Estimation of mean and covariance of stochastic multi-class OD demands from classified traffic counts

This paper proposes a new model to estimate the mean and covariance of stochastic multi-class (multiple vehicle classes) origin–destination (OD) demands from hourly classified traffic counts throughout the whole year. It is usually assumed in the conventional OD demand estimation models that the OD demand by vehicle class is deterministic. Little attention is given on the estimation of the statistical properties of stochastic OD demands as well as their covariance between different vehicle classes. Also, the interactions between different vehicle classes in OD demand are ignored such as the change of modes between private car and taxi during a particular hourly period over the year. To fill these two gaps, the mean and covariance matrix of stochastic multi-class OD demands for the same hourly period over the year are simultaneously estimated by a modified lasso (least absolute shrinkage and selection operator) method. The estimated covariance matrix of stochastic multi-class OD demands can be used to capture the statistical dependency of traffic demands between different vehicle classes. In this paper, the proposed model is formulated as a non-linear constrained optimization problem. An exterior penalty algorithm is adapted to solve the proposed model. Numerical examples are presented to illustrate the applications of the proposed model together with some insightful findings on the importance of covariance of OD demand between difference vehicle classes.     

Rolling horizon stochastic optimal control strategy for ACC and CACC under uncertainty

This paper presents a rolling horizon stochastic optimal control strategy for both Adaptive Cruise Control and Cooperative Adaptive Cruise Control under uncertainty based on the constant time gap policy. Specifically, uncertainties that can arise in vehicle control systems and vehicle sensor measurements are represented as normally-distributed disturbances to state and measurement equations in a state-space formulation. Then, acceleration sequence of a controlled vehicle is determined by optimizing an objective function that captures control efficiency and driving comfort over a predictive horizon, constrained by bounded acceleration/deceleration and collision protection. The optimization problem is formulated as a linearly constrained linear quadratic Gaussian problem and solved using a separation principle, Lagrangian relaxation, and Kalman filter. A sensitivity analysis and a scenario-based analysis via simulations demonstrate that the proposed control strategy can generate smoother vehicle control and perform better than a deterministic feedback controller, particularly under small system disturbances and large measurement disturbances.     

Vehicle routing problem and driver behaviour: a review and framework for analysis

Vehicle routing problems (VRPs) whose typical objective is to minimise total travel costs over a tour have evolved over the years with objectives ranging from minimising travel times and distances to minimising pollution and fuel consumption. However, driver behaviour continues to be neglected while planning for vehicle routes. Factors such as traffic congestion levels, monotonous drives and fatigue have an impact on the behaviour of drivers, which in turn might affect their speed-choice and route-choice behaviours. The behaviour of drivers and their subsequent decision-making owing to these factors impact the revenue of transport companies and could lead to huge losses in extreme cases. There have been studies on the behaviour of drivers in isolation, without inclusion of the objectives and constraints of the traditional routing problem. This paper presents a review of existing models of VRP, planner behaviour models in the VRP context and driver behaviour models and provides a motivation to integrate these models in a stochastic traffic environment to produce practical, economic and driver-friendly logistics solutions. The paper provides valuable insights on the relevance of behavioural issues in logistics and highlights the modelling implications of incorporating planner and driver behaviour in the framework of routing problems.     

Weather impact on containership routing in closed seas: A chance-constraint optimization approach

Weather conditions have a strong effect on the operation of vessels and unavoidably influence total time at sea and associated transportation costs. The velocity and direction of the wind in particular may considerably affect travel speed of vessels and therefore the reliability of scheduled maritime services. This paper considers weather effects in containership routing; a stochastic model is developed for determining optimal routes for a homogeneous fleet performing pick-ups and deliveries of containers between a hub and several spoke ports, while incorporating travel time uncertainties attributed to the weather. The problem is originally formulated as a chance-constrained variant of the vehicle routing problem with simultaneous pick-ups and deliveries and time constraints and solved using a genetic algorithm. The model is implemented to a network of island ports of the Aegean Sea. Results on the application of algorithm reveal that a small fleet is sufficient enough to serve network’s islands, under the influence of minor delays. A sensitivity analysis based on alternative scenarios in the problem’s parameters, leads to encouraging conclusions with respect to the efficiency and robustness of the algorithm.     

