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Personalized Pareto-improving pricing-and-routing schemes with preference learning for optimum freight routing
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Personalized Pareto-improving pricing-and-routing schemes with preference learning for optimum freight routing
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Personalized Pareto-Improving Pricing-and-Routing Schemes with Preference Learning for Optimum Freight Routing by Aristotelis Angelos Papadopoulos 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 (ELECTRICAL ENGINEERING) May 2022 Copyright 2022 Aristotelis Angelos Papadopoulos Acknowledgements I would like to thank Prof. Petros Ioannou, my advisor, for his guidance and support during my PhD studies. Even at the hardest times of this PhD journey, Prof. Ioannou would always point me to the right direction. I also want to thank Prof. Maged Dessouky for being my co-advisor in my PhD journey. His guidance and advise at each step of my PhD studies were crucial for the successful completion of this dissertation. Working under the supervision of Prof. Ioannou and Prof. Dessouky has been a unique learning experience for me. Additionally, I want to thank Dr. Ioannis Kordonis for the numerous conversations and research ideas we discussed and for introducing me to the concepts of Game Theory. Lastly, I also want to thank Prof. Paul Bogdan, Prof. Mihailo Jovanovic and Prof. Ashutosh Nayyar for being on my qualification committee and thesis committee. My deepest thanks go to my father Dimitrios Papadopoulos, my mother Maria Georgiadi and my brother Ioannis Papadopoulos. Even if they were thousands of miles away from Los Angeles, their encouragement, support and unconditional love helped me successfully complete this journey. This thesis is dedicated to them. ii TableofContents Acknowledgements ii ListofTables vi ListofFigures viii Abstract ix Chapter1: Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Congestion Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Tradable Credit and Permit Schemes . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Toll-and-Subsidy Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Pareto-Improving Pricing-and-Routing Schemes with Departure Time Choice Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Personalized Pareto-Improving Pricing-and-Routing Schemes for Heterogeneous Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Personalized Freight Route Recommendations with System Optimality Con- siderations: A Utility Learning Approach . . . . . . . . . . . . . . . . . . 10 Chapter2: Pareto-ImprovingPricing-and-RoutingSchemeswithDepartureTimeChoice Considerations 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Routing without Coordination: User Equilibrium . . . . . . . . . . . . . . 17 2.3 Routing with Coordination: Mechanism Design . . . . . . . . . . . . . . . . . . . 19 2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Main Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Algorithmic Procedure of the Mechanism . . . . . . . . . . . . . . . . . . 28 2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 Numerical calculation of the User Equilibrium . . . . . . . . . . . . . . . 30 iii 2.4.2 Numerical calculation of the System Optimum . . . . . . . . . . . . . . . 30 2.4.3 Braess network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.4 Sioux Falls network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter3: PersonalizedPareto-ImprovingPricing-and-RoutingSchemesforHeteroge- neousUsers 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 User Equilibrium (UE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 System Optimum (SO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Pricing-and-Routing Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.1 Optimum Pricing Scheme (OPS) . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.2 Approximately Optimum Pricing Scheme (AOPS) . . . . . . . . . . . . . 50 3.3.3 Congestion Pricing with Uniform Revenue Refunding (CPURR). . . . . . 54 3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.1 Robustness in the Sioux Falls Network . . . . . . . . . . . . . . . . . . . 57 3.4.1.1 Sensitivity to Initial Conditions . . . . . . . . . . . . . . . . . . 57 3.4.1.2 Sensitivity to the Number of Routes . . . . . . . . . . . . . . . . 60 3.4.2 The Effect of the Weighting Factorl . . . . . . . . . . . . . . . . . . . . 61 3.4.3 Deterministic vs Stochastic Demand Scenario . . . . . . . . . . . . . . . . 63 3.4.4 Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter4: PersonalizedFreightRouteRecommendationswithSystemOptimalityCon- siderations: AUtilityLearningApproach 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 User Equilibrium (UE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.3 System Optimum (SO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Personalized route recommendation . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3.3 Utility learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.4 Optimization formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4.1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.2 Clustering and Utility Learning . . . . . . . . . . . . . . . . . . . . . . . 86 4.4.3 Necessity of a Pricing-and-Routing Scheme . . . . . . . . . . . . . . . . . 89 4.4.4 Pricing-and-Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Chapter5: ConclusionandFutureResearch 93 5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ReferenceList 96 iv AppendixA Mathematical Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A.0.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A.0.2 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 A.0.3 Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 AppendixB Discussion on the the Pareto-Improvement and the Truthfulness Properties . . . . . . . . 111 v ListofTables 2.1 Notation used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Simulation Results of the Braess network with 2 Time Intervals . . . . . . . . . . . 32 2.3 Simulation Results of the Braess network with 6 Time Intervals . . . . . . . . . . . 33 2.4 Simulation Results of the Sioux Falls network . . . . . . . . . . . . . . . . . . . . 34 3.1 Notation used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 The minimum, the maximum, the mean, the standard deviation and the coefficient of variation (CV) values for different metrics for the UE, SO, OPS, AOPS and CPURR methods after 10 simulation runs with different initial conditions. The number of OD pairs was 6 and we considered the 10 least congested routes per OD pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3 The expected total truck travel time E[T tr ], the expected total truck monetary cost E[T mon tr ], the expected total network time E[T s ] and the expected total travel time of the passenger vehicles E[T p ] of the UE, SO, OPS, AOPS and CPURR in the case where the truck drivers follow 6 OD pairs, considering the 5, 10 and 15 least congested routes per OD pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 The expected total truck travel time E[T tr ], the expected total truck monetary cost E[T mon tr ], the expected total network time E[T s ] and the expected travel time of the passenger vehicles E[T p ] of the UE, SO, OPS, AOPS and CPURR for different number of OD pairs, considering the 10 least congested routes per OD pair. The ratio of trucks in the network is shown in Table 3.6. . . . . . . . . . . . . . . . . . 66 3.5 The computational time (in seconds) of the simulation experiments presented in Table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6 The ratio of trucks in the network for the experimental results presented in Table 3.4. 68 4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 An example of a route choice question used to learn drivers’ routing preferences. . 85 4.3 AUROC and AUPR for different numbers of clusters K. . . . . . . . . . . . . . . . 88 vi 4.4 Percentage by which the truck drivers could increase their utility in case they de- cided to deviate from the SO solution. . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5 Simulation Results of the Sioux Falls network. . . . . . . . . . . . . . . . . . . . . 91 vii ListofFigures 2.1 The network transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 The Braess network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 (a) The expected total truck travel time, (b) the expected total truck monetary cost, (c) the expected total network time and (d) the total objective value of the UE (magenta), SO (blue), OPS (orange), AOPS (green) and CPURR (red) for different values of the weighting factorl. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 (a) The expected total truck monetary cost and (b) the total objective value of the UE (blue), SO (orange), OPS (green), AOPS (red) and CPURR (purple) for different demand scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1 The Sioux Falls network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 viii Abstract The sharp increase in e-commerce over the last few years has led to an increase in the volume of trucks both in ports and in commercial areas. The efficient use of the road network for freight transport has a big impact on travel times, pollution, and fuel consumption, as well as on the mo- bility of passenger vehicles. The continuously increasing use of navigation apps has led drivers to make their routing decisions in an independent manner in an effort to minimize their own individ- ual travel time, with possible significant deviation from a socially optimum solution. In contrast, a System Optimum (SO) solution is not a practical solution since some drivers may benefit while some others may be harmed compared to the User Equilibrium (UE), raising several equity and fairness issues. In this dissertation, we address the problem of the inefficiency between the UE and the SO through the use of combined pricing-and-routing schemes. Our main goal is the design of pricing- and-routing schemes that are budget balanced on average and make every participant better-off, while concurrently driving the transportation network as close as possible to the SO solution. In the first part of this dissertation, we design coordination mechanisms that can make every user better-off compared to the UE (Pareto-improvement) while concurrently leading the trans- portation network as close as possible to the SO solution. Initially, for the case where the users are asked to declare both their Origin-Destination (OD) pair as well as their desired departure time, we derive sufficient conditions for the existence of Pareto-improving, truthful in equilibrium and revenue-neutral on average mechanisms. Subsequently, assuming the existence of heterogeneous users with distinct Value-of-Time (VOT) and that the drivers are asked to declare both their OD ix pair as well as pick their VOT from a set of N available options, we prove the existence of Pareto- improving and revenue-neutral on average mechanisms that make every driver have an incentive to truthfully declare his/her VOT. This result allows us to design personalized (VOT-based) pricing- and-routing schemes. In the last part of this dissertation, given that there are several factors that determine a driver’s routing decisions, we aim to design a mechanism that provides personalized routing suggestions to truck drivers. To achieve this, we propose to divide the drivers into disjoint clusters and learn the routing preferences of each cluster using drivers’ individual routing decisions. Having learned a utility function that describes the routing preferences of each cluster of drivers, we propose an approach that optimizes over a total system-wide cost through a combined pricing-and-routing scheme that satisfies the budget balance on average property and ensures that every driver has an incentive to participate in the proposed mechanism. x Chapter1 Introduction 1.1 Motivation The transportation sector constitutes an integral part of the global economy. In the United States (U.S.), it comprises 8:9% of the nation’s economic activity as measured by the gross domestic product (GDP) [14] while in Europe the corresponding percentage is about 5% [23]. The efficient movement of people and freight using the road network is a critical factor in US competitiveness. Worldwide container trade has been growing and current forecasts expect US commodity trade to approximately double by 2030 [97]. According to [5], the traffic congestion on the U.S. highways annually costs the trucking industry more than $63 billion in operating expenses, including 996 million hours of lost productivity. Of the GHG emissions coming from transportation related sources, freight movement accounts for 29% of the total; trucks were responsible for emitting 68% of GHG coming from these freight sources [98]. Of all the motor vehicle traffic fatalities reported in 2009, 9:6% (2,987) involved large trucks and over one-third of the crashes took place in urban areas [102]. Of the nearly 20 billion tons of freight moved in 2012, 13 billion moved by truck [13]. Trucks dominate due to shipment size, trip length, and ubiquity of the road network. Currently, freight is routed from origin to destination via air, ocean, rail and road network. The networks that offer the biggest challenge are the urban rail and road networks, as passengers share both networks and capacity is limited. Due to size and differences in dynamics, freight transport has a disproportionate impact on the road network especially in urban areas [73, 72] as they affect traffic 1 flow much more than passenger vehicles especially during turns, merging and changing lanes, stop and go traffic, lower speeds on highways etc. Unlike passenger traffic that is mainly due to daily commute to work, freight traffic can be more flexible with respect to schedules and routes in order to optimize their operations and reduce costs. The lack of coordination and competition between different users and other constraints are ob- stacles in achieving a more efficient balance of freight loads across the road network. Currently, there is no centralized or decentralized freight load balancing system. Intelligent Transportation Systems (ITS) technologies provide the potential for a load balancing of freight, but methodolo- gies and techniques must be developed and evaluated to realize this potential. A more efficient freight system would not only lead to cost savings for shippers, but also reduce congestion, air pollution, and accident related costs. While technology improved the way decisions are made in freight scheduling and routing, the users of the network act independently in an effort to minimize their own individual cost [50, 80] as a consequence of lack of cooperation. This behavior of the network users is known in the transportation literature as User Equilibrium (UE) [116, 7, 26]. The inefficiency of the UE compared to the System Optimum (SO) has been addressed in the literature as the Price of Anarchy (PoA) [54]. Recently, the PoA of realistic transportation networks using traffic data has been also estimated [128], showing the necessity for its reduction. A way to reduce the PoA of the transportation network is by enforcing drivers to follow routes that guarantee system optimality. However, such an approach would not be a practical solution since it would raise several equity and fairness issues. This is mainly because in a SO solution, some drivers might benefit while some others might be harmed compared to the UE. In this dis- sertation, we focus on truck routing and we study methodologies that make users follow routing suggestions that are beneficial for the system optimum while concurrently guaranteeing that each truck driver will have an incentive to follow the provided suggestions since his/her expected cost is going to be lower compared to the case where he/she would act independently. 2 1.2 LiteratureReview 1.2.1 NetworkModels Given an Origin-Destination (OD) matrix describing the demands from different origins to differ- ent destinations of the network, traffic assignment involves routing of users onto the available paths in the transportation network. In this dissertation, we mainly use static traffic assignment models where the link flows and the travel times are assumed to be constant and the network is assumed to be in an equilibrium state. For a comprehensive review of static traffic assignment models, we refer the interested reader to [78] and references therein. Static traffic assignment models can be classified into user equilibrium models and system op- timum models based on the objective of the assignment process and user behavior assumptions. Equilibrium modeling of traffic assignment dates back to 1920 [80]. The same equilibrium princi- ples were introduced by Wardrop in [116] and systematically studied in [7]. It is generally known that the routes chosen by the travelers are those that are individually perceived as being the shortest under the prevailing traffic conditions, i.e., travelers minimize their individual travel time. Although it has been observed that many factors other than travel time influence the drivers’ route choices, travel time remains the main component in the travel cost [78]. User equilibrium (UE) assignment models are based on Wardrop’s first principle [116] which states that no user can further minimize his/her individual cost by unilaterally changing his/her routing decisions. There are mainly two types of UE models studied in the literature. In the deterministic UE, the users are assumed to have complete information about the travel time. In the Stochastic User Equilibrium (SUE) case [29], it is assumed that there exists some error in travelers’ perception of travel times. Then, the perceived travel time is modeled as the actual travel time plus an error term that follows a probability distribution. Logit-based SUE models assume that the costs on alternative routes are independent gumbel variates while probit-based SUE models assume that the perceived travel time on any path is normally distributed with mean equal to the experienced travel time. 3 The system optimum (SO) model is based on the Wardrop’s second principle [116] where users are assigned to routes so that the total system travel time is minimized. In a SO solution, some drivers might follow routes that are costly to them but beneficial for the network. System optimum assignment is not an equilibrium state and is not achieved in reality. Among the traffic assignment models studied in the literature, most of them are based on the assumption of having fixed demands or elastic demands. In the case of elastic demands, the trip rates in the OD pair are modelled as functions of the least travel cost between the origin and the destination. The basic premise behind such a model is that a driver has a range of choices available to them and is motivated by economic considerations in his/her decisions. Viewing the minimum travel cost as a measure of the perceived benefit to the travelers in an OD pair, the incentive to make a trip decreases with an increasing disutility. The first model with elastic demands was proposed by [7]. Another less studied form of traffic assignment models are the models assuming stochastic demands. In [33], models with a single OD pair are considered. In [117] there are several OD pairs, but the proportion of demand among the OD pairs is fixed. In [104], the demands to the different OD pairs could have varying ratios. Note that traffic assignment models with stochastic demands are fundamentally different from SUE models which have deterministic demand and link flows, with the randomness arising from the users perception of link costs. 1.2.2 CongestionPricing One of the most frequently studied techniques that addresses the problem of the inefficiency of the UE compared to the SO is marginal cost pricing, also known as congestion pricing [80]. The idea of congestion pricing, initially proposed by Pigou and later studied by many more researchers [110, 7], states that each driver should pay a fee (toll) that is equal to the additional cost his/her presence causes to the transportation network. One of the main reasons that made congestion pricing attract a lot of attention from the research community is the fact that a congestion pricing scheme applied to all links of a network can decentralize a SO solution, i.e., the tolled UE is also a SO. 4 Congestion pricing schemes have been traditionally divided in the literature into first-best and second-best schemes. Schemes that can decentralize the system optimal flow pattern are called first-best [108, 82] and they usually require to toll all links of the network. On the other hand, second-best schemes consider the problem where not all links of the network can be tolled [109]. A widely-studied case of a second-best pricing scheme is the one with two parallel links where only one can be tolled. The toll is set below the marginal external congestion cost on the tolled link in order to limit congestion on the untolled link [107]. Based on their form, congestion pricing schemes can be classified into four categories. Facility- based schemes are the basic type of congestion pricing where tolls can be imposed either on roads, bridges, tunnels, etc., or on one part of the targeted facilities (e.g. express lanes in US). In zonal schemes, vehicles pay a fee to enter or exit a zone, or to travel within the zone without crossing its boundary. Toll cordons are another form of area-based charging in which vehicles pay a toll to cross a cordon in the inbound direction, in the outbound direction, or possibly in both direc- tions. Last, in distance-based schemes, vehicles are charged based on the distance travelled. For more details related to all four types of congestion pricing schemes, their advantages and their disadvantages, we refer the interested reader to [32, 44]. It is well-known that the Value of Time (VOT) varies among different drivers due to their dif- ferent levels of income and trip emergency. Several research efforts have studied pricing schemes under user heterogeneity. Assuming that there exists a discrete set of VOTs for several user classes, Yang and Huang showed that the cost-based SO flow can be achieved by implementing a marginal cost link toll that is anonymous, i.e., uniform to everyone who uses the same link [121]. Several research efforts have assumed the existence of discrete classes in order to model the heterogene- ity of users in their VOT [123, 46], while others formulate the VOT as a continuously distributed random variable [70]. In the literature, VOT is usually assumed to follow uniform distribution, exponential distribution, or log-normal distribution. Although congestion pricing schemes can theoretically improve traffic conditions, there are several reasons for the public unacceptability of congestion pricing. First, congestion pricing is 5 known to favor individuals with high VOT, while bringing direct losses to individuals whose VOT savings do not outweigh the paid toll, or those who are “tolled off” the road and have to use un- desirable alternatives [95]. Koster et al. showed that congestion pricing schemes under preference heterogeneity can lead to politically unacceptable solutions since low income travelers may have to pay a higher tax [53]. Additionally, there might be the so-called spatial equity issue, namely con- gestion pricing may have different impact on people traveling between different OD pairs [123]. Last, congestion pricing is also considered publicly unacceptable since the users’ surplus is trans- ferred to the government in the form of the toll revenue. To mitigate this issue, several research efforts have been made to minimize the total pricing revenue [82, 8, 126] or to fully refund the collected revenue [46]. 1.2.3 TradableCreditandPermitSchemes Another well studied traffic management system is the idea of tradable travel credits. In such a case, a social planner is assumed to initially distribute a certain number of credits to all eligible drivers. The credits that travelers hold can be used on any link but the amount of credit charge is link-specific. Usually, a free market for credit trading is assumed where travelers can buy or sell credits according to their individual travel needs. One of the main advantages of tradable travel credits is the fact that the government does not interfere in the market as a buyer or seller, but solely acts as a manager to monitor the system. Assuming homogeneous users, Yang and Wang studied two initial credit distribution schemes: a uniform distribution scheme which is simple and fair and an OD-based credit distribution scheme which has the potential to make everyone better-off (Pareto-improvement) [122]. Further research efforts studied properties of tradable credit schemes by assuming a discrete set of VOT for several distinct user classes [115] or by assuming heterogeneous users with a continuously distributed VOT [132]. Shirmohammadi et al. studied the connections between congestion pricing and tradable travel credits [90].More specifically, they showed that in a deterministic environment, for each tradable credit scheme designed to achieve a specific control target there exists a corresponding congestion pricing scheme that is equally 6 effective in achieving the target. However, under demand and capacity uncertainties, congestion pricing fixes the price but is unable to achieve the control target. On the other hand, tradable credit schemes ensure that the target is achieved but the credit prices can be unacceptably high. For a more detailed review of tradable credit schemes, we refer the interested reader to [119] (and references therein). Similar to tradable travel credits, tradable travel permits have been also studied in the literature. Under these schemes, the city authorities initially determine the number of permits for each road- way link or time period. Subsequently, these permits can be freely traded in a competitive market. One of the main differences between tradable permits and tradable credits is that travelers have to acquire all link permits along their respective chosen paths, and the trading prices of permits are link-specific, whilst they are charged different amount of credits for each roadway link under the single price of credits formed in the competitive market [122]. Wang et al. studied OD-based travel permits for bi-modal transportation networks assuming user heterogeneity in their VOT [112]. For networks with a single OD pair, Akamatsu and Wada compared tradable permit schemes with con- gestion pricing and found that the former can be more efficient in the case where the road manager does not have perfect information about the transportation demands [4]. Additionally, similar to tradable travel credits, permits have the advantage that they do not require a net financial flow from road users to a governing authority which makes them more likely to be publicly acceptable compared to congestion pricing [11]. However, their public acceptability is still questioned since the initial distribution of permits can raise several equity and fairness issues. 