基于蚁群劳动分工的空间分配方法求解带平衡约束的圆形装填问题
|
摘要
针对以卫星舱布局为背景的具有NP难度的全局优化问题——带平衡约束的圆形装填问题,提出基于蚁群劳动分工的空间分配方法。该方法将圆形装填问题看作空间分配问题,并借鉴蚁群劳动分工的任务分配来实现圆形装填问题的空间分配。通过中心平移策略和允许干涉策略,将带平衡约束的圆形装填问题由多目标带约束优化转化为单目标无约束优化。从空间的角度出发,建立了蚁群劳动分工与圆形装填问题之间的映射关系。引入蚁群劳动分工中的刺激—响应原理,提出了空间分配方法。该方法为圆形待布物定义了4个占位动作,并设计了相应的刺激和阈值,进而通过刺激—响应方式完成问题求解。通过3组共13个代表性算例的计算结果及与其他算法的比较表明,所提方法找到的圆形容器半径多为最优或者次优,且静不平衡量的精度最高。
Under the background of satellite module layout design,the circular packing problem with equilibrium constraints was researched and a space allocation method based on ant colony's labor division was proposed.The core idea of the method was to treat the circular packing problem as a space allocation problem,and used the task allocation in ant colony's labor division to achieve the space allocation in the circular packing problem.The problem was converted from a multi-objective constrained optimization problem into a single objective unconstrained optimization problem based on the center translation strategy and the allowing overlap strategy.The mapping relation between ant colony's labor division and the circular packing problem was presented from the perspective of space.The Space Allocation Method(SAM)was put forward by introducing the stimulus-response mechanism of ant colony's labor division.SAM defined four actions for the circular objects as well as stimuli and thresholds,and used the stimulusresponse way to solve the problem.The experiments on three sets of benchmarks consisting of thirteen representative instances showed that SAM obtained the first or the second-best container radii and the minimum equilibrium deviations.
Under the background of satellite module layout design,the circular packing problem with equilibrium constraints was researched and a space allocation method based on ant colony's labor division was proposed.The core idea of the method was to treat the circular packing problem as a space allocation problem,and used the task allocation in ant colony's labor division to achieve the space allocation in the circular packing problem.The problem was converted from a multi-objective constrained optimization problem into a single objective unconstrained optimization problem based on the center translation strategy and the allowing overlap strategy.The mapping relation between ant colony's labor division and the circular packing problem was presented from the perspective of space.The Space Allocation Method(SAM)was put forward by introducing the stimulus-response mechanism of ant colony's labor division.SAM defined four actions for the circular objects as well as stimuli and thresholds,and used the stimulusresponse way to solve the problem.The experiments on three sets of benchmarks consisting of thirteen representative instances showed that SAM obtained the first or the second-best container radii and the minimum equilibrium deviations.
引文
[1]LODI A,MARTELLO S,MONACI M.Two-dimensional packing problems:a survey[J].European Journal of Operational Research,2002,141(2):241-252.
[2]SWEENEY P E,PATERNOSTER E R.Cutting and packing problems:a categorized,application-orientated research[J].Journal of Operation Research Society,1992,43(7):691-706.
[3]GUO Yuanyuan,WANG Qian,LIANG Feng.Facility layout design based on particle swarm optimization[J].Computer Integrated Manufacturing Systems,2012,18(11):2476-2484(in Chinese).[郭源源,王谦,梁峰.基于粒子群优化算法的车间布局设计[J].计算机集成制造系统,2012,18(11):2476-2484.]
[4]HUANG Zhendong,XIAO Renbin.Hybird algorithm for the convex polygon packing problem with constraints of equilibrium[J].Journal of Huazhong University of Science and Technology:Natural Science Edition,2014,42(3):47-51(in Chinese).[黄振东,肖人彬.求解带性能约束凸多边形布局的混合算法[J].华中科技大学学报:自然科学版,2014,42(3):47-51.]
[5]LIN Qinglian,LIU Huchun,WANG Duojin,et al.Integrating systematic layout planning with fuzzy constraint theory to design and optimize the facility layout for operating theatre in hospitals[J].Journal of Intelligent Manufacturing,2015,26(1):87-95.
[6]LIU Jingfa,LIU Siyu.Heuristic algorithm for the rectangular packing problem with static non-equilibrium constraint[J].Journal of Software,2018,29(2):283-298(in Chinese).[刘景发,刘思妤.带静不平衡约束的矩形装填问题的启发式算法[J].软件学报,2018,29(2):283-298.]
