具有清洗工艺的单臂组合设备终止暂态调度

随着多品种小批量生产模式的普及,导致了组合设备频繁的暂态加工过程.为了提高组合设备的生产柔性,同时考虑晶圆驻留时间约束和腔室清洗时间约束,研究了单臂组合设备的终止暂态调度问题.首先,提出了 1-周期清洗工艺的暂态调度规则,并采用了面向资源的Petri网对单臂组合设备的终止暂态过程进行建模,引入避免死锁的变迁触发规则;其次,根据系统的终止暂态时间特性并考虑不同的调度情形,建立了终止暂态调度的线性规划模型;最后,通过实例验证了该方法的可行性.实验结果表明,与运用改进拉式策略的虚拟晶圆方案相比,该调度方案可有效地减少组合设备终止暂态的完工时间,并满足晶圆制造的工艺要求.     

腔室清洗的单臂组合设备初始暂态调度

为了提升腔室洁净度,晶圆厂需对组合设备腔室进行清洗操作,从而提高晶圆的加工质量.考虑腔室清洗时间和晶圆驻留时间的约束条件下,本文研究了单臂组合设备的初始暂态调度问题.首先,提出了机械手的初始暂态活动规则,并对机械手活动序列进行描述,实现了系统的初始暂态可调度性;其次,对机械手在初始暂态和稳态的活动时间进行了建模;然后,根据系统的时间特性,建立了初始暂态调度的线性规划模型;最后,通过实例验证了该方法的有效性.与已有的虚拟晶圆方法相比,该调度方法能有效减少初始暂态的完工时间,提高了组合设备的晶圆生产效率.     

Scheduling Dual-Arm Multi-Cluster Tools With Regulation of Post-Processing Time

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.

     

Scheduling a Single-Arm Multi-Cluster Tool With a Condition-Based Cleaning Operation

As wafer circuit widths shrink less than 10 nm, stringent quality control is imposed on the wafer fabrication processes. Therefore, wafer residency time constraints and chamber cleaning operations are widely required in chemical vapor deposition, coating processes, etc. They increase scheduling complexity in cluster tools. In this paper, we focus on scheduling single-arm multi-cluster tools with chamber cleaning operations subject to wafer residency time constraints. When a chamber is being cleaned, it can be viewed as processing a virtual wafer. In this way, chamber cleaning operations can be performed while wafer residency time constraints for real wafers are not violated. Based on such a method, we present the necessary and sufficient conditions to analytically check whether a single-arm multi-cluster tool can be scheduled with a chamber cleaning operation and wafer residency time constraints. An algorithm is proposed to adjust the cycle time for a cleaning operation that lasts a long cleaning time. Meanwhile, algorithms for a feasible schedule are also derived. And an algorithm is presented for operating a multi-cluster tool back to a steady state after the cleaning. Illustrative examples are given to show the application and effectiveness of the proposed method.

     

Scheduling cluster tools in wafer fabrication using candidate list and simulated annealing

This paper presents a new method for scheduling cluster tools in semiconductor fabrication. A cluster tool consists of a group of single-wafer chambers organized around a wafer transport device, or robot. Cluster fabrication system considered in this paper consists of serial cluster tools. Due to constraints imposed by multiple routes of each wafer type and machines with no buffer, it is difficult to find an optimal or near-optimal schedule. In order to determine the sequence of the operations to be released and the assignment of the machine to each operation, the proposed method uses a job requirement table with random keys as a solution representation. Simulated annealing seeks the optimal or near-optimal sequence and machine assignment of the operations. In this paper, the scheduling objective is to find a schedule with minimum makespan. A Gantt chart is obtained as the final schedule. To handle the constraints, the proposed method uses a candidate list. To determine which operation can be scheduled in considering the constraints, a negotiation procedure between the operations in the candidate list and a current state of the system is introduced. To show the effectiveness of the proposed method, scheduling example of a real cluster fabrication system is presented. Scheduling results are compared with those obtained by using several dispatching rules. From the experimental results, it is shown that the proposed method is promising.     

