Business, Sauder School of
DSpace
UBCV
Rezaeinia, Pouya
Master of Science in Business - MScB
University of British Columbia
Tradable permits have been proposed in the literature as a tool for controlling road traffic congestion. Unlike with tolls, travelers in aggregate do not incur net out- of-pocket costs if permits are distributed free. Moreover, if travel conditions do not change, tradable permits can be designed to support the same traffic flows as tolls. However, the two instruments are no longer equivalent if conditions fluctuate and neither permit quantities nor toll levels can be adjusted accordingly. In this thesis, we evaluate the efficiency of tradable permits and road tolls in controlling traffic congestion during a morning peak travel period on a stylized urban road network with daily commuters who travel from their homes to their workplaces. We compare the economic efficiency of the two instruments when road link capacities are susceptible to reductions due to incidents, bad weather, road repairs or other shocks. We apply the dynamic traffic network simulator METROPOLIS in which travelers make mode, departure time and route choice decisions. Fixed permit quantities and toll levels are chosen to control the number of vehicle trips crossing a cordon into the city centre with the goal of maximizing welfare defined by expected social surplus. Simulations are performed for a range of scenarios in which road capacities are correlated or uncorrelated, and suffer reductions of varying magnitude. Most of our analysis suggests that road tolls outperform tradable permits. However, in some instances tradable permits do better; in particular, when the road links that bypass the centre of the network experience capacity fluctuations.
Traffic Congestion Control: A Comparative Analysis ofTradable Permits and Road TollsbyPouya RezaeiniaA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Science in BusinessinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Transportation and Logistics)The University of British Columbia(Vancouver)May 2021© Pouya Rezaeinia, 2021The following individuals certify that they have read and recommended to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Traffic Congestion Control: A Comparative Analysis of Tradable Per-mits and Road Tollssubmitted by Pouya Rezaeinia in partial fulfillment of the requirements for thedegree of Master of Science in Business in Transportation and Logistics.Examining Committee:Robin Lindsey, Professor, Transportation and Logistics, UBCSupervisorAnming Zhang, Professor, Transportation and Logistics, UBCSupervisory Committee MemberWei Li, Associate Professor, Department of Economics, UBCAdditional ExamineriiAbstractTradable permits have been proposed in the literature as a tool for controlling roadtraffic congestion. Unlike with tolls, travelers in aggregate do not incur net out-of-pocket costs if permits are distributed free. Moreover, if travel conditions donot change, tradable permits can be designed to support the same traffic flows astolls. However, the two instruments are no longer equivalent if conditions fluctuateand neither permit quantities nor toll levels can be adjusted accordingly. In thisthesis, we evaluate the efficiency of tradable permits and road tolls in controllingtraffic congestion during a morning peak travel period on a stylized urban roadnetwork with daily commuters who travel from their homes to their workplaces.We compare the economic efficiency of the two instruments when road link ca-pacities are susceptible to reductions due to incidents, bad weather, road repairsor other shocks. We apply the dynamic traffic network simulator METROPOLISin which travelers make mode, departure time and route choice decisions. Fixedpermit quantities and toll levels are chosen to control the number of vehicle tripscrossing a cordon into the city centre with the goal of maximizing welfare definedby expected social surplus. Simulations are performed for a range of scenariosin which road capacities are correlated or uncorrelated, and suffer reductions ofvarying magnitude. Most of our analysis suggests that road tolls outperform trad-able permits. However, in some instances tradable permits do better; in particular,when the road links that bypass the centre of the network experience capacity fluc-tuations.iiiLay SummaryTraffic congestion is growing in many large cities and has significant economic andhealth impacts. For example, in 2019 the average American driver lost nearly 100hours in traffic congestion, causing $88 billion harm to the US economy. Here,we study the performance of road tolls and tradable permits as policy tools forcontrolling traffic congestion during morning rush hours when road capacities varyunpredictably. Tolls have been implemented in cities such as London and Stock-holm. Tradable permits are still under assessment to be implemented in practice.Permits could be distributed to travelers free of charge, and traded in a market.Unlike with tolls, travelers in aggregate would not incur net out-of-pocket costs,which could make tradable permits more acceptable to the public and politiciansthan tolls. We find that in most instances road tolls outperform tradable permitseconomically. However, for certain types of capacity fluctuations tradable permitsdo better.ivPrefaceThis thesis is original and is done by the author, Pouya Rezaeinia. A version of thiswork will appear in the proceedings of the 56th annual conference of the CanadianTransportation Research Forum (2021).vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Road Network and Travelers . . . . . . . . . . . . . . . . . . . . 72.2 Travelersâ Decisions . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Modal and Departure Time Choice . . . . . . . . . . . . . 92.2.2 Route Choice . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Travelersâ Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . 11vi2.3.1 Cost of Public Transit . . . . . . . . . . . . . . . . . . . 112.3.2 Cost of Driving . . . . . . . . . . . . . . . . . . . . . . . 123 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Road Network Topology and Travel Demand . . . . . . . . . . . 133.2 Road Link Capacity Shocks . . . . . . . . . . . . . . . . . . . . . 163.3 Toll and Permit Regulations . . . . . . . . . . . . . . . . . . . . 174 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1 The METROPOLIS Simulator . . . . . . . . . . . . . . . . . . . 204.2 Derivation of Optimal Tolls Using Response Surface Method . . . 214.3 Derivation of Optimal Permit Quantities Using Response SurfaceMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Derivation of Optimal Regulations Using Grid Search . . . . . . . 245 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1 Perfectly Correlated Capacity Shocks . . . . . . . . . . . . . . . 275.2 Independent Capacity Shocks . . . . . . . . . . . . . . . . . . . . 345.3 Shocks to Specific Types of Links . . . . . . . . . . . . . . . . . 376 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.1 Origin-Destination Matrix . . . . . . . . . . . . . . . . . . . . . 49A.2 Batch Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 49viiList of TablesTable 3.1 Specifications of the Links of the Road Network . . . . . . . . 15Table 3.2 Travel Demand Parameter Values . . . . . . . . . . . . . . . . 16Table 5.1 Summary Statistics for No-regulation and Optimal Regulation Equilibria 31Table 5.2 Average Welfare Effects on Good Days by Origin and DestinationWhen Toll Revenue Is Redistributed . . . . . . . . . . . . . . . . 32Table 5.3 Average Welfare Effects on Bad Days by Origin and Destination WhenToll Revenue Is Redistributed . . . . . . . . . . . . . . . . . . . 32Table 5.4 Summary Statistics for Optimal State-independent Regulations on GoodDays and Bad Days . . . . . . . . . . . . . . . . . . . . . . . . 33Table 5.5 Optimal Regulation for Different Capacity Drops . . . . . . . . . . 34Table 5.6 Optimal State-independent Regulation for Different Capacity Dropson Bad Days . . . . . . . . . . . . . . . . . . . . . . . . . . . 35viiiList of FiguresFigure 3.1 Symmetric Circular City Network . . . . . . . . . . . . . . . 14Figure 3.2 In2 (Cordon2) Links - Circular City Network . . . . . . . . . 18Figure 3.3 Timeline of Events and Actions for Tolls . . . . . . . . . . . 18Figure 3.4 Timeline of Events and Actions for Permits . . . . . . . . . . 19Figure 4.1 Algorithm for Finding the Optimal Toll Value . . . . . . . . . 22Figure 4.2 Algorithm for Finding the Optimal Permit Quantity . . . . . . 24Figure 5.1 Welfare Gain vs Toll on Good Days . . . . . . . . . . . . . . 29Figure 5.2 Welfare Gain vs Toll on Bad Days . . . . . . . . . . . . . . . 29Figure 5.3 Welfare Gain vs Permit Quantity on Good Days . . . . . . . . 30Figure 5.4 Welfare Gain vs Permit Quantity on Bad Days . . . . . . . . 30Figure 5.5 Welfare Gain vs Toll for i.i.d. Capacity Shocks . . . . . . . . 36Figure 5.6 Welfare Gain vs Permit Quantity for i.i.d. Capacity Shocks . . 36Figure 5.7 CDFs for Tolls and Permits . . . . . . . . . . . . . . . . . . . 37Figure 5.8 Capacity Shocks to In3 and In4 Links . . . . . . . . . . . . . 40Figure 5.9 Capacity Shocks to In2 Links . . . . . . . . . . . . . . . . . 40Figure 5.10 Capacity Shocks to Out3 and Out4 Links . . . . . . . . . . . 41Figure 5.11 Capacity Shocks to Ring2 Links . . . . . . . . . . . . . . . . 41ixGlossaryLatin CharactersaK : travellersAk: Accessibility of traveller akA = {a1,a2, . . . ,a|A |}: Set of travellersbQ0 ,bQ1 ,bQ2 : Regression parameters for permitsbÏ0,bÏ1,bÏ2: Regression parameters for tollsB: Bad Dayci: Lane capacity of link eiCk(t): Cost of driving for traveller ak departing at time tCËk(t): Traveler kâs estimate of Ck(t)CPTk : Cost of public transit for traveller akdk: Destination of traveller akei: LinksE = {e1,e2, . . . ,e|E |}: Set of links between intersectionsE â² = {eâ²1,eâ²2, . . . ,eâ²|E â²|}: Set of links between intersections and zonesG: Good Dayi: Index of linksj: Iteration counter for the algorithm that finds the optimal permit quantityk: Index of travellerslengthi: Length of link eiL: Welfare loss from not re-optimizing when state changesm: State of road capacitiesM: Number of statesxn: Iteration counter for the algorithm that finds the optimal toll levelok: Origin of traveller akOcci(t): Number of vehicles on link ei at time tp: Probability of Bad DaysPPTk : Fixed penalty of public transit for traveller akP(t)dt: Probability that a traveler departs within the time interval [t, t+dt]qk(t): Cost or profit associated with toll or permit trade for traveller ak departingat time tQ: Permit quantityspeedi: Free flow speed on link eit: Timetâk : Desired time of arrival for traveller akttck(t): Travel time by car for traveller ak when departing at time tttPT (ok,dk): In vehicle travel time traveller ak spends in public transittti(t): Travel time on link ittyH,i(t): Historical travel time on link i on day y at time tttyS,i(t): Simulated travel time on link i on day y at time tttyE,i j(t): Estimated travel time on link j