1.2.4 Toll-and-SubsidySchemes Toll-and-Subsidy Schemes (TSS) are pricing schemes that can take both positive and negative values, e.g., some users may need to pay a toll while some others may receive a payment in case they are suggested to follow an alternative that may be costly for them but beneficial for the social optimum. The main advantages of TSS compared to other traffic management systems are that they are revenue-neutral and can also be Pareto-improving in most of the cases. 7 There are only a few research efforts studying TSS. Bernstein proposed the use of both positive and negative tolls in a bottleneck congestion model [9]. Adler and Cetin proposed a direct distribu- tion approach to congestion pricing, in which fees collected from users on a more desirable route are directly transferred to users on a less desirable route using a single OD pair with two routes network with bottleneck congestion [2]. For a single OD pair with two modes network, Liu et al. studied Pareto-improving and revenue-neutral pricing schemes considering heterogeneous users with a continuous distribution of VOT [3]. For a general transportation network with fixed de- mands and identical travelers, Xiao and Wang found that a Pareto-improving and system optimum TSS might not be self-sustainable [118]. 1.3 OverviewofDissertationContributions This section provides an overview of the contributions of this dissertation. 1.3.1 Pareto-ImprovingPricing-and-RoutingSchemeswithDepartureTime ChoiceConsiderations Previous research efforts have studied pricing schemes in general transportation networks with fixed or elastic demands. Most of these works assume static networks in order to provide some theoretical guarantees. In this study, we focus on truck routing and we define a network model with stochastic demands where the planning horizon is split into discrete non-overlapping time in- tervals. In contrast with the vast majority of literature studying pricing schemes in the form of tolls applied to the drivers, our mechanism differs significantly since it allows that some of the truck drivers are getting paid in order to follow routes that may be costly to them but beneficial for the social optimum. This design allows us to create individual incentives for voluntary participation of the truck drivers (Pareto-improvement). Moreover, since the proposed mechanism also takes into account the different traffic conditions during the day, it may suggest to some truck drivers to receive a payment in order to change their trip time in an effort to alleviate traffic congestion 8 problems during peak hours. However, the fact that the truck drivers know that they may get paid for scheduling their trip during a different time than the one they actually wanted, may lead them in a situation where they declare a different departure time than their preferred one in order to get a reimbursement while concurrently traveling during their desired time interval. To avoid this kind of exploitability of the mechanism, we derive sufficient conditions which prove the existence of Pareto-improving, truthful in equilibrium and budget balanced on average mechanisms. Further- more, we show that removing the budget balance on average property, these conditions guarantee the maximum possible earnings of a company willing to apply such a mechanism while concur- rently satisfying the Pareto-improvement and the truthfulness criteria. To the best of our knowl- edge, no previous work has studied the idea of a toll-and-subsidy scheme (TSS) with departure time choice considerations for a general transportation network. 1.3.2 PersonalizedPareto-ImprovingPricing-and-RoutingSchemesfor HeterogeneousUsers It is well-known that the Value of Time (VOT) varies among different drivers due to their differ- ent levels of income and trip emergency. Taking into account the VOT of each individual driver when designing a traffic management system (congestion pricing, tradable credits or permits) is essential since it can help mitigate fairness and equity issues as well as provide personalized rec- ommendations. Many research efforts have studied traffic management systems for transportation networks in the presence of heterogeneous users. These efforts mostly assume that either there ex- ists a discrete set of VOTs for several user classes or that the VOT follows a uniform distribution, an exponential distribution, or a log-normal distribution. In this study, we design a coordination mechanism that can be applied to truck drivers taking into account the user heterogeneity in their VOT. More specifically, the coordinator asks the truck drivers to declare their desired OD pair and additionally choose their VOT from a set of N avail- able options. After collecting this information, the coordinator provides routing suggestions and 9 additionally designs pricing schemes that are Pareto-improving (make every user better-off com- pared to the UE) and guarantee that every driver will have an incentive to truthfully declare his/her VOT while concurrently leading to a revenue-neutral (budget balanced) on average mechanism. This is in contrast with the previous literature studying pricing schemes that makes assumptions about the distribution that the user heterogeneity might follow. We should note that it is important to guarantee that a user will truthfully declare his/her VOT in order to avoid the exploitability of the designed mechanism. This is mainly because many truck drivers would be willing to declare a high VOT in order to be assigned to the fastest possible route. To the best of our knowledge, no self-reporting scheme where the users directly report their VOT to a central authority has been previously proposed. 1.3.3 PersonalizedFreightRouteRecommendationswithSystemOptimality Considerations: AUtilityLearningApproach Previous research efforts have shown that there are several factors that can affect drivers’ routing decisions. Additionally, each driver values each factor differently and therefore, it is important to design a system that provides routing suggestions based on drivers’ individual routing prefer- ences. Previous works have tried to incorporate personalization in route planning. However, the proposed approaches took into account user optimality only and therefore, the provided solutions were inefficient for the system. In contrast with these works, in this study, we propose an approach that optimizes over a total system-wide cost through a combined pricing-and-routing scheme that satisfies the budget balance on average property and ensures that every truck driver has an incen- tive to participate in the proposed mechanism by guaranteeing that the expected total utility of a truck driver (including payments) in case he/she decides to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she does not participate. Since estimating a util- ity function for each individual truck driver is computationally intensive, we first divide the truck drivers into disjoint clusters based on their responses to a small number of binary route choice 10 questions and we subsequently use a learning scheme based on the Maximum Likelihood Estima- tion (MLE) principle that allows us to learn the parameters of the utility function that describes each cluster. The estimated utilities are then used to calculate a pricing-and-routing scheme with the aforementioned characteristics. To the best of our knowledge, no previous work has studied the idea of personalized route recommendations with system optimality considerations that con- currently guarantee that every driver will be better-off in case he/she decides to follow the provided routing instructions. 11 Chapter2 Pareto-ImprovingPricing-and-RoutingSchemeswith DepartureTimeChoiceConsiderations 2.1 Introduction The transportation sector constitutes an integral part of the global economy. For example, in the United States (U.S.), it comprises 8:9% of the nation’s economic activity as measured by the gross domestic product (GDP) [14] while in Europe the corresponding percentage is about 5% [23]. However, traffic congestion on the U.S. highways annually costs the trucking industry more than $63 billion in operating expenses, including 996 million hours of lost productivity [5]. These reports together with the fact that the impact of trucks on traffic flow is much higher due to their slower dynamics [73], [72] make the truck routing problem of major importance. In the absence of cooperation, the users of the network act independently in an effort to min- imize their own individual cost [50], [80]. This behavior of the network users is known in the transportation literature as User Equilibrium (UE) [116], [26]. On the other hand, the situation where the network users act in a manner which contributes to the minimization of a social welfare cost is known as System Optimum (SO). The inefficiency of the UE compared to the SO has been addressed to the literature as the Price of Anarchy (PoA) [54]. Bounds on the PoA of general networks for different types of latency functions have been calculated [86],[85]. Recently, the PoA 12 of realistic transportation networks using traffic data has been also estimated [128], showing the necessity for its reduction. Several research efforts have been made in order to reduce the PoA in transportation networks where most of them use the idea of congestion pricing [7], [110], in which each agent pays a fee corresponding to the additional cost its presence causes to the total system cost. Positive results on the willingness of drivers to participate in such policies have been presented in [12]. Additional design criteria concerning equity and fairness considerations of congestion pricing applications have been studied in [43, 62, 35, 125, 34]. Related work on the sensitivity of the users with respect to pricing schemes due to their heterogeneity can be found in [21, 49, 99]. The analysis of congestion pricing techniques and other relevant monetary schemes is based on traffic assignment models which can be either static or dynamic. Various static equilibrium models are presented in [78] and methods for solving them in [40]. Several extensions of these models include traffic models with stochastic demands [104], capacity uncertainties [18] or models with incomplete information for the network users [42]. In this chapter, we consider a non-atomic game theoretic model (with a continuum of users) with stochastic demands for the different origin-destination (OD) pairs presented in [51], but we further generalize it to the case where the traffic conditions vary with time. The users (players) of this model (game) are considered to be the truck drivers and their actions consist of the declaration of their desired OD pair as well as their preferred departure time. After proving the existence of a UE for the defined model, we design a coordination mecha- nism whose purpose is the optimum assignment of the trucks into the network in a way that a social welfare cost is minimized. In contrast with the vast majority of literature studying pricing schemes in the form of tolls applied to the drivers, for instance [28, 93, 47] , our mechanism differs signifi- cantly since it also allows that some of the players (truck drivers) are getting paid in order to follow routes that may be costly to them but beneficial for the social optimum. This design allows us to create individual incentives for voluntary participation [74] of the truck drivers. Moreover, since the proposed mechanism also takes into account the different traffic conditions during the day, it 13 may suggest to some players to receive a payment in order to change their trip time in an effort to alleviate traffic congestion problems during peak hours. However, the fact that the players know that they may get paid for scheduling their trip during a different time than the one they actually wanted, may lead them in a situation where they declare a different departure time than their pre- ferred one in order to get a reimbursement while concurrently traveling during their desired time interval. To avoid this kind of exploitability of the mechanism, we derive sufficient conditions which prove the existence of truthful in equilibrium and budget balanced on average mechanisms [91], which are able to create individual incentives for voluntary participation of the truck drivers. Furthermore, we show that removing the budget balance on average property, these conditions guarantee the maximum possible earnings of a company willing to apply such a mechanism while concurrently satisfying the voluntary participation and the truthfulness criteria. Subsequently, in an effort to maximize the efficiency of our design, we propose an algorithmic procedure which generates those conditions which are necessary for the existence of monetary schemes with the aforementioned characteristics. Simulations on the benchmark network of Sioux Falls show that the proposed mechanism can lead to a SO solution. 2.2 MathematicalFormulation 2.2.1 NetworkModel The notation used throughout this Chapter is summarized in Table 2.1. As mentioned earlier, we use a non-atomic game theoretic model with demand uncertainties for the OD pairs in which we take into account that the traffic conditions vary with time. The players of the game are considered to be the truck drivers whose actions include both the OD pair and the departure time selection. To capture the time-varying behavior of traffic, we assume that the coordinator defines a planning horizon and splits it into non-overlapping time intervals such that it guarantees that every truck driver will have sufficient time to complete his/her trip during one time interval. This assumption 14 makes the notion of time and time interval to have the same meaning in our analysis. Let the vari- able t = 1;:::;N denote the time intervals of the planning horizon. For example, if the planning horizon is defined to be a day, t = 1;2;3;4 represents morning, noon, afternoon and night respec- tively. Then, d ¯ t j is a random variable with finite support representing the demand of OD pair j during time interval ¯ t. We further assume that each truck driver knows the distribution of all the random variables d ¯ t j (symmetric information model). The transportation network is described by a graph G=(V;L) where V is the set of nodes and L=(1;:::;m) is the set of links where each link corresponds to a road segment. Each road segment is allowed to serve both passenger and freight transport vehicles. Hence, X t l p represents the number of passenger vehicles traversing road segment l during time interval t and X t lT stands for the corresponding number of trucks. In our model, we assume that the truck drivers make their routing decisions while knowing the number of passenger vehicles at each road segment. This assumption is not restrictive since passenger traffic has a repetitive behavior during the same day and time of the week in the absence of unexpected incidents. Note that in the case where they made their routing decisions based on the distribution of X t l p , we could write the cost of each link as: E " å l2r C lT (X t l p ;X t lT ) # = E " å l2r ˜ C lT (X t lT ) # (2.1) which is a function of X t lT only and where ˜ C lT (X t lT ) is defined as: ˜ C lT (X t lT )= E[C lT (X t l p ;X t lT )jX t lT ] (2.2) Note that C lT (X t l p ;X t lT ) is assumed to be a known nonlinear function which represents the operation cost of a truck driver who traverses road segment l during time interval t. It is assumed to be strictly increasing in both its arguments, has continuous first partial derivatives and¶C lT =¶X t lT is bounded away from zero. 15 Variable Meaning G=(V;L) The graph representing the transportation network N The number of discrete time intervals of a planning horizon v The number of OD pairs m The number of road segments d ¯ t j Demand of traveling in OD pair j and intended time interval ¯ t R j The set of possible routes in OD pair j X t l p Number of passenger vehicles traversing road segment l during time interval t X t lT Number of trucks traversing road segment l during time inter- val t a t;¯ t j;r The fraction of truck drivers with OD pair j and preferred de- parture time interval ¯ t who actually depart at time interval t and follow route r2 R j T p Total cost of passenger vehicle drivers ¯ T tr Total non-fee truck cost (operation cost + delay cost) T tr Total truck cost (operation cost + delay cost + fees) T s Total system cost (sum of total passengers and trucks cost) D(t; ¯ t) Delay cost of a truck driver who wanted to depart during time interval ¯ t but actually departed at t ¯ J M j;t;¯ t;r Operation cost of a truck driver in OD pair j following route r, intended departure time interval ¯ t and actual departure time interval t under mechanism M suggestions ¯ J M;D j;t;¯ t;r Total non-fee (operation + delay) cost of a truck driver ¯ A M;D j;¯ t Average non-fee cost of truck drivers in OD pair j and pre- ferred departure time interval ¯ t t j;t;¯ t;r Fees assigned by the mechanism to a truck driver in OD pair j following route r, intended departure time interval ¯ t and actual departure time interval t J M;D j;t;¯ t;r Total cost of a truck driver (non-fee cost + fees) A M;D j;¯ t Average total cost of truck drivers in OD pair j and preferred departure time interval ¯ t E[ ¯ J M;D i;k ] Expected non-fee cost of a truck driver who wanted to depart during the i time interval but actually announced the k time interval to the mechanism U A measure of unfairness of the mechanism J UE;D j;¯ t Expected cost of a truck driver in OD pair j and preferred de- parture time interval ¯ t at the UE F t;¯ t j;r Expected non-fee cost of a truck driver in OD pair j, preferred departure time interval ¯ t, actual departure time interval t who follows route r Table 2.1: Notation used. 16 Leta t;¯ t j;r be the fraction of truck drivers with OD pair j, intended departure time interval ¯ t, actual departure time interval t who follow route r2 R j . Then, the number of trucks in link l during time interval t can be expressed as: X t lT = v å j=1 N å ¯ t=1 å r2R j :l2r d ¯ t j a t;¯ t j;r (2.3) where v is the total number of OD pairs of the network, N is the total number of time intervals of the planning horizon and d ¯ t j is the demand of OD pair j during time interval ¯ t. In the following subsection, we consider the case where each truck driver makes individual routing decisions. This enables us to define the notion of the User Equilibrium and prove its existence for the network model used. 2.2.2 RoutingwithoutCoordination: UserEquilibrium In the absence of coordination, each truck driver selfishly routes herself/himself in an effort to minimize her/his own individual cost leading to the so called User Equilibrium (UE) case [116] where no user (truck driver) has an incentive to unilaterally change his/her route and departure time interval selections as such a change will lead to a higher individual cost. Let us first define: a =fa t;¯ t j;r : t; ¯ t= 1;:::;N; j= 1;:::;v;r2 R j g (2.4) Using (2.4), let us denote by F t;¯ t j;r (a) the expected cost of a truck driver in the OD pair j with intended departure time interval ¯ t, actual departure time interval t who follows route r. Then, F t;¯ t j;r (a) is given by the following equation: F t;¯ t j;r (a)= E " å l2r C lT (X t l p ;X t lT (a))+ D(t; ¯ t) # (2.5) where X t lT (a) is given by (2.3) and D(t; ¯ t) represents the delay cost of not departing at the desired time interval ¯ t. 17 The condition for an a to be a UE is that for every OD pair j, any desired departure time interval ¯ t and for any actual departure time interval t and choice of route r for which a t;¯ t j;r > 0, it holds: F t;¯ t j;r (a) F t 0 ;¯ t j;r 0 (a) (2.6) for any alternative route r 0 2 R j and any other departure time interval t 0 , where F t;¯ t j;r (a) is given by (2.5). The existence of a UE for the model defined in subsection A is proved by transforming the time dependent network into a static network as explained below. Network Transformation: Given the transportation network, we first consider N copies where each copy corresponds to the traffic condition of the network during a specific time interval. Then, for each OD pair j and each desired departure time ¯ t, we introduce an additional node O j;¯ t . Finally, we add a link connecting node O j;¯ t to the origin of the OD pair j having a constant cost D(t; ¯ t). Note that D(t; ¯ t)= 0 for t= ¯ t. In Figure 2.1, the network transformation described above has been applied to a network with 2 OD pairs where we have assumed that the planning horizon consists of 2 time intervals. Figure 2.1: The network transformation. Using this network transformation, the resulting network is no longer time dependent and hence, the proof of the existence of a UE is established by using the already existing results in [104] and [51]. 18 Having proved the existence of a UE, we know that each truck driver will minimize his/her own expected individual cost. However, it is well known (see e.g. [116]) that these individual routing decisions do not result to the minimum expected total system cost defined by the following equation: E[T s (a)]= E[T p (a)]+ E[ ¯ T tr (a)] (2.7) where E[T p (a)] represents the expected total cost (e.g. total travel time) for passenger vehicle drivers defined as: E[T p (a)]= E " N å t=1 m å l=1 X t l p C l p (X t l p ;X t lT (a)) # (2.8) where m represents the number of road segments of the transportation network and C l p (X t l p ;X t lT ) is a nonlinear function denoting the operation cost of each passenger vehicle traversing road segment l during time interval t. On the other hand, E[ ¯ T tr (a)] in (2.7) corresponds to the expected total non-fee truck cost (operation cost + delay cost) and can be written as: E[ ¯ T tr (a)]= E " N å t=1 m å l=1 X t lT C lT (X t l p ;X t lT (a)) # + E " v å j=1 N å t=1 N å ¯ t=1 å r2R j d ¯ t j a t;¯ t j;r D(t; ¯ t) # (2.9) In the following section, we will examine if and under what conditions we can design mechanisms which minimize the expected total system cost as given by (2.7), by appropriately choosing the decision variablea while concurrently creating individual incentives for voluntary participation of the truck drivers. 