## The Continuous Dynamic Network Loading Problem: A Mathematical Formulation and Solution MethodJia Hao Wu, Yang Chen, Université de Montréal. Centre de recherche sur les transports, Michael Florian, Centre for Research on Transportation (Montréal, Québec) Université de Montréal, Centre de recherche sur les transports, 1995 - Mathematical optimization - 56 pages The continuous dynamic network problem aims to find, on a congested network, temporal network flows, arc travel times, and path travel times given time-dependent path flow rates for a given time period. This problem may be considered as a subproblem of a temporal (dynamic) traffic assignment problem. This paper studies this problem and formulates it as a system of functional equations. For computational purposes, the authors develop a polynomial approximation which is almost equivalent to the original formulation on a set of finite discrete points. The approximation formulation is a finite dimensional system of equations which is solved as an optimization problem. The paper includes several numerical examples to illustrate the approach developed. |

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6-system algorithm analytical approach approximation approximation formulation arc flows arc travel arrives assume assumption Based basic beka clear computational condition is satisfied congestion consideration considered context continuous dynamic network defined definition departing departure depends develop discrete points discuss dynamic network loading efficient examples existing FIFO condition Figure finite dimensional flow on arc formulation functional equations given going along path head of arc hk(t holds implies important increasing indicates interesting later mathematical formulation method network loading problem nodes nonlinear equations nonsmooth numerical Oakliak(t obtain optimization problem otherwise parameters path flow rates path k path travel possible presented Proof properties queuing reality reduce respected simulation Sk(t smooth solution solve stability condition structure system of functional t e Tg Table tak(m temporal Theorem tik(t time-dependent tion traffic assignment TRANSPORTATION travel time functions trigonometric polynomial valt variables vector of arc