k=1,2 (5) 84

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xw,kijtij+w,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj=08(i,j)2Lt,w,k=1,2 (5)Axw,1=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()Ewdw8w (5)Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5) wherewisavectorofLagrangemultipliersassociatedwiththeowbalanceconstraints( 5 )and( 5 ),whicharealsoknownasnodepotentials( Ahujaetal. 1993 )forODpairw.Let'sderivetheequivalenceconditionsforrestrictedusersrst.Restrictedusers(usergroupk=2)areallowedtoaccesstransitlinksonly,therefore,onlyonepairofcomplementarityconstraints( 5 )and( 5 )appliestorestrictedusers.Whenalink(i,j)isutilized,i.e.,xw,2ij>0,constraint( 5 )forcestheequationtij+w,2i)]TJ /F7 11.955 Tf 12.12 0 Td[(w,2j=0tohold.Combingtogetherthisequationforeachlinkonautilizedpathyieldsthefollowing:X(i,j)2w,2rtij=X(i,j)2w,2rw,2j)]TJ /F7 11.955 Tf 11.95 0 Td[(w,2i=w,2d(w))]TJ /F7 11.955 Tf 11.96 0 Td[(w,2o(w) where,forODpairw,w,2risasetcontainingtransitlinksthatareavailabletorestrictedusersonpathrandd(w)ando(w)denotethedestinationandoriginnodesofODpairw.Thus,constraints( 5 )impliesthatthegeneralizedtraveltimeofeveryutilizedpathequalsw,2d(w))]TJ /F7 11.955 Tf 12.24 0 Td[(w,2o(w)forgeneralusersbetweenODpairw.Whenalink(i,j)isnotutilized,i.e.,xw,2ij=0,constraints( 5 )to( 5 )implytheinequalitytij+w,2i)]TJ /F7 11.955 Tf 12.19 0 Td[(w2j0holds.Addingthisinequalityalonganon-utilizedpathyieldsP(i,j)2w,2rtijw,2d(w))]TJ /F7 11.955 Tf 11.99 0 Td[(w,2o(w).Therefore,forrestrictedusers,wehaveallutilizedpathshavethesametraveltimew,2d(w))]TJ /F7 11.955 Tf 12.41 0 Td[(w,2o(w)anditisnolongerthanthoseofallnon-utilizedpaths.Forgeneralusers,therearetwopairsofcomplementarityconstraints( 5 )to( 5 )involvedbecausetheyareallowedtousealllinksinthetransportationnetworkfreely.Similarprocedurescanbeappliedtoconstructtheuserequilibriumconditionsforgeneralusers(usergroup 85

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k=1).LetCUEw,k=w,kd(w))]TJ /F7 11.955 Tf 12.26 0 Td[(w,ko(w),thenCUEw,kistheequilibriumtraveltime(cost)forusersofgroupkbetweenODpairw.Hence,weprovethattheKKTconditionsoftheaboveoptimizationproblemareequivalenttotheuserequilibriumconditionsunderpureroadspacerationingschemes. 5.1.3Pareto-ImprovingPureRoadSpaceRationingProblem Wangetal. ( 2010 )demonstratedthatundercertainrationingratioandnetworkcongurations,itispossibletoachieveParetoimprovementsusingpureroadspacerationing.ThefollowingmathematicalprogramisformulatedtondtheoptimalrationingratioanditscorrespondingowpatterntominimizethetotalsystemdelaywhileensuresthattherationingschemecanleadtoaPareto-improvement.ThePareto-improvingpureroadspacerationingproblemcanbeformulatedasamathematicalprogramwithcomplementarityconstraints(MPCC)asfollows,P2:min(x,v,,)X(i,j)2Ltij(vij)vij (5)s.t.Axw,1=(1)]TJ /F7 11.955 Tf 11.95 0 Td[()Ewdw8w (5)Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5)tij(vij)+w,1i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,1j08(i,j)2Ln,w (5)xw,1ijtij(vij)+w,1i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,1j=08(i,j)2Ln,w (5)tij+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj08(i,j)2Lt,w,k=1,2 (5)xw,kijtij+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj=08(i,j)2Lt,w,k=1,2 (5)(1)]TJ /F7 11.955 Tf 11.95 0 Td[()hw,1d(w))]TJ /F7 11.955 Tf 11.95 0 Td[(w,1o(w)i+hw,2d(w))]TJ /F7 11.955 Tf 11.96 0 Td[(w,2o(w)iCUEw8w (5) whereCUEwistheequilibriumtraveltimeofODpairwbeforethepolicyimplementation,i.e.,statusquo.Theobjectivefunctionminimizesthetotalsystemdelay.Constraints 86