2.3 RoutingwithCoordination: MechanismDesign 2.3.1 Overview In this section we assume that all truck drivers submit their desired OD pair to a central coordinator who issues individual routes such that the expected total system cost given by (2.7) is minimized. Since the routes that minimize this cost do not correspond to optimum individual routes, some 19 truck drivers will benefit and some others will lose. As a result, there is not strong incentive for voluntaty participation. In order to address this issue, we design coordinated mechanisms to address the issue of minimizing the total system cost while making sure that no truck driver incurs a cost higher than his/her corresponding cost at the UE (i.e. without coordination). To achieve this, the mechanism involves the use of fees for those who benefit and payments to those that lose compared to the UE. The overall design of the mechanism consists of two optimization problems. In the first prob- lem, the coordinator of the mechanism calculates the optimum fractions a according to which he/she should distribute the truck drivers into the different routes. The calculation involves the solution of an optimization problem whose objective is the minimization of the expected total cost of the network while simultaneously the expected total non-fee truck cost is less than in the case where the truck drivers follow a UE. Note at this point that we use the UE as a benchmark for our design since we may expect that it is an optimistic version of the real world traffic conditions where all truck drivers have updated information of the traffic conditions through the use of nav- igation apps. This is an assumption that is verified numerically in [100] where it is shown that the travel time of the users of the simulated network of Los Angeles area is increased whenever a part of them is not routed and does not have updated information of the traffic conditions. Hence, enforcing our mechanism to provide a better solution than the UE, we may expect that this solution will be even better compared to the real world traffic conditions. Subsequently, after having calculated the fractionsa, we use them into the second optimization problem through which the coordinator calculates the optimum payments t by minimizing a de- fined measure of unfairness. Furthermore, the constraints of the second optimization problem are written in a way that are equivalent to requiring that the optimum paymentst lead to a truthful and budget balanced on average mechanism which is able to create individual incentives for voluntary participation of the truck drivers. Hence, sufficient conditions to guarantee the feasibility of the second optimization problem are derived. These conditions are written as linear functions of the fractionsa and thus, they can be added as constraints in the first optimization problem. However, 20 since these conditions happen to be only sufficient for the existence of monetary schemes with the desired criteria, we propose an algorithm which progressively adds as constraints in the first optimization problem only those conditions which appear to be necessary for guaranteeing the fea- sibility of the second, in an effort to maximize the efficiency of the overall mechanism and drive its solution as close as possible to the SO. 2.3.2 MainDesign In the previous section, we proved the existence of a UE in the traffic model used. Moreover, it is generally known that a traffic network may have multiple UEs. Therefore, in order to ensure that the designed mechanism is going to provide a lower cost than any UE, it is considered necessary to compare its efficiency with the UE presenting the lowest total truck cost. We consider a mechanism which promotes voluntary participation in which all the players (truck drivers) prefer the application of the mechanism compared to the particular UE. If the total truck cost under the mechanism suggestions is higher than in the UE case, then we may not expect voluntary participation. According to the aforementioned observations, let us define the following optimization problem: minimize a() E[T s (a)] subject to E[ ¯ T tr (a)] E T UE tr N å t=1 å r2R j a c;t;¯ t j;r = 1;8c; j; ¯ t a c;t;¯ t j;r 0 (2.10) where the index c is used in the fractions a in order to denote a particular realization c of the demand d ¯ t c; j . The cost E[T s (a)] is given by (2.7)-(2.9). Moreover, the first constraint of (2.10) guarantees that the expected total cost of the truck drivers will be less under the mechanism sug- gestions than in the UE, the second constraint guarantees that the proportions of truck drivers which are routed at every realization, every OD pair and every desired time interval ¯ t sum up to one while the third constraint guarantees that each fraction is going to be greater or equal to zero. Overall, 21 the optimization problem (2.10) is clearly feasible since a UE satisfies the constraints. In [51], the authors proved its convexity under the assumption that the functions C lT (X t l p ;) and C l p (X t l p ;) are convex. Note also that the solution of (2.10) is a function of the demand indicating that the coordi- nator of the mechanism is able to make a decision having full information of the traffic conditions i.e., knowing the exact realization of the demand d ¯ t c; j . The solution of the optimization problem (2.10) gives us the optimum fractions according to which the coordinator will route the truck drivers into the network. However, (2.10) can only cre- ate incentives for voluntary participation on the collective level, meaning that it can only guarantee that the expected total non-fee truck cost is lower for the routes generated by the coordinator than those generated individually based on the UE assumption. In order to create individual incentives, the mechanism should be able to guarantee that each individual truck driver is guaranteed to have a lower cost if he/she follows the mechanism suggestions than making independent routing de- cisions. Therefore, we need to find ways in order to take into account individual incentives. In addition, we need to consider the issue of truthfulness i.e, the mechanism should take into account the possibility of truck drivers manipulating the system and prevent it. Since the mechanism allows the truck drivers to declare their preferred departure time interval, it should guarantee that no player (truck driver) will have an incentive to declare a different time interval than the one he/she actually wants in order to gain a benefit. Finally, since the mecha- nism allows that some players pay a fee while some others may receive a payment as part of the incentive for participation and fairness, an important step in the overall design is to ensure that the mechanism will be at least weakly budget balanced on average i.e., the expected total payments made and received are greater or equal to zero. Before presenting the optimization problem through which the optimum payments are calcu- lated, let us first define the expected total benefits that the truck drivers earn under the mechanism suggestions compared to the UE expressed by: B(t)= E " v å j=1 N å t=1 N å ¯ t=1 å r2R j d ¯ t j ¯ a t;¯ t j;r (J UE;D j;¯ t J M;D j;t;¯ t;r (t)) # (2.11) 22 where J UE;D j;¯ t is the expected cost of a truck driver in OD pair j with preferred departure time interval ¯ t at the UE, J M;D j;t;¯ t;r (t) is the total cost of a truck driver (operation cost + delay cost + fees) in OD pair j following route r, intended departure time interval ¯ t and actual departure time interval t under mechanism M suggestions and the symbol “bar” abovea denotes the optimum fractions. Using (2.11), a measure of unfairness representing a weighted variance of the distribution of the benefits is defined as: U(t)= E " v å j=1 N å t=1 N å ¯ t=1 å r2R j d ¯ t j ¯ a t;¯ t j;r (J UE;D j;¯ t J M;D j;t;¯ t;r (t)) E[d ¯ t j ¯ A M;D j;¯ t ] E[d ¯ t j ]E[T M tr (t)] B(t) ! 2 # (2.12) where the term E[T M tr (t)] is the total expected cost of the truck drivers (operation cost + delay cost + fees) given by: E[T M tr (t)]= E " N å t=1 m å l=1 X t lT C lT (X t l p ;X t lT ( ¯ a)) # + E " v å j=1 N å t=1 N å ¯ t=1 å r2R j d ¯ t j ¯ a t;¯ t j;r (D(t; ¯ t)+t t;¯ t j;r ) # (2.13) and A M;D j;¯ t is the average cost (operation cost + delay cost + fees) of a truck driver in OD pair j and preferred departure time interval ¯ t. For more details on this measure of unfairness, see [51]. Taking under consideration the aforementioned definitions, we can formulate the following optimization problem: min. t() U(t) s.t. å c N å t=1 å r2R j p c ¯ a c;t;¯ t j;r t c; j;t ¯ t;r G j;¯ t ;8 j; ¯ t f T j;(i;k) t j;(i;k) E[ ¯ J M;D i;k ] E[ ¯ J M;D i;i ];8 j;i;k i f T j;(i;k) t j;(i;k) E[ ¯ J M;D k;i ] E[ ¯ J M;D k;k ];8 j;i;k i å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r t c; j;t;¯ t;r 0 (2.14) 23 where the function G j;¯ t and the inner product f T j;(i;k) t j;(i;k) are given by the following equations: G j;¯ t = å c N å t=1 å r2R j p c ¯ a c;t;¯ t j;r ( ¯ J M;c; j t;¯ t;r + D(t; ¯ t))+ A UE;D c; j;¯ t (2.15) f T j;(i;k) t j;(i;k) = å c N å t=1 å r2R j p c ¯ a c;t;i j;r t c; j;t;i;r ¯ a c;t;k j;r t c; j;t;k;r (2.16) respectively and p c expresses the probabilities of each possible realization of the demands. More- over, the quantity E[ ¯ J M;D i;k ] denotes the expected non-fee cost (operation cost + delay cost) of a truck driver with preferred departure time interval i, but declared departure time interval k. Therefore, the first constraint of (2.14) together with (2.15) express that the expected average cost of each player is going to be lower under the mechanism suggestions than in the UE, while the second and third constraints together with (2.16) express that no player will have an incentive to declare that he/she wants to make his/her trip during a different time interval than the one he/she actually wants since his/her expected cost is going to be higher. Last, the fourth constraint of (2.14) expresses that the expected total sum of the payments made and received by the mechanism is greater or equal to zero (weakly budget balanced on average mechanism). The optimum paymentst c; j;t ¯ t;r calculated by solving the optimization problem (2.14) will define, if it exists, a truthful and weakly budget balanced on average mechanism which is able to create in- dividual incentives for voluntary participation of the truck drivers. In order to address the question of the existence of such payment schemes, let us first define the following two inequalities: J UE;D j;i J UE;D j;k + å c N å t=1 å r2R j p c a c;t;k j;r (D(t;i) D(t;k));8 j;i;k i (2.17) J UE;D j;k J UE;D j;i + å c N å t=1 å r2R j p c a c;t;i j;r (D(t;k) D(t;i));8 j;i;k i (2.18) where J UE;D j;i expresses the expected cost of a truck driver traversing OD pair j with intended departure time interval i at the UE. Now, we are ready to state the following theorem. 24 Theorem2.1 There exists a truthful in equilibrium and weakly budget balanced on average mech- anism which creates individual incentives for voluntary participation of the truck drivers if (2.17) and (2.18) hold. Proof. See Appendix A.0.1. Theorem 2.1 proves that (2.17) and (2.18) are sufficient conditions to guarantee the feasibility of (2.14). However, observe that in (2.14), we have imposed the last constraint to be greater or equal to zero which only guarantees that the mechanism will be weakly budget balanced on average. Moreover, in this case, U(t) is a nonconvex function oft due to the fact that the variable t appears in the denominator through the term E[T M tr (t)] which is clearly undesirable. As it can be seen from the proof of Theorem 2.1, we could enforce (2.10) to provide us with a solution at which E[ ¯ T M tr ( ¯ a)]= E T UE tr and thus, we could guarantee the budget balance on average property. However, we expect that this conversion will reduce significantly the efficiency of our mechanism since it restricts the feasible space of (2.10) and thus, we will avoid it. Therefore, let us convert (2.14) into the following optimization problem: min. t() U(t) s.t. å c N å t=1 å r2R j p c ¯ a c;t;¯ t j;r t c; j;t ¯ t;r G j;¯ t ;8 j; ¯ t f T j;(i;k) t j;(i;k) E[ ¯ J M;D i;k ] E[ ¯ J M;D i;i ];8 j;i;k i f T j;(i;k) t j;(i;k) E[ ¯ J M;D k;i ] E[ ¯ J M;D k;k ];8 j;i;k i å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r t c; j;t;¯ t;r = 0 (2.19) where the only difference between (2.14) and (2.19) is that we have enforced the last constraint to hold as an equality (budget balance on average property). Theorem2.2 The optimization problem (2.19) is feasible if (2.17) and (2.18) hold. Additionally, it is a convex (quadratic) optimization problem. 25 Proof. See Appendix A.0.2. Theorem 2.2 proves that (2.17) and (2.18) are sufficient conditions to guarantee the existence of truthful in equilibrium and budget balanced on average mechanisms which are also able to create individual incentives for voluntary participation of the truck drivers. Furthermore, enforcing the budget balance on average property to hold, converts the problem of the calculation of the optimum payment schemes into a convex (quadratic) optimization problem. Remark 1: We need to mention at this point that in both (2.14) and in (2.19), we could convert the voluntary participation constraint to hold at each individual realization of the demands and without any need to average the cost of the truck drivers. Moreover, (2.17) and (2.18) would still remain sufficient conditions. However, even if these incentives appear to be stronger than the one provided, we avoided their use since the number of constraints increases significantly. Remark 2: The formulation of the optimization problem (2.19) where the payment schemes satisfy the budget balance on average property, is intended for a non-profit organization, a gov- ernment agency etc. On the other hand, in the case where a private company whose objective is the maximization of its earnings, decides to implement the above monetary schemes into the truck drivers of a transportation network, then by removing the last constraint of (2.19), conditions (2.17) and (2.18) guarantee the maximum possible earnings for the company while concurrently the derived payment schemes satisfy both the individual voluntary participation and the truthfulness criteria. More specifically, the company only has to calculate the optimum fractions ¯ a according to which it is going to distribute the trucks into the network ensuring that (2.17) and (2.18) hold. Then, the optimum payment schemes applied to the drivers will be given by (A.2). Furthermore, by the way of construction of the proof of Theorem 2.1, its earnings can be easily calculated to be E T UE tr E[ ¯ T tr ( ¯ a)]. It has become clear from the results of Theorem 2.1 and Theorem 2.2 that if we are able to ensure that (2.17) and (2.18) hold true, then we could design a mechanism satisfying the desired criteria. Observe that (2.17) and (2.18) are linear inequalities of the fractionsa and thus, we can 26 enforce (2.10) to provide us with a solution that will satisfy these conditions without an important effect on its complexity. To this end, let us modify (2.10) as follows: minimize a() E[T s (a)] subject to E[ ¯ T tr (a)] E T UE tr N å t=1 å r2R j a c;t;¯ t j;r = 1;8c; j; ¯ t a c;t;¯ t j;r 0 (2:17);(2:18) (2.20) where the difference between (2.10) and (2.20) is that we have added inequalities (2.17) and (2.18) as constraints. The question that needs to be answered at this point is whether (2.20) is feasible and the answer to this question comes from the following Lemma. Lemma2.3 The optimization problem (2.20) is feasible. Proof. See Appendix A.0.3. Lemma 2.3 together with the results of Theorem 2.1 and Theorem 2.2 allows us to design our mechanism. More specifically, the coordinator can solve the optimization problem (2.20) in order to calculate the optimum fractions ¯ a according to which he/she should distribute the truck drivers into the network. Then, he/she solves the optimization problem (2.19) in order to calculate the optimum payments t ? that will be made or received by the truck drivers. However, in what follows, we will present a modification of this algorithm which is going to improve the efficiency of the overall mechanism. Remark 3: At this point, the necessity of the derivation of (2.17) and (2.18) must be noted. As also mentioned before, (2.17) and (2.18) are sufficient conditions to guarantee the feasibility of (2.19). However, an intuitive question to ask is whether it is necessary to satisfy some specific conditions in order to guarantee the feasibility of (2.19) and the answer to this question is positive. To see this, observe that the second and the third constraints of (2.19) have opposite left-hand 27 side parts. Hence, a necessary condition in order for these two constraints to define a nonempty polyhedral set is to require that the sum of their right-hand sides is greater or equal to zero i.e., å c N å t=1 å r2R j p c (a c;t;k j;r a c;t;i j;r )(D(t;i) D(t;k)) 0;8 j;i;k i (2.21) It is straightforward to see that the sum of (2.17) and (2.18) satisfy (2.21). 2.3.3 AlgorithmicProcedureoftheMechanism Due to the way we construct the proof of Theorem 2.1, inequalities (2.17) and (2.18) are only sufficient conditions to guarantee the feasibility of (2.19). This means that (2.19) could also be feasible even in the case where (2.17) and (2.18) do not hold. Hence, if we solve (2.20) instead of (2.10) in an effort to guarantee the feasibility of (2.19), we may lose some of the efficiency of our mechanism. To this end, we further modify the algorithm to follow the steps mentioned below: Iteration1 Step 1: Solve the optimization problem (2.10) in order to calculate the optimum fractions ¯ a 1 . Step 2: Solve the optimization problem (2.19) in order to calculate the optimum payments t ? . If (2.19) is solved, STOP;The optimum solution of the mechanism is given by the pair( ¯ a 1 ;t ? ). Else, proceed to Step 3. Step 3: Chooset as given by (A.2) and substitute it to the second and the third constraints of (2.19). For those constraints that are violated after this substitution, take the corresponding inequalities which will be a subset of (2.17) and (2.18) and add them as constraints in the optimization prob- lem (2.10), call it (2.10.1). Proceed to the next iteration. Iterationk Step 1: Solve the optimization problem (2.10.k-1) in order to calculate the optimum fractions ¯ a k . Step 2: Solve the optimization problem (2.19) in order to calculate the optimum payments t ? . If 28 (2.19) is solved, STOP;The optimum solution of the mechanism is given by the pair( ¯ a k ;t ? ). Else, proceed to Step 3. Step 3: Chooset as given by (A.2) and substitute it to the subset of the second and the third con- straints of (2.19) which had never been violated in any of the previous k-1 iterations. For those constraints that are violated after this substitution, take the corresponding inequalities which will be a subset of (2.17) and (2.18) and add them as constraints in the optimization problem (2.10.k-1), call it (2.10.k). Proceed to the next iteration. A few comments regarding this algorithm follow. First, the proposed algorithm described above has the advantage that it only adds in (2.10) constraints of the form of (2.17) and (2.18) that are needed in order to guarantee the feasibility of (2.19) and thus, it is for sure that it will be at least as efficient as the one presented in the previous section. Moreover, if the algorithm needs to generate all the constraints of the form of (2.17) and (2.18), then in the last iteration, the feasibility of (2.19) will be guaranteed and the proposed mechanism will be given by the solutions obtained by successively solving the optimization problems (2.20) and (2.19). It should be also noted that letting v denote the number of OD pairs of the network and N denote the number of time intervals, then the maximum number of feasibility cuts that the algorithm is going to generate is 2v N! 2!(N2)! = vN(N 1). Even though this algorithm seems to be more computationally expensive than the one presented in the previous section, our simulation experiments show that most of the times, it converges in just one iteration. 2.4 SimulationResults In the first two subsections, we present the numerical methods we used in order to calculate the UE and the SO solutions while in the last two subsections, we apply the proposed mechanism into the Braess and the Sioux Falls networks and we make a comparison with the UE and the SO solutions in terms of the expected total system cost and the expected total truck cost. 29 2.4.1 NumericalcalculationoftheUserEquilibrium The UE which presents the minimum total truck cost can be calculated by solving the following nonlinear optimization problem with complementarity constraints [1]: minimize a E[T tr (a)] subject to 0a t;¯ t j;r ? F t;¯ t j;r (a)d t;¯ t j;r 0;8 j;t; ¯ t;r N å t=1 å r2R j a t;¯ t j;r = 1;8 j; ¯ t (2.22) where d t;¯ t j;r is a set of free variables and F t;¯ t j;r (a) is given by (2.5). The optimal objective value of (2.22) is denoted by E[T UE tr ] and will be used in the right hand side of the first constraint of (2.10). 2.4.2 NumericalcalculationoftheSystemOptimum The System Optimum (SO) solution is the total cost of the network if routing was centrally planned and is given from the solution of the following optimization problem: minimize a() E[T s (a)] subject to N å t=1 å r2R j a c;t;¯ t j;r = 1;8c; j; ¯ t a c;t;¯ t j;r 0 (2.23) 2.4.3 Braessnetwork In order to validate the theoretical results and show the performance of the proposed mechanism compared to the UE and the SO solutions, we ran some simulations in a small traffic network known as the Braess network [10] which is illustrated in Figure 2.2. This network consists of 2 OD pairs which have three and two possible routes, respectively. 30 Figure 2.2: The Braess network. For the simulations purpose, we considered two different scenarios. In the first scenario, we split the planning horizon into 2 time intervals and we let the demand to take one of the following 4 equiprobable values: d 1 = 2 6 4 3 2 2 1 3 7 5 ; d 2 = 2 6 4 3:4 2:5 2:9 4 3 7 5 ; d 3 = 2 6 4 2:8 5:1 3:7 3:8 3 7 5 ; d 4 = 2 6 4 4:3 3:7 3:3 4:6 3 7 5 where each column of d 1 ,d 2 ,d 3 and d 4 corresponds to an OD pair and each row corresponds to a particular time interval. The link cost functions C l p and C lT are given by the following relations: C 1p = C 1T = 1+(X 1p + X 1T )+(X 1p + X 1T ) 2 C 2p = C 2T = 1+(X 2p + X 2T )+(X 2p + X 2T ) 2 C 3p = C 3T = 0:5(X 3p + X 3T ) 2 C 4p = C 4T = 2+ 0:5(X 4p + X 4T ) 2 C 5p = C 5T =(X 5p + X 5T ) 2 while the number of passenger vehicles at each link of the network was assumed to be constant across all time intervals and possible demand realizations in order to simplify the simulations and was equal to(X 1p ;X 2p ;X 3p ;X 4p ;X 5p )=(4;4;1;3;2). 31 The delay cost function D(t; ¯ t) was considered to have the following form: D(t; ¯ t)=gjt ¯ tj (2.24) whereg > 0 is a constant which is assumed to be common for all the truck drivers and was chosen to be g = 0:8. Note that a delay cost function with the form of (2.24), also adopted in [120], indicates that a driver is equally harmed whether he/she departs earlier or later than his/her desired departure time interval ¯ t. The simulation results are summarized in Table 2.2. UE Mechanism SO E[T tr ] 591:6 584:5 584:5 E[T s ] 1447:3 1438:5 1438:5 Table 2.2: Simulation Results of the Braess network with 2 Time Intervals Table 2.2 shows that our mechanism was able to reach the same total truck cost and total system cost values as the SO solution. More specifically, the mechanism provided a total truck cost value 1:2% lower than the corresponding UE value. Moreover, the total system cost was also decreased by 0:6% compared to the UE solution. Note that during the simulation run, our algorithm did not need to add any feasibility cuts to the optimization problem (2.10) in order to guarantee the feasibility of (2.19) and thus, it terminated in just one iteration. For the second scenario, we ran a simulation in the same network by splitting the planning horizon into 6 time intervals. The variableg took the valueg = 0:5 and we let the demand to take one of the following 4 equiprobable values: d 1 T = 2 6 4 3 2 4:8 3:6 2:7 3:3 2 1 3:5 4:4 3:1 3:9 3 7 5 ; d 2 T = 2 6 4 3:4 2:9 2 3:5 4:7 3:2 2:5 4 3:6 3:5 3:1 4:1 3 7 5 d 3 T = 2 6 4 2:8 3:7 2:7 3:1 2:8 3:9 5:1 3:8 2:6 4:2 3:8 2:9 3 7 5 ; d 4 T = 2 6 4 4:3 3:3 2:4 3:4 3 3:4 3:7 4:6 3:6 3:9 2:5 3:2 3 7 5 32 where again each column of d 1 ,d 2 ,d 3 and d 4 corresponds to an OD pair and each row corresponds to a particular time interval. The number of passenger vehicles at each road segment as well as the link cost functions were assumed to take identical values as in the first scenario. The simulation results are summarized in Table 2.3. UE Mechanism SO E[T tr ] 1815:1 1753:1 1753:1 E[T s ] 4417:2 4341:9 4341:9 Table 2.3: Simulation Results of the Braess network with 6 Time Intervals The results presented in Table 2.3 show that the designed mechanism was able to reach the SO solution by decreasing the expected total truck cost and the expected total system cost by 3:4% and 1:7% respectively compared to the UE. Again, the proposed algorithm did not need to add any feasibility cuts and it terminated in just one iteration. 2.4.4 SiouxFallsnetwork In this section, the simulation results obtained from the application of the proposed mechanism in the Sioux Falls network [57] are presented. The Sioux Falls network consisting of 24 nodes and 76 links constitutes a benchmark problem in the transportation research field. In our simulation, we assume that the cost of each road segment corresponds to travel time and is given by a Bureau of Public Roads (BPR) function [88] of the following form: C t l p (X t l p ;X t lT )= C t lT (X t l p ;X t lT )= v a + v b X t l p + 3X t lT v k ! 4 (2.25) Furthermore, in order to retain computational tractability, we assumed that there are only 6 avail- able OD pairs for the truck drivers, namely j 1 = (n 1 ;n 7 ); j 2 = (n 1 ;n 11 ); j 3 = (n 10 ;n 11 ), j 4 = (n 10 ;n 20 ); j 5 =(n 15 ;n 5 ); j 6 =(n 24 ;n 10 ) and moreover, we assumed that for each OD pair, truck drivers will choose between the 10 least congested routes. The planning horizon of the simulation 33 model was split into two time intervals. The number of passenger vehicles at each link was con- sidered to remain constant in the two time intervals and the values v a ;v b ;v k were chosen similar to the ones adopted in [51]. The demands of the OD pairs at each time interval were chosen to take one of the following two equiprobable values: d= 8 > > < > > : d 1 w.p. 0:5 d 2 w.p. 0:5 where d 1 = 2 6 4 3 4:5 6 3 14 3:6 1 2:8 5:4 7 9 2 3 7 5 ; d 2 = 2 6 4 5 1:8 3:9 15 6:4 2:4 6 5:5 1:8 6:5 11 6 3 7 5 where each column of d 1 and d 2 corresponds to an OD pair and each row corresponds to a particular time interval. The simulation results are summarized in Table 2.4. UE Mechanism SO E[T tr ] 15100:3 13372:1 13372:1 E[T s ] 83924:8 79245:2 79245:2 Table 2.4: Simulation Results of the Sioux Falls network Table 2.4 shows that our mechanism reached the same total truck cost and total system cost values as the SO solution. More specifically, it provided a total truck cost value 11:4% lower than the corresponding UE value while concurrently reducing the total system cost by 5:6% compared to the UE solution. Note also that during this simulation run, the proposed algorithm terminated again in just one iteration. 