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Simulation metamodel development using uniform design and neural networks for automated material handling systems in semiconductor wafer fabrication
Simulation Modelling Practice and Theory 15 (2007) 1002–1015 www.elsevier.com/locate/simpat
Simulation metamodel development using uniform design and neural networks for automated material handling systems in semiconductor wafer fabrication Yiyo Kuo
a,*
, Taho Yang
b,1
, Brett A. Peters c, Ihui Chang
d
a
c
Department of Technology Management, Hsing Kuo University of Management, Tainan 709, Taiwan b Institute of Manufacturing Engineering, National Cheng Kung University, Tainan 701, Taiwan Department of Industrial and Systems Engineering, Texas A&M University College Station, TX 77843-3131, USA d Industrial Engineering Department, Taiwan Semiconductor Manufacturing Company, Ltd., Hsinchu, Taiwan Received 28 July 2006; received in revised form 18 April 2007; accepted 30 May 2007 Available online 16 June 2007
Abstract Simulation is very time consuming, especially for complex and large scale manufacturing systems. The process of collecting adequate sample data places limitations on any analysis. This paper proposes to overcome the problem by developing a neural network simulation metamodel that requires only a comparably small training data set. In the training data set, the configuration of all input data is generated by uniform design and the corresponding output data are the result of simulation runs. A dispatching problem for a complex simulation model of an automated material handling system (AMHS) in semiconductor manufacturing is introduced as an example. In the example, there are 23 4-levels factors, resulting in a total of 423 possible configurations. However, by using the method proposed in this paper, only 28 configurations had to be simulated in order to collect the training data. The results show that the average prediction error was 3.12%. The proposed simulation metamodel is efficient and effective in solving a practical application. 2007 Elsevier B.V. All rights reserved. Keywords: Neural network; Semiconductor manufacturing; Simulation metamodel; Uniform design
1. Introduction Simulation is a widely accepted tool for the design and analysis of manufacturing systems. It can model non-linear and stochastic problems and allow examination of the likely behavior of a proposed manufacturing system under selected conditions. Through simulation analysis, many details and constraints can be considered in the evaluation of a manufacturing system. *
1
Corresponding author. Tel.: +886 6 2870923. E-mail addresses: [email protected] (Y. Kuo), [email protected] (T. Yang). Tel.: +886 6 2090780; fax: +886 6 2085334.
1569-190X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2007.05.006
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However, simulation modeling and analysis is often challenging and time consuming [23], particularly, when it is used for modeling a complex manufacturing system such as semiconductor manufacturing fabrication. In this case, model building itself is a challenging task and often requires an experienced programmer, even when a high level programming language is used. The analysis of the simulation output from a validated model is another challenge that can take a significant amount of computing time. To overcome these two problems, the use of a metamodel has been proposed to address the drawbacks and represents an emerging research direction [34]. A simulation metamodel can suggest a functional relationship between selected decision variables and system responses. It provides an approach to predicting simulation results, allowing some extrapolation from the simulated range of system conditions and therefore potentially offering some assistance in the decision making process [46]. Let Xj denote a factor j influencing the outputs of a real-world system (j = 1,2, . . . , s), and let Y denote a system response vector and Y = {y1,y2, . . . , yw}. Kleijnen [20] defined the metamodel as follows. The relationship between the response variable Y and the inputs Xj of the system can be represented as Eq. (1): Y ¼ f1 ðX 1 ; X 2 ; . . . ; X s Þ
ð1Þ
A simulation model is an abstraction of the real system, in which we consider only a selected subset of the input variables {Xjjj = 1,2, . . . , r} where r is significantly smaller than the unknown s. The response of the simulation Y 0 is then defined as a function f2 of this subset and a vector of random numbers v representing the effect of the excluded inputs, as shown in Eq. (2): Y 0 ¼ f2 ðX 1 ; X 2 ; . . . ; X r ; vÞ
ð2Þ
Similarly, a metamodel is a further abstraction, in which we select a subset of the simulation input variables {Xjjj = 1,2, . . . , m} and m 6 r. The system can then be represented by Eq. (3) where e denotes a fitting error. Y 00 ¼ f3 ðX 1 ; X 2 ; . . . ; X m Þ þ e
ð3Þ
Metamodels are often constructed in two primary stages. The first stage uses a factorial experimental design to collect a structured data set that is then used to find the functional relationship between decision variables and responses [16,21,26]. Regression analysis and neural networks are the two commonly used methods for finding this functional relationship. In particular, the neural network approach has been shown to be a promising modeling tool [5,14,16,18,30,32]. After a metamodel is constructed, the what-if analysis can be obtained without the further consumption of expensive computing resources. Furthermore, it may be used for system optimization. For example, the response surface method (RSM) can be used to optimize a regression metamodel [35,37]. The present study proposes a simulation metamodel for automated material handling system (AMHS) in semiconductor manufacturing which represents a very complex system for simulation analysis. Due to its complexity, an analytical model is not a viable approach. The objectives of the present study are as follows: • to build a simulation model for AMHS in semiconductor manufacturing, • to use an experimental design to efficiently collect the metamodeling data, and • to use a neural network to build a simulation metamodel. The remainder of the paper is organized as follows. Section 2 provides a review of the relevant literature leading to the modeling and analysis approach employed in the present research. The details for the case study and its associated simulation model are presented in Section 3. The neural network based metamodel is presented in Section 4. Finally, the conclusions and future research direction are presented in Section 5. 2. Literature review A semiconductor fabrication facility is a very large investment. For example, a 300 mm fabrication unit costs more than US$2 billion. Therefore, any non-appropriate production decision always brings about
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serious waste. Among the various production decisions that have to be made, dispatching decisions play a critical role in optimizing system performance in semiconductor fabrication. An automated material handling system (AMHS) of semiconductor manufacturing is illustrated in Fig. 1. In general, there are two types of systems: one is the intrabay system, which transports material within one process bay; the other one is interbay system, which transports material between process bays. The interbay system contains double loops, the vehicles move clockwise on one loop and move counterclockwise in the other loop. In addition, there are buffers between the interbay system and any intrabay system, called a ‘stocker’. When a material has to be transported from one machine to another machine located in another process bay, it has to wait for a vehicle from the intrabay system to transport material to the stocker first. Then the system has to select one loop of the interbay system and wait for a vehicle which is on the selected loop to transport the material to another stocker. Finally, it will be transported to the target machine by the vehicle of another intrabay system. All material in semiconductor manufacturing fabrication is transported by the AMHS. Different vehicle dispatching rules for each intrabay and interbay system will affect the waiting time and total cycle time for the material. Therefore, optimizing the dispatching rules for all interbay and intrabay systems will be important for the decision support system. There is some literature on the optimization of dispatching rules using simulation analysis. In a recent study, Lin et al. [25] considered three decision points in an event-driven dispatching strategy used to control
Legend
: Interbay system
: Automated vehicle
: Intrabay system
: Stocker
Fig. 1. AMHS of semiconductor manufacturing system.
