The Stack Loading Problem With Load-Bearing Limit

The stack loading problem has been studied in recent years for its great impact on the container loading and unloading operations. Among different objectives of the problem considered, minimizing the total number of unordered stackings and minimizing the total number of used stacks are the two important ones, which ensure efficient loading and unloading schedules, as well as reduce storage costs, respectively. The load-bearing setting, where each container has its own weight and bearing weight, is frequently considered in box packing operations but rarely in the existing studies on the stack loading problem. However, the load-bearing constraint on containers is very important for stack loading, because safety is of paramount importance. This paper is the first study on the stack loading problem with the load-bearing constraint with an aim to minimize the number of stacks and the number of unordered stackings. We show that this problem is strongly $\mathcal{NP}$ -hard even when the number of stacks is given and equals $2$ . For the case where the number of stacks is given and jobs on the bottom tiers are fixed, we show that the problem can be solved by dynamic programming in pseudo-polynomial time. For the general problem, based on a two-index integer linear programming formulation and a tabu search heuristic, we develop a binary-search based matheuristic. Our experimental results demonstrate the efficiency and effectiveness of the newly developed matheuristic.