Mode choice in strategic freight transportation models: a constrained Box–Cox meta-heuristic for multivariate utility functions

Abstract

Modal choice models used for freight transportation studies covering inter-regional or international areas are difficult to set up because of the dearth of information about explanatory factors. While cost and transit time are known as being important explanatory variables, they are generally correlated to each other, and their coefficient computed with a Logit model can have unexpected signs. 

Box-Cox transformations (BCT) of the independent variables can help to overcome this problem. If solutions to identify the BCT parameter  that maximises the likelihood of a model are well known, the process is not straightforward once it must respect the constraints that the variables’ coefficient estimators take the expected signs.

This paper presents a shotgun hill climbing meta-heuristic with backtracking capabilities, able to quickly identify Box-Cox λ parameters to use when multiple variables must be transformed. The algorithm appears to be efficient and effective and produces stable and statistically valid solutions.

KEYWORDS:

• Freight transport
• modal choice
• Multinomial Logit
• Box–Cox transforms
• hill climbing
• heuristic

Acknowledgment

The author would like to thank Michel Beuthe for the many constructive discussions during the development phase of this research. He also thanks the three anonymous reviewers who carefully read the submitted manuscript and suggested several improvements.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See Appendix 1 for a description of the content of each category.

2 The ‘spatial distribution’ of the valid solutions (spread and position of the dots) is very different for each group of commodities. Unfortunately, there is no place to publish the 10 figures in this paper, but they are available on request.

3 Note that for values of lambda(s) ≥$|2|$ numerical errors in the computation of the (approximated) Hessian needed during the estimation of the logit sometimes occur.

4 This can be implemented as a recursive function (called from itself several times, once for each ‘dimension’ of the combination of λ's). See Appendix 2.

5 While the heuristic can be applied to the univariate case, its usefulness is limited.

6 One matrix per NST/R, each one containing the aggregated demand for the 3 modes.