A multi-commodity flow network design problem

There exist systems which can be usefully described by a network containingarcs through which a commodity of one type flows. This paper is concerned with finding a solution procedure for a particular multi-commodity flow network design problem. The problem is to identify a set of arcs in the network such that if travel is prohibited in them all flow travels by feasible paths and its total cost is minimal. The total flow in each arc may not exced its capacity, which is a known constant. Each arc and each node of the network has a non-negative constant unit traversal cost. Between each pair of distinct nodes there is a given non-negative rate of flow from the first vertex to the second which may be split up among a number of paths according to some constant traversal cost flow assignment process. The optimality criterion is the total traversal cost of all flow, which is to be minimized. Previous work on network design problems of this type is surveyed. The principal contribution of this paper is the presentation of a solution procedure for the above problem based on branch and bound enumeration. An illustrative numerical example is included. Computational experience gained in using the procedure with a FORTRAN IV program on an IBM 370 is favourable.     

Convexity and robustness of dynamic traffic assignment and freeway network control

We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA) problem to design optimal traffic flow controls for freeway networks as modeled by the Cell Transmission Model, using variable speed limit, ramp metering, and routing. We consider two optimal control problems: the DTA problem, where turning ratios are part of the control inputs, and the Freeway Network Control (FNC), where turning ratios are instead assigned exogenous parameters. It is known that relaxation of the supply and demand constraints in the cell-based formulations of the DTA problem results in a linear program. However, solutions to the relaxed problem can be infeasible with respect to traffic dynamics. Previous work has shown that such solutions can be made feasible by proper choice of ramp metering and variable speed limit control for specific traffic networks. We extend this procedure to arbitrary networks and provide insight into the structure and robustness of the proposed optimal controllers. For a network consisting only of ordinary, merge, and diverge junctions, where the cells have linear demand functions and affine supply functions with identical slopes, and the cost is the total traffic volume, we show, using the Pontryagin maximum principle, that variable speed limits are not needed in order to achieve optimality in the FNC problem, and ramp metering is sufficient. We also prove bounds on perturbation of the controlled system trajectory in terms of perturbations in initial traffic volume and exogenous inflows. These bounds, which leverage monotonicity properties of the controlled trajectory, are shown to be in close agreement with numerical simulation results.     

A dual decomposition method for sector capacity constrained traffic flow optimization

An aggregate air traffic flow model based on a multicommodity network is used for traffic flow management in the National Airspace System. The problem of minimizing the total travel time of flights in the National Airspace System of the United States, subject to sector capacity constraints, is formulated as an Integer Program. The resulting solution achieves optimal delay control. The Integer Program implemented for the scenarios investigated has billions of variables and constraints. It is relaxed to a Linear Program for computational efficiency. A dual decomposition method is applied to solve the large scale Linear Program in a computationally tractable manner. A rounding algorithm is developed to map the Linear Program solution to a physically acceptable result, and is implemented for the entire continental United States. A 2-h traffic flow management problem is solved with the method.     

Location configuration design for Dynamic Message Signs under stochastic incident and ATIS scenarios

This paper proposes a methodology for deploying permanent Dynamic Message Signs (DMS) in a vehicular traffic network. Of particular interest is the planning problem to optimize the number of DMS to deploy in conjunction with Advanced Traveler Information Systems (ATIS), operating and maintenance cost of DMS, and incident-related user cost under random traffic incident situations. The optimal DMS location design problem discussed herein is formulated as a two-stage stochastic program with recourse (SPR). A Tabu search algorithm combined with dynamic traffic simulation and assignment approaches are employed to solve this problem. A case study performed on the Fort-Worth, Texas network highlights the effectiveness of the proposed framework and illustrates the affect factors such as demand, network structure, DMS response rate, and incident characteristics have on the solution. The numerical results suggest that designing and deploying DMS and ATIS jointly is more cost-effective and efficient than the sequential build-out of the two from the system management perspective.     

