# Patent

## Title
Scheduling start-up process for time-constrained single-arm cluster tools

## Abstract

Due to the trend of using larger wafer diameter and smaller lot size, cluster tools need to switch from processing one lot of wafers to another frequently. It leads to more transient periods in wafer fabrication. Their efficient scheduling and control problems become more and more important. It becomes difficult to solve such problems, especially when wafer residency time constraints must be considered. This work develops a Petri net model to describe the behavior during the start-up transient processes of a single-arm cluster tool. Then, based on the model, for the case that the difference of workloads among the steps is not too large and can be properly balanced, a scheduling algorithm to find an optimal feasible schedule for the start-up process is given. For other cases schedulable at the steady state, a linear programming model is developed to find an optimal feasible schedule for the start-up process.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of U.S. Provisional Patent Application No. 62/221,034, filed on Sep. 20, 2015, which is incorporated by reference herein in its entirety.


LIST OF ABBREVIATIONS
CP control policy
LL loadlock
LPM linear programming model
PM process module
PN Petri net
BACKGROUND
Field of the Invention
The present invention generally relates to scheduling a cluster tool, where the cluster tool has a single-arm robot for wafer handling, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint. In particular, the present invention relates to a method for scheduling a start-up process for a single-arm cluster tool with wafer residency time constraints.
List of References
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.

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Description of the Related Art
In semiconductor manufacturing, wafers are processed in cluster tools with a single-wafer processing technology. Such technology allows manufacturers to process wafers one by one at each process module (PM) in cluster tools. These tools can provide a reconfigurable, flexible and efficient environment, leading to better quality control and reduced lead time [Bader et al., 1990; and Burggraaf, 1995]. In a cluster tool, there are several process modules (PMs), an aligner, a wafer handling robot, and loadlocks (LLs) for wafer cassette loading/unloading. All these modules are mechanically linked together in a radial way and computer-controlled. The robot in the center of the tool can have a single arm or dual arms, thus resulting in a single- or a dual-arm cluster tool as respectively shown in FIG. 1A and 1B.
With two LLs, a cluster tool can be operated consecutively without being interrupted such that it can operate in a steady state for most of time. Great efforts have been made in its modeling and performance evaluation [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]. It is found that, under the steady state, a cluster tool operates in two different regions: transport and process-bound ones. For the former, its robot is always busy and the robot task time in a cycle determines its cycle time; while for the latter, its robot has idle time in a robot task cycle and thus the processing time of its PMs dominates its cycle time. Since the robot moving time from one PM to another is much shorter than wafer processing time [Kim et al., 2003], a backward scheduling is optimal for single-arm cluster tools [Lee et al., 2004; and Lopez and Wood, 2003]. For a dual-arm cluster tool, a swap strategy is efficient [Venkatesh et al., 1997] for it can simplify robot tasks and thus reduces cycle time.
For some wafer fabrication processes, a strict constraint on the wafer sojourn time in a PM called residency time constraint must be considered in scheduling a cluster tool [Kim et al., 2003; Lee and Park, 2005; Rostami et al., 2001; and Yoon and Lee, 2005]. Such a constraint requires that a wafer should be unloaded from a PM within a limited time after being processed; otherwise, the wafer would be damaged due to the high temperature and residual chemical gas in the PM. However, no buffer between PMs in a cluster tool makes it complicated to schedule the tool to satisfy wafer residency time constraints. Methods are presented in [Kim et al., 2003; Lee and Park, 2005; and Rostami et al., 2001] to solve this scheduling problem and find an optimal periodic schedule for dual-arm cluster tools. Necessary and sufficient schedulability conditions are proposed for both single- and dual-arm cluster tools and if schedulable, closed-form scheduling algorithms are derived to find the optimal cyclic schedules [Wu et al., 2008a; and Wu and Zhou, 2010b].
Due to the trends of larger wafer diameter and smaller lot sizes, cluster tools need to switch from processing one lot of wafers to another one frequently. This leads to more transient periods in wafer fabrication, which includes start-up and close-down processes. Their efficient scheduling and control problems become more and more important. They become very difficult to solve especially when wafer residency time constraints must be considered. Although most existing studies [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; Qiao et al., 2012a and 2012b; Qiao et al., 2013; and Lee et al., 2014] aim at finding an optimal periodical schedule, few researches focus on scheduling for transient states [Lee et al., 2012 and 2013; Kim et al., 2012, 2013a, 2013b, and 2013c; and Wikborg and Lee, 2013] despite their increasing importance. In [Kim et al., 2012], with a given robot task sequence, the transient period for the start-up and close-down processes is minimized for a dual-arm cluster tool. In [Kim et al., 2013a, and Wikborg and Lee, 2013], scheduling methods are proposed for noncyclic scheduling problem for single-arm cluster tools. With small batch, lot switching occurs frequently. Thus, studies are conducted and techniques are developed for scheduling lot switching processes for both single and dual-arm cluster tools [Lee et al., 2012 and 2013; and Kim et al., 2013b and 2013c].
However, all the above studies about scheduling a transient process in a cluster tool are not applicable for a single-arm cluster tool with wafer residency time constraints, which are not considered in [Lee et al., 2012 and 2013; Kim et al., 2013a, 2013b, and 2013c; and Wikborg and Lee, 2013]. Such constraints can make an optimal schedule for a transient process without residency time constraints considered infeasible. With wafer residency time constraints, Kim et al. [2012] propose scheduling methods to minimize the transient period for the start-up and close-down processes for dual-arm cluster tools. Since different scheduling strategies are required to schedule single-arm cluster tools. Their research results cannot be used to find an optimal feasible transient process for residency time-constrained single-arm cluster tools.
There is a need in the art to derive a solution to this optimal feasible transient process and to develop a method for scheduling a single-arm cluster tool based on the derived optimal solution.
SUMMARY OF THE INVENTION
The present invention provides a method for scheduling a cluster tool. The cluster tool comprises a single-arm robot for wafer handling, a LL for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint.
The method includes scheduling a start-up process for the cluster tool. The start-up process is developed based on Scheduling Algorithm 1 and the LPM model detailed below.
Preferably, the method further includes scheduling a steady-state process according to results obtained in the start-up process.
Other aspects of the present invention are disclosed as illustrated by the embodiments hereinafter.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1A depicts an example of a cluster tool having a single-arm robot.
FIG. 1B depicts an example of a cluster tool having a dual-arm robot.
FIG. 2 depicts a Petri net (PN) for the steady state of a single-arm cluster tool according to [Wu et al., 2008].
FIG. 3 depicts a Petri net for the start-up process of a single-arm cluster tool.
FIG. 4 shows the simulation results for Example 1.
FIG. 5 shows the simulation results for Example 2.