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( 5 )to( 5 )aretheuserequilibriumowconditionsunderapureroadspacerationingpolicy.Constraint( 5 )guaranteesthatwhenaveragedacrossrestrictedandfreedays,nouserismadeworseoffcomparedwiththestatusquo.Asformulated,theaboveproblemisaMPCC,whichisaclassofproblemthatisdifculttosolveformainlytworeasons.OneisbecauseMPCCviolatescertainconstraintqualicationandtheotherisduetothefactthatthefeasibleregionisnon-convex. 5.2Pareto-ImprovingHybridPolicy Themajordrawbackofapureroadspacerationingpolicyisthatasaregulatorydemandmanagementstrategyitrequiresuserstobehaveaccordingtocertaincompulsoryrules(takingtransitwhentheyarerestricted)thatapplytoeveryoneinthesamemanner. Downs ( 2004 )pointedoutitmaynotthemosteffectiveoverallstrategy.Thehybridpolicyweproposedintegratesregulatoryandmarket-baseddemandmanagementstrategies. Inthepresenceofahybridpolicy,roadlinksetLncanbefurtherdividedintotwomutuallyexclusivesets:agenerallinksetLgandarestrictedlinksetLr.WethushaveLn=fLg[Lrg.Eachdayafractionofroadusers(usergroupk=1),100(1)]TJ /F7 11.955 Tf 12.73 0 Td[()%ofthetotaldemand,areentitledtotraveltheentireroadnetworkforfree,whereiscalledrestrictionratio.Theremainingrestrictedusers(usergroupk=2),however,arenotforcedtotaketransitservice.Theyareprovidedwiththeopportunitytopaytollstoaccesstherestrictedlinks,Lrandalsotravelonthegenerallinksforfree.Nonnegativecongestiontolls,ij,areonlychargedonrestrictedlinks,(i,j)2Lr,forusersontheirrestricteddays.Todesignahybridpolicy,transportationagenciesneedtospecifythreecrucialcomponents:restrictionratio,locationstosetuprestrictedlinksandtheircorrespondingtollratesij. 87

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5.2.1UserEquilibriumunderHybridPolicies Theuserequilibriumowdistributioninthepresenceofahybridpolicycanbeestimatedbysolvingthefollowingmathematicalprogram.P3:min(x,v)X(i,j)2LnZvij0tij(!)d!+X(i,j)2Ltvijtij+X(i,j)2Lrijv2ij (5)s.t.Axw,1=(1)]TJ /F7 11.955 Tf 11.95 0 Td[()Ewdw8w (5)Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5) whereconstraints( 5 )to( 5 )describeowbalanceconstraintsforgeneralandrestrictedusers.Forrestrictedusers,however,theyhavetopayijiftheychoosetouserestrictedlinks(i,j)2Lr.Constraints( 5 )to( 5 )essentiallyarethefeasibleowregionofamulticlassmultimodalnetwork,whichisconvex.Theobjectivefunctionisstrictlyconvexwithrespecttotheaggregatelinkowvij.Therefore,problemP3isastrictlyconvexoptimizationproblemandthereisauniquesolutionoftheuserequilibriumlinkowdistribution.Toshowtheabovemathematicalprogramisequivalenttotheuserequilibriumunderahybridpolicy,theKKTconditionsoftheaboveoptimizationproblemcanbestatedasfollows.Forastrictlyconvexmathematicalprogram,theKKTconditionsarebothnecessaryandsufcientforoptimality.tij(vij)+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj08(i,j)2Lg,w,k=1,2 (5)xw,kijtij(vij)+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj=08(i,j)2Lg,w,k=1,2 (5)tij+w,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj08(i,j)2Lt,w,k=1,2 (5)xw,kijtij+w,ki)]TJ /F7 11.955 Tf 11.96 0 Td[(w,kj=08(i,j)2Lt,w,k=1,2 (5)tij(vij)+w,1i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,1j08(i,j)2Lr,w (5)xw,1ijtij(vij)+w,1i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,1j=08(i,j)2Lr,w (5) 88