Modeling and solving discrete network design problem with stochastic user equilibrium

In this paper, we address the discrete network design problem, which determines the addition of new roads to existing transportation network to optimize the transportation system performance. Road users are assumed to follow the traffic assignment principle of stochastic user equilibrium. A mixed‐integer nonlinear nonconvex problem is developed to model this discrete network design problem with stochastic user equilibrium. The original problem is relaxed into a convex mixed‐integer nonlinear program, whose solution provides a lower bound of the original problem. The relaxed problem is then embedded into two proposed global optimization solution algorithms to obtain the global optimal solution of the problem. Copyright © 2016 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.     

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.     

Tolling traffic links under stochastic assignment: Modelling the relationship between the number and price level of tolled links and optimal traffic flows

The classical road-tolling problem is to toll network links such that under the principles of Wardropian User Equilibrium Assignment a System Optimising (SO) flow pattern is obtained. Stochastic assignment methods are accepted to be more realistic than deterministic and it is of interest to examine the potential for optimal tolling in the case of Stochastic User Equilibrium (SUE). In examining the case of Stochastic User Equilibrium the ‘desired flow pattern’ to be created must first be determined. The classical economics solution of replacing unit-cost flow functions with marginal-cost flow functions which under deterministic assignment produces the System Optimal solution (where Total Network Travel Cost (TNTC) is minimised) does not generally result in TNTC being minimised in the Stochastic Case. Instead such tolls produce a ‘Stochastic System Optimal’ (SSO) solution where the Total Perceived Network Travel Cost (TPNTC) is minimised.This paper examines and compares link-based tolling solutions to achieve both the SSO (TPNTC minimised) and true SO (TNTC minimised) under SUE and illustrates the concept with numerical examples. Such link-based tolling schemes produce network benefit by re-routing rather than traffic suppression as opposed to the cordon-based charging schemes which have been implemented in practice. Equity issues relating to charging schemes are discussed and the desirability of zero-toll routes is highlighted associated with greater potential political acceptability of charging schemes that do not impose excessive charges upon users (such as minimal or low revenue tolls). A heuristic is developed to toll network links in such a way as to balance the number of links tolled against the revenue required to produce a desired reduction in TNTC such that optimal network flow patterns are approached.     

A computational model for the probit‐based dynamic stochastic user optimal traffic assignment problem

This paper focuses on computational model development for the probit‐based dynamic stochastic user optimal (P‐DSUO) traffic assignment problem. We first examine a general fixed‐point formulation for the P‐DSUO traffic assignment problem, and subsequently propose a computational model that can find an approximated solution of the interest problem. The computational model includes four components: a strategy to determine a set of the prevailing routes between each origin–destination pair, a method to estimate the covariance of perceived travel time for any two prevailing routes, a cell transmission model‐based traffic performance model to calculate the actual route travel time used by the probit‐based dynamic stochastic network loading procedure, and an iterative solution algorithm solving the customized fixed‐point model. The Ishikawa algorithm is proposed to solve the computational model. A comparison study is carried out to investigate the efficiency and accuracy of the proposed algorithm with the method of successive averages. Two numerical examples are used to assess the computational model and the algorithm proposed. Results show that Ishikawa algorithm has better accuracy for smaller network despite requiring longer computational time. Nevertheless, it could not converge for larger network. Copyright © 2010 John Wiley & Sons, Ltd.     

