New lower bounds for the three-dimensional finite bin packing problem

The three-dimensional finite bin packing problem (3BP) consists of determining the minimum number of large identical three-dimensional rectangular boxes, bins, that are required for allocating without overlapping a given set of three-dimensional rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. We propose new lower bounds for the problem where the items have a fixed orientation and then we extend these bounds to the more general problem where for each item the subset of rotations by 90° allowed is specified. The proposed lower bounds have been evaluated on different test problems derived from the literature. Computational results show the effectiveness of the new lower bounds.     

A cutting-plane approach for the two-dimensional orthogonal non-guillotine cutting problem

The two-dimensional orthogonal non-guillotine cutting problem (NGCP) appears in many industries (like wood and steel industries) and consists in cutting a rectangular master surface into a number of rectangular pieces, each with a given size and value. The pieces must be cut with their edges always parallel or orthogonal to the edges of the master surface (orthogonal cuts). The objective is to maximize the total value of the pieces cut.In this paper, we propose a two-level approach for solving the NGCP, where, at the first level, we select the subset of pieces to be packed into the master surface without specifying the layout, while at a second level we check only if a feasible packing layout exists. This approach has been already proposed by Fekete and Schepers [S.P. Fekete, J. Schepers, A new exact algorithm for general orthogonal d-dimensional knapsack problems, ESA 97, Springer Lecture Notes in Computer Science 1284 (1997) 144–156; S.P. Fekete, J. Schepers, On more-dimensional packing III: Exact algorithms, Tech. Rep. 97.290, Universität zu Köln, Germany, 2000; S.P. Fekete, J. Schepers, J.C. van der Veen, An exact algorithm for higher-dimensional orthogonal packing, Tech. Rep. Under revision on Operations Research, Braunschweig University of Technology, Germany, 2004] and Caprara and Monaci [A. Caprara, M. Monaci, On the two-dimensional knapsack problem, Operations Research Letters 32 (2004) 2–14]. We propose improved reduction tests for the NGCP and a cutting-plane approach to be used in the first level of the tree search to compute effective upper bounds.Computational tests on problems derived from the literature show the effectiveness of the proposed approach, that is able to reduce the number of nodes generated at the first level of the tree search and the number of times the existence of a feasible packing layout is tested.     

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.     

New upper bounds for the two-dimensional orthogonal non-guillotine cutting stock problem

The two-dimensional orthogonal non-guillotine cutting stockproblem (NGCP) appears in many industries (e.g. the wood andsteel industries) and consists of cutting a rectangular mastersurface into a number of rectangular pieces, each with a givensize and value. The pieces must be cut with their edges alwaysparallel to the edges of the master surface (orthogonal cuts).The objective is to maximize the total value of the pieces cut. New upper bounds on the optimal solution to the NGCP are described.The new bounding procedures are obtained by different relaxationsof a new mathematical formulation of the NGCP. Various proceduresfor strengthening the resulting upper bounds and reducing thesize of the original problem are discussed. The proposed newupper bounds have been experimentally evaluated on test problemsderived from the literature. Comparisons with previous boundingprocedures from the literature are given. The computationalresults indicate that these bounds are significantly betterthan the bounds proposed in the literature.     

MIP models for two-dimensional non-guillotine cutting problems with usable leftovers

In this study we deal with the two-dimensional non-guillotine cutting problem of how to cut a set of larger rectangular objects to a set of smaller rectangular items in exactly a demanded number of pieces. We are concerned with the special case of the problem in which the non-used material of the cutting patterns (objects leftovers) may be used in the future, for example if it is large enough to fulfill future item demands. Therefore, the problem is seen as a two-dimensional non-guillotine cutting/packing problem with usable leftovers, also known in the literature as a two-dimensional residual bin-packing problem. We use multilevel mathematical programming models to represent the problem appropriately, which basically consists of cutting the ordered items using a set of objects of minimum cost, among all possible solutions of minimum cost, choosing one that maximizes the value of the usable leftovers, and, among them, selecting one that minimizes the number of usable leftovers. Because of special characteristics of these multilevel models, they can be reformulated as one-level mixed integer programming (MIP) models. Illustrative numerical examples are presented and analysed.     

Algorithms for Unconstrained Two-Dimensional Guillotine Cutting

In this paper we consider the unconstrained, two-dimensional, guillotine cutting problem. This is the problem that occurs in the cutting of a number of rectangular pieces from a single large rectangle, so as to maximize the value of the pieces cut, where any cuts that are made are restricted to be guillotine cuts.We consider both the staged version of the problem (where the cutting is performed in a number of distinct stages) and the general (non-staged) version of the problem.A number of algorithms, both heuristic and optimal, based upon dynamic programming are presented. Computational results are given for large problems.     