DETAILED DESCRIPTION
A PN model is developed for the start-up process of a single-arm cluster tool in Section A. Section B recalls the schedulability conditions and scheduling analysis for single-arm cluster tools [Wu et al., 2008]. Then, a scheduling algorithm and a linear programming model are developed for the start-up transient process scheduling in Section C.
Hereinafter, the notation Nn, n being a positive integer, denotes a set containing positive integers from 1 to n, i.e. Nn={1, 2, . . . , n}.
A. Petri Net Modeling and Control
A.1. Finite Capacity Petri Nets
As an effective tool, PNs are widely used in modeling, analysis, and control of discrete-event systems, process industry, and robotic control systems [Zhou and DiCesare, 1991; Zhou et al., 1992 and 1995; Tang et al., 1995; Simon et al., 1998; Caloini et al., 1998; Zhou and Jeng, 1998; Wu and Zhou, 2001 and 2004; Liao et al., 2004; Ferrarini and Piroddi, 2008; Jung and Lee, 2012; Wu et al., 2008b; and Liu et al., 2013]. Following Zhou and Venkatesh [1998], the present work 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≠∅ and P∩V=∅; 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 number of tokens in places with M0 being the initial marking; and K:P→N\{0} is a capacity function where K(p) represents the largest number 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:pεP and I(p, t)>0}. Its postset is the set of all output places from t, i.e., t●={p:pεP and O(p, t)>0}. Similarly, p's preset ●p={tεT:O(p, t)>0} and postset p●={tεT: 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 work, it is assumed that there are n≧2 steps in a cluster tool and only one PM serves for each step. Let (PM1, PM2, PM3) denote the wafer flow pattern, where PMi, iεNn, represents a process model being used to process wafers at Step i. Thus, a wafer needs to be processed at PM1−PMn, sequentially before it is completed. Wu et al. [2008a] developed a PN model and conducted the steady periodical scheduling analysis for a single-arm cluster tool with wafer residency time constraints. We briefly introduce their PN model next.
In such a PN model, Step i is modeled by timed place pi with K(pi)=1, iεNn. The LLs are treated just as a processing step called Step 0. Since the LLs 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, iεNn, a wafer is being processed in the PM for Step i. 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, iεNn, 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. Note that the explicit representation of a robot wait as a place is critically important to deal with residency time constraints. Non-timed place zij is used to model the state at which it is ready to load a wafer to Step i or the wafer unloading from Step i ends. Transitions are used to model the robot tasks. Timed ti1, iεNn, models loading a wafer into Step i, and t01 models loading a completed wafer into a LL. Timed ti2, iεNn, models unloading a wafer from Step i, and t02 models unloading a raw wafer from a LL. Timed transition yi, iεNn-2∪{0}, represents the robot's moving from Steps i+2 to i without carrying a wafer; while transitions yn−1 and yn represent the robot's moving from a LL to Step n-1 and Steps 1 to n, respectively. Timed transition xi, iεNn−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's by . Then, the PN model for a single-arm cluster tool is shown in FIG. 2.
At the steady state, every process module has one wafer being processed, i.e., Σi=12 K(pi) wafers are being processed. For the PN model in FIG. 2, consider marking M with M(pi)=1, iεNn, and M(r)=1. At this marking, y0 is enabled and firing y0 leads the PN to a dead marking, or the PN is deadlock-prone. Thus, according to [Wu et al., 2008], a control policy is proposed to make it deadlock-free.
Control Policy 1 (CP1): At any M of the PN model in FIG. 2, yi, iεNn−1∪{0}, is said to be control-enabled if M(pi+i)=0; and yn is said to be control-enabled if M(pi)=1, iεNn.
Before a cluster tool reaches its steady state, it must experience a start-up process. For a single-arm cluster tool, because the processing time is much longer than the robot task time, a backward strategy is found to be optimal [Lee et al., 2004; and Lopez and Wood, 2003]. Thus, a backward strategy is also used to operate the single-arm cluster tool for the start-up process. At the initial state, there is no wafers being processed in the tool, or the tool is empty. Let Ms0 denote the initial state. When the tool starts to work, the robot unloads a wafer from the LLs, moves to Step 1, and loads this wafer into Step 1. Let Ms1 denote the state of the system when the robot finishes the robot task of loading the wafer into Step 1. Then, the robot should wait there till this wafer is completed. After the wafer is processed, the robot unloads this wafer from Step 1 as soon as possible, moves to Step 2, loads this wafer into Step 2, returns to the LLs and unloads a raw wafer from the LLs, moves to Step 1, and loads the raw wafer into Step 1. At this time, Step 1 and Step 2 both have one wafer being processed. Thus, let Ms2 denote the state of the system at this time. In the following operations of the system, the tool would reach a state that the Step i, iεNd and d<n, has one wafer being processed and Step i, d<i≦n, is empty. To model this state, a PN model is developed shown in FIG. 3.
The places in the PN model in FIG. 3 have the same meanings as the ones in FIG. 2. Transitions ti1, iεNd+1, d<n and d≧2, ti2, iεNd∪{0},d<n and d≧2,xi, iεNd∪{0}, d<n and d≧2, and yi, iεNd−1∪{0}, d<n and d≧2, in the PN model in FIG. 3 also have the same meanings as the ones in FIG. 2. Transition yd represents the robot's moving from Steps 1 to d. Because Step i, iεNd, d<n and d≧2, has one wafer being processed and Step i, d<i≦n, is empty, we have M(pi)=K(pi), iεNd, d<n and d≧2, and M(r)=1. At the marking shown in FIG. 3, y0 is enabled and can fire. It can be seen that firing y0 leads the PN to a dead marking, or the PN is deadlock-prone. Then, a control policy is introduced to make it deadlock-free.
Control Policy 2 (CP2): For the PN model in FIG. 3, yi, iεNd∪{0}, d<n and d≧2, is said to be control-enabled if M(pi+1)=0.
With CP2, the start-up process could be described by running the PN model in FIG. 3 shown as follows. At the state Ms2, Steps 1 and 2 both have one wafer being processed and the robot stays at Step 1. To describe the state Ms2, we can set d=2 in the PN model and Ms2(pi)=K(pi), iεN2, and Ms2(r)=1 holds. According to CP2, the following transitions firing sequence is: firing y2 (moving to Step 2)→firing t22 (unloading a wafer from Step 2)→firing x2 (moving from Steps 2 to 3)→firing t31 (loading the wafer into Step 3)→firing yi (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)→firing y0 (moving from Steps 2 to 0)→firing t02 (unloading a wafer from Step 0)→firing x0 (moving from Steps 0 to 1)→firing t11 (loading the wafer into Step 1). At this time, the system reaches state Ms3 such that Ms3(pi)=K(pi), iεN3, and Ms3(r)=1 hold. Then, we can set d=3 in the PN model. According to CP2, the PN model can evolve to state Ms4 with Ms4(pi)=K(pi), iεN4, and Ms4(r)=1. Similarly, with the PN model in FIG. 3 and CP2, the PN model can evolve to state Msn with Msn(pi)=K(pi), iεNn, and Msn(r)=1. At this time, the cluster tool is full of wafers and it reaches the steady state.
A.3. Activity Time Modeling
In the PN models in FIGS. 2 and 3, to describe the temporal aspect of a cluster tool, both transitions and places are associated with time. We use t to denote the time for the robot task of moving with or without carrying a wafer. Time ai is used to denote the time taken for completing a wafer at a step. It is assumed that the time taken for the robot's unloading a wafer from a step and loading a wafer into a step/LL is same and denoted by α. Transition t02 models the robot's unloading a wafer from the LLs and aligning a wafer. Therefore, the time associated with t02 is α0 that is different from α. The robot's waiting time (denoted by ωi) in qi is determined by a schedule and can be a real number in [0, ∞), or ωiε[0, ∞). The detailed explanation of temporal features is summarized in Table 1.







TABLE 1







The Time Durations Associated with Transitions 


and Places









Trans-

Allowed


ition

time


or place
Actions
duration





ti1
Robot loads a wafer into Step i, 
α



i ∈ Nn ∪ {0}



ti2
Robot unloads a wafer from Step i, 




i ∈ Nn



t02
Robot unloads a wafer from a LL 
α0 



and aligns it



yi
Robot moves from a step to another 
μ



without carrying a wafer



xi
Robot moves from a step to another 




with a wafer carried



pi
A wafer being processed in pi, 
ai



i ∈ Nn



qi
robot waits before unloading a 
ωi



wafer from Step i, i ∈ Nn ∪ {0}



zij
No activity is associated









With wafer residency time constraints, the deadlock-freeness does not mean that the PNs shown in FIGS. 2 and 3 are live, because a token in pi cannot stay there beyond a given time interval. Let τi be the sojourn time of a token in pi and δi the longest time for which a wafer can stay in pi after it is processed. Then, the liveness of the PN model is defined as follows.
Definition 1: The PN models in FIGS. 2 and 3 for single-arm cluster tools with residency time constraints are live, if 1) at any marking with a token in pi, ∀iεNn, and when ti2 fires ai<τi<ai+δi holds; 2) at any marking with a token in pi, iεNd and d <n, and when ti2 fires ai<τi<ai+δi holds, respectively.
B. Schedulability Conditions
Before scheduling the start-up process, we recall the necessary and sufficient schedulability conditions of a single-arm cluster tool with wafer residency time constraints under the steady state derived in [Wu et al., 2008].
B.1. Timeliness Analysis for the Steady State
It follows from [Wu et al., 2008] that, to complete the processing of a wafer at Step i, iεNn−1, it takes τi+4α+3μ+ωi−1 time units, where τi should be within [ai, ai+δi]. With only one PM at Step i, iεNn, we have that the lower permissive cycle time at Step i is

