Two-dimensional Packing Research Articles

Irregular packing: MILP model based on a polygonal enclosure

This paper addresses the two-dimensional irregular packing problem, also known as the nesting problem. This is a subset of cutting and packing problems of renowned practical and theoretical relevance. A mixed integer-linear programming formulation is proposed to optimize the packing of particular polygonal shapes, convex forms with 3–8 sides, since their opposite sides are parallel. The model can be used to pack enclosures of general irregular shapes, generating upper bounds to the optimal solutions. The model was tested with 270 mass generated instances of small dimensions.

Softening of stressed granular packings with resonant sound waves.

We perform numerical simulations of a two-dimensional bidisperse granular packing subjected to both a static confining pressure and a sinusoidal dynamic forcing applied by a wall on one edge of the packing. We measure the response experienced by a wall on the opposite edge of the packing and obtain the resonant frequency of the packing as the static or dynamic pressures are varied. Under increasing static pressure, the resonant frequency increases, indicating a velocity increase of elastic waves propagating through the packing. In contrast, when the dynamic amplitude is increased for fixed static pressure, the resonant frequency decreases, indicating a decrease in the wave velocity. This occurs both for compressional and for shear dynamic forcing and is in agreement with experimental results. We find that the average contact number Zc at the resonant frequency decreases with increasing dynamic amplitude, indicating that the elastic softening of the packing is associated with a reduced number of grain-grain contacts through which the elastic waves can travel. We image the excitations created in the packing and show that there are localized disturbances or soft spots that become more prevalent with increasing dynamic amplitude. Our results are in agreement with experiments on glass bead packings and earth materials such as sandstone and granite and may be relevant to the decrease in elastic wave velocities that has been observed to occur near fault zones after strong earthquakes, in surficial sediments during strong ground motion, and in structures during earthquake excitation.

A novel characterization of organic molecular crystal structures for the purpose of crystal engineering.

A novel analytical approach is proposed for the characterization of organic molecular crystal structures where close packing is an important factor. It requires the identification of a unique reference axis within the crystal, along which three-dimensional space is divided into close-packed blocks (CPB) and junction zones (JZ). The degree of close packing along the reference axis is quantified by a two-dimensional packing function, ϕ2D, of symmetry determined by the space group. Values of ϕ2D reflect the degree of area-filling in planes perpendicular to this axis. The requirement of close packing within CPB allows the planar structures perpendicular to the reference axis to be analysed as tessellations of area-filling molecular-based cells (MBC), which are generally hexagonal. The form of these cells reflects the molecular shape in the cross-section, since their vertices are given by the centres of the voids between molecules. There are two basic types of MBC, Type 1, of glide or pseudo-glide symmetry, and Type 2, which is formed by lattice translations alone and generally requires a short unit-cell axis. MBC at layers of special symmetry are used to characterize the structures in terms of equivalent ellipses with parameters aell, bell and χell. The ratio aell/bell allows the established α, β, γ classification to be integrated into the current framework. The values of parameters aell and bell arising from all the structures considered, polynuclear aromatic hydrocarbons (PAH), substituted anthracenes and anthraquinones (SAA) and 2-benzyl-5-benzylidene (BBCP) are mapped onto a universal curve. The division of three-dimensional space into CPB and JZ is fundamentally useful for crystal engineering, since the structural perturbations brought about by substitution at hydrogen positions located within JZ are minimal. A contribution is also made to ongoing debate concerning the adoption of polar space groups, isomorphism and polymorphism.

Hybrid approach for the two‐dimensional bin packing problem with two‐staged patterns

AbstractThis paper presents a two‐phase heuristic approach for the two‐dimensional bin packing problem with two‐staged patterns and nonoriented items. A solution is generated in each phase and the better one is selected. Residual problems are solved by column generation in the first phase, where a partial admitting procedure is used to admit some of the patterns into the phase‐1 solution. The second solution is obtained from solving an integer linear programming problem over the set of all patterns generated in the first phase, where a time limit is used and subsequently the solution may not be optimal over the pattern set. The computational results indicate that the approach yields the best solution quality among the heuristics that use two‐staged or more complex patterns.