: Machine
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the vehicles efficiently in a double loop interbay material handling system. A full factorial simulation experiment with four loop selection rules, one cassette-initiated dispatching rule and two vehicle-initiated dispatching rules were then used to find the best strategy. However, only interbay systems were considered. Tyan et al. [41] present an integrated tool and vehicle dispatching strategy for semiconductor manufacturing. A two level 1/4 fractional factorial design was used to investigate the main effects and identify the most favorable dispatching combination. However, in a 300 mm semiconductor fabrication the total number of tool types is more than forty, and any category may still contain many tool types. Accordingly, a simulation model of a real semiconductor manufacturing AMHS system is not only complex and difficult to develop but also very time consuming for analysis. For example, the model built by Tyan et al. [41] takes more than seven hours to collect the outputs of one simulation replication. Therefore, a speedy model to explain the relationship between the input and output may be required. Artificial neural networks (ANNs) are composed of interconnected, usually adaptive, elements, which are intended to respond to stimuli in a manner not unlike the human nervous system [22]. There are three basic elements of an ANN model [10], which are illustrated in Fig. 2. They are synapses, adder and activation function. Synapses are the connecting links between neurons. All the input signals (Xi) transferred by synapses have to be multiplied by its own weighting (Wi). Then the adder sums the bias (b) and all the signals, which have been multiplied, to generate a value (u). Finally, the value (u) will be translated to become the output (Y) of the neuron by the activation function. The adder, bias and activation function form a node. Then the output can be the input of other nodes. Through a learning algorithm and given a set of training examples (including inputs and corresponding desired outputs), all the weights (Wi) can be iteratively modified to minimize the difference between outputs (Y) and desired outputs. There are many kinds of neural network models. Multilayer perceptron, which only feeds forward, multilayered and fully connected networks, were considered in the present study. It consists of three or more layers, including one input layer, one or more hidden layers and one output layer. Fig. 3 illustrates an example of a network with three layers. In Fig. 3, every circle is a node, and every connection between nodes is a synapse. There are two types of learning: supervised and unsupervised. For supervised learning, a set of training input vectors with a corresponding set of target vectors is used to adjust the weights in a neural network. For unsupervised learning, a set of input vectors is proposed, but no target vectors are specified. In this paper, the corresponding target vector of every input vector was generated by a simulation run. The input set and the corresponding output set can be collected before neural network training. Therefore supervised learning was used in the proposed methodology. A supervised learning algorithm named back error propagation (BEEP) (also called the ‘generalized delta rule’), which was presented by Rumelhard et al. [33], is also used in the present study. In contrast with backpropagation (or named delta rule), BEEP accelerates and stabilizes training by including a momentum term [17,43], and it is the most widely used supervised learning algorithm for Multilayer perceptrons [10]. A simulation metamodel using a neural network used the results from a simulation run under different configurations to be the training data, whereby these different configurations were used as input data and the
Fig. 2. Model of a neuron.
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... ...
... Input layer
Hidden layer
Output layer
Fig. 3. A Multilayer perceptron neural network.
corresponding simulation results were used as output data. The trained neural network was then used to quickly predict the simulation results of new configurations. Hurrion [15] developed a neural network simulation metamodel and then optimized the number of kanbans in a manufacturing system. Badiru and Sieger [2] used a neural network as a simulation metamodel in an economic system, and then accelerated the analysis of the future performance of the investment project. Kilmer et al. [19] used a supervised neural network as a metamodel in an inventory simulation system to estimate the confidence interval for estimates of total cost. Fonseca and Navaresse [9] constructed a neural network simulation metamodel to estimate the manufacturing lead time for orders simultaneously processed in a four-machine job shop. However, there is still one disadvantage when using a neural network for developing a simulation metamodel, especially, when the time spent is too long for one simulation run. Since simulation is time consuming, it is still impossible to collect a comprehensive configuration of results by a large number of simulation runs for the training data. For example, in the foregoing literature, Hurrion [15] used 128 simulation results for training data; Kilmer et al. [19] and Fonseca and Navaresse [9] used 900 simulation results for training data. 128 or 900 simulation runs may still be too many to be practicable in some complex simulation cases. Uniform design (UD) was proposed by Fang [8] and Wang and Fang [42]. Its most important merit is that it reduces the number of experimental configurations, especially when the experimental region has many factors and multiple levels. UD is a space filling design and it requires that experimental points be uniformly scattered on the domain [8]. For an example with two factors with 8 levels, such as the one illustrated in Fig. 4 [24], the total number of possible experimental points is 82 = 64. If a set with only 28 experimental points is to be selected, then the set of selected points will be scattered uniformly, when the absolute value of the ratio of the number of selected points lying in the rectangle to the total number of selected points of the set minus the area of the rectangle is small. This absolute value is called the discrepancy. In Fig. 4a the number
1.0
1.0
0.4
0.4
0.75
1.0
Fig. 4. A uniform design example.
0.75
1.0
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of selected points lying in the rectangle is 10, the area of the rectangle is 0.4 · 0.75. Therefore, the discrepancy is about j10/28 0.4 · 0.75j = 0.0571. In Fig. 4b the discrepancy is about j15/28 0.4 · 0.75j = 0.2357. Because the discrepancy in Fig. 4a is very small, the points are considered to be scattered uniformly in the domain when the type of rectangle is 0.4 · 0.75. However, the number of rectangle types in the domain is unlimited, and the selection of a rectangle is a continuous problem. When the number of experimental points is big enough, seeking these points will be an NP-hard problem [24]. Winker and Fang [44] used a stochastic global optimization algorithm called the threshold accepting (TA) algorithm to search the points, and the results, called a uniform design table, can be found at the web site address: http://www.math.hkbu.edu.hk/UniformDesign. Developing a neural network metamodel by using a uniform design table is rare in the literature. Shan et al. [36] used a uniform design table to generate a training data set, and then developed a neural network metamodel. Using the metamodel the pH value and solvent composition in high performance liquid chromatography (HPLC) were optimized. In the study, two seven-level factors were considered. The results showed the good predicting ability of the metamodel that was constructed using this method. 3. Semiconductor manufacturing automated material handling systems In this paper, a commercial software program, eM-Plant [7], was used as the simulation tool. All dispatching rules are coded using the eM-plant embedded programming language (called Method) for customized applications. Its syntax is similar to the programming language C++. The simulation model contains 23 intrabay systems and one double-loop interbay system, and the number of vehicles for every intrabay and interbay system are shown in Appendix A1 and A2 respectively. There are 277 machines, which divide into 43 types, and all wafers need to be processed through 316 operation steps. All the machine data such as location, downtime and buffer size, and process data such as process step, tool name, processing time, maximal size per batch and batching time are not presented in this paper, because of the huge quantity of the data. However, readers who wish to follow up more details of the case can find the relevant data in [3]. In addition, the capacity of all stockers was assumed to be infinite. The releasing policy was 22,000 wafers per month. According to the results of pilot runs, the simulation needed to run for 11 months with 1 six-month warm up period and 5 one-month batches. It took about six simulation hours using a computer with a Celeron 2.3 GHz processor. Therefore, a computationally efficient method to replace the simulation model was needed so that it would be possible to conduct even more analysis in order to optimize the system. This paper aims to develop a neural network simulation metamodel which can replace a simulation to predict the performance of the AMHS when given a set of vehicle dispatching rules for all 23 intrabay systems. Cycle time (CT), which is an important performance index, will be the prediction target. As regards the tool dispatching rule, vehicle dispatching rule of the interbay system and loop selection rule, the first in first out (FIFO), nearest vehicle (NV), and shortest distance (SD) rules, which have proven good performance in AMHS [25], were used respectively. The following vehicle dispatching rules are the candidates for every intrabay system. Least slack (LS): This is a dispatching policy that effectively improves the due date commitment. When a vehicle has just finished, it will search all outputs of machines which have unassigned material and then select the one whose slack is the least. The slack is due date minus current time and remaining processing time. First encounter first service (FEFS): This aims to minimize the travel time of empty vehicles. According to the direction of motion, a vehicle will search for the nearest output of a machine, which has unassigned material and then move to load it. Longest waiting time (LWT): When a vehicle has just finished, it will search all the unassigned material on the output of machines and then select the one which has been waiting longest for transport. High value first (HVF): All materials were divided into three priority values. A vehicle will select an unassigned item of material with the highest priority value on the output of a machine, and then move to load it. The resulting example of the model interface is shown in Fig. 5.
