论文阅读2018-10-13

论文阅读2018-10-13

• Addressing the minimum fleet problem in on-demand urban mobility 原文及翻译

Addressing the minimum fleet problem in on-demand urban mobility 原文及翻译

Information and communication technologies have opened the way to new solutions for urban mobility that provide better ways to match individuals with on-demand vehicles.However,a fundamental unsolved problem is how best to size and operate a fleet of vehicles,given a certain demand for personal mobility.Previous studies either do no provide a scalable solution or require changes in human attitudes towards mobility.Here we provide a network-based solution to the following ‘minimum fleet problem’, given a collection of trips (specified by origin, destination and start time), of how to determine the minimum number of vehicles needed to serve all the trips without incurring any delay to the passengers. By introducing the notion of a ‘vehicle-sharing network’, we present an optimal computationally efficient solution to the problem,as well as a nearly optimal solution amenable to real-time implementation. We test both solution on a dataset of 150 million taxi trips taken in the city of New York over one year. The real-time implementation of the method with near-optimal service levels allow a 30 per cent reduction in fleet size compared to current taxi operation.Although constraints on driver availability and existence of abnormal trip demands may lead to a relatively larger optimal value for the fleet size than that predicted here,the fleet size remains robust for a wide range of variations in historical trip demand.These predicted reductions in fleet size follow directly from a reorganization of taxi dispatching that could be implemented with a simple urban app; they do not assume ride sharing, nor require changes to regulations,business models,or human attitudes towards mobility to become effective. Our results could become even more relevant in the years ahead as fleets of networked,self-driving cars become commonplace.

信息和通信技术为城市交通的新解决方案开辟了道路，为人们提供了更好的方式来与按需车辆相匹配。然而一个仍未解决的基本问题是如何在考虑个人的出行需求的情况下，最好地扩大和运营一支车队。之前的研究要么没有提供可靠的解决方案要么要求人们改变对出行的态度。这里我们提出一个基于网络的‘最小车队’解决方案，给定出行的集合（由原点、目的地和起始时间指定），如何在不引入延迟的情况下，决定能够服务所有要出行的乘客的最小的车辆数量。通过引入‘车辆共享网络’，我们提供了这个问题的最优计算效率解决方案，以及一个可用于实时实现的最优解决方案。我们在New York市一年内 的1亿5千万出租车出行数据集上测试所有的解决方案。该方法的实时实现，其服务水平接近最佳，与当前的出租车运营相比，其车队规模减少了30%。尽管对驾驶员可用性的限制和不常见的旅行需求的存在，可能会导致比这里预测的更大的机队规模的最佳价值，但在历史出行需求的不同变化中，车队规模仍然很大。这不需要假设共享出行，不要去改变规则，商业模式，或者人们对出行的态度来变的更有效率。我们的成果在未来几年在网络化的车队中会更加收到重视，当无人驾驶汽车更加司空见惯。

Two trends—the rise of the autonomous and connected car, and the emergence of a ‘sharing economy’ of transportation-seem poised to revolutionize the way personal mobility needs will be addressed in cities. The way current modes of transportation such as the private car, taxi or bus operate will be challenged and increasingly replaced by personalized, on-demand mobility systems operated by vehicle fleets, similar to what companies like Uber and Lyft offer. If such trends continue, they could lead to the displacement, or eventual disappearance, of jobs for bus and taxi drivers. Along with these possible societal costs, the transportation revolution could also offer immense benefits, including opportunities to resolve existing inefficiencies in individual urban mobility, thereby reducing traffic,whose carbon footprint currently accounts for about 23% of global greenhouse gas emissions.

两种趋势——自主和联网汽车的兴起，以及交通运输的“共享经济”的出现，似乎将彻底改变城市的个人出行需求。目前的交通方式，如私家车、出租车或公共汽车，将受到挑战，并越来越多地由汽车车队运营的个性化的按需出行系统取代，类似于Uber和Lyft这样的公司提供的服务。如果这种趋势继续下去，可能会导致公交车和出租车司机的失业或最终消失。除了这些可能的社会成本之外，交通运输革命还可能带来巨大的好处，包括解决城市交通中现有的低效率的机会，从而减少交通，其碳足迹目前约占全球温室气体排放量的23%。

To turn these opportunities into tangible environmental and societal benefits,autonomous and on-demand mobility systems need to be designed and optimized for efficiency, and integrated with carbon-efficient public transport. This requires the definition of models and algorithms for the evaluation of shared mobility systems that are both computationally efficient and accurate. The former property is mandated by the need to cope with hundreds of thousands (or sometimes millions) of trips routinely occurring in a large city. The latter property determines the relevance of the model results to the real world.

