A Pareto-improving and revenue-neutral scheme to manage mass transit congestion with heterogeneous commuters

•The hybrid fare scheme (HFS) consists of a fare-reward and a uniform fare scheme.•We capture the heterogeneity in commuter’s arrival time flexibility interval.•The optimal hybrid fare scheme can be achieved with multiple solutions.•Individual trip costs reduction depends on the degree of the heterogeneity.•All commuters are no worse off and the transit operator is revenue neutral.