JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 979862 10.1155/2013/979862 979862 Research Article Internal Due Date Assignment in a Wafer Fabrication Factory by an Effective Fuzzy-Neural Approach http://orcid.org/0000-0001-7683-857X Chen Toly Nasseri Hadi Department of Industrial Engineering and Systems Management Feng Chia University 100 Wenhwa Road Seatwen, Taichung City 407 Taiwan fcu.edu.tw 2013 6 11 2013 2013 26 11 2012 26 08 2013 08 09 2013 2013 Copyright © 2013 Toly Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Owing to the complexity of the wafer fabrication, the due date assignment of each job presents a challenging problem to the production planning and scheduling people. To tackle this problem, an effective fuzzy-neural approach is proposed in this study to improve the performance of internal due date assignment in a wafer fabrication factory. Some innovative treatments are taken in the proposed methodology. First, principal component analysis (PCA) is applied to construct a series of linear combinations of the original variables to form a new variable, so that these new variables are unrelated to each other as much as possible, and the relationship among them can be reflected in a better way. In addition, the simultaneous application of PCA, fuzzy c-means (FCM), and back propagation network (BPN) further improved the estimation accuracy. Subsequently, the iterative upper bound reduction (IUBR) approach is proposed to determine the allowance that will be added to the estimated job cycle time. An applied case that uses data collected from a wafer fabrication factory illustrates this effective fuzzy-neural approach.

1. Introduction

Internal due date assignment is to quote an attractive but attainable due date for an arriving customer order. However, the completion time of an order is highly uncertain. It is therefore difficult to accurately forecast the completion time. For this reason, an allowance has to be added to the estimated completion time to reduce the risk .

Wafer fabrication is the most technologically complex step in semiconductor manufacturing, which exacerbates the difficulties of internal due date assignment . In theory, this problem is NP-hard. That is why wafer fabrication is investigated in this study. Internal due date assignment in a wafer fabrication factory is difficult because of the following reasons.

Shop floor control in a wafer fabrication factory is a nontrivial task owing to the complexity of wafer fabrication. Some wafer fabrication processes are repeated processes. Thus, wafers need to visit a machine multiple times. An average job cycle time is several months with hundreds of hours of standard deviation. Many studies have shown that accurately predicting the cycle/completion times for such large systems is very difficult [1, 3, 4].

In addition, the completion time predicted using existing approaches is generally unbiased. This means that if the internal due date is set to be equal to the mean of the estimated completion time, then the probability of on-time delivery is only about 50% on average. To reduce the risk, an allowance or fudge factor has to be added to the estimated completion time . The due date allowance factor is determined on the basis of the feedback information about the factory status at the time a job arrives at the factory.

Due date assignment, release control, and buffer control affect each other. Make-to-order wafer fabrication factories are confronted with both due date quotation and production scheduling problems at the same time . If due date assignment and factory scheduling are processed separately by two systems, the overall performance is unlikely to be satisfactory because the two tasks are actually interrelated. Therefore, the interaction between due date assignment methods and scheduling rules in a wafer fabrication factory needs to be investigated.

To tackle these problems, some treatments have been carried out in the literature. First, various research works have been dedicated to estimate the cycle time using hybrid approaches. For example, Gupta and Sivakumar  presented look-ahead batch scheduling for the real-time control of due date objectives. Chen  proposed the look-ahead self-organization map (SOM)-fuzzy back propagation network (FBPN) approach for this purpose. A set of fuzzy inference rules were also developed to evaluate the achievability of a cycle time forecast. Subsequently, Chen et al.  added a selective allowance to the cycle time estimated using the look-ahead SOM-FBPN approach to determine the internal due date. Further, Chen  showed that the combination of SOM and FBPN could be improved by a minor adjustment of the classification results with the estimation error. Chen et al. [10, 11] proposed a postclassification fuzzy-neural approach in which a job was not preclassified but rather postclassified after estimating the cycle time. Experimental results showed that the postclassification approach was better than the preclassification approaches in some cases. To balance the influence of the preclassification results with that of the postclassifying results, Chen  proposed a bidirectional classifying approach, in which jobs are not only preclassified but also postclassified. Ankenman et al.  proposed a metamodeling approach, which integrates discrete-event simulation, adaptive statistical methods, and analytical queueing analysis to quantify the cycle time-throughput relationship. Chien et al.  used nonlinear regression equations and then related the forecasting error to some factory conditions and job attributes with a back propagation network (BPN) to improve the forecasting accuracy. The major disadvantage of statistical analysis is the lack of forecasting accuracy .

Second, in traditional due date setting rules, the fudge factor is usually equal to a multiple of the standard deviation of the predicted cycle time . Recently, Chen et al.  proposed a selective allowance policy in which the allowance was only assigned to some preselected jobs. In this way, the sum of the allowances added to all jobs was controlled. However, even though the probability of on-time delivery in Chen et al.’s study was only 77% for the testing data, showing that improving the probability of on-time delivery while controlling the fudge factor is a real challenge. In addition, the allowances that were assigned to the chosen jobs in this study were equal, leaving room for improvement. Another way of taking this issue into account is to construct a confidence interval containing the actual completion time . The upper confidence limit sets the internal due date. However, the probability of a job delivered on time is only 99.7% for the testing data, under the assumption that residuals follow a normal distribution. From another point of view, Chen and Wang  incorporated the fuzzy c-means (FCM)-BPN approach with a nonlinear programming (NLP) model to construct the inclusion interval of the predicted completion time. Similarly, the upper inclusion limit sets the internal due date. An inclusion interval is narrower than a confidence interval, and the probability of a job delivered on time is 100%, at least for the training data. Chen and Lin  modified this approach by gathering a group of experts in related fields to set the due date in a collaborative way. Fuzzy intersection is applied to combine the due dates into a representative value.

