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Differentiated Congestion Pricing of Transportation Networks

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Title:
Differentiated Congestion Pricing of Transportation Networks
Creator:
Zangui, Mahmood
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (114 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Civil Engineering
Civil and Coastal Engineering
Committee Chair:
YIN,YAFENG
Committee Co-Chair:
LAWPHONGPANICH,SIRIPHONG
Committee Members:
ELEFTERIADOU,AGELIKI
WASHBURN,SCOTT STUART
SRINIVASAN,SIVARAMAKRISHNAN
GUAN,YONGPEI
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Prices ( jstor )
Pricing ( jstor )
Sensors ( jstor )
Tolls ( jstor )
Traffic congestion ( jstor )
Transportation ( jstor )
Travel ( jstor )
Travel costs ( jstor )
Travel time ( jstor )
Travelers ( jstor )
Civil and Coastal Engineering -- Dissertations, Academic -- UF
congestion -- differentiation -- path -- privacy -- sensor -- traffic
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Civil Engineering thesis, Ph.D.

Notes

Abstract:
Congestion pricing is a market-based approach for congestion mitigation that uses tolls to encourage travelers to use a road network more efficiently. In the literature, congestion tolls are typically uniform or anonymous, implying that toll amounts do not depend on the characteristics of users or trips, and tolls can be collected anonymously, without requiring any identification. One exception to that is toll differentiation with respect to vehicle type, which charges passenger cars and trucks differently. It has been demonstrated that differentiated tolls can be more flexible and reduce congestion in more efficient manner. However, to be practical, the criteria for toll differentiation should be directly observable. For instance, vehicle type is easy to observe but personal income is not. So, it is difficult to differentiate tolls based on income. This dissertation considers differentiated schemes that price tolls based on the trip characteristics such as the paths used in completing trips and the individual trip's origin and/or destination. Recent advances in vehicle tracking technologies make such schemes possible. Without such technologies, differentiating tolls based on trip characteristics is not feasible and, consequently, the literature on such tolls is lacking. To make a case for the proposed schemes and demonstrate their performances, we implement them on sample networks and present the results. Our results reveal very significant improvements in performance measures compared to anonymous tolls. We then proceed by investigating the key issues regarding design and implementation of differentiated schemes. Designing an optimal differentiated pricing scheme can be computationally challenging. However, we develop effective methods for designing differentiated tolls by taking advantage of the problem's special structures and properties. We then show the performance of our methods by applying them on real-size networks. We also propose an innovative method to implement the new schemes that uses devices such as toll-tag readers and license-plate scanners, which have been broadly deployed for implementing anonymous tolls and for other traffic surveillance applications. We demonstrate that implementing differentiated tolls using this approach can be less costly, yet more effective than anonymous tolls. Finally, we address a privacy concern associated with implementation of differentiated tolls. This issue arises as collecting travel characteristics may encroach upon users' location privacy. We address this concern by an incentive program that is designed such that the users who are sensitive about their privacy can remain anonymous, while the ones comfortable with revealing their location will pay lower tolls. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: YIN,YAFENG.
Local:
Co-adviser: LAWPHONGPANICH,SIRIPHONG.
Statement of Responsibility:
by Mahmood Zangui.

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UFRGP
Rights Management:
Copyright Zangui, Mahmood. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
969977048 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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DIFFERENTIATEDCONGESTIONPRICINGOFTRANSPORTATIONNETWORKSByMAHMOODZANGUIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014MahmoodZangui

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Tomywife

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ACKNOWLEDGMENTS Firstandforemost,IwouldliketothankmyadvisorDr.YafengYinforhisguidance,support,andencouragementthroughoutmyPhDstudies.Hehasbeenaknowledgeableteacher,patientmentor,andsupportivefriend.Hetaughtmehowtothink,doresearch,andpresentmyideas.Iwouldalsoliketoexpressmygratitudetowardmyco-advisorDr.SiriphongLawphongpanichforhisguidanceandsupport.Hehasbeenaninvaluablesourceofknowledgeandexperience,whohelpedmebecomeabetterresearcher.Thisdissertationwouldnothavebeencompletedwithouthishelp.IwouldliketoexpressmysinceregratitudetoDr.LilyElefteriadou,Dr.SivaramakrishnanSrinivasan,andDr.ScottWashburnforteachingmedierentaspectsoftransportationengineeringandmakingmystudyatUFrichandrewarding.Ialsothankthemforservingasmycommitteemembers,andfortheirfeedbacksandcomments.Iwouldalsoliketothankmyothercommitteemember,Dr.YongpeiGuan,forreviewingthedissertationmanuscriptandforhisfeedbacks.Iamalsoverygratefultomyfriends,RoosbehNowrouzian,NimaShirmohammadi,DanialDavarnia,andSeckinOzkulformakingmylifeatUFhappierandmorejoyfulthroughouttheseyears.Finally,Iwouldliketothankmywife,ourparents,andsistersfortheircontinuoussupportandencouragements. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1Background ................................... 12 1.2DissertationObjectives ............................. 14 1.3DissertationOutline .............................. 14 2LITERATUREREVIEW .............................. 15 2.1TracDistributions .............................. 15 2.1.1Notation ................................. 15 2.1.2UserEquilibriumandSystemOptimum ................ 16 2.2CongestionPricingModels ........................... 17 2.2.1MarginalSocialCostPricing ...................... 18 2.2.2First-bestandSecond-bestCongestionPricing ............ 19 2.3PriceDierentiation .............................. 19 2.3.1PriceDierentiationinMonopolisticMarkets ............. 20 2.3.2PriceDierentiationinCongestionPricing .............. 21 2.3.2.1BetterEstimateoftheExternalities ............. 21 2.3.2.2NonlinearPricing ....................... 22 2.3.2.3DierentiationBasedonValueofTime ........... 23 2.4ContributionsofThisDissertation ....................... 23 3DIFFERENTIATEDCONGESTIONPRICINGSCHEMES ........... 25 3.1Overview .................................... 25 3.2DierentiatedTolls ............................... 26 3.3ModelFormulations ............................... 27 3.3.1Notation ................................. 27 3.3.2First-bestCondition ........................... 28 3.3.3Second-bestCondition ......................... 29 3.4IllustrativeExamples .............................. 31 3.5Summary .................................... 35 5

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4DESIGNINGPATH-DIFFERENTIATEDCONGESTIONPRICINGSCHEMES 36 4.1Overview .................................... 36 4.2Background ................................... 37 4.3IllustrationofPath-basedPricing ....................... 39 4.4MinimumRevenueTollPricingProblem ................... 44 4.4.1Link-basedPricing ........................... 45 4.4.2Path-basedPricing ........................... 46 4.5AlternativeFormulations ............................ 50 4.5.1Mixed-integerOptimizationProblems ................. 51 4.5.2ConcaveMinimizationProblem .................... 53 4.5.3BilinearOptimizationProblem ..................... 56 4.6NumericalResults ................................ 57 4.7Summary .................................... 61 5IMPLEMENTATIONOFDIFFERENTIATEDPRICINGSCHEMES ...... 62 5.1Overview .................................... 62 5.2Background ................................... 64 5.2.1Notation ................................. 64 5.2.2Path-dierentiatedPricing ....................... 65 5.3PathObservation ................................ 67 5.3.1ModelFormulation ........................... 67 5.3.2Illustration ................................ 69 5.4ImplementingPath-dierentiatedTollsUsingAVISensors ......... 69 5.4.1ModelFormulation ........................... 71 5.4.2Illustration ................................ 71 5.4.3AdjustingTolls ............................. 72 5.5Designingpath-dierentiatedschemeforgivensensorlocations ....... 76 5.5.1ModelFormulation ........................... 76 5.5.2Illustration ................................ 77 5.6SimultaneousDesignofTollsandSensorLocations ............. 78 5.6.1MinimumSensorLocationsforLink-basedTolls ........... 78 5.6.2MinimumSensorLocationsforPath-dierentiatedTolls ....... 79 5.6.3Illustration ................................ 81 5.6.4HeuristicSolutionAlgorithm ...................... 83 5.6.5NumericalExample ........................... 86 5.7ANoteonPathGeneration .......................... 88 5.8SummaryandExtensions ........................... 88 6PRIVACYISSUEOFDIFFERENTIATEDPRICINGSCHEMES ........ 90 6.1Overview .................................... 90 6.2LocationPrivacy ................................ 91 6.2.1ValueofPrivacy ............................. 91 6.2.2ModelingPrivacy ............................ 92 6

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6.2.3PrivacyCostAnalysisofDierentiatedSchemes ........... 94 6.3AddressingPrivacyConcerns ......................... 95 6.3.1DesignofIncentiveProgram ...................... 96 6.3.1.1First-bestcondition ...................... 97 6.3.1.2Second-bestcondition .................... 98 6.3.2NumericalExamples .......................... 98 6.4SummaryandDiscussion ............................ 101 7CONCLUSION .................................... 103 REFERENCES ....................................... 105 BIOGRAPHICALSKETCH ................................ 112 7

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LISTOFTABLES Table page 3-1Illustrationofdierentiatedtolls .......................... 26 3-2Dierentiatedpricingfornine-nodenetworkwithalllinkstollable ........ 31 3-3Second-bestdierentiatedpricingfornine-nodenetwork ............. 33 3-4First-bestdierentiatedpricingforSiouxFallsnetwork .............. 34 3-5Second-bestdierentiatedpricingforSiouxFallsnetwork ............. 34 4-1MINREVlinktollsforthefour-nodenetwork(revenue=10) ........... 40 4-2PathtollsinducedbyMINREVlinktollsinTable4-2 ............... 40 4-3MINREVpathtollsforthefour-nodenetwork(revenue=5) ........... 41 4-4MINREVlinktollsforthethree-nodenetwork(revenue=1) ........... 43 4-5Minimumrevenuepath-dierentiatedtollsforthethree-nodenetwork(revenue=0) .......................................... 44 4-6NumberofODpairswithuniqueandmultipleMMCpaths ............ 59 4-7ResultsforAnaheim ................................. 60 4-8ResultsforSiouxFalls ................................ 61 4-9Benetsofpathinformation ............................. 61 5-1Sensorlocationsforthenine-nodenetwork ..................... 70 5-2Sensorlocationsforagivenpathtollschemefornine-nodenetwork ....... 72 5-3Sensorlocationsforagivenpathtollschemefornine-nodenetwork ....... 75 5-4Indistinguishablepathsfornine-nodenetwork ................... 78 5-5Resultsofsolvingmodel MSL-P fornine-nodenetwork .............. 82 5-6Optimaltollingschemefornine-nodenetwork ................... 82 6-1Percentageoftravelerswhobenetfromorigin-specicpricingonnine-nodenetwork ........................................ 95 6-2Comparisonofdierentschemesonnine-nodenetwork(alllinkstollable) .... 99 6-3Comparisonofdierentschemesonnine-nodenetwork(twotollablelinks) ... 99 6-4Second-besthybridschemesonSiouxFallsnetwork ................ 100 8

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LISTOFFIGURES Figure page 3-1Nine-nodenetwork .................................. 31 3-2SiouxFallsnetwork .................................. 32 4-1Fournodenetwork .................................. 40 4-2Threenodenetwork ................................. 43 4-3Illustrationofconcavity ............................... 56 5-1Nine-nodenetworkwithoptimalsensorlocationsforfullpathobservability ... 69 5-2SensorlocationsforSiouxFallsnetwork ...................... 88 5-3Usablepathwithcycle ................................ 89 6-1Uniformandexponentialdistributionswithsamemean,E()=2 ........ 93 6-2Expectedprivacycost ................................ 94 6-3Anillustrativenetwork ................................ 102 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDIFFERENTIATEDCONGESTIONPRICINGOFTRANSPORTATIONNETWORKSByMahmoodZanguiAugust2014Chair:YafengYinCochair:SiriphongLawphongpanichMajor:CivilEngineeringCongestionpricingisamarket-basedapproachforcongestionmitigationthatusestollstoencouragetravelerstousearoadnetworkmoreeciently.Intheliterature,congestiontollsaretypicallyuniformoranonymous,implyingthattollamountsdonotdependonthecharacteristicsofusersortrips,andtollscanbecollectedanonymously,withoutrequiringanyidentication.Oneexceptiontothatistolldierentiationwithrespecttovehicletype,whichchargespassengercarsandtrucksdierently.Ithasbeendemonstratedthatdierentiatedtollscanbemoreexibleandreducecongestionsinmoreecientmanner.However,tobepractical,thecriteriafortolldierentiationshouldbedirectlyobservable.Forinstance,vehicletypeiseasytoobservebutpersonalincomeisnot.So,itisdiculttodierentiatetollsbasedonincome.Thisdissertationconsidersdierentiatedschemesthatpricetollsbasedonthetripcharacteristicssuchasthepathsusedincompletingtripsandtheindividualtrip'soriginand/ordestination.Recentadvancesinvehicletrackingtechnologiesmakesuchschemespossible.Withoutsuchtechnologies,dierentiatingtollsbasedontripcharacteristicsisnotfeasibleand,consequently,theliteratureonsuchtollsislacking.Tomakeacasefortheproposedschemesanddemonstratetheirperformances,weimplementthemonsamplenetworksandpresenttheresults.Ourresultsrevealverysignicantimprovementsinperformancemeasurescomparedtoanonymoustolls.Wethenproceedbyinvestigatingthekeyissuesregardingdesignandimplementation 10

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ofdierentiatedschemes.Designinganoptimaldierentiatedpricingschemecanbecomputationallychallenging.However,wedevelopeectivemethodsfordesigningdierentiatedtollsbytakingadvantageoftheproblem'sspecialstructuresandproperties.Wethenshowtheperformanceofourmethodsbyapplyingthemonreal-sizenetworks.Wealsoproposeaninnovativemethodtoimplementthenewschemesthatusesdevicessuchastoll-tagreadersandlicense-platescanners,whichhavebeenbroadlydeployedforimplementinganonymoustollsandforothertracsurveillanceapplications.Wedemonstratethatimplementingdierentiatedtollsusingthisapproachcanbelesscostly,yetmoreeectivethananonymoustolls.Finally,weaddressaprivacyconcernassociatedwithimplementationofdierentiatedtolls.Thisissuearisesascollectingtravelcharacteristicsmayencroachuponusers'locationprivacy.Weaddressthisconcernbyanincentiveprogramthatisdesignedsuchthattheuserswhoaresensitiveabouttheirprivacycanremainanonymous,whiletheonescomfortablewithrevealingtheirlocationwillpaylowertolls. 11

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CHAPTER1INTRODUCTION 1.1BackgroundTraccongestionisaseriousproblemthatpeopleinmetropolitanareasarefacing.Accordingto Schranketal. ( 2012 ),in2011,peoplelivingin498largecitiesintheUnitedStatesspentextra5.5millionhoursoftraveltimeand2.9billiongallonsofgasbecauseofcongestionintransportationnetworks.Inthelastdecades,constructingnewroadsoraddingcapacitytoexistingfacilitiestoalleviatetraccongestionprovedimpractical.Between1980and1999,addedcapacityincreasedtheroutemilesofUShighwaysby1.5percent,whiletheincreaseinvehiclemilesoftravelwas76percent.1Asaresult,currentresearchinvestigatesthewaystouseexistingroadsmoreecientlytoalleviatecongestion,ratherthanconstructingnewcapacities.Congestionpricingisamethodoftracmanagementthathasalongandrichhistoryinresearchandiscurrentlyusedinpractice.Infact,FederalHighwayAdministration,suggeststollingisoneofthehigh-priorityeortsforcongestionmitigation.Tollscanreducecongestionbymanagingdemandandchangingtracdistribution,bothspatiallyandtemporally.Therehavebeenseveralsuccessfulimplementationsofcongestionpricingaroundtheworld,e.g.,London,Stockholm,andSingapore,whichprovedtollinganecientmeasureforcongestionmitigation.In1975,Singaporeintroducedapricingscheme,calledAreaLicensingScheme(ALS),thatreducedthecongestionlevelby45%.In1998,theALSsystemwasreplacedbyElectronicRoadPricing(ERP),whichfurtherreducedthetracvolumeby17%.InLondon,sixmonthsaftertheimplementationofcongestiontolls,tracdelayinsidethechargingzonedecreasedbyaround30%( Wu , 2011 ). 1http://www.fhwa.dot.gov/congestion/ 12

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Despitethesesuccessfuldeploymentsofcongestiontollsandtheirpotentialformitigatingcongestion,congestionpricinghasbeenimplementedinjustafewcitiesaroundtheworld,partlybecauseofpublicopposition.Tomakecongestionpricingmoreappealing,manyresearchershaveattemptedtodesignmoreecienttollingschemes.Onewaytodothisisbytakingadvantageofpricedierentiation.Pricedierentiationisaneconomicconceptthatreferstosituationswheresimilargoodsorservicesaresoldfordierentprices.Sellingdiscountedparkticketstolocalsandgivingrebatesforbuyinginbulkaretwoexamplesofpricedierentiation.Thisstrategyhasbeenusedbyvendorsindierentmarketstogainmoreexibilityinpricinggoods,whichcantranslateintomorerevenueorhighersocialwelfare.Applicationsofpricedierentiationintransportationmarketsareabundant,rangingfrompricingofairfaretocongestiontolls.Inthelatter,instancescanbefoundthatdierentiatetollsbasedonvehicleclass,inwhichpassengercarsarechargeddierentlythantrucksandbuses.However,exceptforaverylimitedimplementationonclosedfreewaysystems,noresearchhasbeenpublishedwhichdierentiatesuserswithrespecttotheirtravelcharacteristics.Thereasonforthismaybethelimitedavailabilityofvehicletrackingtechnologiesthatareneededtocollecttravelinformation,whichinturnarerequiredforimplementingdierentiatedtolls.InadditiontoGPS,afamilyoftracsensorsknownasautomaticvehiclesidentication(AVI)canbeusedtotrackvehicles.Sensorsarealsousedforelectronictollcollection(ETC),whicheliminatetheneedformanualtollcollection.Theavailabilityofvehicletrackingtechnologieshasmotivatedthisresearchtostudydierentiatedpricingschemes.Itcanbeshownmathematicallythattheperformanceofadierentiatedpricingschemeisatleastasgoodasitsanonymouscounterpart.However,withoutanalysis,itisnotyetobvioushowmuchthosedierentiatedschemescanperformbetter.Therearealsovarioustechnicalissuesregardingthedesign,implementation,andprivacyconcernsoftheschemesthatneedtobeaddressed. 13

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1.2DissertationObjectivesAsmentionedearlier,thecurrenttechnologyofvehicleidenticationhasmadeitpossibletotrackvehiclesastheytravelinatransportationnetwork.Thecoreideaofthisdissertationistodevelopnewpricingschemesthatutilizethistrackinginformationtodierentiatetravelersandchargethemaccordingly.Afterpresentingthisideainmoredetail,weprovideanswersthefollowingquestions: Whatarethepotentialsoftheproposeddierentiatedpricing?Inparticularweareinterestedincomparingtheperformancesofdierentiatedpricingschemeandtheanonymouscounterparts. Howcananoptimaldierentiatedpricingschemebedesigned?Tokeepthediscussionbrief,wefocusonpath-dierentiatedschemes.Weanalyzethestructureofthetolldesignproblemandformulateitasamodelthatiseasiertosolve. Howcanadierentiatedpricingschemebeimplemented?WestudythepossibilityofusingcurrentETCtechnologiestoimplementadierentiatedpricingscheme. Howcanprivacyissuesbeaddressed?Dierentiatedpricingschemesrequiretrackingvehicles,whichmaycauseprivacyconcernsforusers.Weaddressthisissuebydesigninganincentiveprogram.Eachoftheaboveproblemsisaddressedinaseparatechapterasexplainednext. 1.3DissertationOutlineThenextchapter,Chapter2,reviewsthefundamentalliteratureoftracassignment,congestionpricing,pricedierentiation,anditsapplicationsincongestionpricing.Chapter3introducesdierentiatedcongestionpricingschemesandillustratestheirperformanceonsamplenetworks.Chapter4investigatesthepropertiesofpath-dierentiatedpricingandusethesepropertiestoformulatetolldesignproblemsthatareeasiertosolve.AnideaforimplementingdierentiatedpricingschemesispresentedandstudiedinChapter5.TheprivacyissueassociatedwithdierentiatedpricingschemesismodeledandasolutionisproposedinChapter6.Finally,Chapter7summarizesandconcludesthedissertation. 14

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CHAPTER2LITERATUREREVIEWThischapterprovidesabriefreviewofthestudiesrelevanttodierentiatedcongestionpricing.Toavoidfragmentation,onlytheliteraturesrelatedtogeneralconceptsarepresentedhere.Whenneeded,moredetailedreviewsforsub-topicsareprovidedthroughoutthedissertation.Thischapterconsistsoffoursections.Thenextsectionpresentsashortreviewontracdistributions.Section 2.2 brieysummarizestheliteratureofcongestionpricing.SomeoftherelatedworksonpricedierentiationarereviewedinSection 2.3 .Thelastsectionhighlightstheresearchgapintheliteratureandunveilsthecontributionsofthisdissertation. 2.1TracDistributionsInthissection,werstintroducethenotationsthatareusedthroughthedissertation.Wethendeneuserequilibriumandsystemoptimumastwotypesoftracdistributionsthatarefrequentlyreferredtointhisdissertation. 2.1.1NotationConsideraroadnetworkdenotedbyG(N;A),whereNandAarethenodesandlinksofthenetwork.Foreachlink,a2A,wedenoteitsow,traveltime,andtollbyxa,ta,anda.Wealsodenexandtasvectorsofxaandta,respectively.Inthisdissertation,alower-caseletterwithoutasubscriptrepresentsavectorofvariablesdenotedbythesameletterwithasubscript,e.g.,dandarevectorsofdwandp,respectively.Also,double-strucklettersinupper-caserepresentsets.Forexample,Wdenotesthesetofallorigin-destination(OD)pairs.Forw2W,dwrepresentsitstraveldemandandPwisthesetofsimplepaths(i.e.,oneswithoutanycycle)fromtheoriginofwtoitsdestination.WeassumethatPw6=,8w2WtoensurefeasibilityandletP=Sw2WPwdenotethesetofsimplepathsinthenetwork.Forp2P,fp,cpandpdenoteitsow,traveltime,andtollamount,respectively.Also,letpaequaloneiflinkaisonpathp.Otherwise,pa=0. 15

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Usingtheabovenotation,thesetofallfeasiblepath-owvectorsordistributions,denotedasF,canbedenedasfollows:F=(f2RjPj+Xp2Pwfp=dw;8w2W)Intheaboveexpression,jPjisthecardinalityofPandtheplussign(+)indicatesthateverycomponentoff(i.e.,fp)isnonnegative.Theequationenclosedbythebracketsensuresthatthedemandforwismet. 2.1.2UserEquilibriumandSystemOptimumUserequilibrium(UE)andsystemoptimum(SO)aretwotracconditionsdenedby Wardrop ( 1952 )astwoprinciplesofequilibrium.HedenedUEasthetraccondition,whereforeveryODpair\Thejourneytimesinallroutesactuallyusedareequalandlessthanthosewhichwouldbeexperiencedbyasinglevehicleonanyunusedroute."Thiscondition,whichrepresentstheselshbehavioroftravelerswhenmakingroutechoice,waslaterformulatedby Beckmannetal. ( 1956 )asthefollowingmathematicalprogram: (UE)minXa2AZxa0ta(z)dz (2{1a)s:t:f2F (2{1b)xa=Xp2Ppafp8a2A (2{1c)wherexaistheowonlinka. 16