SIGACT News Online Algorithms Column 26

I mentioned in a previous column [22] that the best known lower bound for the two-dimensional online bin packing problem is 1.907 by Blitz, van Vliet, Woeginger [2], which is an unpublished (and now lost [24]) manuscript. I have realized since then that even the penultimate result, 1.856, was published only in Andre van Vliet’s Ph.D. thesis [23] and is not readily available. It therefore seems like a good idea to describe his methods here, though not in full detail. This will include a discussion of the three-dimensional case. I will also survey other results in multidimensional packing, that were left out in my previous column. I would like to invite more contributions to this column, be it surveys, conference reports, or technical articles related to online algorithms and competitive analysis. If you are considering becoming a guest writer, don’t hesitate to mail me at rvs4@le.ac.uk.

Models and algorithms for packing rectangles into the smallest square

We consider the problem of determining the smallest square into which a given set of rectangular items can be packed without overlapping. We present an ILP model, an exact approach based on the iterated execution of a two-dimensional packing algorithm, and a randomized metaheuristic. Such approaches are valid both for the case where the rectangles have fixed orientation and the case where they can be rotated by 90°. We computationally evaluate the performance and the limits of the proposed approaches on a large set of instances, including a number of classical benchmarks from the literature, for both cases above, and for the special case where the items are squares.

A New Search Procedure for the Two-dimensional Orthogonal Packing Problem

In this paper we propose a new exact procedure for the two-dimensional orthog- onal packing problem, based on F. Clautiaux et al. approach (Clautiaux et al. Eur. J. Oper. Res. 183(3), 1196-1211, 2007). The principle consists in searching first the positions of the items on the horizontal axis, so as that, at each position, the sum of the heights of the items does not exceed the height of the bin. Each time a valid placement of all the items is encountered, another procedure determines if it can be extended to a solution of the pack- ing problem, searching the positions of the items on the vertical axis. Novel aspects of our approach include a simple and efficient search procedure, which only generates restricted placements, at least in a first stage, in order to reduce the search space, and the memoriza- tion of unsuccessful configurations, which are then used to detect dead-ends. We tested our implementation on a selection of orthogonal packing problems and strip packing problems, and we compared our results with those of recent successful approaches.

Shape-controlled percolation transition in 2D random packing of asymmetric dimers

In this paper, we report on an experimental investigation of a shape-controlled percolation transition in two-dimensional (2D) amorphous packing of dimers without long-range order. In the maximally random jammed (MRJ) packing of asymmetric dimers consisting of head and body, a dramatic increase in the connectivity of heads upon increasing the head-to-body size ratio γ leads to a percolation transition of the heads at the well-defined percolation threshold. In comparison with binary disks, the existence of a bond in dimers causes the heads to be homogeneously distributed over a system by inhibiting the local segregation. Interestingly, we found, however, that the cluster structure at the percolation threshold is insensitive to the bond, even though the existence of the bonds affects the percolation threshold as well as the head distribution. The fractal dimensions at the percolation threshold obey the universal law of the 2D percolation theory independently of the existence of bonds. Our finding can provide us with a new perspective of interesting applications of randomly assembled binary composites by using the homogeneous particle distribution and the sensitively tunable connectivity under particle shape control.

Correlation Among Crystal Structure, Mechanical Behavior, and Tabletability in the Co-Crystals of Vanillin Isomers

Tuning mechanical performance of molecular materials is currently attractive owing to their practical applications in pharmaceutical, food, and fine chemical industries and optoelectronics. Here we employed a crystal engineering approach to transform four food flavouring agents, vanillin isomers, from brittle to soft solids by forming co-crystals with 6-chloro-2,4-dinitroaniline (cda). The series includes vanillin (van), ethylvanillin (evan), iso-vanillin (ivan), as well as a Schiff base of ortho-vanillin (ovan) with ethylene diamine (sb-ovan). All the co-crystals adopt flat two-dimensional (2D) layer packing, except the sb-ovan:cda, which adopts a corrugated layer packing with the presence of slip planes. The mechanical properties of the co-crystals were studied by (1) a qualitative method, (2) nanoindentation, and (3) powder compaction techniques, which allowed for successfully establishing the relationship among crystal structure, mechanical properties, and tablet tensile strength. The simple qualitati...

