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Models and algorithms for pricing and routing in ride-sharing

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Models and algorithms for pricing and routing in ride-sharing

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Content Models and Algorithms for Pricing and Routing in Ride-Sharing by Shichun Hu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (INDUSTRIAL AND SYSTEM ENGINEERING) May 2022 Copyright 2022 Shichun Hu Dedication This paper is dedicated to my family and friends whose constant support and understanding helped my completion of this dissertation. First, to my parents who supported my decision on doctoral study and have never interfered with my choice of major. They never tried to make me a big fish and are always prioritizing my health. In particular, I would like to thank my mother for her selfless love since my childhood and her sense of justice that have large impact on who I am today. Second, to my girlfriend who gave me enormous patience on how to become a better person. I would not be able to better understand myself without her help. The road toward graduation was full of challenges and I am grateful for her companion along the way. Last but not the least, to my long-time friend Zha who has been there with me since day one in the United States. Our friendship says it all. Now that I graduate first, I wish him a smooth journey to gradu- ation in the future. ii Acknowledgements I would like to thank my advisor, Professor Maged M. Dessouky, for helping me and guiding me through- out this dissertation. Having him as my advisor is such a fortunate experience for my doctoral study. Professor Dessouky is always patient and never judge his students. He is open to different research ideas and encourages me and other students to explore and implement them. I would also want to thank my committee members (qualification and defense): Professor Geneviene Giuliano, Professor Phebe Vayanos, Professor John Carlsson and Professor Petros Ioannou. Their con- structive feedback helped me to improve the dissertation. iii TableofContents Dedication ii Acknowledgements iii ListofTables vi ListofFigures vii Abstract viii Chapter1: Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter2: Introduction 6 2.1 The Cost-Sharing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The Vehicle Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Dynamic Pickup and Delivery Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Research Gap and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter3: MechanismsforRide-Sharing 15 3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Ride-Sharing Mechanism Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 The Proposed Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.1 Driver-out-of-Coalition Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.2 Passengers Predicting Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2.1 Robust Optimization Method for Totalα Value Estimation . . . . . . . . 30 3.3.3 Driver-in-Coalition Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 An Example of the Proposed Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter4: MechanismsforRide-SharingwithTimeWindows 43 4.1 Basic Discount Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Inconvenience Cost Based Discount Method . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 iv Chapter5: Ride-SharingRoutingwithWalking 58 5.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Dynamic Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2.1 The Routing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.2 The Location Selecting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter6: ConclusionandFutureWork 77 Bibliography 80 v ListofTables 3.1 The Trajectories ofβ π (k),t under the Different Mechanisms . . . . . . . . . . . . . . . . . . 40 3.2 Average Performance Measures for the Different Mechanisms . . . . . . . . . . . . . . . . 42 4.1 Simulation Settings for the Different Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Effect of Loss of Ex-post Incentive Compatibility Property . . . . . . . . . . . . . . . . . . 55 4.3 Average Performance Measures for the Discount Methods in Scenario 1 . . . . . . . . . . . 55 4.4 Average Performance Measures for the Discount Methods in Scenario 2 . . . . . . . . . . . 56 4.5 Average Performance Measures for the Discount Methods in Scenario 3 . . . . . . . . . . . 56 5.1 Performance Measures for HOV Experimental Group . . . . . . . . . . . . . . . . . . . . . 75 5.2 Performance Measures for Meeting Points Experimental Group . . . . . . . . . . . . . . . 76 5.3 Average IVT Ratio with HOV Lanes and Meeting Points . . . . . . . . . . . . . . . . . . . 76 5.4 Percentage Requests Served with HOV Lanes and Meeting Points . . . . . . . . . . . . . . 76 vi ListofFigures 3.1 Origins and Destinations of the Driver and the Passengers . . . . . . . . . . . . . . . . . . 40 4.1 Process Flow Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Map of Sensors in the Studied Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 The effect of willingness-to-pay-level on passengers’ cost and drivers’ cost . . . . . . . . . 57 4.4 The effect of willingness-to-pay-level on % served and % of no-passenger vehicles . . . . . 57 5.1 The Overall Solution Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 The One Circle Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 Ellipse Tangent to Circle A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.4 The Two-Circle Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5 The Solution Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.6 Distance Information of the Studied Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 vii Abstract In this research, we dedicate to the ride-sharing problem where ride-sharing drivers are travelling toward their own destinations while making detours to serve passengers with flexible pickup and drop-off loca- tions. We decompose this problem into the cost-sharing problem and the routing problem. The cost-sharing problem addresses how the total cost of a ride-sharing operation is distributed among participants. In order to solve this problem, one needs to design a cost-sharing mechanism. In the first part of this dissertation, we identify the properties that a desirable cost-sharing mechanism should have and develop a general framework that can be used to create specific cost-sharing mechanisms. We propose specific mechanisms and analyze their relative advantages and disadvantages so that service providers can select a mechanism according to their different needs. In addition, we incorporate the value of time by allowing passengers to have inconvenience costs due to extra travel time caused by detours for picking up the passengers and provide discount methods to compensate for these costs. We evaluate our approach using real traffic data from the downtown Los Angeles area. Our results show that each proposed mech- anism has its unique advantages and that the discount methods can successfully reduce the number of no-passenger vehicles for a large ride-sharing system. The routing problem on the other hand addresses how to route the ride-sharing routes and match passengers with drivers. Especially, we study under the context where drivers can utilize the HOV (High Occupancy Vehicle) lane and passengers have flexible pickup and drop-off locations. We developed a two- stage routing algorithm to solve the routing problem in real time. This two-stage algorithm consists of viii an insertion heuristic to solve the pickup and delivery problem and a second-stage algorithm that can solve the meeting points problem optimally. Using randomly generated data, we designed experiments to evaluate the performance of the HOV lanes and meeting points. The results showed that the existence of HOV lanes reduces the average in-vehicle time (IVT) ratio compared to the situation with no HOV lanes since they provide a boost in driving speed. They also help increase the percentage of requests served in the system since the time saved in using HOV lanes can be utilized to serve more requests and serving more requests increases the possibility of using the HOV lanes. The existence of meeting points increases the utilization of the ride-sharing system and is capable of reducing total miles travelled while maintaining the same level of system usage. ix Chapter1 Introduction 1.1 Motivation In recent years, the world has seen rapid development in cities as well as growth in urban population. According to [7], there are 56% of the world population living in urban areas which usually have high population density. This growing urban population will result in an increase in traffic congestion. Not only is it generating more Green House Gases (GHG) that contributes to global warming, traffic congestion is also increasing the cost for commuters. According to the 2021 Urban Mobility Report by [60], the total cost of congestion in 2019 was $190 billion in the U.S. and the total amount of delayed time was 8.7 billion hours with an extra usage of 3.5 billion gallons of fuel. As a comparison, these measures were $101 billion, 4.3 billion and 1.7 billion in 2020 when the government ordered stay-at-home restrictions due to the pandemic. These two sets of data can best show how the decrease in number of vehicles on the road can alleviate the traffic congestion problem and how less traffic congestion can result in significant reduction (around 50%) in social costs, and improvements in social efficiency and emissions. Moreover, the Harvard Center for Risk Analysis (HCRA) at the School of Public Health conducted a research study in 83 urban areas to evaluate the public health impacts of traffic congestion [34]. The results indicated that traffic congestion led to 4,000 premature deaths with a public health cost of around $31 billion in 2000. If no effective methods 1 are taken by 2030, it is projected that there will be 1,900 premature deaths and $17 billion in social costs annually. In order to tackle traffic congestion, there are two common directions: 1) build more traffic infrastruc- ture to accommodate the increasing traffic demand, and 2) resort to demand management methods which are usually categorized into market based policies and rules based policies. The first direction requires large investments in money, time and resources, none of which is easy to obtain. In certain cases, even if the government increases the capacity of the traffic infrastructure (by a small amount), the problem in traffic congestion is not alleviated. A study by [61] found that the removal of a freeway bottleneck in California increased the traffic volume on that particular freeway. As for the second direction, many public and private transit agencies have used the market based policy, such as congestion pricing, to raise funds and mitigate the congestion. This policy may reduce the overall traffic cost but faces concerns and controversies over inequality and distribution of the funds. Rules based policies like travel restrictions and HOV lanes are also adopted by government agencies but have their own disadvantages. For example, travel restrictions like enforcing vehicles to travel on alternative days based on the license plate number may reduce daily traffic but may cause unfairness. Since the travel demand is still the same, rich people tend to buy multiple cars to get rid of the restriction while poor people have no choice but to follow the rule. HOV lanes, on the other hand, require less resources and do not increase commuters’ costs as much. However, to make the most use of this policy, a transportation mode that supports high vehicle occupancy is required. From the above, we can see that a cost-efficient and system-efficient transportation mode that can help mitigate the congestion issue without adding financial pressure to road service providers and commuters is badly in need. Ride-sharing is one such transportation mode in the ascendant. By sharing a vehicle among multiple riders, it can help with the congestion issue since this concept utilizes significant amount 2 of unused vehicle capacities in the transportation network. Although the first employee ride-sharing (car- pooling) was organized back in 1970s [47], it was not fully embraced by the public. With the development of technology, especially the popularization of smart phones and the effective use of mobile technology, shared mobility services such as UberPool and LyftShared have emerged and changed the travel behavior of individuals. However, the increase in drivers providing these services to earn income may generate more traffic due to a number of factors, such as deadhead miles [22]. Therefore, the biggest motivation for this dissertation is to focus on a ride-sharing context where the drivers pick up passengers on their way to work, home, or while running errands, where the deadhead miles are minimal. 1.2 Background In the context of ride-sharing, we encounter two important research issues: (1) how to determine the routes and schedules of the vehicles, including how to assign passengers to vehicles in the presence of conflicting objectives, such as maximizing the number of serviced passengers, or minimizing the operating cost; and (2) how to allocate the operating cost among the passengers and drivers such that they are incentivized to participate in the ride-sharing operation. The first (optimization) and second (cost-sharing) problems are highly interrelated, because the optimized vehicle routes determine the operating cost that needs to be shared. Conversely, the cost-sharing mechanism imposes constraints on how to optimize the routes; for instance, the cost shared by a passenger should not exceed the price initially quoted to them at any time during the operation. The situation becomes more complex when the routing becomes dynamic; that is, the passengers request service during the ride-sharing operation instead of requesting it ahead of time. Both the optimization problem and the cost-sharing problem have to deal with uncertainty in demand and travel time which makes finding the optimal or near-optimal solutions even harder. 3 The optimization problem has received considerable attention in the literature and is often solved as a vehicle routing problem with pickup and delivery (PDP) (see survey papers by [16, 9]). There has been some research to extend the PDP to consider special features of the ride-sharing problem. [2, 1] developed an optimization-based approach to match drivers and riders in dynamic ride-sharing systems. Their results showed that even at relatively small participation rates, dynamic ride-sharing may have the potential to succeed with a sustainable ride-share population forming over time. [68] considered the problem of ride- sharing routing when there are special dedicated lanes such as High Occupancy Vehicle (HOV) lanes to reduce the travel time when there are multiple individuals sharing a trip. In this case, there is an incentive to take a detour to pick up extra passengers to qualify for driving in an HOV lane. The cost-sharing problem on the other hand has largely been neglected in the literature of studying ride-sharing and is the focus of this research. For general cost-sharing problems in which the set of players and the cost function are both known and deterministic, Moulin mechanisms [43] and acyclic mechanisms [40] are among the most studied families of cost-sharing mechanisms. Even though these two mechanisms served as the frameworks for many cost-sharing mechanisms, they have not been applied to settings rel- evant to ride-sharing, such as the vehicle routing problem (VRP). To the best of our knowledge, [26] were among the earliest authors to touch upon the cost allocation problem in the VRP context. Thus, there is little research in this area. Therefore, in this dissertation, we design a ride-sharing system that can be applied in real-time. This ride-sharing system considers drivers as part of the commuters and aims to provide both the drivers and passengers a cost-effective way of commuting. Note that, it is not restricted to commuting trips and can be implemented in trips like running errands as well. We first design the cost-sharing mechanisms that provide ride-sharing drivers ways to allocate their cost among passengers that incentivize both passengers 4 and drivers to participate in the ride-sharing operation. Next, we provide a routing algorithm for ride- sharing that takes into consideration policies like HOV lanes and personal preferences like flexible pickup and drop-off locations to make the most use of the ride-sharing system. The remainder of the proposal is structured as follows. Chapter 2 reviews the related literature and describes our contribution to the literature. Chapter 3 describes the cost-sharing problem in detail and presents a general solution framework followed by the presentation of the three mechanisms developed under this framework. An individual experimental section is included in the chapter as well to explore the pros and cons of the proposed mechanisms. Chapter 4 extends the mechanisms in Chapter 3 to consider the value of time. Two discount methods are proposed and experimental results are presented using real traffic data. Chapter 5 describes our routing model and algorithm as well as the corresponding experimental results. Finally, in Chapter 6, we conclude our work and discuss possible future work directions. 5 Chapter2 Introduction As we mentioned in the Introduction chapter, the ride-sharing problem mainly concerns two problems: (1) the optimization problem that determines the vehicle routes and the pairing among drivers and passengers; and (2) the cost-sharing problem that determines how the cost is shared among the participants of the ride- sharing operation. In this section, we first dive into the research history of cost-sharing mechanisms and its application in transportation, and then review the papers on solving the VRP and DVRP. 2.1 TheCost-SharingProblem Unlike the optimization part of the ride-sharing problem, the cost-sharing problem, to the best of our knowledge, does not have as rich of a literature as the VRP. For general and static cost-sharing problems where the cost function and the participants are known and deterministic, Moulin mechanisms [43] and acyclic mechanisms [40] are among the original work in this area. They have built the foundation for de- signing truthful, and approximately budget-balanced cost-sharing mechanisms with economic efficiency. In the area of stock exchange and ad exchange, [28] and [44] represented research that facilitate real-time matching between sellers and buyers by providing opportunities for them to trade items at certain prices. [42] proposed an efficient and almost budget balance cost-sharing method and identified several charac- teristics of the cost function so the cost-sharing methods for that cost function satisfy desirable properties. 