34 Chapter3 PersonalizedPareto-ImprovingPricing-and-RoutingSchemes forHeterogeneousUsers 3.1 Introduction Measuring the contribution to the United States (U.S.) economy as the share of all expenditures in transportation-related final goods and services, the transportation sector contributed 8:9% to U.S. Gross Domestic Product (GDP) [15] while in the European Union (EU) it accounts for almost 5% of the GDP [37]. In EU, road transport has the largest share of EU freight transport accounting for 76.7% of the total inland freight transport [39]. Hooper found that the trucking industry expe- rienced nearly 1.2 billion hours of delay on the National Highway System (NHS) of the U.S. as a result of traffic congestion making the operational costs incurred by the trucking industry due to traffic congestion to be $74:5 billion per year [5]. These statistics demonstrate that an optimized routing system is essential and could significantly contribute to the global economy. Drivers usually make their routing decisions using GPS routing apps in an effort to minimize their individual travel time or cost objective. This phenomenon is known as User Equilbrium (UE) or the first Wardrop Principle [116]. However, it is known that UE deviates from an optimized road usage [7, 80] and it is a sub-optimal behavior compared to the socially optimum policy that could be achieved through a centrally coordinated system [124]. Recent studies [71, 127] estimated the Price Of Anarchy (POA) [54], i.e. the inefficiency between a selfish routing strategy and a 35 system optimum policy in realistic transportation networks using real traffic data, demonstrating the necessity for its reduction. Based on the idea of Connected Automated Vehicles (CA Vs) [83, 129], Zhang et al. proposed to reduce the POA by recommending socially optimum routes to all drivers [127]. However, such a strategy would raise several fairness and equity issues since in a System Optimum (SO) solution, some drivers may benefit while some others may be harmed compared to the UE. One of the most common techniques addressing the problem of the inefficiency between the UE and the SO solutions is congestion pricing [82, 110, 7, 80] where each driver is assigned a fee corresponding to the additional cost his/her presence causes to the network. Several other works have studied congestion pricing under user heterogeneity in VOT, e.g. [121, 123], the problem of management of the revenue collected from the application of congestion pricing [46, 94] and the impact of congestion pricing schemes on emissions of freight transport [19]. London [56], Stockholm [36], Singapore and Milan [58] are some of the cities that have already introduced congestion pricing, while recent studies [6, 20] also explore the benefits from applying congestion pricing to more major cities. Recently, there is also a growing research interest for studies related to pricing schemes in the presence of autonomous vehicles [55, 69, 92, 103]. Another well studied set of strategies addressing the problem of the inefficiency between an equilibrium flow pattern and the SO are the applications of Tradable Credit Schemes (TCS) [61, 122] or tradable travel permits [111, 11, 4] among the drivers of the network. In this case, a central coordinator initially distributes a certain number of credits (or permits) to all eligible drivers and free credit (or permit) trading is allowed among travelers. Wang et al. [115] and Zhu et al. [132] studied the application of TCS under user heterogeneity in VOT, while Wang et al. studied OD-based travel permits in the presence of heterogeneous users [113]. Recently, Xiao et al. studied a Cyclic Tradable Credit Scheme (CTCS), where the credits never expire but circulate within the system, and derived a sufficient condition for the existence of a Pareto-improving CTCS in a general network [119]. For a more comprehensive review of credit- and permit-based schemes, we refer the interested reader to [59]. 36 In this Chapter, we address the problem of the inefficiency of an equilibrium flow pattern by studying pricing schemes under a centrally coordinated freight routing system that can alleviate traffic congestion and drive the network as close as possible to a SO solution. We focus our study on pricing-and-routing schemes that can be specifically applied on trucks. Given that truck drivers routinely use varying routes for the same journey depending on the traffic conditions [51] and the fact that their travel time is already a commodity, make trucks form an ideal candidate subclass of vehicles for coordinated routing. To this end, we consider a non-atomic game theoretic model whose users are the truck drivers and their demand is assumed to be stochastic. In the case where the planning horizon is split into discrete non-overlapping time intervals and the drivers choose both their OD pair as well as their desired departure time interval, Papadopoulos et al. derived sufficient conditions for the existence of revenue-neutral (budget balanced) and Pareto- improving pricing schemes that can additionally provide individual incentives to the drivers to truthfully declare their desired departure time [75, 76]. In this work, we take into account the user heterogeneity in the VOT. For the single OD case, using a bottleneck model [110] and assuming two classes of users with distinct VOT, Sun et al. explored the possibility of adopting the instrument of incentives to shift commuters’ departure times in a single morning bottleneck situation [96]. For the fixed demand case, [46] derived sufficient conditions for the existence of Pareto-improving and revenue-neutral pricing schemes. However, since they could not find a way to identify the VOT of each user, they proposed class-anonymous pricing schemes based on the idea of Congestion Pricing with Uniform Revenue Refunding (CPURR). Zheng and Geroliminis [130] and Zheng et al. [131] argued that VOT-based pricing schemes can increase the feasibility of implementation since they take into account the vulnerable user groups. However, most of the existing literature, e.g. [64, 101], makes assumptions about the distribution that the VOT of the drivers might follow and to the best of our knowledge, no self- reporting scheme where the users directly report their VOT to a central authority has been previ- ously proposed. Note that under such a scheme, it would be important to provide incentives to the users to truthfully report their VOT in order to avoid the exploitability of the mechanism. This is 37 mainly because many users would be willing to declare a high VOT in order to be assigned to the fastest possible route. In this work, we design a coordination mechanism for the truck drivers where the central coordinator asks the users to declare their desired OD pair and additionally pick their VOT from a set of N available options. Under this structure, we prove the existence of revenue- neutral and Pareto-improving pricing schemes in the sense that the expected cost of the users at the time they make their decision is less than or equal to their corresponding cost at the UE, that can additionally provide incentives to the drivers to truthfully declare their VOT. This additional information enables us to design personalized (VOT-based) pricing-and-routing schemes. More specifically, we propose an Optimum Pricing Scheme (OPS) that can be calculated by solving a nonconvex optimization problem. To reduce the computational time needed to calculate the OPS, we propose a second pricing-and-routing scheme called Approximately Optimum Pricing Scheme (AOPS) and we prove that it satisfies the desired properties. The simulation experiments demon- strate that both OPS and AOPS provide a much lower expected total travel time and expected total monetary cost to the users compared to the CPURR scheme, while concurrently approaching the SO solution. 3.2 ProblemFormulation The notation used in this Chapter is shown in Table 3.1. Let G=(V;L) denote a transportation network, where V is the set of nodes and L is the set of links in the network. Let C lT (X l p ;X lT (a)) be a known nonlinear function representing the travel time of a truck driver traversing road segment l when there exist X l p passenger vehicles and X lT (a) trucks on it, where a is a set of variables defined as follows: a =fa j w;r : w= 1;:::;N; j= 1;:::;v;r2 R j g (3.1) where j is the index corresponding to a specific Origin-Destination (OD) pair, w is the index corresponding to a class of users with VOT s w , r2 R j denotes a specific route among the set of available routes R j connecting OD pair j, N is the number of distinct classes of users and v is 38 the number of OD pairs in the network. Therefore, a j w;r expresses the proportion of truck drivers belonging to class w with a desired OD pair j who choose route r for their trip. Additionally, we assume that the OD demand of the truck drivers is stochastic and follows a probability distribution with finite support. Let d j;w be random variables denoting the number of truck drivers belonging to the class w with desired OD pair j and let d c j;w be their corresponding values during the demand realization c. Then, the number of trucks traversing the road segment l is given by: X lT (a)= v å j=1 N å w=1 å r2R j :l2r d c j;w a j w;r (3.2) where on the left side of (3.2), we omitted the index c to simplify the notation. We consider a model with a continuum of users. We assume that truck drivers know the number of passenger vehicles at each road segment of the transportation network. In case they do not possess this information and they make their routing decisions based on the probability distribution of X l p , we can write the travel time of each road segment l as: E " å l2r C lT (X l p ;X lT (a)) # = E " å l2r E h C lT (X l p ;X lT (a))jX lT i # (3.3) which is a function of X lT only, leading to a similar analysis. Note that in (3.3), the inner ex- pectation is taken with respect to X l p and the outer expectation is taken with respect to X lT . For the rest of the analysis, we assume that the number of passenger vehicles in the network X l p is a deterministic quantity and is known by the truck drivers. Additionally, we assume that the truck drivers know the probability distribution of the OD demand for the rest of the truck drivers but not the exact realization of the demand. This is a symmetric information model since each truck driver has the same amount of information. A similar model was also used in [51]. In this Chapter, we extend this model by considering user heterogeneity in VOT of the truck drivers. The above formulation is used in subsequent sections to study the User Equilibrium (UE) and System Optimum (SO) flow patterns as well as the existence of Pareto-improving pricing-and- routing schemes. 39 Variable Meaning G The transportation network as a graph V Set of nodes in the network L Set of links in the network R j Set of available routes connecting OD pair j m Number of road segments in the network d c j;w Demand of truck drivers belonging to the class w with desired OD pair j during the demand realization c a c; j w;r Proportion of truck drivers belonging to class w with desired OD pair j who follow route r during the demand realization c s w Value of Time (VOT) of a truck driver belonging to class w X l p Number of passenger vehicles in the road segment l X lT Number of trucks in road segment l C lT Travel time of a truck driver traversing road segment l F w j;r Expected travel time of a truck driver of class w with desired OD pair j who follows route r E[T tr ] Expected total travel time of the truck drivers in the network E[T mon tr ] Expected total monetary cost of the truck drivers p c Probability of the demand realization c a UE j;w;r Proportion of truck drivers belonging to class w with a desired OD pair j who follow route r at the UE J UE c; j;w;r Travel time of a truck driver belonging to class w with an OD pair j who follows route r during the demand realization c at the UE E[T p ] Expected total travel time of the passenger vehicles E[T s ] Expected total travel time of the network l Weighting factor of the objective function m Weighting factor of the objective function N Number of classes with different VOT J M;c; j w;r Travel time of a truck driver belonging to class w with OD pair j who follows route r during the demand realization c under the mechanism suggestions M p c; j w;r Payments of truck drivers of class w with desired OD pair j who follow route r during the demand realization c A UE c; j Average travel time of a truck driver with OD pair j during the demand realization c at the UE F CP j;w;r Expected total cost (travel time + payments expressed in time units) of a truck driver belonging to class w with desired OD pair j who follows route r under the CPURR scheme J CP c; j;w;r Travel time of a truck driver belonging to class w with OD pair j who follows route r during the demand realization c under the CPURR scheme Table 3.1: Notation used. 40 3.2.1 UserEquilibrium(UE) In the absence of pricing schemes, the drivers are trying to minimize their own individual travel time, e.g. through the usage of GPS routing apps. This behavior drives the network to a state called User Equilibrium [116] where no driver has an incentive to unilaterally change his/her routing decision since he/she is not going to benefit from such a change. In a transportation network with heterogeneous users, the equilibrium conditions can be either calculated in time units or in cost units [46]. Since the equilibrium conditions expressed in cost units can be obtained by multiplying each class’ travel time by its corresponding VOT, we for- mulate the UE problem in time units without any loss of generality. Therefore, let F w j;r (a) be the expected travel time of a truck driver with VOT belonging to the class w, travelling in OD pair j and following route r. Then, F w j;r (a) is given by: F w j;r (a)= E " å l2r C lT (X l p ;X lT (a)) # (3.4) where X lT (a) is given by (3.2). Note that in a UE solution, it holds that F w j;r (a)= F w 0 j;r (a);8w6= w 0 , i.e. the equilibrium travel time is identical for all user classes between the same OD pair. Additionally, in an equilibrium condition, it holds that: a j w;r > 0 =) F w j;r (a) F w j;r 0(a);8r 0 6= r (3.5) where r;r 0 2 R j . Inequality (3.5) states that in an equilibrium condition, drivers are choosing the route r that minimizes their individual expected travel time. 41 It has been shown by [51] that there are possibly many non-equivalent UE solutions. In this work, we calculate an equilibrium solution by solving an optimization problem with complemen- tarity constraints [40] which is a nonconvex optimization problem. Before formulating the prob- lem, let us first define the expected total travel time of the truck drivers in the network as: E[T tr (a)]= E " m å l=1 X lT (a)C lT (X l p ;X lT (a)) # (3.6) where m is the number of road segments in the transportation network and X lT (a) is given by (3.2). Under the assumption that the demand of the truck drivers follows a probability distribution with finite support, we can define the expected total monetary cost of the truck drivers in the network as: E[T mon tr (a)]= å c v å j=1 N å w=1 å r2R j p c d c j;w a UE j;w;r s w J UE c; j;w;r (a) (3.7) where c and p c correspond to a specific realization of the demand d c j;w and its associated probability, respectively. Note that since we assumed that the demand of the truck drivers follows a probability distribution with finite support, as it is common in probability theory [45], we use the term ‘demand realization’ to describe the observed value that the demand of the truck drivers takes. Moreover, a UE j;w;r is the proportion of truck drivers belonging to class w with a desired OD pair j who follow route r at the UE, s w is the VOT of the class w and J UE c; j;w;r is the travel time of a truck driver with an OD pair j who follows route r during the demand realization c at the UE and is given by the following equation: J UE c; j;w;r (a)= å l2r C lT (X l p ;X lT (a)) (3.8) where X lT (a) is given by (3.2). Using (3.8), we can rewrite (3.4) as F w j;r (a)= E[J UE c; j;w;r (a)]. Note that in a UE solution, it holds that J UE c; j;w;r (a)= J UE c; j;w 0 ;r (a);8w6= w 0 . At the UE, drivers make their own independent routing decisions. Therefore, in our formula- tion, given the assumption that truck drivers only know the probability distribution of the demand for the rest of the truck drivers and not the exact realization of it, their routing decisions a UE j;w;r 42 do not depend on the exact demand realization c. Given the aforementioned definitions, we can formulate the optimization problem through which we can calculate a UE solution as follows: minimize a;z lE[T tr (a)]+(1l)E[T mon tr (a)] subject to 0a j w;r ? F w j;r (a)z j w 0;8 j;w;r å r2R j a j w;r = 1;8 j;w (3.9) where z j w is a set of free variables that are used in order to solve the equilibrium optimization problem (3.9) and F w j;r (a) is given by (3.4). Additionally, the notation? means that eithera j w;r = 0 or F w j;r (a)z j w = 0 and finally, l is a weighting factor such that l2[0;1]. Therefore, in the equilibrium optimization problem (3.9), among the possibly nonequivalent UE solutions, we are looking for the one that minimizes a weighted combination of the expected total travel time and the expected total monetary cost of the truck drivers. Setting l = 1 in the objective function of (3.9), we can calculate the UE with the minimum expected total travel time. Viewing the expected total travel time of the truck drivers as a uniformly weighted expected total cost and given the fact that E[T mon tr (a)] is equal to the expected total travel time of the truck drivers weighted by the corresponding VOT of each class w, the overall objective of (3.9) can be expressed in cost units. The reasoning behind choosing the UE solution which minimizes the objective function of (3.9) is the following. First, as also mentioned in [46], E[T tr (a)] has long been accepted as a standard index of system performance in a transportation context while E[T mon tr (a)] is a more appropriate system measure from an economic viewpoint. Second, we use the solution of (3.9) as a benchmark for designing Pareto-improving pricing-and-routing schemes. Note that in order to create a Pareto-improving pricing scheme, i.e. a pricing scheme that can make everyone better-off compared to the UE, we first need to guarantee that the expected total travel time and the expected total monetary cost of the truck drivers using the proposed pricing scheme are lower than their best possible corresponding values at the UE. 43 Recently, to study how close a real traffic scenario is to a UE, for the static traffic assignment problem [78], Cabannes et al. defined the average marginal regret as the expected time-saving drivers have in the network if they change their path to an optimal one [17]. They proved that as the number of routing apps used is increased, the observed traffic assignment converges to a UE. The simulation results using real data for the whole Los Angeles network showed that the minimum travel time of a driver can be achieved whenever the ratio of GPS routing app users reaches 100%. In this case, the network converges to the UE. In this work, we use the UE as a benchmark for our design since we may expect that it is an optimistic version of the real world traffic conditions where all truck drivers have updated information of the traffic conditions through the use of GPS routing apps. 3.2.2 SystemOptimum(SO) In a System Optimum (SO) solution, drivers are making routing decisions in a manner that con- tributes to the minimization of a social cost compared to the UE where they minimize their own individual travel time. Letting E[T p (a)] denote the expected total travel time of the passenger vehicles in the network, we define the expected total travel time of the network as: E[T s (a)]= E[T p (a)]+ E[T tr (a)] (3.10) Note that in a SO solution, the routing decisions of the truck drivers depend on the exact realization of the OD demands and therefore, we define the expected total monetary cost of the truck drivers as: E[T mon tr (a)]= å c v å j=1 N å w=1 å r2R j p c d c j;w a c; j w;r s w J c; j;w;r (a) (3.11) 44 where the main difference between (3.7) and (3.11) is the fact that in (3.11), the routing decisions of the truck driversa c; j w;r depend on the demand realization c. Note that J c; j;w;r has a similar definition as in (3.8), i.e.: J c; j;w;r (a)= å l2r C lT (X l p ;X lT (a)) (3.12) where X lT (a) is given by: X lT (a)= v å j=1 N å w=1 å r2R j :l2r d c j;w a c; j w;r (3.13) where the main difference between (3.13) and (3.2) is the fact that in (3.13), the routing decisions of the truck driversa c; j w;r depend on the demand realization c. Since J c; j;w;r (a) expresses travel time, it holds that J c; j;w;r (a)= J c; j;w 0 ;r (a);8w6= w 0 . Using the aforementioned definitions, we calculate the SO solution of the network by solving the following optimization problem: minimize a() l(mE[T tr (a)]+(1m)E[T p (a)])+(1l)E[T mon tr (a)] subject to å r2R j a c; j w;r = 1;8c; j;w a c; j w;r 0;8c; j;w;r (3.14) where l;m2[0;1]. In (3.14), we minimize a weighted combination of the expected total travel time of the truck drivers, their expected total monetary cost and the expected total travel time of the passenger vehicles. The reasoning behind the selection of this objective function is that even though we are providing routing suggestions only to the truck drivers, simultaneously, we want to make sure that the passenger vehicles will not be significantly affected. The parameter m can be used to adjust the weight put on each category of vehicles. In all of the experiments, we set m = 0:9. 45 3.3 Pricing-and-RoutingSchemes. In a UE solution, every driver makes his/her own individual routing decisions which leads to an inefficient road usage. On the other hand, in a SO solution, some drivers may benefit while some others may be harmed compared to the UE solution, providing no incentives to drivers to follow the SO solution in practice. In this section we study pricing-and-routing schemes that are Pareto- improving, i.e. they can make every user better-off compared to the UE while at the same time, they can drive the network as close as possible to the SO solution. Note that even though the proposed schemes are pricing-and-routing schemes, we often refer to them as pricing schemes. We design a coordination mechanism that can be applied to truck drivers taking into account the user heterogeneity in their VOT. More specifically, the coordinator asks the truck drivers to declare their desired OD pair and additionally choose their VOT from a set of N available options. We assume that a monitoring system is in place and thus the truck drivers may not lie about their declared OD pair. This assumption is not restrictive thanks to the advancements in GPS tech- nologies. Additionally, we assume that GPS tracking is used to make sure that the truck drivers obey the routing suggestions. After collecting this information, the coordinator provides routing suggestions and additionally designs pricing schemes that are Pareto-improving and guarantee that every driver will have an incentive to truthfully declare his/her VOT while concurrently leading to a revenue-neutral (budget balanced) on average mechanism. This is in contrast with the previous literature studying pricing schemes, e.g. [64, 101], that makes assumptions about the distribution that the user heterogeneity might follow. We should note that it is important to guarantee that a user will truthfully declare his/her VOT in order to avoid the exploitability of the designed mechanism. This is mainly because many truck drivers would be willing to declare a high VOT in order to be assigned to the fastest possible route. In the next two subsections, we design pricing-and-routing schemes that mathematically satisfy the property of truthfulness. In the following sections, we propose two pricing-and-routing schemes, the Optimum Pric- ing Scheme (OPS) and the Approximately Optimum Pricing Scheme (AOPS). In both of these schemes, after asking the drivers to declare their OD pair j and pick their VOT w from a set of N 46 available options, we solve an optimization problem, we calculate a way to route the drivers into the network as well as the corresponding pricing scheme and finally, we randomly assign the truck drivers who declared the same OD pair j and class w into routes r. 3.3.1 OptimumPricingScheme(OPS) Letp c; j w;r be the payment (made or received) by a truck driver belonging to the class w with an OD pair j who follows route r during demand realization c. We calculate the optimum way to route the truck driversa as well as the the optimum pricing schemep by solving the following nonconvex optimization problem: minimize a();p() l(mE[T tr (a)]+(1m)E[T p (a)])+(1l)E[T mon tr (a)] subject to å c å r2R j p c a c; j w;r (J M;c; j w;r + 1 s w p c; j w;r ) å c p c A UE c; j ;8 j;w å c å r2R j p c a c; j i;r (J M;c; j i;r + 1 s i p c; j i;r ) å c å r2R j p c a c; j k;r (J M;c; j k;r + 1 s i p c; j k;r );8 j;i;k å c v å j=1 N å w=1 å r2R j p c d w c; j a c; j w;r p c; j w;r = 0 å r2R j a c; j w;r = 1;8c; j;w a c; j w;r 0;8c; j;w;r (3.15) where J M;c; j w;r is the travel time of a truck driver belonging to class w with a desired OD pair j who follows route r during the demand realization c under the mechanism routing suggestions M and is defined according to (3.12). Furthermore, A UE c; j is the average travel time of a truck driver with OD pair j during the demand realization c at the UE and is given by the following equation: A UE c; j = å r2R j a UE j;w;r J UE c; j;w;r (3.16) 47 Note that in (3.16), we omit the index w from A UE c; j since at the UE, the average travel times of truck drivers belonging to different classes of VOT are identical. Additionally, it holds that å c p c A UE c; j =å c p c J UE c; j;w;r ;8 j;w;r. Based on the aforementioned definitions, the first constraint of (3.15) guarantees that the expected cost (in time units) of the drivers at the time they make their decision is less than or equal to their corresponding cost at the UE (Pareto-improvement). Note that this definition of the Pareto-improvement property is similar to the one used in [46]. In our case, we extend this definition to the case where the demand is stochastic. The second constraint of (3.15) guarantees that a truck driver which belongs to class i and truthfully declares class i to the coordinator will be better-off on average compared to the case where he/she originally belongs to class i but declares class k to the coordinator. Therefore, under the assumption that truck drivers are rational and will be constantly seeking to minimize their individual aggregated time (travel time + payments in time units), the second constraint of (3.15) guarantees that every user will have an incentive to truthfully declare his/her VOT. Last, the third constraint of (3.15) guarantees that the expected total payments made and received by the coordinator are equal to zero and therefore, the resulting mechanism satisfies the budget balanced on average property. Note that in the third constraint of (3.15), we implicitly assume that the coordinator