Shi-Chung Chang Bo-Jiun Liao Argon Chen scchang@cc.ee.ntu.edu.tw r93546017@ntu.edu.tw achen@ntu.edu.tw
Graduate Institute of Industrial Engineering, National Taiwan University No.1, Sec. 4, Roosevelt Rd., Taipei, Taiwan 10617
Phone: +886 -223625187 Fax: +886-23638247 Abstract–This paper develops modeling methods and fab
behavior models with predictability and scalability to capture variability and manufacturing service differentiation in semiconductor supply chain management. A novel, hybrid decomposition approximation-based priority queueing network model is designed for fab behavior modeling. The model characterizes the relationship between output performance metrics of cycle time, wafer-in-process, throughput, and capacity utilization and input factors of priority mix, wafer release, capacity allocation and machine characteristics. Model evaluation results over some fab models demonstrate that the hybrid decomposition approximation-based network model yields very quick and good quality estimations of mean and variability of tool group and fab performance metrics.
I. INTRODUCTION
A supply chain is a system of nodes that provides manufacturing services—in fact, a variety of services.
Services differentiation, namely, prioritization, is common in operations of semiconductor supply chains (SSC, Figure 1).
It affects how to allocate resources and charge prices. Such a new paradigm of manufacturing services requires new methods of operation control. The grand challenges will be scalability and predictability with respect to differentiation of services and variability that are exacerbated by rapidly increasing product varieties and process variations in the chains.
To provide service with differentiable ensures that the quality of service (QoS), allocation of manufacturing capacity and pricing of services have to be dependent of and differentiated by QoS requirements. Product, process and operation variability affects the performance of individual service nodes such as tool groups and fabs and chains/networks of service nodes. In order to predict the behavior of the SSC that provide differentiated services, research is needed in follow aspects: predictable and scalable performance metrics with respect to the chain structure, and fundamental understanding of the behavior of service nodes and chains under variability.
Among the SSC service nodes, fab is the most expensive, complicated, and important. So this study is aimed at the behavior modeling of a fab and provides a cornerstone model for supply chain management. In behavior modeling, the output performance metrics consist of cycle time, wafer-in-process (WIP), throughput, and capacity utilization.
And input options to a fab include priority mix, wafer release, capacity allocation and machine characteristics. The behavior model describes the relationship between output performance metrics and inputs, which are characterized by not only mean
values but also variability.
In this paper, we develop behavior models and modeling methods that enhance the scalability and predictability of the semiconductor supply chains with respect to varieties, and differentiability of services. We aim at fab behavior modeling that provides a cornerstone model for SSC management. The network-based fab behavior models describe how priority, resource allocation and sources of variations affect fab performance metrics such as mean and variability of cycle time, wafer-in-process, throughputs, and machine utilizations as shown in Figure 2. The input/output relationship of a fab will be modeled as a network of priority service nodes [1].
Such a priority network captures factors and effects of variations throughout the network model. The model is scalable and allows chain/network performance metrics to be decomposed into per node and per priority metrics. It has also predictability that allows very quick evaluation of mean and variability of both nodal and system output performance metrics with various priority input options.
II. FAB BEHAVIOR MODELING BY HYBRID DECOMPOSITION OF PRIORITY NETWORK Consider a fab with multiple part priorities, failure prone machines, and re-entrant process flows. We model the fab as a failure-free, batch-free, deterministic-feedback and priority open queueing network (OQN) [4]. Then we design and develop an innovative, hybrid decomposition approximation-based approach for such a priority OQN with a focus on capturing operation priority and variations in a fab.
Key ingredients of the hybrid decomposition approach for modeling a priority fab OQN are as follows:
1. Decompose the fab network into many independent service nodes and their networking relationship by adopting the decomposition approximation of queueing network analyzer (QNA, [3]).
2. Model single service node behavior by sequential decomposition approximation (SDA) of coupling interactions among priorities in one node ([2], Figure 3;
3. Combine the networking relationship among service nodes with a fixed point iteration to approximate re-entrant flow line performance.
More details will be given in the following subsections.