6 However, it is not straightforward to apply these systems to the ride-sharing context. Typically, ride- sharing passengers have motivations such as sharing travel costs, saving travel time (using HOV lanes), and mitigating environmental concerns (e.g. reducing GHG emission) rather than making profits. In addi- tion, since the final form of ride-sharing is determined only when the last passenger in a vehicle is decided, participants cannot evaluate the value of ride-sharing at the time of order submission unlike the general and static cost-sharing problems. One of the early works studying cost-sharing in the transportation context is by [26]. They proposed an effective algorithm on solving the vehicle routing problem and suggested evenly distributing the cost among passengers. [24] compared a few cost-sharing methods from the perspective of cooperative game theory and propose a new method aiming at equalizing the relative profits of participants as much as pos- sible. As a result, this proposed method could generate a saving of around 9% in the total operating cost. Later, [33] proposed a ride-sharing mechanism based on parallel auctions that is adaptive to individual preferences of the passengers. This mechanism is proven to be incentive compatible, and it allows making trade-offs between reduction in vehicle miles travelled and the probability of pairing drivers and passen- gers. [48] studied the cost-sharing problem in a vendor managed inventory (VMI) context. In the VMI context, suppliers are responsible for managing the inventory level of their customers and decides when and how much to replenish each customer. The research focuses on how to calculate the cost-to-serve for each customer. [25] developed a cost-sharing mechanism for demand-responsive transport (DRT) systems. This new mechanism, named Proportional Online Cost Sharing (POCS), was shown theoretically to satisfy a list of desirable properties including online fairness, incentive compatibility, budget balance and etc. The main idea of this mechanism is to combine the traditional incremental sharing and proportional sharing methods while maintaining their advantages. However, all the above works are either unlikely to apply to the ride-sharing context directly, assume a static vehicle routing environment, or separate the cost allocation mechanism from the routing of vehicles. 7 The work of [58] is among the first studies to consider cost allocation in a dynamic vehicle routing envi- ronment. The authors described a new policy for dynamic DARP that features a non-myopic (considering social welfare) pricing component, assuming elastic passenger demand. The pricing method was shown to be socially optimal and improved social welfare by 10 to 20%. [73] proposed a new method called Hybrid Proportional Online Cost Sharing (HPOCS) to solve the cost-sharing problem in the context of DVRP. This POCS-based method inherited POCS’s advantage of satisfying the desirable properties. HPOCS assumed a known probability of requesting service and a known distribution of service request time for each passen- ger. It was shown to be efficient when the request probability is high but failed to be economically efficient when the request probability is low. The reason for less literature in the dynamic setting is because of the added layer of complexity brought by uncertainty. Some desirable properties for a good cost-sharing mechanism, such as the Ex-post Incen- tive Compatibility property (also known as strategy proof), may not hold in a dynamic setting. This then makes assessing the performance of ride-sharing mechanisms harder, because when certain properties fail to be satisfied, we lack tools of determining whether a ride-sharing mechanism is good or not. To solve this problem, instead of providing a 100% guarantee on the properties, a robust analysis on the performance of the mechanism is usually performed. [5] proposed an alternative method to model uncertainty in the arrivals and services such that closed form upper bounds on the system time in the worst case are obtained. Their expressions generalized the light-tailed distributions of arrivals and services to the case of heavy- tailed behavior which could provide a distribution free probabilistic analyzing framework on ride-sharing requests. 8 2.2 TheVehicleRoutingProblem VRP was first proposed by [18] to solve the gasoline truck delivery problem. It then became a popular topic in the optimization world and has many sub-topics: capacitated vehicle routing problem (CVRP) which deals with vehicles that have limited loading capacity, VRP with time windows (VRPTW) which adds the constraints of time windows in visiting the locations, VRP with pickup and delivery (VRPPD or simply PDP) which considers pickup locations and delivery locations in routing and etc [66]. Among all these different sub-topics, PDP is the most relevant topic to ride-sharing due to the fact that a ride- sharing operation often involves picking up the passenger and delivering the passenger. The PDPs are categorized into three groups: the one-to-one problem, the one-to-many-to-one problem and the many- to-many problem [9]. Ride-sharing lies in the range of the first group which is the one-to-one PDP. In the one-to-one PDP, each object has a pickup and a delivery location. When the object is substituted with a passenger, it is usually called the dial-a-ride problem (DARP) [9]. The problems can be further categorized into two groups: single vehicle and multi-vehicle problems [12]. For the single vehicle problem, researchers often use the dynamic programming approach. Psaraftis (1980) was known as one of the first researchers to solve the problem optimally using dynamic programming in the case of immediate- request with small instances. Later, [19] applied dynamic programming to solve this problem on larger instances. The PDPs are NP-hard problems which means as the size of the problem gets larger, it may be very difficult to solve it optimally. Therefore, many researchers are developing heuristics algorithms to solve this problem efficiently rather than solving it optimally. [30] tackled the PDP with time windows using a heuristic approach based on an intelligent neighborhood move. For the multi-vehicle problem, when searching for the optimal solution, the branch-and-cut algorithm is often used in small instances. [37] formulated the problem as a 0-1 integer-programming problem, and a branch-and-cut algorithm is used to optimally solve the problem. [13] and [57] provided an alternative formulation and applied a 9 branch-and-cut algorithm to optimally solve the problem which is outperformed by a new branch-and- cut-and-price algorithm in their later paper [56]. More recently, [4] introduced two new formulations of the problem and managed to obtain optimality for 41 instances from existing benchmark problem sets that contain up to 100 nodes. Even though researchers have improved the exact algorithms over the years to solve the PDP prob- lem with more nodes, the size of the problem that can be solved optimally is rather small compared to reality. Therefore, another research direction is to develop heuristics and metaheuristics to effectively solve large instances. Construction insertion heuristics is a popular choice. [32] developed a heuristic procedure in which users can only specify either the pickup time or the delivery time. Later, [53] mod- ified this heuristic and added two new phases which obtained better results. A parallel regret insertion heuristic was proposed by [20]. [69] proposed a heuristic including parallel insertions, reinsertions and exchanges. [39] introduced an insertion based heuristic that accounts for operational requirements such as different passenger needs and different specifications of the objective function. Metaheuristics are also widely implemented in solving the PDP problem with large instances. Tabu search has been one of the most commonly used metaheuristics [14]. The Tabu search heuristic developed by [15] proposed a neigh- borhood evaluation procedure to minimize route duration and ride times. This early work in Tabu search has since inspired many recent researchers to extend their work to satisfy more complex and real-life con- straints [29]. Additionally, many other metaheuristics have been applied to the PDP. [63] used the particle swarm optimization algorithm for PDP with multiple depots. [50] proposed a competitive variable neigh- borhood search-based heuristic for the static multi-vehicle DARP. [10] applied the ant colony optimization algorithm and [72] applied simulated annealing to solve the DARP with multiple objectives. 10 2.3 TheDynamicPickupandDeliveryProblem Most of the research works above deal with the static PDP and DARP where the passengers’ service re- quests are known before the operation starts. When the requests are allowed to come in during the op- eration, the problem becomes dynamic and the routing decisions are too complex to solve for an optimal solution. As a result, researchers studying dynamic PDP and DARP use more heuristics methods or hy- brid methods rather than exact methods. [8] proposed a hybrid method combining the Tabu search meta- heuristic with constraint programming which outperformed the two individual methods according to their experiment. Additionally, researchers tend to consider more on multiple objectives rather than a single objective. In [29], 63% of the papers studying dynamic DARP have multiple objectives while only 31% of the papers studying static DARP have multiple objectives. Some of them resort to simulation methods to assess the performance measures and their relationships. For example, [55] investigated the effect of different operating policies on performance measures such as vehicle miles travelled and deadhead miles. If we take a closer look at the literature, we can get a better idea of how researchers address the dynamism of the dynamic PDP. According to [9], [54] represented one of the early studies on dynamic PDP. He considered a single vehicle problem with the objective to minimize a weighted function of total service time and passenger dissatisfaction. The solution approach was based on first establish an algorithm for the static case and then adapt it to the dynamic case where the author used re-optimization every time a new request arrives. This method of addressing the dynamism in the problems is popular among the literature. These papers all consider new requests as a trigger to the re-optimization of the current solution which is often constructed statically at the beginning of the operation. The difference lies in the methods they use to search for better solutions as new requests come in. [3] developed a parallel algorithm which first constructed a static solution based on known requests and then used an insertion algorithm and tabu search to reoptimize the current solution whenever there is a new request. [17] proposed an algorithm that maintains a solution repository. The algorithm then chooses from the repository to insert 11 new requests. [27] assessed the power of ruin and recreate methods in the re-optimization phase of a dynamic PDP. They found that ruin methods based on removal of sequences of requests can obtain better solution quality. Additional to the efforts in developing a better route search algorithm, other research focuses on tuning the re-optimization frequency during the planning horizon to find better solutions [2, 33, 73]. In this approach, one separates the time horizon into small time segments and re-optimizes at the end of each segment. To find the best solution, researchers tune the length of each time segment and adjust the wait time a driver can spend after picking up passengers. Instead of working on different route searching algorithms and re-optimization frequencies, [58] incorporate pricing decisions in determining how a new request is accommodated. Note that all the above papers studying the dynamic PDP in a deterministic environment where infor- mation of requests and the system are known at the time of the decision. However, reality often contains extra complexities such as traffic jam, vehicle breakdowns, request cancellations and passenger no-shows. This new stream of dynamic and stochastic PDPs have attracted more researchers despite their complexi- ties. [70] studied this problem and proposed a fast heuristic to re-optimize the route. This heuristic consists of a local search strategy and a secondary objective function to drive the search out of local optima. [59] studied a dynamic and stochastic PDP where outbound trips can possibly trigger inbound trips. They proposed four different metaheuristics and their results indicated that using the stochastic information on return transport leads to an average improvement of around 15%. [46] used a multi-objective predic- tive control method to predict the future scenarios to obtain optimal control in a stochastic environment. Most recently, [67] considered a restaurant meal delivery problem with random ready times. They pro- posed a route-based Markov decision process (MDP) to model the problem and an anticipatory customer assignment policy to address the stochasticity. Additionally in this dissertation, we are interested in the PDP with the option of walking to a pickup or drop-off location. In public transport, we are used to the idea of walking since a public transit system 12 usually has a fixed transportation route and utilizes the benefits of massive transit [41, 23]. It is a well studied topic in public transport [31, 11, 65, 38, 51] and can provide some intuition in that there are many good solution approaches to jointly optimize the costs of service providers and passengers. In the context of private transportation services, we encounter walking scenarios as well: (1) when passengers request a ride in a limited access area such as universities, and (2) when passengers try to find more taxis at a larger intersection. A study by [45] showed that taxi services that require walking achieves higher efficiency than those that do not under high-demand conditions. Furthermore, in the context of ride-sharing, according to [64], the idea of applying walking in ride-sharing can improve the matching rate and mileage savings. A recent study by [49] also indicated that ride-sharing with exact locations is an obstacle to mass carpooling utilization. Therefore, there exists a need to incorporate walking into ride-sharing systems. Compared to ride-sharing with exact locations, there is not much literature investigating how to integrate walking in a ride-sharing system. [35] proposed a novel dynamic programming method to search for the optimal route with multiple meeting points in an on-demand ride-sharing system. [36] formulated the problem as a mixed integer linear program (MILP) and proposed a Tabu-based meta-heuristic algorithm. Specifically, their results show that introducing meeting points to ride-sharing system saves the total travel time by 2.7%-3.8% for a small-scale ride-sharing system. [71] used a space-time network representation on ride- sharing problems with flexible pickup and drop-off locations and proposed a Lagrangian relaxation inspired solution approach. [62] studied the same problem but they also consider the problem of how to decide which users act as drivers in a ride-sharing system. 2.4 ResearchGapandContribution As we can see through the literature, the optimization part of routing has attracted a significant amount of researchers’ attention. However, the ride-sharing context has yet been fully explored. This is mainly 13 because a typical ride-sharing operation system usually concerns capacitated vehicles, rather strict time windows, dynamic service requests and stochastic information altogether which makes the problem diffi- cult to solve. The cost-sharing part of ride-sharing has received less attention even though it is crucial in the ride-sharing operation since how the system allocates cost among drivers and passengers will affect the performance measures of the system, e.g. number of passengers served in the system. In this dissertation, we extend on the current dynamic ride-sharing landscape to consider an online centralized matching system that shares the cost among passengers and determines the compensation from the passengers to the driver. In our proposed system, drivers are not professional, instead, they are private citizens who choose to share their idle vehicle capacity through the system. The system is designed for remote access and for passengers and drivers to negotiate routes and prices. The intention for an open system such as the one that is proposed is to facilitate entry to the marketplace so as to significantly reduce idle vehicle capacity in the transportation network which in turn mitigates the increasingly severe traffic issues we face nowadays. Our contribution is then threefold: 1) we develop an alternative solution to the cost-sharing prob- lem and integrating it with the routing problem to provide a wholistic solution to ride-sharing; 2) in this aspect, we first build a general cost-sharing mechanism framework that one can easily use to develop de- tailed mechanisms that adapt to different requirements. We then introduce three mechanisms in detail and extend these mechanisms to consider time windows; 3) we propose a routing algorithm which take into consideration the HOV lanes and passengers’ preference of walking. Our solution for the cost-sharing problem and the routing problem are tested by applying them to a similar ride-sharing operation system setting. 14 Chapter3 MechanismsforRide-Sharing In this chapter, we focus on the cost-sharing part of our ride-sharing problem. Recall, our ride-sharing context involves drivers that have their personal origin and destination. More formally, we have drivers who want to share their own direct trip cost with a limited number of passengers in exchange for their service. The organization of this chapter is to first describe the cost-sharing problem and then gradually build up our mechanism framework. Based on the framework, we propose three different mechanisms in detail to share the cost among passengers and drivers. At the end of the chapter, we present our simulation results. 3.1 ProblemDescription In this section, we describe the ride-sharing problem and briefly introduce the POCS mechanism [25]. Unlike services provided by companies like Uber and Lyft, our ride-sharing context does not involve a professional driver. We focus on the scenario where the drivers have their own origins and destinations. More formally, we have drivers who want to share their total costs with a limited number of passengers in exchange for their service. Each passenger who makes a request for the service has a demand (the distance between their origin and destination) and a willingness-to-pay level for the ride-sharing service. We assume passengers make requests dynamically but before the driver starts the operation. At the time 15 of the request, a passenger is immediately given an initial quote and they also immediately decide whether to join the operation or not by comparing the initial quote with their willingness-to-pay level. At the end of the operation, passengers share the total cost of the trip, possibly also with the driver. Their shared costs may be lower than their initial quotes due to the matching of subsequent requests to the vehicle. Our goal is to design cost-sharing mechanisms that compute these initial quotes and shared costs for the passengers as new requests arrive to the system. An important point is that we assume the drivers have the same spatial pattern as the passengers in origins and destinations. With this assumption in mind, different origin and destination patterns indeed affect the allocation of the costs among passengers and drivers in that certain cost-sharing mechanisms may generate favorable results for some passengers such as low shared costs for passengers with high demand in certain patterns. Nevertheless, in this paper, we focus on designing mechanisms that are fair for all passengers regardless of the origin and destination pattern. To achieve this goal, we will aim to satisfy some desirable properties introduced in this section. We first present some notation that we use throughout the paper. We use t∈N to represent the discrete time points at which requests come in, ∗ andΠ ⊆ N to represent the set of passengers. We useπ ∈Σ(Π) to represent the order of submitted requests by passengers inΠ , which is a permutation of setΠ , andk∈N to represent the time point when a ride request is submitted. So, π (k) is the passenger who submits a ride request at timek under submit orderπ . We letπ (0) represent the driver. We letα π (k) be the demand (quantifying the direct distance) of passengerπ (k)’s ride request and is positive by assumption. We letF be the driver’s direct trip cost,c π,t is the total cost to drive at timet under submit orderπ andc d π,t :=c π,t − F is the total detour cost at timet under submit orderπ . The willingness-to-pay level of passengerπ (k) is denoted asW π (k) . ∗ Since passengers come in sequence and the system is updated only when a new passenger requests, we use discrete time instead of continuous time to facilitate the understanding of notations and proofs of the theorems and propositions. The discrete time points are evenly distributed. 