Dynamic journeying under uncertainty

We introduce a journey planning problem in multi-modal transportation networks under uncertainty. The goal is to find a journey, possibly involving transfers between different transport services, from a given origin to a given destination within a specified time horizon. Due to uncertainty in travel times, the arrival times of transport services at public transport stops are modeled as random variables. If a transfer between two services is rendered unsuccessful, the commuter has to reconsider the remaining path to the destination. The problem is modeled as a Markov decision process in which states are defined as paths in the transport network. The main contribution is a backward induction method that generates an optimal policy for traversing the public transport network in terms of maximizing the probability of reaching the destination in time. By assuming history independence and independence of successful transfers between services we obtain approximate methods for the same problem. Analysis and numerical experiments suggest that while solving the path dependent model requires the enumeration of all paths from the origin to the destination, the proposed approximations may be useful for practical purposes due to their computational simplicity. In addition to on-time arrival probability, we show how travel and overdue costs can be taken into account, making the model applicable to freight transportation problems.     

Ranking paths in stochastic time-dependent networks

In this paper we address optimal routing problems in networks where travel times are both stochastic and time-dependent. In these networks, the best route choice is not necessarily a path, but rather a time-adaptive strategy that assigns successors to nodes as a function of time. Nevertheless, in some particular cases an origin–destination path must be chosen a priori, since time-adaptive choices are not allowed. Unfortunately, finding the a priori shortest path is an NP-hard problem.     

On exact solutions to the Kolmogorov–Feller equation

The integrodifferential Kolmogorov–Feller equation describing the stochastic dynamics of a system subjected to a regular “force” and a random external disturbance in the form of short pulses with random “amplitudes” and occurrence times is considered. The equation is written in differential form. A method for finding the regular force from a given stationary probability distribution is described. The method is illustrated by examples.     

The Safest Escape problem

In this paper, the Safest Escape (SEscape) problem is defined for providing evacuation plans for emergency egress from large buildings or a geographical region. The objective of the SEscape problem is to determine the set of paths and number of evacuees to send along each path such that the minimum probability of arrival at an exit for any evacuee is maximized. Such paths minimize the risk incurred by the evacuees who are forced to take the greatest risk. The problem is considered in a dynamic and time-varying network, where arc capacities are recaptured over time, arc traversal times are time-varying and arc capacities are random variables with probability distribution functions that vary with time. An exact algorithm, the SEscape algorithm, is proposed to address this problem.     

Computing small solutions of Burgers' equation backwards in time

We construct and analyze an algorithm for the numerical computation of Burgers' equation for preceding times, given an a priori bound for the solution and an approximation to the terminal data. The method is based on the “backward beam equation” coupled with an iterative procedure for the solution of the nonlinear problem via a sequence of linear problems. We also present the results of several numerical experiments. It turns out that the procedure converges “asymptotically,” i.e., in the same manner in which an asymptotic expansion converges. This phenomenon seems related to the “destruction of information,” at t = 0, which is typical in backwards dissipative equations. We derive a priori stability estimates for the analytic backwards problem, and we observe that in many numerical experiments, the distance backwards in time where significant accuracy can be attained is much larger than would be expected on the basis of such estimates. The method is useful for small solutions. Problems where steep gradients occur require considerably more precision in measurement. The algorithm is applicable to other semilinear problems.     

Application of a simulation-based dynamic traffic assignment model

The evaluation of on-line intelligent transportation system (ITS) measures, such as adaptive route-guidance and traffic management systems, depends heavily on the use of faster than real time traffic simulation models. Off-line applications, such as the testing of ITS strategies and planning studies, are also best served by fast-running traffic models due to the repetitive or iterative nature of such investigations. This paper describes a simulation-based, iterative dynamic equilibrium traffic assignment model. The determination of time-dependent path flows is modeled as a master problem that is solved using the method of successive averages (MSA). The determination of path travel times for a given set of path flows is the network-loading sub-problem, which is solved using the space-time queuing approach of Mahut. This loading method has been shown to provide reasonably accurate results with very little computational effort. The model was applied to the Stockholm road network, which consists of 2100 links, 1191 nodes, 228 zones, representing and 4964 turns. The results show that this model is applicable to medium-size networks with a very reasonable computation time.     