为了将这些机会转化为有形的环境和社会效益，需要设计和优化自主和按需移动系统，以提高效率，并与低碳高效的公共交通相结合。这就需要定义模型和算法来评估共享的移动系统的计算效率和准确性。前一种属性的是因为，系统需要处理成千上万（有时甚至是数百万）的出行，这些出行经常发生在一个大城市。后一种属性决定了模型结果与现实世界的相关性。

In what follows, we solve the ‘minimum fleet problem’ for the general case of on-demand mobility, and show that is solution for a specific case-taxi trips-could lead to breakthroughs in operational efficiency. To the best of our knowledge, no publicly available solution currently exists to address this minimum fleet-size problem at the urban scale for on-demand mobility in both private and public sectors. On the one hand, accurate methods based on mathematical programming (as traditionally used in the design of transportation systems) can handle only a few thousand trips or vehicles at most, which is well below the hundreds of thousands or even millions of trips or vehicles routinely operating in large cities. On the other hand, city-scale studies are obtained using a model of transportation based on aggregated mobility data and Euclidean spatial assumptions, and hence lack the resolution necessary to estimate the urban-scale benefits of vehicle sharing accurately.

在下文中，我们对于一般情况下按需出行的情况，解决‘最小车队问题’，并显示这种针对出租车出行的方案可能会在运营效率上取得突破。据我们所知，目前还没有公开可用的解决方案来解决这个在城市规模的最小车队问题，即私人和公共部门的按需出行。一方面，基于数学编程的精确方法（就像在传统应用在运输系统设计上的）最多只能处理几千次旅行或车辆，这远远低于在大城市日常运营的数十万甚至数百万的出行或车辆。另一方面，城市规模的研究基于聚合移动数据和欧几里得空间假设的交通模型，因此缺乏准确估计车辆共享的城市规模效益的必要解决方案。

We start form the notion of the shareability network introduced in ref.7, which did not focus on the dispatching of vehicles. The type of shareability network introduced here is profoundly different from the type studied previously: it models the sharing of vehicles, whereas previous networks modeled the sharing of rides.The main methodological contribution of this Letter is to show how this vehicle-sharing network can be translated into an exact formulation of the minimum fleet problem as a minimum path cover problem on directed graphs,thus establishing a connection to the rich applied mathematics and computer science field of graph algorithms. Besides revealing a structural property of computationally efficient algorithms for optimal vehicle deployment and dispatching. Although optimally solving the minimum fleet requires offline knowledge of daily mobility demand, in the following we also present a near-optimal,online version of the algorithm that can be executed in real time knowing only a small amount of the trip demand.

我们从不聚焦于车辆分配的‘可共享的网络’的介绍开始，
这里介绍的可共享的网络与之前的几种研究是截然不同的：**它模拟车辆的共享，而以前的网络则模拟出行的共享。这篇文章的主要贡献是展示如何将这个车辆共享网络转化为最小车队问题的形式，有向图的最小路径覆盖问题，从而建立的丰富应用数学和计算机科学领域的图形算法的连接。**除了揭示了计算效率算法的结构特性，以实现最优的车辆部署和调度。尽管最优解决最小舰队需要离线的日常出行数据，但在下面我们也提供了一个近乎最优的在线版本算法，可以实时执行，只知道少量的出行需求。

We are given a collection T of individual trips representing a portion of urban mobility demand during a certain time interval, such as a day. Each tip Ti ∈ T is defined as a tuple$(t^p_i,t^d_i,l^p_i,l^d_i)$(tip​,tid​,lip​,lid​), where$t^p_i$tip​ represents the desired pick-up time, $l^p_i$lip​ the pick-up location,$t^d_i$tid​ the drop-off time, and $l^d_i$lid​ the drop-off location. Here, the pick-up time means the earliest time $t^p_i$tip​ at which passenger can be picked up at location $l^p_i$lip​. The drop-off time means the estimated time of dropping off the passenger, calculated using a travel-time estimation model and assuming the passenger leaves the pick-up location at time $t^p_i$tip​. IN contrast to ref.17, travel times here are computed using the actual road network, and using global positioning system (GPS)-based estimations derived from the taxi trip dataset that account for hourly variations in traffic, as in ref.7. If the set T is extracted from a real-world dataset (for example, taxi trips), the times $t^p_i$tip​ and $t^d_i$tid​ represent the actual times at which a passenger is picked up and dropped off,respectively.

