Patent Grant
10001772
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The present invention relates to a method for scheduling a process for single-arm cluster tools. More particularly, the present invention relates to a method for scheduling close-down process for single-arm cluster tools with wafer residency time constraints.
The following references are cited in the specification. Disclosures of these references are incorporated herein by reference in their entirety.
Cluster tools are widely used as wafer fabrication equipment by semiconductor manufacturers. Each consists of multiple processing modules (PMs), one wafer delivering robot, an aligner and two cassette loadlocks (LLs) for loading/unloading wafers. As shown in
Substantial efforts have been made for a cluster tool's modeling and performance analysis [Chan et al., 2011; Ding et al., 2006; Perkinston et al., 1994; Perkinston et al., 1996; Venkatesh et al., 1997; Wu and Zhou, 2010a; Yi et al., 2008; Zuberek, 2001; and Lee et al., 2014]. These studies show that in the steady state a cluster tool operates in one of two regions: process or transport-bound ones. The robot is always busy in the former and its task time dominates the cycle time of the tool. It has idle time in the latter and the processing time at PMs determines the cycle time. Since the robot moving time from one PM to another is much shorter than the wafer processing time, a backward scheduling strategy is optimal for single-arm cluster tools [Lee et al., 2004; and Lopez and Wood, 2003].
All aforementioned studies are conducted with the assumption that a wafer can stay in a PM for unlimited time after it is processed. However, strict residency constraints should be considered for some wafer fabrication processes, such as low-pressure chemical vapor deposition and rapid thermal processing. For them, a wafer's surface would be detrimental if it stays in a PM for too long time after being processed [Kim et al., 2003; Lee and Park, 2005; Rostami et al., 2001; and Yoon and Lee, 2005]. With such constraints, it is much more complicated to schedule a cluster tool. In order to solve it, the studies [Kim et al., 2003; Lee and Park, 2005; and Rostami et al., 2001] are conducted to find an optimal periodic schedule for dual-arm cluster tools with wafer residency time constraints. This problem is further investigated in [Wu et al., 2008; Wu and Zhou, 2010; and Qiao et al., 2012] for both single and dual-arm cluster tools by using Petri nets. Schedulability conditions are proposed to check if a cluster tool is schedulable. If so, closed-form algorithms are given to find the optimal cyclic schedule.
Majority of the existing studies [Chan et al., 2011; Ding et al., 2006; Perkinson et al., 1994; Perkinson et al., 1996; Venkatesh et al., 1997; Wu and Zhou, 2010a; Yi et al., 2008; Zuberek, 2001; Qiao et al., 2012a and 2012b; Qiao et al., 2013; Zhu et al., 2013a, 2013b, 2014, and 2015; and Lee et al., 2014] focus on finding an optimal periodical schedule for the steady state. However, a cluster tool should experience a start-up process before it reaches a steady state. Then, after the steady state, it will go through a close-down process. Very few studies are done for scheduling the transient process including the start-up and close-down process [Lee et al., 2012 and 2013; Kim et al., 2012, 2013a, 2013b, and 2013c; and Wikborg and Lee, 2013]. In semiconductor manufacturing, recent trends are product customization and small lot sizes. Thus, the transient processes are more common due to product changeover and setups. Therefore, it becomes more important to optimize such transient processes. For a dual-arm cluster tool, Kim et al. [2012] propose methods to minimize the transient period based on a given robot task sequence. For a single-arm cluster tool, non-cyclic scheduling methods are developed in [Kim et al., 2013a, and Wikborg and Lee, 2013]. Due to small batch production, a cluster tool needs to switch from processing one lot to another different one frequently. Thus, techniques are developed to schedule lot switching processes for both single and dual-arm cluster tools in [Lee et al., 2012 and 2013; and Kim et al., 2013b and 2013c].
However, the above work on the transient process scheduling does not consider the wafer residency time constraints. Thus, their method in [Lee et al., 2012 and 2013; Kim et al., 2013a, 2013b, and 2013c; and Wikborg and Lee, 2013] cannot be applied to schedule a cluster tool with such constraints. It is these constraints that make an optimal schedule for a transient process without considering residency time constraints infeasible. Considering such constraints, Kim et al. [2012] develop scheduling methods to optimize the start-up and close-down processes for dual-arm cluster tools. Owing to a different scheduling strategy used to schedule a dual-arm cluster tool with residency constraints, the scheduling methods in [Kim et al., 2012] cannot be used to optimize the transient processes of a single-arm cluster tool. For a single-arm tool with such constraints, based on a developed Petri net (PN) model, Qiao et al. [2014] develops a scheduling algorithm and a liner programming model to find an optimal schedule for the start-up process. However, for the close-down process, there is no scheduling method provided in [Qiao et al., 2014]. Although a close-down process is a reverse of a start-up one, their PN models are totally different. Consequently it requires different scheduling methods.
