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PAGE 1 1 DESIGNING MORE ACCEPTABLE AND EQUITABLE CONGESTION PRICING SCHEMES FOR MULTIMODAL TRANSPORTATION NETWORKS By DI WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Di Wu PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS First of all, I would like to express my gratitude to my advisor, Dr. Yafeng Yin, for his constant support and encouragement. He has been a wonderful advisor, teacher, colleague and friend since the first day I stepped into the campus of University of Flor ida. It is such an honor working with him. I would like to thank Dr. Siriphong Lawphongpanich for serving as my committee member, but also for providing numerous valuable comments and suggestions on my research. I am also very grateful to Dr. Lily Elefteri adou, Dr. Scott Washburn and Dr. Sivaramakrishnan Srinivasan for serving as my committee members. Their support and guidance are also motivations for my Ph.D. study. I would also like to thank all the fellow students and my colleagues in Transportation Res earch Center for the discussions and their friendship. They really made my life in UF full of joy and happiness. Last but certainly not least, I would thank my family for their selfless support. I would have never made any achievement without them. I would specially thank my wife, Xue, for her selfless love and support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ ........ 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 1.1 Background ................................ ................................ ................................ ....... 13 1.2 Objectives ................................ ................................ ................................ ......... 17 1.3 Organization of This Dissertation ................................ ................................ ...... 19 2 LITERATURE REVIEW ................................ ................................ .......................... 20 2.1 Traffic Assignment Problem ................................ ................................ .............. 20 2.1.1 User Equilibrium and System Optimal Condition s ................................ ... 20 2.1.2 Alternative Formulations of User Equilibrium Traffic Assignment Problem ................................ ................................ ................................ ......... 23 2.1.3 Traffic Assignment with Elastic Demand ................................ ................. 24 2.1.4 Stochastic Traffic Assignment Problem ................................ ................... 25 2.1.5 Traffic Assignment Problem on Multimodal Networks ............................. 28 2.2 Congestion Pricing ................................ ................................ ............................ 29 2.2.1 Marginal Cost Pricing ................................ ................................ .............. 29 2.2.1.1 Canonical model o f marginal cost pricing ................................ ....... 29 2.2.1.2 Extensions to general transportation networks .............................. 31 2.2.1.3 First best toll set ................................ ................................ ............. 32 2.2.2 Second Best Road Pricing ................................ ................................ ...... 33 2.2.3 Network Toll Design Problem ................................ ................................ .. 34 2.2.4 Congestion Pricing in Multimodal Networks ................................ ............ 36 2.3 Empirical Results for Public Acceptance of Congestion Pricing ........................ 36 2.3.1 Effectiveness of Congestion Pricin g ................................ ........................ 37 2.3.2 Equity Issues ................................ ................................ ........................... 37 2.3.3 Revenue Distribution ................................ ................................ ............... 38 2. 4 Equity Effects of Congestion Pricing ................................ ................................ 39 2.4.1 Social Welfare and Equity ................................ ................................ ....... 39 2.4.1.1 Definitions of equity ................................ ................................ ........ 40 2.4.1.2 Measures of equity ................................ ................................ ......... 41 2. 4.2 Equity Effects of Congestion Pricing ................................ ........................ 41 2.5 Design of More Equitable and Acceptable Congestion Pricing Schemes ......... 44 2.6 Contributions of This Dissertation ................................ ................................ ..... 46 PAGE 6 6 3 PARETO IMPROVING CONGESTION PRICING ON MULTIMODAL TRANSPORTATION NETWORK ................................ ................................ ........... 51 3.1 Motivation ................................ ................................ ................................ ......... 51 3.1.1 Unattractiveness of First Best Pricing ................................ ...................... 51 3.1.2 Pareto Improvement and Pareto Improving Congestion Pricing ............. 52 3.2 Problem Description ................................ ................................ .......................... 54 3.2. 1 Underlying Network Structure ................................ ................................ .. 54 3.2.2 Strategy Based Transit Route Choice ................................ ..................... 56 3.2.3 Feasible Region ................................ ................................ ...................... 58 3.3 Multimodal Traffic Assignment Models ................................ ............................. 60 3.3.1 Multimodal User Equilibrium Model ................................ ......................... 60 3.3.2 Existence of Multimodal User Equilibrium ................................ ............... 65 3.3.3 Multimodal System Optimal Model ................................ .......................... 66 3.4 Multimodal Pareto Improving Toll Problem ................................ ....................... 67 3.5 Solution Algorithm ................................ ................................ ............................. 70 3.6 Numerical Example ................................ ................................ ........................... 73 3.7 Summary ................................ ................................ ................................ .......... 74 4 MULTIMODAL PARETO IMPROVING CONGESTION PRICING WITH TRIP CHAINING ................................ ................................ ................................ .............. 83 4.1 Tour Based Network Equilibrium Analysis Models ................................ ........... 84 4.1.1 Activity Based Traffic Assignm ent ................................ ........................... 84 4.1.2 Representing Trip Chaining ................................ ................................ ..... 84 4.1.3 Feasible Region ................................ ................................ ...................... 85 4.1.4 Tour Based Multimodal User Equilibrium Model ................................ ..... 86 4.1.5 System Optimum Model ................................ ................................ .......... 88 4.2 Tour Based Model for Pareto Improving Strategies ................................ .......... 88 4.2.1 Model Formulation ................................ ................................ ................... 88 4.2. 2 Solution Algorithm ................................ ................................ ................... 91 4.2.3 Optimum Policy Model ................................ ................................ ............. 93 4.3 Numerical Example ................................ ................................ ........................... 94 4.3.1 Example Network ................................ ................................ .................... 94 4.3.2 Optimal and Pareto Improving Policies ................................ ................... 94 4.3.3 Benefits of Capacity Expansion ................................ ............................... 95 4.3.4 Comparison of Tour Based and Trip Based Models ................................ 96 4.4 Summary ................................ ................................ ................................ .......... 96 5 DESIGN OF MORE EQUITABLE CONGESTION PRICING SCHEME BY CONSIDERING INCOME EFFECTS ................................ ................................ .... 104 5.1 Problem Description ................................ ................................ ........................ 105 5.2 Mathematical Model and Solution Algorithm ................................ ................... 106 5.2.1 Feasible Region ................................ ................................ .................... 107 5.2.2 Multimodal User Equilibrium ................................ ................................ .. 108 5.2.3 Welfare and Equity Measures ................................ ............................... 113 PAGE 7 7 5. 2 3 .1 Individual welfare measure ................................ .......................... 113 5. 2 3 .2 Social benefit measure ................................ ................................ 114 5. 2 3 3 Equity measure ................................ ................................ ............ 115 5.2.4 Model Formulation ................................ ................................ ................. 116 5.2.5 Solution Algorithm ................................ ................................ ................. 116 5. 3 N umerical Examples ................................ ................................ ....................... 117 5.3.1 Nine Node Network ................................ ................................ ............... 117 5.3.2 Seattle Network ................................ ................................ ..................... 118 5.3.2.1 Network characteristic ................................ ................................ .. 118 5.3.2.2 User b ehavior ................................ ................................ ............... 118 5.3.2. 3 Demand estimation ................................ ................................ ...... 119 5.3.2. 4 Existing traffic condition and social welfare ................................ .. 120 5.3.2. 5 Optimal pricing policies ................................ ................................ 120 5.4 Summary ................................ ................................ ................................ ........ 123 6 EQUITABLE AND TRADABLE CREDIT SCHEMES ................................ ............ 143 6.1 Introduction to Tradable Credit Schemes ................................ ........................ 143 6.1.1 Alternative Demand Management Schemes ................................ ......... 143 6.1.2 Tradable Credit Schemes and Its Public Acceptability .......................... 145 6.1.3 The Tradable Credit Distribution and Charging Scheme ....................... 146 6.2 Mathematical Model ................................ ................................ ........................ 147 6.2. 1 Representation of User Utility under Tradable Credit Scheme .............. 147 6.2.2 Single Modal User Equilibrium under the Tradable Credit Sc heme ...... 148 6.2.3 User Equilibrium with Heterogeneous Users and Nonlinear User Utility Function ................................ ................................ ................................ ...... 150 6.2.4 Multimodal Stochastic User Equilibrium under Tradable Credit Scheme ................................ ................................ ................................ ....... 152 6.2.4.1 Problem Description ................................ ................................ ..... 152 6.2.4.2 Mathematical Model ................................ ................................ ..... 153 6.3 Tradable Credit Scheme Design Problem ................................ ....................... 155 6.4 Solution Algorithm ................................ ................................ ........................... 158 6.5 Numerical Example ................................ ................................ ......................... 158 6.6 Su mmary ................................ ................................ ................................ ........ 161 7 CONCLUSIONS AND FUTURE RESEARCH ................................ ....................... 167 7.1 Conclusions ................................ ................................ ................................ .... 167 7.2 Future Research ................................ ................................ ............................. 169 LIST OF REFERENCES ................................ ................................ ............................. 171 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 185 PAGE 8 8 LIST OF TABLES Table page 2 1 Existing research on Pareto improving pricing schemes ................................ .... 49 3 1 System optimal, user equilibrium and Pareto improving conditions .................... 76 3 2 Link characteristics of Sioux Falls network ................................ ......................... 78 3 3 Summary of results in Sioux Falls network ................................ ......................... 79 4 1 Link parameters of five node network ................................ ................................ 98 4 2 Comparisons of user equilibrium, optimal and Pareto improving policies ........... 99 4 3 Pareto improving and optimal policies without capacity expansion .................. 101 4 4 Trip based Pareto improving pricing scheme ................................ ................... 102 4 5 Comparison of system performances ................................ ............................... 102 5 1 Link p arameters ................................ ................................ ................................ 125 5 2 O rigin destination d emands ................................ ................................ .............. 125 5 3 Utility f unction p arameters ................................ ................................ ................ 126 5 4 Results of Nine Node n etwork ................................ ................................ .......... 127 5 5 Link p arameters of Seattle network ................................ ................................ .. 128 5 6 Parameters for the Translog utility function ................................ ...................... 129 5 7 Demand matrix ................................ ................................ ................................ 130 5 8 User equilibrium condition without toll ................................ .............................. 131 5 9 Results for Seattle network ................................ ................................ ............... 132 5 10 Mode specific demand for Seattle network ................................ ....................... 133 5 11 Toll rates for Seattle network ................................ ................................ ............ 134 5 12 Results for policies with negative revenues for Seattle network ....................... 134 5 13 Mode specific demand for policies with nega tive revenues for Seattle network ................................ ................................ ................................ ............. 135 PAGE 9 9 5 14 Toll rates for policies with negative revenues for Seattle network .................... 135 6 1 Results of optimal tradable credit schemes ................................ ...................... 163 6 2 Optimal credit distribution and charging schemes ................................ ............ 164 PAGE 10 10 LIST OF FIGURES Figure page 2 1 Economics foundation of marginal cost pricing ................................ .................. 50 3 1 A five link network ................................ ................................ .............................. 80 3 2 Auto and transit network ................................ ................................ ..................... 80 3 3 Common lines in a transit network ................................ ................................ ...... 80 3 4 Nine node network ................................ ................................ ............................. 81 3 5 Sioux Falls network ................................ ................................ ............................ 82 4 1 Five node network ................................ ................................ ............................ 103 5 1 Ni ne node n etwork ................................ ................................ ........................... 136 5 2 Relation of social benefit and Gini coefficient of the results .............................. 136 5 3 Seattle sketch n etwork ................................ ................................ ..................... 137 5 4 Seattle surrounding area ................................ ................................ .................. 138 5 5 Compositions of productions for the centroid nodes ................................ ......... 139 5 6 Compositions of attractions for the centroid nodes ................................ ........... 139 5 7 ................................ ................... 140 5 8 group ................................ ...... 140 5 9 Relation of social benefit and Gini coefficient of the results .............................. 141 5 10 Changes in equivalent income ................................ ................................ .......... 141 5 11 Relation of social benefit and Gini coefficient of the results .............................. 142 5 12 Changes in equivalent income ................................ ................................ .......... 142 6 1 Pareto frontier of tradable credit schemes ................................ ........................ 165 6 2 Changes in equivalent income ................................ ................................ .......... 165 6 3 Comparison of tradable credit schemes and congestion pricing schemes ....... 166 PAGE 11 11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DESIGNING MORE ACCEPTABLE AND EQUITABLE CONGESTION PRICING SCHEMES FOR MULTIMODAL TRANSPORTATION NETWORKS By Di Wu August 2011 Chair: Yafeng Yin Major: Civil Engineering C ongestion pricing is to impose tolls on transportation facilities to influence alleviate traffic congestion Although recent successful implementation s around the world ha ve gained more support for the policy from transportation authorities and government officials congestion pricing is still facing strong objection among the general public. This dissertation explores technical approaches to improve the public acceptability of congestion pricing and develops more acceptable and equitable pricin g schemes on general multimodal transportation network s The models proposed in this dissertation provide good tools for government agencies to develop congestion mitigation policies that proactively address the concerns of the general public and thus are more likely to gain their supports. Two pricing approaches are investigated in this dissertation. The first approach designs Pareto improving pricing scheme s that make nobody worse off compared to the no toll condition. Two pricing models are proposed base d on trip based and tour based demand models, respectively. The resulting schemes are expected to be more attractive to travelers because the travel time saving s will be more than enough to justify the toll rates charged from them In contrast, conventional congestion pricing schemes PAGE 12 12 e.g. marginal cost pricing may result in much higher toll rate s for less travel time saving s In the second approach, we aim to optimize pricing scheme s against an equity measure. This approach considers the effe cts of income on travelers choices of trip generation, mode and route in the presence of toll and explicitly captur es the impacts of pricing schemes across different income and geographic groups A pricing model is proposed to seek for a balance between e fficiency and equity. Lastly, this dissertation investigates the equity implications of tradable credit schemes. Tradable credit schemes are alternatives to congestion pricing, which charge credits instead of money. The credits are distributed to the trav elers by the government and can be freely traded among travelers. A model is proposed in this dissertation for designing a credit distribution and charging scheme that achieve s a balance between equity and efficiency. The scheme is demonstrated to have mor e potential than congestion pricing scheme in achieving superior performance on both equity and efficiency measures. PAGE 13 13 CHAPTER 1 INTRODUCTION 1.1 Background Traffic congestion has becom e one of the most serious social problems in modern societies. In its 2009 Annual Urban Mobility Report, Texas Transportation Institute (TTI) reports 36 hours of average annual delay per peak traveler as of year 2007 and $87.2 billion of delay and fuel cost in total due to traffic congestion in the U S (Schrank and Lomax, 2009) Yet congestion has grown in cities around the world of every size in t he past ten years in extent, du ration and intensity In August 2011 China National Highway 110 experienced a c omplete gridlock for more than 11 days and stretched more than (Hickman, 2010) In general, congestion can be tackled from both the supply and demand side s Supply side tactics expand the means tha t travelers can use for travel, such as building more roads to increase roadway capacit ies and providing more transit services Demand side approaches manage the total number of vehicles traveling during peak periods via tra vel demand managem ent (TDM) or m ixed land use development, etc Supply side tactics although very straightforward, are su b ject to many spatial and financial constraints Furthermore, providing more road space has been proven to be self defeating in congested areas because the increased capacity will soon be absorbed by the induced travel demand (Goodwin and Noland, 2003 ; Hansen and Huang, 1997) On the demand side, TDM has gained increasing attention s as a more effective and cost efficient solution to the congestion problem Congestion pricing can be viewed as a major component in the TDM system. The concept of congestion pricing can be traced back to the 1920s when Pigou (1920) and PAGE 14 14 Knight (1924) proposed the idea of charging tolls to internalize the externalities travelers imposed to the others in order to better distribute the traffic flow. Although being long advocated, congestion pricing was not adopted until recent advance in technologies that made electronic tolling feasible. E xisting im plementations around the world e.g Singapore, London and Oslo, have proved that congestion pricing is able to significantly reduce traffic congestion within the tolling area. For example, f or the first six months after the introduction of congestion charges in London, the traffic dela ys inside the charging zone reduce d by about 30% (Transport for London, 2003) In Singapore, t he initial drop in traffic in the tolling area was 45% after the implementation of the Area Licensing Scheme (ALS) in 1975. The traffic volume dropped by 17% in 1 998 after Electronic Road Pricing (ERP) replaced ALS in Singapore (Goh, 2002) I n the United States a lthough toll roads commonly existed during the nineteenth c entury, road pricing only started to gain attentions as an instrument for travel demand management recently. A prevalent form of road pricing in the U.S. is high occupancy/toll (HOT) lanes or express toll lanes with the first implementation in 1995 on State Route 91 in Orange County, California. In May 2006, the U.S. Department of Tra nsportation launched a national congestion relief initiative to further promote congestion pricing implementations (U.S. Department of Transportation, 2006) Now, HOT lane or express lane facilities are already operational in more than seven states and man y more are being planned. E valuations of the existing HOT lane projects indicate that they can significantly reduce traffic delay and improve travel time reliability despite that they are different in form from other congestion pricing implementations aro und the world (Burris and Sullivan, 2006 ; Sullivan and Burris, 2006) PAGE 15 15 Despite the proven ability of congestion pricing in reducing congestion delays and the increasing supports from governments and transportation agencies, opposit ion from the general public remain s strong. In 2007, an online petition against road pricing in UK attracted over 1.8 million signatures within three months (Roberts, 2007) Hong Kong drew back after a pilot test on an electronic congestion pricing system between 1983 and 1985 (Hau, 1990) Residents in Cambridge (England) and Edinburgh also voted against congestion pricing schemes proposed for their cities. F or Stockholm, where congestion