Patent Application
20170115651
The present invention relates to a system and method for determining an optimized schedule of a production line and particularly, although not exclusively, to a system and method for determining an optimized schedule of a semiconductor production line with a hybrid multi-cluster tool having both single arm tools and dual arm tools.
Cluster tools have been widely used to process objects such as semiconductor wafers for fabricating microelectronic components. Generally, a cluster tool is a robotic processing system that includes loading and unloading modules and a number of different processing modules (PM) placed around a central automated handling unit with one or more robotic arms. It operates by using the robotic arm to transfer the object to be processed from the loading/unloading loadlocks to different processing modules in sequence so as to perform different mechanical or chemical processing, and then back to the loading/unloading loadlocks. The time for the object to stay in the processing module depends on the time needed for a specific processing procedure.
Multi-cluster tools formed by interconnecting cluster tools have also been used for processing and manufacturing objects. Comparing with a single cluster tool, a multi-cluster tool may provide more PMs for manufacturing products of high degree of complexity. In some cases, these tools may also simultaneously process more than one product to provide improved processing efficiency.
Although multi-cluster tools appear to offer some advantages over single-cluster tools, they may include combinations of different types of single-cluster tools (e.g., both single-arm cluster tools and dual-arm cluster tools) which are of different constructions and different scheduling mechanisms. As a result, it would be difficult to determine an optimal scheduling of these tools for maximum operation efficiency.
In accordance with a first aspect of the present invention, a method is provided for determining an optimized production schedule of a production line, the production line comprises a hybrid multi-cluster tool formed by a plurality of single-arm tools and dual-arm tools interconnected with each other; wherein each single-arm tool includes one robotic arm for manipulating an object and at least one processing module for processing the object or a buffering module for holding the object, and each dual-arm tool includes two robotic arms for manipulating an object and at least one processing module for processing the object or a buffering module for holding the object; single-arm tools and dual-arm tools are connected with each other through at least one buffering module; the method comprising the steps of: determining time for individual operations of the robotic arm and the processing module in the plurality of single-arm tools and dual-arm tools; determining robot waiting time of the single-arm tools and dual-arm tools based on the time for individual operations and different connection relationships of the plurality of single-arm tools and dual-arm tools; determining whether the optimized production schedule exists using the determined robot waiting time, wherein the optimized production schedule only exists if the hybrid multi-cluster tool is process-dominant where robot activity time of the plurality of single-arm tools and dual-arm tools is substantially shorter than processing time at the processing module; and determining the optimized production schedule if the optimized production schedule exists.
In one embodiment of the first aspect, the step of determining the time for individual operations of the robotic arm and the processing module in the plurality of single-arm tools and dual-arm tools comprises the step of determining one or more of: a time required for the robotic arm of the single-arm tool to load or unload the object; a time required for the robotic arms of the dual-arm tool to swap; a time required for the robotic arm of the single-arm tool or the dual-arm tool to move while holding the object; a time required for the robotic arm of the single-arm tool to move without holding an object; a time required for processing the object in the processing module of the single-arm tool or the dual-arm tool; a time required for resting the object in the processing module of the single-arm tool or the dual-arm tool; a time required for the robotic arm of the single-arm tool to wait before unloading the object; a time required for the robotic arms of the dual-arm tool to wait at the processing module of the dual-arm tool; and a time required for the robotic arms of the dual-arm tool to wait during swap at the processing module of the dual-arm tool.
In one embodiment of the first aspect, the different connection relationships comprise: an upstream downstream relationship that includes an upstream single-arm tool and downstream single-arm tool connection, an upstream single-arm tool and a downstream dual-arm tool connection, an upstream dual-arm tool and a downstream single-arm tool connection, or an upstream dual-arm tool and a downstream dual-arm tool connection; and a number relationship that includes a number of adjacent single-arm or the dual-arm tools of which the respective single-arm tool or the dual-arm tool is connected to.
In one embodiment of the first aspect, the step of determining whether the optimized production schedule exists using the time for individual operations and the robot waiting time comprises: calculating an activity time of each of the single-arm tools and dual-arm tools in a production cycle without robot waiting using the time for individual operations; calculating a fundamental period of each of the single-arm tools and dual-arm tools using the time for individual operations; determining a maximum fundamental period from the calculated fundamental periods; determining a robot waiting time of each of the plurality of single-arm tools and dual-arm tools using the maximum fundamental period; evaluating the robot waiting time determined so as to determine if the optimized production schedule exists.
In one embodiment of the first aspect, the step of determining the optimized production schedule comprises setting an optimal robot waiting time for each of the plurality of single-arm tools and dual-arm tools based on the determination results without interfering with the operation of the buffering modules.
In one embodiment of the first aspect, the hybrid multi-cluster tool has a non-cyclic treelike structure with at least one of the single arm tools and dual arm tools being connected with three or more adjacent single arm tools and dual arm tools.
In one embodiment of the first aspect, the optimized production schedule comprises a shortest time for completion of a cycle of production of the object.
