Approximating the likelihood of historical airline actions to evaluate airline delay cost functions

Delay cost functions that quantify the cost of delay to airlines are essential to air traffic management research. Seventeen delay cost functions from previous research are evaluated with airline actions in Airspace Flow Programs. Airlines are assumed to solve a minimum cost perfect matching problem when matching flights to slots. Unobserved aspects of airline costs are accounted for by adding a noise term to the cost functions. The goal of this research is to find the cost function and corresponding noise parameters that maximize the likelihood of airline actions during 32 Airspace Flow Programs in the summer of 2006. A heuristic is developed that finds cost noise parameters that maximize an approximation of the log-likelihood of the airline actions. When applied to sample estimation problem instances generated by solving linear programming problems with known noise parameters, the heuristic can more accurately estimate noise parameters than a simple simulation-based approach. Validation efforts based on synthetic airline action data generated with known delay cost functions and noise parameters demonstrate that the heuristic is in many cases able to correctly identify as most likely the delay cost function that was in fact used to generate the synthetic data. However, the heuristic also under-estimates the magnitude of the cost noise variance on these estimation problem instances. Delay costs that are proportional to the length of delay, but with larger proportionality constants for flights bound for hub airports, maximize the approximation of the log-likelihood of the historical airline actions. The estimated standard deviations of the cost noise, expressed as a fraction of the average assignment cost for the historical matchings, ranged from 0.1 to 0.7 for cost functions that achieved relatively large approximate log-likelihoods.