Approximate Packing Circles in a Rectangular Container: Valid Inequalities and Nesting

 

I. Litvinchev^1,2 and E.L. Ozuna*²

 

¹ Complex Systems Department, Computing Center of Russian Academy of Sciences, Moscow, Russia. * luceroozuna@gmail.com

² Facultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, Monterrey, Nuevo León, México.

 

ABSTRACT

A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles 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 the local search procedures. A new formulation is proposed based on using a regular grid approximated the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by the commercial software. Two families of valid inequalities are proposed to strengthening the formulation. Nesting circles inside one another is also considered. Numerical results are presented to demonstrate the efficiency of the proposed approach.

Keywords: Circle Packing, Integer Programming, Large Scale Optimization.

 

RESUMEN

Se considera el problema de empaquetar un número limitado de círculos de radios diferentes en un contenedor rectangular de dimensiones fijas. El objetivo es maximizar el número (ponderado) de círculos dentro del contenedor o minimizar el desperdicio de espacio dentro del mismo. Este problema tiene numerosas aplicaciones dentro de la logística, incluyendo la producción y empaquetado para la industria textil, naval, automotriz, aeroespacial y la industria de alimentos. Frecuentemente, el problema es formulado como un problema de optimización continua no convexo que es resuelto con técnicas heurísticas combinadas con procedimientos de búsqueda local. Se propone una nueva formulación basada en el uso de una malla regular que cubre el contenedor y donde se considera a los nodos de la malla como posiciones potenciales para la asignación de centros de los círculos. El problema de empaquetamiento se escribe entonces, como un problema de optimización 0-1 a gran escala y es resuelto con software comercial. Resultados numéricos son presentados para demostrar la eficiencia del enfoque propuesto y realizar una comparación con los resultados conocidos.

 

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