Patent Application
20170083009
BM buffer module
FP fundamental period
EST extended sub-tree
LB lower bound
O2CS optimal one-wafer cyclic schedule
OSLB one-wafer cyclic schedule achieving the LB of cycle time
PM process module
PN Petri net
ST sub-tree
The present invention generally relates to scheduling a treelike hybrid multi-cluster tool. In particular, the present invention relates to a method for generating an optimal one-wafer cyclic schedule with minimal cycle time for this multi-cluster tool when no one-wafer cyclic schedule that achieves a LB of cycle time exists.
There follows a list of references that are occasionally cited in the specification. Each of the disclosures of these references is incorporated by reference herein in its entirety.
There follows a list of patent(s) and patent application(s) that are occasionally cited in the specification.
As a kind of integrated equipment that implements single-wafer processing technology, cluster tools play a significant role in the semiconductor manufacturing industry [Yi et al., 2008; and Chan et al., 2007]. Generally, it integrates a wafer handling robot, two loadlocks, and a few process modules (PMs). A tool is called a single or dual-arm tool if its corresponding robot has one arm and two arms, respectively. The two loadlocks are for wafer cassette loading and unloading. Through them, raw wafers are loaded into the system with a cassette-by-cassette way and processed by PMs in a pre-specified order, called a recipe. At each step, they should stay in a PM for a certain of time to be processed and finally returns to the loadlock where it came from [Wu and Zhou, 2010b].
Multi-cluster tools are increasingly adopted to accommodate the industrial demands [Chan et al., 2007]. They are composed of several, individual cluster tools connected by buffering modules (BMs) with a linear or treelike topology. It is called a hybrid K-cluster tool if it contains K (≧2) individual cluster tools including both single and dual-arm tools. A treelike hybrid 9-cluster tool linked by BMs is illustrated in
For effectively operating cluster tools, extensive research efforts have been made on the modeling and performance evaluation of them [Venkatesh et al., 1997; Perkinson et al., 1994 and 1996; Zuberek, 2001; Wu et al., 2011, 2013a, and 2013b; and Wu and Zhou, 2010a, 2012a, and 2012b]. These studies show that, with two loadlocks, a cluster tool often operates under a steady state. In the steady state, if the robot has idle time and processing time in PMs decides the cycle time of the system, it is process-bound. In practice, the robot activity time can be treated as a constant and is much shorter than wafer processing time [Kim et al., 2003]. Hence, often a tool is process-bound. Under such a case, for dual-arm tools, a swap strategy is effective [Venkatesh et al., 1997], while for single-arm ones, a backward strategy is optimal [Lee et al., 2004; Lopez and Wood, 2003].
For scheduling a multi-cluster tool, Jevtic [2001] proposes a heuristic algorithm. However, it is hard to evaluate its performance. By ignoring the robot moving time, the multi-cluster tool is modeled by an event graph combined with a network model in [Ding et al., 2006]. Then, the method finds all optimal periodical schedules based on a simulation approach. In [Yi et al., 2008], a decomposition method is presented. With the robot moving time taking into account, a polynomial algorithm is proposed to find an optimal cyclic schedule for both single-arm 2-cluster tools and m-serial-cluster tool in a treelike M-cluster tool with M>m. 2, in [Chan et al., 2011a and 2011b]. An m-serial-cluster tool is an m-cluster tool with linear topology. By the methods proposed in [Chan et al., 2011a and 201 ib], generally, the obtained cyclic schedule is a multi-wafer schedule, i.e. more than one wafer is produced during every period [Sechi et al., 1992].
Due to that a one-wafer cyclic schedule is easy to understand, implement, and control, it is the most desired one in practice, assuming that the same maximum throughput is achieved. A multi-cluster tool with its bottleneck tool being process-bound is said to be process-dominant in [Zhu et al., 2013, 2015]. It is shown that, for a process-dominant single-arm multi-cluster tool with linear topology, there is always a one-wafer cyclic schedule and a polynomial method is proposed to find an optimal one-wafer cyclic schedule (O2CS), in [Zhu et al., 2013]. A multi-cluster tool composed of both single and dual-arm tools is called a hybrid multi-cluster tool. Similar to a process-dominant linear single-arm multi-cluster tool, there is a one-wafer cyclic schedule for process-dominant linear hybrid multi-cluster tools and efficient algorithms are presented to find an O2CS in [Yang et al., 2014a and 2014b]. Structurally, a treelike hybrid multi-cluster tool is much more complex than a linear multi-cluster tool and thus it is more challenging to schedule. In our previous work [Yang et al., 2015], conditions under which a one-wafer cyclic schedule exists to reach the lower bound (LB) of cycle time for a process-dominant treelike hybrid K-cluster tool are presented, and also, an efficient algorithm is proposed to find such a schedule if it exists. However, if it does not exist, it is open that whether there is an O2CS and how it can be found if yes.
There is a need in the art to determine if the O2CS exists and, if it exists, to develop and apply the O2CS for scheduling a treelike hybrid multi-cluster tool.
An aspect of the present invention is to provide a computer-implemented method for scheduling a treelike hybrid K-cluster tool to generate a one-wafer cyclic schedule. The treelike hybrid K-cluster tool has K single-cluster tools denoted as C1, C2, . . . , CK, with C1 being a head tool of the treelike hybrid K-cluster tool. The single-cluster tool Ck, k∈K, has a robot Rk for wafer handling. The method comprises given a value of cycle time, generating a part of the schedule for a section of the K-cluster tool by performing a generating algorithm where the section of the K-cluster tool is either an EST or a ST.
The generating algorithm for ESTk or STk, with Ci being a downstream adjacent tool of Ck and with Θ being the given value of cycle time for ESTi, STi or Bi, comprises the following steps. The notations and symbols used in the steps are defined hereinafter in the specification.
In Step (S1), perform Steps (S1.1), (S1.2) and (S1.3) if Θ is not an optimal cycle time for ESTk or STk, and if k∈F and Ai(n[i])≠0.
In Step (S1.1), calculate Δm's for m∈Si according to
Δm=Am(n[m])/Σp∈S
and
where Condition 1 is that
and S-S case is considered,
Condition 2 is that
and S-S case is considered, and
Condition 3 is that D-S case is considered.
In Step (S1.2), check whether Δ=min{Δp|p∈Si}=ΔI is satisfied. If it is satisfied, then perform Steps (S1.2.1), (S1.2.2) and (S1.2.3) with:
Y=Φi(S,S)−Ai(n[i]) when Condition 1 is satisfied;
when Condition 2 is satisfied; and
when Condition 3 is satisfied.
Step (S1.2.1) is as follows. In ESTi or Bi, for p∉Si, if p∉L, then set ωp((b[p]_1)-1)=Ap((b[p]_1)-1)+Δ, else set ωp0=Ap0+Δ.
Step (S1.2.2) is as follows. In ESTi or Bi, if p∈Si and p∉F, then set:
ωpj=Apj+Y for j∈D[p], or for j∈n[p]\{n[p]} if p∈L;
ωp((b[p]_1)-1)=Ap((b[p]_1)-1)+Σq∈S
ωp(n[p])=Ap(n[p])−Σq∈S
Step (S1.2.3) is as follows. In ESTi, if p∈Si and p∈F, then set:
ωpj=Apj+Y,j∈D[p];
and
ωp(n[p])=Ap(n[p])−Σq∈S
In Step (S1.3), if Δ=min{Δp|p∈Si}=Δf≠Δi, then perform Steps (S1.2.1), (S1.2.2) and (S1.2.3) with Y=Δf followed by repeating Steps (S1.1), (S1.2) and (S1.3).
