A goal-driven approach to the 2D bin packing and variable-sized bin packing problems

In this paper, we examine the two-dimensional variable-sized bin packing problem (2DVSBPP), where the task is to pack all given rectangles into bins of various sizes such that the total area of the used bins is minimized. We partition the search space of the 2DVSBPP into sets and impose an order on the sets, and then use a goal-driven approach to take advantage of the special structure of this partitioned solution space. Since the 2DVSBPP is a generalization of the two-dimensional bin packing problem (2DBPP), our approach can be adapted to the 2DBPP with minimal changes. Computational experiments on the standard benchmark data for both the 2DVSBPP and 2DBPP shows that our approach is more effective than existing approaches in literature.     

Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints

The paper examines a new problem in the irregular packing literature that has many applications in industry: two-dimensional irregular (convex) bin packing with guillotine constraints. Due to the cutting process of certain materials, cuts are restricted to extend from one edge of the stock-sheet to another, called guillotine cutting. This constraint is common place in glass cutting and is an important constraint in two-dimensional cutting and packing problems. In the literature, various exact and approximate algorithms exist for finding the two dimensional cutting patterns that satisfy the guillotine cutting constraint. However, to the best of our knowledge, all of the algorithms are designed for solving rectangular cutting where cuts are orthogonal with the edges of the stock-sheet. In order to satisfy the guillotine cutting constraint using these approaches, when the pieces are non-rectangular, practitioners implement a two stage approach. First, pieces are enclosed within rectangle shapes and then the rectangles are packed. Clearly, imposing this condition is likely to lead to additional waste. This paper aims to generate guillotine-cutting layouts of irregular shapes using a number of strategies. The investigation compares three two-stage approaches: one approximates pieces by rectangles, the other two approximate pairs of pieces by rectangles using a cluster heuristic or phi-functions for optimal clustering. All three approaches use a competitive algorithm for rectangle bin packing with guillotine constraints. Further, we design and implement a one-stage approach using an adaptive forest search algorithm. Experimental results show the one-stage strategy produces good solutions in less time over the two-stage approach.     

Fuzzy bin packing problem

This paper deals with the fuzzy bin packing problem that is a packing problem of non-rigid rectangles into an open rectangular bin. This problem is different from the conventional bin packing problem, which considers only rigid rectangles. The goal of the fuzzy bin packing problem is to minimize both the height of a packing and the extra cost due to the reduction of each piece. The total cost of the problem is represented as the sum of the height cost and the extra cost due to reductions of the pieces, which is called reduction cost. Because the conventional bin packing problem itself is an NP-hard problem, the presented optimization method assumes that an initial packing for non-reduced pieces has already been found. A closed form solution is presented for fuzzy bin packing problems, in which fuzzy numbers are triangular and the reduction cost is given by a quadratic function.     

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.     

Dynamic multi-dimensional bin packing

A natural generalization of the classical online bin packing problem is the dynamic bin packing problem introduced by Coffman et al. (1983) [7]. In this formulation, items arrive and depart and the objective is to minimize the maximal number of bins ever used over all times. We study the oriented multi-dimensional dynamic bin packing problem for two dimensions, three dimensions and multiple dimensions. Specifically, we consider dynamic packing of squares and rectangles into unit squares and dynamic packing of three-dimensional cubes and boxes into unit cubes. We also study dynamic d-dimensional hypercube and hyperbox packing. For dynamic d-dimensional box packing we define and analyze the algorithm NFDH for the offline problem and present a dynamic version. This algorithm was studied before for rectangle packing and for square packing and was generalized only for multi-dimensional cubes. We present upper and lower bounds for each of these cases.     

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

A hybrid grouping genetic algorithm for bin packing

The grouping genetic algorithm (GGA) is a genetic algorithm heavily modified to suit the structure of grouping problems. Those are the problems where the aim is to find a good partition of a set or to group together the members of the set. The bin packing problem (BPP) is a well known NP-hard grouping problem: items of various sizes have to be grouped inside bins of fixed capacity. On the other hand, the reduction method of Martello and Toth, based on their dominance criterion, constitutes one of the best OR techniques for optimization of the BPP to date.In this article, we first describe the GGA paradigm as compared to the classic Holland-style GA and the ordering GA. We then show how the bin packing GGA can be enhanced with a local optimization inspired by the dominance criterion. An extensive experimental comparison shows that the combination yields an algorithm superior to either of its components.     

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.     

The master bay plan problem: a solution method based on its connection to the three-dimensional bin packing problem

This paper addresses the problem of determining stowage plansfor containers in a ship, referred to as the Master Bay PlanProblem (MBPP). MBPP is NP-complete. We present a heuristic method for solvingMBPP based on its relation with the three-dimensional bin packingproblem (3D-BPP), where items are containers and bins are differentportions of the ship. Our aim is to find stowage plans, takinginto account structural and operational constraints relatedto both the containers and the ship, that minimize the timerequired for loading all containers on board. A validation of the proposed approach with some test casesis given. The results of real instances of the problem involvingmore than 1400 containers show the effectiveness of the proposedapproach for large scale applications.     

TSPACK: A two-level tabu search for the three-dimensional bin packing problem

Three-dimensional orthogonal bin packing is a problem NP-hard in the strong sense where a set of boxes must be orthogonally packed into the minimum number of three-dimensional bins. We present a two-level tabu search for this problem. The first-level aims to reduce the number of bins. The second optimizes the packing of the bins. This latter procedure is based on the Interval Graph representation of the packing, proposed by Fekete and Schepers, which reduces the size of the search space. We also introduce a general method to increase the size of the associated neighborhoods, and thus the quality of the search, without increasing the overall complexity of the algorithm. Extensive computational results on benchmark problem instances show the effectiveness of the proposed approach, obtaining better results compared to the existing ones.     