θiL=ai+4α+3μ+ωi−1, iεNn\{1}.  (1)

The upper permissive cycle time at Step i is

θiU=ai+4α+3μ+ωi−1+δi, iεNn\{1}.  (2)

For Step 1, the lower one is

θiL=a1+3α+α03μ+ω0.  (3)

Its upper one is

θiU=a1+3α+α03μ+ω0+δ1.  (4)

It follows from (1)-(4) that the robot waiting time ωi, iεNn−1∪{0}, affects the permissive wafer sojourn time. Thus, by carefully regulating them, one can change the permissive range among the steps. By removing them from the above expressions, we obtain the lower and upper workloads with no robot waiting for each step as follows:

hd iL=ai+4α+3μ, iεNn\{1},  (5)

hd iU=ai+4α+3μ+δi, iεNn\{1},  (6)

hd iL=a1+3α+α0+3μ,  (7)

and

hd iU=a1+3α+α0+3μ+δ1,  (8)

where jL, and jU are the lower and the upper workloads, respectively, for Step j,jεNn.

To schedule a single-arm cluster tool with residency time constraints, one has to ensure ai≦τi≦ai+δi. Hence, we need to know how τi is calculated. According to [Wu et al., 2008], we have that

τi=2(n+1)μ+(2n+1)α+α0+Σd=0nωd−(4α+3μ+ω0)=Ψ−(4α+3μ+ωi−1), iεNn\{1}  (9)

and

τi=2(n+1)μ+(2n+1)α+α0+Σd=0nωd−(3α+α0+3μ+ω0)=Ψ−(3α+α0+3μ+ω0).  (10)

The robot cycle time is given by

Ψ=2(n+1)μ+(2n+1)α+α0+Σd=0nωd=Ψ1+Ψ2  (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. It should be pointed out that Ψ is independent of the ωi's. Let θ1=τ1+3α+α0+3μ+ω0 and θi=τi+4α+3μ+ωi−1, iεNn−{1}, denote the cycle time for Step i, iεNn. 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. For a periodic schedule in a steady state, we have

θ=θ1=θ2=. . . =θn=Ψ.  (12)

In (11), μ, α, and α0 are all deterministic, only ωd, dεNn∪{0}, are changeable, i.e., Ψ1 is deterministic and Ψ2 can be regulated. Thus, based on the PN model shown in FIG. 2, one can schedule a single-arm cluster tool with residency time constraints by appropriately regulating ωd, dεNn∪{0}, such that (12) holds and at the same time the wafer residency time constraints are fully satisfied.
B.2. Schedulability Conditions for the Steady State Scheduling
To find a feasible cyclic schedule, the key is to know under what conditions there exists θ such that the system is schedulable. Notice that, in (5)-(8), iL and iU denote the lower and upper bounds of θi, respectively. Let max=max {iL, iεNn}. Then, Wu et al. [2008] establish the following schedulability conditions.
Theorem 1: If max≦iU and Ψ1≦iU, iεNn, a single-arm cluster tool with residency time constraints is schedulable.
For this case, when max≦iU and Ψ1≦max, iεNn, the tool is process-bound. When iL≦Ψ1≦iU, iεNn, a tool is transport-bound. With max≦iU, iεNn, the difference of the workloads among the steps is not too large. Thus, by properly setting ωi's, the workloads among the steps can be balanced such that there is a feasible cyclic schedule. It follows from [Wu et al., 2008] that, in this case, one can simply set ωi=0, iεNn−1∪{0}, and ωn=max{max−Ψ1, 0} such that Ψ=max{max, Ψ1} holds. In this way, a feasible schedule is found. Further, it is optimal in terms of cycle time.
By Theorem 1, to make the tool schedulable requires that the workloads among the steps are not too large, i.e. [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅. However, sometimes we have [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅. In this case, let E={i|iεNnU<max} and F=Nn\E. It follows from [Wu et al., 2008] that the time for completing a wafer at Step i can be increased by setting ωi−1>0 without changing sojourn time τi. Hence, a cluster tool may be made schedulable even if the workloads among the steps are not well balanced. To do so, we balance the workloads among the steps by setting ωi−1's as follows:




  
    
      
        
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Theorem 2: If [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅,U<max, iεE≠∅, iU≧max, iεF, and ΣiεEωi−1+Ψ1≦max, a single-arm cluster tool with residency time constraints is schedulable with ωi−1, iεNn, being set by (13).
In this case, with the robot waiting time ωi−1, iεNn, being set by (13), without changing τiθi for completing a wafer at Step i can be increased such that the workload among the steps can be properly balanced. Notice that, by (13), the robot waiting time ωi−1, iεNn, is set, and then let ωn=max−(Ψ1+ΣiεEωi−1) such that Ψ=max holds. Thus, a feasible schedule is obtained and the cycle time is optimal. According to [Wu et al., 2008], the conditions given by Theorems 1 and 2 are the necessary and sufficient schedulability conditions for a single-arm cluster tool with residency time constraints. In the next section, we conduct the start-up process scheduling analysis for the system.
C. Start-up Process Scheduling
C.1. Temporal Properties in Start-up Process
At the initial state denoted by Ms0, the cluster tool is idle. When the tool starts to work, the robot unloads a wafer from the LLs, moves to Step 1, and loads this wafer into Step 1. At this time, Ms1 is reached. From states Ms0 to Ms1, it takes (α0+μ+α) time units. Then, the robot should wait there for a1 time units before the wafer in Step 1 is completed. Then, (α0+3α+3μ) time units would be taken for performing the following robot task sequence: unloads this wafer from Step 1 as soon as possible, moves to Step 2, loads this wafer into the Step 2, returns to the LLs and unloads a raw wafer from the LLs, moves to Step 1, and loads the raw wafer into Step 1. At this time, both Steps 1 and 2 have one wafer being processed and Ms2 is reached. From Ms1 to Ms2, it takes (a1+α0+3α+3μ) time units.
Observing the PN model shown in FIG. 3, we have that Msd(pi)=K(pi), iεNd and 2≦d≦n−2, and Msd(r)=1. With CP2, the PN model can evolve to state Ms(d+1) with Ms(d+1)(pi)=K(pi), iεNd+1, and Ms(d+1)(r)=1, and then to Ms(d+2) with Ms(d+2)(pi)=K(pi), iεNd+2, and Ms(d+2)(r)=1. During the evolution from Ms(d+1) to Ms(d+2), the robot should sequentially go to Steps (d+1), d, . . . , and 1 for unloading the processed wafers. Thus, with wafer residency time constraints, it is necessary to know how much time is needed to complete the processing of a wafer at Step i, iεNd+1. Notice that the wafers unloaded from Step i, iεNd+1, during the period from Ms(d+1) to Ms(d+2) are all the ones loaded into Step i, iεNd+1, during the period from Msd to Ms(d+1). Thus, from the PN model in FIG. 3, CP2, and the period from Msd to Ms(d+2), to complete the processing of a wafer in Step i, iεNd+1, the following transition firing (activities) sequence must be executed: Firing ti2 (time α)→xi (time μ)→t(i+1)1 (time α)→yi−1 (time μ)→robot waiting in qi−1 (time ωi−1)→t(i−1)2 (time α)→xi−1 (time μ)→t(i+1)1 (time α)→processing a wafer at Step i (time τi)→ti2 (time α) again. In this way, a cycle is completed and it takes (τi+4α+3μ+ωi−1) time units to complete a wafer. In fact, for a wafer unloaded from Step i, iεNd, during the period from Msd to Msd(d+1), 2≦d<n, it also takes (τi+4α+3μ+ i−1) time units to complete this wafer. Notice that τi should be within [ai, ai+δi]. When τi=ai, we have the lower permissive time to complete a wafer at Step i, iεNd\{1}, which equals to the one given by (1). When τi=ai+δi, we have the upper permissive time to complete a wafer at Step i, which equals to the one given by (2). For Step 1, the lower and upper permissive time to complete a wafer equals to the ones in (3) and (4), respectively. Then, if the robot waiting time is removed from (1)-(4), the lower and upper workloads without robot waiting for each step can be obtained via (5)-(8), respectively.
EQNS. (5)-(8) present the workload balance information that affects the existence of a feasible schedule. It follows from (2) and (6) that θiU>iU if ωi−1>0. It implies 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 such that the permissive wafer sojourn time ranges among the steps are balanced to some extent to guarantee the feasibility. To do so, we need to know how τi should be calculated. The wafer sojourn time at pi depends on the robot tasks and the workloads of the steps. From the PN model shown in FIG. 3 and CP2, during the evolutions from Ms(d−1) to Msd, a wafer (W1) is loaded by the robot into Step i: Firing yd−1→waiting in qd−1→firing t(d−1)2→firing xd−1→firing td1→firing yd−2→waiting in qd−2→firing t(d−2)2→firing xd−2→firing t(d−1)1→ . . . →firing y1→waiting in qi−1→firing ti2 with unloading a wafer from Step i→firing xi→firing t(i+1)1→firing yi−1→waiting in qi−1→firing t(i−1)2→firing xi−1→firing ti1 to load wafer W1 into Step i → . . . →y1→waiting in q1→firing t12→x1→firing t21→firing y0→waiting in q0→firing t02→firing x0→firing t11. Then, from the PN model shown in FIG. 3 and CP2, during the evolutions from Msd to Ms(d+1), wafer W2 is unloaded by the robot from Step i: Firing yd→waiting in qd→firing td2→firing xd→firing t(d+1)1→firing y(d−1)→waiting in q(d−1)→firing t(d−1)2→firing xd−1→firing td1→ . . . →firing yi→waiting in qi→firing ti2 to unload wafer W1 from Step i→firing xi→firing t(i+1)1→firing yi−1→waiting in qi−1→firing t(i−1)2→firing xi−1→firing ti1→ . . . →y1 waiting in q1→firing t12→x1→firing t21→firing y0→waiting in q0→firing t02→firing x0→firing t11. Thus, from the above PN evolutions we have that, during the evolution from Msd to Ms(d+1), 2≦d<n, the wafer sojourn time in pi, iεNd, is given by