16 Next, we briefly introduce the POCS mechanism [25], a mechanism that was developed for a shuttle service to support a demand responsive transportation system. This mechanism is summerized in Algo- rithm 1. As shown in Algorithm 1, when a new passenger π (k) requests service, POCS calculates the marginal costc m π (k) for this passenger based on the selected routing strategy. It then updates the coalition perα value (CCPA)c a π (i,k) fori=1,...,k. Finally, it computes the total shared costc s π (l),k forl =1,...,k by first looking at the new CCPA values computed with the new passenger and then comparing them with the previous CCPA values (line 5 in Algorithm 1). This puts the passengers into coalitions: POCS charges everyone within the same coalition proportionally to their direct travel distance from their origin to des- tination point; i.e., their unit prices are the same. Algorithm1: The POCS Mechanism Input : Information for a new passengerπ (k): origin, destination andα π (k) CCPA values of previous passengers: c a π (k 1 ,k 2 ) for1≤ k 1 ≤ k 2 ≤ k− 1 Output:c s π (i),k fori=1,...,k 1 c m π (k) ← c π,k − c π,k − 1 2 fori=1,...,k do 3 c a π (i,k) ← P k j=i c m π (j) P k j=i α π (j) 4 forl =1,...,k do 5 c s π (l),k ← α π (l) min l≤ j≤ k max 1≤ i≤ j c a π (i,j) We make the following two major assumptions. Assumption 3.1.1. The total cost (both for the driver and for the passengers) is non-decreasing over time. That is, for all timest andt ′ witht≤ t ′ and all submit ordersπ ,c π,t ≤ c π,t ′. Assumption 3.1.2. The total cost (both for the driver and for the passengers) at time t is independent of the submit order. That is, for all times t ≥ 1 and submit orders π and π ′ such that π (1),...,π (t) and π ′ (1),...,π ′ (t) contain the same set of passengers,c π,t =c π ′ ,t . These two assumptions are satisfied, for example, when the total cost is defined as the minimal oper- ating cost. Their existences are closely related to the satisfaction of some desired properties. As originally 17 described and defined in [25], POCS satisfies the five desirable properties below under these two assump- tions. Definition3.1.1. OnlineFairnessProperty . The shared costs perα value of a passenger is never higher than those of passengers who submit their ride requests after them. That is, for all timesk 1 ,k 2 andt with 1≤ k 1 ≤ k 2 ≤ t and all submit ordersπ , it holds that c s π (k 1 ),t α π (k 1 ) ≤ c s π (k 2 ),t α π (k 2 ) . Definition3.1.2. ImmediateResponseProperty . Passengers are provided immediately after their ride request submissions with (ideally low) upper bounds on their shared costs at any future time. That is, for all timesk,t 1 andt 2 with1≤ k≤ t 1 ≤ t 2 and all submit ordersπ , it holds that c s π (k),t 1 ≥ c s π (k),t 2 . Definition3.1.3. IndividualRationalityProperty . The shared costs of passengers who accepted their quotes never exceed their willingness-to-pay at any future time. That is, for all timesk andt with1≤ k≤ t and all submit ordersπ , we have c s π (k),t ≤ W π (k) . Definition 3.1.4. Budget Balance Property . The total cost equals the sum of the shared costs of all passengers. That is, for all timest≥ 1 and all submit ordersπ , we have t X j=1 c s π (j),t =c π,t . Definition3.1.5. Ex-PostIncentiveCompatibility . The best strategy of every passenger is to submit their ride requests truthfully. That is, given that all other passengers do not change their submit times and 18 whether they accept or decline their quotes, a passenger cannot decrease their shared costs by delaying their ride request submissions. That is, for all times k 1 ,k 2 and t with 1 ≤ k 1 ≤ k 2 ≤ t and all submit ordersπ andπ ′ such that π ′ (k)=                      π (k+1) ifk 1 ≤ k <k 2 , π (k 1 ) ifk =k 2 , π (k) otherwise, we have c s π (k 1 ),t ≤ c s π ′ (k 2 ),t . In other words, consider any time t, any submit order π , and any passen- gerπ (k 1 ). Assume that the passengerπ (k 1 ) delays their ride request and submits thek 2 th instead of the k 1 th ride request, with all other passenger requests in the same order. Then, the shared cost c s π ′ (k 2 ),t at timet under the new submit orderπ ′ should not be lower than the shared costc s π (k 1 ),t at timet under the previous submit orderπ . 3.2 TheRide-SharingMechanismFramework In this section, we introduce a general Ride-Sharing Mechanism Framework that satisfies certain desirable properties. We first discuss the issues and challenges of directly applying POCS to our ride-sharing context. Al- though POCS satisfies the five desirable properties in Definitions 3.1.1 – 3.1.5, it is meant to be applied to a shuttle service, a different context from our ride-sharing operation. To adapt POCS to a ride-sharing operation, one could simply include the entire driver’s direct trip cost into the total cost for the passengers at any time. In this simple mechanism, when|Π | = 0, the driver has no passengers and covers their own direct trip costF . When|Π |≥ 1, the driver pays nothing. Then, we can apply POCS toc π,t , the total cost to drive, to obtain the total shared costc s π (k),t for each passenger. 19 It is easy to see that for this mechanism, all five desirable properties hold (which is proved in [25]). However, the first passenger faces the chance of paying 100% of the driver’s direct trip cost F if there ends up being only one passenger served, which may result in a high initial quote that could deter the first passenger from participating in the ride-sharing operation. To solve this issue, we need to strike a balance between compensating the driver and reducing the burden on the first passenger. We would also like to distribute the driver’s direct trip cost F among the passengers proportionally to their α values. This is desirable because in the ideal case, where all passengers submit their requests at the same time and their origin and destination locations are known, this is arguably the most fair and natural way to distribute F among the passengers. However, when we do not know how many passengers are going to submit requests, this property can be difficult to satisfy. These issues and challenges serve as the motivation to develop our general mechanism framework for ride-sharing. Let c s 1 π (k),t and c s 2 π (k),t be the total detour cost c d π,t and the direct trip cost of the driver F shared by passengerπ (k) at timet under submit orderπ , respectively. Letβ π (k),t be the fraction ofF that will be covered by passengerπ (k) at timet. We want to design a mechanism that satisfies the following properties: • FiveOriginalDesirableProperties: Online Fairness, Immediate Response, Individual Rationality, Budget Balance and Ex-Post Incentive Compatibility in Definitions 3.1.1 – 3.1.5. • ReducedBurdenfortheFirstPassengerProperty: the initial quote for the first passenger π (1) should not have a highβ value. In particular we wantβ π (1),1 < 1. Ifβ π (1),1 = 1, passengerπ (1) may not have the incentive to join the ride-sharing operation. 20 • FairnessinSharingDriver’sCostProperty: the final share of F paid by the passengers should be proportional to theirα values. Since we do not know in advance which passenger is going to be the last one, we require for all timest, β π (1),t α π (1) = β π (2),t α π (2) =...= β π (t),t α π (t) . (3.1) POCS applied directly to the total cost c π,t = c d π,t + F cannot guarantee that the five desirable prop- erties and the Fairness in Sharing Driver’s Cost Property hold at the same time, because all passengers are not guaranteed to be in the same coalition. We derive a new cost-sharing mechanism framework for ride-sharing that shares the total detour cost c d π,t and the driver’s trip cost F separately through 2 sub- mechanisms. This means that a passenger π (k)’s shared cost consists of two parts: (1) c s 1 π (k),t , the total detour cost shared by this passenger at timet; and (2)c s 2 π (k),t , the driver’s direct trip costF shared by this passenger at timet. Next, we formally introduce the Ride-Sharing Mechanism Framework and analyze its properties. We define the framework as follows: For all times k,t withk≤ t and all submit ordersπ , the total shared cost for passengerπ (k) at timet under submit orderπ is c s π (k),t =c s 1 π (k),t +c s 2 π (k),t =c s 1 π (k),t +β π (k),t F (3.2) where the allocation of the total costc π,t consists of two parts: (1)c s 1 π (k),t to share the total detour costc d π,t , which is calculated by POCS applied to the total detour costc d π,t through the same manner as in POCS (line 5 in Algorithm 1); and (2) c s 2 π (k),t := β π (k),t F to share the driver’s direct trip cost F , where the β values are determined by a specific sub-mechanism. In this framework, we constrain the β values to satisfy two properties: the Fairness in Sharing Driver’s Cost property (as shown in Equation (3.1)) and the Immediate 21 Response property. In particular, for the Immediate Response property, we constrain theβ π (k),t values to be non-increasing int; that is, for all timesk,t 1 ,t 2 withk≤ t 1 ≤ t 2 and all submit ordersπ , we regulate β π (k),t 2 ≤ β π (k),t 1 . (3.3) We consider specific methods of computing β π (k),t in Sections 3.3.1 – 3.3.3. We next show that this Ride-Sharing Mechanism Framework satisfies the five original desirable prop- erties except for the Budget Balance property. The Budget Balance property does not hold because the constraints on the β values in the Ride-Sharing Mechanism Framework do not guarantee that all the β values sum up to 1 for all times t and all submit orders π . We first prove that if c s 1 π (k),t and c s 2 π (k),t both satisfies the five original desirable properties, then c s π (k),t must also satisfy the five original desirable prop- erties. Note thatc s 1 π (k),t andc s 2 π (k),t can be the shared costs from any two sub-mechanisms. Then, to show the Ride-Sharing Mechanism Framework satisfies the four properties, we only need to prove that, under (3.1) and (3.3),c s 2 π (k),t satisfies the four properties. Proposition 3.2.1. Provided that both c s 1 π (k),t and c s 2 π (k),t satisfy the Online Fairness, Immediate Response, Individual Rationality, Budget Balance and Ex-Post Incentive Compatibility properties, c s π (k),t satisfies these five properties as well. Proof. For the Online Fairness property, we need to show that for all timesk 1 ,k 2 ,t with1≤ k 1 ≤ k 2 ≤ t and submit ordersπ : c s π (k 1 ),t α π (k 1 ) ≤ c s π (k 2 ),t α π (k 2 ) . We know that both c s 1 π (k),t and c s 2 π (k),t satisfy the Online Fairness property. That is, for all times k 1 ,k 2 ,t with1≤ k 1 ≤ k 2 ≤ t and submit ordersπ , c s 1 π (k 1 ),t α π (k 1 ) ≤ c s 1 π (k 2 ),t α π (k 2 ) and c s 2 π (k 1 ),t α π (k 1 ) ≤ c s 2 π (k 2 ),t α π (k 2 ) . 22 Therefore, c s π (k 1 ),t α π (k 1 ) = c s 1 π (k 1 ),t α π (k 1 ) + c s 2 π (k 1 ),t α π (k 1 ) ≤ c s 1 π (k 2 ),t α π (k 2 ) + c s 2 π (k 2 ),t α π (k 2 ) = c s π (k 2 ),t α π (k 2 ) . Thus, the Online Fairness property is satisfied. For the Immediate Response property, we need to prove that for all timesk,t 1 ,t 2 with1≤ k≤ t 1 ≤ t 2 and submit ordersπ : c s π (k),t 1 ≥ c s π (k),t 2 . We know that bothc s 1 π (k),t andc s 2 π (k),t satisfy the Immediate Response property. That is, for all timesk,t 1 ,t 2 with1≤ k≤ t 1 ≤ t 2 and submit ordersπ , c s 1 π (k),t 1 ≥ c s 1 π (k),t 2 and c s 2 π (k),t 1 ≥ c s 2 π (k),t 2 . Therefore, c s π (k),t 1 =c s 1 π (k),t 1 +c s 2 π (k),t 1 ≥ c s 1 π (k),t 2 +c s 2 π (k),t 2 =c s π (k),t 2 . Thus, the Immediate Response property is satisfied. For the Individual Rationality property, we need to prove that for all times k,t with 1 ≤ k ≤ t and submit ordersπ : c s π (k),t ≤ W π (k) . We know that bothc s 1 π (k),t andc s 2 π (k),t satisfy the Individual Rationality property. That is, for all timesk,t with1≤ k≤ t and submit ordersπ : c s 1 π (k),t ≤ W 1 π (k) and c s 2 π (k),t ≤ W 2 π (k) . 23 SinceW π (k) =W 1 π (k) +W 2 π (k) † , we have: c s π (k),t =c s 1 π (k),t +c s 2 π (k),t ≤ W 1 π (k) +W 2 π (k) =W π (k) . Thus, the Individual Rationality property is satisfied. For the Budget Balance property, we need to prove that for all timest≥ 1 and submit ordersπ : t X j=1 c s π (j),t =c p π,t . We know that bothc s 1 π (k),t andc s 2 π (k),t satisfy the Budget Balance property. That is, for all timest≥ 1 and submit ordersπ : t X j=1 c s 1 π (j),t =c d π,t and t X j=1 c s 2 π (j),t =F π,t , whereF π,t is the amount of the driver’s direct trip cost recovered at timet under submit orderπ . We then have: t X j=1 c s π (j),t = t X j=1 (c s 1 π (j),t +c s 2 π (j),t )=c d π,t +F π,t =c p π,t . Thus, the Budget Balance property is satisfied. For the Ex-Post Incentive Compatibility property, we need to prove that for all times k 1 ,k 2 ,t with 1≤ k 1 ≤ k 2 ≤ t and submit ordersπ,π ′ with π ′ (k)=                      π (k+1) ifk 1 ≤ k <k 2 , π (k 1 ) ifk =k 2 , π (k) otherwise, † This does not mean that the passenger has two willingness to pay levels. They each only have one level that isW π (k) = W 1 π (k) +W 2 π (k) . And the passengers should never worry about howW π (k) is split intoW 1 π (k) andW 2 π (k) . The existence ofW 1 π (k) and W 2 π (k) is only to prove that when the sub-mechanisms of the Ride-Sharing Mechanism Framework satisfy the Individual Rationality property, the Ride-Sharing Mechanism Framework satisfies the property as well. 24 we havec s π (k 1 ),t ≤ c s π ′ (k 2 ),t . We know that bothc s 1 π (k),t andc s 2 π (k),t satisfy the Ex-Post Incentive Compati- bility property. That is, fix any submit orders π,π ′ that satisfy the conditions above, as well as any times k 1 ,k 2 ,t with1≤ k 1 ≤ k 2 ≤ t; We have: c s 1 π (k 1 ),t ≤ c s 1 π ′ (k 2 ),t and c s 2 π (k 1 ),t ≤ c s 2 π ′ (k 2 ),t . Therefore, c s π (k 1 ),t =c s 1 π (k 1 ),t +c s 2 π (k 1 ),t ≤ c s 1 π ′ (k 2 ),t +c s 2 π ′ (k 2 ),t =c s π ′ (k 2 ),t . Thus, the Ex-Post Incentive Compatibility property is satisfied and the proposition is proven. Theorem3.2.1. UndertheRide-SharingMechanismFramework,c s 2 π (k),t satisfiestheOnlineFairness,Imme- diate Response, Individual Rationality, and Ex-Post Incentive Compatibility properties. Proof. For the Online Fairness property, we need to prove that c s 2 π (k 1 ),t α π (k 1 ) ≤ c s 2 π (k 2 ),t α π (k 2 ) . By (3.1), we have β π (k 1 ),t F β π (k 2 ),t F = α π (k 1 ) α π (k 2 ) or equivalently, β π (k 1 ),t F α π (k 1 ) = β π (k 2 ),t F α π (k 2 ) , which implies c s 2 π (k 1 ),t α π (k 1 ) = c s 2 π (k 2 ),t α π (k 2 ) . For the Immediate Response property, we need to prove that for all timesk,t 1 ,t 2 with1≤ k≤ t 1 ≤ t 2 and submit orders π : c s 2 π (k),t 1 ≥ c s 2 π (k),t 2 . Directly by (3.3), we have β π (k),t 1 F ≥ β π (k),t 2 F which implies c s 2 π (k),t 1 ≥ c s 2 π (k),t 2 . For the Individual Rationality property, we need to prove that for all times k,t with 1 ≤ k ≤ t and submit ordersπ :c s 2 π (k),t ≤ W 2 π (k) . By satisfying the Immediate Response property, we know thatc s 2 π (k),t ≤ c s 2 π (k),k . We also know thatc s 2 π (k),k ≤ W 2 π (k) since the passenger accepted the fare quote and so the property is satisfied. 25 For the Ex-Post Incentive Compatibility property, we need to prove that for all times k 1 ,k 2 ,t with 1≤ k 1 ≤ k 2 ≤ t and submit ordersπ,π ′ with π ′ (k)=                      π (k+1) ifk 1 ≤ k <k 2 , π (k 1 ) ifk =k 2 , π (k) otherwise, we havec s 2 π (k 1 ),t ≤ c s 2 π ′ (k 2 ),t . By (3.1), for both submit ordersπ andπ ′ , theβ values are proportional to their α values; that is: β π (1),t α π (1) =...= β π (k 1 − 1),t α π (k 1 − 1) = β π (k 1 ),t α π (k 1 ) and β π ′ (1),t α π ′ (1) =...= β π ′ (k 2 − 1),t α π ′ (k 2 − 1) = β π ′ (k 2 ),t α π ′ (k 2 ) . Since we have π ′ (k)=                      π (k+1) ifk 1 ≤ k <k 2 , π (k 1 ) ifk =k 2 , π (k) otherwise, we also haveα π ′ (k) =α π (k) fork <k 1 andα π ′ (k 2 ) =α π (k 1 ) . Therefore, β π (1),t β π ′ (1),t =...= β π (k 1 − 1),t β π ′ (k 2 − 1),t = β π (k 1 ),t β π ′ (k 2 ),t . (3.4) Theβ values for passengersπ (k) = π ′ (k) fork < k 1 will be determined the same way: β π (k),t = β π ′ (k),t fork < k 1 . Then, combined with (3.4), we haveβ π (k 1 ),t = β π ′ (k 2 ),t which impliesβ π (k 1 ),t F = β π ′ (k 2 ),t F , and soc s 2 π (k 1 ),t =c s 2 π ′ (k 2 ),t . 26 3.3 TheProposedMechanisms In this section, we propose three different mechanisms that fall under the Ride-Sharing Mechanism Frame- work and analyze their advantages and disadvantages. 3.3.1 Driver-out-of-CoalitionMechanism In this mechanism, 100% of the driver’s direct trip costF is guaranteed to be transferred to the passengers and shared proportionally according to the passengers’α values. Definition3.3.1. Under the Ride-Sharing Mechanism Framework, theDriver-out-of-CoalitionMech- anism specifies for all times k, t withk ≤ t and all submit ordersπ , the total shared cost for passenger π (k) at timet under submit orderπ is c s π (k),t =c s 1 π (k),t +β π (k),t F whereβ π (k),t := α π (k) P t i=1 α π (i) . We can see that theβ values in the above definition satisfies Equations (3.1) and (3.3). Therefore, this Driver-out-of-Coalition Mechanism naturally satisfies four out of the five original desirable properties. We have the following theorem for the Budget Balance property. Theorem3.3.1. UndertheDriver-out-of-CoalitionMechanism,c s 2 π (k),t satisfiestheBudgetBalanceproperty. That is, for all timest≥ 1 and submit ordersπ , P t k=1 c s 2 π (k),t =F. Proof. Follows immediately from Definition 3.3.1, we have: t X k=1 c s 2 π (k),t =F t X k=1 β π (k),t =F P t k=1 α π (k) P t k=1 α π (k) =F . 27 The advantages of this mechanism are: (1) all five original desirable properties hold; and (2) the Fair- ness in Sharing Driver’s Cost property holds. The disadvantage of this mechanism is that it fails to reduce the burden of π (1). Proposition 3.3.1 below describes the fact that the Fairness in Sharing Driver’s Cost property and the Reduced Burden for the First Passenger property are contradictory under certain circum- stances. Proposition3.3.1. When the driver’s direct trip costF is fully recovered by the passengers, the Fairness in Sharing Driver’s Cost property and the Reduced Burden for the First Passenger property cannot hold at the same time without breaking one of the five original desirable properties. Proof. Notice that we only consider the case where|Π | ≥ 2 since β π (1),t = 1 for all t when|Π | = 1 and the Reduced Burden of the First Passenger property is always lost. Suppose the first passenger π (1) requests a ride at time 1, receives a quote of the entire driver’s direct trip costF and we want to reduce this burden from 1 (i.e. 100%) tox (wherex<1), and maintain the Fairness in Sharing Driver’s Cost property. Since theα values of the passengers can take arbitrary positive values, it is possible that α π (1) α π (1) +α π (2) >x, which would lead to the loss of the Immediate Response property when the second passenger submits request at time 2. And when the Immediate Response property is lost, the Individual Rationality property is lost ‡ . If we find a way to determine β π (1),t that the Immediate Response property holds, for example, letβ π (1),t = β π (1),t− 1 − 0.1 andβ π (1),1 = 0.9, then we may face the possibility of losing the Fairness in Sharing Driver’s Cost property: β π (1),2 β π (2),2 = β π (1),2 1− β π (1),2 = 0.8 0.2 ̸= α π (1) α π (2) since α π (1) α π (2) may take any value and we let it be 5 here. Or if we satisfy the Fairness in Sharing Driver’s Cost property, We may lose the Budget Balance property: β π (2),2 +β π (1),2 =β π (1),2 α π (1) +α π (2) α π (1) =0.8× ( 6 5 )=0.96≤ 1. When there is more than one passenger, and we are given the maximum and minimumα values, one possible solution to make all seven properties hold is to letβ π (1),1 = α max α min +α max and set theβ values for ‡ SinceW 2 π (k) can take any value greater or equal toc s 2 π (k),k , and failing to satisfy the Immediate Response property means c s 2 π (k),t >c s 2 π (k),k fort>k, therefore, we may havec s 2 π (k),t >W 2 π (k) which leads to the loss of the Individual Rationality property. 