Methodology to Solve Multi-Dimentional Sphere Packing Problems

This paper discusses the problem of optimally packing spheres of various dimensions into containers of arbitrary geometrical shapes. According to the international classification, this problem belongs to Sphere Packing Problems (SPPs). The aim of this work is to create an integrated methodology for solving SPPs.В статті розглядається задача оптимального розміщення куль різної розмірності в контейнерах довільних геометричних форм. Згідно з міжнародною класифікацією ця задача належить до класу SPP (Sphere Packing Problems). Метою даної роботи є створення єдиної методології розв’язання задач SPP.В статье рассматривается задача оптимального размещения шаров различной размерности в контейнерах произвольных геометрических форм. Согласно международной классификации эта задача относится к классу SPP (Sphere Packing Problems). Целью данной работы является создание единой методологии решения задач SPP

Balanced Circular Packing Problems with Distance Constraints

The packing of different circles in a circular container under balancing and distance conditions is considered. Two problems are studied: the first minimizes the container’s radius, while the second maximizes the minimal distance between circles, as well as between circles and the boundary of the container. Mathematical models and solution strategies are provided and illustrated with computational results

Formulation space search for two-dimensional packing problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The two-dimension packing problem is concerned with the arrangement of items without overlaps inside a container. In particular we have considered the case when the items are circular objects, some of the general examples that can be found in the industry are related with packing, storing and transportation of circular objects. Although there are several approaches we want to investigate the use of formulation space search. Formulation space search is a fairly recent method that provides an easy way to escape from local optima for non-linear problems allowing to achieve better results. Despite the fact that it has been implemented to solve the packing problem with identical circles, we present an improved implementation of the formulation space search that gives better results for the case of identical and non-identical circles, also considering that they are packed inside different shaped containers, for which we provide the needed modifications for an appropriate implementation. The containers considered are: the unit circle, the unit square, two rectangles with different dimension (length 5, width 1 and length 10 width 1), a right-isosceles triangle, a semicircle and a right-circular quadrant. Results from the tests conducted shown several improvements over the best previously known for the case of identical circles inside three different containers: a right-isosceles triangle, a semicircle and a circular quadrant. In order to extend the scope of the formulation space search approach we used it to solve mixed-integer non-linear problems, in particular those with zero-one variables. Our findings suggest that our implementation provides a competitive way to solve these kind of problems.This study was funded by the Mexican National Council for Science and Technology (CONACyT)