On regular and random two-dimensional packing of crosses

Packing problems, even of objects with regular geometries, are in general non-trivial. For few special shapes, the features of crystalline as well as random, irregular two-dimensional (2D) packing structures are known. The packing of 2D crosses does not yet belong to the category of solved problems. We demonstrate in experiments with crosses of different aspect ratios (arm width to length) which packing fractions are actually achieved by random packing, and we compare them to densest regular packing structures. We determine local correlations of the orientations and positions after ensembles of randomly placed crosses were compacted in the plane until they jam. Short-range orientational order is found over 2 to 3 cross lengths. Similarly, correlations in the spatial distributions of neighbors extend over 2 to 3 crosses. There is no simple relation between the geometries of the crosses and the peaks in the spatial correlation functions, but some features of the orientational correlations can be traced to typical local configurations.

2DPackLib: a two-dimensional cutting and packing library

Two-dimensional cutting and packing problems model a large number of relevant industrial applications.The literature on practical algorithms for such problems is very large. We introduce the , a library on two-dimensional orthogonal cutting and packing problems. The library makes available, in a unified format, 25 benchmarks from the literature for a total of over 3000 instances, provides direct links to surveys and typologies, and includes a list of relevant links.

Improved interval methods for solving circle packing problems in the unit square

In this work computer-assisted optimality proofs are given for the problems of finding the densest packings of 31, 32, and 33 non-overlapping equal circles in a square. In a study of 2005, a fully interval arithmetic based global optimization method was introduced for the problem class, solving the cases 28, 29, 30. Until now, these were the largest problem instances solved on a computer. Using the techniques of that paper, the estimated solution time for the next three cases would have been 3–6 CPU months. In the present paper this former method is improved in both its local and global search phases. We discuss a new interval-based polygon representation of the core local method for eliminating suboptimal regions, which has a simpler implementation, easier proof of correctness, and faster behaviour than the former one. Furthermore, a modified strategy is presented for the global phase of the search, including improved symmetry filtering and tile pattern matching. With the new method the cases $$n=31,32,33$$ n = 31 , 32 , 33 have been solved in 26, 61, and 13 CPU hours, giving high precision enclosures for all global optimizers and the optimum value. After eliminating the hardware and compiler improvements since the former study, the new proof technique became roughly about 40–100 times faster than the previous one. In addition, the new implementation is suitable for solving the next few circle packing instances with similar computational effort.

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

Research and application of the crow search algorithm for geometric covering optimization problems

The problem of geometric covering is a special case of the optimal design problem and belongs to the class of cutting and packing problems. The challenge is to position some geometric objects on the surface to be coated so that the entire surface is covered. The complexity of the problems under consideration is due to their belonging to the class of NP-hard problems, which excludes the possibility of solving them by exact methods and requires the development of approximate optimization methods and algorithms. This article discusses the problem of geometric covering of an area with circles from a given set of radii. To solve the problem of geometric covering, a hexagonal grid coverage method with optimization by a metaheuristic algorithm is used. The crow search algorithm is such an algorithm, which is a relatively new metaheuristic algorithm based on the intelligent behavior of crows in a flock. The crow search algorithm includes two control parameters: the awareness probability and the flight length. To study the solution method and check the efficiency, a problem was modeled on the basis of a real design of automatic irrigation systems, and the results of experiments with different values of control parameters were presented.

Convex Polygonal Hull for a Pair of Irregular Objects

Introduction. Optimization placement problems are NP-hard. In most cases related to cutting and packing problems, heuristic approaches are used. The development of analytical methods for mathematical modeling of the problems is of paramount important for expanding the class of placement problems that can be solved optimally using state of the art NLP-solvers. The problem of placing two irregular two-dimensional objects in a convex polygonal region of the minimum size, which is a convex polygonal hull of the minimum area or perimeter, is considered. Continuous rotations and translations of non-overlapping objects are allowed. To solve the problem of optimal compaction of a pair of objects, two algorithms are proposed. The first is a sequentially search for local extrema on all feasible subdomains using a solution tree. The second algorithm searches for a locally optimal extremum on a single subdomain using a "good" feasible starting point. Purpose of the paper. Show how to construct a minimal convex polygonal hull for two continuously moving irregular objects bounded by circular arcs and line segments. Results. A mathematical model is constructed in the form of a nonlinear programming problem using the phi-function technique. Two algorithms are proposed for solving the problem of placing a pair of objects in order to minimize the area and perimeter of the enclosing polygonal area. The results of computational experiments are presented. Conclusions. The construction of a minimal convex polygonal hull for a pair of two-dimensional objects having an arbitrary spatial shape and allowing continuous rotations and translations makes it possible to speed up the process of finding feasible solutions for the problem of placing a large number of objects with complex geometry. Keywords: convex polygonal hull, irregular objects, phi-function technique, nonlinear optimization.

COMPACT: Concurrent or Ordered Matrix-Based Packing Arrangement Computation Technique

Despite their versatility in treating irregular geometries, the raster methods have received limited attention in solving packing problems involving rotatable objects. In addition, raster approximation allows the use of unique performance metrics and indirect consideration of constraints, which have not been exploited in the literature. This study presents the Concurrent or Ordered Matrix-based Packing Arrangement Computation Technique (COMPACT). The method allows the objects to be rotated by arbitrary angles, unlike the right-angled rotation restrictions imposed in many existing packing optimization studies based on raster methods. The raster approximations are obtained through loop-free operations that improve efficiency. Additionally, a novel performance metric is introduced, which favors efficient filling of the available space by maximizing the overall contact within the domain. Moreover, the objective functions are exploited to discard the overlap and overflow constraints and enable the use of unconstrained optimization methods. The results of the case studies demonstrate the effectiveness of the proposed technique.