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Fig. 5. Simulation interface.
4. Developing a neural network simulation metamodel In this section a proper uniform design table was used to select scenarios for the proposed simulation model to generate a set of training data. Then a neural network was trained using the data set. Finally some tests and analyzes were conducted. The details were as follows: 4.1. Training data set generation In this case, there are 23 factors with 4 levels. The total number of possible configurations is 423. Traditionally, using the Taguchi method or orthogonal array, the minimal number of experiments are 70 and 80 (see Appendix B for more detail). However, according to the uniform design table of the form Un(423), which was taken from the web site cited above (n is the desired number of experiments) the number of experiments can be in the range from 24 to 30. Because n should be the common multiple of all the levels of variables, n can be 24 or 28. This research selected the uniform design table U28(423) and is shown in Appendix C. In the table, the numbers from 1 to 7 were set as LS, the numbers from 8 to 14 were set as FEFS, the numbers from 15 to 21 were set as LWT and the numbers from 22 to 28 were set as HVF. The simulation results of all 28 configurations are also shown in Appendix C. For comparison purposes, another training data set of 96 configurations was generated by orthogonal array L96(248) for comparison. For further examination of the method using an orthogonal array of this type L96(248), the reader is referred to [13]. The metamodel trained by the uniform design configuration data set was denoted by UDM, and the one trained by the orthogonal array configuration data set was denoted by OAM.
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The proposed uniform design and the orthogonal array use a structured experimental design to collect data for the construction of the metamodel. This is a fractional factorial experiment. It does not reduce variables but significantly reduces the required experimental treatments, which is the power of an orthogonal array [27,31,39]. Note that the existence of the noisy data is commonly found in the pattern recognition problems. The reduced sample size is one of the advantages to adopt the UDM. Due to the resulting small sample size, we will not further explore the noisy data issue. We do not feel the noisy data will be a problem for the present study since the later model validation step will illustrate an unstable training if there are significant noisy data. 4.2. Training data representation Data representation is a critical issue that directly affects the resulting neural network architecture [9]. Each bay will adopt one out of the four candidate dispatching rules—LS, FEFS, LWT, and HVF which are represented by the permutation of +1 and 1. Specifically, LS, FEFS, LWT, and HVF are represented by (1, 1), (1, +1), (+1, 1), and (+1, +1), respectively. Accordingly, there are 46 (=23 · 2) data fields to represent the different dispatching rules for the 23 bays. As regards the output data, the simulation result cycle time of ith configuration was scaled into the range between 1 and +1 by using Eq. (4). oi ¼ 1 þ
2 ðy i yÞ y y
i ¼ 1; 2; . . . ; 24
ð4Þ
In Eq. (4) oi is the corresponding output data of ith input data, yi is the simulation result cycle time of ith configuration, y is the maximum value of all yi, and y is the minimum value of all yi. A neural network with 46 inputs, one output and one hidden layer was then used. The number of nodes in the hidden layer was decided by the method shown in the following section. 4.3. Neural network building and training The training data sets contain 28 simulation results. The configuration of the neural network has 46 input nodes and 1 output node. The number of parameters (connections) of the network is more than the number of training data. For this instance, the trained network will potentially give a very close fit to the data; this is referred to as memorization [12]. This is also known as over-fitting [40]. To overcome this problem, Hagan et al. [11] suggested using another data set for validating the result of training process. If the mean square error (MSE) of validation data is not acceptable, then the trained neural network is over-fit and is not adequate for the application problem. In the present study, the 28 training data were split into two data sets, 17 (60%) for training and 11 (40%) for validation. The back-propagation training algorithm including a momentum term was used. Each neural network was trained for 1000 epochs, so that the training results would not be sensitive to the set-up parameters after so many iterations. This research sequentially determined the network parameters, such as the number of hidden layer nodes, learning rate and momentum constant. This approach has been widely applied to the determination of parameters in a neural network, as is reported in [1,6,28,38,45]. Initially the learning rate was set to 0.1 and the momentum constant was set to 0.9. Then different neural networks with 20, 30, 40, and 50 nodes in the hidden layer were tested respectively. Because initial weights were generated randomly, and training data set is very small, the training results of the same neural network were quite variable between different trainings; moreover, the MSE of the training data set were always far lower than the validation data set in the prior testing. Therefore, every structure of neural networks was trained ten times with different initial weights, and the final training result of every structure was chosen to be the one whose MSE of validation was lowest among the ten training results. By using neural network software NeuralSolution [29], the training results of UDM were as shown in Table 1. In Table 1 the MSE value of the training set and the validation set were lowest when the network had 30 nodes in the hidden layer. Then the learning rate was change to 0.05, 0.01and 0.005 respectively, and the momentum constant was still set to 0.9. The training results of the network with 30 nodes in the hidden layer were as shown in Table 2.
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Table 1 Training results with different number of nodes in the hidden layer Number of nodes in hidden layer
MSE (training set)
MSE (validation set)
20 30* 40 50
0.000053 0.000020* 0.000001 0.000004
0.330768 0.306489* 0.375323 0.371402
Table 2 Training results with different learning rates Learning rate
MSE (training set)
MSE (validation set)
0.1 0.05 0.01* 0.005
0.000020 0.000001 0.006399* 0.005232
0.306489 0.310198 0.261681* 0.349236
In Table 2 the MSE value of the training set and the validation set were lowest when learning rate was 0.01. Then the momentum constant was change to 0.95 and 0.8 respectively. The training results of the network with 30 nodes in the hidden layer and 0.01 learning rate were as shown in Table 3. In Table 3 the MSE value of the training set and the validation set were lowest when the momentum constant was 0.9. Finally the neural network which contained 30 nodes in the hidden layer, training with 0.01 training rate and 0.9 momentum was selected to be the final training result of UDM. The training result of OAM was developed using a similar approach and it is contained 20 nodes in the hidden layer, training with a training rate of 0.005 and momentum of 0.9. 4.4. Neural network testing and analysis Another data set with 10 configurations was generated randomly for testing the neural network simulation metamodel. The prediction time for all testing data was less than one second. The MSE value is 0.483095 for UDM and 0.488047 for OAM and the difference between them is only 1.01% Therefore, the accuracy of UDM and OAM are similar. The test results of UDM were as shown in Table 4. Table 4 show the average prediction error of UMD is 3.12%. However, there is still one prediction error that exceeds 5%. Even so, the UDM is still useful, because, for UDM, the prediction time is less than one second, and it can predict a lot of configurations in a very short time. When analyzing or optimizing with the simulation model, only a few additional simulation runs are needed for confirmation, as proposed in [15]. The main object in developing the simulation metamodel is acceleration of analysis and optimization. The normal method for optimization is predicting all possible configurations with the metamodel [4,14]. However, the total number of possible configurations in this paper was 423, which is too big, and there was a need for other methods. In this paper, only some common, practical configurations, in which all intrabay systems have the same vehicle dispatching rule, were analyzed. The results were as shown in Table 5. In Table 5, from second row to fifth row show the predicted results when the vehicle dispatching rule for every intrabay system were all LS, FEFS, LWT and HVF respectively. The last row shows the best predicted result of the UDM from the testing data set. Although the result in the last row is the best on the testing data, it is not the optimal result. But obviously, the cycle time is much shorter than testing all configurations with a single vehicle dispatching rule. Table 3 Training results with different momentum constant Momentum constant
MSE (training set)
MSE (validation set)
0.95 0.9* 0.8
0.000020 0.006399* 0.003742
0.344452 0.261681* 0.268490
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Table 4 Testing results of UDM Testing no.