Oligothiophenes with the naphthalene core for organic thin-film transistors: variation in positions of bithiophenyl attachment to the naphthalene

Relationship between molecular structure of organic material and its semiconducting properties is far from being trivial. Hence, a complex study based on systematic variation of molecular structure can contribute to understanding of this relationship. Herein we describe the synthesis, characterization and field-effect transistor (OFET) properties of a consistent series of naphthalene derivatives with hexylbithiophenyl chains attached at positions 1 and 4 (H2T14N), 1 and 5 (H2T15N), and 2 and 6 (H2T26N). The almost planar derivative H2T26N with the narrowest HOMO-LUMO band gap has good film forming ability, resulting in the remarkably high hole mobility (0.12cm2V−1s−1). Although both the H2T14N and H2T15N derivatives have almost the same twist of thiophene unit from the naphthalene plane, the latter has weaker conjugation between the bithiophenyl arms but exhibited higher charge mobility (0.016cm2V−1s−1). Apparently, a proper combination of molecular structure prerequisites and film-forming properties of semiconducting molecular material is essential for good OFET performance. It has been shown that also molecular symmetry can play an important role for effective two-dimensional molecular packing of the organic material.

A variable neighborhood descent approach for the two-dimensional bin packing problem

In this paper, we propose a new variable neighborhood descent (VND) algorithm for the two-dimensional bin packing problem. A knapsack packing heuristic is performed to get the initial solution. The proposed VND involves three neighborhood structures in order to disrupt the actual solution and provide a better exploration of the search space. Computational experiments on a standard benchmark show that our algorithm is competitive with the existing heuristics and metaheuristics in literature in terms of both the quality of the solution and the computing time.

Single-crystal field-effect transistors of new Cl₂-NDI polymorph processed by sublimation in air.

Physical properties of active materials built up from small molecules are dictated by their molecular packing in the solid state. Here we demonstrate for the first time the growth of n-channel single-crystal field-effect transistors and organic thin-film transistors by sublimation of 2,6-dichloro-naphthalene diimide in air. Under these conditions, a new polymorph with two-dimensional brick-wall packing mode (β-phase) is obtained that is distinguished from the previously reported herringbone packing motif obtained from solution (α-phase). We are able to fabricate single-crystal field-effect transistors with electron mobilities in air of up to 8.6 cm(2) V(-1) s(-1) (α-phase) and up to 3.5 cm(2) V(-1) s(-1) (β-phase) on n-octadecyltriethoxysilane-modified substrates. On silicon dioxide, thin-film devices based on β-phase can be manufactured in air giving rise to electron mobilities of 0.37 cm(2) V(-1) s(-1). The simple crystal and thin-film growth procedures by sublimation under ambient conditions avoid elaborate substrate modifications and costly vacuum equipment-based fabrication steps.

A variable neighborhood search for the capacitated vehicle routing problem with two-dimensional loading constraints

Abstract This paper addresses the capacitated vehicle routing problem with two-dimensional loading constraints (2L-CVRP), which is a generalized capacitated vehicle routing problem in which customer demand is a set of two-dimensional, rectangular, weighted items. The objective is to design the route set of minimum cost for a homogenous fleet of vehicles, starting and terminating at a central depot, to serve all the customers. All the items packed in one vehicle must satisfy the two-dimensional orthogonal packing constraints. A variable neighborhood search is proposed to address the routing aspect, and a skyline heuristic is adapted to examine the loading constraints. To speed up the search process, an efficient data structure (Trie) is utilized to record the loading feasibility information of routes, but also to control the computational effort of the skyline spending on the same route. The effectiveness of our approach is verified through experiments on widely used benchmark instances involving two distinct versions of loading constraints ( unrestricted and sequential versions). Numerical experiments show that the proposed method outperforms all existing methods and improves or matches the majority of best known solutions for both problem versions.