pricing was tested and implemented permanently, only 5 1 % of voters voted manent implementation while 4 6 (Stockholms stad, 2006) While there are many reasons for the public opposition, m uch of this opposition centers on the perceived inequalities of congestion pricing (Taylor et al., 2010) S everal empirical studie s e.g. Jaensirisak et al. (2005) clearly indicate that one of the main concerns with congestion pricing is the possible loss of welfare associated with, e.g., having to switch to less desirable routes, departure times or transportation modes because the more desirable ones are no longer a ffordable. For example, a long with other schemes, marginal cost pricing (Walters, 1961) a pricing scheme ubiquitous in the transportation literature, generally charges a toll on every link and significantly increases tr avel costs of many road users. In ot her words, marginal cost pricing makes many users worse off when compared to the situation without pricing (Hau, 2005) New York State Assemblyman Richard Brodsky released a report in July 2007 arguing t hat congestion pricing would be r egressive, dispropor tionately burdening working and middle class residents (Berger, 2008) PAGE 16 16 Various efforts have been made in designing more acceptable congestion pricing schemes in the literature. Dial (2000) considered an alternative pricing scheme that charges minimum tolls while minimizing system travel times, which would make road users better off compar ed to marginal cost pricing. Small (1992) proposed a package of revenue uses that can create net benefits for a wide spectrum of people and interest groups through travel allowances and tax reductions, and uses the remaining to improve transportation throughout the area Yang and Zhang (2002) explicitly considered social and spatial equity by imposing an equity constraint in thei r toll design problem. Lawphongpanich and Yin (2010) proposed a Pareto improving pricing scheme that will make nobody worse off comp ar ed to the no toll condition. However, most studies in the literature a) rely on oversimplified assumptions and small size networks and ignores the synergy among different travel mod es, espec ially transit services; b) lack of systematical approaches to provide comprehensive polices that include multiple instruments, such as tolling, revenue refunding, subsidies and developmen t of highway system and transit services; c) ignore the effects of income on travelers choice of trip generation, travel mode and route and not able to capture the impacts of congestion pricing on different income and geographical groups T his dissertation attempts to overcome the shortcomings of the existing literature and address the acceptability problem of congestion pricing More specifically, this dissertation develops more acceptable and equitable policies in general multimodal transporta tion networks that can either 1) make nobody worse off i.e. Pareto improvement, compar ed to the no toll scenario, or 2) achieve a balance between equity and efficiency by directly optimizing equity measures. PAGE 17 17 1.2 Objectives The objective of this dissertati on is to analyze the impact of congestion pricing on the transportation systems and develop pricing schemes that are more acceptable, equitable and efficient in general multimodal transportation networks. In particular, we will investigate three types of schemes They are Pareto improving pricing schemes, toll optimization against equity measures and tradable credit schemes. Pareto improving pricing Pareto improv ing pricing scheme maximizes the social benefit while guaranteeing that nobody will be made worse off in term of the generalized travel cost i.e. the cost of travel time and toll charge s, under the toll scheme when compar ed to the no toll condition. Lawphongpanich and Yin (2007; 2010) and Song et al. (2009) investigated the existence of such a pricing scheme in general networks and provided models to design such schemes in various network settings. However, their numerical results show that although the Pareto improving scheme usually exists, it may not lead to significant improvement of system performance. One reason for this is the lack of revenue redistribution in their designs. With a similar objective, Guo and Yang (2010) Liu et al. (2009) and Nie and Liu (2010) investigated the existence of Pareto improving pricing schemes under di fferent revenue redistribution schemes and users value of time distributions Their research offers valuable insights to the development of such schemes that incorporate revenue usage, but their analys is either rely on oversimplified network representations or do not consider the interactions among travel modes. In this dissertation w e will investigate the possibility of achieving Pareto improvement under more general multi modal networks where regular toll lanes HOT lanes and transit services exist. Furthermore, PAGE 18 18 we will consider multiple instruments, such as tolling, revenue refunding, subsidy, transit and highway development using the toll revenue. Toll optimization against equity measures Although equity has long be en viewed as an important issue in congestion pricing, there are only a limited number of studies that explicitly consider equity measures in their pricing models e.g. Yang and Zhang, 2002; Connors et al., 2005; Sumalee, 2003 etc None of the ex isting research is able to capture the distribution effects of congestion pricing on groups with different incomes. Therefore, they are not able to address vertical equity in a sensible way. In this dissertation we focus on the pricing scheme that explici tly takes into consideration the effects of income on the choices of trip generation, mode and route and captures the distributional impacts of c ongest ion pricing across different income and geographic groups. We will investigate how to incorporate equity measures into the design of pricing models and try to develop a pricing strategy that can maximize social b enefit while being progressive Equitable and t radable credit scheme. T radable credit scheme is an alternative approa ch to manage traffic congestion, originating from the idea of emission credit systems A tradable credit scheme charges travelers credits instead of money to travel on the road network The credits are distribu ted to the travelers by the government and can be traded among travelers at a market determined price. Under tradable credit schemes, the traffic volume can be managed by the government by carefully determining the total number of credits to be distributed D rivers do not necessarily need to pay anything under such schemes to use the roadways as they receive credits from the government Travelers only need to pay out of pocket money when their travel PAGE 19 19 needs require more credits than they are initially distri buted. Thus such schemes are likely to gain more support from the general public. 1.3 Organization of This Dissertation T his dissertation is organized as follows. Chapter 2 reviews the literature on the fundamental s of congestion pricing and transportation network modeling as well as their acceptability issues. Chapter 3 presents a model for the Pareto improving congestion pricing scheme on multimodal transportation networks. Chapter 4 extends the Pareto improving pricing model by considering the tour based mode choice behavior Chapter 5 considers toll optimizations against equity measures. The model presented in this chapter aims to design more equitable congestion pricing schemes by capturing their distributional effects across different income and geogra phic groups A n alternative approach to congestion pricing, more specifically the tradable credit scheme, is investigated in Chapter 6 A summary of the major contributions of this dissertation is presented in Chapter 7 PAGE 20 20 CHAPTER 2 LITERATURE REVIE W There is an extensive literature on congestion pricing and its public acceptability from both empirical and theoretical studies. In this chapter, we will start from reviewing existing models of traffic assignment problems under different settings which is fundamental to the design and analysis of road pricing. We will then briefly summarize the economics principle s of congestion pricing and review the congestion pricing design problems. We then turn our focus to analyze the factors that may affec t the public acceptability of congestion pricing and review the existing models that aim to design more acceptable pricing schemes. 2.1 Traffic Assignment Problem Traffic assignment is the process that allocates a given set of trip s to a specific transport ation network or system (Patriksson, 1994) Traffic assignment problem is the problem that use s given origin destination (OD) demand to estimate traffic conditions such as traffic flow, t ravel time or cost on each road link in the transportation system. Traffic assignment problem is an essential part of transportation system analysis as it serves as the basis for various analysis needs In this section, we briefly review the history of the traffic assignment problem development and existing traffic assig nment models. 2.1.1 User Equilibrium and System Optimal Condition s Wardrop (1952) introduced two principles in tra vel behavior in transportation systems: : The journey times on all the routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route. PAGE 21 21 : The average journey time is a minimum. corporative route choice behavior, where all users t ry to minimize their own travel time s in the network. This principle will result in a state that the travel time on all utilized paths are equal and less than that on any unused paths for any specific OD pair. Under such a condition, no user can further re duce their travel time by unilaterally switching their travel paths. This situation is believed to be most close to reality and is usually used to predict the real traffic condition s choice behavior of travelers in the network such that all users decide their travel routes in order to achieve minimum total system travel time This condition is the best system performance that can be ever achieved under a certain demand level, but is un likely to happen without the coordination or intervention from a central government or regulator The traffic conditions under these two principles are later termed as user equilibrium (UE) and system optimum (SO). Beckmann et al. (1956) in their seminal work fo rmulated the traffic assignment problems as convex nonlinear optimization problems with linear constraints. Let and denote the set s of nodes and links in a general transportation network. Let be the set of OD pairs. For each travel link in the network, denotes the OD specific flows for OD pair is the aggregate link flow The travel time on link depends on the aggregate link flow and can be represented by a link travel time function is the link node incidence matrix and is a with elements which has exactly two non zero elements: one has a value 1 corresponding with the origin node of PAGE 22 22 the OD pair and the other has a value of 1 for the destination node. As suming the OD demand is given and fixed for every OD pair We have the following UE flow assignment problem: P UE: (2 1 ) (2 2 ) (2 3 ) where is the vector of all link flows for each OD pair Beckmann et al. (1956) proved the equivalence between the optimality condition of when the link travel time function is continuous and non decreasing with respect to the aggregate link flow the optimal solution to the above UE assignment will be the unique UE flow distribution for the network (Patriksson, 1994) Similarly, the SO flow assignment problem can be formulated as: P SO: where is the feasible region of link flow defined by C onstraints (2 1 ) (2 2 ) and (2 3 ) PAGE 23 23 2. 1 .2 Alternative Formulations of U ser Equilibrium Traffic Assignment Problem In this section, we will present two alternative mathematical formulations, i.e. nonlinear complementarity problem and variational inequality problem (Patriksson, 1994) of UE traffic assignment problem other than P UE. These models are commonly used in the literature to solve UE assignment problems. We will also demonstrate the equivalence of the three models as well as their relationship to the Wardrop s first principle. UE VI: (2 4 ) The problem UE VI is a variational inequality (VI) problem (Facchinei and Pang, 2003) The VI formulation of traffic assignment problem was first proposed by Smith (1979) and independently developed by Dafermos (1980) and Florian and Spiess (1982) It can be easily shown that t he Karush Kuhn Tucker (KKT) conditions of UE VI are equivalent to the KKT condition of P UE. Thus, the solution that satisfy condition (2 4 ) is a valid UE flow distribution. The following problem is proposed by Aashtiani and Magnanti (1981) as an alternative formulation to the VI for general traffic assignment problem. UE NCP: (2 5 ) (2 6 ) PAGE 24 24 where is called the node potential (Ahuja et al., 1993) for node and OD pair The difference of two node potential s for different node s indicates the shortest travel time between the two nodes for the specific OD pair. and are sets of links start from node and end with node respectively. The above system is a nonlinea r complementarity problem (NCP). C onstraint (2 6 ) guarantees that the travel time for any link in the network is greater or equal to the shortest travel time between the starting point and end point of the link of any directed route as defined by the difference of the node potentials. Further, Constraint (2 5 ) states that there will be positive flow on a certain link only when the travel time of that link is equal to the shortest travel time between the two nodes These two conditions are inciple. Therefore, a feasible solution to the system UE NCP is a UE distribution (Patriksson, 1994) Moreover, UE NCP is exactly the KKT conditions of problem P UE, which again shows that the optimal solution to P 2.1. 3 Traffic Assignment with Elastic Demand In transportation system analysis, the traffic assignment problem is sometimes formulated as a problem with elastic (or variable) demands in order to capture the impact of travel time on trip generation. Assume travel demand follows a demand function with respect to the smallest travel time between OD pair define the inverse demand function as T he UE problem with elastic demand is formulated as follows: PAGE 25 25 UE ED: demand for the network is assumed to be fixed. In order to apply a similar concept to elastic demand case, the SO condition is extended as the traffic flow distribution that maximizes total social benefit. The mathematical program for SO condition can be then formulated as: SO ED: The elastic demand UE model UE ED is similar in structure with the fixed demand case. Actually, it is possible to transform the elastic demand problem to a fixed demand problem by introducing artificial excess demand links (Gartner, 1980) The excess deman d links connect the origin and destination of each OD pair. An overestimate of demand is defined as the fixed demand of each OD pair. The excess demand is carried on the excess demand link whose travel time function is obtained from the inverse demand func tion for each OD pair. 2.1. 4 Stochastic Traffic Assignment Problem The aforementioned UE problems assume that all travelers have accurate information about travel time on each link and all travelers being uniform and rational in PAGE 26 26 accurately predicted. Such models are therefore known as deterministic user equilibrium (DUE) models. Howeve r, such assum ptions may not hold in reality, e.g. mes may subject to variations In 1977, Daganzo and Sheffi (1977) At stochastic user equilibrium (SUE), no user believes he can imp rove his travel time by unilaterally changing routes. SUE describes the traffic condition as a result of that travelers do not necessarily have accurate travel time information. Assume the perceived travel time for a traveler in the network is: (2 7 ) where is the perceived travel time for path connecting OD pair is the actual path travel time. is a random variable that represents the perception error of the travelers. Let be the probability that route is perceived as the shortest path for OD pair and be the traffic flow on path for OD pair The user equilibrium condition can be characterized by the following equation: The corresponding SUE model can be formulated as the following unconstraint mathematical program (Sheffi and Powell, 1982) : P SUE: where is the expectation of the smallest perceived travel time for all paths connecting OD pair PAGE 27 27 DUE condition is a special case of SUE condition by letting the variance of the random variable to be zero. The SUE problem is one application of the discrete choice model (Ben Akiva, 1985) The most commonly used SUE model is t he multinomial logi t model which is based on the assumption that the random residuals in E quation (2 7 ) are independently and identically distributed (i.i.d.) as Gumbel random variables with zero mean and scale parameter Based on this assumption, the probability that a route is chosen can be written as: The use of logit model in traffic assignment problem was first introduced by Dial (1971) for the case of constant travel times. In 1980, Fisk (1980) presented the following logit based SUE assignment model: SUE L: where is the link path incidence. when link is on path otherwise. When problem SUE L reduces to P UE, which suggests that the DUE assignment problem is a special case of the logit based SUE problem. PAGE 28 28 2.1. 5 Traffic Assignment Problem on Multimodal Networks There is a long history of developing multimodal transportation network models F or a review, see e.g. Uchida et al. (2007) De Cea et al. (2005) and Ceder (2007) Early developments of multimodal traffic assignment problem mainly focused on the representation of the static strategic route choice of transit passengers (Chriqui and Robillard, 1975; Nguyen and Pallottino, 1988; Spiess and Florian, 1989; De Cea and Fe rnandez, 1989) This problem is usually addressed as the common line problem. Such problem arises from the fact that transit users usually hav e more complicated route choice behaviors with the opportunities of transfer and waiting Instead of choosing a si ngle shortest path, transit passengers may pre routes arrived that serves one of those attractive routes In order to tackle this problem, Spiess and Florian (1989) and Nguyen and Pallottino (19 88) introduced the notion of strategi route choice is described as a set of possible routes. Later developments in transit models mainly focus on capturing congestion effects of the transit service. De Cea and Fernandez (199 3) and Wu et al. (1994) considered congested networks where waiting time and in vehicle costs (crowding effect) are functions of passenger flows. Lam et al. (1999) considered explicit capacity constraints of transit lines and estimated the additional waiti ng times by examining the Lagrangian (Cepeda et al., 2006) Congestion at stops does not only i ncrease the waiting times but also affects the flow splits among the attractive lines. The models that capture the latter effect are (Cepeda et al., 2006) Examples of the full congested models PAGE 29 29 include Bouzaiene Ayari et al. (2001) Cominetti and Correa (2001) and Cepeda et al. (2006) In the latter two, an effective frequency function is introduced, which decreases with the passenger flows and vanishes when flows exceed the capacity of the bus line. In the mid 1990 researchers started to consider schedule based transit assignment problem where the timetable of transit services are given (Hickman and Bernstein, 1997) Hamdouch and Lawphongpanich (2008; 2010) recently developed a model for schedule based transit assig nment problem that considers the capacity of transit service. 2.2 Congestion Pricing The principle of congestion pricing was first introduced by Pigou (1920) and Knight (1924) in the early 1920s. Their work was later elaborated and extended by many researc hers, e.g. Walters (1961) Mohring and Harwitz (1962) Bechmann et al. (1956) and Vickrey (1969) The original idea of congestion pricing is based on the fact that road use has negative externality. Thus by charging the marginal cost to internalize the ex ternalities, social benefit is able to be maximized (Button, 1993) 2.2.1 Marginal Cost Pricing 2.2.1.1 Canonical model o f marginal cost pricing The concept of c ongestion pricing mainly arose from the field of economics Assuming a single road network with homogenous travelers and elastic demand Figure 2 1 explains the effect of marginal cost pricing. The MC curve represents the marginal social cost with respect to tr affic flow. The marginal social cost is the additional cost of adding one more traveler to the road, which includes the cost for the new ly added traveler herself and the extra cost experienced by other travelers already on the road. The AC curve is the ave rage private cost that only considers the cost of the new traveler PAGE 30 30 herself As the addition of a traveler will always cause extra delay to the others, the MC curve is always higher than AC curve. The D curve is the inverse demand function. The social benef it can be calculated as the area between the inverse demand (D) curve and marginal cost (MC) curve (Hau, 2005) As road users do not perceive the marginal social cost but rather the average private cost when they make their travel decisions, a n equilibrium point in this condition exists when the AC curve intersects with the D curve (point a) with traffic flow h and average travel cost f where total social benefit is the area of bde minus area of abc However, the social optimal point is at point b, the intersection of MC curve and D curve, with a total social benefit as the area of bde where flow is I and average travel cost is j The idea of marginal cost pricing is to charge the road users with a t oll rate set as the difference of the marginal cost and average cost. In the case shown in Figure 2 1 the toll rate will be set to be the value of g j the difference of marginal cost and average cost at point b Under such a tolling scheme, the average cost curve will be moved up b As a result, a new equilibrium will be established at point b which maximizes the social benef it (Walters, 1961) Let denote the travel cost function of the toll road, where is the traffic flow. The marginal cost is defined as the change of total travel cost with respect to flow and can be calculated as PAGE 31 31 The marginal cost toll is the difference between marginal cost and average cost Therefore, the marginal cost toll is calculated as (2 8 ) 2.2.1.2 Extensions to general transportation networks Now, we come back to the mathematical formulation of the traffic assignment problems and examine the effect of marginal cost pricing scheme. We first present the tolled UE problem. The tolled UE condition is the traffic condition such that every traveler minimizes his/her own generalized travel cost For each travel link, the generalized travel cost is the sum of travel time and toll charge, which can be represented as: With the above definition, the tolled UE problem can be formulated as: TUE MP: Problem TUE MP is a simple modification of the original UE assignment problem P UE by replacing the link travel time function with the generalized link travel cost As P UE generates the UE flow distribution associated with the travel time function the optimal solution of problem TUE MP will be UE flow distribution subject to the generalized travel cost. Therefore, TUE MP is the tolled UE assignment problem. PAGE 32 32 By setting the toll rate to the marginal cost toll as in Equation (2 8 ) i t can be easily shown that t he tolled UE assignment problem TUE MP generates the same optimal solution as the SO problem. This confirms mathematically the fact that marginal cost pricing will drive the system from UE to SO condition 2.2. 1.3 First b est t oll s et Under the marginal cost pricing, the flow unde r the tolled UE condition is identical to the SO condition. This pricing scheme is called first best pricing scheme in the literature, as it drives the traffic condition to the best possible one (SO) However, as pointed out by Hearn and Ramana (1998) mar ginal cost pricing defined by Equation (2 8 ) is not the only pricing scheme that can achieve SO condition. Actually, marginal cost pricing is only one element of the so called first best toll set which contains all the first best pricing schemes. To derive the first best toll set, one may consider the problem as to find a toll vector such that the SO flow satisfies the UE conditions. More specifically, Conditions (2 1 ) (2 3 ) where is the traffic flow in SO condition. This formulat ion can be further simplified by summing over all the OD pairs and links: PAGE 33 33 where is the set of all paths between OD pair 2.2. 