28 all k ≥ 1 and all t > 1 according to Definition 3.3.1. This way of determining β π (1),1 guarantees that β π (1),t ≤ β π (1),1 for all t > 1. Therefore, the Immediate Response property holds when the Reduced Burden of the First Passenger property and the Fairness in Sharing Driver’s Cost property hold. On the other hand, this approach may not sufficiently reduce the burden of π (1) if α max α min +α max is close to 1. 3.3.2 PassengersPredictingMechanism One way to reduce the initial quote of the first passenger is to estimate how many additional passengers will be sharing this trip and use this estimate to proportionally share F in the quote. That is, in this mechanism, we deal with the uncertainty of the total number of passengers and the totalα values of the passengers using prediction. We estimate the total α value in order to normalize the share of F that is allocated to each passenger to reduce the initial quote for the first passenger. We use a robust optimization based approach to bound the total α value so that the desirable properties hold. We first introduce our proposed mechanism and then describe in detail our robust estimation method. The totalα value is formally defined in below: Definition 3.3.2. For all times t and all submit orders π , the total α value for all passengers by time t under submit orderπ is: A:= t X i=1 α π (i) . Now, we formally define the Passengers Prediction Mechanism. Definition 3.3.3. Under the Ride-Sharing Mechanisms Framework, the Passengers Prediction Mech- anism specifies for all times k, t withk ≤ t and all submit ordersπ , the total shared cost for passenger π (k) at timet under submit orderπ is c s π (k),t =c s 1 π (k),t +β π (k),t F 29 where β π (k),t = α π (k) ˜ A . (3.5) We let ˜ A represent the estimate ofA using the proposed robust method in Section 3.3.2.1. It can be seen thatβ π (k),t defined in Definition 3.3.3 are proportional to the passengers’ α values and are non-increasing int. Therefore, the Passengers Predicting Mechanism satisfies the requirements of the Ride-Sharing Mechanism Framework, thus satisfying the Online Fairness, Individual Rationality, Immedi- ate Response and Ex-Post Incentive Compatibility properties. The advantage of this mechanism is that it satisfies all of the properties listed in Section 3.2 except for the Budget Balance property. However, this mechanism will be closer to satisfying the Budget Balance property as the prediction accuracy increases. 3.3.2.1 RobustOptimizationMethodforTotalα ValueEstimation We are interested in estimating the quantiles of the distribution of the sum of α values associated with passengers that arrived on or prior to some given timet∈R + , after which no more requests are accepted. Since both passenger arrival times and α values are uncertain, even if the distributions associated with these quantities are perfectly known, this is a hard problem in applied probability which, to the best of our knowledge, does not have an analytical solution. If the distributions of α values and inter-arrival times are perfectly known, asymptotically accurate estimates for the quantiles can be obtained by simulation. Unfortunately, estimating these distributions accurately is often not possible due to lack of data. Thus, we propose an approach inspired by modern robust queuing theory [5, 6] to obtain estimates for the quantiles of the distribution of the sum of α values of those passengers that arrived on or prior to time t. Our approach yields estimates that are robust to ambiguity in the distributions of the uncertain parameters. Our proposed approach proceeds as follows. First, we design uncertainty sets for the passenger inter- arrival times and for theα values. The sets are parameterized byΓ values andτ values which are chosen by the users so that the uncertain sets are guaranteed to materialize with certain probability. Second, we 30 formulate an optimization problem whose optimal objective value we refer to as “robust sum ofα values” and corresponds to an estimate of the desired quantile of the sum of α values. Then, we show that the optimal objective value of this problem can be easily computed in closed form. LetT i ∈R + ,i=1,2,...,n denote the (uncertain) interarrival time between thei th passenger and the (i− 1) th passenger. Thus, the sequence{T 1 ,T 2 ,...,T n } contains the interarrival times of all passengers. Suppose thatλ is the arrival rate of passengers in the system. Similarly, letα i ∈R + ,i=1,2,...,n denote the (uncertain)α value of thei th passenger to arrive and let ¯α denote the expectation of the distribution ofα values. The parametern here is a number large enough to simulate the population of the passengers, and by the Central Limit Theorem, largern yields a smaller standard deviation. The choice ofn depends ont. We propose to adapt the model of uncertainty from [5] which is used to bound partial sums over the interarrival times and theα values. We restrict theα values to lie in the uncertainty set adapted U a := ( (α 1 ,...,α n )∈R n + : − Γ a ≤ P i ℓ=1 α ℓ − i¯α (i) 1 τ a ≤ Γ a , ∀i=1,...,n ) , (3.6) where Γ a is a budget of uncertainty parameter and τ a ∈ (1,2] is a parameter modeling heavy-tailed probability distributions. Accordingly, we restrict the interarrival times to lie in the set U t := ( (T 1 ,...,T n )∈R n + : − Γ t ≤ P i ℓ=1 T ℓ − i λ (i) 1 τ t ≤ Γ t , ∀i=1,...,n ) , (3.7) whereΓ t is a budget of uncertainty parameter andτ t ∈(1,2] is a parameter modeling heavy-tailed prob- ability distributions. The parametersΓ a andΓ t are chosen to guarantee that with some prescribed prob- abilityp,α := (α 1 ,...,α n ) andT := (T 1 ,...,T n ) will materialize in these regions. For example, if the α values are normally distributed, and we are looking to estimate thep = 95 th quantile of the sum ofα 31 values of passengers that arrived before timet, then, we can setτ a = 2 andΓ a = 1.64, corresponding to the 95 percentile of the truncated normal distribution since ourα values are non-negative. We define the robust sum of α values as the maximum value that the sum of allα ’s associated with pas- sengers that arrived prior to timet can take, subject toα andT both lying in their respective uncertainty sets. In other words, it is the optimal value of the optimization problem max n X i=1 α i I i X ℓ=1 T ℓ ≤ t ! s.t. α ∈U a ,T ∈U t . (3.8) For the case of general convex uncertainty sets, our problem can be formulated as the following mixed- integer convex program max n X i=1 α i y i s.t. y∈{0,1} n , α ∈U a ,T ∈U t i X ℓ=1 T ℓ − t≤ M(1− y i ) ∀i=1,...,n, (3.9) whereM is a “big-M” constant. Note that this problem is an MILP if the uncertainty sets are both polyhe- dral. In this formulation, we use the auxiliary binary variablesy i fori=1,...,n. The big-M constraints in Problem (3.9) ensure that, at an optimal solution,y i =1 if and only if P i ℓ=1 T ℓ ≤ t so that theith passen- ger has arrived on or before timet. We have the following proposition stating that these two formulations are equivalent. Proposition 3.3.2. Problems (3.8) and (3.9) are equivalent. In particular, the two problems have the same optimalobjectivevalueandanyfeasiblesolutiontoProblem (3.8)(resp. (3.9))canbeusedtoconstructafeasible solution to Problem (3.9) (resp. (3.8)) with the same cost. 32 Proof. Let(α ⋆ ,T ⋆ ) be feasible in Problem (3.8) and define y ⋆ through y ⋆ i :=I i X ℓ=1 T ⋆ ℓ ≤ t ! . We show that the triple(α ⋆ ,T ⋆ ,y ⋆ ) is feasible in Problem (3.9) with the same cost. Fixi ′ ∈{1,...,n}. If P i ′ ℓ=1 T ⋆ ℓ ≤ t, then,y ⋆ i ′ =1 and it follows that i ′ X ℓ=1 T ⋆ ℓ − t ≤ 0 = M(1− y ⋆ i ′). If, on the other hand, P i ′ ℓ=1 T ⋆ ℓ >t, then,y ⋆ i ′ =0 and, forM sufficiently large, it holds that i ′ X ℓ=1 T ⋆ ℓ − t ≤ M =M(1− y ⋆ i ′). Since the choice ofi ′ was arbitrary, we conclude that(y ⋆ ,α ⋆ ,T ⋆ ) is feasible in Problem (3.9). Moreover, by definition of y ⋆ , it holds that n X i=1 α ⋆ i y ⋆ i = n X i=1 α ⋆ i I i X ℓ=1 T ⋆ ℓ ≤ t ! . We have constructed a feasible solution to Problem (3.9) that attains the same cost as that attained by (α ⋆ ,T ⋆ ) in Problem (3.8). Since the choice of(α ⋆ ,T ⋆ ) was arbitrary, Problem (3.9) upper bounds Prob- lem (3.8). For the converse, let(y ⋆ ,α ⋆ ,T ⋆ ) be optimal in Problem (3.9). SinceU a ⊆ R n + , we may assume without loss of generality that y ⋆ i =1 ∀i such that i X ℓ=1 T ℓ ≤ t. Indeed, if there exists some i ′ such that P i ′ ℓ=1 T ℓ ≤ t and y ⋆ i ′ = 0, we can always increase y ⋆ i ′ to 1 and remain feasible and optimal, since the objective value will not decrease in the process. From the feasibility 33 of y ⋆ in Problem (3.9), it must hold that y ⋆ i = 0 for all i such that P i ℓ=1 T ℓ > t. We conclude that y ⋆ i =I P i ℓ=1 T ⋆ ℓ ≤ t . Therefore,(α ⋆ ,T ⋆ ) is feasible in Problem (3.8) and attains the same cost as that attained by(y ⋆ ,α ⋆ ,T ⋆ ) in Problem (3.9). We conclude that the two problems are equivalent. The following proposition and theorem then state that this problem admits an analytical solution for the specific choices of uncertainty sets in (3.6) and (3.7). Proposition3.3.3. An optimal solution(α ⋆ ,T ⋆ ) to Problem (3.8) is given by: T ⋆ i :=max 0, i λ − Γ t (i) 1 τ t − i− 1 X ℓ=1 T ⋆ ℓ ! ∀i=1,...,n (3.10) and α ⋆ i =i¯α +Γ a (i) 1 τ a − i− 1 X ℓ=1 α ⋆ ℓ ∀i=1,...,n. (3.11) Proof. The proof is in two parts. First, we show that for any fixed α feasible in Problem (3.8),T ⋆ results in an objective value no smaller than that attained by any other feasibleT . Second, we show that for any fixed T feasible in Problem (3.8), α ⋆ results in an objective value no smaller than that attained by any other feasibleα . Fix ˜ α ∈U a in Problem (3.8). Then, Problem (3.8) reduces to max n X i=1 ˜ α i I i X ℓ=1 T ℓ ≤ t ! s.t. T ∈U t . (3.12) 34 We show thatT ⋆ defined in the premise of the proposition is optimal in Problem (3.12). First, we show thatT ⋆ is feasible in Problem (3.12). According to Equation (3.10), for alli∈{1,...,n}, we have i X ℓ=1 T ⋆ ℓ =T ⋆ i + i− 1 X ℓ=1 T ⋆ ℓ = i− 1 X ℓ=1 T ⋆ ℓ +max 0, i λ − Γ t (i) 1 τ t − i− 1 X ℓ=1 T ⋆ ℓ ! (3.13) =max i− 1 X ℓ=1 T ⋆ ℓ , i λ − Γ t (i) 1 τ t ! . We now show by induction that P i ℓ=1 T ⋆ ℓ satisfies the constraints in the definition of U t , i.e., thatT ⋆ ∈U t . We first show the base case. Since Γ t is positive, we have 1 λ − Γ t (1) 1 τ t ≤ T ⋆ 1 = max 0, 1 λ − Γ t (1) 1 τ t ≤ 1 λ +Γ t (1) 1 τ t Next, we show the induction step. Fixi∈{1,...,n} and suppose that i− 1 λ − Γ t (i− 1) 1 τ t ≤ i− 1 X ℓ=1 T ⋆ ℓ ≤ i− 1 λ +Γ t (i− 1) 1 τ t . Then, i X ℓ=1 T ⋆ ℓ = max i− 1 X ℓ=1 T ⋆ ℓ , i λ − Γ t (i) 1 τ t ! , (3.14) and there are two cases. If P i− 1 ℓ=1 T ⋆ ℓ ≤ i λ − Γ t (i) 1 τ t , then it immediately follows that i λ − Γ t (i) 1 τ t = i X ℓ=1 T ⋆ ℓ ≤ i λ +Γ t (i) 1 τ t 35 On the other hand, if P i− 1 ℓ=1 T ⋆ ℓ > i λ − Γ t (i) 1 τ t , we then have i λ − Γ t (n) 1 τ t < i− 1 X ℓ=1 T ⋆ ℓ ≤ i− 1 λ +Γ t (i− 1) 1 τ t < i λ +Γ t (i) 1 τ t . where the second inequality comes from Equation (3.14) and the last inequality holds due to the fact that f(i)= i λ +Γ t (i) 1 τ t is increasing ini. Thus we have proven thatT ⋆ by Equation (3.10) is a feasible solution to Problem (3.12). We then show that no other feasibleT will result in a better objective value than the T ⋆ constructed above. For anyT that is within the uncertainty setU t , we have − Γ t (i) 1 τ t + i λ ≤ i X ℓ=1 T ℓ ≤ Γ t (i) 1 τ t + i λ ∀i=1,...,n . Suppose there exists a feasibleT ′ ∈U t that yields a strictly better objective value. Define i ⋆ andi ′ as i ⋆ := ( max i=1,...,n i : I i X ℓ=1 T ⋆ ℓ ≤ t ! =1 ) (3.15) i ′ := ( max i=1,...,n i : I i X ℓ=1 T ′ ℓ ≤ t ! =1 ) . Then the existence ofT ′ implies thati ′ ≥ i ⋆ +1 which in turn results in i ⋆ X ℓ=1 T ⋆ ℓ ≤ t and i ⋆ +1 X ℓ=1 T ⋆ ℓ >t, i ⋆ X ℓ=1 T ′ ℓ ≤ t and i ⋆ +1 X ℓ=1 T ′ ℓ ≤ t. We then have i ⋆ +1 X ℓ=1 T ′ ℓ ≤ t< i ⋆ +1 X ℓ=1 T ⋆ ℓ . 36 SinceT ⋆ is constructed such that P i ⋆ +1 ℓ=1 T ⋆ ℓ = i+1 λ − Γ t (i+1) 1 τ t , we have i ⋆ +1 X ℓ=1 T ′ ℓ < i+1 λ − Γ t (i+1) 1 τ t , which contradicts withT ′ being feasible. Similarly, we show thatα ⋆ constructed in Proposition 3.3.3 is an optimal solution for Problem (3.8). Fix ˜ T ∈U a in Problem (3.8). Then, Problem (3.8) reduces to max n X i=1 α i I i X ℓ=1 ˜ T ℓ ≤ t ! s.t. α ∈U a . (3.16) First, we show thatα ⋆ defined in the premise of the proposition is feasible in Problem (3.16). According to Equation (3.11), for alli=1,...,n, we have α ⋆ i = ¯α +Γ a i 1 τ a − (i− 1) 1 τ a ≥ 0 and i X ℓ=1 α ⋆ i =i¯α +Γ a (i) 1 τ a which is within the uncertainty setU a given by Equation (3.6). Thusα ⋆ constructed by Equation (3.11) is a feasible solution to Problem (3.16). We next show that no other feasibleα can achieve a strictly higher objective value than theα ⋆ constructed. Assume there exists aα ′ that is feasible and achieves a strictly better objective value than our constructedα ⋆ . Then we have i ⋆ X i=1 α ′ i > i ⋆ X i=1 α ⋆ i =i ⋆ ¯α +Γ a (i ⋆ ) 1 τ a , 37 which contradicts with the assumption thatα ′ is feasible. Theorem3.3.2. TheanalyticaloptimalsolutiontoProblem (3.8)withU a andU t definedasin (3.6)and (3.7), respectively, is given by A ⋆ =Γ a · √ i ⋆ +i ⋆ ¯α , (3.17) wherei ⋆ is defined as the non-negative integer (or the nearest round-down integer) solution of the equation: t= i ⋆ λ − Γ t (i ⋆ ) 1 τ t . Proof. By Proposition 3.3.3, we have the optimal solutionsT ⋆ andα ⋆ ready to computeA ⋆ . According to Proposition 3.3.2,i ⋆ is uniquely defined by T ⋆ . To solve fori ⋆ , we follow Definition (3.15) and we have: i ⋆ X ℓ=1 T ⋆ ℓ ≤ t⇒− Γ t (i ⋆ ) 1 τ t + i ⋆ λ ≤ t . By letting− Γ t (i ⋆ ) 1 τ t + i ⋆ λ =t, we can solve fori ⋆ . This is exactly the same as the procedures provided in the Theorem. Withi ⋆ calculated, we shall have: A ⋆ = i ⋆ X ℓ=1 α ⋆ i ⋆ =i ⋆ ¯α +Γ a (i ⋆ ) 1 τ a . To better understand how the closed form solution is calculated, we use τ t = 2 as an example, then we have: i ⋆ = j Γ 2 t + p Γ 4 t +4µ 2 t 2 2µ 2 k whereµ = 1 λ . TheA ⋆ is then obtained sinceΓ a and ¯α are known. 38 3.3.3 Driver-in-CoalitionMechanism In this mechanism, we do not enforce that the driver’s direct trip costF is fully recovered by the passengers. Instead, we have the driver share a portion of the direct trip costF . Definition 3.3.4. For all timesk andt with0≤ k ≤ t and all submit ordersπ , the fractionβ π (k),t ofF that will be shared byπ (k) (π (0) being the driver) at timet under submit orderπ is: β π (k),t := α π (k) P t i=0 α π (i) . The analysis is the same as the analysis in Section 3.3.1, so the Driver-in-Coalition Mechanism sat- isfies the five original desirable properties. The difference between this mechanism and the Driver-out- of-Coalition Mechanism is the involvement of the driver in paying F . By involving the driver into the coalition for covering F , the initial quote for the first passenger π (1) becomes more reasonable in that β π (1),1 is bounded above by α π (1) α π (0) +α π (1) . As a result,β π (1),1 is high only whenπ (1)’sα value is too high compared to the driver’s α value. The advantages of the Driver-in-Coalition Mechanism are: (1) all five original desirable properties hold; (2) the Fairness in Sharing Driver’s Cost property holds; and (3) it pro- vides some reasonable reduction on the burden of the first passenger π (1). The disadvantage of the Driver- in-Coalition Mechanism is that the driver will have to cover some portion ofF no matter how large|Π | is. 3.4 AnExampleoftheProposedMechanisms In this section, we present a simple example on how the proposed mechanisms in Section 3.3 calculate the shared costs. As shown in Figure 3.1, suppose there are four passengers and one driver with all origins on the left and all destinations on the right. The dotted line suggests a possible ride-sharing tour. Theα value 39 Figure 3.1: Origins and Destinations of the Driver and the Passengers for the driver isα π (0) =6 while theα values for the passengers areα π (1) =α π (3) =4, α π (2) =α π (4) =2. We assume that these α values are from a truncated normal distribution with known expectation and standard deviation, and that the passenger predicting mechanism estimates the total α value to be 14, which is calculated according to Theorem 3.3.2 with Γ a = 1.64, i ⋆ = 4, and ¯α = 2.68. The β values calculated as passenger requests come in are shown in Table 3.1. The table is divided into 3 different parts, illustrating the trajectories ofβ π (k),t for each mechanism. The columns are the time steps whent requests are made, each row represents the trajectory of β values for a particular passenger. Next we illustrate how theβ values are calculated. For instance, in the Driver-Out-of-Coalition mechanism, passengerπ (1) starts with aβ value of 1 and drops to0.67 = 4 4+2 when 2 passenger requests are made and continues to drop to0.33 = 4 4+2+4+2 when all 4 passenger requests are made. In the Driver-in-Coalition mechanism, passenger 1 starts with aβ value of0.4= 4 4+6 and ends with0.22= 4 6+4+2+4+2 . Finally, in the Passengers Predicting mechanism, theβ value for passenger 1 stays the same at0.29= 4 14 . Table 3.1: The Trajectories ofβ π (k),t under the Different Mechanisms Driver-Out-of-Coalition Driver-in-Coalition PassengersPredicting No. t=1 t=2 t=3 t=4 t=1 t=2 t=3 t=4 t=1 t=2 t=3 t=4 1 1 0.67 0.4 0.33 0.4 0.33 0.25 0.22 0.29 0.29 0.29 0.29 2 0.33 0.2 0.17 0.17 0.125 0.11 0.14 0.14 0.14 3 0.4 0.33 0.25 0.22 0.29 0.29 4 0.17 0.11 0.14 40 3.5 ExperimentalResults In this section, we compared the mechanisms in sharing the driver’s direct trip cost F from Section 3.3 using simulations on a randomly generated data set. Our results show that the robust Passenger Prediction mechanism best balances the driver and passenger costs. We tested the mechanisms on a randomly generated data set on a 40 by 40 grid. Each vehicle has four passengers, each with a unique origin and destination. We assumed that the cost per mile is $1, and the α value is the same as the direct distance. We chose a clustered spatial pattern in which the origins were generated randomly within a cluster of size 10 by 10 at the bottom left corner of the grid and the destinations were generated randomly within a cluster of size 10 by 10 at the top right of the grid. We chose this spatial pattern because ride-sharing operations in daily life usually occur during a driver’s commute to or from work, and business (industrial) areas and residential areas are usually clustered. We performed 100 replications. In each replication, because the number of passengers is small per vehicle, the optimal routing cost can be easily determined quickly through enumeration. We used this optimal routing cost as the total cost to be shared. Then, we used the three proposed mechanisms to allocate a portion of the total cost to the passengers, namely the Driver-out-of-Coalition (DooC), the Driver-in-Coalition (DiC), and the Passenger Prediction (PP) mechanisms. Notice that for the PP mechanism, the mean and standard deviation of theα values are required; we estimated them using 10,000 random samples of the origin and destination distributions. Table 3.2 compares the average performance of the three mechanisms. The first two rows are the total cost of the operation and the driver’s direct trip costF , respectively. Note these two values are not impacted by the different mechanisms and depend on how the origin and destinations are generated. The mechanisms differ in how they allocate the total cost among the driver and passengers. The third row shows the average final shared cost per α value among all the passengers. The fourth row shows the percentage of the absolute error of the budget balance violation: note that the PP mechanism does not 41 necessarily satisfy the Budget Balance property. The fifth row shows how much of the driver’s direct costF is recovered. As we have shown, the DooC mechanism results in a high initial quote for the first passenger. The last row shows the percentage reduction of this initial quote, comparing the initial quote of the first passenger in the corresponding mechanism