1 2 3 4 5 6 7 8 9 10
Prediction result
Error %
Metamodel
Simulation model
19.063 19.862 19.890 19.686 20.088 19.679 18.243 18.205 18.857 20.151
18.986 19.188 19.756 19.092 19.144 19.350 19.516 17.824 18.220 19.240
Average
0.40 3.51 0.68 3.11 4.93 1.70 6.52 2.14 3.50 4.74 3.12
Table 5 Analysis results for all intrabay systems with same vehicle dispatching rule Vehicle dispatching rule
LS FEFS LWT HVF The best of testing
Prediction result
Error %
Metamodel
Simulation model
18.212 18.974 18.966 20.088 18.205
19.874 19.006 19.130 20.122 17.824
8.36 0.17 0.85 0.17 2.14
5. Conclusions In generally, using neural network, the training data set should be large enough when comparing to the network size. However, the main difficulty of the proposed simulation case is the time cost of data collection, and it is difficult to collect sufficient training data. Uniform design have been proofed can explain the domain space by quite experimental point. Therefore, this paper propose a way to develop a neural network simulation metamodel using a very small training data set, in which uniform design was used for generating the configurations of input data. A simulation model of an AMHS in a semiconductor manufacturing system with 23 intrabay systems was introduced. For every intrabay system there were four alternative vehicle dispatching rules and the total number of possible configurations was 423. However, in the experiment, only 28 simulation configurations were run for the training data set, and this is far fewer than would be required for the Taguchi method or orthogonal array. The results showed that the training data set generated by uniform design performs similarly to the much larger set generated by orthogonal array. Some error between the metamodel prediction and the simulation run still existed. However, when considering the trade-off between prediction accuracy and speed, the proposed method was encouraging. Moreover, this paper also analyzes some configurations in which the vehicle dispatching rules of all intrabay systems were the same. The results show that finding suitable vehicle dispatching rules for every intrabay system will be an opportunity for system improvement in the future. Since the neural network simulation metamodel can accelerate the prediction operation from several hours to less than a second, it will be possible to find improved and perhaps near optimal configurations by combining some global optimizing algorithms. Although the concept of a metamodel itself is not novel, the use of a small data set to model a complex problem is worth investigating. As discussed in this paper, the modeling of an automated material handling system in semiconductor wafer fabrication is a complex problem. The simulation model building itself is challenging. In addition, the use of uniform design is not a straightforward matter. The combined use of uniform design and neural networks is an effective and efficient method for the construction of the metamodel for the
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proposed problem. In comparison to other methods such as the Taguchi orthogonal array, the proposed methodology illustrated its efficiency, which is an important factor for practice, particularly, for those readers who place an emphasis on practical application. This paper only developed a simulation metamodel to predict the cycle time when given vehicle dispatching rules for every intrabay system. In an AMHS, other performance indices, such as work in process (WIP), service level, utilization, and so on are also important and should be considered in future work. Acknowledgement This work is supported in part by the National Science Council of Taiwan, Republic of China under Grant NSC 94-2213-E-432-002 and NSC 93-2917-I-006-016. Appendix A1 The number of vehicle for every intrabay system Intrabay code
Number of vehicles Intrabay code
Number of vehicles Intrabay code
Number of vehicles
A B C D E F G H
2 4 5 4 4 3 6 3
2 4 3 3 2 4 2 2
2 2 1 1 2 4 1
I J K L M N O P
Q R S T U V W
Appendix A2 The number of vehicles for every loop of interbay systems Loop direction
Number of vehicles
Clockwise Counterclockwise
15 15
Appendix B
Minimum number of simulation runs Total simulation run time (days)
Uniform design
Taguchi method
Orthogonal array
24 6
70 17.5
80a 20
The second row of the above table shows the minimum requirement of experimental treatment for every experimental design. For generating the training data set, one treatment means one simulation run, and takes about 6 h. Then the third row shows the corresponding total simulation run time. Accordingly, developing a neural network simulation metamodel by using uniform design is more effective then using either the Taguchi method or orthogonal array. a According to [13], when the number of factors (k) is 23 and number of levels (s) is 4, the minimum number of experiments (N) will equal to k Æ st. Meanwhile, t (strength) is 2 and k is between 5 and 8. Therefore, N will be least and equal to 80 (5 Æ 42 = 80). Notes, the orthogonal array is still unknown when AO(N, k, s, t) = AO(80, 23, 4, 2), and it just can be sure that N will not be less than 80.
Appendix C Uniform design table U28(423)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Factors 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Responses cycle time (days)
22 8 23 18 6 20 17 26 27 10 1 14 24 28 16 9 3 21 12 17 25 15 4 13 5 2 11 19
27 12 20 25 23 6 13 28 2 10 16 4 15 1 26 19 11 18 17 9 8 7 21 5 24 3 14 22
15 18 20 8 16 4 26 1 21 5 2 17 13 19 10 22 23 28 27 12 7 9 6 25 25 14 2 24
18 27 21 2 19 15 6 25 7 5 26 16 20 28 4 14 22 8 23 12 11 24 10 17 9 1 3 13
3 17 21 15 14 12 4 28 8 26 11 10 5 18 9 7 6 27 2 23 1 22 16 24 19 20 13 25
10 14 25 18 26 23 9 3 27 24 20 6 5 11 13 8 19 17 21 2 28 7 12 15 4 16 1 22
28 23 4 3 11 26 10 17 20 18 2 19 12 1 8 6 25 9 14 15 13 16 24 7 21 27 5 22
6 14 5 3 21 16 9 13 11 10 12 1 17 23 27 19 26 28 7 8 18 25 2 4 20 24 15 22
20 25 6 18 17 12 26 10 28 3 24 5 19 1 8 2 7 23 9 11 15 22 4 27 21 16 13 14
1 26 22 8 3 7 23 24 10 2 6 27 14 12 18 9 16 15 19 4 17 5 13 20 11 21 28 25
2 27 18 23 17 5 15 11 16 14 8 4 21 19 28 7 9 1 26 24 13 25 20 3 10 22 12 6
11 8 7 24 6 10 9 13 18 15 16 4 28 17 12 21 25 2 23 22 1 5 3 26 20 19 14 27
14 8 16 2 25 27 15 13 18 6 10 1 9 23 12 3 20 24 19 26 7 4 22 21 17 11 28 5
2 12 18 15 22 16 26 21 20 3 10 14 4 1 27 25 23 6 5 19 11 24 9 28 8 17 7 13
5 4 8 17 3 27 25 14 15 2 23 18 9 24 28 1 11 12 26 13 16 21 20 10 19 7 6 22
26 25 18 11 7 4 5 1 28 6 21 10 8 14 27 15 20 12 3 23 16 17 19 13 9 2 22 24
14 3 7 2 27 1 20 16 11 21 19 13 28 17 18 6 12 10 5 22 26 8 25 23 9 15 4 24
14 26 3 27 25 5 4 24 15 16 8 17 2 20 19 9 18 13 21 11 28 10 7 22 23 1 12 24
5 21 19 17 12 16 15 14 11 27 20 25 24 13 26 9 23 28 4 22 3 6 7 8 18 1 2 10
8 25 18 5 3 27 6 17 22 19 11 2 28 1 20 21 7 14 15 9 16 13 23 26 24 12 10 4
9 16 3 13 19 14 5 28 24 6 27 21 17 8 15 23 2 16 22 18 1 7 20 11 4 25 10 12
22 14 24 5 15 18 14 2 1 12 28 19 6 16 21 17 4 9 13 26 27 8 3 20 25 23 7 11
13 3 20 8 26 4 6 10 25 17 16 27 22 12 23 5 9 7 18 1 2 21 11 15 24 14 28 19
18.916 19.134 20.002 18.628 19.658 18.742 19.820 20.070 19.434 18.800 18.190 20.048 19.690 19.436 19.728 19.816 19.362 20.124 19.274 19.056 18.626 19.230 18.898 19.652 19.920 18.944 18.600 18.396
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Simulation metamodel development using uniform design and neural networks for automated material handling systems in semiconductor wafer fabrication Yiyo Kuo
a,*
, Taho Yang
b,1
, Brett A. Peters c, Ihui Chang
d
a
c
Department of Technology Management, Hsing Kuo University of Management, Tainan 709, Taiwan b Institute of Manufacturing Engineering, National Cheng Kung University, Tainan 701, Taiwan Department of Industrial and Systems Engineering, Texas A&M University College Station, TX 77843-3131, USA d Industrial Engineering Department, Taiwan Semiconductor Manufacturing Company, Ltd., Hsinchu, Taiwan Received 28 July 2006; received in revised form 18 April 2007; accepted 30 May 2007 Available online 16 June 2007
Abstract Simulation is very time consuming, especially for complex and large scale manufacturing systems. The process of collecting adequate sample data places limitations on any analysis. This paper proposes to overcome the problem by developing a neural network simulation metamodel that requires only a comparably small training data set. In the training data set, the configuration of all input data is generated by uniform design and the corresponding output data are the result of simulation runs. A dispatching problem for a complex simulation model of an automated material handling system (AMHS) in semiconductor manufacturing is introduced as an example. In the example, there are 23 4-levels factors, resulting in a total of 423 possible configurations. However, by using the method proposed in this paper, only 28 configurations had to be simulated in order to collect the training data. The results show that the average prediction error was 3.12%. The proposed simulation metamodel is efficient and effective in solving a practical application. 2007 Elsevier B.V. All rights reserved. Keywords: Neural network; Semiconductor manufacturing; Simulation metamodel; Uniform design
1. Introduction Simulation is a widely accepted tool for the design and analysis of manufacturing systems. It can model non-linear and stochastic problems and allow examination of the likely behavior of a proposed manufacturing system under selected conditions. Through simulation analysis, many details and constraints can be considered in the evaluation of a manufacturing system. *
1
Corresponding author. Tel.: +886 6 2870923. E-mail addresses: [email protected] (Y. Kuo), [email protected] (T. Yang). Tel.: +886 6 2090780; fax: +886 6 2085334.
