An efficient and exact event-based algorithm for solving simplified first order dynamic network loading problems in continuous time

In this paper a novel solution algorithm is proposed for exactly solving simplified first order dynamic network loading (DNL) problems for any generalised network. This DNL solution algorithm, termed eLTM (event-based Link Transmission Model), is based on the seminal Lighthill–Witham–Richards (LWR) model, adopts a triangular fundamental diagram and includes a generalised first order node model formulation. Unlike virtually all DNL solution algorithms, eLTM does not rely on time discretisation, but instead adopts an event based approach. The main advantage of this approach is the possibility of yielding exact results. Furthermore, an approximate version of the same algorithm is introduced. The user can configure an a-priori threshold that dictates the approximation error (measurable a-posteriori). Using this approximation the computational effort required decreases significantly, making it especially suitable for large scale applications. The computational complexity is investigated and results are demonstrated via theoretical and real world case studies. Fixed periods of stationary demands are included adopting a matrix demand profile to mimic basic departure time demand fluctuations. Finally, the information loss of the approximate solution is assessed under different configurations.     

On the existence of pricing strategies in the discrete time heterogeneous single bottleneck model

In this paper, we study the pricing strategies in the discrete time single bottleneck model with general heterogeneous commuters. We first prove that in the system optimal assignment, the queue time must be zero for all the departures. Based on this result, the system optimal problem is formulated as a linear program. The solution existence and uniqueness are discussed. Applying linear programming duality, we then prove that the optimal dual variable values provide an optimal toll with which the system optimal solution is also an equilibrium solution. Extensive computational results are reported to demonstrate the insights gained from the formulations in this paper. These results confirm that a system optimal equilibrium can be found using the proposed approach.     

A steady-state solution for the optimal pavement resurfacing problem

This paper presents a solution approach for the problem of optimising the frequency and intensity of pavement resurfacing, under steady-state conditions. If the pavement deterioration and improvement models are deterministic and follow the Markov property, it is possible to develop a simple but exact solution method. This method removes the need to solve the problem as an optimal control problem, which had been the focus of previous research in this area. The key to our approach is the realisation that, at optimality, the system enters the steady state at the time of the first resurfacing. The optimal resurfacing strategy is to define a minimum serviceability level (or maximum roughness level), and whenever the pavement deteriorates to that level, to resurface with a fixed intensity. The optimal strategy does not depend on the initial condition of the pavement, as long as the initial condition is better than the condition that triggers resurfacing. This observation allows us to use a simple solution method. We apply this solution procedure to a case study, using data obtained from the literature. The results indicate that the discounted lifetime cost is not very sensitive to cycle time. What matters most is the best achievable roughness level. The minimum serviceability level strategy is robust in that when there is uncertainty in the deterioration process, the optimal condition that triggers resurfacing is not significantly changed.     

Congestion-aware system optimal route choice for shared autonomous vehicles

We study the shared autonomous vehicle (SAV) routing problem while considering congestion. SAVs essentially provide a dial-a-ride service to travelers, but the large number of vehicles involved (tens of thousands of SAVs to replace personal vehicles) results in SAV routing causing significant congestion. We combine the dial-a-ride service constraints with the linear program for system optimal dynamic traffic assignment, resulting in a congestion-aware formulation of the SAV routing problem. Traffic flow is modeled through the link transmission model, an approximate solution to the kinematic wave theory of traffic flow. SAVs interact with travelers at origins and destinations. Due to the large number of vehicles involved, we use a continuous approximation of flow to formulate a linear program. Optimal solutions demonstrate that peak hour demand is likely to have greater waiting and in-vehicle travel times than off-peak demand due to congestion. SAV travel times were only slightly greater than system optimal personal vehicle route choice. In addition, solutions can determine the optimal fleet size to minimize congestion or maximize service.     

Constraint reformulation and a Lagrangian relaxation-based solution algorithm for a least expected time path problem

Using a sample-based representation scheme to capture spatial and temporal travel time correlations, this article constructs an integer programming model for finding the a priori least expected time paths. We explicitly consider the non-anticipativity constraint associated with the a priori path in a time-dependent and stochastic network, and propose a number of reformulations to establish linear inequalities that can be easily dualized by a Lagrangian relaxation solution approach. The relaxed model is further decomposed into two sub-problems, which can be solved directly by using a modified label-correcting algorithm and a simple single-value linear programming method. Several solution algorithms, including a sub-gradient method, a branch and bound method, and heuristics with additional constraints on Lagrangian multipliers, are proposed to improve solution quality and find approximate optimal solutions. The numerical experiments investigate the quality and computational efficiency of the proposed solution approach.     

