集成电路cluster tool设备调度策略

To satisfy the requirements of high-mix integrated-circuit chip production, multiple wafer types are grouped into a lot to be fabricated concurrently in a multi-cluster tool system. However, the existing studies do not solve the scheduling problem of multi-cluster tools with multiple wafer types being processed and wafer residency time constraints being imposed. This study aims to find a feasible steady-state schedule for a process-dominant multi-cluster tool that processes two wafer types concurrently. A novel robot cycle, called one-wafer-per-type cycle, and a two-swap robot sequence are defined, under which the minimal cycle time can be achieved. Further, the schedulability for individual cluster tools with two wafer types is proved upon closed-form derivations. The schedulability conditions of a one-wafer-per-type schedule for the system are proposed. With such conditions, an efficient algorithm is established to obtain a schedule in a two-swap sequence. Several instances show the application and effectiveness of the proposed method.

Scheduling single-robotic-arm cluster tools subject to wafer residency time constraints has received much attention. Compared to some scheduling strategies that use all processing modules (PMs) to process wafers, it is much more challenging to schedule a more general case whose optimal scheduling strategy is not limited to the case of using all PMs. The strategy of only adjusting the robot’s waiting time may fail to produce a desired schedule. When a tool using all PMs is not schedulable, it may become schedulable if only some PMs of a type are used. Therefore, it is very important to select an appropriate number of PMs to process wafers. This work studies the cyclic scheduling problem of wafer-residency-time-constrained single-robotic-arm cluster tools by simultaneously adjusting the number of PMs and robot waiting time. We build a virtual cell that includes an appropriate number of PMs to process wafers with the maximal productivity. We establish sufficient and necessary conditions under which the system is schedulable. The schedulability conditions are less conservative than the state-of-the-art one. A polynomial algorithm is developed to find the optimal cyclic schedule, a virtual cell’s configuration, and robot waiting time. We illustrate the practicability of the proposed algorithm via several examples, and its superiority over the existing one. Note to Practitioners—This paper addresses the optimal cyclic scheduling problem of wafer-residency-time-constrained single-robotic-arm cluster tools that are used in every wafer fabrication factory. This work for the first time simultaneously adjusts the number of PMs and the robot waiting time given such a cluster tool such that it can be scheduled with the highest productivity. It presents the optimal scheduling method with polynomial complexity. We confirm that the proposed approach can improve schedulability of single-robotic-arm cluster tools over existing methods by using multiple examples. It can thus be readily applied to industrial wafer fabrication.

Optimal cyclic scheduling problems of wafer-residency-time-constrained dual-arm cluster tools in wafer fabrication are challenging and remain to be fully solved. Existing studies assume that all processing modules (PMs) of a required type are used to process the same type of wafers. This sometimes brings unneeded conservativeness to scheduling results, because we may be able to make a tool schedulable by reducing the number of PMs in some steps if the original one is not. In some cases, we may use fewer PMs to reach the same result if the original one is schedulable, thus saving energy and other production resources. This work selects a proper number of PMs of needed types to process wafers while ensuring the highest productivity of a wafer-residency-time-constrained dual-arm cluster tool. It proposes the necessary and sufficient conditions under which a tool is schedulable. It then develops a polynomial-complexity algorithm that finds an optimal cyclic schedule. Examples are given to show its superiority over existing ones, thus advancing this field of cluster tool scheduling greatly and helping semiconductor producers to realize the green manufacturing of wafers.

With the rapid changes and diversity of market demand, fabs have to produce wafers in many varieties and small batches. This brings a great challenge to the scheduling of wafer-residency-time-constrained cluster tools that concurrently process multiple types of wafers. Existing practice tends to employ all processing modules (PMs) of a required type to process wafers, and adopt a virtual module technology to balance PM workload for different types of wafers. However, since the virtual module technology may fail to well balance the natural workload of each processing step of a type of wafers, the corresponding scheduling results can be too conservative for the cluster tools to reach their optimal productivity. Can we improve their schedulability by choosing the most appropriate number of PMs to process wafers instead of all PMs? Under what conditions, may we use fewer PMs in a cluster tool to reach the same cycle time reached by using all PMs, thus saving resources and reducing energy consumption? This work presents a PM dual-layer configuration method. The first configuration is performed to decide the proper number of PMs, and the second one is done for so-called virtual PMs that are not physically existing PMs. It derives necessary and sufficient conditions for the schedulability of a cluster tool, which are more relaxed than all the existing ones. Furthermore, this work presents a polynomial-complexity algorithm to find an optimal cyclic schedule, and its superiority over existing ones is shown via several examples. Note to Practitioners—This paper addresses the cyclic scheduling problem of wafer-residency-time-constrained single-arm cluster tools handling multiple wafer types, and gives an efficient method to configure PMs for processing wafers. Through the method, we can balance not only the PM workload for different types of wafers, but also the PM workload for each step of a type of wafers. Based on it, a cyclic scheduling approach is presented, which is readily applied to industrial wafer manufacturing. By using our method, the schedulability of the tool can be greatly improved, and some energy and resource consumption can be eliminated to make wafer fabrication greener than the existing methods.

To ensure wafer quality, engineers have to impose wafer residency time constraints and chamber cleaning operations on cluster tools; this has been widely used in semiconductor manufacturing. Wafer residency time constraints and chamber cleaning operations make the scheduling problem of cluster tools more challenging. This work aims to solve such a scheduling problem for single-arm cluster tools and presents a novel method based on the use of virtual wafers. Under a one-cyclic schedule obtained for single-arm cluster tools without chamber cleaning requirements, virtual wafers are loaded into the tool such that when a process module (PM) processes virtual wafers, a chamber cleaning operation is performed in practice. The key to solve this scheduling problem is to find a wafer loading sequence with the highest performance in terms of cycle time. With this idea, this work constructs a genetic algorithm to search for such a solution. Since the obtained solution is a periodical wafer loading sequence based on a one-wafer cyclic schedule, it can be easily implemented. Therefore, this work has high practical value to numerous semiconductor manufacturers. Experiments were performed to show the efficiency and effectiveness of the proposed method.

In semiconductor manufacturing, an extremely stringent quality control is enforced for wafer fabrication processes. Consequently, cleaning operations that clear residual chemical gases and heat in the chambers are frequently performed. As one kind of cleaning operations, a purge operation which executes a cleaning operation each time a wafer is removed from a chamber is widely adopted in leading semiconductor fabrication plants. Such an operation can definitely improve the quality of wafer. However, it results in lower productivity. To make a tradeoff between quality and productivity, a condition-based chamber cleaning operation that performs a cleaning operation with the consideration of the actual state of a chamber is introduced in practice. Aiming to address the scheduling problem of time-constrained single-armed cluster tools with condition-based chamber cleaning operations, efficient scheduling approaches are proposed in this work for the first time and, algorithms for searching for a feasible schedule are also derived. Two illustrative examples are given to show the power of the approach.