In one embodiment of the first aspect, the object is a semiconductor and the production line is a semiconductor manufacturing line.
In accordance with a second aspect of the present invention, there is provided a computerized system arranged to determine an optimized production schedule of a production line, the production line comprises a hybrid multi-cluster tool formed by a plurality of single-arm tools and dual-arm tools interconnected with each other; wherein each single-arm tool includes one robotic arm for manipulating an object and at least one processing module for processing the object or a buffering module for holding the object, and each dual-arm tool includes two robotic arms for manipulating an object and at least one processing module for processing the object or a buffering module for holding the object; each single-arm tools and dual-arm tools are connected with each other through at least one buffering module; the computerized system comprises a Petri-Net (PN) model computation module arranged for: determining time for individual operations of the robotic arm and the processing module in the plurality of single-arm tools and dual-arm tools; determining the robot waiting time of the single-arm tools and dual-arm tools based on the time for individual operations and different connection relationships of the plurality of single-arm tools and dual-arm tools; determining whether the optimized production schedule exists using the determined robot waiting time, wherein the optimized production schedule only exists if the hybrid multi-cluster tool is process-dominant where the robot activity time of the plurality of single-arm tools and dual-arm tools is substantially shorter than processing time at the processing module; determining the optimized production schedule if the optimized production schedule exists.
In one embodiment of the second aspect, the step of determining the time for individual operations of the robotic arm and the processing module in the plurality of single-arm tools and dual-arm tools comprises determining one or more of: a time required for the robotic arm of the single-arm tool to load or unload the object; a time required for the robotic arms of the dual-arm tool to swap; a time required for the robotic arm of the single-arm tool or the dual-arm tool to move while holding the object; a time required for the robotic arm of the single-arm tool to move without holding the object; a time required for processing the object in the processing module of the single-arm tool or the dual-arm tool; a time required for resting the object in the processing module of the single-arm tool or the dual-arm tool; a time required for the robotic arm of the single-arm tool to wait before unloading the object; a time required for the robotic arms of the dual-arm tool to wait at the processing module of the dual arm tool; and a time required for the robotic arms of the dual-arm tool to wait during swap at the processing module of the dual arm tool.
In one embodiment of the second aspect, the different connection relationships comprise: an upstream downstream relationship that includes an upstream single-arm tool and downstream single-arm tool connection, an upstream single-arm tool and a downstream-dual arm tool connection, an upstream dual-arm tool and a downstream single-arm tool connection, or an upstream dual-arm tool and a downstream dual-arm tool connection; and a number relationship that includes a number of adjacent single-arm or the dual-arm tools of which the respective single-arm tool or the dual-arm tool is connected to
In one embodiment of the second aspect, the Petri-Net (PN) model computation module is arranged to determine whether the optimized production schedule exists using the time for individual operations and the waiting time by: calculating an activity time of each of the single-arm tools and dual-arm tools in a production cycle without robot waiting using the time for individual operations; calculating a fundamental period of each of the single-arm tools and dual-arm tools using the time for individual operations;
determining a maximum fundamental period from the calculated fundamental periods; determining the robot waiting time of each of the plurality of single-arm tools and dual-arm tools using the maximum fundamental period; evaluating the waiting time determined so as to determine if the optimized production schedule exists.
In one embodiment of the second aspect, the Petri-Net (PN) model computation module is arranged to determine the optimized production schedule by setting an optimal robot waiting time for each of the plurality of single-arm tools and dual-arm tools based on the determination results without interfering with the operation of the buffering modules.
In one embodiment of the second aspect, the hybrid multi-cluster tool has a non-cyclic treelike structure with at least one of the single-arm tools and dual-arm tools being connected with three or more adjacent single-arm tools and dual-arm tools.
In one embodiment of the second aspect, the optimized production schedule comprises a shortest time for completion of a cycle of production of the object.
In one embodiment of the second aspect, the object is a semiconductor and the production line is a semiconductor manufacturing line.
It is an object of the present invention to address the above needs, to overcome or substantially ameliorate the above disadvantages or, more generally, to provide a.
Embodiments of the present invention will now be described, by way of example, with reference to the accompanying drawings in which:
Multi-cluster tools generally comprise a number of single-cluster tools connected by buffering modules with a capacity of one or two, and they may have linear or non-cyclic treelike topology. A multi-cluster tool with K (≧2) individual cluster tools is generally referred to as a K-cluster tool. Furthermore, if these individual tools include both single and dual-arm tools, then the multi-cluster tool is of a hybrid type.
