Tour-Based Entropy Maximization Formulations of Urban Freight Demand

One of the major obstacles to urban freight demand forecasting is the lack of aggregate-level models that consider commercial vehicle tours. To bridge the gap, this paper describes two variants of entropy maximization formulations that are aimed to estimate the tour flows of commercial vehicles given the number of trips produced by or attracted to each node, and the impedance to travel. The first and second order conditions are derived to gain insight from the entropy maximization formulations. The first-order conditions show that the flow of commercial vehicles traveling along a given tour is a function of the Lagrange multiplier associated with the number of trips produced by each node along that tour, and the tour impedance. The second-order conditions indicate the convexity of the formulations. A case study in the Denver metropolitan area shows the efficiency of this approach: the estimated tour flows closely match the observations with the mean absolute percentage error as 6.71% and 6.61% for the two formulations respectively. Based on the findings, the paper discusses two possible ways of applying this approach to forecast urban freight demand.