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.     

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.     

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.     

Iterative approaches for solving a multi-objective 2-dimensional vector packing problem

In this paper, we address a bi-objective 2-dimensional vector packing problem (Mo2-DBPP) that calls for packing a set of items, each having two sizes in two independent dimensions, say, a weight and a height, into the minimum number of bins. The weight corresponds to a “hard” constraint that cannot be violated while the height is a “soft” constraint. The objective is to find a trade-off between the number of bins and the maximum height of a bin. This problem has various real-world applications (computer science, production planning and logistics). Based on the special structure of its Pareto front, we propose two iterative resolution approaches for solving the Mo2-DBPP. In each approach, we use several lower bounds, heuristics and metaheuristics. Computational experiments are performed on benchmarks inspired from the literature to compare the effectiveness of the two approaches.     

Two-dimensional on-line bin packing problem with rotatable items

In this paper, we consider a two-dimensional version of the on-line bin packing problem, in which each rectangular item that should be packed into unit square bins is “rotatable” by $\text{90°}$. Two on-line algorithms for solving the problem are proposed. The second algorithm is an extension of the first algorithm, and the worst-case ratio of the second one is at least 2.25 and at most 2.565.     

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.     

Corner occupying theorem for the two-dimensional integral rectangle packing problem

This paper proves a corner occupying theorem for the two-dimensional integral rectangle packing problem,stating that if it is possible to orthogonally place n arbitrarily given integral rectangles into an integral rectangular container without overlapping,then we can achieve a feasible packing by successively placing a rectangle onto a bottom-left corner in the container.Based on this theorem,we might develop e?cient heuristic algorithms for solving the integral rectangle packing problem.In fact,as a vague conjecture,this theorem has been implicitly mentioned with different appearances by many people for a long time.     

New heuristics for one-dimensional bin-packing

Several new heuristics for solving the one-dimensional bin packing problem are presented. Some of these are based on the minimal bin slack (MBS) heuristic of Gupta and Ho. A different algorithm is one based on the variable neighbourhood search metaheuristic. The most effective algorithm turned out to be one based on running one of the former to provide an initial solution for the latter. When tested on 1370 benchmark test problem instances from two sources, this last hybrid algorithm proved capable of achieving the optimal solution for 1329, and could find for 4 instances solutions better than the best known. This is remarkable performance when set against other methods, both heuristic and optimum seeking.Scope and purposePacking items into boxes or bins is a task that occurs frequently in distribution and production. A large variety of different packing problems can be distinguished, depending on the size and shape of the items, as well as on the form and capacity of the bins (H. Dyckhoff and U. Finke, Cutting and Packing in Production and Distribution: a Typology and Bibliography, Springer, Berlin, 1992). Similar problems occur in minimising material wastage while cutting pieces into particular smaller ones and in the scheduling of identical processors in order to minimise total completion time. This work addresses the basic packing problem, known as the one-dimensional bin packing problem, where it is required to pack a number of items into the smallest possible number of bins of pre-specified equal capacity. Even though this problem is simple to state, it is NP hard, i.e., it is unlikely that there exists an algorithm that could solve every instance of it in polynomial time. Solution of more general realistic packing problems is probably contingent upon the availability of effective and computationally efficient solution procedures for the basic problem. In this work we present several heuristics capable of doing that. Extensive computational testing attests to the power of these heuristics, as well as to their computational efficiency.     

Three insertion heuristics and a justification improvement heuristic for two-dimensional bin packing with guillotine cuts

The problem of packing two-dimensional items into two-dimensional bins is considered in which patterns of items allocated to bins must be guillotine-cuttable and item rotation might be allowed (2BP|?|G)$(2\mathrm{BP}|?|\mathrm{G})$. Three new constructive heuristics, namely, first-fit insertion heuristic, best-fit insertion heuristic, and critical-fit insertion heuristic, and a new justification improvement heuristic are proposed. All new heuristics use tree structures to represent guillotine-cuttable patterns of items and proceed by inserting one item at a time in a partial solution. Central to all heuristics are a new procedure for enumerating possible insertions and a new fitness criterion for choosing the best insertion. All new heuristics have quadratic worst-case computational complexity except for the critical-fit insertion heuristic which has a cubic worst-case computational complexity. The efficiency and effectiveness of the proposed heuristics is demonstrated by comparing their empirical performance on a standard benchmark data set against other published approaches.     

Modified subset sum heuristics for bin packing

We analyze the worst-case ratio of natural variations of the so-called subset sum heuristic for the bin packing problem, which proceeds by filling one bin at a time, each as much as possible. Namely, we consider the variation in which the largest remaining item is packed in the current bin, and then the remaining capacity is filled as much as possible, as well as the variation in which all items larger than half the capacity are first packed in separate bins, and then the remaining capacity is filled as much as possible. For both variants, we show a nontrivial upper bound of 13/9 on the worst-case ratio, also discussing lower bounds on this ratio.     

