A best-first branch and bound algorithm for the pallet-loading problem

In this study, we proposed an optimal algorithm for the pallet-loading problem (PLP). The PLP involves determining a loading pattern that can load the most identical boxes with rectangular shapes onto a large rectangular pallet. A new branch and bound algorithm, including a sub-algorithm that cheques the feasibility of the solutions, was proposed to solve a relaxed mathematical model, which did not consider the exact position of the boxes. Because the algorithm constructed the layout row-by-row and used effective branching strategies and improved bounds, it found a solution very quickly. Whenever a solution was obtained for the relaxed problem, another branch and bound algorithm (as a sub-algorithm) was used to check whether the solution obtained was feasible and to determine the resulting layout. The branching strategies in the sub-algorithm used the solution from the relaxation problem to quickly find the layout of the solution. The computational results demonstrated the superiority of the proposed algorithm. The algorithm proposed in this study solved all three million problems presented with an area ratio bound of less than 101 boxes in a minute, except for one. Solutions to all of the problems were obtained within 90 s, while other existing exact algorithms required more than an hour for some problems.