with that of the DooC mechanism. Table 3.2: Average Performance Measures for the Different Mechanisms Mechanisms DooC DiC PP TotalCostoftheOperation 69.61 69.61 69.61 Driver’sDirectTripCost 42.46 42.46 42.46 AveragePassengerCost 17.40 15.26 17.17 %ofAbsoluteBudgetBalanceError 0 0 2.2 %ofDriver’sCostRecovered 100 80.01 97.79 %ofReducedBurdenfortheFirstPassenger 0 39.91 60.05 As we can see from Table 3.2, the DiC mechanism produced the lowest average passenger cost because the driver is included in sharing the direct costF . The DooC mechanism recovered all ofF but had the highest burden for the first passenger. The PP mechanism had an overall balanced performance in that it reduced the burden for the first passenger significantly and recovered most of the drivers’ costs while maintaining the second lowest average passenger cost. 42 Chapter4 MechanismsforRide-SharingwithTimeWindows In this chapter, we discuss how to extend our mechanisms in Chapter 3 to satisfy time constraints. Now, we assume that ride-sharing drivers have a limit on how much extra time they are willing to spend driving in order to provide the ride-sharing service. We also assume ride-sharing passengers would like to spend as little time as possible in the vehicle. First, we introduce the additional notation we use in this section. We denoteT tot t as the total operation time period at timet. Note that this is the total travel time needed to finish the operation according to the passenger requests received by time t. We next denote T π (k) as the maximum length of time passenger π (k) can spend in the vehicle andL π (k),t as the in-vehicle time of passengerπ (k) at timet. Letc ic π (k),t be the inconvenience cost of passengerπ (k) at timet (c ic π (1),1 =0), and∆ c ic π (k),t :=c ic π (k),t − c ic π (k),t− 1 be the marginal inconvenience cost of passengerπ (k) at timet. Extending the framework in Section 3.2, we let c s 3 π (k),t be the cost of discounts shared by passengerπ (k) under a discount method at timet. We letc dis π (k),t represent the discount amount provided to passengerπ (k) at timet andc da π (k 1 ,k 2 ) represent the coalition discount cost perα value of passengersπ (k 1 ),...,π (k 2 ). In our extension of the Ride-Sharing Mechanism Framework in Section 3.2, passengers receive a dis- count for the time-based inconvenience caused by other passengers. We have two basic rules when insert- ing a new passengerπ (k) withk≥ 1: (1)T tot t does not exceedT π (0) ; and (2)c s π (k),k , the initial quote price 43 Figure 4.1: Process Flow Diagram of passenger π (k), does not exceed W π (k) . If either one of the rules is violated, passenger π (k) will not join the ride-sharing operation. Notice that if no discount is provided in this time constrained scenario, this is a trivial extension from the previous mechanisms: the driver simply rejects any new passenger that causes the in-vehicle time to exceedT π (0) . Therefore, we study the more interesting case where we allow discounts to be provided. To better illustrate the procedure, see the process flow diagram in Figure 4.1. From Figure 4.1 we can see that the two processes marked in yellow are the ones we need to design. We need to derive a way to provide discounts to passengers to compensate for their inconvenience costs brought by passengerπ (k)’s service request. We also do the same for the new passengerπ (k). In order to determine the discount, we need to measure how passengers value their time spent in the vehicle. We use a non-decreasing convex functionf π (k) L π (k),t to quantify the inconvenience cost for passengerπ (k). We next introduce our discount providing solutions: the Basic Discount Method and the Inconvenience Cost Based Discount Method. Each method outputs the discount costc s 3 π (k),t which serves as another cost 44 component in the general Ride-Sharing Mechanism Framework when calculating the total shared cost c s π (k),t . In other words, Equation (3.2) becomes: c s π (k),t =c s 1 π (k),t +c s 2 π (k),t +c s 3 π (k),t . (4.1) We first formally define the discount amount provided to an existing passenger as well as the new passenger whenever there is a new service request. Definition4.0.1. For all timesm,k,t with1≤ m<k≤ t and all submit ordersπ , the discount amount provided to passengerπ (m) at timek is: c dis π (m),k :=min 0, c s 1 π (m),k− 1 +c s 2 π (m),k− 1 − c s 1 π (m),k +c s 2 π (m),k − ∆ c ic π (m),k . (4.2) 4.1 BasicDiscountMethod In this method, the driver does not pay any portion of the discount. Instead, all the burden falls on the new passenger whose request may increase the existing passengers’ inconvenience costs. Therefore, no dis- count is provided to this new passenger at the time of the request. The procedure is described in Algorithm 2. We next examine if this discount method satisfies the properties listed in Section 3.2. The Fairness in Sharing Driver’s Cost and the Reduced Burden for the First Passenger properties hold because the discount componentc s 3 π (k),t is independent fromc s 2 π (k),t . For the five original desirable properties, since the discount method serves as another additive component under the Ride-Sharing Mechanism Framework, based on Theorem 3.2.1, we only need to examine if thec s 3 π (k),t values generated by this discount method satisfy the five original desirable properties. As shown in the theorem below, the Basic Discount Method satisfies three 45 Algorithm2: The Basic Discount Method Input : Information for a new passengerπ (k): origin, destination,c ic π (k),t andα π (k) Information for previous passengers: c s j π (i),k− 1 fori=1,...,k− 1 andj =1,2,3 Output:c s π (i),k fori=1,...,k 1 c s 3 π (k),k ← 0 2 form=1,...,k− 1do 3 if ∆ c ic π (m),k >0then 4 c s 3 π (m),k ← c s 3 π (m),k− 1 +c dis π (m),k 5 c s 3 π (k),k ← c s 3 π (k),k − c dis π (m),k 6 c s π (k),k ← c s 1 π (k),k +c s 2 π (k),k +c s 3 π (k),k of the five original desirable properties, with the Online Fairness and the Ex-Post Incentive Compatibility properties no longer holding. Theorem 4.1.1. Under the Basic Discount Method, c s 3 π (k),t satisfies the Immediate Response, the Individual Rationality, and the Budget Balance properties. Proof. For the Immediate Response property, we need to prove that for all times k,t 1 ,t 2 with 1 ≤ k ≤ t 1 ≤ t 2 and submit ordersπ : c s 3 π (k),t 1 ≥ c s 3 π (k),t 2 . Directly by Algorithm 2, we have that passengerπ (k) has a non-negativec s 3 π (k),k for all1≤ k≤ t and thatc dis π (k),t ≤ 0. Sincec s 3 π (k),t =c s 3 π (k),t− 1 +c dis π (k),t for allk <t, we have c s 3 π (k),t 2 =c s 3 π (k),t 1 + t 2 X t=t 1 +1 c dis π (k),t ≤ c s 3 π (k),t 1 . For the Individual Rationality property, we need to prove that for all timesk,t with1≤ k≤ t and submit ordersπ :c s 3 π (k),t ≤ W 3 π (k) ∗ . By satisfying the Immediate Response property, we know thatc s 3 π (k),t ≤ c s 3 π (k),k . We also know that c s 3 π (k),k ≤ W 3 π (k) since the passenger accepted the fare quote and so the property is satisfied. ∗ This does not mean that the passenger has another willingness to pay level. They each only have one level that isW π (k) = W 1 π (k) +W 2 π (k) +W 3 π (k) . The existence ofW 3 π (k) is only to facilitate the proof denotation. 46 For the Budget Balance property, we need to prove that for all times t ≥ 1 and submit orders π : P t i=1 c s 3 π (i),t = 0. This means that the discounts provided to the passengers are generated within the system. Directly by Algorithm 2, we have that all the discounts provided to the existing passengers when a new passenger requests service are covered by the new passenger. This means that: t X i=1 c s 3 π (i),t = t X i=1   c s 3 π (i),i + t X j=i+1 c dis π (i),j   = t X i=1 c s 3 π (i),i + t− 1 X i=1 t X j=i+1 c dis π (i),j = t X i=1 c s 3 π (i),i + t X j=2 j− 1 X i=1 c dis π (i),j = t X i=2 c s 3 π (i),i + t X j=2 j− 1 X i=1 c dis π (i),j = t X i=2 c s 3 π (i),i − t X j=2 c s 3 π (j),j =0 , where the fourth equality holds because there are no existing passengers when the first passenger π (1) requests service, thus we havec s 3 π (1),1 =0. The advantages of this discount method are: (1) the Fairness in Sharing Driver’s Cost and the Reduced Burden for the First Passenger properties still hold; and (2) the cost is easy to calculate and passengers are not responsible for the inconvenience costs that are not caused by them. The disadvantage of this discount method is that the Online Fairness property and the Ex-Post Incentive Compatibility property are lost. This can be seen through the following counterexamples. 47 For the Online Fairness property, suppose at time t = 2, both passengers π (1) and π (2) have no inconvenience costs and α π (1) = α π (2) . In addition, suppose when passenger π (3) requests service at timet = 3, both passengers’ in-vehicle time increase and their inconvenience costs go from 0 to positive valuesf π (1) L π (1),3 andf π (2) L π (2),3 respectively. Iff π (1) L π (1),3 <f π (2) L π (2),3 , then∆ c ic π (1),3 < ∆ c ic π (2),3 , and this results in c dis π (1),3 > c dis π (2),3 assuming that their total shared costs before receiving any discount decrease by the same amount. Therefore we havec s 3 π (1),3 >c s 3 π (2),3 , which implies c s 3 π (1),3 α π (1) > c s 3 π (2),3 α π (2) , thus contradicting the Online Fairness property. For the Ex-Post Incentive Compatibility property, suppose passengersπ (3) and π (4) share the same origin and destination (this also meansα π (3) =α π (4) ). Suppose the request ofπ (3) increases the inconve- nience costs for the existing passengers and soc s 3 π (3),3 >0 . Sinceπ (4) has the same origin and destination asπ (3), the participation of this passenger will not increase in-vehicle times for passengersπ (1),π (2) and π (3). Thus, c s 3 π (4),4 = 0. Therefore, it is beneficial for π (3) to delay their request submission until right afterπ (4)’s. Under this new submit orderπ ′ ,π ′ (4)=π (3),π ′ (3)=π (4) andc s 3 π ′ (4),4 =0<c s 3 π (3),3 . 4.2 InconvenienceCostBasedDiscountMethod Similarly, in the Inconvenience Cost Based Discount Method, the driver is not responsible for providing discounts. All the inconvenience costs are shared by all the passengers in a way that is similar to POCS, namely, they form into coalitions to share the total inconvenience cost. They then obtain their discounts based on their inconvenience costs. In other words, c s 3 π (k),t consists of two parts: (1) the amount of the total inconvenience cost passengerπ (k) accounts for; and (2) the discounts provided to passengerπ (k) to compensate for the extra in-vehicle time. 48 Formally, for all timesk andt and all submit ordersπ withk≤ t, the cost of passengerπ (k) calculated by the Inconvenience Cost Based Discount Method at timet under submit orderπ is: c s 3 π (k),t :=α π (k) min k≤ j≤ t max 1≤ i≤ j c da π (i,j) + − c ic π (k),t (4.3) wherec da π (i,j) is the coalition discount cost perα value (CDPA) of passengersπ (i),...,π (j) at timet (i≤ j≤ t) under submit orderπ . The CDPA value is calculated in the same manner as the CCPA value in POCS except with the inconvenience costsc ic π (k),t which results inc da π (i,j) = P j l=1 c ic π (l),j − P i l=1 c ic π (l),i P j l=i α π (l) . Suppose the new passenger is π (k) who requests service at time k. At the time of the request, the procedure of this discount method is shown in Algorithm 3. Algorithm3: The Inconvenience Cost Based Discount Method Input : Information for a new passengerπ (k): origin, destination,c ic π (k),t andα π (k) CDPA values of previous passengers: c da π (k 1 ,k 2 ) for1≤ k 1 ≤ k 2 ≤ k− 1 Output:c s π (i),k fori=1,...,k 1 fori=1,...,k do 2 calculatec ic π (i),k based on a selected routing strategy 3 c da π (i,k) ← P k j=1 c ic π (j) − P i j=1 c ic π (j) P k j=i α π (j) 4 forl =1,...,k do 5 c s 3 π (l),k ← α π (l) min l≤ j≤ k max 1≤ i≤ j c da π (i,j) + − c ic π (k),t 6 c s π (k),k ← c s 1 π (k),k +c s 2 π (k),k +c s 3 π (k),k We next examine if this discount method satisfies the properties described in Section 3.2. The Fair- ness in Sharing Driver’s Cost and the Reduced Burden for the First Passenger properties hold because the discount component c s 3 π (k),t is independent from c s 2 π (k),t . For the five original desirable properties, since the discount method serves as another additive component under the Ride-Sharing Mechanism Frame- work, based on Theorem 3.2.1, we only need to examine if the c s 3 π (k),t values generated by this discount method satisfy the five original desirable properties. The first part of c s 3 π (k),t can be shown to satisfy the five marginal desirable properties using the same arguments as these for POCS: the total inconvenience 49 cost for all the passengers satisfies Assumptions 3.1.1 and 3.1.2, and the CDPA values and the first part of (4.3) are basically the same as that in POCS. So, we can focus our attention on the second part,c ic π (t),t . As shown in the theorem below,c s 3 π (k),t satisfies four of the five original desirable properties, with only the Online Fairness property not holding. As a result, the Inconvenience Cost Based Discount Method satisfies the same properties as well. Theorem 4.2.1. Under the Inconvenience Cost Based Discount Method, c s 3 π (k),t satisfies the Immediate Re- sponse, Individual Rationality, Budget Balance and Ex-Post Incentive Compatibility properties. Proof. For the Immediate Response property, we have that the first part of c s 3 π (k),t already satisfies this property. For the second part, sincec ic π (k),t =f π (k) T tot t , andf is a non-decreasing convex function, we have that− c ic π (k),t 1 ≥− c ic π (k),t 2 becauset 1 ≤ t 2 . And this leads to: c s 3 π (k),t 1 =α π (k) min k≤ j≤ t 1 max 1≤ i≤ j c da π (i,j) + − c ic π (k),t 1 ≥ α π (k) min k≤ j≤ t 2 max 1≤ i≤ j c da π (i,j) + − c ic π (k),t 2 =c s 3 π (k),t 2 . For the Individual Rationality property, by satisfying the Immediate Response property, we know that c s 3 π (k),t ≤ c s 3 π (k),k . We also know thatc s 3 π (k),k ≤ W 3 π (k) since the passenger accepted the fare quote, and so the property is satisfied. For the Budget Balance property, since α π (k) min k≤ j≤ t max 1≤ i≤ j c da π (i,j) satisfies the Budget Balance property, then we have t X k=1 α π (k) min k≤ j≤ t max 1≤ i≤ j c da π (i,j) = t X k=1 c ic π (k),t . 50 In other words, the summation of the value for passengers in the same coalition is the marginal inconve- nience cost for that coalition. Summing this value over all the passengers equals to summing the marginal inconvenience cost for all the coalitions which equals to the total inconvenience cost. Then, we have: t X i=1 c s 3 π (i),t = t X k=1 c ic π (k),t − t X k=1 c ic π (k),t =0 . For the Ex-Post Incentive Compatibility property, recall that the total inconvenience cost for all the passengers satisfy Assumptions 3.1.1 and 3.1.2. This means that P t k=1 c ic π (k),t is independent of the submit order and is non-decreasing in time. And sinceπ (k 1 ) andπ ′ (k 2 ) both refer to the same passenger, then we havec ic π (k 1 ),t = c ic π ′ (k 2 ),t ⇒− c ic π (k 1 ),t ≤− c ic π ′ (k 2 ),t . Combined with the fact that the first part of c s 3 π (k),t satisfies this property, we have: c s 3 π (k 1 ),t =α π (k 1 ) min k 1 ≤ j≤ t max 1≤ i≤ j c da π (i,j) + − c ic π (k 1 ),t ≤ α π ′ (k 2 ) min k 2 ≤ j≤ t max 1≤ i≤ j c da π (i,j) + − c ic π ′ (k 2 ),t =c s 3 π ′ (k 2 ),t . The advantage of this discount method is that the Fairness in Sharing Driver’s Cost and the Reduced Burden for the First Passenger properties still hold. The disadvantages of this discount method are: (1) the Online Fairness property does not hold. The same counter-example as in Section 4.1 can be used to show that the property does not hold; and (2) passengers who have high tolerance for in-vehicle time may not get any discounts while being responsible for a portion of the total inconvenience cost. 51 Figure 4.2: Map of Sensors in the Studied Region 4.3 ExperimentalResults In this section, we compared the two proposed discount methods. These experiments are performed on a large data set that is based on travel demand of an area around downtown Los Angeles. As shown in Section 3.5, the PP mechanism had an overall balanced performance. Therefore, we chose to test the PP mechanism for sharingF when comparing the discount methods. When comparing the discount methods, we remark that if the number of passengers the driver picks up on their way is small, the probability of passengers having inconvenience costs is low. Therefore, fewer discounts are generated within the system, which makes the case less interesting. Taking this into con- sideration, a more ideal experiment setting for testing the discount methods is to have multiple drivers and lots of passengers in the system and observe how different discount methods differ in assigning pas- sengers to drivers. As a result, we extend our basic experiment setting in Section 3.5 to a larger setting that involves hundreds of drivers and passengers. Whenever a passenger requests service, the correspond- ing cost-sharing mechanism is executed for each driver which provides to the passenger different initial quotes to choose from. Note that in this multi-driver context, since every passenger must be provided an initial quote upon submitting their request, Assumption 3.1.2 is difficult to satisfy, which the Ex-post In- centive Compatibility property relies upon. One way to satisfy Assumption 3.1.2 would be to re-optimize 52 the vehicle routes each time a request comes in. However, when the number of passengers and drivers is large, re-optimization is a time consuming approach for routing, making it difficult to apply. In our exper- iment, we use a heuristic approach (cheapest insertion) instead that does not guarantee optimal vehicle routes. This results in the loss of the Ex-post Incentive Compatibility property. In addition to compar- ing the performance of the different discount methods, we also investigate “how much" Ex-Post Incentive Compatibility we lose when we use a routing approach that breaks Assumption 3.1.2. We tested the discount methods on a data set with road sensor data, provided by Los Angeles Metro and archived by researchers at the University of Southern California. They developed the Archived Data Management System (ADMS) that collects, archives, and integrates a variety of transportation data sets from Los Angeles, Orange, San Bernardino, Riverside, and Ventura Counties. ADMS includes access to real-time traffic data with 9500 highway and arterial loop detectors providing data on traffic counts and speeds approximately every 1 minute. We selected a region within Los Angeles County that includes 33 sensors on 7 freeways: I-5, I-10, I-105, I-110, I-710, SR-60, and SR-101 (see Figure 4.2). Based on these traffic flows, we generated an origin-destination (OD) matrix of demand, and based on this demand matrix, we generated an OD probability matrix. We can randomly select an OD using a random number between 0 and 1 and the OD probability matrix. Because the sensors are on the freeway which represents the cluster center of locations rather than the specific locations, we randomly generated origins and destinations within three miles of the sensors. Our simulation settings are as follows. We assumed that the average speed of all the drivers is 36 miles per hour. Each passenger has a different sensitivity towards in-vehicle time modeled with different linear slopes of their inconveniences when their in-vehicle time exceeds their direct travel time. The slopes were generated randomly between 0 and 1. The maximum in-vehicle time T π (k) for passengers and drivers was set to be either 1.5 or 2 times their direct travel time. We evaluated the system with 1,000 passenger requests and 300 or 500 ride-sharing drivers, and a willingness-to-pay-level of 1.5, 2 and 3 times (W -factor) the passengers’ direct cost. The complete set of evaluated scenarios is 53 shown in Table 4.1 where theT in the “Time Limit" column represents the direct travel time of passengers and drivers. Table 4.1: Simulation Settings for the Different Scenarios Scenarios NumberofRequests