Railway line planning with passenger routing: Direct-service network representations and a two-phase solution approach

The railway line planning problem (LPP) plays a crucial role in determining the quality of services provided to passengers, as well as operation costs borne by railway companies. In periodic railway LPPs, it is common to consider passenger transfers between train lines to realize a general passenger travel cost setting in the railway system. While detecting passenger transfers requires incorporating passenger routing into mathematical formulations, thereby significantly complicating the problem. Studies on transfer-included LPPs are generally based on the Change&Go network that is constructed based on a pre-given line pool, which however is usually non-exhaustive due to computational intractability. To efficiently include passenger transfers in large-scale railway LPPs, this paper proposes a novel extended direct-service network representation of LPP, where lines are dynamically generated within the optimization process, and part of passenger transfers between lines can be precisely captured without the need for explicit modeling of passengers' distribution on specific lines. A two-phase solution approach based on the representation is designed. The first phase formulates LPP with part of transfers as a path-based service network design model, solved using a branch-price-and-cut algorithm. The second phase conducts a neighborhood search around the first-phase solution to seek better ones when considering all passenger transfers. Numerical results showcase the good performance of the two-phase solution approach. It delivers optimal solutions in 18 out of 24 test instances for a small case and achieves optimality gaps within 2.85% across all small instances. The large case study of China's Shandong high-speed railway network whose line pool size reaches millions demonstrates the scalability of the approach and its advantage over the traditional Change&Go method with partial line pools and an exact model developed in the paper.