incurs no cost from operating the pricing-and-routing mechanism. This is a common assumption both in a game-theoretic context [91] and in a transportation context, e.g. [46]. At this point, note that the UE solution, where no pricing scheme is applied to the users, satisfies the constraints of (3.15) and therefore, a solution to (3.15) always exists. Hence, the desired properties hold regardless of the objective function that we choose to minimize in (3.9). At this point, let us comment on the Pareto-improvement and the truthfulness properties of OPS. As far as it concerns the Pareto-improvement property, as can be seen from the first constraint of (3.15), on the left side of the inequality, we calculate the average over the routes r and the expectation over the different demand realizations c. Theoretically, it is possible that for some demand realizations, some drivers are given a route with a higher total travel time (travel time + payments expressed in time units) compared to the average travel time at the UE. However, due 48 to the fact that individual drivers only know the probability distribution of the demand and not the exact realization of it and hence they have incomplete information of the traffic conditions, they will be willing to participate in OPS since at the time they make their decision, their expected cost will be lower under the mechanism routing suggestions M than in the UE. Additionally, if users repeatedly participate in OPS, then using randomization in order to assign the truck drivers who declared the same OD pair j and class w into routes r, the first constraint of (3.15) guarantees that every truck driver will be better-off compared to the UE (Pareto-improvement). In Appendix B, we prove the existence of pricing-and-routing schemes that guarantee an even stronger version of the Pareto-improvement property. More specifically, we prove that it is possible to design a scheme that guarantees that at every demand realization c and for every route r that a driver might be assigned to, his/her total travel time (travel time + payments expressed in time units) is going to be lower under the mechanism suggestions compared to the UE. As far as it concerns the truthfulness property of OPS, as can be seen from the second constraint of (3.15), we calculate the average over the routes r and the expectation over the different demand realizations c. Again, using the fact that individual drivers only know the probability distribution of the demand and not the exact realization of it and hence they have incomplete information of the traffic conditions, it is guaranteed that they will be willing to truthfully declare their VOT since at the time they make their decision, their expected cost in the case they are truthful is lower than their corresponding cost in the case where they declared a different VOT than their actual one. In Appendix B, we prove the existence of pricing-and-routing schemes that guarantee an even stronger version of the truthfulness property since we force the second constraint of (3.15) to hold for every demand realization c. Even though the schemes presented in Appendix B can guarantee even stronger versions of the Pareto-improvement and the truthfulness properties, we decided not to include them in our analysis for three main reasons. First, OPS is still sufficient to guarantee that the drivers will have an incentive to participate in the mechanism and truthfully declare their VOT since at the time they make their decision, their expected cost in the case they are truthful is lower than their 49 corresponding cost in the case where they declared a different VOT than their actual one. Second, by forcing the second constraint of (3.15) to hold for every possible demand realization c, the number of constraints would increase and the final scheme would become less computationally efficient. Finally, one can expect that the optimum solution would be less efficient since the size of the feasible region over which we optimize would be smaller. In order to reduce the dimensionality of (3.15), in the following subsection, we present an Approximately Optimum Pricing Scheme (AOPS) and show that we can assign routes to the drivers so that the proposed pricing scheme meets the desired goals. 3.3.2 ApproximatelyOptimumPricingScheme(AOPS) For a given routing decisiona, let us define the following pricing scheme: p AOPS c; j;w;r = s w (A UE c; j J M;c; j w;r )+ s w å N l=1 s l E T mon;M tr E T mon;UE tr å v j=1 d w c; j (3.17) The pricing scheme given by (3.17) initially makes each driver pay (or receive a payment) such that his/her travel time under the mechanism routing suggestions J M;c; j w;r becomes equal to his/her average travel time at the UE A UE c; j . Then, after calculating the expected total monetary benefits of the truck drivers E T mon;M tr E T mon;UE tr obtained from the application of the mechanism, it distributes those benefits to the different classes proportionally to the VOT that each class has. Finally, each class benefits are uniformly shared among the truck drivers of the class. 50 Let us now formulate the following optimization problem: minimize a() l(mE[T tr (a)]+(1m)E[T p (a)])+(1l)E[T mon tr (a)] subject to E[T mon tr (a)] E T mon;UE tr H j i;k (a) N j i;k (a);8 j;i;k å r2R j a c; j w;r = 1;8c; j;w a c; j w;r 0;8c; j;w;r (3.18) where E T mon;UE tr is the expected total monetary cost of the truck drivers at the UE and H j i;k (a) and N j i;k (a) are given by the following equations: H j i;k (a)= 1 s k s i å c p c A UE c; j + 1 å N w=1 s w å c p c E[T mon;M tr ] E T mon;UE tr å v j=1 d i c; j (3.19) N j i;k (a)= 1 s k s i å c å r2R j p c a c; j k;r J M;c; j k;r + s k s i 1 å N w=1 s w å c p c E[T mon;M tr ] E T mon;UE tr å v j=1 d k c; j (3.20) Note that a solution to the optimization problem (3.18) always exists since the UE satisfies all of its constraints. Let us call the optimum solution of the optimization problem described by (3.18)- (3.20) asa AOPS . Now, we are ready to state the following theorem. Theorem3.1 The pair (a AOPS , p AOPS c; j;w;r ) makes the expected cost of the drivers at the time they make their decision to be less than or equal to their corresponding cost at the UE, guarantees that every user will have an incentive to truthfully declare his/her VOT and leads to a budget balanced on average mechanism. Proof. To prove the statement of the theorem, we equivalently prove thatp AOPS c; j;w;r is Pareto-improving, guarantees that every user will have an incentive to truthfully declare his/her VOT and creates a budget balanced on average mechanism if and only if the first and the second constraint of (3.18) 51 together with (3.19)-(3.20) hold. Note that a user will be better-off compared to the UE if the first constraint of (3.15) holds. Therefore, substituting (3.17) into the first constraint of (3.15), we get: å c å r2R j p c a c; j w;r J M;c; j w;r + A UE c; j J M;c; j w;r + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d w c; j å c p c A UE c; j , , 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å c p c å v j=1 d w c; j 0 which holds true if and only if E[T mon tr (a)] E T mon;UE tr which is equivalent to the first constraint of (3.18). Additionally, a user will have an incentive to truthfully declare his/her VOT if the second constraint of (3.15) holds. Therefore, substituting (3.17) into the second constraint of (3.15), we get: å c å r2R j p c a c; j i;r J M;c; j i;r + A UE c; j J M;c; j i;r + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d i c; j å c å r2R j p c a c; j k;r J M;c; j k;r + s k s i A UE c; j J M;c; j k;r + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d k c; j , , 1 s k s i å c p c A UE c; j + 1 å N l=1 s l å c p c E[T mon;M tr ] E T mon;UE tr å v j=1 d i c; j 1 s k s i å c å r2R j p c a c; j k;r J M;c; j k;r + s k s i 1 å N l=1 s l å c p c E[T mon;M tr ] E T mon;UE tr å v j=1 d k c; j where the last inequality is equivalent to the second constraint of (3.18). Last, a mechanism is budget balanced on average if the third constraint of (3.15) holds. Substituting (3.17) into the third constraint of (3.15), we get: å c v å j=1 N å w=1 å r2R j p c d w c; j a c; j w;r s w (A UE c; j J M;c; j w;r )+ s w å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d w c; j = E T mon;UE tr E T mon;M tr + å c v å j=1 N å w=1 p c d w c; j s w å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d w c; j å r2R j a c; j w;r 52 =E T mon;UE tr E T mon;M tr + E T mon;M tr E T mon;UE tr å c p c 1 å N l=1 s l N å w=1 s w 1 å v j=1 d w c; j v å j=1 d w c; j = E T mon;UE tr E T mon;M tr + E T mon;M tr E T mon;UE tr = 0 where in the second equality, we used the feasibility constraint which is given by the third con- straint of (3.18). Since the UE satisfies the constraints of (3.18), a solution to (3.18) always exists. Therefore, we have proved that by solving the optimization problem described by (3.18)-(3.20), we can calculatea AOPS such that the pricing schemep AOPS c; j;w;r given by (3.17) satisfies all the statements of Theorem 3.1 and this concludes the proof. Theorem 3.1 states that one can get a sub-optimal solution to the original optimization problem (3.15) by solving the optimization problem described by (3.18)-(3.20) in order to assign routes to the truck drivers and by subsequently applying the pricing scheme given by (3.17). We call this method Approximately Optimum Pricing Scheme (AOPS). The main advantage of this approach is the fact that we significantly reduce the dimensionality of the problem by calculating a pricing scheme using a simple algebraic equation. As we will later show experimentally, AOPS achieves a significant improvement compared to the UE and provides a solution close to the SO. In the first part of the proof of Theorem 3.1, we proved that the condition E[T mon tr (a)] E T mon;UE tr is a necessary and sufficient condition to guarantee that the pricing scheme (3.17) is Pareto-improving. For the deterministic demand case, a similar property was also proved in [46] for the case of revenue refunding schemes. At this point, let us comment on the Pareto-improvement and the truthfulness properties of AOPS. Similar to OPS, it is theoretically possible that for some demand realizations, some drivers are given a route with a higher total travel time (travel time + payments expressed in time units) compared to the average travel time at the UE. However, due to the fact that individual drivers only know the probability distribution of the demand and not the exact realization of it and hence they have incomplete information of the traffic conditions, they will be willing to participate in AOPS since at the time they make their decision, their expected cost will be lower under the mechanism 53 routing suggestions M than in the UE. Additionally, if users repeatedly participate in AOPS, then using randomization in order to assign the truck drivers who declared the same OD pair j and class w into routes r, it is guaranteed that every truck driver will be better-off compared to the UE (Pareto-improvement). As far as it concerns the truthfulness property, AOPS guarantees that the users will be willing to truthfully declare their VOT since at the time they make their decision, their expected cost in the case they are truthful is lower than their corresponding cost in the case where they declared a different VOT than their actual one. In Appendix B, we prove the existence of pricing-and-routing schemes that can guarantee even stronger versions of the Pareto-improvement and the truthfulness properties. However, we decided not to include them in our analysis for the reasons explained in Section 3.3.1. In the following subsection, we present a class-anonymous pricing scheme, namely, the Con- gestion Pricing with Uniform Revenue Refunding (CPURR) scheme. This is going to be used later as one of the baselines in the simulation experiments. 3.3.3 CongestionPricingwithUniformRevenueRefunding(CPURR). Under a congestion pricing scheme, each driver is assigned a fee depending on the OD pair and the route he/she follows. Guo and Yang proposed to combine Congestion Pricing with a Uniform Rev- enue Refunding (CPURR) scheme, i.e. the fees collected from congestion pricing are uniformly distributed among the participant drivers irrespective of the class they belong [46]. Therefore, the whole scheme is class-anonymous. The original CPURR scheme proposed in [46] is OD-based. However, since our approach focuses on route-based pricing schemes, in this work, we calculate the route-based equivalent of 54 the CPURR scheme. Therefore, the CPURR scheme can be calculated by solving the following optimization problem with complementarity constraints: minimize a;p;z l(mE[T tr (a)]+(1m)E[T p (a)])+(1l)E[T mon tr (a)] subject to 0a j w;r ? F CP j;w;r (a;p)z j w 0;8 j;w;r å r2R j a j w;r = 1;8 j;w å c v å j=1 N å w=1 å r2R j p c d w c; j a j w;r p j r = 0 (3.21) where z j w is a set of free variables that are used in order to solve the equilibrium optimization problem (3.21) and F CP j;w;r (a;p) is given by the following equation: F CP j;w;r (a;p)= å c p c J CP c; j;w;r (a)+ 1 s w p j r (3.22) where J CP c; j;w;r (a) is defined according to (3.8). Note that under a congestion pricing scheme, the network users make their own individual routing decisions while taking into account the fees corre- sponding to each route. Since in our model the truck drivers only know the probability distribution of the demand for the rest of the truck drivers and not its exact realization, the way that the drivers choose their routes a j w;r does not depend on the exact realization c. Additionally, note that the variablesp j r corresponding to the pricing scheme do not depend on w and c. The independence of w can be justified by the fact that the CPURR scheme is class-anonymous. On the other hand, the coordinator of the CPURR scheme who is responsible for assigning fees to each route and then uniformly distribute the collected revenue back to the participant drivers, could design a pricing scheme that depends on the exact realization c of the demand. However, even in that case, since none of the constraints of (3.21) depends on c, the optimum solution of (3.21) would not change. In the next section, CPURR is used as one of the baselines in the simulation experiments. There are several reasons that justify this choice. First, as mentioned earlier, congestion pricing is one of the most widely studied methods for addressing the problem of the inefficiency between 55 the UE and the SO solutions. Additionally, CPURR is a class-anonymous pricing scheme. This fact allows us to directly demonstrate the benefits of applying class-specific (VOT-based) pricing schemes such as OPS and AOPS. 3.4 ExperimentalResults In this section, we demonstrate our approach conducting simulation experiments based on the Sioux Falls network [57]. The Sioux Falls network consists of 24 nodes and 76 links and consti- tutes a benchmark in the transportation research field. The experimental results section is divided into four subsections: In the first subsection, we test the sensitivity of the solutions of the proposed methods to the initial conditions and to the number of routes considered per OD pair. Using these results, in the second subsection, we compare the UE, SO, OPS, AOPS and the CPURR scheme in terms of the expected total travel time of the truck drivers E[T tr ], their expected total monetary cost E[T mon tr ], the expected total travel time of the network E[T s ] and the total objective value by varying the weighting factorl and we show that the VOT-based pricing schemes (OPS and AOPS) outperform class-anonymous pricing schemes like CPURR. Subsequently, in the third subsection, we experimentally show that OPS and AOPS can be efficiently used under both a determinis- tic and a stochastic demand scenario since they outperform the CPURR scheme in terms of the expected total monetary cost of the truck drivers E[T mon tr ] and the total objective value by concur- rently achieving a superior solution compared to the UE in both scenarios. In the last subsection, we increase the number of OD pairs that the truck drivers use in the Sioux Falls network and we experimentally show that both OPS and AOPS need a lower computational time compared to the CPURR scheme. Additionally, we show that AOPS remains computationally tractable even in the case where a large number of OD pairs is used by the truck drivers. For all of the experiments, the fmincon optimization solver implemented in the MATLAB Optimization Toolbox [22] was used. Since fmincon solves optimization problems with local optimality guarantees, in this section, we compare local minima between the approaches. 56 3.4.1 RobustnessintheSiouxFallsNetwork In our experiments, we assumed that the cost of each route corresponds to travel time and can be described by a Bureau of Public Roads (BPR) function [88] of the form: C lT (X l p ;X lT )=e a +e b X l p + 3X lT e c ! 4 wheree a ;e b ande c are constants and their values were chosen similar to the ones adopted in [75] 1 . We further assumed that the number of passenger vehicles at each link of the Sioux Falls network is constant 2 . These numbers have been calculated by solving an equilibrium assignment problem for the passenger vehicles. To retain computational tractability, we further assumed that there are 6 available OD pairs for the truck drivers, namely(O 1 ;D 7 );(O 1 ;D 11 );(O 10 ;D 11 );(O 10 ;D 20 );(O 15 ;D 5 ) and(O 24 ;D 10 ) and that the truck drivers choose their class among two available options with VOT s 1 = 200 $ hr and s 2 = 50 $ hr , respectively. The demand of the truck drivers was assumed to be stochastic and take one of the following two equiprobable values: d 1 = 2 6 4 3 4:5 6 3 14 3:6 1 2:8 5:4 7 9 2 3 7 5 ;d 2 = 2 6 4 5 1:8 3:9 15 6:4 2:4 6 5:5 1:8 6:5 11 6 3 7 5 where each column of d 1 and d 2 corresponds to the demand of truck drivers for each OD pair and each row denotes a different class of users. The ratio of trucks in the network was 5:16% in the case of d 1 and 5:96% in the case of d 2 . The values of the weighting factors were chosen to be l =m = 0:9. 3.4.1.1 SensitivitytoInitialConditions To test the sensitivity of the solutions of the optimization problems of the proposed methods to different initial conditions, in this section, we designed the following test scenario. We randomly 1 These values can be found in this link. 2 These values can be found in this link. 57 initialized the initial conditions for the UE problem. After solving the UE problem, we used its solution as an initial condition to the rest of the methods, i.e. SO, OPS, AOPS and CPURR. In our experiments, we assumed that the truck drivers follow the 10 least congested routes per OD pair. To calculate the least congested routes for each OD pair, we followed the following procedure. Before assigning any truck driver into the network (X lT = 0), we calculated the cost of each route by substituting the number of passenger vehicles at each road segment l into the BPR function. Therefore, for each OD pair, we could calculate the least congested routes, i.e. the routes with the lowest travel time as defined by the BPR function when X lT = 0. For the UE problem described by (3.9), we picked the initial values for the set of free variables z j w according to a uniform distributionU[0;100]. The values for the variables a j w;r were picked according to a uniform distributionU[0;1]. The interior point method provided in the MATLAB Optimization Toolbox [22] was used in order to solve (3.9). Subsequently, this solution was used as an initial condition for the optimization problems of the other methods. Note that this solution is a feasible solution for all of the methods. This experiment was repeated 10 times. In Table 3.2, we report the minimum, the maximum, the mean, the standard deviation and the Coefficient of Variation (CV) values for different metrics for the UE, SO, OPS, AOPS and the CPURR methods. The results of Table 3.2 demonstrate that both OPS and AOPS approaches outperform the CPURR scheme and can approach the SO solution. Additionally, the solutions of OPS and AOPS are shown to be robust since the standard deviation values of these methods are significantly lower compared to the ones of the CPURR scheme. To further explore the robustness of the solutions to the optimization problems of different methods, we conducted some additional experiments. More specifically, we considered solving all the optimization problems with random initial conditions instead of using the UE solution as an initial condition. Note that since the interior point method requires an initial condition that is close to primal and dual feasibility, the random initial conditions for this method were created by adding a random component to the UE solution. Furthermore, we explored solving the problems with the Sequential Quadratic Programming (SQP) method provided in the MATLAB Optimization 58 Method Metric Min Max Mean Std CV(%) UE E[T tr ] 20043.3 20190.7 20117.8 48.6 0.242 E[T mon tr ] 43235.8 43567.8 43414.8 128.0 0.295 E[T s ] 330235.8 330826.2 330534.7 192.7 0.058 E[T p ] 310192.5 310635.5 310416.9 144.1 0.046 Objective 48476.0 48667.1 48574.4 64.5 0.133 SO E[T tr ] 19682.2 19683.1 19682.5 0.3 0.002 E[T mon tr ] 41727.8 41777.7 41741.3 13.3 0.032 E[T s ] 329109.4 329129.5 329125.6 5.7 0.002 E[T p ] 309426.6 309447.3 309443.1 5.8 0.002 Objective 47966.2 47969.2 47966.9 0.8 0.002 OPS E[T tr ] 19680.8 19683.3 19682.9 0.7 0.004 E[T mon tr ] 41729.1 41791.0 41759.4 20.9 0.050 E[T s ] 329111.6 329130.3 329121.2 6.9 0.002 E[T p ] 309428.5 309447.1 309438.3 7.0 0.002 Objective 47966.6 47970.5 47968.5 1.3 0.003 AOPS E[T tr ] 19698.8 19729.4 19716.8 10.1 0.051 E[T mon tr ] 41737.2 41821.7 41783.2 28.5 0.068 E[T s ] 329274.0 329423.7 329378.9 41.9 0.013 E[T p ] 309575.1 309719.1 309662.0 36.6 0.012 Objective 47997.3 48027.0 48018.6 8.3 0.017 CPURR E[T tr ] 19853.4 20070.8 19889.2 62.2 0.313 E[T mon tr ] 42632.8 43195.7 42737.9 162.0 0.379 E[T s ] 329554.6 330565.8 329777.7 270.8 0.082 E[T p ] 309681.4 310494.9 309888.5 210.2 0.068 Objective 48227.5 48521.5 48274.0 84.9 0.176 Table 3.2: The minimum, the maximum, the mean, the standard deviation and the coefficient of variation (CV) values for different metrics for the UE, SO, OPS, AOPS and CPURR methods after 10 simulation runs with different initial conditions. The number of OD pairs was 6 and we considered the 10 least congested routes per OD pair. 59 Toolbox. The experimental results showed that both SO, OPS and AOPS can be efficiently solved under various initial conditions both with the interior point and the SQP methods. On the other hand, the CPURR problem could not achieve a significantly better solution than the UE in the case where the SQP optimization method was used. Additionally, in case the CPURR method was initialized with a random initial condition, the interior point method would fail to converge. Last, we need to mention that for all of the experiments of this section and for the rest of this Chapter, the step tolerance used in the fmincon solver was 10 6 . For UE, SO, OPS and AOPS, the constraint tolerance was set to 10 6 . However, due to the difficulties on calculating the optimum solution of the CPURR problem, the constraint tolerance for this problem was set to 5 10 5 . For the rest of this Chapter, the solutions of the optimization problems have been calculated using the following procedure. We randomly initialized the initial conditions for the UE problem. After solving the UE problem, we used its solution as an initial condition to the rest of the methods, i.e. SO, OPS, AOPS and CPURR. In the next subsection, we test the the sensitivity of the solution to the number of routes considered per OD pair in the Sioux Falls network. 3.4.1.2 SensitivitytotheNumberofRoutes In this section, we test the sensitivity of the solutions of the optimization problems of the proposed methods to different number of routes considered per OD pair. In Table 3.3, we calculate the expected total truck travel time, the expected total truck monetary cost, the expected total network time and the expected total travel time of the passenger vehicles of the UE, SO, OPS, AOPS and CPURR for the case where the truck drivers use 6 OD pairs of the Sioux Falls network, considering the 5, 10 and 15 least congested routes per OD pair. The rest of the parameters were chosen similar to the ones used in Section 3.4.1. As can be observed from the results presented in Table 3.3, the more routes we consider per OD pair, the lower the values we can achieve in all four evaluation metrics. However, as the number of routes considered per OD pair increases, the computational time also increases. Our simulations showed that by considering 10 routes per OD pair, we can achieve a good balance 60 # of routes Metric UE SO OPS AOPS CPURR 5 E[T tr ] 21680.4 21582.6 21581.6 21580.4 21598.7 E[T mon tr ] 46870.0 46250.6 46254.7 46307.6 46667.7 E[T s ] 334508.2 334258.0 334262.3 334256.2 334279.4 E[T p ] 312827.8 312675.4 312680.7 312675.8 312680.7 10 E[T tr ] 20043.3 19682.2 19683.1 19698.8 19857.9 E[T mon tr ] 43235.8 41740.1 41745.0 41794.7 42633.1 E[T s ] 330235.8 329127.6 329124.7 329274.0 329685.7 E[T p ] 310192.5 309445.3 309441.0 309575.1 309827.7 15 E[T tr ] 19521.8 18982.9 18982.8 19097.5 19400.1 E[T mon tr ] 41996.2 40168.8 40177.4 40439.1 41683.9 E[T s ] 326222.4 325966.3 325970.7 325971.5 325925.8 E[T p ] 306700.6 306983.4 306987.9 306828.3 306525.8 Table 3.3: The expected total truck travel time E[T tr ], the expected total truck monetary cost E[T mon tr ], the expected total network time E[T s ] and the expected total travel time of the passen- ger vehicles E[T p ] of the UE, SO, OPS, AOPS and CPURR in the case where the truck drivers follow 6 OD pairs, considering the 5, 10 and 15 least congested routes per OD pair. between network efficiency and computational time in the Sioux Falls network. Therefore, for the rest of this Chapter, we only consider the 10 least congested routes per OD pair. 3.4.2 TheEffectoftheWeightingFactorl In this section, we conduct additional simulation experiments in order to demonstrate the effect of the weighting factorl in the solutions of the UE, SO, OPS, AOPS and CPURR approaches and to show that the VOT-based pricing schemes (OPS and AOPS) outperform both the CPURR (a class- anonymous pricing scheme) and the UE solution. For the Sioux Falls network, the configurations were chosen similar to the ones used in Section 3.4.1. In Figure 3.1(a-d), we plot the expected total travel time of the truck drivers E[T tr ], their ex- pected total monetary cost E[T mon tr ], the expected total travel time of the network (passenger vehi- cles + trucks) and the total objective value, respectively, for different values of the weighting factor l2[0;1]. 