In today’s semiconductor manufacturing industry, wafer foundries often face the challenge of producing a variety of integrated circuit chip products using a single manufacturing line. To address this, multicluster tools have become a popular choice for processing multiple wafer types simultaneously. Operating such tools involves coordinating the robots in adjacent individual tools to transport multitype wafers through a shared buffer. This study aims to develop a scheduling method for the concurrent fabrication processes of two wafer types, performed by a multicluster tool with wafer residency time constraints. The proposed approach presents a two-backward sequence, based on a backward strategy of a single wafer type, to convert a one-wafer cyclic schedule into a one-wafer-per-type cyclic schedule while revealing its temporal properties. To ensure a smooth operation of a single-arm multicluster tool system and synchronize multiple robots, several necessary and sufficient conditions are derived for the first time. Two efficient algorithms are then proposed to determine the feasibility of a periodic schedule and obtain a schedule that achieves the lower-bound cycle time under a two-backward strategy, maximizing the productivity of such a multicluster tool. Finally, numerical simulations and two practical examples are presented to demonstrate the applications and performance of the proposed approach.

Cluster tools play a significant role in the entire process of wafer fabrication. As the width of circuits in semiconductor chips shrinks down to less than 10nm, strict operational constraints are imposed on the operations of cluster tools in order to ensure the quality of processed wafers. Particularly, wafer residency time constraints and chamber cleaning requirements are commonly seen in etching, chemical vapor deposition, coating processes, etc. They make the scheduling problem of cluster tools more challenging. This work aims to provide a solution for dual-arm cluster tools with wafer residency time constraints and chamber cleaning requirements. To do so, it proposes a novel virtual wafer-based scheduling method. By this method, under a steady state, a PM processes either a real or virtual wafer at a time. When a PM processes a virtual one, its chamber can perform a cleaning operation. In this way, we can meet not only the strict residency time constraints for real wafers, but also innovatively meet chamber cleaning requirements. Based on such a novel scheduling method, an efficient binary integer programming model is established to optimize the throughput of cluster tools. Finally, experiments are performed to show the efficiency and effectiveness of the proposed method. Note to Practitioners—To ensure wafer quality in semiconductor manufacturing, engineers have to impose wafer residency time constraints and chamber cleaning requirements on the operations of cluster tools. In order to tackle their scheduling problem with these constraints, this work proposes a novel method based on the use of virtual wafers. Under a one-cyclic schedule obtained for time-constrained cluster tools without chamber cleaning requirements, virtual wafers are loaded into the tool such that when a PM processes a virtual wafer, a chamber cleaning operation can be performed in practice. The key to solve this scheduling problem is to find a wafer loading sequence with the highest performance in terms of cycle time. To do so, this work establishes an efficient binary integer programming model to search for such a solution. Since the obtained solution is a periodical wafer loading sequence based on a one-wafer cyclic schedule, it can be easily implemented. Therefore, this work has a high practical value to numerous semiconductor manufacturers.

A cluster tool, widely used for semiconductor wafer fabrication, consists of several single-wafer processing chambers and a wafer handling robot. A chamber for critical processes is periodically cleaned for removing chemical impurities to reduce quality risk due to extreme circuit width shrinkage. We examine the problem of determining a cleaning plan for dual-armed cluster tools with general chamber cleaning periods $k_{i} > 1$ for chamber $i$ using the popular swap sequence to minimize the cycle time. We derive conditions for which the popular swap sequence minimizes the tool cycle time regardless of the cleaning plan. For the other cases, we develop a cleaning rule named DGC(Dispersing and Gathering Cleaning) that disperses cleaning operations along the robot cycle and gathers the cleaning operations that are interlaced along the robot cycle. For parallel flows with a single process step, we develop a closed-formula for the cycle time and prove that DGC minimizes the cycle time for the swap sequence. We also present conditions under which the swap sequence with the cleaning rule achieves the minimum cycle time compared to all other sequences and cleaning plans. For serial wafer flows, we show that DGC minimizes the cycle time when the cleaning period and the cleaning time are the same for all process steps. We also show by experiments that the proposed DGC effectively reduces the cycle time for the other general cases. Note to Practitioners—Our proposed cleaning rule can significantly improve the tool cycle time and regulate wafer flow times. It is easily applied or adapted to and effective for most wafer flow patterns, sequences, and tool architectures.

We examine a scheduling problem for a dual-armed cluster tool that processes multiple similar wafer types concurrently. It has been recently proved that the well-known swap sequence, which is widely used for single wafer type processing, also minimizes the cycle time for concurrent processing. In this article, we wish to minimize wafer delays in a process chamber, which are critical to wafer quality degradation, while maintaining the minimum cycle time. In particular, we show that concurrent processing of wafers with different processing times complicates the analysis of wafer delays significantly, and the wafer delays can be remarkably reduced by finding a proper cycle plan which is the release sequence of different wafer types. We first characterize wafer delays for a given cycle plan by analyzing the circuits of the timed event graph (TEG) model. From this, we prove that concurrent processing of wafers may cause a significant workload imbalance between parallel chambers of a process step, and hence the wafer delays increase substantially. We present that the wafer delays are minimized by a cycle plan that evenly balances workloads between parallel chambers. We also propose how wafer loading task at each process step has to be postponed to meet wafer delay constraints while maintaining the minimum cycle time. Note to Practitioners—Wafer quality control has become an essential fab operational problem in semiconductor manufacturing industry. In cluster tools, which are dominantly being used for diverse wafer fabrication stages, it has been proven that the wafer delays within process chambers have a crucial impact on the wafer quality. Accordingly, modern fabs have introduced stringent quality control to regulate wafer delays in cluster tools. In this research, we propose a scheduling strategy to minimize the wafer delays when a cluster tool concurrently processes multiple wafer types. We first show that the release sequence of different wafer types significantly impacts the wafer delays under concurrent processing, and we then propose how these wafer delays can be minimized by finding the optimal wafer release sequence.