There is a need in the art for a method for scheduling close-down process for single-arm cluster tools with wafer residency time constraints.
An aspect of the present invention is to provide a method for scheduling a cluster tool for a close-down process with wafer residency time constraints.
According to an embodiment of the present claimed invention, a computer-implemented method for scheduling a cluster tool for a close-down process, the cluster tool comprising a single-arm robot for wafer handling, loadlocks for wafer cassette loading and unloading, and n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, i∈{1, 2, . . . , n}, is used for performing Step i of the n wafer-processing steps for each wafer, the method comprising:
According to an embodiment of the present claimed invention, a computer-implemented method for scheduling a cluster tool for a close-down process, the cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, i∈{1, 2, . . . , n}, is used for performing Step i of the n wafer-processing steps for each wafer, the method comprising:
Embodiments of the present invention are described in more detail hereinafter with reference to the drawings, in which:
In the following description, a method for scheduling close-down process for single-arm cluster tools with wafer residency time constraints is set forth as preferred examples. It will be apparent to those skilled in the art that modifications, including additions and/or substitutions may be made without departing from the scope and spirit of the invention. Specific details may be omitted so as not to obscure the invention; however, the disclosure is written to enable one skilled in the art to practice the teachings herein without undue experimentation.
The present invention aims to schedule a close-down process of a single-arm cluster tool with wafer residency time constraints, which was not addressed yet. According to the specification of the present invention, Section A presents a Petri net model for the close-down process of a single-arm cluster tool. The schedulability results for single-arm cluster tools [Wu et al., 2008] are reviewed in Section B. Then, a closed-form algorithm and a linear programming model are developed to schedule the close-down process in Section C. Section D presents illustrative examples.
A.1 Finite Capacity Petri Nets (PNs)
As an effective modeling tool, Petri nets are widely used in modeling, analysis, and control of discrete event systems [Zhou and DiCesare, 1991; Zhou et al., 1992; Zhou et al., 1995; Wu and Zhou, 2001, 2004, 2005, and 2007; Zhou and Jeng, 1998; Liao et al., 2004; Ferrarini and Piroddi, 2008; Jung and Lee, 2012; Liu et al., 2013; Wu et al., 2013c]. Following Zhou and Venkatesh [1998] and Murata [1989], the present invention adopts a finite capacity PN to model a single-arm cluster tool. It is defined as PN=(P, T, I, O, M, K), where P={p1, p2, . . . , pm} is a finite set of places; T={t1, t2, . . . , tn} is a finite set of transitions with P∪T≠Ø and P∩T=Ø; I: P×T→N={0, 1, 2, . . . } is an input function; O: P×T→N is an output function; M: P→N is a marking representing the count of tokens in places with M0 being the initial marking; and K: P→N\{0} is a capacity function where K(p) represents the count of tokens that p can hold at a time.
The preset of transition t is the set of all input places to t, i.e., •t={p: p∈P and I(p, t)>0}. Its postset is the set of all output places from t, i.e., t•={p: p∈P and O(p, t)>0}. Similarly, p's preset •p={t∈T: O(p, t)>0} and postset p•={t∈T: I(p, t)>0}. The transition enabling and firing rules can be found in [Wu and Zhou, 2009].
A.2 PN Model for Cluster Tools
In the present invention, (m1, m2, . . . , mn) is used to describe the wafer flow pattern in a cluster tool, where n is the number of processing steps and mi is the number of parallel PMs configured for Step i, i∈Nn={1, 2, . . . , n}. It is assumed that there is only one PM at each step. Thus, one has the wafer flow pattern is (m1, m2, . . . , mn) where mi=1, i∈Nn. Based on the PN model, the scheduling analysis of a single-arm cluster tool operating under the steady-state has been well conducted in [Wu et al., 2008]. Then, one briefly introduces the PN model for the steady-state process as developed in [Wu et al., 2008].