61 (a) Expected total truck travel time. (b) Expected total truck monetary cost. (c) Expected total network time. (d) Total objective value. Figure 3.1: (a) The expected total truck travel time, (b) the expected total truck monetary cost, (c) the expected total network time and (d) the total objective value of the UE (magenta), SO (blue), OPS (orange), AOPS (green) and CPURR (red) for different values of the weighting factorl. As can be observed in Figure 3.1(a), as the value of the weighting factor l increases, the ex- pected travel time of the truck drivers E[T tr ] decreases for both SO, OPS and AOPS solutions. It is worth mentioning that for all values of l, the OPS solution closely follows the SO solution. Ad- ditionally, the AOPS solution can significantly decrease E[T tr ] compared to the CPURR solution, especially forl > 0:5. In Figure 3.1(b), we observe that the expected total monetary cost of the truck drivers E[T mon tr ] increases as the value of the weighting factorl increases for both SO, OPS and AOPS solutions. On the other hand, we observe that E[T mon tr ] does not significantly change for the CPURR solution. Note also that AOPS has a smaller increase rate compared to the SO and the OPS solutions. This 62 can be explained by the fact that asl increases, SO and OPS put more emphasis on minimizing the expected total travel time of the network while on the other hand, AOPS applies a pricing scheme during which the expected total monetary benefits E[T mon;M tr ] E[T mon;UE tr ], are shared to the users proportionally to the VOT of the class they belong. Therefore, a truck driver with higher VOT will get reimbursed with a bigger amount of money compared to a truck driver with lower VOT in case both of the drivers are assigned to a slower route. This behavior makes AOPS better contribute to the minimization of the expected total monetary cost of the truck drivers. In Figure 3.1(c), it is shown that as the value of the weighting factorl increases, the expected total travel time of the network E[T s ] decreases. Furthermore, it can be observed that OPS makes E[T s ] closely follow its corresponding value at the SO solution while at the same time, both OPS and AOPS can reduce the expected total travel time of the network compared to the CPURR scheme forl > 0:75. Finally, in Figure 3.1(d), we plot the total objective value l(mE[T tr ]+(1m)E[T p ])+(1 l)E[T mon tr ] and we observe that OPS closely approaches the SO solution. Additionally, AOPS provides a solution close to the SO and the OPS solutions, while constantly outperforming the CPURR scheme. 3.4.3 DeterministicvsStochasticDemandScenario In this subsection, we show that OPS and AOPS can be efficiently used under both a deterministic and a stochastic demand scenario. The network configurations were chosen similar to the ones used in Section 3.4.1. 63 For the purpose of this simulation experiment, we ran two distinct scenarios. In the first sce- nario (deterministic), we assumed that the truck drivers know the exact number of both the passen- ger vehicles and the rest of the trucks in the network. More specifically, the demand vector for the truck drivers was assumed to be: d= 2 6 4 3 4:5 6 3 14 3:6 1 2:8 5:4 7 9 2 3 7 5 The ratio of truck drivers in the network was 5:16% for this scenario. In the second scenario (stochastic), we assumed that the truck drivers know the number of passenger vehicles in the net- work and only the probability distribution of the demand for the rest of the truck drivers. More specifically, the demand of the truck drivers was assumed to take one of the two equiprobable val- ues, d 1 (ratio of trucks: 5:16%) and d 2 (ratio of trucks: 5:96%), as given in Section 3.4.1. The values of the weighting factors were chosen to bel =m = 0:9. In Figure 3.2, we show the expected total truck monetary cost and the total objective value for the two scenarios. As can be observed from the results of Figure 3.2, OPS and AOPS approach the SO solution and outperform the UE and the CPURR scheme both in the deterministic and in the stochastic demand scenarios. Additionally, the benefits of applying OPS and AOPS rather than the CPURR scheme become greater in the stochastic demand scenario. The latter result was expected since OPS and AOPS provide routing suggestions a and calculate payment schemes p that both depend on the exact realization c of the demand of the truck drivers as can be seen from the optimization problems (3.15) and (3.18), respectively, while the corresponding quantities under the CPURR scheme do not depend on c as can be seen from (3.21). In other words, under the OPS and the AOPS schemes, the coordinator provides routing suggestions having complete information of the traffic demand. On the other hand, under the CPURR scheme, the truck drivers make individual routing decisions having incomplete information of the traffic demand. Based on the simulation results presented in Figure 3.2, it can be observed that even in the case where the truck drivers know the exact realization of the demand for the rest of the truck drivers, 64 (a) Expected total truck monetary cost. (b) Objective value. Figure 3.2: (a) The expected total truck monetary cost and (b) the total objective value of the UE (blue), SO (orange), OPS (green), AOPS (red) and CPURR (purple) for different demand scenarios. the expected total monetary cost and the total objective value are lower in the case where OPS and AOPS are applied rather than in the case where the CPURR scheme is applied. These results demonstrate that OPS and AOPS can be efficiently used in both a deterministic and a stochastic demand scenario. 3.4.4 ComputationalTime In this subsection, we gradually increase the number of OD pairs that the truck drivers follow in the Sioux Falls network and we measure the computational time needed to solve the UE, the SO, the OPS, the AOPS and the CPURR problems. For the experiments of this subsection, we assumed that the cost of each link as well as the number of passenger vehicles at each link of the network are identical to the ones used in Section 3.4.1. We further assumed that the truck drivers choose their class among two available options with VOT s 1 = 200 $ hr and s 2 = 50 $ hr , respectively. The weighting factors were chosen to bel =m = 0:9. The demand of the truck drivers was assumed to be stochastic and take one of the two equiprobable values, namely d 1 and d 2 3 . In Table 3.4, we show the expected total truck travel time E[T tr ], the expected total truck mon- etary cost E[T mon tr ], the expected total travel time of the network E[T s ] and the expected total travel 3 The values of d 1 and d 2 can be found in this link. 65 OD pairs Metric UE SO OPS AOPS CPURR 4 E[T tr ] 10545.7 10505.2 10505.5 10504.8 10520.7 E[T mon tr ] 23435.6 23105.2 23107.0 23139.3 23365.7 E[T s ] 261346.5 261446.1 261444.2 261429.7 261337.4 E[T p ] 250800.8 250940.9 250938.7 250924.9 250816.7 8 E[T tr ] 29463.3 28798.1 28801.9 28925.9 29111.5 E[T mon tr ] 62499.2 60282.2 60264.3 60459.3 61626.4 E[T s ] 393917.8 393062.0 393055.2 392995.1 392876.2 E[T p ] 364454.5 364263.9 364253.3 364069.1 363764.7 12 E[T tr ] 56222.9 54732.2 54719.0 55445.5 55392.5 E[T mon tr ] 114578.8 110117.8 110343.3 113907.9 112265.4 E[T s ] 554185.9 549492.6 549476.0 551882.0 551676.9 E[T p ] 497963.0 494760.4 494757.0 496436.5 496284.4 16 E[T tr ] 60264.6 58852.1 58847.2 59443.1 C.I. E[T mon tr ] 122731.0 118475.3 118599.2 121315.8 C.I. E[T s ] 575701.4 571333.9 571307.1 573482.9 C.I. E[T p ] 515436.8 512481.8 512459.9 514039.8 C.I. 20 E[T tr ] 84675.5 82863.7 C.I. 83715.5 C.I. E[T mon tr ] 169191.0 163705.0 C.I. 168081.6 C.I. E[T s ] 701513.8 696210.3 C.I. 699002.9 C.I. E[T p ] 616838.3 613346.6 C.I. 615287.5 C.I. Table 3.4: The expected total truck travel time E[T tr ], the expected total truck monetary cost E[T mon tr ], the expected total network time E[T s ] and the expected travel time of the passenger vehi- cles E[T p ] of the UE, SO, OPS, AOPS and CPURR for different number of OD pairs, considering the 10 least congested routes per OD pair. The ratio of trucks in the network is shown in Table 3.6. time of the passenger vehicles E[T p ] of the UE, SO, OPS, AOPS and CPURR for different number of OD pairs, considering the 10 least congested routes per OD pair. In Table 3.5, we show the corresponding computational times. As can be seen from the results of Table 3.4, OPS always provides a solution close to the SO while AOPS outperforms both the CPURR scheme and the UE solution. Additionally, we observe that the CPURR scheme is the slowest approach since it becomes Computationally Intractable (C.I.) when 16 OD pairs are considered, while the OPS be- comes C.I. when the truck drivers are assumed to follow 20 OD pairs. As far as it concerns the AOPS, it can be observed that even though it is not as efficient as the OPS in the evaluation met- rics used, it remains computationally tractable for a larger number of OD pairs. This result was expected since OPS calculates both the optimum routing strategy a OPS and the optimum pricing 66 OD pairs UE SO OPS AOPS CPURR 4 6.9 8.9 32.5 10.0 33.9 8 30.7 24.9 337.4 59.1 1362.7 12 81.3 55.4 1389.0 128.2 6311.0 16 151.2 105.7 12163.5 179.4 C.I. 20 278.3 169.8 C.I. 446.4 C.I. Table 3.5: The computational time (in seconds) of the simulation experiments presented in Ta- ble 3.4. scheme p OPS through the solution of the optimization problem (3.15). On the other hand, under the AOPS, in the optimization problem (3.18), we only need to find the optimum routing strategy a AOPS since the approximately optimum pricing schemep AOPS c; j;w;r can be easily calculated from the algebraic equation (3.17). This approximation makes AOPS significantly faster compared to OPS as can be seen from the results of Table 3.5. However, as the number of OD pairs increases, the computational time increases as well and after a certain number of OD pairs, even AOPS will be- come computationally intractable. Therefore, for realistic transportation networks with hundreds or thousands of OD pairs, more computationally efficient solutions need to be studied. We next show that the effect of the proposed methods on passenger vehicles is minimal, whereas for the trucks, it can be impactful. First, as can be seen in Table 3.6, the number of trucks in the network is relatively small compared to the number of passenger vehicles. Addition- ally, as can be observed from the results of Table 3.4, the effect of truck routing schemes in the expected total travel time of the passenger vehicles is insignificant. More specifically, in the case of 4 OD pairs, OPS increases the expected travel time of the passenger vehicles in the network by 0:05% compared to the UE. In the rest of the cases, both OPS and AOPS decrease the expected travel time of the passenger vehicles in the network. However, this reduction can be still consid- ered small since it reaches up to 0:6% in the case of 12 and 16 OD pairs. Therefore, we expect that the passenger vehicles will not react to such a minimal change of their traffic environment. On the other hand, the proposed pricing-and-routing schemes can offer significant benefits to trucks since they can reduce the expected total travel time of the truck drivers by 2:7% and their corresponding 67 OD pairs Realization Ratio(%) 4 d 1 2.82 d 2 3.89 8 d 1 6.36 d 2 6.96 12 d 1 8.64 d 2 8.98 16 d 1 10.33 d 2 10.40 20 d 1 11.88 d 2 12.00 Table 3.6: The ratio of trucks in the network for the experimental results presented in Table 3.4. expected total monetary cost by 3:7% compared to the UE as can be observed from the results of Table 3.4. 68 Chapter4 PersonalizedFreightRouteRecommendationswithSystem OptimalityConsiderations: AUtilityLearningApproach 4.1 Introduction Traffic congestion is a major problem in urban areas. According to statistics, in 2017, traffic con- gestion costed urban Americans an extra 8.8 billion hours and an extra 3.3 billion gallons of fuel to travel, for a total congestion cost of $166 billion. Not surprisingly, the major congested points are in metropolitan areas where truck traffic mixes with other traffic and along major interstate highways connecting major metropolitan areas [16]. In the United States (U.S.), transportation’s total estimated contribution to U.S. Gross Domestic Product (GDP) was $1,298.1 billion in 2019. Trucking contributed the largest amount of all the freight modes, at $368.9 billion [105]. In the European Union (EU), transportation sector is also a major contributor to the economy, represent- ing more than 9% of EU gross value added in 2016 [38]. Therefore, it becomes clear that efficient route planning could have a large positive impact on the global economy. The Traffic Assignment Problem (TAP) [78] is a key problem for efficient planning in trans- portation networks. Based on the objective of the assignment process and user behavior assump- tions, many assignment models can be classified as User Equilibrium (UE) or System Optimum (SO) [116]. In a UE, drivers act independently in an effort to maximize their own individual util- ity. On the other hand, in a SO, drivers follow coordinated routing instructions that minimize the 69 expected total cost of the network. It is well known that user optimality does not imply system optimality. The inefficiency between the UE and the SO has been addressed in the literature as the Price of Anarchy (POA) [54]. Recent research efforts have tried to estimate the POA [127, 71] demonstrating that realistic transportation networks suffer from this problem. Many previous works have tried to address the problem of the inefficiency between an equilib- rium flow pattern and a SO solution through pricing schemes. Congestion pricing [80, 110, 7, 41] is the most frequently studied among these methods. In a congestion pricing scheme, drivers are asked to pay a fee (toll) corresponding to the additional cost their presence causes to the network. Other strategies include the applications of Tradable Credit Schemes (TCS) [122, 61] or tradable travel permits [111] among the drivers of the network. In this Chapter, we propose a pricing-and- routing scheme that has a toll-and-subsidy form, similar to [75, 51, 77]. This scheme has three main characteristics. First, it is budget balanced on average. Second, it guarantees that every truck driver has an incentive to participate in the proposed mechanism and lastly, it drives the network close to the SO solution. To ensure that every truck driver has an incentive to participate in the proposed mechanism, we first need to estimate the utility function that describes the routing preferences of each driver. A large amount of research has been conducted to study discrete choice modeling under the Random Utility Maximization (RUM) framework [68]. In the RUM framework, the utility to the decision maker of each alternative is not completely known. It consists of a deterministic component which is a function of the attributes of the alternative and the characteristics of the decision-maker and a random error term that follows a probability distribution. Depending on whether the error terms are assumed to be multivariate normal or independently and identically Type I extreme value (gumbel) distributed, we get the Multinomial Probit (MNP) model [27] and the Multinomial Logit (MNL) model [67], respectively. For further information related to route choice modeling, we refer the in- terested reader to [81] and references therein. Recently, there is also a growing interest in studying the connection between traditional discrete choice models and modern machine learning methods, e.g. Deep Neural Networks (DNNs) [114]. Inspired by these research efforts, in this Chapter, we 70 use a learning scheme based on the Maximum Likelihood Estimation (MLE) principle that allows us to learn the utility of each cluster of truck drivers using a set of binary route choice questions. A basic characteristic of our method is the use of a model that calculates the difference between the utilities of two alternatives, thus guaranteeing that the transitivity property is satisfied. The transi- tivity property is important since by only using a binary model and doing a pairwise comparison between 2 routes, we can accurately calculate the utility of each alternative even if the total num- ber of routes per OD pair is greater than 2. Accurately learning the utility function that describes the routing preferences of each cluster of truck drivers helps us provide more personalized route recommendations. Previous research efforts have tried to incorporate personalization in route planning. In [84], Rogers and Langley used a linear perceptron to learn the drivers’ routing preferences and subse- quently solved the routing problem using Dijkstra’s shortest path algorithm [24]. In [60], Letchner et al. designed a route planner that used real-world GPS data to estimate both time-dependent road speeds and individual driver preferences. In [25], Cui et al. used historical GPS trajectories and a collaborative filtering approach [52] in order to provide personalized travel route recommen- dations. Using trajectory data and considering three commonly used travel costs, namely travel distance, travel time and fuel consumption, [30] proposed a method to recommend personalized routes to individual drivers. Another work proposed to solve a multi-criteria optimization prob- lem to find the optimal route considering air quality, travel time, and fuel consumption from the source to the destination [87]. Recently, multi-modal transportation recommendations have been also studied [63]. Most of the aforementioned works take into account user optimality only and therefore, the provided solutions may be inefficient for the network. On the other hand, in this Chapter, we propose a method to learn truck drivers’ individual routing preferences and we design a pricing-and-routing scheme that recommends routes that are beneficial for the system optimum. Perhaps closest to our work is the work of [106]. In their work, Vayanos et al. built a survey that they distributed to drivers of passenger vehicles. Using the results of the survey, they first clustered the drivers into disjoint clusters, and then using the assumption that the utility of the drivers can 71 be described by a linear function, they used a Mixed Integer Programming (MIP) formulation to learn the parameters of the utility of each cluster. Having learned a linear utility for each cluster, they solved a deterministic traffic assignment problem with an additional constraint to assign the drivers into routes that they will likely follow, i.e. the assigned route will make the utility of the driver not to be much lower than the utility he/she would have if he/she made his/her own routing decisions. In this Chapter, we study pricing-and-routing schemes that can be specifically applied on trucks. Our main goal is the design of pricing-and-routing schemes that drive the network as close as possible to the SO solution and concurrently guarantee that the expected total utility of a truck driver (including payments) in case he/she decides to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she does not participate. Note that the participa- tion to the mechanism is voluntary and therefore, the mechanism can operate even if some drivers do not participate. Additionally, we prove that the resulting pricing scheme is self-sustainable and the expected total payments made or received by the coordinator of the mechanism, are equal to zero. It is worth mentioning that estimating a utility function for each individual truck driver is computationally intensive. To overcome the computational complexity of this problem, we first divide the truck drivers into disjoint clusters based on their responses to a small number of binary route choice questions and we subsequently use a learning scheme based on the MLE principle that allows us to learn the parameters of the utility function that describes each cluster. The es- timated utilities are then used to calculate a pricing-and-routing scheme with the aforementioned characteristics. Let us now comment on the main differences between the current work and [106]. First, in their work, Vayanos et al. deal with passenger vehicles while on the other hand, we focus on truck drivers. Second, Vayanos et al. assumed a deterministic model. In our work, we analyze stochastic models and thus the system coordinator has different information from the truck drivers which creates additional opportunities for coordination. Third, instead of using a MIP formulation to learn the utility of each cluster, we use a learning scheme based on the MLE principle. This 72 learning scheme satisfies the transitivity property and allows us to learn utility functions of any form. In contrast, the MIP formulation is limited to the linear case only. Lastly, in their work, Vayanos et al. solved a traffic assignment problem with an additional constraint based on which the assigned route will make the utility of the driver not to be much lower than the utility he/she would have if he/she made his/her own routing decisions. However, in such a solution some drivers may have a lower utility than the one they would have if they made their own routing decisions, which reduces the incentives for their participation. On the other hand, in our approach, we propose a pricing-and-routing scheme that is budget balanced on average and guarantees that the expected total utility of a truck driver (including payments) in case he/she decides to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she does not participate. Consequently, every truck driver has an incentive to participate in the proposed mechanism. 4.2 MathematicalFormulation 4.2.1 Problemformulation The notation used throughout this Chapter is summarized in Table 4.1. Let G=(V;E) denote a transportation network, where V is the set of nodes and E is the set of road segments of the network, respectively. We assume that the Origin-Destination (OD) demand of the truck drivers is stochastic and follows a probability distribution with finite support. Additionally, we assume a symmetric information model where all truck drivers have the same amount of information and know the number of passenger vehicles at each road segment of the network. Previous research efforts, e.g. [65], have shown that there are several factors that can affect drivers’ routing decisions. For instance, previous studies showed that on average, drivers take the fastest route for only 35% of their journeys [60]. In this work, we aim to propose a methodol- ogy that learns how different factors affect truck drivers’ routing decisions and subsequently use this information to efficiently route the truck drivers into the network. The proposed approach is general and can be applied for any number of attributes that can possibly affect drivers’ routing 73 Variable Meaning G(V;E) Network with nodes V and links E R w The set of all paths connecting OD pair w2 W v The number of OD pairs in the network L The number of OD road segments in the network J i Utility of cluster i x r Vector of attributes of the route r q i Learned weights of the attributes of the utility function of cluster i p c Probability of the demand realization c a c;w i;r Proportion of truck drivers belonging to cluster i who want to travel in OD pair w and are assigned in route r during demand realization c p c;w r Payment (made or received) of truck drivers who want to travel in OD pair w and are assigned in route r during demand realization c d c w;i Demand of truck drivers belonging to the cluster i who want to travel in OD pair w during demand realization c F w i;r Expected utility of a truck driver belonging to cluster i who wants to travel in OD pair w using route r X l p Number of passenger vehicles traversing the road segment l X lT Number of trucks traversing the road segment l C lT Travel time of a truck driver traversing road segment l E[T tr ] Expected total travel time of the truck drivers E[T p ] Expected total travel time of the passenger vehicles E[T S ] Expected total travel time of the network E[U tr ] Expected total utility of the truck drivers O(a) Objective function to be minimized. Weighted combination of the expected total travel time and the negative of the expected total utility l Weighting factor of the objective function r Weighting factor of the objective function L(q i ;x;y) Binary cross-entropy loss function s(q i ;x m ) Probability according to which we predict that a truck driver belonging to cluster i will pick route 1 or route 2 K The number of clusters in the K-means algorithm Table 4.1: Notation 74 decisions. For illustrative purposes, for the rest of this Chapter, we consider four different factors that could affect the routing decision of a truck driver namely, the distance, the number of free- way interchanges, the travel time and the 80th percentile of the travel time of a route. Let the vector x r contain these four attributes of route r. In this Chapter, we assume that different drivers value these four factors differently and therefore, we aim to cluster the truck drivers into K distinct groups. Note that we could have picked any g th percentile to represent the variability of travel time, without loss of generality. Under the assumption that the OD demand of the truck drivers follows a discrete probability distribution with finite support, we define the g th percentile of travel time as the value of u for which P(U u) is greater than or equal to g 100 and P(U u) is greater than or equal to 1 g 100 . In this Chapter, we assume that the OD demand of the truck drivers is the only source of uncertainty that affects the variability of travel time. Let C lT (X l p ;X lT (a)) be a known nonlinear function representing the travel time of a truck driver traversing the road segment l when there exist X l p passenger vehicles and X lT (a) trucks on it, wherea is a set of variables defined as follows: a =fa w i;r : w= 1;:::;v;i= 1;:::;K;r2 R w g (4.1) where w is the index corresponding to a specific OD pair, i is the index corresponding to a specific cluster of truck drivers, r2 R w denotes a specific route among the set of available routes R w con- necting OD pair w, K is the number of clusters of truck drivers and v is the number of OD pairs in the network. Therefore,a w i;r expresses the proportion of truck drivers belonging to cluster i with a desired OD pair w who choose route r for their trip. Based on the assumption that the demand of truck drivers is stochastic, let d w;i be random variables denoting the number of truck drivers belonging to cluster i with desired OD pair w and 75 let d c w;i be their corresponding values during demand realization c. Then, the number of trucks traversing the road segment l is given by: X lT (a)= v å w=1 K å i=1 å r2R w :l2r d c w;i a w i;r (4.2) where on the left side of (4.2), we omitted the index c to simplify the notation. Therefore, the expected total travel time of the truck drivers in the network is given by: E[T tr (a)]= E " L å l=1 X lT (a)C lT (X l p ;X lT (a)) # (4.3) where L is the number of road segments in the transportation network and X lT (a) is given by (4.2). 