An Effective Hybrid Algorithm for the Circles and Spheres Packing Problems

Liu, Jingfa; Yao, Yonglei; Zheng, Yu; Geng, Huantong; Zhou, Guocheng

doi:10.1007/978-3-642-02026-1_12

Cited by 17 publications

(8 citation statements)

References 16 publications

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“…Although the problem appears rather simple and in spite of its practical applications in production and packing for the textile, apparel, naval, automobile, aerospace, food industries, and so forth [54], the CPP received considerable attention in the "pure" mathematics literature but only limited attention in the operations research literature [55]. As it is proven to be a NP-hard problem [53,[56][57][58] and cannot be effectively solved by purely analytical approaches [59][60][61][62][63][64][65][66][67][68][69], a number of heuristic techniques were proposed solving the CPP [52,53,[70][71][72][73][74][75][76][77][78][79][80][81][82]. Most of these approaches address the CPP in limited ways, such as close packing of fixed and uniform sized circles inside a square or circle container [53,[59][60][61][62][63][64][65][66][67][68][69][70]<...>…”

Section: The Circle Packing Problem (Cpp)mentioning

confidence: 99%

A Probability Collectives Approach with a Feasibility‐Based Rule for Constrained Optimization

Kulkarni

Tai

2011

Applied Computational Intelligence and Soft Computing

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This paper demonstrates an attempt to incorporate a simple and generic constraint handling technique to the Probability Collectives (PC) approach for solving constrained optimization problems. The approach of PC optimizes any complex system by decomposing it into smaller subsystems and further treats them in a distributed and decentralized way. These subsystems can be viewed as a Multi-Agent System with rational and self-interested agents optimizing their local goals. However, as there is no inherent constraint handling capability in the PC approach, a real challenge is to take into account constraints and at the same time make the agents work collectively avoiding the tragedy of commons to optimize the global/system objective. At the core of the PC optimization methodology are the concepts of Deterministic Annealing in Statistical Physics, Game Theory and Nash Equilibrium. Moreover, a rule-based procedure is incorporated to handle solutions based on the number of constraints violated and drive the convergence towards feasibility. Two specially developed cases of the Circle Packing Problem with known solutions are solved and the true optimum results are obtained at reasonable computational costs. The proposed algorithm is shown to be sufficiently robust, and strengths and weaknesses of the methodology are also discussed.

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Section: The Circle Packing Problem (Cpp)mentioning

confidence: 99%

A Probability Collectives Approach with a Feasibility‐Based Rule for Constrained Optimization

Kulkarni

Tai

2011

Applied Computational Intelligence and Soft Computing

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“…So, some heuristic methods are generally proposed to solve it. For the circular packing problem in a rectangular container, the solving algorithms include the heuristic tabu search algorithm [1], the heuristic algorithm based on bounded enumeration strategy for evaluating corner placement [2], et al For the problem of packing circles into a larger circular container, some authors have developed various heuristic algorithms to generate approximate solutions, including the quasi-physical quasi-human algorithm [3][4][5], the complete quasi-physical algorithm [6], the Pruned-Enriched Rosenbluth Method (PERM) [7], the dynamic adaptive local search algorithm [8], the simulated annealing (SA) [9] and its improved algorithms [10,11], the beam search algorithm [12] and the adaptive beam search algorithm [13], the energy landscape paving (ELP) method [14,15], the global optimization algorithm based on the quasi-physical method [16], and others.…”

Section: Introductionmentioning

confidence: 99%

A New Heuristic Algorithm for Packing Equal Circles into A Larger Equilateral Triangular Container

Liu¹,

Zhang²,

Liu³

et al. 2012

IJACT

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This paper studies the problem of packing equal circles into a larger equilateral triangular container, which is concerned with how to pack these circles into the container without overlapping. Its aim is to minimize the edge length of the container as much as possible. By incorporating some heuristic configuration update strategies, a local search strategy based on the gradient method and the dichotomous search (DS) strategy into the simulated annealing (SA) algorithm, a new heuristic algorithm, the heuristic simulated annealing algorithm based on dichotomous search (HSADS), is put forward for solving this problem. In HSADS, the heuristic configuration update strategies are used to produce new configurations, and the gradient method is used to search for lower-energy minima near newly generated configurations, and the dichotomous search strategy is used to gain the minimal edge length of the equilateral triangular container. By testing six equal circles instances from the literature, the proposed algorithm improves the best known results of them so far. The experimental results show that HSADS is an effective algorithm for the problem of packing equal circles into a larger equilateral triangular container.

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“…A hybrid algorithm for the circles and spheres packing problems based on a combination of the improved energy landscape paving method with a gradient descent procedure is proposed in [13]. Paper [8] reviews the most efficient models and methods for packing circles and spheres.…”

mentioning

confidence: 99%

Packing congruent hyperspheres into a hypersphere

Stoyan

Yaskov

2011

J Glob Optim

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The paper considers a problem of packing the maximal number of congruent nD hyperspheres of given radius into a larger nD hypersphere of given radius where n = 2, 3, . . . , 24. Solving the problem is reduced to solving a sequence of packing subproblems provided that radii of hyperspheres are variable. Mathematical models of the subproblems are constructed. Characteristics of the mathematical models are investigated. On the ground of the characteristics we offer a solution approach. For n ≤ 3 starting points are generated either in accordance with the lattice packing of circles and spheres or in a random way. For n > 3 starting points are generated in a random way. A procedure of perturbation of lattice packings is applied to improve convergence. We use the Zoutendijk feasible direction method to search for local maxima of the subproblems. To compute an approximation to a global maximum of the problem we realize a non-exhaustive search of local maxima. Our results are compared with the benchmark results for n = 2. A number of numerical results for 2 ≤ n ≤ 24 are given.

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An Effective Hybrid Algorithm for the Circles and Spheres Packing Problems

Cited by 17 publications

References 16 publications

A Probability Collectives Approach with a Feasibility‐Based Rule for Constrained Optimization

A Probability Collectives Approach with a Feasibility‐Based Rule for Constrained Optimization

A New Heuristic Algorithm for Packing Equal Circles into A Larger Equilateral Triangular Container

Packing congruent hyperspheres into a hypersphere

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