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Traffic assignment models for a ridesharing transportation market

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Traffic assignment models for a ridesharing transportation market

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Content TRAFFIC ASSIGNMENT MODELS FOR A RIDESHARING TRANSPORTATION MARKET by Huayu Xu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (INDUSTRIAL AND SYSTEMS ENGINEERING) June 2014 Copyright 2014 Huayu Xu Acknowledgments It has been an unforgettable journey pursuing a doctoral degree. Without the guidance and assistance from many others, I would not have come to this point. First of all, I cannot convey my gratitude by words to my advisor, Dr. Maged Dessouky. I feel honored and lucky from the bottom of my heart that I met such an intelligent, knowledgeable and kind-hearted advisor. Without his firm support and con- stant belief in me, I might have lost my confidence and hope on this path. He guided me with my academic studies. He enlightened me with my research works. He comforted me whenever I felt frustrated. He encouraged me to pursue higher goals. He pushed me to become even better. I feel blessed to have such a wise advisor and a lifetime friend. I would also express my sincere thanks to my co-advisor, Dr. Fernando Ordonez. I learned a lot from him. He led me step by step into this topic. He showed me great patience and diligence. He helped me overcome all kinds of tough problems I encoun- tered. He is such a talented, hard-working, yet easy-going person. His half-advisor-half- big-brother role made the whole Ph.D. studies much more enjoyable. His intelligence and humor are always the warmest and strongest support. Sometimes I wonder what good I have done to deserve the guidance from not only two, but three top professors in our field. I am so grateful that the esteemed professor, Dr. Jong-Shi Pang, also advised me on my research. He sacrificed his weekends and hol- idays to help me enhance the theoretical part of my model. He spent hours patiently to ii show me the guideline of a mathematical proof. He opened a special course to improve the mathematical background for me and my classmates. I was so moved and touched that Dr. Pang would do so much in guiding me. Apart from the above three professors, I would also thank all the other professors in both my qualifying and defense committee: Dr. Genevieve Giuliano, Dr. Alejandro Toriello and Dr. James Moore. Their comments and advice broadened my visions and inspired me with potential research directions. Especially thanks to Dr. Giuliano, I was able to jump out of my small circle and think about the value of my work at a higher level. I thank the chair, Dr. Julia Higle, and the staff in our department, Evelyn Felina, Georgia Lum, Rick Scott, Mary Ordaz, and Norma Perry. I feel proud to graduate from such a wonderful department full of care and energy. I would also like to thank all my lab-mates. First of all, special thanks to Chen Wang, who had set a very high standard for her lab-mates in terms of intelligence and diligence. As a senior member in the lab, she helped me adapt quickly to the new living environment, guided me with my coursework and research, and shared me with her precious experience in both work and life. I thank all the other lab-mates, Shi Mu, Xiaoqing Wang, Christine Nguyen, Michael Poremba, Lunce (Chris) Fu, Weihong Hu, Han Zou, Yihuan (Ethan) Shao, Wentao Zhang, and Liang Liu, for all the supports they gave me and all the happy moments we shared together. I also thank Professor Martha Townsend and my classmates, Wanqing Yu, Xiaolu Han, Xu Qiu, Da Tong, Teng Ma and Dianxun Hou, from the course of academic English. They provided me with valuable advices to improve my writing and presenta- tion skills for this dissertation. I learned a lot from them in English communication in general. iii Last but not least, I wish to express my greatest gratitude to my family. I thank my father, Hanjiu Xu, for always being strict with me as I grew up so that I become who I am today. I thank my mother, Shuzhen Zhang, for constantly being my most loyal friend and listener whom I share everything with at all times. I thank my husband, Su Chen, for accompanying me by my side during all the years of my Ph.D. studies, and giving me great comforts and supports whenever I reach my lowest point. This thesis is dedicated to them all. iv Contents Acknowledgments ii List of Figures viii List of Tables x Abstract xii 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Literature Review 7 2.1 Ridesharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Traffic Assignment Problem . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Complementarity Problem . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Path-based Solution . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 TAP Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Research Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Same OD Pairs (Model I): Formulation 16 3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Elastic Demand TAP with Ridesharing Prices . . . . . . . . . . . . . . 21 3.4 Determining Ridesharing Prices Including Traffic Congestion . . . . . . 25 3.5 Combining the Pricing Model with Elastic Demand TAP . . . . . . . . 27 3.5.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5.2 Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 29 v 4 Same OD Pairs (Model I): Computational Results 36 4.1 Frank-Wolfe Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Optimality Check . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 Overall Results . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.3 Congestion and Disutility Levels . . . . . . . . . . . . . . . . . 43 4.2.4 Ridesharing Prices . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2.5 Number of Travelers . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Large-scaled Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.1 Heuristic Improvement . . . . . . . . . . . . . . . . . . . . . . 48 4.3.2 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . 50 4.3.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5 Different OD Pairs (Model II): Formulation 53 5.1 Network Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Additional Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.2 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 A TAP Model with Capacity Constraints . . . . . . . . . . . . . . . . . 64 5.3.1 Arc Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3.2 Path Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3.3 Generalized User Equilibrium . . . . . . . . . . . . . . . . . . 67 5.4 Mixed Complementarity Problem . . . . . . . . . . . . . . . . . . . . 71 5.5 Variational Inequality Formulation . . . . . . . . . . . . . . . . . . . . 74 5.6 Solution Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . 75 5.6.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.6.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6 Different OD Pairs (Model II): Computational Results 82 6.1 Arc Formulation of the MiCP . . . . . . . . . . . . . . . . . . . . . . . 82 6.1.1 Reformulation w.r.t.x . . . . . . . . . . . . . . . . . . . . . . 82 6.1.2 Formulation Equivalency . . . . . . . . . . . . . . . . . . . . . 85 6.2 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.1 Test Case #1 – Three-node . . . . . . . . . . . . . . . . . . . . 86 6.2.2 Test Case #2 – Braess . . . . . . . . . . . . . . . . . . . . . . 90 6.2.3 Test Case #3 – Sioux-Falls . . . . . . . . . . . . . . . . . . . . 93 6.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.1 Changing Arc Capacityc a . . . . . . . . . . . . . . . . . . . . 99 6.3.2 Changing Inconvenience Parameters d ; d ; p ; p . . . . . . . 101 6.3.3 Changing Pricing Parameters;v;w . . . . . . . . . . . . . . . 103 6.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 vi 7 Conclusions and Future Work 109 7.1 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 A 119 A.1 Positive Semidefiniteness . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.2 Integral Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.3 Proof of Non-negativity . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.4 Model Equivalency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A.5 KNITRO Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 A.5.1 Sioux Falls, 149 OD pairs . . . . . . . . . . . . . . . . . . . . 132 vii List of Figures 3.1 Interaction among congestion, number of travelers and price . . . . . . 17 3.2 Traffic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Economic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Changing pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1 Optimality check – travel cost . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Optimality check – equilibrium condition . . . . . . . . . . . . . . . . 41 4.3 Congestion levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Sensitivity analysis of disutilities . . . . . . . . . . . . . . . . . . . . 44 4.5 Price distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.6 Sensitivity analysis of average prices . . . . . . . . . . . . . . . . . . 46 4.7 Sensitivity analysis of number of drivers . . . . . . . . . . . . . . . . . 47 4.8 Sensitivity analysis of number of passengers . . . . . . . . . . . . . . 48 4.9 Algorithm comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1 Original graph,jN 0 j = 3;jA 0 j = 6 . . . . . . . . . . . . . . . . . . . . 54 5.2 Node duplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 Arc duplication (extended network) . . . . . . . . . . . . . . . . . . . 56 5.4 Arc duplication (extended network) . . . . . . . . . . . . . . . . . . . 57 5.5 Generalized user equilibrium: example . . . . . . . . . . . . . . . . . . 70 6.1 Example with trips and arc capacities for Three-node . . . . . . . . . . 87 viii 6.2 Example with trips and arc capacities for Braess . . . . . . . . . . . . 90 6.3 Proportions of solo drivers, ridesharing drivers and passengers for Sioux- Falls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4 Average proportions of solo, ridesharing drivers and passengers for Sioux- Falls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.5 The original network of Sioux-Falls . . . . . . . . . . . . . . . . . . . 97 6.6 Average proportion of solo drivers changing with arc capacity . . . . . 100 6.7 Average proportion of ridesharing drivers changing with arc capacity . 101 6.8 Average proportion of passengers changing with arc capacity . . . . . . 101 6.9 Proportion of solo drivers changing with the inconvenience cost . . . . 104 6.10 Proportion of ridesharing drivers changing with the inconvenience cost 104 6.11 Proportion of passengers changing with the inconvenience cost . . . . . 104 6.12 Proportion of solo drivers changing with the price . . . . . . . . . . . . 106 6.13 Proportion of ridesharing drivers changing with the price . . . . . . . . 106 6.14 Proportion of passengers changing with the price . . . . . . . . . . . . 106 ix List of Tables 4.1 Constant settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Overall results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Results for Anaheim. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5 Results for Winnipeg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 Results for Barcelona. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.1 Parameter settings for Three-node . . . . . . . . . . . . . . . . . . . . 87 6.2 Computational results forx for Three-node . . . . . . . . . . . . . . . 88 6.3 Computational results fory and the costs for Three-node . . . . . . . . 89 6.4 Parameter settings for Braess . . . . . . . . . . . . . . . . . . . . . . . 91 6.5 Computational results for Braess . . . . . . . . . . . . . . . . . . . . . 92 6.6 The costs of paths for Braess . . . . . . . . . . . . . . . . . . . . . . . 92 6.7 KNITRO – final statistics for Sioux-Falls . . . . . . . . . . . . . . . . 94 6.8 Selectedx k a on the first two original arcs for Sioux-Falls . . . . . . . . 98 6.9 Ridesharing proportions with arc capacity changes (%) . . . . . . . . . 100 6.10 Parameter settings and constraint checks for inconvenience cost changes 102 6.11 Ridesharing proportions with inconvenience cost changes (%) . . . . . 103 6.12 Parameter settings and constraint checks for pricing changes . . . . . . 105 6.13 Ridesharing proportions with pricing changes (%) . . . . . . . . . . . 105 x 6.14 The impact of parameter changes on ridesharing proportions . . . . . . 107 7.1 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 xi Abstract A nascent ridesharing industry is being enabled by new communication technologies and motivated by its many possible benefits, such as reduction in travel cost, pollution, and congestion. Understanding the complex relations between ridesharing and traffic congestion is a critical step in the evaluation of a ridesharing enterprise or of the use- fulness of regulatory policies or incentives to promote ridesharing. In this research, we propose two new traffic assignment models that explicitly represent ridesharing as a mode of transportation. The objective is to analyze how ridesharing impacts traffic congestion, how people can be motivated to participate in ridesharing, and conversely, how congestion influences ridesharing, including ridesharing prices and the number of drivers and passengers. The first model considers the scenario where drivers and passengers sharing the same ride must travel from the same origin and to the same destination. This model is built by combining a ridesharing market model with a classic elastic demand Wardrop traffic equilibrium model. It is formulated as a convex optimization problem. The Frank- Wolfe algorithm is adopted to solve the model and a heuristic approach is applied using the equilibrium condition. Our computational results show that: (1) the ridesharing base price influences the congestion level, (2) within a certain price range, an increase in the price may reduce the traffic congestion, and (3) the utilization of ridesharing increases as the congestion increases. xii The second model drops the constraint of the same origin-destination (OD) pair. In this model, drivers may pick up or drop off any passenger in the middle of their trips, and they may even detour from a seemingly shortest path. In order to describe this scenario, we extend the network by doubling the nodes and tripling the arcs in size. A generalized user equilibrium is defined to represent the new network and the new constraints. The generalized user equilibrium can be formulated as a mixed complementarity problem (MiCP), and equivalently a variational inequality. It is proved that there exists one and only one solution to this model. The KNITRO solver is adopted to solve the MiCP and the computational results are promising. It can be concluded from the results that when the congestion cost decreases or the ridesharing inconvenience cost increases, more travelers would become solo drivers and thus less people would participate in ridesharing. On the other hand, when the ridesharing price increases, more travelers would become ridesharing drivers. In conclusion, the traffic congestion, the ridesharing cost, and the number of trav- elers interact with each other closely in both models. Understanding their relationships enables planners to develop policies to draw more people to participate in ridesharing and thus to reduce traffic congestion. xiii Chapter 1 Introduction 1.1 Background With rapid population growth and city development, traffic congestion has become an important issue, especially in large cities. The 2012 Annual Urban Mobility Report developed by the Texas Transportation Institute estimates that (a) The amount of delay endured by the average commuter was 38 hours, up from 16 hours in 1982, and (b) the cost of congestion is more than $120 billion, nearly $820 for every commuter in the U.S. (Schrank et al., 2012). At the same time, there is no public support for increased spending on infrastructure capacity expansion. Thus, there is a need for innovative transportation modes that can be implemented to improve transportation conditions in a cost-efficient manner. Ridesharing appears as one such transportation mode that could at least help mitigate the congestion increase, as it can tap into the significant amount of unused capacity in transportation networks. Ridesharing occurs when individuals share a personal vehicle for a trip and split travel costs such as gas, toll, and parking fees, with others that have similar itineraries and time schedules (Furuhata et al., 2013). Benefits of ridesharing include travel cost savings, reducing travel time, mitigating traffic congestion, conserv- ing fuel, and reducing air pollution (Chan and Shaheen, 2011; Ferguson, 1997; Kelley, 2007; Morency, 2007). However, ridesharing is still not a regular transportation alter- native and is considered an informal and disorganized activity. The lack of efficient 1 methods to coordinate itineraries and schedules is an important factor that inhibits the wide adoption of ridesharing. Recently, technological advances including global positioning systems (GPS) and mobile devices have greatly enhanced the communication capabilities of travelers, facil- itating the creation of ridesharing in real-time. Taking advantage of this opportunity, a number of companies, such as Avego (Carma), SideCar, flinc, CarpoolWorld, etc., have emerged to develop systems where travelers (including both drivers and passengers) can be matched in real time via web browsers and mobile apps (Furuhata et al., 2013). In a sense these companies are establishing a marketplace for drivers to offer up their empty seats to other travelers. In these newly developed systems the drivers receive a com- pensation for participating, which can be in the form of smaller travel times, reduced tolls, or direct payment that help mitigate the travel costs. The essential difference of such ridesharing systems from traditional public transit systems is that they do not hire professional drivers and they function as a matching agency that pairs passengers with “citizen” drivers. 1.2 Problem Description However, any benefit of ridesharing will be limited by its degree of adoption. There has been controversy over security, payment and regulations. Even if these practical issues are taken care of, it is not easy to determine how attractive ridesharing will be to users in a given city, due to the complex interconnection among traffic congestion, the ridesharing cost (price) and the number of travelers participating in ridesharing. For instance, to decide whether to participate in ridesharing, drivers may weigh the loss of privacy and increased time against the compensation they may earn for taking on passengers, while passengers would compare the inconvenience and security to a 2 saving in the travel time and cost of a shared ride. These tradeoffs would balance in an equilibrium that would determine how many people are participating in ridesharing as drivers and passengers. On the other hand, an increase in the ridesharing price would make it more attractive for drivers and hence more people would like to become a ridesharing driver. At the same time, an increased price would reduce the number of potential passengers leading to an increase in the potential drivers (either solo drivers or ridesharing drivers) and also an increase in the traffic congestion. This means there will be a larger supply of ridesharing trips which will reduce the cost/price of ridesharing. Therefore there would be a balance point for the ridesharing price that both drivers and passengers would agree on and both the number of drivers and passengers and the traffic congestion would be stable. Thirdly, an increase in the adoption of ridesharing could lead to a reduction in the traffic congestion, while the decrease in the traffic congestion would result in a decline in the ridesharing activities. Therefore, all these three factors–the traffic congestion, the ridesharing price, and the number of travelers–are interacting with each other and trading off with one or the other. Understanding how ridesharing would influence traffic congestion is fundamental in the evaluation of a ridesharing enterprise or in assessing the convenience of regulatory policies or incentives to promote ridesharing. 1.3 Motivation In this thesis, we study a transportation system where ridesharing has the ability of capturing a significant portion of travel demand via a real-time matching agency. In this system we assume that passengers will pay drivers for ridesharing services to share 3 the travel cost, and we also assume that such fees are negotiable between drivers and passengers. We refer to this system as the Transportation Market. In the Transportation Market, travelers are categorized by their roles: solo drivers, ridesharing drivers and ridesharing passengers. Any traveler may choose to become a ridesharing driver or passenger, or even drive alone (not participating in ridesharing activities), according to their own decisions after comparing the advantages and disadvantages of different roles. We note that the ridesharing price is an abstraction to represent compensation that drivers take into account in their decision to participate in ridesharing, such as a reduction in the travel time or toll costs that will occur by being able to use HOV lanes. The Transportation Market is envisioned as an advanced ridesharing system based on auction mechanisms and game theoretic models that will allow for the automated nego- tiation of routes and prices between passengers and drivers in real-time. Furthermore we assume that the Transportation Market operates as an open marketplace, where the price for ridesharing is negotiated between both drivers and passengers. A potential passenger would identify the origin-destination and time constraints desired and drivers could bid on these trips based on their utilities. An automated auction mechanism would match acceptable bids with the proper requests where the passenger could then decide which provider to accept based on its own utility function. The ridesharing prices discussed in this research are considered the payment a passenger pays to a driver for the rideshar- ing service. The ridesharing price may include the extra costs of drivers for taking on passengers, but also can represent other benefits that drivers perceive by participating in ridesharing, such as reduced travel time and moral well-being. These benefits are represented through the travelers’ utility functions. The purpose of this research is to determine how attractive ridesharing will be to travelers in a given city where the Transportation Market exists. The key decision factors would include the traffic congestion, the ridesharing cost (price) and the number of 4 travelers, including the number of solo drivers and the number of drivers and passengers that participate in ridesharing. To achieve this goal we propose two new traffic assignment models based on the user equilibrium assumption that would take into consideration the unique characteris- tics of ridesharing. The two models are designed for two different assumptions: (a) the drivers and passengers sharing the same car must travel from the same origin and to the same destination, or (b) otherwise. Such models allow us to analyze how ridesharing and traffic congestion would impact each other, and also to determine the impact that different regulatory interventions could have on ridesharing, and hence on traffic. More specifically, the existing traffic equilibrium models have to be extended to consider the specific characteristics of ridesharing, where 1) the cost/price of ridesharing is deter- mined by the number of people participating, and 2) the offer for shared rides (capacity of the transportation mode) varies with the traffic congestion and the price. Note that although the models are based on the user equilibrium assumption, i.e. each traveler is minimizing his/her own travel cost, the models are still proposed from a system point of view. The purpose of this work is to provide system administrators or city planners an analytic and planning tool that helps to evaluate the performance of the ridesharing Transportation Market. By changing the parameters in the models, planners can draw conclusions on how ridesharing activities interact with the traffic congestion and use these results to set policies to encourage ridership such as congestion pricing. 1.4 Organization of the Thesis The thesis is organized as follows. The literature review is presented in Chapter 2, where we examine previous studies in both ridesharing and traffic equilibrium problems, and stress the differences between the equilibrium models with ridesharing activities from 5 previous multi-modal and elastic demand models. Chapters 3 and 4 focus on the model where drivers and passengers sharing a ride must travel on the same OD pair. While Chapter 3 mainly discusses the model formulation and analysis, Chapter 4 focuses on the algorithm design and the discussion of computational results. Similarly in Chapters 5 and 6, we study the model formulation and discuss the test cases for the problem where drivers and passengers sharing a ride may travel on different OD pairs. Chapter 7 summarizes the entire work and identifies potential future research opportunities. 