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tij(vij)+ij+w,2i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,2j08(i,j)2Lr,w (5)xw,2ijtij(vij)+ij+w,2i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,2j=08(i,j)2Lr,w (5)Axw,1=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()Ewdw8w (5)Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5) wherewisavectorofLagrangemultipliersassociatedwiththeowbalanceconstraints( 5 )and( 5 ),whicharealsoknownasnodepotentials( Ahujaetal. 1993 )forODpairw.Therearefourpairsofcomplementarityconstraints( 5 )to( 5 ).Let'sderivetheequivalenceconditionsforgeneralusersrst.Generalusers(usergroupk=1)areallowedtoaccessalllinks,i.e.,fLg[Lt[Lrginthetransportationnetworkforfree.Whenalink(i,j)isutilized,i.e.,xw,1ij>0,constraints( 5 ),( 5 )and( 5 )forcetheequationtij(vij)+w,1i)]TJ /F7 11.955 Tf 11.39 0 Td[(w,1j=0tohold.Combingtogetherthisequationforeachlinkonautilizedpathyieldsthefollowing:X(i,j)2w,1rtij(vij)=X(i,j)2w,1rw,1j)]TJ /F7 11.955 Tf 11.95 0 Td[(w,1i=w,1d(w))]TJ /F7 11.955 Tf 11.96 0 Td[(w,1o(w) where,forODpairw,w,1risasetcontaininglinksthatareavailabletogeneralusersonpathrandd(w)ando(w)denotethedestinationandoriginnodesofODpairw.Thus,constraints( 5 ),( 5 )and( 5 )implythatthegeneralizedtraveltimeofeveryutilizedpathequalsw,1d(w))]TJ /F7 11.955 Tf 12.44 0 Td[(w,1o(w)forgeneralusersbetweenODpairw.Whenalink(i,j)isnotutilized,i.e.,xw,1ij=0,constraints( 5 )to( 5 )implytheinequalitytij(vij)+w,1i)]TJ /F7 11.955 Tf 12.57 0 Td[(w1j0holds.Addingthisinequalityalonganon-utilizedpathyieldsP(i,j)2w,1rtij(vij)w,1d(w))]TJ /F7 11.955 Tf 12.6 0 Td[(w,1o(w).Therefore,wehavetheuserequilibriumconditionsforgeneralusers.Similarprocedurescanbeappliedtocomplementarityconstraints( 5 )to( 5 ),( 5 )and( 5 )toconstructthetolleduserequilibriumconditionsfor 89

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restrictedusers(usergroupk=2).Hence,theKKTconditionsofoptimizationproblemP1areequivalenttotheuserequilibriumconditionswhenahybridpolicyisinplace. TheaboveformulationalsoprovidesawaytoachievetheUEsolutionunderapureroadspacerationingscheme.InsteadofsolvingproblemP1directly,wecanputanarbitrarylargetollonallroadlinksforrestricteduserstopreventthemfromusingroadlinksinproblemP3.Undersuchsettings,bysolvingproblemP3,theresultingowdistributionshouldcoincidewiththesolutiontoproblemP1. 5.2.2Pareto-ImprovingHybridPolicyProblem Similartocongestionpricingandotherdemandmanagementinstruments,theutmostgoalofahybridpolicyistoimprovetransportationsystemefciency.WhileforaPareto-improvingpolicy,wealsohavetoensurethepolicyleadtoaPareto-improvementoverthestatusquo.Todesignahybridpolicy,therestrictionratio,locationsofrestrictedlinks,i.e.,partitioningofLgandLr,andtheircorrespondingtollratesforrestrictedusersneedtobespeciedtominimizetotalsystemdelay.AMPCCisformulatedasfollow,P4:min(x,v,,,)X(i,j)2Lntij(vij)vij+X(i,j)2Lttijvij (5)s.t.Axw,1=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()Ewdw8w (5)Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5)tij(vij)+w,1i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,1j08(i,j)2Ln,w (5)xw,1ijtij(vij)+w,1i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,1j=08(i,j)2Ln,w (5)tij(vij)+ij+w,2i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,2j08(i,j)2Ln,w (5)xw,2ijtij(vij)+ij+w,2i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,2j=08(i,j)2Ln,w (5)tij+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj08(i,j)2Lt,w,k=1,2 (5) 90