[7]WANG Jinmin,WANG Yuxin,ZHA Jianzhong.Classification and representation of constraints in packing problem[J].Journal of Computer Aided Design and Conputer Graphics,2000,12(5):349-354(in Chinese).[王金敏,王玉新,查建中.布局问题约束的分类及表达[J].计算机辅助设计与图形学学报,2000,12(5):349-354.]
[8]WANG Huaiqing,HUANG Wenqi,ZHANG Quan,et al.An improved algorithm for the packing of unequal circles within a larger containing circle[J].European Journal of Operational Research,2002,141(2):440-453.
[9]LIU Jingfa,LI Jian,LYU Zhipeng.A quasi-human strategybased improved basin filling algorithm for the orthogonal rectangular packing problem with mass balance constraint[J].Computers&Industrial Engineering,2017,107:196-210.
[10]WANG Yingcong,XIAO Renbin,WANG Huimin.A flexible labour division approach to the polygon packing problem based on space allocation[J].International Journal of Production Research,2017,55(11):3025-3045.
[11]HUANG Zhendong,XIAO Renbin.Multi-stage dual neighborhood artificial bee colony algorithm forcollaborative solving satellite module coupling layout optimization problem[J].Science China:Information Sciences,2016,46(2):193-211(in Chinese).[黄振东,肖人彬.多阶段协同求解卫星舱耦合布局优化问题的双邻域人工蜂群算法[J].中国科学:信息科学,2016,46(2):193-211.]
[12]LIU Jingfa,LI Gang.Basin filling algorithm for the circular packing problem with equilibrium behavioral constraints[J].Science China:Information Sciences,2010,40(3):423-432(in Chinese).[刘景发,李刚.求解带平衡性能约束的圆形装填问题的吸引盘填充算法[J].中国科学:信息科学,2010,40(3):423-432.]
[13]TANG Fei,TENG Hongfei.A modified genetic algorithm and its application to layout optimization[J].Journal of Software,1999,10(10):1096-1102(in Chinese).[唐飞,腾弘飞.一种改进的遗传算法及其在布局优化中的应用[J].软件学报,1999,10(10):1096-1102.]
[14]QIAN Zhiqin,TENG Hongfei,SUN Zhiguo.Human-Computer interactive genetic algorithm and its application to constrained layout optimization[J].Chinese Journal of Computers,2001,24(5):553-559(in Chinese).[钱志勤,滕弘飞,孙治国.人机交互的遗传算法及其在约束布局优化中的应用[J].计算机学报,2001,24(5):553-559.]
[15]YU Yang,ZHA Jianzhong,TANG Xiaojun.Learning based GA and application in packing[J].Chinese Journal of Computers,2001,24(12):1242-1249(in Chinese).[于洋,查建中,唐晓君.基于学习的遗传算法及其在布局中的应用[J].计算机学报,2001,24(12):1242-1249.]
[16]LI Ning,LIU Fei,SUN Debao.A study on the particle swarm optimization with mutation operator constrained layout optimization[J].Chinese Journal of Computers,2004,27(7):897-903(in Chinese).[李宁,刘飞,孙德宝.基于带变异算子粒子群优化算法的约束布局优化研究[J].计算机学报,2004,27(7):897-903.]
[17]ZHOU Chi,GAO Liang,GAO Haibing.Particle swarm optimization based algorithm for constrained layout optimization[J].Control and Decision,2005,20(1):36-40(in Chinese).[周驰,高亮,高海兵.基于粒子群优化算法的约束布局优化[J].控制与决策,2005,20(1):36-40.]
[18]LEI Kaiyou,QIU Yuhui.A study of constrained layout optimization using adaptive particle swarm optimizer[J].Journal of Computer Research and Development,2006,43(10):1724-1731(in Chinese).[雷开友,邱玉辉.基于自适应粒子群算法的约束布局优化研究[J].计算机研究与发展,2006,43(10):1724-1731.]
[19]CAO Xukang,WANG Wenhu,JIANG Ruisong,et al.Optimization method for circular multi-components layout problem[J].Computer Integrated Manufacturing Systems,2017,23(1):17-24(in Chinese).[曹旭康,汪文虎,蒋睿嵩,等.圆形多组件装填布局的优化求解方法[J].计算机集成制造系统,2017,23(1):17-24.]