Flexible manufacturing systems (FMSs), which can easily adapt to changes in job types, have been widely used in manufacturing areas. Scheduling of FMSs is a variant of a flexible job shop with transport robots and no buffer, and it is extremely hard as it involves determining the job processing sequence on machines and the sequence of robot tasks, with various jobs with different processing flows and the risk of deadlocks. Due to these characteristics, many existing studies have focused on developing heuristic algorithms. However, an optimal solution is still crucial to minimize FMSs makespan. Therefore, we propose a mixed integer programming (MIP) and a branch and bound (B&B) algorithm with a timed Petri net (TPN) to achieve optimal scheduling of FMSs. FMSs are first modeled with a TPN, and tight lower bounds are proposed based on bottleneck machines and sophisticated ready times. In addition, the search space is effectively reduced by the transition index marking (TIM)-based dominance rule and various deadlock prevention conditions based on the TPN. The experimental results show that our proposed B&B algorithm outperforms mathematical formulation and previous algorithms in various FMS instances. Note to Practitioners—Flexible manufacturing systems (FMSs) can often be found in various manufacturing areas composed of machines, material-handling robots, and controlling computers. With the increase in customer demand for diverse products manufactured in small quantities, the importance of FMSs has grown even more prominent in the manufacturing sector. In this paper, we develop a mixed integer programming (MIP) and a branch and bound (B&B) algorithm for optimal scheduling of FMSs with a timed Petri net (TPN). The proposed B&B can handle various FMS layouts by modeling with a TPN, regardless of the number of product routes, machines, and robots. Three lower bounds and dominance rules are proposed, and they have the great advantage of reduced computational time and search spaces. Our proposed method can also be implemented as a rolling horizon approach when jobs arrive in batches. Additionally, the algorithm can be adapted for use in other types of manufacturing systems, such as cluster tools, robotic cells, and track systems, which can be modeled using TPNs.

We address a robotic flow shop scheduling problem where two part types are processed on each given set of dedicated machines. A single robot moving on a fixed rail transports one part at a time, and the processing times of the parts vary on the machines within a given time interval. We use a reinforcement learning (RL) approach to obtain efficient robot task sequences to minimise makespan. We model the problem with a Petri net used for a RLenvironment and develop a lower bound for the makespan. We then define states, actions, and rewards based on the Petri net model; further, we show that the RL approach works better than the first-in-first-out (FIFO) rule and the reverse sequence (RS), which is extensively used for cyclic scheduling of a robotic flow shop; moreover, the gap between the makespan from the proposed algorithm and a lower bound is not large; finally, the makespan from the RL method is compared to an optimal solution in a relaxed problem. This research shows the applicability of RL for the scheduling of robotic flow shops and its efficiency by comparing it to FIFO, RS and a lower bound. This work can be easily extended to several other variants of robotic flow shop scheduling problems.

The demand for efficient Industry 4.0 systems has driven the need to optimize production systems, where effective scheduling is crucial. In smart manufacturing, robots handle material transfers, making precise scheduling essential for seamless operations. However, research often oversimplifies the Robotic Flexible Job Shop problem by focusing only on transportation time, ignoring resource allocation and robot diversity. This study addresses these gaps, tackling a Multi-Robot Flexible Job Shop (MRFJS) scheduling problem with limited buffers. It involves non-identical parallel machines and robots with varying capabilities overseeing material handling under blocking conditions. The case study is based on a real Industry 4.0 scenario, where the layout restricts each robotic arm’s access, requiring strategic buffer placement for part transfers. A Mixed-Integer Programming (MILP) model aims to minimize makespan, followed by a new Genetic Algorithm (GA) using Roy and Sussman’s Alternative Graph. Computational tests on various scales and real data from a manufacturing plant demonstrate the GA’s efficacy in solving complex scheduling problems in real-world production settings. Based on the data, the Proposed Genetic Algorithm (PGA), with an average Relative Deviation (ARD) of 0.25%, performed approximately 34% better compared to the Basic Genetic Algorithm (BGA), with an average ARD of 0.38%. This percentage indicates that the PGA significantly outperforms in solving complex scheduling problems.

… tool requirements, it is desirable to minimize tool changes that incur significant time costs. We approach this problem as a multi-robot … tasks to avoid unnecessary tool changes. To avoid …

… integrated tools such as cluster tools, … the robot task sequence is determined, all work cycles are determined. Most academic works on cluster tool scheduling consider cyclic scheduling. …

Robotized manufacturing systems consisting of several processing machines and a robot for transporting jobs between the machines have been widely used in mechanical and electronic manufacturing industries. The sequence of robot tasks in such a robotized manufacturing system affects its productivity significantly, which also has the large impact on the overall production line consisting of multiple robotized manufacturing systems. This article addresses the scheduling problem in a single-gripper robotic cell, one of a robotized manufacturing systems. The objective is to minimize the makespan. To achieve this, a novel reinforcement learning method is proposed, which combines a look-ahead search (LAS) to improve decisionmaking using more accurate estimated makespan. Experimental results demonstrate the superior performance of the proposed method compared to existing approaches. Moreover, the method is applicable in dynamic environments with uncertain processing times.

Resource allocation systems (RASs) belong to a kind of discrete event system commonly seen in the industry. In such systems, available resources are allocated to concurrently running processes to optimize some performance criteria. Search strategies in the reachability graph (RG) of a timed Petri net (PN) attracted much attention in the past decades to cope with RAS scheduling problems (RSPs), since PNs are very suitable to model and analyze RASs and their RGs fully reflect systems’ behavior. However, there has been no existing related survey and review paper till now. In this work, we present a tutorial and comprehensive literature survey of RG-based RSP methods. Many state-of-the-art RG-based RAS scheduling strategies are reviewed and summarized. First, we present a framework of RSPs and classify RSPs and their PNs in terms of resource usage and net structures. The differences and relations among the PNs are also given. Then, we introduce timed PN construction methods for RSPs and scheduling objectives and search strategies for RG-based RSPs. Next, we summarize different heuristic functions adopted in a frequently used A* search to solve RG-based RSPs. Finally, we discuss some important future research directions and open issues.

Many scheduling problems, especially for automated manufacturing systems, have been addressed with a timed Petri net (TPN) by developing optimal or heuristic algorithms based on it. However, most of them have focused on cyclic scheduling, in which a firing sequence of transitions in a TPN is repeated, to minimize the cycle time. A few studies have considered a noncyclic scheduling problem but focused on developing a heuristic search method. In this study, a noncyclic scheduling problem for automated manufacturing systems is considered with a TPN to obtain an optimal solution with the makespan measure. Specifically, a scheduling problem of a dual-gripper robotic flow shop with a given job sequence is considered to determine an optimal robot task sequence. We first define resource marking and resource ready times for the dual-gripper robotic flow shop based on a TPN and develop dominance properties based on them. Then, an optimal algorithm that eliminates dominated nodes in a reachability tree by using those properties is proposed. The experimental results show that the proposed algorithm significantly outperforms the existing studies.

Nowadays, cluster tools tend to concurrently process multiple types of wafers with similar recipes in order to improve their utilization and flexibility in semiconductor manufacturing. Different wafer types may have different wafer flow patterns, resulting in that cluster tools are deadlock-prone. It is challenging to develop a general method to solve the deadlock problem of cluster tools without restriction on the wafer types. This work aims at solving such a challenging problem for single-arm cluster tools. To do so, a general Petri net model is developed for single-arm cluster tools. Given the wafer flow patterns of all wafer types to be processed in a single-arm cluster tool, such a Petri net model can be easily obtained by defining the relationship between places and transitions. Then, a control method by using self-loops is presented to prevent the model from deadlocks during the evolutions from the initial state to the final state. Furthermore, such a control method is proved to be optimal. Illustrative examples are given to verify the proposed method at last.