Bicriterion shortest path problem with a general nonadditive cost

A bicriterion shortest path problem with a general nonadditive cost seeks to optimize a combination of two path costs, one of which is evaluated by a nonlinear function. This paper first identifies a number of emerging transportation applications for which such a shortest path problem might be considered a core subproblem. We propose to first approximate the general nonlinear cost function with a piecewise linear counterpart, and then solve each linear subproblem sequentially. A specialized algorithm is developed to solve the subproblems, which makes use of the efficient path set (or the convex hull) to update upper and lower bounds of the original problem. Conditions under which the solution to a subproblem must belong to the efficient path set are specified. Accordingly, we show that the optimal path must be efficient if the nonlinear cost function is concave. If the optimal path to a subproblem is not efficient, partial path enumeration, implemented using a simple K-shortest path ranking procedure, is conducted to close the gap. The proposed algorithm includes strategies aiming to expedite path enumeration by using upper bounds derived from the efficient path set. Numerical experiments are conducted to demonstrate correctness and effectiveness of the proposed algorithm.     

Second best toll pricing within the framework of bounded rationality

The network design problem is usually formulated as a bi-level program, assuming the user equilibrium is attained in the lower level program. Given boundedly rational route choice behavior, the lower-level program is replaced with the boundedly rational user equilibria (BRUE). The network design problem with boundedly rational route choice behavior is understudied due to non-uniqueness of the BRUE. In this study, thus, we mainly focus on boundedly rational toll pricing (BR-TP) with affine link cost functions. The topological properties of the lower level BRUE set are first explored. As the BRUE solution is generally non-unique, urban planners cannot predict exactly which equilibrium flow pattern the transportation network will operate after a planning strategy is implemented. Due to the risk caused by uncertainty of people’s reaction, two extreme scenarios are considered: the traffic flow patterns with either the minimum system travel cost or the maximum, which is the “risk-prone” (BR-TP-RP) or the “risk-averse” (BR-TP-RA) scenario respectively. The upper level BR-TP is to find an optimal toll minimizing the total system travel cost, while the lower level is to find the best or the worst scenario. Accordingly BR-TP can be formulated as either a min –min or a min –max program. Solution existence is discussed based on the topological properties of the BRUE and algorithms are proposed. Two examples are accompanied to illustrate the proposed methodology.     

Computing 4D near-optimal trajectories for dynamic air traffic flow management with column generation and branch-and-price

Abstract

The current air traffic system faces recurrent saturation problems. Numerous studies are dedicated to this issue, including the present research on a new dynamic regulation filter holding frequent trajectory optimisations in a real-time sliding horizon loop process. We consider a trajectory optimisation problem arising in this context, where a feasible four-dimensional (4D) trajectory is to be built and assigned to each regulated flight to suppress sector overloads while minimising the cost of the chosen policy. We model this problem with a mixed integer linear programme and solve it with a branch-and-price approach. The pricing sub-problem looks for feasible trajectories in a dynamic three-dimensional (3D) network and is solved with a specific algorithm based on shortest path labelling algorithms and on dynamic programming. Each algorithm is tested on real-world data corresponding to a complete traffic day in the European air traffic system; experimental results, including computing times measurement, validate the solution process.     

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.     

Intermodal freight transport planning – A receding horizon control approach

This paper investigates intermodal freight transport planning problems among deep-sea terminals and inland terminals in hinterland haulage for a horizontally fully integrated intermodal freight transport operator at the tactical container flow level. An intermodal freight transport network (IFTN) model is first developed to capture the key characteristics of intermodal freight transport such as the modality change phenomena at intermodal terminals, physical capacity constraints of the network, time-dependent transport times on freeways, and time schedules for trains and barges. After that, the intermodal freight transport planning problem is formulated as an optimal intermodal container flow control problem from a system and control perspective with the use of the proposed IFTN model. To deal with the dynamic transport demands and dynamic traffic conditions in the IFTN, a receding horizon intermodal container flow control (RIFC) approach is proposed to control and to reassign intermodal container flows in a receding horizon way. This container flow control approach involves solving linear programming problems and is suited for transport planning on large-sized networks. Both an all-or-nothing approach and the proposed RIFC approach are evaluated through simulation studies. Simulation results show the potential of the proposed RIFC approach.     

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.     