  
    
      
        
          
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Let Ψsd(d+1) and Ψsd(d+1)1 denote the robot task time for transferring the tool from states Msd to Ms(d+1) with and without robot waiting time considered, respectively. Thus, we have




  
    
      
        
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and

Ψsd(d+1)1=2(d+1)μ+(2d+1)α+α0.  (17)

It follows from (14)-(17) that to schedule the transient process of a residency-time constrained single-arm cluster tool is to appropriately regulate ωj, jεNd∪{0}, such that the wafer residency time constraints at each step are all satisfied.

C.2. Scheduling for Start-up Process
Feasibility is an essential requirement for scheduling a transient process of a cluster tool. As we have mentioned that, at initial state Ms0, the cluster tool is idle. When the tool reaches states Ms1 to Msn, Steps 1 to n have one wafer being processed, respectively. For the start-up process, the robot tasks are determined. Thus, we just need to determine the robot waiting time during the period from Ms0 to Msn to find a feasible schedule for the start-up process. Then, we have the following schedulability proposition.
Proposition 1: A start-up process of a single-arm cluster tool with wafer residency time constraints is schedulable if there exists the robot waiting time setting during the period from Ms0 to Msn such that the wafer residency time constraint at each step is satisfied.
With Proposition 1, we know that it is necessary to propose a method to regulate the robot waiting time during the period from Ms0 to Msn such that the cluster tool can enter the desired steady state from the initial state without violating the wafer residency time constraints.
In a cluster tool, it is reasonable to assume that there are more than one processing step. For the tool with two processing steps, the start-up process from Ms0 to Ms2 could be described by a robot task sequence σ1: Unloading a raw wafer (W2) from the LLs (time α0)→moving to Step 1 (time μ)→loading wafer W2 into Step 1 (time α)→waiting at Step 1 for ω1=a1 time units→unloading wafer W2 from Step 1 (time α)→moving to Step 2 (time μ)→loading wafer W2 into Step 2 (time α)→moving to the LLs (time μ)→waiting at the LLs for ω0 time units→unloading a raw wafer (W3) from the LLs (time α0)→moving to Step 1 (time μ)→loading wafer W3 into Step 1 (time α). At this time, the system reaches state Ms2. In σ1, only robot waiting time ω0 is unknown. Let |σ1| denote the time needed to perform sequence ν1. Thus, |σ1|=2α0+4α+4μ+a1+ω0. Therefore, for the single-arm cluster tool, to obtain a feasible start-up schedule is to determine the robot waiting time ω0. For the single-arm cluster tool with n>2 processing steps, during the process from Ms0 to Ms2, the robot task sequence is also σ1. Then, the system keeps working according to the PN model in FIG. 3 and CP2 till it reaches state Msn. For this case, the robot waiting time is also unknown during the process from Ms0 to Msn. Let dmax=max{iL, iεNd}. To solve the scheduling problem, the following algorithm is developed.
Scheduling Algorithm 1: If max≦iU and Ψ1≦iU, iεNn, the robot waiting time is set as follows.

    
    
        Situation 1: A single-arm cluster tool has two steps:
        1) During the process from Ms0 to Ms2, the tool operates according to the robot task sequence σ1 and set ω0=0 and ω1=a1.
        Situation 2: A single-arm cluster tool has n steps, n>2:
        1) During the process from Ms0 to Ms2, the performance of the tool is same as the one regulated by 1) in Situation 1;
        2) During the process from Msd to Ms(d+1), 2≦d<n, the tool operates according to the PN model in FIG. 3 and CP2 and set ωi=0, iεNd−1∪{0}, and ωd=max{dmax−Ψsd(d+1)1, 0}.
    
    