2 Second Best Road Pricing While the first best pricing schemes can achieve the maximum system efficiency, there usually are certain restrictions for congestion pricing schemes for political reasons or due to the difficulties of real world implementation tha t prevent the implementation of first best pricing For example, first best pricing requires charg ing on every link of the network, which may be practically difficult and inefficient. For the case of heterogeneous travelers first best pricing scheme may require to charge different toll rates for different type s of travelers, who may not be practically distinguishable (Arnott and Kraus, 1998) The existence of these restrictions prevents the tolling scheme to achieve the optimal (or first best) traffic con dition. T he pricing schemes that maximize social benefit under such restrictions are usually called second best pricing schemes as they cannot achieve the best condition (Florian and Hearn, 2003) Practically, all pricing schemes are second best. Examples of second best pricing schemes include area based pricing e.g. Maruyama and Sumalee (2007) and cordon pricing e.g. Zhang and Yang (2004) HOT lanes are also second best implementations of congestion pricing as only part of all the road users are char ged. Parking charges can also be considered second best pricing as parking space can be considered as toll links (Verhoef, 2002) PAGE 34 34 2.2. 3 Network T oll D esign P roblem The network toll design problem has received numerous interests in the transportation science community, especially in the past two decades. T he toll design problem is usually formulated as a bi level mathematical program (Clegg et al., 2001; Yang and Bell 1997) A general form of the bi level formulation is presented below: TDP BL : (2 9 ) solves TUE MP The objective is some system performance measure, such as social benefit (Yang and Lam, 1996; Ferrari, 1999; Verhoef, 2002) toll revenue (Dial, 1999) or the total number of toll roads (Hearn and Ramana, 1998) Constraint s (2 9 ) are some restriction s to the pricing scheme e.g. only certain links in the network can be tolled (Yang and Lam, 1996) or tolls can be charged for only a certain type of road travelers such as solo drivers on HOT lanes The resulting pricing scheme will be first best if no such constraint exists in the model (Hearn and Ramana, 1998) The lower level problem is the tolled user equilibrium problem TUE MP or other more general UE assignment problem s e.g. SUE We can use the alternative formulation i.e. NCP, of the tolled UE prob lem to replace the mathematical programming model TUE MP and the formulation for the toll design problem is changed to (Lawphongpanich and Yin, 2010) : PAGE 35 35 TDP MPCC: Constraint (2 9 ) Problem TDP MPCC is a mathematical program with equilibrium constraints (MPEC) (Luo et al., 1996) Although TDP MPCC appears to be a single level problem, the existence of the equilibrium constraints still makes the problem hard to solve (Friesz et al., 19 90) In particular, TDP MPCC violates the Magasarian Fromovitz constraint qualification (MFCQ) This renders standard algorithms for nonlinear programs ineffective. In order to solve the problem, researchers have developed various algorithms. Most of the a lgorithms are heuristic s, including sensitivity analysis based approach (Yang and Lam, 1996; Ying and Yang, 2005) pattern search based algorithm (Abdulaal and LeBlanc, 1979) linearization method (LeBlanc and Boyce, 1986) relaxation method (Ban et al., 2006) simulated annealing method (Friesz et al., 1992) genetic algorithms based approach (Yin, 2000; Shepherd and Sumalee, 2004) cutting plane method (Hearn and Yildirim, 2001) heuristic method based on location index (Verhoef, 2002) cutting constraint method (Lawphongpanich and Hearn, 2004) and manifold suboptimization (Lawphongpanich and Yin, 2010) PAGE 36 36 2.2. 4 Congestion Pricing in Multimodal Networks Multimodal network equilibrium modeling has been well studied in the literature, e.g., Florian (1977) ; Abdulaal and Leblanc (1979) ; Florian and Nguyen (1978) ; Garcia and Marin (2005) For a recent overview, see, e.g., De Cea et al. (2005) Although the networks do not lend themselves for multimodal transportation networks. The asymmetric nature of travel costs in multimodal transportation networks imposes a great challenge for both model formulation and solution algorithm. Among existing studies, Gentil e et al. (2005) presented a multiclass equilibrium model for multimodal networks and compare cordon pricing and rationing policies in an effort to improve social welfare. Hamdouch et al. (2007) rk to multimodal transportation networks and demonstrate its usefulness in estimating the impact on the transit ridership due to toll pricing and transit fare adjustment. 2.3 Empirical Results for Public Acceptance of Congestion Pricing Although congestion pricing has been demonstrated both theoretically and empirically to be able to successfully reduce traffic congestion and are gaining more and more support from politicians and transportation officials, the general public is still quite skeptical about road pricing (Jaensirisak et al., 2005; Schade and Baum, 2007) Research suggests that acceptability is essential to guarantee the success of congestion prici ng and has identified acceptability as the main obstacle for any implementation of road pricing (Sikow Magny, 2003) In this section, we will briefly review the factors that may affect public acceptability of congestion pricing suggested by empirical studi es (for more extensive review, see e.g. Jaensirisak et al. 2005 ). PAGE 37 37 2.3.1 Effectiveness of C ongestion P ricing The effectiveness of pricing schemes is viewed by travelers as how congestion is reduced or how they can benefit from the implementation. Steg (20 03) identified the perceived effectiveness of the pricing measure as one of the important factors that affect public acceptance. In addition, Rienstra et al. (1999) found that the acceptance of policy measures increases if people are more convinced about t he effectiveness of these measures based on a survey conducted in Netherlands on transportation policy It is also found that the higher the toll charges are, as expected, the less acceptable the pricing scheme is (Ubbels and Verhoef, 2006) So it is prefe rred that the congestion pricing scheme can improve system efficiency by charging the minimum toll rates. However, a traveler usually considers the benefit of himself from the pricing scheme more than the society as a whole (Jaensirisak et al., 2003) This is because the individual usually does not see himself/herself as responsible for causing the problem of congestion (Schade and Schlag, 2003) It can be described as the so (Hardin, 1968) This suggests that in orde r to make congestion pricing more acceptable, it is very important to increase every individual s utility besides the total social benefit. 2.3.2 Equity I ssue s Equity issues are considered one of the most relevant objections to congestion pricing. Many equ ity problems have been raise d in practice concern ing the impact of pricing to low income travelers and the use of toll revenue. Jakobsson et al. (2000) and Fujii et al. (2004) analyzed responses from car owners in European and Asian countries and found equ ity (or fairness) plays an important role in the acceptance of road pricing. Other research also found that the perceived or real unfairness due to income and PAGE 38 38 regional differences may also result in a low level of acceptance (Emmerink et al., 1995; Langmyhr, 1997) Weinstein and Sciara (2006) analyzed equity concerns associated with HOT lane projects and conclude d that equity issues may arise at any stage of the project development, thus continuous monitoring of equity implications of the projects is required both before and after the opening of the HOT lanes. 2.3.3 Revenue D istribution How to use the toll revenue also plays a significant role in the public acceptability of road pricing. Verfhoef (1996) surveyed morning commuters for their opinion on road pricing. An overwhelming majority (83%) stated that his/her opinion depends on the allocation of toll revenues. Jaensirisak et al. (2005) summarized previous studies on congestion pricing acceptability and found the mean acceptability was 35 percent w hen there was no explicit mention of revenue usage and 55 percent when some specific plan to use the revenue is presented. Among all the possible usage of the toll revenue, enhancing the performance of alternative travel modes is one of the most favorable options by travelers. Jones (1991) found acceptability increase d from 30 percent where there was no explicit mention of the use to which the revenue would be put to 57 percent when it was stated that the revenue would be used for improving public transport and facilities for pedestrians and cyclists. In another empirical study on the public acceptability of different pricing strategies in four European cities: Athens, Como, Dresden and Oslo (Schade and Schlag, 2003) it was found that the use of toll revenu e for improvements of public transportation is favored by the vast majority of respondents. PAGE 39 39 2.4 Equity Effects of Congestion Pricing The previous section reviews empirical results on the public acceptance of congestion pricing. The empirical evidences show that the acceptability of a congestion pricing proposal centers on its perceived equity. In this section, we elaborate on the equity issues within the transportation system, particularly on congestion pricing. 2.4.1 Social Welfare and Equity S ocial welfare is an economic ter m that can be traced back over one and a half century (Button, 1979) Welfare refers to the overall well being of people, either as individuals or collectively. In a recent review, Levinson (2010) gave the following description of social welfare 1 : Social welfare comprises both efficiency, a measure of the degree to which system outputs achieve a theoretical maximum (minimum) using the same level of inputs, and equity, a measure of the distribution of outputs (or inputs) across some population. For transportation system, efficiency is a relatively well defined concept A n efficient system means less delay, lower travel cost or higher utility level. However, equity is more subjective and may be viewed different ly from different perspe ctives A particular decision may seem equitable when evaluated one way but inequitable when evaluated another way. Moreover, equity is more of a descriptive and normative term. It is generally considered based on comparing of outcomes of different policie s, i.e. policy A is more equitable than policy B rather than saying policy A is equitable and policy B is not. 1 Social welfare is distinguished from social benefit. In this dissertation social benefit only concerns efficiency of the society. PAGE 40 40 2.4.1.1 Definition s of equity Various types of equity have been defined in the literature (for review, see, e.g. Levinson 2010 Banister 1994 ). The definitions that are more relevant to transportation and road pricing are listed below: Horizontal e quity (fairness) : The distribution of impacts between similar (in e.g. income, gender, ability, need) individuals and groups. According to this defi nition, individuals and groups within the same class should receive equal sh a res of resources and opportunities bear equal costs and in other ways be treated similarly. A policy is horizontally equitable if similar individuals are provided with equal oppo rtunities or are made equally well off under the policy. Vertical e quity (justice): The distribution of impacts between different individuals and groups. This definition says that individuals and groups with different backgrounds should be treated the same way (given same opportunities, social resources, etc.). In other words, primary social goods (liberty, opportunity and wealth) should be distributed equally or to favor le s s advantaged people (Rawls, 1971) Consequently, i t is generally considered vertica lly equitable if a policy gives more benefit to economically and socially disadvantaged (i.e. low income) individuals and groups Other equity definitions also exist such as spatial equity, describes how benefits are distributed over space e.g. different OD pairs, (Viegas, 2001) ; temporal equity (longitudinal or intergenerational equity) describes how benefits are distributed to the present or the future (Viegas, 2001) ; market equity, describes how benefits are distributed proportional to the p rice paid (Levinson, 2010) ; social equity, describes how benefits are allocated proportional to need (Jones, 2003) The different types of equity often overlap and conflict with each other. For example, spatial equity requires individuals at different loca tion be treated similar ly which may be considered a s one kind of vertical equity as individuals can be grouped by location s Horizontal e quity requires that everybody bear the costs of their services, but vertical equity often requires subsidies for disad vantaged people. PAGE 41 41 It may depend on what type of equity is used when deciding if a policy is equitable or not. Actually, demanding equality in one space implies inequality in some other space (Sen, 1992) For example, consider the transportation financing ap proach by using tax revenue As roads are financed using local tax revenue, non local users basically get the service for free. Therefore, financing the road using toll revenue is considered more equitable as everybody using the service pays. However, such toll charges will create a larger burden for low income users thus hurt the low income users more than the others and are less equitable when considering vertical equ ity 2.4.1.2 Measures of equity Ramjerdi (2006) summarized a number of potential measures of inequality: range, variance, coefficient of variation, relative mean deviation, logarithmic variance, variance of logarithms, Gini coefficient, Theil s entropy, Atkinson index, Kolm s measure. Other equity meas ures may also consider factors other than cost such as accessibility (Feng et al., 2010) and environmental impact. However, there is no consensus among researchers that which dimension of equity should be considered or which equity measure should be used when analyzing equity effects. In fact, most studies in the literature use their own measures depend ing on their objectives and definitions of equity. It is difficult, if not impossible, for one single study to consider all of the dimensions and all measur es for each dimension (Levinson, 2010) 2.4. 2 Equity E ffect s of C ongestion P ricing First of all, congestion tolls can be a substantial part of household expenditure, therefore significantly affect ing the welfare of travelers. T he U.K. Government Statistical Survey (1992) reported that the average household in U.K. spent 15.4 percent of their weekly expendable income to transportation for an average of £ 39.70. A toll charge of PAGE 42 42 £ 3 5 thus can substantially affect the household budg et. Moreover, the toll charge will have more significant impact for families with lower income. For a £ 4 daily toll price, the transportation expenditure would account for more than 30 percent of household expenditure for low income groups, while the toll charges may only be less than 5 percent of household budget for high income families (Banister, 1994) This suggests that even for a relatively low toll rate, the impact of congestion pricing on the travelers can be significant and may substantially affect the equity of the society. The re are two major equity concerns for congest ion pricing : Distribution of benefit among road users and the rest of the community (horizontal equity) : If the toll revenue is spent outside the transportation system, it can be considered as the road users are cross subsidizing the other part of the society. And the net benefit of the pricing scheme may be compromised. Thus it is generally suggested that th e toll revenue to be spent within the transporta tion system such as on improv ing transit service and increas ing highway capacity. Distribution of welfare between users with different income s (vertical equity) : This is generally referred to as the progres sivity and regressiv ity of congestion pricing. A transportation policy is progressive if it favors the disadvantaged travelers (e.g. travelers with lower income), while it is regressive if it imposes more burden to the disadvantaged travelers Numerous ef forts have been made to analyze the equity effect of congestion pricing. Niskanen (1987) analyzed the consumer welfare of road users with different valuations of time and concluded that uncompensated congestion charge may unambiguously worsen the road users aggregate position. Small (1983) and Hau (2005) PAGE 43 43 also found that the average commuters under congestion pricing would be somewhat worse off without revenue redistribution policies. However, the r esults by Santos and Rojey (2004) show that effect of congestion pricing are location specific and depend on where people live, where people work and what mode of transport they use, and the average impact can be beneficial even without redistribution of t oll revenue. Moreover, uncompensated pricing schemes are usually regressive. Layard (1977) analyzed the equity effect of congestion pricing and suggested that the toll charges deter travel of low income travelers and encourage high income travelers to trav el more, thus it becomes more likely that the pricing scheme would be regressive. The majority of existing studies ( e.g. Glazer 1981 ; Small 1983 ; Jones 1992 ; Hau 2005 ; Arnott et al. 1994 ; Evans ,1992 and many more ) suggested that uncompensated user charge can only benefit users with high value of travel time and l ow income car owners will suffer the greatest detrimental impact, particularly if adequate public transport is not available. Giuliano (1992) further pointed out by citing the work of Gomez Ibanez (1992) that the loser groups to congestion pricing are potentially large and represent a wide range of income levels. At the same time, Foster (1974) viewed congestion pricing as progressive by arguing that richer people generally paid more as the y produced more trips, which is later challenged by Richardson (1974) Schweitzer and Taylor (2008) compared congestion pricing and sales taxes for financing transportation facilities and concluded that congestion pricing was more equitable than sales taxes as low income residents group, on average, pay more out of pocket with sales taxes. PAGE 44 44 Although ther e is still no definite answer for the equity impact of congestion pricing (due to the different ways equity is framed and evaluated) researchers do share the same belief that equity can be improved by redistributing the toll revenue Small (1983) suggeste d that every income class can benefit from congestion pricing under certain revenue redistribution methods He (Small, 1992) later showed a lump sum refunding scheme (an equal travel allowance for all commuters) can make congestion pricing progressive Arn ott et al. (1994) also concluded that although uncompensated pricing is regressive, the effect of pricing with revenue refunding can be progressive. 2. 5 D esign of M ore E quitable and Acceptable C ongestion P ricing S chemes Most of the pricing models in the literature focus on maximizing social benefit or the revenue of the toll road operators However, such schemes are generally lack of public support s due to equity concerns. In order to account for this sho rtcoming, various studies have implicitly or explicitly incorporated equity measures in their design models. Dial (1999; 2000) considers an alternative pricing scheme that minimize s revenue as well as the system travel times. Daganzo (1995) proposed a hybrid scheme that in volves both rationing the right to travel on the highway for free and charging the users being rationed He concluded, by analyzing a bottleneck model, that such hybrid scheme can benefit everyone even if the collected revenues are not returned to populati on. However, a later study by Nakamura and Kockelman (2002) s uggest ed that it is hard to find such a hybrid scheme in an appl ication to the San Francisco Bay Bridge corridor. Adler and Cetin (2001) considers a revenue redistribution scheme that directly tr ansfers toll revenue from users on more desirable routes to users on less desirable routes. By analyzing a simple p arallel link network they claim ed that this scheme can PAGE 45 45 reduce travel cost for all travelers while maintain ing the total system efficiency. Eliasson (2001) based on a similar setting, suggest ed that any toll reform that reduces aggregate travel time and redistributes the toll revenues equally to all users makes everyone better off. Bernstein (1993) proposed a revenue neutral pricing scheme t hat charges for peak hour while subsidiz ing users travelin g during the off peak. He state d that this approach will correct the distributional impacts of congestion pricing. More recently, Lawphongpanich and Yin (2007) investigated Pareto improving tolls on general road networks that will make nobody worse off compar ed to the no toll condition. Their computational experiments (Lawphongpanich and Yin, 2010; Song et al., 2009) suggest that such schemes are relatively prevalent, although they may not lead to a significant level of improvement. Guo and Yang (2010) Liu et al. (2009) and Nie and Liu (2010) further investigated the existence of Pareto improving schemes with revenue redistribution in networks with multiple user classes or with travelers with continu ously distributed value of time (VOT). Their results suggest ed that a Pareto improving pricing scheme usually exists However, this scheme may re quire class specific refunding i.e. refunding differently to users with different value of time or from differ ent OD pairs. The existence of anonymous refunding schemes usually requires more restrictive conditions e.g. the difference of travel time changes among OD pairs should not be too large (Guo and Yang, 2010) or the VOT distributions of the users are concave (Nie and Liu, 2010) In addition to the aforementioned effort s various network toll design model s have been proposed to explicitly consider social and spatial equity (impact of pricing between PAGE 46 46 different OD pairs) e.g. Yang and Zhang (2002) using the Theil measure of equity e.g. Connors et al. (2005) or Gini coefficient e.g. Sumalee (2003) Szeto and Lo (2006) developed a general time dependent design model based on a gap function that measures the degree of intergeneration equity to design the equity based optimal tolls. Ho and Sumalee (2010) proposed a continuum model for designing pricing strategies that are equitable, acceptable and socially beneficial. 