In semiconductor fabs, many key processes adopt cluster tools to process wafers. For some processes, such as atomic layer deposition and plasma enhanced chemical vapor deposition, wafers with reentrant flows should be processed in cluster tool. In such cases, it is necessary to find a way to correctly operate cluster tools such that wafers can be completed according to wafer processing recipe (including the processing parameters and wafer flows) and the productivity of the tools can be maximized. To do so, this work presents a Petri net model for cluster tools with dual-arm robots and multiple-time reentrant flows. Based on the model, control policies are proposed such that several novel scheduling methods can be realize in practice. Note that such control policies can be embedded into the scheduler or controller of cluster tools. Finally, case studies are given to show how cluster tools operates based on the control policies.

We address the scheduling problem of single-armed cluster tools that concurrently process two wafer types without assuming cyclic scheduling. These cluster tools, consisting of multiple processing modules and a transport robot, are commonly used in semiconductor manufacturing processes, such as etching, deposition, and lithography. To optimize the tool’s throughput, we propose a reinforcement learning approach for determining both the robot task sequence and the release sequence of wafer types. By incorporating an adaptive search, our method intelligently explores future states to gather crucial information, enabling the selection of the best action. Extensive experiments demonstrate that our proposed method outperforms the well-known optimal robot task sequence for cyclic scheduling in single-armed cluster tools with concurrent processing. These findings underscore the effectiveness and superiority of our approach in optimizing the throughput of single-armed cluster tools, without relying on cyclic scheduling assumptions. Note to Practitioners—Single-armed cluster tools, comprising multiple processing modules and a transport robot, are commonly employed in semiconductor manufacturing processes. In recent times, there has been a trend towards concurrent processing of multiple wafer types within the fab, aimed at improving the PMs’ utilization. The concurrent backward sequence (CBS) has emerged as an efficient approach for such scenarios, assuming cyclic scheduling. However, the CBS relies on a cycle plan that specifies the number of wafers from each wafer type to be produced and their release sequence in a cycle. While the CBS provides an optimal schedule that maximizes the throughput of the tool in some cases under the cyclic scheduling assumption, its performance has not been guaranteed for other general cases. We propose a highly efficient reinforcement learning approach for the scheduling problem including the robot task sequence and release sequence of wafer types in a single-armed cluster tool. Our method outperforms the CBS, demonstrating its superior performance. By implementing our proposed approach, we believe that practitioners can effectively maximize the throughput of single-armed cluster tools and significantly enhance the fab productivity.

… The remaining sections of the paper are organized as follows: In the next section, we provide a detailed review of literature related to the scheduling of cluster tools and wet stations in …

In semiconductor manufacturing, a multifunctional process module (MPM) can perform multiple processing steps by adjusting its functional settings. This enhances the reconfigurability of cluster tools and allows them to flexibly adapt to diverse production requirements. However, the different function settings of the MPM change the number of processing modules and generate multiple alternative processing routes. Deadlocks occur more frequently in wafer manufacturing processes with flexible routes. The flexible configuration of MPM function leads to a highly complex and large-scale model. Proper configuration of MPM can optimize lot scheduling and improve processing efficiency. Thus, based on the functional setting of MPM, process-oriented Petri nets (POPNs) are established to describe the transient and steady state processing of the system, and control explanations are developed to avoid the system deadlock. Then, based on the evolving mechanism of the Petri nets, the temporal properties of the system under the earliest starting strategy (ESS) are analyzed. An algorithm based on ESS is developed to compute the makespan of wafers in a lot and optimize the settings of the MPM function. Experimental results demonstrate that for scheduling problems unsolvable by the mixed-integer programming (MIP) model, the algorithm can adaptively minimize system lot completion time by reasonably setting the function of MPM.

… -arm cluster tool with purge operations and wafer residency time constraints by using resource-oriented Petri net (… processes of a single-arm cluster tool with purge operations. Then, a …

Cluster tools have been widely used in the semiconductor industry. In previous work, singe-arm and dual-arm cluster tools are dealt with separately. When considering residency time constraints, algorithms for scheduling robot moving sequences are mainly developed based on backward and swap strategies, respectively. How about the performance of other robot moving sequences is still an open problem. The present work addresses general conditions by considering all feasible robot moving sequences—beyond backward- and swap-based strategies. In addition, both dual-arm and single-arm cluster tools with reentrance are handled simultaneously. To solve this, the problem is first divided into two parts—the basic one without reentrance and reentrant operations. Based on detailed analysis of all available robot operations in a cluster tool, mixed-integer programming formulations are developed for both parts. Subsequently, both illustrative examples and randomly generated instances are tested to validate the efficiency of the proposed approach. Results of one example show improvement comparing to previous work. Another two examples—not scheduled previously—are scheduled well by the proposed approach. Meanwhile, the flexibility of operations obtained by the proposed model is demonstrated by an example with six processing modules and 12 stages in a dual-arm cluster tool. Additionally, more randomly generated instances are tested more to validate the proposed approach and analyze computation time.

… For this open problem of a dual-arm multi-cluster tool, we propose two algorithms to … deadlock avoidance at a buffer module and 2) minimize the time for feasibly shutting down the tool …

This work addresses the scheduling problem of deadlock-prone automated manufacturing systems (AMSs) modeled by a class of Petri nets called systems of simple sequential processes with resources (S3PRs), and proposes a novel hybrid heuristic search (HHS) algorithm to minimize the makespan. First, based on place-timed Petri net models of AMSs, a timed state space (TSS) composed of timed states is defined. TSS-based breadth-first search and A* algorithms are developed, through which the optimal solutions for small-scale problems can be obtained. Then, in order to effectively solve the scheduling problems of AMSs with different scales, TSS is condensed, and HHS is designed to search the condensed state space (CSS). In HHS, a duplicate detection policy is proposed for ensuring that only one transition path is reserved for each state in CSS. A pruning policy is proposed for pruning unpromising states to achieve the goal of searching a small part of CSS only, and three different cost estimation functions are developed for evaluating states. A hybrid search strategy is defined to achieve high search efficiency and find better solutions. The integration of the proposed duplicate detection policy, pruning policy, and hybrid search strategy ensures the high optimization performance and computational efficiency of HHS. Experimental results on benchmark AMS instances demonstrate the superiority of the proposed algorithm over the existing ones. The applicability of HHS in solving different industrial problems and manufacturing scheduling problems is also verified. Note to Practitioners—Automated manufacturing systems (AMSs) are computer-controlled manufacturing systems. They exhibit a high degree of resources sharing and route flexibility and can be highly adaptable to various production plans. Solving their scheduling problems is of great significance to manufacturers. When designing a scheduling algorithm for such systems, engineers face two challenges: how to deal with deadlocks caused by jobs competing for limited resources, and how to maintain high solution ability when solving large-scale problems. Existing algorithms for scheduling deadlock-prone AMSs have to rely on specific deadlock handling strategies to ensure their feasibility, and are unable to find high-quality solutions for large-scale problems. This article presents a hybrid heuristic search (HHS) algorithm for minimizing the makespan of deadlock-prone AMSs, in which a duplicate detection policy, a pruning policy, and a hybrid search strategy are specially designed. The combination of these policies ensure that HHS can find a high-quality solution in a short computation time, even if no deadlock handling strategy is used in the algorithm. Experimental tests and comparisons show that HHS significantly outperforms existing algorithms. The proposed HHS can be applied to other AMS scheduling problems, and can be used as an online or real-time scheduling method due to its high computational efficiency.