Preferably, the generating algorithm further comprises Steps (S2), (S3), (S3.1), (S3.2) and S(3.3) as follows.
Step (S.2) is performed if Θ is not an optimal cycle time for ESTk or STk, and if k∉F and Δi(n[i])=0. If this condition is satisfied, then Θ is updated with a value computed by Θ+Δi=Θ+Φl(S, S) for S-S case, or by Φi(D, S) for D-S case. Based on the updated value of Θ, the robot waiting times for the robots in ESTk or STk are recomputed, so that the part of the schedule for ESTk or STk is generated and thereby the performing of the generating algorithm is completed.
In Step (S.3), it is first checked if Θ is not an optimal cycle time for ESTk or STk, and if k∈F holds. If both answers are yes, then Steps (S3.1), (S3.2) and (S3.3) are performed.
Step (S3.1) is to find the optimal cycle time Θ+Δq for Ck with ESTk_q or STk_q, q∈f[k], by performing Steps (S.1) and (S.2), where ESTk_q or STk_q is the EST or the ST having the single-cluster tool Ci and a branch thereof, Bi_q.
Step (S3.2) is to update Θ with a value computed by Θ+max{Δ1, Δ2, . . . , Δf[k]}.
Step (S3.3) is to find the part of the schedule for STk by recomputing the robot waiting times with the updated cycle time Θ.
With the generating algorithm, one can find an O2CS for the K-cluster tool as follows. The O2CS for the smallest ST, say, STj with j=maxl∈F{l} and its ESTs is obtained first by the generating algorithm. Afterwards, the ST that is larger than STj is processed with the generating algorithm. This process is continued until it is done for the K-cluster tool to obtain a complete O2CS.
It follows that the method as disclosed herein may advantageously include a procedure as follows. First identify, in the treelike hybrid K-cluster tool, STj with j=maxl∈F{l}, and one or more ESTs of STj. The one or more ESTs are ESTj-1, ESTj-2 down to ESTi such that an upstream adjacent tool of Ci is a fork tool. A first part of the schedule for STj is determined by performing the generating algorithm. Then a second part of the schedule for ESTj-1 is determined based on the first part of the schedule. Repeat the determination of one part of the schedule ESTj-m based on a determined part of the schedule for ESTj-m+1 until the one or more ESTs are scheduled.
Other aspects of the present invention are disclosed as illustrated by the embodiments hereinafter.
Based on the results obtained in [Yang et al., 2015], the present work confirms that an O2CS exists for a treelike hybrid multi-cluster tool. To do so, a Petri net (PN) model is developed to describe the dynamic behavior of the system. By this model, one shows that there is always a one-wafer cyclic schedule. Then, algorithms are designed to obtain the O2CS by setting the robot waiting time. The method as disclosed herein is shown to be computationally efficient.
Hereinafter, the notation K, K being a positive integer, denotes a set containing positive integers from 1 to K, i.e.
K={1, 2, . . . , K}. Let ΩK={0}∪
K.
By following Chan et al. [2011a and b], for the system, one assumes that: 1) the capacity of a BM is one and has no processing function; 2) only one PM is configured for each step, that is, there is no parallel module, and only one wafer can be processed in a PM at a time; 3) only one type of wafers that have the identical recipe is processed, and they visit a PM only once except entering a BM at least twice; 4) the activity time is a known constant; and 5) in each individual tool except the fork and leaf (to be defined later) tool, besides BM(s), there is at least one PM. The fork tool may have no PM and the leaf tool has at least two PMs.
Let Ci, i∈K, denote the i-th cluster tool, where C1 that has two loadlocks is called the head tool; Ci, i≠1, is called a leaf tool if it connects to only one tool; and if Ci, i∈
K, connects to at least three adjacent tools, it is called a fork tool. Denote Ri as the robot in Ci. As shown in
A BM connecting Ck and Ci is seen as the outgoing module (OM) for Ck and the incoming module (IM) for Ci, respectively, and this IM is numbered as the Step 0 for C7. Let n[i] be the index for the last step in Ci, i∈K. Then, there are n[i]+1 steps in Ci, including the OM(s) and IM(s). Let f[i], 1≦f[i]≦n[i], denote the number of OM(s) in Ci, i∉L. If f[i]>1, C1 is a fork tool; otherwise if f[i]=1, it is not. Let PS10 denote Step j (except the BM(s)) in Ci, i∈
K and j∈
n[i], with PS10 denoting the loadlocks in C1. Further, let Si(b[i]_1), Si(b[i]_2), . . . , and Si(b[i]_f[i]) with b[i]_1<b[i]_2< . . . <b[i]_f[i], denote the OM(s). Note that b[i]_1, b[i]_2, . . . and b[i]_f[i] are not necessary in a consecutive order, in other words, b[i]_2=b[i]_1+1 may not hold. The IM for Ci is denoted as PSi0. Then, the n[i]+1 steps in Ci are denoted as PSi0, PSi1, . . . , Si(b[i]_1), . . . , Si(b[i]_2), . . . , PSi(n[i]), respectively. Notice that b[i]_1=1 if it is Step 1 and b[i]_f[i]=n[i] if it is the last step. By this way, the route of a wafer in
The present work extends the resource-oriented PN developed in [Wu, 1999; and Wu and Zhou, 2001 and 2009] to model a treelike hybrid K-cluster tool. The basic concept of the PN used in the present work is based on [Zhou and Venkatesh, 1998; Wu, 1999; and Wu and Zhou 2009]. As a kind of finite capacity PN, it is defined as PN=(P, T, I, O, M, ), where P={p1, p2, . . . , pm} is a finite set of places and T={t1, t2, . . . , tn} is a finite set of transitions; I/O is the input/output functions; M(p) is a marking representing the number of tokens in place p with M0(p) being the initial marking in p; and
is a capacity function with
(p) being the largest number of tokens that p can hold at a time. Transition t's preset is the set of all input places to t, namely, t={p: p∈P and I(p, t)>0}. Its postset is the set of all output places from t, namely, t={p: p∈P and O(p, t)>0}. Similarly, one has p's preset p={t∈T: O(p, t)>0} and postset p={t∈T: I(p, t)>0}. For the transition enabling and firing rules, one can refer to [Wu, 1999; Wu and Zhou, 2001 and 2010c].
To model a hybrid K-cluster tool, one adopts the PN models developed in [Yang et al., 2015] with the backward and swap strategies for the single and dual-arm tool as shown in
For Ci, i∈K, regardless of whether it is a single or dual-arm tool, transition xij,j∈Ωn[i], models Ri's moving from Steps j to j+1 (or Step 0 if j=n[i]) with a wafer hold, and places ri and pij, model Ri and PSij, j∈Ωn[i], respectively. Note that
(pij)=1 indicating that there is only one PM for a step; and
(ri)=1 if Ci is a single-arm tool and
(ri)=2 if Ci is a dual-arm tool. For a single-arm tool Ci, i∈
K, as shown in
(qij)=1 models Ri's waiting before unloading a wafer (token) from pij, j∈Ωn[i]. Places dij and zij with
(dij)=
(zij)=1 model that Ri holds a wafer for moving from Steps j to j+1 (or Step 0 if j=n[i]) and loading a wafer into pij, j∈Ωn[i], respectively. Transitions uij and tij model that Ri removes a wafer from pij, and drops a wafer into pij, j∈Ωn[i], respectively. Transition yij, j∈Ωn[i]\{0, 1}, models Ri's moving from Steps j+2 to j (or to n[i]−1 if j=0, or to n[i] if j=1), without carrying a wafer.