2.6 Contributions of This Dissertation In the previous sections, we reviewed several factors that have been identified in the literature to have significant impact on the public acceptability of congestion pr icing. Although efficiency is viewed as a very important factor, a successful congestion pricing scheme cannot only consider the overall efficiency of the transportation system. The literature suggests that a traveler usually considers the benefit of himse lf from the pricing scheme more than the society as a whole. Thus it is important to increase every t for a congestion pricing design Based on this idea researchers have proposed Pareto improving pri cing schemes. Such schemes maximizes the social benefit while guarantees that nobody will be worse off in term of generalized travel cost (travel time plus toll charge) under the pricing scheme when compar ed to the no toll condition. Table 2 1 shows the existing literature on the design of Pareto improving pricing schemes. Previous studies on Pareto improving pricing schemes mainly focus on analyze the existence of such a tolling scheme under various network conditions. Their research either relies on the analysis of either a rather small network or a single travel mode and none of them have considered using the toll revenue to expand the capacity of the existing networ k in their design models Their results can be only used as guidelines for PAGE 47 47 congestion pricing development but are not able to develop a comprehensive congestion pricing strategies that include multiple instruments, such as tolling, revenue refunding, subsi dies and development of transit and highway networks. In this dissertation, we try to fill this gap by considering multiple travel modes, subsidizing users and possible network capacity expansions using the toll revenue in the design of Pareto improving pr icing strategies for general transportation networks. Another factor that substantially affects public acceptability on congestion pricing is the equity effect. Although equity problem has been raised for a long time, only a limited number of studies have explicitly considered equity measures in their pricing design models. Moreover, all of these models assume constant value of travel time for each individual traveler and ignore the impact of incomes. As a result, their equity analysis can only measure the spatial equity and cannot capture the distribution effects of congestion pricing on groups with different incomes. According to many empirical studies, e.g. Jara Diaz and Videla (1989) and Franklin (2006) the ignorance of income effect may lead to an und erestimate of the regressivity of the pricing schemes. Therefore, these models cannot address vertical equity in a sensible way. Furt hermore, most of these analyses are based on small networks and usually do not consider instruments other than pricing, suc h as re venue refunding and transit development. The transit service, as an alternative to auto networks, is also critical in evaluating the equity effect of any pricing schemes. So, in this dissertation, we develop a more equitable pricing scheme that dire ctly takes into account the effects of income on the choices of trip generation, mode and route, and explicitly captures the distribution of impacts of pricing schemes across different income and geographic groups. This scheme PAGE 48 48 considers the nonlinear incom e effects of user utilities, thus can accurately address the vertical equity among travelers with different incomes PAGE 49 49 Table 2 1 Existing research on Pareto improving pricing schemes Study Travel mode Heterogeneous users Network size Revenue redistribution Network expansion Lawphonpanich and Yin (2007,2010) Auto only No Large No No Song et al. (2009) Auto only Yes Large No No Guo and Yang (2010) Auto only Yes N / A Yes No Liu et al. (2009) Auto and transit Yes Small Yes No Nie and Liu (2010) Auto and transit Yes Small Yes No PAGE 50 50 Figure 2 1 Economics foundation of marginal cost pricing PAGE 51 51 CHAPTER 3 PARETO IMPROVING CONGESTION PRICING ON MULTIMODAL TRANSPORTATION NETWORK 3.1 Motivation 3.1.1 Unattractiveness of First Best Pricing First best pricing can increase social welfare to the maximum level, which is beneficial to the society. However, the standard arguments in favor of first best pricing schem es tend to ignore the distributional effects, by concentrating on the benefits to the society as a whole (Banister, 1994) This means such schemes do not guarantee that every traveler in the network is made better off and consequently makes the pricing sch eme less equitable. Actually, it is possible that the only stakeholder that gains from the pricing scheme is the government who collects toll revenues. The road users may be made worse off because they have to pay more than the travel time they save or hav e to switch to less desirable routes or modes to avoid the toll charges. Hau (2005) pointed out that marginal cost pricing (or more general first because of the increased travel cost for travel ers. The following example demonstrates how marginal cost pricing can increase road (Lawphongpanich and Yin, 2007) Consider the network in Figure 3 1 where there is only one OD pair (node 1 to node 4) with a total travel demand of 3.6. The link travel time functions are listed besides each travel link. By solving the UE assignment problem for this network the equilibrium travel time under UE conditi on without toll intervention is 71.06 with only two paths used (1 3 4, 1 3 2 4). The total travel time is 255.82. The system is not efficient under UE condition as the capacity of link 1 2 is not utilized. The system performance can be improved using conge stion pricing. A fter introducing marginal cost pricing, the total travel time reduces to 227.11 PAGE 52 52 with all three paths (1 3 4, 1 3 2 4, 1 2 4) utilized. However, considering the toll charges, the travel cost for each traveler increase s to 101.70. In this cas e, the toll charges make the travelers pay an extra 139.02. Therefore, the marginal cost pricing actually makes all traveler s worse off. As a result, the travelers are very unlikely to accept the implementation of such a pricing scheme even it can signific antly reduce traffic congestion. Yang and Zhang (2002) in their paper also provided another simple example but considering multiple user classes and OD pairs that showed marginal cost pricing would may also provide spatial inequality In other words, users from certain OD pair might suffer from the pri cing while others benefit from the reduced congestion. 3.1.2 Pareto Improvement and Pareto Improving Congestion Pricing Pareto optimality (or Pareto efficiency) is a n economic concept named after Italian economist Vilfredo Pareto Pareto optimality refers to the condition that no one can be made better off without making anybody else worse off. Correspondingly, Pareto improvement means the result of a policy change that makes at least one individual better off without making any other individual worse off. Pareto optimality is one of the key concepts in social welfare analysis in social sciences (Feldman and Serrano, 2006) A Pareto improving congestion pricing scheme is a congestion pricing scheme that improves the performance of the transportation system w ithout making any individual, including the government, worse off. The Pareto improving congestion pricing scheme to overcome the drawbacks of marginal cost pricing which make such schemes more at tractive to the general public PAGE 53 53 In the literature many researchers have investigated the possibility of achieving Pareto improvement of the transportation system using congestion pricing approaches. One of the early efforts on this topic is by Daganzo (19 95) and Daganzo and Garcia (2000) who proposed a Pareto improving hybrid scheme on a bottleneck network using both the instruments of congestion pricing and rationing. More recently, Guo and Yang (2010) Liu et al. (2009) and Nie and Liu (2010) analyzed Pareto improving pricing scheme s with revenue refunding. Their approaches use the toll revenue to subsidize the travelers on less favorable routes and OD pairs in order to achieve Pareto improvement. Lawphongpanich and Yin (2010) and Song et al. (2009) f rom a different perspective, focused their research on Pareto improving schemes in general networks that require neither rationing nor revenue redistribution These schemes are derived based on the fact that the original Wardropian user equilibrium flow di stribution may not be strongly Pareto optimal (Hagstrom and Abrams, 2002) There computational experiments suggest that such Pareto improving tolls are relatively prevalent, although they may not lead to a significant level of improvement. Although these studies provide valuable insights in the development of Pareto improving pricing schemes, they either focus on small network s or only consider single travel mode. To achieve higher levels of improvement through the synergy among different modes, the goal o f this study is to determine Pareto improving tolls for general multimodal transportation networks that includes, e.g., transit services, high occupancy/toll (HOT) and general purpose or regular lanes. To achieve a Pareto improvement, our approach adjusts transit fares and charges tolls on highway links to better distribute travel demands among the available modes of transportation. Revenue PAGE 54 54 cross subsidizing between transit and vehicle users is another key component in our schemes for achieving Pareto impro vement. 3.2 Problem Description We consider in this study three modes of transportation: single occupancy vehicle (SOV), high occupancy vehicle (HOV) and transit. The travel models herein assume the following: The total demand for each or igin destination ( O Travelers are homogenous. represented using a multinomial logit model. and capacity. 3.2.1 Underlying Network Structure The overall multimodal network consists of two (sub) networks, auto and transit network. The links in the auto network represent freeways or highways. When a freeway segment contains both regular and HOV/HO T lanes, they are represented as two parallel links. Therefore, the auto network contains two sets of links, regular and HOV/HOT, and they are denoted as and respectively. Henceforth, the superscripts and denote the regular (or SOV mode) a nd HOV/HOT links (or HOV mode), respectively. We assume that every link is tollable. All vehicles on the regular links pay the same amount of toll while only SOVs need to pay a toll on HOT links. Note them HOV only links. In addition to tolling on the auto network, transportation authority can also adjust the fares on various transit lines of the transit network. Transit network is derived from the auto network and allows for passengers to board, alight wait and transfer between PAGE 55 55 various transit lines. Figure 3 2 displays an example of a multi modal network with three transit lines 1 2 3 (line a), 1 2 (line b), and 2 nodes are added for all the bus stops (e.g., 1a, 2a and 1b) as well as embarking (e.g., link 1 1a) and alighting links (e.g., link 2a 2). Buses run on the transit links, and the passengers have to use the embar king and alighting links to access and egress. Let denote transit and embarking/alighting links (henceforth we use or superscript to represent variables associated with transit). In our model, we assume transit vehicles share the right of way wi th other vehicles. Therefore, a transit link has the same in vehicle travel times as the corresponding regular or HOV/HOT link that the transit service uses. For example, transit link 1a 2a and 1b 2b has the same travel time as link 1 2. When integrated to gether, the transit and auto network form a multimodal network where is the set of nodes and stops and is set of auto links and transit lines. Each link is associated with an in vehicle travel time. For each link in there is an associa ted waiting time that travelers have to spend in waiting for the transit service. For transit and alighting links, the waiting time is zero. The embarking link is assigned a er arrival times to transit stations follow exponential distribution and passenger arrival rate is uniform, the average waiting time for transit line is where is the frequency of line l (e.g., Lam et al., 1999). For links in the auto sub ne twork, the waiting time is set to available PAGE 56 56 3.2.2 Strategy B ased T ransit R oute C hoice Unlike travelling on the auto network, transit users may have more complicated route choices. Instead of choosing a single shortest path, travelers may pre select a set attractive lines. Such a behavior is addressed as the common lines problem in the li terature of transit assignment e.g., Chriqui and Robillard, (1975) Alternatively, recent studies have introduced the notion of strategies or hyperpaths e.g., Spiess and Florian, (1989) ; Nguyen and Pallottino, (1988) I n this notion, passengers choose an optimal strategy by specifying at each node a set of immediate successor nodes that allow them to reach to their destinations with a minimum expected total travel time. Consider a transit network in Figure 3 3 where each link is a transit line. Then a strategy for travelers with O D pair ( node 1 to node and 4, and the successors of node 2 as nodes 3 and 4. With such a strategy, the bus arrived is for line 1 2, the travelers will have to transfer at node 2 to either Line 2 3 or 2 4, depending whichever strategy may specify node 4 as the only suc cessor of node 1. Consequently the travelers will only take line 1 4 at node 1. Generally, different strategies lead to different expected waiting and in vehicle travel times. An optimal strategy is the one that minimizes the expected total travel time. Assuming that each transit user chooses the optimal strategy, the user equilibrium transit assignment problem can be equivalently formulated using the following link based formulation (Spiess and Florian, 1989) : PAGE 57 57 where is the set of O D pairs; represents the total expected waiting time experienced by passengers of OD pair at node ; D pair on link and is the supply or demand of node for O D pair is the set of outgoing links from no de and is the total passen and is the travel time or cost of link Capturing congestion effects is very critical in transit assignment. In the above, the travel times or costs of transit l inks are assumed to be constant. In order to capture the Florian (1989) introduced such crowding functions and formulated another equilibrium transit model. However, the model may underestimate the waiting times because Wu et al. (1994) extended the model to con sider both waiting time and in vehicle cost (1999) considered explicit capacity constraints of transit lines and estimate the additional waiting times by examining the Lagrangian multipliers associated with th e capacity constraints. Both models can be PAGE 58 58 (Cepeda et al., 2006) Congestion at stops does not only increase the waiting times but ctive lines. The models that capture the latter (Cepeda et al., 2006) Examples of the fu ll congested models include Bou zaiene Ayari et al. (2001) Cominetti and Correa (2001) and Cepeda et al. (2006) In the latter two, an effective frequency function is introduced, which the bus line. In this study we focus on deriving optimal pricing strategies for multimodal transportation networks. The p roblem of interest is generally larger and more complicated than transit assignment. Our consideration of transit network modeling is more in line with Spiess and Florian (1989) and Lam et al. (1999) strategy based route choice behav ior is considered and congestion effects are captured by incorporating explicit capacity constraints for transit lines. Although the resulting transit model is more a semi congested version and applies in a more restrictive setting, it offers computational advantages over the full congested ones (Cepeda et al., 2006) 3.2.3 Feasible Region ow models. Let be th for O D pair by mode ; be the node link incidence matrix and be the set of transportation modes, i.e., We further denote the tra vel demand between O D pair by mode as To specify the origin and destination of OD pair nts, let be its has exactly two non zero components: one has PAGE 59 59 a value of 1 in the component corresponding the origin node and the other is 1 in the component for the destination. Let be the total dem and for O D pair and be the transit service capacity on link The feasible region can be represented as the f ollowing linear system: ( 3 1 ) ( 3 2 ) (3 3 ) (3 4 ) (3 5 ) (3 6 ) (3 7 ) Equation ( 3 1 ) Constraint ( 3 2 ) ensures that the (3 3 ) originates from the strategy based transit assignment problem (Spiess and Florian, 1989) Equation (3 4 ) implies that the sum of OD demand of each mode is the total demand, which is gi ven as a constant. The C onstraint (3 5 ) requires that the mode should be nonnegative, and the last two constraints ensure that transit and auto users use the corresponding facilities. There is redundancy in these three constraints because Constraint (3 5 ) is only nec Equation s (3 6 ) and (3 7 ) Note that, if there are HOV only lanes in the network, another additional con straint, should be included for those links. We further note that with the PAGE 60 60 assumption that trav formu lated as a multinomial logit model, is always strictly positive. can be added, where is a small positive constant. We denote the set of all fea sible to the above system as which is a polyhedron. 3.3 Multimodal Traffic Assignment Models 3.3.1 Multimodal User Equilibrium Model models for the multimodal networks. We assume that link travel time increases mono follows: where and vehicle) travel time and vehicular capacity of link respectively; which is the sum of SOV, HOV As previously mentioned, the frequency of where is the average occupancy of HOVs and is the set of transit lines sharing the right of way with auto link Therefore, link travel time is an increasing function of its Note t hat the function is asymmetric, e.g., Patriksson (1994) PAGE 61 61 We assume that users behave in a manner consistent with the Wardropian user equilibrium principle within the same transportation mode, and thus the user equilibrium conditions for each mode can be expressed as where is the set of paths (or hyperpaths) between OD pair for mode ; is the (expected) travel time on path for OD pair and mode and is the is the smallest or equilibrium travel time for OD pair by mode multinomial logit model as follows: where and are parameter s that need to be calibrated. is a mode constant, which can be set as 0 for one of the three modes. It follows from the above equation that: (3 8 ) Theorem 3 1. The solution of the following variational inequality (VI) librium conditions in the multi modal network. PAGE 62 62 MUE VI : Proof The KKT conditions for MUE VI are as follows: ( 3 9 ) ( 3 10 ) ( 3 11 ) ( 3 12 ) ( 3 13 ) ( 3 14 ) (3 15 ) PAGE 63 63 where constraint ( 3 1 ) ; and are the multipliers associated with Equations ( 3 2 ) to (3 4 ) respectively. Furthermore, can be interpreted as the extra waiting time on link when the demand for the transit line exceeds its service capacity (Lam et al., 1999) Sim ilarly, is the multiplier associated with Equations (3 5 ) to (3 7 ) Assume that the above conditions hold for a solution For any utilized path the path must be pos itive, i.e., if (where equals one, indicating that link is part of path and zero otherwise). According to Equation (3 15 ) Thus, s umming Equation ( 3 9 ) along the path yields: (3 16 ) Therefore, every utilized path for SOV or HOV has the same travel time and it equals the minimum travel time between a given OD pair Using a similar argument, it follows from Equation ( 3 13 ) to (3 15 ) and ( 3 10 ) tha t every utilized hyperpath between O ( 3 17 ) F rom Eq uation ( 3 14 ) it follows: Summing the above equation over all links and using Equation ( 3 12 ) yields: PAGE 64 64 The right side of the above is the total waiting time for OD pair Therefore, such that is the expected total travel time of the utilized hyperpath. Equation ( 3 17 ) implies that each utilized hyperpath for the same O D pair has the same minimum expected travel time According to Equation (3 16 ) and ( 3 17 ) it is straightforward to show that Equation ( 3 11 ) is equivalent to Equation (3 8 ) i.e., the demand distribution follows the multimodal logit model. Th erefore, the solution modal user equilibrium conditions. As the travel time function is asymmetric, MUE VI cannot be readily reformulated as a mathematical program. Although the solution algorithms proposed for solving VI in the literature, e.g., Lawphongpanich and Hearn (1984) can be applied to solve MUE VI although (Aghassi et al., 2006) In this paper, we apply a new technique developed by Aghassi et al. (2006) that reformulates an asymmetrical VI as a nonlinear optimi zation problem. Applying Theorem 2 in Aghassi et al. (2006) MUE VI can be reformulated as the following optimization problem: MUE NLP: PAGE 65 65 where , are auxiliary variables and is the th column of the incidence matrix The above program is a regular nonlinear optimization problem and can be solved using commercial nonlinear optimization solvers. If the optimal objective value is zero, one part of the optimal solution vector to MUE NLP, will solve MUE VI. 3.3.2 Exi stence of Multimodal User Equilibrium Theorem 3 2 The variational inequality problem for multimodal user equilibrium (MUE VI) has at least one solution. Proof VIR) to the original MUE VI problem by imposing an upper bound for variable as follows: where is large enough to ensure that the feasible region is nonempty. PAGE 66 66 With the upper bound constraint, the feasible region for MUE VIR problem is a compact and convex region. Given that all the functions are continuous, the MUE VIR problem has at least one solution see, e.g., Harker and Pang (1990) to the MUE VIR problem also solves the MUE VI problem because for any and the additional upperbound constraint in MUE VIR does not affect variables and Therefore, the solution to MUE VI exists. 3.3. 3 Multimodal S ystem O ptimal M odel computed as (3 18 ) Small and Rosen (1981) and Ying and Yang (2005) Correspondingly the multimodal system optimum problem can be formulated as MSO: PAGE 67 67 similarly as in Yang (1999) Maher et al. (2005) and Ying and Yang (2005) MSO is a nonlinear convex program that is easy to solve. 3.4 Multimodal Pareto Improving Toll Problem This section discusses the use of congestion pricing to improve the social welfare in the multimodal network. It is assumed tha t a transportation authority determines tolls for auto links and adjusts transit fares for transit lines. The toll is nonnegative while the transit fare adjustment is unrestricted. A negative adjustment means that transit users are subsidized. As previousl y mentioned, both SOVs and HOVs need to pay tolls on regular links while only the former pay tolls on HOT links. The feasible toll set can where and are tolls on link for HOVs and SOVs respectively. Note that if a HOT link is not tolled, it becomes a regular link. HOT links in this paper essentially provide a mean for discriminatory tolling. Similarly, is transit fare adjustment. Given a toll vector (in the u nit of time), the tolled multi modal user equilibrium problem can be formulated as the following nonlinear complementarity problem: PAGE 68 68 TMUE: ( 3 19 ) ( 3 20 ) ( 3 21 ) ( 3 22 ) ( 3 23 ) ( 3 24 ) ( 3 25 ) ( 3 26 ) ( 3 27 ) ( 3 28 ) TMUE is derived from adding a toll vector to the KKT condition of MUE VI. In determining a Pareto improving toll vector, each individual user cannot be made worse off when compared to the situation under MUE. To ensure this, we require the tolled PAGE 69 69 equilibri um travel cost between each OD pair for each mode to be less than or equal to the equilibrium travel cost without toll. If the condition holds, the users who remain on the same mode will not be made worse off and those who switch to other modes will be bet ter off. We further require the total revenue to be nonnegative (or greater than a certain value to cover the toll operation cost) to ensure that the expense of transportation authority does not increase. Mathematically, the above two requirements can be w ritten as ( 3 29 ) ( 3 30 ) where is the equilibrium travel time without toll. Note that we do not consider the economic behaviors of transit operator in this study However, because we assume after the tolling. On the other hand, the transit operator is more likely to have more revenue due to a higher transit ridership. The multimodal Pareto improving toll problem can be formulated as follows: MPIT: ( 3 19 ) to ( 3 30 ) PAGE 70 70 There are two parts in the objective function. The first term is the user benefit while the second term is the total toll revenue representing the benefit for the government (or the producer benefit). As formulated, the MPIT problem is always feasible. In particular, the solution to MUE VI and a zero toll vector is feasible to MPIT. On the other hand, Pareto improving tolls may not exist. If the optimal objective value of MPIT is greater than the the optimal toll vector will be Pareto improving. 3.5 Solution Algorithm As formulated, MPIT is a mathematical problem with complementarity constraints This paper applie s a manifold suboptimization algorithm developed by Lawphongpanich and Yin (2010) to solve MPIT. The basic idea of the algorithm is to solve a sequence of restricted nonlinear optimization problems to obtain a strongly stationary solution to the original M , and , The restricted MPIT (or R MPIT) can thus be formula ted as follows: ( 3 1 ) (3 4 ) ( 3 21 ) ( 3 22 ) ( 3 25 ) ( 3 26 ) ( 3 29 ) ( 3 30 ) PAGE 71 71 (3 31 ) (3 32 ) (3 33 ) where and are auxiliary variables. PAGE 72 72 The manifold suboptimization algorithm solves a sequence of R MPIT and use the associated KKT multipliers to update the index sets The procedure can be described as follows: Step 1: Solve the problem MUE NLP to a zero optimal objective value to obtain a user equilibrium solution. Step 2.1: Let and be the link demand and node waiting time from Step 1. Set and Step 2.2: Let solve the relaxed problem R MPIT. Step 2.3: Let and where and are the multipliers associated with (3 31 ) (3 33 ) respectively. If and are all empty for all and stop and is strongly stationary (La wphongpanich and Yin, 2010) Otherwise, do the following and go to Step 2.2: (a) Set , (b) Set (c) Set Lawphongpanich and Yin (2010) proved that the above algorithm will stop after a MPIT are unique. PAGE 73 73 3.6 Numerical E xample We solved MUE, MSO and MPIT using GAMS (Brooke et al., 2005) and CONOPT (Drud, 1995) on a nine node network shown in Figure 3 4 : The two And there are 10 transit lines: 1 5, 2 6, 5 6, 5 7, 6 5, 6 8, 7 3, 7 8, 8 4, 8 7. The service frequency is 1 for each transit line. The BPR function is used to calculate the in vehicle travel time for each link in the example. Table 3 1 reports the Pareto improving tolls condition with UE and SO. The travel time or cost in the table includes transit waiting time, and travel costs under SO includes the marginal cost tolls, which are listed under the column of SO. The Pareto to 73.5% of the maximum possible level and reduce the total system travel time by 21.4%. At the same time, no individual user is made worse off and a majority is better off. The transportation authority also gains some positive revenue. Pareto improving tolls encourage users to switch to transportation modes with higher occupancy. In contrast, although marginal cost pricing achieve a maximum improvement of social Also note that marginal cost pricing requires discriminatory tolling for all links. The models were also implemented on the Sioux Falls network as shown in Figure 3 5 is assumed to have a transit s are listed in Table 3 2 The OD demands are the ones reported in LeBlanc et al. (1975) multiplied by 0.2. PAGE 74 74 In Table 3 3 the Pareto improving tolls in this example with mild congestion The total vehicle travel improved. However, the total passenger travel time remains approximately the same. This is because SOV and HOV users switch to transit due to the t olling and transit subsidy and will experience additional transit waiting time. The reduction of in vehicle travel time is only large enough in this example to compensate the increase of transit waiting time, leading to approximately the same total passeng er travel time. We expect more severe or (b) the transit operator reacts adaptively to increase the transit frequency to serve higher level of ridership or (c) individua l users can be made worst off to a limited level (by 3.21% in this particular example). On the other hand, marginal cost some users may increase. 