In the PN model, Step i is modeled by timed place pi with K(pi)=1, i∈Nn. The loadlocks are treated just as a processing step called Step 0. Because the loadlocks can hold all the wafers in a tool, they are modeled by p0 with K(p0)=∞. The robot is modeled by place r with K(r)=1, meaning that it has only one arm and can hold one wafer at a time. When M(r)=1, it represents that the robot arm is available. When M(pi)=1, i∈Nn, a wafer is being processed in a PM at Step i. In the following discussions, a token in a place, or a wafer in a place, refers to a wafer in its modeled PM when no confusion arises. When the robot arrives at Step i for unloading a wafer, the wafer may be under way. Then, it has to wait there for some time. Timed place qi, i∈Nn, is added to model the robot's waiting at Step i before unloading a wafer there and M(qi)=1 means that the robot is waiting at Step i. Non-timed place zij is used to model the state that it is ready to load a wafer into Step i or the wafer unloading from Step i ends. Transitions are used to model the robot tasks. Timed ti1, i∈Nn, models loading a wafer into Step i, and t01 models loading a completed wafer into a loadlock. Timed ti2, i∈Nn, models unloading a wafer from Step i, and t02 models unloading a raw wafer from a loadlock. Transition yi, i∈Nn−2∪{0}, represents the robot's moving from Steps i+2 to i without carrying a wafer. Transitions yn−1 and yn represent the robot's moving from a loadlock to Step n−1 and Steps 1 to n, respectively. Transition xi, i∈Nn−1∪{0}, models the robot's moving from Steps i to i+1 with a wafer held, and xn models the robot's moving from Steps n to 0. Pictorially, pi's and qi's are denoted by , zij's by ◯, and r by
. Then, the PN model for a single-arm cluster tool is shown in
In the steady state, there Σi=1n mi wafers being concurrently processed. This means that mi wafers are being processed at Step i, i∈Nn. For the PN model in
Control Policy 1 (CP1):
At any M, transition yi, i∈Nn−1∪{0} is said to be control-enabled if M(pi+1)=0; and yn is said to be control-enabled if M(pi)=1, i∈Nn.
After the steady state, the cluster tool enters the close-down process. Thus, a Petri net model shown in
Control Policy 2 (CP2):
For the PN model in
With CP2, the close-down process could be described by running the PN model in firing yn (moving to Step n)→firing tn2 (unloading a wafer from Step n)→firing xn (moving from Steps n to 0)→firing t01 (loading the wafer into Step 0)→firing yn−1 (moving from Steps 0 to n−1)→firing t(n−1)2 (unloading a wafer from Step n−2)→firing x(n−1) (moving from Steps n−2 to n−1)→firing tn1 (loading the wafer into Step n)→ . . . →firing y1 (moving from Steps 3 to 1)→firing t12 (unloading a wafer from Step 1)→firing x1 (moving from Steps 1 to 2)→firing t21 (loading the wafer into Step 2)
. At this time, let Mc2 denote the state of the system with Mc2(pi)=K(pi), i∈Nn\{1}, Mc2(p1)=0, and Mc2(r)=1. Then, one can set d=2 in the PN model in
A.3 Modeling Activity Time
For the purpose of scheduling, the temporal aspect of a cluster tool should be described in the PN models in
With wafer residency time constraints, the deadlock-freeness does not mean that the PNs shown in
Definition 2.1:
If the PN models for single-arm cluster tools with residency time constraints are live, one has: 1) at any marking with a token in pi, ∀i∈Nn, and when ti2 fires, ai≤τi≤ai+δi holds for the net in
For a single-arm cluster tool with wafer residency time constraints, before discussing how to schedule its close-down process, one recalls the scheduling analysis results for its steady state process [Wu et al., 2008].
B.1 Timeliness Analysis for Steady State
It follows from [Wu et al., 2008] that, to complete the processing of a wafer at Step i, i∈Nn\{1}, it takes τi+4α+3μ+ωi−1 time units, where τi should be within [ai, ai+δi]. With one PMs at Step i, i∈Nn, one has that the lower bound of permissive cycle time at Step i is
θiL=ai+4α+3μ+ωi−1, i∈Nn\{1} (3.1)
The upper bound of permissive cycle time at Step i is
θiU=ai+4α+3μ+ωi−1+δi, i∈Nn\{1} (3.2)
For Step 1, the lower one is
θiL=a1+3α+α0+3μ+ω0 (3.3)
Its upper one is
θ1U=a1+3α+α0+3μ+ω0+δ1 (3.4)
It follows from (3.1)-(3.4) that the permissive wafer sojourn time can be affected by the robot waiting time ωi. By removing it from the above expressions, one can obtain the lower and upper workloads of each step as follows.
