A greedy search for the three-dimensional bin packing problem: the packing static stability case

We suggest a greedy search heuristic for solving the three-dimensional bin packing problem (3D-BPP) where in addition to the usual requirement of minimum amount of bins being used, the resulting packing of items into the bins must be physically stable. The problem is NP-hard in the strong sense and imposes severe computational strain for solving it in practice. Computational experiments are also presented and the results are compared with those obtained by the Martello, Pisinger and Vigo (2000) heuristic.     

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.     

一种新的多约束尺寸可变的装箱问题

多约束尺寸可变的装箱问题作为经典装箱问题的扩展,具有极为广泛的应用背景。在以货车运输为主的物流公司的装载环节中,运输成本不仅仅由车厢的空间利用率决定。分析了该类装箱问题与传统的集装箱装载问题的区别,并据此给出了一种新的尺寸可变装箱问题的定义。除了经典装箱问题中物品体积这一参数,还引入了物品类型、箱子类型等参数,建立了数学模型,将经典的FFD（First Fit Decreasing）算法进行了推广,提出了新的算法MFFD,并分析了相关的算法复杂性。最后对FF、FFD以及MFFD算法进行了模拟实验,实验结果表明,在相关参数符合均匀分布的条件下,MFFD算法效果较好。     

局内装箱算法综述

系统地介绍了局内装箱算法,归纳了其发展过程中的各种改进如数据分配模型、箱的划分等。阐述了该算法在工作分配、任务调度以及日常生活中的计划、包装、调度等计算机工程领域的应用。最后,对局内装箱算法提出了进一步的研究方向。     

Solving the variable size bin packing problem with discretized formulations

In this paper we study the use of a discretized formulation for solving the variable size bin packing problem (VSBPP). The VSBPP is a generalization of the bin packing problem where bins of different capacities (and different costs) are available for packing a set of items. The objective is to pack all the items minimizing the total cost associated with the bins. We start by presenting a straightforward integer programming formulation to the problem and later on, propose a less straightforward formulation obtained by using a so-called discretized model reformulation technique proposed for other problems (see [Gouveia L. A 2n constraint formulation for the capacitated minimal spanning tree problem. Operations Research 1995; 43:130–141; Gouveia L, Saldanha-da-Gama F. On the capacitated concentrator location problem: a reformulation by discretization. Computers and Operations Research 2006; 33:1242–1258]). New valid inequalities suggested by the variables of the discretized model are also proposed to strengthen the original linear relaxation bounds. Computational results (see Section 4) with up to 1000 items show that these valid inequalities not only enhance the linear programming relaxation bound but may also be extremely helpful when using a commercial package for solving optimally VSBPP.     

求解入库堆垛问题的改进约束满足算法

入库堆垛问题普遍存在于堆场作业管理中,是在货物数目和出库顺序已知的前提下,要求较长（重）的货物置于较短（轻）的货物下方,目标是实现占用垛位数最少。通过问题分析,将其归结为一类带顺序约束的A形装箱问题,并建立了约束满足模型,设计了嵌入经典装箱启发式的约束满足求解算法。实验表明,该算法对于求解复杂约束下的大规模堆场问题较现有的装箱启发式有一定程度的改善。     

互联网信息组织和规划中的带拒绝装箱问题

讨论如下定义的带拒绝装箱问题：设有许多等长的一维箱子，给定一个物品集，每个物品有两个参数：长度和罚值．物品可以放入箱子也可被拒绝放入箱子．如果将物品放入箱子，则使该箱剩余长度减少．一旦需将某一物品放入某一箱中，而该箱的剩余长度不够时，则需启用新箱子．如果物品被拒绝放入任何箱中，则产生惩罚．问怎样安排物品使所用箱子数与未装箱的物品总罚值之和最小．该问题是一个新的组合优化问题，来源于内部互联网的信息组织和规划．该文首先给出一个最优解值的下界估计，它可用于分枝定界法求最优解．由于该问题是强NP-难的，该文进一步研究它的离线和在线近似算法的设计与分析．文中给出一个离线算法，其绝对性能比为2；同时给出一个在线算法，其绝对性能比不超过3，渐近性能比为2，还对算法性能比的下界进行了讨论．     

Variable neighbourhood search for the variable sized bin packing problem

The variable sized bin packing problem is a generalisation of the one-dimensional bin packing problem. Given is a set of weighted items, which must be packed into a minimum-cost set of bins of variable sizes and costs. This problem has practical applications, for example, in packing, transportation planning, and cutting. In this work we propose a variable neighbourhood search metaheuristic for tackling the variable sized bin packing problem. The presented algorithm can be seen as a hybrid metaheuristic, because it makes use of lower bounding techniques and dynamic programming in various algorithmic components. An extensive experimentation on a diverse set of problem instances shows that the proposed algorithm is very competitive with current state-of-the-art approaches.     