1569-190X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2007.05.006
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However, simulation modeling and analysis is often challenging and time consuming [23], particularly, when it is used for modeling a complex manufacturing system such as semiconductor manufacturing fabrication. In this case, model building itself is a challenging task and often requires an experienced programmer, even when a high level programming language is used. The analysis of the simulation output from a validated model is another challenge that can take a significant amount of computing time. To overcome these two problems, the use of a metamodel has been proposed to address the drawbacks and represents an emerging research direction [34]. A simulation metamodel can suggest a functional relationship between selected decision variables and system responses. It provides an approach to predicting simulation results, allowing some extrapolation from the simulated range of system conditions and therefore potentially offering some assistance in the decision making process [46]. Let Xj denote a factor j influencing the outputs of a real-world system (j = 1,2, . . . , s), and let Y denote a system response vector and Y = {y1,y2, . . . , yw}. Kleijnen [20] defined the metamodel as follows. The relationship between the response variable Y and the inputs Xj of the system can be represented as Eq. (1): Y ¼ f1 ðX 1 ; X 2 ; . . . ; X s Þ
ð1Þ
A simulation model is an abstraction of the real system, in which we consider only a selected subset of the input variables {Xjjj = 1,2, . . . , r} where r is significantly smaller than the unknown s. The response of the simulation Y 0 is then defined as a function f2 of this subset and a vector of random numbers v representing the effect of the excluded inputs, as shown in Eq. (2): Y 0 ¼ f2 ðX 1 ; X 2 ; . . . ; X r ; vÞ
ð2Þ
Similarly, a metamodel is a further abstraction, in which we select a subset of the simulation input variables {Xjjj = 1,2, . . . , m} and m 6 r. The system can then be represented by Eq. (3) where e denotes a fitting error. Y 00 ¼ f3 ðX 1 ; X 2 ; . . . ; X m Þ þ e
ð3Þ
Metamodels are often constructed in two primary stages. The first stage uses a factorial experimental design to collect a structured data set that is then used to find the functional relationship between decision variables and responses [16,21,26]. Regression analysis and neural networks are the two commonly used methods for finding this functional relationship. In particular, the neural network approach has been shown to be a promising modeling tool [5,14,16,18,30,32]. After a metamodel is constructed, the what-if analysis can be obtained without the further consumption of expensive computing resources. Furthermore, it may be used for system optimization. For example, the response surface method (RSM) can be used to optimize a regression metamodel [35,37]. The present study proposes a simulation metamodel for automated material handling system (AMHS) in semiconductor manufacturing which represents a very complex system for simulation analysis. Due to its complexity, an analytical model is not a viable approach. The objectives of the present study are as follows: • to build a simulation model for AMHS in semiconductor manufacturing, • to use an experimental design to efficiently collect the metamodeling data, and • to use a neural network to build a simulation metamodel. The remainder of the paper is organized as follows. Section 2 provides a review of the relevant literature leading to the modeling and analysis approach employed in the present research. The details for the case study and its associated simulation model are presented in Section 3. The neural network based metamodel is presented in Section 4. Finally, the conclusions and future research direction are presented in Section 5. 2. Literature review A semiconductor fabrication facility is a very large investment. For example, a 300 mm fabrication unit costs more than US$2 billion. Therefore, any non-appropriate production decision always brings about
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serious waste. Among the various production decisions that have to be made, dispatching decisions play a critical role in optimizing system performance in semiconductor fabrication. An automated material handling system (AMHS) of semiconductor manufacturing is illustrated in Fig. 1. In general, there are two types of systems: one is the intrabay system, which transports material within one process bay; the other one is interbay system, which transports material between process bays. The interbay system contains double loops, the vehicles move clockwise on one loop and move counterclockwise in the other loop. In addition, there are buffers between the interbay system and any intrabay system, called a ‘stocker’. When a material has to be transported from one machine to another machine located in another process bay, it has to wait for a vehicle from the intrabay system to transport material to the stocker first. Then the system has to select one loop of the interbay system and wait for a vehicle which is on the selected loop to transport the material to another stocker. Finally, it will be transported to the target machine by the vehicle of another intrabay system. All material in semiconductor manufacturing fabrication is transported by the AMHS. Different vehicle dispatching rules for each intrabay and interbay system will affect the waiting time and total cycle time for the material. Therefore, optimizing the dispatching rules for all interbay and intrabay systems will be important for the decision support system. There is some literature on the optimization of dispatching rules using simulation analysis. In a recent study, Lin et al. [25] considered three decision points in an event-driven dispatching strategy used to control
Legend
: Interbay system
: Automated vehicle
: Intrabay system
: Stocker
Fig. 1. AMHS of semiconductor manufacturing system.