3.7 Summary In order to dev elop congestion pricing schemes more appealing to the public, this study determines a Pareto improving charging scheme on a multimodal transportation and transportation au thority worse off when compared to a situation before pricing. Pareto improvement is achieved by appropriately charging tolls on highway links and adjusting the fares of transit lines. To better distribute travel demands among the available transportation modes, the revenue from tolling highway links is used to subsidize the fare adjustments on transit lines. The Pareto improving scheme can be PAGE 75 75 obtained by solving a mathematical program with complementarity constraints using a manifold suboptimization techni que. PAGE 76 76 Table 3 1 System optimal user equilibrium and Pareto improving conditions UE Pareto improving SO Social benefits 2603.27 2068.95 1875.90 System travel time 2851.63 2241.59 1994.13 Revenue 0.00 70.22 513.97 OD SOV HOV Transit SOV HOV Transit SOV HOV Transit Demand 1 3 20.80 7.65 1.55 10.97 12.60 6.42 7.79 13.56 8.65 1 4 21.51 7.91 0.58 16.38 7.30 6.32 14.67 11.56 3.77 2 3 21.49 7.91 0.61 13.29 15.26 1.44 10.95 17.85 1.20 2 4 27.74 10.20 2.06 16.85 19.35 3.80 16.58 21.59 1.83 Demand change (%) 1 3 47.25 64.67 315.80 62.56 77.25 459.63 1 4 23.83 7.74 987.35 31.80 46.17 548.26 2 3 38.14 93.08 138.68 49.03 125.73 98.78 2 4 39.24 89.65 84.32 40.21 111.54 11.15 Travel cost 1 3 22.40 22.40 25.40 21.22 15.53 13.90 24.59 16.82 14.07 1 4 22.22 22.22 30.27 19.90 18.94 14.66 23.22 19.41 20.02 2 3 20.34 20.34 28.19 18.65 12.96 19.75 21.91 14.47 22.95 2 4 22.07 22.07 25.07 22.07 16.37 19.52 23.38 17.06 24.40 Travel cost change (%) 1 3 5.27 30.68 45.29 9.80 24.91 44.60 1 4 10.45 14.76 51.58 4.51 12.64 33.88 2 3 8.29 36.27 29.93 7.71 28.87 18.58 2 4 0.00 25.79 22.14 5.94 22.70 2.67 PAGE 77 77 Table 3 1. CONTINUED UE Pareto improving SO Link SOV HOV Transit SOV HOV Transit SOV HOV Transit Toll 1 5 0.72 0.72 4.71 1.71 0.68 0.00 1 6 0.00 0.00 1.74 2.39 0.96 0.00 2 5 0.00 0.00 0.77 1.15 0.46 0.00 2 6 0.00 0.00 0.00 0.12 0.05 0.00 5 6 0.00 0.00 3.25 0.00 0.00 0.00 5 7 7.92 7.92 6.54 9.84 3.94 0.00 5 9* 0.00 0.00 0.00 0.50 0.20 0.00 6 5 0.00 0.00 1.30 0.00 0.00 0.00 6 8 0.39 0.39 2.37 1.28 0.51 0.00 6 9* 0.00 0.00 2.29 0.00 0.00 0.00 7 3 0.00 0.00 3.52 1.41 0.56 0.00 7 4 0.79 0.79 0.76 0.84 0.34 0.00 7 8 0.00 0.00 0.64 0.00 0.00 0.00 8 3 0.69 0.69 11.40 0.00 0.00 0.00 8 4 0.00 0.00 7.20 0.50 0.20 0.00 8 7 0.13 0.13 6.36 0.00 0.00 0.00 9 7* 0.00 0.00 1.44 0.25 0.10 0.00 9 8* 0.00 0.00 2.05 0.00 0.00 0.00 1 5* 0.00 0.00 5.42 2.07 0.83 0.00 2 6* 0.00 0.00 0.06 0.12 0.05 0.00 5 7* 5.69 0.00 4.31 10.56 4.22 0.00 6 8* 0.78 0.00 3.18 1.31 0.52 0.00 7 3* 1.60 0.00 3.52 1.74 0.70 0.00 8 4* 0.18 0.00 7.71 0.55 0.22 0.00 HOT link. PAGE 78 78 Table 3 2 Link characteristics of Sioux Falls network Link No. Free flow travel time Capacity Link No. Free flow travel time Capacity Link No. Free flow travel time Capacity 1 3.6 6.02 31 3.6 9.82 61 2.4 6.05 2 2.4 9.01 32 3 .0 20 .00 62 3.6 10.12 3 3.6 12.02 33 3.6 9.82 63 3 .0 10.15 4 3 .0 15.92 34 2.4 9.75 64 3.6 10.12 5 2.4 36.81 35 2.4 36.81 65 1.2 10.46 6 2.4 24.22 36 3.6 9.82 66 1.8 9.77 7 2.4 36.81 37 1.8 51.8 0 67 2.4 20.63 8 2.4 25.82 38 1.8 51.8 0 68 3 .0 10.15 9 1.2 18.25 39 2.4 10.18 69 1.2 10.46 10 3.6 9.04 40 2.4 9.75 70 2.4 10 .00 11 1.2 36.85 41 3 .0 10.26 71 2.4 9.85 12 2.4 13.86 42 2.4 9.85 72 2.4 10 .00 13 3 .0 10.52 43 3.6 17.02 73 1.2 10.16 14 3 .0 9.92 44 3 .0 10.26 74 2.4 11.38 15 2.4 9.9 0 45 2.4 9.64 75 1.8 9.77 16 1.2 21.62 46 2.4 20.63 76 1.2 10.16 17 1.8 15.68 47 3 .0 10.09 77 2.4 10 .00 18 1.2 36.81 48 3 .0 10.27 78 2.4 10 .00 19 1.2 9.8 0 49 1.2 10.46 79 2.4 10 .00 20 1.8 15.68 50 1.8 29.36 80 1.2 10 .00 21 2 .0 10.1 0 51 4.2 9.99 81 1.2 10 .00 22 3 .0 10.09 52 1.2 10.46 82 1.2 10 .00 23 3 .0 20 .00 53 1.2 9.65 83 1.8 10 .00 24 2 .0 10.1 0 54 1.2 36.81 84 1.8 10 .00 25 1.8 17.83 55 1.8 29.36 85 3.6 10 .00 26 1.8 17.83 56 2.4 8.11 86 2.4 10 .00 27 3 .0 20 .00 57 2.4 4.42 87 3.6 10 .00 28 3.6 17.02 58 1.2 9.65 88 1.8 10 .00 29 3 .0 10.27 59 2.4 10.01 89 1.2 10 .00 30 4.2 9.99 60 2.4 8.11 90 1.8 10 .00 PAGE 79 79 Table 3 3 Summary of results in Sioux Falls network SO UE Pareto improving Social benefit 3890.3 4147.4 4084.6 Total vehicle travel time 2907.8 3859.2 3691.0 Total passenger travel time 4173.5 4406.3 4397.0 Revenue 463.8 0.0 30.4 SOV HOV Transit SOV HOV Transit SOV HOV Transit Max travel cost increase (%) 3.21 0.66 0.00 0.00 0.00 0.00 Max travel cost decrease (%) 27.51 27.51 49.08 0.68 0.68 39.78 PAGE 80 80 Figure 3 1 A five link network Figure 3 2 Auto and transit network Figure 3 3 Common lines in a transit network 1 2 3 Regular link HOV/HOT link 1a 1b 2b 2a 3a Transit link Embarking & alighting link 2c 3c 1 2 3 4 PAGE 81 81 Figure 3 4 Nine node network (5, 12) 1 2 5 6 Regular link HOT link 7 8 9 (5, 12) (2, 11) (2, 11) (3, 25) (3, 12) (9, 35) (9, 12) (6, 33) (6, 11) (6, 43) (6, 11) (6, 18) (3, 35) (9, 20) (4, 11) (2, 19) (4, 36) (8, 39) (6, 24) (8, 26) ( 4 26) ( 7 32 ) (8, 30 ) O D demand: 1 3: 30 1 4: 30 2 3: 30 2 4: 40 3 4 PAGE 82 82 Figure 3 5 Sioux Fall s network PAGE 83 83 CHAPTER 4 MULTIMODAL PARETO IMPROVING CONGESTION PRICING WITH TRIP CHAINING Trip chaining is a phenomenon that we know exists but rarely consider in pricing models (Adler and Ben Akiva, 1979) For multimodal transportation networks, m odeling trip chaining is particularly relev ant in determination of pricing policies b ecause: 1) a trip chain or tour refers to a sequence of individual trips between a pair of anchor activities (Primerano et al., 2008) For mode choice, travelers typically decide which mode to take for the entire tour, taking into account all the trip segments. Constrained by the schedules, routes and uncertainty associated with transit service, travelers may be more likely to use private automobiles to pursue multiple activities in a single journey (Frank et al., 2008; Ye et al., 2007) A pricing model without considering trip chaining may overestimate the mode shift to transit and thus the benefit of pricing; 2) For certain forms of pricing, such as area based pricing, a tour based modeling framework is essential (Maruyama and Sumalee, 2007) Area based pricing charges travelers for an entry permit (e.g., per day), i.e., a tour is charged a toll (or daily usage fee) only once regardless of the number of times the restricted zone is accessed during the entire tour. If individual trips are used as the basic unit of analysis, the impact of the pricing may be miscalculated. Based on these considerations, t his chapter extends Chapter 3 by developing a model to represent trip chaining in multimodal transportation networks. W e further consider using the toll revenue to subsidize transit passengers and expand highway capacity Such arrangements could help to achieve even higher level of improvement in social benefit without making anyone worse off. PAGE 84 84 4. 1 Tour Based Network Equilibrium Analysis Models 4. 1 .1 Activity Based Traffic Assignment Activity based travel demand models are developed based on the recognition of based travel deman d or desire to fulfill certain activities (Bhat and Koppelman, 2003) Different from trip based models, activity based traffic analysis models are based on the analysis of a complete tour inc luding several trips for different activities by constructing a trip chain. Although activity based travel demand models are gaining more attention in the literature, such models are rarely considered in traffic assignment models. Lam and Yin (2001) propos ed a dynamic user equilibrium model that integrates route choice and sequential activity choice. Maruyama and Harata (2006) and Maruyama and Sumalee (2007) proposed a path based model with an explicit representation of trip chains to analyze the effect of cordon based and area based congestion pricing schemes. 4. 1 2 Representing Trip C haining It is assumed that travelers can be classified into groups and each group has a specific tour. Travelers in the same group begin at the origin of their trip chain s travel to each node in the chain consecutively, and terminate their trip at the last node in the chain, i.e., their destination. The demand for each group is known and fixed. But the travelers can freely choose any available travel modes to fulfill their travel needs. Let denote a trip chain or tour, denote an OD pair and represent the set of OD pairs of all the trips or legs in the trip chain. For example, if tour is home work shopping home, then PAGE 85 85 We further introduce an index to facilitate converting the tour demand to OD demand. If OD pair belongs to ; otherwise, In a network with trip chaining, it is assumed that user equilibrium will be achieved if for each to ur the travel costs of all utilized paths (or hyperpaths) are equal and not greater than those of unutilized ones. Mor eover, we assume that travelers will choose a single travel mode for their whole tours and the travel mode choice can be formulated as a m ultinomial Logit model: (4 1 ) where is the demand of tour using mode ; is the total travel demand for tour ; is the equilibrium travel cost of tour by mode ; and are parameters to be calibrated. is a mode specific constant and can be set as 0 for one of these three modes. Similar to the Chapter 3, a nother formulation equivalent to (4 1 ) can be written as : (4 2 ) 4. 1 3 Feasible Region The feasible region of our tour based model is defined as: (4 3 ) PAGE 86 86 (4 4 ) The above feasible region is similar to the one defined in Chapter 3 with C onstraints (4 3 ) and (4 4 ) replacing the total demand C onstraint (3 4 ) Constraint (4 3 ) requires that the sum of tour demand of each mode is the total demand. Constraint (4 4 ) converts tour demand to OD demand. All the other variables are defined in Chapter 3. The se t of all feasible to the above system is denoted as 4. 1 4 Tour Based Multimodal User Equilibrium Model A tour based multimodal user equilibrium features the following conditions: 1) the flow distribution is feasible; 2) in the auto modes (SO V and HOV), the travel costs of all utilized paths for each tour are equal and not greater than those of unutilized ones; 3) transit passengers choose cheapest hyperpaths for each tour and 4) the mode split for each tour is the function of the equilibrium tour costs of those three modes. The user equilibrium flow distribution can be obtained by solving the following variational inequality (VI) problem. Theorem 4 1 The solution of the following VI problem satisfies the tour based user equilibrium conditions in the multimodal network. PAGE 87 87 The above theorem can be proved by analyzing the KKT conditions of the VI problem and comparing them to the tour based user equilibrium condition. The proof is similar to the proof of Theorem 3 1, thus is omitted in this section. To solve the VI problem, we adopt the technique proposed by Aghassi et al. (2006) The problem is converted to the following regular nonlinear mathematical program using a gap function. PAGE 88 88 Where , and are auxiliary variables and is the th column of the indicence matrix The reformulated problem is a regular nonlinear optimization program. Commercial nonlinear optimization solvers can be used to solve the above program. If the optimal objective value reaches zero, one portion of the optimal solution, i.e., will solve the VI problem. 4. 1 5 System Optimum Model The system optimum model of the tour based assignment can be written as follows : TSO 4. 2 Tour Based Model for Pareto Improving Strategies 4. 2 .1 Model Formulation In this study, a n improving strategy refers to a policy for tolling roads and highways, adjusting fares on various transit lines and expanding highway link capacities using toll revenues For mathematical formulation, the toll rates should be nonnegative while the adjustments to transit fares are unrestricted. A negative adjustment implies that transit PAGE 89 89 passengers are subsidized. The to ll revenue can also be used to expand existing highway links to further reduce congestion. It is further assumed that the transportation authority is able to leverage the stream of toll revenue to finance the proposed capacity expansion through the financi ng market. Denote as the amortized cost for providing additional unit of capacity to link and the cost includes the construction and financial costs. The Pareto improving strategies can be obtained by solving the following mathematical program: T PIT: (4 5 ) (4 6 ) (4 7 ) (4 8 ) (4 9 ) PAGE 90 90 (4 10 ) (4 11 ) (4 12 ) (4 13 ) (4 14 ) (4 15 ) (4 16 ) (4 17 ) (4 18 ) (4 19 ) (4 20 ) where is the toll rate or fare adjustment on link for mode ; is the equilibrium tour cost without any tolling intervention; is the capacity on link for travel mode and indicates the existing capacity. In the above, the objective function consists of three components. The first component is a logsum measurement of user benefits (Ying and Yang, 2005) The second and third components represent the producer benefits, i.e., the profit received by the government. More specifically, the second component is the total toll revenue PAGE 91 91 and the last one represents the total cost for capacity expansion. Constraint (4 5 ) to (4 15 ) are the tour based multimodal tolled user equilibrium conditions, which can be easily deriv ed from the KKT condition of the VI problem in Theorem 4 1 by adding tolls to travel costs. Constraint s (4 16 ) and (4 17 ) are Pareto improving constraints. The former ensures that the tour cost after policy implementation is no greater than the cost without the tolling intervention. If travelers remain at the same mode, they will n ot be made worse off. If they voluntarily switch to another mode, they will be better off. Constraint (4 17 ) ensures that the toll revenue is enough to cover the cost for capacity expansion such that the government will not be worse off. Of the remaining constraints, C onstraint (4 18 ) ensures that capacity expansion should be nonnegative; C onstraint s (4 19 ) and (4 20 ) require the charging scheme to be feasible. For example, C onstraint (4 20 ) guarantees a uniform toll for both SOVs and HOVs on a regular link. 4. 2 .2 Solution Algorithm The problem stated above is a mathematical programming problem with complementarity constraints (MPCC), a class of problems difficult to solve. W e use a relaxation approach to solve the MPCC (Scholtes, 2001; Ban et al., 2006) The basic idea of t his algorithm is to relax the complementarity constraints, which make the problem hard to solve, to regular nonlinear constraints. The approach solves a sequence of relaxed problems where one has a tighter relaxation and thus generates a better solution than its predecessor. The approach is sketched below: Step 1: Initialization Choose an initial nonnegative value for the auxiliary variable Set the iteration limit updating factor and set Step 2: Solving the following relaxe d problem: PAGE 92 92 R T PIT: (4 7 ) to (4 9 ) (4 12 ) to (4 20 ) Step 3: Parameter updating If set , go to step 2. Otherwise, go to step 4. Step 4: Solving the original R PIT Solve the exact formulation of R T PIT as a regular nonlinear program using the solution from step 2 as initial solution. If successful, an exact solution is obtained; otherwise, an approximate solution is obtained from the last iteration of step 2. The relaxed problem R T PIT is a regular nonlinear program, easier to solve as compared to the original MPCC. The optimal solution will approximate better the PAGE 93 93 solution to the original MPC C as the relaxation parameter decreases. The relaxed problem becomes the original MPCC problem when the relaxation parameter reaches zero. Ralph and Wright (2004) proved that under certain conditions the relaxation scheme can solve op timally the original MPCC. 4. 2 .3 Optimum Policy Model For the comparison purpose, the section presents an optimal policy model without Pareto improving constraints. We note that the system optimal pricing scheme can be derived from the optimality condition of the system optimum model TSO However, the toll scheme is differentiated and will require charging different toll rates to different vehicles, even at regular links. For fair comparison, the solution to the following optimal policy model is used: (4 5 ) to (4 15 ) (4 18 ) to (4 20 ) The optimal policy problem maximizes the total social benefit subject to tolled user equilibrium conditions. The problem has a similar structure as T PIT, and can be solved using the same relaxation approach. In fact, the absence of Pareto improving constraints makes it much easier to obtain a feasible solution to th is problem, thereby expediting the solution process. PAGE 94 94 4. 3 Numerical Example 4. 3 .1 Example Network The models are solved for the toy network shown in Figure 4 1 The ne twork consists of five nodes and sixteen links. Among all the links, fourteen are regular links with uniform toll rates and the others are links for HOVs only. The BPR function is used as the link travel time function and the free flow travel time and capa city of each link are listed in Table 4 1 The average occupancy of the HOV mode is assumed to be 2.5. There are five tours in the network, i.e., {(1, 3), (3, 1)}, {(1, 3), (3, 5), (5, 1)}, {(1, 3), (3, 4), (4, 1)}, {(2, 4), (4, 2)} and {(2, 4), (4, 5), (5, 2)}. The demands are 10, 10, 9, 20 and 14 correspondingly. There is one transit service on each link in the network. The service frequency for each line is 0.5 wi th a capacity of 15. The mode choice parameters are ; ; and The amortized cost for providing unit capacity expansion is 5 and 10 for regular and HOV links respectively. 4. 3 .2 Optimal and Pareto Improving Policies Table 4 2 compares three scenarios, i.e., user equilibrium without any policy implementation, optimal policy, and Pareto improving policy. In addition to the values re ported in the table, the optimal and Pareto improving policies use a portion of the toll revenue to expand Link 5 by 6.72 and 7.35 respectively. The total social benefit user travel time, vehicle travel time and net toll revenue of those three scenarios a re reported in Table 4 2 Under the optimal policy, SOV and HOV users of tour 4 and 5 will suffer from an increase in travel cost while Pareto improving policy makes every one better off. At the same time, the Pareto improving policy significantly increases the social benefit by 10.9% (as the social benefit is a PAGE 95 95 relative value, the percentage is only meaningful for the particular reference point in the example), and re duces the total user and vehicle travel times by 13.1% and 20.1% respectively. We further note that in the above example the toll revenue is not used by full amount to fund capacity expansion. This is because t he benefit incurred by additional capacity may decrease as congestion decreases. At a certain level, the cost of the additional capacity will exceed its benefit and thus adding more capacity will result in a loss in social benefit 4. 3 .3 Benefits of Capaci ty Expansion Table 4 3 presents the system performances under the Pareto improving and optimal pricing (policy without capacity expansion). Compared to the results pr esented in Table 4 2 allowing capacity expansion reduces the total user travel time from 1564.13 to 1518.28, a 2.9% reduction. However, total vehicle travel time inc reases by 4.8% from 1160.72 to 1215.96. This is because when additional capacity is provided, more travelers switch to SOV mode, thereby increasing the total vehicle time. Note that with capacity expansion, the social benefit increases by 2.3% as compared to the pure pricing policy. Another observation worth noting is that without capacity expansion, the optimal pricing policy tends to charge more tolls on highway links in order to redistribute traffic flow and force travelers to switch to travel modes with higher occupancy. Consequently, the trip cost for auto modes will become much higher than before, implying that the auto travelers are made worse off substantially when compared to the original condit ion without toll intervention. In contrast, the Pareto improving pricing scheme can still maintain the same or even less travel cost, which makes it much more appealing to the PAGE 96 96 public. Indeed, Pareto improving pricing is generally more appealing in the situation where providing extra capacity is impossible or e xpensive. 4. 3 .4 Comparison of Tour Based and Trip Based Models We now compare the tour based model with the trip based counterpart where users are assumed to make their mode choice based on travel cost of each individual O D trip. On the same 5 node networ k, the trip based Pareto improving pricing model (without capacity expansion) leads to the toll matrix shown in Table 4 4 with the social benefit of 1450.40. With the same toll structure and travel demand, the flow distribution from the tour based will achieve a social benefit of 1600.46, much smaller than the trip based estimate. The comparison of the results of both models is presented in Table 4 5 The values given in Table 4 5 are results of these two different models based on the same tol l matrix in Table 4 4 The toll matrix is an optimal Pareto improving pricing scheme with the trip based model and is estimated to increase the total social benefits b y 11.9%. However, with the tour based model, the performance of the pricing scheme is greatly reduced. The total social benefit is 1600.46, little improvement from the user equilibrium value before the policy implementation ( 1646.24 as shown in Table 4 2 ). Moreover, it will result in a deficit of 109.53, making the government worse off. 4. 