downstream of link i at time tTi: Free flow travel time on link eivn: Intersection nV = {v1,v2, . . . ,v|V |}: Set of intersectionsx: Link capacity reduction level in percentagey: Index of daysW Ïm: Welfare gain from toll Ï in state mW Qm : Welfare gain from permit quantity Q in state mZ = {z1,z2, . . . ,z|Z |}: Set of zonesGreek Charactersαk: Unit cost of travel time by automobile for traveller akαPTk : Unit cost of travel time by public transit for traveller akβk: Unit cost of early arrival for traveller akη : Binary choice model probabilityxiγk: Unit cost of late arrival for traveller akÎ: Set of feasible departure timesδk: Width of on-time window for traveller akµd : Departure time logit scaleµm: Mode choice logit scaleµr: Route choice logit scaleÏi: Toll level on link eiÏi: Dummy variable for permit requirement on link eixiiAcknowledgmentsI would like to express my deep gratitude to my thesis advisors, Professors RobinLindsey and Anming Zhang at the UBC Sauder School of Business, for their ex-pertise, supervision and encouragement at every stage of this thesis.My sincere thanks to Professor Andre´ de Palma at the CY Cergy Paris Uni-versite´ for his contribution. His willingness to give his time so generously is verymuch appreciated.I would like to thank Professor Wei Li for her careful examination of this thesis.I would also like to thank Lucas Javaudin at the CY Cergy Paris Universite´ forhis help in improving the METROPOLIS user interface and answering questions,and the organizers of the CYU seminars on Transport and Urban Economics forhosting my presentation on this thesis at their seminars.xiiiChapter 1Introduction1.1 MotivationTraffic congestion is growing in many large cities around the world. Accordingto TomTom (2019)1, in twenty four major cities in North America, on weekdayscommuters spent more than 30 minutes extra time during morning and eveningrush hours combined.2 Inrix reports that in 2019, on average, American driverslost nearly 100 hours in traffic congestion, which cost them $1,377, amounting tonearly $88 billion harm to the US economy.3 To put this into perspective, NASAâsbudget for the same fiscal year was about $20 billion.4 In addition to its economicimpacts caused by travel delay and higher fuel consumption, traffic congestionincreases emissions of toxic pollutants that have significant negative environmentaland health impacts.There are two types of traffic congestion, namely recurring and non-recurring.The former, also known as peak-hour traffic congestion, is a consequence of howsociety operates. It originates from limited road capacities as well as the high de-mand and desire of people to commute to their workspace at certain times of theday. The latter, also known as temporary disruption, happens as a result of tempo-rary conditions such as bad weather, crashes and other incidents, construction, etc.1https://www.tomtom.com/en gb/traffic-index/2TomTomâs traffic index looks at 416 urban areas in 57 countries.3https://inrix.com/press-releases/2019-traffic-scorecard-us/4https://www.nasa.gov/sites/default/files/atoms/files/fy19 nasa budget estimates.pdf1Traffic congestion can be treated at the supply level by building new roadsand expanding existing ones. However, this can result in induced demand, a phe-nomenon in which traffic volume increases with the construction and expansion ofroads [1]. Traffic congestion can also be tackled at the demand level in many ways:by banning cars from central business districts, imposing license-plate-numberbased restrictions, limiting the number of registered vehicles, creating high occu-pancy vehicle lanes, raising gasoline taxes, expanding or subsidizing public trans-portation, and other methods. The goal of both supply and demand approaches isto alleviate traffic congestion, thereby reducing travel time and potentially improv-ing safety and reducing emissions (although emissions of some pollutants, such asNOx and CO2, increase at high speeds). In this work, we study two other instru-ments for controlling congestion: congestion pricing and tradable permits.1.2 Literature ReviewCongestion pricing, which is a specific form of road tolling, has been widely pro-moted as one of the most efficient means of mitigating traffic congestion. It can in-fluence individualsâ travel choices (i.e., transportation mode, departure time, route,destination and so on) and encourages them to make decisions that are more ef-ficient in terms of social welfare [2]. This is done by imposing tolls on certainroads or by charging fees based on distance or time. Such pricing schemes werefirst studied in the 1950s by Buchanan [3] and Walters [4] for managing trafficcongestion in London, and Vickrey [5] for application in Washington D.C. Morerecently, Eliasson et al. [6] and Bo¨rjesson and Kristoffersson [7], [8], studied theeffects of operating congestion pricing schemes in Sweden. Lehe [9] provides acomprehensive review of urban congestion pricing schemes.Congestion pricing in central business districts has been implemented in Sin-gapore (1975), London (2003), Stockholm (2006), Milan (2008), and Gothenburg(2013), by charging a fee or requiring the purchase of a daily license as a meansto discourage driving to the central business district during peak-hours. It has alsobeen proposed for Lyon in 1977 [10], Hong Kong in 1995 [9] and many other cities,but was rejected due to privacy concerns and lack of public acceptance. Congestionpricing still remains quite limited in practice.2In the United States, New York is set to be the first city to impose congestionpricing (in Manhattan) to mitigate traffic congestion and also to generate revenueto pay for mass transit improvements. Before the emergence of the COVID-19pandemic, the proposed congestion pricing was anticipated to start in early 2021.Electronic road pricing schemes are also expected to be adopted in the near futurein places such as Jakarta and Manchester, and Singapore is planning to upgrade itsElectronic Road Pricing (ERP) system to employ satellite technology. Congestionpricing has never been applied in Canada due to lack of public and political support.Tradable permits are another, quantity-based, demand management tool thathas been studied for application to roads. A predetermined number of driving per-mits are distributed among a selected group of travelers, and only travelers with apermit can access specific road links. Permits can be freely traded, thereby allow-ing the price to be determined in a market and the travelers with the highest will-ingness to pay for a trip to acquire the permits and obtain usage rights. Users cantrade the permits until their marginal benefits are equalized and maximum aggre-gate benefit is achieved. As technology advances, the administrative, transactionand compliance costs of tradable permits are declining. In addition, some of theadministration cost is relieved by allowing users to trade their permits. Althoughtradable permits have been implemented in air transport to manage take-off andlanding slots in so-called coordinated airports such as London Heathrow Airport,they have never been used to control traffic congestion on roads.There is an extensive literature on the choice between price and quantity con-trols to internalize environmental externalities when the benefits and costs are un-certain, and neither tolls nor permit quantities can be varied day-to-day for legal,technical or other reasons.5 In his pioneering work, Dales [11] introduced tradablepermits for regulating usage rights as an alternative to a complex system of pricesto control water pollution. In a seminal paper, Weitzman [12] showed that if costsfluctuate and neither price nor quantity controls can be varied, quantity controlis more efficient than price control if the marginal cost curve is steeper than themarginal benefit curve. Conversely, price control is more efficient if the marginalbenefit curve is steeper. Weitzmanâs work spawned an extensive literature compar-5Non-adaptive regulations provide simple criteria for traffic congestion control and are morelikely to be translated into policy.3ing taxes and tradable permits for reducing emissions; see [13] for a review. Later,Goddard [14] presented tradable permits as a politically acceptable demand reduc-tion strategy for vehicle emission control. Verhoef et al. in [15] were the first toassess the potential of tradable permits to regulate road transport externalities in amore socially acceptable way than Pigovian taxes.Congestion control requires a different approach from emission control be-cause congestion directly affects travel demand, whereas in the absence of regula-tions pollution does not affect the demand for products and activities that create it.In the past decade, transportation researchers have been studying tradable permitsas an alternative to road pricing. Tradable permits have the advantage that they arerevenue neutral: if permits are distributed free, users in aggregate do not incur anet out-of-pocket cost, which is likely to enhance public acceptability.Quantitative analysis through modelling and simulations is required to evaluateand compare the performance of the two instruments in road congestion control.Yang and Wang [16] examined a tradable travel credit system in a network withhomogeneous travelers, where the credits are initially distributed among the trav-elers, and certain link-specific credit requirements are imposed. They providedconditions under which the flow pattern and the credit price are unique both forfixed demand and elastic demand. In addition, the authors characterized the so-cially optimum flow pattern as well as the credit price.Xiao et al. [17] studied tradable permits for morning commute congestion con-trol with time-varying permit regulations under the assumption that late arrival isnot permitted. They established the existence of the Pareto-optimal permit scheme,when full information on the commutersâ values of time is available. Shirmoham-madi et al. [18] considered a small road network when travel demand and capacityare uncertain. In their model, a ceiling is imposed on the price of tradable per-mits, and if the price hits the ceiling the regulator sells extra credits at the ceilingprice. Shirmohammadi et al. focused on the degree of volatility in permit prices,and did not evaluate the efficiency of tradable permits or compare them with a fee.Akamatsu and Kentaro [19] studied a quantity control management scheme theycall tradable bottleneck permits (TBP), for a general network. They established theequivalence of TBP and congestion pricing in the case of full information aboutthe demand, and illustrated the advantage of TBP in the case of partial information4about demand when demand is constant.de Palma et al. [20] compare the performance of price and quantity regulationswhen there are demand and capacity shocks by using a transportation network withmultiple parallel routes and a public transport alternative. The authors assume amixed logit demand and a static supply for each route. Consistent with