The existing approaches have the following problems.

Some factors used to forecast the job cycle time are dependent on each other, which may cause problems in classifying jobs and in fitting the relationship between the job cycle time and these factors.

In Chen and Wang  and Chen and Lin , NLP models are solved to determine the upper bound of the job cycle time. However, the NLP models involve complicated constraints and therefore are difficult to solve. The NLP models will become too huge if many jobs are to be considered.

To tackle these problems, an effective fuzzy-neural approach is proposed in this study to improve the performance of internal due date assignment in a wafer fabrication factory. The literature provides probabilistic (stochastic) and fuzzy methods that can consider the uncertainty or randomness in the completion time. However, the longest average cycle time exceeds three months with a variation of more than 300 hours. Fitting the cycle time within a future month with a distribution function is not easy, implying that a stochastic approach might not be applicable. That is why a fuzzy approach is proposed in this study.

The effective fuzzy-neural approach has the following innovative characteristics.

Variable replacement using principal component analysis (PCA): PCA uses orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables to reflect information in a better way.

Updating the upper bound of the job cycle time using the iterative upper bound reduction (IUBR) approach: the IUBR approach is proposed to determine the upper bound of the completion time forecast. A tight upper bound means that the allowance assigned to a job is minimized.

Some recent works in this field are relevant. The differences between the proposed methodology and these methods are summarized in Table 1.

The differences between this study and some recent references.

Method Data preprocessing method Forecasting method Upper bound Optimization method Computation complexity
Chen et al. (2008)  SOM FBPN Yes Simulation + fuzzy rules High
Chien et al. (2011)  No Nonlinear regression No BPN Low
Chen and Lin (2011)  FCM BPN Yes NLP Very high
Chen and Wang (2013)  PCA + FCM BPN No An iterative process to reduce outliers Low
The proposed methodology PCA + FCM BPN Yes IUBR Low

NLP: nonlinear programming.

The remainder of this paper is organized as follows. Section 2 introduces the proposed methodology which is composed of four steps. A practical example is used to validate the effectiveness of the proposed methodology. The performance of the proposed methodology is evaluated and compared with those of some existing approaches. Finally, the concluding remarks and some directions for future research are given in Section 4.

2. Methodology

The operating procedure of the effective fuzzy-neural approach consists of several steps that will be described in the following sections.

Step 1.

Forming new variables by constructing linear combinations of the original variables using PCA.

Step 2.

Classifying jobs using fuzzy c-means (FCM).

Step 3.

Forecasting the cycle times of jobs in each category using a BPN.

Step 4.

Determining the upper bound of the cycle time using the IUBR approach.

A flow chart of the proposed methodology is shown in Figure 1.

The flowchart of the proposed methodology.

2.1. Step <xref ref-type="statement" rid="step1">1</xref>: Forming New Variables Using PCA

First, PCA is used to replace the inputs to the BPN. PCA was invented by Pearson  as an analogue of the principal axes theorem in mechanics; it was later independently developed by Hotelling . In the literature, there are more advanced applications of PCA. For example, Jaiswal et al.  used a hybrid of PCA and partial least squares for face recognition. In Mohtasham et al. , linear and exponential weighted PCA techniques based on spectral similarity were employed to predict the dye concentration in coloured fabrics. The operating procedure of PCA consists of several steps that are illustrated in Figure 2.

The PCA process.

The references on the combination of PCA, FCM, and BPN are still very limited [17, 22, 23].

2.2. Step <xref ref-type="statement" rid="step2">2</xref>: Classifying Jobs Using FCM

After employing PCA, examples are then classified using FCM. FCM is one of the most popular fuzzy clustering techniques because it is efficient, straightforward, and easy to implement. However, FCM is sensitive to initialization and is easily trapped in local optima.

The objective function of FCM is to minimize the weighted sum of squared distances such that the jobs in a category will be similar (or related) to one another and different from (or unrelated to) the jobs in other categories. In FCM, the Euclidean distance between two jobs is measured: (1)ej(k)=q=1p(zjp-z-(k)p)2,j=1~n,k=1~K, where (2)z-(k)={z-(k)q},k=1~K,z-(k)q=j=1nμj(k)mzjqj=1nμj(k)m,k=1~K,q=1~p. The weight of a job is a function of its membership: (3)μj(k)=1g=1K(ej(k)/ej(g))2/(m-1),j=1~n,  k=1~K. However, FCM requires prior knowledge about the number of clusters in the data, which may not be known for new data. Then, fuzzy clustering is carried out through an iterative optimization of the objective function (see Figure 3). The clustering process stops when the maximum number of iterations is reached or the improvement in the objective function becomes negligible with more iterations. In addition, the S-index proposed by Xie and Beni  is used to give the ideal number of categories automatically: (4)S=Jmn×emin2, where (5)Jm=k=1Kj=1nμj(k)mej(k)2,emin2=mink1k2(all  p(x-(k1)p-x-(k2)p)2). Chen and Wang  found the empirical relationship between the S-index and the estimation performance.

The FCM procedure.