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Itcanbeshownthatanoptimalsolutionof UE hastosatisfythefollowingconditions:fp(tp(f))]TJ /F3 11.955 Tf 11.96 0 Td[(uw)08p2Pw;w2W;f2Ftp(f))]TJ /F3 11.955 Tf 11.95 0 Td[(uw08p2Pw;w2W;f2FwhereuwistheequilibriumtraveltimeofODpairw,orequivalently,thetraveltimeofshortest(fastest)pathofw.ThesetwoconstraintsarefrequentlyusedinthemodelstoensuretheUEconditionsaresatised.Inchapter 4 ,weformulateUEconditionasvariationalinequalities,whichisanalternativetotheaboveformulations.Wardrop'ssecondprincipledenesSOastheconditionwhere\theaveragejourneytimeisminimum."Thisconditiondescribesthetracdistributionthathastheminimumsystemtraveltime,andrequiresuserstobecooperative,asopposedtoselsh,whenchoosingroutefortheirtravel.TheSOconditioncanbeformulatedasthefollowingmathematicalprogram: (SO)minXa2Axata(xa) (2{2a)s:t:f2F (2{2b)xa=Xp2Ppafp8a2A (2{2c)wheretheobjectivefunctionistominimizetotalsystemtraveltime. 2.2CongestionPricingModelsTheideaofcongestionpricingstartedwiththeseminalworksof Pigou ( 1932 ), Knight ( 1924 ),and Walters ( 1961 ).Theyrealizedthatunregulatedtravelchoicesleadtoasub-optimaluseoftheroads.Theysubsequentlytriedtousecongestiontollstobetter 17

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allocateroadspace,asalimitedresource,amongthetravelers.Theproblemofndingthebestvaluesoftollsthatyieldthedesiredtracconditionsiscalledcongestionpricingortolldesignproblem.Thegeneralobjectiveofcongestionpricingistomaximizesocialwelfare.Throughthisdissertationweassumedemandisxed,whichisacommonassumptionwhenmodelingpeakhourtrac.Underthisassumption,maximizingsocialwelfareisequaltominimizingthetotaltravelcost.Furthermore,weassumecostoftraveliscomprisedoftraveltimeandtoll.Thisassumptionisnotrestrictive,becausemostofothercostscanbeintegratedintraveltime.Notethattollisnotasocialcost,itisratheratransferofmoneyfromtravelerstotransportationsystem. 2.2.1MarginalSocialCostPricingSupposethetracvolumeonlinka2Ais^xa,thenthetotaltraveltimeofvehiclesonlinkais^xata(^xa).Ifanotherdriverdecidestousethislink,thetravelvolumeincreasesto^xa+1andthetotaltraveltimeofthevehiclesincreasesto(^xa+1)ta(^xa+1).Usingtherstdegreeapproximation,onecanreplaceta(^xa+1)withta(^xa)+dta(^xa)=d^xa.Then,thetotalincreaseintraveltimeofvehiclesonlinkacanbeapproximatedbyta(^xa+1)+^xadta(^xa)=d^xa,whichiscalledmarginaltraveltime.Onlyaportionofthisincreasethatisequaltota(^xa+1)isperceivedbythenewdriver,andtherest,whichisequalto^xadta(^xa)=d^xa,isimposedonotherdriversusinglinka.Thelatteriscalledtheexternalcostofatrip.Theprincipleofmarginalcostpricingstatesthatifdriversoneverylinkarechargedwithtolls,eachequaltheexternalcostoftravelonthelink,theresultingequilibriumwillreplicateSOlinkows.Inadditiontotraveltime,thesocialcostofatripalsoincludesenvironmentaleects,roadsurfacedeterioration,costofaccidents,etc.Onecandenemarginaltravelcostandexternalitiessuchthattheyincludeallthesecosts.However,sincethemajorcostoftravelisthetraveltime,thisdissertationdoesnotconsiderothercosts. 18

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2.2.2First-bestandSecond-bestCongestionPricingInthisdissertation,itisassumedthatODdemandsarexedand,exceptinChapter 6 ,theonlycostthattransportationsystemexperiencesistraveltime.Undertheseassumptions,marginalcostpricingisnottheonlyschemethatcanmaximizesocialwelfare,orequivalentlyminimizetotaltraveltime.Ingeneral,thereareinnitenumberoftollingschemesthatcanreplicateSOtracdistribution.Suchapricingschemeiscalledrst-best.Assumingthelinktraveltimefunctionsarestrictlyincreasing, Hearn&Ramana ( 1998 )showedthatthesetofallsuchtolls,calledtollset,canbedescribedbyasystemoflinearconstraints.Asaresult,rst-bestcongestionpricingproblemscanusuallybeformulatedaslinearprogramsthatarerelativelyeasytosolve.Whenrestrictionsonthetollingschemeprecludeachievingtheminimumtraveltime,thenextbeststrategythatmeetsthoserestrictioniscalledsecond-bestscheme.Theserestrictionsareusuallyintheformofrequiringsomelinkstobetoll-free,introducinganupperboundontheamountoftoll,equityconstraint,orenvironmentalconsiderations.Inadditiontothoserestrictions,userheterogeneityisalsoanissueforachievingmaximumsocialwelfare( Chu&Tsai , 2004 ).Section 2.3.2.1 discussesthedierenceinexternalitieswhenusersareheterogeneous.Sincetheequilibriumlinktraveltimesdependontheamountoftolls,second-bestpricingproblemsaretypicallymodeledasbi-levelprogramsormathematicalprogramswithequilibriumconstraints.Intheformer,thelowerlevelistondequilibriumlinktraveltimes,whiletheupperleveloptimizestheobjectiveofinterest( Yan&Lam ( 1996 )and Brotcorneetal. ( 2001 )).Forthelatter,theequilibriumconditionsareguaranteedbyeithercomplementarityconstraints( Lawphongpanich&Hearn , 2004 )orvariationalinequalities. 19

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2.3PriceDierentiationThisdissertationtriestomakeacaseforapplyingpricedierentiationincongestionpricing.Tothisend,werstreviewthefundamentalresearchesfromtheeldofeconomics,whichintroducedanddevelopedtheideaofpricedierentiation.Afterwards,wepresentsomeapplicationsofpricedierentiationintransportationandcongestionpricing. 2.3.1PriceDierentiationinMonopolisticMarketsMonopolyandperfectcompetitionarethetwoextremesofmarketstructures.Inmonopoly,onlyonermisoeringaproductanditdoesnothaveanyclose-substitute,whileinperfectcompetitionseveralrmsaresellingthesameproduct.Firmsunderperfectcompetitiondonothaveanymarketpower,i.e.ifarmtriestoselltheproductforhigherthanmarketprice,itwillloseallofitscustomers.Inmonopoly,ontheotherhand,rmhashighmarketpower,soitcanraisethepriceoftheproductandloseonlysomeofitscustomers,butnotallofthem( Varian , 1992 ).Moreover,monopolistscansellsameproductsfordierentprices.Thesituationwhereidenticalproductsaresoldfordierentpricesiscalledpricediscriminationordierentiation,whichisaneconomicconceptdenedby Dupuit ( 1894 ).Asnotedearlier,pricedierentiationisrelevantinmonopolymarkets,becauseotherwisethepricesaredeterminedbyequilibriumandsellerdoesnothavemarketpowertochangethem. Pigou ( 1932 )laterclassiedpricedierentiationintothreelevels,as: 1. First-degreeorperfectpricedierentiationisthecasewhereeveryonepayshisorhermaximumwillingness-to-payfortheproduct.Inotherwords,themonopolistsellseveryunitoftheproductforthemaximumpriceacustomeriswillingtopay. 2. Second-degreepricedierentiationstatesthepriceperunitofproductasafunctionofthequantitybeingpurchased.However,individualsbuyingthesameamountsofproductpaythesameprice,andthepricedoesnotdependonthecustomer. 20

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3. Third-degreepricedierentiationisthesituation,wherepriceoftheproductcanbedierentfordierentgroupsofusers.Third-degreedierentiationisthemostcommonandmostpracticalamongthethree.Pricediscriminationisnotuncommoninthetransportationmarket.Agoodexampleforsecond-degreediscriminationistransitfare,when,e.g.,atwo-wayticketischeaperthantwoone-waytickets,orthepriceofadailypassisindependentofthenumberofridestakenbyapassengerwithinoneday.Moreover,sometransitagenciesdierentiatetravelersbyageandcollectdierentfaresforkids,students,adultsandelderpeople,whichisanexampleofthird-degreediscrimination. 2.3.2PriceDierentiationinCongestionPricingPricedierentiationhasbeenappliedincongestionpricing,anditisknowntobeawayofdesigningmoreecientschemes. dePalma&Lindsey ( 2011 )suggestedthatoptimalcongestiontollsshouldbedierentiatedbasedonvehicletype,roadsection,timeofday,tracconditions,purposeofthetrip,andpricingofothermodesoftransportation.However,astheypointedout,itisnotpossibletodoso,duetopracticalissues.Others,e.g., Small&Yan ( 2001 ), Yang&Zhang ( 2002 ), Yang&Huang ( 2004 ),and Yin&Yang ( 2004 ),alsostudieddierentiatingusersbasedontheirvaluesoftraveltime.Aspointedoutby Pigou ( 1932 ),third-degreepricediscriminationgenerallyrequiresanabilitytodistinguishdierentcustomergroups,i.e.,theremustbesomeobservableattributesassociatedwitheachgroup,unlessthepricingschemepossessesaself-selectionmechanism.Giventhatthevalueoftimeisnotdirectlyobservable,itisnotsurprisingtondlittlepracticeofpricedierentiationwithrespecttothevalueoftime.Thestudiesondierentiatedpricingcanbecategorizedintothreegroups.Therstcategorycomprisedofschemeswhichdierentiateusersbasedontheamountofexternalcosttheyareimposingonotherdrivers.Thesecondgroupisthenonlinearpricingschemes,whereusersarediscriminatedbasedonhowmuchtheyusethenetwork.Theschemesinthelastgroupdierentiateusersbasedontheirvalueoftime. 21

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2.3.2.1BetterEstimateoftheExternalitiesTheprincipleofmarginalcostpricingrequiresuserstopaytheexternalcostoftheirtravelastoll.Theamountofexternalities,however,dependsonseveraluserandtripcharacteristics.Fromtheliterature,twogeneralresultsregardingrst-bestpricingschemecanbeconcluded.First,thetruemaximumsocialwelfareisonlyachievedwhentollsaresucientlydierentiated.Second,tollscanbethesame(undierentiated)foragroupofusersonlyiftheirexternalitiesarethesame( dePalma&Lindsey , 2004 ).Thecriteriausedforpricedierentiationwiththegoalofachievingabetterestimateofexternalitiescanbesummarizedasfollows: 1. Vehicleclass:Theexternalitiesofdierentclassesofvehicles,e.g.,passengercars,trucks,andbuses,areclearlynotequal.Thecongestioneectofoneextratruckcanbeequaltoseveralpassengercars.Becauseofthisphenomena,congestiontollfordierentclassesofvehiclesshouldbedierent. Holgun-Veras&Cetin ( 2009 )studiedthedesignofoptimalclass-dierentiatedtolls.Theyusedsimulationtondtheeectsofdierenttrucktypesontraccongestion. 2. Timeofdayortraccondition:Ifweconsidertravelatdierenttimesofdayassimilarproducts,thenchargingdierenttollsfordierenttimesofdayisessentiallyadierentiatedpricingscheme. 3. Trippurpose:Thetrueexternalcostsofatripcandependonthetravelpurpose,i.e.commutetoworkornon-worktrip.Congestionpricingmayhaveanadverseeectonthesocialwelfarethroughdistortionofthelabormarket.Dierentiationbasedontrippurposemaynotbepractical;insteadapricingschemewithtollrefundthroughemployercandierentiatedbetweenworktripsandnon-worktrips( Calthrop&Leuven , 2001 ). 4. Weight:Heavyvehiclestravelingonaroadgraduallydestroythepavement,whichimposesadditionalcoststothesociety.Theamountofthisdeteriorationiscloselyrelatedtotheweightofthevehicles( Conway&Walton , 2009 ).Dierentiatedpricingwithrespecttoweightofvehiclescanhelptochargeatollthatisamoreaccurateestimateofexternalities. 5. Speed:Vehiclesdrivingwithhighspeedordoingriskybehaviorsincreasetheriskofaccident,whichimposesadditionalcostsonotherdrivers.Asaresult,theyhavehigherexternalcosts,andshouldpayhighertolls( dePalma&Lindsey , 2011 ).Noticethatalloftheseschemescanbeconsideredasthird-degreepricedierentiation. 22

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2.3.2.2NonlinearPricingNonlinearcongestionpricingisamoreexibleversionofVMTfee. Wangetal. ( 2011 )and Lawphongpanich&Yin ( 2012 )investigatedthisproblem,whichisessentiallyaninstanceofsecond-degreedierentiation,wheretheamountoftolldependson,notstrictlyproportionalto,thedistancetraveledinsideatollingarea.Inpractice,severalcongestionpricingschemes,includingSingapore,London,andStockholm,chargeatollthatisnotproportionaltothedistancetraveledinsidethetolledarea.Forexample,inStockholmthereisamaximumamountoftollperday.Consequently,whenadriverpaysthemaximumtoll,itwillnotbechargedanymoreonthesameday. 2.3.2.3DierentiationBasedonValueofTimeThecostoftravelincludesthetraveltimewhichismeasuredinunitsoftime,andtollwhichismeasuredinmonetaryunits.Valueoftimeistheparameterthatconvertstimetomoney,sothatthetwocostscanbeaddedtogether.Hence,oneofthemostimportantparametersincongestionpricingisthevaluetravelersconsiderfortheirtime.Valueoftimeisnotauniquevaluebutadistribution.Itnotonlydependsonthetraveler'ssocialcharacteristics,e.g.income,butalsoonotherfactors,suchastrippurpose,duration,timeofday,andtracconditions( Calfee&Winston ( 1998 ), Wardman ( 2001 ),and dePalma&Lindsey ( 2004 )).Valueoftimeisnotanobservablecharacteristicofthetravelers,soitisnotpracticallypossibletodierentiateusersbasedonvalueoftime,unlessthereisaselfselectionmechanisminplace.Transportationservicesaresometimesoeredatdierentlevels,whereahigherlevelisassociatedwithhigherprice.Forexample,anexpresstolledlaneprovidesbetterservicetothosewhoarewillingtopayforthevalueofthisservice,whileotherscanuselessconvenientlanesforfree.Sincetheusershaveanoptiontopayforhighervalueservice,thisschemeiscalledvaluepricing. Small&Yan ( 2001 )showedthatthebenetsofvaluepricingareunderestimatedwhenuserdierentiationbasedonvalueoftimeisignored. 23

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Thetracdistributionthatminimizessocialcostbasedonmonetaryvaluecanbedierentthantheonethatminimizessocialcostbasedontimeunit.Inaddition,tollsaecttravelersdierentlybasedontheirvalueoftime.However, Arnott&Kraus ( 1998 )showedthatrst-bestanonymoustollsexist,evenwhenusersareheterogeneousintermsofvalueoftraveltime. 2.4ContributionsofThisDissertationPricedierentiationhasbeenwell-studiedintheliteratureofeconomics.Aspresentedabove,thisconcepthasbeenusedtodierentiatecongestiontollsbasedonseveralcriteria.However,dierentiationbasedontravelcharacteristicsremainstobeinvestigated.Therstcontributionofthisdissertationisrecognizingthisgapintheliteratureandexploringtheadvantagesanddisadvantagesofthesenewschemes.Sincethedierentiatedpricingschemeshavenotbeenstudiedbefore,thereislittleknownabouthowtodesignandimplementsuchtolls.Thesecondandthirdscontributionsofthisresearcharetotacklethesetwoproblems.Finally,werecognizedaprivacyconcernregardingimplementationofdierentiatedschemes,whichhasnotbeenstudiedinthecontextofcongestionpricing.Weproposedaninnovativemethodtoaddressthisconcern. 24

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CHAPTER3DIFFERENTIATEDCONGESTIONPRICINGSCHEMES 3.1OverviewPricedierentiationandsomeofitsapplicationsincongestionpricinghavebeenreviewedinSection 2.3.2 .Anewclassofcongestiontollsthatbelongstothethird-degreepricedierentiationisintroducedinthischapterandstudiedthroughtheentiredissertation.Theproposedschemesdierentiatetravelerswithrespecttotheirtravelcharacteristics,i.e.,origin,OD,orpath.Althoughsimilarschemesmayhavebeenimplementedinclosednetworks,e.g.,tolledfreeways,toourbestknowledge,ithasnotbeenexploredinanopen,urbannetworkenvironmentforthepurposeofcongestionmitigation.Wenotethattheadvancementsofvehicletrackingandidenticationtechnologieshavetechnicallyenabledsuchapricedierentiation.Aninnovativemethodofimplementingpath-dierentiatedschemesisproposedinChapter 5 .Thecontributionsofthischapteraretwofold.First,weusenumericalexamplestodemonstratethepotentialsofpricedierentiationwithrespecttotravelcharacteristics.Theexamplesshowthatinarst-bestnetworkcondition,whereallthelinksaretollable,dierentiatedpricingcansubstantiallyreducetravelers'nancialburden;inasecond-bestenvironmentwhereonlysomelinksaretollable,ithelpsachievealowerlevelofcongestion.Second,weformulateoptimizationmodelstodetermineoptimaldierentiatedpricingschemesforgeneralnetworks.Theremainderofthischapterisorganizedasfollows.Section 3.2 introducesthethreetypesofdierentiatedpricing.MathematicalformulationoftheseschemesfortwogeneraltypesofcongestionpricingsituationsarethenpresentedinSection 3.3 SomenumericalexamplesareprovidedinSection 3.4 toshowtheperformanceofdierentiatedpricingschemesandcomparethemtoundierentiatedcounterparts.Finally,Section 3.5 summarizesthechapter. 25

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3.2DierentiatedTollsDierentiatedpricingschemeswediscussinthischapterincludeorigin-specic,OD-specicandpath-specic.Astheirnamessuggest,travelersonthesamelinkwillbechargeddierently,withrespecttotheirorigin,ODpair,orpath.Intuitively,theseschemesaremoreexiblethantraditionalanonymoustolling.Mathematically,theycanbeviewedasdierentlevelsofrelaxationtoanonymousschemes.Tofacilitatethepresentation,welabelthedierentiationlevelofanonymouspricingaszero,andsubsequentlythelevelsofdierentiationfororigin-specic,OD-specicandpath-specicpricingasone,twoandthree,respectively. Table3-1. Illustrationofdierentiatedtolls PathL0L1L2L3 . . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 9292;394;8;9;11 . . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 9191;392;8;9;11 . . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 9191;492;8;9;12 . . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 9191;491;6;9;12 26

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DierentiatedtollsasshowninTable 3-1 arelink-based,i.e.,tollisspeciedforlinks.Asanalternativetopath-specictoll,itismoreconvenienttodenepath-basedtollthatisasinglevaluerepresentingtheamountoftollfortheentirepath.Path-basedtollsarealsocalledpath-dierentiatedorsimplypathtollsthroughoutthisdissertation. 3.3ModelFormulationsAsaforementioned,path-basedtollshavethehighestlevelofdierentiation,becausetheoriginordestinationofatripcanbeuniquelydeterminedfromthepathutilizedbythetrip.Hence,ageneralpath-basedformulationisusedinthischaptertodescribeallthreedierentiationschemes.Recallthatorigin-specicandOD-specicpricingarelink-basedschemes,andthusthetollofeachpathisthesumoftollsonlinkscomprisingthepath.Incontrast,path-basedtollsaredeterminedforspecicpaths. 3.3.1NotationLetG(N;A)denoteatransportationnetwork,whereNisthesetofnodesandAisthesetofdirectedlinks.Indexaisusedtodenotealink,whichisalsorepresentedbyitsendnodesi;j2N,i.e.,(i;j)=a.Forlinka,xaandaareitsaggregateowandtoll,respectively.Thelatterisexpressedintheunitoftimeforthesakeofsimplicity.LetWNNbethesetofODpairswithstrictlypositivedemand,wbetheindexofitselementsanddwbethedemandofODpairw.ForeveryODpairw2W,o(w)representsitsorigin.ThesetofallpathsconnectingODpairwisdenotedbyPwwithitselementsbeingindexedbyp.Abinaryparameterrepresentsthelink-pathincidence,i.e.,iflinkaisonpathp,thenpaisone;otherwisezero.Foreverypathp,fpandpdenoteitsowandtoll,respectively.Again,thetollisrepresentedintheunitoftime.Also,tp(:)andta(:)arethetraveltimeforpathpandlinka,respectively.Forsecond-bestpricing,thesetoftollablelinksisdenotedby,anditscomplementsetincludesalltheuntollablelinks. 27

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3.3.2First-bestConditionWerstdiscussarst-bestnetworkconditionwherealllinksaretollable.Insuchanenvironment,evenwiththelowestlevelofpricedierentiation,i.e.,anonymoustolling,congestionpricingisabletoinducesystemoptimumandreplicatesystemoptimumlinkows(e.g.,( Hearn&Ramana , 1998 ; Louetal. , 2010 )).Consequently,thebenetofpricedierentiationcanonlybereectedonasecondaryobjective.Inthischapter,wechooserevenueminimizationasthesecondaryobjectivebecauseitrepresentsanancialburdentothetravelingpublic.Below,weformulateaprogramforndingarst-bestpath-basedpricingschemetominimizethetotaltollrevenue: (MINR-L)minXw2WXp2Pwpfp (3{1a)s:t:Xp2Pwfp=dw8w2W (3{1b)fp(tp(f)+p)]TJ /F3 11.955 Tf 11.95 0 Td[(uw)=08p2Pw;w2W (3{1c)tp(f)+p)]TJ /F3 11.955 Tf 11.96 0 Td[(uw08p2Pw;w2W (3{1d)fp08p2Pw;w2W (3{1e)p08p2Pw;w2W (3{1f)Xw2WXp2Pwpafp=xa8a2A (3{1g)wherexaisthesystemoptimumlinkowonlinka.Intheabove,theobjectivefunctionistominimizetotaltollrevenue.Constraint( 3{1b )istoensureowconservation;Constraints( 3{1c )and( 3{1d )aretolleduserequilibriumconditions;Constraints( 3{1e )and( 3{1f )specifynon-negativepathowandtoll,andthelastconstraintrequireslinkowstoreplicatesystemoptimumlinkows. 28

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Theaboveformulationcanbeeasilymodiedfortheothertwodierentiatedschemes.Inorigin-specicandOD-specicschemes,tollsareimposedonlinks,butcanbedierentfordierentoriginsorODpairs.Inourformulation,weassociateasuperscripttotollvariables,,todierentiatetolls.Subsequently,addingthefollowingconstraintstotheabovemodelyieldsaformulationfororigin-specicpricing:p=Xa2Apao(w)a8p2Pw;w2W (3{2)o(w)a08a2A;w2W (3{3)whereo(w)aisthetollamountforlinkaforusersfromoriginofw,o(w).Similarly,theformulationforOD-specicpricingcanbeobtainedbyaddingthefollowingconstraints:p=Xa2Apawa8p2Pw;w2Wwa08a2A;w2WwherewaisthetollonlinkaforusersofOD-pairw.ItisworthmentioningthatConstraint( 3{1f )becomesredundantandcanberemovedfromtheformulationsforbothorigin-andOD-specicschemes. 3.3.3Second-bestConditionWenowconsiderasecond-bestnetworkconditionwherenotallthelinksaretollable.Inthiscase,anonymoustollingmaynotinducesystemoptimumandthuspricedierentiationprovidesadditionalexibilitytofurtherreducesystemtraveltime.Belowwepresentaformulationtoobtainasecond-bestorigin-specicpricingschemethatminimizessystemtraveltime: 29