[20]HUANG Wenqi,CHEN Mao.Note on:an improved algorithm for the packing of unequal circles within a larger containing circle[J].Computers&Industrial Engineering,2006,50(3):338-344.
[21]XIAO Renbin,XU Yichun,AMOS M.Two hybrid compaction algorithms for the layout optimization problem[J].BioSystems,2007,90(2):560-567.
[22]LIU Jian,HUANG Wenqi.A fast local search algorithm for solving circles packing problem with constraints of equilibrium[J].Journal of Image and Graphics,2008,13(5):991-997(in Chinese).[刘建,黄文奇.求解带平衡约束圆形Packing问题的快速局部搜索算法[J].中国图象图形学报,2008,13(5):991-997.]
[23]WANG Yishou,SHI Yanjun,TENG Hongfei.An improved scatter search for circles packing problem with the equilibrium constraint[J].Chinese Journal of Computers,2009,32(6):1214-1221(in Chinese).[王奕首,史彦军,滕弘飞.用改进的散射搜索算法求解带平衡约束的圆形Packing问题[J].计算机学报,2009,32(6):1214-1221.]
[24]LIU Jingfa,LI Gang,CHEN Duanbin,et al.Two-dimensional equilibrium constraint layout using simulated annealing[J].Computers&Industrial Engineering,2010,59(4):530-536.
[25]LI Gang,LIU Jingfa.Heuristic algorithm based on tabusearch for the circular packingproblem with equilibrium constraints[J].Science China:Information Sciences,2011,41(9):1076-1088(in Chinese).[李刚,刘景发.基于禁忌搜索的启发式算法求解带平衡约束的圆形装填问题[J].中国科学:信息科学,2011,41(9):1076-1088.]
[26]LIU Jingfa,JIANG Yucong,LI Gang,et al.Heuristic-based energy landscape paving for the circular packing problem with performance constraints of equilibrium[J].Physica A:Statistical Mechanics and its Applications,2015,431:166-174.
[27]HE Kun,MO Danzeng,XU Ruchu,et al.A quasi-physical algorithm based on coarse and fine adjustment for solving circles packingproblem with constraints of equilibrium[J].Chinese Journal of Computers,2013,26(6):1224-1234(in Chinese).[何琨,莫旦增,许如初,等.基于粗精调技术的求解带平衡约束圆形Packing问题的拟物算法[J].计算机学报,2013,26(6):1224-1234.]
[28]HE Kun,MO Danzeng,YE Tao.A coarse-to-fine quasiphysical optimization method for solving the circle packing problem with equilibrium constraints[J].Computers&Industrial Engineering,2013,66(4):1049-1060.
[29]HE Kun,YANG Chenkai,HUANG Menglong,et al.Quasi-Physical algorithm based on action space for solving the circles Packing problem with equilibrium constraints[J].Journal of Software,2016,27(9):2218-2229(in Chinese).[何琨,杨辰凯,黄梦龙,等.动作空间带平衡约束圆形Packing问题的拟物求解算法[J].软件学报,2016,27(9):2218-2229.]
[30]ROBINSON G E.Regulation of labor division in insect societies[J].Annual Review of Entomology,1992,37(1):637-665.
[31]XIAO Renbin,TAO Zhenwu.Modeling and simulation of ant colony's labor division for virtual enterprises[J].Journal of Management Sciences in China,2009,12(1):57-69(in Chinese).[肖人彬,陶振武.面向虚拟企业的蚁群劳动分工建模与仿真[J].管理科学学报,2009,12(1):57-69.]
[32]LOW K H,LEOW W K,ANG M H.Autonomic mobile sensor network with self-coordinated task allocation and execution[J].IEEE Transactions on Systems,Man,and Cybernetics,Part C:Applications and Reviews,2006,36(3):315-327.
[33]DOS SANTOS F,BAZZAN A L C.Towards efficient multiagent task allocation in the RoboCup rescue:a biologicallyinspired approach[J].Autonomous Agents and Multi-Agent Systems,2011,22(3):465-486.
[34]XIAO Renbin,YU Tongyang,GONG Xiaoguang.Modeling and simulation of ant colony's labor division with constraints for task allocation of resilient supply chains[J].International Journal on Artificial Intelligence Tools,2012,21(3):1240014-1-1240014-19.
