65 research outputs found
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
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Some applications of continuous variable neighbourhood search metaheuristic (mathematical modelling)
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search
metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided
A Vitual-Force Based Swarm Algorithm for Balanced Circular Bin Packing Problems
Balanced circular bin packing problems consist in positioning a given number
of weighted circles in order to minimize the radius of a circular container
while satisfying equilibrium constraints. These problems are NP-hard, highly
constrained and dimensional. This paper describes a swarm algorithm based on a
virtual-force system in order to solve balanced circular bin packing problems.
In the proposed approach, a system of forces is applied to each component
allowing to take into account the constraints and minimizing the objective
function using the fundamental principle of dynamics. The proposed algorithm is
experimented and validated on benchmarks of various balanced circular bin
packing problems with up to 300 circles. The reported results allow to assess
the effectiveness of the proposed approach compared to existing results from
the literature.Comment: 23 pages including reference
Discretization-Based Solution Approaches for the Circle Packing Problem
The problem of packing a set of circles into the smallest surrounding
container is considered. This problem arises in different application areas
such as automobile, textile, food, and chemical industries. The so-called
circle packing problem can be cast as a nonconvex quadratically constrained
program, and is difficult to solve in general. An iterative solution approach
based on a bisection-type algorithm on the radius of the larger circle is
provided. The present algorithm discretizes the container into small cells and
solves two different integer linear programming formulations proposed for a
restricted and a relaxed version of the original problem. The present algorithm
is enhanced with solution space reduction, bound tightening and variable
elimination techniques. Then, a computational study is performed to evaluate
the performance of the algorithm. The present algorithm is compared with BARON
and Gurobi that solve the original nonlinear formulation and heuristic methods
from literature, and obtain promising results
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3D packing of balls in different containers by VNS
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel UniversityIn real world applications such as the transporting of goods products, packing is
a major issue. Goods products need to be packed such that the smallest space is
wasted to achieve the maximum transportation efficiency. Packing becomes more challenging and complex when the product is circular/spherical. This thesis focuses
on the best way to pack three-dimensional unit spheres into the smallest spherical and cubical space. Unit spheres are considered in lieu of non-identical spheres because the search mechanisms are more difficult in the latter set up and any improvements will be due to the search mechanism not to the ordering of the spheres. The two-unit sphere packing problems are solved by approximately using a variable neighborhood search (VNS) hybrid heuristic. A general search framework belonging to the Artificial Intelligence domain, the VNS offers a diversification of the search space by changing neighborhood structures and intensification by thoroughly investigating each neighborhood. It is exible, easy to implement, adaptable to both continuous and discrete optimization problems and has been use to solve a variety of problems including large-sized real-life problems. Its runtime is usually lower than other meta heuristic techniques. A tutorial on the VNS and its variants along with recent applications and areas of applicability of each variant. Subsequently, this thesis considers several variations of VNS heuristics for the two problems at hand, discusses their individual efficiencies and effectiveness, their convergence rates and studies their robustness. It highlights the importance of the hybridization which yields near global optima with high precision and accuracy, improving many best- known solutions indicate matching some, and improving the precision and accuracy of others. Keywords: variable neighborhood search, sphere packing, three-dimensional packing, meta heuristic, hybrid heuristics, multiple start heuristics
Some applications of continuous variable neighbourhood search metaheuristic (mathematical modelling)
In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
An Encoder-Decoder Approach for Packing Circles
The problem of packing smaller objects within a larger object has been of
interest since decades. In these problems, in addition to the requirement that
the smaller objects must lie completely inside the larger objects, they are
expected to not overlap or have minimum overlap with each other. Due to this,
the problem of packing turns out to be a non-convex problem, obtaining whose
optimal solution is challenging. As such, several heuristic approaches have
been used for obtaining sub-optimal solutions in general, and provably optimal
solutions for some special instances. In this paper, we propose a novel
encoder-decoder architecture consisting of an encoder block, a perturbation
block and a decoder block, for packing identical circles within a larger
circle. In our approach, the encoder takes the index of a circle to be packed
as an input and outputs its center through a normalization layer, the
perturbation layer adds controlled perturbations to the center, ensuring that
it does not deviate beyond the radius of the smaller circle to be packed, and
the decoder takes the perturbed center as input and estimates the index of the
intended circle for packing. We parameterize the encoder and decoder by a
neural network and optimize it to reduce an error between the decoder's
estimated index and the actual index of the circle provided as input to the
encoder. The proposed approach can be generalized to pack objects of higher
dimensions and different shapes by carefully choosing normalization and
perturbation layers. The approach gives a sub-optimal solution and is able to
pack smaller objects within a larger object with competitive performance with
respect to classical methods
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