II.1 Nodal Model: Sequential Priority Decomposition The behavior modeling of priority single service node provides a cornerstone for the fab behavior modeling. We develop the nodal behavior model with priority by treating each service node independently as a GI/G/1 non-preemptive
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priority queue, and then adopt the sequential decomposition approximation (SDA) proposed by [2] among priorities for our behavior modeling. SDA decomposes the coupling among priorities into approximately independent queues for individual priorities as shown in Figure 4 by the notion of equivalent service time.
SDA determines “equivalent” service parameters for each queue by taking the interactions with other queues into consideration. The most important feature to consider in each individual queue is that the service time of a job arriving into an empty queue differs from one arriving into a non-empty queue. If a job arrives into an empty queue, the equivalent service time is measured from its arrival; else if a job arrives into a non-empty queue, its equivalent service time is measured from the departure of the previous job having the same priority to its departure. Given the equivalent service time parameters and part release parameters, the means and variances of the overall nodal performances and the departure process of individual priorities can then be obtained.
II.2 Decomposition Approximation-based Queueing Network Modeling of Fab
In combining SDA with QNA [3], a priority open queueing network (OQN) [4] is first developed for a fab with multiple part types, multiple priorities, failure prone machines, and re-entrant process flows. This re-entrant OQN is analyzed by using a class of approximate decomposition methods. For better handling of uncertainties, the aspect of variability, i.e., second order statistics, as well as mean values is adopted to model the characteristics of this fab system. The decomposition methods decompose an OQN into individual network nodes and use two types of parameters to characterize the stochastic arrival, service and departure processes of each node: one describing the rate and the other describing the variability. Various stationary network performance measures, such as cycle time, WIP, and machine utilization, can then be derived based on these two types of parameters.
There is an OQN for each priority. OQNs of individual priorities are coupled through competition of service node resources. The priority coupling is handled by application of the SDA procedure to sequentially solving the equivalent service times from the highest priority in a node. We apply QNA to deal with interactions among nodes: splitting, merging, and deterministic feedback with priority. If a fab has no re-entrant flows, this fab network is tandem queue with flow always in a single direction from one node to the other. Figure 4 depicts a two node-example. The departure parameters of one node are equal to the arrival parameters of next node in tandem queues. We can directly use QNA to separate nodes in the network, and apply SDA to sequentially analyze the performances of each priority in each node. But in a fab, there are re-entrant flows and the arrival parameters of one node are affected by the departure parameters of many other nodes. Figure 5 depicts a simple re-entrant example of two nodes and two priorities. We deal with the re-entrant flows in a priority fab network by combining fixed point iteration [] over SDA and QNA.
III. MODEL EVALUATION
To evaluate the hybrid priority network model, we consider the small example of Figure 5. Figure 6 contrasts the cycle times obtained by hybrid SDA+QNA and simulation and there are good fits. Although there exists some difference in standard deviation for priority 2, the difference does not increase as the number of nodes increases.
Numerical experiments are also conducted on simple but full-scale fab models [6] to examine the efficiency, accuracy and application potential of hybrid decomposition approximation-based queueing network modeling. Also discrete event simulations are developed for validation of the fab behavior model. These fab models have two parts with two priorities classes, and the numbers of processing steps of P1 and P2 are 32 and 60, respectively shown in Figures 7 and 8. In a special fab model (SFM), all the service times of a node have exponential distributions. Other service node data is the same that shown in Table 1. Node and system level cycle times of the SFM are listed in Figure 9 and Table 2 respectively. The differences of mean cycle times of two priorities and the cycle time standard deviation of P2 are mostly within 3%. Although the difference of cycle time standard deviation of P1 is high (up to 45%), the absolute error is only 1.627.
In a general fab model (GFM), the service times of individual nodes have general distributions, such as uniform, erlang and exponential (see Table 1). Node and system level cycle times of the GFM are given in Figure 10 and Table 3 respectively.
The differences of mean cycle times of two priorities and the cycle time standard deviation of P2 are mostly within 10%.
The cycle time standard deviation of P1, has a high relative error of 55%. But again, the absolute error, 1.214, is still very small as compared to the mean.
Comparisons of numerical results with simulation in these two fab models show that our network modeling methodology has good approximations in most nodal and system performances. However, application of hybrid decomposition approximation to each model (listed in Table 4) only requires less than 4 seconds of CPU time on a 2.8 GHz personal computer. This leads to fast calculation with respect to changes of system input options. Consequently, both the accuracy and computing efficiency of hybrid decomposition approximation-based network modeling support its potential for applications of real fab with service differentiation.
