Strip Packing Problem Research Articles

Bidirectional best-fit heuristic considering compound placement for two dimensional orthogonal rectangular strip packing

The two dimensional orthogonal rectangular strip packing problem is a common NP-hard optimisation problem whereby a set of rectangular shapes must be placed on a fixed width stock sheet with infinite length in such a way that wastage is minimised and material utilisation is maximised. The bidirectional best-fit heuristic is a deterministic approach which has previously been shown to outperform existing heuristic methods as well as many metaheuristics from the literature. Here, we propose a modification to the original bidirectional best-fit heuristic whereby combinations of pairs of rectangles are considered generating improved results over standard benchmark sets.

A Particle Swarm Optimization Algorithm for a 2-D Irregular Strip Packing Problem

Two-Dimensional Irregular Strip Packing Problem is a classical cutting/packing problem. The problem is to assign, a set of 2-D irregular-shaped items to a rectangular sheet. The width of the sheet is fixed, while its length is extendable and has to be minimized. A sequence-based approach is developed and tested. The approach involves two phases; optimization phase and placement phase. The optimization phase searches for the packing sequence that would lead to an optimal (or best) solution when translated to an actual pattern through the placement phase. A Particle Swarm Optimization algorithm is applied in this optimization phase. Regarding the placement phase, a combined algorithm based on traditional placement methods is developed. Competitive results are obtained, where the best solutions are found to be better than, or at least equal to, the best known solutions for 10 out of 31 benchmark data sets. A Statistical Design of Experiments and a random generator of test problems are also used to characterize the performance of the entire algorithm.

A Harmonic Algorithm for the 3D Strip Packing Problem

In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box $B$. The goal is to pack all the items in $B$ such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of $B$ and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any $\epsilon>0$, achieves an approximation of $T_{\infty}+\epsilon\approx1.69103+\epsilon$, where $T_{\infty}$ is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance $I$ and the strip packing solutions for the corresponding instance obtained from $I$ by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the $T_{\infty}+\epsilon$ approximation for 2D bin packing due to Caprara. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for 2D strip packing due to Kenyon and Rémila.

Heuristic for the rectangular strip packing problem with rotation of items

This paper presents a heuristic algorithm for the rectangular strip packing problem, where a set of rectangular items are packed orthogonally into a strip of definite width and infinite height, so as to minimize the required height. The items cannot overlap and rotation by 90 degrees is allowed. The solution contains several sections. The algorithm is based on a sequential grouping and value correction procedure that considers multiple candidate solutions. It generates each next section using a subset of the remaining items and then corrects the values of the included items. The algorithm is used to solve 13 groups of benchmark instances. It is able to improve the solution quality for all groups.

Heuristics for the strip packing problem with unloading constraints

This article addresses the Strip Packing Problem with Unloading Constraints (SPU). In this problem, we are given a strip of fixed width and unbounded height, and n items of C different classes. As in the well-known two-dimensional Strip Packing problem, we have to pack all items minimizing the used height, but now we have the additional constraint that items of higher classes cannot block the way out of lower classes items. This problem appears as a sub-problem in the Two-Dimensional Loading Capacitated Vehicle Routing Problem (2L-CVRP), where one has to optimize the delivery of goods, demanded by a set of clients, that are transported by a fleet of vehicles of limited capacity based at a central depot. We propose two approximation algorithms and a GRASP heuristic for the SPU problem and provide an extensive computational experiment with these algorithms using well know instances for the 2L-CVRP problem as well as new instances adapted from the Strip Packing problem.

MINLP formulations for solving strip packing problems in LCD mother glass production

Mother glass production for liquid crystal displays involves the cutting of small rectangles from a large rectangle (strip) to minimize the length of the strip used. This is called the strip packing problem. The main concept for formulating this problem is to generate cutting patterns that can be used to produce items from the strip. This problem is a non-convex mixed integer nonlinear programming (MINLP) problem due to the bilinear terms in the demand and objective function. To obtain global optimal solutions, this problem should be transformed into several linear forms. Numerical examples based on the strip packing problem in LCD mother glass production are provided. Different objective functions for two problems are presented and compared.

A reduction approach for solving the rectangle packing area minimization problem

In the rectangle packing area minimization problem (RPAMP) we are given a set of rectangles with known dimensions. We have to determine an arrangement of all rectangles, without overlapping, inside an enveloping rectangle of minimum area. The paper presents a generic approach for solving the RPAMP that is based on two algorithms, one for the 2D Knapsack Problem (KP), and the other for the 2D Strip Packing Problem (SPP). In this way, solving an instance of the RPAMP is reduced to solving multiple SPP and KP instances. A fast constructive heuristic is used as SPP algorithm while the KP algorithm is instantiated by a tree search method and a genetic algorithm alternatively. All these SPP and KP methods have been published previously. Finally, the best variants of the resulting RPAMP heuristics are combined within one procedure. The guillotine cutting condition is always observed as an additional constraint. The approach was tested on 15 well-known RPAMP instances (above all MCNC and GSRC instances) and new best solutions were obtained for 10 instances. The computational effort remains acceptable. Moreover, 24 new benchmark instances are introduced and promising results are reported.