The time-dependent vehicle routing problem with soft time windows and stochastic travel times

This paper studies a vehicle routing problem with time-dependent and stochastic travel times. In our problem setting, customers have soft time windows. A mathematical model is used in which both efficiency for service as well as reliability for customers are taken into account. Depending on whether service times are included or not, we consider two versions of this problem. Two metaheuristics are built: a Tabu Search and an Adaptive Large Neighborhood Search. We carry out our experiments for well-known problem instances and perform comprehensive analyses on the numerical results in terms of the computational time and the solution quality. Experiments confirm that the proposed procedure is effective to obtain very good solutions to be performed in real-life environment.     

Adaptive route choices in risky traffic networks: A prospect theory approach

This paper deals with route choice models capturing travelers’ strategic behavior when adapting to revealed traffic conditions en route in a stochastic network. The strategic adaptive behavior is conceptualized as a routing policy, defined as a decision rule that maps from all possible revealed traffic conditions to the choices of next link out of decision nodes, given information access assumptions. In this paper, we use a specialized example where a variable message sign provides information about congestion status on outgoing links. We view the problem as choice under risk and present a routing policy choice model based on the cumulative prospect theory (CPT), where utility functions are nonlinear in probabilities and thus flexible attitudes toward risk can be captured.In order to illustrate the differences between routing policy and non-adaptive path choice models as well as differences between models based on expected utility (EU) theory and CPT, we estimate models based on synthetic data and compare them in terms of prediction results. There are large differences in path share predictions and the results demonstrate the flexibility of the CPT model to represent varying degrees of risk aversion and risk seeking depending on the outcome probabilities.     

International experience with speed limits during and prior to the energy crisis of 1973–4

This paper presents an artificial neural network (ANN) based method for estimating route travel times between individual locations in an urban traffic network. Fast and accurate estimation of route travel times is required by the vehicle routing and scheduling process involved in many fleet vehicle operation systems such as dial‐a‐ride paratransit, school bus, and private delivery services. The methodology developed in this paper assumes that route travel times are time‐dependent and stochastic and their means and standard deviations need to be estimated. Three feed‐forward neural networks are developed to model the travel time behaviour during different time periods of the day‐the AM peak, the PM peak, and the off‐peak. These models are subsequently trained and tested using data simulated on the road network for the City of Edmonton, Alberta. A comparison of the ANN model with a traditional distance‐based model and a shortest path algorithm is then presented. The practical implication of the ANN method is subsequently demonstrated within a dial‐a‐ride paratransit vehicle routing and scheduling problem. The computational results show that the ANN‐based route travel time estimation model is appropriate, with respect to accuracy and speed, for use in real applications.     

The split-demand one-commodity pickup-and-delivery travelling salesman problem

This paper introduces a new vehicle routing problem transferring one commodity between customers with a capacitated vehicle that can visit a customer more than once, although a maximum number of visits must be respected. It generalizes the capacitated vehicle routing problem with split demands and some other variants recently addressed in the literature. We model the problem with a single commodity flow formulation and design a branch-and-cut approach to solve it. We make use of Benders Decomposition to project out the flow variables from the formulation. Inequalities to strengthen the linear programming relaxation are also presented and separated within the approach. Extensive computational results illustrate the performance of the approach on benchmark instances from the literature.     

