Integrated optimization of train formation plan and rolling stock scheduling under Time-dependent demand

Abstract

Both train formation plan (TFP) and rolling stock scheduling (RSS) have significant impacts on train operation. However, traditional sequential optimization could lead to infeasible RSS for the predetermined TFP, especially when the number of available rolling stocks is limited. This paper focuses on integrally optimizing TFP and RSS, where each train service can utilize different train formations and multiple turnaround operations occur in the depot. The key to solving the problem is to determine how many formations are needed for each train service and how train services are connected by rolling stocks. The multiple turnaround operations include immediate turnaround, turnaround with entering/exiting depot operations, and turnaround with coupling/decoupling operations. Considering the fixed costs and operating costs, we propose a multi-objective mixed-integer nonlinear programming (MINLP) model to minimize the number of rolling stocks (NR), the number of formations (NF), and the number of coupling/decoupling operations (NC) based on the time–space network. The model is further reformulated into a single-objective mixed-integer linear programming (MILP) model by the linearization method and fuzzy programming, and the reformulated model can be effectively solved by the MILP solver CPLEX. We test the model on realistic instances of the Batong Line in Beijing Subway Network to verify its effectiveness. The results demonstrate that the integrated model effectively solves the shortage of rolling stocks when the number of available rolling stocks is limited. Furthermore, in the integrated model, the decrease in the number of the rolling stocks and coupling/decoupling operations are at the cost of more formations used to balance the conflicts among objectives.