A heuristic algorithm for the strip packing problem

The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.

A simple randomized algorithm for two-dimensional strip packing

Two-dimensional strip packing problem is to pack given rectangular pieces on a strip of stock sheet having fixed width and infinite height. Its aim is to minimize the height of the strip such that non-guillotinable and fix orientation constraints are meet. In this paper, an improved scoring rule is developed and the least waste priority strategy is introduced, and a randomized algorithm is presented for solving this problem. This algorithm is very simple and does not need to set any parameters. Computational results on a wide range of benchmark problem instances show that the proposed algorithm obtains a better or matching performance as compared to the most of the previously published meta-heuristics.

From time representation in scheduling to the solution of strip packing problems

We propose two mixed-integer linear programming based approaches for the 2D orthogonal strip packing problem. Using knowledge from the alternative forms of time representation in scheduling formulations, we show how to efficiently combine three different concepts into the x- and y-dimensions. One model features a discrete representation on the x-axis (strip width) and a continuous representation with general precedence variables on the y-axis (strip height). The other features a full continuous-space representation with the same approach for the y-axis and a single non-uniform grid made up of slots for the x-axis. Through the solution of a set of twenty nine instances from the literature, we show that the former is a better approach, even when compared to three alternative MILP models ranging from a pure discrete-space to a pure continuous-space with precedence variables in both dimensions. All models are available in www.minlp.org.

An exact strip packing algorithm based on canonical forms

Given a set of rectangles and a rectangular container with a fixed width, called a strip, the two-dimensional strip packing problem (2SP) requires all the given rectangles to be placed orthogonally without overlap within the strip so as to minimize the height of the strip. 2SP and its variants have many applications in steel and textile industries, including an indirect application in scheduling problems. However, 2SP is known to be NP-hard.In this paper, we propose an exact algorithm to 2SP with and without rotations of 90°. The algorithm is designed by a branch-and-bound method that solves subproblems represented by g-staircase placements to 2SP with fixed height. We derive a new lower bound on the optimal value to 2SP by relaxing it to the problem of partitioning a set of integers into two subsets with the same total sum (PARTITION). We also design a new exact algorithm to the one-dimensional contiguous bin packing problem with fixed height (1CBPFH) using a g-staircase-placement-based branch-and-bound method. To reduce the search space, we introduce several new ideas to the branch-and-bound method. For example, we define canonical forms of feasible solutions to 2SP and 1CBPFH to limit the search space by looking for only a canonical solution.Our computational experiments on benchmark instances indicate that the proposed algorithm is competitive with the other algorithms. Our algorithm succeeded to find the optimal values for most of the instances in a practical time. Especially, it determined that the optimal values of instances “gcut02” and “cgcut02” (without rotations) are 1187 and 64, respectively, which have not been obtained by any of the other existing algorithms.

An algorithm for the strip packing problem using collision free region and exact fitting placement

The irregular shape packing problem is approached. The container has a fixed width and an open dimension to be minimized. The proposed algorithm constructively creates the solution using an ordered list of items and a placement heuristic. Simulated annealing is the adopted metaheuristic to solve the optimization problem. A two-level algorithm is used to minimize the open dimension of the container. To ensure feasible layouts, the concept of collision free region is used. A collision free region represents all possible translations for an item to be placed and may be degenerated. For a moving item, the proposed placement heuristic detects the presence of exact fits (when the item is fully constrained by its surroundings) and exact slides (when the item position is constrained in all but one direction). The relevance of these positions is analyzed and a new placement heuristic is proposed. Computational comparisons on benchmark problems show that the proposed algorithm generated highly competitive solutions. Moreover, our algorithm updated some best known results.

A tree search algorithm for solving the multi-dimensional strip packing problem with guillotine cutting constraint

The article presents a tree search algorithm (TRSA) for the strip packing problem in two and three dimensions with guillotine cutting constraint. In the 3D-SPP a set of rectangular items (boxes) and a container with fixed width and height but variable length are given. An arrangement of all boxes within the container has to be determined so that the required length is minimised. The 2D-SPP is analogously defined. The proposed TRSA is based on a tree search algorithm for the container loading problem by Fanslau and Bortfeldt (INFORMS J. Comput. 22:222–235, 2010). The TRSA generates guillotine packing patterns throughout. In a comparison with all recently proposed 3D-SPP methods the TRSA performs very competitive. Fine results are also achieved for the 2D-SPP.

A reference length approach for the 3D strip packing problem

In the three-dimensional strip packing problem (3DSP), we are given a container with an open dimension and a set of rectangular cuboids (boxes) and the task is to orthogonally pack all the boxes into the container such that the magnitude of the open dimension is minimized. We propose a block building heuristic based on extreme points for this problem that uses a reference length to guide its solution. Our 3DSP approach employs this heuristic in a one-step lookahead tree search algorithm using an iterative construction strategy. We tested our approach on standard 3DSP benchmark test data; the results show that our approach produces better solutions on average than all other approaches in literature for the majority of these data sets using comparable computation time.