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(MINR-O)minXw2WXp2Pwtp(f)fp (3{4a)s:t:( 3{1b ),( 3{1c ),( 3{1d ),( 3{1e ),( 3{2 ),( 3{3 )o(w)a=08w2W;a2 (3{4b)TheOD-specicformulationcanbedevelopedsimilarly.Noticethatbecausepath-basedpricingdoesnotimposetollsonlinks,itbecomesirrelevantinthesecond-bestnetworkcondition,whichisconsideredinthischaptertobethesituationwherenotalllinksaretollable.However,aconditionsimilartosecond-bestisdenedforpath-basedtollsinChapter 5 .Comparingtheabovewiththerst-bestformulations,Equation( 3{1g )isnolongerincludedbecausesystemoptimumlinkowsmaynotbeachievable.Inaddition,becauseonlyspeciclinkscanbetolled,Constraint( 3{4b )isadded.Wefurthernotethatlink-basedformulationsfortheorigin-specicandOD-specicschemesexist,butwedonotpresentthemtokeepthediscussionconcise.BecauseofConstraints( 3{1c )-( 3{1e ),alloftheformulationspresentedabovebelongtotheclassofmathematicalprogramswithcomplementarityconstraints(MPCC).Theseproblemsarenon-convexandstandardstationaryconditions,i.e.,KKTconditions,maynotholdforthem,becausetheydonotsatisfyMagasarian-Fromovitzconstraintqualication( Scheel&Scholtes , 2000 ).ManysolutionalgorithmshavebeenproposedforMPCC(see,e.g. Luo ( 1996 )andreferencescitedtherein).However,someonlyworkwellforsmallandmediumproblemswhileothers,especiallythosebasedonsolvingequivalentnonlinearprograms(e.g., Lawphongpanich&Yin ( 2010 )),canhandlelargerproblems.Moreeectivealgorithmsmaybedevelopedtosolvetheaboveformulationsbyexploringspecialpropertiesorstructuresthattheymaypossess. 30

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. . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . (5,12) . (6,18) . (3,35) . (9,35) . (2,11) . (8,26) . (6,33) . (7,32) . (4,26) . (8,30) . (3,25) . (6,24) . (8,39) . (6,43) Figure3-1. Nine-nodenetwork Inthischapter,weapproximatecomplementarityconstraintstobenonlinearinequalityconstraintsandsolveaMPCCasanonlinearprogram( Scheel&Scholtes , 2000 ).Sincethenonlinearprogramisnon-convex,wesolveitwithmultipleinitialsolutionsandpresentthebest-obtainedsolution.Somemathematicalmodelsfordesigningpath-basedtollsarepresentedinnextchapterthatareeasiertosolve. 3.4IllustrativeExamplesWenowdemonstratethepotentialsofdierentiatedpricingschemesonanine-nodenetwork.Figure 3-1 showsthenetworkwithfourODpairs[1;3],[1;4],[2;3],and[2;4],whosedemandsare10,20,30and40,respectively.Thelinkperformancefunctionsareofthefollowingform:ta(xa)=Ta 1+0:15xa ba4!whereTaandbaareprovidedinFigure 3-1 as(Ta;ba)neareachlink. Table3-2. Dierentiatedpricingfornine-nodenetworkwithalllinkstollable TollingschemeRevenueamountRevenuereductionu[1;3]u[1;4]u[2;3]u[2;4] Userequilibrium--24.923.824.325.1Anonymous887.60%30.629.233.031.6Origin-specic311.665%23.429.325.824.4OD-specic295.667%23.422.025.827.6Path-based263.670%23.422.029.024.4 31

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. . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20 . 21 . 22 . 23 . 24 Figure3-2. SiouxFallsnetwork 32

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Table 3-2 presentstheresults1ofdierentlevelsofdierentiationwhenalllinksaretollable.Thesecondandthirdcolumnsshowtheminimumtollrevenueofeachscheme,andthepercentreductionascomparedtotheanonymousscheme.Itcanbeobservedthatthetollrevenuesforalldierentiatedschemesaresubstantiallylowerthanthatofanonymouspricing.Moreover,asthelevelofdierentiationincreases,therevenuedecreases.Particularly,pricedierentiationwithrespecttopathyieldsa70%reductioninrevenue.ThelastfourcolumnspresenttheequilibriumtravelcostforeachODpair.Observethatthetravelcostsunderdierentiatedschemesarelessthanthoseundertheanonymousschemewithonlyoneexception,suggestingthatdierentiatedpricingmaybemoreappealingtoindividualtravelersinthisnetwork. Table3-3. Second-bestdierentiatedpricingfornine-nodenetwork TollingschemeTotaltraveltimeTraveltimesavingu[1;3]u[1;4]u[2;3]u[2;4] Userequilibrium2455.90%24.923.824.325.1Anonymous2361.246.9%25.824.925.125.9Origin-specic2306.174.2%24.324.227.125.7OD-specic2281.786.2%24.422.926.825.3Systemoptimum2253.9100%---Table 3-3 presentstheresultsofsolvingsecond-bestdierentiatedpricingforthenine-nodenetwork,whenonlylinks(5,7)and(7,3)aretollable.Inthistable,thesecondcolumnshowsthetotalsystemtraveltimeundereachtollingscheme.Knowingthatsystemoptimumyieldsthesmallestsystemtraveltime,wepresentthethirdcolumnastheratiobetweentraveltimereductionfromuserequilibriumandthemaximumpossiblereduction,i.e.,thedierenceintraveltimesofuserequilibriumandsystemoptimum.Itisevidentthatpricedierentiationleadstoadditionaltraveltimereduction.Specically,evenwithonlytwolinksbeingtollable,theOD-specictollingschemeachieves86.2%of 1Resultsarethebest-obtainedones,butlikelylocaloptima.Thisnoteappliestoothertablesinthischapter. 33

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themaximumpossiblereduction,areductionachievedbyarst-bestpricingschemethatmaytollalllinks.WealsosolvedfordierentiatedschemesontheSiouxFallsnetworkfrom( LeBlancetal. , 1975 )asshowninFigure 3-2 andtheresultsarepresentedinTables 3-4 and 3-5 .Forsecond-bestpricing,onlythedashedlinksinFigure 3-2 areassumedtobetollable.Table 3-4 showsthatrst-bestdierentiatedpricingyieldsasubstantialreductionintollrevenue,whileminimizingsystemtraveltime.Usingsystemoptimumasthebenchmark,theadditionaltraveltimeunderuserequilibriumis2.859.Similarly,asshowninTable 3-5 thesecond-bestpricingschemesoerpromisingresults.Forexample,origin-specicschemecanreducetheadditionaltimeto1.117,whichisequivalenttoa60.93%reduction.Comparedtoanonymoustolling,origin-specictollscanachieveapproximatelytwicetraveltimereduction. Table3-4. First-bestdierentiatedpricingforSiouxFallsnetwork TollingschemeMinimumrevenueRevenuereductionpercentage Anonymous20.6660.00%Origin-specic0.75096.37%OD-specic0.61697.02%Path-based0.18299.12% Table3-5. Second-bestdierentiatedpricingforSiouxFallsnetwork TollingschemeTraveltimeSavingpercentage UserEquilibrium74.8020.00%Anonymous74.04326.55%Origin-specic73.06060.93%OD-specic72.99763.13%SystemOptimum71.943100.00% Ingeneral,forboththerst-bestandsecond-bestconditions,higherlevelsofdierentiationleadtomorefavorableresults.Ontheotherhand,dierentiatedpricingschemesaremorediculttoimplement.Oneneedstoconsidersuchatrade-otodeterminewhetherahigherlevelofdierentiationisworthimplementingornotonaparticularnetwork. 34

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3.5SummaryThischapterexploredanewclassoftollingschemesthatchargeusersfromdierentorigins,ODs,orpathswithdierentamountsoftoll.Theseschemesprovidemoreexibilitythantraditionalanonymouspricing.Thenumericalexamplesinthischapterdemonstratedthattheadditionalexibilitycanhelptoreducethenancialburdenonmotoristsinarst-bestnetworkconditionorleadtomoretraveltimesavinginasecond-bestcondition.Thenextchapterprovidesamoredetailedstudyofpath-dierentiatedschemes,andproposesmethodstosolvethetolldesignproblem. 35

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CHAPTER4DESIGNINGPATH-DIFFERENTIATEDCONGESTIONPRICINGSCHEMES 4.1OverviewPreviously,wearguedthatdesigninganoptimaldierentiatedcongestionpricingschemecanbechallenging,becauseofthemodelstructure.Thischapterfocusesondesigningpath-dierentiatedschemes,asaninstanceofdierentiatedtolls.Weinvestigatethepropertiesandstructureoftheproblem,andusethemtodevelopmodelformulationsthatareeasiertosolve.Acongestionpricingschemecanbedesignedfordierentgoals.Asoneofthemostgeneralobjectives,weattempttodesignpathtollsthatleadtoadesiredtargetlinkowdistribution,e.g.,onethatminimizestotaldelay.Apathtollingschemeisvalidifitinducesordecentralizes1atargetlink-owdistribution.Thesetofvalidtollingschemesaretypicallyinniteandasecondaryobjectiveisoftenusedtoselectoneforimplementation.Tolessenthenancialburdenonmotorists,thesecondaryobjectiveweselectistominimizetollrevenue.(See Hearn&Ramana ( 1998 ),forothersecondaryobjectives.)Asshowninpreviouschapter,whencomparedtolinktolls(i.e.,tollschargedforusageofindividualroadsorlinks)pathtollsaremoreexiblebecausepathsaremorenumerousthanlinks.Addingtogetherlinktollsonagivenpathalsoyieldsatollforthepath.So,avalidlink-basedschemealwaysinducesasetofvalidpathtolls.Theconverseisgenerallyfalse.Beingmoreexible,pathtollscanbedesignedtoimposelessnancialburdenonmotoristsinthattheypermitlessrevenuetobecollectedthanlinktolls,whilereplicatingthesametargetlink-ow.Weviewthereductionintollrevenuethatpathtollsoerasawayofappraisingthemonetaryvalueoftheinformationconcerningtravelroutes.Performingthisappraisalrequiresndinglinkandpathtolls 1Thistermismorecommonineconomicliterature. 36

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thatgeneratetheleastrevenue.Wheneverylinkistollableandalltollsarenonnegative, Hearn&Ramana ( 1998 )showthatndingvalidlinktollswiththeminimumrevenuecanbeformulatedasalinearprogram,aproblemrelativelyeasytosolve.Findingpathtollswithminimumrevenue,ontheotherhand,ismoredicultandcanbeformulatedasmathematicalprogramwithcomplementarityorequilibriumconstraints.However,alternativeformulationsexistandsomearemoreadvantageouscomputationally.Fortheremainder,Section 4.2 describesournotationanddenesowdistributionssuchasuserequilibrium(UE)forlaterreferences.Section 4.3 illustratespathtollsandtheirbenetswithsmallexamples.Section 4.4 denestollpricingproblemswhoseobjectivesaretominimizerevenuecollected.Section 4.5 exploresalternativeformulationsforthepath-dierentiatedpricingproblemintroducedintheprevioussection.Section 4.6 presentsanddiscussesresultsfromourexperimentsusingdatafromtheroadnetworksinAnaheim,CaliforniaandSiouxFalls,SouthDakota.Thissectionshowsthatthereductionsintollrevenuesforthesenetworksaresubstantial.Finally,Section 4.7 concludesthechapterbysummarizingourndings. 4.2BackgroundThissectionintroducesournotationanddenesowdistributionssuchasUEwithandwithouttolls.Afterward,apropertyconcerningpathtollsisdiscussed.LetG(N;A)denoteanetworkofroads,whereNandAarethesetsofnodesandlinks,respectively.Foreachlinka2A,xarepresentsitsowandta()isitstraveltimefunction.Wealsoletxandt()bevectorsofxaandta(),respectively.(Inthischapter,alower-caseletterwithoutasubscriptrepresentsavectorofvariablesdenotedbythesameletterwithasubscript,e.g.,dandarevectorsofdwandp,respectively.Double-strucklettersinupper-caserepresentsets.)Additionally,Wdenotesthesetofallorigin-destination(OD)pairs.Forw2W,dwrepresentsitstraveldemandandPwisthesetofsimplepaths(i.e.,oneswithoutanycycle)fromtheoriginofwtoitsdestination.WeassumethatPw6=,8w2WtoensurefeasibilityandletP=Sw2WPwdenotethe 37

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setofsimplepathsinthenetwork.Forp2P,fpandpdenoteitsowandtollamount,respectively.Toavoidintroducingafactorforvalueoftimeandsimplifyourpresentation,tollsareinunitsoftime.Usingtheabovenotation,thesetofallfeasiblepath-owvectorsordistributions,denotedasF,canbedenedasfollows:F=(f2RjPj+Xp2Pwfp=dw;8w2W)Intheaboveexpression,jPjisthecardinalityofPandtheplussign(+)indicatesthateverycomponentoff(i.e.,fp)isnonnegative.Theequationenclosedbythebracketsensuresthatthedemandforwismet.TodeneUEdistribution,letpaequaloneiflinkaisonpathp.Otherwise,pa=0.Thetraveltimeassociatedwithpathpiscp(f)=Pa2Apata(x),wherexisavectorofxa=Pp2Ppafp.Then,^fisaUEpathdistributionif^fsolvesthevariationalinequalityc(^f)>(f)]TJ /F1 11.955 Tf 14.69 3.15 Td[(^f)0;8f2F.Similarly,^xisaUElinkdistributionifitsolvesthevariationalinequalityt(^x)>(x)]TJ /F1 11.955 Tf 12.71 0 Td[(^x)0;8x2X,whereX=nxx=Pp2Ppafp;f2Fo.Henceforth,wealsorefertothetwovariationalinequalitiesmorecompactlyasVI[c;F]andVI[t;X].Asdened,^fsatisesWardropsrstprincipleofequilibrium(see,Wardrop,1952),anotionthatisrelatedtoequilibriumsolutionsinnon-atomiccongestiongames(Coleetal.,2006)andNashequilibrium( Nash ( 1951 )and Roughgarden ( 2005 )).Whent(x)isstrictlymonotone(i.e.,when(t(x1))]TJ /F3 11.955 Tf 11.95 0 Td[(t(x2))>(x1)]TJ /F3 11.955 Tf 12.03 0 Td[(x2)>0;8x16=x2),theUElinkdistribution^xisunique(see,e.g., Facchinei&Pang ( 2003 )).Ingeneral,thesameisnottruewithUEpathdistributions.TherearetypicallymanysolutionstoVI[c;F].However,thestrictmonotonicityassumptionimpliesthatallUEpathdistributionsmustyieldthesameUElinkdistribution.Mathematically,ift(x)isstrictlymonotoneandf1andf2solveVI[c;F],thenthelinkdistributionsassociatedwithf1andf2mustbethesameastheUElinkdistributionor^x=x1=x2,wherex1a=Pp2Ppaf1pandx2a=Pp2Ppaf2p. 38

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Whenpath(p)orlink(a)tollsarepresent,fandxdenotethetolledUEdistributions,i.e.,theysolveVI[c+;F]andVI[t+;X],respectively.Whent(x)isstrictlymonotone,thefollowinginequalitiesdemonstratethats(x)=t(x)+isalsostrictlymonotone:(s(x1))]TJ /F3 11.955 Tf 11.96 0 Td[(s(x2))>(x1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2)=(t(x1)+)]TJ /F3 11.955 Tf 11.95 0 Td[(t(x2))]TJ /F3 11.955 Tf 11.95 0 Td[()>(x1)]TJ /F3 11.955 Tf 11.95 0 Td[(x2)=(t(x1))]TJ /F3 11.955 Tf 11.95 0 Td[(t(x2))>(x1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2)>0Thus,xisalsounique.Asbefore,fistypicallynot.Whenf1andf2solveVI[c+;F],thecorrespondinglinkdistributionsmustbethesame,i.e.,x1=x2,wherex1=Pa2Ppaf1pandx2=Pa2Ppaf2p.Italsofollowsfromtheuniquenessthatx=x1=x2.Theabovediscussionisformalizedinthefollowingtheoremwhoseproofisstraightforwardandfollowsdirectlyfromthestrictmonotonicityofs(:). Theorem4.1. Letf1andf2solveVI[c+;F].Ift(:)isstrictlymonotone,thenx1=x2,wherexia=Pp2Ppafip;fori=1;2. 4.3IllustrationofPath-basedPricingThissectionpresentstwoexamplesthatillustratethebenetsofpath-basedtollsovertheiranonymouscounterparts.TherstexampleisasingleODpairnetworkwhichshowsthepathtollsaremoreexibleevenwhenthereisonlyoneODpair.Thesecondonedemonstratesthatpathtollscaninduceatargetlinkowdistributionwithoutcollectinganytollbymakingsomepathstoocostlytouse.Intheliterature, Abrams&Hagstrom ( 2006 )addresstheproblemofreducingcongestionbyclosingroads(orequivalently,imposinginnitetolls)anddoingsogeneratesnotollrevenue. Knockaertetal. ( 2010 )alsodiscussbarringmotoristsfromusingtheroadinVickerysbottleneckmodelinsteadofcollectingtolls. Songetal. ( 2013 )mitigatecongestionviaroad-spacerationing,anapproachthatcollectsnotoll. 39

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Considerthefour-nodenetworkinFigure 4-1 .Nodesarelabeledaso,a,b,andd.Linksarenumberedfrom1to7asindicatedbythesubscriptofthelinkperformancefunctionadjacenttoeachlink.ThereisonlyoneODpair,(o;d),anditsdemandisve. . . o . a . b . d . t1(x1)=2+0:33x1 . t2(x2)=1+0:33x2 . t3(x3)=1+0:33x3 . t4(x4)=7 . t5(x5)=4 . t6(x6)=6 . t7(x7)=1+x7 Figure4-1. Fournodenetwork Table4-1. MINREVlinktollsforthefour-nodenetwork(revenue=10) Linkxata(xa)mra 1331232133214170514061607121 Table 4-1 providesthetargetlinkdistributionxandtheminimum-revenue(MINREV)linktolls,mr,thatgeneratearevenueof10units.Suchlinktollscanbedeterminedby,e.g.,solvingalinearprograminthenextsection.Fortheabovefour-nodenetwork,mrainthetablealsoequalsxadsa(xa)=dxaorthePigouviantaxforarca.ItisalsoeasytoverifythevalidityofthelinktollsinTable 4-1 becausethegeneralizedcost(timeplustolls)ofeverypossiblepathinthenetworkis10(seeTable 4-2 ).Thus,thelinktollsdonotprovideuserswithanyincentivetoswitchtoroutesthatdonotsupportx.Table 4-3 givesMINREVpathtollswheretherstvepathsareutilizedandtheremainingthreearenot.WhilethereareseveralpathdistributionsthatinduceorsupportxinTable 4-1 ,thedistributionwiththeleasttollrevenueislistedinthesecondcolumnofTable 4-3 .Assigningoneunitofowtotherstvepathsandzeroowtotheremaining 40

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Table4-2. PathtollsinducedbyMINREVlinktollsinTable4-2 PathLinksinpathcppcp+p 11!2!3731025!2!3821034!3911041!2!7731051!6911065!2!7821074!7911085!610010 threepathssatisesdemandandsupportsx.TheMINREVpathtollsareinthecolumnlabeledmr.(Subsequentsectionsdiscussalgorithmsforcomputingmr.)Fromthelastcolumn,thegeneralizedcostofeveryutilizedpathisthesame(9units)andnolargerthanthosenotutilized.Thus,mrisvalidand,asshowninthelastrowofTable 4-3 ,generatesonly5unitsofrevenue.So,pathtollsinthisexamplereducetherevenueby50%(from10unitsto5)whencomparedtolinktolls. Table4-3. MINREVpathtollsforthefour-nodenetwork(revenue=5) PathLinksinpathfpcpmrp9 11!2!3172925!2!3181934!3190941!2!7172951!6190965!2!7081974!7090985!6010010 Theaboveexampleillustratesthefollowingcharacteristicsaboutlinkandpathtollswhentheproblemhasonlyoneoriginor,equivalently,onedestination. Dial(1999)considersthesubnetworkinducesbythearcsetA+=f(i;j)2Ajxij>0g.Inwords,thisisthesubnetworkofow-bearingarcs.Tominimizerevenue,heobservesthatthelongestpathbetweenoriginanddestinationinthissubnetworkmusthavezerotoll.HismethodfordeterminingtheMINREVlinktollsmakesthegeneralizedcostofeverypath,utilizedornot,inthissubnetworkequaltothelengthofthelongestone.(FromTable 4-2 ,path8isthelongestpathwithalength(c8)of10.)Thus,therearetollsoneverypathexceptthelongest.Usingmr,thetollsoneachoftherstsevenpathsinTable 4-2 varyfrom1to3andequaltothedierence 41

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betweenitslengthandthatofthelongest.Beingthelongest,thetollonpath8iszero.(Whentherearemultipleoriginsanddestinations,thelongestpathsinthesubnetworkmayhavetolls.) Whenusinglinktolls,makingthegeneralizedcostofeverypathintheabovesubnetwork,A+,thesameisnecessarybecausetherearemanypathdistributionsthatsupportthetargetlinkdistributioninTable 4-1 .InadditiontotheoneinTable 4-3 ,pathdistributions(f1;f2;f3;f4;f5;f6;f7;f8)=(3;0;0;0;0;0;1;1)and(2;1;0;0;1;0;1;0)alsoyieldthetargetlinkdistribution.Withrespecttothesubnetwork,linktollsensurethatallpathdistributionsthatsupportxareintolleduserequilibriumbymakingthegeneralizedcostsofeverypathinTable 4-2 thesame.Ontheotherhand,pathtollsselectonlyonepathdistribution(i.e.,theoneyieldstheleastrevenue)andmakeitsatisfythetolledUEconditions. Tominimizerevenue,pathtollsuseasimilarstrategyas Dial ( 1999 ),butonlyonutilizedpaths.Givenapathdistribution,pathtollsinTable 4-3 aredeterminedsothatthegeneralizedcostofeveryutilizedpathequalstothelengthofthelongestutilizedpath.(ThisobservationalsoappliestoeachODindividuallywhentherearemultipleODpairs.)Toguaranteetolleduserequilibrium,placingasucientlylargetolloneverynon-utilizedpathmakeseachoneunattractivewithoutgeneratinganyrevenue.InTable 4-3 ,itsucestoplacetwounitsoftollonpaths6,7and8.Becausethelengthofthelongestpathamongallpossiblepathsisgenerallylongerthantheoneamongonlytheutilizedones,pathtollsneededforeachofthelatteraregenerallysmallerandthusgeneratelessrevenuethanlinktolls.Forthefour-nodenetwork,therevenueassociatedwithmrishalfofthatassociatedwithmr.Inasubsequentsection,weusetheaboveobservationinaformulationforpricingpathtolls.Intermsofpolicies,makingthelongestutilizedpathstoll-freeseemsreasonablebecauseusersonthesepathsareworse-othanthoseonshorterpathsandshouldnotbefurtherpenalizedbyhavingtopaytollsaswell.Thispolicyisnotguaranteedwithlinktollsinmultipleoriginanddestinationsettings.Thenextexampleillustratesacaseinwhichpathtollsinducethetargetdistributionatnocostorwithoutcollectinganyrevenuewhilelinktollsareunabletodoso.Considerthethree-nodenetworkinFigure 4-2 withtwoODpairs,(a;c)and(b;c)whosedemandsare2and1,respectively.Thenetworkattributesareasshowninthegure.Table 4-4 providesthetargetdistributionxandMINREVlinktollsmrwitharevenueofoneunit. 42