ϑiL=ai+4α+3μ, i∈Nn\{1} (3.5)
ϑiU=ai+4α+3μ+δi, i∈Nn\{1} (3.6)
ϑ1L=a1+3α+α0+3μ (3.7)
ϑ1U=a1+3α+α0+3μ+δ1 (3.8)
To schedule a single-arm cluster tool with residency time constraints, one has to ensure ai≤τi≤ai+δi. Hence, one needs to know how τi is calculated. According to [Wu et al., 2008], one has
τi=2(n+1)μ+(2n+1)α+α0+Σd=0nωd−(4α+3μ+ωi−1)=ψ−(4α+3μ+ωi−1), i∈Nn\{1} (3.9)
τ1=2(n+1)μ+(2n+1)α+α0+Σd=0nωd−(3α+α0+3μ+ω0)=ψ−(3α+α0+3μ+ω0) (3.10)
The robot cycle time is
ψ=2(n+1)μ+(2n+1)α+α0+Σd=0nωd=ψ1+ψ2 (3.11)
where ψ1=2(n+1)μ+(2n+1)α+α0 is a known constant and ψ2=Σd=0nωd is to be decided by a schedule.
Let θ1=τ1+3α+α0+3μ+ω0 and θi=τi+4α+3μ+ωi−1, i∈Nn\{1}, denote the cycle time for step i, i∈Nn. Then, it can be seen that, by making ωi−1>0, the cycle time of Step i is increased without increasing the wafer sojourn time. Thus, it is possible to adjust the robot waiting time to balance the wafer sojourn time among the steps such that a feasible schedule can be obtained. Following Wu et al. [2008], for a periodic schedule for the steady state, one has
θ=θ1=θ2= . . . =θn=ψ (3.12)
In (3.11), μ, α, and α0 are constants, only ωd's d∈Nn∪{0}, are variables, i.e., ψ1 is deterministic and ψ2 can be regulated. Thus, with the PN model in
B.2 Schedulability Conditions and Scheduling for Steady State
To find a feasible cyclic schedule, the key is to know under what conditions there exist θ such that the system is schedulable. It is known that, in (3.5)-(3.8), ϑiL and ϑiU denote the lower and upper bounds of θi, respectively. Let ϑmax=max{ϑiL, i∈Nn}. Then, Wu et al. [2008] developed the sufficient and necessary schedulability conditions shown below.
Theorem 3.1 [Wu et al., 2008]:
If ϑmax≤ϑiU and ψ1≤ϑiU, i∈Nn, a single-arm cluster tool with residency time constraints is schedulable.
For this case, if ϑmax≤ϑiU and ψ1≤ϑmax, i∈Nn, the tool is process-bound. If ϑiL≤ψ1≤ϑiU, i∈Nn, it is transport-bound. With ϑmax≤ϑiU, i∈Nn, the difference of the workloads among the steps is not too large. Thus, with ωi's being set appropriately, the workloads among the steps can be balanced such that a feasible schedule can be found. It follows from [Wu et al., 2008] that, in this case, one can simply set ωi=0, i∈Nn−1∪{0}, and ωn=max{ϑmax−ψ1, 0} such that ψ=max{ϑmax, ψ1} holds. In this way, a feasible schedule is determined. Further, it is optimal in terms of cycle time.
Theorem 3.1 shows that the difference of the workloads among the steps is not too large, i.e., ∩j∈N
Theorem 3.2 [Wu et al., 2008]:
If ∩j∈N
In this case, with the obtained ωi−1, i∈Nn, by (3.13), the workload among the steps can be well balanced. Notice that, by (3.13), the robot waiting time ωi−1, i∈Nn, is set, and then set ωn=ϑmax−(ψ1+Σi∈Eωi−1) such that ψ=ϑmax holds. Thus, a feasible schedule is obtained. Further, the cycle time of the tool is optimal. According to [Wu et al., 2008], the schedulability conditions given by Theorems 3.1 and 3.2 are the sufficient and necessary for the steady state scheduling. Based on them, in the next section, one conducts the scheduling analysis for the close-down process.