A cell-based Merchant-Nemhauser model for the system optimum dynamic traffic assignment problem

A cell-based variant of the Merchant-Nemhauser (M-N) model is proposed for the system optimum (SO) dynamic traffic assignment (DTA) problem. Once linearized and augmented with additional constraints to capture cross-cell interactions, the model becomes a linear program that embeds a relaxed cell transmission model (CTM) to propagate traffic. As a result, we show that CTM-type traffic dynamics can be derived from the original M-N model, when the exit-flow function is properly selected and discretized. The proposed cell-based M-N model has a simple constraint structure and cell network representation because all intersections and cells are treated uniformly. Path marginal costs are defined using a recursive formula that involves a subset of multipliers from the linear program. This definition is then employed to interpret the necessary condition, which is a dynamic extension of the Wardrop’s second principle. An algorithm is presented to solve the flow holding back problem that is known to exist in many discrete SO-DTA models. A numerical experiment is conducted to verify the proposed model and algorithm.     

Traffic signal timing problems with environmental and equity considerations

Traffic signal timings in a road network can not only affect total user travel time and total amount of traffic emissions in the network but also create an inequity problem in terms of the change in travel costs of users traveling between different locations. This paper proposes a multi‐objective bi‐level programming model for design of sustainable and equitable traffic signal timings for a congested signal‐controlled road network. The upper level of the proposed model is a multi‐objective programming problem with an equity constraint that maximizes the reserve capacity of the network and minimizes the total amount of traffic emissions. The lower level is a deterministic network user equilibrium problem that considers the vehicle delays at signalized intersections of the network. To solve the proposed model, an approach for normalizing incommensurable objective functions is presented, and a heuristic solution algorithm that combines a penalty function approach and a simulated annealing method is developed. Two numerical examples are presented to show the effects of reserve capacity improvement and green time proportion on network flow distribution and transportation system performance and the importance of incorporating environmental and equity objectives in the traffic signal timing problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Network traffic flow optimization under performance constraints

In this paper, a model-based perimeter control policy for large-scale urban vehicular networks is proposed. Assuming a homogeneously loaded vehicle network and the existence of a well-posed Network Fundamental Diagram (NFD), we describe a protected network throughout its aggregated dynamics including nonlinear exit flow characteristics. Within this framework of constrained optimal boundary flow gating, two main performance metrics are considered: (a) first, connected to the NFD, the concept of average network travel time and delay as a performance metric is defined; (b) second, at boundaries, we take into account additional external network queue dynamics governed by uncontrolled inflow demands. External queue capacities in terms of finite-link lengths are used as the second performance metric. Hence, the corresponding performance requirement is an upper bound of external queues. While external queues represent vehicles waiting to enter the protected network, internal queue describes the protected network’s aggregated behavior.By controlling the number of vehicles joining the internal queue from the external ones, herewith a network traffic flow maximization solution subject to the internal and external dynamics and their performance constraints is developed. The originally non-convex optimization problem is transformed to a numerically efficiently convex one by relaxing the performance constraints into time-dependent state boundaries. The control solution can be interpreted as a mechanism which transforms the unknown arrival process governing the number of vehicles entering the network to a regulated process, such that prescribed performance requirements on travel time in the network and upper bound on the external queue are satisfied. Comparative numerical simulation studies on a microscopic traffic simulator are carried out to show the benefits of the proposed method.     

Network design for personal rapid transit under transit-oriented development

This study addresses guideway network design for personal rapid transit (PRT) favoring transit-oriented development. The guideway network design problem seeks to minimize both the guideway construction cost and users’ travel time. In particular, a set of optional points, known as Steiner points, are introduced in the graph to reduce the guideway length. The model is formulated as a combined Steiner and assignment problem, and a Lagrangian relaxation based solution algorithm is developed to solve the optimal solution. Numerical studies are carried on a real-sized network, and illustrate that the proposed model and solution algorithm can solve the PRT guideway network design problem effectively.