An exemplary treelike hybrid 9-cluster tool 100 linked by buffering modules 106 is illustrated in
In the present invention, a Petri net (PN) model has been developed for determining and evaluating an optimized schedule of a production line formed by a treelike hybrid multi-cluster tool, such as that shown in
In one embodiment, the present invention provides a method for determining an optimized production schedule of a production line, the production line comprises a hybrid multi-cluster tool formed by a plurality of single-arm tools and dual-arm tools interconnected with each other; wherein each single-arm tool includes one robotic arm for manipulating an object and at least one processing module for processing the object or a buffering module for holding the object, and each dual-arm tool includes two robotic arms for manipulating an object and at least one processing module for processing the object or a buffering module for holding the object; the single-arm tools and dual-arm tools are connected with each other through at least one buffering module; the method comprising the steps of: determining time for individual operations of the robotic arm and the processing module in the plurality of single-arm tools and dual-arm tools; determining waiting time of the single-arm tools and dual-arm tools based on the time for individual operations and different connection relationships of the plurality of single-arm tools and dual-arm tools; determining whether the optimized production schedule exists using the determined waiting time, wherein the optimized production schedule only exists if the hybrid multi-cluster tool is process-dominant where the robot activity time of the plurality of single-arm tools and dual-arm tools is substantially shorter than processing time at the processing module; and determining the optimized production schedule if the optimized production schedule exists.
The following assumptions are made in the embodiment of the present invention:
1) a buffering module has no processing function and its capacity is one;
2) for each step, only one process module is configured for product (e.g., semiconductor wafer) processing and, only one product can be processed in a process module at a time;
3) only one type of product is processed with an identical recipe, and they visit a process module only once except entering a buffering module at least twice;
4) the robots' task time is constant; and
5) besides buffering module, there is at least one processing step in each individual tool except the fork and leaf tool. The fork tool may have no processing step and the leaf tool has at least two process modules.
Let n={1, 2, . . . , n}, Ωn={0}∪
n, and Ci and Ri, i∈
K, denote the i-th cluster tool and its robot, respectively, where C1 with load-locks is the head tool, Ci, i≠1 is the leaf tool if it connects with only one adjacent tool, and Ci, i≠1 is the fork tool if it connects at least three adjacent tools. Further, let L={i|Ci is a leaf tool} be the index set of leaf tools. For the tool shown in
K\{1}, with Ck/Ci being the upstream/downstream one where i>k. It should be noted that, however, k and i are not necessary in a consecutive order as shown in
K and j∈
r.
A buffering module connecting Ck and Ci can be considered as the outgoing module for Ck and the incoming module for Ci respectively. Let n[i] represent the index for the last step in Ci, i∈K. Then, the number of steps in Ci, including the incoming and outgoing steps, is n[i]+1. Further, let f[k] (1≦f[k]≦n[k]) denote the number of outgoing modules in Ck, k∉L, and these outgoing modules are denoted as PSk(b[k]_1), PSk(b[k]_1), . . . , and PSk(b[k]_f[k]) with b[k]_1<b[k]_2< . . . <b[k]_f[k]. When f[k]>1, Ck is a fork tool, otherwise it is not. It should be pointed out that b[k]_1, b[k]_2, . . . and b[k]_f[k] may not be in a consecutive order, in other words, b[k]_2=b[k]_1+1 may not hold. The incoming module for Ci is denoted as PSi0. In this way, the n[k]+1 steps in Ck are denoted as PSk0, PSk1, . . . , PSk(b[k]_1), PSk((b[k]_1)+1), . . . , PSk(b[k]_2), . . . , PSk(n[k]), respectively. Notice that b[k]_1 can be 1 if it is Step 1 and b[k]_f[k] can be n[k] if it is the last step. Hence, the route of a product in
To model a hybrid K-cluster tool, the behavior of individual tools and buffering modules need to be modeled. Since the behavior of a single-arm tool is different from that of a dual-arm tool, their models are presented separately.
For Ci, regardless of whether it is a single or dual-arm tool, places ri and pij model Ri and PSij, i∈K, j∈Ωn[i], respectively. For a single-arm tool Ci, as shown in
For a dual-arm tool Ci, as shown in
For a buffering module that links Ck and Ci, k∉L, i∈K\{1}, with Ci being the downstream one, there are four different cases: 1) both Ck and Ci are dual-arm tools (D-D); 2) Ck and Ci are single and dual-arm tools, respectively (S-D); 3) Ck and Ci are dual and single-arm tools, respectively (D-S); and 4) both Ck and Ci are single-arm tools (S-S). As discussed above, the buffering module connecting Ck and Ci can be denoted as PSk(b[k]_q) for Ck, or the q-th OM, 1≦q≦f[k], in Ck. This buffering module can be also denoted as PSi0 in Ci. Then, it is modeled by pk(b[k]_q) and pi0 for Ck and Ci, respectively, with pk(b[k]_q)=pi0 and K(pk(b[k]_q))=K(pi0)=1. The PN models of these four different cases are shown in
With the PN structure developed, by putting a 17-token representing a virtual product (not real one), the initial marking M0 of the PN model is set as follows.
If C1 is a single-arm tool, set M0(r1)=0; M0(p10)=n, representing that there are always products to be processed; M0(p1(b[1]_1)=0 and M0(p1j)=1, j∈n[1]\{b[1]_1}; M0(z1j)=M0(d1j)=0, j∈Ωn[1]; M0(q1j)=0, j∈Ωn[1]\{(b[1]_1)−1}; and M0(q1((b[1]_1−1))=1, meaning that R1 is waiting at PS1((b[1]_1)−1) for unloading a product there.
