Prescriptive analytics of electric bus battery allocation optimization based on the Plackett-Luce model

Hybrid charging stations for electric buses (EBs) face a critical decision in choosing between battery swapping and plug-in charging. This study introduces a prescriptive analytics approach, termed “Predict, then Optimize” to address this challenge. The approach first predicts the state of charge (SOC) of EBs and then optimizes battery allocation. Given the inherently ranking-based nature of this problem, where allocation decisions depend on the ranking of SOC values, this study extends the framework into a semi-smart “Predict, then Optimize” (semi-SPO) paradigm, incorporating ranking characteristics into the SOC prediction model. The Plackett-Luce (PL) probability model is employed to represent the ranking distribution of SOC values in a continuous and differentiable manner. Acknowledging the significance of higher-ranked SOC values, a top-k strategy is implemented to prioritize critical rankings, thereby enhancing both prediction accuracy and computational efficiency. An artificial neural network (ANN) is utilized to predict SOC, with the Kullback-Leibler (KL) divergence between the true and predicted PL ranking distributions serving as the loss function. The proposed model is validated through a real-world case study in Nanjing, China. Compared to a conventional neural network, it achieves a significantly lower regret across battery allocation decisions, with the optimal configuration reducing regret from 0.373 to 0.047—representing an 87% improvement in allocation reliability under realistic constraints.