On Guillotine Separable Packings for the Two-Dimensional Geometric Knapsack Problem

In two-dimensional geometric knapsack problem, we are given a set of n axis-aligned rectangular items and an axis-aligned square-shaped knapsack. Each item has integral width, integral height and an associated integral profit. The goal is to find a (non-overlapping axis-aligned) packing of a maximum profit subset of rectangles into the knapsack. A well-studied and frequently used constraint in practice is to allow only packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts that do not intersect any item of the solution. In this paper we study approximation algorithms for the geometric knapsack problem under guillotine cut constraints. We present polynomial time (1+?)-approximation algorithms for the cases with and without allowing rotations by 90 degrees, assuming that all input numeric data are polynomially bounded in n. In comparison, the best-known approximation factor for this setting is 3+? [Jansen-Zhang, SODA 2004], even in the cardinality case where all items have the same profit. Our main technical contribution is a structural lemma which shows that any guillotine packing can be converted into another structured guillotine packing with almost the same profit. In this packing, each item is completely contained in one of a constant number of boxes and ?-shaped regions, inside which the items are placed by a simple greedy routine. In particular, we provide a clean sufficient condition when such a packing obeys the guillotine cut constraints which might be useful for other settings where these constraints are imposed

Pattern Generation for Three Dimensional Cutting Stock Problem

We consider the problem of three-dimensional cutting of a large block that is to be cut into some small block pieces, each with a specific size and request. Pattern generation is an algorithm that has been used to determine cutting patterns in one-dimensional and two-dimensional problems. The purpose of this study is to modify the pattern generation algorithm so that it can be used in three-dimensional problems, and can determine the cutting pattern with the minimum possible cutting residue. The large block will be cut based on the length, width, and height. The rest of the cuts will be cut back if possible to minimize the rest. For three-dimensional problems, we consider the variant in which orthogonal rotation is allowed. By allowing the remainder of the initial cut to be rotated, the dimensions will have six permutations. The result of the calculation using the pattern generation algorithm for three-dimensional problems is that all possible cutting patterns are obtained but there are repetitive patterns because they suggest the same number of cuts.

An anytime tree search algorithm for two-dimensional two- and three-staged guillotine packing problems

[libralesso_anytime_2020] proposed an anytime tree search algorithm for the 2018 ROADEF/EURO challenge glass cutting problem (https://www.roadef.org/challenge/2018/en/index.php). The resulting program was ranked first among 64 participants. In this article, we generalize it and show that it is not only effective for the specific problem it was originally designed for, but is also very competitive and even returns state-of-the-art solutions on a large variety of Cutting and Packing problems from the literature. We adapted the algorithm for two-dimensional Bin Packing, Multiple Knapsack, and Strip Packing Problems, with two- or three-staged exact or non-exact guillotine cuts, the orientation of the first cut being imposed or not, and with or without item rotation. The combination of efficiency, ability to provide good solutions fast, simplicity and versatility makes it particularly suited for industrial applications, which require quickly developing algorithms implementing several business-specific constraints. The algorithm is implemented in a new software package called PackingSolver

A beam search approach to solve the convex irregular bin packing problem with guillotine cuts

This paper presents a two dimensional convex irregular bin packing problem with guillotine cuts. The problem combines the challenges of tackling the complexity of packing irregular pieces, guaranteeing guillotine cuts that are not always orthogonal to the edges of the bin, and allocating pieces to bins that are not necessarily of the same size. This problem is known as a two-dimensional multi bin size bin packing problem with convex irregular pieces and guillotine cuts. Since pieces are separated by means of guillotine cuts, our study is restricted to convex pieces.A beam search algorithm is described, which is successfully applied to both the multi and single bin size instances. The algorithm is competitive with the results reported in the literature for the single bin size problem and provides the first results for the multi bin size problem

Generating Optimized Cutting Layouts of Drywall Panels Using Building Information Modeling

