A Primal-Dual Algorithm for Link Dependent Origin Destination Matrix Estimation

Origin-Destination Matrix (ODM) estimation is a classical problem in transport engineering aiming to recover flows from every Origin to every Destination from measured traffic counts and a priori model information. In addition to traffic counts, the present contribution takes advantage of probe trajectories, whose capture is made possible by new measurement technologies. It extends the concept of ODM to that of Link dependent ODM (LODM), keeping the information about the flow distribution on links and containing inherently the ODM assignment. Further, an original formulation of LODM estimation, from traffic counts and probe trajectories is presented as an optimisation problem, where the functional to be minimized consists of five convex functions, each modelling a constraint or property of the transport problem: consistency with traffic counts, consistency with sampled probe trajectories, consistency with traffic conservation (Kirchhoff's law), similarity of flows having close origins and destinations, positivity of traffic flows. A primal-dual algorithm is devised to minimize the designed functional, as the corresponding objective functions are not necessarily differentiable. A case study, on a simulated network and traffic, validates the feasibility of the procedure and details its benefits for the estimation of an LODM matching real-network constraints and observations

A linear programming based heuristic framework for min-max regret combinatorial optimization problems with interval costs

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems

Two new methods for solving the path‐based stochastic user equilibrium problem

In this paper, we present two new methods for the path-based logit stochastic user equilibrium problem, and investigate their convergence properties. First, a two level partial linearization method is proposed. Second, a dual method is developed. Both of these two methods use second order approximation of the objective function. Our novel methods are compared to Damberg's partial linearization method (Damberg, 1996), which is known to be one of the best performing methods. Numerical results on the Sioux Falls and Winnipeg networks show that, if properly scaled, our new methods can significantly improve the performance of Damberg’s method

Traffic matrix estimation in IP networks

An Origin-Destination (OD) traffic matrix provides a major input to the design, planning and management of a telecommunications network. Since the Internet is being proposed as the principal delivery mechanism for telecommunications traffic at the present time, and this network is not owned or managed by a single entity, there are significant challenges for network planners and managers needing to determine equipment and topology configurations for the various sections of the Internet that are currently the responsibility of ISPs and traditional telcos. Planning of these sub-networks typically requires a traffic matrix of demands that is then used to infer the flows on the administrator's network. Unfortunately, computation of the traffic matrix from measurements of individual flows is extremely difficult due to the fact that the problem formulation generally leads to the need to solve an under-determined system of equations. Thus, there has been a major effort from among researchers to obtain the traffic matrix using various inference techniques. The major contribution of this thesis is the development of inference techniques for traffic matrix estimation problem according to three different approaches, viz: (1) deterministic, (2) statistical, and (3) dynamic approaches. Firstly, for the deterministic approach, the traffic matrix estimation problem is formulated as a nonlinear optimization problem based on the generalized Kruithof approach which uses the Kullback distance to measure the probabilistic distance between two traffic matrices. In addition, an algorithm using the Affine scaling method is developed to solve the constrained optimization problem. Secondly, for the statistical approach, a series of traffic matrices are obtained by applying a standard deterministic approach. The components of these matrices represent estimates of the volumes of flows being exchanged between all pairs of nodes at the respective measurement points and they form a stochastic counting process. Then, a Markovian Arrival Process of order two (MAP-2) is applied to model the counting processes formed from this series of estimated traffic matrices. Thirdly, for the dynamic approach, the dual problem of the multi-commodity flow problem is formulated to obtain a set of link weights. The new weight set enables flows to be rerouted along new paths, which create new constraints to overcome the under-determined nature of traffic matrix estimation. Since a weight change disturbs a network, the impact of weight changes on the network is investigated by using simulation based on the well-known ns2 simulator package. Finally, we introduce two network applications that make use of the deterministic and the statistical approaches to obtain a traffic matrix respectively and also describe a scenario for the use of the dynamic approach