4 Summary The model presented in this chapter extended Chapter 3 by incorporating a activity based travel demand model. by considering the fact that travelers make their decision based on their entire tours instead of single trips. In contrast to the trip based counterpart, the tour based model captures more realistic travel behaviors, thus is expected to be able to more accurately estimate the socioeconomic impacts that transportation policies may incur. Results from PAGE 97 97 t he numerical example demonstrate that Pareto improvement is achievable by adjusting transit fares, charging tolls on highway links and financing additional capacity to the highway ne twork through the toll revenue. PAGE 98 98 Table 4 1 Link parameters of five node network Link Free flow travel time Capacity 1 7 18 2 4 18 3 10 35 4 4 15 5 10 16 6 7 11 7 4 26 8 13 21 9 6 13 10 7 32 11 2 25 12 5 24 13 4 19 14 6 19 15 7 10 16 10 10 PAGE 99 9 9 Table 4 2 Comparisons of user equilibrium, optimal and Pareto improving policies User e quilibrium Optimal p olicy Pareto i mproving p olicy Social benefits 1646.25 1451.83 1467.03 Total user travel time 1748.12 1502.36 1518.28 Total vehicle travel time 1522.46 1188.10 1215.96 Net toll revenue 0.00 161.39 98.56 Tour SOV HOV Transit SOV HOV Transit SOV HOV Transit Demand 1 7.26 2.67 0.07 4.85 4.93 0.22 5.23 4.61 0.17 2 7.29 2.68 0.03 4.91 4.99 0.10 5.28 4.65 0.08 3 6.56 2.41 0.03 4.41 4.48 0.11 4.75 4.19 0.06 4 14.51 5.34 0.15 13.72 6.10 0.18 14.06 5.21 0.73 5 10.18 3.75 0.07 9.63 4.28 0.09 9.96 3.69 0.35 Tour cost 1 28.85 28.85 31.85 25.94 20.85 21.29 24.76 20.39 22.02 2 36.92 36.92 43.92 34.02 28.94 33.31 34.25 29.89 35.31 3 42.70 42.70 49.70 39.92 34.84 38.50 38.07 33.70 39.89 4 23.20 23.20 26.20 24.18 23.24 25.80 23.09 23.05 17.87 5 23.20 23.20 28.20 24.18 23.24 27.35 23.09 23.05 19.81 PAGE 100 100 Table 4 2. CONTINUED User e quilibrium Optimal p olicy Pareto i mproving p olicy Link SOV HOV Transit SOV HOV Transit SOV HOV Transit Toll 1 4.78 4.78 0.00 4.22 4.22 7.26 2 5.97 5.97 0.00 3.43 3.43 0.00 3 0.94 0.94 0.00 0.00 0.00 0.00 4 0.94 0.94 0.00 0.00 0.00 0.00 5 2.71 2.71 0.00 2.26 2.26 7.00 6 0.38 0.38 0.00 0.04 0.04 0.00 7 0.03 0.03 0.00 1.44 1.44 3.99 8 0.00 0.00 0.00 0.00 0.00 0.88 9 0.40 0.40 0.00 0.04 0.04 1.18 10 0.00 0.00 0.00 0.00 0.00 0.00 11 0.00 0.00 0.00 0.00 0.00 0.00 12 0.88 0.88 0.00 0.12 0.12 0.88 13 0.03 0.03 0.00 0.00 0.00 5.21 14 0.00 0.00 0.00 0.00 0.00 0.00 15 0.00 0.00 0.00 0.00 16 0.00 0.00 0.00 10.01 PAGE 101 101 Table 4 3 Pareto improving and optimal policies without capacity expansion Pareto i mproving p olicy Optimal p olicy Social benefits 1502.26 1482.53 Total user travel time 1564.13 1539.77 Total vehicle travel time 1160.72 1135.99 Net toll revenue 46.10 322.85 Tour SOV HOV Transit SOV HOV Transit Demand 1 3.14 5.86 1.00 3.38 5.94 0.68 2 3.32 6.19 0.48 3.51 6.17 0.32 3 2.97 5.53 0.50 3.15 5.53 0.31 4 14.27 5.25 0.48 13.72 6.10 0.18 5 10.07 3.70 0.23 9.63 4.28 0.09 Tour cost 1 28.47 20.36 14.21 34.41 26.60 22.43 2 36.50 28.38 26.16 43.46 34.65 34.44 3 41.75 33.64 30.64 48.00 40.19 39.55 4 23.20 23.20 20.12 24.18 23.24 25.80 5 23.20 23.20 22.06 24.18 23.24 27.35 PAGE 102 102 Table 4 4 Trip based Pareto improving pricing scheme Link SOV HOT Transit 1 0 .00 0 .00 7.67 2 0 .00 0 .00 0 .00 3 0 .00 0 .00 6.2 0 4 0 .00 0 .00 1.84 5 3.71 3.71 11.89 6 0 .00 0 .00 2 .00 7 0 .00 0 .00 0 .00 8 0 .00 0 .00 0 .00 9 0.05 0.05 0.53 10 0 .00 0 .00 0 .00 11 0 .00 0 .00 0 .00 12 0 .00 0 .00 0 .00 13 0.01 0.01 0.35 14 0 .00 0 .00 4.75 15 0 .00 7.03 16 0 .00 6.16 Table 4 5 Comparison of system performances Trip based Tour based Social benefits 1450.40 1600.46 Total travel time 1648.57 1593.00 Total vehicle travel time 1315.52 1144.22 Toll revenue 0.00 109.53 PAGE 103 103 Figure 4 1 Five node network 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 PAGE 104 104 CHAPTER 5 DESIGN OF MORE EQUITABLE CONGESTION PRICING SCHEME BY CONSIDERING INCOME EFFECTS The previous two chapters discussed the implementation of Pareto improving pricing strategies in transportatio n networks. The idea of Pareto improving tolls originates from the fact that travelers in the networks are selfish and only care their own travel cost instead of the society as a whole. Pareto improving pricing schemes guarantee that the travel cost will n ot increase for any travelers, thus can be more attractive to the general public. In this chapter, we will investigate another pricing scheme that will not only improve the overall efficiency of the transportation systems but also be able to improve the eq uity within the society. This scheme can successfully address the equity issue of congestion pricing, thus will have higher acceptability then conventional pricing schemes. T he design of more equitable congestion pricing scheme has been well studied in the literature. For instance, Yang and Zhang (2002) developed an optimal pricing model for multiclass networks with social and spatial equity constraints. Lawphongpanich and Yin (2010) Guo and Yang (2010) and Wu et al. (2011) adopted Pare to improving approaches to ensure that none is worse off in the presence of tolls. This study differs from previous research in the following aspects: (a) it directly takes into account the effects of income on choices of trip generation, mode and route in the presence of tolls ; and (b) it explicitly captures the impacts of pricing schemes across different income and geographic groups. In the literature, Franklin (2006) considered the income effects on travel choice behaviors and estimated the welfare impac ts of tolling on the SR520 Bridge in Seattle, Washington. Bureau and Glachant (2008) simulated and compared PAGE 105 105 the distributional effects of nine toll scenarios in Paris. However, both studies focused on evaluation of pricing schemes instead of design. 5. 1 P r oblem Description We consider a general multimodal transportation network that includes three types of facilities, i.e., transit services, high occupancy/toll (HOT) and regular toll lanes. In this setting, a pricing scheme refers to a strategy for tolling roads as well as adjusting fares on various transit lines. The toll on a regular toll lane is anonymous or uniform while single occupancy vehicles (SOV) and high occupancy vehicles (HOV) may be charged different toll rates on HOT lanes. Tolls on highway se gments are nonnegative while adjustments to transit fares are unrestricted. Herein, a negative fare adjustment means that transit users are subsidized by the toll revenue. Therefore, a component of the pricing scheme in this paper can be viewed as revenue recycling. We further consider a discrete set of user groups with different incomes and preferences among four travel modes, i.e., no travel, transit, drive alone and car pool. It whose utility functions are nonlinear in income. This specification allows us to more accurately capture the income effects on the choice behavior in the presence of tolls. According to Jara Diaz and Videla (1989) Franklin (2006) and Bureau and Glachant (2008) the traditional specification with constant marginal utility of income may lead to an underestimate of the regressivity of a pricing scheme. The equivalent variation is adopted in this paper as the benefit measure. It is the dollar amount that has to be taken away from a traveler in the no tolling scenario to leave him as well off as he would be in the tolling scenario (Nicholson, 1998) Intuitively, if the traveler benefits from the pricing scheme, the equivalent variation will be negative PAGE 106 106 The equivalent income defined as the original in come minus the equivalent variation, can thus serves as a welfare measure. It is a measure of how wealthy the traveler feels under the pricing scheme. As the user utility is monotonically increasing with income, an equivalent income higher than the origina l income of the traveler implies that the pricing income is difficult in the random utility framework if the utility is nonlinear function of income. In this chapter we adopt the formul a derived by Dagsvik and Karlstrom (2005) to compute the expected equivalent income Although there is no consensus definition of equity in the context of congestion pricing, this chapter deems a pricing scheme to be more equitable if it leads to a more un iform distribution of wealth across different groups of population categorized by income and geographic locations. Such a notion of equity combines some aspects of both vertical and spatial equity discussed in the literature. The Gini coefficient (Gini, 19 12) is adopted to measure the equi ty effect of a pricing scheme The coefficient measures the inequality of income distribution across a population, a value of 0 expressing complete equality and a value of 1 implying complete inequality. With the above co nsideration, we intend to design a pricing scheme to maximize the social benefit and minimize the inc ome inequality simultaneously. Since these two objectives are often conflicting, we seek for a balance between them. 5. 2 M athematical Model and Solution Algorithm The notations used in this chapter are slightly different from the previous chapters and will be redefined at the time of using. PAGE 107 107 5. 2 .1 Feasible Region Let be a directed transportation network where is the set of nodes and the set of directed links. As aforementioned, there are three types of facilities in the network, i.e., regular (toll) lanes, HOT lanes and transit lines. There are three travel modes, i.e., SOV, HOV and transit, as well as a no travel option for each traveler to cho ose. Hereinafter, we use the superscript and to denote regular lanes (or SOV mode), HOT lanes (or HOV mode) and transit lines (or transit mode) and the su perscript t o denote the no travel option. Each link has an associated travel time that depends on link flow and link capacity The transit services are assumed to share the roadway with the other auto modes, and thus have the same trave l times. However, transit users may experience additional waiting and tr ansferring times at bus stops. Let be the set of all traveler groups and denote the set of origin destination (OD) pairs. The demand of traveler group for OD pair is denoted as The demand is assumed to be fixed With the above notations, the set of all feasible flow distributions for the network can be described as follows: (5 1 ) (5 2 ) (5 3 ) where is the flow of user group on path of mode between OD pair is the demand of user group between OD pair using mode and is the set of all paths connecting OD pair for mode Constraints (5 1 ) and (5 2 ) PAGE 108 108 ensure that demands are satisfied and Constraint (5 3 ) requires path flow variables to be nonnegative. The feasible toll set can be represented as: (5 4 ) (5 5 ) (5 6 ) (5 7 ) where is the toll rate or fare adjustment for mode on link and is the path link incident matrix, i.e., if path uses link and otherwise. Constraint (5 4 ) requires that the total toll revenue is nonnegative. Constraints (5 5 ) and (5 7 ) ensure that the tolls for auto modes are nonnegative and the toll for the no travel option must be zero. 5. 2 .2 Multimodal User Equilibrium model where travelers first decide their travel modes and then choose the routes among those available for the selected modes. The utility function for a group traveler betw een OD pair by mode on route is defined as: where is the utility for the traveler; is the deterministic observable portion of the utility and is a random error representing the unobservable factors As aforementioned, the deterministic utility function is assumed to be nonlinear, and its specification can be, e.g., the generalized Leontief model (Diewert, 1971) : PAGE 109 109 Alternatively, the Translog model (Christensen et al., 1971) : where is the income of the traveler; , , and are parameters to be calibrated; and are the travel time and toll of path between OD pair by mode They can be calculated as follows: where is the expected waiting and transferring time for transit users on path between OD pair The transit frequency is assumed to be fixed and known, and thus the expected waiting and transferring time can be estimated beforehand for each transit path. F or the no travel mode (mode ), the travel time and toll are always zero. As per the nested logit model, the probability for the group traveler between OD pair to choose mode and path is given by the following: PAGE 110 110 where is the expected utility for choosing mode and calculated as follows: (5 8 ) where is a measure of the degree of independence in the error terms of the utility function for different paths of mode for OD pair and user group (Train, 2003) The above behavioral considerations lead to a multimodal user equilibrium where the perceived utility for each traveler is maximized. The corresponding flow distribution can be obtained by solving the following vari ational inequality (VI) problem. Theorem 5 1 The solution of the following VI problem is in user equilibrium. (5 9 ) Proof The optimality conditions of the above IV problem include the following : (5 10 ) PAGE 111 111 (5 11 ) where and are multipliers associated with constraints (5 1 ) and (5 2 ) From (5 10 ) (5 12 ) Summing (5 12 ) for all paths for the same and and consider ing constraint (5 1 ) we have : (5 13 ) Comparing (5 8 ) and (5 13 ) yields : (5 14 ) Substitute (5 14 ) into (5 11 ) and use C ons traint (5 2 ) we obtain : Therefore : PAGE 112 112 The above two equations demonstrate that the solution to the VI i.e., the demand flow distribution follows the nested Logit based user equilibrium condition. The above VI can be solved by a variety of existin g algorithms in the literature. In this dissertation we reformulate it as a regular nonlinear programming using a gap function introduced by Aghassi et al. (2006) If the reformulated program can be solved with the optimal objective value of zero, part of the solution, i.e. will solve the VI problem. The reformulated program is as follows: PAGE 113 113 5. 2 .3 Welfare and Equity Measures 5. 2 3 .1 Individual welfare measure To represent the welfare change of each individual traveler under the pricing scheme, we use the equivalent variation which is the dollar amount that the traveller would be indifferent about accepting in lieu of the toll charge. More specifically, it is the change in his or her wealth that would be equivalent to the toll charge in terms of its welfare impact. Let be the utility for a traveler in group between OD pair with income in the presence of toll rate The e quivalent variation i.e., is defined as: Based on the equivalent variation, we can calculate the equivalent inco me level that allows the individual to experience the same level of utility before tolling as the original income does after tolling The e quivalent income for the individual is defined as: It follows that The e quivalent income measures the impact of congestion pricing on each indivi dual traveler in monetary term. For a traveler, if the equivalent expenditure is higher than his or her original income, the pricing scheme is considered to be beneficial to the traveller Due to the existence of the random error term in the utility function, the exact equivalent income cannot be calculate d for each individual traveler. Instead, we adopt the method proposed by Dagsvik and Karlstrom (2005) to compute the expected PAGE 114 114 equiva lent income Although the formula derived by Dagsvik and Karlstrom calculates the expected compensating expenditure, it can be easily modified to calculate the expected equivalent income Let be the deterministic portion of the utility for users in group traveling between OD pair on mode and path with income level in the pre sence of toll level The expected equivalent income for a traveler in the group can be calculated as: (5 15 ) where and are defined as: Equation (5 15 ) provides a way to calculate the expected equivalent income for each individual traveler without knowing his or her exact mode choice. 5. 2 3 .2 Social benefit me asure The sum of the expected equivalent income of all travellers can serve as a benefit measure as it represents, at an aggregate level, how wealthy travelers feel under a pricing scheme. In designing a pricing scheme, we attempt to maximize the social benefit, which consists of two components: the first is the total expected equivalent PAGE 115 115 inco me of all travelers in the network while the second is the total toll revenue collected calculated as: A measure of net benefit can also be defined, which differs from the above by the total original income, a constant. Both measures thus lead to the same solution. 5. 2 3 3 Equity measure We use the Gini coefficient to represent the equity impact of a pricing scheme. The coefficient measures the inequality of a wealth distribution. A Gini coefficient value of 0 means total equality and a value of 1 expresses maximal inequality. Given the expected equivalent income of travelers in the presenc e of a pricing scheme, it is straightforward to compute the Gini coefficient as follows: where is the average expected equivalent expenditure for all travelers. The Gini coefficient is calculated based on equivalent income A more equitable pricing scheme will lead to a smaller value of the Gini coefficient. Moreover, if the value is smaller than t he Gini coefficient without tolling, the pricing scheme is considered progressive Otherwise, it is regressive. PAGE 116 116 It is worth noting that our modeling framework does not impose any restriction on which equity measure can be used. Other equity measure such as Atkinson index can be used instead. 5. 2 .4 Model Formulation We now formulate a mathematical program to determine a pricing scheme that improves both the equity and social surplus. As these two objectives are often conflicting, we seek for a balance between them. The formulation maximizes a weighted sum of the social benefit and the Gini coefficient as follows: (5 1 ) to (5 7 ) and (5 9 ) where both terms of the objective function are normalized by the corresponding measures without tolling and is a positive weighting parameter. As formulated, the problem is a mathematica l programming program with equilibrium constraint s (MPEC) a class of problem difficult to solve. Compounding to the difficulty is that Equation (5 15 ) involves numeri cal integration. 5. 2 5 Solution Algorithm We adopt the compass search algorithm (Kolda et al., 2003) to sol ve the above toll optimization model The algorithm solves a series of multimodal user equilibrium problems, i.e., the VI formulation (5 9 ) with different toll vectors. The algorithm varies the toll rates one at a time by a step size, and evaluates the resulting objective function in order to find a better feasible solution. If no better solution is found, the step size is then halved and the evaluation procedure will be repeated. The process continues until the step size is sufficiently small. We implement the algorithm in GAMS (Brooke et al., PAGE 117 117 2005) in conjunction with Matlab (Wong, 2009) and use CONOPT (Drud, 1995) as the solver for the multimodal user equilibrium problem. 5. 3 N umerical Example s We solve the toll optimization model for two networks, the nine node network and the Seattle regional netw ork. 5. 3 .1 Nine Node Network Figure 5 1 illustrates the network which consists of 9 nodes and 4 OD pairs. The link travel time is the BPR function and the free flow travel time and capacity of each link are displayed in Table 5 1 There are four groups of travelers in the network and their demands for each OD pair are listed in Table 5 2 The following Generalized Leonief utility function is used in this example: The values of these parameters are reported in Table 5 3 Under the no toll condition, the total income Gini coefficient and total travel time of the network are 12765 0.1328 and 3041.5 respectively. It is assumed that only ten links in the network can be tolled. They are link 1, 2, 3, 4, 6, 9, 11, 12, 14 and 15. Four pricing schemes are obtained using the proposed algorithm wi th different weighting factors. Policy I focus es on maximizing the social benefit with the parameter equal to one In contrast, Policy IV solely minimizes the Gini coefficient. Policy II and III are intermediate with the weighing parameter equal to 0.91 and 0.80 respectively. Table 5 4 compares these four schemes. The net benefit and Gini coefficient for each pricing policies as well as those in the no toll condition are plotted in Figure 5 2 PAGE 118 118 In Figure 5 2 Policy I, the pricing scheme that maximizes the social benefit, is regressive and does harm the poor. However, all the other policies are progressive. In fact they improve the system in all three aspects, i.e., social benefit, system travel time and equity. We further note that the system travel time does not change with the weighting fact or in a monotone fashion As shown in Table 5 4 Policy I the most efficient pricing scheme yields a total travel time of 2196.1 while Policy II, a more equitable scheme, understandably leads to a higher system travel time of 2242.8. However, Policy III, which is more equitable than Policy II, produces a smaller total travel time of 2196.3. 5. 3 .2 Seattle Network 5. 3 .2.1 Network characteristic The network shown in Figure 5 3 is a sketch of the regional freeway network of the Seattle area in Washington. The network consists of 16 nodes, 44 links and 30 OD pairs. It is assumed the freeway links (solid lines in the figure) have travel time functions in the form of the BPR function The free flow travel time and capacity of all the freeway links are listed in Table 5 5 as calibrated by Boyles (2009) The link travel times for all the other links are assumed to be fixed at 5 minutes. It is further assumed that the transit services share the sam e roadway s with auto user s and thus have the same in vehicle travel time. However an extra 10 minutes of waiting and transfer time is added to all transit paths. 5. 3 .2.2 User b ehavior The user utility function is assumed to be a Translog function as follo w: PAGE 119 119 where is the daily income for user group converts daily income to annual income. Franklin (2006) has calibrated the above parameters for auto and transit mode (NuStats Research and Consulting, 1999) Based on these parameters we set the parameters for the Translog utility function a s shown in Table 5 6 The travel time for the no travel option is undefined. To calculate its utility, we use the following formulation: where is the shortest free flow travel time for OD pair The parameter is selected that approximately 15% of the total demand would choose the no travel option in the multimodal user equilibrium condition with no toll charges. There is not enough information to calibrate the parameter s for the HOV mode, so only one auto mode is considered in the example. 5. 3 .2. 3 Demand estimation The OD demand matrix is generated in order to match the PM peak traf fic flow count on the freeway links in the Seattle area as reported by WSDOT (2004) To further specify the demand for each user groups, additional information regarding household income and wages in the surrounding area is obtained from County Business Pa tterns (U.S. Census Bureau, 2004) and 2000 Decennial Census (U.S. Census Bureau, 2002) The surrounding region of Seattle is divided to six areas by zip codes to their nearby demand nodes as shown in Figure 5 4 The zip code based data are then used to calculate the income and wage distribution for the six areas. The user group specified PAGE 120 120 demand is estimated based on the assumption that the income of the travelers should have similar distribution as the region they depart from or arrive to. The travellers of the network are aggregated into four groups based on their annual household incomes i.e., $ 20 000, $ 40 000, $ 70 000 and $ 120 000, respectively. The estimated demand matrix is listed in Table 5 7 Figure 5 5 and Figure 5 6 present the production and attraction of each traffic analysis zone, including Everett, Seattle, Bellevue, Tacoma, Lynnwood and Renton 5. 3 .2. 4 Existing traffic condition and social welfare Table 5 8 reports the multimodal user equilibrium condition for the existing no toll condition. Figure 5 7 and Figure 5 8 the no toll condition by origin and by income group. For different origins, the about 80% of the total population choosing to drive and 4% for transit service. However, there is an obvious trend in Figure 5 8 that as the income increasing, a higher percentage of travelers will choose auto modes. 5. 3 .2. 