Joint optimization of pricing and routing for flexible bus services under stochastic demand

Flexible Bus (FB) systems have emerged as a promising solution for connecting transport hubs with urban areas. However, existing studies often decouple tactical pricing from operational routing decisions, which can lead to suboptimal system performance. To address this issue, we propose an integrated optimization framework that jointly optimizes pricing, maximum detour limits, and vehicle routing to maximize platform profit under demand uncertainty. At the tactical level, we design a detour-based pricing rule in which the fare is dynamically adjusted based on the realized detour. Consequently, we utilize a logit framework to capture the passenger service choice behavior based on both monetary incentives and detour-based service guarantees. At the operational level, routing decisions are formulated over a hyper-dimensional space–time–state network, and interact with the pricing mechanism to maximize profit. To solve this model, we propose the Iterative Pricing-Routing Algorithm (IPRA), which integrates three main components: (i) grid search for tactical parameters, (ii) Sample Average Approximation (SAA) for stochastic demand, and (iii) an Alternating Direction Method of Multipliers (ADMM)-based algorithm to optimize routing. Numerical studies on the Solomon dataset demonstrate that the detour-based pricing approach increases average platform profit by 45% compared to uniform pricing. Moreover, the results confirm that the algorithm can achieve near-optimal solutions within a reasonable timeframe. Finally, a large-scale case study using Shanghai Hongqiao hub data is conducted to evaluate the performance of the proposed solution algorithm in practical applications.