Flow-Balanced Scheduled Routing and Robust Refueling for Inland LNG-Fuelled Liner Shipping

Inland LNG-fuelled liner shipping is emerging as a significant trend, yet limited refueling infrastructure presents operational challenges. The complexity of inland navigation requires frequent speed adjustments to meet scheduled arrivals, which directly affects fuel consumption and refueling strategies. Additionally, imbalances in domestic and foreign trade container flows further increase operating costs for liner shipping companies. Given estimated weekly demands, considering navigational restrictions such as water depth and bridge clearance, as well as streamflow velocity, port time windows, empty container repositioning, port selection, speed adjustment, and uncertain fuel consumption, two novel models based on empty container arc variables and node variables are formulated, aiming to maximize voyage profit. These models are extended from divisible demand to indivisible demand cases. The explicit expression for the maximum fuel consumption under the worst-case speed deviation is derived, and an external linear approximation algorithm is proposed to linearize the nonlinear models while controlling approximation errors. Furthermore, the NP-hardness of the problem, the strict equivalence of the two modeling approaches, and the solution properties are proved. A case study of LNG-fuelled liner shipping on the Yangtze River shows the following: (1) for divisible demand, both models achieve optimal solutions within seconds, while for indivisible demand, the node-variable model outperforms the arc-variable model; (2) tactical strategies should be flexibly adjusted based on seasonal water depth, fuel prices, carbon taxes, speed deviations, and expected lock passage times; and (3) increasing fuel prices and carbon taxes generally reduce port calls and sailing speeds, suggesting that stricter fuel price and carbon tax policies can support the transition to green shipping. This study provides both theoretical guidance and managerial insights, supporting shipping companies in optimizing operations and promoting the development of sustainable inland shipping.