Weitzmanâsanalysis [12], they establish the equivalence of price and quantity regulation fordemand and supply shocks when the two instruments can be adapted to the state.They also show that when the instruments cannot be adapted, quantity regulationachieves higher social welfare when the marginal external congestion function hasa steeper slope than the demand function.de Palma and Lindsey [21] study a general congestible facility that is proneto demand and supply shocks. They show that when demand and cost curves arelinear, price regulation outperforms quantity regulation for several combinations ofadditive and/or multiplicative demand and cost shocks. More generally, the rankingof the two instruments depends on the nature and magnitude of shocks, the shape ofthe cost function, and whether the permit requirement always binds. Other studiesare reviewed in [10], [21], [22].In this work, we study the performance of road tolls and tradable permits incontrolling traffic congestion during a morning peak travel period on a stylizedurban road network given uncertainty about road link capacities. We apply thedynamic traffic network simulator METROPOLIS in which travelers make mode,departure time and route choice decisions. We investigate the aggregate impacts ofthe two instruments on traffic flows and welfare as well as the welfare-distributionalconsequences on individual travelers when the level of tolls and the number ofpermits distributed are fixed.de Palma et al. [23] developed METROPOLIS6 in 1997 to simulate automobiletraffic in urban areas. It has been used to study congestion pricing in a numberof studies [24], [25], [26], [27], [28]. For instance, de Palma and Lindsey [26]used METROPOLIS to study the impact of road pricing on morning peak trafficcongestion in Ile-de-France, and Vosough et al. [28] investigate the effectivenessof tolls to target emissions and congestion externalities. We use METROPOLIS6A web-interface was developed in 2018 to make METROPOLIS accessible to the public.5to study the efficiency of tolls and tradable permits in managing traffic congestionand compare the performance of the two instruments in the presence of stochasticcapacity shocks.METROPOLIS allows tolls to be set on any subset of road links. The currentversion of the software does not make it possible to restrict directly the numberof automobile trips on road links. However, the number of trips can be controlledindirectly by adjusting tolls until a target volume is met. Quantity constraints canbe effectively implemented in this way.In studying the relative performance of fixed tolls and permits under uncer-tainty, we restrict attention to capacity shocks (supply fluctuations), assuming thatthe travelers are informed of the realized capacities before making trip decisions.Demand fluctuations can be modeled with METROPOLIS by varying the origin-destination matrix and treating each instance as a state with a given probability.However, predictable demand fluctuations such as weekday vs. weekend, and sea-sonal variations can be treated by adjusting the regulations, and do not fit withinthe scope of this thesis.7 Also, it is not clear how to implement daily demand fluc-tuations by varying the origin destination matrix in a plausible way. If all flows arevaried proportionally, the results will be similar to varying capacities proportion-ally.The remainder of this thesis is organized as follows. Chapter 2 contains thepreliminaries, the related useful graph theory concepts, and a description of travel-ersâ decisions and travel costs. Chapter 3 provides a formal characterization of thestatement of the problem under consideration. Chapter 4 includes the simulationmethods. We present our results in Chapter 5. Chapter 6 offers conclusions, andsuggests future directions for research.7Note that, unlike capacity shocks considered here, systematic demand fluctuations such as week-day versus weekend are known well in advance by the planner as well as travelers. Adjusting regu-lations for such demand fluctuations is likely to be viable in practice.6Chapter 2PreliminariesIn this chapter, we present the preliminaries to set the stage for the statement of theproblem under study. First, we describe the model of road network and travelersimplemented in METROPOLIS [29], which is a stylized model of an urban areawith daily commuters who travel from their origins to their destinations around thetime of a peak congestion period. Each origin represents a commuterâs residence,and each destination represents a workplace, school, etc. For each traveler thereare two modes of transportation available, automobile and public transit. Next, wedescribe the travelersâ three decisions as well as the costs that are induced by thosedecisions.2.1 Road Network and TravelersWe start by defining the network describing the intersections and road links of anurban area. Let V = {v1,v2, . . . ,v|V |} and E = {e1,e2, . . . ,e|E |} denote the sets ofintersections (nodes) and directed links (edges) of the network, respectively. Eachlink in E connects two intersections in V uni-directionally, i.e., for each ei â Ethere exist vm,vn â V such that ei = (vm,vn). Each link ei is characterized by acertain length, number of lanes, lane capacity denoted by ci, free-flow speed, con-gestion function describing the travel time function of ei, Ïi denoting the toll leveland Ïi denoting whether ei requires a permit, where Ïi = 1 if a permit is requiredand Ïi = 0 otherwise. These links and intersections represent the main arteries of7a city network and not lower-speed narrow routes and back allies. To describe thetravelersâ origins and destinations, we define zones. Let Z = {z1,z2, . . . ,z|Z |} de-note the set of zones, which are the location points that represent the origins anddestinations of the travelers. We assume that each zone is connected to an intersec-tion via a link. Let E â² = {eâ²1,eâ²2, . . . ,eâ²|E â²|} denote the links between the zones andintersections.Travelers are treated as daily commuters who start their trips at home in themorning. We denote the set of travelers by A = {a1,a2, . . . ,a|A |}. Let ok anddk denote the origin and destination of traveler ak, respectively. As noted above,ok,dk â E â². Each traveler can choose to drive to her1 destination using a sequenceof links, or ride public transit. Each traveler ak is described by a set of parametersâ¢ tâk denoting the desired time of arrival of ak at her destination,â¢ δk denoting the full width of the on-time window, which is the permissibleamount of deviation from the desired time of arrival for which there is nopenalty for ak,â¢ αk and αPTk denoting the value of time spent driving and in public transit,respectively,â¢ βk denoting the penalty incurred per unit of time for early arrival,â¢ γk denoting the penalty incurred per unit of time for late arrival,â¢ and PPTk denoting the fixed penalty associated with using public transit.2.2 Travelersâ DecisionsTravel times and congestion conditions depend on the time of the day, travel de-mand, road link capacities and the travelersâ decisions. We are interested in thesteady state travel times and congestion conditions. To reach an (approximate)steady state, METROPOLIS follows an iterative adjustment process. Starting fromiteration 1, each traveler makes three decisions and seeks to minimize her trip costusing the information available to her in each iteration. In iteration 1, a traveler who1Throughout this thesis, we use she/her as the gender-neutral pronoun.8opts to drive chooses the shortest path that would exist under free-flow conditions(i.e., with no congestion). From the second iteration onward, travelers update theirdecisions based on their experience and new information. After a series of severaldozen iterations, the model reaches an approximate steady state for a given traveldemand and link capacity condition. Individual travel preferences are described bya nested logit model; see [25, Figure 1]. In each iteration, travelers first make abinary choice between taking transit and driving. Those that choose to drive thendecide on their departure time and a route.2.2.1 Modal and Departure Time ChoiceEach traveler chooses between driving to her destination and taking public transit.If the traveler chooses to drive, then she incurs a cost that is determined by herdesired time of arrival, her value of time, congestion, toll level or permit price, etc.We denote the cost of driving for traveler ak by Ck(t), when she departs her originat time t. If the traveler decides to ride public transit, she incurs a cost, denotedby CPTk , that depends on her origin, destination, her value of time spent in publictransit, and the fixed cost of using public transit. The cost functions CPTk and Ck(t)will be specified later in this chapter.Public transit operates either by rail or by bus on dedicated lanes that are im-mune to congestion such as bus lanes and high occupancy vehicle lanes, and hasa given speed. Therefore, the travel times are fixed for all the origin-destinationpairs. If the traveler rides public transit, she departs her origin at a time such thatshe reaches her destination at her desired time. However, if she decides to driveher travel time and consequently her cost depend on her departure time. Departuretime is assumed to be governed by a continuous logit model. Let Î denote the setof feasible departure times. The probability that traveler ak departs within the timeinterval [t, t+dt] isP(t)dt =exp(âCËk(t)/µd)â«Î exp(âCËk(u)/µd)dudt,where µd is a scale parameter measuring the degree of heterogeneity in departure9time choice, and CËk(t) is the travelerâs estimate of Ck(t). Define accessibility asAk = µd lnâ«Îexp(âCËk(u)/µd)du.Traveler ak uses automobile with the probability given by a binary choice model asη =exp(Ak/µm)exp(Ak/µm)+ exp(âCPTk /µm),and uses public transit with probability 1âÎ· . Next, we describe the route choicedecisions.2.2.2 Route ChoiceIf a traveler chooses to drive, she needs to decide on a route. There are two typesof route decisions: pre-trip and en-route. Pre-trip decisions are made before depar-ture, using historical travel times on the links. As mentioned earlier, travelers learnthe historical link travel times using an iterative adjustment process. We denotethe historical travel time on link ei at time t in iteration y by ttyH,i(t), where t isthe moment at which the traveler reaches the entrance to the link. When a travelerreaches an intersection, she observes the realized (simulated) travel times of thedownstream links of that intersection. The en-route decisions are made sequen-tially at each intersection based on the estimated travel times, which consist of thesimulated and the historical travel times. We denote the simulated travel time oflink ei when the traveler reaches the entrance to the link at time t in iteration y byttyS,i(t).The historical travel time on link ei is adjusted in iteration y+1 using the linearMarkov processtty+1H,i (t) =yy+1ttyH,i(t)+1y+1ttyS,i(t).Note that the weight on the simulated travel times declines as