Strip Packing Problem Research Articles

A STRIP PACKING SOLVING METHOD USING AN INCREMENTAL MOVE BASED ON MAXIMAL HOLES

When handling 2D packing problems, numerous incomplete and complete algorithms maintain a so-called bottom-left (BL) property: no rectangle placed in a container can be moved more left or bottom. While it is easy to make a rectangle BL when it is added in a container, it is more expensive to maintain all the placed pieces BL when a rectangle is removed. This prevents researchers from designing incremental moves for metaheuristics or efficient complete optimization algorithms. This paper investigates the possibility of violating the BL property. Instead, we propose to maintain the set of maximal holes, which allows incremental additions and removals of rectangles. To validate our alternative approach, we have designed an incremental move, maintaining maximal holes, for the strip packing problem, a variant of the famous 2D bin-packing. We have also implemented a metaheuristic, with no user-defined parameter, using this move and standard greedy heuristics. We have finally designed two variants of this incomplete method. In the first variant, a better first layout is provided by a hyperheuristic proposed by some of the authors. In the second variant, a fast repacking procedure recovering the BL property is occasionally called during the local search. Experimental results show that the approach is competitive with the best known incomplete algorithms.

Offline and Online Aspects of Defragmenting the Module Layout of a Partially Reconfigurable Device

Modern generations of field-programmable gate arrays (FPGAs) allow for partial reconfiguration. In an online context, where the sequence of modules to be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of modules leads to progressive fragmentation of the available space, making defragmentation an important issue. We address this problem by proposing an online and an offline component for the defragmentation of the available space. We consider defragmenting the module layout on a reconfigurable device. This corresponds to solving a 2D strip packing problem. Problems of this type are NP-hard in the strong sense, and previous algorithmic results are rather limited. Based on a graph-theoretic characterization of feasible packings, we develop a method that can solve 2D defragmentation instances of practical size to optimality. Our approach is validated for a set of benchmark instances. We also discuss a simple strategy for dealing with online scenarios, called ldquoleast-interference fitrdquo (LIF); we give a number of analytic results that allow a comparison of LIF with the best offline solution, and demonstrate that it works well on benchmark instances of moderate size.

Recursive algorithm for generating layer patterns for strip packing

The strip packing problem was studied.The algorithm divided the strip into layers with horizontal cuts.The width of a layer was equal to the strip width,and the height was the same as that of the leftmost item in the layer.The algorithm used a recursive procedure to arrange the layers on the strip,selected the leftmost item for each layer,and determined the item layout on each layer.Branch-and-bound techniques were used to shorten the searching space.The computational results indicate that the algorithm is more efficient than several recent algorithms.

A recursive algorithm for the rectangular guillotine strip packing problem

This article presents a recursive heuristic algorithm to generate cutting patterns for the rectangular guillotine strip packing problem in which a set of rectangular items must be cut from the strip such that the consumed strip length is minimized. The strip is placed with its length along the horizontal direction, and is divided into several segments with vertical cuts. The length of a segment is determined by the item placed at the bottom. Orthogonal cuts divide the segments into blocks and finished items. For the current block considered, the algorithm selects an item, puts it at the bottom-left corner of the block, and divides the unoccupied region into two smaller blocks with an orthogonal cut. Rotation of the items by 90 is allowed. Both lower and upper bounds are used to prune unpromising branches. The computational results indicate that the algorithm performs better than several recently published algorithms.

A branch and bound algorithm for the strip packing problem

We propose a new branch and bound algorithm for the two dimensional strip packing problem, in which a given set of rectangular pieces have to be packed into a strip of given width and infinite length so as to minimize the required height of the packing. We develop lower bounds based on integer formulations of relaxations of the problem as well as new bounds based on geometric considerations, and reduce the tree search with some dominance criteria. An extensive computational study shows the relative efficiency of the bounds and the good performance of the exact algorithm.

A heuristic for the three-dimensional strip packing problem

The contribution presents a heuristic for the three-dimensional strip packing problem (3D-SPP) with rectangular pieces (boxes). The considered 3D-SPP can be formulated as follows: for a given set of boxes and a given longitudinal open container, determine an arrangement of all boxes within the container so that the required container length is minimized. The presented heuristic was derived from a branch-and-bound approach for the container loading problem (CLP) that was recently proposed by Pisinger. Two approaches are investigated to adapt the CLP method to the 3D-SPP. The first approach looks directly at a container of practically infinite length. The second one solves an SPP instance by computing a series of CLP instances with descending container lengths. The best parts of previously best solutions are reused systematically within the second approach that proves to be more successful. The method is tested by means of 800 benchmark instances with up to 1000 boxes. A comparison to other methods from the literature shows the high performance of the 3D-SPP heuristic.