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xw,kijtij+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj=08(i,j)2Lt,w,k=1,2 (5)(1)]TJ /F7 11.955 Tf 11.96 0 Td[()hw,1d(w))]TJ /F7 11.955 Tf 11.96 0 Td[(w,1o(w)i+hw,2d(w))]TJ /F7 11.955 Tf 11.95 0 Td[(w,2o(w)iCUEw8w (5)ij08(i,j)2Ln (5) Constraints( 5 )to( 5 )areequilibriumowconditionsassumingallroadlinksarepotentialrestrictedlinks.Constraint( 5 )guaranteesthatwhenaveragedacrossrestrictedandfreedays,nouserismadeworseoffcomparedwiththestatusquo.Constraint( 5 )requiresthetollvectortobenonnegative.Intheaboveformulation,therestrictionratioandtollratesaredecisionvariablesandallroadlinksaretreatedaspotentialrestrictedlinks.Thedesignationofrestrictedlinkscanbedeterminedbyreferringtotheoptimaltollrates.Iftheresultingtollrateforrestrictedusersisstrictlypositiveonlink(i,j),thelinkbelongstoLr,otherwise,(i,j)2Lg. NotethattheaboveformulationcanalsobeusedtoobtainthesolutiontoaPareto-improvingpurerationingproblem(P2)bysettinganarbitrarylargetollonallroadlinksforrestricteduserssothattheychoosenottousetheroadnetworkanyhow.SincetheoptimalsolutiontothePareto-improvingpurerationingproblem(P2)isalwaysafeasiblesolutiontotheaboveformulation,wecanconcludethatthesolutiontoproblemP2shouldbedominatedbytheoptimalsolutiontotheproblemP4intheory. ItisalsoworthmentioningthatParetoimprovementsinaboveexampleareattainedbeforeredistributionoftollrevenues.Inotherwords,wecouldachieveevenbetterimprovementsifcertainrevenueredistributionplanisintegratedinthemodel.Asanexample,let'sconsideratransitsubsidyplan,whichdirectlytransferstollrevenuestosubsidizetransitfares.ThehybridpolicywithtransitsubsidyproblemcanbeformulatedasaMPCCasfollows:P5:min(x,v,,,)X(i,j)2Lntij(vij)vij+X(i,j)2Lttijvij (5)s.t.Axw,1=(1)]TJ /F7 11.955 Tf 11.96 0 Td[()Ewdw8w (5) 91

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Axw,2=Ewdw8w (5)vij=XwXkxw,kij (5)x0 (5)tij(vij)+w,1i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,1j08(i,j)2Ln,w (5)xw,1ijtij(vij)+w,1i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,1j=08(i,j)2Ln,w (5)tij(vij)+ij+w,2i)]TJ /F7 11.955 Tf 11.95 0 Td[(w,2j08(i,j)2Ln,w (5)xw,2ijtij(vij)+ij+w,2i)]TJ /F7 11.955 Tf 11.96 0 Td[(w,2j=08(i,j)2Ln,w (5)(tij)]TJ /F4 11.955 Tf 11.95 0 Td[(sij)+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj08(i,j)2Lt,w,k=1,2 (5)xw,kij(tij)]TJ /F4 11.955 Tf 11.95 0 Td[(sij)+w,ki)]TJ /F7 11.955 Tf 11.95 0 Td[(w,kj=08(i,j)2Lt,w,k=1,2 (5)(1)]TJ /F7 11.955 Tf 11.96 0 Td[()hw,1d(w))]TJ /F7 11.955 Tf 11.96 0 Td[(w,1o(w)i+hw,2d(w))]TJ /F7 11.955 Tf 11.95 0 Td[(w,2o(w)iCUEw8w (5)X(i,j)2LtsijvijX(i,j)2LnXw2Wijxw,2ij (5)sij,ij08(i,j)2Ln (5) wheresijisananonymousnonnegativesubsidytotransitusersusinglink(i,j).Nonnegativetolls,ij,areimposedonrestricteduserswhochoosetouserestrictedlinksandsubsidies,sij,areprovidedtousersusingthetransitlinks.Constraint( 5 )makessurethetotalsubsidytotransitusersshouldbelessthantotaltollrevenuescollected.Constraint( 5 )requiresbothtollsandsubsidiesarenonnegative. 5.3NumericalExamples ConsideramultimodalnetworkinFigure 5-1 ,whichcontainsfourODpairs:(1,3),(1,4),(2,3)and(2,4).Itconsistsof18autolinksand4transitlinks.ThelinkperformancefunctionsassociatedwithautolinksareassumedtofollowtheBPRfunction:tij(vij)=t0ijh1+0.15(vij=bij)4i,wheret0ijisthefree-owtraveltimeandbijisthecapacityofthatlink.Theparenthesisnearautolink(i,j)intheguredenotes 92