iterations proceed.Although we only consider the linear updating process, other learning processescan be considered such as exponential, quadratic and genetic processes [29]. Theseadjustment processes are different in terms of the rate at which they guide travel10decisions towards a stationary state.2When a traveler takes the link ei and arrives at its intersection with downstreamlinks ei1 ,ei2 , . . . ,eiJ , she observes the realized travel times on these links.3 LetttyE,i j(t) denote the estimated travel time for a traveler to arrive at her destinationwhen she enters the link ei j at time t. She updates her estimates of the travel timesas4ttyE,i j(t) = ttyS,i j(t)+ ttyH,R j(t+ ttyS,i j(t)),where R j is the travelerâs route to her destination when she uses the link ei j , andttyH,R j(t+ ttyS,i j(t)) is the sum of the historical travel times of the links on that route.The traveler chooses the link ei j according to a discrete choice model with theprobability given asexp(âttyE,i j(t)/µr)âJj=1 exp(âttyE,i j(t)/µr) ,where µr is a scale parameter measuring the degree of heterogeneity in preferencesfor route. When µr = 0, the traveler chooses the shortest path with probability one.Each traveler incurs a cost that depends on her modal, route and departure timechoices. These costs are specified in the following.2.3 Travelersâ Costs2.3.1 Cost of Public TransitThe cost of riding public transit for traveler ak isCPTk (ok,dk) = αPTk · ttPT (ok,dk)+PPTk ,2Note that convergence to a stationary state is not guaranteed. Iterations oscillate in perpetuityabout a stationary state. This is true not only of METROPOLIS, but also other dynamic trafficsimulators.3Here we do not consider ei as a downstream link.4The estimated travel time for each traveler depends on the travelerâs destination, but to avoidburdensome notation we omit the traveler index here.11where ttPT (ok,dk) is the in-vehicle travel time that ak spends in public transit, andPPTk is the fixed penalty of taking public transit, which includes the cost of walkingto the station, the waiting time, the fare, the discomfort of public transit, etc. Publictransit travel times are exogenously given for each origin-destination pair, and donot depend on the traveler and the number of individuals who take public transit.Thus, transit crowding is assumed away.2.3.2 Cost of DrivingUnlike the cost of public transit, which only depends on origin and destination,the cost of driving depends on departure time, arrival time and route, and is onlyrealized when the traveler reaches her destination. If traveler ak departs at time tand drives to her destination, she incurs the costCk(t) =αk · ttc(t)+βk · [(tâk âÎ´k/2)â (t+ ttc(t))]++ γk · [(t+ ttc(t))â (tâk +δk/2)]++qk(t), (2.1)where ttc(t) is the realized travel time. [ ]+ denotes the positive part function,i.e., [x]+ = max{0,x} for x â R. The terms on the right-hand side are the traveltime cost, the penalty associated with early arrival, the penalty associated with latearrival, and the cost of traversing regulated road links, respectively. If tolls areused, qk(t) denotes the toll paid by traveler ak when departing at time t. If tradablepermits are used, qk(t) denotes either the cost of buying a permit, or the proceedsfrom selling one (in which case qk(t)< 0).12Chapter 3Problem StatementWe consider a network of road links and morning commuters, as described in thepreliminaries. Each traveler attempts to make decisions to her benefit. It is clearthat since travelers attempt to minimize their individual costs, the outcome is inef-ficient when congestion occurs. Implementing road tolls or tradable permits willassist in raising the cost of using (certain) links and discouraging travelers fromdriving or taking shorter routes. If tolls and permit quantities are suitably chosen,this will boost social welfare.We are interested in investigating the efficiency of tolls and tradable permitswhen link capacities fluctuate. We use a stylized circular city network in whichtrips are made between multiple origins and destinations, and congestion is worsenear the city centre. We evaluate the two instruments according to expected socialwelfare: the sum of consumersâ surplus and (in the case of tolls) toll revenue.1 Wealso study the robustness of each instrument to variations in road link capacities.3.1 Road Network Topology and Travel DemandWe consider the symmetric circular city network shown in Figure 3.1. In thisnetwork, the origin-destination demands and link capacities are the same in radial1Welfare is measured using social surplus which assigns equal weight to consumersâ surplus andtoll revenue. Policymakers might opt for unequal weights. Concern for equity and acceptabilitymilitates in favour of putting a higher weight on consumersâ surplus. Budgetary constraints anda positive marginal cost of public funds militate in favour of putting a higher weight on revenuegeneration.13Figure 3.1: Symmetric Circular City Networkdirections and invariant to 45-degree rotations. This network was used in [24], [25],[27], and [28] to study and compare the efficiency of various road pricing schemessuch as flat and time-varying step tolls on parts, or all, of the road network. For thesake of completeness, here we provide a full description of the road network andthe travel demand.The road network consists of 33 intersections that are connected by four con-centric bidirectional ring roads (Rings 1-4) and eight bidirectional arterial links.The ring roads are spaced 4 kilometres apart, and the arterial roads are uniformlyorientated around the compass. The sections of the arterial links between the ringroads are labeled and numbered as in Figure 3.1. The number of lanes, the flow ca-pacity of each lane (vehicles per lane per hour), and the free-flow speed (kilometresper hour) of the links are specified in Table 3.1. All arterial links are 4 kilometreslong. Hence, the ring roads are multiples of 8pi in length.In the absence of congestion, vehicles can traverse a link at its free-flow speed.14Number of Flow Capacity Free-flow SpeedLanes (Veh. per Lane/hr) (km/hr)In1, Out1 1 3,000 50In2-4, Out2-4 2 2,000 70Ring Roads 1 2,000 50Table 3.1: Specifications of the Links of the Road NetworkCongestion occurs if the arrival rate of vehicles exceeds the flow capacity of a link.Vehicles then accumulate in a queue at the downstream end of the link. Queuesare âverticalâ; they do not extend âhorizontallyâ upstream. The delay is computedas the number of vehicles in the queue divided by the capacity of the link. Let Tidenote free-flow travel time on link ei, and Occi(t +Ti) the number of vehicles onthe link. Travel time can then be computed as follows:1. when Occi(t + Ti) â¤ lanesià cià lengthi/speedi, the travel time tti(t) of avehicle entering link ei at time t istti(t) = Ti = lengthi/speedi (hours).2. Otherwise,tti(t) = Ti+(Occi(t+Ti)â lanesià cià lengthi/speedi).Public transportation is assumed to use dedicated lanes (underground, bus lanesor high occupancy vehicle lanes), and operates free of congestion at a speed of40 km/h. Travelers are assumed to have the same unit costs of travel. They dif-fer in terms of their origins, destinations, desired arrival times and idiosyncraticpreferences for mode and departure time. Origins and destinations are locatedat zones that are connected to the network intersections by connectors. An origin-destination matrix describing the origin-destination pairs of 264,000 (8,000 travelerresidences per zone) travelers is given.2 The number of trips between an origin-destination pair is an exponentially decreasing function of the free-flow travel time2The origin-destination matrix is described in the appendix.15between them. Consequently, destinations are concentrated towards the city centre.Parameter values are listed in Table 3.2. These values are the same as the parametervalues in [24] and [25].Parameter Description ValueMode µm Logit Scale $5µd Departure Time Logit Scale $2µr Route Choice Logit Scale $0tâ Desired Arrival Time Uniformly Distributed(Mean = 8:00AM, SD = 20 mins)Auto δ Width of on-time Window 10 minsβ Unit Cost of Early Arrival $6/hrγ Unit Cost of Late Arrival $25/hrα Unit Cost of Travel Time $10/hrPT PPT Fixed Penalty $10αPT Unit Cost of Travel Time $15/hrTable 3.2: Travel Demand Parameter Values3.2 Road Link Capacity ShocksCapacity shocks can be either perfectly correlated or independent. Perfect cor-relation is descriptive of conditions such as bad weather in which all road linksare similarly affected. Independent shocks are descriptive of crashes and otherincidents, construction or other temporary conditions that affect road links individ-ually. In either case, each link experiences one of two states each day: A Goodday or a Bad day. On Good days, which happen with probability 1â p, the roadlink operates at its design capacity. On Bad days, which happen with probabilityp, the linkâs capacity is reduced by x%. The state of the network is determined bythe states of the links. When road link capacities are perfectly correlated there areonly two network states. When capacities are independent, there are 2|E | states.As noted in Chapter 1, the congestion control instruments are not adaptive tothe state of the network and are set based on the values of p and x. Travelersare informed of the realized capacities on a given day before making decisions.We compute the toll level and the permit quantity that yield the optimal expected16welfare gain in the presence of these shocks.3.3 Toll and Permit RegulationsA fixed toll or permit requirement is imposed on the eight In2 links extendingfrom Ring2 to Ring1 shown in Figure 3.2 to form a cordon that will be calledcordon2. Neither the toll nor the permit requirement depends on the time of day.Hence the instruments are âflatâ. This regulatory scheme was selected for severalreasons. First, time-varying instruments take considerably more time to optimizewith METROPOLIS than flat instruments. Second, drivers might have difficultytiming their trips precisely so that they traverse the cordon while their permit isvalid.3 Similarly, imposing tolls or permit requirements on more than one cordonwould complicate traveler decisions. Third, in [24] a toll on cordon2 yielded ahigher welfare gain than a toll on either cordon1 or cordon3 4 or tolls on individuallinks. Since the network is invariant to 45-degree rotations, we assume the tolllevels and permit requirements are the same on all the cordon2 links. In addition,we assume that the permit market is competitive so that all permits trade at thesame price. We focus on user equilibrium in which no traveler can reduce hertravel cost by modifying her decisions.The sequences of events and decisions for tolls, depicted in Figure 3.3, is asfollows:1. The regulator imposes a common and state-independent toll on each of theeight cordon2 links.2. The state (i.e., type of day) on each link is realized.3. Travelers decide whether to drive or take public transit.4. Travelers who choose to drive select a departure time and make a provisional3There is also