A Mixed Integer Linear Programming Formulation for Subway Timetable Optimization Problems with Passenger Waiting and Energy Saving Objectives

 Subway timetabling problems are important but difficult discrete optimization problems that are usually solved with strict computational time requirements in order to minimize passenger waiting time while maximizing energy-saving profits. To obtain high-quality solutions, a promising direction is to formulate the problems as Mixed Integer Linear Programming (MILP) problems, thereby taking advantage of widely available MILP methods such as Branch-and-Cut (B&C). However, based on existing MILP formulations, the computational efforts required by MILP methods are high because of the large numbers of decision variables and constraints. In this paper, we develop a novel MILP formulation for optimizing subway timetables. In the formulation, train arrival times are innovatively formulated by using decision variables of two types: continuous variables for formulating energy-saving profits and binary variables for formulating passenger waiting time. Compared to existing formulations, the proposed formulation has far fewer decision variables and constraints. Numerical testing results show that high-quality solutions are efficiently obtained using B&C based on the proposed formulation in reasonable time. Thus, the proposed formulation paves the way for efficiently solving subway problems using MILP methods.