# A mixed-integer linear programming method for time-dependent line planning in passenger railway systems

## TL;DR

This paper introduces a new method for optimizing train and passenger schedules in railways by considering changing travel demands over time.

## Contribution

A novel mixed-integer linear programming approach is developed to optimize time-dependent train and passenger planning.

## Key findings

- The proposed model efficiently integrates train operations and passenger choices in a directed graph.
- An extended time-dimension method transforms a non-linear problem into a solvable MILP model.
- The case study shows improved operational efficiency and better handling of time-dependent travel preferences.

## Abstract

This paper addresses a line planning problem (LPP) that simultaneously optimizes both train and passenger times in passenger railway systems, considering time-dependent origin-destination-period demand and passenger train choice. The problem is clearly and flexibly modeled in a physical infrastructure-based directed graph, which efficiently integrates the train operation choice and the passenger train choice. The problem is first formulated as a mixed-integer, non-concave, and non-linear programming model aimed at minimizing both the total operating cost of trains and the total travel cost of passengers. To solve the problem, an extended time-dimension method is proposed to transform the non-concave and non-linear model into a mixed-integer linear programming (MILP) model that can be solved using a commercial solver. Additionally, a set of simplification strategies is introduced to reduce the computational complexity while ensuring the global optimality of the linear model. A case study of a busy Chinese railway line demonstrates that the optimized time-dependent line plan enhances operational efficiency and accommodates the diversified travel preferences driven by time-dependent demand.