6 Chapter 2 Literature Review 2.1 Ridesharing Ridesharing is a joint-trip of at least two participants that share a vehicle and requires coordination with respect to itineraries (Furuhata et al., 2013). Ridesharing has drawn much interest in both industry and academic fields in recent years. According to Furuhata et al. (2013), ridesharing activities can be divided into three main types: (1) unorganized ridesharing, (2) semi-organized ridesharing, and (3) organized ridesharing. Unorganized ridesharing, which involves family, colleagues, neighbors, and friends, has a long history, yet is limited scaled due to inefficient communication methods. Semi- organized ridesharing services occur spontaneously among individual travelers moti- vated by access to faster HOV (High-Occupancy Vehicle) lanes or reduced toll. Exam- ples of this type of ridesharing service are casual carpooling (Burris and Winn, 2006; Kelley, 2007), and slugging, which formed in the Washington D.C. area free of charge to the participants (LeBlanc, 1999; Spielberg and Shapiro, 2000). These services run on their own momentum; they are not started or run by a public or private entity (Levofsky and Greenberg, 2001). Therefore they are limited to specific locations or circumstances and are difficult to replicate elsewhere. Organized ridesharing is operated by agencies that provide ride-matching opportuni- ties for participants without regard to any previous historical involvement (Dailey et al., 1999). With innovative technologies inhibitors of ridesharing can be overcome and a number of private matching agencies have emerged during the last decade (Dailey et al., 7 1999; Heinrich, 2010; Amey, 2010; Ghoseiri et al., 2011; Chan and Shaheen, 2011). By introducing mobile technologies like smart phones as well as global positioning systems (GPS), ridesharing systems can be implemented in a real-time fashion and its degree of adoption will increase when agencies can help match any two participants with their travel itineraries and current locations. However, there are still a limited number of papers that deal with issues of dynamic/real-time ridesharing services (Agatz et al., 2012). While most of the existing papers focus on the matching mechanisms of real-time ridesharing services, the pricing problem has received less attention in the literature. Pricing specifies the amount of money transferred between the involved parties (drivers and passengers), including how to share the costs of gas, toll, and parking, and how to charge transaction fees by the matching agencies (Furuhata et al., 2013). One type of pricing for dynamic ridesharing is based on the auction mechanism, where drivers or passengers specify their least or highest preferred price, respectively (Kleiner et al., 2011). A related example can be found in e-Bay, where multiple sellers offer the same commodity with different deadlines and the clearing prices are not iden- tical. Another type of pricing is called cost-sharing, where a price is determined by a cost calculation formula specified by a matching agency. Several different cost-sharing mechanisms have been designed and they are applicable to share the transportation cost in a static setting (Winter and Nittel, 2006; Frisk et al., 2010). This type of pricing is dif- ficult to implement in dynamic ridesharing because it should involve the consideration of fairness when splitting the cost with multiple passengers who may be picked up or dropped off at any time during the trip. A third type of pricing is more negotiable, where the matching agencies are not involved in pricing. The price is negotiated between the potential participants (either drivers or passengers) while they determine whether or not to participate in the ridesharing activity (Furuhata et al., 2013). 8 There are few papers discussing the relationship or the interaction between rideshare pricing and traffic congestion. Yang and Huang (1999) discussed the carpooling behav- ior and the optimal congestion pricing in a multilane highway with or without HOV lanes where the first-best pricing and the second-best pricing models were formulated and compared. The models, however, were limited to identical commuters (single origin and single destination, SOSD) and the number of passengers in each carpooling vehicle is fixed to one. Later Qian and Zhang (2011) studied the morning commute problem with three modes: transit, driving alone and carpool. They analyzed the interactions among the three modes and how different factors affect their mode shares and network performance. Again, the model is limited to a SOSD network and does not consider the tremendous interactions of rideshare pricing among different origin-destination (OD) pairs. 2.2 Traffic Assignment Problem To study the effects of multiple OD pairs, one classic model is the Traffic Assignment Problems (TAP), which evaluates the distribution of travelers among different routes and OD pairs. There are many methods to assign traffic to paths, a standard assumption in a TAP is the user-equilibrium (UE) assumption, also known as Wardrop’s user traffic equilibrium law (Wardrop, 1952). According to this assumption, the travel times (con- gestion costs) in all the used paths are equal and less than those which would be expe- rienced by a single vehicle on any unused path. In other words, no traveler can reduce their travel cost/congestion by switching to another route at the equilibrium (Patriks- son, 1994). However, by traveling every individual causes congestion and helps deter- mine the travel cost for everyone else. Therefore the traffic assignment problem with the user equilibrium assumption establishes a model that predicts how travelers choose 9 their routes given a road network. In this thesis the TAP refers to the TAP with UE assumption. There could exist several fastest routes in this assumption, as long as they have the same travel (time) cost. 2.2.1 Convex Optimization There exist many solution approaches to the traffic assignment problems. One of them is to formulate it as a convex optimization problem (Patriksson, 1994; Babonneau, 2006; Bar-Gera, 2006, 2010), and apply general nonlinear programming (NLP) methods to solve the TAP. One of the most widely used NLP methods is the Frank-Wolfe method (Frank and Wolfe, 1956), which works as a reduced gradient method. The method was originally designed for the solution of quadratic convex problems. The main idea is to find a descending direction by solving a linear programming subproblem. For most TAP models, which have convex objective functions and linear constraints, the Frank- Wolfe algorithm is always an easy solution. In a basic TAP model, instead of solving a linear programming, the shortest paths under current flow distribution can be used as the descending direction. The Frank-Wolfe algorithm generally makes good progress towards the optimum during the first few iterations, but convergence often slows down substantially when close to the optimal solution. This method was efficient back in the 1970s (LeBlanc, 1975) and has been popular for UE models for several decades. Another well-used solution approach is the Analytic Center Cutting Plane Method (ACCPM) (Goffin et al., 1992), which is related to the Dantzig-Wolfe decomposition method. It is based on a column generation technique defining a sequence of primal linear programming maximization problems. This method is efficient for solving UE models with fixed demands. Later Babonneau and Vial (2008) extended ACCPM to solve UE models with elastic demands. They showed that ACCPM is capable of solving large instances at a high level of accuracy. 10 2.2.2 Complementarity Problem A more generalized form of convex optimization is the complementarity problem. The KKT conditions of the convex optimization problem include complementary slackness conditions, which together with variable nonnegativity form a complementarity prob- lem. Facchinei and Pang (2003) in their book “Finite-Dimensional Variational Inequal- ities and Complementarity Problems, V olume I”, pages 41–46, gave the detailed for- mulation of the user equilibrium as a complementarity problem. They formulated the problem in both path- and arc-based formulations, and also proved the equivalency of the complementarity problem and the variational inequality. The benefit of formulating the user equilibrium as a complementarity problem instead of a convex optimization is that it does not require an objective function. In some cases, it is hard for some types of TAPs to have an explicit objective function. There are also existing algorithms and solvers to solve complementarity problems, like PATH (Ferris and Munson, 1999) or KNITRO (Byrd et al., 1999, 2006). Agdeppa et al. (2007) studied the traffic equilibrium problem with nonadditive costs, i.e. the cost of a path does not equal the sum of the costs of the arcs belonging to the path. The paper formulates the problem as a monotone mixed complementarity problem under appropriate conditions, and hence the existence and uniqueness of the solution can be proved. 2.2.3 Path-based Solution With rapid urban development and dramatic information explosion, the efficiency and the scalability have both become significant issues and stimulate researches to find better algorithms. Another type of solution approach for TAPs is based on the network features of the problem. Researchers have being focusing on specialized algorithms making 11 use of the actual paths of the traffic network. For example, Jayakrishnan et al. (1994) proposed a path-based algorithm and their results show that their method converges in 1/10 iterations than the conventional Frank-Wolfe algorithm. Later Bar-Gera (2002) utilized the acyclic subnetworks rooted at origins and designed a new approach called the origin-based algorithm. Nie (2012) pointed out that this algorithm may generate negative second-order derivative, leading to a “wrong search direction”, and then fixed the algorithm by approximating the second-order derivative with an upper bound. Bar-Gera (2006) presented a general consistency condition that is satisfied by any set of minimum-cost routes, and showed how it can be used in choosing a set of routes that is likely to be similar to the set of user-equilibrium routes. The proposed consis- tency condition is also essential for finding the entropy-maximizing route flows solution, which may be regarded as the most likely one, and an efficient method for finding the entropy-maximizing solution is presented. In a following paper, Bar-Gera (2010) intro- duced an efficient solution method called the Traffic Assignment by Paired Alternative Segments (TAPAS) algorithm. Instead of comparing two entire paths, the TAPAS algo- rithm focuses on pairs of alternative segments. These algorithms are efficient but are only designed to solve the basic traffic assignment model with fixed demands. To the best of our knowledge they have not been applied to models with elastic demands or multiple modes. 2.3 TAP Variants One of the most important variations is the traffic assignment problems with elastic demands. By introducing a “utility” (or “disutility”) function, the problem decides not only how people will choose their paths, but also how many people would travel given certain congestion conditions. This kind of model serves to illustrate those cases where 12 people might not travel when the traffic is highly congested. The problem can also be formulated as a convex optimization problem with a decreasing convex utility function for a tradeoff between congestion and demand. The elastic TAP is well studied in the literature (Gartner, 1980a,b). Early studies can also be found in (Florian and Nguyen, 1974; Fukushima, 1984; Hearn and Yildirim, 2002; LeBlanc and Farhangian, 1981). Babonneau and Vial (2008) used a variant of the analytic center cutting plane method (ACCPM) to solve this problem. They tested the method on elastic demand instances with the Bureau of Public Roads congestion function and different demand functions (constant elasticity and linear), and showed that ACCPM is capable of solving large instances at a high level of accuracy. Yildirim and Hearn (2005) presented a toll pricing framework for a general variable demand traffic assignment problem with side constraints, where the demand between an origin destination pair is a function of the least total travel cost for making the trip. They modeled the problem from both the system and the user points of view. Another variation of the traffic assignment is the multi-mode model, where two or more transit modes, say private vehicle and public transit, are being studied simultane- ously. The multi-mode variant also includes another characteristic of traffic equilibrium: multi-class model, where travelers are divided into several classes, say high and low income (Aashtiani and Magnanti, 1981; Florian et al., 2002). Aashtiani and Magnanti (1981) formulated a multi-mode model as its optimal con- ditions. In their model, the total demand of travelers in all modes is a constant and the well-known logit model is applied in deciding the demand for each mode. They also proved the existence and uniqueness of the solution to their model. LeBlanc and Abdulaal (1982) studied a combined mode split assignment problem, where there are two groups (classes) of travelers and two modes (public transit and auto mobile). The cost function is designed as a quartic function of the total auto demand plus 13 a linear function of the total transit demand, where the number of buses and the number of passengers on each bus are not specified. Only the total number of trips (passengers) is in consideration and their cost is approximated linearly. In addition to a fixed total demand model, they also introduced a multi-mode model with elastic demand. Similar to a utility function, they defined a gravity property, where the demand of each group of travelers is exponentially decreasing with the congestion cost. Boyce and Bar-Gera (2003) described the formulation, estimation and validation of combined models for making detailed urban travel forecasts. Their model was based on a large-scale, multiclass model of peak period urban travel (Chicago region). Other models and methods of multi-mode traffic assignment problems can be found in Beck- mann et al. (1955); Abdulaal and LeBlanc (1979); LeBlanc and Farhangian (1981). 2.4 Research Gap Although there is rich literature on elastic demand TAPs and multi-modal TAPs, these models cannot easily be adapted to ridesharing due to the existence of an endogenous modal price. Although vehicle tolls and transit fares have been considered in the prior literature (Boyce and Bar-Gera 2003), these costs are treated as exogenous, known ahead of time and independent of how users occupy the transportation network. Ridesharing prices on the other hand could conceivably be determined by the amount of ridesharing that occurs due to traffic conditions. For instance, in the presence of high congestion (or high transportation costs due to high gas prices) it is plausible that more drivers are will- ing to participate in ridesharing, thus lowering the ridesharing price. A low ridesharing price would increase the number of people willing to be passengers reducing conges- tion. The presence of this endogenous transportation mode price, whose dependence on congestion is not straightforward, is a novel feature of the TAP studied here. Finally, we 14 note that few papers have considered the impact that, even an exogenous, ridesharing cost would have on traffic congestion. 15 Chapter 3 Same OD Pairs (Model I): Formulation In this chapter, a model is constructed to describe the interaction between traffic conges- tion and ridesharing activities in the Transportation Market, especially for the case where both ridesharing drivers and passengers sharing the same vehicle must travel between the same OD pair. We consider the case where the number of travelers is not fixed, i.e. an elastic demand traffic assignment model under the user equilibrium assumption is the main structure. Suppose travelers are travelling according to their own decisions. They can decide to travel or not depending on both traffic congestion and ridesharing features (prices, number of travelers, etc.). If they choose to travel, they need to choose one mode of traveling: to drive alone, to drive as a ridesharing driver taking on other passengers, or to travel as a ridesharing passenger. For example, a traveler may observe the congestion condition and ridesharing prices before traveling. If it is too congested, he/she may not travel at all. If one decides to travel, he/she will estimate the pros and cons of ridesharing and then decide in which way he/she would like to travel. If the inconvenience of ridesharing costs one too much, he/she may decide to drive alone. If the benefit of taking on other passengers overcomes the inconvenience, one may choose to be a ridesharing driver. If the inconvenience cost and the price to pay for ridesharing appear less than driving, one may prefer to be a ridesharing passenger. The following model captures the above decision activities and the interactive impact among traffic congestion, the number of travelers, and the price paid for ridesharing 16 services. Such interactions include, as shown in Figure 3.1, (a) traffic congestion would impact the number of travelers and the ridesharing price; (b) the number of travelers would determine traffic congestion directly, and also will influence the price; (c) the ridesharing price would also impact the number of travelers. Figure 3.1: Interaction among congestion, number of travelers and price 3.1 Problem Description Consider a transportation network represented by a graph with nodes and arcs, where nodes could be origins, destinations or intermediate stops, and arcs are direct roads that connect two nodes. Each individual travels from an origin to a destination, which is called an origin-destination (OD) pair. For each OD pair, there exist multiple paths that start from the same origin and end at the same destination. The congestion cost of each path is evaluated by the travel time along that path, which is a summation of travel times on each arc that builds up the path. The travel time of each arc is determined by the number of vehicles (drivers) travelling on that link/arc. Therefore, each arc may be shared by multiple paths (may or may not be for the same OD pair), and conversely each path of a certain OD pair may encounter drivers for other OD pairs. This fact is essential since a slight change in the number of drivers on one arc may impact the travel 17 times or congestion costs of many paths. A user equilibrium is a state where for each OD pair, the travel times of all used paths are equal or less than those which would be experienced by a single vehicle/driver on any unused path (Wardrop, 1952). In addition to user equilibrium, to include specific features of ridesharing it is also assumed that: Elastic demand: the total number of travelers of any OD pair is not fixed. It is determined by people’s willingness to travel, i.e. the utility function. A driver may decide to drive or not travel at all according to this utility function. Same utility function: the utility function of drivers is identical for all drivers across OD pairs. For this model, one utility function of driving is employed for all drivers, including both solo drivers and ridesharing drivers. It is determined by traffic congestion and ridesharing prices. Same OD pair: ridesharing drivers would only be willing to take on passengers that travel between the same OD pair, i.e. drivers would not pick up or drop off any passenger in the middle of their route, even if no detour is required. Congestion cost: the travel time or the congestion cost is calculated only by the total number of vehicles/drivers (including both solo drivers and rideshar- ing drivers). That is to say, the number of passengers in each vehicle does not contribute to the congestion cost. Inconvenience cost: the passenger pick-up and drop-off times will be treated as part of the inconvenience cost of ridesharing drivers. The ridesharing prices: passengers would pay ridesharing drivers some fee for the trip. The price is determined by the availability of ridesharing vehicles and 18 requests of passengers for each OD pair. The fee can be seen as a form of com- pensation to the drivers for the additional cost and inconvenience, and turns out to limit the number of passengers in the network. Conversely, ridesharing passengers would save money when compared to driving alone. Therefore ridesharing could attract some solo drivers to become passengers, mitigating traffic congestion. Unlimited capacity: the vehicle capacity, i.e. the number of passengers per vehicle is unlimited. Under this assumption, we do not distinguish different types of drivers in the model, since all passengers can be squeezed into one vehicle or may be dispatched evenly among all available vehicles. It is reasonable from the perspective of total travel cost: a driver can be either driving alone or sharing a ride only when the costs of the two are equal. In other words, suppose the travel cost of a solo driver is only the congestion cost, while the travel cost of a ridesharing driver is the congestion cost plus the inconvenience cost minus the ridesharing income. In this case, the inconvenience cost of ridesharing will be canceled by the income (or profit) of sharing a ride. Otherwise, all drivers would prefer the lower total cost: either driving alone or sharing a ride. Given the above assumptions, the main interactions between travelers, ridesharing prices and traffic congestion can be summarized as below (also see Figure 3.1). The traffic congestion cost is calculated by the total number of all drivers. The total number of all drivers is determined by traffic congestion and the ridesharing prices according to the drivers’ willingness to travel on the road. Similarly the number of passengers is also determined by traffic congestion and the ridesharing prices according to the passengers’ willingness to participate in ridesharing. 19 The ridesharing prices are determined by the interaction between ridesharing drivers and passengers, and are also dependent on the congestion cost. Our goal is to derive a model that reflects the above interactions and provides the number of travelers (both drivers and passengers), ridesharing prices, and congestion cost (travel time) for each OD pair at the equilibrium. 3.2 Notations Below is a list of notations that we use to formulate an elastic demand TAP model that includes ridesharing. N set of nodes. A set of arcs. ON set of origins. DN set of destinations. K set of OD pairs,KOD. k2K OD pair, wherek = (o k ;d k );o k 2O;d k 2D. a2A arc. p k ridesharing price for each passenger of OD pairk2K. q k number passengers for OD pairk2K. (0) k free flow time (fixed) for OD pairk2K. k congestion cost for OD pairk2K, k (0) k . k total number of drivers for OD pairk. x k a amount of flow (number of drivers) for OD pairk2K on arca2A. y a total amount of flow on arca2A,y a = P k2K x k a . 20 vector with components k ,2R jKj . x k vector with componentsx k a with respect tok,x k 2R jAj . x vector with componentsx k a ,x2R jKjjAj . y vector with componentsy a ,y2R jAj . 