For a dual-arm tool Ci, i∈K, as shown in
(dij)=
(zij)=1 model the state that a swap ends and Ri's waiting before unloading a wafer from pij, j∈Ωn[i], respectively. Place qij with
(qij)=2 models that both arms of Ri hold a wafer and wait during swap at pij, j∈Ωn[i]. Transitions tij and uij model Ri's loading and unloading a wafer into and from pij, j∈Ωn[i], respectively. By firing uij, two tokens enter qij, implying that both arms hold a wafer and wait. For more details, the readers can refer to [Wu et al., 2008] and [Wu and Zhou, 2010b].
For a BM that connects Ck and Ci, k∉L, i∉{1}, with Ck being the upstream one, there are four different cases:
1) both Ck and Ci are dual-arm tools (herein referred to as a D-D case in short);
2) Ck is a single-arm tool and Ci is a dual-arm one (herein referred to as a S-D case in short);
3) Ck is a dual-arm tool and Ci is a single-arm one (herein referred to as a D-S case in short); and
4) both are single-arm tools (herein referred to as a S-S case in short).
As the BM linking Ck and Ci can be treated as a step denoted by PSk(b[k]_q) for Ck, or the q-th OM, 1≦q≦f[k], in Ck, and it is also seen as PSi0 in Ci. Hence, this BM is modeled by pk(b[k]_q) and pi0 for Ck and Ci, respectively, with pk(b[k]_q)=pi0 and (pk(b[k]_q))=
(pi0)=1. The PN models of the four cases are shown in
With the developed PN, the initial marking M0 of the PN model can be set by putting a V-token representing a virtual wafer (not real one) as follows.
For C1, set M0(p10)=n, representing that there are always wafers to be processed. If C1 is a single-arm tool, set M0(r1)=0; M0(p1(b[i]_1))=0 and M0(p1j)=1, j∈n[1]\{b[1]_1}; M0(z1j)=M0(d1j)=0, j∈Ωn[1]; M0(q1j)=0, j∈Ωn[1]\{(b[1]_1)−1}, and M0(q1((b[1]_1)-1))=1, implying that R1 is waiting at PS1((b[1]_1)-1)) for unloading a wafer there. If C1 is a dual-arm tool, set M0(r1)=1; M0(p1j)=1, j∈
n[1]; M0(qij)=M0(d1j)=0, j∈Ωn[1]; M0(z1j)=0, j∈Ωn[1]\{[b1]_1} and M0(z1(b[1]_1))=1, implying that R1 is waiting at PS1(b[i]_1) for unloading a wafer there.
For a single-arm tool Ci, i∈K\{1}, M0(ri)=0; M0(zij)=M0(dij)=0, j∈Ωn[i]; M0(pij)=0 and M0(pij)=1, j∈
n[i]\{1}; M0(qij)=0, j∈
n[i], and M0(qi0)=1, representing that Ri is waiting at PSi0 for unloading a wafer there. For a dual-arm tool Ci, i∈
K\{1}, set M0(ri)=1; M0(pij)=1, j∈
n[i]; M0(qij)=M0(dij)=0, j∈Ω[n]; M0(zij)=0, j∈
n[i], and M0(zi0)=1, representing that Ri is waiting at PSi0 for unloading a wafer there. It should be pointed out that, at M0, for any adjacent Ck and Ci, the token in pi0 is assumed to enable uk(b[k]_q), but not ui0.
In
Definition 1:
The color of a transition ti is defined as C(ti)={ci}.
By Definition 1, the colors for ui0 and uk(b[k]_q) are ci0 and ck(b[k]_q), respectively. Then, one can define the color for a token as follows.
Definition 2:
If a token in p∈ti enables ti, it has the same color of ti, i.e. {ci}.
With Definition 2, a token entering pi0 by firing tk(b[k]_q) has color ci0 and enables ui0 only, and the token entering pi0 by firing ti0 has color ck(b[k]_q) and enables uk(b[k]_q) only. By this way, the PN becomes conflict-free. One more issue is that although the model for a dual-arm tool is deadlock-free, the model for a single-arm one is deadlock-prone [Yang et al. 2015]. A control policy defined in [Yang et al. 2015] is applied to solve this problem.
Definition 3
[Yang et al. 2015]: In the PN model of a treelike hybrid K-cluster tool, for a single-arm tool Ci, i∈K, at marking M, yij, j∈Ωn[i]\{n[i], (b[i]_q)−1} and q∈
f[i], is control-enabled if M(pi(j+1))=0; yi(n[i]) is control-enabled if transition ti1 has just been executed, and yi((b[i]_q)-1) is control-enabled if transition ti((b[i]_q)+1) has just been executed.
Thereafter, it is assumed that the PN is always controlled by the control policy given in Definition 3 such that the model is deadlock-free.
In the developed PN model, the time related activities are modeled by both transitions and places and time should be associated with both of them. As pointed out by Kim et al. [2003] and Lee and Lee [2006], for both single (S) and dual-arm (D) tools, the time for the robot to move from one step to another with or without carrying a wafer is identical, so is the time for the robot to unload/load a wafer from/into a PM. By following [Wu et al., 2008; and Wu and Zhou, 2010b], the activity time is modeled as shown in TABLE 1.
With the above developed PN model, this section presents the temporal properties for individual tools to parameterize a schedule by robot waiting time. According to Wu et al. [2008], for a single-arm tool Ci, i∈K, the time taken for processing a wafer at Step j, j∈
n[i], is
θij=αij+4λi+3μi+ωi(j-1) (1)
For Step 0, as αi0=0, one has
θi0=4λi+3μi+ωi(n[i]) (2)
With the robot waiting time being removed, the time needed for completing a wafer at Step j is
ξij=αij+4λi+3μi,j∈Ωn[i]. (3)
To be feasible, a wafer should stay in PMij for τij (≧αij) time units. By replacing αij with τij, the cycle time at Step j in Ci is computed by
πij=+4A+3μi+ωi(j-1),j∈Nn[i] (4)
and
πij=τij+4λi+3μi+ωi(j0-1),j∈Ωn[i]. (4)
The robot cycle time for a single-arm tool Ci is
where ψij=2(n[i]+1)(λi+μi) is the robot's activity time in a cycle without waiting and it is a known constant, while ψi2=Σj=0n[i]ωij is the robot waiting time in a cycle.
For a dual-arm tool Ci, i∈K, according to Wu et al. [2010a], the time needed for processing a wafer at Step j, j∈Ωn[i], is
ξij=αij+λi. (7)
Similarly, by replacing αij with τij in (7), the cycle time at Step j, j∈Ωn[i], is
πij=τij+λi. (8)
The robot cycle time for a dual-arm tool Ci is
where ψi1=(n[i]+1)(λi+μi) is the robot cycle time in a cycle without waiting and ψi2=Σj=0n[i]ωij is the robot's waiting time in a cycle.
Since the production process is serial for each tool Ci, i∈K, the productivity for each step is identical. Thus, Ci should be scheduled such that
πi=πi0=πi1= . . . =πi(n[i])=ψi. (10)
Due to the fact that both πi and ψi are functions of ωij's, the schedule for each tool Ci, i∈K, is parameterized by robots' waiting time. With this observation in mind, the key for scheduling a treelike hybrid K-cluster tool is to determine ωij's such that the activities of the multiple robots are coordinated to act in a paced way.
Let Πi=max{ξi0, ξi1, . . . , ξi(n[i]), ψ(n[i])} denote the fundamental period (FP) of Ci, i∈K. If Πi=max{ξi0, ξi1, . . . , ξi(n[i])}, Ci is process-bound. Let Π=max{Π1, Π2, . . . ΠK}=Πh, 1≦h≦K, or Ch is the bottleneck tool. Since, by assumption, a treelike hybrid K-cluster tool addressed here is process-dominant, Ch must be process-bound. Let Θ be the cycle time of the system. Then, with πi being the cyclic time of Ci, to obtain a one-wafer cyclic schedule for the system, each individual tool must have an identical cycle time that is equal to Θ for the entire system, i.e.