A recursive algorithm for constrained two-dimensional cutting problems

This paper presents a recursive algorithm for constrained two-dimensional guillotine cutting problems of rectangular items. The algorithm divides a stock plate into a sequence of small rectangular blocks. For the current block considered, it selects an item, puts it at the left-bottom corner of the block, and determines the direction of the dividing cut that divides the unoccupied region of the block into two smaller blocks for further consideration. The dividing cut is either along the upper edge or along the right edge of the selected item. The upper bound obtained from the unconstrained solution is used to shorten the searching space. The computational results on benchmark problems indicate that the algorithm can improve the solutions, and is faster than other algorithms.     

A branch-and-price algorithm for the two-stage guillotine cutting stock problem

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their widths and others down their heights. To solve this model, we propose an exact branch-and-price algorithm. To the best of our knowledge, this is the first contribution with regard to obtaining integer optimal solutions to Gilmore and Gomory model. Extensive results, on a set of real-world problems, indicate that the proposed algorithm delivers optimal solutions for instances with up to 809 items and that the hybrid cutting strategy often yields improved solutions. Furthermore, our computational study reveals that the proposed modelling and algorithmic strategy outperforms a recently proposed arc-flow model-based solution strategy.     

A MIP-CP based approach for two- and three-dimensional cutting problems with staged guillotine cuts

This work presents guillotine constraints for two- and three-dimensional cutting problems. These problems look for a subset of rectangular items of maximum value that can be cut from a single rectangular container. Guillotine constraints seek to ensure that items are arranged in such a way that cuts from one edge of the container to the opposite edge completely separate them. In particular, we consider the possibility of 2, 3, and 4 cutting stages in a predefined sequence. These constraints are considered within a two-level iterative approach that combines the resolution of integer linear programming and constraint programming models. Experiments with instances of the literature are carried out, and the results show that the proposed approach can solve in less than 500 s approximately 60% and 50% of the instances for the two- and three-dimensional cases, respectively. For the two-dimensional case, in comparison with the recent literature, it was possible to improve the upper bound for 16% of the instances.

     

A two-dimensional strip cutting problem with sequencing constraint

We study a strip cutting problem that arises in the production of corrugated cardboard. In this context, rectangular items of different sizes are obtained by machines, called corrugators, that cut strips of large dimensions according to particular schemes containing at most two types of items. Because of buffer restrictions, these schemes have to be sequenced in such a way that, at any moment, at most two types of items are in production and not completed yet (sequencing constraint). We show that the problem of finding a set of schemes of minimum trim loss that satisfies an assigned demand for each item size is strongly NP-hard, even if the sequencing constraint is relaxed. Then, we present two heuristics for the problem with the sequencing constraint, both based on a graph characterization of the feasible solutions. The first heuristic is a two-phase procedure based on a mixed integer linear programming model. The second heuristic follows a completely combinatorial approach and consists of solving a suitable sequence of minimum cost matching problems. For both procedures, an upper bound on the number of schemes (setups) is found. Finally, a computational study comparing the quality of the heuristic solutions with respect to an LP lower bound is reported.     

Reducing the number of cuts in generating three-staged cutting patterns

Three-staged guillotine patterns are widely used in the manufacturing industry to cut stock plates into rectangular items. The cutting cost often increases with the number of cuts required. This paper focuses on the rectangular two-dimensional cutting stock problem, where three-staged guillotine patterns are used, and the objective is to minimize the sum of plate and cutting costs. The column generation framework is used to solve the problem. It uses a pattern-generation procedure to obtain the patterns. The cutting cost is considered in both the pattern-generation procedure and the objective of the linear programming formulation. The computational results indicate that the approach can reduce the number of cuts, without increasing the plate cost.     

Two-dimensional packing problems: A survey

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. We discuss mathematical models, and survey lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches. The relevant special cases where the items have to be packed into rows forming levels are also discussed in detail.     

The two-dimensional finite bin packing problem. Part I: New lower bounds for the oriented case

The Two-Dimensional Finite Bin Packing Problem (2BP) consists of determining the minimum number of large identical rectangles, bins, that are required for allocating without overlapping a given set of rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. In this paper we describe new lower bounds for the 2BP where the items have a fixed orientation and we show that the new lower bounds dominate two lower bounds proposed in the literature. These lower bounds are extended in Part II (see Boschetti and Mingozzi 2002) for a more general version of the 2BP where some items can be rotated by . Moreover, in Part II a new heuristic algorithm for solving both versions of the 2BP is presented and computational results on test problems from the literature are given in order to evaluate the effectiveness of the proposed lower bounds.     