If C1 is a dual-arm tool, set M0(r1)=1; M0(P10)=n; M0(p1j)=1, j∈n[1]; M0(q1j)=M0(d1j)=0, j∈Ωn[1]; M0(z1j)=0, j∈Ωn[i]\{b[1]_1}; and M0(z1(b[1]_1))=1, meaning that R1 is waiting at PS1(b[1]_1) for unloading a product there.
For Ci, i∈K\{1}, if it is a single-arm tool, set M0(ri)=0; M0(zij)=M0(dij)=0,j∈Ωn[i]; M0(pi1)=0 and M0(pij)=1, j∈
n[i]\{1}; M0(qij)=0, j∈
n[i]; and M0(qi0)=1, implying that Ri is waiting at PSi0 for unloading a product there. If Ci, i∈
K\{1}, is a dual-arm tool, set M0(ri)=1; M0(pij)=1,j∈
n[i]; M0(qij)=M0(dij)=0,j∈Ωn[i]; M0(zij)=0, j∈
n[i]; and M0(zi0)=1, implying that Ri is waiting at PSi0 for unloading a product there. It should be pointed out that, for any adjacent Ck and Ci, k∈
K-1, at M0, it is assumed that the token in pi0 enables uk(b[k]_q), but not ui0.
In
Definition 2.1: Define the color of a transition ti as C(ti)={ci}. From Definition 2.1, the colors for ui0 and uk(b[k]_q) are ci0 and ck(b[k]_q), respectively. Then, the color for a token can be defined as follows.
Let ti be the set of places of transition ti. We present the following definition.
Definition 2.2: A token in p∈ti that enables ti has the same color of ti's, i.e., {ci}. For example, a token that enters pi0 by firing tk(b[k]_q) has color ci0, while the one that enters pi0 by firing ti0 has color ck(b[k]_q). In this way, the PN is made conflict-free.
Based on the above discussion, the PN model for a dual-arm fork tool Ci is deadlock-free but the PN model for a single-arm non-fork tool Ci is deadlock-prone. To make the model deadlock-free, the following control policy is introduced.
Definition 2.3: For the PN model of a single-arm tool Ci, i∈K, at marking M, transition yij, j∈Ωn[i]\{n[i], (b[i]_q)−1}, q∈
f[i], is control-enabled if M(pi(j+1))=0; yi((b[i]_q)−1) is control-enabled if transition t(i((b[i]_q)−1) has just been executed; yi(n[i]) is control-enabled if transition ti1 has just been executed.
By using this control policy, the PN for a single-arm non-fork tool Ci is deadlock-free. For a single-arm fork tool Ci, assume that, at M, ui0 is enabled. After ui0 fires, Ri performs the following activities: <xi0→ti1→yi(n[i])→ui(n[i])→xi(n[i])→ti0→yi(n[i]−1)M(pi(n[i]))=0)→ . . . →yi((b[i])_q) (M(pi((b[i]_q)+1)))=0)→ui((b[i]_q)→xi((b[i]_q)→ti((b[i]_q)+1)→yi((b[i]_q)−1) ti((b[i]_q)+1) has just been executed)→ . . . yi0(M(pi1)=0)→ui0→xi0>. In this way, a cycle is completed and this process can be repeated. Thus, there is no deadlock. This implies that, by the control policy, the PN for the system is made deadlock-free.
In the PN model developed above, with transitions and places representing activities that take time, time is associated with both transitions and places. Since the activity time taken by single-arm and dual-arm tool is different, it should be modeled for both tools. For both types of tools, the robot activity time is modeled such that the time for the robot to move from one step to another is same, so is the time for the robot to load/unload a product into/from a process module. In the present invention, the activity time is modeled as shown in Table I.
With the PN model, the following presents the temporal properties of individual tools such that a schedule can be parameterized by robot waiting time. For a single-arm tool Ci, i∈K, the time taken for processing a product at Step j, j∈
n[i], is
θij=αij+4λi+3μi+ωi(j−1). (3.1)
For Step 0, as αi0=0,
αi0=αi0+4λi+3μi+ωi(n[i])=4λi+3μi+ωi(n[i]) (3.2)
With the robot waiting time being removed, the time taken for completing a product at Step j is:
ξij=αij+4λi+3μi, j∈Ωn[i]. (3.3)
To make a schedule feasible, a product should stay at process module PMij for τij (≧αij) time units and, by replacing αij with τij, the cycle time at Step j in Ci is:
πij=τij+4λi+3μi+ωi(j−1), j∈n[i]. (3.4)
and
πi0=τi0+4λi+3μi+ωi(n[i]). (3.5)
The robot cycle time for a single-arm tool Ci is:
ψi=2(n[i]+1)(λi+μi)+Σj=0n[i]ωij=ψi1+ψi2. (3.6)
where ψi1=2(n[i]+1) (λi+μi) is the robot's activity time in a cycle without waiting and ψi2=Σj=0n[i]ωij is the robot waiting time in a cycle.