Θ=πi≧Π,∀i∈K. (11)
From [Yang et al. 2015], each individual tool can be scheduled to be paced, if and only if, for any adjacent pair Ck and Ci, k∉L and i∉{1}, linked by PSk(b[k]_q), 1≦q≦f[k], at any marking M, one has: 1) whenever Ri (Rk) is scheduled to load a token into pi0 (pk(b[k]_q)), M(pi0)=0 (M(pk(b[k]_q))=0); and 2) whenever Ri (Rk) is scheduled to unload a token from pi0 (pk(b[k]_q)), M(pi0)=1 (M(pk(b[k]_q))=1). In [Yang et al. 2015], the existence of a one-wafer cyclic schedule with the LB of cycle time, i.e. OSLB for short, is discussed and conditions under which an OSLB exists are derived. Algorithm is developed to find it if it exists. This work conducts a study on the case that there is no OSLB and presents a method to find an O2CS with Θ>Π.
For a process-dominant treelike hybrid K-cluster tool, the conditions under which an OSLB can be found are given as follows [Yang, et al., 2015].
Lemma 1: For a process-dominant treelike hybrid K-cluster tool, an OSLB exists, if and only if, for any adjacent pair Ck and Ci, k∉L, i∉{1}, connected by PSk(b[k]_q), the following conditions are satisfied by determining ωkj's and ωil's, j∈Ωn[k] and l∈Ωn[i] such that:
πkj=πil=Π,j∈Ωn[k] and l∈Ωn[i]; (12)
if Ck and Ci are D-S case,
Π−4λk≧4λi+3μi+ωi(n[i]); (13)
and if Ck and Ci are S-S case,
Π−(4λk+3μk+ωk((b[k]_q)-1))≧4λi+3μi+ωi(n[i]); (14)
Based on Lemma 1, an efficient algorithm is developed in [Yang, et al., 2015] to test the existence of an OSLB and find it if it exists. In the sequel, for a process-dominant treelike K-cluster tool, one shows that there is always a one-wafer cyclic schedule and present an efficient algorithm to find the minimal cycle time Θ>Π and O2CS. In the following discussion, one focuses on the situation that there is no OSLB for the system.
Let Bi denote a linear multi-cluster tool in a treelike K-cluster tool called a branch that starts from Ci and ends at Cm, m∈L. For simplicity, one assumes that the tools in Bi is labeled consecutively as Ci, Ci+1, . . . , Cm. Assume that, with Θ=Π as the cycle time, the conditions given in Lemma 1 are violated for a pair of Ci and Ci+1 in Bi. Then, let Akj=ωkj, k=i+1, i+2, . . . , m, j∈Ωn[k], where ωkj is Rk's waiting time obtained by the algorithm given in [Yang, et al., 2015] with cycle time Θ=Π. Define Φi+1(S, S)=4λi+1+3μi+1+A(i+1)(n[i+1])+4λi+3μi, when Ci and Ci+1 are S-S, and Φi+1(D, S)=4λi+1+3μi+1+A(i+1)(n[i+1])−Θ+λi, when Ci and Ci+1 are D-S. If A(i+1)(n[i+1])=0, it follows from [Yang et al., 2014b] that an O2CS with cycle time Θ=Π+Φi+1(S, S) (or Π+Φi+1(D, S) if Ci is a dual-arm tool) can be found for Bi. Otherwise, if A(i+1)(n[i+1])≠0, assume that, for g∈[i+1, m], Ak+1)(n[k+1])>0, i≦k≦g−1, and A(g+1)(n[g+1])=0, or Cg+1 is a dual-arm one, or g∈L. Define D[k]={PMkj|j≠(b[k]_1)−1 and j≠n[k]} be the set of steps in Ck, and B[k]=n[k]−1, k=i+1, i+2, . . . , m. Then, let Δg=Δg(n[g])/B[g], Δg−1=A(g−1)(n[g−1])/(B[g−1]+B[g]), Δg−2=A(g−2)(n[g−2])/(B[g−2]+B[g−1]+B[g]), . . . , Δi+2=A(i+2) (n[i+2])/(B[i+2]+B[i+3]+ . . . +B[g]), Δi+1 is defined by (15) below if Ci is a single-arm tool, where Γ=(B[i+1]+B[i+2]+ . . . +B[g]).
If Ci is a dual-arm tool, Δi+1=Φi+1(D, S)/(Γ+1). Then, one has Lemmas 2-3 [Yang, et al., 2014b].
Lemma 2: For Bi that is composed of two tools, there is an O2CS and its minimal one-wafer cycle time is Θ=Π+Δ, where Δ is determined by
With Lemma 2, a method is proposed to find the minimal cycle time and the O2CS for a Bi composed of two tools in [Yang, et al., 2014b].
Lemma 3: For a Bi in a treelike hybrid K-cluster tool, if, by the algorithm in [Yang, et al., 2015], a one-wafer cyclic schedule with cycle time Θ (≧Π) is found, a feasible one-wafer cyclic schedule is obtained with cycle time (Θ+Δ) by setting ωg((b[g]_1)-1)=Ag((b[g]_1)+Δ, g∈[i, m−1], and ωm0=Δm0+Δ where m∈L and Δ>0.
Let STi denote a sub-tree in a treelike hybrid K-cluster tool that starts from Ci with Ci being a fork tool. Let STi and STj be two STs, and STi can be contained by STj, and such a relation is denoted as STi<STj or STj>STi. In a treelike hybrid K-cluster tool, the individual tools can be numbered such that STj>STi only if j<i. However, j<i does not necessarily mean STj>STi, for they may be independent of each other. Such a relation is denoted as STjvSTi. Then, one can discuss the one-wafer cyclic scheduling problem. One has the following result.
Theorem 1: For an STi composed of three tools, there is an O2CS and its minimal cycle time is Θ=Π+Δ=Π+max{Δ1, Δ2}, where Δ1 and Δ2 are obtained by the method given in Lemma 2.
Proof:
An STi composed of three tools must contain a fork tool denoted by Ci and two downstream tools denoted by Ci+1 and Ci+2. It follows from Lemma 2 that one can schedule both Ci and Ci+1 such that their cycle time is Θ=Π+Δ1 to obtain an O2CS for tool pair of Ci and Ci+1. Similarly, both Ci and Ci+2 can be scheduled such that their cycle time is Θ=Π+Δ2 to obtain an O2CS for tool pair of Ci and Ci+2. Thus, for STi composed of Ci, Ci+1, and Ci+2, one can schedule the three tools such that their cycle time is Θ=max{Π+Δ1, Π+Δ2}=Π+max{Δ1, Δ2} to obtain an O2CS.
The ST discussed in Theorem 1 is the simplest one. A fork tool Ci with f[i] adjacent downstream tools Ci_1, Ci_2, and Ci_f[i] may contain no ST but only branches and these branches are Bi_1, Bi_2, . . . , Bi_f[i]. By the method presented in [Yang, et al., 2014b], one can find O2CSs for Bi_q, q∈f[i]. Assume that the cycle time for the branch composed of Ci and Bi_q is Π+Δq. Then, based on Theorem 1, one has the following corollary immediately.
Corollary 1:
For an STi composed of Ci and its downstream branches Bi_1, Bi_2, . . . , Bi_f[i], there is an O2CS and its minimal one-wafer cycle time is Θ=Π+max{Δ1, Δ2, . . . , Δf[i]}, where Π+Δq is the cycle time for the branch composed of Ci and Bi_q, q∈f[i].