具有多品种晶圆混合加工的单臂组合设备调度

为了提高晶圆制造中组合设备的生产效率,在考虑晶圆驻留时间约束条件下,研究没有共享加工模块的多品种晶圆混合加工的单臂组合设备调度问题.首先,采用面向资源的Petri网模型描述多种晶圆产品的混合加工过程,引入控制变迁避免模型的死锁,采用赋时库所和赋时变迁模拟系统资源的活动时间.其次,通过虚拟加工的方法平衡工序的负载,基于系统Petri网模型和拉式调度策略,推导出单臂组合设备在多品种晶圆混合加工情形下的可调度性判定条件,并以解析形式描述.最后,提出了系统稳态调度求解算法并以实例验证了算法的有效性和可行性.     

Cycle time analysis for wafer revisiting process in scheduling of single-arm cluster tools

Some wafer fabrication processes performed by cluster tools require revisiting. With wafer revisiting, a cluster tool is very difficult to be scheduled due to a large number of possible schedules for the revisiting process. Atomic layer deposition (ALD) is a typical process with wafer revisiting that should be performed by cluster tools. This paper discusses the scheduling problem of single-arm cluster tools for the ALD process. In scheduling such a system, the most difficult part is to schedule the revisit...     

晶圆重入加工的组合设备终止暂态的调度与分析

晶圆重入是半导体组合设备加工中典型的复杂加工工艺,分析和优化暂态加工过程对于晶圆重入加工具有重要意义.为了满足加工需求和提高组合设备的加工性能,基于稳态重入加工的双臂组合设备Petri网模型和1-晶圆周期调度策略,采用虚拟晶圆的加工模式分析了系统的终止暂态过程,讨论了系统终止暂态的加工时间分布,并给出相应的解析式进行描述.利用eM-Plant仿真平台建立了重入加工的双臂组合设备终止暂态的仿真模型,并用例子验证了1-晶圆周期调度的可行性及解析式的有效性,为研究晶圆重入加工过程的优化提供了有效方法和手段.     

基于Petri网的组合设备建模与调度综述

组合设备是半导体晶圆制造的核心装备, 其调度与控制优化是半导体制造领域极具挑战性的课题. Petri网因其强大的建模能力和简约的图形化表达优势, 被广泛地应用于组合设备的建模与调度. 对基于Petri网的组合设备建模与调度方法进行综述, 归纳总结了组合设备的结构类型、晶圆流模式、调度策略及Petri网建模方法, 并系统阐述组合设备的7类典型调度问题, 包括驻留时间约束、作业时间波动、晶圆重入加工、多品种晶圆加工、加工模块(Process module, PM)故障、PM清洗和组合设备群. 最后, 讨论了当前组合设备调度存在的挑战及后续可能的研究方向.     

应用离散事件系统控制理论求解生产调度的新方法

众所周知, 生产调度问题属组合优化问题, 一般来说不存在求得精确最优解的多项式算法. 因此, 对于大规 模调度问题, 人们应用启发式算法和元启发式算法以企求得满意解. 在实际的应用中, 许多工业过程需要满足严格 的工艺约束. 对于这类过程的调度问题, 很难应用启发式算法和元启发式算法, 因为这些方法难于保证所求得调度 的可行性. 为了解决这一问题, 本文以半导体芯片制造中组合设备的调度问题作为例子, 介绍了一种基于离散事件 系统控制理论的生产调度新方法. 利用Petri网建模, 任何违反约束的状态均被描述为非法状态, 而使非法状态出现 的调度则是不可行调度. 通过可行调度的存在性分析, 该方法获得可行解空间并将调度问题转化为连续优化问题, 从而可以有效求解. 并且指出, 该方法可以应用于其他应用领域.     

Theoretical proof of edge search strategy applied to power plantstart-up scheduling

Power plant start-up scheduling is aimed at minimizing the start-up time while limiting maximum turbine rotor stresses. This scheduling problem is highly nonlinear and has a number of local optima. In our previous research, we proposed an efficient search model: genetic algorithms (GAs) with enforcement operation to focus the search along the edge of the feasible space where the optimal schedule is supposed to stay. Based on a nonlinear dynamic simulation and a linear inverse calculation with the iteration method, the enforcement operation is applied to move schedules generated by GA toward the edge. We prove that the optimal schedule lies on the edge, ensuring that searching along the edge instead of the entire space can improve the search efficiency significantly without missing the optimum. Furthermore, we provide a theoretical setting equation for the inverse enforcement gains of the linear inverse calculation, intended to move schedules closer to the edge at each iteration of the enforcement operation. The theoretical setting equation is verified and discussed with the test results. We propose the theoretical setting equation with the test results as a guideline for the use of our proposed search model: GA with enforcement operation.     