3.3 Elastic Demand TAP with Ridesharing Prices In the elastic demand traffic assignment problem, the objective function consists of two components: the sum of the integrals of the congestion cost over all arcs and the sum of the integrals of the utility function over all OD pairs. We adopt the classic BPR (Bureau of Public Roads) function (Dafermos and Spar- row, 1969) to calculate the congestion cost on each arc, i.e. tt a (y a ) =t a 1 + y a c a 4 ! wheret a is the free-flow travel-time of arca,c a is the capacity of arca and is a fixed coefficient. It is a strictly increasing function with respect to the amount of traffic flow. In the objective function, an integral of congestion cost is calculated on each arc so that the KKT conditions will give us the travel cost experienced by each driver. The utility function of drivers is denoted by k ( k ;p k ), which is a function of the total number of drivers k and ridesharing pricep k for OD pairk. It provides the worst congestion cost the drivers could endure given the number of vehicles and the rideshar- ing price. Therefore it represents an aggregate utility for all drivers (including both solo drivers and ridesharing drivers) according to certain traffic congestion and ridesharing prices. The above utility function looks like one in the standard elastic demand model, 21 except that it includes the ridesharing price as a second variable. It is trivial in elastic demand models that this driver utility function decreases with k as more drivers (more congestion) makes a trip less appealing. The dependence on the ridesharing price will capture the fact that a payment for taking on passengers can be a form of compensation to drivers. We therefore assume that the utility function increases with the rideshar- ing price, i.e. drivers may accept worse traffic condition if there is an increase in their compensation of providing ridesharing services while the number of drivers stays the same. From the above definitions, we have the following relationship between the num- ber of drivers and the traffic congestion (see Figure 3.2): when the number of drivers (or the amount of traffic flow) k (or y a ) increases, the congestion cost tt a (y a ) would increase whereas the utility k ( k ;p k ) would decrease. Also, the utility k ( k ;p k ) would increase if the ridesharing price p k goes up. The traffic equilibrium is attained balancing these two relationships. The ridesharing price for every OD pair should be determined by the balance between supply and demand for shared rides in the Transportation Market. The eco- nomic equilibrium system (Hubbard and O’Brien, 2012) is adopted to describe the tradeoff between the ridesharing price and the number of drivers and passengers, where drivers are the supply and passengers are the demand. Let us denote byS k (q k ) the aggre- gate supply function and byD k (q k ) the aggregate demand function for each OD pairk. These functions represent for the supply (resp. demand) the price at which drivers are willing to offer (resp. passengers are willing to pay) given the number of available ridesharing seats (the number of passengers)q k . The pricep k and the number of passen- gersq k is determined when these two functions are equal, that is,S k (q k ) =D k (q k ) =p k for each OD pairk. 22 Figure 3.2: Traffic equilibrium In the next subsection we will give explicit form to these supply and demand func- tions. In particular we include the fact that the willingness of drivers to participate in ridesharing is increasing with the congestion cost, making it possible for drivers to ask for a lower price. Therefore the supply function is decreasing in the congestion level k , giving a supply function of the formS k (q k ; k ). 23 In sum, the elastic demand traffic assignment model for the Transportation Market is given as below: min x;y X a2A Z ya 0 tt a (s) ds X k2K Z k 0 k (r;p k ) dr (3.1) s.t. Nx k k k = 0; 8k2K (3.2) X k2K x k a y a = 0; 8a2A (3.3) S k (q k ; k ) =D k (q k ) =p k ; 8k2K (3.4) x k a 0; 8a2A;8k2K (3.5) Constraints (3.3) represents the flow decomposition constraint, i.e. the total amount of flow on each arc equals the sum of flows over all OD pairs on that arc. Constraint (3.4) describes the demand-supply balancing constraint from above, where k equals the con- gestion that k drivers create. Constraint (3.5) is the non-negativity constraint of vari- ables. Constraint (3.2) depicts the flow conservation constraint in a compact form, where N and k are coefficient matrix and vector.N = [(i;a)] i2N;a2A is anjNjjAj matrix with element (i;a) = 8 > > > > > < > > > > > : 1; nodei2N is the tail of arca2A, i.e.a = (j;i) 1; nodei2N is the head of arca2A, i.e.a = (i;j) 0; otherwise 24 and k = ( k i ) i2N is a vector inR jNj for anyk2K, with element k i = 8 > > > > > < > > > > > : 1; i =d k 2D 1; i =o k 2O 0; otherwise In other words, for each OD pairk, constraint (3.2) is a compact form ofjNj constraints, each of which is a flow conservation constraint at each node: (a) if it is a demand (destination) node, all incoming flows minus all outgoing flows should be equal to the demand of OD pair k; (b) if it is a supply (origin) node, the difference should be the negative value of the demand; or (c) if otherwise, the difference must be zero. Model (3.1) (3.5) shows how the ridesharing price influences drivers’ willingness to travel and the traffic congestion. Next we illustrate how the traffic congestion impacts the ridesharing prices. 3.4 Determining Ridesharing Prices Including Traffic Congestion The ridesharing prices, as a type of compensation to the drivers, are determined by the number of drivers and passengers participating in the ridesharing activities. For a given OD pair k, suppose p k is the price for each passenger and q k is the number of passengers in total. We assume that the drivers will have a joint utility given byU(p k ;q k ; k ) = p k q k W (q k ; k ), which represents a revenue ofp k q k minus an additional inconvenience cost W (q k ; k ) for providing q k ridesharing seats in total when the congestion cost is k . Assume that W (q k ; k ) is convex and quadratic in terms ofq k . By setting @U @q k (p k ;q k ; k ) = 0, the maximum utility would be attained at 25 p k = @W @q k (q k ; k ) =S k (q k ; k ), which gives us the supply function. We consider that S k (q k ; k ) is increasing in q k and decreasing in k , given the fact that, from the per- spective of drivers, the price should increase with the number of passengers, q k , but should decrease with the traffic congestion (in order to mitigate the congestion for faster travel). Therefore the inconvenience costW (q k ; k ) is also decreasing with respect to the congestion cost k . This is applicable, since the drivers are more likely to take on passengers under higher congestion cost. Suppose picking up passengers takes a driver 5 minutes, no matter how much the congestion cost is. Therefore the inconvenience cost would become relatively less when the congestion cost is 100 minutes, compared to a cost of 10 minutes, since both drivers and passengers are traveling on the same OD pair. As a result, it is reasonable that both the price and the inconvenience cost should decrease with congestion, from the perspective of drivers. On the other hand, we wish to maximize the benefit to the passengers as well. Sup- poseu(q k ) is the utility function of passengers, i.e. the benefit one can obtain being a passenger. With a total cost p k q k , the profit of all passengers is u(q k )p k q k . Hence assuming that u(q k ) is concave and quadratic with respect to q k , the maximal utility of passengers will be received atp k = du dq k (q k ) =D k (q k ), which gives us the demand function in the economic equilibrium model. In addition we consider that D(q k ) is decreasing inq k , since as the price increases the number of passengers should decrease. Given the above assumptions, we should have the following patterns (Figure 3.3): when the number of passengersq k increases, drivers (supply) would increase the price p k while passengers (demand) would decrease it. Also the price from the drivers would drop if the traffic congestion k increases. The economic equilibrium that determines the ridesharing price is obtained at the intersection. The above is true for all OD pairs. 26 Figure 3.3: Economic equilibrium 3.5 Combining the Pricing Model with Elastic Demand TAP Combining the two equilibria together (Figure 3.2 and Figure 3.3), we can see the fol- lowing changing pattern depicted in Figure 3.4: when increasing the congestion cost k , the ridesharing pricep k would decrease and the number of passengersq k would increase (from the economic equilibrium, Figure 3.3). When decreasing the price p k , both the congestion cost k and the number of drivers k would decrease (from the traffic equi- librium, Figure 3.2); and vice versa. Therefore the two equilibrium models influence each other and they will try to balance these interactions in a common equilibrium. To formulate a tractable optimization problem that represents the combined equilib- ria described above and stated in general terms in Model (3.1) (3.5), we consider spe- cific functional forms for the supply and demand functions of the economic equilibrium, as well as specific driver utility and congestion functions of the traffic equilibrium. The following result presents a tractable formulation for the combined equilibrium model 27 Figure 3.4: Changing pattern using generic quadratic functions for the economic utilities and linear function for the driver’s utility function. 3.5.1 Model Formulation Consider the following definition of the cost/utility functions: Congestion cost:tt a (y a ) =t a 1 + ya ca 4 . Driver utility function: k ( k ;p k ) = k p k k k . Driver inconvenience cost:W (q k ; k ) =b k q 2 k + d k k q k +e k ; b k > 0;d k 0; k > 0. Passenger utility function:u(q k ) =f k q 2 k +g k q k +h k ; f k > 0. 28 Then the combined equilibrium model generated from (3.1) (3.5) can be formulated as below. min x;y X a2A Z ya 0 tt a (s) ds X k2K Z k 0 k (r) dr (3.6) s.t. Nx k k k = 0; 8k2K (3.7) X k2K x k a y a = 0; 8a2A (3.8) 0 k k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k ; 8k2K (3.9) x k a 0; 8a2A;8k2K (3.10) where k ( k ), k 2 k + k ( k ) 2V k + k b k g k 2V k ; (3.11) k ( k ), q [ k V k ] 2 2 k 2 k k b k g k V k k + [4 k d k f k V k + ( k b k g k ) 2 ]; (3.12) V k ,b k +f k ; (3.13) for allk2K. Note thatp k can also be expressed as a function of k , thus k ( k ;p k ) is replaced by k ( k ), i.e. from a two-variable function to a one-variable function. 3.5.2 Model Analysis Proposition 3.1. Model (3.6) (3.10) is generated from Model (3.1) (3.5), substi- tutingp k (and its corresponding constraint (3.4)) by k from the result of the economic equilibrium. Proof. According to the definition, the driver’s ridesharing utility function, i.e. the driver’s profit by taking on passengers, is given byU(p k ;q k ; k ) = p k q k W (q k ; k ). 29 First order optimality condition gives usp k = S k (q k ; k ) = @W @q k (q k ; k ) = 2b k q k + d k k . Henceq k = 1 2b k p k d k k . Similarly, by optimizing the passenger’s profitu(q k )p k q k , we havep k =D(q k ) = du dq k (q k ) =2f k q k +g k . Henceq k = 1 2f k (g k p k ). According to constraint (3.4), we have the system p k = 2b k q k + d k k =2f k q k +g k ; q k = 1 2b k p k d k k = 1 2f k (g k p k ); we have p k = b k g k +d k f k = k b k +f k ; (3.14) q k = g k d k = k 2(b k +f k ) : (3.15) According to the result at the equilibrium of an elastic demand TAP, the congestion cost equals the utility function of drivers. We define the parameter k = k ( k ;p k ) = k p k k k . It can then be substituted into Eq. (3.14), and thus p k = b k g k +d k f k = k b k +f k = b k g k +d k f k = ( k p k k k ) b k +f k ; ) k (b k +f k )p 2 k [ k (b k +f k ) k + k b k g k ]p k + ( k b k g k k d k f k ) = 0: Therefore by solving the above quadratic equation in terms ofp k , we have ( k ),p k = k 2 k k + b k g k 2 (b k +f k ) ( k ) 2 k (b k +f k ) ; (3.16) 30 where ( k ) = q [ k (b k +f k )] 2 2 k 2 k k b k g k (b k +f k ) k + 4 k d k f k (b k +f k ) + ( k b k g k ) 2 : Thusp k can be expressed as a function of k . Moreover, the negative root can be discarded. The plus sign is chosen to guarantee the convexity of the objective function, i.e. making k ( k ) a decreasing function (see Proposition 2). In sum, we have p k = k 2 k k + b k g k 2 (b k +f k ) + ( k ) 2 k (b k +f k ) : Substituting the above the utility function k ( k ;p k ), we have k ( k ) = k 2 k + k b k g k 2 (b k +f k ) + ( k ) 2 (b k +f k ) : Proposition 3.2. If the parameters satisfy the following, b k ;f k > 0 andd k 0, g k d k = (0) k ; 8k2K, k d k f k b k +f k h (0) k i 2 ; 8k2K then Model (3.6) (3.10) is a convex optimization problem. Proof. Constraints (3.7) (3.10) are all linear. It is sufficient to show that the objective function (3.6) is a convex function. The first part of the objective function is convex since it is an increasing function with respect toy a (or equivalently k ). To show the convexity of the second part, it is sufficient to prove that k ( k ) is a decreasing function of k . 31 First of all, consider the fact that k ( k ;p k ) can never be negative (it is treated as the congestion cost that k drivers could bear and thus should always be positive, see Babonneau and Vial (2008)). Hence we must have k = max n 0; 1 k ( k p k k ) o . Substituting in (3.14), 0 k k k b k g k +d k f k = k b k +f k k k = k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 k k k k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k ; i.e. constraint (3.9). The last inequality holds because k (0) k , for allk2K. On the other hand, substituting (3.16) into k ( k ;p k ), we have k ( k ), k ( k ;p k ) = k ( k ) k k = k 2 k + k b k g k 2 (b k +f k ) ( k ) 2 (b k +f k ) : Note that the symmetric axis of ( k ) is k = k b k g k k (b k +f k ) . Therefore k ( k ) is decreasing only when (a) k ( k ) = k 2 k + k b k g k 2 (b k +f k ) + ( k ) 2 (b k +f k ) and k k b k g k k (b k +f k ) ; or (b) k ( k ) = k 2 k + k b k g k 2 (b k +f k ) ( k ) 2 (b k +f k ) and k k b k g k k (b k +f k ) : Combining constraint (3.9), we must have k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k k b k g k k (b k +f k ) ; or k b k g k k (b k +f k ) 0: 32 Since all the parameters are nonnegative and some are strictly positive, the latter will never hold. Therefore we must have k ( k ) = k 2 k + k b k g k 2(b k +f k ) + ( k ) 2(b k +f k ) and k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k k b k g k k (b k +f k ) , i.e. k d k f k b k +f k h (0) k i 2 for all OD pairk2K. Moreover, to assure the non-negativity of the number of passengers in (3.15), we must haveg k d k = (0) k , for allk2K. In sum, given the above conditions for parameter settings, k ( k ) is a decreasing function of k in domain (3.9). And thus Model (3.6) (3.10) is a convex optimization problem. Proposition 3.3. There exists an optimal solution to Problem (3.6) (3.10) due to convexity. Moreover, the optimal solution is a user equilibrium. Proof. Let k , a , k 0, k 0 and k a 0 be multipliers to constraints (3.7) (3.10), where k is a vector, i.e. k = ( k i ) i2N 2R jNj . The KKT conditions are tt a (y a ) a = 0; 8a2A (3.17) k j k i + a k a = 0; 8a2A;8k2K (3.18) k ( k ) + k o k k d k k + k = 0; 8k2K (3.19) k k = 0; 8k2K (3.20) k k k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k ! = 0; 8k2K (3.21) k a x k a = 0; 8a2A;8k2K (3.22) plus primal constraints (3.7) (3.10) and all non-negativities. 33 In (3.18),i andj correspond to arca, i.e. a = (i;j); similarly in (3.19),o k andd k correspond to the origin and the destination ofk, i.e.k = (o k ;d k ). Also note that (3.17) plus (3.18) gives us tt a (y a ) = k i k j + k a : For any pathl of a given OD pairk, the travel time (congestion cost) or the path is the summation of costs on consecutive arcs that consist of pathl, i.e. X a2l tt a (y a ) = X a2l k i k j + k a = k o k k d k + X a2l k a = k ( k ) + k k : The second equality holds because the arcs are consecutive and thus k i ’s are canceled out except for the origin and the destination. The third equality holds due to (3.19). Note that k ( k ) represents the least travel congestion cost a driver could bear given k vehicles on the road. If (3.9) strictly holds, i.e. 0 < k < k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k : we must have k = k = 0, due to the complementary slackness (3.20) and (3.21). This indicates that all vehicles (drivers) are traveling with the same congestion cost, i.e. k = X a2l tt a (y a ) = k ( k ); 8l (3.23) If k = 0: we must have k = 0 and 0. Therefore P a2l tt a (y a ) k ( k ), indicating that the congestion cost is larger than what drivers could bear and thus there is no driver for OD pairk, i.e. k = 0. 34 If k = k b k g k k (b k +f k ) + k d k f k k (b k +f k ) 1 (0) k (0) k k : we shall have k = 0 and k 0. This illustrates the congestion cost is less than drivers could bear, and thus all potential drivers are traveling on the road. 35 Chapter 4 Same OD Pairs (Model I): Computational Results In this chapter we propose an algorithm that will solve the model introduced in Chapter 3. The Frank-Wolfe algorithm is one of the classical and commonly used algorithms in solving the Traffic Assignment Problems (TAP). Generally speaking it is a reduced gradient method that iteratively solves nonlinear programming problems. Unlike some other efficient algorithms for TAP, which are more sensitive to some specific features of the problem, the Frank-Wolfe algorithm is more generalized and flexible to deal with complicated variants of TAP and thus can be easily applied to solve our problem. 4.1 Frank-Wolfe Algorithm As introduced in Chapter 3, (3.6) (3.10) is a convex optimization problem with a con- vex objective function and linear constraints. Letz denote the solution triplet (x;y;). LetF (z) andZ denote respectively the objective function and the feasible space defined by constraints (3.7) (3.10), i.e. F (z) =F (x;y;), X a2A Z ya 0 tt a (s) ds X k2K Z k 0 k (r) dr; Z,fz = (x;y;)j(x;y;) satisfying (3.7) (3.10)g: 36 The core idea of the Frank-Wolfe algorithm is to replace the objective function by its first-order Taylor expansion, i.e. F (z 0 )F (z) +rF (z) T (z 0 z); wherez;z 0 2Z. Therefore minimizing F (z 0 ) can be approximated by minimizing F (z) +rF (z) T (z 0 z), wherez is fixed. The Frank-Wolfe algorithm for Problem (3.6) (3.10) works as follows. Step 1. Initilization Set (0) k as the given initial demand D (0) k , for all k2K. Find the initial feasible solutionx (0) andy (0) using all or nothing assignment (i.e. for each OD pairk, send all demand (0) k to the least cost path, which is calculated by free-flow times). Setz 0 = (x (0) ;y (0) ; (0) ) andt = 0. Step 2. Finding Descending Direction Minimize the first order approximation subproblem min z2Z F (z t ) +rF (z t ) T (zz t ) (4.1) wherez t is given from the last iteration and is considered fixed in (4.1). Therefore the constant parts can be discarded and the problem is simplified as min z2Z rF (z t ) T z (4.2) (4.2) is a linear optimization problem, and thus can be solved by linear programming (LP) solvers. The solution of (4.2) is denoted byz t , argmin z2Z rF (z t ) T z. Step 3. Finding Step Size Apply a line search to find the step size, i.e.! t , argmin !2[0;1] F ((1!)z t +!z t ). 37 Step 4. Update Solution Letz t+1 , (1! t )z t +! t z t . Sett =t + 1 and go to Step 2. The algorithm stops after a given fixed number of total steps or until the solution stays at the same level after a certain number of iterations. 4.2 Computational Results The goal of the computational experiments is to analyze how the parameter settings will impact the numerical outcome, so that we can choose appropriate settings to obtain a reasonable range of the congestion level and the amount of passengers and drivers for ridesharing. The network tested in this proposal is the classic Sioux Falls (Babonneau and Vial, 2008; Bar-Gera, 2010). All the constant settings are listed in Table 4.1. Table 4.1: Constant settings Constant Value k D k 1 or 10 b k ;f k 1=D k d k (0) k ; = 1; 2; 4 g k (0) k ; = 1; 2; 4 According to the above settings, we have p k = b k g k +d k f k = k b k +f k = 1 2 g k + d k k 2 1 2 (0) k ; 1 2 (0) k + (4.3) q k = g k d k = k 2(b k +f k ) = D k 4 g k d k k 2 D k 4 (0) k ; D k 4 (0) k (4.4) So the range of price p k for OD pair k is determined by the free flow time (0) k and coefficients and; it is reasonable that the ridesharing price for each OD pair is 38 proportional to the free flow time and can fluctuate in a small range due to congestion levels. Also, the range of the number of passengers q k is proportional to the given demandD k and has slight fluctuations with the congestion. The computation was conducted on a PC (Windows OS, CPU 3.10 GHz, RAM 4GB) and all the codes were written in C++, using a CPLEX solver for finding the descending direction in the Frank-Wolfe algorithm. The program is terminated after 100 iterations, when the solution remains nearly unchanged for 20 iterations. The following subsec- tions illustrate an overall summary of the computational results, a sensitivity analysis of congestion, price and the number of passengers and drivers. 4.2.1 Optimality Check Before any further analysis, we first check the optimality of our computational results. There are two different ways available for this. One intuitive thought is to check the definition of user equilibrium, i.e. if travelers of the same OD pair are experiencing the same travel cost. Figure 4.1: Optimality check – travel cost 39 Figure 4.1 gives the travel costs as well as the actual demands of drivers for all OD pairs. In Figure 4.1 (a) the “Travel cost difference” value is calculated, for each OD pair, as the percentage difference between the maximum and minimum travel cost divided by the average travel cost. The travel costs of different route for each OD pair are obtained by applying a Breadth First Search (BFS). These differences are then sorted from minimum to maximum, together with their corresponding final demands of drivers in Figure 4.1 (b). From Figure 4.1 (a) we may see that over 80% of the OD pairs have almost zero travel cost difference, i.e. travelers from the same OD pair are sharing the same travel cost. For the remaining 20%, however, when we observe Figure 4.1 (b), we may find that these OD pairs have nearly no travel on the road. This does not mean that our algorithm is not converging to optimality. Actually, these OD pairs are suffering high congestion and thus very few people show up on the road. The high value of difference comes from the accuracy when we calculate the travel cost on a path. In our program, the travel cost of a path is recorded if the amount of flow for that OD pair is greater than 0.01. And observing the tail of the demand figure, the values are all greater than 0.01 but far less than 1, which could be neglected compared to the amount of other demands (from around 10 2 to 10 4 ). Another more theoretical way is to check the equilibrium condition Eq. (3.23), and see if the actual travel cost equals to the utility function of the demand for each OD pair. This can be examined in Figure 4.2, where part (a) gives the sorted percentage difference between the actual travel cost and the theoretical cost calculated by Eq. (3.23), i.e. k = P a2l tt a (y a ) = k ( k ). The actual travel cost (or the minimum travel cost) of each OD pair is obtained by applying the Dijkstra’s algorithm. Figure 4.2 (b) shows the corresponding demands. Similar to the previous analysis, about 80% of the OD pairs have reached the theoretical optimal travel cost, while the remaining 20% with higher differences have little demand. 40 Figure 4.2: Optimality check – equilibrium condition 4.2.2 Overall Results Table 4.2 summarizes the outputs of all the 18 combinations of , d k and g k settings. Note that we change the values ofd k andg k by changing and, according to Table 4.1. The fourth to the sixth columns list the averages over all OD pairs. p, 1 jKj P k2K p k , q , 1 jKj P k2K q k and , 1 jKj P k2K k are the average price, the average number of passengers, and the average number of drivers, respectively. The last two columns are the two parts of the objective function (3.6), whereF 1 = P a2A R ya 0 tt a (s) ds is the sum of the integrated congestion costs, andF 2 = P k2K R k 0 k (r) dr shows the sum of the integrated disutility (negative utility). Table 4.2: Overall results p q F 1 F 2 1 1 1 5.55 1934.63 1790.38 1:73 10 8 1:46 10 9 1 1 2 5.57 1930.04 1795.37 1:63 10 8 1:46 10 9 1 1 4 5.59 1920.67 1799.67 1:59 10 8 1:45 10 9 41 1 2 1 11.08 3876.44 2557.01 8:11 10 8 3:68 10 9 1 2 2 11.09 3874.95 2621.43 7:09 10 8 3:79 10 9 1 2 4 11.10 3872.08 2595.07 7:92 10 8 3:69 10 9 1 4 1 22.16 7755.17 3614.55 3:11 10 9 5:46 10 9 1 4 2 22.16 7754.56 3631.62 2:85 10 9 5:54 10 9 1 4 4 22.17 7753.50 3607.73 2:95 10 9 5:27 10 9 10 1 1 5.96 1790.77 302.56 1:87 10 6 1:98 10 8 10 1 2 6.37 1649.38 315.08 1:93 10 6 1:98 10 8 10 1 4 7.18 1365.41 321.57 1:97 10 6 1:98 10 8 10 2 1 11.33 3791.73 602.27 5:55 10 6 7:92 10 8 10 2 2 11.53 3718.84 603.07 5:29 10 6 7:92 10 8 10 2 4 11.96 3568.80 593.07 5:17 10 6 7:92 10 8 10 4 1 22.20 7739.90 1151.04 3:65 10 7 3:16 10 9 10 4 2 22.24 7725.99 1150.35 3:72 10 7 3:16 10 9 10 4 4 22.32 7699.12 1150.81 3:88 10 7 3:16 10 9 From Table 4.2, we can draw the following conclusions: 1. The base price, i.e. g k or , influences the number of drivers as a higher price makes it more attractive to become a driver. 2. Within a certain price range, i.e. fixingg k or, an increase in price (d k or) may reduce the traffic congestion. The following subsections provide further examination and analysis on the important system parameters. 