For a dual-arm tool Ci, i∈K, is the time needed for completing a product at Step j, j∈Ωn[i], in Ci is:
ξij=αij+λi (3.7)
Similarly, by replacing αij with τij in (3.7), the cycle time at Step j, j∈Ωn[i], in Ci is:
πij=τij+λi. (3.8)
The robot cycle time for a dual-arm tool Ci is:
ψi=(n[i]+1)(λi+μi)+Σj=0n[i]ωij=ψi1+ψi2. (3.6)
where ψi1=(n[i]+1)(λi+μi) is the robot cycle time without waiting and ψi2=Σj=0n[i]ωij is the robot's waiting time in a cycle.
As the manufacturing process in each Ci, i∈K, is serial, in the steady state, the productivity for each step must be same. Thus, Ci should be scheduled such that
πi=πi0=πi1= . . . =πi(n[i])=ψi. (3.10)
Notice that both πi and ψi are functions of ωij's, which means that the schedule for Ci, i∈K, is parameterized by robots' waiting time. Based on the schedule for Ci, i∈
K, to schedule a treelike hybrid K-cluster tool, the key is to determine ωij's such that the activities of the multiple robots are coordinated to act in a paced way.
Let Πi=max{ξi0, ξi1, . . . , ξi(n[i]), ψi1} be the fundamental period (FP) of Ci, i∈K. If Πi=max{ξi0, ξi1, . . . , ξi(n[i])}, Ci is process-bound; otherwise it is transport-bound. Let Π=max{Π1, Π2, . . . , ΠK} and assume that Π=Πh, 1≦h≦K, or Ch is the bottleneck tool. As mentioned above, Ch is process-bound in a process-dominant treelike hybrid K-cluster tool. Let Θ denote the cycle time for the system. With πi being the cycle time of Ci, to obtain a one-product cyclic schedule for a process-dominant treelike hybrid multi-cluster tool, every individual tool must have the same cycle time and it should be equal to the cycle time of the system, i.e.,
Θ=πi≧Π, ∀i∈K. (4.1)
Based on (4.1), to find a one-product cyclic schedule is to schedule the individual tools such that they can act in a paced way. Since both πi and ψi are functions of ωij's, given Θ(≧Π) as cycle time, a one-product cyclic schedule can be obtained by determining ωij's for each tool Ci, i∈K, j∈
n[i]. The individual tools are scheduled to be paced, if and only if for any adjacent pair Ck and Ci, k∉L, i∉{1}, linked by PSk(b[k]_q), 1≦q≦f[k], at any marking M: 1) whenever Ri (Rk) is scheduled to load a product (token) into pi0 (pk(b[k]_q)), ti0 (tk(b[k]_q)) is enabled; and 2) whenever Ri (Rk) is scheduled to unload a product (token) from pi0 (pk(b[k]_q), μi0 (μk(b[k]_q)) is enabled. Given Θ=Π, if a one-product cyclic schedule is found, the lower bound of cycle time is achieved.
B. Existence of a One-Product Cyclic Schedule with the Lower Bound of Cycle Time (OSLB)
For a process-dominant linear hybrid K-cluster tool, the conditions under which an OSLB exists are given as follows.
Lemma 4.1: For a process-dominant linear hybrid K-cluster tool, an OSLB exists, if and only if for any adjacent tool pair Ci and Ci+1, i∈K-1, the following conditions are satisfied by determining ωij's and ω(i+1)l's, j∈Ωn[i] and l∈Ωn[i+1].
πij=π(i+1)\=Π, j∈Ωn[i] and l∈Ωn[i+1]. (4.2)
If C and Ci+1 are D-S case
Π−λi≧4λi+1+3μi+1+ω(i+1)(n[i+1]). (4.3)
If C and Ci+1 are S-S case
Π−(4λi+3μi+ωi((b[i]_1)−1))≧4λi+1+3μi+1ω(i+1)(n[i+1]). (4.4)
Lemma 4.1 states that, to check the existence of an OSLB, one needs to examine the D-S and S-S cases only. By Lemma 4.1, given Θ=Π, if conditions (4.2)-(4.4) are satisfied, a linear hybrid K-cluster tool can be scheduled such that the individual tools are paced. Notice that Condition (4.2) says that the cycle time of each individual tool is same, while Conditions (4.3) and (4.4) involve the operations of buffering modules only. Hence, although a treelike hybrid K-cluster tool is structurally different from a linear hybrid K-cluster tool, the similar conditions can be obtained for the existence of an OSLB. The following result are obtained.
