A new look at departure time choice equilibrium models with heterogeneous users

•A systematic and unified approach to Vickrey’s single bottleneck problem with heterogeneity is presented.•The equilibrium can be constructed from a linear programming (LP) problem.•The LP problem is analytically solvable exploiting the theory of optimal transport with a decomposition.•The essential condition for emerging equilibrium sorting patterns in the literature is the Monge property of the schedule delay functions. This paper presents a systematic approach for analyzing the departure-time choice equilibrium (DTCE) problem of a single bottleneck with heterogeneous commuters. The approach is based on the fact that the DTCE is equivalently represented as a linear programming problem with a special structure, which can be analytically solved by exploiting the theory of optimal transport combined with a decomposition technique. By applying the proposed approach to several types of models with heterogeneous commuters, it is shown that (i) the essential condition for emerging equilibrium “sorting patterns,” which have been known in the literature, is that the schedule delay functions have the “Monge property,” (ii) the equilibrium problems with the Monge property can be solved analytically, and (iii) the proposed approach can be applied to a more general problem with more than two types of heterogeneities.