A parallel multi-objective algorithm for two-dimensional bin packing with rotations and load balancing

Bin packing problems are NP-hard combinatorial optimization problems of fundamental importance in several fields, including computer science, engineering, economics, management, manufacturing, transportation, and logistics. In particular, the non-guillotine version of the single-objective two-dimensional bin packing problem with rotations is a highly complex scheduling problem that consists in packing a set of items into the minimum number of bins, where items can be rotated 90° and are characterized by having different heights and widths. Recently, some authors have proposed multi-objective formulations that also consider additional objectives, such as the balancing the bin load in order to increase its stability. The load imbalance minimization, which depends on the distribution of the items packed in them, is a critical point in many real applications. This paper analyzes how to solve two-dimensional bin packing problems with rotations and load balancing using parallel and multi-objective memetic algorithms that apply a set of search operators specifically designed to solve this problem. Results obtained using a set of test problems show the good performance of parallel and multi-objective memetic algorithms in comparison with other methods found in the literature.     

Robust hyper-heuristic algorithms for the offline oriented/non-oriented 2D bin packing problems

The offline 2D bin packing problem (2DBPP) is an NP-hard combinatorial optimization problem in which objects with various width and length sizes are packed into minimized number of 2D bins. Various versions of this well-known industrial engineering problem can be faced frequently. Several heuristics have been proposed for the solution of 2DBPP but it has not been possible to find the exact solutions for large problem instances. Next fit, first fit, best fit, unified tabu search, genetic and memetic algorithms are some of the state-of-the-art methods successfully applied to this important problem. In this study, we propose a set of novel hyper-heuristic algorithms that select/combine the state-of-the-art heuristics and local search techniques for minimizing the number of 2D bins. The proposed algorithms introduce new crossover and mutation operators for the selection of the heuristics. Through the results of exhaustive experiments on a set of offline 2DBPP benchmark problem instances, we conclude that the proposed algorithms are robust with their ability to obtain high percentage of the optimal solutions.     

Sensitivity analysis of the setup knapsack problem to perturbation of arbitrary profits or weights

The setup knapsack problem can be viewed as a more complex version of the well‐known classical knapsack problem. An instance of such a problem may be defined by a set of n items that is divided into m different classes For each class, only one item is considered as a setup item. The aim of the problem is to pack a subset of items in a knapsack of a predefined capacity that maximizes an objective function. In this paper, we analyze the sensitivity of an optimal solution depending on the variation of the profits or weights of arbitrary items. The optimality of the solution at hand is guaranteed by establishing the sensitivity interval that is characterized by both lower and upper values (called limits). First, two cases are distinguished when varying the profits: perturbation of the profit of an item (either ordinary or setup item) and, variation of the profits of a couple of items containing both setup and ordinary items belonging to the same class. Second, two cases are studied, where the perturbation concerns the weights: the variation is relied on the weight of an item and, the case of the variation of the weights of a subset of items. The established results are first illustrated throughout a didactic example, where both approximate and exact methods are used for analyzing the quality of the proposed results. Finally, an extended experimental part is proposed in order to evaluate the effectiveness of the proposed limits.     

Self-adaptive metaheuristics for solving a multi-objective 2-dimensional vector packing problem

In this paper, a multi-objective 2-dimensional vector packing problem is presented. It consists in packing a set of items, each having two sizes in two independent dimensions, say, a weight and a length into a finite number of bins, while concurrently optimizing three cost functions. The first objective is the minimization of the number of used bins. The second one is the minimization of the maximum length of a bin. The third objective consists in balancing the load overall the bins by minimizing the difference between the maximum length and the minimum length of a bin. Two population-based metaheuristics are performed to tackle this problem. These metaheuristics use different indirect encoding approaches in order to find good permutations of items which are then packed by a separate decoder routine whose parameters are embedded in the solution encoding. It leads to a self-adaptive metaheuristic where the parameters are adjusted during the search process. The performance of these strategies is assessed and compared against benchmarks inspired from the literature.     

半在线入库堆垛问题的动态求解算法

堆场垛位优化问题一直是仓储管理的难点和焦点之一,垛位优化可以保证物料装卸和出入库的高效率,同时对保证合同交货期也起着至关重要的作用。针对仓储和生产一体化下的入库堆垛问题,本文通过分析将其归结为一类半在线的A型装箱问题,并依据问题的特点,建立了最小化总倒垛次数的优化模型。根据货场天车在相邻入库过程中存在空闲作业量的特点,设计了一种前序货物允许移动的动态堆垛策略,结合堆垛约束后嵌入到经典装箱启发式算法中,最后通过仿真算例验证了该策略的有效性。     

Comments on the hierarchically structured bin packing problem

We study the hierarchically structured bin packing problem. In this problem, the items to be packed into bins are at the leaves of a tree. The objective of the packing is to minimize the total number of bins into which the descendants of an internal node are packed, summed over all internal nodes. We investigate an existing algorithm and make a correction to the analysis of its approximation ratio. Further results regarding the structure of an optimal solution and a strengthened inapproximability result are given.     