[35]WU Husheng,LI Hao,XIAO Renbin,et al.Modeling and simulation of dynamic ant colony's labor division for task allocation of UAV swarm[J].Physica A:Statistical Mechanics and its Applications,2018,491:127-141.
[36]AKEB H,HIFI M,M'HALLAH R.A beam search algorithm for the circular packing problem[J].Computers&Operations Research,2009,36(5):1513-1528.
[37]BONABEAU E,THERAULAZ G,DENEUBOURG J L.Quantitative study of the fixed threshold model for the regulation of division of labour in insect societies[J].Proceedings of the Royal Society of London.Series B:Biological Sciences,1996,263(1376):1565-1569.
[38]THERAULAZ G,BONABEAU E,DENUEBOURG J L.Response threshold reinforcements and division of labour in insect societies[J].Proceedings of the Royal Society of London.Series B:Biological Sciences,1998,265(1393):327-332.
[39]HUANG Wenqi,YE Tao.Quasi-physical global optimization method for solving the equalcircle packing problem[J].Science China:Information Sciences,2011,41(6):686-693(in Chinese).[黄文奇,叶涛.求解等圆Packing问题的拟物型全局优化算法[J].中国科学:信息科学,2011,41(6):686-693.]
[40]HUANG Wenqi,FU Zhanghua,XU Ruchu.Tabu search algorithm combined with global perturbation for packing arbitrary sized circles into a circular container[J].Science China Information Sciences,2013,56(9):1-14.
[2]SWEENEY P E,PATERNOSTER E R.Cutting and packing problems:a categorized,application-orientated research[J].Journal of Operation Research Society,1992,43(7):691-706.
[3]GUO Yuanyuan,WANG Qian,LIANG Feng.Facility layout design based on particle swarm optimization[J].Computer Integrated Manufacturing Systems,2012,18(11):2476-2484(in Chinese).[郭源源,王谦,梁峰.基于粒子群优化算法的车间布局设计[J].计算机集成制造系统,2012,18(11):2476-2484.]
[4]HUANG Zhendong,XIAO Renbin.Hybird algorithm for the convex polygon packing problem with constraints of equilibrium[J].Journal of Huazhong University of Science and Technology:Natural Science Edition,2014,42(3):47-51(in Chinese).[黄振东,肖人彬.求解带性能约束凸多边形布局的混合算法[J].华中科技大学学报:自然科学版,2014,42(3):47-51.]
[5]LIN Qinglian,LIU Huchun,WANG Duojin,et al.Integrating systematic layout planning with fuzzy constraint theory to design and optimize the facility layout for operating theatre in hospitals[J].Journal of Intelligent Manufacturing,2015,26(1):87-95.
[6]LIU Jingfa,LIU Siyu.Heuristic algorithm for the rectangular packing problem with static non-equilibrium constraint[J].Journal of Software,2018,29(2):283-298(in Chinese).[刘景发,刘思妤.带静不平衡约束的矩形装填问题的启发式算法[J].软件学报,2018,29(2):283-298.]
[7]WANG Jinmin,WANG Yuxin,ZHA Jianzhong.Classification and representation of constraints in packing problem[J].Journal of Computer Aided Design and Conputer Graphics,2000,12(5):349-354(in Chinese).[王金敏,王玉新,查建中.布局问题约束的分类及表达[J].计算机辅助设计与图形学学报,2000,12(5):349-354.]
[8]WANG Huaiqing,HUANG Wenqi,ZHANG Quan,et al.An improved algorithm for the packing of unequal circles within a larger containing circle[J].European Journal of Operational Research,2002,141(2):440-453.
[9]LIU Jingfa,LI Jian,LYU Zhipeng.A quasi-human strategybased improved basin filling algorithm for the orthogonal rectangular packing problem with mass balance constraint[J].Computers&Industrial Engineering,2017,107:196-210.
[10]WANG Yingcong,XIAO Renbin,WANG Huimin.A flexible labour division approach to the polygon packing problem based on space allocation[J].International Journal of Production Research,2017,55(11):3025-3045.
[11]HUANG Zhendong,XIAO Renbin.Multi-stage dual neighborhood artificial bee colony algorithm forcollaborative solving satellite module coupling layout optimization problem[J].Science China:Information Sciences,2016,46(2):193-211(in Chinese).[黄振东,肖人彬.多阶段协同求解卫星舱耦合布局优化问题的双邻域人工蜂群算法[J].中国科学:信息科学,2016,46(2):193-211.]