A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations

In this paper, we propose a link-node complementarity model for the basic deterministic dynamic user equilibrium (DUE) problem with single-user-class and fixed demands. The model complements link-path formulations that have been widely studied for dynamic user equilibria. Under various dynamic network constraints, especially the exact flow propagation constraints, we show that the continuous-time dynamic user equilibrium problem can be formulated as an infinite dimensional mixed complementarity model. The continuous-time model can be further discretized as a finite dimensional non-linear complementarity problem (NCP). The proposed discrete-time model captures the exact flow propagation constraints that were usually approximated in previous studies. By associating link inflow at the beginning of a time interval to travel times at the end of the interval, the resulting discrete-time model is predictive rather than reactive. The solution existence and compactness condition for the proposed model is established under mild assumptions. The model is solved by an iterative algorithm with a relaxed NCP solved at each iteration. Numerical examples are provided to illustrate the proposed model and solution approach. We particularly show why predictive DUE is preferable to reactive DUE from an algorithmic perspective.     

Design framework of large-scale one-way electric vehicle sharing systems: A continuum approximation model

This paper proposes a Continuum Approximation (CA) model for design of a one-way Electrical Vehicle (EV) sharing system that serves a metropolitan area. This model determines the optimal EV sharing station locations and the corresponding EV fleet sizes to minimize the comprehensive system cost, including station construction investment, vehicle charging, transportation and vehicle balancing, under stochastic and dynamic trip demands. This is a very complex problem due to the NP-hard nature of location design, the large number of individual users, and the stochasticity and dynamics of generated trips. Further, the considerable charging time required by EVs distinguishes this problem from traditional car sharing problems where a vehicle is immediately available for pickup after being dropped at a station. We find that the CA approach can overcome these modeling challenges by decomposing the studied area into a number of small neighborhoods that each can be approximated by an Infinite Homogeneous Plane (IHP). We find that the system cost of an IHP is a unimodal function of the station service area size and can be efficiently solved in a sub-linear time by the bisection algorithm. Then integrating the solutions of all IHPs yields an approximate solution to the original heterogeneous area. With numerical experiments, we show that the CA solution is able to estimate the total system cost of the discrete counterpart solution efficiently with good accuracy, even for large-scale heterogeneous problems. This implies that the proposed CA approach is capable of providing a near-optimum solution to the comprehensive design of a practical large-scale EV sharing system. With this model, we also conduct sensitivity analysis to reveal insights into how cost components and system design vary with key parameter values. As far as the author’s knowledge, this study is the first work that addresses design of an EV sharing system considering both longer-term location and fleet size planning and daily vehicle operations. The proposed CA model also extends the CA methodology literature from traditional location problems with stationary demand, single-facility based service to EV sharing problems considering dynamic demands, OD trips, and nonlinear vehicle charging times.     

A heterogeneous reliable location model with risk pooling under supply disruptions

This paper investigates a facility location model that considers the disruptions of facilities and the cost savings from the inventory risk-pooling effect and economies of scale. Facilities may have heterogeneous disruption probabilities. When a facility fails, its customers may be reassigned to other surviving ones to hedge against lost-sales costs. We first develop both an exact and an approximate expression for the nonlinear inventory cost, and then formulate the problem as a nonlinear integer programming model. The objective is to minimize the expected total cost across all possible facility failure scenarios. To solve this problem, we design two methods, an exact approach using special ordered sets of type two (SOS2) and a heuristic based on Lagrangian relaxation. We test the model and algorithms on data sets with up to 150 nodes. Computational results show that the proposed algorithms can solve the problem efficiently in reasonable time. Managerial insights on the optimal facility deployment, customer assignments and inventory control strategies are also drawn.     