A network wide simulation strategy of alternative fuel vehicles

This paper presents an integrated simulator “CUIntegration” to evaluate routing strategies based on energy and/or traffic measures of effectiveness for any Alternative Fuel Vehicles (AFVs). The CUIntegration can integrate vehicle models of conventional vehicles as well as AFVs developed with MATLAB-Simulink, and a roadway network model developed with traffic microscopic simulation software VISSIM. The architecture of this simulator is discussed in this paper along with a case study in which the simulator was utilized for evaluating a routing strategy for Plug-in Hybrid Electric Vehicles (PHEVs) and Electric Vehicles (EVs). The authors developed a route optimization algorithm to guide an AFV based on that AFV driver’s choice, which included; finding a route with minimum (1) travel time, (2) energy consumption or (3) a combination of both. The Application Programming Interface (API) was developed using Visual Basic to simulate the vehicle models/algorithms developed in MATLAB and direct vehicles in a roadway network model developed in VISSIM accordingly. The case study included a section of Interstate 83 in Baltimore, Maryland, which was modeled, calibrated and validated. The authors considered a worst-case scenario with an incident on the main route blocking all lanes for 30 min. The PHEVs and EVs were represented by integrating the MATLAB-Simulink vehicle models with the traffic simulator. The CUIntegration successfully combined vehicle models with a roadway traffic network model to support a routing strategy for PHEVs and EVs. Simulation experiments with CUIntegration revealed that routing of PHEVs resulted in cost savings of about 29% when optimized for the energy consumption, and for the same optimization objective, routing of EVs resulted in about 64% savings.     

A reinforcement learning framework for the adaptive routing problem in stochastic time-dependent network

Most previous work in addressing the adaptive routing problem in stochastic and time-dependent (STD) network has been focusing on developing parametric models to reflect the network dynamics and designing efficient algorithms to solve these models. However, strong assumptions need to be made in the models and some algorithms also suffer from the curse of dimensionality. In this paper, we examine the application of Reinforcement Learning as a non-parametric model-free method to solve the problem. Both the online Q learning method for discrete state space and the offline fitted Q iteration algorithm for continuous state space are discussed. With a small case study on a mid-sized network, we demonstrate the significant advantages of using Reinforcement Learning to solve for the optimal routing policy over traditional stochastic dynamic programming method. And the fitted Q iteration algorithm combined with tree-based function approximation is shown to outperform other methods especially during peak demand periods.     

The load-dependent vehicle routing problem and its pick-up and delivery extension

The present paper examines a Vehicle Routing Problem (VRP) of major practical importance which is referred to as the Load-Dependent VRP (LDVRP). LDVRP is applicable for transportation activities where the weight of the transported cargo accounts for a significant part of the vehicle gross weight. Contrary to the basic VRP which calls for the minimization of the distance travelled, the LDVRP objective is aimed at minimizing the total product of the distance travelled and the gross weight carried along this distance. Thus, it is capable of producing sensible routing plans which take into account the variation of the cargo weight along the vehicle trips. The LDVRP objective is closely related to the total energy requirements of the vehicle fleet, making it a credible alternative when the environmental aspects of transportation activities are examined and optimized. A novel LDVRP extension which considers simultaneous pick-up and delivery service is introduced, formulated and solved for the first time. To deal with large-scale instances of the examined problems, we propose a local-search algorithm. Towards an efficient implementation, the local-search algorithm employs a computational scheme which calculates the complex weighted-distance objective changes in constant time. Solution results are presented for both problems on a variety of well-known test cases demonstrating the effectiveness of the proposed solution approach. The structure of the obtained LDVRP and VRP solutions is compared in pursuit of interesting conclusions on the relative suitability of the two routing models, when the decision maker must deal with the weighted distance objective. In addition, results of a branch-and-cut procedure for small-scale instances of the LDVRP with simultaneous pick-ups and deliveries are reported. Finally, extensive computational experiments have been performed to explore the managerial implications of three key problem characteristics, namely the deviation of customer demands, the cargo to tare weight ratio, as well as the size of the available vehicle fleet.     