42 4.2.3 Congestion and Disutility Levels It can be seen from Table 4.2 that the setting of has a major impact on the congestion levels. That is, ColumnF 1 shows that = 1 has much higher congestion than = 10. From Figure 4.3 (where the horizontal axis labels “#xy” represent the cases where =x and =y. For example case “12” means is set to 1 and is set to 2. This same labeling is used in the following figures), it can also be seen that g k = (0) k has a significant impact on the congestion level when is fixed. The congestion increases when the price parametersg k increase. This explains that when the prices go up, more people are attracted to become drivers. Figure 4.3: Congestion levels However, the impact of the parametersd k on the congestion between the different values is quite the opposite. At = 1 (Figures 4.3 (a)), where the congestion scale is high,d k has a negative impact on the congestion. When both the price and the conges- tion are high, a slight change in the price contributes more to the people’s inconvenience (more congestion) than the drivers’ benefits (i.e., the prices are not high enough to com- pensate for their congestion cost). Therefore the number of drivers decreases and thus 43 the congestion level goes down. At = 10 (see Figures 4.3 (b)), where the congestion scale is not so high, increasingd k will have a similar outcome to increasingg k . Both of these are price parameters and increasing the price attracts more drivers to the traffic network system. The above analysis can also be verified when observing the changes to the disutility level. In Figure 4.4, it can be concluded that both andg k have a significant impact on the disutility levels. At = 1, when the congestion scale is high, the disutility levels are lower than for = 10, when the congestion scale is much smaller. In other words, the utility levels (negative values of disutility values) are higher when the congestion levels are higher because the traffic equilibrium is a balance between the congestion and the utility functions. Consequently, there must be enough of a benefit to attract travelers using this mode of transportation; otherwise, people will choose alternative modes due to high congestion. Figure 4.4: Sensitivity analysis of disutilities Similarly, a higherg k leads to smaller disutility levels (or higher utility levels), since g k dominates the price values (see the next section) and a higher price gives larger utility values. d k has little impact on the disutility levels, especially when = 10 and the 44 congestion scale is low. Increasing prices in a small range is not enough to draw people out under low congestion due to lack of passengers. When the congestion scale is high, increasing prices a little bit, especially when the price base is high (a largerg k ), does not show enough of an attraction to the drivers but causes more inconvenience. Therefore the disutility levels would increase asd k increases. It is consistent with the impact on congestion levels. In summary, (1) mainly decides the scale of the congestion levels and a smaller leads to higher congestion levels and lower disutility levels; (2)g k (base price parame- ters) has a significant influence on congestion and disutility levels and a largerg k results in higher congestion levels and lower disutility levels; (3)d k has a small impact on the congestion and disutility levels. 4.2.4 Ridesharing Prices From Eq. (3.14) it is known that the ridesharing pricep k is mainly determined by the parameters in the economic equilibrium model (see Chapter 3 Section 3.4), on which the government agencies and market can influence. Furthermore, the price may decrease in a reasonable range when the congestion goes up, so as to encourage people to share their vehicles. Figure 4.5 shows one example of the distribution of price among all the 528 OD pairs and their corresponding free flow time, where = 10,d k = 4 (0) k andg k = (0) k . According to Eq. (4.3),p k = 1 2 g k + d k k = 1 2 (0) k + 4 (0) k k , i.e. the ridesharing price is slightly above half the free-flow time. It can be seen in Figure 4.5 that the ridesharing price is close to half of the free flow time for each OD pair. While the free flow times are sorted in ascending order, the price is not strictly increasing with them. This serves to illustrate the impact of congestion on the prices. The closer congestion k is to the free flow time (0) k , the more difference the pricep k is above the half free-flow time. 45 Figure 4.5: Price distribution Figure 4.6 compares the average prices of all the 18 combinations of,d k andg k . From the Figure 4.6 it can be seen that there is not as much as a difference between cases = 1 and = 10 because does not have a direct impact on prices but mainly on the congestion. Figure 4.6 shows that prices are mainly determined byg k , with slight changes in prices by varyingd k . Also from Section 4.2.3 we know that = 1 has much higher congestion than = 10, making it hard to have an impact on prices. Figure 4.6: Sensitivity analysis of average prices 46 4.2.5 Number of Travelers Figures 4.7 plots the number of drivers for all 18 cases. The number of drivers is con- sistent with the congestion levels discussed in Section 4.2.3, since the congestion is a quartic function of arc flows, which are linear combinations of the number of drivers. From Figures 4.7, mainly decides the scale of the number of drivers. A smaller corresponds to more drivers and a largerg k also increases the number of drivers.d k has little impact on the number of drivers. Figure 4.7: Sensitivity analysis of number of drivers Figures 4.8 plots the number of passengers for all 18 cases. The number of pas- sengers behaves similarly to the ridesharing prices, since both of them are primarily determined by the economic equilibrium model. From Figures 4.8 it can be concluded that there is no significant difference between = 1 and = 10. As we can also see from Eq. (3.15), the number of passengers increases withg k but decreases withd k . When the congestion cost k is high ( = 1), the impact ofd k can be neglected. 47 Figure 4.8: Sensitivity analysis of number of passengers 4.3 Large-scaled Datasets For large-scaled problem instances, solving an LP in (4.2) may not be computationally efficient. It could be polynomial time or even exponential time. Therefore heuristic methods are required to improve the performance in Step 2. 4.3.1 Heuristic Improvement Observing the solutionz = (x;y;), it can be seen thatx andy are linear combination of. It would be much easier to getx andy if could be determined. Note that at the equilibrium we have (see (3.23)), k = P a2l tt a (y a ) = k ( k ), for most OD pairsk whose demand k is strictly within its given domain. Hence this condition is adopted as heuristic improvement in Step 2. The full updated Frank-Wolfe algorithm is described as follows. Step 1. Initilization Set (0) k as the given initial demand D (0) k , for all k2K. Find the initial feasible solutionx (0) andy (0) using all or nothing assignment (i.e. for each OD pairk, send all 48 demand (0) k to the least cost path, which is calculated by free-flow times). Setz 0 = (x (0) ;y (0) ; (0) ) andt = 0. Step 2. Finding Descending Direction For each OD pairk, apply the Dijkstra’s algorithm (Dijkstra, 1959) to find the least cost path based on the current flow distributiony (t) . The minimum congestion cost is denoted by (t) k . According to (3.23), calculate the corresponding demand by the inverse of the utility function, i.e. ~ (t) k , 1 k (t) k ; 8k2K: If ~ (t) k does not satisfy constraint (3.9), set ~ (t) k to the closest boundary. Next reassign the flows on each arc proportionally according to the ratio of the new demand to the previous one, i.e. ~ x k;(t) a , ~ (t) k (t) k x k;(t) a ; 8a2A;8k2K ~ y (t) a , X k2K ~ x k;(t) a = ~ (t) k (t) k y (t) a ; 8a2A It can be verified easily that ~ z (t) , (~ x (t) ; ~ y (t) ; ~ (t) ) is still feasible, i.e. it satisfies con- straints (3.7) (3.10). Step 3. Finding Step Size Similarly to the previous Frank-Wolfe algorithm, apply a line search to find the step size, i.e.! t , argmin !2[0;1] F (1!)z (t) +!~ z (t) . Step 4. Update Solution Letz t+1 , (1! t )z t +! t z t . Sett =t + 1 and go to Step 2. In sum, the main change in this heuristic method is (in Step 2) to calculate using y from the previous iteration and updatex andy by the new. In other words, we 49 calculate and (x;y) iteratively through each other, while in the LP we solve all of them at the same time. The complexity of the Dijkstra’s algorithm isO(jNj 3 ), compared to solving the LP in polynomial time or even worse, exponential time, with respect to the number of variables,O(jKjjAj)O(jNj 2 jAj), which is much more thanO(jNj). 4.3.2 Algorithm Comparison In this section we show that the heuristic approach is an efficient procedure and also provides solutions close to the Frank-Wolfe method. Figure 4.9 shows the convergence of the objective functions for both algorithms, where the solid line represents the Frank-Wolfe method using a CPLEX solver (labeled by “cplex”, and the dashed line represents the heuristic approach (labeled by “heur”). Both algorithms ran 1000 iterations and shared the same settings ( = = = 1). The first ten iterations (where the values of the two methods are very close) are omitted in the figure for a clear observation. Figure 4.9 shows that the heuristic converges faster than the Frank-Wolfe method (see the first 200 iterations) and the values both algorithms converge to are very close. The running time of the heuristic method is also very competitive. It takes around one minute to run a 1000-iteration case. The Frank-Wolfe method, however, due to the interaction with CPLEX, takes 2-3 minutes to run a single iteration. In sum, our heuristic outperforms the Frank-Wolfe method in the speed of conver- gence, and the running time. Also the heuristic solves the problem without much loss of solution accuracy. 4.3.3 Test Cases The datasets that are tested in this work are listed in Table 4.3. All of them can be downloaded from http://www.bgu.ac.il/ bargera/tntp/ (accessible on 03/05/2014). 50 Figure 4.9: Algorithm comparison Table 4.3: Test cases Instance Z jNj jAj jKj Sioux-Falls 24 24 76 528 Anaheim 38 416 914 1,406 Winnipeg 147 1,052 2,836 4,344 Barcelona 110 1,020 2,522 7,922 where Z is the number of “Zones” – similar to setO orD, i.e. Z = O(jOj) = O(jDj). We have analyzed the results of ”Sioux-Falls” in the previous sections. Table 4.4, Table 4.5 and Table 4.6 give the results for “Anaheim”, “Winnipeg”, and “Barcelona”, respectively. According to Table 4.4 Table 4.6, we can draw similar conclusions as before: the traffic congestion, or the total number of vehicles (drivers) on the network is mainly determined by the base price for ridesharing, i.e. g k or , as we can see that a higher price makes it more attractive to become a driver. 51 Table 4.4: Results for Anaheim. F 1 F 2 P k k 0.1 0.1 2.2983E+05 -6.9108E+05 1.8064E+04 0.25 0.25 4.8956E+07 -4.4478E+08 4.4047E+05 0.5 0.5 2.8999E+08 -1.4113E+09 6.5051E+05 1 1 1.3192E+09 -3.2373E+09 9.0339E+05 1 2 1.3207E+09 -3.2321E+09 9.0505E+05 1 5 1.3208E+09 -3.2196E+09 9.1147E+05 1 10 1.3231E+09 -3.2086E+09 9.1955E+05 1 100 1.3559E+09 -3.4093E+09 1.0350E+06 2 1 4.7458E+09 -1.9263E+09 1.2136E+06 2 2 4.7461E+09 -1.9124E+09 1.2134E+06 2 5 4.7459E+09 -1.8971E+09 1.2155E+06 Table 4.5: Results for Winnipeg. F 1 F 2 P k k 0.1 0.1 1.5093E+05 -6.3410E+03 1.1186E+04 0.25 0.25 7.5005E+05 -8.5712E+05 5.1800E+04 0.5 0.5 1.9555E+06 -3.7564E+06 1.0938E+05 1 1 4.7855E+06 -1.1314E+07 1.7267E+05 1 5 5.1200E+06 -1.2585E+07 1.8665E+05 1 10 5.4756E+06 -1.5919E+07 2.0110E+05 2 1 1.2872E+07 -2.1980E+07 2.4077E+05 2 5 1.3053E+07 -2.1656E+07 2.4669E+05 5 1 5.5685E+07 5.9859E+07 3.5549E+05 Table 4.6: Results for Barcelona. F 1 F 2 P k k 0.1 0.1 2.5477E+05 -2.5742E+05 3.5466E+04 0.25 0.25 9.4428E+05 -2.4664E+06 1.2383E+05 0.5 0.5 2.2939E+06 -1.0454E+07 2.7148E+05 1 1 5.8245E+06 -4.0879E+07 5.1196E+05 1 2 6.1851E+06 -4.1617E+07 5.4603E+05 1 5 7.1299E+06 -4.4987E+07 6.2824E+05 1 10 8.4177E+06 -5.2121E+07 7.2892E+05 1 100 2.0875E+07 -2.3174E+08 1.4162E+06 2 1 1.9165E+07 -1.4659E+08 8.0559E+05 2 2 1.9386E+07 -1.4651E+08 8.1841E+05 2 5 2.0043E+07 -1.4751E+08 8.5368E+05 52 Chapter 5 Different OD Pairs (Model II): Formulation In this chapter and the next, we are going to study the case where drivers and passen- gers sharing a ride may travel on different OD pairs. Drivers may pick up or drop off passenger(s) at any time in the middle of their trips and they may even detour to do so if needed. In Chapters 3 and 4, the model does not explicitly describe how drivers choose their paths. To describe the path information, we need to expand the travel network in order to capture the drivers’ decision to pick up or drop off passengers. A driver may travel alone in the first half of his/her trip and then pick up some passenger(s) and share the ride till the end of their trip. That means, the driver’s path is a mix of “driving alone” and “taking on passenger(s)”. Therefore we need to construct an extended network that may capture the information of mixed paths. 5.1 Network Reconstruction Suppose the original graph isG 0 = (N 0 ;A 0 ), whereN 0 andA 0 represent the original node set and arc set, respectively. In order to describe the fact that ridesharing drivers and passengers may travel on different OD pairs,G 0 will be expanded to a larger graph G = (N;A) in the following way. To illustrate this construction, we consider a complete 53 graph with 3 nodes and 6 directed arcs, as shown in Figure 5.1, whereN 0 =f1; 2; 3g, A 0 =f(1; 2); (2; 1); (1; 3); (3; 1); (2; 3); (3; 2)g. Figure 5.1: Original graph,jN 0 j = 3;jA 0 j = 6 Step 1. For each nodei2N 0 , definei 0 =dup(i) as its duplicate. Denote N 0 0 , [ i2N 0 fdup(i)g; N ,N 0 [N 0 0 : Hence the new extended node setN is twice big as the originalN 0 , i.e.jNj = 2jN 0 j. In the example of Figure 5.1, we haveN 0 0 =f1 0 ; 2 0 ; 3 0 g, see Figure 5.2. The original node i 2 N 0 represents the driver node that only drivers (solo or ridesharing drivers) can travel through. The duplicated node i 0 2N 0 0 represents the passenger node that only passengers can travel through. 54 Figure 5.2: Node duplication Step 2. For each arc a = (i;j) 2 A 0 , where i;j 2 N 0 ;i 6= j, construct two duplicatesa 0 , dup(a) = (i;j) anda 00 , (i 0 ;j 0 ), wherei 0 = dup(i) andj 0 = dup(j) (see Figure 5.3). Define A 1 , [ (i;j)2A 0 f(i;j)g =A 0 ; A 2 , [ a2A 0 fdup(a)g = [ (i;j)2A 0 f(i;j)g; A 3 , [ (i;j)2A 0 f(dup(i);dup(j))g = [ (i;j)2A 0 f(i 0 ;j 0 )g; A,A 1 [A 2 [A 3 : HencejA 1 j =jA 2 j =jA 3 j =jA 0 j andjAj = 3jA 0 j. Figure 5.3 shows the new network extended up to Step 2 from the example of Figure 5.1, from which we can tell that, A 1 =f(1; 2); (2; 1); (1; 3); (3; 1); (2; 3); (3; 2)g; A 2 =f(1; 2); (2; 1); (1; 3); (3; 1); (2; 3); (3; 2)g; A 3 =f(1 0 ; 2 0 ); (2 0 ; 1 0 ); (1 0 ; 3 0 ); (3 0 ; 1 0 ); (2 0 ; 3 0 ); (3 0 ; 2 0 )g: 55 Figure 5.3: Arc duplication (extended network) The arcs a 1 2A 1 ;a 2 2A 2 and a 3 2A 3 represent the arcs for the solo drivers, the ridesharing drivers, and the passengers, respectively. In other words, the flow that travels on arca 1 (a 2 anda 3 , respectively) is counted as solo drivers (ridesharing drivers and passengers, respectively). It looks as if we simply duplicate the original graph to two separate ones. As a matter of fact, however, these two “separate graphs” are connected by the demands, as shown in Figure 5.4. Imagine we link both nodesi andi 0 to a third nodei 0 , which is called the decision node. Before leaving, each traveler would make a decision on whether to driver a car or take a ride. If one decides to be a driver, he/she will travel from nodei 0 toi; otherwise to be a passenger, he/she will travel from nodei 0 toi 0 . Hence nodesi andi 0 are linked together by the demand. In other words, for each OD pairk and its demandD k , we do not know how much of it would start ato k and how much would start ato 0 k . The only certain thing is that the amount of flow starting ato k plus the amount of flow starting ato 0 k equals toD k , by linking both nodes to the “ultimate original node”i 0 . However, in the model formulation and calculation, the amount of flow on arcs (i 0 ;i) and (i 0 ;i 0 ) 56 Figure 5.4: Arc duplication (extended network) can be substituted by other flows, i.e. the sum of flows starting ato k ando 0 k . And also we assume that the costs on these arcs, i.e. (i 0 ;i) and (i 0 ;i 0 ), are zero. Therefore we do not include them in the model formulation. It only helps us to understand how these seemingly separate graphs can be linked together. In sum we obtain the expanded networkG = (N;A), where Nodei2N 0 represents the driver node, which can be used by both solo drivers and ridesharing drivers. These drivers share the same node since they may change roles in the middle of their trips, i.e. they may pick up or drop off passengers anywhere along their trips. Nodei 0 2N 0 0 represents the passenger node, which can be used only by passen- gers. They are strictly separated from the driver nodes since they can never switch roles during their trips, i.e. passengers remain passengers throughout their trips, and so do drivers. 57 Arca 1 2A 1 represents the arc for solo drivers, i.e. flows on these arcs are treated as solo drivers. Arca 2 2A 2 represents the arc for ridesharing drivers, i.e. flows on these arcs are treated as ridesharing drivers. Arca 3 2A 3 represents the arc for ridesharing passengers, i.e. flows on these arcs are treated as ridesharing passengers. Therefore, travelers may start at either nodei if they wish to drive, or nodei 0 if they want to take a ride. If they pick the driver nodei to start, they can travel on arca 1 2A 1 and/or arc a 2 2A 2 before they reach the destination node j2N 0 . For example, in Figure 5.3, a traveler may be driving along the solid arc (1; 2) and then the dashed arc (2; 3). This means that he/she drives alone from node 1 to 2 and then picks up some passenger(s) to share a ride from 2 to 3 (and drop off the passenger(s) at 3). But drivers cannot travel to passenger arcs (dotted arcs) once they choose to start from a driver node. It is because we assume that drivers cannot leave their vehicles at a node other than their destinations. Similarly, once travelers start at a passenger node i 0 , they can only travel on arcs a 3 2A 3 before they reach their destination, which is also a passenger nodej 0 2N 0 0 . There exists no arc connecting a passenger node inN 0 0 to a driver node inN 0 . Also this is because we assume that travelers cannot start driving a vehicle from a node other than their origin. 58 5.2 Additional Assumptions Since the network is almost triple in size and thus is more complicated than Model I in Chapters 3 and 4, other assumptions are added to keep the model tractable and more realistic. Fixed demand: in this chapter and the next we assume that the total number of travelers of a given OD pair is fixed. Therefore the drivers’ utility function is not required in this model. Limited capacity: since the solo drivers and ridesharing drivers are separable in this model, the vehicle capacity constraint can be added. It is more realistic to assume that there is a limit on how many passengers each vehicle can hold. Pickup and Drop Off : it is assumed that drivers may pick up and drop off any passenger(s) at any time. They can do so as many times as they want. Explicit cost functions: unlike Model I from Chapters 3 and 4, we treat all types of cost/profit as a function of the number of travelers in different roles. We consider the congestion cost, the inconvenience cost and the ridesharing price (cost for passengers and income for drivers). In this new model, the ridesharing price is no longer a decision variable but a function in terms of the number of drivers and passengers. The definitions of these functions are given in Section 5.2.2. 5.2.1 Notation New notation that is used in this chapter and the next is given below. G 0 = (N 0 ;A 0 ) original network with the node setN 0 and the arc setA 0 . G = (N;A) extended network with the node setN and the arc setN . 59 O; DN 0 original sets of origins and destinations, respectively. O;DN extended sets of origins and destinations, respectively, whereO, O[ O 0 andD, D[ D 0 . K set of OD pairs,KOD. k2K OD pair, k =f(o k ;d k ); (o 0 k ;d 0 k )g, o k 2 O;d k 2 D;o 0 k 2 O 0 ;d 0 k 2 D 0 , including both driver and passenger OD pairs. A 1 ;A 2 ;A 3 arcs of solo drivers, ridesharing drivers and passengers, respectively,A =A 1 [A 2 [A 3 . a 1 ;a 2 ;a 3 arcs representing solo drivers, ridesharing drivers and pas- sengers, respectively,a j 2A j ;j = 1; 2; 3. D k total demand of travelers (including all drivers and passen- gers) for OD pairk, assumed to be a fixed constant. C vehicle capacity,C > 1. x k a amount of flow for OD pairk2K on arca2A. y a total amount of flow on arca2A,y a = P k2K x k a . x k vector with componentsx k a with respect tok,x k 2R jAj . x vector with componentsx k a ,x2R jKjjAj . y vector with componentsy a ,y2R jAj . IN(i), OUT(i) sets of arcs entering and leaving nodei2N , respectively. T j (a 0 ) corresponding arca j 2A j of typej that is generated from the original arca 0 2A 0 ,j = 1; 2; 3.T j :A 0 !A j . T 0 (a) corresponding original arca 0 2A 0 that generates arca, for anya2A.T 0 :A!A 0 . p a path that consists of one or several consecutive arcs. P k set of paths for OD pairk, i.e. set of all paths that starts at o k oro 0 k and ends atd k ord 0 k . 60 P = S k2K P k set of all paths in the network (for all OD pairs). h p amount of flow on pathp, i.e. the amount of flow that travels fromo k (oro 0 k ) tod k (ord 0 k ), for anyk andp2P k . h vector with componentsh p . x?y x;y2 R n are perpendicular vectors, that x T y = 0. Spe- cially, 0 x? F (x) 0 is equivalent tox 0;F (x) 0;x T F (x) = 0 for any vector x2 R n and vector-valued functionF :R n !R n . 5.2.2 Cost Functions The cost functions of each arca in this model are defined as below. Congestion cost for drivers (experienced by every solo or ridesharing driver): The classic BPR function is adopted here, where the amount of flow on arca is the sum of the number of both solo and ridesharing drivers. The cost is given by tt a (y),t a 1 +b y a 1 +y a 2 c a 4 ! ; a2A 1 [A 2 (5.1) wherea 1 =T 1 (T 0 (a)) anda 2 =T 2 (T 0 (a)) are the corresponding arcs for solo drivers and ridesharing drivers, respectively. The values t a and c a are constants with respect to the original arcT 0 (a)2A 0 andb is a common constant. This is exactly the same as the BPR function, calculated by the total number of drivers on the arc. Congestion cost for passengers (experienced by every passenger): Passengers should also experience a congestion cost when they travel. Strictly speaking, the 61 congestion cost is not the same as the travel time, but more as a measure of peo- ple’s tolerance to travel times. The more the travel time is, the more people have to endure. The passengers, however, are relatively less intolerant to the congestion than the drivers who are traveling on the same (original) arc/path. They do not need to worry about other vehicle-related costs, such as gas cost, especially when it is congested. Therefore the congestion cost of passengers is different from, and most likely less than, that of drivers. We define this cost as tt p a (y),t a 1 +b 0 y a 1 +y a 2 +ey a 3 c a 4 ! ; a2A 3 (5.2) wherea j =T j (T 0 (a)) is the corresponding arc of typej,j = 1; 2; 3. Similarly, t a andc a are the same as above, andb 0 ;e are positive constants. Inconvenience cost for ridesharing drivers: Besides the congestion costs, ridesharing drivers will also experience the inconvenience for taking on passen- gers. Since the congestion costs are the same for both solo and ridesharing drivers, it does not include the cost of picking up, dropping off, or even waiting for pas- sengers. These costs will be taken into the inconvenience cost, which is given by I d a (y), d y a 2 + d y a 3 ; a2A 2 (5.3) wherea 2 =T 2 (T 0 (a)) = a anda 3 =T 3 (T 0 (a)) are the corresponding arcs for ridesharing drivers and passengers, respectively. The values d and d are both positive constants. Inconvenience cost for ridesharing passengers: Similarly, passengers would also experience some inconvenience for taking a ride. The inconvenience cost includes but not limited to waiting for drivers to pick them up, or possibly having to make 62 a detour together with the driver in order to pick up or drop off other passengers. The cost is defined as I p a (y), p y a 2 + p y a 3 ; a2A 3 (5.4) wherea 2 =T 2 (T 0 (a)) anda 3 =T 3 (T 0 (a)) = a are the corresponding arcs for ridesharing drivers and passengers, respectively, and the values p and p are both positive constants. Ridesharing cost for each passenger (paying to drivers): In our model, the key point that benefits drivers to participate in ridesharing activities is that they can receive compensation that will cover part of their driving costs. The compensation is in the form of a price paid by each passenger, which is given by R p a (y),t a vy a 2 +wy a 3 ; a2A 3 (5.5) where a 2 = T 2 (T 0 (a)) and a 3 = T 3 (T 0 (a)) = a are the corresponding arcs for ridesharing drivers and passengers, respectively. t a is the same as in (5.1) and (5.2).;v;w are positive constants. Ridesharing income for each driver (paid by passengers): Intuitively, the driver’s income should equal the sum of the passengers’ ridesharing costs (prices) in his/her car. The actual number of passengers in each vehicle is not determined, but belongs to the range [1;C], i.e. at least one passenger and at mostC passen- gers in each vehicle. Hence for simplicity we set the income of each driver to a fixed constant times the price paid by each passenger, i.e. R d a (y),R p a (y) = (t a vy a 2 +wy a 3 ); a2A 2 (5.6) 63 where 1C. In sum, each traveler on arca2A is experiencing a total cost of f a (y) = 8 > > > > > < > > > > > : tt a (y); a2A 1 tt a (y) +I d a (y)R d a (y); a2A 2 tt p a (y) +I p a (y) +R p a (y); a2A 3 (5.7) 5.3 A TAP Model with Capacity Constraints We formulate the model constraints in both arc-based and path-based forms to provide a clear description of the model. The arc formulation is easy for computation purposes and the path formulation enables us to clearly depict the user equilibrium. 