2.3. Step <xref ref-type="statement" rid="step3">3</xref>: Forecasting the Cycle Times of Jobs in Each Category with a BPN

Subsequently, the jobs/examples of a category are learned with the same BPN. BPN is a popular tool with applications in a variety of fields. Nevertheless, different problems may require different parameter settings for a given network architecture. In the literature, researchers have used BPNs for estimating cycle times and assigning due dates. The configuration of the BPN is established as follows.

Inputs: the new factors determined by PCA associated with the jth example/job. These factors have to be partially normalized so that their values fall within [0.1,0.9] [10, 11].

Single hidden layer: generally one or two hidden layers are more beneficial for the convergence property of the BPN .

The number of neurons in the hidden layer: 1 to 2K. An increase in the number of hidden-layer nodes lessens the output errors for the training examples but increases the errors for novel examples. Such a phenomenon is often called “overfitting”. There exist many different approaches such as the pruning algorithm, the polynomial time algorithm, the canonical decomposition technique, and the network information criterion for finding the optimal configuration of a BPN . In addition, there has been some research considering the relation among the complexity of a BPN, the performance for the training data, and the number of examples, for example, using Akaike’s information criterion (AIC)  or the minimum description length (MDL) .

Activation/transformation function: there are a number of common activation/transformation functions, such as identity function, binary step function, bipolar step function, sigmoid functions (binary sigmoid function and bipolar sigmoid function), and ramp function. In the proposed methodology, the binary sigmoid function is used: (6)f(x)=11+e-x. Therefore, the output ranges between 0 and 1.

Output (oj): the (normalized) cycle time forecast of the example. oj is compared with the normalized cycle time N(CTj), for which root mean squared error (RMSE) is calculated: (7)RMSE=j=1n(oj-N(CTj))2n.oj is derived by transforming the signal transferred to the output layer: (8)oj=11+e-njo, where (9)njo=Ijo-θo,Ijo=l=1Lwlohjl. Similarly, hjl is derived by transforming the signal transferred to the hidden layer: (10)hjl=11+e-njlh, where (11)njlh=Ijlh-θlh,Ijlh=q=1pwqlhzjq.

Some algorithms are applicable for training a BPN in the backward phase, such as the gradient descent algorithms, the conjugate gradient algorithms, and the Levenberg-Marquardt algorithm. In this study, the Levenberg-Marquardt algorithm is applied. The Levenberg-Marquardt is the most widely used optimization algorithm. It outperforms simple gradient descent and other conjugate gradient methods in a wide variety of problems. The Levenberg-Marquardt algorithm uses approximation and updates the network parameters in a Newton-like way, as described below.

The network parameters are placed in vector β. The network output oj can be represented with f(xj,β). The objective function of the BPN is to minimize RMSE or equivalently the sum of squared error (SSE): (12)SSE(β)=j=1n(N(CTj)-f(xj,β))2.

The Levenberg-Marquardt algorithm is an iterative procedure. In the beginning, the user should specify the initial values of the network parameters. In each step, the parameter vector is replaced by a new estimate, and the network output by its linearization. When the network converges, the gradient of the objective function will be zero. It should be noted that while the Levenberg-Marquardt method is in no way optimal but is just a heuristic, it works extremely well in practice.

2.4. Step <xref ref-type="statement" rid="step4">4</xref>: Establishing the Upper Bound for the Job Cycle Time Using the IUBR Approach

In order to apply the BPN obtained at the previous step to determine the internal due date of a job, the parameter values in the BPN must be adjusted. To this end, in Chen and Wang  and Chen and Lin , the NLP model is constructed to adjust the connection weights and thresholds in the BPN, which is not easy to solve. In the IUBR approach, only the threshold of the output node will be adjusted in an iterative way. This way is much simpler and can also achieve satisfactory results.

Substituting (9) into (8),(13)oj=11+e-njo=11+e-(Ijo-θo)=11+eθo-Ijo. Therefore, (14)ln(1oj-1)=θo-Ijo. So (15)Ijo=θo-ln(1oj-1).

Assume that the adjustment made to the threshold of the output node is indicated as Δθo. After adjustment, the output from the new BPN, oj, determines the upper bound of the cycle time: (16)oj=11+e-njo, where (17)njo=Ijo-θo=Ijo-(θo+Δθo). Substituting (17) into (16), (18)oj=11+e-(Ijo-θo-Δθo). Substituting (15) into (18), (19)oj=11+e-(θo-ln((1/oj)-1)-θo-Δθo)=11+eln((1/oj)-1)+Δθo=11+eΔθo((1/oj)-1). Obviously, the maximum of Δθo establishes the lowest upper bound.

Since oj is the upper bound of the cycle time, ojN(CTj), (20)11+eln((1/oj)-1)+ΔθoN(CTj),(21)Δθoln(1N(CTj)-1)-ln(1oj-1). Equation (21) holds for all jobs, so (22)Δθominj(ln(1N(CTj)-1)-ln(1oj-1)). According to (19), the optimal value of Δθo should be set to the maximum possible value: (23)Δθo*=minj(ln(1N(CTj)-1)-ln(1oj-1)).

Then the optimization results of the BPN are sensitive to the initial conditions and may be different for each iteration. Assume that the optimal value of oj in the tth iteration is indicated with oj(t). After some iterations, (24)oj(all  iterations)=mintoj(t). In this way, the upper bound of the cycle time is decreased gradually (see Figure 4). Another merit of the IUBR approach is that it does not rely on the parameters of the BPN.

The upper bound is reduced in an iterative manner.