C.1 Temporal Properties in Close-Down Process
Let Mc0 denote the state with Mc0(pi)=K(pi), i∈Nn, Mc0(r)=1, and there is only one raw wafer in the loadlocks. In other words, the robot task sequence from Mc0 to Mc1 is the last robot task cycle for the steady state. Then, the system operates according to the PN model in
During the evolution from Mcd to Mc(d+1), the robot should sequentially move to Steps n, n−1, . . . , d+1, and d to unload the processed wafers. Thus, with wafer residency time constraints considered, it is necessary to determine how long a wafer visits Step i, i∈Nn\Nd−1. With the PN model in
Let ωid, d≤i≤n, 0≤d≤n−1, and ωnn denote robot waiting time in places qi and qn during the evolutions from Mcd, to Mc(d+1) and Mcn to Mce, respectively. At Mc0), there is only one raw wafer in the loadlocks and let W1 denote it. During the evolution from Mc0 to Mc1, according to the model in
τ1=[2(n+1)μ+(2n+1)α+α0]+Σ1nωj1−(3α+α0+3μ) (4.1)
Similarly, when the robot arrives at Step d during the evolution from Mcd to Mc(d+1), 2≤d≤n−1, the wafer sojourn time at Step d is
τd=[2(n−d+2)μ+2(n−d+2)α]Σdnωjd−(4α+3μ) (4.2)
When the robot arrives at Step i, 2≤i≤n, during the evolution from Mc1 to Mc2, the wafer sojourn time at Step i is
τi=[2(n+1)μ+(2n+1)α+α0]+Σ0i−2ωj0+Σinωj1−(4α+3μ) (4.3)
When the robot arrives at Step i, d+1≤i≤n, during the evolution from Mcd to Mc(d−1), 2≤d≤n−1, the wafer sojourn time at Step d is
τi=[2(n−d+2)μ+2(n−d+2)α]+Σd−1i−2ωjd−1+Σinωjd−(4α+3μ) (4.4)
During the evolution from Mcn to Mce, the wafer sojourn time at Step n is
τn=ωnn (4.5)
Due to ψ1=2(n+1)μ+(2n+1)α+α0, expressions (4.1) and (4.3) can be respectively rewritten as
τ1=ψ1+Σ1nωj1−(3α+α0+3μ) (4.6)
τi=ψ1+Σ0i−2ωj0+Σinωj1+(4α+3μ) (4.7)
Let ψc(d−1), 2≤s≤n, and ψcn denote the robot task time for transferring the tool from Mc(d−1) to Mcd and Mcn to Me without considering the robot waiting time, respectively. Then, one has
ψc(d−1)=2(n−d+2)μ+2(n−d+2)α, 2≤d≤n (4.8)
ψcn=μ+2α (4.9)
Thus, from (4.8), expressions (4.2) and (4.4) can be rewritten as
τd=ψc(d−1)+Σdnωjd−(4α+3μ) (4.10)
τi=ψc(d−1)+Σd−1i−2ωjd−1+Σinωjd−(4α+3μ) (4.11)
Then, one discusses how to regulate the robot waiting time such that the residency time constraints at all steps are satisfied.
C.2 Scheduling for Close-down Process
Feasibility is an essential requirement for scheduling a transient process of a cluster tool. From the above analysis, one knows that the robot task sequence during the evolution from Mc1 to Mce is determined. Thus, it is very important to determine the robot waiting time during the close-down process such that the residency time constraints are met at each step. Thus, one has the schedulability results next.
Proposition 4.1:
A cluster tool with wafer residency constraints in a close-down process is schedulable if the robot waiting time during the period from Mc1 to Mce can be found such that the constraint at each step is satisfied.
Generally, a cluster tool has not less than two steps. By the PN model in
Scheduling Algorithm 4.1:
If ϑmax≤ϑiU and ψ1≤ϑiU, i∈Nn, the robot waiting time is set as follows:
1) Let ψc0=ψ1. During the period from Mcd to Mc(d−1), 1≤d≤n−1, the tool operates according to the model in
2) During the period from Mcn to Mce, let ωnn=an.
