# Application of Machine Learning Methods for Pallet Loading Problem
## 2 Literature Review  
The PLP is used to enhance performance measures, such as the number of boxes per pallet (or pallet utilization) and stability, by arranging boxes on the basis of their dimensions and weights on a rectangular pallet with known dimensions and weight limits. In the case of a 3D manufacturer's PLP for assigning the maximum number of boxes per pallet, an optimized solution depends upon the pallet stability on the number of horizontal layers per pallet and the layout pattern of the boxes per layer [10]. In the literature, there are multiple methods for solving the PLP, such as classical mathematical methods, evolutionary methods, other intelligent optimization methods, and hybrid methods. [PERSON] [18] developed a pallet optimization system to place boxes on pallets, which identified an optimal solution to the PLP within \(\sim\)5 min. However, the proposed solution was not efficient for larger problems. In another study, the binary integers-based formulation was used to develop a mathematical model to arrange multiple pallets and materials compatibility with those pallets [19]. In this model, the box dimensions were not necessarily integers. Similarly, another mathematical model based on a polynomial time algorithm was developed for the 2-dimensional (2D) guillotine cutting stock problems [20]. In another similar study, [PERSON] et al. [21] developed a mathematical model with a staircase structure that had geometrical properties. The proposed model provided reasonable efficiency in the PLP by identifying an optimal solution. Contrary to all these classical solutions to the PLP, [PERSON] et al. [22] optimized the stack stability in addition to the stack volume. An important feature of this work was minimizing the number of products of various customers on the same pallet. However, the proposed model can optimally solve only small-scale problems, and it becomes computationally expensive for large problems. Furthermore, [PERSON] et al. [23] developed a two-phase intelligent Decision Support-System (DSS) for the Air-Cargo Loading Problem. It is an efficient solution for variable-sized (or shaped) pallets to be uploaded on the Air-Cargo. This particular solution has been effectively utilized in airport cargo systems, and it successfully provides over 90% volume utilization. The common factor in all these methods is that they are based on classical mathematical models. It is evident that these models should address the PLP by performing an exhaustive search of all possibilities, which become impractical for large problem sizes. Hence, the researchers looked for alternate solutions, such as genetic algorithms or heuristics, to find faster and sub-optimal solutions. The heuristic algorithms were also utilized in the literature to solve the PLP until the mid-1990s. It was ascertained whether they can provide optimal results with the new upper-bound methods, which are based on some structural constraints [24]. For instance, [PERSON] et al. [25] used a new heuristic algorithm to solve distributors' pallet packing problems efficiently and achieved the natural balance. Another distributor's pallet packing problem was solved with the heuristic methods by [PERSON] et al. [1]. In the study, a layer-wise loading strategy with the optimal 2D loading patterns was employed in the 3D solution method. Similarly, [26] developed an effective heuristic algorithm that addressed the issue of the 2D PLP. Their novel method could readily identify complicated solutions with five-block structures. Although heuristic algorithms are utilized extensively, there are several other intelligent optimization and search methods as well. For instance, the new branch-and-bound algorithms are integer programming-based methods, which are employed to optimize the number of boxes to be loaded on a pallet. It is an efficient method that has solved over three million problems for an area ratio bound of less than 101 boxes by executing the optimal solution in less than a minute [21].