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(t0ij,bij).FourstransitlinksdirectlyconnecttheoriginanddestinationnodeofeachODpair.AggregatedemandofeachODpairisalsoshownintheFigure 5-1 Mathematicalprogramsformulated(P2,P4andP5)areMPCCproblems,whichisaclassofproblemsthatisdifculttosolveusingcommercialsoftware.Inthischapter,werevisedthemanifoldsuboptimizationalgorithmproposedby LawphongpanichandYin ( 2010 )tondstronglystationarysolutions.ThealgorithmisimplementedusingGAMS( Brookeetal. 2003 )andnon-linearsub-problemsinvolvedaresolvedbyacommercialnonlinearprogrammingsolvercalledCONOPT.Asanoteonthesolutionprocedure,itisgenerallynoteffectivetousethemanifoldsuboptimizationalgorithmtosolvethePareto-improvinghybridpolicyproblem(P4)directly.Theresultingowdistributionofthepurerationingscheme(P2)isusedasaninitialsolutiontothehybridpolicyproblembecauseitisalwaysafeasiblesolutiontothelater. Fortherstscenario,weassumethattraveltimesonalltransitlinksare150%oftheircorrespondingtraveltimesundertheoriginalUEcondition.BysolvingproblemP2,P4andP5,thesetwoPareto-improvingstrategiesarecompared,andresultsaresummarizedinTable 5-1 and 5-2 Whentransittraveltimeis150%theUEequilibriumtravelcost,totalsystemdelayis4904.0961and5883.8004respectivelyunderSOandUEconditions.Forcomparisonpurposes,resultsfortransitlinksarelistedintherstfourrowsofbothtables.Asshowninthetables,boththepurerationingandhybridpoliciesgenerateParetoimprovements.Wecanobservethatalthoughtheaverageequilibriumtravelcostunderthehybridpolicyisslightlyhigherthanthatunderpurerationingscheme,thehybridpolicyprovidessubstantiallybettersystemperformance.Moreimportantly,wecanobservethatusersthatareforcedtousetransitlinks,e.g.,link,(2,3)underpurerationingschemechoosetopaytollstoaccessroadnetworksunderthehybridpolicy,whichindicatesthatusersareenjoyingtheexibilityprovidedbythehybridpolicy.Iftollrevenuesareusedtosubsidizetransitusers,weobservethatsystemefciencycanbeimprovedevenfurther. 93