2.5. Ensemble Learning

Ensemble learning is based on the notion of perturbing and combining. An ensemble consists of a collection of ANNs and combines their predictions to obtain a final prediction. In FCM, a job can be classified into several categories to different degrees. In theory, the BPNs of all categories can be applied to predict the cycle time of a job. The forecasts obtained by using the BPNs may not be the same and need to be aggregated. To this end, some treatments have been carried out in the literature.

Linear aggregation : (25)CTEj=k=1K(μj(k)·CTEj(k))k=1Kμj(k)=k=1K(μj(k)·CTEj(k)), where k=1Kμj(k)=1. CTEj(k) is the cycle time of job j estimated by the BPN of category k.

BPN aggregation : the membership and cycle time forecast of a job are fed into another BPN to be aggregated. Consider (26)CTEj=BPN(μj(1),CTEj(1),,μj(K),CTEj(K)).

Generalized average method : in FCM, the error is proportional to the distance to the center. For this reason, a natural way to aggregate the forecasts is (27)CTEj=k=1K1/μj(k)2/(m-1)·CTEj(k)k=1K1/μj(k)2/(m-1).

3. Application and Analyses

To demonstrate the application of the proposed methodology, a real case with the data of 40 jobs from a wafer fabrication factory located in Taichung City Scientific Park, Taiwan (see Table 2), was used, where xj1~xj6 stand for the job size, factory utilization, the queue length on the route, the queue length before the bottleneck, the work in progress (WIP), and the average waiting time. The wafer fabrication factory produces more than ten products and has a monthly capacity of 20,000 wafers. The wafer fabrication processes include photolithography, thermal processes, implantation, chemical vapor deposition, etching, physical vapor deposition, chemical mechanical polishing, process diagnostics and control, and cleaning. The production characteristic of “reentry,” which is highly relevant to the semiconductor industry, is clearly reflected in this problem. It also shows the difficulties facing production planners and schedulers who attempt to provide an accurate due date for a product with a very complicated routing.

An example.

j x j 1 x j 2 x j 3 x j 4 x j 5 x j 6 CT j
1 24 1261 181 781 112 0.92 935
2 24 1263 181 762 127 0.90 958
3 24 1220 176 761 127 0.89 1047
4 23 1282 178 802 127 0.94 1011
5 23 1303 180 780 175 0.93 1068
6 23 1281 183 782 175 0.93 1143
7 23 1242 184 741 163 0.89 1103
8 24 1262 182 681 139 0.86 1250
9 22 1260 182 701 98 0.86 1181
10 22 1260 179 700 257 0.87 1194
11 24 1301 163 722 99 0.84 1260
12 22 1221 184 641 131 0.82 1240
13 23 1323 159 740 247 0.87 1180
14 24 1362 181 782 191 0.95 1227
15 24 1261 181 762 219 0.91 1236
16 23 1321 177 801 219 0.96 1215
17 22 1343 180 822 219 0.97 1228
18 24 1321 177 762 54 0.93 1266
19 25 1343 179 781 54 0.96 1285
20 25 1300 180 740 54 0.92 1272
21 22 1320 181 721 54 0.91 1310
22 24 1321 182 742 49 0.92 1265
23 23 1262 165 680 201 0.80 1308
24 22 1240 161 722 103 0.82 1331
25 23 1183 183 661 53 0.82 1294
26 23 1282 184 701 53 0.88 1314
27 22 1202 177 680 248 0.84 1321
28 23 1202 178 681 248 0.85 1353
29 24 1202 185 701 82 0.86 1226
30 23 1202 158 721 98 0.81 1301
31 24 1343 181 760 67 0.94 1280
32 24 1381 185 801 67 0.97 1286
33 22 1362 156 780 67 0.91 1252
34 23 1282 179 782 223 0.92 1214
35 23 1320 180 782 176 0.93 1251
36 25 1340 176 801 462 0.97 1222
37 23 1320 182 781 168 0.95 1187
38 22 1361 181 781 141 0.94 1205
39 22 1381 179 781 95 0.97 1120
40 23 1363 178 802 179 0.97 1133

The standard deviations of the six inputs are compared in Figure 5. Note that the variability in xj2, xj4, and xj5 is substantially higher than that in the remaining variables.

The comparison of the standard deviations of the inputs.

Subsequently, we standardize the data (see Table 3) and obtain the correlation matrix as (28)R=[0.970.100.160.21-0.030.250.100.980.010.70-0.010.780.160.010.980.05-0.070.370.210.700.050.980.150.86-0.03-0.01-0.070.150.980.100.250.780.370.860.100.98]. The eigenvalues and eigenvectors of R are then calculated. Based on them, the variance contribution rates can be derived as (29)η1=46%,η2=20%,η3=16%,  η4=14%,η5=4%,η6=0%. Summing up ηq’s, we obtain (30)ηΣ(1)=46%,ηΣ(2)=65%,ηΣ(3)=81%,ηΣ(4)=95%,ηΣ(5)=100%,ηΣ(6)=100%. After conducting a Pareto analysis, p is chosen as 3 to meet the requirement ηΣ(p)85%~90%. The first three principal components explain roughly 80% of the total variability in the standardized data, so that it might be a reasonable way to reduce the dimensions in order to visualize the data.