Cluster tools, which are extensively used in semiconductor and display manufacturing, offer the capability to perform multiple processing steps within a single tool. Many companies have recently been reducing wafer lot sizes due to diversified customer demands and circuit width reduction, resulting in more frequent transient and non-cyclic periods. We therefore present a novel mixed integer programming (MIP) model that can handle these non-cyclic scheduling problems of cluster tools. Our model is specifically designed to efficiently handle various wafer flows, accommodating both single- and dual-armed robots, while ensuring a shorter computation time, compared to the previous formulations. We first model cluster tools with a timed Petri net (TPN) and develop several precedence relations between transitions by analyzing the characteristics of the scheduling problems. These precedence relations are incorporated into a TPN by introducing additional places and arcs. This modification helps reduce the overall number of decision variables involved in the scheduling problem. We then develop an MIP model which specifies the precedence relations between transitions, as opposed to the position-based decision variables used in the previous studies. The proposed MIP model is capable of handling various flow scenarios encountered in cluster tools, including serial, parallel, concurrent, and re-entrant flows with time window constraints. Note to Practitioners—In response to the diverse demands of customers, the use of larger wafer sizes, and the need for smaller circuit widths, operating cluster tools in a cyclic manner has become increasingly challenging. Cyclic scheduling, where the robot repeats a fixed sequence while assuming identical wafers, is no longer viable in such scenarios. Unfortunately, previous research on cluster tool scheduling has predominantly focused on cyclic scheduling, failing to address the growing importance of non-cyclic scenarios. To address this gap, we propose an innovative and efficient mixed integer programming (MIP) model capable of handling non-cyclic scheduling problems in cluster tools. Our model accommodates various flow types, including serial, parallel, concurrent, and re-entrant flows. Through comparative analysis, we demonstrate that our proposed model outperforms the well-known MIP model that relies on position-based decision variables. We believe that our proposed model holds practical value as it exhibits relatively short computation times. Additionally, widely-used commercial solvers, such as CPLEX or Google OR Tools, can seamlessly implement the model, making it easily accessible and applicable in real-world scenarios.

… deadlock avoidance condition based on the Petri net cannot be implemented explicitly. Besides, depending on the nature of the problem, some action masks should be set according to …

With the change of wafer fabrication mode to multi-species and small lot, urgent order insertion occurs frequently in the actual production process, and the wafer flow patterns of the different wafer types may be different such that cluster tools are deadlock-prone. To improve the production flexibility of cluster tools, this paper investigates the scheduling problem of single-arm cluster tools facing urgent order insertion under the consideration of wafer residency time. Firstly, the processing process of emergency insertion order is analyzed, and the scheduling rules of the robot for single-arm cluster tools are proposed to realize the schedulability of the processing process. Secondly, the linear programming model for solving the robot waiting time is given. Finally, the optimal scheduling algorithm is proposed for the cluster tools when facing emergency order insertion, and the effectiveness of the scheduling method is verified by means of example verification and comparative analysis.

This paper studies the scheduling problem of single-arm cluster tools that mixedly process two different kinds of wafers without sharing and revisiting processing modules (PMs). We balance internal workloads by adjusting the number of PMs used to process wafers, and balance the external workloads by configuring virtual PMs. We derive the scheduling conditions for single-arm cluster tools, which are more relaxed than the existing ones. We can also use less PMs to get the same production cycle time as the existing literature using configuration of virtual PMs only. We give some examples to show the application and power of the theory.

Recent advances in AI/ML have significantly increased the demand for various semiconductor products. To meet the growing demand and improve productivity, semiconductor companies now produce multiple wafer types simultaneously and, in some cases, even transport them in a single cassette. These changes in manufacturing practices have increased the complexity of cluster tool scheduling, making efficient scheduling algorithms a key focus for both researchers and manufacturers. In this paper, we propose a deep reinforcement learning (DRL)-based scheduling approach that effectively addresses the challenges of scheduling multiple wafer types in cluster tools. The proposed model considers both input sequencing and robot move sequencing to handle various wafer types across different cluster tool configurations. It is structured around an encoder–decoder framework: the encoder captures wafer recipes (i.e., processing times and flow), and the decoder uses dynamic scheduling information to determine the robot’s next actions while avoiding deadlock situations. Through extensive experiments that consider multiple cluster tool structures and wafer types, our model consistently outperforms conventional robot move sequences and metaheuristic methods, without relying on cyclic assumptions. These results demonstrate that our approach can derive efficient robot sequences without requiring expert knowledge or extensive manual analysis, across single-arm, dual-arm, serial-parallel flow, concurrent flow, and skip flow configurations, while also considering purge operations within cluster tools. As such, our model is highly adaptable and can be effectively applied to a wide range of cluster tool scheduling problems. We provide benchmark datasets to support further research and practical applications: https://github.com/yeoneee/ClusterToolSchedulingRL.git Note to Practitioners—Semiconductor cluster tools are deployed in a variety of structural configurations depending on the company and the specific processes in use. For instance, robots can be configured as single-arm or dual-arm handlers, processing modules (PMs) are generally arranged in a serial configuration, or in cases of certain process steps, multiple PMs may be utilized to form a serial-parallel structure. In addition to these tool configuration factors, the complexity of cluster tool scheduling problems increases significantly when multiple wafer types are processed simultaneously to enhance productivity. For instance, some wafer types are processed using dedicated concurrent flows for each type, and certain wafer types may utilize skip flows, bypassing specific processes. These production scenarios, involving the simultaneous handling of diverse wafer types, not only demand additional decision-making regarding wafer release sequences but also require careful planning of robot move sequences within the tool, thereby further amplifying the complexity of the scheduling problem. The DRL method proposed in this paper offers a practical, scalable solution for operating cluster tools. Through reinforcement learning, the model autonomously derives effective robot move rules without relying on expert defined heuristic sequences or cyclic assumptions. This allows engineers to apply the DRL model in various tool configurations, eliminating the need for trial-and-error approaches. Our method improves throughput by consistently generating wafer-type input sequences and efficient robot move sequences across various tool setups. This framework can reduce the scheduling burden on engineers, which can also allow them to maximize productivity.