A 17/10-approximation algorithm fork-bounded space on-line variable-sized bin packing

A version of thek-bounded space on-line bin packing problem, where a fixed collection of bin sizes is allowed, is considered. By packing large items into appropriate bins and closing appropriate bins, we can derive an algorithm with worst-case performance bound 1.7 fork≥3. This research is supported by the Science Foundation under State Education Committee of China. The earlier version was done in Institute of Applied Mathematics, Academia Sinica.     

Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.     

New classes of fast lower bounds for bin packing problems

The bin packing problem is one of the classical NP-hard optimization problems. In this paper, we present a simple generic approach for obtaining new fast lower bounds, based on dual feasible functions. Worst-case analysis as well as computational results show that one of our classes clearly outperforms the previous best “economical” lower bound for the bin packing problem by Martello and Toth, which can be understood as a special case. In particular, we prove an asymptotic worst-case performance of 3/4 for a bound that can be computed in linear time for items sorted by size. In addition, our approach provides a general framework for establishing new bounds. Received: August 11, 1998 / Accepted: February 1, 2001?Published online September 17, 2001     

A 3-approximation algorithm for two-dimensional bin packing

In the classical two-dimensional bin packing problem one is asked to pack a set of rectangular items, without overlap and without any rotation, into the minimum number of identical square bins. We give an approximation algorithm with absolute worst-case ratio of 3.     

An agent-based approach to the two-dimensional guillotine bin packing problem

The two-dimensional guillotine bin packing problem consists of packing, without overlap, small rectangular items into the smallest number of large rectangular bins where items are obtained via guillotine cuts. This problem is solved using a new guillotine bottom left (GBL) constructive heuristic and its agent-based (A–B) implementation. GBL, which is sequential, successively packs items into a bin and creates a new bin every time it can no longer fit any unpacked item into the current one. A–B, which is pseudo-parallel, uses the simplest system of artificial life. This system consists of active agents dynamically interacting in real time to jointly fill the bins while each agent is driven by its own parameters, decision process, and fitness assessment. A–B is particularly fast and yields near-optimal solutions. Its modularity makes it easily adaptable to knapsack related problems.     

基于图着色模型的冲突装箱问题启发式算法

带有冲突关系装箱问题的优化目标是在满足货物冲突关系的前提下,使用数量最少的货箱完成货物装箱的目的。本文分析了冲突装箱问题的数学模型,提出了基于图着色模型的启发式算法进行求解。首先,使用冲突图来描述货物之间的冲突关系;其次,基于冲突图,采取图着色的方式将货物进行分组,并且组内的货物之间不存在冲突关系;最后,采取改进FFD算法对每组的货物进行装箱操作。实验表明,本文提出的启发式算法能够快速有效地找到问题的可行解,为此类装箱问题的求解提供了新思路。     

A hybrid heuristic algorithm for the 2D variable-sized bin packing problem

In this paper, we consider the two-dimensional variable-sized bin packing problem (2DVSBPP) with guillotine constraint. 2DVSBPP is a well-known NP-hard optimization problem which has several real applications. A mixed bin packing algorithm (MixPacking) which combines a heuristic packing algorithm with the Best Fit algorithm is proposed to solve the single bin problem, and then a backtracking algorithm which embeds MixPacking is developed to solve the 2DVSBPP. A hybrid heuristic algorithm based on iterative simulated annealing and binary search (named HHA) is then developed to further improve the results of our Backtracking algorithm. Computational experiments on the benchmark instances for 2DVSBPP show that HHA has achieved good results and outperforms existing algorithms.     

Resource augmented semi-online bounded space bin packing

We study on-line bounded space bin-packing in the resource augmentation model of competitive analysis. In this model, the on-line bounded space packing algorithm has to pack a list L of items with sizes in (0, 1], into a minimum number of bins of size b, b≥1. A bounded space algorithm has the property that it only has a constant number of active bins available to accept items at any point during processing. The performance of the algorithm is measured by comparing the produced packing with an optimal offline packing of the list L into bins of size 1. The competitive ratio then becomes a function of the on-line bin size b. Csirik and Woeginger studied this problem in [J. Csirik, G.J. Woeginger, Resource augmentation for online bounded space bin packing, Journal of Algorithms 44(2) (2002) 308-320] and proved that no on-line bounded space algorithm can perform better than a certain bound ρ(b) in the worst case. We relax the on-line condition by allowing a complete repacking within the active bins, and show that the same lower bound holds for this problem as well, and repacking may only allow one to obtain the exact best possible competitive ratio of ρ(b) having a constant number of active bins, instead of achieving this bound in the limit. We design a polynomial time on-line algorithm that uses three active bins and achieves the exact best possible competitive ratio ρ(b) for the given problem.     

A linear time algorithm for restricted bin packing and scheduling problems

In this paper we are concerned with the subproblem of bin packing, where the set of possible weights of elements is finite. In [5] it was mentioned that this problem could be solved by an exhaustive search procedure in polynomial time, but the degree of the polynomial is high and increases as the cardinality of the set of weights increases. However, we will show that a more careful analysis of the problem leads to a linear time algorithm. The impact of this result on task scheduling is discussed.     

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.