Accurate bending analysis of rectangular plates with two adjacent edges free and the others clamped or simply supported based on new symplectic approach

The symplectic geometry approach is introduced for accurate bending analysis of rectangular thin plates with two adjacent edges free and the others clamped or simply supported. The basic equations for rectangular plates are first transferred into Hamilton canonical equations. Using the symplectic approach, the analytic solution of rectangular thin plate with two adjacent edges simply supported and the others slidingly supported is derived. Accurate bending solutions of title problems are then obtained using the superposition method. The approach used in this paper eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results are presented to demonstrate the validity and efficiency of the approach as compared with those reported in other literatures.     

Integer linear programming models for 2-staged two-dimensional Knapsack problems

 We are given a unique rectangular piece of stock material S, with height H and width W, and a list of m rectangular shapes to be cut from S. Each shape's type i (i = 1, ..., m) is characterized by a height , a width , a profit , and an upper bound ub _i indicating the maximum number of items of type i which can be cut. We refer to the Two-Dimensional Knapsack (TDK) as the problem of determining a cutting pattern of S maximizing the sum of the profits of the cut items. In particular, we consider the classical variant of TDK in which the maximum number of cuts allowed to obtain each item is fixed to 2, and we refer to this problem as 2-staged TDK (2TDK). For the 2TDK problem we present two new Integer Linear Programming models, we discuss their properties, and we compare them with other formulations in terms of the LP bound they provide. Finally, both models are computationally tested within a standard branch-and-bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries. Received: October 17, 2000 / Accepted: December 19, 2001 Published online: September 27, 2002 Key words. packing – cutting – integer linear programming     

A new dynamic programming procedure for three-staged cutting patterns

Three-staged patterns are often used to solve the 2D cutting stock problem of rectangular items. They can be divided into items in three stages: Vertical cuts divide the plate into segments; then horizontal cuts divide the segments into strips, and finally vertical cuts divide the strips into items. An algorithm for unconstrained three-staged patterns is presented, where a set of rectangular item types are packed into the plate so as to maximize the pattern value, and there is no constraint on the frequencies of each item type. It can be used jointly with the linear programming approach to solve the cutting stock problem. The algorithm solves three large knapsack problems to obtain the optimal pattern: One for the item layout on the widest strip, one for the strip layout on the longest segment, and the third for the segment layout on the plate. The computational results indicate that the algorithm is efficient.     

一种高效快速的裁剪算法

本文的裁剪策略是，巧妙的利用窗口与线段的两种不同数学描述，将有效交点的判定、求交运算及包含性检验，归结为三个条件的判别。     

Approximation algorithms for the oriented two-dimensional bin packing problem

Given a set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins, we consider the problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized. The problem is strongly NP-hard and finds practical applications in cutting and packing. We discuss a simple deterministic approximation algorithm which is used in the initialization of a tabu search approach. We then present a tabu search algorithm and analyze its average performance through extensive computational experiments.     

A Well-Conditioned,Nonconforming Nitsche's Extended Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a nonconforming Nitsche's extended finite element method (NXFEM) for elliptic interface problems on unfitted triangulation elements. The solution on each side of the interface is separately expanded in the standard nonconforming piecewise linear polynomials with the edge averages as degrees of freedom. The jump conditions on the interface and the discontinuities on the cut edges (the segment of edges cut by the interface) are weakly enforced by the Nitsche's approach. In the method, the harmonic weighted fluxes are used and the extra stabilization terms on the interface edges and cut edges are added to guarantee the stability and the well conditioning. We prove that the convergence order of the errors in energy and $L^2$ norms are optimal. Moreover, the errors are independent of the position of the interface relative to the mesh and the ratio of the discontinuous coefficients. Furthermore, we prove that the condition number of the system matrix is independent of the interface position. Numerical examples are given to confirm the theoretical results.     

弹性矩形薄板受迫振动的功的互等定理法(Ⅰ)——四边固定的矩形板和三边固定的矩形板

本文将功的互等定理法(MRT)推广于求解在简谐干扰力作用下矩形板的稳态响应.给出了各种边界条件矩形板的一系列封闭解并提供了一些有实用价值的图表.功的互等定理法(MRT)是求解在各种简谐干扰力作用下的矩形板稳态响应的一个简便、通用的方法.本文包括三部分:(Ⅰ)四边固定的矩形板和三边固定的矩形板;(Ⅱ)二邻边固定的矩形板;(Ⅲ)悬臂矩形板.我们准备分三次陆续发表它们.