Multi-cluster tools are widely adopted in semiconductor manufacturing. When a process module (PM) at a step fails, a multi-cluster tool cannot complete the process recipe of work-in-process wafers and must be forced to enter a closedown process to be empty. To increase the throughput of a wafer fab, it is economically significant to shorten the failure-closedown process. However, due to the wafer residency time constraints, it is highly challenging to respond to a PM failure and find a corresponding optimal schedule. By assuming no parallel PM at a step, this work is the first to study this important issue for scheduling multi-cluster tools. We analyze the task sequences and synchronization conditions for robot activities to avoid deadlock in a shared buffer module. Upon these analyses, for process-dominant multi-cluster tools whose optimal steady-state schedule is known, algorithms are proposed to synthesize the proper sequences for robots in case of PM failures, then, a nonlinear program model is proposed to find an optimal schedule for the corresponding closedown process or decide no feasible solutions. The proposed model can uniformly deal with different scenarios of PM failures. Examples are given to illustrate the application of the proposed method. Note to Practitioners—In a wafer fab, there are hundreds, even thousands, of cluster tools. It is common that a failure of a processing module happens in a cluster tool. How to intelligently respond to such a random failure in a multi-cluster tool is an important issue in real-world production. This paper studies the scheduling problem of a multi-cluster tool in case of a failure at a PM. For a multi-cluster tool in case of a failure module, the proposed method can significantly reduce the loss of work-in-process wafers and shorten the closedown process.

Traditional rule-based cluster tool scheduling in semiconductor manufacturing faces significant limitations, including inflexibility, reliance on domain-specific expertise, and suboptimal performance in dynamic and complex environments. These methods often struggle to adapt to varying process conditions and equipment configurations, which are common in modern fabrications (fabs). Furthermore, previous research has typically relied on simplified simulators of cluster tools, failing to capture the full complexity of real-world semiconductor manufacturing equipment. To address these challenges, this study examines the potential of deep reinforcement learning (DRL) for optimizing cluster tool scheduling. This research presents a comprehensive simulation environment that models a cluster tool system, including both vacuum (VTM) and atmospheric (ATM) robots. This study progressively evaluates DRL agents, starting with a single-agent deep Q-network (DQN) and advancing to a multi-agent DQN (MADQN) framework to schedule the combined VTM-ATM system. Experimental results demonstrate that the proposed DRL agents consistently outperform traditional rule-based methods in terms of productivity and adaptability. In the complex multi-agent environment, the MADQN agent demonstrated robust performance across all tested configurations, achieving a productivity improvement of up to 8.9% over standard rule-based scheduling methods. These findings highlight the potential of DRL to overcome the limitations of existing scheduling methods and significantly enhance productivity in semiconductor manufacturing.

Cluster tools are extensively used for processing wafers in semiconductor manufacturing. To enhance the throughput of cluster tools, concurrent processing of multiple types of wafers has become a popular approach. In practice, due to mass customization and the shrinking of chips’ feature sizes, the lot size of wafers tends to become smaller, leading to more frequent transient and non-cyclic periods. Therefore, the scheduling problem of single-armed cluster tools that concurrently process two types of wafers from start-up to close-down period should be urgently addressed. To do so, a mixed integer programming (MIP) model is established to optimize the release sequence of wafer types and robot task sequence so that the total completion time of the two wafer types can be minimized. Since solving the MIP model for large-sized instances is hard, a two-combined backward sequence (2-CBS) is further proposed. Experimental results show that the 2-CBS can save a significant amount of computing time and increase the total completion time by only 1.09% on average. However, by quickly determining the completion time of each wafer lot, the throughput of cluster tools could be enhanced to some extent, as the idle time caused by the arrival delay of overhead hoist transports can be reduced. Note to Practitioners—When considering the concurrent processing of multiple wafer types, almost all existing studies focus on cyclic scheduling and thus fail to address the growing importance of non-cyclic scenarios. To fill this gap, the present work develops an MIP model to optimize the release sequence of wafer types and robot task sequence such that the total completion time of two wafer types can be minimized. Such a model is efficient for small-sized instances. Meanwhile, a 2-CBS strategy is developed for large-sized instances, which can be solved quickly and results in an increase in the total completion time by only 1.09%, demonstrating its effectiveness and applicability.

Cluster tools play a significant role in the entire process of wafer fabrication. Wafer residency time constraints and chamber cleaning requirements are commonly seen in etching, chemical vapor deposition, coating processes, etc. They make the scheduling problem of cluster tools more challenging. This work aims to provide a solution for dual-arm cluster tools with wafer residency time constraints and chamber cleaning requirements. To do so, it proposes a novel virtual wafer-based scheduling method. By this method, under a steady state, a process module (PM) processes either a real or virtual wafer at a time. When a PM processes a virtual one, its chamber performs a cleaning operation. In this way, we can meet not only the strict residency time constraints for real wafers, but also innovatively performs chamber cleaning operations as required. Based on such a novel scheduling method, an efficient binary integer programming model is established to maximize the throughput of cluster tools. Finally, experiments are performed to show the efficiency and effectiveness of the proposed method.

Cluster tools are the key equipment in semiconductor manufacturing systems. They have been widely adopted for many wafer fabrication processes, such as chemical and physical vapor deposition processes. Reentrant wafer flows are commonly seen in cluster tool operations for deposition processes. It is very complicated to schedule cluster tools with reentrant processes. For a dual-arm cluster tool with two-time reentering, the existing studies point out that a one-wafer periodical (1-WP) schedule can be found, and it is optimal in terms of productivity. However, for some wafer fabrication processes, wafers should be processed at some PMs more than two times. This gives rise to a question of whether there still exists a 1-WP schedule for dual-arm cluster tools with the number of reentering times being more than two such that the cycle time of a tool can reach the lower bound. This problem is still open, and this is what this work wants to tackle. For a dual-arm cluster tool with the number of reentering times being k (>2) times, if there does not exist a value f ∈ {1, 2 …} such that k = 3f, theoretical proofs are given to show that a 1-WP schedule can be found, otherwise it does not exist. For cases with a 1-WP schedule, the cycle time can be obtained by analytical expressions. For the cases without a 1-WP schedule, two new methods for a three-wafer periodical schedule are proposed to improve the system productivity by comparing it with an existing three-wafer periodical schedule. The applications of the obtained results are demonstrated by examples. Wafer residency time constraints are required for some wafer fabrication processes. Note that the results obtained in this work cannot be directly applied to cluster tools with both reentrant wafer flows and wafer residency time constraints. Nevertheless, schedulablity and scheduling analyses for that applications can be conducted based on the obtained results in this work.