a practical constraint to adjusting tolls to real-time demand and supply conditions.To date, adaptive tolls have only been used on high-occupancy toll (HOT) lanes, where the travelerscan instantly choose to take a HOT lane and pay the toll, or take the toll-free lanes that run in parallelonly a few metres away. Changing a cordon toll in real time would not be as effective. The travelerscould switch to a longer route, or park their vehicle outside the cordon and complete the journeyusing some other mode. However, the travelers might arrive at their final destinations extremely late.4Similar to cordon2 links, cordonx links for x = 1,3 are the eight Inx links.17Figure 3.2: In2 (Cordon2) Links - Circular City Networkchoice of route that may be revised en route if travelers experience traveltimes that differ from what they expected.TollRegulationState isRealizedDrive orPublic TransitDeparture &Route DecisionsFigure 3.3: Timeline of Events and Actions for TollsIn the case of permits, three types of decisions are made. First, the regulatordecides on the number of permits to distribute, the number of permits required totraverse the cordon, and the allocation of permits to travelers. Second, travelersdecide whether to buy or sell permits. Third, they decide on a mode, departuretime and route. The sequence of events and decisions for permits, depicted inFigure 3.4, is as follows:1. The regulator imposes a common and state-independent permit requirement18on each of the eight cordon2 links. We assume that permit quantities arenormalized so that traversing each cordon2 link requires one permit.2. The regulator allocates the permits to travelers.3. The state (i.e., type of day) on each link is realized.4. Travelers buy or sell their permits in a free market while they plan their trips.The price of permits is determined endogenously in the market. All tradesare made at the same price.5. Travelers decide whether to drive or take public transit.6. Travelers who choose to drive select a departure time and make a provisionalchoice of route that may be revised en route.PermitRegulationPermitAllocationState isRealizedPermitTradeDrive orPublic TransitDeparture &Route DecisionsFigure 3.4: Timeline of Events and Actions for PermitsAs noted, toll and permit regulations are the same each day, but travelershave full information about the state from sources such as traffic websites, mo-bile phones, etc.19Chapter 4Simulation Methods4.1 The METROPOLIS SimulatorWe use METROPOLIS to study the efficiency of tolls and tradable permits in alle-viating road traffic congestion when link capacities fluctuate. The current versionof METROPOLIS does not support quantity constraints on road link usage. Yet,by taking advantage of the equivalence of the two instruments when travel condi-tions do not change (i.e., road link capacities are fixed ), we can replicate tradablepermits by imposing tolls that support target traffic flows. As mentioned in theprevious chapter, tolls and permit regulations are imposed on the cordon2 links. Inthe case of tolls, travelers that traverse these links pay the toll. In the case of per-mits, only travelers with a permit can cross these links. At the user equilibrium, allpermits trade at the same price. This equilibrium price matches the toll value, andthe permit quantity matches the total number of trips that pass through the cordon2links.At least two methods can be used to identify optimal toll levels and permitquantities: the response surface method (RSM) [30] and grid search (GS). Al-though in this thesis we have used GS for the simulations we describe the twomethods for several reasons:1. RSM was used in the earlier METROPOLIS studies as well as in [28].2. Since computation time is a concern, we experimented with the two methods20and found that RSM required fewer iterations and was superior in terms ofspeed. Note that each METROPOLIS simulation, which is run on the UBCserver, takes approximately 15 minutes to be completed. GS on the otherhand, required more iterations and was superior in terms of accuracy.3. GS is not always practical, especially when the optimization problem is de-fined over multiple parameters.In this work, we used an Application Programming Interface (API) that allowed usto run multiple simulations simultaneously. The API is provided in the Appendix.The goal is to find the toll level or permit quantity that maximizes social surplus.4.2 Derivation of Optimal Tolls Using Response SurfaceMethodWe start with the procedure to find the optimal state-independent toll. We assumem= 1, . . . ,M possible states for the road link capacities, with state m occurring withprobability pm. The RSM is an iterative procedure. In each iteration n = 1,2, . . . ,N,we set a trial toll Ïn and find the welfare gain W Ïm(Ïn) for all m. Then we calculatethe expected welfare gain asE {W Ï(Ïn)}=Mâm=1pm Â·W Ïm(Ïn).We start with three arbitrary toll values, and from the third iteration onward,we calculate (or update) our estimate of the optimal toll ÏË using the RSM. For eachstate, by iteratively fitting a quadratic surface of the formbÏ0 +bÏ1Ï+bÏ2Ï2,to the (toll, welfare gain) pairs, we update our estimate of the optimal toll asÏË =â bÏ12bÏ2 , the maximizer of the surface, and record the corresponding expectedwelfare gain.1 We continue this procedure by each time setting ÏË as the new trial1Since the expected welfare gain is unlikely to be a quadratic function of the toll level globally,to derive more accurate values we apply the RSM in a local neighbourhood of the optimal toll . Weidentify this neighbourhood by running simulations for a number of toll levels and observing the21toll value until the difference between the estimates of the optimal toll in two con-secutive iterations is within the desired precision, which we define to be a fractionalchange of eÏ . We set the toll value in the last iteration as the optimal toll Ïâ andthe corresponding expected welfare gain as the optimal expected welfare gain W Ïâ.This procedure is outlined in the flowchart in Figure 4.1. To find the optimal tollfor each state m = 1, . . . ,M, we follow a similar procedure by assuming only onestate m, and replacing E{W Ï(Ïn)} by W Ïm(Ïn).Figure 4.1: Algorithm for Finding the Optimal Toll Valuecorresponding expected welfare gains.224.3 Derivation of Optimal Permit Quantities UsingResponse Surface MethodWe start with the procedure to find the optimal state-independent permit quantity.As noted earlier in this section, we implement the operation of tradable permits byimposing quantity constraints which is done by adjusting toll levels. The M statesare the same as for optimizing the toll. In iteration j = 1,2, . . . ,J, we set a trialpermit quantity Q j and find the welfare gains WQm (Q j) for all m. Then we calculatethe expected welfare gain asE{W Q(Q j)}=Mâm=1pm Â·W Qm (Q j).We start with three arbitrary permit quantities, and from the third iteration onward,we fit a quadratic response surface of the formbQ0 +bQ1 Q+bQ2 Q2,to the (permit quantity, welfare gain) pairs. We then update our estimate of the op-timal permit quantity as QË =â bQ12bQ2and record the corresponding expected welfaregain.2 We continue this procedure by each time setting QË as the new trial permitquantity until the difference of the estimates of the optimal permit quantity in twoconsecutive iterations is within the desired precision, which we define to be a frac-tional change of eQ. We set the permit quantity in the last iteration as the optimalpermit quantity Qâ and the corresponding expected welfare gain as the optimal ex-pected welfare gain W Qâ. This procedure is outlined in the flowchart in Figure 4.2.Note that the subroutine that finds the tolls that support Q j for state m can be im-plemented using binary search (also known as half-interval search) or other searchalgorithms. To find the optimal permit quantity for each state m= 1, . . . ,M, we fol-low a similar procedure by assuming only one state m, and replacing E{W Q(Qn)}by W Qm (Qn).2Similar to tolls there is no guarantee that the expected welfare gain is a quadratic function ofthe permit quantity. To derive more accurate results we apply the response surface methodology in alocal neighbourhood of the optimal permit quantity.23Figure 4.2: Algorithm for Finding the Optimal Permit Quantity4.4 Derivation of Optimal Regulations Using GridSearchAnother method to find optimal toll levels or permit quantities is grid search, alsoknown as parameter sweep, which is an exhaustive search through a specifiedrange of parameter values. Here, the parameters are the two congestion controlinstruments. The grid search algorithm to find the optimal state-dependent andstate-independent tolls is as follows. We choose the smallest trial toll value Ï024and the largest trial toll value ÏN , and the increment to the trial toll âÏ defin-ing our level of precision. For each state m = 1, . . . ,M, and trial toll valuesÏn = Ï0,Ï0+âÏ,Ï0+2âÏ, . . . ,ÏN , we run simulations, and for each one we recordthe welfare gain and the corresponding permit quantity.3 To find the optimal tolllevel, for each state m we fit a quadratic surface of the formW Ïm(Ï) = bÏm,0+bÏm,1 · Ï+bÏm,2 · Ï2,to the (toll, welfare gain) pairs. We take the maximizing toll value and the corre-sponding welfare gain of each surface as the optimal toll level and welfare gain forthat state. To find the optimal state-independent toll level we fit a quadratic surfaceof the formW Ï(Ï) = bÏ0 +bÏ1 · Ï+bÏ2 · Ï2,to the (toll, expected welfare gain) pairs obtained as(Ïn,âMm=1 pm Â·W Ïm(Ïn)). Again,we take the maximizing toll value and the corresponding welfare gain of the surfaceas the optimal state-independent toll level and welfare gain.In addition, this procedure provides us with data points with measurements onpermit quantities and the corresponding welfare gains for each state.Similar to tolls, to find the optimal state-dependent and state-independent per-mit quantities, we choose the smallest trial permit quantity Q0 and the largest trialpermit quantity QJ , and the increment to the trial quantity âQ defining our level ofprecision. Given the (permit quantity, welfare gain) measurements obtained fromthe procedure above, and by using linear interpolation, we compute the welfaregain for trial permit quantities Qn = Q0,Q0+âQ,Q0+2âQ, . . . ,QJ for each statem. Then, we fit a quadratic surface of the formW Qm (Q) = bQm,0+bQm,1 Â·Q+bQm,2 Â·Q2,to the (permit quantity, welfare gain) pairs. We take the maximizing permit quan-tity and the corresponding welfare gain of each surface as the optimal permit quan-3As mentioned in Chapter 3, the permit quantity matches the number of trips that pass throughIn2 links.25tity and welfare gain for that state. To find the optimal state-independent permitquantity we fit a quadratic surface of the formW Q(Q) = bQ0 +bQ1 Â·Q+bQ2 Â·Q2,to the (permit quantity, expected welfare gain) pairs obtained as(Q j,âMm=1 pm Â·W Qm (Q j)). We take the maximizing permit quantity and thecorresponding welfare gain of the surface as the optimal state-independent