[12]LIU Jingfa,LI Gang.Basin filling algorithm for the circular packing problem with equilibrium behavioral constraints[J].Science China:Information Sciences,2010,40(3):423-432(in Chinese).[刘景发,李刚.求解带平衡性能约束的圆形装填问题的吸引盘填充算法[J].中国科学:信息科学,2010,40(3):423-432.]
[13]TANG Fei,TENG Hongfei.A modified genetic algorithm and its application to layout optimization[J].Journal of Software,1999,10(10):1096-1102(in Chinese).[唐飞,腾弘飞.一种改进的遗传算法及其在布局优化中的应用[J].软件学报,1999,10(10):1096-1102.]
[14]QIAN Zhiqin,TENG Hongfei,SUN Zhiguo.Human-Computer interactive genetic algorithm and its application to constrained layout optimization[J].Chinese Journal of Computers,2001,24(5):553-559(in Chinese).[钱志勤,滕弘飞,孙治国.人机交互的遗传算法及其在约束布局优化中的应用[J].计算机学报,2001,24(5):553-559.]
[15]YU Yang,ZHA Jianzhong,TANG Xiaojun.Learning based GA and application in packing[J].Chinese Journal of Computers,2001,24(12):1242-1249(in Chinese).[于洋,查建中,唐晓君.基于学习的遗传算法及其在布局中的应用[J].计算机学报,2001,24(12):1242-1249.]
[16]LI Ning,LIU Fei,SUN Debao.A study on the particle swarm optimization with mutation operator constrained layout optimization[J].Chinese Journal of Computers,2004,27(7):897-903(in Chinese).[李宁,刘飞,孙德宝.基于带变异算子粒子群优化算法的约束布局优化研究[J].计算机学报,2004,27(7):897-903.]
[17]ZHOU Chi,GAO Liang,GAO Haibing.Particle swarm optimization based algorithm for constrained layout optimization[J].Control and Decision,2005,20(1):36-40(in Chinese).[周驰,高亮,高海兵.基于粒子群优化算法的约束布局优化[J].控制与决策,2005,20(1):36-40.]
[18]LEI Kaiyou,QIU Yuhui.A study of constrained layout optimization using adaptive particle swarm optimizer[J].Journal of Computer Research and Development,2006,43(10):1724-1731(in Chinese).[雷开友,邱玉辉.基于自适应粒子群算法的约束布局优化研究[J].计算机研究与发展,2006,43(10):1724-1731.]
[19]CAO Xukang,WANG Wenhu,JIANG Ruisong,et al.Optimization method for circular multi-components layout problem[J].Computer Integrated Manufacturing Systems,2017,23(1):17-24(in Chinese).[曹旭康,汪文虎,蒋睿嵩,等.圆形多组件装填布局的优化求解方法[J].计算机集成制造系统,2017,23(1):17-24.]
[20]HUANG Wenqi,CHEN Mao.Note on:an improved algorithm for the packing of unequal circles within a larger containing circle[J].Computers&Industrial Engineering,2006,50(3):338-344.
[21]XIAO Renbin,XU Yichun,AMOS M.Two hybrid compaction algorithms for the layout optimization problem[J].BioSystems,2007,90(2):560-567.
[22]LIU Jian,HUANG Wenqi.A fast local search algorithm for solving circles packing problem with constraints of equilibrium[J].Journal of Image and Graphics,2008,13(5):991-997(in Chinese).[刘建,黄文奇.求解带平衡约束圆形Packing问题的快速局部搜索算法[J].中国图象图形学报,2008,13(5):991-997.]
[23]WANG Yishou,SHI Yanjun,TENG Hongfei.An improved scatter search for circles packing problem with the equilibrium constraint[J].Chinese Journal of Computers,2009,32(6):1214-1221(in Chinese).[王奕首,史彦军,滕弘飞.用改进的散射搜索算法求解带平衡约束的圆形Packing问题[J].计算机学报,2009,32(6):1214-1221.]
[24]LIU Jingfa,LI Gang,CHEN Duanbin,et al.Two-dimensional equilibrium constraint layout using simulated annealing[J].Computers&Industrial Engineering,2010,59(4):530-536.