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InTable 5-2 ,transitsubsidyontransitlink(2,4)is4.06andsystemdelayreductionis87.82%ofthemaximumpossiblereduction.Also,asanotetothePareto-improvingpricingproblemproposedby LawphongpanichandYin ( 2010 ),noParetoimprovementisobservedunderpurePareto-improvingpricingschemes. TheabovenumericalexampledemonstratesthatthehybridpolicycanleadtoParetoimprovementsandachievemuchbettersystemefciencythanpureroadspacerationingschemes.TofurtherinvestigateandcomparethreePareto-improvingstrategiesmentionedinthischapter,threeproblems(P2,P4andP5)aresolvedforanotherscenariowithhighertransittraveltime(200%ofequilibriumtraveltimesunderUE).TheresultsareshowninTable 5-3 Whentraveltimesontransitlinksare150%ofequilibriumtraveltimesunderUE,asshownintherstscenario,thehybridpolicycanprovidebettersystemefciency.And,iftollrevenuesareusedtosubsidizetransitusers,systemefciencycanbefurtherimprovedasexpected.Whentraveltimesontransitlinksarehigher,whichcouldhappeninreality,wendthatpurerationingschemefailstogenerateaParetoimprovement.Ontheotherhand,thehybridpolicyisstillabletoachieveaParetoimprovementandattain62.68%reductionofthemaximumpossiblereductioninsystemperformance.Inthisscenario,puttingmoneybacktosubsidizetransitusersdoesnotimprovesystemefciencyfurtherbecausetraveltimesoftransitservicearesolongthanthetransitmodeisnotattractiveevenifsubsidyisprovidedtousers.TheaboveexampledemonstratesthattheproposedhybridpolicyismorerobustthanpurerationingpolicyinbalancingsystemefciencyandParetoimprovements,whichisanimportantfeaturerequiredforreal-worldimplementations. 5.4Summary ThischapterproposesanewhybridPareto-improvingpolicythatcombinesmultiplepolicyinstruments.Theproposedhybridpolicyprovidesgreaterexibilitythanpureroadspacerationing.Generallyrationingisenactedonthewholenetworkandrationing 94

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ratioisuniformforallODpairs.Traveldemandsaredivertedtotransitlinksatthesameratioregardlessofthenetworktrafcconditions,whichmaycauseinefcientallocationofroadresources.Intheproposedhybridpolicy,althoughrestrictionratioisstillthesameforallODpairs,tollsareintroducedtoadjustroadwaydemandlevelsofdifferentODpairs.ForsomeODpairs,usersmaybebetteroffpayingtollstoentertheroadnetworkonrestricteddaysinsteadoftakingtransitservice.Furthermore,althoughtollsarechargedonrestrictedusersonly,theirroutechoicesmayinuenceroutechoicesofnon-restrictedusersandachievebettersystemperformance. ThereasonsthattheproposedpolicyismoreprominentinleadingtoaParetoimprovementaretwofolds.First,onlyacertainportionofusershavetopaytollstouserestrictedlinkswhileallusersarerequiredtopayanonymoustollsiftheychoosetousetolledlinksinpurecongestionpricing.Essentially,thehybridpolicycanbeviewedasarealizationofdifferentiatedpricingschemes,althoughtollsbeingchargedarestillanonymous.Second,byintroducingtimedimension,aPareto-improvingsolutionismorelikelytoexist.ToachieveanoverallParetoimprovement,usersdonotnecessarilyhavetobebetteroffeverysingleday,aslongastheaveragetraveltimeofrationingandnon-rationingdaysisbetterthantheoriginalequilibriumtraveltime,i.e.,thestatusquo. 95

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Figure5-1. Amultimodaltransportationnetwork 96

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Table5-1. UserequilibriumandPareto-improvingpurerationingproblems Link UE Purerationingpolicy Userow Generaluserow Restricteduserow (1,3) 0.00 0.00 9.00 (1,4) 0.00 0.00 9.00 (2,3) 0.00 0.00 12.00 (2,4) 0.00 0.00 15.00 (1,5) 22.94 14.50 0.00 (1,6) 37.06 27.50 0.00 (2,5) 48.11 43.76 0.00 (2,6) 41.90 19.24 0.00 (5,6) 0.00 0.00 0.00 (5,7) 31.65 28.31 0.00 (5,9) 39.39 29.95 0.00 (6,5) 0.00 0.00 0.00 (6,8) 59.12 46.74 0.00 (6,9) 19.83 0.00 0.00 (7,3) 45.75 42.50 0.00 (7,4) 28.32 15.76 0.00 (7,8) 0.00 0.00 0.00 (8,3) 24.25 6.50 0.00 (8,4) 51.68 40.24 0.00 (8,7) 0.00 0.00 0.00 (9,7) 42.42 29.95 0.00 (9,8) 16.80 0.00 0.00 ODpairs Equilibriumcost Averageequilibriumcost (1,3) 45.63 40.50 (1,4) 45.33 39.45 (2,3) 35.22 34.07 (2,4) 34.92 33.52 Optimalratio 0.00 0.30 Totaldelay 5883.80 5437.17 Systemdelayreduction 0% 46.00% 97