An evolutionary local search for the capacitated vehicle routing problem minimizing fuel consumption under three-dimensional loading constraints

This study introduces a new practical variant of the combined routing and loading problem called the capacitated vehicle routing problem minimizing fuel consumption under three-dimensional loading constraints (3L-FCVRP). It presents a meta-heuristic algorithm for solving the problem. The aim is to design routes for a fleet of homogeneous vehicles that will serve all customers, whose demands are formed by a set of three-dimensional, rectangular, weighted items. Unlike the well-studied capacitated vehicle routing problem with 3D loading constraints (3L-CVRP), the objective of the 3L-FCVRP is to minimize total fuel consumption rather than travel distance. The fuel consumption rate is assumed to be proportionate to the total weight of the vehicle. A route is feasible only if a feasible loading plan to load the demanded items into the vehicle exists and the loading plan must satisfy a set of practical constraints.To solve this problem, the evolutionary local search (ELS) framework incorporating the recombination method is used to explore the solution space, and a new heuristic based on open space is used to examine the feasibility of the solutions. In addition, two special data structures, Trie and Fibonacci heap, are adopted to speed up the procedure. To verify the effectiveness of our approach, we first test the ELS on the 3L-CVRP, which can be seen as a special case of the 3L-FCVRP. The results demonstrate that on average ELS outperforms all of the existing approaches and improves the best-known solutions for most instances. Then, we generate data for 3L-FCVRP and report the detailed results of the ELS for future comparisons.     

Inter-city bus routing and timetable setting under stochastic demands

Vehicle fleet routing and timetable setting are essential to the enhancement of an inter-city bus carrier’s operating cost, profit, level of service and competitiveness in the market. In past research the average passenger demand has usually served as input in the production of the final fleet routes and timetables, meaning that stochastic disturbances arising from variations in daily passenger demand in actual operations are neglected. To incorporate the stochastic disturbances of daily passenger demands that occur in actual operations, in this research, we established a stochastic-demand scheduling model. We applied a simulation technique, coupled with link-based and path-based routing strategies, to develop two heuristic algorithms to solve the model. To evaluate the performance of the proposed model and the two solution algorithms, we developed an evaluation method. The test results, regarding a major Taiwan inter-city bus operation, were good, showing that the model and the solution algorithms could be useful in practice.     

Congestion avoidance in city traffic

The number of vehicles on the road (worldwide) is constantly increasing, causing traffic jams and congestion especially in city traffic. Anticipatory vehicle routing techniques have thus far been applied to fairly small networked traffic scenarios and uniform traffic. We note here a number of limitations of these techniques and present a routing strategy on the assumption of a city map that has a large number of nodes and connectivity and where the vehicles possess highly varying speed capabilities. A scenario of operation with such characteristics has not previously been sufficiently studied in the literature. Frequent short‐term planning is preferred as compared with infrequent planning of the complete map. Experimental results show an efficiency boost when single‐lane overtaking is allowed, traffic signals are accounted for and every vehicle prefers to avoid high traffic density on a road by taking an alternative route. Comparisons with optimistic routing, pessimistic routing and time message channel routing are given. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation of origin-destination matrices from link counts and sporadic routing data

Estimation of origin-destination (OD) matrices from link count data is a challenging problem because of the highly indeterminate relationship between the observations and the latent route flows. Conversely, estimation is straightforward if we observe the path taken by each vehicle. We consider an intermediate problem of increasing practical importance, in which link count data is supplemented by routing information for a fraction of vehicles on the network. We develop a statistical model for these combined data sources and derive some tractable normal approximations thereof. We examine likelihood-based inference for these normal models under the assumption that the probability of vehicle tracking is known. We show that the likelihood theory can be non-standard because of boundary effects, and provide conditions under which such irregular behaviour will be observed in practice. For regular cases we outline connections with existing generalised least squares methods. We then consider estimation of OD matrices under estimated and/or misspecified models for the probability of vehicle tracking. Theoretical developments are complemented by simulation experiments and an illustrative example using a section of road network from the English city of Leicester.     