我们得到了一个个人出行的集合T代表一定时间间隔内的城市交通需求的一部分，比如一天之内。每一个出行Ti属于T被定影成一个元组$(t^p_i,t^d_i,l^p_i,l^d_i)$(tip​,tid​,lip​,lid​),，其中$t^p_i$tip​ 表示接到乘客的时间$l^p_i$lip​是接乘客的地点,$t^d_i$tid​是乘客下车时间, and $l^d_i$lid​ 是下车地点. 这里, 接乘客的时间t $t^p_i$tip​ 表示乘客在上车点 $l^p_i$lip​能被接到的最早时间.下车时间表示计算的下车时间，通过一个计算模型计算，假设乘客在 $t^p_i$tip​离开上车点. ref.17相比, 这里的旅行时间是用实际的道路网络计算的，并使用全球定位系统（GPS）基于出租车旅行数据集的估计，这些数据记录了每小时的流量变化，如果从真实世界的数据集（例如，出租车出行）中提取出集合 T，那么时间$T^p_i$Tip​和$T^d_i$Tid​代表了一个乘客上车和下车的实际时间。

The minimum fleet problem is formally defined as follows:'find the minimum number of vehicles needed to serve all trips in T, given that a vehicle is available at each $l^p_i$lip​ on or before $t^p_i$tip​'. A service designed around this problem is ideal from a passenger’s perspective,since a vehicle is guaranteed to be available at the desired location and time, On the other hand, the above problem formulation might entail substantial inefficiencies for the operator and the environment. Consider two consecutive trips $T_A$TA​ and $T_B$TB​ served by a single vehicle, and call the time needed to connect them the (trip) connection time, formally $t_{AB} = t^p_B -t^d_A$tAB​=tBp​−tAd​. If this time is very long, say, a few hours, it is trivially possible to connect trips that occur at distant location or times. Hence, an excessively large connection time leads to inefficiencies for the operator (longer traveled distances,lower vehicle occupancy ratio) and the citizens (a lot of emissions and traffic just to connect trips). we therefore re-formulate the problem as follows:‘find the minimum number of vehicles needed to serve all trip in T,under the assumptions that (1) a vehicles is available at each $l^p_i$lip​ on or before $t^p_i$tip​ and (2) the connection time is at most $\delta$δ minutes’, where the upper bound $\delta$δ on the connection time is a problem parameter.

最小车队问题的正式定义如下：‘找到最小所需的车辆来服务所有集合T中的出行，给出一辆在时间$t^p_i$tip​ 之前，在 $l^p_i$lip​可用的车’。以此定义的服务是完全从乘客的角度来看的，因为一辆车保证能在预定的时间和地点使用。另一方面，上述问题的制定可能会导致操作人员和环境的效率低下。考虑两个出行 $T_A$TA​和 $T_B$TB​ 被同一辆车服务，把连接他们的时间成为连接时间，正式的，$t_{AB} = t^p_B -t^d_A$tAB​=tBp​−tAd​。如果这个时间很长，假如，几个小时，在很远的地方或较长的时间上连接出行也是可能的。因此，过多的连接时间会导致操作员（长途旅行距离，车辆占用率较低）和市民（大量的排放和交通只是为了连接旅行）的效率低下。因此，我们将如下问题重新表述如下：‘找到最小所需的车辆来服务所有集合T中的出行，并满足（1）车辆在时间$t^p_i$tip​ 之前，在 $l^p_i$lip​可用 （2）连接时间最大不超过$\delta$δ 分钟’，连接时间中的上限$\delta$δ是问题的一个从参数。

figure 1 illustrates the construction of the vehicle-shareability network that enables the minimum fleet problem to be optimally solved with parameter $\delta$δ . This is a directed network defined as V = (N,E), where node $n_i$ni​ $\in$∈ N corresponds to trip Ti $\in$∈ T and the directed edge ($n_i$ni​,$n_j$nj​) $\in$∈E if and only if ($t^d_i +t_{ij} ) \le t^p_j$tid​+tij​)≤tjp​ (which accounts for assumption (1) above) and $t^p_j -t^d_i \le \delta$tjp​−tid​≤δ (which accounts for assumption (2) above). Here $t_{ij}$tij​ represents the estimated travel time between $l^d_i$lid​ and $l^p_j$ljp​. The existence of a link in the network indicates that the two incident trips can be consecutively served by a single vehicle, and a path in V corresponds to a sequence of trips that can be served by a single vehicle- that is, a dispatch. There for, solving the minimum fleet problem is equivalent to finding the number of paths(vehicles) in the minimum path cover of V. The solution also gives the optimal dispatching strategy, that is, a sequence of trips to be served for each vehicle in the minimum fleet. The problem of finding the minimum path cover on general graphs is NP-hard, but it can be solved efficiently on directed acyclic graphs$^{19}$19. The acyclic nature of time guarantees that any vehicle shareability network is a directed acyclic graph, and the minimum fleet problem can be efficiently and optimally solved; see Methods for formal proofs

