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

Constructive procedures to solve 2-dimensional bin packing problems with irregular pieces and guillotine cuts

This paper presents an approach for solving a new real problem in cutting and packing. At its core is an innovative mixed integer programme model that places irregular pieces and defines guillotine cuts. The two-dimensional irregular shape bin packing problem with guillotine constraints arises in the glass cutting industry, for example, the cutting of glass for conservatories. Almost all cutting and packing problems that include guillotine cuts deal with rectangles only, where all cuts are orthogonal to the edges of the stock sheet and a maximum of two angles of rotation are permitted. The literature tackling packing problems with irregular shapes largely focuses on strip packing i.e. minimizing the length of a single fixed width stock sheet, and does not consider guillotine cuts. Hence, this problem combines the challenges of tackling the complexity of packing irregular pieces with free rotation, guaranteeing guillotine cuts that are not always orthogonal to the edges of the stock sheet, and allocating pieces to bins. To our knowledge only one other recent paper tackles this problem. We present a hybrid algorithm that is a constructive heuristic that determines the relative position of pieces in the bin and guillotine constraints via a mixed integer programme model. We investigate two approaches for allocating guillotine cuts at the same time as determining the placement of the piece, and a two phase approach that delays the allocation of cuts to provide flexibility in space usage. Finally we describe an improvement procedure that is applied to each bin before it is closed. This approach improves on the results of the only other publication on this problem, and gives competitive results for the classic rectangle bin packing problem with guillotine constraint

A Quasiphysical and Dynamic Adjustment Approach for Packing the Orthogonal Unequal Rectangles in a Circle with a Mass Balance: Satellite Payload Packing

Packing orthogonal unequal rectangles in a circle with a mass balance (BCOURP) is a typical combinational optimization problem with the NP-hard nature. This paper proposes an effective quasiphysical and dynamic adjustment approach (QPDAA). Two embedded degree functions between two orthogonal rectangles and between an orthogonal rectangle and the container are defined, respectively, and the extruded potential energy function and extruded resultant force formula are constructed based on them. By an elimination of the extruded resultant force, the dynamic rectangle adjustment, and an iteration of the translation, the potential energy and static imbalance of the system can be quickly decreased to minima. The continuity and monotony of two embedded degree functions are proved to ensure the compactness of the optimal solution. Numerical experiments show that the proposed QPDAA is superior to existing approaches in performance

Two-Dimensional Bin Packing Problem with Guillotine Restrictions

This thesis, after presenting recent advances obtained for the two-dimensional bin packing problem, focuses on the case where guillotine restrictions are imposed. A mathematical characterization of non-guillotine patterns is provided and the relation between the solution value of the two-dimensional problem with guillotine restrictions and the two-dimensional problem unrestricted is being studied from a worst-case perspective. Finally it presents a new heuristic algorithm, for the two-dimensional problem with guillotine restrictions, based on partial enumeration, and computationally evaluates its performance on a large set of instances from the literature. Computational experiments show that the algorithm is able to produce proven optimal solutions for a large number of problems, and gives a tight approximation of the optimum in the remaining cases

Un algoritmo FFD-Eficiente para resolver el problema de corte de guillotina con demanda no unitaria de requerimientos sobre stock de tamaño variado

Resuelve el problema Guillotine Cutting Stock Problem with Demand on Varied Stock (GCSP-DVS) a través de un algoritmo FFD-Eficiente variado (FFD-E 2DGV). Además, demuestra la capacidad del algoritmo propuesto para incidir en el ahorro significativo a través del reúso de materia prima reciclable para el proceso industrial de corte bidimensional. Asimismo, compendia los resultados del algoritmo propuesto aplicado al GCSP-DVS y los resultados comparativos entre el FFD y el FFD-E aplicado al GCSP-D; generando un banco inédito para instancias de cortes 2 dimensiones de tipo guillotina sobre stock de tamaño variado y otra de demostraciones numéricas comparativas del FFD-E respecto al FFD, respectivamente. Finalmente, implementa un sistema computacional parametrizable que ejecute el algoritmo propuesto y arroje reportes de solución del citado problema GCSP con demanda sobre stock variado (GCSP-DVS).Tesi

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