LP bounds in various constraint programming approaches for orthogonal packing

We consider the 2D orthogonal feasibility problem (OPP-2) and the 2D strip packing problem (SPP-2). Given a set of rectangular items, the OPP-2 is to decide whether all items can be orthogonally packed into the given rectangular container; the SPP-2 is to find a packing of all items occupying the minimal height of the given semi-infinite strip. Both OPP-2 and SPP-2 are considered without items rotation. We investigate the known Constraint Programming (CP) approaches for OPP-2, in particular the dichotomy and disjunctive branching strategies and adapt 1D relaxation bounds based on linear programming (LP) into the constraint propagation process of the CP. Using the dichotomic search procedure the developed methods for OPP-2 are transformed for the case of SPP-2. Numerical results demonstrate the efficiency of the proposed strategies and of the combination of CP and LP-based pruning rules.     

Approximation algorithms for orthogonal packing problems for hypercubes

Orthogonal packing problems are natural multidimensional generalizations of the classical bin packing problem and knapsack problem and occur in many different settings. The input consists of a set I={_r1,…,_rn}$I=\{r_1,\dots ,r_n\}$ of d$d$-dimensional rectangular items _ri=(_ai,1,…,_ai,d)$r_i=(a_{i,1},\dots ,a_{i,d})$ and a space Q$Q$. The task is to pack the items in an orthogonal and non-overlapping manner without using rotations into the given space. In the strip packing setting the space Q$Q$ is given by a strip of bounded basis and unlimited height. The objective is to pack all items into a strip of minimal height. In the knapsack packing setting the given space Q$Q$ is a single, usually unit sized bin and the items have associated profits _pi$p_i$. The goal is to maximize the profit of a selection of items that can be packed into the bin.     

New bin packing fast lower bounds

In this paper, we address the issue of computing fast lower bounds for the Bin Packing problem, i.e., bounds that have a computational complexity dominated by the complexity of ordering the items by non-increasing values of their volume. We introduce new classes of fast lower bounds with improved asymptotic worst-case performance compared to well-known results for similar computational effort. Experimental results on a large set of problem instances indicate that the proposed bounds reduce both the deviation from the optimum and the computational effort.     

A new constraint programming approach for the orthogonal packing problem

The two-dimensional orthogonal packing problem (2OPP) consists in determining if a set of rectangles can be packed in a larger rectangle of fixed size. We propose an exact method for 2OPP, based on a new constraint-based scheduling model. We provide a generalization of energetic reasoning techniques for the problem under investigation. Feasibility tests requiring the solution of subset-sum problems are described. Computational results confirm the efficiency of our method compared to others in the literature.     

Efficient lower bounds and heuristics for the variable cost and size bin packing problem

We consider the variable cost and size bin packing problem, a generalization of the well-known bin packing problem, where a set of items must be packed into a set of heterogeneous bins characterized by possibly different volumes and fixed selection costs. The objective of the problem is to select bins to pack all items at minimum total bin-selection cost. The paper introduces lower bounds and heuristics for the problem, the latter integrating lower and upper bound techniques. Extensive numerical tests conducted on instances with up to 1000 items show the effectiveness of these methods in terms of computational effort and solution quality. We also provide a systematic numerical analysis of the impact on solution quality of the bin selection costs and the correlations between these and the bin volumes. The results show that these correlations matter and that solution methods that are un-biased toward particular correlation values perform better.     

求解二维正交矩形布局问题的动态填空启发式算法

为更高效解决二维正交矩形布局问题,建立该问题的数学模型,改进BL算法规则;为寻找布局过程中的空余平面,建立了新颖的图形矩阵化理论。最后提出一种动态填空（DFB）启发式算法,制定了四条动态调整机制,结合遗传算法对该问题进行求解。大量算例测试显示：DFB算法可达到100%的平面利用率,极大提高了BL算法的效率,并且可以适用于大规模布局问题。     

An efficient placement heuristic for three-dimensional rectangular packing

By embodying the spirit of “gold corner, silver side and strawy void” directly on the candidate packing place such that the searching space is reduced considerably, and by utilizing the characteristic of weakly heterogeneous problems that many items are in the same size, a fit degree algorithm (FDA) is proposed for solving a classical 3D rectangular packing problem, container loading problem. Experiments show that FDA works well on the complete set of 1500 instances proposed by Bischoff, Ratcliff and Davies. Especially for the 800 difficult strongly heterogeneous instances among them, FDA outperforms other algorithms with an average volume utilization of 91.91%, which to our knowledge is 0.45% higher than current best result just reported in 2010.     

Distributive relaxations for the spectral stokes operator

The technique of distributive relaxations is presented for the spectral Stokes operator. The sample iterative method is described and its equivalence to the Uzawa algorithm is shown. Furthermore the occurrence of the spectral Pseudo-Laplace operator is demonstrated. Numerical results are presented which show the good performance of the distributive relaxations.     

