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. Note to Practitioners - This paper is motivated by the stack loading problem and is the first study on the load-bearing limit case. The load-bearing limit is a fundamental constraint but has not been taken into account in studies in the stack loading problem. Based on ISO Standard 1496, the corner posts and corner fittings of ISO Series I containers can bear a certain amount of weight. If the total weight of the containers above exceeds the load-bearing limit of the lower container, it will hazard the load-bearing safety. This paper proposes two problem formulations: three-index formulation and two-index formulation. The three-index formulation adds the load-bearing limit to the existing stack loading problem formulation. It turns out that the traditional three-index formulation of the stack loading problem is not efficient when being used in solving the problem with load-bearing constraints. Therefore, we propose a new two-index formulation. Apart from the theoretical results, this paper proposes a matheuristic solution framework: firstly, using binary search with greedy matheuristic for feasibility checking to minimize the number of stacks, and secondly, using tabu search matheuristic to minimize the number of unordered stackings. In future research, we will apply the matheuristic to different types of container scenarios and the parallel stack loading case.