Highlights • We propose an improved learning-and-optimization method that needs less information to address the commuting congestion of train stations located in central business district. • Use inexact information to address a bi-objective transportation demand management problem. • We rigorously prove that the convergence rate of the improved learning-and-optimization method is exponential. • The proposed method is applicable to other general practical problems.

Abstract This study proposes an improved learning-and-optimization train fare design method to deal with the commuting congestion of train stations at the central business district (CBD). The conventional learning-and-optimization scheme needs accurate boarding/alighting demand to update the train fare in each trial. However, when congestion happens, the observed boarding/alighting demand will be larger than the actual boarding/alighting demand due to the delays and the longer dwelling time. Thus, the actual boarding/alighting demand is not available in practice. The improved algorithm deals with this issue by using inexact and less information to determine the new trial fare during the iteration. Namely, the improved method bypasses the conditions that may lead to biased results so as to significantly enhance the reliability of the learning-and-optimization method. The simplified algorithm also makes this method more practical. The convergence property of the proposed algorithm is rigorously proved and the convergence rate is demonstrated to be exponential. Numerical studies are performed to demonstrate the efficiency of the improved learning-and-optimization method.