Perturbation bounds for orthogonal canonical forms and numericalcontrollability analysis

A first-order perturbation analysis of the orthogonal canonical forms of linear multivariable systems is presented. To avoid ill posedness, a numerical system structure which is invariant to small perturbations in the data is defined. It is shown that the orthogonal canonical forms may be, like the system controllability matrix, unreliable as a tool for numerical controllability analysis     

Lagrangian bounds for just-in-time job-shop scheduling

We study the job-shop scheduling problem with earliness and tardiness penalties. We describe two Lagrangian relaxations of the problem. The first one is based on the relaxation of precedence constraints while the second one is based on the relaxation of machine constraints. We introduce dedicated algorithms to solve the corresponding dual problems. The second one is solved by a simple dynamic programming algorithm while the first one requires the resolution of an NP-hard problem by branch and bound. In both cases, the relaxations allow us to derive lower bounds as well as heuristic solutions. We finally introduce a simple local search algorithm to improve the best solution found. Computational results are reported.     

求解online packing problem的F-B绝对近似算法

FF算法由于其在线特性在处理在线装箱问题得到广泛使用,但它无法预测后面达到物品造成装箱率低,提出一种预留一定比例的各类未装满箱体的装箱算法。首先对未装满箱体分类并给出相应的数据结构,接着设计一种绑定配对策略来预留各类未装满箱体数目,并引入间隔函数控制新箱体的启用,最后基于FF算法结合预留策略对物品进行装箱来保证装箱的绝对近似比。提出了一种预留绑定配对策略为后续输入物品提供预测空间,特别的是F-B算法能得到5/3的绝对近似比。     

Asymptotic fully polynomial approximation schemes for variants of open-end bin packing

We consider three variants of the open-end bin packing problem. Such variants of bin packing allow the total size of items packed into a bin to exceed the capacity of a bin, provided that a removal of the last item assigned to a bin would bring the contents of the bin below the capacity. In the first variant, this last item is the minimum sized item in the bin, that is, each bin must satisfy the property that the removal of any item should bring the total size of items in the bin below 1. The next variant (which is also known as lazy bin covering is similar to the first one, but in addition to the first condition, all bins (expect for possibly one bin) must contain a total size of items of at least 1. We show that these two problems admit asymptotic fully polynomial time approximation schemes (AFPTAS). Moreover, they turn out to be equivalent. We briefly discuss a third variant, where the input items are totally ordered, and the removal of the maximum indexed item should bring the total size of items in the bin below 1, and show that this variant is strongly NP-hard.     

Models and algorithms for packing rectangles into the smallest square

We consider the problem of determining the smallest square into which a given set of rectangular items can be packed without overlapping. We present an ILP model, an exact approach based on the iterated execution of a two-dimensional packing algorithm, and a randomized metaheuristic. Such approaches are valid both for the case where the rectangles have fixed orientation and the case where they can be rotated by 90°. We computationally evaluate the performance and the limits of the proposed approaches on a large set of instances, including a number of classical benchmarks from the literature, for both cases above, and for the special case where the items are squares.     

A heuristic algorithm for cube packing with time schedule

Packing problem has been proved to be an NP-hard problem. Many algorithms such as simulation annealing algorithm, genetic algorithm and other heuristic algorithms have been proposed to solve twodimensional and three-dimensional packing problem. To solve the cube packing problem with time schedule, this paper first introduces some concepts such as packing level, space distance and average neighbor birth order and then proposes a greedy algorithm. The algorithm tries every feasible corner greedily to calculat...     

Tree-decomposition based heuristics for the two-dimensional bin packing problem with conflicts

This paper deals with the two-dimensional bin packing problem with conflicts (BPC-2D). Given a finite set of rectangular items, an unlimited number of rectangular bins and a conflict graph, the goal is to find a conflict-free packing of the items minimizing the number of bins used. In this paper, we propose a new framework based on a tree-decomposition for solving this problem. It proceeds by decomposing a BPC-2D instance into subproblems to be solved independently. Applying this decomposition method is not straightforward, since merging partial solutions is hard. Several heuristic strategies are proposed to make an effective use of the decomposition. Computational experiments show the practical effectiveness of our approach.     

矩形件优化排样的混合启发式方法

提出一种启发式递归与遗传算法相结合的混合启发式算法求解矩形件优化排样问题。首先给出一种启发式递归算法,利用该算法逐个从待排矩形件中生成局部利用率高的条料,直到所有待排矩形件均生成条料;利用遗传算法全局搜索能力强的特点,对这些条料序进行搜索重组,使其所用的板材数最少;最后再次利用遗传算法,对条料生成之前的矩形件种类序进行全局最优搜索,使总的板材利用率达到了最大。对两个典型实际算例进行计算,并与相关文献比较,结果表明了该算法的有效性。     

A new heuristic algorithm for rectangle packing

The rectangle packing problem often appears in encasement and cutting as well as very large-scale integration design. To solve this problem, many algorithms such as genetic algorithm, simulated annealing and other heuristic algorithms have been proposed. In this paper, a new heuristic algorithm is recommended based on two important concepts, namely, the corner-occupying action and caving degree. Twenty-one rectangle-packing instances are tested by the algorithm developed, 16 of which having achieved optimum solutions within reasonable runtime. Experimental results demonstrate that the algorithm developed is fairly efficient for solving the rectangle packing problem.     

A faster parameterized algorithm for set packing

We present an efficient parameterized algorithm for the (k,t)-set packing problem, in which we are looking for a collection of k disjoint sets whose union consists of t elements. The complexity of the algorithm is O(²O(t)nNlogN). For the special case of sets of bounded size, this improves the O(k(ck)n) algorithm of Jia et al. [J. Algorithms 50 (1) (2004) 106].