图1描述了能够解决带参数$\delta$δ的最小车队问题最优解决的车辆共享网络的结构，这个有向网络的定义是 V=（N，E），其中节点$n_i \in$ni​∈ N 相当于 出行记录 Ti $\in$∈ T 并且 有向边（$n_i,n_j) \in$ni​,nj​)∈E 当且仅当($t^d_i +t_{ij} ) \le t^p_j$tid​+tij​)≤tjp​ (依据假设（1）) ，并且$t^p_j -t^d_i \le \delta$tjp​−tid​≤δ （依据假设（2））。这里，$t_{ij}$tij​ 代表计算出来的从 $l^d_i$lid​ 到 $l^p_j$ljp​的行驶时间。网络中存在的一个连接表明两个不同的出行可以被同一辆车连续的服务，V 中的路径相当于一辆车行驶的顺序，这就是分配。这样，解决最小车队问题和找到覆盖V的最小的路径条数是等价的。这个解决方案还提供了最优调度策略，也就是说，在最小车队中，每辆车都要进行一系列的行驶。在一般图上找到最小路径覆盖的问题是NP-hard，但是它可以在有向无环图中有效地解决，通过使用 Hopcroft-Karp算法计算二分图匹配。时间的无环特性保证了任何车辆的可共享网络都是一个有向无环图，最小车队问题可以得到有效和最优的解决;正式的证明看Methods。

We have tested our methodology on a dataset of over 150 million trips performed in the city of New York in the year 2011. This dataset has been selected form a number of available datasets because it is publicly available and, thanks to taxi statistics published by the New York Taxi and Limousine Commission$^6$6, it is possible to compare our methodology directly with current taxi operation. The data have been sliced into daily datasets Ti, each of which is an input to the minimum fleet size problem.

我们在2011年纽约城市进行的超过1.5亿次的出行数据上测试了我们的方法。这一数据集从为许多可用的数据集被选出，是因为它是公开可用的，而且，由于纽约出租车和豪华轿车委员会发布的出租车统计数据，我们可以直接将我们的方法与当前的出租车运营进行比较。这些数据按天数来进行分割，每一个都是对最小车队大小问题的输入。

Next,we discuss how to set the parameter $\delta$δ. When $\delta$δ is decreased to 0, we approach a situation in which each trip is served by a dedicated vehicle: a solution with maximal vehicle utilization that is also optimal for traffic- under the assumption that vehicles materialize at the origin and dematerialize at the destination of the served trip- but incurring prohibitive cost for the mobility operator. On the other hand, when $\delta$δ grows excessively, the fleet size is reduced, but the operational and traffic efficiency problems described previously occur. Thus, the setting of $\delta$δ is an important design choice that is left to mobility operators,traffic authorities and policy makers. In this study, we set $\delta$δ = 15 min, as explained in Methods. The results of our method with different values of $\delta$δ are reported in Methods (see Extended Data Fig.1)

接下来，我们将讨论如何设置参数$\delta$δ。$\delta$δ下降到0时,我们的方法中,每个出行是由一个专门的车辆负责:一个解决方案与最大车辆利用率也适合交通——假设下车辆出现在原点,消失在目的地的旅行,但移动运营商承担高昂的成本。另一方面，当$\delta$δ过度增长时，车队的大小会减少，但是前面描述的操作和交通效率问题出现了。因此，$\delta$δ的设置是一个重要的设计选择，留给移动运营商、交通部门和政策制定者。在这项研究中，我们设置了$\delta$δ=15分钟，正如在方法中所解释的那样。我们的方法在不同的$\delta$δ值下的结果在Methods中被报告（见Extended Data Fig.1）