permitquantity and welfare gain.26Chapter 5Simulation ResultsAs mentioned earlier, capacity shocks are assumed to be either perfectly correlatedor independent. First, we provide results for perfectly correlated shocks, and thenindependent shocks. We conclude with robustness analysis for the two instruments.5.1 Perfectly Correlated Capacity ShocksWe find the optimal (i.e., social-surplus maximizing) state-dependent and state-independent toll levels and permit quantities for two states, Good days and Baddays, and the corresponding welfare gains relative to no regulation. Bad days en-tail capacity reductions of x = 20%. Studies on freeways in Tokyo [31] and Hous-ton [32] have found that rain can reduce freeway flow capacity by up to 14â16%.The percentage reduction is rounded up here to 20% so that potential differencesbetween tolls and permits are easier to detect.1We also investigate the welfare-distributional effects on the travelers based on their origins and destinations. Thenwe find the state-independent toll level and permit quantity, and their correspond-ing expected welfare gains, and compare the efficiency of the two instruments.In addition, we present summary statistics for the optimal state-independent reg-ulations for each state, and examine how much they differ to assess how muchstate-independent regulation falls short of being optimal for each state.To find the optimal toll in each state, we apply a grid search by incrementing1Severe snow storms and other extreme weather conditions can reduce road capacity by morethan 20%, but such conditions are typically infrequent.27the toll in steps of $0.20 from $3 to $7.80, and find the user equilibrium at each tolllevel and record the welfare gain. Then we fit a second-order polynomial curve tothe set of (toll, welfare gain) trial points, and find the maximum of the curve. OnGood days, when road links operate at their design capacities, the optimal toll levelis $4.71 and the welfare gain is $62,656.2 On Bad days, when road link capacitiesdrop by 20%, the optimal toll level is $6.23 and the welfare gain is $57,309. The(toll, welfare gain) pairs and the fitted curves for Good days and Bad days areshown in Figure 5.1 and Figure 5.2, respectively.Note that the maximum number of individuals that can travel without conges-tion increases with capacity. Since road link capacities drop on Bad days whiletravel demand remains unchanged, congestion begins to emerge at a lower level ofusage than on Good days and boosts the optimal toll on Bad days above the toll onGood days.We again use a grid search to compute the optimal permit quantity. We incre-ment the permit quantity in steps of 550 from 27,050 to 40,250. As explained inSection 4.4, by using interpolation and the measurements obtained in the optimaltoll derivation process, we find the (permit quantity, welfare gain) pairs and thenfit a second-order polynomial curve to these trial points, and find the maximum ofthe curve.On Good days, the optimal permit quantity is 37,016 and the welfare gain is$62,463. On Bad days, the optimal permit quantity is 28,752 and the welfare gainis $57,127.3 The (permit quantity, welfare gain) pairs and the fitted curves forGood days and Bad days are shown in Figure 5.3 and Figure 5.4, respectively.For each state, we record the properties of the no-regulation equilibrium andthe user equilibrium when the optimal regulation (toll or permit) is employed. Ta-ble 5.1 presents summary statistics in each state for the toll regulations. Since tollsand permits both support optimal usage under stationary conditions, the statistics2The maximum relative standard deviation for all the welfare gains reported throughout this chap-ter is 1.68%. Standard deviation for each simulation is computed for the last 50 iterations of that sim-ulation. Standard deviations for the expected welfare gains are computed as the standard deviationof the weighted average of the states and by assuming independence.3Note that in theory the optimal welfare gains on Good days and Bad days must be identicalfor tolls and permits. However, since by fitting a second order polynomial we find the approximatewelfare gains, these values are slightly different for the two instruments.28Figure 5.1: Welfare Gain vs Toll on Good DaysFigure 5.2: Welfare Gain vs Toll on Bad Daysfor permits are the same and hence omitted. When either of the regulations is im-posed, the percentage of trips made by automobile, the length of the peak period,the congestion index4, the average auto travel time and the schedule delay cost (i.e.,the cost of arriving early or late) decrease. In addition, the average driving speedand more importantly the social surplus increase. Therefore, regulations alleviateoverall traffic congestion and improve social welfare. Nevertheless, the averagedriving distance of auto trips, the average free-flow travel cost and late arrivalsincrease as some travelers choose to take longer trips to avoid the regulated links.Permits can be uniformly distributed among the travelers. Since one permit is4This congestion index is the same as the one used by TomTom.29Figure 5.3: Welfare Gain vs Permit Quantity on Good DaysFigure 5.4: Welfare Gain vs Permit Quantity on Bad Daysrequired to cross the cordon, and only a fairly small fraction of all travelers crossthe cordon, each person receives only a fraction of a permit. In aggregate, tradablepermits leave travelers better off if they receive permits free of charge. Such is notthe case with the toll since average consumerâs surplus drops. However, the regula-tor can offset the loss by rebating toll revenues. If a rebate is distributed in the formof a uniform lump sum to each traveler, the net welfare effect on each person is thesame as for tradable permits if permits are also allocated uniformly to each person.Nevertheless, every person does not necessarily end up better off because the ef-fects of regulation vary by origin, destination, individual preferences and whetherthe person needs a permit to traverse one of the cordon2 links. Table 5.2 and Ta-30Good Day Bad Day (20% Reduction)Statistic No Reg. Optimal Reg. No Reg. Optimal Reg.Auto share 69.8% 66.2% 66.6% 62.0%Free-flow travel cost $2.63 $2.71 $2.70 $2.80Peak period length 1.34 h 1.23 h 1.40 h 1.30 hCongestion index 35.4% 20.7% 52.3% 33.3%Avg. auto mileage 14.9 km 15.0 km 15.1 km 15.4 kmAvg. auto speed 41.7 km/h 45.9 km/h 36.8 km/h 41.2 km/hAvg. travel time 21.4 min 19.6 min 24.6 min 22.3 minTravel cost $5.04 $5.58 $5.71 $6.22Consumerâs surplus $-4.38 $-4.80 $-4.76 $-5.21Early arrivals 59.6% 57.6% 59.8% 57.7%On-time arrivals 29.3% 30.5% 28.7% 29.7%Late arrivals 11.1% 11.9% 11.5% 12.6%Schedule delay cost $1.47 $1.32 $1.6 $1.41Welfare gain â $62,656 â $57,309Auto share: percentage of trips made by autoFree-flow travel cost: average auto free-flow travel time costPeak period length: duration of peak period, starting when 10% of drivers have reachedtheir destination, and ending when 90% have reached their destinationCongestion index: congestion delay as percentage of free-flow travel time on same routeAverage auto mileage: average kilometers per auto tripAverage auto speed: average auto travel speedAverage travel time: average auto travel timeTravel cost: average travel cost for auto including the price of regulationsConsumerâs surplus: average user surplusEarly arrivals: Percentage of drivers arriving earlyOn-time arrivals: Percentage of drivers arriving within their desired arrival-time windowLate arrivals: Percentage of drivers arriving lateSchedule delay cost: Mean cost of arriving by auto early or lateWelfare gain: the increase in social surplus when a regulation is imposedTable 5.1: Summary Statistics for No-regulation and Optimal Regulation Equilibriable 5.3 show the net average welfare effects on individual travelers by origin anddestination on Good days and Bad days, respectively. A positive value indicatesthat a group is better off, and a negative value that it is worse off. All but six groupsend up better off. Those that lose start their trips on or outside Ring2, and end theirtrips inside it. This is not surprising since these groups cannot avoid paying thetoll or using a permit if they choose to drive. All the welfare effects are larger inmagnitude for Bad days than Good days.31OriginDestinationCBD Ring1 Ring2 Ring3 Ring4CBD 0.74 0.76 0.76 0.75Ring1 1.45 1.34 1.19 1.17 1.13Ring2 -1.30 -1.47 0.17 0.31 0.31Ring3 -1.08 -1.25 0.43 0.37 0.45Ring4 -0.98 -1.14 0.47 0.49 0.53Table 5.2: Average Welfare Effects on Good Days by Origin and Destination WhenToll Revenue Is RedistributedOriginDestinationCBD Ring1 Ring2 Ring3 Ring4CBD 0.74 0.77 0.80 0.80Ring1 1.58 1.37 1.28 1.29 1.24Ring2 -1.75 -1.93 0.32 0.47 0.47Ring3 -1.46 -1.60 0.59 0.50 0.57Ring4 -1.30 -1.43 0.64 0.60 0.60Table 5.3: Average Welfare Effects on Bad Days by Origin and Destination WhenToll Revenue Is RedistributedNext, we study state-independent regulations when the probability of a Badday is 0.25. The optimal state-independent toll level is $4.82. It yields a socialwelfare gain of $62,593 on Good days and $54,742 on Bad days, for an expectedgain of $60,630. The optimal state-independent permit quantity is 36,233, whichyields a social welfare gain of $62,075 on Good days and $48,710 on Bad days,for an expected gain of $58,734. Permits trade at a price of $4.94 on Good days,and $3.72 on Bad days. The price is lower on Bad days because the cost of drivingis higher, and travelers have a lower willingness to pay for a permit.In summary, when road link capacities are perfectly correlated, and regulationsare non-adaptive, tolls yield an expected welfare gain about 3% higher than per-32mits. Tolls also outperform permits in each state. 