Then, an O2CS for a treelike hybrid K-cluster tool can be found as follows. Let Cj be a fork tool. There must be a STj. Assume that there are tools Ci, Ci+1, . . . , Cj-2, Cj-1, and none of them is a fork tool. Then, one calls the multi-cluster tool formed by Ci and all its downstream tools an extended sub-tree denoted as ESTi. Note that Ci is not a fork tool. If ESTi contains STj, one calls ESTi is an extended sub-tree of STj. Let F={I/Ci is a fork tool} be the index set of fork tools. Assume that j∈F has the largest value, i.e. j=maxl∈F{l}. Then, STj must contain no ST and its O2CS can be found based on Corollary 1. There may be more than one EST of STj. Thus, with the O2CS for STj found, one needs to find the O2CSs for all its ESTs. Then, let F←F\{j}. If F≠Ø, find h such that it has the largest value in F, i.e., h=maxl∈F{l}. If SThvSTj, or they are independent of each other, it can be dealt with just as STj. Otherwise, STj can be contained in STh or one has STh>STj. In this case, one needs to find the O2CS for STh and its ESTs. Then, let F←F\{h} and repeat the above procedure. In this way, with the decreasing order of the indices in F, one finds the O2CSs for all the STs and their ESTs. When C1 is reached, one finds the O2CS for the treelike hybrid K-cluster tool. The key is how to find an O2CS for a ST or an extended one. By the definition of an ESTk, if Bi_q in Corollary 1 is replaced by ESTi_q, the result must also hold.
In the following, one first shows how to find the O2CS for an EST that contains only one fork tool and then present a method for finding the O2CS for the whole system. Assume that j=maxl∈F{l} and ESTi is an extended sub-tree of STj. Further, assume that for tool pair Ci and Ci+1, i+1≦j, there is no one-wafer cyclic schedule with cycle time Θ. Notice that, based on the O2CS for STj that can be found based on Corollary 1, Condition (13) or (14) is not satisfied for tool pair Ci and Ci+1, i+1≦j, only.
Let Apj=ωpj, p∈K\Ni, j∈Ωn[p], where ωp's are obtained by the algorithm in [Yang, et al., 2015] with cycle time Θ, and define D[l]={PM1j|j≠n[l] and j≠(b[l]_d)−1, d∈f[l]} and B[l]=n[l]−1, l∈
K. Let Ωi+1(S, S)=4λi+1+3μi+1+A(i+1)(n[i+1])−Θ+4λi+3μi for the S-S case and Ωi+1(D, S)=4λi+1+3μi+1+A(i+1)(n[i+1])=Θ+4λi+3μi for the D-S case. It follows from [Yang, et al., 2014b] that, if A(i+1)(n[i+1])=0, to make (13) or (14) satisfied for Ci and Ci+1, Θ needs to be increased by Φi+1(D, S) and Ωi+1(S, S) time units, respectively. If A(i+1)(n[i+1])≠0, one needs to check if Aj(n[j])=0. If yes, the optimal cycle time can still be found by the methods in [Yang, et al., 2014b]. If Aj(n[j])≠0 and there is a dual-arm tool in the tools Ci+1, Ci+2, Cj, or there is at least one Ap(n[p])=0, i+1<p<j, one can still find the optimal cycle time by the methods in [Yang, et al., 2014b]. However, when Ap(n[p])>0, i+1 one cannot find the optimal cycle time by the methods in [Yang, et al., 2014b]. In this situation, one discusses how to find the optimal cycle time as follows.
Assume that Cj has f[j] downstream tools such that ST, has f[j] branches. Then, for Cj_q in Bj_q, q∈f[j] one first checks whether A(j_q)(n[q])>0. If so, find g such that A((j_q)+z)(n[(j_q)+z])>0, 0≦z≦g, and A(j_q)+z+1)(n[(j_q)+z+1])>0 or C(j_q)+g+1 is a dual-arm one, or (j_q)+g∈L. Then, let A((j_q)+g)=A((j_g)(n[(j_q)+g])/B[(j_q)+g], Δ((j_q)+g−1)=A((j_q)+g−1)(n[j_q)+g−1])/(B[(j_q)+g−1]+B[(j_q)+g]), Δ(j_q)+g−2)=A((j_q)+g−2)(n[j_q)+g−2])/B[(j_q)+g−2]+B[(j_q)+g−1]+B[(j_q)+g]), . . . , and Δj_q)=A(j_q)(n[j_q])/B[j_q]+B[(j_q)+1]+ . . . +B[(j_q)+g]). Further, let Υj_q=(B[j_q]+B[(j_q)+1]+ . . . +B[(j_q)+g]), q∈
f[j], and Υj=Υj_1+Υj
f[i] and 0≦z≦g, where Δi+1 is defined by
Then, one has the following results.
Theorem 2:
Assume that: 1) j=maxl∈F {l}; 2) for an ESTi of STj, by the algorithm in [Yang, et al., 2015] with cycle time Θ, Condition (14) is violated for tool pair Ci and Ci+1, i+1≦j, only; and 3) Δ=Δκ=Ai+1. Then, for ESTi, an O2CS with cycle time Θ+Δ can be found, where Δ is given by (17).
Proof:
Since (14) is violated for tool pair Ci and Ci+1 only, they should be S-S. It follows from (17) that if 0<A(i+1)(n[i+1])≦Λ×Ωi+1(S, S)/(Λ+1) and Δ=Δi+1=Ωi+1(S, S)−A(i+1)(n[i+1]), one has (Λ+1)×A(i+1)(n[i+1])≦Λ×Ωi+1(S, S)A(i+1)(n[i+1])<Λ×(Φi+1(S, S)−A(i+1)(n[i+1]))
A(i+1)(n[i+1])/Λ<Φi+1(S, S)−A(i+1)(n[i+1])=Δ. By A(i+1)(n[i+1])>0, one has Δ>0, or it is feasible in the sense of that the time is set to be positive value. Let Θ←Θ+Δ. Then, for B(j_q)+g+1) (if it exists) in Bj_q q∈
f[i], reset the robots' waiting time as given by Lemma 3. Meanwhile, for C((j_q)+g), set ω((j_q)+g)l=A((j_q)+g)l+A(i+1)(n[i+1])/Λ,l∈D[(j_q)+g] (or l∈
