An Island-Based Parallel Biased Random-Key Genetic Algorithm for the Three-Dimensional Trailer Loading Problem

The online three-dimensional bin packing problem (3D-BPP) is a longstanding challenge in logistics and industrial palletizing. Recent learning-based methods use a learned policy to select among feasible candidate placements. Performance depends on the candidate generator and representation, especially in industrial settings where packings must be space-efficient, stable, compact, and balanced. However, prior work has mainly optimized the policy, while candidate generation and representation remain largely geometry-driven. We address this gap with OPAL, an operationally guided placement-aware learning framework for industrial online 3D-BPP which combines an Operationally Guided Empty-Maximal-Space generator (OG-EMS), an operational representation for each candidate placement, and a masked ranking policy trained with proximal policy optimization. OG-EMS evaluates multiple anchors within each free-space region and prioritizes low, well-supported, compact, and spatially diverse placements. An xLSTM-based Placement Encoder models dependencies among geometric and operational candidate attributes, while a lightweight recurrent core combines the resulting embeddings with the current item and pallet state to rank feasible actions. On the BED-BPP benchmark, OPAL achieves a mean space utilization of 0.49, with improvements of 15.1% from operationally guided candidate generation and 6.3% from learned ranking, while maintaining robust inference-time performance.

3D bin packing rectangular items into standardised containers to maximise space utilisation under geometric shipping automation. Loading a furniture purchase into a personal vehicle is the same task, but under more complex conditions that standard container loading algorithms ignore. This paper addresses the physically stable placement under these realistic conditions with heterogeneous boxes (e.g. varying dimensions and weights) and occupied containers (e.g. groceries). This paper provides a real-world benchmark dataset and baseline model for the Heterogeneous furniture-in-vehicle packing task. The dataset uses real furniture company flat-pack packaging data covering a large number of catalogue products via family-level extrapolation with diversity length, widths, heights, and weights. We also propose a PackingGPT framework for packing as a sequential placement inspired by the Lego assembly process, where heterogeneous boxes of varying dimensions (bricks) are placed step-by-step into the irregular remaining cargo space (creations). Five baseline packing methods were tested on our dataset without considering the Centre-of- Mass (CoM) constraints. In sedan car simulations, 10-40% of placed boxes failed the stability check on average. When the LLP model was trained on packing sequences with CoM constraints enforced during placement, the failure rate dropped to 0.67% (SUV-500).

Multi-agent combinatorial optimization problems are notoriously challenging due to their NP-hard nature. Recent parallel autoregressive neural solvers improve inference efficiency by allowing agents to make decisions simultaneously, but their performance often degrades on large-scale instances. This is largely attributable to weak modeling of local geometric structures and the fact that conflicting task selections are handled only after action generation. To address these limitations, we propose GeoPAR, a geometry-guided parallel autoregressive reinforcement learning framework for scalable multi-agent combinatorial optimization. GeoPAR integrates three key components: (1) a projection-window sparse geometry mechanism that builds lightweight local candidate neighborhoods through multi-directional projections, (2) sparse edge-biased attention that injects these geometric relations into node representations, and (3) cache-guided conflict-aware assignment that reuses the geometric cache during decoding to suppress duplicate selections of exclusive tasks. Experiments on heterogeneous vehicle routing and open multi-depot pickup-and-delivery problems show that GeoPAR improves large-scale zero-shot generalization while substantially reducing rollout steps and maintaining efficient inference.