An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem

The irregular strip packing problem is a combinatorial optimization problem that requires to place a given set of two-dimensional polygons within a rectangular container so that no polygon overlaps with other polygons or protrudes from the container, where each polygon is not necessarily convex. The container has a fixed width, while its length can change so that all polygons are placed in it. The objective is to find a layout of the set of polygons that minimizes the length of the container. We propose an algorithm that separates overlapping polygons based on nonlinear programming, and an algorithm that swaps two polygons in a layout so as to find their new positions in the layout with the least overlap. We incorporate these algorithms as components into an iterated local search algorithm for the overlap minimization problem and then develop an algorithm for the irregular strip packing problem using the iterated local search algorithm. Computational comparisons on representative instances disclose that our algorithm is competitive with other existing algorithms. Moreover, our algorithm updates several best known results

07112 Abstracts Collection -- Cutting, Packing, Layout and Space Allocation

From 13.03. to 16.03.2007, the Dagstuhl Seminar 07112 ``Cutting, Packing, Layout and Space Allocation\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Voxel-Based Solution Approaches to the Three-Dimensional Irregular Packing Problem

Research on the three-dimensional (3D) packing problem has largely focused on packing boxes for the transportation of goods. As a result, there has been little focus on packing irregular shapes in the operational research literature. New technologies have raised the practical importance of 3D irregular packing problems and the need for efficient solutions. In this work, we address the variant of the problem where the aim is to place a set of 3D irregular items in a container, while minimizing the container height, analogous to the strip packing problem. In order to solve this problem, we need to address two critical components; efficient computation of the geometry and finding high-quality solutions. In this work, we explore the potential of voxels, the 3D equivalent of pixels, as the geometric representation of the irregular items. In this discretised space, we develop a geometric tool that extends the concept of the nofit polygon to the 3D case. This enables us to provide an integer linear programming formulation for this problem that can solve some small instances. For practical size problems, we design metaheuristic optimisation approaches. Because the literature is limited, we introduce new benchmark instances. Some are randomly generated and some represent realistic models from the additive manufacturing area. Our results on the literature benchmark data and on our new instances show that our metaheuristic techniques achieve the best known solutions for a wide variety of problems in practical computation times

Dotted-board Model Dan Extended Local Search Untuk Optimalisasi Tata Letak Pola Busana Pada Bahan Bermotif Dengan Mempertimbangkan Aturan Keserasian Motif

Permasalahan peletakkan pola busana penting dilakukan untuk memperoleh efisiensi dalam penggunaan bahan kain.Tidak hanya itu, waktu pemrosesan dengan memperhatikan keserasian motif juga masih menjadi masalah yang belum terselesaikan. Permasalahan ini dikenal dengan irregular strip packing problem (SPP). Penelitian irregular SPP menggunakan bahan bermotif pernah dilakukan sebelumnya, namun tidak memperhatikan keserasian isi motif. Penelitian ini diusulkan untuk menyelesaikan irregular SPP pada bahan bermotif dengan mempertimbangkan keserasian isi motif. Metode yang diusulkan adalah Dotted Board Model (DBM) yang dikombinasikan dengan Extended Local Search (ELS). Pada tahap awal pola busana dibagi menjadi dua kelompok. Kelompok pola busana yang memiliki aturan keserasian mo-tif, dan kelompok pola busana yang tidak memiliki aturan keserasian motif. Selanjutnya, inisialisasi tata letak awal dil-akukan pada kelompok pola busana yang memiliki aturan keserasian motif menggunakan DBM. Selebihnya, pola busana tanpa aturan keserasian motif akan dioptimalisasi dengan menggunakan ELS. Setiap aturan keserasian memiliki poin yang digunakan sebagai tolak ukur keserasian motif. Berdasarkan ujicoba, kombinasi terbaik ELS+DBM terdapat pada resolusi 3 piksel dengan iterasi local search ke 5. Nilai efisiensi dan waktu ELS+DBM adalah 57% dan 381 detik. Waktu komputasi ELS+DBM lebih cepat dengan selisih waktu komputasi 392,7detik dibandingkan tanpa DBM. Hal ini menun-jukkan bahwa metode ELS+DBM lebih unggul dibandingkan ELS tanpa DBM, karena metode ELS+DBM memiliki waktu yang lebih singkat untuk mencapai nilai efisiensi yang hampir sama

2D multi-objective placement algorithm for free-form components

This article presents a generic method to solve 2D multi-objective placement problem for free-form components. The proposed method is a relaxed placement technique combined with an hybrid algorithm based on a genetic algorithm and a separation algorithm. The genetic algorithm is used as a global optimizer and is in charge of efficiently exploring the search space. The separation algorithm is used to legalize solutions proposed by the global optimizer, so that placement constraints are satisfied. A test case illustrates the application of the proposed method. Extensions for solving the 3D problem are given at the end of the article.Comment: ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego : United States (2009

Crowdsourcing solutions to 2D irregular strip packing problems from Internet workers