A Hybrid Demon Algorithm for the Two-Dimensional Orthogonal Strip Packing Problem

This paper develops a hybrid demon algorithm for a two-dimensional orthogonal strip packing problem. This algorithm combines a placement procedure based on an improved heuristic, local search, and demon algorithm involved in setting one parameter. The hybrid algorithm is tested on a wide set of benchmark instances taken from the literature and compared with other well-known algorithms. The computation results validate the quality of the solutions and the effectiveness of the proposed algorithm.

Hydrogen bond patterns in 3-carboxypiperidinium semi-oxalatemonohydrate

The crystal structure of compound C6H12NO2+·C2HO4−·H2O, or NIPE+·OXAL−·HYDR, is an ionic ensemble assisted by hydrogen bonds established between NIPE+ cations, OXAL− semi-oxalate anions and water molecules. The molecules are connected by O--H···O and N--H···O hydrogen bonds, forming linear ∙∙∙semi-oxalate∙∙∙semi-oxalate∙∙∙ chains extending along the a axis and described by the graph-set motif C(5). These chains interact with the NH2+ group of NIPE+ and the water molecules building a two-dimensional crystal packing of rings described by graph-set R21(5), R24(8) and R55(19), with corrugated layers running parallel to the (001) plane, and separated by hydrophobic interactions at c/2. Analyzes of the structures of the individual components and the hydrogen-bond network of the crystal structure are given.

A single machine scheduling problem with two-dimensional vector packing constraints

Abstract We consider a scheduling problem where jobs consume a perishable resource stored in vials. It leads to a new scheduling problem, with two-dimensional jobs, one dimension for the duration and one dimension for the consumption. Jobs have to be finished before a given due date, and the objective is to schedule the jobs on a single machine so that the maximum lateness does not exceed a given threshold and the number of vials required for processing all the jobs is minimized. We propose a two-step approach embedding a Recovering Beam Search algorithm to get a good-quality initial solution reachable in short time and a more time consuming matheuristic algorithm. Computational experiments are performed on the benchmark instances available for the two-dimensional vector packing problem integrated with additional due dates to take into account the maximum lateness constraints. The computational results show very good performances of the proposed approach that remains effective also on the original two-dimensional vector packing instances without due dates where 7 new bounds are obtained.

Constructive procedures to solve 2-dimensional bin packing problems with irregular pieces and guillotine cuts

This paper presents an approach for solving a new real problem in cutting and packing. At its core is an innovative mixed integer programme model that places irregular pieces and defines guillotine cuts. The two-dimensional irregular shape bin packing problem with guillotine constraints arises in the glass cutting industry, for example, the cutting of glass for conservatories. Almost all cutting and packing problems that include guillotine cuts deal with rectangles only, where all cuts are orthogonal to the edges of the stock sheet and a maximum of two angles of rotation are permitted. The literature tackling packing problems with irregular shapes largely focuses on strip packing i.e. minimizing the length of a single fixed width stock sheet, and does not consider guillotine cuts. Hence, this problem combines the challenges of tackling the complexity of packing irregular pieces with free rotation, guaranteeing guillotine cuts that are not always orthogonal to the edges of the stock sheet, and allocating pieces to bins. To our knowledge only one other recent paper tackles this problem. We present a hybrid algorithm that is a constructive heuristic that determines the relative position of pieces in the bin and guillotine constraints via a mixed integer programme model. We investigate two approaches for allocating guillotine cuts at the same time as determining the placement of the piece, and a two phase approach that delays the allocation of cuts to provide flexibility in space usage. Finally we describe an improvement procedure that is applied to each bin before it is closed. This approach improves on the results of the only other publication on this problem, and gives competitive results for the classic rectangle bin packing problem with guillotine constraints.

