Demand-driven multi-period dual-cost rebalancing for bike-sharing systems

Variable and uncertain bike-sharing demand presents significant scheduling challenges for urban multimodal systems. This paper develops a multi-period dual-cost optimisation framework that links variable fleet-size to rebalancing, employing a distributionally robust optimisation method to manage demand variability. Firstly, we construct a model without fixed resource constraints, measuring service efficiency through the demand gap rate and a fairness criterion, dynamically identifying zonal demand priorities while minimising dynamic dual costs. Subsequently, we introduce a Wasserstein distance-based distributionally robust optimisation framework with expectation constraints for demand uncertainty and develop an adaptive distributionally robust optimisation with dual expectation by leveraging the linear decision rule approximation for multi-period decision-making. This approach captures the nonlinearity of demand and reflects distributional characteristics. Compared to deterministic and stochastic models, the proposed method improves robustness by 54% and 53%, respectively, and reduces conservatism by 16% compared to robust optimisation. In addition, the adaptive method we proposed further reduces the conservativeness in large-scale instances compared with the traditional adaptive distributionally robust optimisation framework that utilises linear decision rules. It accurately identifies high-demand spatiotemporal hotspots and balances average performance and robustness. The method is also scalable, with parameters that can be adjusted to match the traffic and user behaviour of different cities, thereby enhancing resource utilisation and user experience.