Theorem 4.1: For a process-dominant treelike hybrid K-cluster tool, an OSLB exists, if and only if for any adjacent pair Ck and Ci, k∉L, i∉{1}, linked by PSk(b[k]_q), the following conditions are satisfied by determining ωkj's and ωil's, j∈Ωn[k] and l∈Ωn[i] such that
πkj=πil=Π, j∈Ωn[k]and l∈Ωn[i]. (4.5)
If Ck and Ci are D-S case
Π−λk≧4λi+3μi+ωi(n[i]). (4.6)
If Ck and Ci are S-S case
π−(4λk+3μk+k+ωk((b[k]_q)−1)≧4λi+3μi+ωi(n[i]). (4.7)
Proof: With the fact that a linear hybrid K-cluster tool is a special case of a treelike hybrid K-cluster tool, it follows from Lemma 4.1 that Conditions (4.5)-(4.7) must be necessary. Thus, it is necessary to show the “if” part only. If a Ck in a treelike hybrid K-cluster tool is not a fork, it acts just as an individual tool in a linear hybrid K-cluster tool. Thus, it is only necessary to examine the fork tools.
Assume that Ck is a single-arm fork tool and Ck_1, Ck_2, . . . , Ck_f[k] are the set of its adjacent downstream tools. Based on M0, it can be assumed that the tool is at marking M with M(rk)=0; M(zkj)=M(dkz)=0, j∈Ωn[k]; M(pk(b[k]_1))=0 and M(pkj)=1, j∈n[k]\{b[k]_1}; M(qkj)=0,j∈Ωn[k]\{(b[k]_1)−1}; and M(qk((b[k]_1)−1))=1, implying that Rk is waiting at PSk((b[k]_1)−1) for unloading a product there. Then, if (4.5) holds, the tool can be scheduled as follows. For Ck, Rk unloads (firing μk((b[k]_1)−1)) a product from PSk((b[k]_1)−1)), and then moves (xk((b[k]_1)−1)) to PSk(b[k]_1) and loads (tk(b[k]_1)) the product into it. After firing tk(b[k]_1), u(k_1)0 fires immediately. Then, with a backward strategy, after some time, Rk comes to PSk((b[k]_f[k])−1) and uk((b[k]_f[k])−1) fires to unload the product in PSk((b[k]_f[k])−1). Then, Rk moves (xk((b[k]_f[k])−1) to PSk(b[k]_f[k]) and loads (tk(b[k]_f[k])) the product into it. After firing tk(b[k]_f[k]), uk_f[k])0 fires immediately. Similarly, the tool can be scheduled such that after firing tk(b[k]_q), q=2, 3, . . . f[k]−1, ends, uk_q) 0 fires immediately. In this way, when Rk comes to a qk(b[k]_1) again, if Ck_1 is a single-arm tool and (4.7) is satisfied, Ck
Notice that the conditions given in Theorem 4.1 are the functions of robots' waiting time of single-arm tools and have nothing to do with dual-arm tools. Thus, to schedule a treelike hybrid K-cluster tool is to determine ωij's for single-arm tools Ci, i∈K, j∈Ωn[i], only. By this observation, for a dual-arm tool Ci, i∈
K, simply set ωi0=ψi2=Π−ψi1. Observe (4.6) and (4.7), it can be concluded that, for a single-arm tool Ci, i∈
K; to make the conditions given in Theorem 4.1 satisfied, it is necessary to set ωij's such that ωi((b[i]_q)−1), q∈
f[i], and ωi(n[i]) are as small as possible. With this as a rule, the following discussion covers how to determine ωij's for the single-arm tools sequentially from the leaves to the head one to find an OSLB.
To ease the presentation, let h(z) be an arbitrary function and define Σxyh(z)=0 if x>y. It follows from the discussion of the last section that, for a single-arm tool Ci, it is necessary to assign Ri's idle time ψi2=Π−2(n[i]+1)(μi+λi) into ωij's. For a dual-arm tool Ci, i∈L, set ωij=0,j∈n[i], and ωi0=ψi2=Π−ψi1. For a single-arm tool Ci, i∈L, set ωij=min{Π−(4λi+3μi+αi(j+1)), Π−ψi1−Σd=0j−1 ωid}, j∈Ωn[i]−1 such that, ωi(n[i])=Π−ψi1−Σj=0n[i]−1ωij is minimized. Then, for the adjacent upstream tool Ck of Ci with Ck and Ci being linked by PSk(b[k]_1), there are altogether four cases.
For Case 1), Ck is a dual-arm non-fork tool, check if Π−λk−4λi−3μi−ωi(n[i])≧0. If not, there is no OSLB, otherwise for j∈n[k], set ωkj=0 and ωk0=ψk2=Π−ψk1.
For Case 2), Ck is a single-arm non-fork tool, check if Π−(4λk+3μk)−(4λi+3μi+ωi(n[i]))≧0. If not, there is no OSLB, otherwise set ωk[(b[k]_1)−1]=min{Π−(4λk+3μk)−(4λi+3μi+ωi(n[i])), Π−ψk1}, ψkj=min{Π−(4λk+3μk+αk(j+1)), Π−ψk1−ωk((b[k]_1)−1)−Σd=0j−1ωkd (d≠(b[k]_1)−1)}, j∈Ωn[k]\{n[k], (b[k]_1)−1}, and ωk(n[k])=Π−ψk1−ωk((b[k]_1)−1)−Σj=0n[k]−1 ωkj (j≠(b[k]_1)−1).
