One-dimensional Bin Packing Problem Research Articles

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

We consider the one-dimensional bin packing problem under the discrete uniform distributions $U\{j,k\}$, $1 \leq j \leq k-1$, in which the bin capacity is $k$ and item sizes are chosen uniformly from the set $\{1,2,\ldots,j\}$. Note that for $0 < u = j/k \leq 1$ this is a discrete version of the previously studied continuous uniform distribution $U(0,u]$, where the bin capacity is 1 and item sizes are chosen uniformly from the interval $(0,u]$. We show that the average-case performance of heuristics can differ substantially between the two types of distributions. In particular, there is an online algorithm that has constant expected wasted space under $U\{j,k\}$ for any $j,k$ with $1 \leq j < k-1$, whereas no online algorithm can have $o(n^{1/2})$ expected waste under $U(0,u]$ for any $0 < u \leq 1$. Our $U\{j,k\}$ result is an application of a general theorem of Courcoubetis and Weber that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of $n$ items must be either $\Theta (n)$, $\Theta (n^{1/2} )$, or $O(1)$, depending on whether certain ``perfect'' packings exist. The perfect packing theorem needed for the $U\{j,k\}$ distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper. We also survey other recent results comparing the behavior of heuristics under discrete and continuous uniform distributions.

A SCANLINE-BASED ALGORITHM FOR THE 2D FREE-FORM BIN PACKING PROBLEM

This paper describes a heuristic algorithm for the two-dimensional free-form bin packing (2D-FBP) problem, which is also called the irregular cutting and packing, or nesting problem. Given a set of 2D free-form bins, which in practice may be plate materials, and a set of 2D free-form items, which in practice may be plate parts to be cut out of the materials, the 2D-FBP problem is to lay out items inside one or more bins in such a way that the number of bins used is minimized, and for each bin, the yield is maximized. The proposed algorithm handles the problem as a variant of the one-dimensional bin-packing problem; i.e., items and bins are approximated as sets of scanlines, and scanlines are packed. The details of the algorithm are given, and its application to a nesting problem in a shipbuilding company is reported. The proposed algorithm consists of the basic and the group placement algorithms. The basic placement algorithm is a variant of the first-fit decreasing algorithm which is simply extended from the one-dimensional case to the two-dimensional case by a novel scanline approximation. The group placement algorithm is an extension of the basic placement algorithm with recombination of input items. A numerical study with real instances shows that the basic placement algorithm has sufficient performance for most of the instances, however the group placement algorithm is required when items must be aligned in columns. The qualities of the resulting layouts are good enough for practical use, and the processing times required for both algorithms are much faster than those by manual nesting.

Variable-Sized Bin Packing: Tight Absolute Worst-Case Performance Ratios for Four Approximation Algorithms

In this paper we consider a one-dimensional bin packing problem where the bins do not have identical sizes, or a variable-sized bin packing problem, to minimize the bin consumption, i.e., the total size of the opened bins. The identical size problem has been extensively studied in the literature both for static and dynamic settings. The worst-case or average-case performance has been analyzed. Our problem setting particularly arises in metal cutting industries. Therefore, it presents a great practical relevance. Four greedy approximation algorithms based on a construction approach called largest object first with least absolute waste (LFLAW), largest object first with least relative waste (LFLRW), least absolute waste (LAW), and least relative waste (LRW) are examined. Their absolute worst-case performances are analyzed. The worst case bounds are 2 for LFLAW and LFLRW, 3 for LAW, and 2+ln 2 for LRW. We also show that these worst-case bounds are tight.

A note on an open-end bin packing problem

We consider a variant of the classical one-dimensional bin packing problem, which we call the open-end bin packing problem. Suppose that we are given a list L=(p1,p2,…,pn) of n pieces, where pj denotes both the name and the size of the jth piece in L, and an infinite collection of infinite-capacity bins. A bin can always accommodate a piece if the bin has not yet reached a level of C or above, but it will be closed as soon as it reaches that level. Our goal is to find a packing that uses the minimum number of bins. In this article, we first show that the open-end bin packing problem remains strongly NP-hard. We then show that any online algorithm must have an asymptotic worst-case ratio of at least 2, and there is a simple online algorithm with exactly this ratio. Finally, we give an offline algorithm that is a fully polynomial approximation scheme with respect to the asymptotic worst-case ratio. Copyright © 2001 John Wiley & Sons, Ltd.