5 Optimal p ricing p olicies We assume that 12 links can be charged They are link 3, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17 and 20 which are marked as red in Figure 5 4 The maximum link toll is $20 and the maximal subsidy for transit users i s also $20 for each link. By varying the weighing fact or we solve the toll optimiz ation model to obtain a Pareto frontier for the network, as shown in Figure 5 9 The corresponding results and tolling schemes are summarized in Table 5 9 Table 5 10 and Table 5 11 It can be observed that policy 1 is the most efficient scheme, providing a net benefit of $97.1 million annually. However, it increases the Gini coefficient by 4.4% at the same time. In PAGE 121 121 fact, only government benefits from the scheme, collecting $ 180.5 million in toll revenue annually All the travelers experience a decrease in the equivalen t income. The poorest group, i.e., Group 1, suffers the most loss of 7.5% or $1500 annually. In contrast, the richest group, i.e. group 4, experiences a negligible loss of 0.06% or $72 annually. It is thus safe to say that policy 1 achieves the maximal sys tem efficiency at the price of low income travelers, thereby compromising the equity. The basic mechanism of policy 1 is to charge a high toll to discourage travel and reduce traffic congestion. The reduction in travel time is more valuable to the high inc ome groups, which is the reason why the richest travelers are almost not harmed by the policy. However, the high toll prices off a significant portion of low income travelers. As shown in Table 5 10 the trips made by the lowest income group decrease by 30.0% where auto trips reduce by 44.6% and the transit trips are almost tripled. In the meanwhile, the travel demand of group 4 remains the same due to the negligible c hange in their utility level. When the equity is more favored, the optimal pricing policies tend to charge less and subsidize more. As the poor being the largest transit rider group, this mechanism acts as a mean to offer more benefit to low income travele rs. Consequently, the toll revenue for the government becomes lower. Under policy 2 and 3, although the rich still suffers less than the poor, the difference in loss is smaller as compared to policy 1. Note that the richest people in group 1 actually benef it from the pricing policies, as shown in Figure 5 10 With the equity measure given enough weight in the objective function, policy 4 exhibits a different and interes ting pattern. The lowest income travelers become the ones that enjoy the most benefit. Moreover, as shown in Figure 5 10 the low and high PAGE 122 122 income travelers obtain more benefit than those mid income travelers, from different sources though. The former benefits more from the transit subsidy while the latter value s more the reduced travel time. All user groups are better off compared to the no toll condition, suggesting that policy 4 is Pareto improving. Unfortunately, our computation experiments do not find a pricing policy that reduces the Gini coefficient. All t he policies discussed previously are essentially regressive and lead to a more uneven distribution of social wealth. By carefully examining the results, we find that the constraint that restricts the total toll revenue to be nonnegative is the one that pre venting the pricing scheme from achieve higher equity levels. Thus we try to further investigate the performance of the pricing schemes by eliminating the nonnegative revenue constraint. Figure 5 11 shows the results of optimal policies obtained by varying the weighing factor and allowing the toll revenue to be negative. This is done by relaxing C onstraint (5 4 ) When efficiency is favored, i.e., the value of is large, ignoring C onstraint (5 4 ) will not change the solution. However, when more weight is put towards equity, the resulting optimal pricing scheme s (Policy N1 and Policy N2) yields negative total toll revenue s as shown in Table 5 12 which suggest that the government subsidizes transit users from additional external sources other than the toll revenue to further improve equity and effi ciency. The negative toll revenue may help achieve a better overall performance by providing more subsidy to transit riders who are more likely to be low income travelers and encouraging more people to switch to transit services These new pricing policies are reported in Table 5 12 Table 5 13 and Table 5 14 It is easy to observe that policy N1 and N2 are both progressive with lower Gini coeff icients. PAGE 123 123 Moreover, Policy N1 improves the network in all three aspects, i.e., the Gini coefficients, social welfare and total system travel time. Policy N2 charges no toll at all but offers the maximal amount of subsidies allowed to the transit links. This certainly cannot be done without significant external support. Figure 5 12 shows the changes in average equivalent income for different income groups under differen t pricing policies. Although the results display similar trend as with the nonnegative toll revenue constraint in effect where high income travelers get the most benefit when the policy weights efficiency higher and both high income and low income traveler s gain more than the mid income groups when the equity weights more, the low income travelers becomes the single user groups that can obtain significantly higher benefit under policies N1 and N2 This is because policies N1 and N2 offers substantial subsid ies for transit service, and the benefit from the subsidy significantly outweighs the benefit from congestion reduction which valued the most by the high income traveler. 5. 4 Summary We have presented a toll optimization model to design efficient and equit able pricing schemes for general multimodal transportation networks. The model considers explicitly captures the impacts of pricing schemes on different income and geographi c groups. The model is formulated as a MPEC and solved by a derivative free algorithm. Two numerical examples are reported in the paper to illustrate the model. The first example is on the nine node network where optimal pricing schemes can be found to ach ieve higher efficiency and better equity. In the second example, we find it difficult to design a progressive pricing policy. However, by allowing a certain shortfall in the total PAGE 124 124 toll revenue, we obtain a pricing policy that reduce traffic congestion, inc rease social welfare and improve the equity. Our examples demonstrate that the model provides a valuable mean to design pricing policies with a balance between efficiency and equity. The proposed model is very flexible in accommodating a variety of welfar e and equity measures. PAGE 125 125 Table 5 1 Link p arameters Link Free f low t ravel t ime Capacity 1 5 12 2 6 18 3 3 35 4 9 35 5 9 20 6 2 11 7 8 26 8 4 11 9 6 33 10 7 32 11 3 25 12 6 24 13 2 19 14 8 39 15 6 43 16 4 36 17 4 26 18 8 30 Table 5 2 O rigin destination d emands OD 1 3 1 4 2 3 2 4 Group 1 2 2 20 35 Group 2 1 3 15 25 Group 3 2 10 10 5 Group 4 10 20 5 10 PAGE 126 126 Table 5 3 Utility f unction p arameters SOV HOV Transit No t ravel Group 1 9.0 0.2 0.0 1.5 2.1 8.0 Group 2 10.0 0.3 0.0 1.5 2.6 9.0 Group 3 10.0 0.4 0.0 1.8 3.5 9.5 Group 4 10.0 0.4 0.0 2.5 4.0 10.0 PAGE 127 127 Table 5 4 Results of Nine Node n etwork Policy I Policy II Policy III Policy I V 1.00 0.91 0.80 0.00 Net b enefit 613.5 604.6 562.7 417.7 Gini c oefficient 0.1338 0.1316 0.1291 0.1272 Total t ravel t ime 2196.1 2242.8 2196.3 2210.0 Toll r evenue 89.8 81.5 43.9 0.7 Link SOV HOV Transit SOV HOV Transit SOV HOV Transit SOV HOV Transit Toll 1 0.00 0.00 0.00 1.00 1.00 0.75 2.25 2.50 1.25 3.00 6.00 2.00 2 0.50 0.00 0.00 2.00 0.75 0.25 3.25 1.75 0.00 6.00 5.25 2.00 3 0.00 0.00 0.00 0.00 0.00 1.25 0.00 0.00 3.00 0.00 0.00 2.00 4 0.00 0.00 0.00 0.00 0.00 1.50 0.00 0.00 3.50 0.00 0.00 4.00 6 6.00 0.00 0.00 6.50 0.00 0.00 6.50 0.00 0.00 7.50 0.00 0.00 9 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 11 0.75 0.00 0.00 0.25 0.00 0.25 0.00 0.00 1.00 0.00 1.50 0.00 12 1.00 0.00 0.00 0.25 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.00 14 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.00 0.00 2.00 15 0.75 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.50 PAGE 128 128 Table 5 5 Link p arameters of Seattle network Link Number Link Free f low t ravel t ime Capacity 1 (1,2) 7 6864 2 (2,1) 7 6864 3 (2,3) 14 6864 4 (2,4) 16 5895 5 (3,2) 14 6864 6 (3,4) 10 3825 7 (3,5) 4 6864 8 (4,2) 16 5722 9 (4,3) 10 3825 10 (4,6) 3 5895 11 (5,3) 4 6864 12 (5,6) 10 6614 13 (5,7) 13 8762 14 (6,4) 3 5722 15 (6,5) 10 4944 16 (6,8) 9 5609 17 (7,5) 13 7577 18 (7,8) 2 5994 19 (7,9) 12 7449 20 (8,6) 9 5994 21 (8,7) 2 5609 22 (8,10) 10 4079 23 (9,7) 12 7449 24 (10,8) 10 3950 PAGE 129 129 Table 5 6 Parameters for the Translog utility function Parameter Auto Transit No t ravel 0.227 0.227 9.014 9.014 9.014 1.512 1.512 1.512 0.56 0 0.56 0 0.56 0 * See Section 5. 3 .2.2 for more details PAGE 130 130 Table 5 7 Demand matrix User group 1 Income: 20000 Everett Seattle Bellevue Tacoma Lynnwood Renton Everett 0 953 288 900 470 466 Seattle 625 0 1027 1764 479 1588 Bellevue 50 806 0 492 50 50 Tacoma 828 1889 739 0 669 751 Lynnwood 596 877 320 866 0 649 Renton 358 1263 50 1148 415 0 User group 2 Income: 40000 Everett Seattle Bellevue Tacoma Lynnwood Renton Everett 0 437 321 468 352 311 Seattle 729 0 1324 1597 625 1697 Bellevue 80 999 0 770 50 50 Tacoma 610 1313 713 0 492 538 Lynnwood 391 314 307 389 0 449 Renton 357 904 50 875 456 0 User group 3 Income: 70000 Everett Seattle Bellevue Tacoma Lynnwood Renton Everett 0 510 850 533 428 473 Seattle 1066 0 2249 2057 1096 2256 Bellevue 1086 2137 0 1899 651 691 Tacoma 462 1298 1152 0 479 611 Lynnwood 263 319 768 385 0 543 Renton 536 1215 581 1177 769 0 User group 4 Income: 120000 Everett Seattle Bellevue Tacoma Lynnwood Renton Everett 0 50 491 50 50 50 Seattle 50 0 2550 432 400 959 Bellevue 865 2559 0 1389 1070 509 Tacoma 50 50 646 0 50 50 Lynnwood 50 50 295 50 0 50 Renton 50 258 619 50 50 0 PAGE 131 131 Table 5 8 User equilibrium condition without toll Total Equivalent Income : $ 19.2230 million Gini Coefficient: 0.31288 Total Travel Time: 3 104767 min/day Link Flow Travel t ime v/c Ratio Toll 1 6733 7.97 0.98 2 7305 8.35 1.06 3 6063 15.28 0.88 4 4766 17.02 0.81 5 7114 16.42 1.04 6 5333 15.67 1.39 7 2846 4.02 0.41 8 5343 17.82 0.93 9 5106 14.76 1.33 10 2210 3.01 0.37 11 2603 4.01 0.38 12 7072 11.96 1.07 13 12031 19.93 1.37 14 2347 3.01 0.41 15 6066 13.40 1.23 16 7560 13.45 1.35 17 9563 17.95 1.26 18 4257 2.08 0.71 19 7913 14.29 1.06 20 6271 10.62 1.05 21 4717 2.15 0.84 22 5526 15.05 1.35 23 6057 12.79 0.81 24 4697 13.00 1.19 PAGE 132 132 Table 5 9 Results for Seattle network Policy 1 Policy 2 Policy 3 Policy 4 No Toll 1 .00 0.91 0.83 0.5 0 0 .00 Net benefit (million $) 0.3884 0.3783 0.3298 0.2233 0 Gini coefficient 0.32664 0.32358 0.31854 0.31341 0.31288 Total travel time (hour) 45,511 46,331 47,863 49,009 51,746 Toll revenue ($) 721,878 593,198 321,787 40 0 Gain in income Group 1 7.54% 5.33% 1.35% 3.46% 0.00% Group 2 4.36% 3.11% 1.19% 0.72% 0.00% Group 3 1.83% 1.12% 0.05% 0.96% 0.00% Group 4 0.06% 0.31% 0.86% 1.43% 0.00% PAGE 133 133 Table 5 10 Mode specific demand for Seattle network Policy 1 Policy 2 Policy 3 Policy 4 No Toll Auto Transit Auto Transit Auto Transit Auto Transit Auto Transit Group demand Group 1 9225 3160 9988 4239 10603 6025 10206 8116 16642 939 Group 2 11997 1501 12257 1708 12636 2038 12703 2520 14434 690 Group 3 22194 1581 22336 1690 22615 1849 22764 2063 23517 984 Group 4 11663 542 11667 562 11711 583 11757 599 11724 418 Travel demand 55078 6785 56249 8200 57565 10495 57430 13298 66317 3030 PAGE 134 134 Table 5 11 Toll rates for Seattle network Policy 1 Policy 2 Policy 3 Policy 4 Link Auto Transit Auto Transit Auto Transit Auto Transit 3 5.00 1.00 0.00 1.00 0.00 5.00 0.00 15.00 4 5.00 1.00 1.00 0.00 0.00 4.25 0.00 15.00 5 10.25 0.00 8.25 2.00 3.75 10.00 2.25 16.00 6 13.00 1.00 13.00 1.00 11.50 4.00 12.50 1.00 8 10.00 0.50 8.25 2.50 3.25 8.00 0.00 15.50 9 13.00 0.25 12.75 2.25 10.75 3.25 11.00 0.25 12 10.00 0.00 10.00 0.00 7.75 3.75 6.00 0.00 13 20.00 2.00 20.00 6.00 16.00 12.00 12.00 18.00 15 11.50 0.00 11.00 1.75 9.00 2.75 5.50 0.00 16 15.00 2.00 15.00 6.00 11.75 10.00 7.00 18.00 17 15.00 0.25 11.25 5.50 6.75 11.75 2.00 20.00 20 8.00 0.50 4.25 4.50 0.75 10.25 0.00 16.50 Table 5 12 Results for policies with negative revenues for Seattle network Policy N1 Policy N2 0.67 0 Net benefit (million $) 0.1496 0.0003 Gini coefficient 0.30882 0.30478 Total travel time (hour) 50 706 52 524 Toll revenue ($) 348 060 711 219 Gain in income Group 1 8.47% 13.27% Group 2 2.69% 4.38% Group 3 1.99% 2.81% Group 4 1.99% 2.39% PAGE 135 135 Table 5 13 Mode specific demand for policies with negative revenues for Seattle network Policy N1 Policy N2 Auto Transit Auto Transit Group demand Group 1 9885 9410 9677 10183 Group 2 12788 2898 12909 3106 Group 3 22908 2262 23066 2355 Group 4 11797 636 11803 660 Travel demand 57368 15206 57456 16304 Table 5 14 Toll rates for policies with negative revenues for Seattle network Policy N1 Policy N2 Link Auto Transit Auto Transit 3 0 .00 20 .00 0 .00 20 .00 4 0 .00 20 .00 0 .00 20 .00 5 0 .00 20 .00 0 .00 20 .00 6 7 .00 11 .00 0 .00 20 .00 8 0 .00 19 .00 0 .00 20 .00 9 5 .00 9 .00 0 .00 20 .00 12 2.25 10.75 0 .00 20 .00 13 6 .00 20 .00 0 .00 20 .00 15 3.25 8.5 0 0 .00 20 .00 16 5.25 20 .00 0 .00 20 .00 17 0 .00 20 .00 0 .00 20 .00 20 0 .00 20 .00 0 .00 20 .00 PAGE 136 136 Figure 5 1 Nine node n etwork Figure 5 2 Relation of social benefit and Gini coefficient of the results Policy I Policy II Policy III Policy IV No Toll 0.126 0.127 0.128 0.129 0.13 0.131 0.132 0.133 0.134 0.135 0 100 200 300 400 500 600 700 Gini Coefficient Net Benefit PAGE 137 137 Figure 5 3 Seattle sketch n etwork PAGE 138 138 Figure 5 4 Seattle surrounding area PAGE 139 139 Figure 5 5 Compositions of productions for the centroid nodes Figure 5 6 Compositions of attractions for the centroid nodes 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Group 1 Group 2 Group 3 Group 4 0 1000 2000 3000 4000 5000 6000 7000 Everett Seattle Bellevue Tacoma Lynnwood Renton Group 1 Group 2 Group 3 Group 4 PAGE 140 140 Figure 5 7 choice percentage by origin Figure 5 8 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% Auto Transit No Travel 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% Group 1 Group 2 Group 3 Group 4 Auto Transit No Travel PAGE 141 141 Figure 5 9 Relation of social benefit and Gini coefficient of the results Figure 5 10 Changes in equivalent income Policy 1 Policy 2 Policy 3 Policy 4 No toll 0.312 0.314 0.316 0.318 0.32 0.322 0.324 0.326 0.328 0 0.1 0.2 0.3 0.4 0.5 Gini Coefficient Net Benefit (million USD) 10.0% 8.0% 6.0% 4.0% 2.0% 0.0% 2.0% 4.0% 6.0% Policy 1 Policy 2 Policy 3 Policy 4 Gain in Income Group 1 Group 2 Group 3 Group 4 PAGE 142 142 Figure 5 11 Relation of social benefit and Gini coefficient of the results Figure 5 12 Changes in equivalent income Policy 1 Policy 2 Policy 3 Policy 4 No toll Policy N1 Policy N2 0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.1 0 0.1 0.2 0.3 0.4 0.5 Gini Coefficient Net Benefit (million USD) 10.0% 5.0% 0.0% 5.0% 10.0% 15.0% Policy 1 Policy 2 Policy 3 Policy 4 Policy N1 Policy N2 Gain in Income Group 1 Group 2 Group 3 Group 4 PAGE 143 143 CHAPTER 6 EQUITABLE AND TRADABLE CREDIT SCHEMES 6.1 I ntroduction to Tradable Credit Schemes 6 .1.1 Alternative Demand Management Schemes Although pricing approach has been consistently advocated by economists and transportation researchers for nearly a cent ury, the public objections toward congestion pricing still limit its implementations to a small number of cities worldwide. Given the ubiquitous political resistance and public objection to congestion charges, researchers and planners have turned to other demand management instruments to curtail the unrestricted use of private vehicles. A commonly used quantity control method is road space rationing. One implementation of such a scheme is the plate number based rationing strategy, which has been implemented i n many cities in Latin American, e.g. Mexico City and Sao Paulo and China e.g. Beijing and Guangzhou Such a method restricts cars from using the road space for certain days during the week by their license plate number s Recently, Han et al. (2010) and Wang et al. (2010) analyzed the user equilibrium condition and efficiency of such a plate number based rationing strategy. Their analytical results show that Pareto improvement is achieva ble in both short term and long term equilibrium conditions. In practice, observable congestion reduction and air quality improvement have been reported under short term rationing. However, in long term, pure plate number based rationing strategy tends to promote car ownership as commuters may seek a second car in order to circumvent the restriction. As a result, the effectiveness of such a scheme will be limited in long term. In particular, an increase in the total number of vehicles in circulation and a change toward old, cheap and polluting vehicles are PAGE 144 144 observed in Mexico City after the implementation of the rationing strategy (Davis, 2008; Mahendra, 2008) A hybrid strategy is proposed by Daganzo (1995) by using both pricing and rationing He proved the possibility to achieve Pareto optimum using this strategy without toll redistribution for a single bottleneck. However, analysis of such scheme on the San Francisco Oakland bay bridge by Nakamura and Kockelman (2002) suggest ed that the possibility of achiev ing Pareto improvement for such a scheme is very small if not im possible. Another quantity control method proposed in the literature is the tradable mobility permit (or credit) scheme. Under such a scheme, drivers are require d to process a permit (or certain number of credits) in order to use (a certain area of) the road space. The perm its (or credits) are tradable among road users. Earlier deve lopment of t radable permit schemes usually focus es on environmental objectives p ar tially due to the fact that the permit system gives the regulators direct control over the quantity of environmental impacts (e.g. emissions) (Cropper and Oates, 1992) Dales (1968) was among the earliest researchers on this topic, who proposed the creation of transferable rights to attain water quality targets in a cost effective manner. Existing implementations of tradable permit schemes include the US emissions trading policy, which aim s to reduce emissions by issuing emission reduction credits (Tietenberg, 1994) The Kyoto Protocol and the European Union Emission Trading Scheme also implemented similar schemes (Perrels, 2010) One derivative of such permit schemes for congestion mitigation is the vehicle ownership qu ota system where the governmental authority imposes a limit for the PAGE 145 145 number of new vehicle registrations allowed each year. Such a scheme was implemented in Singapore (Chin and Smith, 1997) and Beijing (Yan and Wills, 2010) 6.1.2 Tradable Credit Schemes a nd Its Public Acceptability A tradable credit scheme for traffic congestion management involves: a) initial distribution of credits; b) credit trading among travelers ; c) credit charging schemes. Under such a scheme, the road users will be charged certain number of credit s to use road facilities. However, users can decide whether to use the credits for their travel or to sell them to other users. Verhoef et al. (1997) analyzed the potential of such a scheme in the regulation of road transportation externalities and discussed many practical applications on both the demand side (user oriented) and the supply side (the automobile and fuel industries). Kockelman and Kalman je (2005) investigated responses from a survey conducted in Austin, Texas. Their analysis suggests that the tradable credit scheme can emerge as a viable, cost effective and strongly supported str ategy for congestion mitigation. S uch a scheme promises subs tantial benefits for both network efficiency and equity, and addresses key issues that can undermine other methods A personal tradable permit system that aims to reduce carbon emission is evaluated by Wadud (2010) The results showed that the policy can b e progressive if the allocation strategies are appropriately selected (such as the per adult based uniform allocation) Mathematically, tradable credit schemes have been formulated as variational inequalities by Nagurney and Dhanda (1996) and Nagurney (200 0) Recently, Yang and Wang (2011) proposed a simple mathematical program to model the equilibrium condition under the tradable credit scheme where the price of the credit is determined by the market. The model is briefly introduced in Section 6. 2 .2 PAGE 146 146 6. 1.3 The Tradable Credit Distribution and Charging Scheme We assume that the credits are universal for all links but link specific in the amount of credit charge. The design of tradable credit distribution and charging scheme consists of three major parts: a) initial distribution of credits by the government ; b) the determination of the number of credits to be charged for each link in the network; c) design of the trading mechanism that enables transfer of credits among road users. It is assumed that the gov ernment will distribute the credits to eligible travelers free of charge. However, the credits can only be used for a specific time period (a month, for example), thus no one can gain by banking or stocking credits for future use. The government needs to d etermine the total number of credits to be distributed and how to distribute the credits among all the road users. The distribution can be uniform, but it may be more beneficial to implement more complicated schemes based on the objective of the government The credits that travelers received can be used for travel on any link in the network. However, the number of credits needed may vary from link to link. In order to travel on the network, the traveler must submit the required number of credits for all th e links he or she visited. The credit charging scheme is designed by the government in order to manage traffic condition in the network. If the available credits for a traveler are less than the number required the traveler cannot travel on the specific l ink unless additional credits are obtained. These extra credits can be purchased from other travelers who have leftover credits or simply prefer to receive the money in exchange for travel right s It is assumed that the government does not interfere with t he credit trading market but solely acts as a manager to monitor the system. It is also assumed the transaction cost is minimal that it can be ignored in PAGE 147 147 the model. The price of the credit, on the other hand, is market determined and solely based on the su pply and demand on the market. 6. 2 Mathematical Model 6. 2 .1 Representation of User Utility under Tradable Credit Scheme In the conventional pricing schemes, travelers need to directly pay a certain amount of money to use the roadways. Thus their income level is directly affected by the amount of toll charges they paid. For a utility function where is the travel tim e and is the income level, the utility under a pricing scheme can be easily calculated by adjusting the user income level with the toll rates i.e. where is the total toll cost for the user. However, under a tradable credit scheme, road u sers do not pay directly to access the roadway. But their effective income levels are still affected by the charging scheme through the realized value of the tradable credits i.e. the market determined credit price For example, if a traveler needs more c redits than the ones he/she was distributed, he/she needs to buy those credits from the market and pay the corresponding price. T he travelers can also sell their extra unused credits to increase their income. Thus, the effective income for the travelers ar e decided by their initial income, the number of credits initially distributed to them, the number of credits required for their travel and also the market determined price of the tradable credits. Mathematically, where is the effective income level under the tradable credit scheme, is the market determined credit price, is the initial number of credits distributed, is the total number of credits charged for the traveler. PAGE 148 148 The user utility level then can be calculated using the effective income level. The following equations are some commonly used utility functions : Linear In Income: Generalized Leonief: Translog: For the linear in income utility function, as the initial income level is a constant for every traveler, the removal of decision making. Thus the utility function can be simplified as: 6. 2 2 Single Modal User Equilibrium under the Tradable Credit Scheme Now we introduce the single modal user equilibrium formulation proposed by Yang and Wang (2011) This model considers one single travel mode and fixed OD demand. A linear utility function is used that the travelers will choose the path with the smallest ge neral travel cost (travel time plus the cost of credits). Given the initial distribution and the charging scheme, the feasible flow and credit price set can be represented as: PAGE 149 149 where is the set of all OD pairs, is the set of all links, is the set of all paths connecting OD pair is the flow on path for OD pair is the aggregate flow on link is the demand for OD pair is the path link incident matrix. is the total number of credits distributed and is the number of credits to be charged on link Based on the feasible region the user equilibrium condition can be represented as: (6 1 ) (6 2 ) (6 3 ) (6 4 ) where is the travel time on link is the smallest generalized travel cost for OD pair is the market determined credit price. Constraint s (6 1 ) and (6 2 ) equil ibrium condition, which state that the paths with positive flows must be the ones PAGE 150 150 with the lowest generalized travel cost Constraint s (6 3 ) and (6 4 ) state the relationship between the credit price and the total number of credits being charged. Because the credit can be freely traded without transaction cost the price will only be positive if all the credits distributed are charged back by the government. The above equilibrium conditions can be formulated as a nonlinear mathematical programming problem as follow: For the proof of the equivalence of the above minimization problem and the credit based user equilibrium conditions, please refer to Yang and Wang (2011) 6. 