According to Algorithm 4.1, during the period from Mcd to Mc(d+1), 1≤d≤n−1, Step i with 1≤i≤d−1, is empty. Thus, ϑdmax depends on the bottleneck step from steps d to n. With ωi=0, i∈Nn\Nd−1, and ωn=max{ϑdmax−ψc(d−1)1, 0}, the residency time constraints at Steps d to n are satisfied and the time to complete each Step i∈Nn\Nd−1, is expected to be shortest in the permissive range. Finally, during the period from Mcn to Mce, after the robot loads a wafer into Step n, it only waits there for the end of wafer processing and unloads the wafer immediately. One can show that this is feasible by the following theorem.
Theorem 4.1:
For a single-arm cluster tool with wafer residency time constraints, if ϑmax≤ϑiU, ψ1≤ϑiU, i∈Nn, a schedule obtained by Algorithm 4.1 for the close-down process is feasible.
Proof:
With the PN model in
When the robot arrives at Step d during the evolution from Mcd to Mc(d+1), 2≤d≤n−1, for unloading a wafer, based on Rule 1) in Algorithm 4.1, it follows from expressions (4.2) and (4.10) that the wafer sojourn time at Step d is τd=ψc(d−1)+Σdnωjd−(4α+3μ)=ψc(d−1)+max{ϑdmax−ψc(d−1), 0}−(4α+3μ). If ψc(d−1)≥ϑdmax leading to max{ϑdmax−ψc(d−1), 0}=0, from (3.5), (3.6), and the assumption of ϑmax≤ϑiU, ψ1≤ϑiU, i∈Nn, one has that ad≤ϑdL−(4α+3μ)≤ϑdmax−(4α+3μ)≤τd=ψc(d−1)−(4α+3μ)<ψ1−(4α+3μ)≤ϑdU−(4α+3μ)≤ad+δd. If ψc(d−1)<ϑdmax leading to max{ϑdmax−ψc(d−1), 0}=ϑdmax−ψc(d−1), from (3.5), (3.6), and the assumption of ϑmax≤ϑiU, ψ1≤ϑiU, i∈Nn, one has that ad≤ϑdL−(4α+3μ)≤τd=ϑdmax−(4α+3μ)≤ϑdU−(4α+3μ)≤ad+δd. Thus, when the robot arrives at Step d during the evolution from Mcd to Mc(d+1), 2≤d≤n−1, for unloading a wafer, the wafer residency time at Step d is not violated. Similarly, based on Rule 1) in Algorithm 4.1, (3.5), (3.6), (4.4), (4.11), and the assumption of ϑmax≤ϑiU, ψ1≤ϑiU, i∈Nn, one has that ai≤τi≤ai+δ1, d<i≤n, when the robot arrives at Step i, d<i≤n, for unloading a wafer during the evolution from Mcd to Mc(d+1), 2≤d≤n−1. This means that when the robot arrives at Step i, d<i≤n, for unloading a wafer during the evolution from Mcd to Mc(d+1), 2≤d≤n−1, the wafer residency time at Step i is not violated.
During the period from Mcn to Mce, based on Rule 2) in Algorithm 4.1 and expression (4.5), one has τn=an. Hence, from all the above analysis, during the close-down process from Mc1 to Mce, the wafer residency time constraints are all satisfied, or the theorem holds.
During the period from Mc1 to Mc2, ωi1=ωi0, i∈Nn, it is obvious that residency constraints are satisfied. ϑmax≤ϑiU, ψ1≤ϑiU, i∈Nn, implies the workloads among the steps are properly balanced. During the period from Mcd to Mc(d+1), 2≤d≤n−1, ψc(d−1), decreases as d increases. If ϑdmax≥ψc(d−1), the cluster tool operates in a process-bound region. If ϑdmax<ψc(d−1), it operates in a transport-bound region. Due to the varied ψc(d−1) as d increases, the cluster tool may operate in a process-bound region in the next state. Thus, one has to adjust the robot waiting time dynamically to meet the residency constraints by Algorithm 4.1, which assigns the robot waiting time to the last step. Theorem 4.1 guarantees that the obtained schedule by Algorithm 4.1 is feasible to satisfy the residency constraints. Further, one has the following theorem to show its optimality.
Theorem 4.2:
For a single-arm cluster tool with residency time constraints, if ϑmax≤ϑiU, ψ1≤ϑiU, i∈Nn, a schedule obtained by Algorithm 4.1 is optimal for the close-down process.