n[j_q)+g]\{n[(j_q)+g]} if (j_q)+g∈L), ω(j_q)+g)((b[(j_q)+g]_1)-1)=A(j_q)+g)((b[(j_q)+g]_1)-1)+Δ (or ω(j_q)+g)0=A((j_q)+g)0+Δ if (j_q)+g∈L), and ω(j_q)+g)(n[j_q)+g])=A(j_q)+g)(n[j_q)+g])−B[(j_q)+g]×A(i+1)(n[i+1])/Λ>A((j_q)+g)(n[j_q)+g])−B[(j_q)+g]×Δ>A((j_q)+g)(n[(j_q)+g])−B[(j_q)+g]×Δ(j_q)+g=0. It follows from the discussion in the last section that it is feasible for C(j_q)+g. For C(j_q)+g−1, set ω((j_q)+g−1)((b[(j_q)+g−1]_1)-1)=A((j_q)+g−1)((b[(j_q)+g−1]_1)-1)+B[(j_q)+g]×A(i+1)(n[i+1])/Λ+Δ, ω((j_q)+g−1)l=A((j_q)+g−1)l+A(i+1)(n[i+1])/Λ, l∈D[(j_q)+g−1], and ω(j_q)+g−1)(n[(j_q)+g−1])=A(j_q)+g−1)(n[(j_q)+g−1])−(B[(j_q)+g]+B[(j_q)+g−1])×A(i+1)(n[i+1])/Λ>0. It is also feasible for C(j_q)+g−1. Then, set the robot waiting time for C(j_q)+g−2, . . . , C(j_q)+2, C(j_q)+1, and Cj_q in a similar way. For Cj_q, one has ω(j_q)((b[j_q]_1)-1)=A(j_q)((b[j_q]_1)-1)+(B[(j_q)+g]+B[(j_q)+g−1]+ . . . +B[(j_q)+1])×A(i+1)(n[i+1])/Λ+Δ, ω(j_q)l=A(j_q)l+A(i+1)(n[i+1])/Λ, l∈D[j_q], and ω(j_q)(n[j_q])=A(j_q)(n[j_q])−(B[(j_q)+g]+B[(j_q)+g−1]+ . . . +B[j_q])×A(i+1)(n[i+1])/Λ>0. It is feasible for Cj_q too. For the fork tool Cj, set ωj((b[j]_d)-1)=Aj((b[j]_d)-1)+(B[(j_q)+g]+B[(j_q)+g−1]+ . . . +B[j_q]+1)×A(i+1)(n[i+1])/Λ, d∈{2, 3, . . . , f[j]}; ωj((b[j]_1)-1)=Aj((b[j]_1)-1)+(B[j_q)+g]+B[(j_q)+g−1]+ . . . +B[j_q])×A(i+1)(n[i+1])+Δ; and ωjl=Ajl+A(i+1)(n[i+1])/Λ, l∈D[j]. At last, set ωj(n[j])=Aj(n[j])−(Υj+B[j])×A(i+1)(n[i+1])/Λ>0. Similarly, set the robot waiting time for Cj-1, Cj-2, . . . , till to Ci+1. In this way, for Ci+1, one has ω(i+1)((b[i+1]_1)-1)=A(i+1)(b[i+1]_1)-1)+(Λ−B[i+1])×A(i+1)(n[i+1])/Λ+Δ, ω(i+1)l=A(i+1)l+A(i+1)(n[i+1])/Λ, j∈D[i+1], and ω(i+1)(n[i+1])=A(i+1)(n[i+1])−Λ×Δ(i+1)(n[i+1])/Λ=0. By doing so, one has Θ+Δ−(4λi+1+3μi+1)=Θ+4λi+1+3μi+1+A(i+1)(n[i+1])−Θ+4λi+3μi)−A(i+1)(n[i+1])−(4λi+1+3μi+1)=4λi+3μi. This implies that a one-wafer cyclic schedule is found for ESTi with cycle time Θ←Θ+Δ. Notice that one has Θ+Δ−(4λi+1+3μi+1)=4λi+3μi, implying that any decrease of cycle time Θ would result in Θ+Δ−(4λi+1+3μi+1)<4λi+3μi, or the schedule obtained is optimal in the sense of cycle time.
If A(i+1)(n[i+1])≧Λ×Ωi+1(S, S)/(Λ+1) and Δ=Φi+1(S, S)/(Λ+1)>0. Let Θ←Θ+Δ, and set the robot waiting time as done above with both A(i+1)(n[i+1])/Λ and Δ=Φi+1(S, S)−A(i+1)(n[i+1]) being replaced by Δ=Φi+1(S, S)/(Λ+1). Eventually, for Ci+1, one has ω(i+1)((b[i+1]_1)-1)=A(i+1)((b[i+1]_1)-1)+(Λ−B[i+1])×Δ+Δ, ω(i+1)l=A(i+1)l+Δ, l∈D[i+1], and ω(i+1)(n[i+1])=A(i+1)(n[i+1])−Λ×Δ≧0. Similar to the above case, the obtained schedule can be shown to be a feasible and optimal one-wafer cyclic schedule.
The result given by Theorem 2 can be illustrated by
If Ci and Ci+1 are D-S, one has the following result.
Theorem 3:
Assume that: 1) j=maxl∈F{l}; 2) for an ESTi of STj, by the algorithm in [Yang, et al., 2015] with cycle time Θ, Condition (13) is violated for adjacent tool pair Ci and Ci+1, i+1≦j, only; and 3) Δ=Δκ=Δi+1. Then, for ESTj, an O2CS with cycle time Θ+Δ can be found, where Δ is given by (17).
Proof:
Since (13) is violated for adjacent tool pair Ci and Ci+1 only, it should be D-S. It follows from (17) that Δ=Ωi+1(D, S)/(Λ+1)>0. Let Θ←Θ+Δ, and set the robot waiting time as done for Case 1 in Theorem 2 with both A(i+1)(n[i+1])/Λ and Δ=Ωi+1(S, S)−A(i+1)(n[i+1]) being replaced by Δ=Ωi+1(S, S)/(Λ+1)>0. At last, for Ci+1, set ω(i+1)((b[i+1]_1)-1)=A(i+1)((b[i+1]_1)-1)+(Λ−B[i+1])×Δ+Δ, ω(i+1)l=A(i+1)l+Δ, l∈D[i+1], and ω(i+1)(n[i+1])=A(i+1)(n[i+1])−Λ×Δ=A(i+1)(n[i+1])−Λ×Ωi+1(D, S)/(Λ+1). By the assumption in Section A, for each tool Cp, p∈K, n[p]≧2 holds. Hence, for single-arm tool Ci+1, one has Θ=ψi=Σl=0n[i]Akl+(n[i]4λi+1+3μi+1<4(λi+1+μi+1)≦(2/3)Θ. For dual-arm tool Ci, one has Θ=ψi=Σl=0n[0]Akl+(n[i]+1)(λi+μi)≧Σl=0n[i]Akl+3(λi+μi)
3λi<Θ
2Θ<3Θ−3λi
(2/3)Θ<Θ−λi. Thus, one has 4λi+1+3μi+1<(2/3)Θ<Θ−λi
4λi+1+3μi+1−(Θ−λi)+A(i+1)(n[i+1])<A(i+1)(n[i+1])
A(i+1)(n[i+1])>Ωi+1(D, S)=4λi+1+3μi+1−(Θ−λi)+A(i+1)(n[i+1]). This implies that ω(i+1)(n[i+1])=A(i+1)(n[i+1])−Λ×Ωi+1(D, S)/(Λ+1)>0, i.e. it is feasible. Similar to Theorem 2, the schedule obtained is optimal.
The idea of Theorem 3 is the same as that of Theorem 2. In Theorems 2 and 3, it is assumed that κ=i+1. When κ=argmin{|Ai+1, Δi+2, . . . , Δj, Δ(j_q)+z)}≠i+1, q∈f[i], 0≦z≦g, the following result presents how to find an O2CS for ESTi.
Theorem 4:
Assume that: 1) j=maxl∈F{l}; 2) in ESTi of STj, by the algorithm in [Yang, et al., 2015] with cycle time Θ, Condition (13) or (14) is violated for tool pair Ci and Ci+1, i+1≦j, only; 3) Δ=Δκ≠Δi+1. Then, there is an O2CS with cycle time Θ+Δ for ESTi, where Δ is defined as above.
Proof:
Assume κ=p, or Δ=min{Δi+1, Ai+2, . . . , Δj, Δ(j_q)+z)}=Δp, q∈f[j], 0≦z≦g, and p∈[j_1, (j_1)+g]. Let Θ←Θ+Δ and, for ESTi, set the robots' waiting time as done for Case 1 in Theorem 2 with both A(i+1)(n[i+1])/Λ and Δ=Ωi+1(S, S)−A(i+1)(n[i+1]) being replaced by Δ=Δp such that ωp(n[p])=Ap(n[p])−(B[p]+B[p+1]+ . . . +B[g])×(Ap(n[p])/(B[p]+B[p+1]+ . . . +B[g]))=0. Then let g=p−1 and calculate A again. If Δ=Δq, q∈[j_1, (j_1)+g], let Θ←Θ+Δ and repeat the above process. If p∈[j_q, (j_q)+g], q=2, 3, . . . , f[j], or p∈argmin{Δi+1, Ai+2, . . . , Δj}, set the robots' waiting time for ESTi in the same way. In this way, finally, Δ=Ai+1 must occur and then an O2CS for ESTi is found by using Theorem 2 (or Theorem 3 when Ci is a dual-arm one).