: Machine
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the vehicles efficiently in a double loop interbay material handling system. A full factorial simulation experiment with four loop selection rules, one cassette-initiated dispatching rule and two vehicle-initiated dispatching rules were then used to find the best strategy. However, only interbay systems were considered. Tyan et al. [41] present an integrated tool and vehicle dispatching strategy for semiconductor manufacturing. A two level 1/4 fractional factorial design was used to investigate the main effects and identify the most favorable dispatching combination. However, in a 300 mm semiconductor fabrication the total number of tool types is more than forty, and any category may still contain many tool types. Accordingly, a simulation model of a real semiconductor manufacturing AMHS system is not only complex and difficult to develop but also very time consuming for analysis. For example, the model built by Tyan et al. [41] takes more than seven hours to collect the outputs of one simulation replication. Therefore, a speedy model to explain the relationship between the input and output may be required. Artificial neural networks (ANNs) are composed of interconnected, usually adaptive, elements, which are intended to respond to stimuli in a manner not unlike the human nervous system [22]. There are three basic elements of an ANN model [10], which are illustrated in Fig. 2. They are synapses, adder and activation function. Synapses are the connecting links between neurons. All the input signals (Xi) transferred by synapses have to be multiplied by its own weighting (Wi). Then the adder sums the bias (b) and all the signals, which have been multiplied, to generate a value (u). Finally, the value (u) will be translated to become the output (Y) of the neuron by the activation function. The adder, bias and activation function form a node. Then the output can be the input of other nodes. Through a learning algorithm and given a set of training examples (including inputs and corresponding desired outputs), all the weights (Wi) can be iteratively modified to minimize the difference between outputs (Y) and desired outputs. There are many kinds of neural network models. Multilayer perceptron, which only feeds forward, multilayered and fully connected networks, were considered in the present study. It consists of three or more layers, including one input layer, one or more hidden layers and one output layer. Fig. 3 illustrates an example of a network with three layers. In Fig. 3, every circle is a node, and every connection between nodes is a synapse. There are two types of learning: supervised and unsupervised. For supervised learning, a set of training input vectors with a corresponding set of target vectors is used to adjust the weights in a neural network. For unsupervised learning, a set of input vectors is proposed, but no target vectors are specified. In this paper, the corresponding target vector of every input vector was generated by a simulation run. The input set and the corresponding output set can be collected before neural network training. Therefore supervised learning was used in the proposed methodology. A supervised learning algorithm named back error propagation (BEEP) (also called the ‘generalized delta rule’), which was presented by Rumelhard et al. [33], is also used in the present study. In contrast with backpropagation (or named delta rule), BEEP accelerates and stabilizes training by including a momentum term [17,43], and it is the most widely used supervised learning algorithm for Multilayer perceptrons [10]. A simulation metamodel using a neural network used the results from a simulation run under different configurations to be the training data, whereby these different configurations were used as input data and the
Fig. 2. Model of a neuron.
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... ...
... Input layer
Hidden layer
Output layer
Fig. 3. A Multilayer perceptron neural network.
corresponding simulation results were used as output data. The trained neural network was then used to quickly predict the simulation results of new configurations. Hurrion [15] developed a neural network simulation metamodel and then optimized the number of kanbans in a manufacturing system. Badiru and Sieger [2] used a neural network as a simulation metamodel in an economic system, and then accelerated the analysis of the future performance of the investment project. Kilmer et al. [19] used a supervised neural network as a metamodel in an inventory simulation system to estimate the confidence interval for estimates of total cost. Fonseca and Navaresse [9] constructed a neural network simulation metamodel to estimate the manufacturing lead time for orders simultaneously processed in a four-machine job shop. However, there is still one disadvantage when using a neural network for developing a simulation metamodel, especially, when the time spent is too long for one simulation run. Since simulation is time consuming, it is still impossible to collect a comprehensive configuration of results by a large number of simulation runs for the training data. For example, in the foregoing literature, Hurrion [15] used 128 simulation results for training data; Kilmer et al. [19] and Fonseca and Navaresse [9] used 900 simulation results for training data. 128 or 900 simulation runs may still be too many to be practicable in some complex simulation cases. Uniform design (UD) was proposed by Fang [8] and Wang and Fang [42]. Its most important merit is that it reduces the number of experimental configurations, especially when the experimental region has many factors and multiple levels. UD is a space filling design and it requires that experimental points be uniformly scattered on the domain [8]. For an example with two factors with 8 levels, such as the one illustrated in Fig. 4 [24], the total number of possible experimental points is 82 = 64. If a set with only 28 experimental points is to be selected, then the set of selected points will be scattered uniformly, when the absolute value of the ratio of the number of selected points lying in the rectangle to the total number of selected points of the set minus the area of the rectangle is small. This absolute value is called the discrepancy. In Fig. 4a the number
1.0
1.0
0.4
0.4
0.75
1.0
Fig. 4. A uniform design example.
0.75
1.0
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of selected points lying in the rectangle is 10, the area of the rectangle is 0.4 · 0.75. Therefore, the discrepancy is about j10/28 0.4 · 0.75j = 0.0571. In Fig. 4b the discrepancy is about j15/28 0.4 · 0.75j = 0.2357. Because the discrepancy in Fig. 4a is very small, the points are considered to be scattered uniformly in the domain when the type of rectangle is 0.4 · 0.75. However, the number of rectangle types in the domain is unlimited, and the selection of a rectangle is a continuous problem. When the number of experimental points is big enough, seeking these points will be an NP-hard problem [24]. Winker and Fang [44] used a stochastic global optimization algorithm called the threshold accepting (TA) algorithm to search the points, and the results, called a uniform design table, can be found at the web site address: http://www.math.hkbu.edu.hk/UniformDesign. Developing a neural network metamodel by using a uniform design table is rare in the literature. Shan et al. [36] used a uniform design table to generate a training data set, and then developed a neural network metamodel. Using the metamodel the pH value and solvent composition in high performance liquid chromatography (HPLC) were optimized. In the study, two seven-level factors were considered. The results showed the good predicting ability of the metamodel that was constructed using this method. 3. Semiconductor manufacturing automated material handling systems In this paper, a commercial software program, eM-Plant [7], was used as the simulation tool. All dispatching rules are coded using the eM-plant embedded programming language (called Method) for customized applications. Its syntax is similar to the programming language C++. The simulation model contains 23 intrabay systems and one double-loop interbay system, and the number of vehicles for every intrabay and interbay system are shown in Appendix A1 and A2 respectively. There are 277 machines, which divide into 43 types, and all wafers need to be processed through 316 operation steps. All the machine data such as location, downtime and buffer size, and process data such as process step, tool name, processing time, maximal size per batch and batching time are not presented in this paper, because of the huge quantity of the data. However, readers who wish to follow up more details of the case can find the relevant data in [3]. In addition, the capacity of all stockers was assumed to be infinite. The releasing policy was 22,000 wafers per month. According to the results of pilot runs, the simulation needed to run for 11 months with 1 six-month warm up period and 5 one-month batches. It took about six simulation hours using a computer with a Celeron 2.3 GHz processor. Therefore, a computationally efficient method to replace the simulation model was needed so that it would be possible to conduct even more analysis in order to optimize the system. This paper aims to develop a neural network simulation metamodel which can replace a simulation to predict the performance of the AMHS when given a set of vehicle dispatching rules for all 23 intrabay systems. Cycle time (CT), which is an important performance index, will be the prediction target. As regards the tool dispatching rule, vehicle dispatching rule of the interbay system and loop selection rule, the first in first out (FIFO), nearest vehicle (NV), and shortest distance (SD) rules, which have proven good performance in AMHS [25], were used respectively. The following vehicle dispatching rules are the candidates for every intrabay system. Least slack (LS): This is a dispatching policy that effectively improves the due date commitment. When a vehicle has just finished, it will search all outputs of machines which have unassigned material and then select the one whose slack is the least. The slack is due date minus current time and remaining processing time. First encounter first service (FEFS): This aims to minimize the travel time of empty vehicles. According to the direction of motion, a vehicle will search for the nearest output of a machine, which has unassigned material and then move to load it. Longest waiting time (LWT): When a vehicle has just finished, it will search all the unassigned material on the output of machines and then select the one which has been waiting longest for transport. High value first (HVF): All materials were divided into three priority values. A vehicle will select an unassigned item of material with the highest priority value on the output of a machine, and then move to load it. The resulting example of the model interface is shown in Fig. 5.
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Fig. 5. Simulation interface.
