袁庆欣等：带有限缓冲区的混合流水车间多目标调度 1497 600 500 580 (a) 450 四 560 400 540 350 e 300 480 250 460 200 440 420 50 400 510 1520253035404550 100101520n253035404550 Buffer volume Buffer volume 3.0 (c) S25 -Model 2 -Model 1 1.5 1.0 V 0 101520.253035404550 Buffer volume 图6模型1与模型2对比.(a)完工时间：(b)运输时间：(c)缓神区均衡指数 Fig.6 Comparison of model I and model 2:(a)completion time;(b)transportation time;(c)buffer equilibrium index 衡指数分别趋于相等 [2]Huang J Z,Li A P,Liu X M,et al.Optimal design of production line layout considering buffer allocation.J Tongji Univ Nat Sci, 4结论 2015,43(7):1075 (I)同时采用NSGA-II与NSGA-I两个算法 (黄君政，李爱平，刘雪梅，等.考虑缓冲区配置的生产线布局优 化设计.同济大学学报（自然科学版），2015,43(7)：1075) 对3目标调度模型进行优化，得出在处理3个目标 [3] Papadopoulos H T,Vidalis M I.A heuristic algorithm for the 的问题时，NSGA-I在收敛性和生成一级个体数 buffer allocation in unreliable unbalanced production lines 量上要好于NSGA-Ⅱ的结论 Comput Ind Eng,2001,41(3):261 (2)求解3目标带有限缓冲区的混合流水车间 [4] Gupta J N D.Two-stage,hybrid flowshop scheduling problem.J 调度模型，不断增加缓冲区容积，得出缓冲区容积 0 per Res Soc,.1988,39(4:359 越大，完工时间基本不变，运输时间越短和缓冲区 5 Djellab H,Djellab K.Preemptive hybrid flowshop scheduling 占用率更均衡的结论 problem of interval orders.EurJOper Res,002,137(1):37 (3)若不考虑缓冲区占用率均衡，在缓冲区容 [6] Bolat A,Al-Harkan I,Al-Harbi B.Flow-shop scheduling for three 积较小时模型1可以得到与模型2相比更小的运 serial stations with the last two duplicate.Comput Oper Res,2005, 32(3):647 输时间，但会导致缓冲区占用率均衡性较差，缓冲 [7] Guirchoun S,Martineau P,Billaut J C.Total completion time 区容积增大后，模型1的缓冲区占用率均衡性逐 minimization in a computer system with a server and two parallel 渐得到改善，则得出在缓冲区容积较小时，考虑缓 processors.Comput Oper Res,2005,32(3):599 冲区占用率均衡更具备研究意义 [8] Brah S A.A comparative analysis of due date based job sequencing rules in a flow shop with multiple processors.Prod 参考文献 P1 an Control,1996,7(4):362 [1]Khosla I.The scheduling problem where multiple machines [9]Brah S A,Wheeler G E.Comparison of scheduling rules in a flow compete for a common local buffer.Eur J Oper Res,1995,84(2): shop with multiple processors:A simulation.Simulation,1998, 330 71(5):302衡指数分别趋于相等. 4    结论 （1）同时采用 NSGA-II 与 NSGA-III 两个算法 对 3 目标调度模型进行优化，得出在处理 3 个目标 的问题时，NSGA-III 在收敛性和生成一级个体数 量上要好于 NSGA-II 的结论. （2）求解 3 目标带有限缓冲区的混合流水车间 调度模型，不断增加缓冲区容积，得出缓冲区容积 越大，完工时间基本不变，运输时间越短和缓冲区 占用率更均衡的结论. （3）若不考虑缓冲区占用率均衡，在缓冲区容 积较小时模型 1 可以得到与模型 2 相比更小的运 输时间，但会导致缓冲区占用率均衡性较差，缓冲 区容积增大后，模型 1 的缓冲区占用率均衡性逐 渐得到改善，则得出在缓冲区容积较小时，考虑缓 冲区占用率均衡更具备研究意义. 参    考    文    献 Khosla I. The scheduling problem where multiple machines compete for a common local buffer. Eur J Oper Res, 1995, 84（2）: 330 [1] Huang J Z, Li A P, Liu X M, et al. Optimal design of production line layout considering buffer allocation. J Tongji Univ Nat Sci, 2015, 43（7）: 1075 （黄君政, 李爱平, 刘雪梅, 等. 考虑缓冲区配置的生产线布局优 化设计. 同济大学学报 (自然科学版), 2015, 43（7）：1075） [2] Papadopoulos H T, Vidalis M I. A heuristic algorithm for the buffer allocation in unreliable unbalanced production lines. Comput Ind Eng, 2001, 41（3）: 261 [3] Gupta J N D. Two-stage, hybrid flowshop scheduling problem. J Oper Res Soc, 1988, 39（4）: 359 [4] Djellab H, Djellab K. Preemptive hybrid flowshop scheduling problem of interval orders. Eur J Oper Res, 2002, 137（1）: 37 [5] Bolat A, Al-Harkan I, Al-Harbi B. Flow-shop scheduling for three serial stations with the last two duplicate. Comput Oper Res, 2005, 32（3）: 647 [6] Guirchoun S, Martineau P, Billaut J C. Total completion time minimization in a computer system with a server and two parallel processors. Comput Oper Res, 2005, 32（3）: 599 [7] Brah S A. A comparative analysis of due date based job sequencing rules in a flow shop with multiple processors. Prod Plan Control, 1996, 7（4）: 362 [8] Brah S A, Wheeler G E. Comparison of scheduling rules in a flow shop with multiple processors: A simulation. Simulation, 1998, 71（5）: 302 [9] 600 580 560 540 520 500 480 460 440 420 400 5 10 15 20 25 30 35 Buffer volume (a) Time/min 40 45 50 500 450 400 350 300 250 200 150 100 5 10 15 20 25 30 35 Buffer volume (b) Time/min 40 45 50 3.0 Model 2 Model 1 2.5 2.0 1.5 1.0 0.5 0 5 10 15 20 25 30 35 Buffer volume (c) Buffer equilibrium index/10 4 40 45 50 图 6 模型 1 与模型 2 对比. （a）完工时间；（b）运输时间；（c）缓冲区均衡指数 Fig.6 Comparison of model 1 and model 2: (a) completion time; (b) transportation time; (c) buffer equilibrium index 袁庆欣等： 带有限缓冲区的混合流水车间多目标调度 · 1497 ·