Scheduling algorithms for a semiconductor probing facility

This paper focuses on a scheduling problem in a semiconductor wafer probing facility. In the probing facility, wafer lots with distinct ready times are processed on a series of workstations, each with identical parallel machines. We develop a heuristic algorithm for the problem with the objective of minimizing total tardiness of orders. The algorithm employs a bottleneck-focused scheduling method, in which a schedule at the bottleneck workstation is constructed first and then schedules for other workstations are constructed based on the schedule at the bottleneck. For scheduling wafer lots at the bottleneck workstation, we consider prospective tardiness of the lots as well as sequence-dependent setup times required between different types of wafer lots. We also present a rolling horizon method for implementation of the scheduling method on a dynamic situation. The suggested methods are evaluated through a series of computational experiments and results show that the methods work better than existing heuristic methods.     

Optimization of microorganisms growth processes

Microorganisms growth processes are encountered in many biotechnological applications. For an increased economic benefit, optimizing their productivity is of great interest. Often the growth is inhibited by the presence in excess of other components. Inhibition determines the occurrence of multiple equilibrium points, which makes the optimal steady state reachable only from a small region of the system state space. Thus dynamic control is needed to drive the system from an initial state (characterized by a low concentration of microorganisms) to the optimal steady state. The strategy presented in this paper relies on the solutions of two optimization problems: the problem of optimal operation for maximum productivity in steady state (steady state optimization) and the problem of the start-up to the optimal steady state (transient optimization). Steady state optimization means determining the optimal equilibrium point (the amount of microorganisms harvested is maximum). The transient optimization is solved using the maximum principle of Pontryagin.The proposed control law, which drives the bioreactor from an initial state to the optimal steady state while maximizing the productivity, consists of switching the manipulated variable (dilution rate) from the minimum to the maximum value and then to the optimal value at well defined instants. This control law substantially increases the stability region of the optimal equilibrium point. Aside its efficiency, the strategy is also characterized by simplicity, being thus appropriate for implementation in real-life systems. Another important advantage is its generality: this technique may be applied to any microorganisms growth process which involves only one biochemical reaction. This means that the sequence of the control levels does not depend on the structure and parameters of the reaction kinetics, the values of the yield coefficients or the number of components in the bioreactor.     

Dynamic supply chain scheduling

Based on a combination of fundamental results of modern optimal program control theory and operations research, an original approach to supply chain scheduling is developed in order to answer the challenges of dynamics, uncertainty, and adaptivity. Both supply chain schedule generation and execution control are represented as an optimal program control problem in combination with mathematical programming and interpreted as a dynamic process of operations control within an adaptive framework. Hence, the problems and models of planning, scheduling, and adaptation can be consistently integrated on a unified mathematical axiomatic of modern control theory. In addition, operations control and flow control models are integrated and applicable for both discrete and continuous processes. The application of optimal control for supply chain scheduling becomes possible by formulating the scheduling model as a linear non-stationary finite-dimensional controlled differential system with the convex area of admissible control and a reconfigurable structure. For this model class, theorems of optimal control existence can be used regarding supply chain scheduling. The essential structural property of this model are the linear right parts of differential equations. This allows applying methods of discrete optimization for optimal control calculation. The calculation procedure is based on applying Pontryagin’s maximum principle and the resulting essential reduction of problem dimensionality that is under solution at each instant of time. The gained insights contribute to supply chain scheduling theory, providing advanced insights into dynamics of the whole supply chains (and not any dyadic relations in them) and transition from a partial “one-way” schedule optimization to the feedback loop-based dynamic and adaptive supply chain planning and scheduling.     