4. Developing a neural network simulation metamodel In this section a proper uniform design table was used to select scenarios for the proposed simulation model to generate a set of training data. Then a neural network was trained using the data set. Finally some tests and analyzes were conducted. The details were as follows: 4.1. Training data set generation In this case, there are 23 factors with 4 levels. The total number of possible configurations is 423. Traditionally, using the Taguchi method or orthogonal array, the minimal number of experiments are 70 and 80 (see Appendix B for more detail). However, according to the uniform design table of the form Un(423), which was taken from the web site cited above (n is the desired number of experiments) the number of experiments can be in the range from 24 to 30. Because n should be the common multiple of all the levels of variables, n can be 24 or 28. This research selected the uniform design table U28(423) and is shown in Appendix C. In the table, the numbers from 1 to 7 were set as LS, the numbers from 8 to 14 were set as FEFS, the numbers from 15 to 21 were set as LWT and the numbers from 22 to 28 were set as HVF. The simulation results of all 28 configurations are also shown in Appendix C. For comparison purposes, another training data set of 96 configurations was generated by orthogonal array L96(248) for comparison. For further examination of the method using an orthogonal array of this type L96(248), the reader is referred to [13]. The metamodel trained by the uniform design configuration data set was denoted by UDM, and the one trained by the orthogonal array configuration data set was denoted by OAM.
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The proposed uniform design and the orthogonal array use a structured experimental design to collect data for the construction of the metamodel. This is a fractional factorial experiment. It does not reduce variables but significantly reduces the required experimental treatments, which is the power of an orthogonal array [27,31,39]. Note that the existence of the noisy data is commonly found in the pattern recognition problems. The reduced sample size is one of the advantages to adopt the UDM. Due to the resulting small sample size, we will not further explore the noisy data issue. We do not feel the noisy data will be a problem for the present study since the later model validation step will illustrate an unstable training if there are significant noisy data. 4.2. Training data representation Data representation is a critical issue that directly affects the resulting neural network architecture [9]. Each bay will adopt one out of the four candidate dispatching rules—LS, FEFS, LWT, and HVF which are represented by the permutation of +1 and 1. Specifically, LS, FEFS, LWT, and HVF are represented by (1, 1), (1, +1), (+1, 1), and (+1, +1), respectively. Accordingly, there are 46 (=23 · 2) data fields to represent the different dispatching rules for the 23 bays. As regards the output data, the simulation result cycle time of ith configuration was scaled into the range between 1 and +1 by using Eq. (4). oi ¼ 1 þ
2 ðy i yÞ y y
i ¼ 1; 2; . . . ; 24
ð4Þ
In Eq. (4) oi is the corresponding output data of ith input data, yi is the simulation result cycle time of ith configuration, y is the maximum value of all yi, and y is the minimum value of all yi. A neural network with 46 inputs, one output and one hidden layer was then used. The number of nodes in the hidden layer was decided by the method shown in the following section. 4.3. Neural network building and training The training data sets contain 28 simulation results. The configuration of the neural network has 46 input nodes and 1 output node. The number of parameters (connections) of the network is more than the number of training data. For this instance, the trained network will potentially give a very close fit to the data; this is referred to as memorization [12]. This is also known as over-fitting [40]. To overcome this problem, Hagan et al. [11] suggested using another data set for validating the result of training process. If the mean square error (MSE) of validation data is not acceptable, then the trained neural network is over-fit and is not adequate for the application problem. In the present study, the 28 training data were split into two data sets, 17 (60%) for training and 11 (40%) for validation. The back-propagation training algorithm including a momentum term was used. Each neural network was trained for 1000 epochs, so that the training results would not be sensitive to the set-up parameters after so many iterations. This research sequentially determined the network parameters, such as the number of hidden layer nodes, learning rate and momentum constant. This approach has been widely applied to the determination of parameters in a neural network, as is reported in [1,6,28,38,45]. Initially the learning rate was set to 0.1 and the momentum constant was set to 0.9. Then different neural networks with 20, 30, 40, and 50 nodes in the hidden layer were tested respectively. Because initial weights were generated randomly, and training data set is very small, the training results of the same neural network were quite variable between different trainings; moreover, the MSE of the training data set were always far lower than the validation data set in the prior testing. Therefore, every structure of neural networks was trained ten times with different initial weights, and the final training result of every structure was chosen to be the one whose MSE of validation was lowest among the ten training results. By using neural network software NeuralSolution [29], the training results of UDM were as shown in Table 1. In Table 1 the MSE value of the training set and the validation set were lowest when the network had 30 nodes in the hidden layer. Then the learning rate was change to 0.05, 0.01and 0.005 respectively, and the momentum constant was still set to 0.9. The training results of the network with 30 nodes in the hidden layer were as shown in Table 2.
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Table 1 Training results with different number of nodes in the hidden layer Number of nodes in hidden layer
MSE (training set)
MSE (validation set)
20 30* 40 50
0.000053 0.000020* 0.000001 0.000004
0.330768 0.306489* 0.375323 0.371402
Table 2 Training results with different learning rates Learning rate
MSE (training set)
MSE (validation set)
0.1 0.05 0.01* 0.005
0.000020 0.000001 0.006399* 0.005232
0.306489 0.310198 0.261681* 0.349236
In Table 2 the MSE value of the training set and the validation set were lowest when learning rate was 0.01. Then the momentum constant was change to 0.95 and 0.8 respectively. The training results of the network with 30 nodes in the hidden layer and 0.01 learning rate were as shown in Table 3. In Table 3 the MSE value of the training set and the validation set were lowest when the momentum constant was 0.9. Finally the neural network which contained 30 nodes in the hidden layer, training with 0.01 training rate and 0.9 momentum was selected to be the final training result of UDM. The training result of OAM was developed using a similar approach and it is contained 20 nodes in the hidden layer, training with a training rate of 0.005 and momentum of 0.9. 4.4. Neural network testing and analysis Another data set with 10 configurations was generated randomly for testing the neural network simulation metamodel. The prediction time for all testing data was less than one second. The MSE value is 0.483095 for UDM and 0.488047 for OAM and the difference between them is only 1.01% Therefore, the accuracy of UDM and OAM are similar. The test results of UDM were as shown in Table 4. Table 4 show the average prediction error of UMD is 3.12%. However, there is still one prediction error that exceeds 5%. Even so, the UDM is still useful, because, for UDM, the prediction time is less than one second, and it can predict a lot of configurations in a very short time. When analyzing or optimizing with the simulation model, only a few additional simulation runs are needed for confirmation, as proposed in [15]. The main object in developing the simulation metamodel is acceleration of analysis and optimization. The normal method for optimization is predicting all possible configurations with the metamodel [4,14]. However, the total number of possible configurations in this paper was 423, which is too big, and there was a need for other methods. In this paper, only some common, practical configurations, in which all intrabay systems have the same vehicle dispatching rule, were analyzed. The results were as shown in Table 5. In Table 5, from second row to fifth row show the predicted results when the vehicle dispatching rule for every intrabay system were all LS, FEFS, LWT and HVF respectively. The last row shows the best predicted result of the UDM from the testing data set. Although the result in the last row is the best on the testing data, it is not the optimal result. But obviously, the cycle time is much shorter than testing all configurations with a single vehicle dispatching rule. Table 3 Training results with different momentum constant Momentum constant
MSE (training set)
MSE (validation set)
0.95 0.9* 0.8
0.000020 0.006399* 0.003742
0.344452 0.261681* 0.268490
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Table 4 Testing results of UDM Testing no.
