A Convex Formulation of the Multi-Commodity Dynamic Traffic Assignment

TL;DR

This paper introduces a convex formulation for multi-commodity dynamic traffic assignment on the Cell Transmission Model, enabling efficient optimal control of traffic flow with multiple destinations using convex relaxations.

Contribution

It extends the single-commodity convex relaxation approach to multi-commodity scenarios, allowing for multiple destinations and commodity-specific control policies.

Findings

Established a tight convex relaxation for multi-commodity DTA on CTM.

Demonstrated the convexity of the optimal control problem with commodity-specific demand and supply functions.

Enabled tractable traffic management with variable speed limits and dynamic routing for multiple commodities.

Abstract

We consider a multi-commodity Dynamic Traffic Assignment (DTA) problem formulated as a network flow control problem on the Cell Transmission Model (CTM). The objective is to design optimal control policies using variable speed limits, ramp metering, and dynamic routing to regulate traffic evolution over time on a given limited-capacity transportation network. Even simple instances of DTA problems on the CTM are known to give rise to non-convex optimal control formulations. Nevertheless, a single-commodity DTA formulation has recently been proposed that admits a tight convex relaxation, thereby enabling tractable optimal control synthesis. The single-commodity formulation, however, is structurally restrictive, as it effectively allows only a single destination. To address this limitation, we develop a multi-commodity CTM model in which each commodity is associated with potentially distinct sets of off-ramps. By extending the convexification approach developed for the single-commodity case, we establish a tight convex relaxation of the multi-commodity DTA problem on the CTM model. This relaxation relies on concave, commodity-specific demand functions and concave aggregate supply functions for every cell, which ensure convexity of the resulting optimal control problem. Our proposed formulation requires commodity-dependent implementation of variable speed limits and dynamic routing policies.