Performance Bounds on Multiprocessor Scheduling Strategies for Chain Structured Programs

In multiprocessors with static allocation of processes to processors, scheduling can be done locally for each processor. The scheduling strategy may have dramatic effect on the execution time of a parallel program. It is NP-hard to find an optimal schedule, and very little is known on how close the heuristic solutions get. In order to obtain nontrivial performance bounds, this study focuses on a restricted class of parallel programs, viz., chain structured programs. The major result is a theorem stating that if certain program parameters are known the execution time of the optimal schedule can be calculated within a factor of 2, even though the optimal schedule is unknown. Using a previously developed tool, one can extract the necessary parameters from a parallel program. This technique makes it possible to compare the execution time for different scheduling strategies with the optimal case. The technique used for calculating the performance bounds gives important hints on how to design efficient scheduling algorithms for chain structured programs.     

Simultaneous Scheduling of Replication and Computation for Data-Intensive Applications on the Grid

Managing large datasets has become one major application of Grids. Life science applications usually manage large databases that should be replicated to scale applications. The growing number of users and the simple access to Internet-based application has stressed Grid middleware. Such environment are thus asked to manage data and schedule computation tasks at the same time. These two important operations have to be tightly coupled. This paper presents an algorithm (Scheduling and Replication Algorithm, SRA) that combines data management and scheduling using a steady-state approach. Using a model of the platform, the number of requests as well as their distribution, the number and size of databases, we define a linear program to satisfy all the constraints at every level of the platform in steady-state. The solution of this linear program will give us a placement for the databases on the servers as well as providing, for each kind of job, the server on which they should be executed. Our theoretical results are validated using simulation and logs from a large life science application. This work was supported in part by the ACI GRID and Grid5000 projects of the French Department of Research.     

Theory and practice of optimal control in continuous fermentation process

In this paper a mathematical model of aminoacid fermentation using bacteria is built based on the kinetic study of experimental data of the pilot plant. This model makes it possible to formulate mathematically two optimal control problems in the actual process: (1) the problem of optimal start-up to a desired steady state, involving the optimization of the steady state, and (2) the problem of dynamical operation aiming at the maximum production of the amino-acid within a specified operation period. Both problems are solved by the maximum principle and Green's theorem. The numerical results suggest one of the most practical patterns of operation.     

Scheduling strategies for master-slave tasking on heterogeneous processor platforms

We consider the problem of allocating a large number of independent, equal-sized tasks to a heterogeneous computing platform. We use a nonoriented graph to model the platform, where resources can have different speeds of computation and communication. Because the number of tasks is large, we focus on the question of determining the optimal steady state scheduling strategy for each processor (the fraction of time spent computing and the fraction of time spent communicating with each neighbor). In contrast to minimizing the total execution time, which is NP-hard in most formulations, we show that finding the optimal steady state can be solved using a linear programming approach and, thus, in polynomial time. Our result holds for a quite general framework, allowing for cycles and multiple paths in the interconnection graph, and allowing for several masters. We also consider the simpler case where the platform is a tree. While this case can also be solved via linear programming, we show how to derive a closed-form formula to compute the optimal steady state, which gives rise to a bandwidth-centric scheduling strategy. The advantage of this approach is that it can directly support autonomous task scheduling based only on information local to each node; no global information is needed. Finally, we provide a theoretical comparison of the computing power of tree-based versus arbitrary platforms.     

Application of improved optimal regulator theory to optimal efficiency control of an electrical drive system

A new method to synthesize a speed control system for a dc-motor which attains maximum efficiency is proposed on the basis of an improved optimal regulator theory. Almost maximum efficiency is attained with negligible steady-state error except at the transient state. Considerable improvements on the efficiency at the steady state are obtained, compared to the usual control method in which only armature impressed voltage is controlled.     

A rigorous solution of the infinite time interval LQ problem with constant state tracking

A linear quadratic constant state tracking problem is considered over an infinite time horizon. It is shown that the solution cannot be obtained as a limit from a finite time horizon problem, as in general the limiting problem is ill-posed. To obtain a rigorous solution, the problem is split in two natural well-posed subproblems. One optimal control problem addresses the transient and the other optimal control problem concerns the steady state behavior. It is shown that the transient problem and the steady state problem are solved by the same control law.