4.2.2 UserEquilibrium(UE) In the absence of cooperation, the users of the network act independently in an effort to maximize their own individual utility. This behavior drives the network to a situation called User Equilibrium (UE). In this Chapter, after clustering the truck drivers into K distinct groups, we aim to learn a utility function J c;w i;r (q i ;x r (a)) for each cluster i. Note that J c;w i;r (q i ;x r (a)) represents the utility of a truck driver belonging to cluster i who wants to travel in OD pair w and is assigned to route r during demand realization c. Additionally,q i represents the parameters of the utility function of cluster i that we aim to learn. In case J c;w i;r (q i ;x r (a)) is a linear function of the route attributes x r , it can be written in the following form: J c;w i;r (q i ;x r (a))=q i1 x 1r +q i2 x 2r +q i3 x 3r (a;c)+q i4 x 4r ( ¯ a) (4.4) where x 1r and x 2r are the distance and the number of freeway interchanges of route r, respectively. Additionally, x 3r (a;c) and x 4r ( ¯ a) are the travel time and the 80th percentile of travel time of route r during the demand realization c when the vehicles are routed according toa, respectively. Note 76 that we use the notation ¯ a for x 4r ( ¯ a) to denote that the 80th percentile of travel time depends on all the demand realizations and consequently depends on all the values ofa, for all the realizations c. To simplify the notation, we omit using the notation ¯ a when defining J c;w i;r (q i ;x r (a)). Based on the assumption that the truck drivers only know the probability distribution of the demand for the rest of the truck drivers and not the exact realization of it, their routing decisions a UE w;i;r do not depend on the exact demand realization c. Additionally, it has been shown that there possibly exist many non-equivalent UE solutions [51]. In this work, we calculate a UE solution that minimizes a weighted combination of the expected total travel time of the truck drivers and the negative of their expected total utility. Given the aforementioned, we can calculate a UE solution by solving the following optimization problem with complementarity constraints [40]: minimize a;d lE[T tr (a)](1l)E[U tr (a)] subject to 0a w i;r ?d w i F w i;r (a) 0;8w;i;r å r2R w a w i;r = 1;8w;i (4.5) wherel2[0;1] is a weighting factor,d w i is a set of free variables, the notation? means that either a w i;r = 0 ord w i F w i;r (a)= 0 and finally, F w i;r (a) is the expected utility of a truck driver belonging to cluster i who wants to travel in OD pair w using route r and is given by: F w i;r (a)= å c p c J c;w i;r (q i ;x r (a)) (4.6) where p c is the probability of demand realization c. Additionally, E[U tr (a)] represents the ex- pected total utility of the truck drivers and at the UE, it is given by: E[U tr (a)]= å c v å w=1 å i å r2R w p c d c w;i a UE w;i;r J c;w i;r (q i ;x r (a)) (4.7) where c and p c correspond to a specific realization of demand d c w;i and its associated probability, respectively. Note that by settingl = 1 in the objective function of (4.5), we can calculate the UE 77 with the minimum expected total travel time while on the other hand, by setting l = 0, we can calculate the UE with the maximum expected total utility of the truck drivers. 4.2.3 SystemOptimum(SO) In a System Optimum (SO) solution, drivers make routing decisions in an effort to minimize a total system cost compared to the UE where they are acting in a manner that maximizes their own individual utility. Before presenting the optimization problem through which we calculate the SO solution, let us first define some terms. The number of trucks traversing the road segment l is given by: X lT (a)= v å w=1 K å i=1 å r2R w :l2r d c w;i a c;w i;r (4.8) where the main difference between (4.2) and (4.8) is that in the latter, a c;w i;r depends on the exact demand realization c which is known by the coordinator. Therefore, the expected total travel time of the truck drivers in the network can be calculated by substituting (4.8) into (4.3). Additionally, the expected total total travel time of the passenger vehicles is given by: E[T p (a)]= E " L å l=1 X l p C l p (X l p ;X lT (a)) # (4.9) where it holds that C l p (X l p ;X lT (a))= C lT (X l p ;X lT (a)). In this work, we aim to minimize the expected total travel time of the truck drivers drivers and maximize their expected total utility. Therefore, we define the following objective: O(a)=l(rE[T tr (a)]+(1r)E[T p (a)])(1l)E[U tr (a)] (4.10) wherel;r2[0;1] are weighting factors and E[U tr (a)] is given by: E[U tr (a)]= å c v å w=1 å i å r2R w p c d c w;i a c;w i;r J c;w i;r (q i ;x r (a)) (4.11) 78 where the main difference between (4.7) and (4.11) is that in the latter,a c;w i;r depends on the exact demand realization c. Note that the objective function of (4.12) is a weighted combination of the expected total travel time of the truck drivers, the negative of their expected total utility and the expected total travel time of the passenger vehicles. This objective function aims to guarantee that by only routing the trucks, the expected total travel time of the passenger vehicles will not be significantly affected. Based on the aforementioned definitions, we calculate the SO solution of the network by solv- ing the following optimization problem: minimize a() O(a) subject to å r2R w a c;w i;r = 1;8c;w;i a c;w i;r 0;8c;w;i;r (4.12) where O(a) is given by (4.10). 4.3 Personalizedrouterecommendation 4.3.1 Overview A System Optimum (SO) solution is not a practical solution since as will be also experimentally shown later, drivers have an incentive to deviate from this solution in order to increase their in- dividual utility. On the other hand, a UE solution is inefficient for the network. To mitigate this issue, we propose to initially learn individual drivers’ utilities. Subsequently, using these utilities, we calculate a pricing-and-routing scheme that minimizes a weighted combination of the expected total travel time and the negative of the expected total sum of utilities while guaranteeing that every truck driver has an incentive to participate in such a mechanism. The participation to the mech- anism is voluntary. Note also that the designed mechanism is self-sustainable since the expected total payments made or received by the coordinator are equal to zero. 79 The proposed approach is based on the following steps: • Step 1: Based on drivers’ past routing choices, we divide the drivers into disjoint clusters. In our experiments, we use the K-means algorithm. However, other clustering algorithms can be also used. In case a truck driver participates in the mechanism for the first time, he/she will be asked to answer a small number of binary route choice questions. • Step 2: For each cluster, we learn a utility that is a function of distance, travel time, 80th percentile of travel time and number of freeway interchanges of a route. • Step 3: Having learned a utility function for each cluster of truck drivers, we solve an op- timization problem that calculates a pricing-and-routing scheme. This scheme is budget balanced on average and minimizes a weighted combination of the expected total travel time and the negative of the expected total utility of the truck drivers while guaranteeing that every truck driver has an incentive to participate in such a mechanism. In the following sections, we describe each step of the proposed approach in detail. 4.3.2 Clustering Using drivers’ past routing choices, we cluster them into disjoint clusters. Several algorithms can be applied for clustering. In this Chapter, we decided to use the K-means algorithm due to its simplicity and speed. The K-means algorithm divides a set of Q samples into K disjoint clusters C, each described by the mean n i of the samples in the cluster. The K-means algorithm aims to create the clusters and choose their centroids (clusters’ centers) by minimizing the within-cluster sum-of-squares criterion: argmin C K å i=1 å q2C i kqn i k 2 (4.13) 80 where q is the vector containing the responses of a truck driver to a set of binary route choice questions. Initially, the algorithm selects K cluster centers. Subsequently, the algorithm alternates between the two following steps: • Step 1: Each sample (in this case truck driver) is assigned to its closest cluster center. • Step 2: Each cluster center is updated to be the mean of all of the samples (in this case truck drivers) assigned to each previous centroid. At each iteration, we calculate the difference between the old and the new centroids and the algo- rithm stops when this value becomes less than a threshold. Having clustered the truck drivers into disjoint clusters, in the next section, we show how we can learn a utility function for each cluster of the truck drivers. 4.3.3 Utilitylearning For each cluster i of truck drivers, we solve the following optimization problem: minimize q i L(q i ;x;y) (4.14) whereL(q i ;x;y) is the binary cross-entropy loss function given as follows: L(q i ;x;y)= 1 M M å m=1 y m log(s(q i ;x m ))+(1 y m )log(1 s(q i ;x m )) (4.15) where q i are the learned parameters of the utility J i of cluster i, M is the total number of truck drivers in cluster i multiplied by the number of route choices they have made in the available dataset, y m is a binary variable that represents the route choice that a driver made and can either take value 0 for route 1 or 1 for route 2. Finally, s(q i ;x m ) denotes the probability according to 81 which we predict that a truck driver belonging to cluster i will pick route 1 or route 2 and is given by the sigmoid function: s(q i ;x m )= 1 1+ exp((J i (q i ;x m1 ) J i (q i ;x m2 ))) (4.16) where x m1 and x m2 are the attributes of route 1 and route 2, respectively. In case we assume a linear model for utility J i of cluster i, then the probability according to which we predict that a truck driver belonging to cluster i will pick route 1 or route 2 takes the following form: s(q i ;x m )= 1 1+ exp(q T i (x m1 x m2 )) (4.17) Note that by taking the difference J i (q i ;x m1 ) J i (q i ;x m2 ) in the denominator of (4.16), we make sure that the transitivity property is satisfied, i.e. if alternative a is preferred to alternative b (a b) and alternative b is preferred to alternative c (b c), then alternative a is also preferred to alternative c (a c). The transitivity property is important since by only using a binary model and doing a pairwise comparison between 2 routes, we can accurately calculate the utility of each alternative even if the total number of routes per OD pair is greater than 2. 4.3.4 Optimizationformulation Having learned the utility J i of each cluster i of truck drivers, we aim to design a mechanism that will provide personalized routing instructions to the truck drivers which at the same time will optimize a total system cost. To achieve this, we introduce a pricing scheme p c;w r to the system. Depending on demand realization c, OD pair w and route r that a truck driver follows, this pricing scheme determines if the truck driver needs to pay or receive a payment by the coordinator. The participation to the mechanism is voluntary. 82 Based on the aforementioned, let us formulate the following optimization problem: minimize a();p() O(a) subject to B w i D w i ;8w;i å c v å w=1 å i å r2R w p c d c;w i a c;w i;r p c;w r = 0 å r2R w a c;w i;r = 1;8c;w;i a c;w i;r 0;8c;w;i;r (4.18) where B w i and D w i are given by the following equations: B w i = å c å r2R w p c a c;w i;r J c;w i;r (q;x r (a))+p c;w r (4.19) D w i = max r2R w å c p c J c;w i;r (q;x r (a)) (4.20) respectively and O(a) is given by (4.10). The first constraint of (4.18) guarantees that the expected total utility of a truck driver (including payments) in case he/she decided to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she did not participate. Additionally, the second constraint of (4.18) guarantees that the total payments made or received by the coordinator are equal to zero and hence, the overall mechanism is budget balanced on average. The following lemma shows that a solution to the optimization problem (4.18) always exists. Lemma4.1 The optimization problem (4.18) is feasible. The first constraint of (4.18) can be equivalently written as: å c å r2R w p c a c;w i;r J c;w i;r (q;x r (a))+p c;w r å c p c J c;w i;r (q;x r (a));8w;i;r (4.21) 83 Letp c;w r = 0 and additionally leta c;w i;r =a UE w;i;r . The above values satisfy (4.21) and this concludes the proof. Lemma 4.1 proves the existence of a pricing-and-routing scheme that is budget balanced on average and guarantees that the expected total utility of a truck driver (including payments) in case he/she decides to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she does not participate. Additionally, it is worth noting that the pricing scheme p c;w r is uniform across clusters and it only depends on demand realization c, OD pair w and the route r. Lastly, note that a pricing-and-routing scheme with the aforementioned characteristics always exists and does not depend on the form of the utility function J c;w i;r (q;x r (a)). In the next section, we experimentally show that the proposed pricing-and-routing scheme provides a solution that is close to the SO. 4.4 SimulationResults As mentioned in section 4.3, our proposed methodology consists of 3 steps. In the first step, using drivers’ past routing decisions, we cluster them into disjoint clusters. Subsequently, in the second step, for each cluster, we learn a utility that is a function of distance, travel time, 80th percentile of travel time and number of freeway interchanges of a route. Lastly, having learned a utility function for each cluster of truck drivers, we solve the optimization problem (4.18). In this section, we run simulations in order to demonstrate the efficiency of the proposed ap- proach. First, we describe how we generated synthetic data. Using the generated data, after split- ting the data into a train and a test set, we cluster the drivers, learn the utility function of each cluster and then decide the appropriate number of clusters to use. Subsequently, we experimen- tally show the degree at which the truck drivers would have an incentive to deviate from a SO solution, demonstrating the necessity for a pricing-and-routing scheme. Lastly, we run simulations in a benchmark transportation network and we compare the proposed pricing-and-routing scheme with the UE and the SO solutions. 84 4.4.1 DataGeneration To cluster the truck drivers into disjoint clusters and learn a utility function for each cluster, we first need to have access to drivers’ past routing decisions. As mentioned in section 4.3, in case a truck driver participates in the mechanism for the first time, he/she will be asked to answer a small number of binary route choice questions. In this work, we use 9 route choice questions for training and 5 questions for test 1 . An example of a route choice question is shown in Table 4.2. Route1 Route2 Distance (miles) 45.0 55.0 Travel time duration (min) 55.0 50.0 80th percentile of travel time (min) 69.0 65.0 # of freeway interchanges 7 4 Table 4.2: An example of a route choice question used to learn drivers’ routing preferences. As a first step in the data generation process, we generate the variables q that describe the drivers’ routing preferences. To do this, we draw samples from a Gaussian mixture model assuming 3 components consisting of isotropic Gaussian distributions. In our experiments, we generated 200 samples per cluster. Note that the term ‘clusters’ here refers to the Gaussian distributions of the Gaussian mixture model. The cluster centers and the standard deviations of the corresponding covariance matrices were assumed to be: m q 1 =[0:08;0:12;0:06;0:02] m q 2 =[0:15;0:15;0:15;0:05] m q 3 =[0:04;0:12;0:12;0:04] 1 The train and test route choice questions can be found in this link: https://bit.ly/3o8fZif 85 and s q 1 =[0:03;0:04;0:02;0:01] s q 2 =[0:03;0:04;0:04;0:01] s q 3 =[0:02;0:04;0:02;0:02] respectively, whereq i are the parameters of cluster i. Note that for each cluster, there are 4 param- eters that correspond to distance, travel time, 80th percentile of travel time and number of freeway interchanges, respectively. The cluster centers were chosen with the following intuition. The first Gaussian distribution includes the truck drivers who consider travel time as the most important factor, followed by the distance of the route. The second Gaussian distribution describes the truck drivers who treat both distance, travel time and the 80th percentile of travel time as equally im- portant factors when making a routing decision. Lastly, the third Gaussian distribution describes the truck drivers who consider the travel time and the 80th percentile of travel time as the most important factors when making a routing decision. Assuming that the truck drivers’ utility can be described by a linear function, e.g. like the one described by (4.4), using the generatedq in (4.17), we can subsequently generate the responses of the drivers to the route choice questions. Given that we generated a total of 600 samples from the Gaussian mixture model and we used 9 route choice questions for training and 5 questions for test, we got a total of 5400 data points for training and 3000 for test. 4.4.2 ClusteringandUtilityLearning Given the responses of the drivers to the route choice questions, we can cluster the drivers into disjoint clusters using the K-means algorithm as described in Section 4.3.2 and then use the learn- ing algorithm as described in Section 4.3.3 in order to learn the parameters ˜ q i of each cluster i. However, an important factor in this procedure is the number of clusters K that we assume in the K-means algorithm. To apply the K-means algorithm, we used the Scikit-learn package in Python [79]. 86 Before describing the way we determine K, let us first mention some additional implementation details regarding the learning algorithm described in Section 4.3.3. Assuming that the function that describes the utility of each cluster of truck drivers is linear, e.g. like the one described by (4.4), we build a linear model that we train using projected gradient descent with a fixed learning rate h = 0:001. The projected gradient descent is implemented as follows: x k = ˜ q k hÑL( ˜ q k ) ˜ q k+1 = Pr R 4 0 (x k ) (4.22) whereL( ˜ q k ) is given by (4.15) and Pr is the projection to the positive orthant. In our case, the projection operator ensures that at each iteration, the learned ˜ q k is non-positive. Additionally, in case there is class imbalance in the data, we modify (4.15) as follows: L(q i ;x;y)= 1 M M å m=1 e 0;i y m log(s(q i ;x m ))+e 1;i (1 y m )log(1 s(q i ;x m )) (4.23) where e 0;i and e 1;i are weights for the classes 0 (Route 1) and 1 (Route 2), respectively. After clustering the truck drivers using the K-means algorithm, for each cluster i, we determine the values ofe 0;i ande 1;i using the following formulas: e 0;i = max(# of samples in class 0;# of samples in class 1) # of samples in class 0 e 1;i = max(# of samples in class 0;# of samples in class 1) # of samples in class 1 Note that the loss function described by (4.23) is the weighted cross-entropy loss function [48]. There are several ways to pick the weights e 0;i and e 1;i . However, a common approach is to give more weight to the minority class. At this point, let us describe how we choose the number of clusters K in the K-means algorithm. First, we cluster the drivers based on their responses to the 9 route choice questions that are used for training, as described in Section 4.4.1. In this work, we experiment with K = 1;2;3;4;5 and 87 10. Subsequently, for each cluster i of truck drivers, we train a linear model using the 9 training route choice questions and we learn the parameters ˜ q i using (4.23) and (4.17). Having learned the parameters ˜ q i for each cluster, we test the performance of our method using the 5 test route choice questions. To measure the performance of our learning approach, we use the Area Under the Receiver Operating Characteristic curve (AUROC) [31] and the Area Under the Precision-Recall curve (AUPR) [66]. The results are presented in Table 4.3. K AUROC AUPR 1 0.75 0.79 2 0.86 0.90 3 0.90 0.94 4 0.89 0.94 5 0.94 0.96 10 0.94 0.96 Table 4.3: AUROC and AUPR for different numbers of clusters K. As can be observed from Table 4.3, choosing K = 5 or K = 10 gives us the highest AUROC and AUPR scores. However, note that as the number of clusters K increases, the computational complexity increases as well for two main reasons. First, for each cluster i, we need to learn a dif- ferent set of parameters ˜ q i . Second, note that the set of decision variablesa() in the optimization problem (4.18) depends on i and therefore, as the number of clusters increases, the computational complexity of the optimization problem (4.18) also increases. Based on these observations, we choose K = 5 as the appropriate number of clusters for the rest of the experiments. Then, for each cluster, we train a linear model using the 9 training route choice questions and we learn the parameters ˜ q i using (4.23) and (4.17). These parameters are shown below: ˜ q 1 =[0:843;0:529;0:636;0:148] ˜ q 2 =[0:686;1:225;1:189;0] ˜ q 3 =[0:770;0:912;0:319;0:537] ˜ q 4 =[0:654;1:310;0:765;0:340] ˜ q 5 =[0;0:868;1:004;1:059] (4.24) 88 where each element of a set of parameters ˜ q i corresponds to distance, travel time, 80th percentile of travel time and number of freeway interchanges. Having clustered the truck drivers into K= 5 disjoint clusters and by using the learned param- eters ˜ q i given by (4.24), in the following sections, we compute the UE, the SO and a Pricing-and- Routing scheme by solving the optimization problems (4.5), (4.12) and (4.18), respectively. 4.4.3 NecessityofaPricing-and-RoutingScheme In this section, we experimentally show the degree that truck drivers have an incentive to deviate from the SO generated routes. In our experiments, we use the Sioux Falls network which is a benchmark in the transportation research field consisting of 24 nodes and 76 links [57]. The Sioux Falls network 2 is shown in Figure 4.1. In our experiments, we assumed that the cost of each route Figure 4.1: The Sioux Falls network. corresponds to travel time and can be described by a Bureau of Public Roads (BPR) function [89] of the form: C lT (X l p ;X lT )=g a +g b X l p + 3X lT g c ! 4 2 The distance of each link can be found in this link: https://bit.ly/3lW8Oah 89 whereg a ;g b andg c are constants 3 . After solving an equilibrium assignment problem for the passenger vehicles, we calculated the number of passenger vehicles at each link of the Sioux Falls network. Similar to [77], these num- bers were assumed to remain constant 4 . For the truck drivers, we assumed that they want to travel in 6 available Origin-Destination (OD) pairs, namely(E 1 ;E 7 );(E 1 ;E 11 );(E 10 ;E 11 );(E 10 ;E 20 );(E 15 ;E 5 ) and (E 24 ;E 10 ) and that they follow the 5 least congested routes per OD pair. Their demand was assumed to take one of the 5 equiprobable values, namely d 1 ;d 2 ;d 3 ;d 4 and d 5 5 . Lastly, the values of the weighting factors of the objective function (4.10) were chosen to bel =r = 0:9. Using the aforementioned, we solved the optimization problem (4.12) that gives us the SO solution. To demonstrate the necessity for a pricing-and-routing scheme, in Table 4.4, we present results that show the percentage by which the truck drivers could increase their utility in case they decided to deviate from the SO solution. ODpair/Cluster 1 2 3 4 5 1 4.0 0.3 6.0 0.3 2.0 2 10.0 0 7.7 3.9 2.5 3 5.0 0 7.0 0.5 3.5 4 8.9 0.1 6.2 4.3 4.6 5 8.3 0 7.6 7.3 7.1 6 9.6 2.8 6.8 6.5 12.0 Table 4.4: Percentage by which the truck drivers could increase their utility in case they decided to deviate from the SO solution. More specifically, Table 4.4 measures: I 1 I 2 I 2 100% 3 The values ofg a ,g b andg c can be found in this link: https://bit.ly/3ibHNi1 4 The number of passenger vehicles at each link of the network can be found in this link: https://bit.ly/3zBJoDy 5 The demand values of the truck drivers can be found in this link: https://bit.ly/3zJ3Ims 90 where I 1 and I 2 are given by: I 1 = max r2R w å c p c J c;w i;r ( ˜ q;x r (a SO )) I 2 = å c å r2R w p c a SO c;w;i;r J c;w i;r ( ˜ q;x r (a SO ))+p c;w r respectively. As can be observed from the results of Table 4.4, most truck drivers have an incentive to deviate from the SO solution regardless of the OD pair or the cluster they belong. Therefore, the SO solution is not a practical solution and a pricing-and-routing scheme is needed in order to guarantee the participation of the truck drivers. 