Inverse optimization with endogenous arrival time constraints to calibrate the household activity pattern problem

A parameter estimation method is proposed for calibrating the household activity pattern problem so that it can be used as a disaggregate, activity-based analog of the traffic assignment problem for activity-based travel forecasting. Inverse optimization is proposed for estimating parameters of the household activity pattern problem such that the observed behavior is optimal, the patterns can be replicated, and the distribution of the parameters is consistent. In order to fit the model to both the sequencing of activities and the arrival times to those activities, an inverse problem is formulated as a mixed integer linear programming problem such that coefficients of the objectives are jointly estimated along with the goal arrival times to the activities. The formulation is designed to be structurally similar to the equivalent problems defined by Ahuja and Orlin and can be solved exactly with a cutting plane algorithm. The concept of a unique invariant common prior is used to regularize the estimation method, and proven to converge using the Method of Successive Averages. The inverse model is tested on sample households from the 2001 California Household Travel Survey and results indicate a significant improvement over the standard inverse problem in the literature as well as baseline prescriptive models that do not make use of sample data for calibration. Although, not unexpectedly, the estimated optimization model by itself is a relatively poor forecasting model, it may be used in determining responses of a population to spatio-temporal scenarios where revealed preference data is absent.