A branch-and-price algorithm for the two-dimensional level strip packing problem

The two-dimensional level strip packing problem (2LSPP) consists in packing rectangular items of given size into a strip of given width divided into levels. Items packed into the same level cannot be put on top of one another and their overall width cannot exceed the width of the strip. The objective is to accommodate all the items while minimizing the overall height of the strip. The problem is \({{\mathcal{NP}}}\) -hard and arises from applications in logistics and transportation. We present a set covering formulation of the 2LSPP suitable for a column generation approach, where each column corresponds to a feasible combination of items inserted into the same level. For the exact optimization of the 2LSPP we present a branch-and-price algorithm, in which the pricing problem is a penalized knapsack problem. Computational results are reported for benchmark instances with some hundreds items.

A recursive branch-and-bound algorithm for the rectangular guillotine strip packing problem

A heuristic recursive algorithm for the two-dimensional rectangular strip packing problem is presented. It is based on a recursive structure combined with branch-and-bound techniques. Several lengths are tried to determine the minimal plate length to hold all the items. Initially the plate is taken as a block. For the current block considered, the algorithm selects an item, puts it at the bottom-left corner of the block, and divides the unoccupied region into two smaller blocks with an orthogonal cut. The dividing cut is vertical if the block width is equal to the plate width; otherwise it is horizontal. Both lower and upper bounds are used to prune unpromising branches. The computational results on a class of benchmark problems indicate that the algorithm performs better than several recently published algorithms.

Reactive GRASP for the strip-packing problem

This paper presents a greedy randomized adaptive search procedure (GRASP) for the strip packing problem, which is the problem of placing a set of rectangular pieces into a strip of a given width and infinite height so as to minimize the required height. We investigate several strategies for the constructive and improvement phases and several choices for critical search parameters. We perform extensive computational experiments with well-known instances which have been previously reported, first to select the best alternatives and then to compare the efficiency of our algorithm with other procedures. The results show that the GRASP algorithm outperforms recently reported metaheuristics.

A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces

Given a set of rectangular pieces and a container of fixed width and variable length, the two-dimensional strip packing problem (2D-SPP) consists of orthogonally placing all the pieces within the container, without overlapping, such that the overall length of the layout is minimised. Until now mainly heuristics, for example genetic algorithms (GA), were proposed for the 2D-SPP which use encoded solutions that are manipulated by standard operators. In this paper a GA for the 2D-SPP is suggested that works without any encoding of solutions. Rather fully defined layouts are manipulated as such by means of specific genetic operators. Two additional constraints, namely the orientation constraint and the guillotine constraint, can be taken into account. The GA is subjected to a comprehensive test using benchmark instances with up to 5000 pieces. Compared to eleven competing methods from the literature the GA performs best.

Solving Irregular Strip Packing problems by hybridising simulated annealing and linear programming

In this paper a hybrid algorithm to solve Irregular Strip Packing problems is presented. The metaheuristic simulated annealing is used to guide the search over the solution space while linear programming models are solved to generate neighbourhoods during the search process. These linear programming models, which are used to locally optimise the layouts, derive from the application of compaction and separation algorithms. Computational tests were run using instances that are commonly used as benchmarks in the literature. The best results published so far have been improved by this new hybrid packing algorithm.

This side up!

We consider two- and three-dimensional bin-packing problems where 90° rotations are allowed. We improve all known asymptotic performance bounds for these problems. In particular, we show how to combine ideas from strip packing and two-dimensional bin packing to give a new algorithm for the three-dimensional strip packing problem where boxes can only be rotated sideways. We propose to call this problem “This side up”. Our algorithm has an asymptotic performance bound of 9/4.