NumberofDrivers TimeLimit W-factor 1 1000 300 1.5T 2 2 1000 300 2T 2 3 1000 500 1.5T 2 4 1000 300 1.5T 1.5 5 1000 300 1.5T 3 To examine the loss of Ex-Post Incentive Compatibility due to using a heuristic routing approach, for every instance we used the setting of Scenario 1 and tested the mechanism when no discount is included. We delayed the first passenger’s request to become the 250 th ,500 th ,750 th and1000 th passenger request. This resulted in altogether 100× 4 = 400 samples. The results are shown in Table 4.2. Each column represents the delayed requests, meaning that the first passenger has delayed to become the 250 th ,500 th , 750 th or the last in the submit order. The first three rows are the percentage of the samples in which the passengers’ final shared costs are better off, not changed and worse, the fourth row is the percentage of the samples in which the passengers are not served since the delayed passengers may experience high initial quote that they are not willing to pay. The last row shows the average change in final price per α value. Note that a positive price change means that the behavior of delaying one‘s request submission time on average results in a higher final price. We can see that although there may be cases where the passenger is better off by delaying their request, on average the passenger is worse off. Furthermore, the possibility of the passengers rejecting the initial quote is higher when they delay their request submission time. These results suggest that the previous mechanisms, although they do not satisfy the Ex-post Incentive Compatibility property when using an insertion based routing approach, still work well on average to discourage passengers in delaying their request time in hopes of getting a better final price. We next present the results of comparing the discount methods. We performed 100 replications. Table 4.3 contains the detailed results for Scenario 1. The first row presents the average direct trip cost per 54 Table 4.2: Effect of Loss of Ex-post Incentive Compatibility Property DelaySlots 250 th 500 th 750 th 1000 th %BetterOff 10.0 6.0 5.0 3.0 %NoChange 20.0 12.0 11.0 9.0 %Worse 56.0 58.0 46.0 43.0 %Unserved 14.0 24.0 38.0 45.0 %ofAveragePriceChange 9.8 22.4 26.3 38.5 vehicle, which is independent of the mechanism used. The second row contains the total cost of the operation per vehicle. The third row shows the shared cost per passenger which is averaged over all the passengers served in one instance. The fourth row presents the average cost paid by the driver. The fifth row shows the percentage of passenger requests that are satisfied by the ride-sharing system. The last row shows the number of drivers that did not pick up any passengers. We compare the Inconvenience Cost Based Discount method (ICBD) and the Basic Discount Method with the case where no discounts are provided. Table 4.3: Average Performance Measures for the Discount Methods in Scenario 1 Mechanisms NoDiscount ICBD BasicDiscount Driver’sDirectTripCost 7.33 7.33 7.33 TotalOperationCostperVehicle 9.54 9.82 9.65 SharedCostPerPassenger 3.10 3.33 3.19 SharedCostPerDriver 2.72 2.48 2.48 %ofRequestsServed 74.67 71.86 75.76 #ofNo-PassengerVehicles 87.34 46.23 62.03 Comparing the discount methods in Scenario 1, we find that both methods reduce the shared costs for the drivers. The ICBD method results in the highest shared cost per passenger as well as the total operation cost per vehicle. The higher shared cost per passenger also causes more passengers to reject their quotes, resulting in fewer served passengers. Combined together with its low number of no-passenger vehicles, we can see that the ICBD method reduces the driver’s cost by spreading the passengers among more vehicles so that more drivers have some form of cost recovery. The Basic Discount method, on the other hand, reduces the driver’s cost by bringing more passengers in the operation and using fewer vehicles (more 55 no-passenger vehicles) compared to the ICBD method. Its increase in the passengers’ shared cost is not as much as the ICBD method which attracts more passengers to the operation. Tables 4.4 presents the results for Scenario 2, where the time limit is higher. When increasing the time limit, the total operation cost per vehicle increases compared to Scenario 1 since a higher time limit allows the drivers to pick up more passengers during their trip. This in turn helps the drivers to recover more of their direct cost and they end up incurring less of a cost. As a result, the number of passengers served in the system increases and the number of no-passenger vehicles decreases. Table 4.5 presents the results for Scenario 3, where the number of drivers is higher. When increasing the number of drivers in the system, the total operation cost per driver decreases because more vehicles are empty and the average number of detours served by a single vehicle is lower. Also, an increase in the number of drivers allows for an increase in the number of passengers served since there is better opportunity for more efficient matching and there is a higher possibility of lower initial quotes. This is because when passengers need to provide discounts for already on-board passengers, their initial quotes are lower when joining a no-passenger vehicle. Consequently, the drivers’ shared cost increases. Table 4.4: Average Performance Measures for the Discount Methods in Scenario 2 Mechanisms NoDiscount ICBD BasicDiscount Driver’sDirectTripCost 7.30 7.30 7.30 TotalOperationCostperVehicle 11.27 11.87 11.35 SharedCostPerPassenger 3.10 3.40 3.15 SharedCostPerDriver 2.75 2.43 2.42 %ofRequestsServed 91.89 86.40 90.95 #ofNo-PassengerVehicles 85.7 29.49 51.54 Table 4.5: Average Performance Measures for the Discount Methods in Scenario 3 Mechanisms NoDiscount ICBD BasicDiscount Driver’sDirectTripCost 7.33 7.33 7.33 TotalOperationCostperVehicle 8.74 9.17 8.88 SharedCostPerPassenger 3.16 3.46 3.28 SharedCostPerDriver 3.70 3.24 3.38 %ofRequestsServed 90.89 90.67 91.63 #ofNo-PassengerVehicles 208.29 115.97 115.00 56 We are also interested in how the willingness-to-pay-level affects the ride-sharing operation system. Using the same setting as in Scenario 1 except for having different willingness-to-pay-levels, we compare the discount methods based on these four performance indicators: shared cost per passenger, shared cost per driver, percentage of requests served and the number of no-passenger vehicles. The results are shown in Figures 4.3 and 4.4. The horizontal axis in each figure represents the W -factor while the vertical axis represents the performance indicator. As we can observe from the figures, the increase in passengers’ willingness-to-pay-level increases the shared cost of passengers for both discount methods since the pas- sengers are willing to pay more. Moreover, this also allows more passengers in the system which decreases the drivers’ cost. The ICBD method benefits the most because this method has fewer no-passenger vehicles which spreads the passengers across the vehicles more efficiently. Figure 4.3: The effect of willingness-to-pay-level on passengers’ cost and drivers’ cost Figure 4.4: The effect of willingness-to-pay-level on % served and % of no-passenger vehicles 57 Chapter5 Ride-SharingRoutingwithWalking In this chapter, we focus on the routing part of our ride-sharing problem. Our ride-sharing context remain the same as in the previous chapters where drivers have their personal origins and destinations. Specifically in the context of vehicle routing, we study the pickup and deliver problem (PDP) with meeting points which means that the passengers have flexible pickup and drop-off locations. We also consider the utilization of high occupancy (HOV) lanes to maximize the savings in time brought by ride-sharing. This chapter first describes the routing problem and establishes the associated models. Then solution algorithms are introduced in detail, followed by experimental results. 5.1 ProblemDescription Although we are interested in solving the dynamic version of the routing problem, we first present a mathematical model of the static version to formalize our problem description. Furthermore, the decom- position of the static formulation is incorporated in our dynamic solution approach which is presented in Section 5.2. Given a set of passengersP ={1,...,P} and a set of driversV ={1,...,V}, each passengerp∈P (driverv∈V) has an originO p (O v ), a destinationD p (D v ). Each driver has a maximum in-vehicle time H v while each passenger has a maximum walking distanceL p to the deviated origin and destination and a 58 maximum wait timeI p before the vehicle arrives. The passengers share the same walking speed ofW . We need to output the deviated origin (O d p ) of a passenger’s request and the deviated destination (D d p ) while satisfying all the constraints provided by the passengers’ information. The objective is to minimize the total travel time. We first create a network G(N,A) withn passengers andm drivers. The node setN ={1,...,2n+ 2m}=O p ∪D p ∪O v ∪D v whereO p ={1,...,n},D p ={n+1,...,2n},O v ={2n+1,...,2n+m} and D v ={2n+m+1,...,2n+2m}. The arc setA={A i,j |i,j∈N}. Note that the incorporation of HOV lanes may generate multiple arcs between two nodes. For simplicity, since HOV lanes are always chosen when possible (due to less travel time), instead of representing them with multiple arcs, we will introduce a constraint to indicate whether the travel time associated with HOV lanes are valid. The deviated locations will not be included in the network graph, no arcs connected to them, but they are among the decision variables. Let c i,j denote the travel time between node i and j, d i,j denote the distance between node i and j, β i,j denote the time discount factor between node i and j when HOV lane is chosen, and H denotes the number of people required to go on a HOV lane. (r x i ,r y i ) denotes the coordinates of node i∈ N where(l x i ,l y i ) denotes the coordinates of the deviated locations of nodei∈ O p ∪D p . Both sets of coordinates are inR 2 space.L i ,i∈O p ∪D p denote the maximum walking distance of each passenger to the deviated locations,E denotes the average driving speed andU denotes the capacity for each vehicle (i.e., the maximum number of passengers in a vehicle). We also denoteg i,v as the load indicator to show whether a passenger is picked up or delivered: g i,v =                      1, ifi∈O p − 1, ifi∈D p 0, otherwise. 59 The formulation of this PDP is below. The three sets of decision variables are: y i,j,v =            1, if vehiclev travels from nodei to nodej 0, otherwise b i,j,v =            1, if nodei is visited before nodej on vehiclev 0, otherwise. α i,j,v =            1, if HOV lane is valid for vehiclev from nodei to nodej 0, otherwise. (l x i ,l y i ) ∀i∈O p ∪D p The mathematical formulation is then: min X v∈V X i∈N X j∈N (1− α i,j,v β i,j )c i,j y i,j,v +M X j∈Op 1− X v∈V X i∈N y i,j,v ! s.t. X v∈V X j∈N y i,j,v ≤ 1 ∀i∈N\D v (5.1) X v∈V X i∈N y i,j,v ≤ 1 ∀j∈N\O v (5.2) X j∈N y i,j,v = X j∈N y j,i,v ∀i∈O p ∪D p ,v∈V (5.3) X j∈N y i,j,i− 2n = X j∈N y j,i+m,i− 2n ∀i∈O v (5.4) b k,i,v ≤ b k,j,v +(1− y i,j,v ) ∀i∈N\D v ,j∈N\O v ,k∈N\{i}andv∈V (5.5) b k,j,v ≤ b k,i,v +(1− y i,j,v ) ∀i∈N\D v ,j∈N\O v ,k∈N\{i}andv∈V (5.6) y i,j,v ≤ b i,j,v ∀A i,j ∈A,v∈V (5.7) b i,i,v =b i,k,v =0 ∀i∈N,k∈O v andv∈V (5.8) 60 b j,i,v =0 ∀v∈V,i∈O p andj =i+n,i∈O v andj =i+m (5.9) b i,j,v =1 ∀v∈V,i∈O v andj =i+m (5.10) X v∈V b i,j,v =1 ∀i∈O p andj =i+n (5.11) b i,k,v =b i+n,k,v ∀v∈V,i∈O p ,k∈D v (5.12) g j,v + X i∈N b i,j,v g i,v ≤ U ∀j∈N,v∈V (5.13) (l x i − r x i ) 2 +(l y i − r y i ) 2 ≤ L 2 i ∀i∈O p ∪D p (5.14) d 2 i,j =(l x i − l x j ) 2 +(l y i − l y j ) 2 ∀i,j∈O p ∪D p (5.15) d 2 i,j =(r x i − l x j ) 2 +(r y i − l y j ) 2 ∀i∈O v ,j∈O p (5.16) d 2 i,j =(l x i − r x j ) 2 +(l y i − r y j ) 2 ∀i∈D p ,j∈D v (5.17) c i,j E =d i,j ∀A i,j ∈A (5.18) X k∈N b k,i,v g k,v ≥ H− M(1− α i,j,v ) ∀i,j∈N,v∈V (5.19) X i∈N X j∈N (1− α i,j,v β i,j )c i,j y i,j,v ≤ H v ∀v∈V (5.20) α i,j,v =0or 1 (5.21) y i,j,v =0or 1 (5.22) b i,j,v =0or 1 (5.23) The objective is to minimize the total travel cost (first term) plus the minimization of unserved passen- gers (second term) whereM is a weighting factor. M is set to a large number when it is desired to serve as many requests as possible and for that solution to minimize the travel cost. Constraint sets (21) and (22) are network flow constraints imposing that one passenger is served by one driver or no driver. Constraint set (23) ensures that the origin and destination of a passenger be assigned 61 to the same driver. Constraint set (24) ensures that the origin and destination of a driver is assigned to the same driver. Constraint sets (25) and (26) ensure that if nodei is immediately before nodej (y i,j,v = 1), then we haveb k,i,v =b k,j,v for allk∈N\i,v∈V . Similarly, constraint set (27) enforces that ify i,j,v =1, b i,j,v = 1 and if b i,j,v = 0, y i,j,v = 0. Constraint sets (28) - (32) are prior constraints that enforce the deviated origins to be ahead of the deviated destinations. They also enforce the drivers’ origins are ahead of their corresponding destinations. Constraint set (33) is the capacity constraint. Constraint set (34) ensures that the deviated locations are within the passengers’ walking ranges. Constraint sets (35) - (37) describe how the actual distance costs between nodes are calculated; that is, even though we determine the pickup and delivery sequence based on nodesO p andD p , we calculate the distances using the deviated locations. Constraint set (38) describes the relationship between distance and time to travel from nodei to nodej. Note if HOV lane is eligible to be taken from nodei to nodej then a discount is applied in the objective function. Constraint set (39) ensures the time discount factor for an arc is activated only when the HOV eligibility threshold is reached. Constraint set (40) ensures that for all the vehicles, the time a vehicle spends in the operation does not exceed its maximum in-vehicle timeH v . Since constraint set (40) may result in certain passengers not being served by any vehicle, we add an extra term in the objective function to avoid the infeasibility of the formulation. Therefore, these set of constraints ensure that the objective function is based on the deviated locations that minimizes the total time costs while maximizing the number of passengers served in the system. As we can see from the above formulation, the PDP and the location selection problem are simultane- ously solved. The location selection problem itself is a non-linear problem which causes extra complexity on the NP-hard PDP. In order to solve these two problems, we propose to separately solve them. We delete constraint sets (34) - (38) from the above formulation and simply letc i,j = q (r x i − r x j ) 2 +(r y i − r y j ) 2 /E for alli,j∈N. As a result, we have a model dedicated to solving the multi-vehicle PDP. The output of this model is a list of vehicle routes with each passenger assigned to a vehicle. Then for each vehicle routeZ, it 62 contains one driver andq passengers. Let the first node in the route be Z 1 ∈O v ={2n+1,...,2n+m}, the last node beZ 2q+2 ∈ D v ={2n+m+1,...,2n+2m} and the nodes in betweenZ i ∈ O p ∪D p = {1,...,2n} be the pickup and delivery locations of the q passengers. We then establish the following quadratic model: min 2q+1 X i=1 d Z i ,Z i+1 s.t. r x Z i − l x Z i 2 + r y Z i − l y Z i 2 ≤ L 2 Z i ∀i=2,...,2q+1 r x Z 1 − l x Z 2 2 + r y Z 1 − l y Z 2 2 ≤ d 2 Z 1 ,Z 2 r x Z 2q+2 − l x Z 2q+1 2 + r y Z 2q+2 − l y Z 2q+1 2 ≤ d 2 Z 2q+1 ,Z 2q+2 l x Z i+1 − l x Z i 2 + l y Z i+1 − l y Z i 2 ≤ d 2 Z i ,Z i+1 ∀i=2,...,2q 5.2 DynamicSolutionAlgorithm In this section, we describe our solution approach for solving the dynamic version of the above static problem. In the dynamic version, instead of requests being known at the beginning of the day, the requests arrive dynamically throughout the day as well as the driver’s departure time. Figure 5.1 describes how each new passenger request is dealt with from a high level point of view. Once a request is received by the ride- sharing system, it first generates a feasible set of vehicles based on the personal preferences of the request. That is, starting at the vehicle’s current location, if it fails to reach any point within the maximum walking circle of the request location, then this vehicle is not feasible for this new request. This procedure provides a basic filter for assigning requests to vehicles. If the feasible set turns out to be empty, the system rejects the request. Otherwise, for each vehicle in the feasible set, the system calls on our routing algorithm to calculate a route and meeting points for this new request. That is, it inserts the request in the current route of the vehicle and determines the meeting points with minimal increase in travel time for that vehicle. 63 Figure 5.1: The Overall Solution Framework During this procedure, if an insertion to a certain vehicle will cause any violation of the driver’s maximum detour timeT v or the maximum waiting timeI p of an existing request on that vehicle, then this vehicle is removed from the feasible vehicle set. After this procedure, if the set is empty, the request is rejected. Otherwise, the request is accepted with the route and meeting points that has the minimal increase in travel time among all the feasible vehicles. Next, we introduce our algorithms in greater detail. We first present our routing algorithm and then the incorporated location selection algorithm that is used to generate the meeting points given a route. 5.2.1 TheRoutingAlgorithm In this section, we introduce how we match the passengers with the drivers and route them. We use an insertion algorithm to determine the ordering of passengers in the route. We use an insertion procedure because it is shown to be fast and effective in solving dynamic routing problems [9, 52, 29]. In order to introduce our algorithm in detail, we first introduce some extra notation. We denote V F p,t as the feasible set of drivers for passengerp at timet. We also denoteP v,t as the set of passengers assigned to driverv at timet,R v,t as the current route of driverv at timet which contains a sequence of deviated 64 locations O d p and D d p for p ∈ P v,t . Then, we describe how a new request from passenger p is routed in Algorithm 4. First, a feasible setV F p,t is first created by checking all vehicles to see if they can reach any point on J O p (a circle with a radius ofL p centered atO p ) within maximum waiting timeI p . Immediately after a passengerp submits a request at timet (see Algorithm 4), passengerp will go through each vehiclev∈V F p,t to see if it can be inserted. For each vehiclev, the algorithm tries to insert passengerp by first checking the capacity constraint. After that, temporary routes are generated with meeting points calculated as well (the next section describes Algorithm 5 to compute the meeting points). Then, for each temporary route, it checks the maximum in-vehicle time constraint for the driver and the maximum wait time constraint for the passengers inP v,t and passengerp as well. Lastly, if a temporary route survives all feasibility checks, it is then added to the potential route setΦ . IfΦ turns out to be empty, then vehiclev will not be added to V F p,t+1 . Otherwise, not only v is added to V F p,t+1 , but also the corresponding R v,t+1 and P v,t+1 are updated. Once all the vehicles inV F p,t are checked, we compare the vehicles inV F p,t+1 and find the v whose corresponding routeR v,t+1 has the minimal increase in travel time and assignp to thatv. 5.2.2 TheLocationSelectingAlgorithm The location selection algorithm determines the meeting points (deviated points) given a route ordering for a vehicle. In order to solve the location selection problem, we first acknowledge that the problem is equivalent to this following problem: given two fixed points, O andD, given2n circles{ J 1 ,..., J 2n } each with its center and radius, and the sequence of connecting the circles, how to determine the 2n points, each within its corresponding circle (including the boundary), such that the total distance connectingO to the circles in the given sequence and then toD is the shortest? 