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. . a . b . c . s1(x1)=4 . s2(x2)=1 . s3(x3)=1+x3 . s4(x4)=3 Figure4-2. Threenodenetwork Table4-4. MINREVlinktollsforthethree-nodenetwork(revenue=1) Linkxata(xa)mra 1140211031214130 Forpathtolls,therearethreepossiblepathsforODpair(a;c)andtwofor(b;c)seeTable 4-5 below.Thepathdistributionfthatsupportsxisinthefourthcolumnofthetable.ForODpair(a;c),onlythersttwopathsareutilized.Itfollowsfromthediscussionabovethatnopathtollisnecessarybecausebothhaveequallength.Theremainingpath,Path3,isnotutilizedandisshorterthantheothertwopaths.ToensuretolleduserequilibriumforODpair(a;c),chargingtwounitsoftollonPath3suces.Doingsogeneratesnorevenue.Thereisonlyoneutilizedpath(Path4)forODpair(b;c).InTable 4-5 ,notollisnecessaryforPath5becauseitisalreadylongerthantheutilizedone(Path4).Thus,theMINREVlinktolls(mr)inTable 4-4 generateoneunitofrevenuewhilemrinTable 4-5 generatesnone,i.e.,mrimposesnonancialburdenontheuserswhileinducingthetargetdistribution.Ingeneral,ifasupportingpathdistributionrequiresonlyoneutilizedpathforeveryODpair,thentheMINREVpathtollsgeneratenorevenue.Asshownlater,apathdistributionthatsupportsthetargetlinkdistributionwithleastdelayfortheSiouxFallsnetworkrequiresonlyoneutilizedpathforapproximately63%ofits528ODpairs.Thisleadstoa95%reductionintollrevenuewhenusingpathtolls.Also,whenthereisonlyonepathconnectinganODpairintheroadnetwork,thepathmustbeutilizedanda 43

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reasonablepolicywouldbetonotchargeanytollonthepath,particularlywhen,e.g.,usersfortheODpairhavenootheralternative(suchaspublictransit)totraveltowork.(Notethatthisisaconsequenceofassumingthattraveldemandsontheroadnetworkarexed.)Fromtheabovediscussion,pathtollssupportthispolicywhilelinktollsmaynot.Therefore,whiletheyrequireprivateinformation,pathtolls(a)generallyimposelessnancialburdenonusers,(b)alwaysmakethelongestutilizedpathstollfreeand(c)chargenotolltousersofanODpairwhichhasonlyonepathinthenetwork.ItisimportanttoemphasizethattheaboveexamplesandsubsequentsectionsassumethattraveldemandsarexedandWardropsequilibrium( Wardrop , 1952 )applies.Whentraveldemandsareelastic, Hearn&Yildirim ( 2002 )and Yin&Lawphongpanich ( 2009 )showthatallvalidlinkandpathtollsmustgeneratethesametollrevenue.Thus,whendemandsareelastic,ndingvalidtollswiththeleasttollrevenueisnotmeaningful.(However,othersecondaryobjectivessuchasminimizingthenumberofrequiredtollfacilitiesmaystillbemeaningful.)Also,Wardropsequilibriumassumesthatmotoristsareperfectlyrationalandalwaysswitchtoshorterorcheaperroutesregardlessofthesaving.Inpractice,motoristsareboundedlyrational(see,e.g., Conlisk ( 1996 ), Mahmassani&Chang ( 1987 ),and Louetal. ( 2010 ))andwouldswitchtoshorterorcheaperroutesonlywhendoingsoyieldsignicantsavings.Thus,thenumberofODpairswithonlyoneutilizedpathinpracticeisprobablylessthanthoseundertheperfectrationalityassumptionorWardropsequilibrium. Table4-5. Minimumrevenuepath-dierentiatedtollsforthethree-nodenetwork(revenue=0) ODPathLinksinpathfpcp~pcp+~p (a,c)111505(a,c)22!41505(a,c)32!30426(b,c)431303(b,c)540404 44

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4.4MinimumRevenueTollPricingProblemInthischapter,tollpricingreferstodecidinghowmuchtolltochargeoneachpathorlink.Atypicalprimaryobjectiveintollpricingistoinduceatargetlinkdistributiondenotedasx.Inthischapter,tollsarevalidif,foreveryODpair,theutilizedpathshavethesamegeneralizedcost(i.e.,traveltimeplustolls)thatislessthanorequaltothosenotutilizedand,whenaggregated,utilizedpathsyieldthetargetlinkdistribution.Usingournotation,linktollvectorisvalidifxsolvesVI[t+;X].Similarly,apathtollvector,,isvalidiffsolvesVI[c+;F]andx=P(p2P)pafp.Ingeneral,thenumberofvalidtollsisinniteandtherearemany(secondary)criteriawithwhichtoselectsuitabletollsforimplementation(see,e.g., Hearn&Ramana ( 1998 )).Tominimizethenancialburdenonmotorists,ourchoiceistondvalidtollsthatgeneratetheleastrevenue(see,e.g., Hearn&Ramana ( 1998 ),forothersecondarycriteriaandgeneralizations,someofwhichcanbeeasilyextendedtopathtolls.Forexample,itisstraightforwardtoallowpathtollstobenegativewhileensuringthattollrevenueequalstotalsubsidy.Negativetollscanbeviewedeitherasrebatesorsubsidies.).Itiswellknownthattheminimum-revenuepricingproblemwithlinktollsorlinktollpricingproblemisalinearprogram(see,e.g., Hearn&Ramana ( 1998 ))thatcanbesolvedecientlyusingthesimplexalgorithm(see,e.g., Bazaraaetal. ( 2011 )).However,thepath-dierentiatedpricingproblemhasnotbeenfullyexplored.Ourliteraturesurveydiscoveredonlyonejournalarticle( Dafermos , 1973 )withasubstantivetreatmentofpathtolls.Inthepaper, Dafermos oersasetoffeasiblelinktollsinvolvingthefamiliarmarginalcost(see,e.g., Arnott&Small ( 1994 ), Lindsey&Verhoef ( 2000 ),and Yin&Lawphongpanich ( 2009 ))andobtainsfeasiblepathtollsinamultipleuser-classsettingbyaddingupthetollsonlinksalongeachpath.Below,thediscussionregardinglinktollpricingproblemispresentedforcompletenessanditissimilartothatin Hearn&Ramana ( 1998 ).Asshowninasubsequentsection,the 45

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path-tollpricingproblemisatopicofinterestinitsownrightandchallengingtosolve,particularlywhencomparedtothelinktollcounterpart. 4.4.1Link-basedPricingLetxbethetargetlinkowdistribution,wherexa=Pp2Ppafpforsomef2F.In Hearn&Ramana ( 1998 ),theauthorsformulatethepricingproblemasfollows: (MINREV-L)min>x (4{1a)s:t: (4{1b)t(x)+A>w8w2W (4{1c)(t(x)+)>x=Xw2WdwE>ww (4{1d)0 (4{1e)In MINREV-L ,thedecisionvariablesareandwandtheobjectivefunctioncomputesthetollrevenueinformofthedotproductbetweenthetollvector()andthetargetdistribution.ThersttwosetsofconstraintsensurethatxsatisestheKKTconditionsassociatedwithV[t+;X],whereXisequivalentlywrittenasX=fxjAxw=Ewdw;xw0;8xw0;8w2Wg.Intheprecedingexpression,Aisthenode-arcincidencematrixoftheroadnetworkandEwisavectorwithtwonon-zeroelementswheretheelementcorrespondingtotheoriginofwequals1andtheonecorrespondingtothedestinationequals-1.Inthismodel,wisavectorofnodepotentials(orKKTmultipliers)forODpairw2W.Inshort,theseKKTconditionsensurethatisvalid.Thelastsetofconstraintsforcestobenonnegative,i.e.,subsidyisnotallowed.Foragiventargetdistributionx, MINREV-L maybeinfeasible,becausenononnegativetollcansupportx.Intheliterature,many( Fleischeretal. ( 2004 ), Yang&Huang ( 2004 ), Baietal. ( 2006 ), Yin&Lawphongpanich ( 2006 ), Guo&Yang ( 2009 ), 46

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Hagstrom&Abrams ( 2010 ),and Chen&Yang ( 2012 ))haveestablishednecessaryandsucientconditionsunderwhich MINREV-L isfeasible. 4.4.2Path-basedPricingRecallfromabovethatforeverylinka,xa=Pp2Ppafpforsomef2F.Thus,F,thesetofallfeasiblepath-owvectorsthatsupportxdenedbelow,isnonempty.Giventhetargetdistributionx,thesetofallfeasiblepath-owvectorsthatsupportx,isdenedasfollows:F=(f2FXp2Ppafp=xa;8a2A)Tondvalidpathtollsthatyieldtheminimumtollrevenue,therearethreesetsofdecisionvariablesforeveryODpairw.Therstset(fp;p2Pw)determineswhichpathtouseandbyhowmanymotorists,thesecond(p;p2Pw)setstollpricesonpathsandthethird,w,representstheminimumgeneralizedcostfortheODpair.Theformulationbelowusescomplementarityconstraints(seethesecondsetofconstraints)tomakethegeneralizedcostofeveryutilizedpathequaltheminimum,w. (MINREV-P)min>f (4{2a)s:t:f2F (4{2b)cp+pw8p2Pw;w2W (4{2c)fp(cp+p)]TJ /F3 11.955 Tf 11.96 0 Td[(w)=08p2Pw;w2W (4{2d)p;w08p2Pw;w2W (4{2e)In MINREV-P ,theobjectiveistominimizethetollrevenuewrittenastheproductofpathtollsandows.Therstconstraintrequireftosupportx.Thesecondsetensuresthatallgeneralizedcostsarenolessthanw,wherecp=Pa2Apata(x)isthetraveltimeforpathpevaluatedatthetargetdistribution.Whenfp>0,thethirdsetofconstraints 47

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ensuresthatcp+p=w.Thus,ifpathpisutilized,itsgeneralizedcost,cp+pmustequalw.Ontheotherhand,theconstraintisautomaticallysatisedwhenfp=0.Intheliterature,constraintsinthissetaretypicallyreferredtoascomplementarityconstraints.Theseconstraintscontaintwononnegativeterms(fpandcp+p)]TJ /F3 11.955 Tf 12.5 0 Td[(winourcase)andrequireatleastoneofthetwotermstobezero.Thelastsetofconstraintsin MINREV-P requirestollsandgeneralizedcoststobenonnegative.Ourearlierassumption(Pw2WPp2Pwpafp=xaforsomef2F)ensuresthat MINREV-P isalwaysfeasible.Onefeasiblesolutioncanbeobtainedbysettingf=fandlettingwtobeasfollows:w=8><>:w)]TJ /F1 11.955 Tf 12.2 0 Td[(cpiffp>0Mwiffp=0Intheabove,w=maxcpfp>0;p2Pwisasucientlargetoll(seethediscussioninSection 4.3 ).Itiseasytoverifythatthesolutionsoconstructedisfeasibleto MINREV-P .Asformulatedabove, MINREV-P isamathematicalprogramwithcomplementarityconstraints(MPCC),aclassofoptimizationproblemsdiculttosolveoptimally.(Becausecomplementarityconstraintsensureequilibrium,MPCCarealsoclassiedasmathematicalprogramswithequilibriumconstraintsorMPEC.)Suchproblemsarenon-convexandviolatetheMangasarian-Fromovitzconstraintqualication(see,e.g.,Gauvin,1977).WhiletherearealgorithmsproposedforMPCC,manyguaranteetondsolutionsthatsatisfynecessaryconditionsforlocaloptimalityandforonlysmalltomediumsizeproblems(see Luo ( 1996 ),and Lawphongpanich&Yin ( 2010 )).Theaboveformulationof MINREV-P seemsnaturalbecausetheconstraintsreecttheuserequilibriumconditions.However,itisnotthemostcompact.Amongthethreedecisionvariablesin MINREV-P ,thegeneralizedtravelcostwimplicitlydeterminesthevaluesoftheremainingtwovariables,fpandp.Givenw,thepathtollscanbedeterminedbythefollowingexpression: 48

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p=maxf0;w)]TJ /F1 11.955 Tf 12.2 0 Td[(cpg8p2Pw (4{3)Forpathp2Pw,ifcp>w,thenthetollforpathporpiszero.Otherwise,p=w)]TJ /F1 11.955 Tf 12.22 0 Td[(cp.Thelattermeansthatthegeneralizedcost(p+cp)ofpathpequalsw,theminimum,andfpcanbepositivewithoutviolatingthecomplementarityconstraintseethesecondsetofconstraintsin MINREV-P .TomeetthedemandforODpairw,f2Fmustfulllthefollowingdemandequation:Xp2Pw(w)fp=dw (4{4)wherePw(w)=fp2Pwjp+cp)]TJ /F3 11.955 Tf 11.96 0 Td[(w=0gisthesetofusablepathsbasedonw.IfPw(w)containsonlyonepath,thenthesolutionto( 4{4 )isunique.Otherwise,thetheorembelowshowsthatallsolutionsto( 4{4 )forallw2Wmustyieldthesametollrevenue. Theorem4.2. Foreachw2Wletwbegivenandpbedeterminedvia( 4{3 ).Iff1andf2areinFandfeasibleto( 4{4 )forallw2Wthentheymustyieldthesamerevenue,i.e.,>f1=>f2. Proof. Thefollowingmusthold:>f1=Xw2WXp2Pw(w)f1pp=Xw2WXp2Pw(w)f1p(w)]TJ /F1 11.955 Tf 12.19 0 Td[(cp)=Xw2WwXp2Pw(w)f1p)]TJ /F8 11.955 Tf 13.27 11.36 Td[(Xw2WXp2Pw(w)f1pcp=Xw2Wwdw)]TJ /F3 11.955 Tf 11.96 0 Td[(t(x)>x 49

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Asdenedabove,Pw(w)denotesthesetofusablepaths.Thus,therstequationexpressestherevenueintermsofthesepaths.Thesecondfollowsbecausecp+p=w;8p2Pw(w).Thethirdexpandstheexpressioninthesecondequationintothedierencebetweentwototals,generalizedcostandtraveltime.Finally,thelastequalityusesthefactthatowsonutilizedpathsmustsatisfy( 4{4 )forallw2Wandreplacesthetotaltraveltimewithanequivalentexpressionthatisconstantwithrespectto MINREV-P .However,theabovesequenceofequationsalsoholdforf2.Thus,>f1=Pw2Wwdw)]TJ /F3 11.955 Tf 11.96 0 Td[(t(x)>x=>f2. FromtheproofofTheorem 4.2 ,theobjectivefunctionof MINREV-P canbereplacedbyPw2Wwdw.Doingsoreduces MINREV-P toalinearprogramwithcomplementarityconstraints(see Huetal. ( 2008 )).Inaddition,( 4{3 )impliesthatthersttwosetsofconstraintsareequivalenttorequiringfp(w)]TJ /F1 11.955 Tf 12.25 0 Td[(cp)0forallp2Pwandw2W.Thus,pcanbeeliminatedand MINREV-P reducestothefollowingproblemwithoutp: minXw2Wwdws:t:fp(w)]TJ /F1 11.955 Tf 12.2 0 Td[(cp)08p2Pw;w2Wf2Fw0w2WSimilarto MINREV-P ,theaboveproblemremainsnon-convexandequallydiculttosolveoptimally.Ontheotherhand,therstconstraintintheaboveimpliesthat,ifpathpisutilized(i.e.,fp>0),thenw)]TJ /F1 11.955 Tf 12.53 0 Td[(cp.Becausethismustholdforallutilizedpaths,wcannotbesmallerthanthelongestutilizedpath.Inthenextsection,thislastobservationisusedinanalternativeformulationof MINREV-P . 50

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4.5AlternativeFormulationsThissectionexploresalternativeformulationsforthepath-tollpricingprobleminaneorttobetterunderstandtheproblemanddevelopbetterformulations,e.g.,onesthataremoreecientcomputationally.Below,threeapproachesareinvestigated.Therstformulatesthepricingproblemasmixed-integeroptimizationproblems.Intheliterature,several,e.g., Wang&Lo ( 2010 ), Luathepetal. ( 2011 ),and Lietal. ( 2012 ),haveusedthisapproachtosolveseveralbi-level,MPCCorMPECproblemsintransportation.Forsmallproblems,theapproachcanbeecientlyimplementedparticularlywhenusingcommercialsoftware.(Thesubsequentsectionprovidesempiricalevidencethattheapproachmaybeecientforlargeproblemsaswell.)Thesecondandthirdexpressthepricingproblemaslinearlyconstrainedoptimizationproblemswithconcaveandbilinearobjectivefunction,respectively.ThepricingproblemwithabilinearobjectiveisNP-Complete.Thissuggeststhatitmaynotbepracticaltosolvelargepricingproblemsoptimally.Minimizingaconcavefunctionoveralinearlyconstrainedregion,ontheotherhand,indicatethatanoptimalsolutionmustbeanextremepointofF. 4.5.1Mixed-integerOptimizationProblemsBelowaretwomixedintegerprogramming(MIP)formulationforthepath-tollpricingproblem.Bothusebinaryvariables.Oneusesthemtoensurethecomplementarityconditionsand,intheother,thebinaryvariablesselectpathstobecomethelongestutilizedpathforeachODpair.RecallfromSection 4.3 thatthelengthofsuchapathbecomeseachODpairsgeneralizedcost,w.From( 4{3 ),thevaluewalsodeterminesthetollsoneverypathconnectingODpairw.Therstformulation( MIP-1 )minimizeslambda>d,thetotalgeneralizedcost.FromTheorem 4.2 ,doingsoisequivalenttominimizing>f,thetollrevenue.Additionally,theformulationusesbinaryvariables,yp,toensurethecomplementarityconditionsin MINREV-P . 51

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(MIP-1)min>ds:t:fpypdw8p2Pw;w2Wcp+p)]TJ /F3 11.955 Tf 11.96 0 Td[(w(1)]TJ /F3 11.955 Tf 11.95 0 Td[(yp)Mw8p2Pw;w2Wcp+pw8p2Pw;w2Wf2Fp0;yp2f0;1gp2PIntheaboveformulation,Mwisasucientlylargeconstant.Forexample,itsucestoletMwbethedierencebetweenthelengthsoflongestandshortestpathofODpairw,i.e.,Mw=maxfcpjp2PwgWhenyp=0,therstsetofconstraintsforcesfptobezeroandthesecondsetofconstraintsbecomesvacuousornon-bindingforpathpbecauseMwislarge.Ontheotherhand,settingyp=1allowspathptobeutilized,i.e.,fp0andforcesitsgeneralizedcosttoequalw,i.e.,cp+p)]TJ /F3 11.955 Tf 12.5 0 Td[(w=0inthesecondsetofconstraints.Theremainingconstraintsaresimilartobefore.ObservethattheobjectivefunctionandconstraintsinMIParelinear.Inparticular,Fasdenedearlierisapolyhedron.ThesecondformulationmakesuseoftherelationshipbetweenthelongestutilizedpathforagivenODpairanditsgeneralizedcostw.In MIP-2 below,thebinaryvariableypequals1ifpisthelongestutilizedpathforsomeODpairand,similarto MIP-1 ,theobjectiveistominimizethetotalgeneralizedcost. 52

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(MIP-2)minXw2W Xp2Pwcpyp!dws:t:Xp2Pwyp=18w2Wfpdw0@Xq2Pw&cqcpyq1A8p2Pw;w2Wf2Fyp2f0;1gp2PIntheobjectivefunction,theexpressionenclosedbytheparenthesesdeterminesthelengthofthelongestpathforODpairwandthesummationrepresentsthetotalgeneralizedcost.TherstsetofconstraintsselectsonepathtobethelongestutilizedforeveryODpair.Thesecondallowspathsnotlongerthanthelongestonetohavepositiveows.Toillustrate,letPw=f1;2;3gandc1
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maxfcpjp2Pw++(f)g,wherePw++(f)=fp2Pwjfp>0g.Inwords,gw(f)isthelengthofthelongestutilizedpathassociatedwithpath-owvectorfforODpairw,inotherwordsgw(f)isthegeneralizedcostforODpairw.Then,apricingproblemthatminimizesthetotalgeneralizedcostcanbeformulatedasfollows: (CMP)minXw2Wdwgw(f)s:t:f2FIn CMP ,thetotalgeneralizedcostisexpressedasapositivelyweightedsumoffunctionsgw().Theorem 4.3 belowshowsthatgw()isconcaveforeveryw2W.Thus, CMP isconcaveminimizationproblem. Theorem4.3. gw(f)isconcavewithrespecttof2F. Proof. Letf1andf2denotetwopathdistributionsinF.Forany2(0;1),f=f1+(1)]TJ /F3 11.955 Tf 12.31 0 Td[()f2alsobelongtoFbecauseitisapolyhedron,asetthatisalwaysconvex.Thus,gw(f)iswelldened.Observethat,iff1p>0,thenfp=f1p+(1)]TJ /F3 11.955 Tf 12.07 0 Td[()f2p>0,i.e.,p2Pw(++)(f1)impliesthatp2Pw(++)(f).Thus,Pw(++)(f1)Pw(++)(f)andgw(f1)=maxcpp2Pw(++)(f1)maxcpp2Pw(++)(f)=gw(f)Similarly,Pw(++)(f2)Pw(++)(f)andgw(f2)gw(f).Then,thefollowingmusthold:gw(f)=gw(f)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()gw(f)gw(f1)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()gw(f2)Thus,gw(f)isconcave. Itfollowsfromtheaboveproofthat,alongthelinesegmentjoiningtwoadjacentextremepoints,f1andf2,gw(f)mustbeconstant,i.e.,gw(f)=maxcpPw(++)(f1)[Pw(++)(f2) 54