By Theorems 2-4, to find an O2CS for an ESTi of STj, it requires that j is the largest one in F, and such an ESTi contains only one fork tool. This is not enough for one to find the O2CS for a treelike hybrid K-cluster tool. However, based on Theorems 2-4, one can obtain the O2CS for an arbitrary ESTk as follows. Assume that for an ESTk that may contain more than one fork tool. Then Condition (13) or (14) is violated for Ck and Ci only with Ci being the downstream adjacent tool of Ck. A path from Ck to a downstream Cl is called a shortest path if there are the fewest nodes (tools) on the path. Then, for an ESTk, one says that Cl is “connected” to Ck if: 1) there are only single-arm tools on the shortest path from Ck to Cl; and 2) for each tool Cp on the path, ωp(n[p])'s>0. By default, Ck is connected to itself. Let Sk={p|Cp is connected to Ck}. Further, as is done above, let Φ1(S, S)=4λi+3μi+Ai(n[i])−Θ+4λk+3μk for the S-S case and Φi(D, S)=4λi+3μi+Ai(n[i])−Θ+λk (for the D-S case. Then, for an arbitrary ESTk with Ci being the downstream adjacent tool of Ck, to find the minimal one-wafer cycle time, Δm can be calculated as follows. One has that
Δm=Am(n[m])/(Σp∈S
and if m=i, then one has
Clearly, (19) is the extension of (17) for an ESTk containing one or more fork tools. Thus, based on (18) and (19), one can find the O2CS for an ESTk if (13) or (14) is violated for Ck and Ci only with Ci being the downstream adjacent tool of Ck. With the O2CS for an ESTj, the O2CS for a STk that contains a number of ESTj's can be obtained by using Corollary 1. One presents the following algorithm.
Algorithm 1:
Given an O2CS with cycle time Θ for ESTi (or ST or Bi), find the O2CS for ESTk (or STk) with C, being the downstream adjacent tool of Ck.
Step 1:
Given cycle time Θ, check if Θ is the optimal cycle time for ESTk (or STk) by the algorithm in [Yang, et al., 2015]. If yes, go to Step 5.
Step 2:
If k∉F and Ai(n[i])=0, then:
2.1. Θ←Θ+Δi=Θ+Φi(S, S) (or Φi(D, S) if it is D-S case);
2.2. Reset the robots' waiting time by the algorithm in [Yang, et al., 2015] with cycle time Θ, then go to Step 5.
Step 3:
If k∉F and Ai(n[i])≠0, then:
3.1. For m∈Si, calculate Δm's, by (18) or (19).
3.2. If Δ=min{Δp|p∈Si}=Δi, do:
3.2.1. If it belongs to the first case in (19), then:
3.2.1.1. In ESTi or Bi, for p∉Si, if p∉L, ωp((b[p]_1)-1)=Ap((b[p]_1)-1)+Δ, otherwise ωp0=Ap0+Δ.
3.2.1.2. In ESTj or Bi, if p∈Si and p∉F, set ωpj=Apj+Ai(n[i])/(Σp∈Sn[p]\{n[p]} if p∈L), ωp((b[p]_1)-1)=Ap((b[p]_1)-1)+Σq∈S
3.2.1.3. In ESTi, if p∉S1and p∉F, set ωpj=Apj+Ai(n[i]/(Σp∈S
3.2.1.4. Go to Step 5.
3.2.2. If it belongs to the second/third case in (19), set the robot waiting time by repeating Steps 3.2.1.1-3.2.1.4, in doing so, one needs to replace Ai(n[i])/(Σp∈S
3.3. If Δ=min{Δp|p∈Si}=Δf≠Δi, set the robot waiting time by repeating Procedures in 3.2.1.1 to 3.2.1.4 with both Ai(n[i])/(Σp∈S
Step 4: If k∈F, then:
4.1. Find the optimal cycle time Θ+Δq for Ck with ESTk_q (or STk_q), q∈f[k], by repeating Steps 2 and 3.
4.2. Θ←Θ+max{Δ1, Δ2, . . . , Δf[k]}.
4.3. Find the O2CS for STk by resetting the robot waiting time with cycle time Θ by using the algorithm in [Yang, et al., 2015].
Step 5:
Stop and output the schedule.
With the above analysis, one can find the O2CS for a process-dominant treelike hybrid K-cluster tool as follows. One finds the O2CS for the smallest ST, say STj with j=maxl∈F{l} and its ESTs first. Then, do that for the ST that is larger than STj. Continue this process until it is done for the K-cluster tool to find the final solution. However, by Algorithm 1, it requires that, for an ESTk, Condition (13) or (14) is violated for tool pair Ck and Ci only. This may not hold for general cases. To solve this problem, the O2CS for the ESTs of STj can be found as follows. Based on the O2CS of STj, for ESTj-1 with Cj-1 being not a fork, the above requirement must be satisfied. Thus, one can find the O2CS for ESTj-1. Thereafter, one can do that for ESTj-2, until to ESTi such that the upstream adjacent tool of Ci is a fork, and in these processes the above requirement is always satisfied. Thus, one presents the following algorithm.
Algorithm 2:
Find an O2CS for a process-dominant treelike hybrid K-cluster tool.
Step 1:
Θ=max{Π1, Π2, . . . , ΠK}.
Step 2:
Check the existence of an OSLB by the algorithm in [Yang et al., 2015]. Then:
2.1. If yes, find the schedule and go to Step 4.
2.2. Otherwise find the set F such that l∈F, find the fork tool Cj with j=maxl∈F{l}, and go to Step 3.
Step 3:
Find the O2CSs for STj. and all of its ESTk's:
3.1. Find the O2CS for STj.
3.2. Find the O2CS for all of the ESTk's of STj:
3.2.1. Find k such that Ck is the upstream adjacent tool of Cj.
3.2.2. If Ck is a fork tool, go to Step 3.3; otherwise proceed to execute Step 3.2.3.
3.2.3. Find the O2CS for ESTk by using Algorithm 1.
3.2.4. Let i=k, if i=1, go to Step 4; otherwise find k such that Ck is the upstream adjacent tool of Ci, and go to Step 3.2.2.
3.3. If F←F\{j}≠Ø, go back to Step 2.2.
Step 4:
End and output the schedule.
Notice that, for a process-dominant treelike hybrid K-cluster tool, the number of Bs+the number of STs+the number of ESTs must be less than K. Also, any B or ST, or EST contains no more than K tools. Given any of B, ST, and EST, one needs to calculate the optimal cycle time at most (K−1) times by increasing Δ each time. Thus, the computational complexity is O(K2). With K being limited in practice, the method presented is efficient and is applicable to industrial cases.
This section uses two examples to show the application of the disclosed method.
It is a treelike hybrid 3-cluster tool, where C1 is a fork tool and its adjacent downstream tools are C2 and C3. Furthermore, C2 and C3 are single-arm tools, and C1 is a dual-arm tool. Their activity times are as follows. For C1, (α10, α11, α12, α13, λ1, μ1)=(0, 77, 0, 0, 23, 1); for C2, (α20, α21, α22, λ2, μ2)=(0, 75, 79, 4, 1); and for C3, (α30, α31, α32, λ3, μ3)=(0, 71, 69, 6, 1).