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Table5-2. Pareto-improvinghybridpolicyproblems Link Hybridpolicy Hybridpolicywithsubsidy General Restricteduser General Restricteduser Flow Flow Toll Flow Flow Toll (1,3) 0.00 10.02 0.00 0.00 11.46 0.00 (1,4) 0.00 5.48 0.00 0.00 1.84 0.00 (2,3) 0.00 0.00 0.00 0.00 0.00 0.00 (2,4) 0.00 10.56 0.00 0.00 9.13 -4.06 (1,5) 12.32 3.60 22.72 8.32 9.62 26.08 (1,6) 27.64 0.94 14.32 28.76 0.00 18.96 (2,5) 43.92 0.00 10.31 41.50 0.00 11.27 (2,6) 16.01 19.50 0.00 14.12 25.25 0.00 (5,6) 0.00 0.00 0.00 0.00 0.00 0.00 (5,7) 28.55 0.00 15.90 28.51 0.00 10.41 (5,9) 27.69 3.60 0.00 21.31 9.62 0.00 (6,5) 0.00 0.00 0.00 0.00 0.00 0.00 (6,8) 43.65 0.00 22.75 42.88 0.00 17.37 (6,9) 0.00 20.45 7.34 0.00 25.25 7.00 (7,3) 38.95 0.00 1.25 33.04 0.00 7.36 (7,4) 17.29 2.77 0.00 16.78 9.12 0.00 (7,8) 0.00 0.00 0.00 0.00 0.00 0.00 (8,3) 7.66 13.36 2.80 10.22 15.28 7.52 (8,4) 35.99 7.92 3.46 32.66 10.47 4.31 (8,7) 0.00 0.24 0.00 0.00 0.00 0.00 (9,7) 27.69 2.52 15.90 21.31 9.12 10.41 (9,8) 0.00 21.52 8.68 0.00 25.75 2.89 ODpairs Averageequilibriumcost Averageequilibriumcost (1,3) 41.90 43.85 (1,4) 41.00 42.87 (2,3) 34.55 34.89 (2,4) 34.92 34.92 Optimalratio 0.33 0.38 Totaldelay 5078.11 5023.47 Systemdelay 80.48% 87.82% reduction Table5-3. SystemdelayreductionsunderdifferentPareto-improvingstrategies Transittime Purerationing Hybridpolicy Hybridpolicywithtransitsubsidy 150%UE 46.00% 80.48% 87.82% 200%UE 0% 62.68% 62.68% 98

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CHAPTER6CONCLUSIONS 6.1SummaryofMajorFindings Despitethesuccessesofpricingprojectsworldwideandgrowinggovernmentsupport,congestionpricingremainslargelyunappealingtothegeneralpublic.Along-standingdilemmafortransportationauthoritiesishowtoenjoytheefciencybenetsofcongestionpricingwhilekeepingthegeneralpublichappy.Inthisdissertation,weproposedthatusingaPareto-improvingcongestionpricingapproachwillbridgethegapbetweenthesetwoseeminglycontradictorygoals.Thisdissertationprovidesanin-depthinvestigationofthe-state-of-the-artofPareto-improvingpricingstrategiesforgeneraltransportationnetworks. ThefundamentalreasonthataPareto-improvingcongestionpricingschemeisachievableisthataUEowdistributionmaynotbePareto-optimal.IfaowdistributionisnotPareto-optimal,theremayexistadominatingowdistributionthatimprovessystemefciencywithoutmakinganyuserworseoff.Sinceadominatingowdistributionisnotinequilibrium,congestionpricingcanbeusedasaninstrumentforinducingsuchaowdistribution.AsshownChapter3,theabsenceofloops(single-andmulti-commodity)isthenecessaryandsufcientconditionfortheexistenceofanonymousnonnegativetollstosupportagivendominatingowdistribution.Also,therelationshipbetweenadominatingowdistributionandanonymousnonnegativePareto-improvingtollvectorswasexamined.WefurtherprovedthatanonnegativeOD-dependentPareto-improvingtollvectorcanalwaysbefoundtoinduceanydominatingowdistributionwithoutasingle-commodityloopasanequilibriumowdistribution. Chapter4discussedPareto-improvingpricingschemesforageneralnetworkwhenusersbelongtoadiscretesetofVOTclasses.MathematicalformulationsforndingaPareto-improvingschemeaswellasadominatingowdistributionweredeveloped. 99

