A Multi-start Local Search Approach to the Multiple Container Loading Problem

Abstract

This paper is concerned with the multiple container loading problem with rectangular boxes of different sizes and one or more containers, which means that the boxes should be loaded into the containers so that the waste space in the containers are minimized or the total value of the boxes loaded into the containers is maximized. Several approaches have been taken to solve this problem (Ivancic et al., 1989; Mohanty et al., 1994; George, 1996; Bortfeldt, 2000; Eley, 2002; Eley, 2003; Takahara, 2005; Takahara, 2006). The aim of this paper is to propose an efficient greedy approach for the multiple container loading problem and to contribute to develop a load planning software and applications. The problem considered in this paper is the three-dimensional packing problem, therefore it is known to NP-hard. This implies that no simple algorithm has been found so far. In this paper this problem is solved by a relatively simple method using a two-stage strategy. Namely, all boxes are numbered and for the sequence of the numbers a greedy algorithm of loading boxes into containers is considered. This greedy algorithm is based on first-fit concept (Johnson et al., 1974) and determines the arrangement of each box. This algorith try to load the boxes all kind of containers and select the best arrangement result. It’s requires a box loading sequence and a set of orientation orders of each box type for each container type. Each box is tried to load according to these dynamic parameters. Moreover, a static parameter overhang ratio is introduced here. This is a percentage of the bottom face area of the loading box that isn’t supported with other boxes that are below. As shown by Takahara (Takahara, 2006), if this parameter is different, the arrangement given this algorithm is different in spite of using a same pair of the loading sequence and the orientation orders. Some initial pairs of solution, that is a box loading sequence and a set of orientation orders of each box type, are selected among some rules, which are given beforehand. This solution is altered by using a multi-start local search approach. The local search approach is used to find a good pair of solutions that is brought an efficient arrangement by the greedy loading algorithm. The arrangement obtained in the iterations is estimated by volume utilization or total value of the loaded boxes. The effectiveness of the proposed approach is shown by comparing the results obtained with the approaches presented by using benchmark problems from the literature. Finally, the layout examples by the application using the proposed approach for container loading are illustrated. O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg