Joint optimization of route and frequency with flexible rail pricing in a container intermodal

This paper focuses on how to optimize road-rail-combined routes and container-train-associated frequencies for effectively delivering freight demands from origins to destinations in a container intermodal network. Unlike common single-rate rail pricing contracts, we employ a flexible rail pricing (FRP) contract which depends on freight carrying volumes and average loading rates within the corresponding rail subnetwork. With the help of the formulated FRP requirements, a bi-objective integer programming model is developed for this problem, aiming to minimize the total freight transportation cost and maximize the total rail net profit. Further, we tackle the nonlinear constraints caused by FRP through the big-M method. The model, with the purpose of enhancing applicability and flexibility, is then extended to incorporate actual route selection rules. By using the ε-constraint method, the obtained bi-objective models are transformed into single-objective forms, which are addressed with state-of-the-art commercial solvers and heuristic strategies. Finally, we perform several different-sized numerical experiments to validate the efficiency and effectiveness of the proposed approach.