An ADMM-based decomposition approach for flexible blending in port operations

Billions of tons of coal, ore, and grain are shipped worldwide through ports each year, making efficient dry-bulk handling within ports critical to industrial production, socioeconomic development, and livelihoods. Yet, due to mismatches between supply and demand and the heterogeneity of bulk materials, port dry-bulk operations frequently suffer delays from cargo shortages. Flexible blending, producing contractual cargo in real time during vessel loading by mixing available materials, has emerged as a promising remedy. However, it introduces substantial operational challenges by embedding production-level decisions into the loading process and exacerbating scheduling complexity. To address these challenges, we propose an optimization model that jointly integrates production planning and reclaiming scheduling within a unified decision framework. The problem is formulated as a coupled space-time network-based integer programming model, which avoids intractable sequencing variables and big-M constraints, while allowing complex logical requirements to be incorporated during network construction. The model exhibits a decomposable structure that isolates computational difficulty in a small set of coupling constraints. Leveraging this structural property, we propose a dual decomposition solution method. With the alternating direction method of multipliers, we linearize and decompose the monolithic problem into a set of least-cost path subproblems. For solving subproblems efficiently, we design a customized bidirectional label-setting algorithm with enhancements including label elimination, dominance relaxation, and a heuristic-based warm start. We test our approach through experiments based on varying-scale instances and the layout of a major coal loading terminal in China. The results demonstrate that the proposed method efficiently produces provably near-optimal solutions even for large-scale scenarios. Managerially, our experiments show that by mitigating cargo shortages and alleviating peak-period pressures, flexible blending not only enhances port efficiency but also generates broader benefits for bulk supply chains and beyond.