For Case 3), Ck is a single-arm fork tool, check if Π−(4λk+3μk)−(4λi+3μi+ωi(n[i]))≧0. If not, there is no OSLB, otherwise set ωk((b[k]_q)−1)=min{Π−(4λk+3μk)−(4λi+3μi+ωi(n[i]), Π−ψk1−Σd=1q−1ωk((b[k]_d)−1)}, ωkj=min{Π−(4λk+3μk+αk(j+1)), Π−ψk1−Σd=1f[k]ωk((b[k]_d)−1)−Σn=0j−1 ωkn (n∉{(b[k]_h)−1|h∈f[k]})}, j∈Ωn[k]\{n[k]}\{(b[k]_h)−1|h∈
f[k]}, and ωk(n[k])=Π−ψk1−Σd=1f(k)ωk((b[k]_d)−1)−Σj=0n[k]−1ωkj (j∉{(b[k]_h)−1|h∈
f[k]}).
For Case 4), Ck is a dual-arm fork tool, check if Π−λk−4λi−3μi−ωi(n[i])≧0. If not, there is no OSLB, otherwise set ωkj=0, j∈n[k], and ωk0=ψk2=Π−ψk1.
Then, do the same for the adjacent upstream tool of Ck and this process is repeated till C1 such that an OSLB is obtained if it exists, or the process terminates at a Ck with no such a schedule. Based on the above discussion, let Q be a binary variable indicating the existence of an OSLB. Then, the present invention utilises the following algorithm to test the existence of an OSLB and find it if it exists.
Algorithm 4.1: Test the existence of an OSLB for a treelike hybrid K-cluster tool
Using Algorithm 4.1, if Q=1 is returned, an OSLB is found for a process-dominant treelike hybrid K-cluster tool, otherwise there is no such a schedule. Based on Algorithm 4.1, to check if LB of cycle time can be achieved for a treelike hybrid K-cluster tool, it is necessary to set the robot waiting time from the leaf to the head one by one. In the worst case when Q=1 is returned, it is only necessary to set the robot waiting time for each individual tool once and check condition 4.6 or 4.7 for each buffering module once. Let H=max(n[1]+1, n[2]+1, . . . , n[K]+1). Since it is necessary need to set ωij for each Step j in Ci, i∈K, j∈Ωn[i], there are at most H×K operations in setting the robot waiting time. Meanwhile, there are K−1 buffering modules for checking Condition 4.6 or 4.7. Hence, there are at most (H+1)×K−1 operations altogether. With H and K being bounded to known constants, the computational complexity of Algorithm 4.1 is bounded by a constant and thus it is very efficient.
Example 1 is a treelike hybrid 3-cluster tool, where dual-arm tool C1 is a fork tool and its adjacent downstream tools are single-arm tools C2 and C3. In this example, their activity time is as follows. For C1, (α10, α11, α12, α13, λ1, μ1) (0, 77, 0, 0, 13, 1); for C2, (α20, α21, α22, λ2, μ2)=(0, 65, 69, 4, 1); for C3, (α30, α31, α32, λ3, μ3)=(0, 61, 55, 6, 1).
For C1, ξ10=13 s, ξ11=90 s, ξ12=13 s, ξ13=13 s, ψ11=(n[1]+1)×(λ1+μ1)=4×14=56 s, Π=90 s and it is process-bound. For C2, ξ20=19 s, ξ21=84 s, ξ22=88 s, ψ21=2(n[2]+1)×(λ2+μ2)=6×5=30 s and Π2=88 s. For C3, ξ30=27 s, ξ31=88 s, ξ32=82 s, ψ31=2(n[3]+1)×(λ3+μ3)=6×7=42 s and Π3=88 s. This 3-cluster tool is process-dominant with Π1>Π2=Π3 and let Π=Π1=π1=π2=π3=90 s. By Algorithm 4.1, the robots' waiting time is set as ω20=6 s, ω21=2 s, and ω22=Π−ψ21−ω20−ω21=90−30−6−2=52 s; ω30=2 s, ω31=8 s, and ω32=Π−ψ31−ω30−ω31=90−42−2−8=38 s. Then, for C1, as Π−(4λ2+3μ2+ω22)−λ1=90−(19+52)−13=6>0 and, Π−(4λ3+3μ3+ω32)−λ1=90−(27+38)−13=12>0, from Algorithm 4.1, a cyclic schedule with lower bound of cycle time can be obtained by setting ω10=34 s, ω11=ω12=ω13=0. Simulation is used to verify the correctness of the schedule as shown in Table II.