4.4.4 Pricing-and-Routing In this section, we run simulations in the Sioux Falls network. More specifically, using the learned parameters ˜ q i from (4.24), we calculate the UE, the SO and the pricing-and-routing scheme by solving the optimization problems (4.5), (4.12) and (4.18), respectively. To solve these problems, the fmincon optimization solver implemented in the MATLAB Optimization Toolbox [22] was used. Since fmincon solves optimization problems with local optimality guarantees, in this section, we compare local minima between the approaches. Note also that the UE solution was obtained with a constraint tolerance of 4 10 3 , while the SO and the pricing-and-routing scheme solutions were obtained with a constraint tolerance of 10 6 . In our experiments, we increased the constraint tolerance when calculating the UE solution compared to the default of 10 6 in order to accelerate the computation of the solution. The results are shown in Table 4.5. UE SO Pricing-and-Routing E[T tr ] 33025.5 30141.3 30169.7 E[U tr ] -69688.2 -60681.5 -60996.0 E[T S ] 51381.4 47557.9 47603.8 O(a) 35371.5 32050.1 32106.2 Table 4.5: Simulation Results of the Sioux Falls network. 91 As can be observed from the results presented in Table 4.5, the pricing-and-routing scheme achieves a significant reduction in the expected total travel time of the truck drivers and the ex- pected total travel time of the network compared to the UE, while simultaneously increasing the expected total utility of the truck drivers. Furthermore, the pricing-and-routing scheme achieves a performance that is close to the SO solution. However, in contrast with the SO, the pricing-and- routing scheme guarantees that the expected total utility of a truck driver (including payments) in case he/she decided to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she did not participate through the first constraint of (4.18). Lastly, as already mentioned, the expected total payments made or received by the coordinator are equal to zero, thus making the mechanism self-sustainable. 92 Chapter5 ConclusionandFutureResearch 5.1 Conclusion In this dissertation, we studied the problem of the inefficiency of an equilibrium flow pattern and the System Optimum (SO) solution in a general transportation network. It is well known that a SO solution is not a practical solution since some drivers may benefit while some others may be harmed compared to the User Equilibrium (UE). To overcome this issue, we proposed the use of combined pricing-and-routing schemes specifically designed for truck drivers that provide individual incentives for participation, satisfy the budget balance on average property and drive the network as close as possible to the SO solution. First, assuming a network model with stochastic Origin–Destination (OD) demands for the truck drivers and that the planning horizon is split into discrete non-overlapping time intervals, we derived sufficient conditions which prove the existence of Pareto-improving, truthful in equilib- rium and budget balanced on average mechanisms. Subsequently, we studied the case with user heterogeneity in Value-Of-Time (VOT) by adopting a multi-class model with stochastic OD de- mands for the truck drivers. A main characteristic of the proposed approach is that the coordinator asks the truck drivers to declare their desired OD pair and pick their individual VOT from a set of N available options, and guarantees that the resulting pricing-and-routing scheme is Pareto- improving, i.e. every truck driver will be better-off compared to the User Equilibrium (UE) and that every truck driver will have an incentive to truthfully declare his/her VOT, while leading to 93 a revenue-neutral (budget balanced) on average mechanism. This approach enabled us to design personalized (VOT-based) pricing-and-routing schemes. Lastly, based on previous research efforts that showed that there are several factors that can affect drivers’ routing decisions and that each driver values each factor differently, we proposed an approach that provides personalized routing suggestions and concurrently optimizes over a total system-wide cost through a combined pricing- and-routing scheme that satisfies the budget balance on average property and ensures that every truck driver has an incentive to participate in the proposed mechanism by guaranteeing that the ex- pected total utility of a truck driver (including payments) in case he/she decides to participate in the mechanism, is greater than or equal to his/her expected utility in case he/she does not participate. 5.2 FutureResearch The main directions for future research are as follows: • Pricing-and-Routing in a Dynamic Environment. In Chapters 2, 3 and 4, we studied pricing- and-routing schemes that provide individual incentives for participation to the truck drivers and concurrently satisfy the budget balanced on average property. A common characteristic of the proposed approaches is the fact that they have been studied for networks that are in equilibrium state and hence, static. However, accidents, major public events, and road closures affect the traffic conditions and could disrupt the network. In such scenarios, the transportation network is no longer in an equilibrium state. Therefore, dynamic pricing- and-routing schemes that change over time need to be studied in order to provide individual incentives for participation to the truck drivers in a dynamic environment. • Scalability. Most of the approaches proposed in Chapters 2, 3 and 4 do not take the scala- bility issue into account. The AOPS method discussed in Chapter 3 proposed to calculate the payment scheme through a simple algebraic equation and solve an optimization problem to optimally route the truck drivers in the network. However, none of the proposed ap- proaches can scale to very large transportation networks with hundreds or thousands of OD 94 pairs. Since the pricing schemes studied in this dissertation are all route-based, a potential remedy to the scalability problem is the study of link-based, OD-based, distance-based etc. pricing schemes that provide individual incentives for participation to the truck drivers and concurrently satisfy the budget balanced on average property. Another potential solution to the scalability problem is the study of distributed optimization methods in order to calculate pricing-and-routing schemes that satisfy the desired properties. 95 ReferenceList [1] H.Z. Aashtiani and T.L. Magnanti. Equilibria on a congested transportation network. 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Let us define: å c N å t=1 å r2R j p c ¯ a c;t;¯ t j;r t c; j;t ¯ t;r = å c N å t=1 å r2R j p c ¯ a c;t;¯ t j;r ( ¯ J M;c; j t;¯ t;r + D(t; ¯ t))+ A UE;D c; j;¯ t (A.1) At this point, chooset c; j;¯ t;t;r such that: t c; j;¯ t;t;r =( ¯ J M c; j;t;¯ t;r + D(t; ¯ t))+ A UE;D c; j;¯ t (A.2) Now, substituting (A.2) to the second constraint of (2.14) and using (2.16), we get: å c N å t=1 å r2R j p c ¯ a c;t;i j;r [( ¯ J M c; j;t;i;r + D(t;i))+ A UE;D c; j;i ]+ å c N å t=1 å r2R j p c ¯ a c;t;i j;r [ ¯ J M c; j;t;i;r + D(t;i)] å c N å t=1 å r2R j p c ¯ a c;t;k j;r [( ¯ J M c; j;t;k;r + D(t;k))+ A UE;D c; j;k ]+ å c N å t=1 å r2R j p c ¯ a c;t;k j;r [ ¯ J M c; j;t;k;r + D(t;i)] (A.3) which can be equivalently written as: å c N å t=1 å r2R j p c ¯ a c;t;i j;r A UE;D c; j;i å c N å t=1 å r2R j p c ¯ a c;t;k j;r A UE;D c; j;k + å c N å t=1 å r2R j p c ¯ a c;t;k j;r [D(t;i) D(t;k)], , å c p c A UE;D c; j;i N å t=1 å r2R j ¯ a c;t;i j;r å c p c A UE;D c; j;k N å t=1 å r2R j ¯ a c;t;k j;r + å c N å t=1 å r2R j p c ¯ a c;t;k j;r [D(t;i) D(t;k)], , J UE;D j;i J UE;D j;k + å c N å t=1 å r2R j p c ¯ a c;t;k j;r (D(t;i) D(t;k)) (A.4) 106 where in the last induction, we made use of the second constraint of (2.10). Working in the same manner and substituting (A.2) to the third constraint of (2.14) and using (2.16), we get: å c N å t=1 å r2R j p c ¯ a c;t;k j;r [( ¯ J M c; j;t;k;r + D(t;k))+ A UE;D c; j;k ]+ å c N å t=1 å r2R j p c ¯ a c;t;k j;r [ ¯ J M c; j;t;k;r + D(t;k)] å c N å t=1 å r2R j p c ¯ a c;t;i j;r [( ¯ J M c; j;t;i;r + D(t;i))+ A UE;D c; j;i ]+ å c N å t=1 å r2R j p c ¯ a c;t;i j;r [ ¯ J M c; j;t;i;r + D(t;k)] (A.5) which can be equivalently written as: å c N å t=1 å r2R j p c ¯ a c;t;k j;r A UE;D c; j;k å c N å t=1 å r2R j p c ¯ a c;t;i j;r A UE;D c; j;i + å c N å t=1 å r2R j p c ¯ a c;t;i j;r [D(t;k) D(t;i)], , å c p c A UE;D c; j;k N å t=1 å r2R j ¯ a c;t;k j;r å c p c A UE;D c; j;i N å t=1 å r2R j ¯ a c;t;i j;r + å c N å t=1 å r2R j p c ¯ a c;t;i j;r [D(t;k) D(t;i)], , J UE;D j;k J UE;D j;i + å c N å t=1 å r2R j p c ¯ a c;t;i j;r (D(t;k) D(t;i)) (A.6) Regarding the fourth constraint of (2.14), using (A.2) we get: å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r t c; j;t;¯ t;r = = å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r [( ¯ J M c; j;t;¯ t;r + D(t; ¯ t))+ A UE;D c; j;¯ t ]= =E[ ¯ T M tr ( ¯ a)]+ å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r A UE;D c; j;¯ t = =E[ ¯ T M tr ( ¯ a)]+ å c v å j=1 N å ¯ t=1 p c d ¯ t c; j A UE;D c; j;¯ t N å t=1 å r2R j ¯ a c;t;¯ t j;r = =E[ ¯ T M tr ( ¯ a)]+ E T UE tr 0 where in the last equality we used the second constraint of (2.10) and the last inequality holds due to the first constraint of (2.10). Note at this point that (A.4) and (A.6) are identical to (2.17) and (2.18) and this concludes the proof. A.0.2 ProofofTheorem2.2 In Theorem 2.1, we proved the feasibility of (2.14) if (2.17) and (2.18) hold. Hence, now, we only need to prove that even if we enforce the last constraint of (2.14) to hold as an equality, the problem will remain feasible. So, suppose we solve (2.14) and we calculate t ? which gives us an optimal 107 objective value U ? . In this case, if the expected total payments sum up to zero, we are done. So, let us assume: å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r t ? c; j;t;¯ t;r = T ? > 0 (A.7) Then, define: ˆ t c; j;t;¯ t;r =t ? c; j;t;¯ t;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j ;8c; j;t; ¯ t;r (A.8) Substituting (A.8) into the last constraint of (2.14) and using (A.7), we get: å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r ˆ t c; j;t;¯ t;r = = å c v å j=1 N å t=1 N å ¯ t=1 å r2R j p c d ¯ t c; j ¯ a c;t;¯ t j;r " t ? c; j;t;¯ t;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j # = 0 Now, observe that the second term of (A.8) is positive and hence ˆ t c; j;¯ t;t;r <t ? c; j;¯ t;t;r ;8c; j;t; ¯ t;r. So, sincet ? c; j;¯ t;t;r satisfied the first constraint of (2.14), ˆ t c; j;¯ t;t;r will also satisfy it. Regarding the first truthfulness constraint, substituting (A.8) into the second constraint of (2.14) and using (2.15) and (2.16), we get: å c N å t=1 å r2R j p c ¯ a c;t;i j;r ˆ t c; j;t;i;r ¯ a c;t;k j;r ˆ t c; j;t;k;r E[ ¯ J M;D i;k ]+ E[ ¯ J M;D i;i ]= = å c N å t=1 å r2R j p c " ¯ a c;t;i j;r t ? c; j;t;i;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j ! ¯ a c;t;k j;r t ? c; j;t;k;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j !# E[ ¯ J M;D i;k ]+ E[ ¯ J M;D i;i ] å c N å t=1 å r2R j p c " ¯ a c;t;k j;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j ¯ a c;t;i j;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j # = = T ? å c p c å v j=1 å N ¯ t=1 d ¯ t c; j N å t=1 å r2R j ¯ a c;t;k j;r ¯ a c;t;i j;r = 0;8 j;k i 108 where in the inequality part we used the fact thatt ? is feasible in (2.14) and in the last equality we made use of the second constraint of (2.10). Now, working in the same manner and substituting (A.8) into the third constraint of (2.14) and using (2.15) and (2.16), we get: å c N å t=1 å r2R j p c ¯ a c;t;k j;r ˆ t c; j;t;k;r ¯ a c;t;i j;r ˆ t c; j;t;i;r E[ ¯ J M;D k;i ]+ E[ ¯ J M;D k;k ]= = å c N å t=1 å r2R j p c " ¯ a c;t;k j;r t ? c; j;t;k;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j ! ¯ a c;t;i j;r t ? c; j;t;i;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j !# E[ ¯ J M;D k;i ]+ E[ ¯ J M;D k;k ] å c N å t=1 å r2R j p c " ¯ a c;t;i j;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j ¯ a c;t;k j;r T ? å v j=1 å N ¯ t=1 d ¯ t c; j # = = T ? å c p c å v j=1 å N ¯ t=1 d ¯ t c; j N å t=1 å r2R j ¯ a c;t;i j;r ¯ a c;t;k j;r = 0;8 j;k i Up to now, we have proved that if at optimality of (2.14) the expected total payments are greater than zero, then we can always find another ˆ t given by (A.8) which is also feasible in (2.14). This proves that the optimization problem (2.14) will remain feasible even if we impose its last constraint to hold as an equality and equivalently proves the feasibility of the optimization problem (2.19). At this point, observe that U(t) is a nonconvex function due to the fact that the variablet appears in the denominator through the term E[T M tr ]. However, due to the last constraint of (2.19), E[T M tr ] will no longer be a function oft converting U(t) into a convex (quadratic) function and this concludes the proof. A.0.3 ProofofLemma2.3 First, it can be easily seen that the UE satisfies the first three constraints of (2.20). Regarding con- straints (2.17) and (2.18), observe at first, that they are equivalent to (A.3) and (A.5) respectively. Now, at the UE, (A.3) and (A.5) can be rewritten as: å c N å t=1 å r2R j p c a UE;c;t i; j;r [ ¯ J UE c; j;t;i;r + D(t;i)] å c N å t=1 å r2R j p c a UE;c;t k; j;r [ ¯ J UE c; j;t;k;r + D(t;i)], , E[ ¯ J UE;D i;i ] E[ ¯ J UE;D i;k ] (A.9) and å c N å t=1 å r2R j p c a UE;c;t k; j;r [ ¯ J UE c; j;t;k;r + D(t;k)] å c N å t=1 å r2R j p c a UE;c;t i; j;r [ ¯ J UE c; j;t;i;r + D(t;k)], , E[ ¯ J UE;D k;k ] E[ ¯ J UE;D k;i ] (A.10) 109 respectively. Inequalities (A.9) and (A.10) express that, at the UE, no player (truck driver) has an incentive to pretend that he/she wants to travel during a different time interval than the one he/she actually wants since his/her expected cost is going to be higher. This is a property that holds true at the UE and this concludes the proof. 110 AppendixB DiscussiononthethePareto-Improvementandthe TruthfulnessProperties In Sections 3.3.1 and 3.3.2, we designed the Optimum Pricing Scheme (OPS) and the Approxi- mately Optimum Pricing Scheme (AOPS) and we guaranteed the Pareto-improvement, the truth- fulness and the budget balance on average properties. As also mentioned in Sections 3.3.1 and 3.3.2, using the formulations of OPS and AOPS, it is theoretically possible that for some demand realizations, some drivers are given a route with a higher total travel time (travel time + payments expressed in time units) compared to the average travel time at the UE. However, due to the fact that individual drivers only know the probability distribution of the demand and not the exact realization of it and hence they have incomplete information of the traffic conditions, at the time they make their decision, their expected cost under the mechanism suggestions M is going to be lower than their corresponding cost at the UE and therefore, they will always be willing to participate in both coordinated schemes. At this point, we should note that it is possible to design an even more robust version for the Pareto-improvement property that guarantees that for every demand realization c and for every route r, the total travel time of each driver (travel time + payments expressed in time units) is going to be lower under the mechanism suggestions compared his/her average travel time at the UE. We call this property robust Pareto-improvement. As far as it concerns the truthfulness property, both OPS and AOPS guaranteed that every driver will have an incentive to truthfully declare his/her VOT since at the time they make their decision, their expected cost in the case they are truthful is lower than their corresponding cost in the case where they declared a different VOT than their actual one. We call this property ex- ante truthfulness. However, note that we are able to guarantee an even stronger version of the truthfulness property by ensuring that at each demand realization c, the average total travel time (travel time + payments expressed in time units) of a truck driver in case they are truthful is lower than their corresponding total travel time in the case where they declared a different VOT than their actual one. We call this property ex-post truthfulness. Based on the aforementioned, we want to design a mechanism with the following properties. • A mechanism is robust Pareto-improving if: J M;c; j w;r + 1 s w p c; j w;r A UE c; j ;8c; j;w;r (B.1) 111 • A mechanism is ex-post truthful if: å r2R j a c; j i;r (J M;c; j i;r + 1 s i p c; j i;r ) å r2R j a c; j k;r (J M;c; j k;r + 1 s i p c; j k;r );8c; j;i;k (B.2) • A mechanism is budget balanced on average if: å c v å j=1 N å w=1 å r2R j p c d w c; j a c; j w;r p c; j w;r = 0 (B.3) For convenience of the reader, we state that the payment scheme that we used in the AOPS formulation is given by: p AOPS c; j;w;r = s w (A UE c; j J M;c; j w;r )+ s w å N l=1 s l E T mon;M tr E T mon;UE tr å v j=1 d w c; j (B.4) Now, let us formulate the following optimization problem: minimize a() l(mE[T tr (a)]+(1m)E[T p (a)])+(1l)E[T mon tr (a)] subject to E[T mon;M tr ] E[T mon;UE tr ] Q c; j i;k (a) U c; j i;k (a);8c; j;i;k å r2R j a c; j w;r = 1;8c; j;w a c; j w;r 0;8c; j;w;r (B.5) where Q c; j i;k (a) and U c; j i;k (a) are given by the following equations: Q c; j i;k (a)= 1 s k s i A UE c; j + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d i c; j (B.6) U c; j i;k (a)= 1 s k s i å r2R j a c; j k;r J M;c; j k;r + s k s i 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d k c; j (B.7) Note that a solution to the optimization problem described by (B.5)-(B.7) always exists since the UE satisfies all of its constraints. Leta AOPSEPT be the optimum solution of the optimization problem described by (B.5)-(B.7). In Proposition 1, we prove that the pricing-and-routing scheme described by (B.4)-(B.7) can guarantee the robust Pareto-improvement, the ex-post truthfulness and the budget balance on aver- age properties. We call this scheme Approximately Optimum Pricing Scheme with Ex-Post Truth- fulness guarantees (AOPS-EPT). Note that the only difference between AOPS-EPT and AOPS is at the second constraint of the corresponding optimization problems. More specifically, observe that the second constraint of (3.18) which corresponds to AOPS, is calculated by taking the expectation over the different demand realizations which makes the truthfulness property hold ex-ante, i.e. at 112 the time they make their decision, the expected cost of the truck drivers in the case they are truthful is lower than their corresponding cost in the case where they declared a different VOT than their actual one. On the other hand, the second constraint of (B.5) which corresponds to AOPS-EPT, holds for every possible demand realization c, making the truthfulness property hold ex-post. Proposition1 The pair (a AOPSEPT ,p AOPS c; j;w;r ) makes every truck driver better-off compared to the UE (robust Pareto-improvement), guarantees that every user will have an incentive (ex-post) to truthfully declare his/her VOT and leads to a budget balanced on average mechanism. Proof. Similar to the proof of Theorem 3.1, substituting (B.4) into (B.1), we get: J M;c; j w;r + A UE c; j J M;c; j w;r + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d w c; j A UE c; j , , 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d w c; j 0 which holds true if and only if E[T mon;M tr ] E[T mon;UE tr ] which is equivalent to the first constraint of (B.5). Additionally, a user will have an incentive (ex-post) to truthfully declare his/her VOT if (B.2) holds. Therefore, substituting (B.4) into (B.2), we get: å r2R j a c; j i;r J M;c; j i;r + A UE c; j J M;c; j i;r + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d i c; j å r2R j a c; j k;r J M;c; j k;r + s k s i A UE c; j J M;c; j k;r + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d k c; j , , 1 s k s i A UE c; j + 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d i c; j 1 s k s i å r2R j a c; j k;r J M;c; j k;r + s k s i 1 å N l=1 s l E[T mon;M tr ] E T mon;UE tr å v j=1 d k c; j where the last inequality is equivalent to the second constraint of (B.5). Last, a mechanism is budget balanced on average if (B.3) holds. In Theorem 3.1, we proved that (B.4) satisfies the budget balance on average property and this concludes the proof. There are three main reasons that justify our choice to use the AOPS instead of the AOPS-EPT in our analysis. First, since the drivers only know the probability distribution of the demand and not the exact realization of it, they have incomplete information of the traffic conditions. AOPS is still sufficient to guarantee that the drivers will have an incentive to participate in the mechanism and truthfully declare their VOT since at the time they make their decision, their expected cost in the case they are truthful is lower than their corresponding cost in the case where they declared a different VOT than their actual one. Second, using AOPS-EPT would be less computationally efficient since the second constraint of (B.5) should hold for every possible demand realization c which significantly increases the number of constraints compared to the optimization problem (3.18). Third, one can expect that the optimum solution calculated by AOPS-EPT would be less 113 efficient compared to the one calculated by AOPS since the size of the feasible region over which we optimize is smaller. 114
Abstract (if available)
Abstract
The sharp increase in e-commerce over the last few years has led to an increase in the volume of trucks both in ports and in commercial areas. The efficient use of the road network for freight transport has a big impact on travel times, pollution, and fuel consumption, as well as on the mobility of passenger vehicles. The continuously increasing use of navigation apps has led drivers to make their routing decisions in an independent manner in an effort to minimize their own individual travel time, with possible significant deviation from a socially optimum solution. In contrast, a System Optimum (SO) solution is not a practical solution since some drivers may benefit while some others may be harmed compared to the User Equilibrium (UE), raising several equity and fairness issues. ❧ In this dissertation, we address the problem of the inefficiency between the UE and the SO through the use of combined pricing-and-routing schemes. Our main goal is the design of pricing-and-routing schemes that are budget balanced on average and make every participant better-off, while concurrently driving the transportation network as close as possible to the SO solution. ❧ In the first part of this dissertation, we design coordination mechanisms that can make every user better-off compared to the UE (Pareto-improvement) while concurrently leading the transportation network as close as possible to the SO solution. Initially, for the case where the users are asked to declare both their Origin-Destination (OD) pair as well as their desired departure time, we derive sufficient conditions for the existence of Pareto-improving, truthful in equilibrium and revenue-neutral on average mechanisms. Subsequently, assuming the existence of heterogeneous users with distinct Value-of-Time (VOT) and that the drivers are asked to declare both their OD pair as well as pick their VOT from a set of N available options, we prove the existence of Pareto-improving and revenue-neutral on average mechanisms that make every driver have an incentive to truthfully declare his/her VOT. This result allows us to design personalized (VOT-based) pricing-and-routing schemes. ❧ In the last part of this dissertation, given that there are several factors that determine a driver's routing decisions, we aim to design a mechanism that provides personalized routing suggestions to truck drivers. To achieve this, we propose to divide the drivers into disjoint clusters and learn the routing preferences of each cluster using drivers' individual routing decisions. Having learned a utility function that describes the routing preferences of each cluster of drivers, we propose an approach that optimizes over a total system-wide cost through a combined pricing-and-routing scheme that satisfies the budget balance on average property and ensures that every driver has an incentive to participate in the proposed mechanism.
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Papadopoulos, Aristotelis Angelos
(author)
Core Title
Personalized Pareto-improving pricing-and-routing schemes with preference learning for optimum freight routing
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Degree Conferral Date
2022-05
Publication Date
12/15/2022
Defense Date
12/02/2021
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
freight routing,OAI-PMH Harvest,Pareto-improvement,preference learning,road pricing,traffic assignment,user equilibrium
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Ioannou, Petros (
committee chair
), Bogdan, Paul (
committee member
), Dessouky, Maged (
committee member
)
Creator Email
aristotelis.a.papadopoulos@gmail.com,aristotp@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-oUC18807291
Unique identifier
UC18807291
Legacy Identifier
etd-Papadopoul-10313
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Papadopoulos, Aristotelis Angelos
Type
texts
Source
20211223-wayne-usctheses-batch-906-nissen
(batch),
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 author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.
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
Repository Email
cisadmin@lib.usc.edu
Tags
freight routing
Pareto-improvement
preference learning
road pricing
traffic assignment
user equilibrium