Figure 2 shows the daily number of vehicles needed to address the entire taxi demand in New York City using our approach.The minimum number of vehicles needed to serve trips is correlated with the number of daily trips (see Fig. 2a), with an overall R$^2$2 value of 0.74.However,for the vast majority of days having between 300,000 and 550,000 trips (inset to Fig. 2a) this correlation becomes much weaker, with an R$^2$2 value of only 0.18. Thus,trip density is a first determinant of fleet size, but trip spatiotemporal patterns are likely to play a large part as well. To investigate this issue further, we have analysed daily vehicle usage in the optimal solution。

图2显示了每天所需的车辆数量来满足纽约市所有的出租车需求。所需要的最少车辆数量与每天的出行需求相关（见图2a），总体的R$^2$2 值为0.74。然而， 在绝大多数的天数中，有300，000到550，000的出行需求，这种相关性就变得更弱了，R$^2$2的值只有0.18。这样，因此，行程密度是车队规模的第一个决定因素，但出行的时空模式也可能发挥很大的作用。为了进一步研究这个问题，我们分析了最优解的日常车辆使用情况。

The vehicle usage analysis reported in Method shows that a fraction of vehicles, ranging between 5% and 10%, are highly underutilized and serve only around 1% of the trips, a lower utilization pattern that occurs especially during the weekend and is probably related to the extra nightly demand. The analysis also highlight clear weekly patterns in vehicle use, consistent with the relatively stable vehicle fleet size across the year.This observed stability can be explained by a simple model for vehicle trip assignment, and is fundamental for mobility operators: it indicates that investment in acquiring an optimal number of vehicles for operation gives consistent yearly returns. The dip in vehicle fleet size occurring at weekends hints also at an opportunity to perform routine vehicle maintenance on a weekly basis.

在Methods中报告的车辆使用分析显示，一小部分车辆，在5%到10%之间，的利用率非常低，只提供了大约1%的行程，较低的利用率经常是在周末发生，可能与夜间额外需求有关。该分析还强调了汽车使用的每周模式，这与全年相对稳定的车队规模一致。这种观察到的稳定性可以用一个简单的车辆旅行分配模型来解释，这对于移动运营商来说是最基本的：它表明，在获得最佳的运营车辆的投资上，每年的回报是一致的。周末发生的车辆数量下降也暗示了每周例行的车辆维修的机会。

A better scaling law relating vehicle fleet size to the daily number of trips can be obtained by defining a metric for fleet sizing that incorporates how long a vehicle is used during a day. We define a ‘full-time equivalent’ vehicles as a vehicle continuously operating 24h a day.(In this case of human-driven vehicles, we can think of having the vehicle operated in three 8-h shifts,for instance.) Figure 2b shows that the scaling law relating the number of daily trips with full-time equivalent vehicles is more accurate than the previous one, with the coefficient of determination R$^2$2 value increased from 0.74 to 0.91, and from 0.18 to 0.70 for trip-intense days reported in the inset.

一个更好的关于车队大小和每天出行次数的标度律可以通过定义一个车队大小的标准来获得，其包含每天使用的车辆的时间相结合。我们定义了一种“全天等效”的车辆，作为一种连续运行24小时的汽车。（在这种情况下，我们可以考虑让汽车在3个8小时的轮班中运行）图2b显示，与前一辆车的每日出行次数有关的标度律比前一项更准确，其确定系数为R$^2$2，从0.74增加到0.91，在出行紧张的日子里为为0.18至0.70，见图。

Figure 3 shows the efficiency breakthrough provided by network-based optimization: when compared to current taxi operation in New York City, the number of circulating taxis can be reduced by an impressive 40%, and kept fairly constant through the day. This improvement is all the more noticeable considering that it is achieved without imposing any delay on customers, nor sharing of rides as in refs.$^{7.9}$7.9. That fleet size can be reduced by as much as 40% without the use of ride sharing and with on delay for passengers has, to the best of our knowledge, not been reported in the literature before, and it is one of the main results of this paper.