The standardized data.

j x j 1 x j 2 x j 3 x j 4 x j 5 x j 6
1 0.88 −0.53 0.40 0.74 −0.40 0.37
2 0.88 −0.49 0.48 0.30 −0.22 −0.05
3 0.88 −1.27 −0.17 0.29 −0.22 −0.31
4 −0.22 −0.15 0.07 1.18 −0.22 0.65
5 −0.22 0.24 0.37 0.71 0.35 0.58
6 −0.22 −0.17 0.78 0.74 0.35 0.45
7 −0.22 −0.87 0.90 −0.14 0.21 −0.19
8 0.88 −0.51 0.53 −1.45 −0.08 −0.87
9 −1.32 −0.55 0.60 −1.01 −0.56 −0.81
10 −1.32 −0.54 0.23 −1.03 1.34 −0.55
11 0.88 0.20 −1.87 −0.57 −0.56 −1.19
12 −1.32 −1.25 0.80 −2.33 −0.18 −1.64
13 −0.22 0.60 −2.40 −0.16 1.22 −0.60
14 0.88 1.31 0.47 0.75 0.55 0.94
15 0.88 −0.53 0.40 0.31 0.88 0.19
16 −0.22 0.57 −0.05 1.17 0.88 1.13
17 −1.32 0.97 0.33 1.62 0.88 1.38
18 0.88 0.56 −0.01 0.31 −1.09 0.62
19 1.97 0.96 0.15 0.74 −1.09 1.06
20 1.97 0.19 0.38 −0.16 −1.09 0.36
21 −1.32 0.55 0.51 −0.57 −1.09 0.10
22 0.88 0.55 0.54 −0.13 −1.16 0.32
23 −0.22 −0.52 −1.59 −1.47 0.67 −2.00
24 −1.32 −0.91 −2.11 −0.57 −0.51 −1.54
25 −0.22 −1.95 0.72 −1.89 −1.11 −1.64
26 −0.22 −0.15 0.89 −1.01 −1.11 −0.36
27 −1.32 −1.59 −0.07 −1.47 1.23 −1.26
28 −0.22 −1.60 0.07 −1.47 1.23 −1.11
29 0.88 −1.60 0.93 −1.03 −0.76 −0.87
30 −0.22 −1.59 −2.57 −0.58 −0.56 −1.86
31 0.88 0.97 0.41 0.27 −0.94 0.70
32 0.88 1.65 1.00 1.16 −0.94 1.27
33 −1.32 1.30 −2.73 0.71 −0.94 0.10
34 −0.22 −0.14 0.16 0.74 0.92 0.33
35 −0.22 0.55 0.34 0.75 0.36 0.54
36 1.97 0.91 −0.23 1.17 3.79 1.28
37 −0.22 0.55 0.61 0.73 0.27 0.91
38 −1.32 1.28 0.42 0.72 −0.05 0.81
39 −1.32 1.66 0.14 0.72 −0.60 1.36
40 −0.22 1.33 0.13 1.18 0.40 1.42

Subsequently, the component scores are calculated (see Table 4), which contain the coordinates of the original data in the new coordinate system defined by the principal components, and will be used as the new inputs to the FCM-BPN.

New inputs to the FCM-BPN.

z j 1 z j 2 z j 3
−0.56 0.91 −0.19
−0.13 0.87 −0.34
0.51 0.57 −0.37
−0.97 −0.10 0.20
−0.87 −0.20 −0.26
−0.75 0.14 −0.51
0.57 0.56 −0.66
1.30 1.18 −0.55
1.55 0.31 0.47
1.37 −0.87 −1.04
1.11 −0.59 0.91
3.04 0.63 −0.20
0.51 −2.44 −0.02
−1.94 0.12 −0.43
−0.30 0.35 −1.29
−1.62 −0.84 −0.48
−2.04 −1.24 −0.17
−0.87 0.77 0.89
−1.92 1.34 0.64
−0.58 1.70 0.34
0.22 0.23 1.29
−0.62 1.31 0.73
2.54 −1.26 −0.16
2.39 −1.64 1.20
3.02 1.57 0.14
0.89 1.21 0.66
2.56 −0.74 −1.19
2.19 −0.13 −1.54
1.61 1.90 −0.42
2.72 −1.23 0.87
−1.27 0.99 0.71
−2.56 1.07 0.78
−0.37 −2.44 2.47
−0.60 −0.51 −0.82
−1.06 −0.27 −0.17
−2.54 −1.36 −3.41
−1.31 −0.02 −0.18
−1.32 −0.63 0.67
−1.77 −0.58 1.32
−2.13 −0.66 0.12

Subsequently, jobs are classified using FCM based on the new variables. The results of the S-test are summarized in Table 5. In this case, the optimal number of job categories was 5. However, there will be some categories with very few jobs. For this reason, the second best solution is used, that is, 4 categories, by setting the threshold of membership to 0.3. The classification results are shown in Table 6.

The results of the S-test.

Number of categories (K) J m e min 2 S
2 1.96 0.14 0.34
3 1.21 0.09 0.34
4 0.86 0.07 0.30
5 0.67 0.06 0.26
6 0.53 0.03 0.43

The classifying results (μL=0.3).

Category Jobs
1 4–6, 14–17
2 10-11, 13, 23-24, 27, 30, 33
3 1, 18–22, 31-32
4 2-3, 7–9, 12, 25-26, 28-29, 34–40

After preclassification, the three-layer BPN of each category was applied to predict the cycle times of jobs belonging to the category according to the new variables. Different network architectures were evaluated to compare the forecasting performance. The best-fitted network which was selected, and, therefore, the architecture which presented the best forecasting accuracy, is composed of three inputs, six hidden and one output neurons.