1 2 3 4 5 6 7 8 9 10
Prediction result
Error %
Metamodel
Simulation model
19.063 19.862 19.890 19.686 20.088 19.679 18.243 18.205 18.857 20.151
18.986 19.188 19.756 19.092 19.144 19.350 19.516 17.824 18.220 19.240
Average
0.40 3.51 0.68 3.11 4.93 1.70 6.52 2.14 3.50 4.74 3.12
Table 5 Analysis results for all intrabay systems with same vehicle dispatching rule Vehicle dispatching rule
LS FEFS LWT HVF The best of testing
Prediction result
Error %
Metamodel
Simulation model
18.212 18.974 18.966 20.088 18.205
19.874 19.006 19.130 20.122 17.824
8.36 0.17 0.85 0.17 2.14
5. Conclusions In generally, using neural network, the training data set should be large enough when comparing to the network size. However, the main difficulty of the proposed simulation case is the time cost of data collection, and it is difficult to collect sufficient training data. Uniform design have been proofed can explain the domain space by quite experimental point. Therefore, this paper propose a way to develop a neural network simulation metamodel using a very small training data set, in which uniform design was used for generating the configurations of input data. A simulation model of an AMHS in a semiconductor manufacturing system with 23 intrabay systems was introduced. For every intrabay system there were four alternative vehicle dispatching rules and the total number of possible configurations was 423. However, in the experiment, only 28 simulation configurations were run for the training data set, and this is far fewer than would be required for the Taguchi method or orthogonal array. The results showed that the training data set generated by uniform design performs similarly to the much larger set generated by orthogonal array. Some error between the metamodel prediction and the simulation run still existed. However, when considering the trade-off between prediction accuracy and speed, the proposed method was encouraging. Moreover, this paper also analyzes some configurations in which the vehicle dispatching rules of all intrabay systems were the same. The results show that finding suitable vehicle dispatching rules for every intrabay system will be an opportunity for system improvement in the future. Since the neural network simulation metamodel can accelerate the prediction operation from several hours to less than a second, it will be possible to find improved and perhaps near optimal configurations by combining some global optimizing algorithms. Although the concept of a metamodel itself is not novel, the use of a small data set to model a complex problem is worth investigating. As discussed in this paper, the modeling of an automated material handling system in semiconductor wafer fabrication is a complex problem. The simulation model building itself is challenging. In addition, the use of uniform design is not a straightforward matter. The combined use of uniform design and neural networks is an effective and efficient method for the construction of the metamodel for the
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proposed problem. In comparison to other methods such as the Taguchi orthogonal array, the proposed methodology illustrated its efficiency, which is an important factor for practice, particularly, for those readers who place an emphasis on practical application. This paper only developed a simulation metamodel to predict the cycle time when given vehicle dispatching rules for every intrabay system. In an AMHS, other performance indices, such as work in process (WIP), service level, utilization, and so on are also important and should be considered in future work. Acknowledgement This work is supported in part by the National Science Council of Taiwan, Republic of China under Grant NSC 94-2213-E-432-002 and NSC 93-2917-I-006-016. Appendix A1 The number of vehicle for every intrabay system Intrabay code
Number of vehicles Intrabay code
Number of vehicles Intrabay code
Number of vehicles
A B C D E F G H
2 4 5 4 4 3 6 3
2 4 3 3 2 4 2 2
2 2 1 1 2 4 1
I J K L M N O P
Q R S T U V W
Appendix A2 The number of vehicles for every loop of interbay systems Loop direction
Number of vehicles
Clockwise Counterclockwise
15 15
Appendix B
Minimum number of simulation runs Total simulation run time (days)
Uniform design
Taguchi method
Orthogonal array
24 6
70 17.5
80a 20
The second row of the above table shows the minimum requirement of experimental treatment for every experimental design. For generating the training data set, one treatment means one simulation run, and takes about 6 h. Then the third row shows the corresponding total simulation run time. Accordingly, developing a neural network simulation metamodel by using uniform design is more effective then using either the Taguchi method or orthogonal array. a According to [13], when the number of factors (k) is 23 and number of levels (s) is 4, the minimum number of experiments (N) will equal to k Æ st. Meanwhile, t (strength) is 2 and k is between 5 and 8. Therefore, N will be least and equal to 80 (5 Æ 42 = 80). Notes, the orthogonal array is still unknown when AO(N, k, s, t) = AO(80, 23, 4, 2), and it just can be sure that N will not be less than 80.
Appendix C Uniform design table U28(423)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Factors 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Responses cycle time (days)
22 8 23 18 6 20 17 26 27 10 1 14 24 28 16 9 3 21 12 17 25 15 4 13 5 2 11 19
27 12 20 25 23 6 13 28 2 10 16 4 15 1 26 19 11 18 17 9 8 7 21 5 24 3 14 22
15 18 20 8 16 4 26 1 21 5 2 17 13 19 10 22 23 28 27 12 7 9 6 25 25 14 2 24
18 27 21 2 19 15 6 25 7 5 26 16 20 28 4 14 22 8 23 12 11 24 10 17 9 1 3 13
3 17 21 15 14 12 4 28 8 26 11 10 5 18 9 7 6 27 2 23 1 22 16 24 19 20 13 25
10 14 25 18 26 23 9 3 27 24 20 6 5 11 13 8 19 17 21 2 28 7 12 15 4 16 1 22
28 23 4 3 11 26 10 17 20 18 2 19 12 1 8 6 25 9 14 15 13 16 24 7 21 27 5 22
6 14 5 3 21 16 9 13 11 10 12 1 17 23 27 19 26 28 7 8 18 25 2 4 20 24 15 22
20 25 6 18 17 12 26 10 28 3 24 5 19 1 8 2 7 23 9 11 15 22 4 27 21 16 13 14
1 26 22 8 3 7 23 24 10 2 6 27 14 12 18 9 16 15 19 4 17 5 13 20 11 21 28 25
2 27 18 23 17 5 15 11 16 14 8 4 21 19 28 7 9 1 26 24 13 25 20 3 10 22 12 6
11 8 7 24 6 10 9 13 18 15 16 4 28 17 12 21 25 2 23 22 1 5 3 26 20 19 14 27
14 8 16 2 25 27 15 13 18 6 10 1 9 23 12 3 20 24 19 26 7 4 22 21 17 11 28 5
2 12 18 15 22 16 26 21 20 3 10 14 4 1 27 25 23 6 5 19 11 24 9 28 8 17 7 13
5 4 8 17 3 27 25 14 15 2 23 18 9 24 28 1 11 12 26 13 16 21 20 10 19 7 6 22
26 25 18 11 7 4 5 1 28 6 21 10 8 14 27 15 20 12 3 23 16 17 19 13 9 2 22 24
14 3 7 2 27 1 20 16 11 21 19 13 28 17 18 6 12 10 5 22 26 8 25 23 9 15 4 24
14 26 3 27 25 5 4 24 15 16 8 17 2 20 19 9 18 13 21 11 28 10 7 22 23 1 12 24
5 21 19 17 12 16 15 14 11 27 20 25 24 13 26 9 23 28 4 22 3 6 7 8 18 1 2 10
8 25 18 5 3 27 6 17 22 19 11 2 28 1 20 21 7 14 15 9 16 13 23 26 24 12 10 4
9 16 3 13 19 14 5 28 24 6 27 21 17 8 15 23 2 16 22 18 1 7 20 11 4 25 10 12
22 14 24 5 15 18 14 2 1 12 28 19 6 16 21 17 4 9 13 26 27 8 3 20 25 23 7 11
13 3 20 8 26 4 6 10 25 17 16 27 22 12 23 5 9 7 18 1 2 21 11 15 24 14 28 19
18.916 19.134 20.002 18.628 19.658 18.742 19.820 20.070 19.434 18.800 18.190 20.048 19.690 19.436 19.728 19.816 19.362 20.124 19.274 19.056 18.626 19.230 18.898 19.652 19.920 18.944 18.600 18.396
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