## 1 Introduction  
Over the past decades, the accelerated development of high-speed railway (HSR) is the outcome of increasing travel demand and social economic development, which poses huge challenges to railway infrastructure managers and operators to produce the most effective operating plans [1, 2, 3]. In practical operation, railway planning generally follows a hierarchical top-down approach, including the strategic level (network design and line planning), tactical level (timetabling, train routing, rolling stock scheduling, and crew scheduling), and operational level (real-time management) [1, 4, 5, 6], where all stages should be coordinated for stable and sustainable operation. In these stages, line planning is the stage to match supply and demand by determining train routes, frequencies, stop patterns, fleet sizes, and capacities, where stop planning is the critical and significant issue to provide the final services for passenger demand [7]. It determines the spatial-temporal distribution of travel services and impacts many aspects including service level, operation organization, capacity utilization, etc. It should not only match the travel demand of passengers, but also involve coordination among train lines for operation safety. The stop planning problem (SPP) has been studied for a long time. There have been many studies that focus on impact factors, setting principles, optimization models, and algorithms for the SPP. Since the SPP is the critical phase in line planning and related to the subsequent stages in railway planning, it is generally studied combined with other transportation plans. The related studies can be mainly classified into three categories. The first category is SPP optimization with line planning, where the final stop plan is a part of the optimized line plan. [PERSON] et al. [1] incorporated time-varying demand into line planning by trading off the benefits of railway companies and passenger demand. The approach was formulated by a bi-level programming model based on Stackelberg game theory that was solved under the framework of Simulated Annealing Algorithm (SAA). Then, the method was extended and applied to daily line planning with consideration of the balance of system cost and operation stability [8]. Given different types of train lines and stations, [PERSON] et al. [9] optimized the halting stations of lines by a mixed integer linear programming model based on a line-event graph. With heterogeneous demand, [PERSON] et al. [10] developed an analytical model to integrate different stop patterns associated with frequencies for a transit route based on capacity and fleet size constraints, in order to optimize the user cost and supplier cost. [PERSON] and [PERSON] [11] integrated the line planning, cyclic timetabling, and routing decisions of passengers to realize the minimization of the total engine time and total passenger travel time, which was solved by a cross-entropy metaheuristic. [PERSON] et al. [12] constructed a bi-level programming model for line planning that was solved by a two-stage heuristic approach, where the higher-classified stop trains were generated from the possible line set and the lower-classified stop trains were added by adding intermediate stops and increasing frequencies of the existing lines. More related studies include [PERSON] et al. [13], [PERSON] et al. [14], [PERSON] et al. [15], and [PERSON] et al. [16; 17; 18]. The second category is SPP optimization with train scheduling to satisfy train line coordination and operation safety. [PERSON] et al. [6] studied the collaborative optimization of train scheduling and train stop planning for a single-track HSR corridor to realize the minimization of the total dwelling time and total delay between the real and expected departure times. [PERSON] et al.