The convergence condition in training networks was established as either the improvement in MSE becomes less than 10-6 with one more epoch or 1000 epochs have already been run. 3/4 of the adopted examples in each category are fed as “training examples” into the BPN. The remaining 1/4 is left for testing. For example, category 3 has 8 jobs; 6 of them are randomly chosen for training the BPN while the remaining 2 jobs are left for testing. The forecasting accuracy can be evaluated with mean absolute error (MAE), mean absolute percentage error (MAPE), and RMSE. The forecasting performances are summarized in Table 7. The forecasting results are shown in Figure 6. The performance of the proposed methodology is compared with those of statistical analysis (i.e., multiple linear regression), BPN, FCM-BPN, and PCA-BPN in Table 8. The nonlinear nature of this problem is obvious since the performance of statistical analysis (a linear approach) is poor. In addition, the simple combination of PCA and BPN does not have much effect. The main effect of PCA is to improve the correctness of job classification, as mentioned in Chen and Wang .

The forecasting performances.

Category MAE (hrs) MAPE RMSE (hrs)
1 20 1.7% 44
2 6 0.5% 14
3 5 0.4% 12
4 8 0.6% 18

Total 11 0.9% 29

Comparison of the forecasting performances.

Category MAE (hrs) MAPE RMSE (hrs)
Statistical analysis 73 6.1% 99
BPN 30 2.4% 69
FCM-BPN 15 1.2% 38
PCA-BPN 27 2.3% 67
PCA-FCM-BPN 11 0.9% 29

The forecasting results using PCA-FCM-BPN.

Subsequently, the IUBR approach is applied to determine the upper bound of the cycle time. In the first iteration, Δθo*(t) is -0.808, and the upper bounds of the cycle times are shown in Figure 7.

The upper bounds of the cycle times.

The process stops after five iterations because the upper bounds remain unchanged after the fifth iteration. The results of the five iterations are summarized in Table 9, from which the allowances which are 25, 33, 54, 48, 56, 57, 58, 44, 54, 53, 42, 46, 55, 48, 47, 50, 48, 41, 36, 39, 30, 41, 31, 24, 34, 29, 27, 13, 37, 24, 47, 15, 34, 50, 44, 49, 53, 50, 53, and 53 added to the cycle times are derived with an average of 42 (hours). The due date of a job is then set to the release time plus the upper bound of the cycle time.

The results of the five iterations.

Iteration 1(Δθo*(t)=-0.808) Iteration 2(Δθo*(t)=-0.669) Iteration 3(Δθo*(t)=-0.446) Iteration 4(Δθo*(t)=-0.446) Iteration 5(Δθo*(t)=-0.446)
987 976 960 960 960
1025 1011 991 991 991
1147 1129 1100 1100 1100
1103 1086 1059 1058 1058
1171 1153 1124 1123 1123
1243 1227 1200 1199 1199
1207 1190 1161 1161 1161
1322 1312 1294 1294 1294
1274 1259 1235 1235 1235
1283 1270 1246 1246 1246
1329 1319 1302 1302 1302
1316 1305 1286 1286 1286
1273 1259 1234 1234 1234
1307 1296 1276 1275 1275
1313 1302 1282 1282 1282
1299 1286 1265 1265 1265
1308 1296 1276 1276 1276
1332 1323 1307 1307 1307
1344 1336 1321 1321 1321
1336 1327 1311 1311 1311
1358 1352 1340 1340 1340
1331 1322 1305 1305 1305
1357 1350 1338 1338 1338
1369 1364 1355 1355 1355
1349 1342 1328 1328 1328
1360 1354 1343 1343 1343
1364 1358 1348 1348 1348
1353 1353 1353 1353 1353
1334 1334 1320 1320 1320
1369 1364 1355 1355 1355
1311 1300 1280 1280 1280
1384 1381 1376 1376 1376
1350 1343 1329 1329 1329
1298 1286 1264 1263 1263
1323 1313 1295 1295 1295
1304 1292 1271 1271 1271
1381 1378 1371 1250 1250
1399 1399 1397 1265 1265
1405 1405 1405 1247 1247
1348 1340 1326 1244 1244

To make a comparison, six other allowance determination policies are also applied to the collected data.

Total work content policy (TWK): in TWK, the due date allowance factor is estimated based on historical data by a regression model. There is another product in the wafer fabrication factory with an average cycle time of 1278 hours. The total processing time and cycle time standard deviation of the product are 317 and 87 hours, respectively. The product was adopted as the comparison basis, and in this case the cycle time forecast and allowance are determined as follows: (31)Cycle  time  forecast=1278317*the  total  processing  time,Allowance=3*87*(the  total  processing  time317).

Gamma distribution fitting method (Gamma): the waiting time of a job is fitted with a Gamma distribution. For example, the waiting time of a job with 24 pieces of wafers is fitted with a Gamma distribution in Figure 8. The 50% and 95% percentiles are 929 and 1160, respectively, and the total processing time is 251 hours. So the cycle time forecast is 1160 + 251 = 1411 hours, and allowance is 1160-929=231 hours.

Constant allowance policy (CON, PCA-FCM-BPN + CON): add three times the RMSE of the prediction approach to the completion time forecasts to determine the due date.

Selective allowance policy (SAP, PCA-FCM-BPN + SAP): add three times the RMSE of the prediction approach to the completion time forecasts of a small quantity of jobs that might encounter difficulties in keeping the internal due date. Such jobs are chosen in the following way: (32)i=16N(xji)0.5·maxji=16N(xji)+0.5·minji=16N(xji). In other words, these jobs are among the 50% percentiles.

