Semi-flexible transit service optimization considering scenario-based demand fluctuations

Semi-flexible transit, integrating fixed-route and on-demand services, offers a demand-adaptive and cost-effective alternative for public transit users, particularly in low-demand conditions. Despite the growing interest in this system, existing approaches have failed to develop comprehensive optimization methods for managing demand fluctuations across distinct scenarios, thereby significantly constraining operational adaptability in semi-flexible transit services. To address this research gap, we propose a scenario-based optimization model that jointly determines the fleet size and master routes at the tactical level as well as sub-routes at the operational level. The objective is to minimize travel costs while ensuring service feasibility under varying passenger demand scenarios, accounting for constraints such as travel time, state changes, time windows, and route consistency. Then, an Augmented Lagrangian Relaxation under Alternating Direction Method of Multipliers (ALR-ADMM) decomposition solution framework is introduced to decouple the proposed integrated problem into three sub-problems, namely master route, sub-route and service planning problems. Numerical experiments on the Sioux-Falls network validate the proposed model and solution approach, achieving a 94.93 % reduction in computation time while maintaining an average optimality difference of 0.57 % compared to the Gurobi optimizer. Sensitivity analysis further examines the effects of vehicle capacity limits, penalty parameters, and demand stop selection, revealing their impact on computational efficiency and operational costs. The applicability of our approach is further assessed through a real-world case study on the West Jordan network, which provides evidence of the ALR-ADMM-based algorithm in terms of both solution quality and computational efficiency. Our findings illustrate the feasibility and potential of the proposed model and algorithm in navigating both the tactical and operational scheme of semi-flexible transit within modern urban transit systems.