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.     

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

In this paper, we consider the three-dimensional orthogonal bin packing problem, which is a generalization of the well-known bin packing problem. We present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature. The comparison is also done concerning asymptotic worst-case performance ratios. The new lower bounds can be more efficiently computed in polynomial time. In addition, we study the non-oriented model, which allows items to be rotated.     

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.     

Three-dimensional bin packing problem with variable bin height

This paper studies a variant of the three-dimensional bin packing problem (3D-BPP), where the bin height can be adjusted to the cartons it packs. The bins and cartons to be packed are assumed rectangular in shape. The cartons are allowed to be rotated into any one of the six positions that keep the carton edges parallel to the bin edges. This greatly increases the difficulty of finding a good solution since the search space expands significantly comparing to the 3D-BPP where the cartons have fixed orientations. A mathematical (mixed integer programming) approach is modified based on [Chen, C. S., Lee, S. M., Shen, Q. S., 1995. An analytical model for the container loading problem. European Journal of Operational Research 80 (1), 68–76] and numerical experiments indicate that the mathematical approach is not suitable for the variable bin height 3D-BPP. A special bin packing algorithm based on packing index is designed to utilize the special problem feature and is used as a building block for a genetic algorithm designed for the 3D-BPP. The paper also investigates the situation where more than one type of bin are used and provides a heuristic for packing a batch of cartons using the genetic algorithm. Numerical experiments show that our proposed method yields quick and satisfactory results when benchmarked against the actual packing practice and the MIP model with the latest version of CPLEX.     

Space defragmentation for packing problems

One of main difficulties of multi-dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed. This study proposes a defragmentation technique to combine the fragmented space into a continuous usable space, which potentially allows the packing of additional items. We illustrate the effectiveness of this technique using the two- and three-dimensional bin packing problem, where the aim is to load all given items (represented by rectangular boxes) into the minimum number of identical bins. Experimental results based on well-known 2D and 3D bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta-heuristic approaches for the problem. In conjunction with a bin shuffling strategy for incremental improvement, our resultant algorithm outperforms all leading meta-heuristic approaches based on the commonly used benchmark data by a significant margin.     

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.     

A new lower bound for the non-oriented two-dimensional bin-packing problem

We propose a new scheme for computing lower bounds for the non-oriented bin-packing problem when the bin is a square. It leads to bounds that theoretically dominate previous results. Computational experiments show that the bounds are tight. We also discuss the case where the bin is not a square.     

New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.     

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     

Consecutive Ones Matrices for Multi-dimensional Orthogonal Packing Problems

The multi-dimensional orthogonal packing problem (OPP) is a well studied decisional problem. Given a set of items with rectangular shapes, the problem is to decide whether there is a non-overlapping packing of these items in a rectangular bin. The rotation of items is not allowed. A powerful caracterization of packing configurations by means of interval graphs was recently introduced. In this paper, we propose a new algorithm using consecutive ones matrices as data structure. This new algorithm is then used to solve the two-dimensional orthogonal knapsack problem. Computational results are reported, which show its effectiveness.     

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.     

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.     

On-line bin packing — A restricted survey

In the classical bin packing problem, one is asked to pack items of various sizes into the minimum number of equal-sized bins. In the on-line version of this problem, the packer is given the items one by one and must immediately and irrevocably assign every item to its bin, without knowing the future items. Beginning with the first results in the early 1970's, we survey — from the worst case point of view — the approximation results obtained for on-line bin packing, higher dimensional versions of the problem, lower bounds on worst case ratios and related results.This work was partially supported by the Christian Doppier Laboratorium für Diskrete Optimierung.     

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.     

Relaxations and exact solution of the variable sized bin packing problem

We address a generalization of the classical one-dimensional bin packing problem with unequal bin sizes and costs. We investigate lower bounds for this problem as well as exact algorithms. The main contribution of this paper is to show that embedding a tight network flow-based lower bound, dominance rules, as well as an effective knapsack-based heuristic in a branch-and-bound algorithm yields very good performance. In addition, we show that the particular case with all weight items larger than a third the largest bin capacity can be restated and solved in polynomial-time as a maximum-weight matching problem in a nonbipartite graph. We report the results of extensive computational experiments that provide evidence that large randomly generated instances are optimally solved within moderate CPU times.     

Fast algorithms for fragmentable items bin packing

Bin packing with fragmentable items is a variant of the classic bin packing problem where items may be cut into smaller fragments. The objective is to minimize the number of item fragments, or equivalently, to minimize the number of cuts, for a given number of bins. Models based on packing fragmentable items are useful for representing finite shared resources. In this article, we present improvements to approximation and metaheuristic algorithms to obtain an optimality-preserving optimization algorithm with polynomial complexity, worst-case performance guarantees and parametrizable running time. We also present a new family of fast lower bounds and prove their worst-case performance ratios. We evaluate the performance and quality of the algorithm and the best lower bound through a series of computational experiments on representative problem instances. For the studied problem sets, one consisting of 180 problems with up to 20 items and another consisting of 450 problems with up to 1024 items, the lower bound performs no worse than 5 / 6. For the first problem set, the algorithm found an optimal solution in 92 % of all 1800 runs. For the second problem set, the algorithm found an optimal solution in 99 % of all 4500 runs. No run lasted longer than 220 ms.     

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.     

Relations between capacity utilization,minimal bin size and bin number

We consider the two-dimensional bin packing problem given a set of rectangular items, find the minimal number of rectangular bins needed to pack all items. Rotation of the items is not permitted. We show for any integer \({k} \ge 3\) that at most \({k}-1\) bins are needed to pack all items if every item fits into a bin and if the total area of items does not exceed \({k}/4\) -times the bin area. Moreover, this bound is tight. Furthermore, we show that only two bins are necessary to pack all items if the total area of items is not larger than the bin area, and if the height of each item is not larger than a third of the bin height and the width of every item does not exceed half of the bin width.     

A note on a selfish bin packing problem

In this paper, we consider a selfish bin packing problem, where each item is a selfish player and wants to minimize its cost. In our new model, if there are k items packed in the same bin, then each item pays a cost 1/k, where k ≥ 1. First we find a Nash Equilibrium (NE) in time O(n log n) within a social cost at most 1.69103OPT + 3, where OPT is the social cost of an optimal packing; where n is the number of items or players; then we give tight bounds for the worst NE on the social cost; finally we show that any feasible packing can be converged to a Nash Equilibrium in O(n ²) steps without increasing the social cost.     

A new exact method for the two-dimensional orthogonal packing problem

The two-dimensional orthogonal packing problem (2OPP) consists in determining if a set of rectangles (items) can be packed into one rectangle of fixed size (bin). In this paper we propose two exact algorithms for solving this problem. The first algorithm is an improvement on a classical branch&bound method, whereas the second algorithm is based on a new relaxation of the problem. We also describe reduction procedures and lower bounds which can be used within enumerative methods. We report computational experiments for randomly generated benchmarks which demonstrate the efficiency of both methods: the second method is competitive compared to the best previous methods. It can be seen that our new relaxation allows an efficient detection of non-feasible instances.