From Table II, it can be seen that, by firing u20 immediately after R1's swap operation at p12, when R2 (R1) is scheduled to unload a product from p20(p12), M(p20)=1(M(p12)=1); when R2(R1) is scheduled to load a product into p20(p12), M(p20)=0(M(p12)=0). Similarly, R3(R1) can act as scheduled without being delayed by a buffering module if u30 fire immediately after R1's swap operation at p13. The Gantt chart for the schedule obtained is shown in
Example 2 is a treelike hybrid 5-cluster tool, where C2 is a fork tool and its adjacent tools are C1, C3 and C5. C5 and C4 that is adjacent to C3 are leaf tools. Furthermore, C1 is a dual-arm tool and the others are all single-arm tools. Their activity time is as follows: for C1, (α10, α11, λ1, μ1) (0, 61.5, 0, 28.5, 0.5); for C2, (α20, α21, α22, λ2, μ2) (0, 0, 0, 10, 1); for C3, (α30, α31, α32, α33, λ3, μ3) (0, 56, 0, 58, 7, 1); for C4, (α40, α41, α42, α43, λ4, μ4) (0, 56, 66, 65, 5, 1); and for C5, (α50, α51, α52, λ5, μ5)=(0, 48, 50, 6, 1).
For C1, ξ10=28.5 s, ξ11=90 s, ξ12=28.5 s, ψ11, ψ11=(n[1]+1)×(λ1+μ1)=3×29=87 s, Π=90 s, and it is process-bound. For C2, ξ20=43 s, μ21=43 s, ξ22=43 s, ψ21=2(n[2]+1)×(λ2+μ2)=2×3×11=66 s, and Π2=66 s. For C3, ξ30=31 s, ξ31=87 s, ξ32=31 s, ξ33=89 s, ψ31=2(n[3]+1)×(λ3+μ3)=2×4×8=64 s, and Π3=89 s. For C4, ξ40=23 s, ξ41=79 s, ξ42=89 s, ξ43=88 s, ψ41=2(n[4]+1)×(λ4+μ4) 2×4×6−48 s, and Π4=89 s. For C5, ξ50=27 s, ξ51=75 s, ξ52=77 s, ψ51=2(n[5]+1)×(λ5+μ5)=2×3×7=42 s, and Π5=77 s. This is shown that it is process-dominant with Π=Π1=90 s. Next, let Π=π1=π2=π3=π4=π5=90 s.
By Algorithm 4.1, for C4, ω40=11 s, ω41=1 s, ω42=2 s, and ω43=Π−ψ41−ω40−ω41−ψ42=90−48−11−1−2=28 s. For C5, ψ50=15 s, ω51=13 s, and ω32=Π−ω51−ω50−ω51=90−42−15−13=20 s. For C3, since Π−(4λ4+3μ4+ω43)−(4λ3+3μ3)=90−(23+28)−31=8>0, set ω31=min{8,Π−ψ31}=min{8,26}=8 s, ω30=min{90−ξ31,Π−ψ31−ω31}=min{3,26−8}=3 s, ω32=min{90−ξ33,Π−ψ31−ω31−ω30}=min{1,26−8−3}=1 s, and ω33=Π−ψ31+ω31−ω32=26−8−3−1=14 s. For C2, Π−(4λ3+μ3+ω33)−(4λ2+3μ2)=90−(31+14)−43=2>0, set ω20=min{2,Π−ψ21}=min{2,24}=2 s. With Π−(4λ5+3μ5+ω32)−(4λ2+3μ2)=90−(27+20)−43=0, set ω21=min{0,Π−ψ21−ω20}=min{0,22}=0. At last, set ω22=Π−ψ21−ω20−ω21=24−2−0=22 s. For C1 and C2, since Π−(4λ2+3μ2+ω22)−λ1=90−(43+22)−28.5=−3.5<0, or condition (4.6) is violated, i.e., there is no one-product cyclic schedule to achieve its lower bound and this result can also be verified by simulation.
The embodiments of the present invention are particularly advantageous as it provides a solution for scheduling a treelike multi-cluster tool with a complex topology which is process-bound. By developing a Petri net model to describe the system based mainly on the buffering modules, the necessary and sufficient conditions under which a one-wafer cyclic schedule with the lower bound of cycle time can be found. The present invention also proposes an efficient algorithm to test whether such a schedule exists and to find it if it exists.
Although not required, the embodiments described with reference to the Figures can be implemented as an application programming interface (API) or as a series of libraries for use by a developer or can be included within another software application, such as a terminal or personal computer operating system or a portable computing device operating system. Generally, as program modules include routines, programs, objects, components and data files assisting in the performance of particular functions, the skilled person will understand that the functionality of the software application may be distributed across a number of routines, objects or components to achieve the same functionality desired herein.
It will also be appreciated that where the methods and systems of the present invention are either wholly implemented by computing system or partly implemented by computing systems then any appropriate computing system architecture may be utilized. This will include stand-alone computers, network computers and dedicated hardware devices. Where the terms “computing system” and “computing device” are used, these terms are intended to cover any appropriate arrangement of computer hardware capable of implementing the function described.
It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.
Any reference to prior art contained herein is not to be taken as an admission that the information is common general knowledge, unless otherwise indicated.
