A two-stage stochastic optimization framework for integrated passenger-freight bus transport under time-varying capacity constraints

This paper develops an optimization framework for integrated passenger-freight bus transport (IPFBT), which leverages the spare capacity of public bus systems to enhance sustainable urban freight delivery. We formulate a two-stage stochastic programming model that jointly optimizes the strategic location of passenger-freight integration stations and operational-level freight routing under time-varying passenger demand. The model innovatively introduces a link availability matrix to characterize dynamic capacity constraints under the passenger-priority principle, and designs temporal shift and spatial shift strategies to address capacity shortage during peak periods. To solve this large-scale, uncertainty-driven problem, we design a decomposition-based algorithm combining a L-shaped framework with Lagrangian relaxation. The master problem is efficiently solved via a greedy set-cover heuristic, while subproblems are decomposed into parallelizable single-commodity shortest path problems. Adaptive multiplier updates and feasibility restoration mechanisms are incorporated to ensure convergence and solution quality. A real-world case study based on the Shijingshan District bus network in Beijing demonstrates the model’s effectiveness and computational scalability. The study reveals phenomena such as obvious economies of scale and “capacity paradox” in the system, and provides implementation recommendations including phased deployment strategies for urban managers.