618 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 14, NO. 2, JUNE 2013 Estimation of Dynamic Origin–Destination Matrices Using Linear Assignment Matrix Approximations Tomer Toledo and Tanya Kolechkina Abstract—This paper presents a general solution scheme for the problem of offline estimation of dynamic origin–destination (OD) demand matrices using traffic counts on some of the network links and historical demand information. The proposed method uses lin- ear approximations of the assignment matrix, which maps the OD demand to link traffic counts. Several iterative algorithms that are based on this scheme are developed. The various algorithms are implemented in a tool that uses the mesoscopic traffic simulation model Mezzo to conduct network loadings. A case study network in Stockholm, Sweden, is used to test the proposed algorithms and to compare their performance with current state-of-the-art methods. The results demonstrate the applicability of the proposed methodology to efficiently obtain dynamic OD demand estimates for large and complex networks and that, computationally, this methodology outperforms existing methods. Index Terms—Assignment matrix, dynamic traffic assignment (DTA), origin–destination (OD) matrix estimation. I. I NTRODUCTION I N RECENT years, there have been significant advances in the development of dynamic traffic assignment (DTA) and traffic simulation models, which predict time-dependent traffic conditions on a road network. An important input to these models is the demand for travel, which is commonly represented by origin–destination (OD) demand matrices. Col- lecting OD information directly by conducting surveys is time consuming and cost expensive. Moreover, the measurements may quickly become outdated. Therefore, OD demand matrices are commonly estimated using traffic counts, collected from the links of the network, and effectively combined with available OD information (e.g., derived from direct measurements or from previous estimates). OD estimation methods for the static problem (e.g., [1]–[5]) predict trip rates over a long period of time (such as a peak period), within which conditions on the network are assumed Manuscript received June 17, 2012; revised October 1, 2012; accepted October 19, 2012. Date of publication November 15, 2012; date of current version May 29, 2013. This work was supported in part by the Israel Science Foundation under Grant 1218/07, by the Israel Ministry of Industry, Trade and Labor, and by the European Commission through the MULTITUDE project European Cooperation in Science and Technology (COST) Action TU0903. The Associate Editor for this paper was L. Li. T. Toledo is with the Faculty of Civil and Environmental Engineer- ing, Technion—Israel Institute of Technology, 32000 Haifa, Israel (e-mail: toledo@technion.ac.il). T. Kolechkina was with the Technion—Israel Institute of Technology, 32000 Haifa, Israel. She is now with the Department of Civil and Environmen- tal Engineering, Carlton University, Ottawa, ON K1S 5B6, Canada (e-mail: Tanya.kolechkina@gmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TITS.2012.2226211 to be stationary. Dynamic OD estimation (DODE) models (e.g., [6]–[9]) relax the assumptions of stationary demand, represent traffic dynamics, and incorporate stochasticity in these ele- ments. The estimated OD matrices are, therefore, more suitable as inputs to DTA and traffic simulation models. While the problem formulation has been well established in the literature, there is still a need for efficient algorithms for its solution in large-scale congested networks. This has been demonstrated, for example, in a recent work by Cipriani et al. [10]. The authors presented a modification to the simultaneous pertur- bation stochastic approximation (SPSA) algorithm, which is a state-of-the-art solution approach to the DODE problem. Their algorithm required 15 hours to estimate an OD demand with four time slices for the Calgary network (734 links, 221 nodes, and 77 centroids). This paper presents a general solution scheme for the DODE problem. A critical construct in DODE is an assignment matrix, which maps OD demand flows to traffic counts at sensor locations. In congested networks, the assignment proportions depend on the unknown time-dependent OD demands. The methods proposed in this paper are based on the use of linear approximations of the assignment matrix in the optimization iterations. Several iterative algorithms, which are based on this scheme, that differ in the search direction they use are devel- oped. The algorithms are tested using the mesoscopic traffic simulation model Mezzo [11] on a network in the Stockholm area. The case study demonstrates the computational efficiency of these algorithms compared with current state-of-the-art ap- proaches for large-scale networks. The rest of this paper is organized as follows. In the follow- ing section, the DODE estimation problem is mathematically formulated. Algorithms for the solution of this problem, which have been proposed in the literature, are presented in Sec- tion III. The solution scheme proposed in this work is presented in Section IV. The details of this algorithm and its implementa- tion are presented in Sections V and VI, respectively. A case study demonstrating the proposed algorithms is presented in Section VII. Finally, conclusions are presented in Section VIII. II. PROBLEM FORMULATION Consider a transportation network represented by a directed graph G(C, L), where C is a set of nodes, and L is a set of links. L  ⊆ L is a subset of links equipped with sensors. The OD matrix X = {x nr } defines the demand for travel for each OD pair n ∈ N in each departure time period r ∈ R. N and R are the number of OD pairs and departure time periods, re- spectively. The data available for the demand estimation include 1524-9050/$31.00 © 2012 IEEE