The min-conflict packing problem

In the classical bin-packing problem with conflicts (BPC), the goal is to minimize the number of bins used to pack a set of items subject to disjunction constraints. In this paper, we study a new version of BPC: the min-conflict packing problem (MCBP), in which we minimize the number of violated conflicts when the number of bins is fixed. In order to find a tradeoff between the number of bins used and the violation of the conflict constraints, we also consider a bi-objective version of this problem. We show that the special structure of its Pareto front allows to reformulate the problem as a small set of MCBP. We solved these two problems through heuristics, column-generation methods, and a tabu search. Computational experiments are reported to assess the quality of our methods.     

A new variable-sized bin packing problem

The problem of BLASTing a genome against a database of DNA sequences to identify potential relationships with other genomes can be divided into subproblems quite naturally. We consider a setting where the problem is distributed to PCs having idle time. This results in a new variant of bin packing, where a rectangle is divided into smaller rectangles that are to be packed in variable-sized bins which arrive on-line. A rectangle fits in a bin, if the sum of its height and width is no more than the size of the bin. The goal is to minimize the total size of the bins used for packing the entire rectangle.     

On the generalized bin packing problem

The generalized bin packing problem (GBPP) is a novel packing problem arising in many transportation and logistic settings, characterized by multiple items and bins attributes and the presence of both compulsory and non‐compulsory items. In this paper, we study the computational complexity and the approximability of the GBPP. We prove that the GBPP cannot be approximated by any constant, unless . We also study the particular case of a single bin type and show that when an unlimited number of bins is available, the GBPP can be reduced to the bin packing with rejection (BPR) problem, which is approximable. We also prove that the GBPP satisfies Bellman's optimality principle and, exploiting this result, we develop a dynamic programming solution approach. Finally, we study the behavior of standard and widespread heuristics such as the first fit, best fit, first fit decreasing, and best fit decreasing. We show that while they successfully approximate previous versions of bin packing problems, they fail to approximate the GBPP.     

A GRASP/Path Relinking algorithm for two- and three-dimensional multiple bin-size bin packing problems

The three-dimensional multiple bin-size bin packing problem, MBSBPP, is the problem of packing a set of boxes into a set of bins when several types of bins of different sizes and costs are available and the objective is to minimize the total cost of bins used for packing the boxes. First we propose a GRASP algorithm, including a constructive procedure, a postprocessing phase and some improvement moves. The best solutions obtained are then combined into a Path Relinking procedure for which we have developed three versions: static, dynamic and evolutionary. An extensive computational study, using two- and three-dimensional instances, shows the relative efficiency of the alternatives considered for each phase of the algorithm and the good performance of our algorithm compared with previously reported results.     

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。     

An efficient heuristic algorithm for arbitrary shaped rectilinear block packing problem

Arbitrary shaped rectilinear block packing problem is a problem of packing a series of rectilinear blocks into a larger rectangular container, where arbitrary shaped rectilinear block is a polygonal block whose interior angle is either 90° or 270°. This problem involves many industrial applications, such as VLSI design, timber cutting, textile industry and layout of newspaper. Many algorithms based on different strategies have been presented to solve it. In this paper, we proposed an efficient heuristic algorithm which is based on principles of corner-occupying action and caving degree describing the quality of packing action. The proposed algorithm is tested on six instances from literatures and the results are rather satisfying. The computational results demonstrate that the proposed algorithm is rather efficient for solving the arbitrary shaped rectilinear block packing problem.     

A new heuristic algorithm for the circular packing problem with equilibrium constraints

The circular packing problem with equilibrium constraints is an optimization problem about simplified satellite module layout design.A heuristic algorithm based on tabu search is put forward for solving this problem.The algorithm begins from a random initial configuration and applies the gradient method with an adaptive step length to search for the minimum energy configuration.To jump out of the local minima and avoid the search doing repeated work,the algorithm adopts the strategy of tabu search.In the pr...     

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.     

Online square and cube packing

In online square packing, squares of different sizes arrive online and need to be packed into unit squares which are called bins. The goal is to minimize the number of bins used. Online cube packing is defined analogously. We show an upper bound of 2.2697 and a lower bound of 1.6406 for online square packing, and an upper bound of 2.9421 and a lower bound of 1.6680 for online cube packing. The upper bound for squares can be further reduced to 2.24437 using a computer proof. These results improve on the previously known results for the two problems. We also show improved lower bounds for higher dimensions.Preliminary versions of different parts of this paper appeared in Proc. Symp. Discr. Alg. 2004 (SODA 2004) and Proc. Eur. Symp. Alg. 2004 (ESA 2004).