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Givenadominatingowdistributionwithheterogeneoususers,wealsoprovidednecessaryandsufcientconditionsfortheexistenceofananonymousnonnegativePareto-improvingtollvectortoinducethedistribution.ThenumericalresultsfromthreenetworksillustratethatmulticlassPareto-improvingpricingschemesarelessprevalentwhencomparedtoitssingle-classcounterpart. AnewhybridPareto-improvingpolicythatcombinescongestionpricingandfree-travel-rightassignmentwasproposedinChapter5.MathematicalformulationsfordevelopingPareto-improvingpureroadspacerationingschemesandhybridpoliciesweregiven.Thehybridpolicyproposedoffersgreaterexibilitythanpureroadspacerationing.NumericalexamplesdemonstratedthattheproposedhybridpolicyismoreprominentinleadingtoaParetoimprovementthanbothpurecongestionpricingandroadspacerationingschemes. Threemajorcontributionsofthisdissertationtotheliteraturecanbesummarizedasfollows: ThisdissertationprovidedasystematicstudyoftheexistenceandpropertiesofPareto-improvingpricingschemesingeneraltransportationnetworks. ItexploredtheproblemofPareto-improvingpricingschemesfornetworkswithheterogeneoususers. Itproposedahybridpolicythattakesadvantageofthesynergisticeffectsbetweencongestionpricingandfree-travel-rightassignment. ThecongestionpricingandhybridstrategiesdevelopedinthisdissertationdemonstratethatthePareto-improvingapproachisaviableandpromisingwayofdesigningefcientdemandmanagementstrategiesthatarealsoappealingtothegeneralpublic.Thendingsmaymakecongestionpricingnolongerahardselltodecisionmakersandthegeneralpublic,whichmayeventuallyleadthenation'stransportationsystemtoamoresustainablefuture. 100

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6.2FutureResearch InChapter4,weassumedthatroadusersarecategorizedintoadiscretesetofVOTclasses.UserswhobelongtothesameuserclassareassumedtohavethesameVOT,whichisaroughapproximationoftheactualcontinuousVOTdistribution. NieandLiu ( 2010 )examinedtheexistenceofaPareto-improvingpricingschemeinabottlenecknetwork(oneODpair)withheterogeneoususerscharacterizedbycontinuousVOTdistributions.ItwouldbeavaluableadditiontotheliteratureofthePareto-improvingcongestionpricingapproach,ifwecanextendthemodeltogeneralroadnetworks. ThehybridpolicyproposedinChapter5combinesregulatoryandmarket-baseddemandmanagementpolicies.Futureresearchcouldintroduceheterogeneoususersintotheproposedformulation.Bydoingso,wemayobtainmoreinsightsintothepolicyimplicationsofthehybridmodel.Also,itmightbeinterestingtoinvestigatethelong-termeffectsofthehybridpolicy.Roadusersmaypurchaseadditionalvehiclestobypassrestrictions.However,whethertheychoosetodosointhelongtermdependsontollratesonrestrictedlinksandtherestrictionratio.Becauseusersareallowedtopaytollstouserestrictedlinksonrestricteddays,itmaynotbebenecialforthemtomaintainasecondvehicletobypasstherestriction.Ontheotherhand,iftherestrictionratioandtollratesarehigh,usersmayndthemselvesbetteroffpurchasingasecondcarinthelongrun. 101

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BIOGRAPHICALSKETCH ZiqiSonggrewupinNanjing,China.HereceivedhisB.Eng.inTransportationEngineeringfromSoutheastUniversity,Chinain2003.Aftergraduation,hewenttoHongKongtopursuehispostgraduatestudyatTheUniversityofHongKongandgraduatedwithM.Phil.degreeinCivilEngineering.In2006,hejoinedtheUniversityofFloridaforhisPh.D.studyintransportationattheDepartmentofCivilandCoastalEngineering.HealsoreceivedhisM.S.inOperationsResearchfromtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.HeworkedatTechnischeUniversitatMunchen,Germanyasaresearchfellowin2011.Hisresearchcoverscongestionpricing,transportationnetworkmodelingandtransportationpolicyanalysis. 108