IV. REMARKS ON APPLICATIONS
Responding to rapidly changing complex SSC requirements, SSC planners/managers need an effective performance evaluation/prediction tool to do what-if analysis between SSC inputs and outputs. Given a set of priority mix, capacity, mean and variability of wafer release, mean service time and variability of tools, the hybrid decomposition approximation-based network model allows very quick evaluation of mean and variability of nodal and fab performance metrics with good accuracy.
Effective evaluation of various input options in terms of capacity allocation, priority mix, wafer release policy, tool
adjustment, etc. may serve as a behavior model for SSC performance optimization. For example, the mean and variability statistics at individual tool groups may also be utilized by six-sigma management for on time and quick delivery, where the mean values can be used to derive control targets while variability values for calculation of control limits.
ACKNOWLEDGMENT
This work was supported in part by the Semiconductor Research Corporation and International Semiconductor Manufacturing Initiative under FORCe-II project task 1214.001 and by the National Science Council, Taiwan, ROC, under grants NSC-93-2213-E-002-043 and 94-2213-E-002-015.
Figure 1: Semiconductor supply chain
Figure 2: Fab behavior modeling
P1 moving
Figure 3: Decomposition approximation-based queueing network modeling
Figure 4: Decomposition among priorities
Figure 5: Two stations with reentrant line
Figure 6: Mean and standard deviation of cycle time for multi-nodes (2~5) with reentrant line
Enter 1 2 10
Figure 7: Process Flow of P1 in 2-PR fab model
Enter 1 2 3 8 10
Figure 8: Process Flow of P2 in 2-PR fab model Table 1: Service processes of nodes for general fab model
Node # of 1 1 Erlang Order 4 0.125 88.38 2 1 Exponential 0.125 85.23
3 1 Uniform 0.25 91.54
4 1 Erlang Order 3 1.8 68.18 5 1 Erlang Order 2 0.9 90.91 6 1 Erlang Order 4 0.6 83.33
7 1 Exponential 1.8 68.18
8 1 Erlang Order 3 0.2 80.81
9 1 Uniform 0.6 83.33
10 1 Erlang Order 2 0.3333 88.38 11 1 Exponential 0.6 83.33
12 1 Uniform 1.25 78.91
Mean Cycle Time
0
Number of Nodes Unit Time
SDA+QNA 1 SDA+QNA 2 Simulation 1 Simulation 2
Cycle Time Standard Deviation
0
Number of Nodes Unit Time
SDA+QNA 1 SDA+QNA 2 Simulation 1 Simulation 2
*Utilization % =
0.2525(# 1 )( )
# 100
0.3788(# 2 )( )
#
of P Visits MPT of Machines
of P Visits MPT of Machines
Figure 9: Mean & std. dev. of nodal cycle times (SFM) Table 2: System level performance comparisons (SFM) Mean Cycle Time Cycle Time
Standard Deviation
SDA+QNA 18.613 5.190
Simulation 18.572±0.014 3.5622±0.008 Absolute Error 0.041 1.627 1
Relative Error 0.22 % 45.68 %
SDA+QNA 172.78 54.674
Simulation 174.695±1.159 53.088±0.899 Absolute Error -1.915 1.586 2
Relative Error -1.10 % 2.99 %
Figure 10: Mean & std. dev. of nodal cycle times (GFM)
Table 3: System level performance comparisons of general fab model
Mean Cycle Time Cycle Time Standard Deviation
SDA+QNA 15.137 3.503
Simulation 15.451±0.009 2.289±0.005 Absolute Error -0.314 1.214 1
Relative Error -2.03 % 53.02 %
SDA+QNA 122.22 36.392
Simulation 112.015±0.668 38.133±0.424 Absolute Error 10.205 -1.740 2
Relative Error 9.11% -4.56%
Table 4: Comparisons of CPU times in fab models Average CPU time (seconds) Two Priorities
Fab Models Hybrid SDA+QNA Simulation (one run)
Special case 3.606 2466
General case 3.622 2484
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