In this case, there are two situations. For Situation 1, there are two steps in a single-arm cluster tool. Then, during the start-up process from Ms0 to Ms2, the tool operates according to the robot task sequence σ1, and the robot waiting time ω0 and ω1 in σ1 can be set as ω0=0 and ω1=a1. With Ms2 being reached, the system reaches its steady state. For Situation 2, a single-arm cluster tool has n steps, n>2. From Ms0 to Ms2, the performance of the tool is same as the one regulated by 1) in Situation 1. Then, during the process from Msd to Ms(d+1), 2≦d<n, the tool operates according to the PN model in FIG. 3 and CP2. At the same time, the robot waiting time is set as ωi=0, iεNd−1∪{0}, and ωd=max{dmax−Ψsd(d+1)1, 0} such that Ψsd(d+1)=max{dmax, Ψsd(d+1)1}. When the tool reaches state Msn, the system reaches its steady state. Then, by the PN model in FIG. 2 and CP1, the system operates with the backward strategy. By Scheduling Algorithm 1, a schedule could be found for the start-up process. Then, the next question is if this obtained schedule is feasible. The following theorem answers it.
Theorem 3: For a single-arm cluster tool with wafer residency time constraints, if max≦iU, Ψ1≦iU, iεNn, a schedule obtained by Scheduling Algorithm 1 is feasible.
Proof: Consider Situation 1. For the start-up process from Ms0 to Ms2, the robot performs the robot tasks σ1. It is easy to find that wafer W2 can be unloaded from Step 1 without violating the residency time constraints. Then, W3 is delivered to Step 2. When Ms2 is reached, the system enters its desired steady state. Consider Situation 2. By 1) for Situation 2 of Algorithm 1, similarly, the robot can perform the robot tasks σ1 such that the cluster tool can reach Ms2 from Ms0 without violating the wafer residency time constraints. Then, by 2) for Situation 2 of Algorithm 1, the tool operates according to the PN model in FIG. 3 and CP2 during the process from Msd to Ms(d+1), 2≦d<n. At the same time, ωi=0, iεNd−1∪{0}, and ωd={dmax−Ψsd(d+1)1, 0}. It follows from (14) and (15) that σ1=2(d+1)μ+(2d+1)α+α0+Σj=0dωj−(3α+α0+3μ+ω0)=2(d+1)α+α0max{dmax−Ψsd(d+1)1, 0}−(3α+α0+3μ) and τi=2(d+1)μ+(2d+1)α+α0+Σj=0dωj−(4α+3μ+ωi−1)=2(d+1)μ+(2d+1)α+α0+max {dmax−Ψsd(d+1)1, 0}−(4α+3μ), 1<i≦d, hold, respectively. Then, there are two cases. Case 1: dmax≧Ψsd(d+1)1. Then, we have τ1=2(d+1)μ+(2d+1)α+α0+dmax−Ψsd(d+1)1−(3α+α0+3μ) and τi=2(d+1)μ+(2d+1)α+α0+dmax−Ψsd(d+1)1−(4α+3μ), 1<i≦d, hold. Then, by (17) and max≦iU, iεNn, we have iL−(3α+α0+3μ)≦τ1=dmax−(3α+α0+3μ)≦iU−(3α+α0+3μ) and iL−(4α+3μ)≦τi=dmax−(4α+3μ)≦iU−(4α+3μ), 1<i≦d. Thus, it follows from (5)- (8) that a1=1L−(3α+α0+3μ)≦τ1≦1U−(3α+α0+3μ)=a1+δ1 and ai=iL−(4α+3μ)≦τ1≦iU−(4α+3μ)=ai+δi, 1<i≦d, hold. Case 2: dmax<Ψsd(d+1)1. Then from (17), we have τ1=2(d+1)μ+(2d+1)α+α0−(3α+α0+3μ)=Ψsd(d+1)1−(3α+α0+3μ) and τi=2(d+1)μ+(2d+1)α+α0−(4α+3μ)=Ψsd(d+1)1−(4α+3μ), 1<i≦d, hold. By the assumption of Ψ1≦iU, iεNn, we have τi=Ψsd(d+1)1−(3α+α0+3μ)<Ψ1−(3α+α0+3μ)≦1U−(3α+α0+3μ) and τi=Ψsd(d+1)1−(4α+3μ)<Ψ1−(4α+3μ)≦iU−(4α+3μ), 1<i≦d. Then, it follows from (5)-(8), max≦iU, iεNn, and dmax<Ψsd(d+1)1 that a1=1L−(3α+α0+3μ)≦dmax−(3α+α0+3μ)<τ1=Ψsd(d+1)1−(3α+α0+3μ)<1U−(3α+α0+3μ)=a1+δ1and ai=iL−(4α+3μ)≦dmax−(4α+3μ)<τi=Ψsd(d+1)1−(4α+3μ)<iU−(4α+3μ)=ai+δi, 1<i≦d. Therefore, during the start-up process from Ms0 to Msn, the wafer residency time constraints at each step are all satisfied. Hence, the theorem holds.
In the case of Situation 1, by Algorithm 1, the robot performs σ1 such that the cluster tool can successfully go through the start-up process from Ms0 to Ms2 without violating any residency time constraints. Also, it takes |σ1|=2α0+4α+4μ+a1 time units for the start-up process. In the case of Situation 2, by Algorithm 1, the schedule is same as the one before Ms2 is reached. Then, by Algorithm 1, we need to dynamically adjust the robot waiting time at step d during the process from Msd to Ms(d+1), 2≦d<n, such that Ψsd(d+1)1=max{dmax, Ψsd(d+1)1}. Thus, it takes |σ1|+Σd=2n−1max(d max, Ψsd(d+1)1) time units for the start-up process. For a single-arm cluster tool with n≧2 steps, when Msn, is reached, the system enters its desired steady state. In the following evolution, the system operates with the backward strategy. Based on Theorem 1, for the steady state scheduling, a feasible and optimal schedule is obtained by setting ωi=0, iεNn−1∪{0}, and ωn=max{max−Ψ1, 0} such that Ψ=max {max, Ψ1} holds. Then, the following theorem proves its optimality.
Theorem 4: For a single-arm cluster tool with wafer residency time constraints, if max≦iU, Ψ1≦iU, iεNn, a schedule obtained by Scheduling Algorithm 1 for the start-up process is optimal.
Proof Situation 1: For the start-up process from Ms0 to Ms2, the robot performs the robot tasks σ1. If there be a schedule better than the one obtained by Algorithm 1, it must be that the robot waiting time ω1 is shortened because of ω0=0. However, if ω1 is less than a1, the wafer being processed at Step 1 cannot be processed. Therefore, for Situation 1, the obtained schedule by Algorithm 1 is optimal. For Situation 2, similar to Situation 1, the obtained schedule by Algorithm 1 for the process from Msd to Ms2 is optimal. It follows from Theorem 3 that during the process from Msd to Ms(d+1), 2≦d<n, we have τi=2(d+1)μ+(2d+1)α+α0+max{dmax−Ψsd(d+1)1, 0}−(3α+α0+3μ) and τi=2(d+1)μ+max{dmax−Ψsd(d+1)1, 0}−(4α+3μ), 1<i≦d, hold, respectively. It is assumed that dmax=kL, 1≦k≦d. Then, we have τ1=2(d+1)μ+(2d+1)α+α0+max{1L−Ψsd(d+1)1, 0}−(3α+α0+3μ) if dmax=1L and τk=2(d+1)μ+(2d+1)α+α0+max{kL−Ψsd(d+1)1, 0}−(4α+3μ) if dmax=kL, 1<k≦d. If dmax≧Ψsd(d+1)1, we have dmax=Ψsd(d+1) by Algorithm 1. By (17) and (5)-(8), we have τ1=1L−(3α+α0+3μ)=a1 if dmax=1L and τk=kL−(4α+3μ)=ak if dmax=kL, 1<k≦d, hold. This means that it takes Ψsd(d+1)=kL time units for the process from Msd to Ms(d+1), 2≦d<n. Thus, for the process from Msd to Ms(d+1), 2≦d<n, the wafer sojourn time just equals to ak at Step k. If there exists a schedule for the process from Msd to Ms(d+1), 2≦d<n, better than the one obtained by Algorithm 1, there must exist Step k where the wafer sojourn time is less than ak. This means the wafer at Step k cannot be processed. If dmax<Ψsd(d+1)1, a better schedule for the process from Msd to Ms(d+1), 2≦d<n, cannot be found because Ψsd(d+1)1 cannot be shortened. Therefore, the obtained schedule by Algorithm 1 for the process from Msd to Ms(d+1), 2≦d<n, is also optimal. Hence, the theorem holds.
By Theorem 3, the workloads among the steps are properly balanced, i.e. [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅. However, there is also another case with [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅. Under the steady state, the cycle time is a constant. Then, a feasible schedule could be found by setting ωi−1>0, iεE, to reduce the wafer sojourn time τi without changing the time for completing a wafer at Step i [Wu et al., 2008]. For the transient process, we have: 1) wafers are processed at Step i, iεE, during the process from Msd to Ms(d+1) and Ms(d+1) to Ms(d+2), 2≦d≦n−2, respectively; 2) the time taken for the process from Msd to Ms(d+1) and Ms(d+1) to Ms(d+2) may be different. Thus, the key to find a feasible and optimal schedule for the process from Msd to Ms(d+2) is to dynamically adjust the robot waiting time ωi−1. However, increasing and decreasing ωi−1 would decrease and increase the wafer sojourn time respectively. This makes it difficult to guarantee the feasibility and optimality at the same time. Thus, a linear programming model is developed to solve this problem. Let tijd and wid denote the time when firing tij completes and the robot waiting at Step i before unloading a wafer during the process from Msd to Ms(d+1), respectively. Then, we have a linear programming model.
Linear Programming Model (LPM): If [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅ and the system checked by Theorem 2 is schedulable under the steady state, then a schedule can be found by the following LPM:




