Combined Models with Hierarchical Demand Choices: A Multi‐Objective Entropy Optimization Approach

Abstract

This article proposes a multi‐objective optimization approach to the formulation of a number of equilibrium problems that typically arise in the transportation planning process. These fall into two classes: combined demand and network equilibrium problems, the latter here called performance‐demand equilibrium problems. The demand formulations are based on entropy maximization while the network equilibrium designs are modelled on Wardrop’s first principle. Both are fully compatible with models based on random utility maximization (multinomial and hierarchical logit). Given the entropy‐maximization aspect of the demand models and the use of symmetric cost functions in the networks, the multi‐objective formulations yield classical single‐objective convex optimization programs. In the past, many such problems have not been obtained deductively, their derivation being based rather on previous knowledge and the modeller’s intuition. Of particular interest, therefore, is the simple deductive method presented here for formulating new problems, one that can accommodate new choices such as departure time and transfer point for combined modes. This novel approach also facilitates a better interpretation of the model parameters. In addition, we suggest a calibration procedure that permits consistent estimation of the proposed model’s parameters.

Acknowledgements

This research was financed by the Pontificia Universidad Catolica de Chile. We are grateful to Juan de Dios Ortúzar for some useful suggestions.

Notes

1. Functions f_k (x), i = 1, 2,…,k, may also be known as objectives, criteria, value functions or costs, depending on the specific application.

2. In the context of demand models, ‘sequential’ refers to decisions made at different hierarchical levels, whereas ‘simultaneous’ decisions are those made at the same level in a hierarchical structure (a decision tree). In the context of performance‐demand equilibrium in transportation networks, the word ‘simultaneous’ indicates consistency among the levels of service in the system and flow values (link flows and O‐D trips) for each stage in a particular transportation problem (also known as combined models).