5.3.1 Arc Formulation According to the problem descriptions and assumptions, the constraints of this model can be formulated as below in terms of arcs. Flow decomposition y a = X k2K x k a ; 8a2A (5.8) Flow conservation X a2IN(i) x k a X a2OUT(i) x k a = 0; 8i2Nnfo k ;d k ;o 0 k ;d 0 k g;8k2K (5.9) Demand requirement X a2IN(d k )[IN(d 0 k ) x k a X a2OUT(d k )[OUT(d 0 k ) x k a =D k ; 8k2K (5.10) 64 Capacity constraint y a 2 y a 3 Cy a 2 ; 8a 0 2A 0 ;a 2 =T 2 (a 0 );a 3 =T 3 (a 0 ) (5.11) Non-negativity constraints x k a 0; 8a2A;8k2K: (5.12) Define Y,fyj9x satisfying (5.9), (5.10), (5.12) s.t. (5.8) and (5.11) holdsg; (y), (f a (y)) a2A ; i.e. : YR jAj !R jAj ; whereY is the set of feasible solutions and (y) is the vector-valued cost function with respect to arcs. 5.3.2 Path Formulation Note that for any pathp2P, it implies an origin, a destination, and a series of connected arcs and nodes. Therefore given p, we can tell which OD pair k it belongs to and all the arcs traversed by the path. Furthermore, the constraints of flow conservation on the nodes and arcs are no longer needed in the path formulation. On the other hand, we have the following relation between the path flowh p and the arc flowy a , y a = X p2P;a2p h p (5.13) 65 i.e. the amount of flow on arca is the sum of the flows of all paths passing through the arc. Therefore the arc-based constraints (5.8) (5.12) can be written equivalently in terms of path flowsh p as below. X p2P k h p =D k ; 8k2K (5.14) X p2P;a 2 2p h p X p2P;a 3 2p h p C X p2P;a 2 2p h p ; 8a 0 2A 0 ;a 3 =T 3 (a 0 );a 2 =T 2 (a 0 ) (5.15) h p 0; 8p2P (5.16) where (5.14) easily expresses the demand requirement, (5.15) describes the vehicle capacity constraint, and (5.16) requires the nonnegativity of flows. Let H,fh satisfying (5.14) (5.16)g be the feasible set of path flows. Sincey a = P p2P;a2p h p , it can be written in the vector form y = h; (5.17) where is the arc-path incidence matrix with entries ap , 8 > > < > > : 1; if pathp traverses arca, i.e.a2p 0; otherwise ; 8p2P;8a2A 66 Therefore we have H =fhjh2Yg: (5.18) Suppose the path cost function is additive, that is, the cost for a path is the sum of the costs on all the arcs traversed by the path, i.e. g p (h) = X a2p f a (y) = X a2p f a (h); (5.19) (h) = T (y) = T (h); (5.20) where () is a vector-valued function with componentsg p (h). 5.3.3 Generalized User Equilibrium Note that the major difference between our problem and the classic user equilibrium is the vehicle capacity constraints on arcs, i.e. Constraint (5.11) in the arc formulation, or Constraint (5.15) in the path formulation. In the classic user equilibrium, the costs of the used paths cannot be reduced. Furthermore, all the used paths of the same OD pair share the same minimum cost. Under the capacity constraints, however, the equalization of travel cost is hard to achieve – the costs of all used paths may vary for each OD pair due to certain constraints. Such constraints are called side constraints (Larsson and Patriksson, 1994, 1999), i.e. a set of convex constraints defined on arc flows other than flow conservation or demand requirement. Given the side constraints, the user equilibrium is generalized as below. Definition 5.1. Assumey andh are arc- and path-flow vectors, respectively, andy = h. We define the following: An arca is said to be saturated above if the flowy a reaches its upper bound. 67 An arca is said to be saturated below if the flowy a reaches its lower bound. An arca is said to be saturated if it is either saturated above or saturated below. A pathp is said to be saturated above if it contains at least one saturated-above arc and no saturated-below arc. A pathp is said to be saturated below if it contains at least one saturated-below arc and no saturated-above arc. A pathp is said to be saturated if it is either saturated above or saturated below. A path p is said to be unchangeable if it contains both saturated-above and saturated-below arcs. In our model we have the vehicle capacity constraints: y a 2 y a 3 Cy a 2 ; 8a 3 2A 3 ;a 2 =T 2 (T 0 (a 3 )) or 1 C y a 3 y a 2 y a 3 ; 8a 2 2A 2 ;a 3 =T 3 (T 0 (a 2 )) Therefore, according to Definition 5.1, an arca 1 inA 1 can never be saturated since it does not have upper or lower bound. an arca 2 inA 2 can be saturated above ify a 2 = y a 3 , or saturated below ify a 2 = 1 C y a 3 ,a 3 =T 3 (T 0 (a 2 )). an arca 3 inA 3 can be saturated above ify a 3 =Cy a 2 , or saturated below ify a 3 = y a 2 ,a 2 =T 2 (T 0 (a 3 )). 68 Note that whenever an arca 2 2A 2 is saturated above (or below), there also exists an arca 3 2A 3 that is saturated below (or above). Given the above definition, the user equilibrium in our scope is generalized as below. Definition 5.2. A feasible OD flow is a generalized user equilibrium if for every OD pairk there exists an unconstrained minimum travel costc k such that 1. the travel cost of any unused path for OD pairk is at leastc k . 2. any used path with smaller cost thanc k is a path that is saturated above. 3. any used path with greater cost thanc k is a path that is saturated below. In the classic user equilibrium, travelers from the same OD pair are always traveling at the same cost. In the generalized user equilibrium, however, some travelers may experience a higher travel cost than others traveling on the same OD pair. In this case, the path with less travel cost must be saturated above or the path with more travel cost must be saturated below. In other words, ifh is a flow vector at the generalized user equilibrium, for any OD pairk and any two pathsp;p 0 2P k , we must have if pathsp andp 0 are not saturated, theng p (h) =g p 0(h); (5.21) ifg p (h)>g p 0(h); then eitherp is saturated below orp 0 is saturated above. (5.22) Note that any unused paths must be saturated below and thus they can have higher costs according to (5.21) (5.22). For example, consider an original graph with only one arc, two nodes and one OD pair, as shown in Figure 5.5. The amount of flowy a and the corresponding travel cost f a (y) are given above each arc. We can see that the arc/path for ridesharing drivers is saturated above (y d = y p = 1). Even with less cost, people can no longer switch roles to become a ridesharing driver. More cases will be discussed in Chapter 6. 69 Figure 5.5: Generalized user equilibrium: example Therefore our problem is, to find the generalized user equilibrium given the con- straints (5.8) (5.12) (arc formulation) or (5.14) (5.16) (path formulation). In the following sections, we will show that this problem can be formulated as a Mixed Com- plementarity Problem (MiCP), and equivalently, a Variational Inequality (VI) problem. And we will also provide a proof of the existence and uniqueness of the solution. 70 5.4 Mixed Complementarity Problem Let k be the multiplier to Constraint (5.14), for allk2K, and + a , a for the first and second inequalities of Constraint (5.15), respectively, for alla2A 0 . Therefore we can formulate the problem as a mixed complementarity problem, i.e. 0h p ? g p (h) ~ p () k 0; 8k2K;8p2P k (5.23) 0 + a ? X p2P;T 3 (a)2p h p X p2P;T 2 (a)2p h p 0; 8a2A 0 (5.24) 0 a ? C X p2P;T 2 (a)2p h p X p2P;T 3 (a)2p h p 0; 8a2A 0 (5.25) k free ? X p2P k h p =D k ; 8k2K (5.26) whereh is the vector of allh p , is the vector with components + a and a , and ~ p (), X a 2 2A 2 \p C T 0 (a 2 ) + T 0 (a 2 ) + X a 3 2A 3 \p + T 0 (a 3 ) T 0 (a 3 ) : According to the definition, (5.23) indicates h p > 0 ) g p (h) ~ p () k = 0; and g p (h) ~ p () k > 0 ) h p = 0: And similar to the rest complementarity constraints (5.24) (5.26). Proposition 5.1. The solution to the MiCP (5.23) (5.26) is a generalized user equi- librium defined by (5.21) (5.22). Proof. Suppose (h;;) is a solution to the MiCP (5.23) (5.26), where is the vector of all k , and k is a constant for all pathsp2P k for any givenk. 71 1. If all paths are NOT saturated (i.e. neither saturated above nor saturated below), according to (5.24) and (5.25), we must have + a = a = 0 for alla2A 0 , i.e. = 0, and thus ~ p () = 0 andg p (h) k . For any pathp withh p > 0, we must haveg p (h) = k . And for anyg p (h)> k , we must haveh p = 0. In this case, for any given OD pairk, all paths withh p > 0 (all paths in use) will have the same cost k , and all paths with greater costsg p (h) > k must have zero flows on it. This is exactly a classic user equilibrium with k as the “least” travel cost, or the “standard” travel cost. 2. If for some pathp, it is saturated or even unchangeable, i.e. there exists some arc a2 p such that either + a > 0 or a > 0. Note that only one of the two can be positive, since givenC > 1, we have either + a > 0 ) y a 2 =y a 3 <Cy a 2 ) a = 0; or a > 0 ) y a 2 <y a 3 =Cy a 2 ) + a = 0: for alla2A 0 anda 2 =T 2 (a);a 3 =T 3 (a). Also note that a path cannot contain both arca 2 2A 2 and arca 3 2A 3 . Therefore ~ p () can be simplified as ~ p () = 8 > > > > > < > > > > > : 0; A 2 \p =A 3 \p =; P a 2 2A 2 \p C T 0 (a 2 ) + T 0 (a 2 ) ; A 2 \p6=;;A 3 \p =; P a 3 2A 3 \p + T 0 (a 3 ) T 0 (a 3 ) ; A 2 \p =;;A 3 \p6=; For any pathp withh p > 0, we must haveg p (h) ~ p () = k . Note that ~ p () could be either positive or negative, or even zero if some + a and a cancel each other. 72 (a) If ~ p () > 0, we have greater costg p (h) > k . IfA 2 \p6=;, there exists at least one arca 2 2A 2 \p such thaty a 2 = 1 C y a 3 , indicating that the path is saturated below (from the driver’s view); ifA 3 \p6=;, there exists at least one arc a 3 2A 3 \p such that + T 0 (a 3 ) > 0 and thus y a 2 = y a 3 , indicating that the path is saturated below (from the passenger’s view). In any case, the path is with greater cost and saturated below, and the amount of flow on this path cannot be reduced. (b) If ~ p () < 0, we have less costg p (h) < k . Similarly, it indicates that the path is saturated above, and other paths cannot switch flows onto this path. (c) If ~ p () = 0, the cost equals the “standard” costg p (h) = k , and also the path must be unchangeable (contains both saturated-above and -below arcs). In sum, the solution to the mixed complementarity problem (5.23) (5.26) is a generalized user equilibrium. Furthermore, the term ~ p () can be treated as the “compensation” cost whenever the path is saturated or unchangeable. In other words, for anyh p > 0, we must have g p (h) ~ p () k ; 8k2K;8p2P k : That is, for a given OD pairk, any pathp2P k in use (h p > 0) would have the same “generalized” cost,g p (h) ~ p (). If we define a generalized cost for each pathp as ~ g p (h),g p (h) ~ p () 73 then the MiCP formulation (5.23) (5.26) will give us a classic user equilibrium in the scope of the generalized cost, i.e. h p > 0 ) ~ g p (h) = k ; 8k2K;8p2P k (5.27) h p = 0 ) ~ g p (h) k ; 8k2K;8p2P k (5.28) i.e. all used paths withh p > 0 will always have the same, minimal “generalized” cost, for any OD pairk. 5.5 Variational Inequality Formulation Note thath is a vector inHR jPj , and (h) is a mapping (or a vector-valued function) fromH toR jPj . A variational inequality problem (Facchinei and Pang, 2003) is to find a solutionh 2H, such that (hh ) T (h ) 0;8h2H (5.29) The problem is denoted by VI(H; ) and its solution set is denoted by SOL(H; ). Note that the constraints of VI(H; ) are the constraints ofh, i.e. H. Since all constraints (5.14) (5.16) are linear, VI(H; ) is also called a linearly constrained VI. Proposition 5.2. A linearly constrained VI is equivalent to an MiCP , i.e.h is a solution in SOL(H; ), (5.29), if and only if (h;;) solves the MiCP (5.23) (5.26). Proof. See Facchinei and Pang (2003), 1.2.1 Proposition and its corollary. Similarly, we can write down a VI formulation on arc flows, i.e. to find ay 2Y such that (yy ) T (y ) 0;8y2Y (5.30) 74 The VI problem and its solution set are denoted by VI(Y; ) and SOL(Y; ), respec- tively. It is trivial that VI(H; ) is equivalent to VI(Y; ), since (yy ) T (y ) = (h h ) T (h ) = (hh ) T T (h ) = (hh ) T (h ) wherey = h andy = h . Therefore, Proposition 5.3. The solutiony to VI(Y; ) is a generalized user equilibrium in terms of arc flows. Proof. For anyy2 SOL(Y; ), there exists anh2H such thaty = h. For any h 0 2H,y 0 = h 0 2Y, (h 0 h) T (h) = (y 0 y) T (y) 0: Henceh2 SOL(H; ). According to Proposition 5.2, h is also a solution to the MiCP (5.23) (5.26). Thereforey andh are flows at the generalized user equilibrium. 5.6 Solution Existence and Uniqueness In this section, we prove the existence and uniqueness of the solution to VI(Y; ), (5.30). 75 5.6.1 Existence Proposition 5.4. (y) is continuous. Furthermore, there exists a solution for the VI formulation (5.30). Proof. It is trivial to check that all cost functions (5.1) (5.6) are continuous in the given domainY. Hence is continuous. Also it is trivial to check that all the constraints are linear, and thusY is closed and convex. According to Facchinei and Pang (2003), 1.2.1 Proposition, there exists a solution for the VI formulation (5.30). Remark. Note that according to the definition, the cost functionf a (y) may not equal zero wheny = 0 or the correspondingy a ’s are zero. This means that even if there exist zero travelers on the road, the cost of each type of arc (or each traveler role) still exists. For example,tt a (y) andtt p a (y) would be equal tot a instead of 0 wheny a 1 =y a 2 = y a 3 = 0, which is the free flow time. AlsoR p a (y) andR d a (y) would be equal tot a and t a , respectively, wheny a 2 = y a 3 = 0. It can be explained in this way: when there is no driver or passenger, there still exists a non-zero, positive cost (fee) for ridesharing, a base price. Therefore the cost on arcs a 2 2A 2 is strictly less than the cost on arcs a 1 2A 1 (sincey a 2 = y a 3 = 0). This will draw drivers from traveling alone to taking on passengers. However, once the number of ridesharing driversy a 2 becomes positive, there will be a positive inconvenience costI d a (y), that could prevent more solo drivers from switching to ridesharing drivers. This change would also be constrained by the number of passengers. If the number of passengers is zero, the number of ridesharing drivers would be forced to be zero. Moreover,R p a (y) andR d a (y) can even be negative on some arca given the negative signs in (5.5) and (5.6). This negative ridesharing fee implies that in some extreme cases, the market would give away money to passengers 76 in order to encourage more people to give up driving, especially when the traffic is very congested. 5.6.2 Uniqueness In this subsection, we show the uniqueness of the VI formulation (5.30). The proof uses the monotonicity of (y), i.e. (y 0 y) T ((y 0 ) (y)) 0; y 0 ;y2Y: (5.31) Also (y) is said to be strictly monotone if (y 0 y) T ((y 0 ) (y))> 0; y 0 ;y2Y: (5.32) Given the above definition, the proof of the uniqueness include four steps: Step 1. Rewrite (5.31) in a quadratic form. Step 2. Prove the positive semidefiniteness of the matrix in the above quadratic form. Step 3. Positive semidefiniteness) monotonicity. Step 4. Positive definiteness) strict monotonicity) uniqueness. The detailed proof of each step is given by the following propositions. Proposition 5.5. (Step 1.) For anyy 0 ;y2Y, there exist a symmetric matrix A = A y 0 ;y , such that (y 0 y) T ((y 0 ) (y)) = (y 0 y) T A(y 0 y): (5.33) 77 Proof. For anyy;y 0 2Y, according to the Mean Value Theorem, (y 0 ) (y) = Z 1 0 J(y +t(y 0 y))dt (y 0 y); whereJ is the Jacobian matrix of function ,J :R jAj !R jAjjAj . Hence (y 0 y) T ((y 0 ) (y)) = (y 0 y) T Z 1 0 J(y +t(y 0 y))dt (y 0 y) = (y 0 y) T J(y 0 y) = (y 0 y) T 1 2 (J + J T ) (y 0 y) = (y 0 y) T A(y 0 y); where J , R 1 0 J(y +t(y 0 y))dt is the integral of the Jacobian matrix, and A , 1 2 (J + J T ) is its symmetric part. Note that both J and A are matrices with respect to the vector pair (y;y 0 ). Proposition 5.6. (Step 2.) If the parameters satisfy 4( d +v)( p +w) ( d w + p v) 2 0; (5.34) and 4ebb 0 (1 +eC) 3 0; (5.35) then the matrix A in (5.33) is positive semidefinite. Furthermore, if at least one of (5.34) and (5.35) holds strictly, A in (5.33) is positive definite. Proof. Note that the Jacobian matrix of function is of sizejAjjAj and it can be written injA 0 jjAj 0 blocks, where each block is a 3 3 square matrix, i.e. 78 J = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B @ 0 B B B @ @fa 1 @ya 1 @fa 1 @ya 2 @fa 1 @ya 3 @fa 2 @ya 1 @fa 2 @ya 2 @fa 2 @ya 3 @fa 3 @ya 1 @fa 3 @ya 2 @fa 3 @ya 3 1 C C C A 0 B B B @ 0 0 0 0 0 0 0 0 0 1 C C C A 0 B B B @ 0 0 0 0 0 0 0 0 0 1 C C C A 0 B B B @ 0 0 0 0 0 0 0 0 0 1 C C C A 0 B B B @ @fa 1 @ya 1 @fa 1 @ya 2 @fa 1 @ya 3 @fa 2 @ya 1 @fa 2 @ya 2 @fa 2 @ya 3 @fa 3 @ya 1 @fa 3 @ya 2 @fa 3 @ya 3 1 C C C A 0 B B B @ 0 0 0 0 0 0 0 0 0 1 C C C A . . . . . . . . . . . . 0 B B B @ 0 0 0 0 0 0 0 0 0 1 C C C A 0 B B B @ 0 0 0 0 0 0 0 0 0 1 C C C A 0 B B B @ @fa 1 @ya 1 @fa 1 @ya 2 @fa 1 @ya 3 @fa 2 @ya 1 @fa 2 @ya 2 @fa 2 @ya 3 @fa 3 @ya 1 @fa 3 @ya 2 @fa 3 @ya 3 1 C C C A 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A jAj 0 jAj 0 : In other words,J = diag(B;:::; B) consist of diagonal blocks, where each block B is a Jacobian sub-matrix with respect to each arc, a 0 2A 0 , in the original graph. More specifically, B = 0 B B B @ @fa 1 @ya 1 @fa 1 @ya 2 @fa 1 @ya 3 @fa 2 @ya 1 @fa 2 @ya 2 @fa 2 @ya 3 @fa 3 @ya 1 @fa 3 @ya 2 @fa 3 @ya 3 1 C C C A = 0 B B B @ 1 1 0 1 1 + d +v d w 2 2 + p v 3 + p +w 1 C C C A ; where 1 , 4bt a c 4 a (y a 1 +y a 2 ) 3 ; 2 , 4b 0 t a c 4 a (y a 1 +y a 3 +ey a 3 ) 3 ; 3 , 4eb 0 t a c 4 a (y a 2 +y a 2 +ey a 3 ) 3 =e 2 : 79 Therefore matrix A can also be divided in such blocks, where each block is the symmetric part of the integral of the above block B, denoted by ~ B, 1 2 R B + R B T . To show that A is positive semidefinite, it is sufficient to show that each such block of A, ~ B, is positive semidefinite. It can be calculated that given (5.34) and (5.35), ~ B is positive semidefinite and thus so is A (see Appendix A.1). Proposition 5.7. (Step 3.) If A is positive semidefinite, (y) is monotone; moreover, if A is positive definite, (y) is strictly monotone. Proof. It is trivial given Proposition 5.5 and Proposition 5.6. Proposition 5.8. (Step 4.) If (y) is strictly monotone, or at least one of (5.34) and (5.35) holds strictly, there exists a unique solution to the VI formulation (5.30). Proof. Suppose there exist two solutions to (5.30),y andy , such that (yy ) T (y ) 0;8y2Y (yy ) T (y ) 0;8y2Y We need to show thaty =y . Since the above is true for anyy2Y, it also holds that (y y ) T (y ) 0; (y y ) T (y ) 0: Therefore (y y ) T ((y ) (y )) 0. 80 According to the monotonicity of (y), (y y ) T ((y ) (y )) 0. Hence we must have (y y ) T ((y ) (y )) = 0: When at least one of (5.34) and (5.35) holds strictly, matrix A is positive definite. According to the previous proof (using Mean Value Theorem), (y y ) T ((y ) (y )) = (y y ) T A (y y ) = 0; if and only ify y = 0, i.e.y =y and the solution is unique. In sum, we have the following theorem. Theorem 5.1. If the parameters satisfy (5.34) and (5.35), and at least one of them holds strictly, then there exists a unique solution to the VI formulation (5.30). 81 Chapter 6 Different OD Pairs (Model II): Computational Results We have shown in Chapter 5 that there exists one and only one solution to the VI(Y; ) in (5.30), and equivalently to the MiCP (5.23) (5.26). The latter (MiCP) is formulated in terms of path flows, which is hard to handle in practice since we need to enumerate all possible paths for each OD pair. Therefore in this chapter we only focus on the arc formulation. To solve VI(Y; ) in (5.30), one of the common methods is to transform it into the corresponding MiCP, in terms of arc flows, and then solve the MiCP using existing solvers. 6.1 Arc Formulation of the MiCP According to Proposition 5.2 a linearly constrained VI is equivalent to an MiCP. VI(Y; ) in (5.30) is linearly constrained, and hence it can be reformulated into an MiCP. 6.1.1 Reformulation w.r.t.x Recall that in the formulation of VI(Y; ), (5.30),Y is the feasible set of vectory, and Y =fyj9x satisfying (5.9), (5.10), (5.12) s.t. (5.8) and (5.11) holdsg: 82 The formulation in terms ofy is convenient to prove the existence and uniqueness of the solution. However, to solve the problem, we need to knowx as well. Hence we can rewrite the VI formulation in terms ofx, since each elementy a ofy can be replaced by y a = P k2K x k a , according to constraint (5.8). Therefore Constraint (5.8) can be dropped and Constraint (5.11) will be replaced by X k2K x k a 2 X k2K x k a 3 C X k2K x k a 2 ; 8a 0 2A 0 ;a 2 =T 2 (a 0 );a 3 =T 3 (a 0 ) (6.1) Therefore the feasible region in terms ofx is given by X ,fxjx satisfying (5.9), (5.10), (5.12) and (6.1).g If we put multipliers to constraints (5.9), (5.10), (5.12) and (6.1), i.e. k i : X a2IN(i) x k a = X a2OUT(i) x k a ; 8i2Nnfo k ;d k ;o 0 k ;d 0 k g;8k2K k d k : X a2IN(d k )[IN(d 0 k ) x k a =D k ; 8k2K + a ; a : X k2K x k a 2 X k2K x k a 3 C X k2K x k a 2 ; 8a2A 0 ;a 2 =T 2 (a);a 3 =T 3 (a): 83 then we can formulate the following MiCP 0x k a ? f a ( x) +! + a + T 0 (a) +! a T 0 (a) k i + k j 0; 8a = (i;j)2A;8k2K (6.2) 0 + a ? X k2K x k a 3 X k2K x k a 2 0; 8a2A 0 ;a 2 =T 2 (a);a 3 =T 3 (a) (6.3) 0 a ? C X k2K x k a 2 X k2K x k a 3 0; 8a2A 0 ;a 2 =T 2 (a);a 3 =T 3 (a) (6.4) k i free ? X a2IN(i) x k a X a2OUT(i) x k a = 0; 8i2Nnfo k ;d k ;o 0 k ;d 0 k g;8k2K (6.5) k d k free ? X a2IN(d k )[IN(d 0 k ) x k a D k = 0; 8k2K (6.6) where! + a and! a are constant coefficients given by ! + a = 8 > > > > > < > > > > > : 0; a2A 1 1; a2A 2 1; a2A 3 ; and ! a = 8 > > > > > < > > > > > : 0; a2A 1 C; a2A 2 1; a2A 3 : and x =y is the compact form ofy a = P k2K x k a . = I 1 ;:::;I jKj 2R jAjjAjjKj is a matrix withjAj rows andjAjjKj columns, and eachI k is anjAjjAj identity matrix. Recall that x = x 1 1 ;:::;x 1 jAj ;:::;x jKj 1 ;:::;x jKj jAj T , or x = x 1;T ;:::;x jKj;T T 2 84 R jAjjKj , wherex k = x k 1 ;:::;x k jAj T 2 R jAj andx k;T = x k T is the transpose of x k . More explicitly, y = x = 0 B B B B B B B @ 1 0 ::: 0 :::::: 1 0 ::: 0 0 1 ::: 0 :::::: 0 1 ::: 0 . . . . . . . . . . . . :::::: . . . . . . . . . . . . 0 0 ::: 1 :::::: 0 0 ::: 1 1 C C C C C C C A 0 B B B B B B B B B B B B B B B B @ x 1 1 . . . x 1 jAj . . . x jKj 1 . . . x jKj jAj 1 C C C C C C C C C C C C C C C C A = 0 B B B @ P k2K x k 1 . . . P k2K x k jAj 1 C C C A : Furthermore, we can substitutey by x in the formulation of VI(Y; ), i.e. (yy ) T (y ) = (xx ) T T ( x ) Define (x) = T ( x); 8x2X Therefore the VI(Y; ) is equivalent to the VI(X; ), i.e. to find a vectorx 2X such that (xx ) T (x ) 0; 8x2X (6.7) 6.1.2 Formulation Equivalency Up to this point we have several formulations regarding the same problem described in Chapter 5, and their relations can be summarized as below. MiCP(h) () VI(H; ) () VI(Y; ) () VI(X; ) ? () MiCP(x) 85 where MiCP(h) represents the MiCP formulation with respect toh in (5.23) (5.26), and MiCP(x) represents the MiCP formulation with respect tox in (6.2) (6.6). We have proved the first three equivalency relations. The last equivalency is also true according to Proposition 5.2. A detailed proof can be found in Appendix A.4. 6.2 Computational Results We have tested MiCP(x) using the solver KNITRO on the NEOS server (Czyzyk et al., 1998; Gropp and Mor´ e, 1997; Dolan, 2001). The computational results are shown below separated by different test cases. 