[25]LI Gang,LIU Jingfa.Heuristic algorithm based on tabusearch for the circular packingproblem with equilibrium constraints[J].Science China:Information Sciences,2011,41(9):1076-1088(in Chinese).[李刚,刘景发.基于禁忌搜索的启发式算法求解带平衡约束的圆形装填问题[J].中国科学:信息科学,2011,41(9):1076-1088.]
[26]LIU Jingfa,JIANG Yucong,LI Gang,et al.Heuristic-based energy landscape paving for the circular packing problem with performance constraints of equilibrium[J].Physica A:Statistical Mechanics and its Applications,2015,431:166-174.
[27]HE Kun,MO Danzeng,XU Ruchu,et al.A quasi-physical algorithm based on coarse and fine adjustment for solving circles packingproblem with constraints of equilibrium[J].Chinese Journal of Computers,2013,26(6):1224-1234(in Chinese).[何琨,莫旦增,许如初,等.基于粗精调技术的求解带平衡约束圆形Packing问题的拟物算法[J].计算机学报,2013,26(6):1224-1234.]
[28]HE Kun,MO Danzeng,YE Tao.A coarse-to-fine quasiphysical optimization method for solving the circle packing problem with equilibrium constraints[J].Computers&Industrial Engineering,2013,66(4):1049-1060.
[29]HE Kun,YANG Chenkai,HUANG Menglong,et al.Quasi-Physical algorithm based on action space for solving the circles Packing problem with equilibrium constraints[J].Journal of Software,2016,27(9):2218-2229(in Chinese).[何琨,杨辰凯,黄梦龙,等.动作空间带平衡约束圆形Packing问题的拟物求解算法[J].软件学报,2016,27(9):2218-2229.]
[30]ROBINSON G E.Regulation of labor division in insect societies[J].Annual Review of Entomology,1992,37(1):637-665.
[31]XIAO Renbin,TAO Zhenwu.Modeling and simulation of ant colony's labor division for virtual enterprises[J].Journal of Management Sciences in China,2009,12(1):57-69(in Chinese).[肖人彬,陶振武.面向虚拟企业的蚁群劳动分工建模与仿真[J].管理科学学报,2009,12(1):57-69.]
[32]LOW K H,LEOW W K,ANG M H.Autonomic mobile sensor network with self-coordinated task allocation and execution[J].IEEE Transactions on Systems,Man,and Cybernetics,Part C:Applications and Reviews,2006,36(3):315-327.
[33]DOS SANTOS F,BAZZAN A L C.Towards efficient multiagent task allocation in the RoboCup rescue:a biologicallyinspired approach[J].Autonomous Agents and Multi-Agent Systems,2011,22(3):465-486.
[34]XIAO Renbin,YU Tongyang,GONG Xiaoguang.Modeling and simulation of ant colony's labor division with constraints for task allocation of resilient supply chains[J].International Journal on Artificial Intelligence Tools,2012,21(3):1240014-1-1240014-19.
[35]WU Husheng,LI Hao,XIAO Renbin,et al.Modeling and simulation of dynamic ant colony's labor division for task allocation of UAV swarm[J].Physica A:Statistical Mechanics and its Applications,2018,491:127-141.
[36]AKEB H,HIFI M,M'HALLAH R.A beam search algorithm for the circular packing problem[J].Computers&Operations Research,2009,36(5):1513-1528.
[37]BONABEAU E,THERAULAZ G,DENEUBOURG J L.Quantitative study of the fixed threshold model for the regulation of division of labour in insect societies[J].Proceedings of the Royal Society of London.Series B:Biological Sciences,1996,263(1376):1565-1569.
[38]THERAULAZ G,BONABEAU E,DENUEBOURG J L.Response threshold reinforcements and division of labour in insect societies[J].Proceedings of the Royal Society of London.Series B:Biological Sciences,1998,265(1393):327-332.
[39]HUANG Wenqi,YE Tao.Quasi-physical global optimization method for solving the equalcircle packing problem[J].Science China:Information Sciences,2011,41(6):686-693(in Chinese).[黄文奇,叶涛.求解等圆Packing问题的拟物型全局优化算法[J].中国科学:信息科学,2011,41(6):686-693.]
[40]HUANG Wenqi,FU Zhanghua,XU Ruchu.Tabu search algorithm combined with global perturbation for packing arbitrary sized circles into a circular container[J].Science China Information Sciences,2013,56(9):1-14.