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forallf=f1p+(1)]TJ /F3 11.955 Tf 10.33 0 Td[()f2pand2(0;1).Thus,theobjectivefunctionof CMP ispiecewiseconstantandgenerallydiscontinuousatextremepoints.Toillustrate,Figure 4-3 displaysgw(f)forthecaseinwhichtherearethreepathswithlengthsc1=3,c2=5,andc3=8.ThefunctionisdisplayedoverthesetS=(ff=1f1+2f2+3f3;3Xi=1i=1;i0)wherefiisavectorinR3with1intheithcomponentandzeroeverywhereelse.Thegureonlyshowsthevaluesfor1and2.Thevalueofcanbedeterminedfromtheothers.Forexample,3=1at(1;2)=(0;0)and3alongthe(diagonal)line1+2=1inthe(1;2)-plane.Observethatgw(f)isnotdierentiableat(1;2)=(1;0),(0;1)andalongdiagonaltheline.Moreover,gw(f)=3:0at(1;2)=(1;0)andgw(f)=5:0at(0;1)andalongthediagonalline.Attheextremepoint(0;0)andintheinteriorofS,thereisnoimprovingfeasibledirection(asdenedin,e.g., Bazaraaetal. ( 2013 ))becausegw(f)=8:0atthesepoints.At(1;2)(1;0);(0;1)andtherelativeinteriorofthelinesegmentjoiningthem,noimprovingfeasibledirectionexistsaswell.Movingawayfrom(1;0)and(0;1)doesnotimprovethevalueofgw(f).Thissuggeststhatthe CMP maybemorediculttosolvethananordinarylinearlyconstrainedconcaveoptimizationproblembecausenoimprovingfeasibledirectionexistsatanyfeasiblepoint.Althoughitiswellknownthatanoptimalsolutionto CMP mustoccuratanextremepoint( Bazaraaetal. , 2013 ),thetheorembelowshowsthateveryextremepointofFmustbelocallyoptimalto CMP ,thusvalidatingoneoftheaboveobservations. Theorem4.4. IffisanextremepointofF,thenfmustbelocallyoptimalto CMP . Proof. Letf1beanextremepointofF,apolyhedron.Then,considerpath-owvectorsoftheform:f=f1+(f2)]TJ /F3 11.955 Tf 12.73 0 Td[(f1),wheref22F(notnecessarilyanextremepoint)andkf2)]TJ /F3 11.955 Tf 12.67 0 Td[(f1k=1.Inotherwords,d=f2)]TJ /F3 11.955 Tf 12.68 0 Td[(f1isafeasibledirectionawayfromf1.Asconstructed,fisfeasibletoFforall2(0;1).Foranyf22Fand2(0;1), 55

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. . 1 . 2 . gw(f) . . . . . . . gw(f)=3 . gw(f)=5 . gw(f)=8 Figure4-3. Illustrationofconcavity f=(1)]TJ /F3 11.955 Tf 12.1 0 Td[()f1+f2anditfollowsfromtheproofofTheorem 4.3 thatgw(f)gw(f1).Thus,thereisnoimprovingfeasibledirectionatf1,i.e.,itmustbelocallyoptimal. 4.5.3BilinearOptimizationProblemWhilethecomplementarityconditionsarewrittenasconstraintsin MINREV-P ,theformulationbelowassessesalargepenaltyintheobjectivewhentheseconditionsdonothold. (BOP)min>d+MXw2WXp2Pwfpps:t:cp)]TJ /F3 11.955 Tf 11.96 0 Td[(wp8p2Pw;w2Wf2Fp0p2P 56

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Asbefore,Misasucientlypositiveconstant.Recallthat,atequilibrium,thegeneralizedcostwmustequalthelengthofthelongestutilizedpathforODpairw.Ifpathpislongerthanthelongestutilizedpath,i.e.,cpw,thenp>0andconsequentlyfpmustequalzerotomaintaincomplementarityanddecreasetheobjectivevalue.Ontheotherhand,whenpathpisshorterthanthelongest,i.e.,cpworcp)]TJ /F3 11.955 Tf 12.12 0 Td[(w0,then,toreducetheobjectivevalue,pmustbezerowhicheectivelyallowsfptobepositive,i.e.,pathpisusable.Thus,thesecondtermintheobjectivefunctionof BOP assesseslargepenaltiesforsolutionsnotsatisfyingthecomplementarityconditionsandmustequalzeroatanoptimalsolution.Consequently,theobjectiveof BOP reducestominimizingthetotalgeneralizedcostterm,>d.Thesecondtermintheobjectiveof BOP isbilinearbecauseholdingfpconstantmakestheexpressionlinearwithrespecttopandviceversa.In Bennett&Mangasarian ( 1993 ),theauthorsestablishthat BOP isNP-Complete.Whileseveral(e.g., Vaish&Shetty ( 1977 ), Sherali&Shetty ( 1980 ), Czochralska ( 1982 ),and White ( 1992 ))haveproposednitelyconvergentalgorithmsfor BOP ,theprecedingcomplexityresultsuggeststhatthesealgorithmsmaynotbeeectiveforlargeproblemsthenumberofiterationsrequiredtosolvetheproblemincreasesexponentiallywithitssize. 4.6NumericalResultsThissectionreportsresultsfromourexperimentswith MIP-1 and MIP-2 formulationsinSection 4.5.1 .Althoughitisanaturalformulationforthepathtollpricingproblem, MINREV-P inSection 4.4.2 isaMPCCproblemthatisdiculttosolveandalgorithmsintheliteraturecannotguaranteeagloballyoptimalsolution,particularlyforlargeproblems.Theconcaveminimization( CMP )andbilinear( BOP )optimizationprobleminSections 4.5.2 and 4.5.3 ,respectively,arenon-convexoptimizationproblems.Algorithmsforsolvinggeneralnon-convexoptimizationproblemsarecopiousintheliterature(see,e.g., Horst&Tuy ( 1996 ), Horstetal. ( 2002 ),andreferencescitedtherein).Specializingthesealgorithmsordesigningnewonesforthepath-tollpricingproblemisasubjectforfutureresearch. 57

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Weimplemented MIP-1 and MIP-2 usingGAMSandsolvedtheresultingmixedintegerprogrammingproblemswithCPEXVersion12.1.0.AllCPUtimesarefroma3.06GHzdualcorepersonalcomputerwith3GBofmemory.OurtestdataconsistoftheroadnetworksfromSiouxFalls,SouthDakotaandAnaheim,California.(Bothareavailablefromhttp://www.bgu.ac.il/bargera/tntp/.)SiouxFallsnetworkcontains24nodes,76links,and528ODpairsandthereare416nodes,914linksand1406ODpairsinAnaheim.Inaddition,wegeneratedfouradditionalproblemscenariosbymultiplyingtheoriginaldemands,capacitiesandlinkfree-owtraveltimeswithrandomnumbersbetween0.75and1.25.Inboththeoriginalandrandomproblemscenarios,thetargetisthesystemoptimaldistribution,i.e.,ourtargetdistributionxsolvesmint(x)>xx2X.DoingsosimpliesthegenerationofpathsthatsupportF.ForODpairw,apath^pcansupportFifithasthesmallestmarginalcostorXa2A^pata(x)+xadta(x) dxa=minpata(x)+xadta(x) dxap2PwPathswithlargermarginalcostscannotbesystemoptimal.Inourexperiments,ndingpathswiththesmallestmarginalcostswasformulatedasashortestpathproblem.Whentheshortestpathalgorithmterminated,weusedatopologicalsearch(see,e.g., Leisersonetal. ( 2001 ))tondpathsbetweeneachODpairinasubnetworkconsistingonlyoflinkswithzeroreducedcosts.Forbothnetworks,thenumbersofODpairswithauniqueminimummarginalcost(MMC)patharesignicant(seeTable 4.6 ).ForSiouxFalls,theaveragenumberofODpairswithauniqueMMCpathisapproximate63%ofthetotalanditisapproximately87%forAnaheim.FromthediscussioninSection 4.3 ,tollsareunnecessaryfortheuniqueMMCpathsconnectingtheseODpairsandtheassociateddecisionvariablescanbeeliminatedfromboth MIP-1 and MIP-2 .AmongthoseODpairswithtwoormoreMMCpaths,theaveragenumberofMMCpathsperODpairis2.79and2.55forSiouxFallsandAnaheim,respectively. 58

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Table4-6. NumberofODpairswithuniqueandmultipleMMCpaths NetworkDataNumberofODwithauniqueMMCpathNumberofODwith2ormoreMMCpaths SiouxFallsOriginaldata327201SiouxFallsRandomscenario1337191SiouxFallsRandomscenario2327201SiouxFallsRandomscenario3348180SiouxFallsRandomscenario4338190AnaheimOriginaldata1138268AnaheimRandomscenario11227179AnaheimRandomscenario21261145AnaheimRandomscenario31276130AnaheimRandomscenario41242164 Amongthetwonetworks,Anaheimiseasiertosolve.Forallveversions,oneusingtheoriginaldataandfourwithrandomdata,600secondswasourlimitforCPUtimeand,asdisplayedinTable 4-7 ,CPLEXwasabletosolveboth MIP-1 and MIP-2 optimally.Exceptfortheproblemwithoriginaldata,thetollrevenueobtainedbyCPEXisessentiallythesameasthelowerbound.Whensolving MIP-1 withtheoriginaldata,CPLEXwasunabletogeneratealowerboundbetterthan1105.93.Thisboundisunreasonablegiventhattollsarenonnegativeandsuggeststhatthelinearrelaxationof MIP-1 isnottight.Computationally, MIP-2 issignicantlyeasiertosolve.ExceptforRandomScenario1, MIP-2 canbesolvedinlessthan3CPUseconds.TheaverageCPUsecondsfor MIP-2 amongtheveversionsofAnaheimis10.02secondswhichisapproximately3.36%oftheaverageCPUtimesfor MIP-1 .Notethat,whensolving MIP-1 withtheoriginalAnaheimdata,CPLEXterminatedbecauseitreachedourtimelimitof600CPUsecondsandwasunabletoverifythattherevenue(2177.87)itobtainedisinfactoptimal.Wesurmisethatthecomputationaleciencyof MIP-2 isduetothefactthatitsformulationreliesonthekeystructureinthepricingproblem.OnesetofdecisionvariablesdesignatesapathtobethelongestutilizedpathforeachODpair.Doingsodeterminestheequilibriumgeneralizedcost(w)andtollsonallMMCpaths.Ontheotherhand, MIP-1 doesnot 59

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makeuseofthisstructureandsimplyusesbinaryvariablestohandlethecomplementarityconstraints. Table4-7. ResultsforAnaheim Data MIP-1 MIP-1 MIP-1 CPU MIP-2 MIP-2 MIP-2 CPUrevenuelowerbound(sec.)revenuelowerbound(sec.) Original2177.87-1105.93600.002177.892177.841.12Scen.11812.431812.35542.281812.361812.3245.11Scen.26825.616825.595.976825.656825.590.48Scen.35107.215107.11138.035107.275107.172.2Scen.43569.893569.82199.843569.883569.821.17 AlthoughtheSiouxFallsnetworkissignicantlysmaller,CPLEXrequiredsubstantiallymoreCPUtimestoobtainsolutionswithlowerquality.ThisisbecausealowerpercentageofODpairsinSiouxFallshaveuniqueMMCpaths.WealsosuspectthatthisineciencyisduetothefactthatMMCpathsbetweenanODpairhavesimilarlengths.ForSiouxFalls,thelongestandshortesttraveltimeofMMCpathsforeachODpairdierbyapproximately15%.Intuitively,thismakestheobjectivefunctionratheratandagloballyoptimalsolutionmorediculttond.ForSiouxFalls,wesettheCPUtimelimitto3600seconds(or1hour)andtheoptimalitycriterionto10%.Thus,CPLEXterminateswhenthetimelimitisreachedorthecurrentbestsolutioniswithin10%ofthebestlowerbound.Inallveversions,CPLEXranoutofmemoryorreachedthetimelimit(seeTable 4-8 )beforendingasolutionto MIP-1 thatsatisestheoptimalitycriterion.Ontheotherhand,CPLEXwereabletondsatisfactorysolutionsto MIP-2 within1hourlimitforallveversions.ForRandomScenario4,CPLEXdidsoin40seconds.Allrevenuesfrompathtollsfor MIP-2 arealsobetterthanthosefrom MIP-1 .Todeterminethereductioninthenancialimpactonusers,wesolved MINREV-L inSection 4.4.1 todeterminetheminimumrevenuefromlinktolls.Therevenuesfrompathtollsarefrom MIP-2 inTables 4-7 and 4-8 .Table 4-9 displaystherevenuesfromlinkandpathtolls.Relativetolinktolls,pathtollsreducethetollrevenuebynearly97%on 60

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Table4-8. ResultsforSiouxFalls Data MIP-1 MIP-1 MIP-1 CPU MIP-2 MIP-2 MIP-2 CPUrevenuelowerbound(sec.)revenuelowerbound(sec.) Original0.203-0.8243185.00.1830.1732871.4Scen.10.095-1.3443505.00.0880.083488Scen.20.086-1.35836000.0850.0813245Scen.30.109-1.16936000.1080.10140Scen.40.075-1.60636000.0690.0663404 average.Thus,usingpathtollscanresultinsubstantialsavingsformotoristsontheseroadnetworks.Webelievethatthesesavingsisasignicantadvantageofpathtollsandshouldbeconsideredfordoinganycomparisonbetweenlinkandpathtolls. Table4-9. Benetsofpathinformation NetworkDataTollrevenueforlink-basedTollrevenueforpath-basedRevenuereductionamountRevenuereductionpercentage AnaheimOriginal59,7662,17857,58896.36%AnaheimScen.156,3811,81254,56996.79%AnaheimScen.265,9596,82659,13389.65%AnaheimScen.377,5985,10772,49193.42%AnaheimScen.467,9843,57064,41494.75%SiouxFallsOriginal20.6660.18320.48399.11%SiouxFallsScen.119.4620.08819.37499.55%SiouxFallsScen.259.0190.08558.93499.86%SiouxFallsScen.327.1020.10826.99499.60%SiouxFallsScen.432.8620.06932.79399.79% 4.7SummaryInthischapterwetackledthetolldesignproblemforarst-bestpath-basedpricingscheme.Westudiedthepropertiesoftheproblemanddevelopedseveralmodelsfordesigningtolls.Themodelsareexaminedintermsofcomputationaladvantagesanddisadvantages.Amongtheproposedmodels,amixed-integerlinearprogram,called MIP-2 ,isselectedasthemostecientone.Wetestedthemodelontworeal-sizenetworks,namelySiouxFallsandAnaheim.Thismodeldemonstratedgoodperformanceandwasabletoproducenearoptimalsolutionsinalimitedtime. 61

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CHAPTER5IMPLEMENTATIONOFDIFFERENTIATEDPRICINGSCHEMES 5.1OverviewDierentiatedpricingschemes,asproposedinChapter 3 ,chargeusersbaseontheirtravelcharacteristics,i.e.,originofthetrip,OD,ortheentirepath.Thefocusofthischapterisonimplementationofpath-dierentiatedschemes.AlthoughthesimilarideacanbeusedtoimplementoriginandODspecicschemes,theyarenotdiscussedheretoavoidredundancy.Inordertoimplementapath-dierentiatedpricingscheme,tollcollectionsystemneedstoidentifyeveryvehicle'spath.Usingon-boardtrackingdevices,suchasGPS,isoneoption.However,sincethisoptionrequiresallvehiclestobeequippedwithtrackingdevices,itmaybeexpensiveanddiculttomaintain.Alternatively,wecanuseautomatedvehicleidentication(AVI)sensorstoidentifyvehicles'paths.TheAVIsensorscanuniquelyidentifynearbyvehiclesandrecordthetimeandlocationofthedetection.Thisinformation,ifcollectedsuciently,canbeusedtoinferthepathofvehicles.AbroadrangeofdevicescanbecategorizedasAVIsensors.Infact,twoofthebasictasksofanytollchargingsystemaretoidentifyvehiclesandrecordtheirlocations,whicharethefunctionalitiesofAVIsensors.Inthecurrentpractice,mostofthetollingfacilitiesuseelectronictollcollectiontechnologiessuchastoll-tagreaders(e.g.,SunPassinFlorida,TxTaginTexas,andE-ZPassin15states)orlicenseplatescanners(e.g.,Toll-By-PlateinFlorida,andLicensePlateTollinColorado).Thesedevicescandetectandidentifyvehicles(see,e.g., dePalma&Lindsey ( 2011 )forareviewonsensortechnologiesusedfortollcollection.),i.e.,theycanserveasAVIsensors.Tracsensors,includingAVIs,havemanyapplicationsintransportationsystemsandeachonegivesrisetoadierentsensorlocationproblem(see,e.g., Gentili&Mirchandani ( 2012 )).Amongthem,andthemostrelevantone,ishowtooptimallylocatesensorsfor 62

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thepurposeoftracowinference,wherethedatafromsensorsisusedtoobtaintheunobservedows1.Thesesensorlocationproblemscanbecategorized,withrespecttothetypeofowtheyattempttoobtain,asfollows: Origin-destination(OD)demand:MostofthestudiesattempttoestimateODdemandsbyminimizingdeviationfromahistoricaltripmatrix( Yangetal. ( 1991 ), Yang&Zhou ( 1998 ), Bianco ( 2001 ), Ehlertetal. ( 2006 ),and Castilloetal. ( 2008c )).Whileothers,e.g., Liou&Hu ( 2009 ),triedtoobtaintheexactvalueofODdemandusingtheleastnumberofsensorsonthenetwork. Flowonallpaths. Gentili&Mirchandani ( 2005 )assumedthatvehiclesareequippedwithdevicesthatcantransmittheirpathinformationtoanearbyroadsidesensor.Theystudiedthelocationproblemforthistypeofsensors.Whileothers,e.g., Castilloetal. ( 2008c )and Minguezetal. ( 2010 )attemptedtondtheoptimallocationsofsensorsforestimatingpathowsandODdemands. Flowonalllinks.Thisproblemhasbeenintroducedby Huetal. ( 2009 ),whoalsoproposedasolutionalgorithmthatusesthelink-pathincidencematrix,thusrequirespathenumeration.Later, Ng ( 2012 )proposedasolutionmethodfortheproblemthatavoidspathenumeration. He ( 2013 )introducedmorevariantsofsensorlocationproblemforobtaininglinkows,andproposedsolutionalgorithmsforthoseproblems.Inadditiontotheabove,amoregeneralformofowobservationproblemhasemerged,whichattemptstoinferdierentowsfromasetofobservedowdata,e.g.,usinglinkandpathowinformationcollectedusingsensorstoobtainowonotherpaths. Castilloetal. ( 2008b ), Castilloetal. ( 2010 ),and Castilloetal. ( 2011 )discussedandproposedsolutionmethodforthesetypesofproblems.ThischapterexplainstheuseofAVIsensorsinidentifyingpathsofvehiclesforthepurposeofimplementingpath-dierentiatedchargingschemes.Weproposeandstudythreedierentsensorlocationproblemsassociatedwithpath-dierentiatedtolls.Thecontributionsofthischapterarethreefold.First,weproposetheideaofusing 1Thereisalsoanotherclassofproblems,whichusethesensorsdatatoestimate,asopposedtoobtain,uncountedows(see,e.g., Gentili&Mirchandani ( 2011 )foracomprehensivereview). 63

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AVIsensorstoimplementpath-dierentiatedtolls.Second,wedevelopmathematicalmodelsfordierentvariantsoftheproblemandinvestigatetheirproperties.Third,throughexamples,wedemonstratetheadvantagesofusingAVIsensorstoimplementpath-dierentiatedtollsratherthantraditionallink-basedschemes.Thenextsectionintroducesthenotationandbrieyreviewstheconceptofpath-dierentiatedpricing.Section 5.3 denesandformulatesconditionsforasetofsensorlocations,capableofidentifyingpathofeveryvehicle.Threedierentversionsofsensorlocationproblemforpath-dierentiatedtollsarethendiscussedinSections 5.4 , 5.5 ,and 5.6 .Therstproblemaimstondoptimalsensorlocationsforimplementingagivenpath-dierentiatedscheme,whereasthesecondoneassumesthesensorsarealreadylocatedonthenetworkandattemptstodesignanoptimalpath-dierentiatedschemethatisimplementableusingtheavailablesensors.Thethirdproblemishowtodesignapath-dierentiatedschemethatinducesagivenlink-owdistributionandrequirestheleastnumberofsensorstoimplement.Section 5.8 thenconcludesthepaper. 5.2BackgroundThissectionintroducesthenotationusedthroughoutthepaperandpresentsareviewonpath-dierentiatedcongestionpricing. 5.2.1NotationLetG=(N;A)representatransportationnetwork,whereNandAarethesetsofnodesandlinks,respectively.ThesetofODpairsisdenotedbyW,anddwrepresentsdemandforODpairw2W.ThesetofpathsconnectingODpairw2WisdenotedbyPw,andtheunionofallthesepathsetsbyP.Foreachpathp,itsow,traveltime,andtollarerepresentedbyfp,tp(f),andp.Toavoidintroducingaparameterforvalueoftime,weassumetollsareintheunitoftime.Inthispaper,vectorsofvariablesaredenotedbythesamelettersbutwithoutsubscript,e.g.,fisthevectorofpathowsfp,anddoublestruckcapitallettersdenotesets.AlsodeneF,asthesetofall 64

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demand-feasiblepathows,i.e.,F=(f2RjPj+Xp2Pwfp=dw;8w2W)wherejPjisthecardinalityofsetPorthetotalnumberofpaths.AndRjPj+isthesetofallnonnegativevectorsinRjPj. 5.2.2Path-dierentiatedPricingInthecontextoftracassignment,userequilibrium(UE)isdenedasthetraccondition,whenthecostofutilizedpathsareequalandarenotgreaterthancostofanyun-utilizedpathforeveryODpair.TheUEconditioncanbewrittenasthefollowingcomplementaritysystem:0fp?tp(f))]TJ /F3 11.955 Tf 11.95 0 Td[(uw08p2Pw;w2W;f2FwhereuwistheequilibriumtravelcostofODpairw,orequivalentlythecostofshortestpathforw.Andaistheamountoftollonlinka.Ingeneral,userequilibriumisnotthemostecienttracdistribution,becausethetotalsystemtraveltimewillnotbeatitslowestlevel.Thetracdistributioncorrespondingtotheminimumsystemtraveltimeiscalledsystemoptimum(SO)anddenedasfollows:f=argminf2Ff>t(f)TheSOtracdistributionusuallyrequiressomeuserstotakepathsotherthantheshortestavailablepath.Becauseofthis,systemoptimumowsmaynotberealizedunderanuncontrolledcondition.Congestionpricingusestollstoalterthecostoftravelondierentpathsandyieldamoreecientequilibriumstate,e.g.,SO.Theuserequilibriumconditionunderalink-basedtollingschemecanbewrittenas:0fp?tp(f)+Xa2Aapa)]TJ /F3 11.955 Tf 11.96 0 Td[(uw08p2Pw;w2W (5{1) 65

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wherepais1ifpathpcontainslinka,and0otherwise.Thus,thesummationincondition( 5{1 )representsthetotaltollatraveleronpathphastopay.Asstated,condition( 5{1 )assumesthateveryusertravelingonlinka2A,mustpaythesame\anonymous"toll,namelya,regardlessofotherpartsofthetrip.Underthisassumption,tollchargedonpathswithcommonlinkscouldbehighlyinterdependent,i.e.,changingtollononepathcannotbedonewithoutalteringtollonotherpaths. Zanguietal. ( 2013 )relaxedthisassumptionbydierentiatingusersbasedontheirtripcharacteristics,e.g.,origin,destination,orentirepath,andchargingthemaccordingly.Particularly,thepath-dierentiatedscheme,whichhasbeenstudiedindetailin Zanguietal. ( 2014 ),providesthemostexiblecongestionpricingscheme,thatispathtollscanbedeterminedindependently.Inthisscheme,insteadoflinks,pathsarebeingcharged,hencetheequilibriumcondition( 5{1 )changesasfollows:0fp?tp(f)+p)]TJ /F3 11.955 Tf 11.96 0 Td[(uw08p2Pw;w2Wwhere,pisthetollchargedonpathp.Aswithanyothertypeofcongestionpricing,path-dierentiatedtollsmaybedesignedtosatisfydierentgoals.Tokeepthediscussionbrief,weonlypresentageneralpath-dierentiatedcongestionpricingschemewiththegoalofminimizingtotaltraveltime. 66