6.2.1 Test Case #1 – Three-node Recall the example in Chapter 5, where the original graph has 3 nodes and 6 arcs.N 0 = f1; 2; 3g, andA 0 = f(1; 2); (2; 1); (1; 3); (3; 1); (2; 3); (3; 2)g. Suppose the capacity of each arc c a is c 1 = c 2 = 259, c 3 = c 4 = 234, and c 5 = c 6 = 149. Note that this is the original network before duplication. The set of OD pairs include all 6 trips, K =f(1; 2); (2; 1); (1; 3); (3; 1); (2; 3); (3; 2)g. For each trip the demand is 100, i.e. there are 100 people traveling from node 1 to node 2, some (or none or all) of them can be solo drivers and some can be ridesharing drivers or passengers. Other parameter settings are given in Table 6.1. Note that all arc parameters are based on the original network, i.e. a2A 0 =f1;:::; 6g in this case. The duplicated arcs will adopt the same parameter settings as the original arcs. It can also be checked that the parameters in Table 6.1 satisfy Constraints (5.34) and (5.35), i.e. 4( d +v)( p +w) ( d w + p v) 2 0; and 4ebb 0 (1 +eC) 3 0: 86 Figure 6.1: Example with trips and arc capacities for Three-node In fact, it can be calculated according to Table 6.1 that both constraints hold strictly and thus there exists a unique solution. Table 6.1: Parameter settings for Three-node Description Constant Value free flow time t a (a = 1;:::; 6) 6; 6; 4; 4; 5; 5 arc capacity (threshold) c a (a = 1;:::; 6) 259; 259; 234; 234; 149; 149 congestion coefficients b;b 0 ;e 0:15; 0:015; 0:3 inconvenience coefficients d ; p 0:1 inconvenience coefficients d ; p 0:01 price coefficients ;v;w 0:5; 0:2; 0:1 vehicle capacity ;C 2; 4 demand D k (k = 1;:::; 6) 100 Note that we setb 0 < b making the congestion cost of passengers less than that of drivers. We also sete< 1, since the contribution of passengers to the congestion cost is considerably less than that of drivers. As for the inconvenience costs, we set d d , under the assumption that drivers contribute more on the inconvenience cost of each 87 driver than passengers. In other words, we assume that the inconvenience cost of adding one passenger is less than adding one ridesharing driver. Similarly, we set p p , assuming that drivers contribute more on the inconvenience cost of each passenger. p (or p ) could be either the same or different from d (or d ). Last but not least, we set v > w under the assumption that adding one passenger will contribute less to the ridesharing price than adding one driver. Given the above settings, the computational results for MiCP(x) (6.2) (6.6) are shown in Table 6.2, where only non-zero values ofx k a are listed. Table 6.2: Computational results forx for Three-node (k,a) x k a (k,a) x k a (k,a) x k a (1, 1) 81.1756 (1, 2) 9.4122 (1, 3) 9.4122 (2, 7) 87.4147 (2, 8) 6.2927 (2, 9) 6.2927 (3, 4) 81.1756 (3, 5) 9.4122 (3, 6) 9.4122 (4, 13) 83.7752 (4, 14) 8.1124 (4, 15) 8.1124 (5, 10) 87.4147 (5, 11) 6.2927 (5, 12) 6.2927 (6, 16) 83.7752 (6, 17) 8.1124 (6, 18) 8.1124 The values of the dual variables are + 1 = + 2 = 3:08221, + 3 = + 4 = 2:04928, + 5 = + 6 = 2:48516, and a = 0, for alla2A 0 =f1;:::; 6g. The values of are omitted. Note that for each arc a2A =f1;:::; 18g, the flow on a comes from only one OD pair, thereforey a equals to the correspondingx k a with the samea. The values ofy a , f a (y) and ~ (x) a;k are given in Table 6.3, where ~ (x) a;k is given by ~ (x) a;k ,f a ( x) +! + a + T 0 (a) +! a T 0 (a) k i + k j ; 8a = (i;j)2A;8k2K Note that we only care about those ~ (x) a;k whose correspondingx k a is non-zero. It is easy to check, from Table 6.2 and Table 6.3, that the solution is feasible, i.e. all constraints (6.2) (6.6) are satisfied. Also the solution is indeed a general user 88 Table 6.3: Computational results fory and the costs for Three-node a y a f a (y) ~ (x) a;k a y a f a (y) ~ (x) a;k 1 81.1756 6.0134 0.0000 10 87.4147 4.0153 0.0000 2 9.4122 2.9312 0.0000 11 6.2927 1.9660 0.0000 3 9.4122 9.0956 0.0000 12 6.2927 6.0646 0.0000 4 81.1756 6.0134 0.0000 13 83.7752 5.1080 0.0000 5 9.4122 2.9312 0.0000 14 8.1124 2.6228 0.0000 6 9.4122 9.0956 0.0000 15 8.1124 7.5931 0.0000 7 87.4147 4.0153 0.0000 16 83.7752 5.1080 0.0000 8 6.2927 1.9660 0.0000 17 8.1124 2.6228 0.0000 9 6.2927 6.0646 0.0000 18 8.1124 7.5931 0.0000 equilibrium. Since every path consists of only one arc, the path cost equals to the cor- responding arc cost. For each OD pair, the generalized cost of each path, which is calculated byf a ( x) +! + a + T 0 (a) +! a T 0 (a) , equals to the same constant. For exam- ple, whenk = 1, there are altogether three paths, using arcsa = 1; 2; 3, respectively. This means that, 81.1756 out of the 100 travelers are driving alone from node 1 to node 2, while 9.4122 are taking on passengers and the rest 9.4122 are traveling as passen- gers. The travel costs experienced by each traveler for the different travel modes are f 2 (y) + + 1 C 1 =f 3 (y) + 1 + 1 =f 1 (y) = 6:0134. Under the generalized user equilibrium, the generalized costs experienced by each traveler are all the same for each OD pair, although their travel costsf a (y) may vary. Moreover, it can be seen that all paths are saturated from the perspective of the rideshar- ing drivers. Therefore even though ridesharing drivers are experiencing the least travel costs, others cannot switch to become a ridesharing driver. From the perspective of pas- sengers, their travel costs are the highest. But since their path is saturated below, people cannot leave this path for cheaper-cost paths. It can also be calculated from Table 6.3 that on average, the proportions of solo drivers, ridesharing drivers and passengers on each arc is 84.12% : 7.94% : 7.94%. 89 6.2.2 Test Case #2 – Braess One interesting test case for the traffic equilibrium problems is the Braess net- work (see Figure 6.2). In the original graph, N 0 = f1; 2; 3; 4g, A 0 = f(1; 3); (1; 4); (3; 2); (3; 4); (4; 2)g (for simplicity, rewriteA 0 =f1;:::; 5g), and there is only one OD pairK =f(1; 2)g with demandD(1; 2) = 6. Figure 6.2: Example with trips and arc capacities for Braess The free flow time is also labeled by each arc in Figure 6.2, i.e. t 1 = t 5 = 0:00000001 0, t 2 = t 3 = 50 and t 4 = 10. Other parameter settings are given in Table 6.4. Note that in the Braess network the power in the BPR function is set to 1, f a (y) =t a h 1 +b ya 1 +ya 2 ca i for the drivers andf a (y) =t a h 1 +b 0 ya 1 +ya 2 +eya 3 ca i for the passengers. This example is an interesting paradox in game theory. Consider the simplest case: suppose all travelers are solo drivers. Every traveler will observe a travel cost of 10 on path “1! 3! 4! 2” before traveling. Since each of them acts selfishly, all of them 90 Table 6.4: Parameter settings for Braess Description Constant Value free flow time t a (a = 1;:::; 5) 0, 50, 50, 10, 0 arc capacity (threshold) c a (a = 1;:::; 5) 1 congestion coefficients b a (a = 1;:::; 5) 1:0 10 9 ; 0:02; 0:02; 0:1; 1:0 10 9 congestion coefficients b 0 a ;e 0:1b a , 0:3 inconvenience coefficients d ; p 0:1 inconvenience coefficients d ; p 0:01 price coefficients ;v;w 0:5; 0:2; 0:1 vehicle capacity ;C 2; 4 demand D 1 (1! 2) 6 will travel on that path. As a result, the cost would increase to 60 due to congestion. If they cooperate, they could reduce the travel cost to 50 for each of them. Now consider our ridesharing case, where travelers may choose to be a solo driver, a ridesharing driver or a passenger. Given the above parameter settings for the network, and the same settings as in 6.2.1 for the ridesharing activities, the computational results are shown in Table 6.5. Note that there is only one OD pair (1! 2), hence ally a equal x a . Table 6.5 shows us that the average proportion of solo driver, ridesharing driver and passenger on each arc is 0% : 20% : 80%. Note that the values of ~ (x) a;k forx a 0 have been labeled by “X”, since these values are of no interest. From the results we can tell that it is a generalized user equi- librium since it satisfies all the constraints in (6.2) (6.6). From Table 6.5 we can also tell that all travelers are choosing the path “1! 3! 4! 2”. No traveler is a solo driver. 1.2 out of 6 of them choose to be a ridesharing driver and the remaining 4.8 of them choose to be passengers. The costs of all the paths are given in Table 6.6. It can be easily seen that, for the used path “1! 3! 4! 2”, the cost of solo drivers (35.2) is the highest, and thus no one would like to drive alone. The cost of 91 Table 6.5: Computational results for Braess a a 0 =T 0 (a) x a =y a f a (y) ~ (x) a;k 1 1 0.0 12.000 X 2 1 1.2 11.688 0.000 3 1 4.8 3.048 0.000 4 2 0.0 50.000 X 5 2 0.0 0.000 X 6 2 0.0 75.000 X 7 3 0.0 50.000 X 8 3 0.0 0.000 X 9 3 0.0 75.000 X 10 4 0.0 11.200 X 11 4 1.2 0.888 0.000 12 4 4.8 15.672 0.000 13 5 0.0 12.000 X 14 5 1.2 11.688 0.000 15 5 4.8 3.048 0.000 Table 6.6: The costs of paths for Braess path 1! 3! 4! 2 1! 3! 2 1! 4! 2 solo driver 35.2 62.0 62.0 ridesharing driver 24.3 11.7 11.7 passenger 21.8 78.0 78.0 passengers (21.8) is the lowest. But since there is a capacity constraint for each vehicle, i.e. each driver can take on at mostC = 4 passengers. No more travelers can switch to be a passenger. This is exactly the case in the generalized user equilibrium, where the costs of the other used paths cannot be reduced since the least cost path is saturated with y a 3 =Cy a 2 . For the other unused paths, the costs of both solo drivers and passengers are much greater than the used paths. Therefore even though the ridesharing drivers may have an even lower cost (11.7) than the current used path, these unused paths are saturated for ridesharing drivers. Since there are zero passengers on these paths, people cannot switch to be a ridesharing driver given the bound 0y a 2 y a 3 = 0. 92 Up to now we have seen from the first two test cases that both the lower bounds and the upper bounds in (5.11) or (5.15) have been reached. These cases are relatively small and simple. In the following subsection we will study the case of Sioux-Falls and see how ridesharing drivers will take on passengers from other OD pairs. 6.2.3 Test Case #3 – Sioux-Falls The data data setset of Sioux-Falls was downloaded from the website called “Transporta- tion Network Test Problems” (http://www.bgu.ac.il/ ˜ bargera/tntp/, accessible on April 16, 2014.) The original data contains a network with 24 nodes and 76 arcs. The set of OD pairs covers almost all 24 23 combinations of node pairs – there are altogether 528 OD pairs. After expanding the network, we shall have 24 2 = 48 nodes and 76 3 = 228 arcs. This gives the total number ofx k a variables = 228 528 = 120; 384, plus the dual variables and the same number of constraints. This is too large a problem to solve using KNITRO. Therefore we need to reduce the problem size by considering only a subset of the OD pairs. When reading the trip file (with 528 OD pairs), an acceptance rate of 30% was applied and thus it randomly generated 149 OD pairs. In this subsection we consider only the case with the same parameter settings shown in Table 6.1, except for “free flow time” and “arc capacity (threshold)” (given by the network file of Sioux-Falls), and “demand” (given by the trip file of Sioux-Falls). Note that the total demand in the network is approximately only 30% out of the original input (149 out of the 528 OD pairs). Hence the arc capacityc a is divided by 10 based on the original input in order to make the arcs more congested, pushing people to participate in ridesharing. First let us check the solution feasibility and optimality. 93 Feasibility and optimality The KNITRO solver provides a solution with accuracies shown in Table 6.7. From this table we can see that the solution is feasible and optimal with errors less than 2:0010 2 . Table 6.7: KNITRO – final statistics for Sioux-Falls Final objective value 0.00e+00 Final feasibility error (abs / rel) 2.70e-04 / 6.16e-08 Final optimality error (abs / rel) 2.00e-02 / 2.00e-02 # of iterations 2697 # of CG iterations 53870 # of function evaluations 2776 # of gradient evaluations 2696 # of Hessian evaluations 2696 Total program run time (secs) 7038.58 ( 7030.38 CPU time) Time spent in evaluations (secs) 167.77 The raw data ofy a , + a and a can be found in Appendix A.5.1, from which we can also see that the vehicle capacity constraint is satisfied for each original arca 0 2A 0 , i.e. y a 2 y a 3 Cy a 2 , for all a 0 2A 0 ;a 2 =T 2 (a 0 );a 3 =T 3 (a 0 ). For most arcs, the above constraints hold strictly with the corresponding + a = a = 0. Only for arc a = 462A 0 , we have + a = 0:3773 > 0 andy a 2 = y a 3 , meaning the arc is saturated for ridesharing drivers or passengers. On this arc, we can see the costs of solo drivers, ridesharing drivers and passengers are 4.3426, 3.9653 and 4.7200, respectively. In other words, the cost of ridesharing driver is the lowest among the three. That is why the flow is saturated for ridesharing drivers on this arc. For all other arcs, the costs of both drivers are always the same, and it is either larger or smaller than the cost for passengers. In particular, Figure 6.3 shows the proportion of each role (solo driver, ridesharing driver or passenger) for each arc in the original graph. From Figure 6.3 we can see that for some arcs, there are very few people participating in the ridesharing activities, while 94 Figure 6.3: Proportions of solo drivers, ridesharing drivers and passengers for Sioux- Falls for some other arcs, the sum of the number of ridesharing drivers and passengers makes up to half of the total number of travelers passing those arcs. Also if we take the average of the amount of flow on each type of arc, we can see (from Figure 6.4) that on average, the proportions of solo drivers, ridesharing drivers and passengers are 70%, 9%, and 21%. In other words, on average 21% of the travelers will become passengers, reducing the traffic congestion by 21% in the amount of travelers. Path selection analysis In the test cases of “Three-node” and “Braess”, it is not clear if ridesharing drivers have been taking on passengers from other OD pairs. It is because either the network is too simple (Three-node) or there is only one OD pair (Braess). Therefore with the network of Sioux-Falls, we may analyze such activities among the different OD pairs. 95 Figure 6.4: Average proportions of solo, ridesharing drivers and passengers for Sioux- Falls Consider only arcs 1; 22A 0 in the original graph (see Figure 6.5), i.e. arc “1! 2” labeled by “1” and arc “1! 3” labeled by “2”. Table 6.8 gives the detailed flow distributionx k a over all OD pairs on these two arcs. In this table, the rows are for OD pairsk2K and the columns are for arcsa2A (after extension). OD pairs with zero flow on these arcs are omitted. From Table 6.8 we can see that all the passengers on arc “1! 2” come from OD pairk = 12, i.e. starting from node 3 and traveling to node 8. The number of passengers (96.12) on this arc is far above the capacity that the ridesharing drivers (4.47) from the same OD pair (k = 12) can take on. On the other hand, there are several other OD pairs, such ask = 2; 7; 66; 67, etc., that do not have passengers, but have ridesharing drivers traveling from node 1 to node 2. These drivers will take on the rest of the passengers that those 4.47 drivers from OD pairk = 12 could not. Similarly, on arc “1! 3”, OD pairs k = 3; 5; 36 contribute a total of 96.06 passengers, which will be spread out to drivers from the other OD pairs. It is also interesting to observe the fact that some seemingly unrelated OD pairs are also using arcs “1! 2” and “1! 3”. For instance, whenk = 88, traveling from node 96 Figure 6.5: The original network of Sioux-Falls 17 to node 2, some travelers will make their path through arc “1! 2”. Note that from the input file the demand, or the total number of travelers, of this OD pair is 200, which is much greater than the number of travelers on arc “1! 2”, i.e. 66:71 + 5:31 = 72:02. This indicates that drivers will make a detour like this due to traffic congestion, or to pick up passengers from other OD pairs for their own benefit. 97 Table 6.8: Selectedx k a on the first two original arcs for Sioux-Falls k o k d k arc “1”2A 0 , 1! 2 arc “2”2A 0 , 1! 3 1 2 3 4 5 6 1 1 5 0.00 0.00 0.00 196.24 3.76 0.00 2 1 6 294.34 5.66 0.00 0.00 0.00 0.00 3 1 11 0.00 0.00 0.00 490.16 3.81 6.03 4 1 13 0.00 0.00 0.00 496.19 3.81 0.00 5 1 14 0.00 0.00 0.00 273.05 3.78 23.16 6 1 19 0.00 0.00 0.00 296.21 3.79 0.00 7 1 20 171.69 5.58 0.00 119.01 3.72 0.00 9 2 9 0.00 0.00 0.00 196.24 3.76 0.00 10 2 11 0.00 0.00 0.00 196.24 3.76 0.00 12 3 8 19.90 4.47 96.12 0.00 0.00 0.00 27 6 13 0.00 0.00 0.00 196.24 3.76 0.00 33 6 21 0.00 0.00 0.00 96.31 3.69 0.00 34 6 22 0.00 0.00 0.00 33.74 3.45 0.00 36 7 3 0.00 0.00 0.00 29.74 3.40 66.86 66 12 16 657.07 5.72 0.00 0.00 0.00 0.00 67 12 18 160.58 5.57 0.00 0.00 0.00 0.00 78 15 2 94.56 5.44 0.00 0.00 0.00 0.00 88 17 2 66.71 5.31 0.00 0.00 0.00 0.00 132 22 2 94.56 5.44 0.00 0.00 0.00 0.00 146 24 6 69.64 5.33 0.00 0.00 0.00 0.00 y a 1629.04 48.53 96.12 2619.35 44.50 96.06 6.3 Sensitivity Analysis As mentioned earlier in Subsection 6.2.3, we did not adopt the original full input of Sioux-Falls, but took only a subset of the OD pairs, and also reduced arc capacities by ten times. These settings can have an impact on the solution, or the distribution of the flows, i.e. how people will choose their paths. Hence we are interested in checking how the solution will change according to different parameter settings. 98 6.3.1 Changing Arc Capacityc a The capacity of each arc c a is essential to the solution of the model, since it helps to determine the congestion cost for every traveler, which will interfere with people’s deci- sion on which type of role to travel. In the BPR functions (5.1) and (5.2), i.e. tt a (y) =t a 1 +b y a 1 +y a 2 c a 4 ! ; a2A 1 [A 2 tt p a (y) =t a 1 +b 0 y a 1 +y a 2 +ey a 3 c a 4 ! ; a2A 3 the arc capacityc a acts as a threshold of the amount of flow on that arc. If the amount of flow is equal to or belowc a , the congestion cost on arca would be close to its free flow time t a . If bigger, however, the congestion cost would increase dramatically as the amount of flow increases. Therefore, ifc a decreases, i.e. the threshold decreases, the arcs would become more congested under the same amount of flow. This would push more travelers to participate in ridesharing activities, meaning an increase in the number of both ridesharing drivers and passengers and thus a decrease in the number of solo drivers. Table 6.9 shows the changes with c a for the different networks. In the table, D denotes the average demand, i.e. D, 1 jKj P k2K D k . D can help understand how many people are traveling on the road. Thus we set c a to be proportional to 0:1 D, D and 10 D, respectively. The triplet in each cell gives the proportions (in %) of solo drivers, ridesharing drivers and passengers. As expected, when c a increases (from 0:1 D to 10 D), the proportion of solo drivers increases while that of ridesharing drivers and passengers decreases. Note that other parameters remain unchanged in this subsection. It can be summarized from Table 6.9 that when the traffic becomes less congested (asc a increases), fewer people would participate in ridesharing. 99 Table 6.9: Ridesharing proportions with arc capacity changes (%) Test case c a 0:1 D c a D c a 10 D Three-node 9.60 31.80 58.61 84.12 7.94 7.94 84.37 7.81 7.81 Braess 0.00 20.00 80.00 0.00 20.00 80.00 20.67 29.33 50.00 Sioux-Falls 15.56 22.82 61.62 70.27 8.38 21.35 99.56 0.22 0.22 Such changes can be more clearly observed in Figures 6.6, 6.7 and 6.8. When c a increases, i.e. the congestion cost decreases, more people would become solo drivers while less people will become ridesharing passengers. Note that as the arc capacity increases, the proportion of ridesharing drivers may increase or decrease, but overall the adoption of ridesharing, i.e. the sum of the proportion of ridesharing drivers and passengers, still decreases. Figure 6.6: Average proportion of solo drivers changing with arc capacity 100 Figure 6.7: Average proportion of ridesharing drivers changing with arc capacity Figure 6.8: Average proportion of passengers changing with arc capacity 6.3.2 Changing Inconvenience Parameters d ; d ; p ; p The convenience costs of ridesharing for both drivers and passengers are defined in Chapter 5 by (5.3) and (5.4), respectively, i.e. I d a (y) = d y a 2 + d y a 3 ; a2A 2 I p a (y) = p y a 2 + p y a 3 ; a2A 3 101 They are both (linearly) increasing functions of the amount of flow (note that all coef- ficients are positive). Intuitively, when the inconvenience cost increases, the cost of ridesharing goes up for both drivers and passengers. Hence less people would like to participate in ridesharing. And vice versa. Note that all the parameters of the inconvenience costs d ; d ; p ; p must satisfy Constraints (5.34) and (5.35) to assure the existence and uniqueness of the solution. Thus define Con1, 4( d +v)( p +w) ( d w + p v) 2 ; Con2, 4ebb 0 (1 +eC) 3 : When changing any parameters involved above, we need to check if any of the two constraints is violated. In changing the arc capacitiesc a , however, there is no need to check these constraints since c a is not involved. In the previous tests, ( d ; d ; p ; p ) is set to be (0.1, 0.01, 0.1, 0.01). To compare with this settings while maintaining the above two constraints, we multiply ( d ; d ; p ; p ) by 0.1 and 10, respectively. The three sets of parameter values are listed in Table 6.10 and the test results are given in Table 6.11. We can see that all three sets of ( d ; d ; p ; p ) satisfy all the parameter constraints (due to positive Con1 and Con2 values). The column title (0.01, 0.001) represents the set ( d = p = 0:01; d = p = 0:001). Table 6.10: Parameter settings and constraint checks for inconvenience cost changes d d p p C v w e Con1 Con2 0.01 0.001 0.01 0.001 2 4 0.5 0.2 0.1 0.3 0.0143 0.1352 0.1 0.01 0.1 0.01 2 4 0.5 0.2 0.1 0.3 0.1359 0.1352 1 0.1 1 0.1 2 4 0.5 0.2 0.1 0.3 0.6300 0.1352 In the result of Table 6.11,c a is set to 0:1 D in the ”Three-node” network, 10 D in the ”Braess” network, and D in the ”Sioux-falls” network. This is because the original 102 Table 6.11: Ridesharing proportions with inconvenience cost changes (%) Test case (0.01, 0.001) (0.1, 0.01) (1, 0.1) Three-node 0.00 37.11 62.89 9.60 31.80 58.61 42.19 11.56 46.25 Braess 17.15 32.85 50.00 20.67 29.33 50.00 25.00 25.00 50.00 Sioux-Falls 45.72 17.92 36.36 70.27 8.38 21.35 89.17 2.17 8.66 settingc a D for both networks will give ridesharing flows that touch their upper/lower bounds (saturated arcs/paths). These are extreme cases for ridesharing activities where the actual travel costs (NOT the generalized cost) of different travelers are not equal at the equilibrium. Hence different c a values are selected for the ”Three-node” network and the ”Braess” network to eliminate this influence. We only consider the situation where the vehicle capacity constraints hold strictly, and thus drivers and passengers may switch roles more willingly. Despite differentc a values for the different networks, all the parameter settings remain fixed except for the inconvenience-cost parameters ( d ; d ; p ; p ). From Table 6.11 we can see that as the inconvenience cost increases, i.e. parameter settings changes from (0.01, 0.001) to (1, 0.1), the proportion of ridesharing decreases (including both drivers and passengers), which meets our expectation. Such changes can be more clearly observed in Figures 6.9, 6.10 and 6.11. When the inconvenience cost increases, more people would become solo drivers while less people will become ridesharing drivers or passengers. 6.3.3 Changing Pricing Parameters;v;w Changing the pricing parameters can be tricky compared to changing other parame- ters. When the price increases, it is appealing for more travelers to become ridesharing drivers. At the same time, however, it may lose passengers as well due to a higher cost 103 Figure 6.9: Proportion of solo drivers changing with the inconvenience cost Figure 6.10: Proportion of ridesharing drivers changing with the inconvenience cost Figure 6.11: Proportion of passengers changing with the inconvenience cost 104 being a passenger. Therefore the proportion of ridesharing can either be increasing or decreasing with the changes of pricing parameters. Same as before, we kept other parameters fixed while changing any pricing param- eters. Note that we are setting c a 0:1 D for ”Three-node” network, c a 10 D for ”Braess” andc a D for ”Sioux-Falls”. Still, we setb 0 = 0:1b for all arcs, all networks. The sets of parameter values are listed in Table 6.12 and we can see that all parameter settings satisfy all the parameter constraints. The test results are given in Table 6.13. Table 6.12: Parameter settings and constraint checks for pricing changes d d p p C v w e Con1 Con2 0.1 0.01 0.1 0.01 2 4 0.05 0.02 0.01 0.3 0.0063 0.1352 0.1 0.01 0.1 0.01 2 4 0.5 0.2 0.1 0.3 0.1359 0.1352 0.1 0.01 0.1 0.01 2 4 5 2 1 0.3 1.4319 0.1352 Table 6.13: Ridesharing proportions with pricing changes (%) (;v;w) (0.05, 0.02, 0.01) (0.5, 0.2, 0.1) (5, 2, 1) Three-node 21.41 15.72 62.87 9.60 31.80 58.61 5.37 38.61 56.02 Braess 0.33 20.07 79.61 20.67 29.33 50.00 12.87 37.13 50.00 Sioux-Falls 68.50 6.30 25.20 70.27 8.38 21.35 70.37 9.90 19.73 For better observation, see Figures 6.12, 6.13 and 6.14 for the changes of different types of travelers. From Table 6.13 or Figures 6.12 6.14, we can see that, when the price goes up there is a significant increase in the number or proportion of ridesharing drivers, while that of passengers decreases as expected. The proportion changes of solo drivers to the price are not obvious. For “Three-node”, the proportion of solo drivers keeps decreasing with the increase of the ridesharing price. This means that the benefit as a ridesharing driver appears to be so compelling that both solo drivers and passengers would switch to be a ridesharing driver. 