Lexicographic bi-objective scheduling under budget constraints in distributed production-transportation-assembly systems

In response to the demand for integrated and cost-efficient production systems, this study addresses a lexicographic bi-objective scheduling problem with budget constraints in distributed production–transportation–assembly flow shop systems. The first objective is to minimise makespan under heterogeneous resource and budget limitations across multiple sites; the second is to minimise total transportation and assembly costs without exceeding the optimal makespan. This yields a two-stage sequential scheduling framework. In Stage 1, a reduced mixed-integer linear programming (MILP) model is first solved to determine the minimum feasible budget, followed by three alternative formulations – Manne-based MILP, position-based MILP, and constraint programming (CP) While exact models solve small instances, they are intractable for realistic scales. To enhance scalability, we develop logic-based Benders decomposition (LBBD) and branch-and-check (BCH) variants with bound-strengthening strategies. In Stage 2, MILP, CP, and LBBD methods minimise costs subject to the makespan. Here, a CP warm start is embedded in LBBD to accelerate convergence. Computational experiments show that decomposition-based methods substantially outperform monolithic models. In Stage 1, LBBD achieves clear advantages over MILP and CP; in Stage 2, LBBD reduces costs by 4.04% and 13.26% on small and large instances, respectively. Sensitivity analyses further provide managerial insights for budget-aware scheduling in distributed networks.