Energy Landscape Paving Algorithm for Solving Circles Packing Problem

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

“…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]<...>…”

“…Moreover, in those of the few cases in which the voting heuristic was required, it was required to be implemented only once in the entire execution of the algorithm. A variant of the voting heuristic was also implemented in conjunction with energy landscape paving algorithm [54,76,77]. The smallest circle was picked and placed randomly at the vacant place producing new configuration.…”

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

“…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.…”

A distributed multi-agent system approach for solving constrained optimization problems using probability collectives

“…[32] the CPP received considerable attention in the 'pure' mathematics literature but only limited attention in the operations research literature [33]. As it is proven to be a NP-hard problem [185][186][187][188] and cannot be effectively solved by purely analytical approaches [189][190][191][192][193][194][195][196][197][198][199], a number of heuristic techniques were proposed solving the CPP [184,185,[200][201][202][203][204][205][206][207][208][209][210][211][212]. 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 [185,[189][190][191][192][193][194][195][196][197][198], close packing of fixed and different sized circles inside a square or circle container…”

“…Moreover, in those of the few runs in which the voting heuristic was required, it was required to be implemented only once in the entire execution of the algorithm. A variant of the voting heuristic was also implemented in conjunction with energy landscape paving algorithm [32,206,207] in which, the smallest circle was picked and placed randomly at the vacant place to produce a new configuration. It was claimed that this heuristic helped the algorithm jump out of the local minima.…”