5 These results correspond fairlyclosely to the model in [21] in which tolls outperform permits with multiplicativeshocks. Summary statistics for the optimal state-independent regulations on GoodDays and Bad Days are provided in Table 5.4. The state-independent regulationsfall short of being optimal for each state, particularly on Bad days. Indeed, thecongestion index on Bad days is roughly double its level on Good days whichhighlights the significant differences in travel conditions on the two types of day.Good Days Bad Days (20% Reduction)Statistic Toll = $4.82 Q = 36,233 Toll = $4.82 Q = 36,233Auto share 66.3% 65.9% 63.0% 63.8%Free-flow travel cost $2.70 $2.70 $2.74 $2.73Peak period length 1.24 h 1.23 h 1.33 h 1.35 hCongestion index 19.61% 18.7% 38.4% 41.0%Avg. auto mileage 15.0 km 15.0 km 15.1 km 15.1 kmAvg. auto speed 46.3 km/h 46.6 km/h 39.8 km/h 39.2 km/hAvg. travel time 19.4 min 19.3 min 22.8 min 23.1 minTravel cost $5.54 $5.57 $6.18 $6.13Consumerâs surplus $-4.81 $-4.82 $-5.15 $-5.09Early arrivals 57.9% 57.9% 58.6% 58.7%On-time arrivals 30.3% 30.3% 29.2% 29.0%Late arrivals 11.8% 11.7% 12.2% 12.3%Schedule delay cost $1.31 $1.30 $1.45 $1.47Welfare gain $62,593 $62,075 $54,742 $48,710Table 5.4: Summary Statistics for Optimal State-independent Regulations on GoodDays and Bad DaysIn our simulations, we have also considered x = 10% and x = 30% capacityreductions on Bad days. The optimal toll values, the optimal permit quantities, andthe corresponding welfare gains are reported in Table 5.5. Again, since the traveldemand stays unchanged, for larger capacity drops congestion begins to emerge atlower levels of usage, which results in higher optimal toll values. Similarly, largercapacity drops result in lower permit quantities. The state-independent optimal tollvalues and permit quantities, and the corresponding expected welfare gains are re-ported in Table 5.6. In all cases, Good days and Bay days are assumed to occurwith probabilities 1â p = 0.75 and p = 0.25, respectively. Tolls uniformly out-5Note, however, that in general one instrument could outperform the other on average, but doworse in particular states.33Capacity ReductionRegulation Parameter x = 0% x = 10% x = 20% x = 30%Tolls Ïâm $4.71 $5.52 $6.23 $6.68W Ïm(Ïâ) $62,656 $61,555 $57,309 $53,436Permits Qâm 37,016 32,719 28,752 25,493W Qm (Qâ) $62,463 $61,460 $57,127 $53,297Ïâm: optimal toll value in state mW Ïm(Ïâ): optimal welfare gain in state m when state-dependent tolls are em-ployedQâm: optimal permit quantity in state mW Qm (Qâ): optimal welfare gain in state m when state-dependent permits areemployedTable 5.5: Optimal Regulation for Different Capacity Dropsperform tradable permits by a margin of 2â5%. Perhaps surprisingly, the optimalstate-independent toll value and the optimal state-independent permit quantity varylittle with the size of the capacity drop.As shown in Table 5.6, the optimal state-independent toll decreases slightlywith the magnitude of capacity drops on Bad days and gets a bit closer to theoptimal toll on Good days. This happens because the number of car trips on Baddays declines as capacity drops. Consequently, welfare on Bad days depends lesson the toll level that is chosen, and more weight is attached on the tollâs effectson Good days. In an extreme case in which capacity dropped to zero, the welfaregain on Bad days would be zero no matter what toll is chosen, and the optimalstate-independent toll would match the optimal toll for Good days. Table 5.6 alsoshows that the optimal state-independent permit quantity increases slightly withthe magnitude of capacity drops on Bad days and gets closer to the optimal permitquantity on Good days. The explanation is similar to that for the toll.5.2 Independent Capacity ShocksIn the independent capacity scenario, each of the 128 unidirectional links on theCircular City network is independently susceptible to a 20% capacity drop with34Capacity Reduction on Bad DaysRegulation Parameter x = 10% x = 20% x = 30%Tolls Ïâ $4.84 $4.82 $4.82W Ïâ$61,952 $60,630 $59,489Permits Qâ 36,184 36,233 36,300W Qâ$60,708 $58,734 $56,783Ïâ: optimal state-independent toll valueW Ïâ: optimal welfare gain when state-independent tolls are em-ployedQâ: optimal state-independent permit quantityW Qâ: optimal welfare gain when state-independent permits are em-ployedTable 5.6: Optimal State-independent Regulation for Different Capacity Drops onBad Daysprobability 0.25. From this distribution, we take 25 random draws.6 As before, tocompute the optimal state-independent toll, for each sample draw we increment thetoll in steps of $0.20 from $3 to $7.80, and find the user equilibria for each toll level.Then we fit a second-order polynomial curve to the (toll, expected welfare gain)pairs to find the optimal toll level and the corresponding expected welfare gain. Thecurve is shown in Figure 5.5. The optimal state-independent toll is $4.88, whichincreases social welfare by $59,328 relative to no regulation. Similarly, to computethe optimal state-independent permit quantity, for each sample draw we incrementthe number of permits in steps of 550 from 27,050 to 40,250, and record the wel-fare gain. Then we fit a second-order polynomial curve to the (quantity, expectedwelfare gain) pairs to find the optimal quantity and the corresponding expectedwelfare gain. The curve is shown in Figure 5.6. The optimal state-independentpermit quantity is 35,936, which increases social welfare by $59,647.Unlike for the case of correlated capacities, with independent capacities thepermit yields an expected welfare gain that is about 0.5% higher than for the toll.6Computation time militated against taking a larger number of draws. Each METROPOLIS sim-ulation takes approximately 15 minutes to complete. Using the grid search method, 25 simulationswith the trial toll values of 3,3.2, . . . ,7.8 are required for each sample draw to find the optimal wel-fare gain. Including pre-processing and post-processing time, this takes nearly 7 hours. Taking alarger number of draws would, therefore, be quite time-consuming.35Figure 5.5: Welfare Gain vs Toll for i.i.d. Capacity ShocksFigure 5.6: Welfare Gain vs Permit Quantity for i.i.d. Capacity ShocksTo compare the frequency distribution of gains of the two instruments for the 25random draws, we apply first- and second-order stochastic dominance. An instru-ment A has first-order stochastic dominance (FOSD) over another instrument B if,for any welfare gain value w, A has a higher probability than B of achieving atleast w. FOSD can be assessed by comparing the cumulative distribution functions(CDFs) of the two instruments. A has FOSD over B if FA(w)â¤ FB(w) for all valuesof w. As shown in Figure 5.7, the CDF for the toll lies mostly above the CDFfor the permits. However, the two CDFs cross at a welfare gain of about $56,000.There is one random draw for which the toll yields a higher welfare gain than thepermit. Hence, neither instrument outperforms the other as far as FOSD.36Hence, we turn attention to second-order stochastic dominance (SOSD). In-strument A has SOSD over instrument B if, for any welfare gain value w,â« wââ[FA(Ï)âFB(Ï)]dÏ â¤ 0, (5.1)with strict inequality for some w. If this condition is satisfied, a risk-averse regula-tor will prefer instrument A as it involves less risk. As shown in Figure 5.7, permitshave SOSD over tolls because the integral under the CDF for the toll exceeds theintegral under the CDF for the permit.Figure 5.7: CDFs for Tolls and PermitsAlthough tolls outperform permits with correlated shocks, tradable permits ap-pear to be more effective than tolls at internalizing congestion externalities for in-dependent shocks. This could be because route-choice decisions are more volatilewhen link capacities vary independently, and tradable permits are more effectivethan tolls at internalizing congestion externalities in these circumstances. Also, theregulations are uniform across the cordon2 links. When capacity shocks are inde-pendent, in most cases the network loses its symmetry. Uniform permit regulationsare likely to be more successful in controlling traffic congestion than uniform tollswhen the network is asymmetric.5.3 Shocks to Specific Types of LinksIn some circumstances it is possible to rank the fixed toll and fixed permit withoutcalculating expected costs for any particular probability or frequency distribution37of states. Suppose that each instrument is set optimally conditional on a particularstate. The instruments can be ranked by calculating the efficiency loss if the instru-ments are not re-optimized when the state changes. For the toll and the tradablepermit, the maximum achievable welfare gains in state m are W Ïm(Ïâm) and WQm (Qâm),respectively. If Ï deviates from Ïâm, the welfare loss for the toll isLÏ(m) =W Ïm(Ïâm)âW Ïm(Ï). (5.2)Similarly, if Q deviates from Qâm, the welfare loss for the permit isLQ(m) =W Qm (Qâm)âW Qm (Q). (5.3)Since the toll and permit are equally efficient when optimized for a particular state,W Ïm(Ïâm) =WQm (Qâm).The efficiency of the toll and permit can be compared using Equations (5.2)and (5.3). Suppose that Ï and Q are set optimally conditional on state m, but statemâ² is realized instead. Total welfare is then W Ïmâ²(Ïâm)with the toll, and WQmâ²(Qâm)withthe permit. The following theorems establish a link between the loss functions andthe relative performance of tolls and permits under variable conditions.Theorem 1 Assume W Ïmâ²(Ïâm)â¥W Qmâ²(Qâm) for all m and mâ². Then, the optimal fixedtoll weakly outperforms the optimal fixed permit for any non-degenerate frequencydistribution of m. If the inequality is reversed, the optimal fixed permit weaklyoutperforms the optimal fixed toll.Proof 1 Let f (m) be a nondegenerate PDF of m, which may be discrete or con-tinuous. Given f , let Ïâ be the optimal fixed toll, and Qâ the optimal fixed permitquantity. Let m¯ be a particular value of m such that Qâ¯m = Qâ. (Note that m¯ may ormay not be in the support of f .) Finally, let E{W Ï(Ïâ)} denote the expected wel-fare gain given the fixed toll of Ïâ, and E{W Q(Qâ)} denote the expected welfaregain given the fixed permit quantity Qâ.38Since Ïâ is optimal given f ,E{W Ï(Ïâ)} â¥ E{W Ï(Ïâm¯)}. (5.4)Furthermore,E{W Ï(Ïâm¯)}âE{W Q(Qâ)}=â«M[W Ïmâ²(Ïâm¯)âW Qmâ²(Qâm¯)]f (mâ²)dmâ² â¥ 0, (5.5)where the inequality follows by assumption with m¯ in place of m. Together, in-equalities (5.4) and (5.5) yieldE{W Ï(Ïâ)} â¥ E{W Q(Qâ)}. QEDTheorem 2 Assume W Ïmâ²(Ïâm)>WQmâ²(Qâm) for all distinct m and mâ². Then, the opti-mal fixed toll strictly outperforms the optimal fixed permit for any non-degeneratefrequency distribution of m. If the inequality is reversed, the optimal fixed permitstrictly outperforms the optimal fixed toll.The proof of Theorem 2 is analogous to the proof of Theorem 1.To compare the performance of the two instruments, we use the welfare lossfunctions. We set the toll and permit optimally for road links with their designcapacities. Four set of links are varied independently, and the losses LÏ and LQ areevaluated for different magnitudes of capacity drop. The results are illustrated inthe following four graphs.Figure 5.8 shows the welfare loss for capacity shocks to In3 and In4 links, thelinks that are used by the travelers with origins on the periphery of the network(Ring3 and Ring4). For the two instruments, the welfare loss remains low withsmall shocks and increases for those of higher magnitudes. For the permit, thiswelfare loss increases significantly to more than 40% when the road links experi-ence capacity drops of 40%.Figure 5.9 shows the welfare loss for capacity shocks to In2 links, which arethe regulated links. Travelers with destinations at the centre of the network (Ring1and the CBD) that choose to drive, inevitably use these links if they drive. Thosewith destinations on the periphery of the network can bypass these links if desired.39Figure 5.8: Capacity Shocks to In3 and In4 LinksSimilar to the previous case, the welfare losses of the two instruments are low forsmall shocks. The welfare loss for the permit increases dramatically when the roadlinks experience capacity drops of over 30%.Figure 5.9: Capacity Shocks to In2 LinksFigure 5.10 shows the welfare loss for capacity shocks to Out3 and Out4 links,the links that are used by travelers with destinations on the periphery of the net-work. Again, the welfare loss of the two instruments is low for