Sara Ali

Abstract
In online three-dimensional packing problems where items are received one by one and require immediate packing decisions without prior knowledge of upcoming items, considering the static stability constraint is crucial for safely packing each arriving item in real time. Unstable loading patterns can result in risks of potential damage to items, containers, and operators during loading/unloading operations. Nevertheless, static stability constraints have often been neglected or oversimplified in existing online heuristic methods in the literature, undermining the practical implementation of these methods in real-world scenarios. In this study, we analyze how different static stability constraints affect solutions’ efficiency and cargo stability, aiming to provide valuable insights and develop heuristic algorithms for real-world online problems, thus increasing the applicability of this research field. To this end, we embedded four distinct static stability constraints in online heuristics, including full-base support, partial-base support, center-of-gravity polygon support, and novel partial-base polygon support. Evaluating the impact of these constraints on the efficiency of a wide range of heuristic methods on real instances showed that regarding the number of used bins, heuristics with polygon-based stabilities have superior performance against those under full-base and partial-base support stabilities. The “static mechanical equilibrium” approach offers a necessary and sufficient condition for the cargo static stability, and we employed it as a benchmark in our study to assess the quality of the four studied stability constraints. Knowing the number of stable items under each of these constraints provides valuable managerial insight for decision-making in real-world online packing scenarios. © 2025 Elsevier Ltd

Abstract
In online three-dimensional packing problems (3D-PPs), unlike offline problems, items arrive sequentially and require immediate packing decisions without any information about the quantities and sizes of the items to come. Heuristic methods are of great importance in solving online problems to find good solutions in a reasonable amount of time. However, the literature on heuristics for online problems is sparse. As our first contribution, we developed a pool of heuristics applicable to online 3D-PPs with complementary performance on different sets of instances. Computational results showed that in terms of the number of used bins, in all problem instances, at least one of our heuristics had a better or equal performance compared to existing heuristics in the literature. The developed heuristics are also fully applicable to an intermediate class between offline and online problems, referred to in this paper as a specific type of semi-online with full look-ahead, which has several practical applications. In this class, as in offline problems, complete information about all items is known in advance (i.e., full look-ahead); however, due to time or space constraints, as in online problems, items should be packed immediately in the order of their arrival. As our second contribution, we presented an algorithm selection framework, building on developed heuristics and utilizing prior information about items in this specific class of problems. We used supervised machine learning techniques to find the relationship between the features of problem instances and the performance of heuristics and to build a prediction model. The results indicate an 88% accuracy in predicting (identifying) the most promising heuristic(s) for solving any new instance from this class of problems.

Abstract
Three-Dimensional Packing Problems (3D-PPs) can be applied to effectively reduce logistics costs in various areas, such as airline cargo management and warehouse management. In general, 3D-PP studies can be divided into two different streams: those tackling the off-line problem, where full knowledge about items is available beforehand; and those tackling the on-line (real-time) problem, where items arrive one by one and should be packed immediately without having full prior knowledge about them. During the past decades, off-line and online 3D-PPs have been studied in the literature with various constraints and solution approaches. However, and despite the numerous practical applications of on-line problems in real-world situations, most of the literature to date has focused on off-line problems and is quite sparse when it comes to on-line solution methods. In this regard, and despite the different nature of on-line and off-line problems, some approaches can be applied in both environments. Hence, we conducted an in-depth and updated literature review to identify and structure various constraints and solution methods employed by researchers in off-line and on-line 3D-PPs. Building on this, by bringing together the two separate streams of the literature, we identified several off-line approaches that can be adopted in on-line environments. Additionally, we addressed relevant research gaps and ways to bridge them in the future, which can help to develop this research field.