A Plan of Lauding the Boxes for a Three Dimensional Bin Packing Model MARINESCU DANIELA, IACOB PAUL, B ˘ AICOIANU ALEXANDRA Transilvania University of Bras ¸ov Department of Computer Science Iuliu Maniu 50, 500091 Bras ¸ov ROMANIA mdaniela@unitbv.ro, iacobpaulcorneliu@yahoo.com, a.baicoianu@unitbv.ro Abstract: We consider the rectangular three dimensional bin packing problem with knapsack, where the bin is packed with a set of rectangular boxes, without gaps or overlapping. Starting from a solution of the three dimen- sional bin packing model, our objective is to determine an order of the loading the boxes in the bin so that a box will be packed in the bin only if there are no empty spaces down to this box and the origin of the box is in a fixed position, determinated by the boxes situated in the West and North neighbourhood. By extension of the previous work regarding the two dimensional covering problem [12] and the three dimensional bin packing problem [15], we define three kind of adjacency relations between two boxes from a packing model, similarly with [13, 14]. Combining these relations we define an acyclic graph representation of the bin packing model. A plan for lauding of the boxes in the bin is obtained using a topological sorting algorithm of the vertices of this acyclic graph. Key–Words: bin packing, topological sorting 1 Introduction In computational complexity theory, the bin-packing optimization problem is a known NP-hard problem, concerns efficiently placing box-shaped objects of ar- bitrary size and number into a box-shaped. Such prob- lems are also referred to as Cutting and Packing prob- lems in [3] or Cutting and Covering in [5]. Since the advent of computer science, bin-packing remains one of the classic difficult problems today. At this time, no optimal polynomial time algo- rithm is known for the bin-packing problem. In other words, finding a perfect solution to one non-trivial in- stance of the bin-packing problem with even the most powerful computer may take months or years. Many kinds of bin-packing problems were con- sidered, one dimensional, two dimensional and three dimensional with many kinds of constrains depending on technological restrictions [9]. The three-dimensional bin-packing problem re- tains the difficulty of lesser dimensional bin-packing problems, but holds unique and important applica- tions. As one would expect, each object and bin exists in three dimensions: width, length, and height. The goal is to minimize the number of bins required to pack all the boxes. Like two dimensional bin-packing, each box must stay orthogonal, or maintains its orien- tation in the container. If two dimensional bin-packing is equivalent to rectangle-to-floor plan packing [5, 17], three dimensional bin-packing is equivalent to box-to- room packing. The three dimensional bin-packing may involve a single bin or multiple bins. The singular bin-packing problem involves only one bin with either definite or infinite volume. Bins with infinite volume are defined with finite length and width, but with height extending to infin- ity. This allows packing solutions to pack until the set of boxes are exhausted. Solutions dealing with in- finitely sized bins aim to include a maximum number of objects. Another way to approach this problem is by con- sidering multiple boxes. Each bin has definite volume. In this way, if the volume of the objects exceeds the volume of the room, an algorithm must make choices of which boxes to include in the packing and which to throw away. This approach is good for deterministic approaches to the bin-packing problem. These problems ask: ”Do the bins hold enough volume to fit these objects?”, and if it does: ”Can we arrange all objects in bins?”. Like all bin-packing problems, extra constraints WSEAS TRANSACTIONS on SYSTEMS Marinescudaniela,Iacobpaul and B˘Aicoianualexandra ISSN: 1109-2777 830 Issue 10, Volume 7, October 2008