Consistent formulation of network equilibrium with stochastic flows

Traffic flows in real-life transportation systems vary on a daily basis. According to traffic flow theory, such variability should induce a similar variability in travel times, but this “internal consistency” is generally not captured by existing network equilibrium models. We present an internally-consistent network equilibrium approach, which considers two potential sources of flow variability: (i) daily variation in route choice and (ii) daily variation in origin–destination demand. We particularly aspire to a flexible formulation that permits alternative statistical assumptions, which allows the best fit to be made to observed variability data in particular applications. Joint probability distributions of route—and therefore link—flows are derived under several assumptions concerning stochastic driver behavior. A stochastic network equilibrium model with stochastic demands and route choices is formulated as a fixed point problem. We explore limiting cases which allow an equivalent convex optimization problem to be defined, and finally apply this method to a real-life network of Kanazawa City, Japan.     

Stochastic optimal path problem with relays

This paper studies the optimal path problem for travelers driving with vehicles of a limited range, such as most battery electric vehicles currently available in the market. The optimal path in this problem often consists of several relay points, where the vehicles can be refueled to extend its range. We propose a stochastic optimal path problem with relays (SOPPR), which aims at minimizing a general expected cost while maintaining a reasonable arrival probability. To account for uncertainty in the road network, the travel speed on a road segment is treated as a discrete random variable, which determines the total energy required to traverse the segment. SOPPR is formulated in two stages in this paper. In the first stage, an optimal routing problem is solved repeatedly to obtain the expected costs and arrival probabilities from any node to all refueling nodes and the destination. With this information, the second stage constructs an auxiliary network, on which the sequence of refueling decisions can be obtained by solving another optimal path problem. Label-correcting algorithms are developed to solve the routing problems in both stages. Numerical experiments are conducted to compare the stochastic and deterministic models, to examine the impact of different parameters on the routing results, and to evaluate the computational performance of the proposed algorithms.     

Integration of vehicle routing and resource allocation in a dynamic logistics network

A shipper plans daily hub-to-hub transports within a hub and spoke network. Since a limited number of swap containers is available for transportation, two problems arise. 1. Swap containers have to be routed as pickup and delivery requests in multi-hub tours. 2. Day-by-day routing may lead to an imbalance of swap containers requiring a dynamic allocation. Neglecting interdependencies between vehicle routing and resource allocation seems inferior. An integration of the two problems overcomes this deficiency. We formulate mathematical models and propose integration approaches. The advantages of these approaches are discussed based on a computational study.     

A simulation model of the bus transitway (TRNSIM)

For planning and design of a bus rapid transit system and for the analysis of multimodal corridors, methodology is required for simulating bus traffic operation on a Transitway. Macroscopic models of vehicle flow are gaining popularity due to their capability to analyze complex operations and yet offer efficiency in development and applications. A macroscopic model is developed for the investigation of travel time, energy and emissions that correspond to bus volume levels on the Transitway. This paper describes the travel time part of the model. The model treats stochastic characteristics of bus traffic and passenger activities. Also, safety regimes in vehicle flow and factors affecting minimum headways in station areas are incorporated. The model is verified by comparing simulated travel time for the Ottawa-Carleton Transitway with actual data.     

Vehicle headway modeling and its inferences in macroscopic/microscopic traffic flow theory: A survey

Vehicle headway distribution models are widely used in traffic engineering fields, since they reflect the fundamental uncertainty in drivers' car-following maneuvers and meanwhile provide a concise way to describe the stochastic feature of traffic flows. This paper presents a systematic review of vehicle headway distribution studies in the last few decades. Since it is impossible to enumerate the merits and drawbacks of all of existing distribution models, we emphasize four advances of headway distribution modeling in this paper. First, we highlight the chronicle of key assumptions on the existing distribution models and explain why this evolution occurs. Second, we show that departure headways measured for interrupted flows on urban streets and headways measured for uninterrupted flows on freeways have common features and can be simulated by a unified microscopic car-following model. The interesting finding helps gather two kinds of headway distribution models under one umbrella. Third, we review different approaches that aim to link microscopic car-following models and mesoscopic vehicle headway distribution models. Fourth, we show that both the point scattering on the density-flow plot and the shape of traffic flow breakdown curve implicitly depend on the vehicular headway distribution. These findings reveal pervasive connections between macroscopic traffic flow models and mesoscopic headway distribution. All these new insights bring new vigor into vehicle headway studies and open research frontiers in this field.