small shocks andincreases for those of higher magnitudes. The two instruments have fairly similarperformance, although the welfare loss is still higher for the permits. Since theOut3 and Out4 links are not as heavily used by the travelers as the In2, In3 and In440links, the welfare loss for the Out3 and Out4 links is smaller.Figure 5.10: Capacity Shocks to Out3 and Out4 LinksFinally, Figure 5.11 shows the welfare loss for capacity shocks to Ring2 links.These links are mainly used by travelers with both origins and destinations on theperiphery of the network. The welfare loss of the toll is low for small shocks anduniformly increases for those of higher magnitudes. The welfare loss of the permitincreases for capacity drops of up to 20%, and appears to plateau for capacity dropsof between 20% and 40%. Note that, here, the percentage welfare loss is less than1%.Figure 5.11: Capacity Shocks to Ring2 LinksIn summary, the toll is more robust than the permit to variations in link capac-41ities for the first three cases. The permit is more robust with respect to capacityshocks to the Ring2 links. This could be because with capacity shocks to Ring2links, more travelers would prefer to traverse the In2 links. A higher toll wouldbe required to discourage them, but the toll is fixed. By contrast, the fixed permitprevents any traffic diversion. The losses for the Ring2 links are small even thoughthey are the closest alternative to crossing the cordon. Shocks to the Ring3 andRing4 links are not considered since they are not as close substitutes, and are lessheavily traveled than the Ring2 links.42Chapter 6ConclusionUsing the dynamic traffic network simulator METROPOLIS, we have studied theabsolute and relative performance of road tolls and tradable permits in control-ling traffic congestion. Tradable permits have been promoted in the literature as arevenue neutral and more publicly acceptable alternative to road tolls. For station-ary travel conditions we established that the two instruments reduce overall trafficcongestion and increase social welfare, although some travelers end up worse offdepending on their origin and destination.We studied the performance of the two instruments when road link capacitiesfluctuate. When these fluctuations are perfectly correlated, the simulation resultsindicate that road tolls outperforms tradable permits. This is consistent with theanalytical results for cost shocks in [21]. A Monte Carlo exercise indicates thatwhen capacity fluctuations are independent, tradable permits yield higher welfaregains than road tolls in most states, and outperform road tolls in terms of second-order stochastic dominance.Finally, we studied the robustness of the two instruments to capacity fluctu-ations incurred by each type of road link. Our results indicate that road tolls aremore robust than tradable permits when the arterial links experience capacity dropsof various magnitudes. On the other hand, tradable permits are more robust whenRing2 links that bypass the central city incur capacity drops.This thesis has not directly addressed some political economy considerationsthat bear on the choice between tolls and tradable permits. Tradable permits are43revenue-neutral in that if the travelers receive the permits for free, in aggregatethey do not incur net out-of-pocket cost. Therefore, tradable permits do not re-quire financial transfers to the government, which increases their public and polit-ical acceptability. However, compared to road tolls, tradable permits have highertransaction costs to the operator and the travelers. They also require designing andimplementing a competitive market that allows permit trades between the travelers.Further analytical and empirical research is needed to verify and consolidatethe findings in this thesis. One option is to study analytically a simpler version ofthe Circular City Network, which might help to provide insights into the relativeperformance of tolls and permits found here. Alternatively, METROPOLIS couldbe applied to the more complex networks of real cities, as in [26]. Another avenuefor future research is investigating the relative performance of the two instrumentswhen regulations depend on the time of the day, for example, imposing one regula-tion for peak period travel and another for off-peak travel. A permit system couldalso be introduced for use of public transit, especially if it is prone to congestion.44Bibliography[1] Phil B. Goodwin. Empirical evidence on induced traffic. Transportation, 23(1):35â54, 1996. â page 2[2] Robin Lindsney and Erik T. 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Does weather affect highway capacity. In Proceedings ofthe 5th International Symposium on Highway Capacity and Quality ofService, volume 1, pages 139â146, 2006. â page 27[32] E Roy Jones and Merrell E Goolsby. The environmental influence of rain onfreeway capacity. Highway Research Record, (321), 1970. â page 27[33] Sriram Krishnamurthy and Kara Maria Kockelman. Propagation ofuncertainty in transportation land use models: Investigation of dram-empaland utpp predictions in austin, texas. Transportation Research Record, 1831(1):219â229, 2003. â page 4948Appendix ASupporting MaterialsA.1 Origin-Destination MatrixWe use the origin-destination matrix in [25]. Trips originate at one of the 33 zones,each generating 8,000 trips. The number of trips from a zone to each of the other32 zones is an exponentially decreasing function of the free-flow automobile traveltime with the decay rate of â0.07/min based on estimates for home-based worktrips in [33, Table 3]. For example, the total number of travelers with origins atthe centre of the network and destinations at each of the zones that are directlyconnected to Ring1 and Ring2 are 347 and 273, respectively. The number of tripsper origin-destination pair has a mean value of 250, and a range of 123â661. Notethat using integer numbers avoids problems with unequal total numbers of travelersfrom simulation to simulation.A.2 Batch SimulationsWe use the open-source web-based automation tool Selenium in Python as an APIto accelerate simulations, and to run multiple simulations simultaneously. Batchsimulations are implemented using the following Python code with only smallmodifications when necessary.# We start by importing modules and libraries.import pandas as pdimport numpy as npimport multiprocessing49import timeimport osimport csvimport xlsxwriterfrom selenium import webdriverfrom selenium.webdriver.common.by import Byfrom selenium.webdriver.support.ui import WebDriverWaitfrom selenium.webdriver.support import expected_conditions as ECfrom selenium.webdriver.common.keys import Keys# The range_float module is defined to allow loops with fractional steps.def range_float(start,stop,step):x = startwhile x <= stop:yield xx = x + step# The toll module is used to generate the road pricing files, to be input to METROPOLIS.def toll(x):data = {âlinkâ: [â41â,â45â,â49â,â53â,â57â,â61â,â65â,â69â],âvaluesâ:[0,0,0,0,0,0,0,0],âtimesâ:[ââ,ââ,ââ,ââ,ââ,ââ,ââ,ââ],âtraveler_typeâ:[ââ,ââ,ââ,ââ,ââ,ââ,ââ,ââ]}df = pd.DataFrame(data)temp = df[âvaluesâ][1]df[âvaluesâ].replace(temp,x,inplace = True)df.to_csv("/**directory**/toll="+str(round(x,2))+".tsv",sep=" ",index =False)def metropolis():# Opening the METROPOLIS website on the UBC server using the Chrome web browser.browser = webdriver.Chrome(executable_path=â/**directory**/chromedriverâ)browser.get(âhttps://metropolis.sauder.ubc.ca/loginâ)# Loging into the METROPOLIS website.browser.find_element_by_id(âid_usernameâ).send_keys(â**insert username**â)browser.find_element_by_id(âid_passwordâ).send_keys(â**insert password**â)browser.find_element_by_id(âid_passwordâ).send_keys(Keys.ENTER)# Opening the âEditâ page.browser.execute_script("arguments[0].click();", browser.find_element_by_link_text(âEditâ))# The simulation module inserts the inputs and runs the simulations. x is the toll level.def simulation(x):# Inserting tolls:browser.execute_script("arguments[0].click();", browser.find_element_by_xpath(â/html/body/div/div[1]/div[4]/div/center/aâ)) # Open the âPoliciesâ page.#For each simulation tolls must be reset first.if browser.find_element_by_xpath("/html/body/div/div[1]/div/div/button").is_enabled():browser.execute_script("arguments[0].click();", browser.find_element_by_xpath(â/html/body/div/div[1]/div/div/buttonâ)) # Click the âReset tollsâ button.browser.execute_script("arguments[0].click();", browser.find_element_by_xpath("//*[@id=âdeleteModalâ]/div/div/div[3]/a")) # Click the âResetâ button.# Wait so that resetting is done.flag = 0t = 0while flag<1:try:myElem = WebDriverWait(browser, 2).until(EC.presence_of_element_located((By.XPATH,"/html/body/div/div[1]/div/div/div[2]/button[2]")))flag = 1except:50t = t+2# Importing tolls with the value of x.browser.execute_script("arguments[0].click();", browser.find_element_by_xpath(â/html/body/div/div[1]/div/div/div[2]/button[2]â)) #click the âImport tollsâ button.browser.find_element_by_id("id_import_file").send_keys("/**directory**/toll="+str(round(x,2))+".tsv")browser.execute_script("arguments[0].click();", browser.find_element_by_xpath("//*[@id=âimportModalâ]/div/div/form/div[2]/button[2]")) #click the import button# Going back to the âEditâ page.browser.execute_script("arguments[0].click();", browser.find_element_by_xpath(â/html/body/div/div[1]/center/aâ))# Running the program.browser.execute_script("arguments[0].click();", browser.find_element_by_xpath(â/html/body/div/div[1]/div[5]/div/center/buttonâ)) # Clicking âRunâbrowser.execute_script("arguments[0].click();", browser.find_element_by_xpath("//*[@id=ârunModalâ]/div/div/form/div[2]/button[2]")) # Clicking âRunâ again!# Waiting until the run is done.flag = 0t = 0while flag<1:try:myElem = WebDriverWait(browser, 10).until(EC.presence_of_element_located((By.XPATH,"/html/body/div/a[3]")))print("Page is ready!")flag = 1except:t = t + 10print("Running simulation! (t = "+str(round(t,4))+") (tolls = "+str(round(x,2))+")")# Read and exporting the output table.tbl = pd.read_html(browser.find_element_by_xpath("/html/body/div/div[1]/div/table").get_attribute(âouterHTMLâ))df = pd.DataFrame(tbl[0]) # Convert the table into a DataFrame# Save the table in the output files.datatoexcel = pd.ExcelWriter("/**directory**/output_toll="+str(round(x,2))+".xlsx",engine = (âxlsxwriterâ))df.to_excel(datatoexcel,sheet_name = âSheet1â)datatoexcel.save()browser.execute_script("arguments[0].click();", browser.find_element_by_xpath(â/html/body/div/center/aâ)) # Back to the edit page.return True# End of the simulation.# Range of toll values for which simulations are run.for x in range_float(0.0,9.9,0.2):toll(round(x,2))simulation(round(x,2))# Multiprocessingp = multiprocessing.Process(target=metropolis)p.start()p.join()51
Thesis/Dissertation
eng
Business Administration - Transportation and Logistics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Lindsey, Robin, 1955-
Traffic congestion control : a comparative analysis of tradable permits and road tolls
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