  
    
      
        min
        ⁢
        
          
            ∑
            
              d
              =
              1
            
            n
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              d
            
            ⁢
            
              ω
              i
              d
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




subject to

t110=α0+μ+α,  (19)

t121=t110+ω11+α,  (20)

ti1d=ti−1)2d+μ+α, 1≦i≦d+1 and 1≦d≦n−1,  (21)

ti2d=ti+2)1d+μ+ωid+α, 1≦i≦d−1 and 1≦d≦n−1,  (22)

t02d=td+1)1d+μ+ω0d+α0, 1≦d≦n−1,  (23)

td2d=t11d−1+μ+ωdd+α, 2≦d≦n−1,  (24)

ti1d=ti−1)2d+μ+α, 1≦i≦n and dε{n, n+1},  (25)

t01d=tn2d+μ+α, dε{n, n+1},  (26)

ti2d=t(i+2)1d+μ+ωid+α, 1≦i≦n−2 and dε{n, n+1},  (27)

t(n−1)2d=t01d+μ+ωn−1d+α, dε{n, n+1},  (28)

tn2d=t11d−1+μ+ωnd+α, dε{n, n+1},  (29)

t02d=t21d+μ+ω0d+α0, dε{n, n+1},  (30)

ωin=ωin+1, 0≦i≦n,  (31)





  
    
      
        
          
            
              ∑
              
                d
                =
                1
              
              n
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  0
                
                d
              
              ⁢
              
                ω
                i
                d
              
            
          
          ≥
          0
        
        ,
      
    
    
      
        (
        32
        )
      
    
  




and

ai≦ti2d−α−ti1d−1≦ai+δi, 1≦i≦d and 1≦d≦n+1.  (33)