一种自适应优化松弛量的装箱算法

装箱问题是物流系统和生产系统中的一个经典而重要的数学优化问题。装箱指把一系列物品按照一定顺序放进具有固定容量的箱子中,并最小化所使用的箱子数量,以最大限度地获取装箱问题的近似最优解。然而,现有的装箱算法存在明显的缺陷。遗传算法计算量过大,甚至无法求出所需解,启发式算法无法处理极端值问题,而现有的改进算法即使在引入松弛量的情况下,也极易陷入局部最小值。文中提出的Adaptive-MBS算法采用自适应权重来改进原有方法,即允许方法有一定的松弛量,并具有捕捉物体样本空间随时间变化的直觉,以使用更好的松弛量策略来装箱。Adaptive-MBS算法首先以当前箱子为中心,使用Adaptive_Search搜索算法迭代找到适合箱子容量的集合中所有物体的子集,Adaptive_Search搜索算法不要求完全装满箱子,而是允许箱子具有一定的松弛量,在训练过程中根据当前状态的变化,实现自动地调整松弛量,在找到完全填满箱子的子集后迭代至下轮搜索直至遍历完成。该方法不易陷入局部最优,具有较强的发现全局最优解的能力。文中使用装箱问题中经典的BINDATA和SCH_WAE数据集进行实验,结果表明,数据集中多达991例问题可以通过Adaptive-MBS算法得到最优解。在没有求解出最优解的实例上,所提算法也在所有对比算法上具有最低的相对偏移量百分比。数值实验结果表明,相较于其他经典的装箱算法,Adaptive-MBS算法有更好的效果,其收敛速度也显著优于其他算法。     

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.     

Graph fragmentation problem: analysis and synthesis

Vulnerability metrics play a key role in the understanding of cascading failures and target/random attacks to a network. The graph fragmentation problem (GFP) is the result of a worst‐case analysis of a random attack. We can choose a fixed number of individuals for protection, and a nonprotected target node immediately destroys all reachable nodes. The goal is to minimize the expected number of destroyed nodes in the network. In this paper, we address the GFP by several approaches: metaheuristics, approximation algorithms, polytime methods for specific instances, and exact methods for small instances. The computational complexity of the GFP is included in our analysis, where we formally prove that the corresponding decision version of the problem is ‐complete. Furthermore, a strong inapproximability result holds: there is no polynomial approximation algorithm with factor lower than 5/3, unless . This promotes the study of specific instances of the problem for tractability and/or exact methods in exponential time. As a synthesis, we propose new vulnerability/connectivity metrics and an interplay with game theory using a closely related combinatorial problem called component order connectivity.     

An analysis of parameter adaptation in reactive tabu search

On‐line parameter adaptation schemes are widely used in metaheuristics. They are sometimes preferred to off‐line tuning techniques for two main reasons. First, they promise to achieve good performance even on new instance families that have not been considered during the design or the tuning phase of the algorithm. Second, it is assumed that an on‐line scheme could adapt the algorithm's behaviour to local characteristics of the search space. This paper challenges the second hypothesis by analysing the contribution of the parameter adaptation to the performance of a state‐of‐the‐art reactive tabu search () algorithm for the maximum clique problem. Our experimental analysis shows that this on‐line parameter adaptation scheme converges to good instance‐specific settings for the parameters, and that there is no evidence that it adapts to the local characteristics of the search space. The insights gained from the analysis are confirmed by further experiments with an algorithm for the quadratic assignment problem. Together, the results of the two algorithms shed some new light on the reasons behind the effectiveness of .     

Polynomial time approximation algorithms for proportionate open‐shop scheduling

This paper deals with the presentation of polynomial time (approximation) algorithms for a variant of open‐shop scheduling, where the processing times are only machine‐dependent. This variant of scheduling is called proportionate scheduling and its applications are used in many real‐world environments. This paper develops three polynomial time algorithms for the problem. First, we present a polynomial time algorithm that solves the problem optimally if , where n and m denote the numbers of jobs and machines, respectively. If, on the other hand, , we develop a polynomial time approximation algorithm with a worst‐case performance ratio of that improves the bound existing for general open‐shops. Next, in the case of , we take into account the problem under consideration as a master problem and convert it into a simpler secondary approximation problem. Furthermore, we formulate both the master and secondary problems, and compare their complexity sizes. We finally present another polynomial time algorithm that provides optimal solution for a special case of the problem where .