A Tour-based Entropy Maximization Formulation of Urban Commercial Vehicle Movements

A Tour-based Entropy Maximization Formulation of Urban Commercial Vehicle Movements


Q Wang, J HolguĂ­n-Veras, Rensselaer Polytechnic Institute, US


This paper discusses a novel formulation of a tour based model based on entropy maximization. This is the first closed-form model of commercial vehicle tours reported in the literature.


In recent years, one of the unique features of urban commercial vehicle movements, i.e., trip chaining behavior, has been receiving more attention. As has been found by some researchers, commercial vehicles tend to make long tours composed of multiple trips, and these trips made on a tour are interrelated according to the underlying logistic decisions. This attribute breaks down the typical assumption of the traditional four-step approach that postulates that trips are made independently and trips between an origin-destination (OD) pair are only related to the zonal attributes and the travel cost of the corresponding OD pair. Therefore, new paradigms of freight transportation models considering tours are needed for freight demand forecasting.

In order to address this feature, a handful of tour-based models have been formulated for urban freight movements. Most of these formulations modeled commercial vehicle tours in the disaggregate level: tours are simulated either by solving a vehicle routing problem or by using the probabilities generated by a set of discrete choice models. These disaggregate models, on one hand, are capable of capturing the underlying decision making processes behind vehicle operations. On the other hand, they have some limitations that hurdle their applications, particularly in large cities. These include the expensive procedures required for collecting tour diary data, the enormous time consumed for computation, and the strong committal to the specific assumptions made for logistic operations. In contrast to the disaggregate approach, the aggregate tour-based approach can be used as an alternative way to forecast urban freight movements, considering its less consumption of input data, less requirement of computational time, and less committal to behavioral assumptions. However, little contribution has been made in this direction.

This research is based on the use of entropy maximization. The approach assumes that when no enough information is available about the behaviour of individual commercial vehicle tours, the individual tours are equally probable unless information is available to the contrary; and of all the feasible ways to distribute tours to the road network, the most probable ones would be those that can be generated in the greatest number of ways under the constraints of the known aggregate information. Based on this assumption, the resulting entropy formulations are aimed to find the most likely set of tours that meet the system?s constraints such as the trip production of each node, the trip attraction of each node and the total costs of the entire network.

For a tour made in a network, its cost is composed of the travel cost in each trip and the handling cost in each stop. The two types of cost components may have the same or different effects on tours generated. Considering this, two entropy maximization formulations are introduced. The first formulation includes only one cost constraint, the total cost which is the summation of the total travel cost and the total handling cost, while the second one uses the total travel cost and the total handling cost separately as the cost constraints.

Two types of conditions, the first and second order conditions, are derived to gain insight from the entropy maximization formulations. The first-order conditions (Karush-Kuhn-Tucker conditions) show that the number of tours in each node sequence is a linear combination of the Lagrange multipliers associated with the trip productions and attractions along that node sequence, and the tour cost factors. The second-order conditions indicate that the formulation is a convex program with a convex objective function and a set of linear constraints. Given the convexity, this paper chooses the primal-dual interior method for optimization programs with convex objectives (PDCO) as the solution algorithm, considering its efficiency in solving large-scale entropy problems.

The proposed formulation is applied to the Denver region. The estimation results indicate that the entropy maximization formulation is a feasible and efficient way to accommodate the commercial vehicle tours into the aggregate freight demand modeling. The estimated tours closely match the observed ones with the mean absolute percentage error (MAPE) as 6.71% for the first formulation and 6.61% for the second formulation. As the consequence of the good match obtained, the estimated tour length distribution resembles the observed one as well.


Association for European Transport