105 Figure 6.12: Proportion of solo drivers changing with the price Figure 6.13: Proportion of ridesharing drivers changing with the price Figure 6.14: Proportion of passengers changing with the price 106 For “Braess”, the proportion of solo drivers increases when the price increases from (0.05, 0.02, 0.01) to (0.5, 0.2, 0.1), because the increase of the price has more of an impact passengers than to drivers, and thus more passengers are switching to solo drivers than to ridesharing drivers. In other words, the decrease of the proportion of passengers overcomes the increase of the proportion of ridesharing drivers. Therefore the overall proportion of ridesharing decreases, which leads to an increase of solo drivers. For “Sioux-Falls”, the proportion of solo drivers keeps increasing with the increase of the ridesharing price. Similar to the first half of the case for “Braess”, The increase in price pushes more passengers to drive alone than to drive with other passengers. 6.3.4 Summary The changes with the parameter settings can be summarized in Table 6.14. Table 6.14: The impact of parameter changes on ridesharing proportions Proportion changes arc capacity" inconvenience cost" ridesharing pricing" solo driver " " undetermined ridesharing (overall) # # undetermined driver undetermined # " passenger # # # It can be observed from Table 6.14 that When the arc capacity increases, more people would become solo drivers and thus less people would participate in ridesharing. Passengers will become solo drivers or ridesharing drivers. 107 When the inconvenience cost increases, more people would become solo drivers and thus less people would participate in ridesharing. In particular, both the num- ber of ridesharing drivers and passengers will decrease due to an increased cost. When the price of ridesharing increases, more people would become ridesharing drivers and less people would become ridesharing passengers. The number of solo drivers is undetermined, that is, it is possible that more passengers are switching to solo drivers, and it is also likely that more solo drivers would become ridesharing drivers. In conclusion, the traffic congestion, the ridesharing price and the number of each type of travelers closely interact with each other. 108 Chapter 7 Conclusions and Future Work 7.1 Model Comparison In conclusion, we have studied two models for the ridesharing Transportation Market under different assumptions, i.e. Model I and Model II in Chapter 3 and Chapter 5, respectively. A comparison between these two models is listed in Table 7.1. Table 7.1: Model comparison Difference Mode I Model II OD pairs same different Demands elastic fixed Vehicle capacity unlimited limited and fixed Model formulation convex optimization VI or MiCP Price variable function of flow amount Inconvenience utility function function of flow amount From Table 7.1 we can see that both models have their own pros and cons. These two models study different situations of ridesharing. For example, Model I has the simpler assumption that restricting both drivers and passengers in the same vehicle to travel on the same OD pair. Also Model I assumes unlimited vehicle capacity. Such simplification helps us to build an easier model that can solve much bigger datasets and also enables us to focus on other important features, such as having the ridesharing price as a variable. A price variable is more flexible and would have more freedom to interact with other variables and parameters in the model. Model II, on the other hand, may look more complex compared to Model I, since it is triple in size of the graph. This largely increases the complexity of model analysis and 109 computation. The biggest pros of Model II is that it allows for more freedom to drivers and passengers. The drivers can choose anyone to pick up or drop off. Travelers in the same vehicle no longer need to be restricted to travel on the same OD pair. And the vehicle capacities are also considered in this model, making the problem closer to real world applications. Compromise has to be made in order to cope with such advances: we only consider the case where the demand of each OD pair is given and fixed. 7.2 Conclusions This dissertation studies a transportation system, called the Transportation Market, where ridesharing has the ability of capturing a significant proportion of the total travel demand. In the Transportation Market it is assumed that drivers and passengers are matched up via a real-time matching agency and that passengers will pay drivers for ridesharing services to share the travel cost. This ridesharing price is assumed to be negotiable between drivers and passengers. The focus of this dissertation is to build models that describe the complex interac- tions among the traffic congestion, the ridesharing price and the number of travelers (including both the total number of solo drivers and also the number of travelers partic- ipating in ridesharing). In Model I, it is assumed that all the drivers and passengers that are sharing the same car (or ride) must travel on the same OD pairs, i.e. from the same origin and to the same destination. In this model, the total demand, or the total number of travelers, is treated as a variable that will change dynamically according to the traffic condition and also the ridesharing prices. The function of such changes is given by a utility function. Hence Model I can be formulated as a traffic assignment problem with elastic demands. At the same time, the relation between the number of ridesharing passengers and the 110 ridesharing price can be formulated as an economic equilibrium, with the drivers as the supply and the passengers as the demand. The traffic congestion is also taken as a variable in the driver’s utility function for the economic equilibrium. The traffic equilib- rium and the economic equilibrium can therefore be combined together via the common variables–the ridesharing price and the traffic congestion. The combined model is a con- vex optimization problem and hence the Frank-Wolfe algorithm can be adopted as the solution approach. The computational results show that (1) the ridesharing base price influences the congestion level, (2) within a certain price range, an increase in price may reduce the traffic congestion, and (3) the utilization of ridesharing increases as the congestion increases. In Model II, the same OD pair constraints have been loosened: the drivers and pas- sengers that are sharing the same car (or ride) may travel on different OD pairs. The drivers may pick up or drop off any passenger(s) anywhere at any time. The model also sets a vehicle capacity constraint. In order to cope with the above two new settings, the travel network has been extended by doubling the size of the node set and tripling the size of the arc set. The total travel demand is considered as a constant in this model for simplicity. The cost functions of solo drivers, ridesharing drivers and passengers are defined separately, where both the inconvenience cost and the ridesharing price are defined for ridesharing in addition to the congestion cost. A generalized traffic equi- librium is defined to cope with the additional vehicle capacity constraints. To describe the generalized user equilibrium mathematically, the model is formulated as a mixed complementarity problem (MiCP) and also equivalently as a variational inequality. The latter is presented to prove the existence and uniqueness of the solution. The KNITRO solver is adopted to solve the MiCP. The computational results are also promising. They show the cases when the upper bound or the lower bound of the vehicle capacity is reached, and also the cases where the ridesharing drivers are actually picking up and 111 dropping off passengers from other OD pairs. From the sensitivity analysis of the com- putational results, it can also be concluded that (1) when the arc capacity increases (the congestion cost decreases), there will be more solo drivers and less ridesharing passen- gers; (2) when the inconvenience cost increases, the number of solo drivers will increase whereas both the number of ridesharing drivers and passengers will decrease due to the higher ridesharing cost; and (3) when the ridesharing price increases, the number of the ridesharing drivers will increase while the number of the ridesharing passengers will decrease. Overall, two models have been formulated and solved to discuss different assump- tions and constraints in the Transportation Market, in order to describe the relations and the interactions among the traffic congestion, the ridesharing prices and the number of travelers. 7.3 Future Work The comparison of the two models have been discussed in Section 7.1. For future stud- ies, it is possible to further combine the strength of both models. Possible new develop- ments are listed below. In Model I, it is possible to include the vehicle capacity constraints as well. This will be more realistic. However, it will greatly increase the complexity in analyz- ing and solving the model. The capacity will not only impact the traffic equilib- rium, but might also play a role in the economic equilibrium as well. In Model II, the total demand D k can be treated as a variable. In other words, consider the generalized user equilibrium with elastic demands. This will give travelers more options to choose from traveling or not, as Model I does. When treated as a variable, it is needed to include a utility function like in Model I 112 to prevent the value of D k from dropping to zero. This will give much more complexity in formulating the mixed complementarity problem or the variational inequality, and thus will be harder to solve. In Model II, the ridesharing price (income) for each driver is calculated by a con- stant times the price for each passenger. It is an estimation of the real income of each driver. In the next step, it is possible to treat as a variable. Unlike the variableD k , turning into a variable will be even more complicated. D k is only a linear combination of the flow variablesx k a and has no involvement in the cost functions. , on the other hand, plays an important role in the cost functions and thus the number of the complementary constraints will be almost doubled. In addition, city planners can combine our models for the use of congestion pric- ing. To do so, a bi-level programming model can be formulated, with the first level minimizing the total social cost and the second level as a traffic equilibrium model. In this way, the equilibrium model serves as an auxiliary tool that helps determine what the traffic congestion would be under certain circumstances (first- level variables). 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Interna- tional Journal of Geographical Information Science, 20:899–916. Yang, H. and Huang, H. (1999). Carpooling and congestion pricing in a multilane highway with high-occupancy-vehicle lanes. Transportation Research Part A: Policy and Practice, 33(2):139–155. Yildirim, M. and Hearn, D. (2005). A first best toll pricing framework for variable demand traffic assignment problems. Transportation Research Part B: Methodologi- cal, 39(8):659–678. 118 Appendix A A.1 Positive Semidefiniteness We continue the proof of Proposition 5.6, that matrix A is positive semidefinite, where A is the symmetric part of the integral of diagonal blocks like B. For each block B, its integral is calculated as below (note that the integral of a matrix is its element-wise integral). Z 1 0 B(y +t(y 0 y))dt = 0 B B B @ R 1 0 1 (t)dt R 1 0 1 dt 0 R 1 0 1 (t)dt R 1 0 1 (t) + d +v dt R 1 0 d w dt R 1 0 2 (t)dt R 1 0 ( 2 (t) + p v) dt R 1 0 ( 3 + p +w) dt 1 C C C A = 0 B B B @ ^ 1 ^ 1 0 ^ 1 ^ 1 + d +v ^ 2 + d w ^ 2 ^ 2 + p v ^ 3 + p +w 1 C C C A ; and its symmetric part is ^ B, 0 B B B @ 2 ^ 1 2 ^ 1 ^ 2 2 ^ 1 2 ^ 1 + 2( d +v) ^ 2 + ( d w + p v) ^ 2 ^ 2 + ( d w + p v) 2 ^ 3 + 2( p +w) 1 C C C A ; 119 where (see A.2 for detailed calculation) ^ 1 , Z 1 0 1 (t)dt = bt a c 4 a [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 ; ^ 2 , Z 1 0 2 (t)dt = b 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 ; ^ 3 , Z 1 0 3 (t)dt = eb 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 =e ^ 2 : wherey 1 ;y 2 ;y 3 andy 0 1 ;y 0 2 ;y 0 3 are components ofy andy 0 , respectively, corresponding to the block (sub matrix) B. For convenience we rewrite ^ B as ^ B = 0 B B B @ a a c a b d c d e 1 C C C A ; thus for any vector (x;y;z) T 2R 3 , which is completely independent of anyy2Y, x y z A 0 B B B @ x y z 1 C C C A = x y z 0 B B B @ a a c a b d c d e 1 C C C A 0 B B B @ x y z 1 C C C A =ax 2 +by 2 +ez 2 + 2axy + 2cxz + 2dyz =a x +y + c a z 2 + (ba)y 2 + e c 2 a z 2 + 2(dc)yz =a x +y + c a z 2 + (ba) y + dc ba z 2 + e c 2 a (dc) 2 ba z 2 120 The third equality sign holds whena6= 0. Ifa = 0, ) ^ 1 = 0; ) y 0 1 +y 0 2 y 1 y 2 6= 0 and bt a c 4 a [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 = 0; which is impossible when all values are non-negative; OR; y 0 1 +y 0 2 y 1 y 2 = 0 and r = 4bt a c 4 a (y 1 +y 2 ) 3 = 0; ) y 0 1 +y 0 2 =y 1 +y 2 = 0; ) y 0 1 =y 0 2 =y 0 3 =y 1 =y 2 =y 3 = 0; due to capacity constraint and flow non-negativity. Therefore as long asy 0 andy do not equal to zero (vector) at the same time, we will always havea6= 0. The fourth equality sign holds whenba6= 0. Ifba = 0, i.e. d +v = 0. This cannot be true since it is required that d > d > 0 and all other parameters are non-negative. Therefore we must haveba> 0. Therefore, A is positive semidefinite when a = 2 ^ 1 0; (always holds) ba 0; (always holds strictly) e c 2 a (dc) 2 ba 0: 121 Note that e c 2 a (dc) 2 ba = 2 ^ 3 + 2( p +w) ^ 2 2 2 ^ 1 ( d w + p v) 2 2 ( d +v) = 4 ^ 3 ^ 1 ^ 2 2 2 ^ 1 + 4( p +w)( d +v) ( d w + p v) 2 2 ( d +v) 0; if (5.34) and (5.35) are true (see Appendix A.3 for details). We have shown that A is positive semidefinite. Next we need to show that A is positive definite if at least one of (5.34) and (5.35) holds strictly. For anyu T Au = 0;u = (x;y;z) T , u T Au =a x +y + c a z 2 + (ba) y + dc ba z 2 + e c 2 a (dc) 2 ba z 2 = 0: Sincea> 0 andba> 0, we must have x +y + c a z = 0; y + dc ba z = 0; e c 2 a (dc) 2 ba z 2 = 0: Note that c = ^ 2 0. When we choose parameters such that e c 2 a (dc) 2 ba > 0, i.e. (5.34) and/or (5.35) holds strictly. Then we must havex =y =z = 0. A is positive definite, with respect to anyy;y 0 2Y. 122 A.2 Integral Calculation Here gives the calculation of each integral element in R 1 0 B(y+t(y 0 y))dt. Also again, y 1 ;y 2 ;y 3 andy 0 1 ;y 0 2 ;y 0 3 are components ofy andy 0 , respectively, corresponding to the block (sub matrix) B. Z 1 0 1 (t)dt = 4bt a c 4 a Z 1 0 (y 1 +t(y 0 1 y 1 ) +y 2 +t(y 0 2 y 2 )) 3 dt = 4bt a c 4 a (y 0 1 y 1 +y 0 2 y 2 ) Z 1 0 (y 1 +t(y 0 1 y 1 ) +y 2 +t(y 0 2 y 2 )) 3 d (t(y 0 1 y 1 +y 0 2 y 2 ) +y 1 +y 2 ) = 4bt a c 4 a [(y 0 1 +y 0 2 ) (y 1 +y 2 )] Z y 0 1 +y 0 2 y 1 +y 2 s 3 ds = bt a c 4 a [(y 0 1 +y 0 2 ) (y 1 +y 2 )] s 4 y 0 1 +y 0 2 y 1 +y 2 = bt a c 4 a [(y 0 1 +y 0 2 ) (y 1 +y 2 )] (y 0 1 +y 0 2 ) 4 (y 1 +y 2 ) 4 = bt a c 4 a [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 ; if (y 0 1 +y 0 2 ) (y 1 +y 2 )6= 0. Otherwise Z 1 0 1 (t)dt = 4bt a c 4 a Z 1 0 (y 1 +y 2 ) 3 dt = 4bt a c 4 a (y 1 +y 2 ) 3 = bt a c 4 a [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 ; is also true when (y 0 1 +y 0 2 ) (y 1 +y 2 ) = 0. 123 Similarly, Z 1 0 ~ 1 (t)dt = 4b 0 t a c 4 a Z 1 0 [y 1 +y 2 +ey 3 +t(y 0 1 +y 0 2 +ey 0 3 y 1 y 2 ey 3 )] 3 dt = 4b 0 t a c 4 a (y 0 1 +y 0 2 +ey 0 3 y 1 y 2 ey 3 ) Z y 0 1 +y 0 2 +ey 0 3 y 1 +y 2 +ey 3 s 3 ds = b 0 t a c 4 a (y 0 1 +y 0 2 +ey 0 3 y 1 y 2 ey 3 ) (y 0 1 +y 0 2 +ey 0 3 ) 4 (y 1 +y 2 +ey 3 ) 4 = b 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 if (y 0 1 +y 0 2 +ey 0 3 ) (y 1 +y 2 +ey 3 )6= 0. Otherwise Z 1 0 ~ 1 (t)dt = 4b 0 t a c 4 a Z 1 0 (y 1 +y 2 +ey 3 ) 3 dt = 4b 0 t a c 4 a (y 1 +y 2 +ey 3 ) 3 = b 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 : Z 1 0 ~ 3 (t)dt = 4eb 0 t a c 4 a Z 1 0 [y 1 +y 2 +ey 3 +t(y 0 1 +y 0 2 +ey 0 3 y 1 y 2 ey 3 )] 3 dt = 4eb 0 t a c 4 a (y 0 1 +y 0 2 +ey 0 3 y 1 y 2 ey 3 ) Z y 0 1 +y 0 2 +ey 0 3 y 1 +y 2 +ey 3 s 3 ds = eb 0 t a c 4 a (y 0 1 +y 0 2 +ey 0 3 y 1 y 2 ey 3 ) (y 0 1 +y 0 2 +ey 0 3 ) 4 (y 1 +y 2 +ey 3 ) 4 = eb 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 =e Z 1 0 ~ 1 (t)dt; 124 if (y 0 1 +y 0 2 +ey 0 3 ) (y 1 +y 2 +ey 3 )6= 0. Otherwise Z 1 0 ~ 3 (t)dt = 4eb 0 t a c 4 a Z 1 0 (y 1 +y 2 +ey 3 ) 3 dt = 4eb 0 t a c 4 a (y 1 +y 2 +ey 3 ) 3 = eb 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 : 125 A.3 Proof of Non-negativity Claim A.1. If (5.34) and (5.35) hold true, we must havee c 2 a (dc) 2 ba 0. Proof. We only need to show 4 ^ 3 ^ 1 ^ 2 2 0, given (5.34) and (5.35). 4 ^ 3 ^ 1 ^ 2 2 = 4 bt a c 4 a [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 eb 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 b 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 2 = b 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 4e bt a c 4 a [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 b 0 t a c 4 a [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 0; if and only if 0 4eb [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 b 0 [(y 0 1 +y 0 2 +ey 0 3 ) + (y 1 +y 2 +ey 3 )] (y 0 1 +y 0 2 +ey 0 3 ) 2 + (y 1 +y 2 +ey 3 ) 2 : 126 Since we havey 3 Cy 2 andy 0 3 Cy 0 2 (according to (5.11)), we have Righthand side of the above inequality 4eb [(y 0 1 +y 0 2 ) + (y 1 +y 2 )] (y 0 1 +y 0 2 ) 2 + (y 1 +y 2 ) 2 b 0 [(y 0 1 +y 0 2 +eCy 0 2 ) + (y 1 +y 2 +eCy 2 )] (y 0 1 +y 0 2 +eCy 0 2 ) 2 + (y 1 +y 2 +eCy 2 ) 2 = 4eb (y 0 1 +y 0 2 ) 3 + (y 0 1 +y 0 2 )(y 1 +y 2 ) 2 + (y 0 1 +y 0 2 ) 2 (y 1 +y 2 ) + (y 1 +y 2 ) 3 b 0 (y 0 1 + (1 +eC)y 0 2 ) 3 + (y 0 1 + (1 +eC)y 0 2 )(y 1 + (1 +eC)y 2 ) 2 + (y 0 1 + (1 +eC)y 0 2 ) 2 (y 1 + (1 +eC)y 2 ) + (y 1 + (1 +eC)y 2 ) 3 = 3 X j=0 4eb(y 0 1 +y 0 2 ) 3j (y 1 +y 2 ) j b 0 (y 0 1 + ~ cy 0 2 ) 3j (y 1 + ~ cy 2 ) j where ~ c = 1 +eC > 1. It is sufficient to show that the last formula is non-negative. Forj = 0: 4eb(y 0 1 +y 0 2 ) 3 b 0 (y 0 1 + ~ cy 0 2 ) 3 = (4ebb 0 )(y 0 1 ) 3 + 3(4ebb 0 ~ c)(y 0 1 ) 2 y 0 2 + 3(4ebb 0 ~ c 2 )y 0 1 (y 0 2 ) 2 + (4ebb 0 ~ c 3 )(y 0 2 ) 3 : We need 4ebb 0 > 4ebb 0 ~ c> 4ebb 0 ~ c 2 > 4ebb 0 ~ c 3 0, which is exactly (5.35). Forj = 1: 4eb(y 0 1 +y 0 2 ) 2 (y 1 +y 2 )b 0 (y 0 1 + ~ cy 0 2 ) 2 (y 1 + ~ cy 2 ) = (4ebb 0 )(y 0 1 ) 2 y 1 + (4ebb 0 ~ c)(y 0 1 ) 2 y 2 + 2(4ebb 0 ~ c)y 0 1 y 0 2 y 1 + 2(4ebb 0 ~ c 2 )y 0 1 y 0 2 y 2 + (4ebb 0 ~ c 2 )(y 0 2 ) 2 y 1 + (4ebb 0 ~ c 3 )(y 0 2 ) 2 y 2 : We once again conclude (5.35). 127 j = 2 andj = 3 are symmetric asj = 1 andj = 0 and thus the proof is trivial. 128 A.4 Model Equivalency Proposition A.1. A solution to VI(X; ) is also a solution to MiCP(x). Proof. The linear constraints ofX , i.e. Constraints (5.9), (5.10), (5.12) and (6.1), can be written in a compact form as following. Flow conservation (5.9) and demand requirement (5.10): N k x k =D k k ; 8k2K whereN k is an (jNj 2)jAj matrix, whose first two rows are for origin and destination, respectively, and the rest (jNj 4) rows are for all non-OD nodes. For each elementn k i;a ofN k , (a) ifi = 1 (for origin), n k 1;a = 8 > > < > > : 1; if arca2A leaves fromo k oro 0 k 0; otherwise (b) ifi = 2 (for destination), n k 2;a = 8 > > < > > : 1; if arca2A entersd k ord 0 k 0; otherwise (c) ifi2Nnfo k ;o 0 k ;d k ;d 0 k g (for origin), n k i;a = 8 > > > > > < > > > > > : 1; if arca2A entersi, i.e.a = (j;i) for somej2N 1; if arca2A leaves fromi, i.e.a = (i;j) for somej2N 0; otherwise 129 and k is a vector inR jNj w.r.t.k2K, with element k i = 8 > > > > > < > > > > > : 1; i = 1 1; i = 2 0; otherwise In sum, the constraints (5.9) and (5.10) can be rewritten as Nx =D (A.1) where N = diag N 1 ;:::;N jKj is a (jNj 2)jKj-by-jAjjKj matrix, D = D 1 T 1 ;:::;D jKj T jKj T is a vector inR (jNj2)jKj . Capacity constraint (5.11): P 2 xP 3 xCP 2 x (A.2) whereP 2 andP 3 projecty = x fromR jAj ontoR jA 0 j , i.e. P 2 = 0 B B B B B B B @ 0 1 0 0 0 0 ::: 0 0 0 0 0 0 0 1 0 ::: 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 ::: 0 1 0 1 C C C C C C C A ; and P 3 = 0 B B B B B B B @ 0 0 1 0 0 0 ::: 0 0 0 0 0 0 0 0 1 ::: 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 ::: 0 0 1 1 C C C C C C C A 130 are matrices inR jA 0 jjAj . Non-negativity constraints (5.12) x 0: (A.3) The MiCP formulation with respect tox is as below. 0x ? (x) +N T + T (P 2 P 3 ) T + T (CP 2 P 3 ) T 0 0 + ? (P 3 P 2 )x 0 0 ? (CP 2 P 3 )x 0 free ? NxD = 0 (A.4) where2R (jNj2)jKj and + ; 2R jA 0 j are multipliers. 131 A.5 KNITRO Outputs A.5.1 Sioux Falls, 149 OD pairs a y a 1 y a 2 y a 3 + a a f a 1 f a 2 f a 3 1 1,629.04 48.53 96.12 0.0000 0.0000 6.1584 6.1584 14.7375 2 2,619.35 44.50 96.06 0.0000 0.0000 5.0071 5.0071 12.2216 3 1,532.91 37.41 66.86 0.0000 0.0000 6.1216 6.1216 12.6270 4 1,207.02 198.53 496.12 0.0000 0.0000 53.4338 53.4338 49.4649 5 2,055.77 44.53 96.12 0.0000 0.0000 4.3892 4.3892 12.1616 6 4,403.41 172.21 432.12 0.0000 0.0000 34.6830 34.6830 39.7440 7 4,694.50 67.60 156.85 0.0000 0.0000 14.2855 14.2855 17.5631 8 4,702.76 169.48 424.94 0.0000 0.0000 43.4471 43.4471 40.1694 9 3,882.73 74.19 184.70 0.0000 0.0000 9.3544 9.3544 16.6759 10 1,185.85 183.23 450.61 0.0000 0.0000 60.4557 60.4557 48.1802 11 3,726.67 53.20 129.47 0.0000 0.0000 8.1239 8.1239 12.5596 12 1,037.39 86.03 205.35 0.0000 0.0000 19.9440 19.9440 21.9591 13 3,068.18 460.13 1,184.55 0.0000 0.0000 121.2320 121.2320 108.8490 14 1,208.29 243.20 613.68 0.0000 0.0000 60.0843 60.0843 59.5661 15 1,174.97 205.44 519.59 0.0000 0.0000 40.3467 40.3467 48.1865 16 1,748.55 676.07 1,768.62 0.0000 0.0000 182.0600 182.0600 169.6770 17 2,328.10 181.63 462.19 0.0000 0.0000 50.2125 50.2125 43.0318 18 3,376.82 123.78 315.21 0.0000 0.0000 3.5017 3.5017 25.4624 19 1,678.71 374.44 974.84 0.0000 0.0000 94.5799 94.5799 88.5590 20 2,398.82 201.78 515.21 0.0000 0.0000 57.4307 57.4307 47.8525 21 951.38 165.56 383.06 0.0000 0.0000 45.8905 45.8905 45.8905 22 1,197.77 354.05 905.39 0.0000 0.0000 72.0958 72.0958 84.4784 132 23 2,991.64 480.24 1,237.46 0.0000 0.0000 113.9730 113.9730 111.9570 24 1,016.22 307.36 756.20 0.0000 0.0000 80.7716 80.7716 80.7716 25 4,046.12 208.74 533.52 0.0000 0.0000 42.3296 42.3296 46.8725 26 3,123.36 70.79 170.49 0.0000 0.0000 15.4911 15.4911 17.5062 27 2,970.51 477.57 1,230.45 0.0000 0.0000 111.0170 111.0170 111.0170 28 4,293.75 822.40 2,132.62 0.0000 0.0000 190.9850 190.9850 190.9850 29 1,322.91 652.21 1,695.29 0.0000 0.0000 168.3590 168.3590 168.3590 30 1,067.71 288.76 717.78 0.0000 0.0000 73.3421 73.3421 73.8603 31 959.45 113.87 268.09 0.0000 0.0000 26.5710 26.5710 29.8486 32 3,051.02 580.80 1,502.12 0.0000 0.0000 135.4850 135.4850 135.4850 33 1,161.31 337.90 857.63 0.0000 0.0000 84.3035 84.3035 84.3035 34 1,329.49 669.18 1,739.96 0.0000 0.0000 173.3090 173.3090 173.3090 35 3,899.37 24.15 42.49 0.0000 0.0000 8.7395 8.7395 8.7395 36 1,178.32 304.09 768.66 0.0000 0.0000 80.8532 80.8532 76.4905 37 4,152.62 139.08 350.21 0.0000 0.0000 6.3925 6.3925 29.4888 38 3,581.49 32.36 69.36 0.0000 0.0000 4.7056 4.7056 9.0684 39 1,523.65 521.08 1,350.21 0.0000 0.0000 160.0960 160.0960 134.5790 40 1,315.36 365.46 940.69 0.0000 0.0000 88.6847 88.6847 88.6847 41 1,325.64 549.02 1,418.48 0.0000 0.0000 139.0040 139.0040 139.0040 42 1,084.33 91.88 220.73 0.0000 0.0000 23.5227 23.5227 23.5227 43 3,636.57 302.57 764.67 0.0000 0.0000 71.0087 71.0087 71.0087 44 1,309.14 380.07 973.87 0.0000 0.0000 93.3406 93.3406 93.3406 45 1,853.05 12.52 17.15 0.0000 0.0000 4.2113 4.2113 5.2573 46 1,253.13 8.46 8.46 0.3773 0.0000 4.3426 3.9653 4.7200 47 1,138.61 204.81 512.65 0.0000 0.0000 42.6854 42.6854 49.2246 48 1,470.19 220.22 558.46 0.0000 0.0000 92.1841 92.1841 58.2786 133 49 716.10 38.37 90.46 0.0000 0.0000 3.2993 3.2993 9.2625 50 4,268.66 53.52 125.06 0.0000 0.0000 13.4697 13.4697 13.9879 51 1,089.16 389.51 982.92 0.0000 0.0000 100.2660 100.2660 100.2660 52 1,789.72 404.87 1,054.93 0.0000 0.0000 95.0164 95.0164 94.4982 53 1,695.39 444.10 1,158.15 0.0000 0.0000 118.0780 118.0780 107.1790 54 3,498.37 78.22 195.33 0.0000 0.0000 3.6364 3.6364 16.8380 55 3,843.30 218.22 558.46 0.0000 0.0000 11.1635 11.1635 45.0689 56 6,326.35 169.27 424.40 0.0000 0.0000 39.6051 39.6051 39.6051 57 4,006.15 205.61 525.29 0.0000 0.0000 34.4664 34.4664 45.3654 58 861.91 10.52 17.15 0.0000 0.0000 5.2094 5.2094 4.1634 59 1,107.55 176.13 442.45 0.0000 0.0000 30.0137 30.0137 40.9127 60 6,754.61 224.51 569.75 0.0000 0.0000 51.4493 51.4493 51.4493 61 1,385.32 360.21 926.88 0.0000 0.0000 92.9352 92.9352 87.9994 62 942.14 99.91 231.34 0.0000 0.0000 22.1889 22.1889 26.5515 63 1,229.47 230.56 580.42 0.0000 0.0000 56.3484 56.3484 56.3484 64 1,224.65 503.24 1,292.73 0.0000 0.0000 128.3870 128.3870 128.3870 65 1,684.28 256.88 665.48 0.0000 0.0000 58.9370 58.9370 58.9370 66 1,494.27 585.01 1,523.72 0.0000 0.0000 150.6660 150.6660 146.3040 67 3,436.99 666.24 1,737.48 0.0000 0.0000 153.2380 153.2380 153.2380 68 1,260.67 284.72 722.94 0.0000 0.0000 69.4498 69.4498 69.4498 69 1,221.47 76.84 191.68 0.0000 0.0000 13.3935 13.3935 17.7562 70 1,375.53 458.09 1,184.45 0.0000 0.0000 112.5210 112.5210 112.5210 71 1,321.24 333.10 855.53 0.0000 0.0000 80.4017 80.4017 80.4017 72 1,319.43 275.79 704.72 0.0000 0.0000 66.1673 66.1673 66.1673 73 1,595.90 224.61 580.55 0.0000 0.0000 51.5392 51.5392 51.5392 74 1,402.69 428.08 1,105.46 0.0000 0.0000 104.3190 104.3190 104.3190 134 75 1,125.57 208.48 532.83 0.0000 0.0000 28.0216 28.0216 46.1986 76 1,247.19 181.52 467.17 0.0000 0.0000 20.7913 20.7913 38.9683 135