For a single-arm cluster tool with two processing steps, the robot task sequence for the start-up process from Ms0 to Ms2 is σ1. Then, the system is operated with the backward strategy based on the PN model in FIG. 2. For a tool with n>2 processing steps, the robot task sequence for the start-up process from Ms0 to Ms2 is also σ1. Then, the system will run according to the PN model in FIG. 3 till it reaches state Msn. In the following operations, the system is operated by the backward strategy based on the PN model in FIG. 2. Notice that the robot task sequence for the start-up process of the two cases is known in advance. However, the robot waiting time is unknown. If the waiting time during the process from Ms0 to Msn is determined, the schedule for the start-up process is determined. Objective (18) in LPM is to minimize the total robot waiting time. Constraints (19) and (21) give the time for completing the robot task of loading a wafer into a step. Constraints (20) and (22)-(24) represent the time for completing the robot task of unloading a wafer from a step. After Msn is reached, the cluster tool enters its steady state and operates by the backward strategy based on the PN model in FIG. 2 and CP1. Then, when the first robot task cycle for the steady state is completed, state Ms(n+1) is reached. When the second robot task cycle for the steady state is completed, state Ms(n+2) is reached. Thus, for the first and second cycles for the steady state, Constraints (25) and (26) mean the time for completing the robot task of loading a wafer into a step, and Constraints (27)-(30) indicate the time for completing the robot task of unloading a wafer from a step. Constraint (31) makes sure that the robot waiting time is same for the different cycle under the steady state. Constraint (32) means that the robot waiting time is no less than zero. With wafer residency time constraints being considered, Constraint (33) is used to guarantee such constraints to be satisfied. In this way, a schedule for the start-up process could be obtained by this model.
For the case with [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅, Theorem 2 gives schedulability conditions to check if the system is schedulable. Thus, it gives rise to a question that, for the case with [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅, if the system checked by Theorem 2 is schedulable, can a feasible schedule be obtained by LPM? To answer it, a schedule can be obtained by setting the robot waiting time as: 1) For the tool with two processing steps, the robot waiting time can be set as ω0d=max{max−1U, 0}, dε{1,2}, ω11=a1, ω12=max{max−2U, 0}, and ω22=max−Ψ1−(ω02+ω12); and 2) For the tool with more than n>2 processing steps, the robot waiting time can be set as ωid=max{−(i+1)U, 0}, 0≦i≦n−1 and 2≦d≦n+1, ω11=a1, and ωdd=max−Ψ1−Σi=0d−1ωid, 2≦d≦n+1. It is easy to verify that this schedule is in the feasible region of LPM. Therefore, if the system is schedulable according to Theorem 2's conditions, a feasible and optimal schedule can be obtained by LPM. For the first and second cycles for the steady state, the robot waiting time ωin and ωin+1, 0≦i≦n, can be determined by LPM. Then, in the following operations of the system under the steady state, the robot waiting time is also set as ωi=ωin, 0≦i≦n−1, and ωn=max−Σi=0n−1ωi. Thus, another question is if the schedule for the steady state is feasible and optimal? The following theorem answers it.
Theorem 5: For a single-arm cluster tool with [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅, with the PN in FIG. 2 and CP1, if a schedule for the steady state is given by setting ωi=ωin,0≦i≦n−1, and ωn=max−Σi=0n−1ωi, ωin,0≦i≦n−1is obtained by LPM, then such a schedule is feasible and optimal.
Proof: By LPM, during the processes from Msn to Ms(n−1) and Ms(n+1) to Ms(n+2), the robot waiting time is ωi=ωin,0≦i≦n. Then, the cycle time for the processes from Msn to Ms(n+1) and Ms(n+1) to Ms(n+2) should be Ψ=max. If there exists a schedule with the cycle time Ψ<max and it is assumed that max=kL, k≠1 holds, it follows from (9) that τk=[2(n+1)μ+(2n+1)α+α0+Σd=0nωd]−(4α+3μ+ωk−1)=Ψ−(4α+3μ+ωk−1)<kL−(4α+3μ+ωk−1)≦kL−(4α+3μ). Then, from (5), we have τk<kL−(4α+3μ)=ak. This means that the wafer at Step k is not completed. Similarly, if there exists a schedule with the cycle time Ψ<max and max=1L holds, we have τ1<a1. Therefore, the cycle time for the processes from Msn to Ms(n+1) and Ms(n+1) to Ms(n+2) should be Ψ=max. This implies that Σi=0nωi=max−Ψ1. Thus, based on LPM and Theorem 2, this theorem holds and the cycle time of the system for the steady state is max.
Up to now, for the case that the workloads among the steps can be properly balanced, i.e. [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅, a scheduling algorithm is proposed to find the optimal schedule for the start-up process such that the single-arm cluster tool can enter its steady state optimally. For the case that the differences of the workloads among the steps are too large such that [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅, a linear programming model is developed to find a feasible and optimal schedule to transfer a single-arm cluster tool from the initial state to a steady one. Notice that Scheduling Algorithm 1 consists of several expressions and LPM is a linear programming model. Therefore, it is very computationally efficient to use the proposed methods to find a feasible and optimal schedule for the start-up process for single-arm cluster tools with wafer residency time constraints.
EXAMPLES
Example 1: The flow pattern is (PM1, PM2, PM3, PM4, PM5). It takes 5 s for the robot to unload a wafer from a PM and to load a wafer to a PM/LL (α=5 s), 10 s to unload a wafer from the LLs and align it (α0=10 s), and 2 s to move between PMs/LLs =2 s). It needs 90 s, 100 s, 100 s, 105 s, and 115 s for a PM at Steps 1-5 to process a wafer (a1=90 s, a2=100 s, a3=100 s, a4=105 s, and a5=115 s), respectively. After being processed, a wafer at Steps 1-4 can stay there for 20 s (δ1=δ2=δ3=δ4=δ520 s).
It follows from (5)-(8) that, we have 1L=121 s, 1U=141 s, 2L=126 s, 2U=146 s, 3L=126 s, 3U=146 s, 4L=131 s, 4U=151 s, 5L=141 s, 5U=161 s, and Ψ1=89 s. By Theorem 1, the single-arm cluster tool is schedulable. For the steady state, an optimal and feasible schedule is obtained by setting ω0=ω1=ω2=ω3=ω4=0 s and ω5=52 s. Then, the cycle time of the system under the steady state is 141 s. For this example, the workloads among the steps are properly balanced, i.e. [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅. Thus, an optimal and feasible schedule can be found by Algorithm 1 for the start-up process. This example belongs to Situation 2 of Algorithm 1. Therefore, the robot waiting time during the start-up process is set as follows: 1) During the process from Ms0 to Ms2, ω0=0 s and ω1=90 s; 2) During the process from Ms2 to Ms3, ω0=ω1=0 s and ω2=79 s; 3) During the process from Ms3 to Ms4, ω0=ω1=ω2=0 s and ω3=65 s; 4) During the process from Ms4 to Ms5, ω0=ω1=ω2=ω3=0 s and ω4=56 s. In this way, an optimal and feasible schedule is obtained for the start-up process. The simulation result is shown in FIG. 4. It shows that it takes 521 s for the start-up process.
In [Wu et al., 2008], a method is proposed to transfer the system to enter its steady state from the initial state. It puts a virtual token (wafer) in places p2- pn and none at p1 in FIG. 2, i.e. M0(pi)=1, iεNn−{1}, and M0(p1)=0 at the initial state. Then, by running the PN model with a schedule obtained by the scheduling algorithm in [Wu et al., 2008], when all the virtual wafers go out of the system, the steady state is reached. It takes 563 s for the start-up process. Therefore, by the method as disclosed herein in the present work, the time taken for the start-up process is reduced by 8.1% compared with the method in [Wu et al., 2008].
Example 2: The flow pattern is (PM1, PM2, PM3, PM4). α=5 s, α0=10 s, μ=2 s, a1=85 s, a2=85 s, a3=110 s, a4=120 s, and δ1=δ2=δ3=δ4=20 s hold.
It follows from (5)-(8) that, we have 1L=116 s, 1U=136 s, 2L=111 s, 2U=131 s, 3L=136 s, 3U=156 s, 4L=146 s, 4U=166 s, and Ψ1=75 s. By Theorem 2, the single-arm cluster tool is schedulable. For the steady state, an optimal and feasible schedule is obtained by setting ω0=10 s, ω1=15 s, ω2=ω3=0 s, and ω4=46 s. Then, the cycle time of the system under the steady state is 146 s. For this example, differences between the workloads among the steps are too large and [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]≠∅ holds. Thus, LPM is used to find an optimal and feasible schedule for the start-up process. With LPM, the robot waiting time during the start-up process is set as follows: 1) During the process from Ms0 to Ms2, ω0=ω01=10 s and ω1=ω11=85 s; 2) During the process from Ms2 to Ms3, ω0=ω02=0 s, ω1=ω12=15 s and ω2=ω22=54 s; 3) During the process from Ms3 to Ms4, ω0=ω03=10 s, ω1=ω13=15 s, ω2=ω23=0 s, and ω3=ω33=60 s. Then, the tool enters its steady state and it is scheduled by setting ω0=ω04=10 s, ω1=ω14=15 s, ω2=ω3=ω24=ω34=0 s, and ω4=ω44=46 s. In this way, an optimal and feasible schedule is obtained for the start-up process. The simulation result is shown in FIG. 5. It shows that it takes 405 s for the start-up process. However, with the existing method in [Wu et al., 2008], it takes 455 s for the start-up process. Thus, the time taken for this start-up process is reduced by 11%.
E. The Present Invention
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 cluster tool. The cluster tool comprises a single-arm robot for wafer handling, a LL 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, iεNn, is used for performing Step i of the n wafer-processing steps for each wafer. Note that although the cluster tool is said to comprise a LL, it is understood that in the present invention, the cluster tool can have one or more LLs.
The method includes scheduling a start-up process for the cluster tool. The start-up process has plural system states Msi, i=0, 1, . . . n−1, where Ms0 is an initial state of system start-up, and Msi, 1≦i≦n−1 denotes that i instances of a wafer unloading from the robot to any one of the n process modules have occurred since the system start-up.
Advantageously, the start-up process is developed based on Scheduling Algorithm 1. When max≦iU and Ψ1≦iU, i=1, 2, . . . n, values of ω0, ω1, . . . , ωd for each of the system states Msd, d=0, 1, ... n−1, are determined. As mentioned above, ωj, Jε{0, 1, . . . , d}, is a robot waiting time used in the state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module. According to Scheduling Algorithm 1, the values of ω0, ω1, . . . , ωd and ωd are determined by: setting ω0=0 and ω1=a1 for the states Ms0 and Ms1; and setting ωi=0, iεNd−1∪{0}, and ωd=max{dmax−Ψsd(d+1)1, 0} for the state Msd, 2≦d≦n−1 when n>2.
As a steady-state process follows the start-up process, preferably the method further includes scheduling the steady-state process based on the results obtained in the start-up process. In particular, values of ω0, ω1, . . . , ωd are determined, in which ωj,jε{0, 1, . . . n}, is a robot waiting time, used in a steady state of the cluster tool, for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module. As indicated in Section B.2 above, one option is to set ωi=0, iεNn−1∪{0}, and ωn=max{max−Ψ1, 0}.
Also advantageously, the start-up process is further developed based on the LPM model. When [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅, values of ω0d, ω1d, . . . , ωdd for each of the system states Msd, d=0, 1, ... , n−1, are determined, where ωjd,jε{0, 1, . . . , d}, is a robot waiting time used in the state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module. The values of ω0d, ω1d, . . . , ωdd =0, 1, . . . , n, are numerically optimized such that (18) is minimized subject to constraints (19)-(33). For the steady-state process, Theorem 5 indicates that one option is to set ωi=ωin, 0≦i≦n−1, and ωn=max−Σi=0n−1ωi for use in the state Msn and thereafter. Similarly, Msn denotes that n instances of a wafer unloading from the robot to any one of the n process modules have occurred since the system start-up.
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 single-arm cluster tool if the cluster tool includes one or more processors. The one or more processors are configured to execute a process of scheduling the 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.

## Claims

1. A computer-implemented method for scheduling a cluster tool, the cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, iε{1, 2, . . . , n}, is used for performing Step i of the n wafer-processing steps for each wafer, and a specialized processor configured to control the robot, the loadlock, and the process modules through execution of the method, the method comprising:
when max≦iU and Ψ1≦iU, i=1, 2, . . . , n, determining, by the specialized processor, values of ω0, ω1, . . . , ωd for each of plural system states Msd, d=0, 1, . . . , n−1, of the cluster tool, where Msi, 1≦i≦n−1 denotes that i instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up, and ωj,jε{0, 1, . . . d}, is a robot waiting time used in the state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module;
wherein the determining of ω0, ω1, . . . , ωd for each of the system states Msd, d=0, 1, . . . , n−1, comprises:
setting, by the specialized processor, ω0=0 and ω1=a1 for the states Ms0 and Ms1;

where:
max=max{iL, iεNn};
Ψ1=2(n+1)μ+(2n+1)α+α0 
iU=ai+4α+3μ+δi, iεNn\{1};
iU=a1+3α+α0+3μ+δ1;
iL=ai+4α+3μ, iεNn\{1};
iL=a1+3α+α0+3μ;
ai, iεNn, is a time that a wafer is processed in the ith process module;
δi, is the wafer residency time constraint of Step i, given by a pre-determined longest time for which a wafer in the ith process module is allowed to stay therein after this wafer is processed;
α is a time of loading a wafer to or unloading the wafer to the robot in Step i;
μ is a time of the robot moving from one wafer-processing step to another;
α0 is a time of the robot unloading a wafer from the loadlock and aligning the same; and
Nm={1, 2, . . . , m} for a positive integer m.