The construction industry generates a substantial amount of solid waste. Cutting building materials into smaller pieces to fit the design is one of the sources of waste in new construction. This type of waste is known as leftover or residual. Drywall leftover is an example of such waste. Currently, contractors do not perform a detailed analysis of how many drywall panels would be required. Moreover, they do not use a consistent system for reusing their scrap and often cut a needed piece from a brand new panel instead of using available scrap. Building Information Model (BIM), as an object-oriented representation of the building contains all the required data and can be utilized to provide drywall crews with layouts indicating how to cut the panels into the required pieces so the leftover could be reduced. Also, some commercially available software applications, such as Autodesk Revit provide a platform to automate processes such as optimization by implementing algorithms through their Application Programming Interface (API). Similar problems have been studied in other fields and industries. Bin packing problem in mathematics and Nesting process in the cutting industry are examples of such research. As the result, automated optimization methods that utilize Evolutionary Algorithms (EA) are introduced to address these problems. There is an opportunity to apply Evolutionary Algorithms to solve a similar problem in the construction industry. This study investigates if it is feasible to implement EA-based optimization methods on a BIM platform to develop an automated optimization tool. In light of available tools and methods, an automated optimization tool is developed as a Revit add-in. It extracts geometrical data from BIM and receives dimensions of available drywall panel(s) from the user. The algorithm, finds the most desirable arrangement of panels and the number of full panels is calculated. The outline of smaller pieces that need to be cut out of full panels are also determined. Then by utilizing an EA-based optimization method, it generates the cutting layouts. The add-in is tested on a certain number of simple models for several iterations and the generated cutting layouts show very optimal leftover. On a very specific model containing twenty pieces that need to be cut out of full panels, the add-in application spent 100 minutes to generate the cutting layouts, which resulted in 36% reduction in the leftover, compared to the layouts generated in the initial iterations. The test proved that the proposed algorithm is able to optimize cutting layouts. It demonstrates that utilizing such optimization algorithm on a BIM platform could be considered as an effective way to reduce the material waste

Three-Dimensional Knapsack Problem with Pre-Placed Boxes and Vertical Stability

A three-dimensional knapsack problem packs a subset of rectangular boxes inside a bin with fixed size such that the total value of packed boxes is maximized. Each box has its own value and size and can be freely rotated into any of the six positions while its edges are parallel to the bin\u27s edges. A Mixed Integer Linear Programming is developed for the 3D knapsack problem, while some practical constraints such as vertical stability are considered. However, the given model can be applied to two dimensional problems as well. The proposed solution methodology is based on the sequence triple. Simulated annealing technique is used to model the heuristic approach. Moreover, the situation where some boxes are pre-placed in the bin is investigated. These pre-placed boxes represent potential obstacles. Numerical experiments are conducted for bins with and without obstacles. The results show that the heuristic approach is successful and can handle different kinds of instances

Automating the packing heuristic design process with genetic programming

The literature shows that one-, two-, and three-dimensional bin packing and knapsack packing are difficult problems in operational research. Many techniques, including exact, heuristic, and metaheuristic approaches, have been investigated to solve these problems and it is often not clear which method to use when presented with a new instance. This paper presents an approach which is motivated by the goal of building computer systems which can design heuristic methods. The overall aim is to explore the possibilities for automating the heuristic design process. We present a genetic programming system to automatically generate a good quality heuristic for each instance. It is not necessary to change the methodology depending on the problem type (one-, two-, or three-dimensional knapsack and bin packing problems), and it therefore has a level of generality unmatched by other systems in the literature. We carry out an extensive suite of experiments and compare with the best human designed heuristics in the literature. Note that our heuristic design methodology uses the same parameters for all the experiments. The contribution of this paper is to present a more general packing methodology than those currently available, and to show that, by using this methodology, it is possible for a computer system to design heuristics which are competitive with the human designed heuristics from the literature. This represents the first packing algorithm in the literature able to claim human competitive results in such a wide variety of packing domains

Moldable Items Packing Optimization

This research has led to the development of two mathematical models to optimize the problem of packing a hybrid mix of rigid and moldable items within a three-dimensional volume. These two developed packing models characterize moldable items from two perspectives: (1) when limited discrete configurations represent the moldable items and (2) when all continuous configurations are available to the model. This optimization scheme is a component of a lean effort that attempts to reduce the lead-time associated with the implementation of dynamic product modifications that imply packing changes. To test the developed models, they are applied to the dynamic packing changes of Meals, Ready-to-Eat (MREs) at two different levels: packing MRE food items in the menu bags and packing menu bags in the boxes. These models optimize the packing volume utilization and provide information for MRE assemblers, enabling them to preplan for packing changes in a short lead-time. The optimization results are validated by running the solutions multiple times to access the consistency of solutions. Autodesk Inventor helps visualize the solutions to communicate the optimized packing solutions with the MRE assemblers for training purposes