Random assignment policy (RAP, PCA-FCM-BPN + RAP): add the extra allowance to the completion time forecasts of the same quantity of jobs that are randomly chosen.

No allowance policy (NAP, PCA-FCM-BPN + NAP): no allowance will be assigned to any job.

Fitting the waiting time with a Gamma distribution (jobs with 24 wafers).

Due date related performances are impacted by the quality of the due date assignment methods. After applying the seven allowance determination policies, the following performance measures are compared:

number of tardy jobs (NT);

mean tardiness (T-);

sum of allowances.

The comparison results are summarized in Table 10. The proposed IUBR approach outperforms the other allowance determination policies.

It guarantees the on-time delivery of the jobs. Both NT and T- are zeros. Among the other allowance determination policies, only Gamma and CON can achieve that at the expense of adding some extra allowance.

The percentage of reduction in the sum of allowances over CON is 52%. The advantages over TWK, Gamma, SAP, and RAP are 79%, 74%, 12%, and 12%, respectively. The percentage of on-time delivery is not derived from a greater buffer on the completion time prediction.

The performance of SAP is not better than that of RAP, which shows it is not easy to anticipate jobs that may delay.

Compared with TWK and Gamma, the other policies effectively reduce the allowances added to the job cycle times, which is due to the forecasting accuracy of the PCA-FCM-BPN approach.

The performances of various allowance determination policies.

TWK Gamma CON SAP RAP NAP (basis) IUBR
Sum of allowances 7982 6479 3480 1914 1914 0 1680
N T 31 0 0 9 7 14 0
T - (hours) 82 0 0 3.9 2.1 3.5 0
4. Conclusions and Directions for Future Research

Owing to the complexity of the wafer fabrication, the due date assignment of each job presents a challenging problem to the production planning and scheduling people. The firm has to offer a price reduction if the due date is far away from the expected one. Conversely, the looser the due date is set, the higher the probability that the job will be completed or delivered on time is. That is very important to maintain a good reputation with the customers. This study explores a new application of fuzzy-neural approaches in the due date assignment problem of the wafer fabrication factory. The proposed methodology decomposes internal due date assignment in a wafer fabrication factory into two subproblems: completion time prediction and allowance determination. To overcome the problems with the existing approaches, two innovative treatments are taken in the proposed methodology. First, PCA is applied to construct a series of linear combinations of the original variables to form a new variable, so that these new variables are unrelated to each other as much as possible, and the relationship among them can be reflected in a better way. The combination of PCA and BPN also reduces the space for storing the input variables in the modeling of the wafer fabrication system. In addition, the simultaneous application of PCA, FCM, and BPN further improved the estimation accuracy. Subsequently, the IUBR approach is proposed to determine the allowance that will be added to the estimated job cycle time. Our result is existentially tight.

The validity that the effective fuzzy-neural approach for internal due date assignment is able to improve on-time delivery has been proved by the case study. Based on the above analysis,

the forecasting accuracy (measured with MAE, MAPE, and RMSE) of the PCA-FCM-BPN was significantly better than those of many existing approaches;

it is easier to determine the allowance in the IUBR method than the method based on NLP;

the bound on the job cycle time is tighter than the bounds by TWK, Gamma, and CON and simpler than the bound by Chen and Wang , which requires NLP optimization.

However, there are two limitations that need to be acknowledged and addressed regarding the present study.

The first limitation concerns the experimental nature of this research. The proposed methodology was studied within a short period of time. There is an apparent danger involved whenever conclusions are drawn from such a limited sample and then applied in the highly dynamic semiconductor manufacturing environment.

The BPN part in the methodology is usually regarded as a black box. To exploit the knowledge embedded in the back box, and to facilitate the practical application of the proposed methodology, some association rules have to be extracted from the estimation results.

The IUBR approach only modifies the threshold of the output node. In future studies, other parameters in the BPN can be modified in similar ways. However, it is a challenge to make the modification results independent of the original parameter values. In addition, the concept of customer satisfaction can be incorporated into the proposed methodology; thereby, the due date can achieve a higher level of customer satisfaction. In contrast, the proposed methodology only guarantees a positive level of customer satisfaction.

Acknowledgment

This study was financially supported by the National Science Council of Taiwan.