65 Algorithm4: The Routing Algorithm Input : Passengerp’s information,O p ,D p ,L p andI p The feasible setV F p,t andR v,t ,P v,t corresponding tov∈V F p,t Output: a vehiclev min that passengerp is assigned to and theR v min ,t+1 ,P v min ,t+1 corresponding to thisv min 1 V F p,t+1 =∅ 2 forv∈V F p,t do 3 create an empty setΦ to save potential routes 4 fori=1,...,|R v,t |+1do 5 insertO p to become thei th inR v,t , generatingR ′ v,t 6 if the insertion will cause capacityU be violated at any point then 7 continue 8 else 9 forj =i,...,|R ′ v,t |+1do 10 insertD p to become thej th inR ′ v,t , generatingR ′′ v,t 11 calculateO d p andD d p using Algorithm 5 and replaceO p ,D p inR ′′ v,t 12 if H v is violated or anyI p forp∈P v,t is violated then 13 continue 14 else 15 addR ′′ v,t to setΦ 16 if Φ ̸=∅then 17 addv toV F p,t+1 18 addp toP v,t+1 19 R v,t+1 = the route inΦ with minimal increase in travel time 20 else 21 reject requestq 22 Letv min ∈V F p,t+1 be the vehicle with the minimal increase in travel time, assignp tov min and outputR v min ,t+1 andP v min ,t+1 66 Figure 5.2: The One Circle Case Let’s first take a look at the one circle case. As shown in Figure 5.2, we want to find a point P within J A (including the boundary) such that the length of|OP| +|PD| is the shortest. It is trivial to see that whenl OD intersects with J A, pointP is any point on the line segment ofl OD that is within J A (this includes the scenario where l OD is tangent to J A). Since there may be multiple points, we could generally provide a single point solution for these two scenarios: pointP is the projection of pointA onto l OD . Ifl OD does not intersect with J A, then pointP is a point such thatl AP is the bisector of∠OPD. In fact, the latter description generalizes all the scenarios and the following theorem describes how to determine pointP . Theorem5.2.1. Giventwofixedpoints OandD,and J A,theoptimalpointP within J Athatminimizes |OP|+|PD| is the point onA such thatl AP is the bisector of∠OPD. Proof. Whenl OD intersects with J A,P is onl OD because the distance among three points is the shortest when they are on the same line. When projecting A onto l OD , we have l AP ⊥ l OD and∠OPD = π . Therefore,P is indeed the point onA wherel AP is the bisector of∠OPD. Whenl OD does not intersect with J A, we know that the distance of any point on an ellipse to the ellipse’s two foci is a constant. As shown in Figure 5.3, suppose O and D are the foci of an ellipse and suppose thatP is on that ellipse, then we have|OP|+|PD|=2a wherea is a constant. We can see that 2a is minimized when P is on l OD and that 2a = |OD|. However, when 2a is minimized, P is not on 67 Figure 5.3: Ellipse Tangent to Circle A J A. Therefore, we increase2a untilP is on J A. As a result, we have an ellipse that is tangent to J A and hasO andD as its foci. Since for an ellipse, the path for a light beam starting at one focus will always travel through a point P on the ellipse boundary and then reach the other focus, by Fermat’s Law, this path is the shortest path fromO toP and then toD. We also know that∠1=∠2. Therefore,l AP is the bisector of∠OPD. We now have arrived at the geometrical property of pointP . Additionally, based on this property, an algebraic solution approach can be found according to [21] (referred as the Eberly Algorithm for the rest of the thesis). The problem of finding P is reduced to solving a unary quadratic equation whose solution is well known and is in closed-form. Therefore, in the one circle case, the optimal meeting pointP can be found withinO(1) time complexity. Next, we show how this one circle case can be extended to the case of multiple circles in our solution approach. We first introduce how it is applied to the two-circle case and then the 2q-circle case where q is the number of passengers. As shown in Figure 5.4, we have a pair of circlesA andB which are the origin and destination of a single passenger. As usual, we have the origin and destination of the driver as well, denoted as pointsO andD. According to Theorem 5.2.1, we find the initial point of P A , denoted as 68 Figure 5.4: The Two-Circle Case P 0 A , given points O, B and J A. Similarly, we find P 0 B , given points A, D and J B, the total distance is denoted as d 0 . Next, we search in the neighbourhood of P 0 A . Given a fixed range and spacing, we pickK points in the neighbourhood ofP 0 A . For each neighbourhood point denoted as(P 0 A ) k , we find the corresponding optimal point(P 0 B ) k (given fixed point (P 0 A ) k , pointD and J B). After that, we calculate the total distanced 0 k associated with(P 0 A ) k and select the neighbour and its correspondingP B who have the shortest total distance to beP 1 A andP 1 B . The corresponding total distance is denoted asd 1 . We also denoteϵ 1 = d 1 − d 0 . Similarly, we search theK neighbourhood points ofP 1 B and obtainP 2 A ,P 2 B andd 2 . We iterate for a finite number of iterations F or until the error is smaller than a given precisionϵ . Then we have found the optimal pair of pointsP ⋆ A andP ⋆ B . We next describe how the above procedures can be applied to the2q-circle case (q passengers). Using the same set of notation as in Section 5.1, Algorithm 5 describes our solution approach. Given a vehicle route Z of q passengers, Z 1 and Z 2q+2 are the origin and destination of the driver which are fixed points. Additionally, we have J Z 2 ,..., J Z 2q+1 representing the circles centered at pointsZ 2 ,...,Z 2q+1 . The idea of Algorithm 5 is to find a pair of P ⋆ A andP ⋆ B at a time whereA=Z i and B =Z 2q+3− i fori=2,...,q+1. When determining a pair ofP ⋆ Z i andP ⋆ Z 2q+3− i , we use exactly the same procedures as the two-circle case we introduced in above. The only difference is that we have an extra step 69 of propagating the P f− 1 Z i+1 k ,..., P f− 1 Z 2q+2− i k during the neighbourhood search in each iteration. The way to do that is to apply Theorem5.2.1 and obtain the optimal points. For example, given J Z i+1 and fixed points P f− 1 Z i k and P f− 1 Z i+2 , we determine P f− 1 Z i+1 k . This procedure terminates after F iterations or if the total distance errorϵ f is smaller than a given error parameterϵ . Since the one circle case can be solved in O(1) time, the time complexity of Algorithm 5 is at most O(KFn 2 ) where K is a given parameter describing the number of neighbourhood searches, F is the number of iterations which is associated with given parameterϵ andn is the total number of passengers in the system. Therefore, we have deviced an algorithm for solving the quadratically constrained quadratic program shown at the end of Section 5.1. We next show in Theorem 5.2.2 that this algorithm is optimal when parameterϵ →0. Theorem5.2.2. When the given error parameterϵ →0, the solution of Algorithm 5 goes to optimal. Proof. We prove by induction. Previously, we have shown that when there is only one circle in the se- quence, there exists a closed-form optimal solution. Therefore, when the number of circles k = 1, the statement holds. When there is only one passenger on vehicle routeZ, that isk =2 andq =1 as shown in Figure 5.4. For every neighbourhood ofP A , the corresponding optimalP B is calculated and vice versa. Therefore, if one of the pointsP ⋆ A andP ⋆ B is found during the neighbourhood search, the other one is found immediately. The quadratic model at the end of Section 5.1 is convex due to the fact that both the objective and the constraints are convex. Therefore, an optimal solution exists. We useP ⋆ A as an example. First, we observe that it is within a bounded region. As shown in Figure 5.5,P ⋆ A is within the range ofP l A andP r A on the circle boundary. P l A and P r A are determined by picking extreme points P 1 B and P 2 B on J B. Since our initial pointP 0 A is obtained by fixed points O,B and J A, it will be within the solution region as well. Then, an original search range radius of± π 4 fromP 0 A would be enough to cover the solution region. Additionally, we observe that, there is no local optima because moving away fromP ⋆ A will result in non-decreasing total 70 Algorithm5: The Location Selecting Algorithm Input : Vehicle routeZ,q number of passengers in routeZ, and location coordinates(r x Z i ,r y Z i ), radiusL Z i for each pointZ i ,i=1,...,2q+2 Output: Meeting pointsP ⋆ Z i =(l x Z i ,l y Z i ) fori=2,...,2q+1 1 fori=2,...,2q+1do 2 letO =Z i− 1 ,D =Z i+1 , andA=Z i 3 solve for initial pointP 0 Z i and calculated 0 by Theorem 5.2.1 and the Eberly Algorithm. 4 fori=2,...,q+1do 5 forf =1,...,F do 6 if f is odd then 7 fork =1,...,K do 8 obtain P f− 1 Z i k 9 calculate the corresponding P f− 1 Z j k points forj =i+1,...,2q+3− i 10 calculate the total distanced f− 1 k 11 pick n P f− 1 Z i k o 2q+3− i i=2 with lowestd f− 1 k and set them as n P f Z i o 2q+3− i i=2 andd f 12 else 13 fork =1,...,K do 14 obtain P f− 1 Z 2q+3− i k 15 calculate the corresponding P f− 1 Z j k points forj =i,...,2q+2− i 16 calculate the total distanced f− 1 k 17 pick n P f− 1 Z i k o 2q+3− i i=2 with lowestd f− 1 k and set them as n P f Z i o 2q+3− i i=2 andd f 18 obtainϵ f =d f − d f− 1 19 ifϵ f <ϵ ,P ⋆ Z i =P f Z i ,P ⋆ Z 2q+3− i =P f Z 2q+3− i ,break 20 iff =F ,P ⋆ Z i =P F Z i ,P ⋆ Z 2q+3− i =P F Z 2q+3− i . 71 Figure 5.5: The Solution Region distances (the route ofO− P A − P B − D). To better illustrate this, see Figure 5.6. The figure on the left is an example of a two-circle case with green crosses being the origin and destination of a driver and the blue dots being the origin and destination of a passenger. The 10 red dots are equally spaced P A on the circumference of J A and they are numbered 1 to 10 from left to right. The plot on the right shows how the total minimum distance of the route changes when we fix P A and obtain the corresponding optimal P B . We can see that it is indeed a convex plot. This guarantees that as long as we are approachingP ⋆ A , we will obtain the optimal objective value. As we calculate the total distance for each neighbourhood point, we can pick the two consecutive points that have a decrease and then an increase in the total distance, and set it to be the search bounds for the next iteration. As a result, if we setϵ →0, we will obtain the optimal pointsP ⋆ A andP ⋆ B . Therefore, when the number of circlesk =2, the statement holds. Assume that the statement holds when k = 2q− 1, then we have that Algorithm 5 goes to optimal when ϵ → 0. Therefore, when k = 2q and ϵ → 0, the 2q circles can be divided into 2q− 1 circles and 1 circle. By assumption, we can find the optimal meeting points for the first 2q− 1 circles for each fixed meeting point of the last circle. And by fixing the (2q− 1) th circle, we can find the optimal meeting point of the last circle. Therefore, the problem when k = 2q has been reduced in the same fashion as when 72 (a) The Two-Circle Example with Points (b) The Total Distance Plot for Selected Points Figure 5.6: Distance Information of the Studied Region k = 2 which we have already shown to be true. As a result, the statement holds when k = 2q and the proof is complete. 5.3 ExperimentalResults In this section, we present our experimental analysis. The main purpose of these experiments is to see how HOV lanes and meeting points contribute to the efficiency of a dynamic ride-sharing system. In general, there will be four different types of settings: 1) the control group which has no HOV lanes and every passenger request is served at the exact location instead of meeting points; 2) the HOV experimental group which has HOV lanes but no meeting points; 3) the meeting points experimental group which has no HOV lanes but meeting points, and 4) the experimental group with both the HOV lanes and the meeting points. All experiments are run on a 20 by 20 grid where OD pairs are randomly generated. For each driver, the normal driving speed on average (without HOV lane) is 36 miles per hour and the vehicle capacity is set to 4 (including the driver). The maximum detour time is set to be 1.5 times the driver’s direct travel time. For each passenger request, it comes with a randomly generated maximum walking time within 0 to 73 20 minutes (for the control setting, this will be 0 for all requests) and they have the same walking speed of 3 miles per hour. The requests submission follows a Poisson process with a mean of 850 requests an hour. We performed 20 replications, each has 300 passengers and 100 drivers. As for the performance measures, there are four major measures: 1) the average time spent in the ride- sharing system for each driver; 2) the average extra time needed for each served request; 3) the average in-vehicle time (IVT) ratio for each request, and 4) the percentage of requests served in the ride-sharing system. The first measure monitors the cost to the drivers. The second and the third measures monitor the cost to the passengers. In detail, the second measure is calculated by summing the extra time needed for all served requests (excluding the direct travel time for the driver) and then divide the summation by the number of served requests. The third measure is obtained by averaging the in-vehicle time ratio among the served requests. The IVT ratio is the ratio between the actual in-vehicle time and the direct travel time (without HOV lanes) of the passenger. The last measure is to check and compare the efficiency of the ride-sharing system under different settings. There are also four other measures that help with understanding the results: 1) the average direct distance of the drivers; 2) the average miles each vehicle travels; 3) the average requests served per loaded vehicle and 4) the percentage of vehicles with passengers. We first present the results for the HOV experimental group. In this group, we change normal lanes to HOV lanes with the requirement of2+ or3+ persons in a vehicle. The driving speed on HOV lanes is around 50 miles per hour if the requirement is2+ persons in a vehicle or around 70 miles per hour if the requirement is3+. Table 5.1 shows the results under different HOV lane requirements. The first column is the control group while the "HOV2" means all roads contain HOV lanes that requires2+ persons in a vehicle. Similarly, "HOV3" means all roads contain HOV lanes that requires3+ persons in a vehicle. As we can see, HOV lanes indeed reduce the average IVT ratio since they provide a boost in driving speed. They also help increase the percentage of requests served in the system since the time saved in using HOV lanes 74 can be utilized to serve more requests and serving more requests increases the possibility of using the HOV lanes. Another observation is that HOV2 outperforms HOV3 in the percentage of requests served. This is due to HOV3 has higher requirements for entering the HOV lanes thus forcing the drivers to pickup more passengers along the way to be eligible for the speed boost, resulting in an increase in average request served per vehicle but a decrease in the percentage of loaded vehicles. Table 5.1: Performance Measures for HOV Experimental Group Original HOV2 HOV3 AvgTimeperVehicle(min) 22.35 21.66 20.80 AvgExtraTimeperRequest(min) 4.03 2.70 2.31 AvgIVTRatio 1.16 0.91 0.85 %RequestsServed 40.50% 51.98% 48.43% AvgDirectDistanceperVehicle(mile) 10.48 10.48 10.48 AvgDistanceTravelledperVehicle(mile) 13.41 15.82 14.93 Avg#ofRequestsServed 2.34 2.52 2.84 %ofLoadedVehicles 52.05% 61.95% 51.30% Next, we present the results for the meeting points experimental group. In this group, we change the maximum walking time from 0 to 10 to 20 minutes. Table 5.2 shows the results. As we increase the maximum walking time, more requests are served in the system. This is due to the fact that the larger walking time allows closer meeting points among locations so more passengers can be served within the same amount of travel distance. Since no HOV lanes are involved in this experimental group, there will not be any speed boost to help save time. But if we take a closer look, we can find that around a 4.2% increase in time spent per vehicle and 4.3% increase in average IVT ratio bring around 10.7% increase in usage of the ride-sharing system. The percentage of requests served in the 20-min-walk case is the same as that of the HOV3 in Table 5.1 with fewer miles travelled per vehicle and fewer empty vehicles. The reduction in miles travelled while maintaining the same level of system usage indicates its ability in reducing detours. Lastly, we want to see how the combination of HOV lanes and meeting points together contribute to the efficiency of the ride-sharing system. As shown in Tables 5.3 - 5.4, we find that increase in maximum walking time always results in higher system usage for ride-sharing while HOV lanes always results in time 75 Table 5.2: Performance Measures for Meeting Points Experimental Group Original 10-min-walk 20-min-walk AvgtimeperVehicle(min) 22.35 23.28 24.28 AvgExtraTimeperRequest(min) 4.03 4.33 4.71 AvgIVTRatio 1.16 1.21 1.28 %RequestsServed 40.50% 44.83% 48.32% AvgDirectDistanceperVehicle(mile) 10.48 10.48 10.48 AvgDistanceTravelledperVehicle(mile) 13.41 13.97 14.57 Avg#ofRequestsServed 2.34 2.59 2.77 %ofLoadedVehicles 52.05% 52.05% 52.40% efficiency which significantly increases the passengers’ in-vehicle experience and reduce the passengers’ cost (due to sharing the ride). Table 5.3: Average IVT Ratio with HOV Lanes and Meeting Points AvgIVTRatio Original HOV2 HOV3 Original 1.16 0.91 0.85 10-min-walk 1.21 0.94 0.86 20-min-walk 1.28 0.97 0.88 Table 5.4: Percentage Requests Served with HOV Lanes and Meeting Points %RequestsServed Original HOV2 HOV3 Original 40.50% 51.98% 48.43% 10-min-walk 44.83% 56.52% 52.88% 20-min-walk 48.32% 59.57% 56.37% 76 Chapter6 ConclusionandFutureWork In this dissertation, we studied the ride-sharing problem which is a combination of the cost-sharing prob- lem and the vehicle routing problem. We provided our wholistic solution to design a ride-sharing system that is differentiated with service provides like Uber, Lyft and etc. The aim of our ride-sharing system is to provide a convenient and applicable tool to daily travellers so that commuters with a car can share a ride with those who does not and recover their travelling costs during the ride-sharing operation. In order to achieve this, we first explored the cost-sharing part of ride-sharing and proposed desirable properties under our ride-sharing scenario and explained why mechanisms from the existing literature fail to satisfy these properties. We proposed a general mechanism framework that satisfies the Online Fair- ness, Immediate Response, Individual Rationality and Ex-Post Incentive Compatibility properties. This framework is flexible and can be adapted to serve different purposes. Based on this general mechanism framework, we then proposed three different mechanisms that each has its advantages and disadvantages. If the driver cares about the five original desirable properties, they may prefer the Driver-out-of-Coalition Mechanism. If the driver cares less about the Budget Balance property but more about the Fairness in Sharing Driver’s Cost and the Reduced Burden for the First Passenger properties, they may prefer the Passengers Predicting Mechanism. If the driver has some flexibility in not recovering all of the direct trip cost, they may prefer the Driver-in-Coalition Mechanism. Next, we incorporated time constraints with 77 the previously proposed mechanisms and analyzed the performances of the proposed methods. Again, the choice of the proposed method to deal with time constraints depends on the preferences of the driver. The results show that both discount methods can reduce the driver’s cost by spreading the same number of passengers onto more vehicles. The Basic Discount method in general outperformed the ICBD method in that it had lower shared cost per passenger and higher percentage of requests served with a competitive driver cost. However, the ICBD method led to a more distributed system with fewer no-passenger vehicles and higher reduction in the driver’s cost when the passengers had a high willingness-to-pay-level. Finally, we provided our solution to the vehicle routing part of ride-sharing. We explored the use of HOV lanes and meeting points. In order to adapt to the dynamic environment, we proposed a two-stage heuristic algorithm which consists of an insertion heuristic to solve the PDP problem and a second stage algorithm that can solve the meeting points problem optimally. The results show that the HOV lanes and meeting points can increase the efficiency of the dynamic ride-sharing system. When we choose a maximum walk- ing time of 10 minutes and a common HOV requirements of 2+ persons in the vehicle, the system could serve 39% more requests while reducing the average in-vehicle time ratio by 19%. Therefore, we can say that a good combination of HOV lanes and meeting points can provide passengers with lower commuting cost and faster commuting experience. For future work, there are two primary directions: (1) design cost-sharing mechanisms in the dynamic case where passengers submit their request after the operation starts, and (2) choose different methods to solve the dynamic ride-sharing problem. Topic two would be under the guidance of the dynamic cost- sharing mechanism so as to maintain the identified desirable properties. In this section, we elaborate on how those two topics could be advised. The first direction is a non-trivial extension of the mechanisms in this proposal. Especially, the following statement may be true: in the dynamic setting, it is not possible for cost-sharing mechanisms to guarantee the Ex-post Incentive Compatibility property and the other four desirable properties at the same time. In other words, in the dynamic setting, a cost-sharing mechanism 78 satisfying the other four desirable properties may fail to satisfy the Ex-post Incentive Compatibility prop- erty. Therefore, one interesting direction could be to develop bounds on how many passengers can game the system for a given mechanism. There are many possible tracks in the second direction, one can use methods such as applying re-optimization in a rolling horizon, forecasting passenger requests through- out the operation or based on historical data, and resorting to Markov Decision Process to generate an on-average good solution facing uncertainties. 