2 3 User Equilibrium with H eterogeneous Users and Nonlinear User Utility Function The above model has a simple structure and easy to solve. However, it is based on the strong assumption that the road users are homogeneous and the user utility function is linear with respect to travel time and credit cost. In this subsection, we extend the user equilib rium model to consider the nonlinear user utility function with heterogeneous users Assuming we have travelers in different income groups traveling on different OD pairs. Let be the demand of users in group of OD pair and be the us er utility for the user s in group of OD pair choosing path which is a function of the path travel time, number of credits to be charged and credit price. The users then will PAGE 151 151 choose their travel paths in order to maximize their utility. The user e quilibrium condition under this situation can be written as: The user equilibrium problem can be reformulated as the following VI problem: where the feasible region is defined as. (6 5 ) Not e that credit price is now a decision variable in the model. The equivalency of the VI formulation and the user equilibrium conditions can be shown by analyzing the KKT condition s of the VI as shown below: PAGE 152 152 (6 6 ) (6 7 ) (6 8 ) (6 9 ) where is the multipl iers associated with Constraint (6 5 ) Constraints (6 6 ) and (6 7 ) ensure the path flow follows the user equilibrium condition that every traveler in the network tries to maximize his/her utility level, where is the maximal utility for travelers in group of OD pair Constraints (6 8 ) and (6 9 ) regulate the credit price and guarantee the total number of credits charged is not more than the amount distributed. 6. 2 4 Multimodal Stochastic User Equilibrium u nder Tradable Credit Scheme 6 2 4 .1 Problem Description With the aforementioned tradable credit scheme, we consider a general multimodal transportati on network that includes three types of facilities, i.e., transit services, high occupancy/toll (HOT) and regular toll lanes. Instead of charging tolls, HOT and regular toll lanes will only charge travelers credits. Further, HOT lanes can charge different number of credits from SOV and HOV users. A tradable credit scheme in this setting involves a credit distribution scheme that distribute the initial credits to the travelers based on certain criteria, and a link based credit charging schemes that PAGE 153 153 specifies the number of credits to be charged on each link for different travel modes. The number of credits being charged must be nonnegative. We further consider multiple user groups with different incomes and preferences among four travel modes, i.e., no travel, transit, drive alone and car pool. It is assumed fun ctions are nonlinear in income. Under the nested logit model, travelers will first choose their travel modes based on the expected utility for those modes and then choose among all the available paths within the selected mode. 6. 2 4 .2 Mathematical Model In the aforementioned setting, the multimodal user equilibrium problem under the tradable credit scheme can be formulat ed as following VI problem: (6 10 ) where is the observable portion of the user utility. T he feasible region is defined as: (6 11 ) PAGE 154 154 (6 12 ) (6 13 ) The KKT condition s of the VI problem can be written as: where and are multipliers associated with C onstraints (6 11 ) and (6 12 ) respectively. I t is straightforward to prove that the above KKT conditions are the multimodal user equilibrium conditions under tradable credit scheme. To solve the problem, we reformulate the VI problem as a nonlinear programming problem (Ag hassi et al., 2006) The reformulated problem is as follow: PAGE 155 155 If the above problem can be solved with a zero objective value, part of the solution, i.e. will solve the VI problem. 6. 3 Tradable Credit Scheme Design Problem The design of a tradable credit scheme involves specifying both the credit distribution scheme and the credit charging scheme. We assume only positive credit number can be charged for each link and travel mode. That is, travelers will not be given the chan ce to obtain more credits by traveling. So that the transportation agency can have strict control over the total number of credits available to the travelers. Further, the credit charges are mode specific that they can be different for different travel modes PAGE 156 156 even on the same link. The initial credit is distributed based on certain criteria such as is to improve both the efficie ncy and equity of the transportation system. The feasible region of the credit scheme can be described as: (6 14 ) (6 15 ) (6 16 ) (6 17 ) where is the number of credits to be charged on link for travel mode and is the number of credits distributed to the travelers in group of OD pair Constraint (6 14 ) ensures the only positive number of credit can be charged, while C onstraint (6 15 ) guarantees travelers will not be charged anything if they decide not to travel. Constraint (6 16 ) says the amount of the initial distributed credits is nonnegative. Constraint (6 17 ) requires the total credits distributed to be a predetermined amount The tradable credit scheme design problem can be formulated as follows: (6 10 ) (6 17 ) where and are social benefit and Gini coefficient under the credit scheme as specified by is a weighting factor We still use equivalent income to calculate the social benefit and Gini coefficient. The equivalent income under tradable credit scheme is defined as: PAGE 157 157 Where is the user utility of user group of OD pair given income level and tradable credit scheme Using the result from Dagsvik and Karlstrom (2005) we can calculate the expected equivalent income for each user group and OD pair in a similar ways as in Chapter 5. Under the tradable credit scheme, the government will not receive or pay any money. Thus the social benefit only includes user benefit, which is the sum of the expected equivalent income of all the travelers. The Gini coefficient, as the equity measure, is also calculated based on the expected equivalent income where is the average expected equivalent expenditure for all travelers under the credit scheme PAGE 158 158 6. 4 Solution Algorithm Similar to the pricing scheme design problem as described in Chapter 5, the tradable credit scheme problem is also a MPCC problem. A derivative free algorithm such as the compass search algorithm discussed in Section 5 2 .5 can be used to solve this problem. However, as the initial credit distribution also need to be determ ined in addition to the credit charging scheme, the number of the decision variables is significantly large r, which renders the compass search algorithm inefficient. In order to efficiently solve the problem, we adopt the SID PSM (A Pattern Search Method G uided by Simplex Derivatives) algorithm (Custdio and Vicente, 2007; Custdio et al., 2010) SID PSM algorithm is a derivative free algorithm for constrained and unconstrained optimization problem. The algorithm use simplex derivatives to approximate the e xact derivatives of the objective function and use them to guide the search directions. The solution algorithm is developed as that the SID PSM algo rithm on the upper lever decides the credit distribution and charging scheme, while the lower level multimodal traffic assignment problem based on any given credit scheme is solved using the nonlinear reformulation. We implement the algorithm in GAMS (Brooke et al., 2005) in conjunction with Matlab (Wong, 2009) and use CONOPT (Drud, 1995 ) as the solver for the multimodal user equilibrium problem. 6. 5 Numerical Example The model and solution algorithm is implemented in the Seattle sketch network as described in Section 5. 3 .2. We use the same network parameters and demand data in this examp le. We assume the initial credits can only be distributed by the origin of each traveler. That is to say, all the travelers from the same origin will be given the same PAGE 159 159 number of initial credits. Such a credit distribution scheme is easy to implement. The g overnment can simply distribute the credits by the travelers home address. The user utility function is assumed to be a form of the Translog utility function as follow: where is the daily income for user group converts daily income to annual income is the total number of credits charged on path for OD pair of travel mode is the number of initial distributed credits and is the market determined credit trading price The values of parameters , and are the same as shown in Figure 5 6 The minimal OD travel time is used in place of for the no travel option when calculating the user utility function. We further assumed that the total number of credits distributed to the travelers are fixed and averages 10 credit per traveler. The problem is solved by varying the value of the weighing factor The results are shown in Table 6 1 Figure 6 1 plots the results of the obtained optimal credit schemes and the original UE result without the credit scheme It can be seen that all the obtained credit schemes can improve the social benefit, including Scheme 6 that only considers the equ ity measure. Those schemes with higher weight in equity measures can also achieve a better level of equity Figure 6 2 shows the changes in equivalent income for different traveler groups under the obtained optimal policies. Under the most efficient credit scheme, travelers from all four groups benefit from the scheme. The ones that enjoy the most benefit PAGE 160 160 increase are the low income and high income travelers. For high income travelers, they obtained most of the benefit from the decrease in the travel time due to the decrease in the demand for the auto travel mode (an 18% decrease from 66317 trips to 54426 trips). The low i ncome travelers, however, benefit the most by giving up traveling by auto mode or just choosing not to travel and selling unused credits to the others. As the credit schemes give more weight to the equity measure, the only gainers of the schemes become the low income travelers. Their gains in equivalent income increase dramatically to 13% of their total income under Scheme 6. At the same time, the high income travelers are made worse off. In this scenario, a significant amount of low income travelers decide not to take their trip and sell their credit to others as this offers them the most value. However, the travel demand from high income travelers is only slightly affect, which means these travelers with higher income are buying credits from others. Table 6 2 reports the credit distribution and charging schemes obtained by solving the credit scheme design problem. In c ontrast to the congestion pricing schemes as discuss ed in Chapter 5, with the total number of credits distributed fixed, a more equitable credit charging scheme generally charges more than the more efficient ones. For the most equitable scheme (Scheme 6), the credit charges on almost all links of auto mode are close to the upper bound (20 credits per link). Moreover, the transit travel mode will also be charge d a lot of credits for certain links, which is the exact opposite of what is observed from the congestion pricing scheme designs. The reason behind th is phenomenon is that the mechanism to achieve higher equity level is different in pricing and credit scheme s In general, in order to improve PAGE 161 161 equity level, subsidies from high income travelers to low income travelers is required. Under congestion pricing, the only way to subsidize the travelers is to subsidize transit service. As transit is the mode that has the highest composition of low income travelers, subsidizing the transit service can indeed improve equity. Under tradable credit scheme, the subsidiz ation appears in a totally different form by the trading of credits. During any credit trading transaction, the credit buyer pays directly to the seller which essentially acts as subsidizing the seller. Because high income travelers value the travel right s (or credit s ) higher, they generally gain by buying credit at the market price. On the other hand, low income travelers are also willing to sell their credits in exchange for money. Thus, the higher the credit price, the more money low income travelers wi ll receive during the transactions. When a credit charging scheme charges more credit for travel, it will increase the demand of credits. As a result, the credit price will be higher, which effectively offers the credit sellers (i.e. low income travelers) more subsidies. Figure 6 3 shows the comparison of the results of tradable credit schemes and congestion pricing schemes. It can be observed that the Pareto frontier of the tradable credit schemes is strictly dominatin g that of congestion pricing schemes. In other words, for whatever the combination of the weighing parameters, there is always a tradable credit scheme that can perform better than the best congestion pricing scheme available. Moreover, tradable credit sch emes can easily provide progressive results while congestion pricing schemes cannot reach equity levels that are better than the original condition without external subsidy. 6. 6 Summary In this chapter, we considered an alternative to congestion pricing, n amely, the tradable credit scheme. Different from congestion pricing scheme, tradable credit PAGE 162 162 scheme charges credits instead of money from road travelers. The credits are distributed to the travelers in a way to maximize system efficiency and/or equity. The credits distributed can be freely traded among travelers at a market determined price. A mathematical model is developed in this chapter for the design of more efficient and equitable tradable credit scheme. The model is formulated as a n MPCC problem and solved using a derivative free algorithm SID PSM. A numerical example is given based on the Seattle sketch network. The results from the solution algorithm show that tradable credit scheme can significantly improve the condition of the transportation syste m by both reducing congestion and improving system equity. Compared to congestion pricing scheme as discussed in Chapter 5, the tradable credit scheme can achieve higher system achievement in both efficiency and equity measures and does not require any kin d of external subsidy. PAGE 163 163 Table 6 1 Results of optimal tradable credit schemes Scheme 1 Scheme 2 Scheme 3 Scheme 4 Scheme 5 Scheme 6 10 0.98 0.91 0.67 0.5 0 Net b enefit (million $) 0.3977 0.3950 0.3843 0.3742 0.3103 0.1499 Gini c oefficient 0.3233 0.3187 0.3127 0.3107 0.3089 0.3072 Total t ravel t ime 45 201 45 298 45 380 45 097 44 458 42 831 Gain in i ncome Group 1 2.62% 5.63% 5.79% 7.48% 10.07% 13.08% Group 2 1.17% 2.06% 2.31% 2.33% 2.30% 1.60% Group 3 2.04% 1.90% 1.53% 1.29% 0.64% 0.69% Group 4 2.34% 1.32% 1.45% 1.14% 0.31% 0.98% Auto Transit Auto Transit Auto Transit Auto Transit Auto Transit Auto Transit Group d emand Group 1 8937 2785 9091 2809 9716 2695 9099 2768 7721 2801 5697 1316 Group 2 11822 1446 11886 1440 11958 1416 11645 1478 10655 1635 9169 1084 Group 3 22035 1567 22050 1567 21975 1576 21720 1636 20818 1811 19697 1366 Group 4 11632 542 11628 543 11581 553 11531 567 11317 615 11161 496 Travel d emand 54426 6340 54655 6359 55230 6239 53995 6448 50511 6863 45723 4261 PAGE 164 164 Table 6 2 Optimal c redit distribution and charging schemes Scheme 1 Scheme 2 Scheme 3 Scheme 4 Scheme 5 Scheme 6 Credit d istributed Everett 0.0 2.0 8.2 15.8 15.3 14.4 Seattle 0.0 0.0 13.1 9.2 6.8 3.4 Bellevue 13.1 0.0 0.0 0.0 0.0 0.0 Tacoma 0.0 0.8 7.0 11.6 15.6 21.8 Lynnwood 0.0 7.9 14.2 18.8 21.5 23.6 Renton 54.1 65.1 19.6 13.8 12.7 11.9 Auto Transit Auto Transit Auto Transit Auto Transit Auto Transit Auto Transit Credit c harged Link 3 11.3 2.6 10.8 1.9 5.5 0.1 5.8 0.3 4.1 0.6 6.6 1.5 Link 4 8.0 2.1 8.4 1.8 8.8 0.2 14.9 1.3 19.9 2.1 19.9 14.5 Link 5 11.8 0.6 12.1 0.6 14.0 1.6 12.3 0.3 11.5 0.8 14.6 4.1 Link 6 12.9 0.0 13.3 0.0 18.6 0.0 19.9 1.7 19.9 2.5 19.9 14.9 Link 8 13.1 0.8 13.4 1.0 15.3 1.6 15.2 0.2 19.9 2.2 19.9 14.4 Link 9 15.8 0.5 15.7 0.2 15.7 0.2 16.8 1.1 20.0 3.5 19.9 15.8 Link 12 9.7 0.0 10.2 0.0 16.1 0.0 16.6 2.0 19.9 3.1 19.9 15.4 Link 13 20.0 0.0 20.0 0.0 20.0 0.0 20.0 0.2 20.0 0.9 20.0 5.0 Link 15 14.3 0.6 13.7 0.2 13.4 0.5 14.9 1.2 18.9 3.6 19.9 15.9 Link 16 16.9 0.0 15.5 0.0 12.3 0.0 12.5 0.0 16.4 0.0 17.9 10.9 Link 17 17.3 0.7 17.5 0.6 14.4 0.9 13.9 0.6 13.4 1.0 16.3 6.4 Link 20 6.3 0.0 5.7 0.0 4.4 0.0 4.5 0.0 7.9 0.0 20.0 11.3 PAGE 165 165 Figure 6 1 Pareto frontier of tradable credit schemes Figure 6 2 Changes in equivalent income Scheme 1 Scheme 2 Scheme 3 Scheme 4 Scheme 5 Scheme 6 Original Condition 0.306 0.308 0.31 0.312 0.314 0.316 0.318 0.32 0.322 0.324 0.326 0 0.1 0.2 0.3 0.4 0.5 Gini Coefficient Net Benefit (million USD) 2% 0% 2% 4% 6% 8% 10% 12% 14% Scheme 1 Scheme 2 Scheme 3 Scheme 4 Scheme 5 Scheme 6 Percentage Change in Income Group 1 Group 2 Group 3 Group 4 PAGE 166 166 Figure 6 3 Comparison of tradable credit schemes and congestion pricing schemes 0.305 0.31 0.315 0.32 0.325 0.33 0 0.1 0.2 0.3 0.4 0.5 Gini Coefficient Net Benefit (million USD) Tradable Credit Congestion Pricing Original Condition PAGE 167 167 CHAPTER 7 CONCLUSION S AND FUTURE RESEARCH 7.1 Conclusions This dissertation focuses on technical approaches to improve the public acceptability of congestion pricing schemes in multimodal transportation networks The network s being consider ed in this dissertation include transit services, high occupancy vehicle/high occupancy toll ( HOV/HOT ) facilities (which can implement differentiating toll charges) and regular toll lanes. W e try to improve the acceptability of congestion pricing in such networks by carefully design ing where to charge and how much to charge and incorporate multiple innovative instruments such as differentiating tolls subsidizing transit service and expandin g highway capacity We proposed design models for t hree types of schemes in this dissertation. They are Pareto improving pricing, toll optimization with explicit equity considerations and tradable credit schemes. A Pareto improving pricing scheme improves the transportation system efficiency with the guarantee that no traveler will be made worse off compar ed to the condition without any pricing intervention. Such a scheme overcomes the drawback of marginal cost pricing and is expected to receive more approval from the general public. In this dissertation, two pricing scheme design models are proposed based on trip based and tour based travel demand models, respectively. Our models also allows the use of toll revenues 1) to subsidize transit servi ce in order to encourage travelers to switch to higher occupancy travel modes, 2) to expand highway capacities to reduce traffic congestion. The se model s are formulated as MPEC problem s and solve d via manifold suboptimization method and relaxation algorith m. Numerical examples indicate Pareto improvement can be achieved using the proposed model and solution algorithm in the PAGE 168 168 example networks The Pareto improving pricing schemes can lead to significant reduction in traffic congestion while guarantee that nob ody is made worse off. The tour based model offers a more realistic representation of the travelers mode choice behavior, thus will provide more accurate results. However, the trip based mod el may be used as an approximat ion when there is insufficient data to support the tour based model. As equity is one of the most important factors that affect public acceptance on congestion pricing, the next part of this dissertation develops a model that explicitly takes into consideration the distributional impacts of congestion pricing across different income and geographic groups. More importantly, this model considers the usually The proposed model provides transportation agencies a powerful tool in congestion pricing design with both efficiency and equity objectives. The model provides a framework for congestion pricing scheme designs that can incorporate varies demand models and equity measures. The model is implemented in a test n etwork as well as a realistic freeway network in the Seattle area. The results show that although a progressive pricing scheme may not be available, our model can always provide pricing schemes that achieve a better balance between efficiency and equity, w hich may be favor ed by the general public. Lastly we investigated an innovative tradable credit scheme as an alternative to traditional congestion pricing scheme s In contrast to congestion pricing, under tradable credit sc heme, travelers do not necessari ly need to pay out of pocket money in order to use the roadway being regulated. Moreover, tradable credit scheme provides another PAGE 169 169 way to subsidize low income travelers. We tested such a scheme in the Seattle area freeway network, the results are very promi sing that tradable credit schemes can achieve better performance than congestion pricing in both efficiency and equity simultaneously. With the advance of electrical tolling technologies, tradable credit scheme could be implemented without much technical d ifficulty This scheme provides a very attractive alternative for congestion pricing. 7.2 Future Research The literature in transit modeling has grown extensively since the in tro duction of the strategy based model as adopted in Chapters 3 and 4. There are more advanced models that can be incorporated in the design models proposed in this dissertation in order to more accurately represent the transit network. Moreover, the models presented in this disserta tion only consider simple travel mode choices that only consist of one mode for a trip or tour. It is worth to extend the models to capture more complicated mode choice behaviors such as park and ride (a tour that includes both transit and auto modes) and carpooling with travelers from different OD pairs (a tour consists of trips with both SOV and HOV modes). For the tour based Pareto improving design models, another important future research direction is to consider the time dimension. Travelers do not mak e all the trips at once. Instead, they take their trips by sequence and their departure time s could be subject to the traffic condition and service availability of public transit at different time s of day. Other factors, such as trip purpose, parking pri ce, propensity for flexibility could also affect travelers mode choice decisions. Further research could inco rporate these considerations in to the pricing design models to provide more accurate and more meaningful analysis of the impact s of pricing schemes. PAGE 170 170 The models for the design of more equitable pricing and tradable credit schemes as proposed in Chapters 5 and 6 are generally hard to solve, especially when the number of toll links is large. The algorithms proposed in this dissertation, although being able to solve the problem s to good solutions, are usually time consuming and the result quality highly depends on the initial solution s given to the algorithm s Future work may explore how to further improve the efficiency and reliability of the algor ithm s The pricing or credit charging schemes considered in this dissertation is link based, where tolls or credits are charged when travelers traveling on certain links of the network. Many real world implementation s however, may appear in different forms such as area based and distance based pricing schemes. Thus, it is meaningful to consider extend the proposed model to accommodate different types of pricing or tradable credit schemes. Lastly, the long team effect of congestion pricing or tradable credit schemes could also be investigated. The models presented in this dissertation mainly focus on short term situations where travelers origins and destinations are assumed to be fixed. In the long term, these congestion mitigation strategies could have substantial impacts to travelers choices of origins and destinations such as residential and work locations. People may prefer to live or work at places where they can avoid the toll charges and their travel pattern may also be significant altered A pricing or tradable credit model that can take into consideration the long term effect could become a very helpful tool for the government agencies in develop ing their traffic management strategies PAGE 171 171 LIST OF REFERENCES Aashtiani, H.Z. and Magnanti, T.L., 1981. Equilibria on a congested transportation network. SIAM Journal on Algebraic and Discrete Methods 2(3), pp.213 226. Abdulaal, M. and LeBlanc, L.J., 1979. Methods for combining modal split and equilibrium assignment models. 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PAGE 185 185 BIOGRAPHICAL SKETCH Di Wu received his Bachelor of Science degree in a utomation from Tsinghua University in 2005 in China. He then joined the m aster s degree program in s pace c enter of Tsinghua University and received his Master of Science degree in a erospace e ngineering in summer 2007. After t hat, Di started to pursue his Ph.D. degree in Department of Civil and Coastal Engineering at University of Florida under the supervision of Dr. Yafeng Yin. Since then, Di has been awarded two Master of Science degrees in c ivil e ngineering and i ndustrial e n gineering at University of Florida, modeling. 