New Approximability Results for Two-Dimensional Bin Packing

We study the two-dimensional bin packing problem: Given a list of $$n$$n rectangles the objective is to find a feasible, i.e. axis-parallel and non-overlapping, packing of all rectangles into the minimum number of unit sized squares, also called bins. Our problem consists of two versions; in the first version it is not allowed to rotate the rectangles while in the other it is allowed to rotate the rectangles by $$90^{\circ }$$90?, i.e. to exchange the widths and the heights. Two-dimensional bin packing is a generalization of its one-dimensional counterpart and is therefore strongly $$\mathcal {NP}$$NP-hard. Furthermore Bansal et al. (Math Oper Res 31(1):31---49, 2006) showed that even an $$\mathcal {APTAS}$$APTAS is ruled out for this problem, unless $$\mathcal {P}=\mathcal {NP}$$P=NP. This lower bound of asymptotic approximability was improved by Chlebik and Chlebikova (J Discrete Algorithms 7(3):291---305, 2009) to values $$1+1/3792$$1+1/3792 and $$1+1/2196$$1+1/2196 for the version with and without rotations, respectively. On the positive side there is an asymptotic 1.69.. approximation by Caprara (Math Oper Res 33:203---215, 2008) without rotations and an asymptotic 1.52... approximation by Bansal et al. (SIAM J Comput 39(4):1256---1278, 2009) for both versions. We give a new asymptotic upper bound for both versions of our problem: For any fixed $$\varepsilon $$? and any instance that fits optimally into $$\mathrm {OPT}$$OPT bins, our algorithm computes a packing into $$(3/2+\varepsilon )\cdot \mathrm {OPT}+69$$(3/2+?)·OPT+69 bins in the version without rotations and $$(3/2+\varepsilon )\cdot \mathrm {OPT}+39$$(3/2+?)·OPT+39 bins in the version with rotations. The algorithm has polynomial running time in the input length. In our new technique we consider an optimal packing of the rectangles into the bins. We cut a small vertical or horizontal strip out of each bin and move the intersecting rectangles into additional bins. This enables us to either round the widths of all wide rectangles, or the heights of all long rectangles in this bin. After this step we round the other unrounded side of these rectangles and we achieve a solution with a simple structure and only few types of rectangles. Our algorithm initially rounds the instance and computes a solution that nearly matches the modified optimal solution.

Packing induced bistable phenomenon in granular flow: analysis from complex network perspective

The effects of packing configurations on the phase transition of straight granular chute flow with two bottlenecks are studied. The granular flow shows a dilute-to-dense flow transition when the channel width is varied, accompanied with a peculiar bistable phenomenon. The bistable phenomenon is induced by the initial packing configuration of particles. When the packing is dense, the initial flux is small and will induce a dense flow. When the packing is loose, the initial flux is large and will induce a dilute flow. The fabric network of granular packing is analyzed from a complex network perspective. The degree distribution shows quantitatively different characteristics for the configurations. A two-dimensional (2D) packing clustering coefficient is defined to better quantify the fabric network.

An Exact Algorithm for the Two-Dimensional Orthogonal Packing Problem with Unloading Constraints

This paper describes an exact algorithm for solving a two-dimensional orthogonal packing problem with unloading constraints, which occurs as a subproblem of mixed vehicle routing and loading problems. The packing considered in this work is basically a feasibility problem involving a single bin. The problem is addressed through a decomposition approach wherein a branch-and-cut algorithm is designed for solving a one-dimensional relaxation of the original problem. When an integer solution is found in the branching tree, a subsidiary problem is solved to identify a two-dimensional packing that does not lead to any overlap and satisfies the unloading constraints. Cuts are added when the subsidiary problem proves to be infeasible. Several preprocessing techniques aimed at reducing the size of the solution space and uncovering infeasibility are also described. A numerical comparison with the best known exact method is reported at the end based on benchmark instances.