Since last decade, the cluster tool has been mainstream in modern semiconductor manufacturing factories. In general, the cluster tool occupies 60% to 70% of production machines for advanced technology factories. The most characteristic feature of this kind of equipment is to integrate the relevant processes into one single machine to reduce wafer transportation time and prevent wafer contaminations as well. Nevertheless, cluster tools also increase the difficulty of production planning significantly, particularly for shop floor control due to complicated machine configurations. The main objective of this study is to propose a short-term scheduling model. The noteworthy goal of scheduling is to maximize the throughput within time constraints. There are two modules included in this scheduling model—arrival time estimation and short-term scheduling. The concept of the dynamic cycle time of the product’s step is applied to estimate the arrival time of the work in process (WIP) in front of machine. Furthermore, in order to avoid violating the time constraint of the WIP, an algorithm to calculate the latest time of the WIP to process on the machine is developed. Based on the latest process time of the WIP and the combination efficiency table, the production schedule of the cluster tools can be re-arranged to fulfill the production goal. The scheduling process will be renewed every three hours to make sure of the effectiveness and good performance of the schedule.

In cluster tools widely used for semiconductor manufacturing, a wafer processed in a chamber should wait until a robot arm unloads it. Such wafer delays degrade the wafer quality due to residual chemicals and heat and hence cause quality variability and even failures. To prevent excessive wafer delays and their variability for single-armed and dual-armed cluster tools, we propose a simple, effective, and robust feedback control method called Kanban feedback control(KFC) that postpones a wafer loading task until a completion event of an associated task triggers it. We model the feedback control design problem as a problem of adding a feedback path between a pair of transitions to regulate token delays at a place in a timed event graph model. We develop closed formulae for wafer delays of the tools with KFC. We prove that KFC minimizes the worst-case wafer delay. We also prove that KFC makes the tool have a unique 1-cyclic schedule with constant wafer delays and ensures strong stability that recovers the same 1-cyclic schedule and the constant wafer delay at each chamber in a few cycles after a time disruption. By experimentation, we verify that KFC significantly reduces wafer delays and robustly regulates wafer delays and even cycle times against persistent time variation and significant sporadic time disruptions. Note to Practitioners—As circuit widths shrink to a few nanometers and chip architectural complexity soars up due to FinFET, GAA, and high-rise circuit stack-ups, quality risk in wafer fabrication processes surges. Therefore, wafer delays within a chamber after processing can cause more serious quality variations and failures. We propose a simple, effective, and robust way of regulating wafer delays. It can significantly contribute to yield enhancement.

The semiconductor manufacturing system operates in lot units comprising wafers with identical recipes. Production predominantly occurs within a cluster tool, encompassing multiple single-wafer processing chambers and a wafer-handling robot. Recently, the increasing popularity of small lot sizes has been driven by the diversification of customer demand and larger wafer diameters. Thus, cluster tools frequently encounter lot switching periods, where lots with different recipes are processed consecutively. Existing research on cluster tool scheduling primarily relies on conventional backward and swap sequences optimized for cyclic robot operations. This approach persists even during lot switching period scheduling, which represents non-cyclic scenarios. However, it remains unclear whether these conventional robot task sequences remain optimal or ensure consistent performance during lot switching periods. Therefore, we introduce a theoretical analysis of the performance of the conventional robot sequences, such as backward and swap sequences, in lot switching periods for the first time. We examine various dominant relations, establishing the conditions under which the conventional sequences provide an optimal schedule. We also show worst-case performance bounds. Finally, given the limited performance of the conventional swap sequence during lot switching periods for dual-armed robots, we propose a new form of the robot task sequence and demonstrate its effectiveness. Note to Practitioners—The conventional robot task sequences, including backward and swap, are widely known to perform optimally in cyclic situations where the same lot is processed consecutively. Therefore, many cluster tool scheduling studies assumed these sequences were the basic movements of robots and proceeded with their research based on them. Such assumptions are applied even during the lot switching period when two different lots are considered. The performance of the robot task sequence is crucial because the tool performance depends on it. However, when the recipes of two consecutive lots differ, it is uncertain whether the conventional backward and swap sequences still offer optimal performance. Hence, this study theoretically verifies whether the conventional sequence still performs well during the lot-switching period. We believe that our findings will provide sufficient theoretical support for utilizing the conventional sequence in non-cyclic scheduling in the future. Furthermore, by proposing a new form of sequence suitable for the non-cyclic situations of dual-armed cluster tools, we suggest the potential for expanding to a more diverse range of sequences beyond the conventional ones.

… : wafer residence time … robot is shorter compared to the wafer handling time of the processing module. Due to the limited capacity of the robot, the scheduling problem of the cluster tool …

… robot action sequences, and regulating wafer postprocessing residency time. For scheduling such a dual-arm cluster tool … specific reinforcement learningbased scheduling method. We …

This study addresses a new scheduling problem for dual-arm cluster tools (CTs) that process two wafer types of different processing priority, each with distinct processing route and time. The goal is to maximize productivity by utilizing the fewest processing modules (PMs) for the wafers of high priority, while allocating available PMs for other wafers of low priority. A simple swap sequence for robot scheduling is introduced, supporting cyclic operations without disrupting priority wafer production. Necessary and sufficient conditions for scheduling CTs are provided for the first, along with the optimal PM configuration. An algorithm is developed for optimal cyclic scheduling. Its applications are demonstrated through a number of examples.

Cluster tools are vital in semiconductor manufacturing, where multifunctional process modules (MPMs) enhance flexibility and efficiency. However, variable MPMs and processing time in dual-arm cluster tools (DACTs) complicate scheduling, as variable MPM allocation patterns yield distinct productivity. This paper proposes a reinforcement learning-based method for DACTs with MPMs. Firstly, an algorithm enumerates all valid MPM allocation patterns. Then, an adaptive deep Q-Network (DQN) with masking techniques efficiently selects the most efficient pattern and generates robot schedules, minimizing makespan and wafer post-processing residency time across diverse DACT configurations. Experiments validate the proposed approach that offers robust, flexible scheduling solutions to boost semiconductor manufacturing productivity.

ABSTRACT In semiconductor manufacturing systems, a time-constrained multi-cluster tool should be scheduled such that a wafer stays in a process chamber in a given time range to satisfy a wafer residency time constraint. In practice, activity time is subject to variation. It could lead to some fluctuation of wafer residency time in a process chamber. Hence, it is crucial to analyze how wafer residency time varies with activity time variation. This issue is especially challenging for multi-cluster tools. This work focuses on determining the exact upper bound of wafer sojourn time delay resulted from activity time variation for dual-arm multi-cluster tools. After discussing their dynamic behaviours, it presents a two-level real-time operational architecture and a real-time control policy. Based on them, this work derives for the first time an efficient algorithm to calculate the exact upper bound of wafer sojourn time delay in a process chamber. As a result, engineers can test whether a given schedule is feasible. Several examples of industrial significance are used to demonstrate the application of the proposed approach.