2. The method of claim 1, wherein the determining of ω0, ω1, . . . , ωd for each of the system states Msd, d=0, 1, . . . , n−1, further comprises:
setting ω1=0, iεNd−1∪{0}, and ωd=max{dmax−Ψsd(d+1)1, 0} for the state Msd, 2≦d≦n−1 when n>2;

where:
dmax=max{iL, iεNd}; and
Ψsd(d+1)1=2(d+1)μ+(2+1)α+α0.

3. The method of claim 2, further comprising:
when max ≦iU and Ψ1≦iU, i=1, 2, . . . , n, setting, by the specialized processor, ω1=0, iεNn−1∪{0}, and ωn=max{max−Ψ1, 0} for use in a state Msn and thereafter, where Msn denotes that n instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up, and ωj,jε{0, 1, . . . n}, is a robot waiting time, used in a steady state of the cluster tool, for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module.

4. The method of claim 3, further comprising:
when [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅, determining, by the specialized processor, values of ω0d, ω1d, . . . , ωdd for each of the system states Msd, d=0, 1, . . . , n−1, where ω1d, jε{0, 1, . . . d}, is a robot waiting time used in the state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module;

wherein the determining of ω0d, ω1d, . . . , ωdd for each of the system states Msd, d=0, 1, . . . , n, comprises:
numerically optimizing the values of ω0d, ω1d, . . . , ωdd such that


Σd=1nΣi=0dωid 


is minimized subject to:
t110=α0+μ+α;
t121=t110+ω11+α;
ti1d=ti−1)2d+μ+α, 1≦i≦d+1 and 1≦d≦n−1;
ti2d=ti+2)1d+μ+ωid+α, 1≦i≦d−1 and 1≦d≦n−1;
t02d=td+1)1d+μ+ω0d+α0, 1≦d≦n−1;
td2d=t11d−1+μ+ωdd+α, 2≦d≦n−1;
ti1d=ti−1)2d+μ+α, 1≦i≦n and dε{n, n+1};
t01d=tn2d+μ+α, dε{n, n+1};
ti2d=t(i+2)1d+μ+ωid+α, 1≦i≦n−2 and dε{n, n+1};
t(n−1)2d=t01d+μ+ωn−1d+α, dε{n, n+1};
tn2d=t11d−1+μ+ωnd+α, dε{n, n+1};
t02d=t21d−1+μ+ω0d+α0, dε{n, n+1};
ωin=ωin−1, 0≦i≦n;
Σd=1nΣi=0dωid≧0; and
ai≦ti2d−α−ti1d−1≦ai+δi, 1≦i≦d and 1≦d≦n+1;


where:
ti1d denotes a time when the robot completes loading a wafer into Step i, iεNn∪{0};
ti2d a time when the robot completes unloading a wafer from Step i, iεNn;
t02d denotes a time when the robot completes unloading the loadlock and aligning the same; and
ωjd, jεNn and dε{n, n+1}, is a robot waiting time used in a state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module, the state Msd denoting that d instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up.

5. The method of claim 4, further comprising:
when [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅, setting, by the specialized processor, ωi=ωin, 0≦i<n−1, and ωn=max−Σi=0n−1ωi for use in the state Msn and thereafter.

6. A computer-implemented method for scheduling a cluster tool, the cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, n process modules each for performing a wafer-processing step with a wafer residency time constraint where the ith process module, iε{1, 2, . . . , n}, is used for performing Step i of the n wafer-processing steps for each wafer, and a specialized processor configured to control the robot, the loadlock, and the process modules through execution of the method, the method comprising:
when [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅, determining, by the specialized processor, values of ω0d, ω1d, . . . , ωdd for each of plural system states Msd, d=0, 1, . . . , n−1, of the cluster tool, where Msi, 1≦i ≦n−1 denotes that i instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up, and ωjd, jε{0, 1, . . . d}, is a robot waiting time used in the state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module;

wherein the determining of ω0d, ω1d, . . . , ωdd for each of the system states Msd, d=0, 1, . . . , n, comprises:
numerically optimizing the values of ω0d, ω1d, . . . , ωdd such that


Σd=1nΣi=0dωid 


is minimized subject to:
t110=α0+μ+α;
t121=t110+ω11+α;
ti1d=ti−1)2d+μ+α, 1≦i≦d+1 and 1≦d≦n−1;
ti2d=ti+2)1d+μ+ωid+α, 1≦i≦d−1 and 1≦d≦n−1;
t02d=td+1)1d+μ+ω0d+α0, 1≦d≦n−1;
td2d=t11d−1+μ+ωdd+α, 2≦d≦n−1;
ti1d=ti−1)2d+μ+α, 1≦i≦n and dε{n, n+1};
t01d=tn2d+μ+α, dε{n, n+1};
ti2d=t(i+2)1d+μ+ωid+α, 1≦i≦n−2 and dε{n, n+1};
t(n−1)2d=t01d+μ+ωn−1d+α, dε{n, n+1};
tn2d=t11d−1+μ+ωnd+α, dε{n, n+1};
t02d=t21d−1+μ+ω0d+α0, dε{n, n+1};
ωin=ωin−1, 0≦i≦n;
Σd=1nΣi=0dωid≧0; and
ai≦ti2d−α−ti1d−1≦ai+δi, 1≦i≦d and 1≦d≦n+1;


where:
iU=ai+4α+3μ+δi, iεNn\{1};
iU=a1+3α+α0+3μ+δ1;
iL=ai+4α+3μ, iεNn\{1};
iU=a1+3α+α0+3μ;
ti1d denotes a time when the robot completes loading a wafer into Step i, iεNn∪{0};
ti2d a time when the robot completes unloading a wafer from Step i, iεNn;
t02d denotes a time when the robot completes unloading the loadlock and aligning the same; and
ωjd, jεNn and dε{n, n+1}, is a robot waiting time used in a state Msd for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module, the state Msd denoting that d instances of a wafer unloading from the robot to any one of the n process modules have occurred since system start-up;
ai, iεNn, is a time that a wafer is processed in the ith process module;
δi, is the wafer residency time constraint of Step i, given by a pre-determined longest time for which a wafer in the ith process module is allowed to stay therein after this wafer is processed;
α is a time of loading a wafer to or unloading the wafer to the robot in Step i;
μ is a time of the robot moving from one wafer-processing step to another;
α0 is a time of the robot unloading a wafer from the loadlock and aligning the same; and
Nm={1, 2, . . . , m} for a positive integer m.

7. The method of claim 6, further comprising:
when [1L, 1U]∩[2L, 2U]∩ . . . ∩[nL, nU]=∅, setting, by the processor, ωi=ωin, 0≦i<n−1, and ωn=max−Σi=0n−1ωi for use in the state Msn and thereafter, where ωj, jε{0, 1, . . . n}, is a robot waiting time, used in a steady state of the cluster tool, for the robot to wait before unloading a wafer in Step j from the robot to the (j+1)th process module.

8. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 1.

9. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 2.

10. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 3.

11. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 4.

12. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 5.

13. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 6.

14. A cluster tool comprising a single-arm robot for wafer handling, a loadlock for wafer cassette loading and unloading, and plural process modules each for performing a wafer-processing step with a wafer residency time constraint, wherein the cluster tool further comprises one or more specialized processors configured to execute a process of scheduling the cluster tool according to the method of claim 7.