79 Bibliography [1] Niels Agatz, Alan Erera, Martin Savelsbergh, and Xing Wang. “Optimization for dynamic ride-sharing: A review”. In: European Journal of Operational Research 223.2 (2012), pp. 295–303. [2] Niels Agatz, Alan L Erera, Martin WP Savelsbergh, and Xing Wang. “Dynamic ride-sharing: A simulation study in metro Atlanta”. In: Procedia-Social and Behavioral Sciences 17 (2011), pp. 532–550. [3] Andrea Attanasio, Jean-François Cordeau, Gianpaolo Ghiani, and Gilbert Laporte. “Parallel tabu search heuristics for the dynamic multi-vehicle dial-a-ride problem”. In: Parallel Computing 30.3 (2004), pp. 377–387. [4] Imadeddine Aziez, Jean-François Côté, and Leandro C Coelho. “Exact algorithms for the multi-pickup and delivery problem with time windows”. In: European Journal of Operational Research (2020). [5] Chaithanya Bandi, Dimitris Bertsimas, and Nataly Youssef. “Robust queueing theory”. In: Operations Research 63.3 (2015), pp. 676–700. [6] Chaithanya Bandi, Nikolaos Trichakis, and Phebe Vayanos. “Robust multiclass queuing theory for wait time estimation in resource allocation systems”. In: Management Science 65.1 (2018), pp. 152–187. [7] World Bank. Urban Population by Year. 2020.url: https://data.worldbank.org/indicator/SP.URB.TOTL.IN.ZS (visited on 09/26/2021). [8] Gerardo Berbeglia, Jean-François Cordeau, and Gilbert Laporte. “A hybrid tabu search and constraint programming algorithm for the dynamic dial-a-ride problem”. In: INFORMS Journal on Computing 24.3 (2012), pp. 343–355. [9] Gerardo Berbeglia, Jean-François Cordeau, and Gilbert Laporte. “Dynamic pickup and delivery problems”. In: European Journal of Operational Research 202.1 (2010), pp. 8–15. [10] Bülent Catay. “Ant colony optimization and its application to the vehicle routing problem with pickups and deliveries”. In: Natural intelligence for scheduling, Planning and packing problems. Springer, 2009, pp. 219–244. 80 [11] Shyue Koong Chang and Paul M Schonfeld. “Multiple period optimization of bus transit systems”. In: Transportation Research Part B: Methodological 25.6 (1991), pp. 453–478. [12] Jean-Françcois Cordeau, Gilbert Laporte, and Stefan Ropke. “Recent models and algorithms for one-to-one pickup and delivery problems”. In: The vehicle routing problem: latest advances and new challenges. Springer, 2008, pp. 327–357. [13] Jean-François Cordeau. “A branch-and-cut algorithm for the dial-a-ride problem”. In: Operations Research 54.3 (2006), pp. 573–586. [14] Jean-Francois Cordeau and Gilbert Laporte. “Tabu search heuristics for the vehicle routing problem”. In: Metaheuristic Optimization via Memory and Evolution. Springer, 2005, pp. 145–163. [15] Jean-François Cordeau and Gilbert Laporte. “A tabu search heuristic for the static multi-vehicle dial-a-ride problem”. In: Transportation Research Part B: Methodological 37.6 (2003), pp. 579–594. [16] Jean-François Cordeau and Gilbert Laporte. “The dial-a-ride problem: models and algorithms”. In: Annals of Operations Research 153.1 (2007), pp. 29–46. [17] Luca Coslovich, Raffaele Pesenti, and Walter Ukovich. “A two-phase insertion technique of unexpected customers for a dynamic dial-a-ride problem”. In: European Journal of Operational Research 175.3 (2006), pp. 1605–1615. [18] George B Dantzig and John H Ramser. “The truck dispatching problem”. In: Management Science 6.1 (1959), pp. 80–91. [19] Jacques Desrosiers, Yvan Dumas, and Francois Soumis. “A dynamic programming solution of the large-scale single-vehicle dial-a-ride problem with time windows”. In: American Journal of Mathematical and Management Sciences 6.3-4 (1986), pp. 301–325. [20] Marco Diana and Maged M Dessouky. “A new regret insertion heuristic for solving large-scale dial-a-ride problems with time windows”. In: Transportation Research Part B: Methodological 38.6 (2004), pp. 539–557. [21] David Eberly. Geometric Tools: Computing a point of reflection on a sphere . 2008.url: https://www.geometrictools.com/Documentation/SphereReflections.pdf. [22] Gregory D Erhardt, Sneha Roy, Drew Cooper, Bhargava Sana, Mei Chen, and Joe Castiglione. “Do transportation network companies decrease or increase congestion?” In: Science Advances 5.5 (2019). [23] Andrés Fielbaum, Sergio Jara-Diaz, and Antonio Gschwender. “Beyond the Mohring effect: Scale economies induced by transit lines structures design”. In: Economics of Transportation 22 (2020), p. 100163. [24] Mikael Frisk, Maud Göthe-Lundgren, Kurt Jörnsten, and Mikael Rönnqvist. “Cost allocation in collaborative forest transportation”. In: European Journal of Operational Research 205.2 (2010), pp. 448–458. 81 [25] Masabumi Furuhata, Kenny Daniel, Sven Koenig, Fernando Ordóñez, Maged Dessouky, Marc Etienne Brunet, Liron Cohen, and Xiaoqing Wang. “Online Cost-Sharing Mechanism Design for Demand-Responsive Transport Systems”. In: IEEE Transactions on Intelligent Transportation Systems 16.2 (2015), pp. 692–707. [26] Robert Geisberger, Dennis Luxen, Sabine Neubauer, Peter Sanders, and Lars Volker. “Fast detour computation for ride sharing”. In: arXiv preprint arXiv:0907.5269 (2009). [27] Carl H Häll and Anders Peterson. “Improving paratransit scheduling using ruin and recreate methods”. In: Transportation Planning and Technology 36.4 (2013), pp. 377–393. [28] Larry Harris. Trading and exchanges: Market microstructure for practitioners. OUP USA, 2003. [29] Sin C Ho, WY Szeto, Yong-Hong Kuo, Janny MY Leung, Matthew Petering, and Terence WH Tou. “A survey of dial-a-ride problems: Literature review and recent developments”. In: Transportation Research Part B: Methodological 111 (2018), pp. 395–421. [30] Manar I Hosny and Christine L Mumford. “The single vehicle pickup and delivery problem with time windows: intelligent operators for heuristic and metaheuristic algorithms”. In: Journal of Heuristics 16.3 (2010), pp. 417–439. [31] VF Hurdle. “Minimum cost locations for parallel public transit lines”. In: Transportation Science 7.4 (1973), pp. 340–350. [32] Jang-Jei Jaw, Amedeo R Odoni, Harilaos N Psaraftis, and Nigel HM Wilson. “A heuristic algorithm for the multi-vehicle advance request dial-a-ride problem with time windows”. In: Transportation Research Part B: Methodological 20.3 (1986), pp. 243–257. [33] Alexander Kleiner, Bernhard Nebel, and Vittorio Amos Ziparo. “A mechanism for dynamic ride sharing based on parallel auctions”. In: Twenty-Second International Joint Conference on Artificial Intelligence. 2011. [34] Jonathan I Levy, Jonathan J Buonocore, and Katherine Von Stackelberg. “Evaluation of the public health impacts of traffic congestion: a health risk assessment”. In: Environmental Health 9.1 (2010), pp. 1–12. [35] Rong-Hua Li, Lu Qin, Jeffrey Xu Yu, and Rui Mao. “Optimal multi-meeting-point route search”. In: IEEE Transactions on Knowledge and Data Engineering 28.3 (2015), pp. 770–784. [36] Xin Li, Sangen Hu, Wenbo Fan, and Kai Deng. “Modeling an enhanced ridesharing system with meet points and time windows”. In: PloS One 13.5 (2018), e0195927. [37] Quan Lu and Maged Dessouky. “An exact algorithm for the multiple vehicle pickup and delivery problem”. In: Transportation Science 38.4 (2004), pp. 503–514. [38] Yan Lyu, Chi-Yin Chow, Victor C.S. Lee, Joseph K.Y. Ng, Yanhua Li, and Jia Zeng. “CB-Planner: A bus line planning framework for customized bus systems”. In: Transportation Research Part C: Emerging Technologies 101 (2019), pp. 233–253.issn: 0968-090X. 82 [39] Nikola Marković, Rahul Nair, Paul Schonfeld, Elise Miller-Hooks, and Matthew Mohebbi. “Optimizing dial-a-ride services in Maryland: benefits of computerized routing and scheduling”. In: Transportation Research Part C: Emerging Technologies 55 (2015), pp. 156–165. [40] Aranyak Mehta, Tim Roughgarden, and Mukund Sundararajan. “Beyond moulin mechanisms”. In: Games and Economic Behavior 67.1 (2009), pp. 125–155. [41] Herbert Mohring. “Optimization and scale economies in urban bus transportation”. In: The American Economic Review 62.4 (1972), pp. 591–604. [42] Hervé Moulin. “An efficient and almost budget balanced cost sharing method”. In: Games and Economic Behavior 70.1 (2010), pp. 107–131. [43] Hervé Moulin. “Incremental cost sharing: Characterization by coalition strategy-proofness”. In: Social Choice and Welfare 16.2 (1999), pp. 279–320. [44] Shanmugavelayutham Muthukrishnan. “Ad exchanges: Research issues”. In: International workshop on internet and network economics. Springer. 2009, pp. 1–12. [45] Yu Marco Nie. “How can the taxi industry survive the tide of ridesourcing? Evidence from Shenzhen, China”. In:Transportation Research Part C: EmergingTechnologies 79 (2017), pp. 242–256. [46] Alfredo Núñez, Cristián E Cortés, Doris Sáez, Bart De Schutter, and Michel Gendreau. “Multiobjective model predictive control for dynamic pickup and delivery problems”. In: Control Engineering Practice 32 (2014), pp. 73–86. [47] Marc Oliphant and Andrew Amey. “Dynamic ridesharing: carpooling meets the information age”. In: Webinar for the American Planning Association, co-hosted by the MIT team 63 (2010). [48] Okan Örsan Özener, Özlem Ergun, and Martin Savelsbergh. “Allocating cost of service to customers in inventory routing”. In: Operations Research 61.1 (2013), pp. 112–125. [49] Panayotis Papoutsis, Safa Fennia, Constant Bridon, and Tarn Duong. “Relaxing door-to-door matching reduces passenger waiting times: a workflow for the analysis of driver GPS traces in a stochastic carpooling service”. In: Transportation Engineering (2021), p. 100061.issn: 2666-691X. [50] Sophie N Parragh, Karl F Doerner, and Richard F Hartl. “Variable neighborhood search for the dial-a-ride problem”. In: Computers & Operations Research 37.6 (2010), pp. 1129–1138. [51] Mingyang Pei, Peiqun Lin, Jun Du, and Xiaopeng Li. “Operational design for a real-time flexible transit system considering passenger demand and willingness to pay”. In: IEEE Access 7 (2019), pp. 180305–180315. [52] Victor Pillac, Michel Gendreau, Christelle Guéret, and Andrés L Medaglia. “A review of dynamic vehicle routing problems”. In: European Journal of Operational Research 225.1 (2013), pp. 1–11. [53] JY Potvin and JM Rousseau. “Constraint-directed search for the advanced request dial-a-ride problem with service quality constraints”. In: Computer Science and Operations Research. Elsevier, 1992, pp. 457–474. 83 [54] Harilaos N Psaraftis. “Dynamic vehicle routing problems”. In: Vehicle routing: Methods and studies 16 (1988), pp. 223–248. [55] Luca Quadrifoglio, Maged M Dessouky, and Fernando Ordóñez. “A simulation study of demand responsive transit system design”. In: Transportation Research Part A: Policy and Practice 42.4 (2008), pp. 718–737. [56] Stefan Ropke and Jean-François Cordeau. “Branch and cut and price for the pickup and delivery problem with time windows”. In: Transportation Science 43.3 (2009), pp. 267–286. [57] Stefan Ropke, Jean-François Cordeau, and Gilbert Laporte. “Models and branch-and-cut algorithms for pickup and delivery problems with time windows”. In: Networks: An International Journal 49.4 (2007), pp. 258–272. [58] Hamid R Sayarshad and Joseph YJ Chow. “A scalable non-myopic dynamic dial-a-ride and pricing problem”. In: Transportation Research Part B: Methodological 81 (2015), pp. 539–554. [59] Michael Schilde, Karl F Doerner, and Richard F Hartl. “Metaheuristics for the dynamic stochastic dial-a-ride problem with expected return transports”. In: Computers & Operations Research 38.12 (2011), pp. 1719–1730. [60] David Schrank, Luke Albert, Bill Eisele, and Tim Lomax. “2021 urban mobility report”. In: Texas A&M Transportation Institute. The Texas A&M University System (2021). [61] Joshua Seeherman and Alexander Skabardonis. “Searching for induced travel: Elimination of a freeway bottleneck and subsequent effects on rail and freeway volumes”. In: Case Studies on Transport Policy 4.3 (2016), pp. 239–247.issn: 2213-624X.doi: https://doi.org/10.1016/j.cstp.2016.06.003. [62] Pieter Smet. “Ride sharing with flexible participants: a metaheuristic approach for large-scale problems”. In: International Transactions in Operational Research 28.1 (2021), pp. 91–118. [63] Pandhapon Sombuntham and Voratas Kachitvichayanukul. “A particle swarm optimization algorithm for multi-depot vehicle routing problem with pickup and delivery requests”. In: World Congress on Engineering 2012. July 4-6, 2012. London, UK. Vol. 2182. Citeseer. 2010, pp. 1998–2003. [64] Mitja Stiglic, Niels Agatz, Martin Savelsbergh, and Mirko Gradisar. “The benefits of meeting points in ride-sharing systems”. In: Transportation Research Part B: Methodological 82 (2015), pp. 36–53. issn: 0191-2615. [65] Alejandro Tirachini. “The economics and engineering of bus stops: Spacing, design and congestion”. In: Transportation Research Part A: Policy and Practice 59 (2014), pp. 37–57. [66] Paolo Toth and Daniele Vigo. The Vehicle Routing Problem. SIAM, 2002. [67] Marlin W Ulmer, Barrett W Thomas, Ann Melissa Campbell, and Nicholas Woyak. “The Restaurant Meal Delivery Problem: Dynamic Pickup and Delivery with Deadlines and Random Ready Times”. In: Transportation Science (2020). 84 [68] Xiaoqing Wang, Maged Dessouky, and Fernando Ordonez. “A pickup and delivery problem for ridesharing considering congestion”. In: Transportation Letters 8.5 (2016), pp. 259–269. [69] Ka-Io Wong and Michael GH Bell. “Solution of the Dial-a-Ride Problem with multi-dimensional capacity constraints”. In:InternationalTransactionsinOperationalResearch 13.3 (2006), pp. 195–208. [70] Zhihai Xiang, Chengbin Chu, and Haoxun Chen. “The study of a dynamic dial-a-ride problem under time-dependent and stochastic environments”. In: European Journal of Operational Research 185.2 (2008), pp. 534–551. [71] Meng Zhao, Jiateng Yin, Shi An, Jian Wang, and Dejian Feng. “Ridesharing problem with flexible pickup and delivery locations for app-based transportation service: mathematical modeling and decomposition methods”. In: Journal of Advanced Transportation 2018 (2018). [72] Issam Zidi, Khaled Mesghouni, Kamel Zidi, and Khaled Ghedira. “A multi-objective simulated annealing for the multi-criteria dial a ride problem”. In: Engineering Applications of Artificial Intelligence 25.6 (2012), pp. 1121–1131. [73] Han Zou. “An Online Cost Allocation Model for Horizontal Supply Chains”. PhD thesis. University of Southern California, 2017. 85

Abstract (if available)

Abstract In this research, we dedicate to the ride-sharing problem where ride-sharing drivers are travelling toward their own destinations while making detours to serve passengers with flexible pickup and drop-off locations. We decompose this problem into the cost-sharing problem and the routing problem. ❧ The cost-sharing problem addresses how the total cost of a ride-sharing operation is distributed among participants. In order to solve this problem, one needs to design a cost-sharing mechanism. In the first part of this dissertation, we identify the properties that a desirable cost-sharing mechanism should have and develop a general framework that can be used to create specific cost-sharing mechanisms. We propose specific mechanisms and analyze their relative advantages and disadvantages so that service providers can select a mechanism according to their different needs. In addition, we incorporate the value of time by allowing passengers to have inconvenience costs due to extra travel time caused by detours for picking up the passengers and provide discount methods to compensate for these costs. We evaluate our approach using real traffic data from the downtown Los Angeles area. Our results show that each proposed mechanism has its unique advantages and that the discount methods can successfully reduce the number of no-passenger vehicles for a large ride-sharing system. ❧ The routing problem on the other hand addresses how to route the ride-sharing routes and match passengers with drivers. Especially, we study under the context where drivers can utilize the HOV (High Occupancy Vehicle) lane and passengers have flexible pickup and drop-off locations. We developed a two-stage routing algorithm to solve the routing problem in real time. This two-stage algorithm consists of an insertion heuristic to solve the pickup and delivery problem and a second-stage algorithm that can solve the meeting points problem optimally. Using randomly generated data, we designed experiments to evaluate the performance of the HOV lanes and meeting points. The results showed that the existence of HOV lanes reduces the average in-vehicle time (IVT) ratio compared to the situation with no HOV lanes since they provide a boost in driving speed. They also help increase the percentage of requests served in the system since the time saved in using HOV lanes can be utilized to serve more requests and serving more requests increases the possibility of using the HOV lanes. The existence of meeting points increases the utilization of the ride-sharing system and is capable of reducing total miles travelled while maintaining the same level of system usage.

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Creator Hu, Shichun (author)

Core Title Models and algorithms for pricing and routing in ride-sharing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2022-05

Publication Date 01/02/2022

Defense Date 10/20/2021

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

Tag cost-sharing mechanisms,OAI-PMH Harvest,ride-sharing,vehicle routing problem

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Dessouky, Maged (committee chair), Giuliano, Geneviene (committee member), Vayanos, Phebe (committee member)

Creator Email hscsy91@gmail.com,shichunh@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC110454554

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