Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting

Litvinchev, Igor; Infante, Luis; Ozuna, Lucero

doi:10.1007/978-3-319-18899-7_9

Cited by 11 publications

(7 citation statements)

References 27 publications

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“…The valid inequalities (8) hold for any norm used to define the circular object. Numerical experiments presented in [18][19][20] demonstrate that aggregating valid inequalities (8) to the original problem (1)-(6) improves significantly the value of the corresponding LP-bound. Moreover, valid inequalities change the structure of the optimal LP-solution.…”

Section: Basic Modelmentioning

confidence: 97%

“…min , , , constraints (4) that the point i can not be a centre of the circle k C if the distance from i to the boundary is less than k R ; pair-wise constraints (5) guarantee that there is no overlapping between the circles; constraints (6) represent the binary nature of variables. More details on the problem (1)-(6) and its properties one can find in [18][19][20]. Note that all constructions proposed above remain valid for any norm used to define a circular-like object.…”

Section: Basic Modelmentioning

confidence: 99%

“…To the best of our knowledge, the idea to use a grid for approximated solution was first implemented by Beasley [4] in the context of cutting problems. Recently this approach was applied in [9,[17][18][19][20][21]29] for packing problems. This work is a continuation of [20].…”

Section: Introductionmentioning

confidence: 99%

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Packing a limited number of unequal circular objects in a rectangular container

Litvinchev¹,

Mosquera²

2017

Proceedings of the First EAI International Conference on Computer Science and Engineering

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A problem of packing a limited number of unequal circular objects in a fixed size rectangular container is considered. The circular object is considered in a general sense, as a set of points that are all the same distance (not necessary Euclidean) from a center. The aim is to maximize the (weighted) number of objects placed into the container or minimize the waste. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace and food industries. Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. We study a linear integer programming formulation based on approximating container by a regular grid. The nodes of the grid are considered as potential positions for assigning centers of the objects thus giving rise to a large scale linear 0-1 optimization problem with binary variables representing the assignment of centers to the nodes of the grid. Recursive packing allowing nesting circles inside one another is also considered. Numerical results are presented to demonstrate the efficiency of the proposed approach.

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Section: Basic Modelmentioning

confidence: 97%

Section: Basic Modelmentioning

confidence: 99%

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Packing a limited number of unequal circular objects in a rectangular container

Litvinchev¹,

Mosquera²

2017

Proceedings of the First EAI International Conference on Computer Science and Engineering

Self Cite

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“…Nonoverlapping of Clusters. Let us describe analytically the placement constraint (2). To this aim we introduce a quasi-phi-function for two objects Λ ( ) and Λ ( ).…”

Section: Tools Of Mathematical Modelingmentioning

confidence: 99%

“…A two-dimensional packing problem of objects in a container is NP-hard [1]. This problem is typical in logistics (transporting rolls of wallpaper, pipes, boxes, paint buckets, etc.,) and also has important applications in computer science, industrial engineering, manufacturing and production processes, healthcare, project portfolio selection, nanophysics, and agriculture (see, e.g., [2] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Optimized Packing Clusters of Objects in a Rectangular Container

Romanova

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Litvinchev

et al. 2019

Mathematical Problems in Engineering

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A packing (layout) problem for a number of clusters (groups) composed of convex objects (e.g., circles, ellipses, or convex polygons) is considered. The clusters have to be packed into a given rectangular container subject to nonoverlapping between objects within a cluster. Each cluster is represented by the convex hull of objects that form the cluster. Two clusters are said to be nonoverlapping if their convex hulls do not overlap. A cluster is said to be entirely in the container if so is its convex hull. All objects in the cluster have the same shape (different sizes are allowed) and can be continuously translated and rotated. The objective of optimized packing is constructing a maximum sparse layout for clusters subject to nonoverlapping and containment conditions for clusters and objects. Here the term sparse means that clusters are sufficiently distant one from another. New quasi-phi-functions and phi-functions to describe analytically nonoverlapping, containment and distance constraints for clusters are introduced. The layout problem is then formulated as a nonlinear nonconvex continuous problem. A novel algorithm to search for locally optimal solutions is developed. Computational results are provided to demonstrate the efficiency of our approach. This research is motivated by a container-loading problem; however similar problems arise naturally in many other packing/cutting/clustering issues.

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