For C1, one has ξ10=23 s, ξ11=100 s, ξ12=23 s, ξ13=23 s, ψ11=(n[1]+1)×(λ1+μ1)=4×24=96 s, and Π1=100 s. For C2, one has ξ20=19 s, ξ21=94 s, ξ22=98 s, ψ21=2(n[2]+1)×(λ2+μ2)=6×5=30 s and Π2=98 s. For C3, one has ξ30=27 s, ξ31=98 s, ξ32=96 s, ψ31=2(n[3]+1)×(λ3+μ3)=6×7=42 s and Π3=98 s. Since all the individual tools are process-bound, this 3-cluster tool is process-dominant, and one lets Θ=π1=π2=π3=Π=Π1=100 s. By the algorithm provided in [Yang, et al., 2015], the robots' waiting time is set as ω20=6 s, ω21=2 s, and ω22=Π−ψ21−ω20−ω21=100−30−6−2=62 s; ω30=2 s, ω31=4 s, and ω32=Π−ψ31−ω30−ω31=100−42−2−4=52 s. Thus, for C1, as Π−(4λ2+3μ2+ω22)−λ1=100−(19+62)−23=−4<0 and Π−(4λ3+3μ3+ω32)−λ1=100−(27+52)−23=−2<0, or (14) is violated and there is no OSLB. Thus one needs to find an optimal cycle time Θ by Algorithm 2.
By Algorithm 2, one has Δ2=Ω2(D, S)/n[2]=4/2=2 s, Δ3=Ω3(D, S)/n[3]=2/2=1 s, Δ=max{Δ2, Δ3}=2 s, and Θ=102 s. Then, let Aij=ωij, i∈3\{1} and j∈Ωi(n[i]), where ωij is obtained by the algorithm provided in [Yang, et al., 2015]. By Algorithm 2, the robot waiting time is set as ω30=A30+Δ=4 s, ω31=A31+Δ=6 s, ω32=A32−Δ=50 s, ω20=A20+Δ=8 s, ω21=A21+Δ=4 s, ω22=A22−Δ=60 s, ω10=A10+Δ=6 s, and ω11=ω12=ω13=0. In this way, the minimal cycle time and optimal one-wafer cyclic schedule is obtained and it is shown by the Gantt chart in
It is from [Yang, et al., 2015]. A treelike hybrid 5-cluster tool with C2 be a fork tool, and its adjacent downstream tools are C3 and C5. The tool C4 is the downstream tool of C3, and C2 is the downstream tool of C1. Furthermore, Ci is a dual-arm tool and the others are single-arm tools. Their activity time is as follows: for C1, one has (α10, α11, λ1, μ1)=(0, 61.5, 0, 28.5, 0.5); for C2, one has (α20, α21, α22, λ2, μ2)=(0, 0, 0, 10, 1); for C3, one has (α30, α31, α32, α33, λ3, μ3)=(0, 56, 0, 58, 7, 1); for C4, one has (α40, α41, α42, α43, λ4, μ4)=(0, 56, 66, 65, 5, 1); and for C5, one has (α50, α51, α52, λ5, μ5)=(0, 48, 50, 6, 1).
From [Yang, et al., 2015], the lower bound of cycle time of the system is Θ=Π=90 s. With Θ=Π=90 s, the robot waiting time is set as follows. For C4, ω40=11 s, ω41=1 s, ω42=2 s, and ω43=28 s are set. For C5, ω50=15 s, ω51=13 s, and ω52=20 s are set. For C3, ω31=8 s, ω30=3 s, ω32=1 s, and ω33=14 s are set. For C2, ω20=2 s, ω21=0, and ω22=22 s are set. Then, for C1, one has Π−(4λ2+3μ2+ω22)−λ1=90−(43+22)−28.5=−3.5<0, or there is no OSLB. Therefore, one needs to find the minimal cycle time Θ by Algorithm 2.
With ω22=22>0, ω52=20>0, ω33=14>0, and ω43=28>0, one has Σp∈S5\{1} and j∈Ωi(n[i]), where ωij's are obtained by the algorithm in [Yang, et al., 2015] with cycle time Θ=Π=90 s as given above. Then, by Algorithm 2, the robot waiting time is set as follows. For C4, ω40=A40+Δ=11.5 s, ω41=A41+Δ=1.5 s, ω42=A42+Δ−2.5 s, and ω43=A43−2Δ−27 s; for C3, ω31=A31+2Δ+Δ=9.5 s, ω30=A30+Δ=3.5 s, ω32=A32+Δ=1.5 s, and ω33=A33−2Δ−2Δ=12 s; for C5, ω50=A50+Δ−15.5 s, ω51=A51+Δ=13.5 s, and ω52=A52+Δ=19.5 s; for C2, Ω20=A20+4Δ+Δ=4.5 s, ω21=A21+Δ+Δ=1 s, and ω22=A22−4Δ−Δ−Δ=19 s; and for C1, ω10=Θ−ψ11=3.5 s and ωil=ω12=0. In this way, an optimal one-wafer cyclic schedule is obtained and its Gantt chart is shown in
The present invention is developed based on the theoretical development in Sections A-C above.
An aspect of the present invention is to provide a computer-implemented method for scheduling a treelike hybrid K-cluster tool to generate a one-wafer cyclic schedule. The K-cluster tool has K single-cluster tools.
The method comprises given a value of cycle time, generating a part of the schedule for a section of the K-cluster tool by performing a generating algorithm. The section of the K-cluster tool is either an EST or a ST. The generating algorithm is based on Algorithm 1 above. In particular, the generating algorithm for ESTk or STk, with Ci being a downstream adjacent tool of Ck and with Θ being the given value of cycle time for ESTi, STi or Bi comprises Step 3 of Algorithm 1 under a condition that the checking result of Step 1 is negative. Preferably, the generating algorithm further comprises Step 2 and Step 4 of Algorithm 1, provided that the checking result of Step 1 is negative.
The method is further elaborated based on Algorithm 2 as follows. First identify STj, with j=maxl∈F{l}, and one or more ESTs of STj in the K-cluster tool. The one or more ESTs are denoted as ESTj-1, ESTj-2 down to ESTi such that an upstream adjacent tool of Ci is a fork tool. A first part of the schedule for STj is first determined by performing the generating algorithm. Then determine a second part of the schedule for ESTj-1 based on the first part of the schedule. Determining one part of the schedule ESTj-m based on a determined part of the schedule for ESTj-m+1 is repeated until the one or more ESTs are scheduled.
The embodiments disclosed herein may be implemented using general purpose or specialized computing devices, computer processors, or electronic circuitries including but not limited to digital signal processors (DSP), application specific integrated circuits (ASIC), field programmable gate arrays (FPGA), and other programmable logic devices configured or programmed according to the teachings of the present disclosure. Computer instructions or software codes running in the general purpose or specialized computing devices, computer processors, or programmable logic devices can readily be prepared by practitioners skilled in the software or electronic art based on the teachings of the present disclosure.
In particular, the method disclosed herein can be implemented in a treelike hybrid K-cluster tool if the K-cluster tool includes one or more processors. The one or more processors are configured to execute a process of scheduling the K-cluster tool according to one of the embodiments of the disclosed method.
The present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiment is therefore to be considered in all respects as illustrative and not restrictive. The scope of the invention is indicated by the appended claims rather than by the foregoing description, and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.
This application claims the benefit of U.S. Provisional Patent Application No. 62/221,038, filed on Sep. 20, 2015, which is incorporated by reference herein in its entirety.
| Number | Date | Country | |
|---|---|---|---|
| 62221038 | Sep 2015 | US |
