Single machine lot scheduling to minimize maximum weighted completion time

The development of artificial intelligence is a significant factor in the surge in demand for micro-products. Consequently, optimizing production scheduling for micro-products has become crucial in improving efficiency, quality, and competitiveness, which is essential for the sustainable development of the industry. In micro-product manufacturing, it is common for manufacturers to receive customized orders with varying quantities and priority levels. This work focuses on situations where orders are processed in lots with unified capacity on a single machine. Each lot has the potential to accommodate multiple orders, and if necessary, any order can be split and processed in consecutive lots. Each order is characterized by its size and weight. The objective of the problem is to minimize the maximum weighted completion time. In order to investigate the differences in the calculation of completion times for split orders, two mixed-integer linear programming models are established, and the optimal characteristics of these problems are subsequently analyzed. Furthermore, in consideration of the inherent unpredictability of order arrival over time in practice, we also explore the potential of online versions of these problems and propose an online algorithm for online problems. Finally, the experimental results assess the efficacy of the proposed optimality rules and the online algorithm and derive several managerial insights.