图3 显示了基于网络的优化在效率上的突破：与当前纽约市的出租车调度相比，路上巡弋的出租车可以检查惊人的40%，并且保持平时的出行条件。考虑到这一改进是在不延误客户的情况下实现的，也不像在refs$^{7,9}$7,9中共享乘车服务那样，这种改进更加引人注目。在不使用拼车服务的情况下，车队规模可以减少40%，而对于乘客来说，在我们的知识中，这是我们所知的最好的情况，这是本文的主要成果之一。

The 40% fleet reduction reported above refers to the model with full knowledge of daily trip demand. If only a portion of trip demand is known, as in current on-demand mobility services where trip requests are collected in real time, we can still achieve near-optimal performance with the online version of the algorithm reported in the Methods. This version collects trip requests for a short time ,for example, one minute, and locally optimizes vehicle dispatching based on this limited knowledge.Figure 4 shows that, with a 30% fleet reduction and using the online version of the algorithm, more than 90% of the trip requests can be successfully served. hitting a performance very close to the 40% fleet reduction possible when the entire daily demand is known beforehand,

上面报告的40%的车队减少是指对每日出行需求有充分了解的模型。如果只知道出行需求的一部分，就像当前按需出行服务那样，在实时收集旅行请求的情况下，我们仍然可以通过在方法中报告的算法的在线版本实现近乎最佳的性能。这个版本在短时间内收集旅行请求，例如，一分钟，根据有限的知识在本地优化车辆调度。图4显示，使用了该算法的在线版本，减少了30%的车队。如果超过90%的出行请求可以被成功的服务。当所有的日常出行请求都事先知道的时候，就能达到接近40%的车队减少的效果，

Our approach assumes that trip requests and vehicle-dispatching decisions are centralized, a model that is radically different from current taxi operation and similar to the one used by online mobility operators. Therefor, the benefits of optimized operation from a fully distributed operation, where the deployment strategy is based on individual driver decisions, to a centralized operation, where dispatching decisions are globally optimized. To some extent, our results can then be seen as a quantification of the well known game-theory notion of the ‘price of anarchy’$^{20}$20 in urban taxi operation. Taking a mobility market perspective, this is a transition from a regulated mobility market with numerous micro-operators (down to the level of the single taxi driver), to a monopolistic market with a single mobility operator with centralized operation. Although optimal from the vehicle operation and environment viewpoint, a monopolistic market is however highly undesirable for many other reasons, most importantly, lack of competition with consequent higher prices for customers. An additional analysis reported in Methods shows that most of the efficiency benefits of centralized vehicle operation are still possible in an oligopolistic market.

我们的方法假定旅行请求和车辆调度决策是集中的，这一模型与当前的出租车操作完全不同，类似于在线移动运营商使用的模式。因此，从完全分布的操作中优化操作的好处，即部署策略基于单个驱动程序决策，到一个集中的操作，在那里调度决策是全局优化的。在某种程度上，我们的研究结果可以被看作是对“无政府主义价格”的一个众所周知的博弈论概念的量化，即“无政府主义的价格”在城市出租车运营中所占的20美元。从流动市场的角度来看，这是一个由监管的流动市场与众多微型运营商（降至单一出租车司机的水平）的过渡，到一个垄断市场，一个单一的移动运营商集中运营。尽管从汽车运营和环境观点来看，垄断市场是最不受欢迎的，但由于许多其他原因，最重要的是，缺乏竞争，从而导致更高的价格。在方法中报告的另一项分析表明，在一个寡头垄断的市场中，集中的车辆操作的大部分效率效益仍然是可能的。

Although the characterization of minimum fleet size reported here is fully representative of an autonomous driving scenario where human operation of vehicles is not necessary, constraints on driver availability and maximum operating hours, shift operation and so on might produce relatively larger values of the minimum fleet requirement than those predicted here. Extending the concept of the vehicle -sharing network to incorporate driver constraints is possible and is left for further analysis.

Boarder effects on traffic are foreseen if our methodology is to be used for optimizing urban ‘on-demand’ mobility services more in general, especially in a future of autonomous vehicles. However, it is well know that an improvement in mobility efficiency is sometimes linked with an increase in demand which, in turn, could reduce the amount of traffic reductions. Evaluating this ‘second-order’ effect of optimized fleet operation on urban traffic requires coupling a micro-level traffic simulation, agent-based passenger models and our network-based methodology, a challenging task which wee leave to future work.

Finally, we observe that, while applied here to taxi trips as a case study, the proposed methodology for optimal vehicle fleet sizing and dispatching is general and can be applied to model any type of point-point mobility. However, the approach presented here focuses on optimizing and dispatching a single fleet of vehicles. Optimization across different fleets and transportation modes is possible by extending our approach to consider multiple coexisting fleets of various types to serve the mobility demand. With the approaching advent of autonomous mobility and the forecast increase in sharing cars (or other autonomous vehicles, such as flying drones), the problem of how to optimize and orchestrate multiple autonomous fleets will come to the forefront, and might be addressed using the scalable and accurate analytical tools presented here for optimal solution of the ‘minimum fleet’ problem.