A mixed-integer model for two-dimensional polyominoes strip packing and tiling problems

Two-dimensional irregular strip packing problem is one of the common cutting and packing problems, where it is required to assign (cut or pack) a set of 2D irregular-shaped items to a rectangular sheet. The sheet width is fixed, while its length is extendable and has to be minimised. In this paper, a new mixed-integer programming (MIP) model is introduced to optimally solve a special case of the problem, where item shapes are polygons with orthogonal edges, named polyominoes. Polyominoes strip packing may be classified as polyominoes tiling; a problem that can also be handled by the proposed model. Reasonable problem sizes (e.g. 45 polyominoes inside a 10 × 25 sheet) are solvable using an ordinary PC. Larger problem sizes are expected to be solvable when using state-of-the-art computational facilities. The model is also verified via a set of benchmark problems that are collected from the literature and provided optimal solution for all cases.

О построении расписаний выполнения параллельных задач на группах кластеров с различной производительностью

A classical problem of scheduling the set of tasks optimizing load balancing for a set of given processors was considered in theory in 1966. Graham's algorithm sending each task to a least loaded machine (processor) was the first example of approximate algorithm with guaranteed constant approximation quality. But all tasks and processors (machines) in Graham's example were simple (only one processor was sufficient for each task). On the other hand Graham's algorithm was an on-line one that is each task was sended to processor without knowing future tasks. With wide use of computational clusters very actual became the problem of scheduling parallel tasks on the set of parallel clusters in the class of on-line algorithms.. This problem has very close relation to a two dimensional packing problem, socalled multiple strip packing problem. Few years ago the author proposed an on-line algorithm for this problem and proved that for arbitrary set of parallel tasks (rectangles) the obtained schedule (packing) is within 2e of the optimal. In this paper the author generalized his on-line algorithm for the case of related parallel machines when all clusters have different speeds of processors in different clusters. An on-line algorithm for scheduling parallel tasks on a group of clusters with different speeds of processors is presented and it was proved that its approximation ratio is 2e for any set of parallel tasks. This result is non-trivial because even the case of one-processor related machines was intensively investigated and some algorithms with constant approximation ratio were presented.

LP bounds in various constraint programming approaches for orthogonal packing

We consider the 2D orthogonal feasibility problem (OPP-2) and the 2D strip packing problem (SPP-2). Given a set of rectangular items, the OPP-2 is to decide whether all items can be orthogonally packed into the given rectangular container; the SPP-2 is to find a packing of all items occupying the minimal height of the given semi-infinite strip. Both OPP-2 and SPP-2 are considered without items rotation. We investigate the known Constraint Programming (CP) approaches for OPP-2, in particular the dichotomy and disjunctive branching strategies and adapt 1D relaxation bounds based on linear programming (LP) into the constraint propagation process of the CP. Using the dichotomic search procedure the developed methods for OPP-2 are transformed for the case of SPP-2. Numerical results demonstrate the efficiency of the proposed strategies and of the combination of CP and LP-based pruning rules.

A global search framework for practical three-dimensional packing with variable carton orientations

This article aims to tackle a practical three-dimensional packing problem, where a number of cartons of diverse sizes are to be packed into a bin with fixed length and width but open height. Each carton is allowed to be packed in any one of the six orientations, and the carton edges are parallel to the bin edges. The allowance of variable carton orientations exponentially increases the solution space and makes the problem very challenging to solve. This study first elaborately devises the packing procedure, which converts an arbitrary sequence of cartons into a compact packing solution and subsequently develops an improved genetic algorithm (IGA) to evolve a set of solutions. Moreover, a novel global search framework (GSF), utilizing the concept of evolutionary gradient, is proposed to further improve the solution quality. Numerical experiments indicate that IGA provides faster and better results and GSF demonstrates its superior performance, especially in solving relative large-size and heterogeneous instances. Applying the proposed algorithms to some benchmarking cases of the three-dimensional strip packing problem also indicates that the algorithms are robust and effective compared to existing methods in the literature.

GREEDY ALGORITHMS FOR PACKING UNEQUAL SPHERES INTO A CUBOIDAL STRIP OR A CUBOID

Given a finite set of spheres of different sizes, we study the three-dimensional Strip Packing Problem (3D-SPP) as well as the three-dimensional Knapsack Problem (3D-KP). The 3D-SPP asks for a placement of all spheres within a cuboidal strip of fixed width and height so that the variable length of the cuboidal strip is minimized. The 3D-KP requires packing of a subset of the spheres in a given cuboid so that the wasted space is minimized. To solve these problems two greedy algorithms were developed which adapt the algorithms proposed by Huang et al. (2005) to the 3D case with some important enhancements. The resulting methods were tested using the instances provided by Stoyan et al. (2003). Additionally, two series of 12 instances each for the 3D-SPP and for the 3D-KP are introduced and results for these new instances are also reported.

A note on the Kenyon–Remila strip-packing algorithm

We show that a modification of the Kenyon–Remila algorithm for the strip-packing problem yields an improved bound on the value of the approximate solution. As a corollary we derive that there exists a polynomial-time algorithm that always finds a solution of value OPT + O ( OPT log OPT ) where OPT is the optimal value.