Chen T. Jeang A. Wang Y. A hybrid neural network and selective allowance approach for internal due date assignment in a wafer fabrication plant International Journal of Advanced Manufacturing Technology 2008 36 5-6 570 581 2-s2.0-43449100389 10.1007/s00170-006-0869-8 Chen T. A flexible way of modelling the long-term cost competitiveness of a semiconductor product Robotics & Computer Integrated Manufacturing 2013 29 3 31 40 Pearn W. L. Chung S. H. Lai C. M. Due-date assignment for wafer fabrication under demand variate environment IEEE Transactions on Semiconductor Manufacturing 2007 20 2 165 175 2-s2.0-34248190308 10.1109/TSM.2007.895215 Chien C. F. Hsu C. Y. Hsiao C. W. Manufacturing intelligence to forecast and reduce semiconductor cycle time Journal of Intelligent Manufacturing 2011 23 6 2281 2294 Wilamowsky Y. Epstein S. Dickman B. Optimal common due-date with completion time tolerance Computers and Operations Research 1996 23 12 1203 1210 2-s2.0-0030395652 10.1016/S0305-0548(96)00020-2 Wang Y. C. Chen T. Lin C. W. A slack-diversifying nonlinear fluctuation smoothing rule for job dispatching in a wafer fabrication factory Robotics & Computer Integrated Manufacturing 2013 29 3 41 47 Gupta A. K. Sivakumar A. I. Optimization of due-date objectives in scheduling semiconductor batch manufacturing International Journal of Machine Tools and Manufacture 2006 46 12-13 1671 1679 2-s2.0-33746357611 10.1016/j.ijmachtools.2005.08.017 Chen T. A hybrid look-ahead SOM-FBPN and FIR system for wafer-lot-output time prediction and achievability evaluation International Journal of Advanced Manufacturing Technology 2007 35 5-6 575 586 2-s2.0-36049051023 10.1007/s00170-006-0741-x Chen T. A SOM-FBPN-ensemble approach with error feedback to adjust classification for wafer-lot completion time prediction International Journal of Advanced Manufacturing Technology 2008 37 7-8 782 792 2-s2.0-42649085767 10.1007/s00170-007-1007-y Chen T. Wang Y. Tsai H. Lot cycle time prediction in a ramping-up semiconductor manufacturing factory with a SOM-FBPN-ensemble approach with multiple buckets and partial normalization International Journal of Advanced Manufacturing Technology 2009 42 11-12 1206 1216 2-s2.0-67650261054 10.1007/s00170-008-1665-4 Chen T. Wu H. Wang Y. Fuzzy-neural approaches with example post-classification for estimating job cycle time in a wafer fab Applied Soft Computing Journal 2009 9 4 1225 1231 2-s2.0-68849091472 10.1016/j.asoc.2009.03.006 Chen T. Job cycle time estimation in a wafer fabrication factory with a bi-directional classifying fuzzy-neural approach International Journal of Advanced Manufacturing Technology 2011 56 9–12 1007 1018 2-s2.0-80053565700 10.1007/s00170-011-3228-3 Ankenman B. E. Bekki J. M. Fowler J. Mackulak G. T. Nelson B. L. Yang F. Simulation in production planning: an overview with emphasis on recent developments in cycle time estimation International Series in Operations Research & Management Science 2011 151 565 591 Gordon V. Proth J.-M. Chu C. A survey of the state-of-the-art of common due date assignment and scheduling research European Journal of Operational Research 2002 139 1 1 25 10.1016/S0377-2217(01)00181-3 MR1888259 ZBL1009.90054 Chen T. Wang Y. Incorporating the FCM-BPN approach with nonlinear programming for internal due date assignment in a wafer fabrication plant Robotics and Computer-Integrated Manufacturing 2010 26 1 83 91 2-s2.0-74249088273 10.1016/j.rcim.2009.04.001 Chen T. Lin Y. A collaborative fuzzy-neural approach for internal due date assignment in a wafer fabrication plant International Journal of Innovative Computing, Information and Control 2011 7 9 5193 5210 2-s2.0-80052494997 Chen T. A collaborative and artificial intelligence approach for semiconductor cost forecasting Computers & Industrial Engineering 2013 66 2 476 484 Pearson K. On lines and planes of closest fit to systems of points in space Philosophical Magazine 1901 2 11 559 572 Hotelling H. Analysis of a complex of statistical variables into principal components Journal of Educational Psychology 1933 24 6 417 441 2-s2.0-58149421595 10.1037/h0071325 Jaiswal A. Kumar N. Agrawal R. K. A hybrid of principal component analysis and partial least squares for face recognition across pose Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications 2012 7441 Berlin, Germany Springer 67 73 Lecture Notes in Computer Science Mohtasham J. Nateri A. S. Khalili H. Textile colour matching using linear and exponential weighted principal component analysis Coloration Technology 2012 128 3 199 203 Chen T. Wang Y. C. Long-term load forecasting by the collaborative fuzzy-neural approach International Journal of Electrical Power and Energy Systems 2012 43 454 464 Chen T. Wang Y. C. An iterative procedure for optimizing the performance of the fuzzy-neural job cycle time estimation approach in a wafer fabrication factory Mathematical Problems in Engineering 2013 2013 15 740478 10.1155/2013/740478 Xie X. L. Beni G. A validity measure for fuzzy clustering IEEE Transactions on Pattern Analysis and Machine Intelligence 1991 13 8 841 847 2-s2.0-0026204245 10.1109/34.85677 Heaton J. Introduction to Neural Networks with Java 2013 St. Louis, Mo, USA Heaton Research Khashei M. Forecasting the Esfahan steel company production price in Tehran metals exchange using artificial neural networks (ANNs) [M.S. thesis] 2005 Isfahan, Iran Isfahan University of Technology Akaike H. A new look at the statistical model identification IEEE Transactions on Automatic Control 1974 19 6 716 723 MR0423716 ZBL0314.62039 2-s2.0-0016355478 Rissanen J. Stochastic complexity and modeling The Annals of Statistics 1986 14 3 1080 1100 10.1214/aos/1176350051 MR856807 ZBL0602.62008 Chen T. Incorporating fuzzy c-means and a back-propagation network ensemble to job completion time prediction in a semiconductor fabrication factory Fuzzy Sets and Systems 2007 158 19 2153 2168 2-s2.0-34547462575 10.1016/j.fss.2007.04.013 MR2405300 Wu H. C. Chen T. A fuzzy-neural ensemble and geometric rule fusion approach for scheduling a wafer fabrication factory Mathematical Problems in Engineering 2013 2013 14 956978 10.1155/2013/956978