Many industrial processes require the nesting of 2D profiles prior to the cutting, or stamping, of components from raw sheet material. Despite decades of sustained academic effort algorithmic solutions are still sub-optimal and produce results that can frequently be improved by manual inspection. However the Internet offers the prospect of novel ‘human-in-the-loop’ approaches to nesting problems, that uses online workers to produce packing efficiencies beyond the reach of current CAM packages. To investigate the feasibility of such an approach this paper reports on the speed and efficiency of online workers engaged in the interactive nesting of six standard benchmark datasets. To ensure the results accurately characterise the diverse educational and social backgrounds of the many different labour forces available online, the study has been conducted with subjects based in both Indian IT service (i.e. Rural BPOs) centres and a network of homeworkers in northern Scotland. The results (i.e. time and packing efficiency) of the human workers are contrasted with both the baseline performance of a commercial CAM package and recent research results. The paper concludes that online workers could consistently achieve packing efficiencies roughly 4% higher than the commercial based-line established by the project. Beyond characterizing the abilities of online workers to nest components, the results also make a contribution to the development of algorithmic solutions by reporting new solutions to the benchmark problems and demonstrating methods for assessing the packing strategy employed by the best workers

3D packing of balls in different containers by VNS

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityIn real world applications such as the transporting of goods products, packing is a major issue. Goods products need to be packed such that the smallest space is wasted to achieve the maximum transportation efficiency. Packing becomes more challenging and complex when the product is circular/spherical. This thesis focuses on the best way to pack three-dimensional unit spheres into the smallest spherical and cubical space. Unit spheres are considered in lieu of non-identical spheres because the search mechanisms are more difficult in the latter set up and any improvements will be due to the search mechanism not to the ordering of the spheres. The two-unit sphere packing problems are solved by approximately using a variable neighborhood search (VNS) hybrid heuristic. A general search framework belonging to the Artificial Intelligence domain, the VNS offers a diversification of the search space by changing neighborhood structures and intensification by thoroughly investigating each neighborhood. It is exible, easy to implement, adaptable to both continuous and discrete optimization problems and has been use to solve a variety of problems including large-sized real-life problems. Its runtime is usually lower than other meta heuristic techniques. A tutorial on the VNS and its variants along with recent applications and areas of applicability of each variant. Subsequently, this thesis considers several variations of VNS heuristics for the two problems at hand, discusses their individual efficiencies and effectiveness, their convergence rates and studies their robustness. It highlights the importance of the hybridization which yields near global optima with high precision and accuracy, improving many best- known solutions indicate matching some, and improving the precision and accuracy of others. Keywords: variable neighborhood search, sphere packing, three-dimensional packing, meta heuristic, hybrid heuristics, multiple start heuristics

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

Nesting Problems : Exact and Heuristic Algorithms

Nesting problems are two-dimensional cutting and packing problems involving irregular shapes. This thesis is focused on real applications on Nesting problems such as the garment industry or the glass cutting. The aim is to study different mathematical methodologies to obtain good lower bounds by exact procedures and upper bounds by heuristic algorithms. The core of the thesis is a mathematical model, a Mixed Integer Programming model, which is adapted in each one of the parts of the thesis. This study has three main parts: first, an exact algorithm for Nesting problems when rotation for the pieces is not allowed; second, an Iterated Greedy algorithm to deal with more complex Nesting problems when pieces can rotate at several angles; third, a constructive algorithm to solve the two-dimensional irregular bin packing problem with guillotine cuts. This thesis is organized as follows. The first part is focused on developing exact algorithms. In Chapter 2 we present two Mixed Integer Programming (MIP) models, based on the Fischetti and Luzzi MIP model. We consider horizontal lines in order to define the horizontal slices which are used to separate each pair of pieces. The second model, presented in Section 2.3, uses the structure of the horizontal slices in order to lift the bound constraints. Section 2.5 shows that if we solve these formulations with CPLEX, we obtain better results than the formulation proposed by Gomes and Oliveira. The main objective is to design a Branch and Cut algorithm based on the MIP, but first a Branch and Bound algorithm is developed in Chapter 3. Therefore, we study different branching strategies and design an algorithm which updates the bounds on the coordinates of the reference point of the pieces in order to find incompatible variables which are fixed to 0 in the current branch of the tree. The resulting Branch and Bound produces an important improvement with respect to previous algorithms and is able to solve to optimality problems with up to 16 pieces in a reasonable time. In order to develop the Branch and Cut algorithm we have found several classes of valid inequalities. Chapter 4 presents the different inequalities and in Chapter 5 we propose separation algorithms for some of these inequalities. However, our computational experience shows that although the number of nodes is reduced, the computational time increases considerably and the Branch and Cut algorithm becomes slower. The second part is focused on building an Iterated Greedy algorithm for Nesting problems. In Chapter 6 a constructive algorithm based on the MIP model is proposed. We study different versions depending on the objective function and the number of pieces which are going to be considered in the initial MIP. A new 11 idea for the insertion is presented, trunk insertion, which allows certain movements of the pieces already placed. Chapter 7 contains different movements for the local search based on the reinsertion of a given number of pieces and compaction. In Chapter 8 we present a math-heuristic algorithm, which is an Iterated Greedy algorithm because we iterate over the constructive algorithm using a destructive algorithm. We have developed a local search based on the reinsertion of one and two pieces. In the constructive algorithm, for the reinsertion of the pieces after the destruction of the solution and the local search movements, we use several parameters that change along the algorithm, depending on the time required to prove optimality in the corresponding MIP models. That is, somehow we adjust the parameters, depending on the difficulty of the current MIP model. The computational results show that this algorithm is competitive with other algorithms and provides the best known results on several known instances. The third part is included in Chapter 9. We present an efficient constructive algorithm for the two dimensional irregular bin packing problem with guillotine cuts. This problem arises in the glass cutting industry. We have used a similar MIP model with a new strategy to ensure a guillotine cut structure. The results obtained improve on the best known results. Furthermore, the algorithm is competitive with state of the art procedures for rectangular bin packing problems