Robust weekly aircraft maintenance routing problem and the extension to the tail assignment problem

In this paper, we study two closely related airline planning problems: the robust weekly aircraft maintenance routing problem (RWAMRP) and the tail assignment problem (TAP). In real life operations, the RWAMRP solution is used in tactical planning whereas the TAP solution is implemented in operational planning. The main objective of these two problems is to minimize the total expected propagated delay (EPD) of the aircraft routes. To formulate the RWAMRP, we propose a novel weekly line-of-flights (LOF) network model that can handle complex and nonlinear cost functions of EPD. Because the number of LOFs grows exponentially with the number of flights to be scheduled, we propose a two-stage column generation approach to efficiently solve large-scale real-life RWAMRPs. Because the EPD of an LOF is highly nonlinear and can be very time-consuming to accurately compute, we propose three lower bounds on the EPD to solve the pricing subproblem of the column generation. Our approach is tested on eight real-life test instances. The computational results show that the proposed approach provides very tight LP relaxation (within 0.6% of optimal solutions) and solves the test case with more than 6000 flights per week in less than three hours. We also investigate the solutions obtained by our approach over 500 simulated realizations. The simulation results demonstrate that, in all eight test instances, our solutions result in less EPDs than those obtained from traditional methods. We then extend our model and solution approach to solve realistically simulated TAP instances.     

Revisiting Jiang’s dynamic continuum model for urban cities

Jiang et al. (Jiang, Y.Q., Wong, S.C., Ho, H.W., Zhang, P., Liu, R.X., Sumalee, A., 2011. A dynamic traffic assignment model for a continuum transportation system. Transportation Research Part B 45 (2), 343–363) proposed a predictive continuum dynamic user-optimaDUO-l to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers. Their modeled region is a dense urban city that is arbitrary in shape and has a single central business district (CBD). However, we argue that the model is not well posed due to an inconsistency in the route-choice strategy under certain conditions. To overcome this inconsistency, we revisit the PDUO-C problem, and construct an improved path-choice strategy. The improved model consists of a conservation law to govern the density, in which the flow direction is determined by the improved path-choice strategy, and a Hamilton–Jacobi equation to compute the total travel cost. The simultaneous satisfaction of both equations can be treated as a fixed-point problem. A self-adaptive method of successive averages (MSA) is proposed to solve this fixed-point problem. This method can automatically determine the optimal MSA step size using the least squares approach. Numerical examples are used to demonstrate the effectiveness of the model and the solution algorithm.     

A model of optimal transport maintenance with demand responsiveness

Maintenance activities can be performed throughout the lifetime of a particular facility or piece of equipment, thereby affecting its quality in a continuous fashion. It is assumed in this paper that quality of a facility is determined by natural factors, rate of use and maintenance investments while demand for the facility is assumed to be a function of its quality. A dynamic optimization model in the form of a simple linear optimal control problem is then developed in order to determine optimal maintenance policies under these circumstances. Bang-bang and singular policies are derived and given economic interpretations. An algorithm is presented for numerical solution of the Pontryagin necessary conditions. Finally, sufficiency conditions are obtained for the model structure considered.     

Equity-oriented skip-stopping schedule optimization in an oversaturated urban rail transit network

In this study, we focus on improving system-wide equity performance in an oversaturated urban rail transit network based on multi-commodity flow formulation. From the system perspective, an urban rail transit network is a distributed system, where a set of resources (i.e., train capacity) is shared by a number of users (i.e., passengers), and equitable individuals and groups should receive equal shares of resources. However, when oversaturation occurs in an urban rail transit network during peak hours, passengers waiting at different stations may receive varying shares of train capacity leading to the inequity problem under train all-stopping pattern. Train skip-stopping pattern is an effective operational approach, which holds back some passengers at stations and re-routes their journeys in the time dimension based on the available capacity of each train. In this study, the inequity problem in an oversaturated urban rail transit network is analyzed using a multi-commodity flow modeling framework. In detail, first, discretized states, corresponding to the number of missed trains for passengers, are constructed in a space-time-state three-dimensional network, so that the system-wide equity performance can be viewed as a distribution of all passengers in different states. Different from existing flow-based optimization models, we formulate individual passenger and train stopping pattern as commodity and network structure in the multi-commodity flow-modeling framework, respectively. Then, we aim to find an optimal commodity flow and well-designed network structure through the proposed multi-commodity flow model and simultaneously achieve the equitable distribution of all passengers and the optimal train skip-stopping pattern. To quickly solve the proposed model and find an optimal train skip-stopping pattern with preferable system-wide equity performance, the proposed linear programming model can be effectively decomposed to a least-cost sub-problem with positive arc costs for each individual passenger and a least-cost sub-problem with negative arc costs for each individual train under a Lagrangian relaxation framework. For application and implementation, the proposed train skip-stopping optimization model is applied to a simple case and a real-world case based on Batong Line in the Beijing Subway Network. The simple case demonstrates that our proposed Lagrangian relaxation framework can obtain the approximate optimal solution with a small-gap lower bound and a lot of computing time saved compared with CPLEX solver. The real-world case based on Batong Line in the Beijing Subway Network compares the equity and efficiency indices under the operational approach of train skip-stopping pattern with those under the train all-stopping pattern to state the advantage of the train skip-stopping operational approach.     

