袁庆欣等：带有限缓冲区的混合流水车间多目标调度 1493· 表1参数及变量设计 Table 1 Design of the parameters and decision variables Parameters Description c Capacity of the kth machine's front buffer in the stage s C Capacity of the kth machine's back buffer N Number of machines at the s processing stage 台 Total number of artifacts in batch a TCkJ Completion time of job j on the kth machine belong to sth stage Tis-(s+l)j Transportation completion time of ith transporter for transports jobj Tikj Starting time of job j that processed on the kth machine belong to sth stage 山 The processing time of jobjon the kth machine belong to sth stage 左-s+时 The transportation time of ith transporter for transports jobj Tis-(stij The leaving time of job j that leaves ith transporter Thk The idle time of the Ath machine belong to sth stage T The leaving time of job jthat leaves the kth machine belong to sth stages Tk8j The leaving time of job jthat leaves back buffer of the kth machine belong to sth stage Taak The leaving time of ath batch that leav ves the kth machine belong to sth stage Tis-(s+lk The arriving time of ith transporter that arrives the kth machine belong to sth stage 咀 The remaining volume of the back buffer of the kth machine belong to sth stage 生 The remaining volume of the front buffer of the Ath machine belong to sth stage T The last time the back buffer of the kth machine has enough room for job j Torlka The last time the front buffer of the th machine has enough room for batcha Tisk The moment that the back buffer of the kth machine has enough room for jobj Taak The moment that the front buffer of the kth machine has enough room for batch a t Production moment Xis-(s+D).j If job j is transported by transporter ith transporter at t,it is equal to 1,otherwise 0 XE加 If job j is processed on the kth machine at t,it is equal to 1,otherwise 0 XjskF If job j is in the front buffer of the kth machine belong to sth stage at t,it is equal to 1,otherwise 0 XiB」 If job j is in the back buffer of the kth machine at t,it is equal to 1,otherwise 0 Xi.s-(s+l)al If ath transported by ith transporter,it is equal to1,otherwise (3) TkB/=maxT-+lmt,Tdk小，，k,ijea,ac0 (10) x.crvnk Tis-(s+Dj=TskB.j+his-(D)j,Vi.jEO (11) (4) Ts-s+l=max(Ti.-s+lT+Dkd小.，i.kjea,ac0 (12) 2 (5) Ta=maxtT when V Ja.T=1 Vs.k (13) TSj=Tkj+Vs.k.jeO (6) TikpTsk.Bj≥0，sk,je0 (14) Tik=maxT-e+DTVs,.k,ije0,s≥1（7） 上述模型中，式(1)~(3)为机器能力、运载设 备运输能力约束，具体为一台机器一次只能加工 Tk=max(TC.T小.Ys,k,je0 (8) 一个工件，一台运载设备一次也只能运输一个批 TR max(TRl,when V >1.TRk=t.Vs.k.jeo 次的工件：式(4)~(5)为缓冲区容积限制；式(6)、 (9) 式(11)分别为加工时间约束和运输时间约束；式∑ J j=1 Xk, j,t ⩽ 1,∀k （3） ∑ J j=1 Xj,s,k,F,t ⩽ C F s,k ,∀s, k （4） ∑ O j=1 Xj,k,B,t ⩽ C B k ,∀k （5） T C s,k, j = T s s,k, j +t p s,k, j ,∀s, k. j ∈ O （6） T s s,k, j = max{T l i,s→(s+1), j ,T i s,k };∀s, k,i, j ∈ O,s ⩾ 1 （7） T l s,k, j = max{T C s,k, j ,T B j,s,k },∀s, k, j ∈ O （8） T B j,s,k = max{T B j,s,k },when V B s,k ⩾ 1,T B j,s,k = t,∀s, k, j ∈ O （9） T l s,k,B, j = max{T a i,s→(s+1),msk ,T l a,s,k },∀s, k,i, j ∈ a,a ⊆ O （10） Ti,s→(s+1), j = T l s,k,B, j +ti,s→(s+1), j ,∀i, j ∈ O （11） T l i,s→(s+1), j =max{Ti,s→(s+1), j ,T F (s+1),k,a },∀s,i, k, j∈a,a⊆O （12） T F s,k,a = max{T B a,s,k },when V F s,k ⩾ Ja,T B a,s,k = t,∀s, k （13） T s s,k, j ,T l s,k,B, j ⩾ 0,∀s, k, j ∈ O （14） 上述模型中，式（1）～（3）为机器能力、运载设 备运输能力约束，具体为一台机器一次只能加工 一个工件，一台运载设备一次也只能运输一个批 次的工件；式（4）～（5）为缓冲区容积限制；式（6）、 式（11）分别为加工时间约束和运输时间约束；式 表 1 参数及变量设计 Table 1 Design of the parameters and decision variables Parameters Description C F s,k Capacity of the kth machine’s front buffer in the stage s C B k Capacity of the kth machine’s back buffer Ns Number of machines at the s processing stage Ja Total number of artifacts in batch a T C s,k, j Completion time of job j on the kth machine belong to sth stage Ti,s→(s+1), j Transportation completion time of ith transporter for transports job j T s s,k, j Starting time of job j that processed on the kth machine belong to sth stage t p s,k, j The processing time of job j on the kth machine belong to sth stage ti,s→(s+1), j The transportation time of ith transporter for transports job j T l i,s→(s+1), j The leaving time of job j that leaves ith transporter T i s,k The idle time of the kth machine belong to sth stage T l s,k, j The leaving time of job j that leaves the kth machine belong to sth stage s T l s,k,B, j The leaving time of job j that leaves back buffer of the kth machine belong to sth stage T l a,s,k The leaving time of ath batch that leaves the kth machine belong to sth stage T a i,s→(s+1),k The arriving time of ith transporter that arrives the kth machine belong to sth stage V B s,k The remaining volume of the back buffer of the kth machine belong to sth stage V F s,k The remaining volume of the front buffer of the kth machine belong to sth stage T B j,s,k The last time the back buffer of the kth machine has enough room for job j T F (s+1),k,a The last time the front buffer of the kth machine has enough room for batch a T B j,s,k The moment that the back buffer of the kth machine has enough room for job j T F a,s,k The moment that the front buffer of the kth machine has enough room for batch a t Production moment Xi,s→(s+1), j,t If job j is transported by transporter ith transporter at t, it is equal to 1, otherwise 0 Xk, j,t If job j is processed on the kth machine at t, it is equal to 1, otherwise 0 Xj,s,k,F,t If job j is in the front buffer of the kth machine belong to sth stage at t, it is equal to 1, otherwise 0 Xj,k,B,t If job j is in the back buffer of the kth machine at t, it is equal to 1,otherwise 0 Xi,s→(s+1),a,t If ath transported by ith transporter t, it is equal to 1, otherwise 0 袁庆欣等： 带有限缓冲区的混合流水车间多目标调度 · 1493 ·