Int J Adv Manuf Technol (1996) 11:425-429 © 1996 Springer-Verlag London Limited The International Journal of Rdvanced manufacturing Technologg 3D Volumetric Pallet-Loading Optimisation J. Arghavani and G. Abdou* Department of Mechanical Engineering, Concordia University, Montreal, Quebec, Canada and *Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology, Newark, USA The paper describes the development of an efficient integer linear programming model for pallet volumetric optimisation. The proposed model optimises the volume utilisation of the pallet with rectangular and~or square boxes of varying dimensions. The resulting optimum solution may be of layered or stacking palletisation. Two different case studies are presented and the results are compared with results for the same problems in the published literature. Keywords: 3D Palletisation; ILP Optimisation 1. Introduction The pallet-loading problem considers loading boxes of multisize and of both rectangular and square shapes on a pallet. Palletisation usually takes place in manufacturing environments and distribution centres to ease the transportation of boxes containing all kinds of goods. The most important aspect in overcoming the loss and damage of goods, hence increasing profit, can be regarded as the planning, storage and placement problems in the distribution/warehouse centres. A common problem experienced by manufacturers and distribution centres is the "optimal" utilisation of the pallet loads. Utilisation means how well the loaded boxes use the space specified by the dimensions (length, width, and height) of the pallet. On the other hand, one may want to determine the optimal layout on the pallet regardless of the location of boxes. The general pallet-loading problem can be viewed as a 2D cutting stock problem, for which a large piece is cut into smaller pieces. Usually cutting problems in 2D involve orthogonal cuts which are either guillotine patterns or nested patterns. The 2D pallet-loading problem is similar to the orthogonal cutting problem, i.e. it is a surface partitioning problem for which two cases can be considered: 1. A single-layer pallet loading with either boxes of the same dimensions Correspondence and offprint requests to: Dr G. Abdou, Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA. 2. Boxes of different base dimensions but with the same height. The 3D pallet-loading problem is more complex since boxes have different dimensions. Since the early 1960s, the pallet-loading problem has prompted researchers to look into many criteria and to develop many procedures for the solution of palletisation problems. Such problems arise in a variety of situations including cutting-stock problems, packing, container loading and placement problems. Many researchers have focused on the modelling and solution of the problem in 2D, 2~-D and 3D. Several approaches, including graph-theoretic, tree-search, and heuristics have been used. Most research has focused on three approaches to obtain optimal solution to the palletisation problem: mathematical programming, dynamic programming and heuristics. The extension of the earlier works on the 2D cutting-stock problems was applied to the 2D loading problems. Gilmore and Gomory [1] worked on the 2D cutting stock problems and considered that the packing problem was essentially the same. They used mathematical programming techniques and discussed the nature of the 3D problem. However, no mathematical procedures were developed. The 2D pallet-loading problems can be found in many operational research publications, for instance: Steudel [2], Smith and De Cani [3], Bischoff and Dowsland [4] and Dowsland [5] presented procedures for the solutions of the pallet-packing problem with boxes of identical dimensions. Brown [6] developed an algorithm which packs small rectangu- lar pieces into a larger rectangular. Tsai et al. [7] used Brown's method and developed an integer linear programming (ILP) model for a 2D palletisation problem, for which boxes of various types might be loaded on the pallet base. This model considered only one layer on the pallet, and assumed that all the boxes were the same height. Chen et al. [8] presented a mathematical model to pack non-uniform box sizes onto a pallet. Their research considers boxes of uniform height, with feasible solutions for only one layer on the pallet which in fact can be considered as a 2D pallet-loading problem with different rectangular pieces. Abdou and Yang [9] developed systematic procedures for the 3D layered-palletisation problem and assumed that boxes can