The nonlinear equation system approach to solving dynamic user optimal simultaneous route and departure time choice problems

Dynamic user optimal simultaneous route and departure time choice (DUO-SRDTC) problems are usually formulated as variational inequality (VI) problems whose solution algorithms generally require continuous and monotone route travel cost functions to guarantee convergence. However, the monotonicity of the route travel cost functions cannot be ensured even if the route travel time functions are monotone. In contrast to traditional formulations, this paper formulates a DUO-SRDTC problem (that can have fixed or elastic demand) as a system of nonlinear equations. The system of nonlinear equations is a function of generalized origin-destination (OD) travel costs rather than route flows and includes a dynamic user optimal (DUO) route choice subproblem with perfectly elastic demand and a quadratic programming (QP) subproblem under certain assumptions. This study also proposes a solution method based on the backtracking inexact Broyden–Fletcher–Goldfarb–Shanno (BFGS) method, the extragradient algorithm, and the Frank-Wolfe algorithm. The BFGS method, the extragradient algorithm, and the Frank-Wolfe algorithm are used to solve the system of nonlinear equations, the DUO route choice subproblem, and the QP subproblem, respectively. The proposed formulation and solution method can avoid the requirement of monotonicity of the route travel cost functions to obtain a convergent solution and provide a new approach with which to solve DUO-SRDTC problems. Finally, numeric examples are used to demonstrate the performance of the proposed solution method.     

A new strategy for minimum usage of external yaw moment in vehicle dynamic control system

Due to the loss of vehicle directional stability in emergency maneuvers, a new complete desired model for vehicle handling based on the linear two-degrees-of-freedom (2DOF) model and tire/road conditions is presented to be tracked by the direct yaw moment control (DYC) system. In order to maintain the vehicle actual motions, yaw rate and side-slip angle, close to the proposed desired responses without excessively large external yaw moment, a complete linear quadratic (LQ) optimal problem is formulated and its analytical solution is obtained. Here, the derived control law is evaluated and its different versions are discussed. It is shown that the side-slip tracking by DYC is more effective than the yaw rate control to stabilize vehicle motions in nonlinear regimes. Also, optimal property of the control law provides the possibility of reducing the external yaw moment as low as possible, at the cost of some admissible tracking errors. Simulation studies of vehicle handling, with and without control, have been conducted using a full nonlinear vehicle dynamic model. The results, obtained during various maneuvers, indicate that when the proposed optimal controller is engaged with the model, improvements in the handling performance through a reduced external yaw moment can be acquired.     

Dynamic system-optimal traffic assignment using a state space model

We propose a new mathematical formulation for the problem of optimal traffic assignment in dynamic networks with multiple origins and destinations. This problem is motivated by route guidance issues that arise in an Intelligent Vehicle-Highway Systems (IVHS) environment. We assume that the network is subject to known time-varying demands for travel between its origins and destinations during a given time horizon. The objective is to assign the vehicles to links over time so as to minimize the total travel time experienced by all the vehicles using the network. We model the traffic network over the time horizon as a discrete-time dynamical system. The system state at each time instant is defined in a way that, without loss of optimality, avoids complete microscopic detail by grouping vehicles into platoons irrespective of origin node and time of entry to network. Moreover, the formulation contains no explicit path enumeration. The state transition function can model link travel times by either impedance functions, link outflow functions, or by a combination of both. Two versions (with different boundary conditions) of the problem of optimal traffic assignment are studied in the context of this model. These optimization problems are optimal control problems for nonlinear discrete-time dynamical systems, and thus they are amenable to algorithmic solutions based on dynamic programming. The computational challenges associated with the exact solution of these problems are discussed and some heuristics are proposed.     