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(PATH)minXp2Pfptp(f) (5{2a)s:t:f2F (5{2b)fp(tp(f)+p)]TJ /F3 11.955 Tf 11.96 0 Td[(w)=08w2W;p2Pw (5{2c)tp(f)+p)]TJ /F3 11.955 Tf 11.96 0 Td[(w08w2W;p2Pw (5{2d)p;w08w2W;p2Pw (5{2e)wherewistheminimumgeneralizedcost,i.e.,traveltimeplustoll,oftravelingbetweenODpairw.Constraint( 5{2b )ensuresthatthepathowsarefeasible.Thetolleduserequilibriumconditionsarerepresentedbyconstraints( 5{2c )and( 5{2d ).Finally,thelastsetofconstraintsguaranteesthattollsandODgeneralizedcostsarenon-negative. 5.3PathObservationAsapreparationformodelingtasks,thissectionillustratestheideaofusingAVIsensorstoobtainpathsofvehicles,whichisalsoknownaspathobservationproblemintheliterature. 5.3.1ModelFormulationAssumesomelinksareequippedwithsensors.Wesaythatpathpisobservablewithagivensetofsensors,ifdatafromthesesensorscanidentifyvehiclesthattraversepathp.Thisrequiresvehiclesonpathptobedetectedbyatleastonesensorandtobedistinguishablefromvehiclestravelingonanyotherpath.Thus,pathpisobservablewhenthefollowingconditionsaresatised:Detectablepath:Pathpiscalleddetectableifatleastoneofitslinkshasasensor.Distinguishablepaths:Pathspandqaredistinguishableifa a) Atleastonelinkonpathpisequipped. 67

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b) Foreveryotherpathk,atleastoneuncommonlinkofthesetwopathsisequipped.Condition(a)guaranteesthateveryvehicleonpathpisdetectedbyatleastonesensor.Condition(b)ensuresthatthedatafromthesensorsissucienttodistinguishvehiclesonpathpfromthosetravelingonanyotherpath.Weformulatethepathobservabilityproblem,i.e.,ndingtheoptimallocationofsensorsthatprovidesfullpathobservabilitywiththeleastnumberofsensors,asfollows,thesamemodelispresentedin Castilloetal. ( 2008a ): (OBSV)minXa2Aza (5{3a)s:t:Xa2Azapa18p2P (5{3b)Xa2Azapka18p;k2P;p6=k (5{3c)za2f0;1g8a2A (5{3d)where,pka=jpa)]TJ /F3 11.955 Tf 12.3 0 Td[(kajisaparameterequaltooneiflinkaislocatedonpathpork,butnotboth;andiszerootherwise.Also,zaisabinaryvariablethatequalsoneiflinkahasasensor,andzerootherwise.Theobjectivefunctionminimizesthenumberofsensors.Constraint( 5{3b )requiresthenumberofsensorsonpathpbeatleast1.While,condition( 5{3c )ensuresatleastonesensorisinstalledonanuncommonlinkofpandk,foreverypairofpathsp6=k.Thefollowingtheoremshowsthissensorlocationproblemisaninstanceofsetcoveringproblem,whichisknowntobeNP-hard( Karp , 1972 ). Theorem5.1. Sensorlocationproblemisaninstanceofsetcoveringproblem. Proof. DeneU1=P,U2=Sp6=kf(p;k)g,andU=U1[U2astheuniversalset.Also,deneSa1=Sp:pa=1fpg,Sa2=S(p;k):pka=1f(p;k)g,andSa=Sa1[Sa2.Inwords,Sa1isthe 68

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setofallpathsthatwillbedetectedbythesensoronlinka,andSa2isthesetofallpairsofpathsthatwillbedistinguishedbythesensoronlinka.ThentheminimumsensorlocationproblemcanbestatedasndingAAwiththeminimumcardinality,suchthatSSa2ASa=U. AlthoughthissensorlocationproblemisNP-hard,someresearchers,e.g., Castilloetal. ( 2008b )and Ng ( 2012 ),proposedsolutionalgorithmsthatcansolvetheproblemforlargenetworks.But,thesealgorithmmaytakeanexponentialamountofCPUtimetondanoptimalsolution. 5.3.2IllustrationFigure 5-1 showsanetwork,simpliedfromtheonein Hearn&Ramana ( 1998 ),wherenodes1and2areorigins,andnodes3and4aredestinations.EachofthefourODpairsisconnectedbysixpaths,asshowninTable 5-1 .Wesolvedmodel OBSV forthenine-nodenetworktondtheminimumnumberofsensorsrequiredforobservingallofthe24paths.TheoptimalsensorlocationsaremarkedbyuinFigure 5-1 .Foreverypath,wedeneitssignature2asasubsetofitslinksthatareequippedwithsensors.Pathsignaturesforthenine-nodenetworkarepresentedinTable 5-1 .SincethesensorsinFigure 5-1 produceauniquesignaturesforeachpath,wecandeterminethepathsthatanyvehiclefollows. 5.4ImplementingPath-dierentiatedTollsUsingAVISensorsImplementingapath-dierentiatedschemerequiresobtainingthepathsofindividualvehiclesbasedoninformationfromAVIsensors.Inthissection,apath-dierentiatedschemeisgiven,andourgoalistondtheleastnumberofsensorsrequiredtoimplementthescheme. 2Apathsignaturehasbeencalleda\scanningmap"in Gentili&Mirchandani ( 2012 ) 69

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. . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . . 1 . . 2 . . 3 . 4 . 5 . . 6 . 7 . . 8 . 13 . 14 . . 9 . 10 . . 11 . . 12 Figure5-1. Nine-nodenetworkwithoptimalsensorlocationsforfullpathobservability Ifthereisnolimitonthenumberofsensorstobeplacedonthenetwork,thesensorlocationproblemreducestothepathobservabilityproblemdiscussedinSection 5.3 .Whenallthepathsareobservable,wecandeterminethepathofeveryvehicle,thenchargethetollaccordingtothepath-dierentiatedscheme.Themathematicalprogramforthisproblemisessentiallythesameasmodel OBSV inSection 5.2 .Implementingapath-dierentiatedpricingschemedoesnotnecessarilyrequirefullpathobservability.First,atoll-freepathisnotrequiredtohaveanyequippedlinks,becauseusersofatoll-freepathdonotneedtobedetectedfortollingpurposes.Second,iftheamountoftollforpathspandkareequal,thereisnoneedtodistinguishthetravelersonthesetwopaths,becausewhethertheyareusingpathporktheyshouldpaythesametoll. 5.4.1ModelFormulationAmathematicalmodelforimplementationproblemcanbederivedfrommodel OBSV bypartiallyrelaxingconstraints( 5{3b )and( 5{3c )asfollows: 70

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Table5-1. Sensorlocationsforthenine-nodenetwork ODPathLinksSignature [1;3]11,5,91,9[1;3]22,7,112,11[1;3]31,6,13,91,6,9[1;3]41,6,14,111,6,11[1;3]52,8,13,92,8,9[1;3]62,8,14,112,8,11[1;4]71,5,101[1;4]82,7,122,12[1;4]91,6,13,101,6[1;4]101,6,14,121,6,12[1;4]112,8,13,102,8[1;4]122,8,14,122,8,12[2;3]133,5,93,9[2;3]144,7,1111[2;3]153,6,13,93,6,9[2;3]163,6,14,113,6,11[2;3]174,8,13,98,9[2;3]184,8,14,118,11[2;4]193,5,103[2;4]204,7,1212[2;4]213,6,13,103,6[2;4]223,6,14,123,6,12[2;4]234,8,13,108[2;4]244,8,14,128,12 (IMPL)minXa2Aza (5{4a)s:t:Xa2Azapa18p2P;p>0 (5{4b)Xa2Azapka18p;k2P;p6=k (5{4c)za2f0;1g8a2A (5{4d)where,pisagiventollforpathp.Constraint( 5{4b )isdenedonlyforpathswith 71

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positivetoll.So,toll-freepathsdonotrequiretohaveequippedlinks.Similarly,constraints( 5{4c )aredenedforpathpairswithdierentamountsoftolls,becausepathsthathaveequaltollsdonothavetobedistinguishable. 5.4.2IllustrationWesolvedmodel IMPL fornine-nodenetwork,andtheresultsareprovidedinTable 5-2 .Thepathtollsweattempttoimplement,aspresentedinTable 5-2 ,aretheonesthatminimizethenancialburdenonusers,i.e.,tollrevenue,whilereplicatingtheSOowdistribution. Table5-2. Sensorlocationsforagivenpathtollschemefornine-nodenetwork ODPathLinksSignatureToll [1;3]11,5,95,98.0[1;3]22,7,11110.0[1;3]31,6,13,990.0[1;3]41,6,14,11110.0[1;3]52,8,13,990.0[1;3]62,8,14,11110.0[1;4]71,5,1054.0[1;4]82,7,120.0[1;4]91,6,13,100.0[1;4]101,6,14,120.0[1;4]112,8,13,100.0[1;4]122,8,14,120.0[2;3]133,5,93,5,915.2[2;3]144,7,114,113.2[2;3]153,6,13,93,97.2[2;3]163,6,14,113,110.0[2;3]174,8,13,94,93.2[2;3]184,8,14,114,110.0[2;4]193,5,103,58.0[2;4]204,7,1240.0[2;4]213,6,13,1030.0[2;4]223,6,14,1230.0[2;4]234,8,13,1040.0[2;4]244,8,14,1240.0 Thelinksthatareappearedinthepathsignaturecolumnaretheonesequippedwithsensors.Thesesensorlocationssupporttheimplementationofthegivenpathtolls.For 72

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example,ifavehicleisonlydetectedbysensor5,itwillbecharged4.0forusingpath7;ifitisdetectedbysensors5and9,thenithastopay8forusingpath1;andifdetectedbysensors3,5,and9,thenitwillbecharged15.2forusingpath13.ByinspectingthepathsinTable 5-2 ,onecantellthatallpathswithpositivetollareobservable.Paths8,9,and11donottraverseanyequippedlink,sotravelersonthosepathsarenotdetectedbyanysensors.However,sincetheyaretoll-freeonecanimplementthegivenpathtollswithoutinstallinganysensoronthoselinks.Also,noticethatimplementingthisschemerequiresinstallingsixsensors,whichistwolessthanthenumberofsensorsneededforfullobservability. 5.4.3AdjustingTollsAlthoughweassumedpathtollsarepredetermined,itremainspossibletoalterthemwithoutaectingtheperformanceofthescheme.Forexample,ifapathissocostlythatitwillneverbeutilized,thenincreasingitstollwillnothaveanyimpactonanyperformancemeasureofthenetwork.Giventhepathtolls,pathsetPcanbepartitionedintotwosetsofusable,P+,andunusablepaths,P0,asdenedbelow:P0=[w2Wfp2Pwjtp+p>wgP+=P)]TJ /F6 11.955 Tf 11.96 0 Td[(P0where,tpandwarethetraveltimeofpathpandgeneralizedcostofODpairw,underimplementationofthepathtolls.Thus,therstconditiondenessetP0asthesetofpathswithtravelcostsgreaterthantheODgeneralizedcost.Usingthesesets,constraint 73

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( 5{4c )canbedividedintothefollowingfoursetsofconstraints: Xa2Azapka18p;k2P+;andp6=k (5{5a)Xa2Azapka18p2P+;k2P0;andpk (5{5c)Xa2Azapka18p;k2P0;andp6=k (5{5d)Ifadjustingtollsisallowed,itmightbepossibletoeliminateconstraints( 5{5c )and( 5{5d )byincreasingtollonunusablepaths.Forexample,ifp2P+,k2P0,andp>k,wecanincreasektothevalueofptoeliminatethecorrespondingconstraint.Althoughonecanincreasethetollsonunusablepathswithoutaectingtheequilibriumstate,decreasingitstollmaymakeanunusablepathusable,andthusimpacttheperformanceofthepricingscheme.Toimplementthisidea,werstmodifymodel IMPL byreplacingconstraint( 5{4c )by( 5{5a )and( 5{5b ),whichisequivalenttorelaxingconstraints( 5{5c )and( 5{5d ).So,theoptimalsolutiontotherelaxedmodelmaynotsatisfyconstraints( 5{5c )and( 5{5d ).Supposep,q,k,andl,arefourpaths,suchthatp;q2P0andk;l2P+.Also,assumethatpisindistinguishablefromq,k,andl,bytheoptimalsensorlocations.Thefollowinglemma,theorem,andcorollaryprovethateverypairsofthesepathsareindistinguishable. Lemma1. Ifzapka=zapqa=0,thenzaqka=0 Proof. 8><>:pka=0)pa=kapqa=0)pa=qa)ka=qa)qka=0;whenza=1za=0)zaqka=0;whenza=0 74

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Theorem5.2. Indistinguishabilityisanequivalencerelation. Proof. Considerthreepaths,p,k,andr,andletrepresenttheindistinguishabilityrelation.Thisrelationhasthefollowingproperties: Reexive:ppa=0;8a2A)Pa2Azappa=0)pp. Symmetric:pka=kpa)Pa2Azapka=Pa2Azakpa)ifpkthenkp. Transitive:Usinglemma 1 ,wehave:pk)Pa2Azapka=0pq)Pa2Azapra=0)Xa2Azakra=0)kr:Sinceindistinguishabilityrelationisreexive,symmetric,andtransitive,itisanequivalencerelation. Corollary1. Indistinguishabilityrelationpartitionsthesetofpathsintoequivalenceclasses,whereeverypairofpathsfromsameclassareindistinguishable.Sincepathsq,k,andlareindistinguishablefromp,allfourpathsbelongtothesameindistinguishableclass.Thus,pandqshouldbelessthanorequaltobothkandl,becauseotherwisethesensorlocationsdonotsatisfyconstraint( 5{5b ).Similarly,kshouldbeequaltolbecauseotherwiseconstraint( 5{5a )willbeviolated.Increasingpandqtobeequalkwouldeliminatethecorrespondingdistinguishabilityconstraints.So,theoptimalsensorlocationsfortherelaxedproblemwouldbecomefeasibletomodel IMPL ifweincreasethetollonunusablepathstothelevelofusablepathsinthesameclass.Theresultsofimplementingtheabovemodicationonthenine-nodenetworkarepresentedinTable 5-3 .Forthisexample,adjustingthetollsonunusablepathsresultedinaschemethatneedsvesensors,areductionofonesensoror17percent,comparedtomodel IMPL whichrequiressixsensors. 75

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Table5-3. Sensorlocationsforagivenpathtollschemefornine-nodenetwork PathLinksSignatureInit.TollAdj.Toll 11,5,95,98.08.022,7,117,110.03.231,6,13,990.03.241,6,14,11110.00.052,8,13,990.03.262,8,14,11110.00.071,5,1054.04.082,7,1270.00.091,6,13,100.00.0101,6,14,120.00.0112,8,13,100.00.0122,8,14,120.00.0133,5,93,5,915.215.2144,7,117,113.23.2153,6,13,93,97.27.2163,6,14,113,110.00.0174,8,13,993.23.2184,8,14,11110.00.0193,5,103,58.08.0204,7,1270.00.0213,6,13,1030.00.0223,6,14,1230.00.0234,8,13,100.00.0244,8,14,120.00.0 5.5Designingpath-dierentiatedschemeforgivensensorlocationsInthissectionitisassumedthatAVIsensorshavebeendeployedforotherapplications,andweattempttoutilizethesesensorstoimplementpath-dierentiatedpricing.Thus,wediscusshowtodesignpathtollsthatcanbeimplementedbythegivensensorlocations. 5.5.1ModelFormulationIftherearealreadyenoughsensorsonthenetworkthatmakeallpathsobservable,anypath-dierentiatedpricingschemecanbeimplemented.Otherwise,aschemehastosatisfythetwoconditionsmentionedinSection 5.3 tobeimplementablebytheavailable 76

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sensors.Theseconditionscanbeenforcedbythefollowingconstraints: p=08p2P;Xa2Azapa=0 (5{6a)p=k8p;k2P;Xa2Azapka=0 (5{6b)where,unliketheprecedingsections,zaisnotavariable,butaknownconstant.Therstsetofconstraintsrequirepathswithoutequippedlinkstobetoll-free.Andthesecondonerequiresindistinguishablepathstohavesametolls.Addingtheaboveconstraintstomodel PATH yieldsaformulationforndinganoptimalschemethatcanbeimplementedwithavailablesensors.Sinceconstraint( 5{6b )isdenedoverpairsofpaths,itssizegrowsapproximatelyproportionaltothesquareofthenumberofpaths,jPj2,whichcanbeverylarge.Usingtheconceptofequivalenceclasses,fromcorollary 1 ,allowsamoreconciseformulationoftheproblem.Pathsineachequivalenceclassareindistinguishableandcanonlybechargedthesameamountoftoll.Hence,wewilldeterminetheamountoftollforequivalenceclassesratherthanindividualpaths,andcallthemtollclasses.Thishelpstoremovethedistinguishibilityconstraintandmakethemodelformulationmoreconcise.Giventhesensorlocations,wecanclassifythepathsaccordinglyandformulatethefollowingoptimizationproblemtondthepath-dierentiatedtollingthatminimizesthetotalsystemtraveltime: 77

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(DES)minXp2Pfptp(f) (5{7a)s:t:f2F (5{7b)fp)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(tp(f)+cl(p))]TJ /F3 11.955 Tf 11.95 0 Td[(uw=08w2W;p2Pw (5{7c)tp(f)+cl(p))]TJ /F3 11.955 Tf 11.95 0 Td[(uw08w2W;p2Pw (5{7d)cl(p)08p2P (5{7e)0=0 (5{7f)where,cl(p)isthetollclassofpathp,andthuscl(p)istheamountoftollforpathp.Wededicateclass0toundetectedpaths,whichshouldbetoll-free.Similartomodel PATH ,model DES isverydiculttosolve,becauseithascomplementarityconstraint. 5.5.2IllustrationToillustratetheidea,assumethatlinks3,6,10,and12ofthenine-nodenetworkareequippedwithsensors.Table 5-4 showstheindistinguishablepaths,andassignanIDtoeachclassofpaths.Thepathsarepartitionedinto11classes,eachchargedierentamountoftoll. Table5-4. Indistinguishablepathsfornine-nodenetwork ClassSignaturePathsToll 0None1,2,5,6,12,14,17,180.00131353.61263,45.033107,11,2311.204128,20,2423.3353,615,165.8463,10194.4876,1097.0086,121017.1393,6,10211.34103,6,122210.00 78

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Table 5-4 alsopresentsanoptimalsolutiontomodel DES .Thispath-dierentiatedschemereducesthetotaltraveltimeofthenetworkfrom2455.9,to2253.92,areductionof201.98units.Incomparison,anoptimalanonymouslink-basedscheme,whichchargetollonlinks3,6,10,and12willreducethetraveltimeto2326.59,areductionof129.31.Thisexampleillustrateshowournewschemeoutperformstheanonymouslink-basedcounterpart. 5.6SimultaneousDesignofTollsandSensorLocationsThegoalofthissectionistodesignapath-dierentiatedschemethatinducesatargetlink-owdistributionwiththeleastnumberofsensors.Theowdistributioncanbeonethatminimizestotaltraveltime,airpollution,oranyotherobjectivethatisafunctionoflinkows.Werstdiscussaboutthelink-basedversionoftheproblem,andthenpresentthepath-dierentiatedcounterpart.Asolutionalgorithmandanumericalexamplearepresentedafterwardstoillustratetheidea. 5.6.1MinimumSensorLocationsforLink-basedTollsAsimilarminimumsensorlocationproblemforlink-basedtollshasbeenintroducedby Hearn&Ramana ( 1998 ),andstudiedbyotherssuchas Bai ( 2004 ), Bai&Rubin ( 2009 ),and Baietal. ( 2010 ).Theproblemdenedin Hearn&Ramana ( 1998 )istondaminimumnumberoftollcollectionfacilities,theirlocations,andtollamountssuchthattheresultinglinkowsreplicateSO.Belowisamathematicalformulationofthisproblem: 79

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(MSL-L)minXa2Aza (5{8a)s:t:f2F (5{8b)Xp2Pfppa=xa8a2A (5{8c)tp+Xa2Aapa)]TJ /F3 11.955 Tf 11.95 0 Td[(uw08w2W;p2Pw (5{8d)Xp2P tp+Xa2Aapa!fp=Xw2Wuwqw (5{8e)0azaM8a2A (5{8f)za2f0;1g8a2A (5{8g)uw08w2W;p2P (5{8h)whereMisthelargestpossibletollandxisthevectorofSOlinkowsthatisthetargetdistribution.The MSL-L formulationaspresentedaboveisalinearmixed-integermodel.Foreverylinka,denotedbyzaisabinaryvariablethatisequalto1ifasensorislocatedonlinka,and0otherwise.Thisconditionisenforcedbyconstraint( 5{8f ).Theobjectivefunctionminimizesthenumberofsensors.Constraints( 5{8d )and( 5{8e )representthetolledUEconditions.Constraint( 5{8f )ensuresalltolledlinksareequippedwithsensors. Bai ( 2004 )showedthisproblemisNP-complete. Bai ( 2004 ), Bai&Rubin ( 2009 ),and Baietal. ( 2010 )proposeddierentanalyticandheuristicsolutionalgorithmsforthisproblem.Despitethesedevelopments,solvingthisproblemforlargesizenetworksisstillchallenging. 80

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5.6.2MinimumSensorLocationsforPath-dierentiatedTollsInthe MSL-L modelpresentedabove,asensorhasasinglefunctionalityofcollectingtoll.However,asdiscussedearlier,sensorsdeployedintollcollectionsystemscanworkasAVIsensors,andallowimplementationofpath-dierentiatedpricing.Weconsiderthisapproachtobeanimprovementover MSL-L ,thatisthemodelpresentedheregenerallyrequireslessnumberofsensorstoachievethesametargetlinkowdistribution.Thisimprovementhasbeenillustratedwithnumericalexampleslaterinthissection.Thefollowingisalinearintegerformulationforthisproblem: (MSL-P)minXa2Aza (5{9a)s:t:f2F (5{9b)Xp2Ppafp=xa8a2A (5{9c)fp(1)]TJ /F3 11.955 Tf 11.96 0 Td[(yp)M8p2P (5{9d)tp+p)]TJ /F3 11.955 Tf 11.96 0 Td[(wypM8w2W;p2Pw (5{9e)tp+p)]TJ /F3 11.955 Tf 11.96 0 Td[(w08w2W;p2Pw (5{9f)pMXa2Azapa8p2P (5{9g)p)]TJ /F3 11.955 Tf 11.95 0 Td[(k)]TJ /F3 11.955 Tf 21.92 0 Td[(MXa2Azapka8p;k2P (5{9h)yp;za2f0;1g8a2A;p2P (5{9i)wheretpisthepathtraveltimeforthegivendistribution,i.e.,tp=Pa2Apata(xa).Thersttwosetsofconstraintsarerestrictingpathowstotheonesthatarefeasibleandyieldthetargetaggregatelikeows.ThenextthreeconstraintsensurethetolledUEconditions.Thesummationontherighthandsideofconstraint( 5{9g )isthenumberofsensorson 81

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pathp.Whenthissummationisequaltozero,pathphastobetoll-free.Thesummationontherighthandsideofconstraint( 5{9h )indicatesthenumberofsensorslocatedonuncommonlinksofthepathspandk.Ifthissummationisgreaterthanzero,thenpathspandkaredistinguishableandcanbechargeddierentamountsoftoll.Otherwise,constraint( 5{9h )indicatesp)]TJ /F3 11.955 Tf 12 0 Td[(k0andk)]TJ /F3 11.955 Tf 12.01 0 Td[(p0,whichmeansthesetwopathswillhavethesameamountsoftoll.Itisimportanttopointoutthatmodel MSL-P ,canbetoolargetosolve.Onemajorreasonforthatisthenumberofpathsgrowexponentiallywiththesizeofthenetwork,andthenumberofconstraints( 5{9h )isproportionaltothesquareofnumberofpaths,i.e.,jPj2.InSection 5.6.5 wepresentaheuristicapproachthatcanndgoodsolutionswithinlimitedtime. 5.6.3IllustrationWesolvedmodel MSL-P forthenine-nodenetwork,andchosetheSOdistributionasthetargetlinkow.Table 5-5 ,showsall24pathsinthenine-nodenetwork,andthecorrespondingtraveltime,toll,totalcostandow.Theequippedlinksare3,5,and12.Noticethateachoneofthetolledpathsincludesatleastoneequippedlink,andeachpairofpathswithdierenttollhaveatleastoneuncommonequippedlink.Theminimumnumberofsensorsrequiredtoreplicatethetargetlink-owsis3,whichshowsa40percentimprovementcomparedtomodel MSL-L thatrequires5sensorstodothesame.Noticethatthecostcolumnshowssomepathswithminimumcostthatarenotutilized,thissuggeststhattheremightbemultiplepathowsolutiontothisproblem.However,asdiscussedin Zanguietal. ( 2014 ),allofthemyieldtothetargetlinkow.Table 5-6 presentsthetollingscheme.Forexample,ifavehicleisonlydetectedbyoneofsensors3or12,itwillbecharged4unitsoftoll,butifitisdetectedbyboththetollwouldbe0.8units.Vehiclesdetectedonlink5willbecharged8,andthosedetected 82

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Table5-5. Resultsofsolvingmodel MSL-P fornine-nodenetwork ODPathLinksSignatureTimeTollCostFlow [1;3]11,5,9515.39823.39[1;3]22,7,1123.3923.39[1;3]31,6,13,923.3923.39[1;3]41,6,14,1130.5930.59[1;3]52,8,13,923.3923.3910[1;3]62,8,14,1130.5930.59[1;4]71,5,10518.01826.019.451[1;4]82,7,121222.01426.017.918[1;4]91,6,13,1026.0126.01[1;4]101,6,14,121229.21433.21[1;4]112,8,13,1026.0126.012.631[1;4]122,8,14,121229.21433.21[2;3]133,5,93,513.751225.753.177[2;3]144,7,1125.7525.7510.342[2;3]153,6,13,9321.75425.7516.481[2;3]163,6,14,11328.95432.95[2;3]174,8,13,925.7525.75[2;3]184,8,14,1132.9532.95[2;4]193,5,103,516.371228.378.725[2;4]204,7,121224.37428.3721.264[2;4]213,6,13,10324.37428.37[2;4]223,6,14,123,1227.570.828.3710.011[2;4]234,8,13,1028.3728.37[2;4]244,8,14,121231.57435.57 onbothlinks3and5willbecharged12.Also,noticethatsomecombinationsofsensors,e.g.,5and12,arenotrelevant,becausenoneofthepathscontainbothoftheselinks. Table5-6. Optimaltollingschemefornine-nodenetwork PathsignatureToll 34.058.0124.03,512.03,120.8 83

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5.6.4HeuristicSolutionAlgorithmAsmentionedearlier,thenumberofconstraints( 5{9h )isproportionaltothesquareofthenumberofpaths,whichcanbeverylarge.Inadditiontothehugenumberofconstraints,model MSL-P hasnumerousvariables,proportionaltothenumberofpaths.Thesolutionmethodproposedhereaddressesbothissues.Recallthatifanuncommonlinkofpathspandkisequippedwithsensors,thesetwopathsaredistinguishableandtheirtollscanbedierent,i.e.,thecorrespondingconstraint( 5{9h )willbeautomaticallysatised.Therefore,apairofpathswithmanyuncommonlinkshaveahighprobabilityofbeingdistinguishablewithanarbitrarysetofsensorlocations,e.g.,optimalsensorlocations.Forexample,assumepathspandkhave4uncommonlinksandsensorsaretobeinstalledon30percentofthelinks.Ifsensorlocationsarecompletelyrandom,thereisaprobabilityof1)]TJ /F1 11.955 Tf 12.55 0 Td[((1)]TJ /F1 11.955 Tf 12.55 0 Td[(0:3)4=0:7599thatthetwopathsbecomedistinguishable.Ourproposedmethodrelaxesconstraint( 5{9h )forpathpairswithhighnumberofuncommonlinks,andthenaddthembacktothemodeliteratively,asneeded.Thisapproachmakesthesolutionprocessmoreecientbyeliminatingsomeoftheconstraints,butregardlessofwhichconstraintsareremovedinitially,thenalsolutionsatisesalltheconstraints.Tofacilitatethepresentationofoursolutionalgorithm,wedenethreenewsets,Zasthesetsofequippedlinksatthecurrentiteration,Uasthesetofpathswithnoequippedlink,andSasthesetofpathpairsthat,withahighprobability,willnotbedistinguishableinthenalsolution.ThemethodinitializesZasanemptyset,andthenaddsensorstoititeratively.ForagivensetZ,Pa2Zpawouldbethenumberofequippedlinksonpathp.So,setUcanberepresentedasU(Z)=p2PPa2Zpa=0.Inaddition,Pa2ApkaandPa2Zpkaarethenumberofuncommonlinksanduncommonequippedlinksofpathspandk,respectively.Asbefore,apairofpathswithuncommonlinkscanhavedierenttolls.Ifapathpairhavemanyuncommonlinks,e.g.,greaterthann,thentheyhaveahighprobabilityofbeingdistinguishablewiththenalsetofsensors. 84

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So,thesetofpathsthatareindistinguishableinthecurrentiterationandhavealowprobabilityofbeingdistinguishableinthenaliterationcanbeshownasfollows:S(Z)=((p;k)2P2Xa2Apkan;andXa2Zpka=0)Usingthethreesetsdenedabove,model MSL-P ismodiedasfollowsandsolvedwithaniterativeapproach. (MSL-R)minXa2Aza (5{10a)s:t:f2F (5{10b)Xp2Ppafp=xa8a2A (5{10c)fp(1)]TJ /F3 11.955 Tf 11.96 0 Td[(yp)M8p2P (5{10d)tp+p)]TJ /F3 11.955 Tf 11.96 0 Td[(wypM8w2W;p2Pw (5{10e)tp+p)]TJ /F3 11.955 Tf 11.96 0 Td[(w08w2W;p2Pw (5{10f)pMXa2Azapa8p2U (5{10g)p)]TJ /F3 11.955 Tf 11.95 0 Td[(k)]TJ /F3 11.955 Tf 21.92 0 Td[(MXa2Azapka8(p;k)2S (5{10h)yp;za2f0;1g8a2A;p2P (5{10i)where,constraints( 5{10g )and( 5{10h )arethepartiallyrelaxedversionsof( 5{9g )and( 5{9h ).SetZintheabovemodelwillbeinitializedasanemptyset,thensensorsareaddedtoitinaniterativemanner.ThevalueofnshouldbechosensuchthatthecardinalityofsetSissmallenoughtomakemodel MSL-R solvable.AlowervalueofnresultsinasmallerjSjandmorerelaxedconstraints.AssensorsareaddedtoZateveryiteration, 85

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morepathpairsbecomedistinguishable,shrinkingthesizeofS.Asaresult,thevalueofncanbeincreasedateveryiteration,withoutdrasticallyincreasingthesizeoftheproblem.Atthenaliteration,thevalueofnshouldbehighenoughthatdoesnotexcludeanypairofpathsfromS.So,inthenaliteration,onlytheconstraintsthatarealreadysatisedwiththesensorsinsetZarerelaxed.Sinceourformulationispath-based,weneedamethodtogeneratepathsinsteadofenumeratingallofthem.Hence,wealsoadoptapathgenerationmethodthataddstherequiredpathsiteratively,insteadofgeneratingallthepossiblepaths.Here,weusemodel PATH-GEN togeneratenewpathsforeveryODpair.Thismodelissimilartotheshortestpathproblem,butithassomeextraconstraints.Constraints( 5{11c ),whicharecalledcanonicalcuts,aretheextraconstraintsandtheyensurethegeneratedpathdoesnotalreadyexistinsetPw.Then,wendtheclassofthenewlygeneratedpath,pnew,anditsassociatedtoll.Ifthetotalcost,i.e.,traveltimeplustoll,ofpnewislessthanitsODgeneralizedtravelcost,thepathwillbeaddedtothepathset.ForeachODpair,wecontinuegeneratingnewpathsuntilthetraveltimeofthenewpathisgreaterthanitsODgeneralizedtravelcost. (PATH-GEN)minXij2Ahijtij(xij) (5{11a)s:t:Xj:kj2Ahkj)]TJ /F8 11.955 Tf 15.11 11.35 Td[(Xi:ir2Ahir=Ersk (5{11b)Xij2phrj)]TJ /F8 11.955 Tf 11.96 11.35 Td[(Xij=2phirnp)]TJ /F1 11.955 Tf 11.95 0 Td[(18p2Pw (5{11c)hij2f0;1g8ij2A (5{11d)where,Ersr=1,Erss=)]TJ /F1 11.955 Tf 9.3 0 Td[(1,andallotherelementsofErsarezero.Also,npisthenumberoflinkscomprisingpathp. 86

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Algorithm1Heuristicalgorithmforsensorlocationproblem 1: ForeachODpair,constructtheinitialsetofpathsPw. 2: SetZ=S=U=andsolvemodel MSL-R . 3: Chooseavalueforn,constructSaccordingly,andsetU=A. 4: Solvemodel MSL-R .LetLBandZrepresenttheoptimalvalueofobjectivefunction,andoptimalsensorlocations. 5: repeat 6: Fixthesensorlocationsinthesolution,Z=Z[Z. 7: Updatevalueofnifneeded. 8: forallOD-pairswdo 9: repeat 10: Solvemodel PATH-GEN tondtheshortestpaththatisnotalreadyinPw,callitpnew,anddenotetheoptimalvalueofobjectivefunctionbyl. 11: Findthetollclassofpnewandthecorrespondingtoll, cl(pnew). 12: ifl+ cl(pnew)uwthen 13: AddpnewtoPw,Pw=Pw[fpnewg. 14: endif 15: untill>uw 16: endfor 17: Re-solvemodel MSL-R andletZrepresenttheoptimalsensorlocations. 18: untilnonewpathisaddedtoanyPwandZ=. AteveryiterationofAlgorithm 1 ,exceptthenalone,atleastonepathisbeinggenerated.Sincethenumberofpathsisnite,thealgorithmwillstopafterlimitednumberofiterations.Theconvergencetooptimalsolutioniscontingentupontheproperwayofupdatingn.Ifthevalueofnatthenaliterationishighenough,e.g.,numberofnodesminusone,thatdoesnotexcludeanypathpairsfromsetS,whenthealgorithmstops,thepathtollsinducethetargetlink-owandareimplementablebythesensorsinZ.So,thealgorithmconvergestoafeasiblesolution. 5.6.5NumericalExampleThissectionpresentstheresultsofsolvingtheminimumsensorlocationproblemforpath-dierentiatedtollsforSiouxFallsnetwork,Figure 5-2 ,usingthesolutionalgorithmproposedintheprecedingsubsection.Resultsofmodels MSL-L and MSL-P arealsocomparedtoshowtheadvantageofimplementingpath-dierentiatedtollsovertheiranonymouslink-basedcounterpart. 87

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Forthisexample,wechosetoreplicatethelink-owsthatminimizetotalsystemtraveltime,i.e.,SO.TheinitialpathsetforeachODpairisconstructedbyaddingallthesystemoptimumpaths,i.e.,pathswithminimummarginaltraveltime,andpathsthatarenotlonger(intermsoftraveltime)thanthelongestSOpath.Thesizeoftheinitialpathsetis1096,i.e.,jPj=1096.Werelaxconstraints( 5{9h )forthepathpairswith3ormoreuncommonlinks,i.e.,n=2.Model MSL-R isthensolved,bycallingGurobi5.5solverinC#environment.Weterminatethesolutionprocessafterndinganintegerfeasiblesolution,whenthebestobtainedsolutionis21,andthebestlower-boundis17.Wethenxthelocationofthese21sensorsonthenetwork,byaddingthemtosetZ.Theviolatedconstraintsarealsoaddedbacktomodel MSL-R .Atotalof199newpathsaregeneratedbysolvingmodel PATH-GEN andaddedtosetP.Fortheseconditeration,thevalueofnisincreasedto23,onelessthanthenumberofnodes,sothatnoneoftheconstraints( 5{9h )isrelaxedexceptforthepathsthatarealreadydistinguishable.BecausesetZhasalready21sensors,mostofthepathpairsaredistinguishableandthesizeofproblemdoesnotincreasedrastically.Pathgenerationandresolvingmodel MSL-R havebeenrepeateduntilnonewpathisaddedtothepathset,P.Forthisexample,wehavetorepeatthisloopsixtimes.Thenalpathsethas1601paths,andthebestobtainedsolutioncontains24sensors.ThesensorlocationsaremarkedwithauinFigure 5-2 .Thenumberofsensorsthatareneededtoachievethesameperformanceusingtheanonymouslink-basedschemesis32.Asdiscussedearlier,thetypeofsensorsthatareusedforimplementinglink-basedschemesareoftenbelongtoAVIcategory,thatisalsothetypeofsensorsweproposeforimplementingpath-dierentiatedtolls.So,ourproposedschemehasreducedthenumberofrequiredtollfacilitybymorethan30percent. 88

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. . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20 . 21 . 22 . 23 . 24 . . . . . . . . . . . . . . . . . . . . . . . Figure5-2. SensorlocationsforSiouxFallsnetwork 5.7ANoteonPathGenerationItisworthmentioningthat,topreventcheating,itiscrucialthatmodel MSL-P containsalltheusablepaths,whethertheycontainloopsornot.Toclarifythis,considerthenetworkinFigure 5-3 ,wherethetraveltimeofeverylinkisequalto1.Thetollforvehiclesdetectedonlyonlink1is3andforthosedetectedonbothlinks1and2is0.Inthiscase,avehicletravelingfromoriginatodestinationbwouldbebetter-oifitpassesnodeb,gotonodecandcomesbacktonodeb.Thisexampleshowstheneedtoconsideranypath,whetherithasloopornot,inmodel MSL-P .Forthisreason,model PATH-GEN doesnothaveanyconstraintforavoidingloops. 5.8SummaryandExtensionsInpreviouschapters,wedemonstratedthatpath-dierentiatedschemescanoeramuchbetterperformancethananonymousschemes.Notingthattollcollectiondevices, 89

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. . a . b . c . 1 . . . 2 . 3 Figure5-3. Usablepathwithcycle e.g.,toll-tagreaders,orlicenseplatescanners,canidentifythevehiclesandcollectlocationinformationthatisrequiredtoobtainthepathsofthevehicles,thischapterproposesamethodofimplementationforpath-dierentiatedschemesusingthesameinfrastructuresasanonymouspricing.Westudieddierentvariantsofthesensorlocationproblem,andimplementedthemonsmallnetwork.WealsousedaheuristicmethodtosolvethenalmodelfortheSiouxFallsnetwork.Theresultsoftheproposedschemearesuperiortothelink-basedcounterpart.Infact,forSiouxFallsnetworkourmethodcanreducetheinfrastructurerequiredtoachievethesamelevelofperformancebymorethan30percent.Thesensorlocationproblemcanbeviewedastheproblemofndingtheleastexpensiveschemetoimplementandmaintain,asitminimizesthenumberofsensors.Inthisstudyweassumedthecostoflocatingasensoronallthelinksareequal,sothesolutionthatminimizesthenumberofsensorsalsominimizethecost.However,itismorereasonabletoassumethecostofinstallingasensorisdierentfordierentroads.Forexampleatwolanearterialcanbelessexpensivethanafourlanefreeway.Furthermore,thecostoflocatingtwosensorsontwodirectionofafreewaycanbelessthanthecostfortwosensorsthatarefarapart.Consideringthisfactscanresultinmoreaccuratemodels.Inaddition,weonlyassumedplacingthesensorsonthelinks,whileitcanbemoreecienttolocatethemonthenodessoonesensorcandetectmultipleroads.Addressingtheseextensionscanbeagoodtopicforfutureresearch. 90

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CHAPTER6PRIVACYISSUEOFDIFFERENTIATEDPRICINGSCHEMES 6.1OverviewImplementingadierentiatedpricingschemerequirestrackingvehicles,whichviolatesmotorists'locationprivacy.Locationprivacyisdenedastheabilitytopreventotherpartiesfromlearningone'scurrentorpastlocation( Beresford&Stajano , 2003 ).Theissueoflocationprivacycommonlyariseswhenoeringaservicerequiressomesortoflocationdata.Theissuehasbeenmostlystudiedforsituationswheremobileapplicationsorcomputerprogramsneedtoknowthelocationofauser(e.g., Cvrceketal. ( 2006 )and Ban&Gruteser ( 2010 )).Thetraditionalwayofmanuallycollectingtollpreserveslocationprivacyalmostcompletely.Needlesstosay,itisnotanecientwaytocollecttoll,asvehicleshavetostopandpay.Electronictollcollection(ETC)systemshavebeenbuilttomaketollcollectionmoreecient,butthewaytheycurrentlyoperatemaycompromisemotorists'privacyrights( Sager , 1998 ).Thesystemsoftenlinkmotorists'accountsandrecordlocationsandtimesoftransactions(e.g.,theSunpassprepaidtollprograminFlorida).Iftollgantriesareubiquitous,therecordedtransactioninformationmayimpingeontheprivacyrightsofmotorists.However,thosewhoareconcernedabouttheirlocationprivacyhavetheoptiontopaythetollbycashandavoidriskofprivacydisclosure.Moreover,foranonymouslink-basedtolling,itispossibletodesignaprivacy-preservingETCsystem(e.g., Cavoukian ( 1998 )and Balaschetal. ( 2010 )).Unfortunately,itisdicult,ifnotimpossible,todesignaprivacy-preservingdierentiatedpricingsystem,becausethesystemrequirestheknowledgeoftravelers'locationinformation,e.g.,theoriginanddestinationofeachtripforanOD-specicscheme. Golle&Partridge ( 2009 )and Krumm ( 2009 )pointedoutthatthehome/worklocationdata,eveniftheyareanonymous,canbeusedtoidentifyindividuals.Inaddition 91

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tothis,thesolefactofbeingtrackedbythetollingsystemcancauseinconvenienceordiscomfort.Alltheseprivacyconcernsneedtobeaddressed.Ontheotherhand,therehavebeensomeindicationsthatmotorists,someataprice,arewillingtoprovidelocationinformationwiththeunderstandingthatitwillnotbepublishedand/ormisused.Forexample,intheTravelChoicesStudyby PugetSoundRegionalCouncil ( 2008 ),eachparticipantwasgivena$1016debitaccountwithaGPS-basedon-boardunitinstalledonhisorhercar.Thisunittracksandrecordswhenandwheretheparticipantsdriveanddeductstollsfromtheaccount.Themoneyremainingineachaccountattheendofthestudywasgiventothestudyparticipant.Inthisexample,locationinformationwascollectedforthepurposeoftollingandwithfullknowledgeofstudyparticipants.Wesurmisethattheparticipantsmaybeattractedtothe$1016incentivewhenjoiningthestudy.Thecontributionsofthischapteraretwofold.First,recognizingthatpricedierentiationwithrespecttotravelcharacteristicsmaycompromisetravelers'locationprivacyandproposinganapproachformodelingprivacycost.Second,designinganincentiveprogramthatprovidesincentivesfortravelerstorevealtheirtravelinformationandvoluntarilyparticipateindierentiatedpricing.Suchanopt-inprogramisdesignedtocreateawin-winsituationforbothtravelersandsociety.Fortheremainderofthischapter,Section 6.2 explainslocationprivacy,associatesacosttoit,andpresentsmathematicalmodelstomeasurethatcost.ThenSection 6.3 proposesanincentiveprogramtoaddresstheprivacyissueandillustrateitbynumericalexamples.Thenalsectionconcludeandsummarizethechapter. 6.2LocationPrivacyThissectionexplainshowpeoplevaluetheirlocationprivacyandpresentsamodelforquantifyingtheviolationofprivacyasacost.Thatmodelisusedafterwardsforevaluatingprivacycostofdierentiatedschemes. 92

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6.2.1ValueofPrivacyEmpiricalexperimentsintheliteraturehaveprovedthatindividualsvaluetheirlocationprivacydierently.Theycanbegroupedintocategoriesofprivacyunconcerned,privacypragmatist,andprivacyfundamentalist( Krumm , 2009 ).Therstgroupdonotcareaboutlocationprivacyandareinsensitivetothenegativeconsequencesoflocationleak.Thesecondgrouparewillingtorevealtheirlocationfora,sometimesverysmall,price,whilethelastgrouphighlyvalueandstrivetoprotecttheirlocationprivacy.Mathematically,wecanuseadistributiontorepresentdierentindividualvaluationsofprivacyacrossthepopulation. Acquistietal. ( 2009 )suggestedaU-shapeddistribution,butcautionedthatthevalueofprivacycanbeverymalleableandmanynon-normativefactorsmayaectitsdistribution(alsosee Cvrceketal. ( 2006 )).Hence,wedonotbaseourmodelsonanyspecic,butageneraldistributionforthevalueofprivacy.Nevertheless,itisimportanttounderstandtheimplicationofaproposeddistribution.Forexample,althoughlogisticdistributionsoercomputationaladvantagesaswehaveseeninthechoicemodeling,suchdistributionsimplythatsomeuserswillhaveanegativevalueofprivacy,anunjustiableassumption.Figure 6-1 illustratesmorereasonableuniformandexponentialdistributions,bothwithameanoftwo.Inthisgure,U(0;4)denotesauniformdistributionbetween0and4,andEXP(0:5)isanexponentialdistributionwithaparameterof0.5.Noticethattheexponentialdistributionismoreclusteredaroundsmallervalues,whichimpliesthatmoreusersvaluetheirprivacyless.However,exponentialdistributionswithhighermeanvaluesbecomemoreevenlydistributed.Alsonoticethatthespanforanexponentialdistributionisallnonnegativerealnumbers,whiletheuniformdistributionisboundedonbothsides.So,uniformdistributionimpliesthatthevalueofprivacyoftravelersisevenlydistributedandhasanupperbound.Ontheotherhand,theexponentialdistributionsuggeststhatsometravelersareextremeprivacyfundamentalistsandwillnotdisclosetheirlocationsatanyprice. 93