Vehicle routing with soft time windows and Erlang travel times

This paper investigates vehicle-routing problems in which the travel times are random variables, and deliveries are made subject to soft time-window constraints. In particular, we model the travel time using a shifted gamma distribution. Penalties are incurred for deviations from the customers' time windows—early or late—and are developed using a fixed cost, a linear cost penalty, and/or a quadratic loss penalty. Alternatively, specifying a given probability of meeting the time-window constraints is considered. A tabu-search metaheuristic is developed, and computational results on test problems from the literature are reported.     

A framework for user equilibrium dynamic traffic assignment

Several analytic approaches have been developed to describe or predict traffic flows on networks with time-varying (dynamic) travel demands, flows and travel times. A key component of these models lies in modelling the flows and/or travel times on the individual links, but as this is made more realistic or accurate it tends to make the overall model less computationally tractable. To help overcome this, and for other reasons, we develop a bi-level user equilibrium (UE) framework that separates the assignment or loading of flows on the time–space network from the modelling of flows and trip times within individual links. We show that this model or framework satisfies appropriate definitions of UE satisfies a first-in-first-out (FIFO) property of road traffic, and has other desirable properties. The model can be solved by iterating between (a) a linear network-loading model that takes the lengths of time–space links as fixed (within narrow ranges), and (b) a set of link flow sub-models which update the link trip times to construct a new time–space network. This allows links to be processed sequentially or in parallel and avoids having to enumerate paths and compute path flows or travel times. We test and demonstrate the model and algorithms using example networks and find that the algorithm converges quickly and the solutions behave as expected. We show how to extend the model to handle elastic demands, multiple destinations and multiple traffic types, and traffic spillback within links and from link to link.     

The quickest path problem with batch constraints

The quickest path problem has been proposed to cope with flow problems through networks whose arcs are characterized by both travel times and flowrate constraints. Basically, it consists in finding a path in a network to transmit a given amount of items from a source node to a sink in as little time as possible, when the transmission time depends on both the traversal times of the arcs and the rates of flow along arcs. This paper is focused on the solution procedure when the items transmission must be partitioned into batches with size limits. For this problem we determine how many batches must be made and what the sizes should be.     

Random walk of a “drunk company”

We study the collective behavior of a system of Brownian agents each of which moves orienting itself to the group as a whole. This system is the simplest model of the motion of a “united drunk company.” For such a system, we use the functional integration technique to calculate the probability of transition from one point to another and to determine the time dependence of the probability density to find a member of the “drunk company” near a given point. It turns out that the system exhibits an interesting collective behavior at large times and this behavior cannot be described by the simplest mean-field-type approximation. We also obtain an exact solution in the case where one of the agents is “sober” and moves along a given trajectory. The obtained results are used to discuss whether such systems can be described by different theoretical approaches.     

Arriving on Time

This research proposes a procedure for identifying dynamic routing policies in stochastic transportation networks. It addresses the problem of maximizing the probability of arriving on time. Given a current location (node), the goal is to identify the next node to visit so that the probability of arriving at the destination by time t or sooner is maximized, given the probability density functions for the link travel times. The Bellman principle of optimality is applied to formulate the mathematical model of this problem. The unknown functions describing the maximum probability of arriving on time are estimated accurately for a few sample networks by using the Picard method of successive approximations. The maximum probabilities can be evaluated without enumerating the network paths. The Laplace transform and its numerical inversion are introduced to reduce the computational cost of evaluating the convolution integrals that result from the successive approximation procedure. We are grateful to the colleagues who responded to this work with questions and comments during the Transportation Science Section session on Urban Transportation Planning Models II at the 2002 Meeting of the Institute for Operations Research and Management Science (INFORMS) in San José, California.     

A discrete survival model with random effects and covariate-dependent selection

In this paper we present a discrete survival model with covariates and random effects, where the random effects may depend on the observed covariates. The dependence between the covariates and the random effects is modelled through correlation parameters, and these parameters can only be identified for time-varying covariates. For time-varying covariates, however, it is possible to separate regression effects and selection effects in the case of a certain dependene structure between the random effects and the time-varying covariates that are assumed to be conditionally independent given the initial level of the covariate. The proposed model is equivalent to a model with independent random effects and the initial level of the covariates as further covariates. The model is applied to simulated data that illustrates some identifiability problems, and further indicate how the proposed model may be an approximation to retrospectively collected data with incorrect specification of the waiting times. The model is fitted by maximum likelihood estimation that is implemented as iteratively reweighted least squares. © 1998 John Wiley & Sons, Ltd.     

Shortest path games

We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.     

基于鲁棒优化的随机时变网络最优路径研究

交通事故、恶劣天气以及偶发的交通拥堵等都会导致道路交通网络中行程时间的不确定性,极大地影响了道路交通系统的可靠性,同时给日常生活中出行计划的制定以及出行路径的选择带来了不便。因此,本次研究将综合考虑道路交通网络中由于交通流量的全天变化所导致的路径行程时间的时变特征,以及由于事故、天气等不确定因素所导致的路径行程时间的随机特征,并以此作为路网环境的假设条件,对出行路径选择问题进行研究。具体地,首先建立行程时间的动态随机变量,并在此基础上模拟构建了随机时变网络。随后,定义了该网络环境下路径选择过程中所考虑的成本费用,并通过鲁棒优化的方法,将成本费用鲁棒性最强的路径视为最优路径。随后,在随机一致性条件下,通过数学推导证明了该模型可以简化为解决一个确定性时变网络中的最短路径问题。最终,具有多项式时间计算复杂度的改进Dijkstra算法被应用到模型的求解中,并通过小型算例验证模型及算法的有效性。结果表明,本研究中所提出的方法可以被高效率算法所求解,并且不依赖于先验行程时间概率分布的获取,因此对后续的大规模实际城市道路网络应用提供了良好的理论基础。此外,由于具有行程时间随机时变特征的交通网络更接近实际道路情况,因此本次研究的研究成果具有较高的实际意义和应用价值。     

Measuring legislative boundaries

We introduce the first family of district compactness measures that can incorporate a wide range of internal geographic features. The measures in this family are the probability that a district contains an admissible path between a randomly selected pair of people. The measure can account for roads, travel time, political boundaries, and prior districts. This family of measures includes the path-based measure of Chambers and Miller (2010).     

Algorithms for the quickest path problem and the reliable quickest path problem

The quickest path problem consists of finding a path in a directed network to transmit a given amount of items from an origin node to a destination node with minimal transmission time, when the transmission time depends on both the traversal times of the arcs, or lead time, and the rates of flow along arcs, or capacity. In telecommunications networks, arcs often also have an associated operational probability of the transmission being fault free. The reliability of a path is defined as the product of the operational probabilities of its arcs. The reliability as well as the transmission time are of interest. In this paper, algorithms are proposed to solve the quickest path problem as well as the problem of identifying the quickest path whose reliability is not lower than a given threshold. The algorithms rely on both the properties of a network which turns the computation of a quickest path into the computation of a shortest path and the fact that the reliability of a path can be evaluated through the reliability of the ordered sequence of its arcs. Other constraints on resources consumed, on the number of arcs of the path, etc. can also be managed with the same algorithms.     

A continuous whole-link travel time model with occupancy constraint

The continuous dynamic network loading problem (CDNLP) aims to compute link travel times and path travel times on a congested network, given time-dependent path flow rates for a given time period. A crucial element of CDNLP is a model of the link performance. Two main modeling frameworks have been used in link loading models: The so-called whole-link travel time (WTT) models and the kinematic wave model of Lighthill–Whitham–Richards (LWR) for traffic flow.In this paper, we reformulate a well-known whole-link model in which the link travel time, for traffic entering a time t, is a function of the number of vehicles on link. This formulation does not require the satisfying of the FIFO (first in, first out) condition. An extension of the basic WTT model is proposed in order to take explicitly into account the maximum number of vehicles that the link can accommodate (occupancy constraint). A solution scheme for the proposed WTT model is derived.Several numerical examples are given to illustrate that the FIFO condition is not respected for the WTT model and to compare the travel time predictions effected by LWR and WTT models.     

Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.     

A Higher-Order Hidden Markov Chain-Modulated Model for Asset Allocation

This paper presents an analysis of asset allocation strategies when the asset returns are governed by a discrete-time higher-order hidden Markov model (HOHMM), also called the weak hidden Markov model. We assume the drifts and volatilities of the asset returns switch over time according to the state of the HOHMM, in which the probability of the current state depends on the information from previous time-steps. The “switching” and “mixed” strategies are studied. We use a multivariate filtering technique in conjunction with the EM algorithm to obtain estimates of model parameter at a given time. This, in turn, aids investors in determining the optimal investment strategy for the next time step. Numerical implementation is applied to data on Russell 3000 value and growth indices. We benchmark the respective performances of portfolio using three classical investment measures.