Demand responsive transit systems with time-dependent demand: User equilibrium,system optimum,and management strategy

The operating cost of a demand responsive transit (DRT) system strictly depends on the quality of service that it offers to its users. An operating agency seeks to minimize operating costs while maintaining the quality of service while users experience costs associated with scheduling, waiting, and traveling within the system. In this paper, an analytical model is employed to approximate the agency's operating cost for running a DRT system with dynamic demand and the total generalized cost that users experience as a result of the operating decisions. The approach makes use of Vickrey's (1969) congestion theory to model the dynamics of the DRT system in the equilibrium condition and approximate the generalized cost for users when the operating capacity is inadequate to serve the time-dependent demand over the peak period without excess delay. The efficiency of the DRT system can be improved by optimizing one of three parameters that define the agency's operating decision: (1) the operating capacity of the system, (2) the number of passengers that have requested a pick-up and are awaiting service, and (3) the distribution of requested times for service from the DRT system. A schedule management strategy and dynamic pricing strategies are presented that can be implemented to manage demand and reduce the total cost of the DRT system by keeping the number of waiting requests optimized over the peak period. In the end, proposed optimization strategies are compared using a numerical example.     

Optimal vehicle speed trajectory on a signalized arterial with consideration of queue

Vehicle speed trajectory significantly impacts fuel consumption and greenhouse gas emissions, especially for trips on signalized arterials. Although a large amount of research has been conducted aiming at providing optimal speed advisory to drivers, impacts from queues at intersections are not considered. Ignoring the constraints induced by queues could result in suboptimal or infeasible solutions. In this study, a multi-stage optimal control formulation is proposed to obtain the optimal vehicle trajectory on signalized arterials, where both vehicle queue and traffic light status are considered. To facilitate the real-time update of the optimal speed trajectory, a constrained optimization model is proposed as an approximation approach, which can be solved much quicker. Numerical examples demonstrate the effectiveness of the proposed optimal control model and the solution efficiency of the proposed approach.     

Dynamic system‐optimal traffic assignment for a city using the continuum modeling approach

This paper presents a continuum dynamic traffic assignment model for a city in which the total cost of the traffic system is minimized: the travelers in the system are organized to choose the route to their destinations that minimizes the total cost of the system. Combined with the objective function, which defines the total cost and constraints such as certain physical and boundary conditions, a continuum model can be formulated as an optimization scheme with a feasible region in the function space. To obtain an admissible locally optimal solution to this problem, we first reformulate the optimization in discrete form and then introduce a heuristic method to solve it. This method converges rapidly with attractive computational cost. Numerical examples are used to demonstrate the effectiveness of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Model and a solution algorithm for the dynamic resource allocation problem for large-scale transportation network evacuation

Allocating movable resources dynamically enables evacuation management agencies to improve evacuation system performance in both the spatial and temporal dimensions. This study proposes a mixed integer linear program (MILP) model to address the dynamic resource allocation problem for transportation evacuation planning on large-scale networks. The proposed model is built on the earliest arrival flow formulation that significantly reduces problem size. A set of binary variables, specifically, the beginning and the ending time of resource allocation at a location, enable a strong formulation with tight constraints. A solution algorithm is developed to solve for an optimal solution on large-scale network applications by adopting Benders decomposition. In this algorithm, the MILP model is decomposed into two sub-problems. The first sub-problem, called the restricted master problem, identifies a feasible dynamic resource allocation plan. The second sub-problem, called the auxiliary problem, models dynamic traffic assignment in the evacuation network given a resource allocation plan. A numerical study is performed on the Dallas–Fort Worth network. The results show that the Benders decomposition algorithm can solve an optimal solution efficiently on a large-scale network.     

Global optimization methods for the discrete network design problem

This paper addresses the discrete network design problem (DNDP) with multiple capacity levels, or multi-capacity DNDP for short, which determines the optimal number of lanes to add to each candidate link in a road network. We formulate the problem as a bi-level programming model, where the upper level aims to minimize the total travel time via adding new lanes to candidate links and the lower level is a traditional Wardrop user equilibrium (UE) problem. We propose two global optimization methods by taking advantage of the relationship between UE and system optimal (SO) traffic assignment principles. The first method, termed as SO-relaxation, exploits the property that an optimal network design solution under SO principle can be a good approximate solution under UE principle, and successively sorts the solutions in the order of increasing total travel time under SO principle. Optimality is guaranteed when the lower bound of the total travel time of the unexplored solutions under UE principle is not less than the total travel time of a known solution under UE principle. The second method, termed as UE-reduction, adds the objective function of the Beckmann-McGuire-Winsten transformation of UE traffic assignment to the constraints of the SO-relaxation formulation of the multi-capacity DNDP. This constraint is convex and strengthens the SO-relaxation formulation. We also develop a dynamic outer-approximation scheme to make use of the state-of-the-art mixed-integer linear programming solvers to solve the SO-relaxation formulation. Numerical experiments based on a two-link network and the Sioux-Falls network are conducted.