New heuristic and interactive approaches to 2D rectangular strip packing

In this paper, we consider the two-dimensional rectangular strip packing problem. A standard simple heuristic, Bottom-Left-Decreasing (BLD), has been shown to perform quite well in practice. We introduce and demonstrate the effectiveness of BLD*, a stochastic search variation of BLD. While BLD places the rectangles in decreasing order of height, width, area, and perimeter, BLD* successively tries random orderings, chosen from a distribution determined by their Kendall-tau distance from one of these fixed orderings. Our experiments on benchmark problems show that BLD* produces significantly better packings than BLD after only 1 min of computation. Furthermore, we also show that BLD* outperforms recently reported metaheuristics. Furthermore, we observe that people seem able to reason about packing problems extremely well. We incorporate our new algorithms in an interactive system that combines the advantages of computer speed and human reasoning. Using the interactive system, we are able to quickly produce significantly better solutions than BLD* by itself.

Two- and three-dimensional parametric packing

We present approximation algorithms for the two- and three-dimensional bin packing problems and the three-dimensional strip packing problem. We consider the special case of these problems in which a parameter m (a positive integer) is given, indicating that each of the dimensions of the items to be packed is at most 1 / m of the corresponding dimension of the recipient. We analyze the asymptotic performance of these algorithms and exhibit bounds that, to our knowledge, are the best known for this special case.

Two- and three-dimensional parametric packing

We present approximation algorithms for the two- and three-dimensional bin packing problems and the three-dimensional strip packing problem. We consider the special case of these problems in which a parameter m (a positive integer) is given, indicating that each of the dimensions of the items to be packed is at most 1/m of the corresponding dimension of the recipient. We analyze the asymptotic performance of these algorithms and exhibit bounds that, to our knowledge, are the best known for this special case.

Average-Case Performance Analysis of a 2D Strip Packing Algorithm?NFDH

The two-dimensional strip packing problem is a generalization of the classic one-dimensional bin packing problem. It has many important applications such as costume clipping, material cutting, real-world planning, packing, scheduling etc. Average-case performance analysis of approximation algorithms attracts a lot of attention because it plays a crucial role in selecting an appropriate algorithm for a given application. While approximation algorithms for two-dimensional packing are frequently presented, the results of their average-case performance analyses have seldom been reported due to intractability. In this paper, we analyze the average-case performance of Next Fit Decreasing Height (NFDH) algorithm, one of the first strip packing algorithms proposed by Coffman, Jr. in 1980. We prove that the expected height of packing with NFDH algorithm, when the heights and widths of the rectangle items are independent and both obey (0, 1] uniform distribution, is about n/3, where n is the number of rectangle items. We also validate the theoretical result with experiments.

Models and Bounds for Two-Dimensional Level Packing Problems

We consider two-dimensional bin packing and strip packing problems where the items have to be packed by levels. We introduce new mathematical models involving a polynomial number of variables and constraints, and show that their LP relaxations dominate the standard area relaxations. We then propose new (combinatorial) bounds that can be computed in O(nlog n) time. We show that they dominate the other bounds, and establish their absolute worst-case behavior. The quality of models and bounds is evaluated through extensive computational experiments.

The two-dimensional cutting stock problem revisited

In the strip packing problem (a standard version of the two-dimensional cutting stock problem), the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed. The k-stage Guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing. We present a complete analysis of the quality of k-stage Guillotine strip packings versus globally optimal packings: k=2 stages cannot guarantee any bounded asymptotic performance ratio. k=3 stages lead to asymptotic performance ratios arbitrarily close to 1.69103; this bound is tight. Finally, k=4 stages yield asymptotic performance ratios arbitrarily close to 1.

Strip-packing using hybrid genetic approach

A hybrid genetic approach is proposed for the two-dimensional strip packing problems. With a combination of genetic algorithm and a novel heuristic allocation method, the packing problem is transformed into a simple permutation problem which can be effectively solved by a genetic algorithm and the searching domain is greatly reduced. The time complexity of the heuristic method is found to be O( n) for the best and average cases. From the simulation results, it is demonstrated that optimal results can be obtained in a reasonably short period of time for problems with different sizes.

Exhaustive approaches to 2D rectangular perfect packings

In this paper, we consider the two-dimensional rectangular strip packing problem, in the case where there is a perfect packing; that is, there is no wasted space. One can think of the problem as a jigsaw puzzle with oriented rectangular pieces. Although this comprises a quite special case for strip packing, we have found it useful as a subroutine in related work. We demonstrate a simple pruning approach that makes a branch-and-bound-based exhaustive search extremely effective for problems with less than 30 rectangles.