Nowadays, cluster tools are extensively used for many wafer manufacturing processes, such as coating, lithograph, developing, etching, deposition, and testing. Traditional process modules in cluster tools can execute a single operation only. With the rapid development of equipment design, multifunctional process modules (MPMs) are equipped to serve for processing multiple operations together just like a single operation. With different wafer processing parameters, MPMs may be set for processing multiple operations together or processing just a single operation to form different schedules so as to maximize the productivity. Thus, it is highly desired to find an efficient scheduling method to quickly adapt to wafer processing parameter changes for productivity maximization by taking the advantages of MPMs. To tackle this issue, a deadlock-free Petri net (PN) model is developed to describe the behavior of a single-arm cluster tool. Based on the evolving mechanism of the PN model, two algorithms are developed to calculate the makespan for completing a given number of wafers. Then, an adaptive scheduling method is presented to set the functions of MPMs to minimize the makespan. Finally, experimental results show the efficiency and effectiveness of the proposed method.

In order to improve quality assurance in wafer manufacturing, there are strict process requirements. Besides the well-known residency time constraints (RTCs), a dual-arm cluster tool also requires each robot arm to execute a specific set of tasks. We call such a tool an arm task-constrained dual-arm cluster tool (ATC-DACT). To do this, one of the arms is identified as the dirty one and the other as the clean one. The dirty one can deal with raw wafers, while the clean one can deal with processed wafers. This requirement raises a new problem for scheduling a cluster tool. This paper discusses the scheduling problem of ATC-DACTs with RTCs. Due to the arm task constraints, the proven, effective swap strategy is no longer applicable to ATC-DACTs, making the scheduling problem difficult. To address this problem, we explicitly describe the robot waiting as an event and build a Petri net (PN) model. Then, we propose a hybrid task sequence (HTS) as an operation strategy by combining the swap and backward strategies. Based on the HTS, the necessary and sufficient conditions for schedulability are established; also, a linear programming model is developed. We then develop an algorithm using these results to optimally schedule the system. Industrial case studies demonstrate the application of this method.

As wafer circuit width shrinks down to less than ten nanometers in recent years, stringent quality control in the wafer manufacturing process is increasingly important. Thanks to the coupling of neighboring cluster tools and coordination of multiple robots in a multi-cluster tool, wafer production scheduling becomes rather complicated. After a wafer is processed, due to high-temperature chemical reactions in a chamber, the robot should be controlled to take it out of the processing chamber at the right time. In order to ensure the uniformity of integrated circuits on wafers, it is highly desirable to make the differences in wafer post-processing time among the individual tools in a multi-cluster tool as small as possible. To achieve this goal, for the first time, this work aims to find an optimal schedule for a dual-arm multi-cluster tool to regulate the wafer post-processing time. To do so, we propose polynomial-time algorithms to find an optimal schedule, which can achieve the highest throughput, and minimize the total post-processing time of the processing steps. We propose a linear program model and another algorithm to balance the differences in the post-processing time between any pair of adjacent cluster tools. Two industrial examples are given to illustrate the application and effectiveness of the proposed method.

In semiconductor manufacturing, chamber cleaning operations (CCOs) are performed from time to time to clear particles and chemical gases that remain in the processing chambers of cluster tools (CTs) to ensure wafer quality. For single-armed CTs with just one CCO that could be completed in one system cycle time, researchers have proposed several scheduling approaches. In practice, more than one CCO may be required concurrently, thus failing the existing ones under such cases. Furthermore, the duration of a cleaning operation may be larger than one system cycle time, which highly complicates the scheduling of such tools. This work addresses this challenge by proposing several scheduling approaches and algorithms for single-armed CTs with several CCOs, each of which exceeds the duration of one system cycle. Suggestions are given for engineers in semiconductor fabrication plants to take the most suitable scheduling approach and algorithm for enhancing the productivity of CTs given their specific scenarios. Finally, two illustrative examples are provided to showcase the power of the proposed concepts and techniques.

In wafer fabrication, it is imperative to minimize the transient process of cluster tools for the sake of on-demand and preventive maintenance. Due to the trend of multi-type and small-batch production, transient processes appear more and more frequently. Thus, the optimization problems of transient processes have gained increasing attention from both industry and academia. The requirement for wafer revisiting tend to complicate this problem significantly. However, only a few studies take such a challenge for cluster tools with wafer revisiting. This paper focuses on the schedule optimization of transient processes for dual-arm cluster tools with wafer revisiting. To accelerate transient processes, including both start-up and close-down ones, we adopt a program evaluation and review technique to analyze and harness a cluster tool’s state evolution. We then propose computationally efficient algorithms to speed up transient processes. Finally, we provide illustrative examples to show their applications and validate their effectiveness.

Multi-cluster tools are extensively utilized in semiconductor manufacturing. As these tools integrate more functionalities and handle increasingly complex processing recipes under numerous constraints, cyclic scheduling strategies become impractical. This poses challenges for non-cyclic scheduling approaches as well. This paper focuses on the non-cyclic scheduling problem of multi-cluster tools with residency time constraints. A generic model is developed to describe dual-armed multi-cluster tools and propose an optimization scheduling algorithm based on Proximal Policy Optimization (PPO). This algorithm is designed to learn effective scheduling policies in environments with small batches of wafers and can scale to handle large batches. Numerical experiments demonstrate that the proposed algorithm enables rapid scheduling, achieving scheduling performance close to optimal and minimizing residency time. Furthermore, comparisons with cyclic scheduling based on implicit enumeration methods show that our approach achieves shorter makespan.

This contribution highlights some concepts and aspects of Petri nets that are frequently neglected, but that the authors consider important or interesting, or that Carl Adam Petri emphasized.

We investigate the problem of parameter synthesis for time Petri nets with a cost variable that evolves both continuously with time, and discretely when firing transitions. More precisely, parameters are rational symbolic constants used for time constraints on the firing of transitions and we want to synthesise all their values such that some marking is reachable, with a cost that is either minimal or simply less than a given bound. We first prove that the mere existence of values for the parameters such that the latter property holds is undecidable. We nonetheless provide symbolic semi-algorithms for the two synthesis problems and we prove them both sound and complete when they terminate. We also show how to modify them for the case when parameter values are integers. Finally, we prove that these modified versions terminate if parameters are bounded. While this is to be expected since there are now only a finite number of possible parameter values, our algorithms are symbolic and thus avoid an explicit enumeration of all those values. Furthermore, the results are symbolic constraints representing finite unions of convex polyhedra that are easily amenable to further analysis through linear programming. We finally report on the implementation of the approach in Romeo, a software tool for the analysis of time Petri nets.