A 5/4 Linear Time Bin Packing Algorithm

In 1985, Martel published a linear time algorithm with a 43 asymptotic worst-case ratio for the one-dimensional bin packing problem. The algorithm is based on a linear time classification of the sizes of the items, and thereafter—according to the number of elements in certain subclasses—pairing the items in a clever way. In his paper Martel mentioned a natural generalization of this algorithm, that suggested a 54 algorithm. Although this seemed to be very simple, the improvement has not been realized until now. In this paper we present an algorithm which uses the ideas of Martel and has a 54 asymptotic worst-case ratio.

Bin Packing with Discrete Item Sizes, Part I: Perfect Packing Theorems and the Average Case Behavior of Optimal Packings

We consider the one-dimensional bin packing problem with unit-capacity bins and item sizes chosen according to the discrete uniform distribution U{j,k}, $1 < j \leq k,$ where each item size in {1/k,2/k,. . .,j/k} has probability 1/j of being chosen. Note that for fixed j,k as $m\rightarrow\infty$ the discrete distributions U{mj,mk} approach the continuous distribution U(0,j/k], where the item sizes are chosen uniformly from the interval (0,j/k]. We show that average-case behavior can differ substantially between the two types of distributions. In particular, for all j,k with j < k-1, there exist on-line algorithms that have constant expected wasted space under U{j,k}, whereas no on-line algorithm has even o(n1/2 ) expected waste under U(0,u] for any $0 < u \leq 1$. Our U{j,k} result is an application of a general theorem of Courcoubetis and Weber [C. Courcoubetis and R.R. Weber, Probab. Engrg. Inform. Sci., 4 (1990), pp. 447--460] that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of n items must be either $\Theta (n)$, $\Theta (n^{1/2} )$, or O(1), depending on whether certain"perfect" packings exist. The perfect packing theorem needed for the U{j,k} distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper.

Maximizing the number of mixed packages subject to variety constraints

We develop a polynomial-time algorithm to optimise a variant of the one-dimensional bin-packing problem with side constraints. We also develop a pseudo-polynomial procedure to actually implement that optimal solution. The specific application is the allocation of excess of a population of various types of cards (e.g., left over from a previous selling season) to fixed-sized “variety packs” which guarantee a given level of variety (i.e., no more than k of any type of card). Some card types with large numbers (perhaps the most unpopular from the previous season) may have to be discarded to preserve the variety constraint. The method developed employs a test for feasibility of a given number of packs and includes a simple allocation procedure. A numerical example is provided along with (worst-case) complexity calculations. In addition, we solve a practical problem in which an organisation marketing Christmas cards sought to determine the impact of pack size and variety level on the level of unallocated cards. Scope and purpose Organisations involved in the stocking and sale of seasonal “style” items such as greeting cards, may periodically face excess stock of various items (e.g., from previous seasons), and may decide to market the items in packs guaranteed to contain a certain degree of variety. Our objective in this paper is to show how to maximise the number of card packs that can be formed from an assortment of excess stock, where there is a marketing-based variety constraint restricting the number of each type of card in each pack. The solution procedure developed is intuitively appealing and can be easily implemented on a spreadsheet. In addition to presenting a numerical example, we provide results from the application and implementation of the method in the card sales operations of a charitable organisation. The method could also have broader application in other settings where variety is sought – such as in groups or teams.

A new heuristic algorithm for the one-dimensional bin-packing problem

We describe a new heuristic algorithm to solve the one-dimensional bin-packing problem. The proposed algorithm is optimal if the sum of requirements of items is less than or equal to twice the bin capacity. Our computational results show that effectiveness of the proposed algorithm in finding optimal or near-optimal solutions is superior to that of the FFD and BFD algorithms, specifically for those so called 'difficult' problems that require an optimal solution to fill most of the bins, if not all, exactly to capacity.

Bison: A fast hybrid procedure for exactly solving the one-dimensional bin packing problem

In this paper, we consider the well-known one-dimensional bin packing problem (BPP-1), which is to pack a given set of items having different sizes into a minimum number of equal-sized bins. For solving BPP-1, an exact hybrid solution procedure, called BISON, is proposed. It favourably combines the well-known meta-strategy tabu search and a branch and bound procedure based on known and new bound arguments and a new branching scheme. Computational results indicate that BISON is very effective and outperforms existing approaches.

An improved lower bound for the bin packing problem

This paper unifies and generalizes the existing lower bounds for the one-dimensional bin packing problem. The generalization is motivated by and based on the work of Martello and Toth (this journal, 1990). The worst-case performance of the unified lower bound is analyzed and two new lower bounds are proposed and compared with existing lower bounds through numerical experiments.

A tight lower bound for optimal bin packing

In this paper, we present an O( n log n) algorithm to compute a tight lower bound for the one-dimensional bin packing problem. We have simulated the algorithm on randomly generated bin packing problems with item sizes drawn uniformly from ( a, b), where 0 ⩽ a < b ⩽ B and B is bin size. Using our lower bound, the average error of BFD is less than 2%. For a + b ⩾ B, the error is less than 0.003%.

Bin packing problems in one dimension: Heuristic solutions and confidence intervals

The one-dimensional bin packing problem has many applications to problems of data management, scheduling and resource allocation. Very large examples of this problem are likely to exceed the limits of computational tractability, and in such circumstances the use of approximate (heuristic) methods is suggested. We adapt a variety of heuristic methods for generating feasible solutions, and the results are then used to generate confidence intervals for the (in practice unknown) value of the optimal solution. A large-scale computational study for randomly generated problems with prespecified optimal values suggests that these intervals are extremely narrow and are also highly likely to contain the optimal solution value. The interval estimation procedure seems to generate much better results for the bin packing problem than for other problems tested in the literature.

A stochastic analysis of the NFD bin-packing algorithm

We consider the Next-Fit-Decreasing (NFD) algorithm for the one-dimensional bin packing problem. The input consists of n pieces of i.i.d. sizes. Bounds on the first two moments of the number of bins used in the packing are established for arbitrary size distributions. These bounds are asymptotically tight. For the uniform distribution we also exhibit numerical results and examine their dependence on the support of the size distribution.

An O(n) bin-packing algorithm for uniformly distributed data

We give a first-fit type algorithm, with running timeO(n), for the classical one-dimensional bin-packing problem, and examine it from a probabilistic point of view. Our main result is that the expected waste for this algorithm isO(n2/3).

Parametric Lower Bound for On-Line Bin-Packing

In this note we give lower bounds for such a one-dimensional bin-packing problem, in which we can use only one-line rules to pack the elements, and the maximal size of the elements are bounded. Our lower bound contains, as a special case, the result given by Liang [Inform. Proc. Lett., 10 (1980), pp. 76–79].

On a dual version of the one-dimensional bin packing problem

The NP-hard problem of packing items from a given set into bins so as to maximize the number of bins used, subject to the constraint that each bin be filled to at least a given threshold, is considered. Approximation algorithms are presented that provide guarantees of 1 2 , 2 3 , and 3 4 the optimal number, at running time costs of O( n), O( nlog n), and O( nlog 2 n), respectively, and the average case behavior of these algorithms is explored via empirical tests on randomly generated sets of items.

A class of simple stochastic online bin packing algorithms

In the one-dimensional bin packing problem a list ofn items has to be packed into a minimum number of unit-capacity bins. A class of linear online algorithms for the approximate solution of bin packing with items drawn from a known probability distribution is presented. Each algorithm depends on the distribution and on a parameter controlling the performance of the algorithm. It is shown that with increasing number of items the expected performance ratio has an arbitrary small deviation from optimum.

Bin packing: Maximizing the number of pieces packed

We consider a variant of the classical one-dimensional bin-packing problem: The number of bins is fixed and the object is to maximize the number of pieces packed from some given set. Both problems have applications in processor and storage allocation in computer systems in addition to a broad application in operations research. It can easily be shown that both problems are NP-complete; our approach will be to propose and analyze very fast heuristics. We consider a class of algorithms and bound the performance of an arbitrary algorithm in that class. Finally we propose an algorithm, the first-fit-increasing algorithm, and analyze its running time and relative performance.