Abstract (if available)

Abstract A nascent ridesharing industry is being enabled by new communication technologies and motivated by its many possible benefits, such as reduction in travel cost, pollution, and congestion. Understanding the complex relations between ridesharing and traffic congestion is a critical step in the evaluation of a ridesharing enterprise or of the usefulness of regulatory policies or incentives to promote ridesharing. In this research, we propose two new traffic assignment models that explicitly represent ridesharing as a mode of transportation. The objective is to analyze how ridesharing impacts traffic congestion, how people can be motivated to participate in ridesharing, and conversely, how congestion influences ridesharing, including ridesharing prices and the number of drivers and passengers. ❧ The first model considers the scenario where drivers and passengers sharing the same ride must travel from the same origin and to the same destination. This model is built by combining a ridesharing market model with a classic elastic demand Wardrop traffic equilibrium model. It is formulated as a convex optimization problem. The Frank‐Wolfe algorithm is adopted to solve the model and a heuristic approach is applied using the equilibrium condition. Our computational results show that: (1) the ridesharing base price influences the congestion level, (2) within a certain price range, an increase in the price may reduce the traffic congestion, and (3) the utilization of ridesharing increases as the congestion increases. ❧ The second model drops the constraint of the same origin‐destination (OD) pair. In this model, drivers may pick up or drop off any passenger in the middle of their trips, and they may even detour from a seemingly shortest path. In order to describe this scenario, we extend the network by doubling the nodes and tripling the arcs in size. A generalized user equilibrium is defined to represent the new network and the new constraints. The generalized user equilibrium can be formulated as a mixed complementarity problem (MiCP), and equivalently a variational inequality. It is proved that there exists one and only one solution to this model. The KNITRO solver is adopted to solve the MiCP and the computational results are promising. It can be concluded from the results that when the congestion cost decreases or the ridesharing inconvenience cost increases, more travelers would become solo drivers and thus less people would participate in ridesharing. On the other hand, when the ridesharing price increases, more travelers would become ridesharing drivers. ❧ In conclusion, the traffic congestion, the ridesharing cost, and the number of travelers interact with each other closely in both models. Understanding their relationships enables planners to develop policies to draw more people to participate in ridesharing and thus to reduce traffic congestion.

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Asset Metadata

Creator Xu, Huayu (author)

Core Title Traffic assignment models for a ridesharing transportation market

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 07/15/2014

Defense Date 05/22/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag convex optimization,mixed complementarity problem,OAI-PMH Harvest,ridesharing,traffic assignment problem,Transportation

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dessouky, Maged M. (committee chair), Ordóñez, Fernando (committee chair), Giuliano, Genevieve (committee member), Pang, Jong-Shi (committee member)

Creator Email cxuhuayu@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-440133

Unique identifier UC11287818

Identifier etd-XuHuayu-2682.pdf (filename),usctheses-c3-440133 (legacy record id)

Legacy Identifier etd-XuHuayu-2682.pdf

Dmrecord 440133

Document Type Dissertation

Format application/pdf (imt)

Rights Xu, Huayu

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA