Unrelated parallel batch processing machine scheduling with time requirements and two-dimensional packing constraints

In various complex manufacturing systems such as wafer fabrication, ceramic processes, and additive manufacturing, a single machine often performs multiple distinct jobs simultaneously, requiring non-overlapping positions in the 2D-plane. As the demand for manufacturing with two-dimensional (2D) packing constraints continues to rise, there is an urgent need to address the integrated scheduling of batch processing machines (BPMs) and 2D packing. However, only a limited number of studies have explored BPM scheduling problems with 2D packing constraints. In this study, we focus on the scheduling of unrelated parallel BPMs to process jobs with non-identical sizes, unequal release dates, and due dates. To model the machine's capacity, we move beyond the traditional one-dimensional knapsack approach and consider it as a 2D rectangle, taking into account the 2D packing constraints. It is crucial to ensure that no overlapping or stacking occurs within the machine's capacity under these constraints. To address this problem, we propose a mixed integer linear programming model as the initial approach. Additionally, we develop an adaptive large neighborhood search (ALNS) heuristic specifically designed for handling large instances. The ALNS incorporates eleven removal operators and seven insertion operators, offering a comprehensive exploration and improvement of solutions. To tackle the packing procedure, we adopt two distinct packing patterns, namely Skyline and Open Space. These patterns play a significant role in optimizing the packing of jobs within the machine's capacity. Furthermore, we conduct extensive computational experiments, generating 48 categories of instances with varying job attributes. These experiments serve to compare the performance of the two packing patterns and validate the effectiveness of our ALNS approach. In conclusion, our research addresses the critical gap in BPM scheduling problems with 2D packing constraints. By introducing a mixed integer linear programming model and developing the ALNS heuristic with diverse operators, coupled with the adoption of different packing patterns, we present a comprehensive approach to tackle this complex problem. Computational experiments are conducted to compare packing patterns and verify the performance of our ALNS method.