A comparative analysis of sequential and integrated optimization models for ship refueling operations

Maritime transportation underpins global trade and supply chains, with bunker fuel comprising a major share of voyage operating costs. Optimizing ship refueling is thus critical for improving energy efficiency and reducing expenses. A common sequential (two-stage) framework first derives a cost-minimal bunkering plan and then maximizes profit from cargo flows under that plan. Yet, this decoupled approach often yields suboptimal solutions, as refueling and container decisions are optimized separately. We develop an integrated one-stage model that simultaneously optimizes refueling and container flows on fixed liner services, comparing it to the two-stage model on stylized, industry-calibrated networks of 3, 8, and 15 ports. Results reveal consistent profit superiority for the integrated model, with gaps expanding alongside network scale and complexity (1.02% for 3 ports to 12.99% for 15 ports). Sensitivity tests on fuel-tank capacity, revenue levels, and local fuel price shocks affirm robustness, with gains amplifying under high demand or capacity constraints. Although the two-stage model provides modular transparency, the integrated formulation better balances fuel costs and cargo revenues, delivering substantial economic benefits in complex, constrained settings. Both models solve to optimality in seconds to minutes on standard hardware for tested instances, underscoring the integrated model's viability as a tactical decision tool. Overall, findings illuminate refueling optimization dynamics, quantify the inefficiency of bunker-first sequencing, and advise adopting integrated methods for larger networks.