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 fuzzyneural 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 cmeans (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 fuzzyneural approach.
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 [
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 [
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 ontime 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 [
Due date assignment, release control, and buffer control affect each other. Maketoorder wafer fabrication factories are confronted with both due date quotation and production scheduling problems at the same time [
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 [
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 [
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 [
To tackle these problems, an effective fuzzyneural 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 fuzzyneural 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
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
The operating procedure of the effective fuzzyneural approach consists of several steps that will be described in the following sections.
Forming new variables by constructing linear combinations of the original variables using PCA.
Classifying jobs using fuzzy cmeans (FCM).
Forecasting the cycle times of jobs in each category using a BPN.
Determining the upper bound of the cycle time using the IUBR approach.
A flow chart of the proposed methodology is shown in Figure
The flowchart of the proposed methodology.
First, PCA is used to replace the inputs to the BPN. PCA was invented by Pearson [
The PCA process.
The references on the combination of PCA, FCM, and BPN are still very limited [
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:
The FCM procedure.
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
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
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:
Output (
Some algorithms are applicable for training a BPN in the backward phase, such as the gradient descent algorithms, the conjugate gradient algorithms, and the LevenbergMarquardt algorithm. In this study, the LevenbergMarquardt algorithm is applied. The LevenbergMarquardt is the most widely used optimization algorithm. It outperforms simple gradient descent and other conjugate gradient methods in a wide variety of problems. The LevenbergMarquardt algorithm uses approximation and updates the network parameters in a Newtonlike way, as described below.
The network parameters are placed in vector
The LevenbergMarquardt 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 LevenbergMarquardt method is in no way optimal but is just a heuristic, it works extremely well in practice.
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 [
Substituting (
Assume that the adjustment made to the threshold of the output node is indicated as
Since
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
The upper bound is reduced in an iterative manner.
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 [
BPN aggregation [
Generalized average method [
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
An example.









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
The comparison of the standard deviations of the inputs.
Subsequently, we standardize the data (see Table
The standardized data.








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
New inputs to the FCMBPN.




−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
The results of the
Number of categories 




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
Category  Jobs 

1  4–6, 14–17 
2  1011, 13, 2324, 27, 30, 33 
3  1, 18–22, 3132 
4  23, 7–9, 12, 2526, 2829, 34–40 
After preclassification, the threelayer 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 bestfitted 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
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 
FCMBPN  15  1.2%  38 
PCABPN  27  2.3%  67 
PCAFCMBPN  11  0.9%  29 
The forecasting results using PCAFCMBPN.
Subsequently, the IUBR approach is applied to determine the upper bound of the cycle time. In the first iteration,
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
The results of the five iterations.
Iteration 1 
Iteration 2 
Iteration 3 
Iteration 4 
Iteration 5 

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:
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
Constant allowance policy (CON, PCAFCMBPN + CON): add three times the RMSE of the prediction approach to the completion time forecasts to determine the due date.
Selective allowance policy (SAP, PCAFCMBPN + 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:
Random assignment policy (RAP, PCAFCMBPN + RAP): add the extra allowance to the completion time forecasts of the same quantity of jobs that are randomly chosen.
No allowance policy (NAP, PCAFCMBPN + 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 (
mean tardiness (
sum of allowances.
The comparison results are summarized in Table
It guarantees the ontime delivery of the jobs. Both
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 ontime 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 PCAFCMBPN 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 

31  0  0  9  7  14  0 

82  0  0  3.9  2.1  3.5  0 
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 fuzzyneural 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 fuzzyneural approach for internal due date assignment is able to improve ontime delivery has been proved by the case study. Based on the above analysis,
the forecasting accuracy (measured with MAE, MAPE, and RMSE) of the PCAFCMBPN 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 [
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.
This study was financially supported by the National Science Council of Taiwan.