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. . . () . EXP(0:5) . 0.25 . 0.50 . U(0;4) . 4 . 2 Figure6-1. Uniformandexponentialdistributionswithsamemean,E()=2 6.2.2ModelingPrivacyTheprivacyissueofthethreedierentiatedschemesproposedinChapter 3 canbemodeledinthesamefashion.Thus,weonlymodelprivacyissuefororigin-specicpricingasaninstance,toavoidredundancy.Denotethetravelcost,i.e.,timeplustoll,betweenODpairwunderimplementationofanorigin-specicschemeasw;1,whichconsistsoftraveltimeandtoll.Sincetheyarebeingtracked,motoristsincuradditionalcostforthelossoftheirlocationprivacy,whichwecallprivacycost.Mathematically,thefullcostforatravelerbetweenODpairwunderorigin-specicpricingisw;1+,whereisarandomvariablerepresentingthevalueofprivacy,whichisexpressedintheunitoftimeforsimplicity.Inthischapter,weassumedistributionofvalueofprivacyisthesamefortravelersfromeveryODpair,butthisassumptioncanbeeasilyrelaxed.Supposevalueofprivacyfollowsaknowndistribution,i.e.,.Dene()=R0(z)dzasthecumulativedistributionfunctionassociatedwiththevalueofprivacy,i.e.,Prob()=().DenotethetravelcostbetweenODpairwunderanonymoustollingasw;0andletw=w;0)]TJ /F3 11.955 Tf 12.58 0 Td[(w;1.Asdened,wisthedierencebetweentravelcostsofODpairwunderanonymousandorigin-specicschemes.Ifwisnegative,travelcostofwunderorigin-specicschemeishigherandnoindividualfromthisODpairpreferstheorigin-specicscheme.Otherwise,ifwispositive,travelerswhovaluetheirprivacylessthanwwouldprefertheorigin-specicschemetotheanonymousone,whilethosewithhighervalueofprivacywouldpreferanonymoustolling.Thepercentageofthe 94

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. . w . E(;w) . . () Figure6-2. Expectedprivacycost formeris(w)whileitis1)]TJ /F1 11.955 Tf 12.64 0 Td[((w)forthelatter.Figure 6-2 illustratesthissituationforahypotheticaldistributionofthevalueofprivacy,wheretheshadedarearepresentsthepercentageoftravelerswhowillbebetteroandthusprefertheorigin-specicscheme.Theirtotalprivacycost,denotedasPCw(w),canbecomputedasfollows:PCw(w)=Rw0dw(z)zdz,wheredwisthetotaldemandbetweenODpairw.DeneE(;w)=Rw0z(z)dz,andtheequationisthuswrittenasPCw(w)=dwE(;w).Thecalculationsof()andE(;)involveintegration.Ifthevalueofprivacyfollowsauniformoranexponentialdistribution,theintegralswillhaveaclosedform.Inageneralcasewheretheintegralsdonothaveaclosedform,weneedtocomputethemvianumericalintegrationmethods,suchasRiemannsumasfollows:()=Z0(z)dz=1 nnXi=1(i n)E(;)=Z0z(z)dz=1 nnXi=1iw n(i n)wherenisthenumberofbarsusedtoapproximatethearea.Choosingalargernwouldresultinahigherprecision. 6.2.3PrivacyCostAnalysisofDierentiatedSchemesInthissection,weexaminetheresultsofthenine-nodenetworkinSection 3.4 fromaprivacyperspective.Table 6-1 calculatesthepercentagesoftravelersbetweeneachOD 95

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pairwhowouldbenetfromorigin-specicpricingafterconsideringprivacycost,i.e.,100(w),underdierenthypotheticaldistributionsforthevalueofprivacy. Table6-1. Percentageoftravelerswhobenetfromorigin-specicpricingonnine-nodenetwork NetworkConditionFirst-bestSecond-bestODpair[1,3][1,4][2,3][2,4][1,3][1,4][2,3][2,4]TravelCostSaving7.2-0.17.27.21.50.7-2.00.2 U(0;4)100.000.00100.00100.0037.5017.500.005.00U(0;8)90.000.0090.0090.0018.758.750.002.50U(0;16)45.000.0045.0045.009.374.380.001.25EXP(0:500)97.270.0097.2797.2752.7629.530.009.52EXP(0:250)83.470.0083.4783.4731.2716.050.004.88EXP(0:125)59.340.0059.3459.3417.108.380.002.47 FromTable 6-1 ,itcanbeobservedthat,eveniftheaveragevalueofprivacyishigh,sometravelersstillbenetfromdierentiatedschemes.However,thepercentagedecreasesastheaveragevalueofprivacyincreases.Alsoobservethatwhentravelcostsavingissmall,exponentialdistributionspredicthigherpercentagesofuserswhowillbenetfromdierentiatedschemes,becausethedistributionsaremoreclusteredaroundsmallervalues.Apparently,thesavingsoftimeandtollthatsometravelersenjoyfromdierentiatedschemesareosetbythelossoftheirprivacy.Section 6.3 presentsawaytotakeadvantageofthepotentialsofdierentiatedpricing,whileallowingthoseconcernedtravelerstomaintaintheirprivacy. 6.3AddressingPrivacyConcernsRecognizingthatsomeusersmaybenetfromdierentiatedschemeswhileotherswithhighervalueofprivacymaybebetterounderanonymoustolling,wedesignanincentiveprogramfortravelerstooptintodierentiatedpricing.Morespecically,ahybridofanonymousanddierentiatedpricingschemeswillbeimplementedonthenetwork.Travelerswhochoosetorevealtheirlocationinformationwillpaydierentiatedtollswhilethosewhodonotdisclosetheirinformationpayanonymoustolls. 96

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Sincetravelcosts(timeplustoll)indierentiatedschemesaregenerallylessthanthoseintheanonymousscheme,thecostsavingscanbeviewedasincentivesfordriverstoparticipateindierentiatedpricing.Althoughotherincentives,suchassubsidiesorcredits,canbeprovided,wefocusondesigninganonymousanddierentiatedtollsinthehybridschemeandallowingforthecostsavingsasincentives.Theoverallgoalofthishybridschemeistocreateawin-winsituationforbothusersandsociety. 6.3.1DesignofIncentiveProgramAsanexample,wedesigntheincentiveprogramforahybridoforigin-specicandanonymoustolls.Theformulationsforotherhybridschemescanbedevelopedwithsomestraightforwardmodications.Hence,wedonotpresentthemtokeepthediscussionbrief.Itisreasonabletoassumeallthemotoristswhoarebetterounderanorigin-specicscheme,i.e.,thosewhovaluetheirprivacylowerthanw;0)]TJ /F3 11.955 Tf 11.35 0 Td[(w;1,willoptintothisscheme.Thus,thenumberofthesemotoristswillbedw;1=(w;0)]TJ /F3 11.955 Tf 12.03 0 Td[(w;1)dw.Travelerswhochoosetheanonymousschemewillnotincuranyprivacycost.Hence,thetotalprivacycostfortravelersbetweenODpairwisequaltoPCw(w;0)]TJ /F3 11.955 Tf 11.96 0 Td[(w;1),asdenedinSection 6.2.2 . 97

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Thefollowingconstraintsdenethefeasibleregionoftheproblem:dw;0+dw;1=dw8w2W (6{1)dw;1=(w;0)]TJ /F3 11.955 Tf 11.95 0 Td[(w;1)dw8w2W (6{2)Xp2Pwfp;c=dw;c8w2W;c2C (6{3)fp;c(tp(f)+p;c)]TJ /F3 11.955 Tf 11.95 0 Td[(w;c)=08p2Pw;w2W;c2C (6{4)tp(f)+p;c)]TJ /F3 11.955 Tf 11.96 0 Td[(w;c08p2Pw;w2W;c2C (6{5)w;0w;18w2W (6{6)fp;c08p2Pw;w2W;c2C (6{7)p;c08p2Pw;w2W;c2C (6{8)p;0=Xa2Apaa8p2Pw;w2W (6{9)a08a2A (6{10)p;1=Xa2Apao(w)a8p2Pw;w2W (6{11)o(w)a08a2A;w2W (6{12)whereC=f0;1g.Constraints( 6{1 )and( 6{2 )splitthedemandforeachODpair.Constraint( 6{3 )ensuresowbalance.ThetolleduserequilibriumisguaranteedbyConstraints( 6{4 )and( 6{5 ).Constraint( 6{6 )requirestravelcost(timeplustoll)intheorigin-specicschemetobelessthanthatintheanonymousscheme.Constraints( 6{9 )and( 6{11 )makethetolloneachpathtobeequaltothesumoflinktolls.Denotethefeasibleregiondenedbytheaboveconstraintsas. 6.3.1.1First-bestconditionWerstdiscusstheproblemofndingtheoptimalhybridschemeintherst-bestnetworksettingwhereallthelinksaretollable.Inthissituation,weareinterestedinreplicatingtheowdistributionwithminimumsystemtraveltime,i.e.,x,aswellas 98

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minimizingtheusercostasasecondaryobjective.Thefollowingisthetotal(full)usercost:Xw2W PCw(w;0)]TJ /F3 11.955 Tf 11.96 0 Td[(w;1)+Xp2Pw(p;0fp;0+p;1fp;1)!+Xa2Axata(xa)Sincex=xoughttobeachievedinrst-bestpricing,thelasttermisaconstantandcanbeomittedfromtheoptimization.Consequently,wehavethefollowingformulationforndinganoptimalhybridschemeinanetworkwithalllinksbeingtollable: (HYB-F)minXw2W PCw(w;0)]TJ /F3 11.955 Tf 11.96 0 Td[(w;1)+Xp2Pw(p;0fp;0+p;1fp;1)! (6{13a)s:t:(f;d;;)2Xw2WXp2Pwpa(fp;0+fp;1)=xa8a2Awherethelastconstraintistoensurethelinkowstobetheleast-system-timeows. 6.3.1.2Second-bestconditionSecond-bestsituationhappenswhenonlysomelinksaretollable.Inthissituation,weattempttominimizetotalsystemcost,whichdiersfromtheabovetotal(full)usercostbythetollrevenue,becausetherevenueisnotacostforthesystembutatransferfromtravelerstothegovernment.Theproblemofndinganoptimalhybridschemecanbeformulatedasfollows: 99

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(HYB-S)minXw2W PCw(w;0)]TJ /F3 11.955 Tf 11.96 0 Td[(w;1)+Xp2Pwtp(f)fp! (6{14a)s:t:(f;d;;)2o(w)a=08w2W;a2Thelastconstraintensuresthatonlytollablelinkscanhavepositiveamountoftoll.Noticethattheaboveformulationsarepath-based,andsolvingthemrequirespathenumeration.However,itispossibletoformulatethemaslink-basedmodels.Weusetheabovepath-basedformulationstofacilitatethepresentation. 6.3.2NumericalExamplesTheproposedmodelsfordesigningincentiveprogramfororigin-specicschemewereimplementedonthenine-nodeandSiouxFallsnetworksfromChapter 3 .Eachmodelwassolvedforbothuniformandexponentialdistributionforthevalueofprivacy,eachwiththreedierentexpectedvalues.Table 6-2 presentstheresultsonthenine-nodenetworkofFigure 3-1 withallthelinksbeingtollable.Theperformancesofthehybridschemesarealsocomparedwiththoseoftheanonymousandorigin-specictollswhenimplementedseparately.Aspointedoutpreviously,origin-specicpricingcanreducethetollrevenuesignicantlyinarst-bestnetworkcondition.However,thisreductioncomeswithapriceofviolatingtravelers'privacy.Sinceorigin-specicschemesrequirealltheuserstorevealtheirorigininformation,theprivacycostisequaltotheexpectedvalueofprivacymultiplyingbythetotaldemand.Theprivacycostincreasesastravelersvaluetheirprivacymore,eventuallycausingthetotalusercostunderorigin-specicpricingtobelargerthanthatunderanonymoustolling,whentheexpectedvalueofprivacyisequalto8.Incontrast,thehybridschemeoersanoptionfortravelersofhighvalueofprivacyto 100

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Table6-2. Comparisonofdierentschemesonnine-nodenetwork(alllinkstollable) PricingSchemeDistributionofE()TollRevenuePrivacyCostTotalUserCost Anonymous--887.600.00887.60Origin-specic-2311.60200.00511.60Origin-specic-4311.60400.00711.60Origin-specic-8311.60800.001111.60HybridU(0;4)2247.8228.46276.28HybridU(0;8)4235.2558.47293.72HybridU(0;16)8218.10113.92332.02HybridEXP(0:500)2249.8417.49267.33HybridEXP(0:250)4236.1634.57270.73HybridEXP(0:125)8210.5969.16279.75 remainanonymous.Suchaself-selectionmechanismleadstomuchsmallerlossofprivacyandsubsequentlylesstotalusercost.Interestingly,inthisexample,thehybridschemesalsoleadtolessamountoftollrevenuethantheirorigin-speciccounterparts.However,thisobservationneednotbegenerallytrue. Table6-3. Comparisonofdierentschemesonnine-nodenetwork(twotollablelinks) PricingSchemeDistributionofE()TravelTimePrivacyCostTotalSystemCost Anonymous--2361.160.002361.16Origin-specic-22306.10200.002506.10Origin-specic-42306.10400.002706.10Origin-specic-82306.10800.003106.10HybridU(0;4)22291.799.132300.92HybridU(0;8)42296.7613.082309.84HybridU(0;16)82304.6317.572322.20HybridEXP(0:500)22291.455.822297.27HybridEXP(0:250)42293.479.562303.04HybridEXP(0:125)82299.1013.302312.40 Wesolvedfortheanonymous,origin-specicandhybridschemesonthenine-nodenetworkwhenonlytwospeciclinks,(5,7)and(7,3),aretollable.Table 6-3 displaystheresultsforeachscheme.Asexpected,ineverycase,thetotalcostunderthehybridschemeislessthanthoseintheanonymousandorigin-specicschemes.Interestingly,thehybridschemesalsoyieldevenlesstotaltraveltimethantheorigin-specicscheme,eventhough 101

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thelatteristominimizetotaltraveltime.Asthefeasibleregionsofthesetwomodelsarenotthesame,oneshouldnotexpectthattheorigin-specicproblemalwaysyieldslesstotaltraveltimethanthehybridcounterpart.Becausethehybridschemeproblemistominimizethetotalsystemcostthatincludestotaltraveltimeasonecomponent,andanyfeasiblesolutiontotheorigin-specicproblemcanbeobtainedfromthehybridschemeproblembysettinganonymoustollstobesucientlyhigh,itcanhappenthatthehybridschemeproblemyieldslesstotaltraveltime. Table6-4. Second-besthybridschemesonSiouxFallsnetwork PricingSchemeDistributionofE()(hr)TravelTime(103hrs)PrivacyCost(103hrs)TotalSystemCost(103hrs) Anonymous--74.0430.00074.043Origin-specic-0.0273.0607.21280.272Origin-specic-0.0473.06014.42487.474Origin-specic-0.0873.06028.848101.908HybridU(0;0:04)0.0273.2940.11873.412HybridU(0;0:08)0.0473.4210.13873.421HybridU(0;0:16)0.0873.5910.16373.753HybridEXP(50:0)0.0273.2720.08673.357HybridEXP(25:0)0.0473.3550.10673.461HybridEXP(12:5)0.0873.4550.16373.618 Todemonstratethemodelsonamorerealisticnetwork,wesolvedthemontheSiouxFallsnetworkwherethetollablelinksarethedashedonesinFigure 3-2 .TheobtainedresultsarepresentedinTable 6-4 .Similartothenine-nodenetwork,theprivacycostandtotalcostincreaseastheexpectedvalueofprivacyincreases.Also,theprivacycostandtotalcostundertheexponentialdistributionsislessthanthoseassociatedwiththeuniformdistributions.Theresultsinthissectionillustratethepotentialsoftheincentiveprogramfororigin-specicpricing.Fortwoextremecases,withthevalueofprivacybeingzeroorinnity,thehybridschemeyieldsthesameresultsasdierentiatedoranonymousscheme,respectively.But,intherealworld,thisvalueshouldbeniteandpositive.Ourresultsindicatethattheperformanceoftheincentiveprogramismuchbetterwhen 102

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theexpectedvalueofprivacyisrelativelylow,i.e.,moreusersarewillingtorevealtheirinformationforasmallamountofmoney(previousempiricalstudiesseemsuggestso).Theincentiveprogramalsodemonstratespromisingresultswhentheexpectedvalueofprivacyisrelativelyhigher.Whilethissectiononlyfocusesonahybridoforigin-specicandanonymoustolls,weexpectotherhybridstoperformfavorablyinasimilarfashion. 6.4SummaryandDiscussionRecognizingthatthedierentiatedpricingmaycompromisetravelers'privacy,wehaveproposedanincentiveprogramtoallowtravelerstooptintothedierentiatedpricing,iftheyndtheamountofincentivetobeworthdisclosingtheirlocationinformation.Thisself-selectionmechanismallowsthetollingagencytotakeadvantagesofthepotentialsofdierentiatedpricingwithoutdoingharmtotravelers'privacyrights.Otherapproachescanbeexploredtomitigateprivacyconcernsassociatedwithdierentiatedpricing.Forinstance,insteadofchargingusersbasedontheirtrueorigins,thetollingagencycandesignateatollingareaandthenchargeusersbasedonwheretheyenterthearea.Becausethetrueoriginsarenotrevealed,thisschememaypartiallymitigatetravelers'privacyconcern.Notethatthisschemeisdierentfromthetraditionalcordonpricinginwhichmotoristspayauniformtolltocrossthecordon.Intherenedscheme,motoristsonalinkwithinthetollingareawillpaydierentamountsoftolldependingonwheretheyenterthetollingarea.Wecallthisschemeasasub-networkorigin-specicpricingscheme.Toillustratetheconcept,considerthenetworkinFigure 6-3 wherethetollingareaconsistsofthedashedlinks.Considerthreedierentpaths,p1:2!6!5!7!3,p2:2!1!5!7!3,andp3:1!5!7!3.Whileintheoriginalorigin-specicscheme,travelersonp1andp2willpaythesameamountoftollonlink(5,7),theymaypaydierentamountsoftollfortraversingthelinkundertherenedscheme,becausetheyenterthetolledsub-networkfromdierentnodes(Nodes6and5respectively).Also,unlikeintheoriginalorigin-specicpricing,motoristsonp2and 103

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. . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 Figure6-3. Anillustrativenetwork p3willpaythesameamountoftollforlink(5,7)becausetheybothenterthesub-networkfromNode5.Weimplementedthesub-networkorigin-specicschemefortheSiouxFallsnetwork(Fig. 3-2 ),wherethetollingareaconsistsofdashedlinksandnodes10,11,14,15,17and19.Thebestdesignyieldsasystemtraveltimeof73.215,whichisslightlygreaterthanthesystemtraveltimeof73.060underthetrueorigin-specicscheme.Inthiscase,therenedsub-networkorigin-specicpricingisverypromising.Aninterestingfuturestudycanbeconductedtoexplorehowtoselectthetollingareatoachieveasimilarperformanceasthetrueorigin-specictolling. 104

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CHAPTER7CONCLUSIONThisdissertationproposedanewclassofcongestionpricingschemesthatdierentiatestravelerswithrespecttotheirorigin,OD,orentirepath,andchargesthemdierenttollamountsaccordingly.Weinvestigatedcriticalissuesassociatedwithdesignandimplementationofthenewschemes.Althoughitisknownthatdierentiationcanenhancetheperformanceofpricingschemes,themagnitudeofthisimprovementdeservedinvestigation.Westudieddierentvariantsofdierentiatedcongestionpricingproblem,allofwhichdemonstratesignicantimprovementsovertheiranonymouscounterparts.Asanexample,whenonlysomeofthelinksaretollable,ourresultsshowthatdierentiatedtollscanreducethetotaltraveltimetoasignicantlylowerlevelthanwhatanonymoustollscan.Modelsfordesigningdierentiatedtollsbelongtoafamilyofoptimizationproblemsthatarechallengingtosolve.Westudiedthepropertiesofpath-dierentiatedtolldesignproblemandprovidedreformulationsbasedonthediscussedpropertiesthatareeasiertosolve.Wesolvedthemforrealsizenetworkstodemonstratedtheirperformance.Implementingadierentiatedpricingschemerequirestrackingvehicles.ThisdissertationproposedtoleverageAVIsensors,thesameinfrastructuresthatareusedforanonymouspricing,totrackvehicles.Wefocusedonimplementationofpath-dierentiatedtollsandstudieddierentvariantsofthesensorlocationproblem.Wehaveshownthatusingsensorstoimplementapath-dierentiatedschemeismuchmoreecientthanimplementinganonymoustollswiththesamesensors.Particularly,moretraveltimesavingcanbeachievedifthesensorsareusedtoimplementanoptimalpath-dierentiatedscheme,insteadoftheanonymouscounterpart.Dierentiatedpricingschemesrequiretrackingvehicles,whichmaycompromisetravelers'locationprivacy.Werecognizedthisissueandproposedamethodtomodeltheeectsofprivacyviolationonusers'travelchoices.Tomitigateprivacyconcerns,we 105

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proposedanincentiveprogramthatusestollrebatetoenticeuserstovoluntarilydisclosetheirlocationinformation,iftheyvaluetheirlocationprivacylessthantheamountofrebate,andstayanonymousotherwise.Thisself-selectionmechanismallowsthetollingagencytotakeadvantagesofthepotentialsofdierentiatedpricingwithoutencroachingupontravelersprivacyrights.Wesuspectthattheonlydrawbackoftheproposedschemesistheircomplexity.Forcommuters,thiswillnotbeanongoingissue,becauseonceacommuterselectedapath,shewillusethesamepathforhereverydaycommute.Inaddition,asmoretravelersusesmartphonesforrouteguidancethecomplexityofschemesbecomelessrelevant.Itisalsopossibletodesignlesscomplicatedschemes,buttherewillbeatrade-obetweeneciencyandsimplicity.Insummary,thisdissertationcoversthemostcriticalproblemsrelatedtodesignandimplementationofdierentiatedcongestionpricingschemes.Wedemonstratedthattheproposedschemeshaveagreatpotentialtoimproveperformanceoftransportationsystems.Theyarenotmuchmorecostlytoimplementcomparedtoanonymouscounterparts.Infact,wedesignedanimplementationmethodfordierentiatedpricingthatutilizesthesameinfrastructurescurrentlyusedforimplementinganonymoustolls.Werecognizedthatthenewschemesmaycauseprivacyconcerns,thenaddressedtheissuebydesigninganincentiveprogram.Basedonthisstudy,wesuggestthattheproposeddierentiatedschemesbeconsideredasaviableoptionforcongestionmitigation.Werecommendurbanareaswithcurrentorfutureplansforcongestionpricingtoevaluatedierentiatedschemes,andpossiblyimplementthemasamoreecientalternativetoanonymoustolls. 106

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BIOGRAPHICALSKETCH MahmoodZanguireceivedhisbachelor'sdegreeinCivilEngineeringfromSharifUniversityofTechnologyin2007.HecontinuedhiseducationinTransportationEngineeringprogramofSharifUniversityandgothismaster'sdegreein2010.Laterthatyear,hejoinedthetransportationengineeringprogramoftheDepartmentofCivilandCoastalEngineeringatUniversityofFloridaasaPhDstudentunderthesupervisionofDr.YafengYinandDr.Siriphong(Toi)Lawphongpanich.HereceivedhisPhDfromtheUniversityofFloridainthesummerof2014.DuringhisPhDstudy,Mahmoodalsoearnedamaster'sdegreefromDepartmentofIndustrialandSystemsEngineeringatUniversityofFlorida.Hisresearchinterestsincludetransportationnetworkmodeling,analysis,andoptimization. 114