Proof:
Without loss of generality, let ϑmax=ϑnL. During the period from Mc1 to Mc2, by (4.3) and Rule 1) of Algorithm 4.1 one has that τn[2(n+1)μ+(2n+1)α+α0]+Σ0n−2ωj0+Σnnωj1−(4α+3μ)=[2(n+1)μ+(2n+1)α+α0]+ωn1−(4α+3μ)=[2(n+1)μ+(2n+1)α+α0]+max{ϑdmax−ψ1, 0}−(4α+3μ)=ψ1+max{ϑmax−ψ1, 0}−(4α+3μ), if ϑmax≥ψ1, τn=ψ1+ϑdmax−ψ1−(4α+3μ)=ϑnL−(4α+3μ)=an. If ϑdmax<ψ1, τn=ψ1−(4α+3μ) cannot be shortened. During the period from Mcd to Mc(d−1), 2≤d≤n−1, τn=[2(n−d+2)μ+2(n−d+2)α]+Σd−1n−2ωjd−1+Σnnωjd−(4α+3μ)=[2(n−d+2)μ+2(n−d+2)α]+ωnd−(4α+3μ)=[2(n−d+2)μ+2(n−d+2)α]+max{ϑmax−ψc(d−1), 0}−(4α+3μ), if ϑdmax≥ψc(d−1), τn=ϑdmax−(4α+3μ)=ϑnL−(4α+3μ)=an. If ϑdmax<ψc(d−1), τn=ψc(d−1)−(4α+3μ) cannot be shortened. Since ωid=0, i∈Nn\Nd−1, 1≤d≤n−1, the period from Mcd to Mc(d+1), 1≤d≤n−1, is determined by τn and minimized. By Rule 2), during the period from Mcn to Mce, τn=an is also minimized. Thus, during the period from Mc1 to Mce, τn is minimized. That is to say, the time span of the close-down process is minimal. Therefore, a schedule obtained by Algorithm 4.1 is optimal for the close-down process.
The conditions in Theorem 4.1 indicates that the workloads among the steps are well balanced, i.e., ∩j∈N
Linear Programming Model (LPM):
If ∩j∈N
Subject to:
ωi1=ωi0, 1≤i≤n (4.13)
βn21=μ+ωn1 (4.14)
β01d=βn2d+α+μ, 1≤d≤n−1 (4.15)
βi22=β(i+1)2d+2(α+μ)+ωid, d≤i≤n−1 and 1≤d≤n−1 (4.16)
βi1d=β(i−1)2d+α+μ, d+1≤i≤n and 1≤d≤n−1 (4.17)
βn1d=β01d+2(α+μ)+ωn−1d, 1≤d≤n−1 (4.18)
βn2d=βd1d−1α+μ+ωnd, 2≤d≤n−1 (4.19)
ωnn=an (4.20)
βn2n=βn1n+α+ωnn (4.21)
ωid≥0, 1≤i≤d and 1≤d≤n (4.22)
ai≤βi2d−βi1d−1−α≤ai+δi, d≤i≤n and 2≤d≤n−1 (4.23)
After reaching Mc1, the cluster tool operates according to the model in
For the case of ∩j∈N
1≤d≤n−1; and 4) ωnn=an. It is easy to verify that this schedule is in the feasible region of LPM. Therefore, if a system is schedulable according to Theorem 3.2, a feasible and optimal schedule can be obtained by LPM.
Up to now, for the case that the workloads among the steps are properly balanced, i.e., [ϑ1L, ϑ1U]∩[ϑ2L, ϑ2U]∩ . . . ∩[ϑnL, ϑnU]≠Ø, a scheduling algorithm is proposed to find an optimal schedule for the close-down process. For the case that such differences are too large such that [ϑ1L, ϑ1U]∩[ϑ2L, ϑ2U]∩ . . . ∩[ϑnL, ϑnU]≠Ø, a linear programming model is developed to find a feasible optimal schedule for a single-arm cluster tool during the close-down operations. Notice that Algorithm 4.1 consists of several expressions and LPM is a linear programming model. Therefore, it is very efficient to use the present proposed methods to find a feasible and optimal schedule for the close-down process for single-arm cluster tools with wafer residency time constraints.
In a single-arm cluster tool, the wafer flow pattern is (1, 1, 1, 1, 1). The activity time is as follows: (a1, a2, a3, a4, a5; α0, α, μ)=(90 s, 100 s, 100 s, 105 s, 115 s; 10 s, 5 s, 2 s). After being processed, a wafer can stay at Steps 1-5 for 20 s (δi=20 s, 1≤i≤5).
By (3.5)-(3.8), one has ϑ1L=121 s, ϑ1U=141 s, ϑ2L=126 s, ϑ2U=146 s, ϑ3L=126 s, ϑ3U=146 s, ϑ4L=131 s, ϑ4U=151 s, ϑ5L=141 s, ϑ5U=161 s, and ψ1=89 s. According to Theorem 3.1, the cluster tool is schedulable. For its steady state, an optimal schedule can be obtained by setting ω0=ω1=ω2=ω3=ω4=0 s and ω5=52 s. Then, its cycle time in the steady state is 141 s. It is easy to verify that the workloads can be balanced among the steps, i.e., [ϑ1L, ϑ1U]∩[ϑ2L, ϑ2U]∩ . . . ∩[ϑnL, ϑnU]≠Ø. By Algorithm 4.1, one can find an optimal feasible schedule for the close-down process. Thus, the robot waiting time is set as follows: 1) During the process from Mc1 to Mc2, ω0=ω1=ω2=ω3=ω4=0 s and ω5=52 s: 2). During the process from Mc2 to Mc3, ω2=ω3=ω4=0 s and ω5=71 s; 3) ω3=ω4=0 s and ω5=85 s; 4) ω4=0 s and ω5=99 s; 5) ω5=115 s. Thus, this robot waiting time determines an optimal feasible schedule for the close-down process. The Gantt chart in
The flow pattern is (1, 1, 1, 1). α=5 s, α0=10 s, μ=2 s, a1=85 s, a2=120 s, a3=110 s, a4=85 s, and δi=20 s, 1≤i≤4.
It follows from (3.5)-(3.8) that, one has ϑ1L=116 s, ϑ1U=136 s, ϑ2L=146 s, ϑ2U=166 s, ϑ3L=136 s, ϑ3U=156 s, ϑ4L=111 s, ϑ4U=131 s, and ψ1=75 s. By Theorem 3.2, the single-arm cluster tool is schedulable. For the steady state, an optimal feasible schedule is obtained by setting ω0=10 s, ω1=ω2=0 s, ω3=15 s, and ω4=46 s. Then, the cycle time of the system under the steady state is 146 s. For this example, [ϑ1L, ϑ1U]∩[ϑ2L, ϑ2U]∩ . . . ∩[ϑnL, ϑnU]=Ø holds since the differences between each step's workload are too large. By the proposed LPM, an optimal feasible schedule is found for the close-down process, during which the robot waiting time is set as follows: 1) From Mc1 to Mc1, ω11=ω21=0, ω31=15, ω41=46 s; 2) From Mc2 to Mc3, ω22=0 s, ω32=35 s and ω4255 s; 3) From Mc3 to Mc4, ω33=5 s and ω43=89 s; 4) From Mc4 to Mce, ω44=85 s. The Gantt chart in
Semiconductor industry has shifted to larger size wafers and smaller lot production. Frequently, the wafer fabrication in the cluster tools switches from one size of wafer lot to another. This leads to many transient switching states, including start-up and close-down process. In some wafer fabrication process, quality products require that a processed wafer should leave the processing module within a given limit time to avoid its excessive exposure to the residual gas and high temperature inside a module. Such time constraints complicate the optimization issue for scheduling a close-down process. The problem and its solution are not seen in the existing research of scheduling cluster tools. This invention develops a Petri net model to analyze the time properties of this close-down process with time constraints. Based on it, the present invention proposes a closed-form algorithm and a linear programming model to regulate the robot waiting time for balanced and unbalanced workload situations, respectively, thereby finding an optimal schedule. The proposed methods are highly efficient.
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 some embodiments, the present invention includes computer storage media having computer instructions or software codes stored therein which can be used to program computers or microprocessors to perform any of the processes of the present invention. The storage media can include, but is not limited to, floppy disks, optical discs, Blu-ray Disc, DVD, CD-ROMs, and magneto-optical disks, ROMs, RAMs, flash memory devices, or any type of media or devices suitable for storing instructions, codes, and/or data.
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. 62/221,027, filed on Sep. 20, 2015, which is incorporated by reference herein in its entirety.
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| Number | Date | Country | |
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| 20170083010 A1 | Mar 2017 | US |
| Number | Date | Country | |
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| 62221027 | Sep 2015 | US |
