A nonlinear programming and artificial neural network approach is presented in this study to optimize the performance of a job dispatching rule in a wafer fabrication factory. The proposed methodology fuses two existing rules and constructs a nonlinear programming model to choose the best values of parameters in the two rules by dynamically maximizing the standard deviation of the slack, which has been shown to benefit scheduling performance by several studies. In addition, a more effective approach is also applied to estimate the remaining cycle time of a job, which is empirically shown to be conducive to the scheduling performance. The efficacy of the proposed methodology was validated with a simulated case; evidence was found to support its effectiveness. We also suggested several directions in which it can be exploited in the future.
This study attempts to optimize the performance of a job dispatching rule in a wafer fabrication factory. The production equation required by a wafer fabrication factory is very expensive and must be fully utilized. For this purpose, to ensure that the capacity does not substantially exceed the demand is a perquisite. Subsequently, how to plan the use of the existing capacity to shorten the cycle time and maximize the turnover rate is an important goal. In this regard, scheduling is undoubtedly a very useful tool.
However, some studies [
Semiconductor manufacturing can be divided into four stages: wafer fabrication, wafer probing, packaging, and final testing. The most important stage is wafer fabrication. It is also the most time-consuming one. In this study, we investigated the job dispatching for this stage. This field includes many different methods, including dispatching rules, heuristics, data-mining-based approaches [
Some advances in this field are as follows. Altendorfer et al. [
In a multiple-objective study, Chen and Wang [
The differences between the proposed methodology and the previous methods.
Rule | Number of objectives | Objectives | Number of adjustable parameters | Optimized? | How to derive the rule? |
---|---|---|---|---|---|
NFSMCT | 1 | Average cycle time | 1 | No | (i) Generalizing FSMCT |
1f-TNFSVCT | 1 | Cycle time standard variation | 1 | No | (i) Generalizing FSVCT |
1f-TNFSMCT | 1 | Average cycle time | 1 | No | (i) Generalizing FSMCT |
2f-TNFSVCT | 1 | Cycle time standard deviation | 2 | No | (i) Generalizing FSVCT |
4f-biNFS | 2 | Average cycle time, cycle time standard deviation | 2 | Yes | (i) Fusing FSVCT and FSMCT |
The proposed methodology | 2 | Average cycle time, cycle time standard deviation | 2 | Yes | (i) Fusing 2f-TFSMCT and 2f-TNFSVCT |
At the same time, Chen [
This study adopts several treatments to further improve Wang et al.’s approach. In nonlinear fluctuation smoothing rules, it is common that some jobs have very large or small slack values, that is the extreme case (see Figure Two objectives, the average cycle time and cycle time standard deviation, are considered at the same time by fusing the results from 2f-TNFSMCT and 2f-TNFSVCT. A nonlinear programming problem is solved to find the optimal values of parameters in 2f-TNFSMCT and 2f-TNFSVCT. On the other hand, the remaining cycle time of a job needs to be estimated in 2f-TNFSMCT and 2f-TNFSVCT. For this reason, we also propose a more effective fuzzy-neural approach to estimate the remaining cycle time of a job. The fuzzy-neural approach is a modification of the fuzzy c-means and back propagation network (FCM-BPN) approach [
The extreme cases.
The differences between the proposed methodology and the previous methods are summarized in Table
The remainder of this paper is arranged as follows. Section
The variables and parameters that will be used in the proposed methodology are defined in the following.
The proposed methodology includes the following seven steps.
Replacing parameters using PCA.
Use FCM to classify jobs. The required inputs for this step are the new variables determined by PCA. To determine the optimal number of categories, we use the
Use the BPN approach to estimate the cycle time of each job. Jobs of different categories will be sent to different three-layer BPNs. The inputs to the three-layer BPN include the new variables of a job, while the output is the estimated cycle time of the job.
Derive the remaining cycle time of each job from the estimated cycle time.
Incorporate the estimated remaining cycle time into the new rule that is composed of two subrules–2f-TNFSMCT and 2f-TNFSVCT.
Find out the optimal value of parameters in the new rule by solving a nonlinear programming problem.
The remaining cycle time of a job being produced in a wafer fabrication factory is the time still needed to complete the job. If the job is just released into the wafer fabrication factory, then the remaining cycle time of the job is its cycle time. The remaining cycle time is an important input for the scheduling rule. Past studies (e.g., [
First, PCA is used to replace the inputs to the FCM-BPN. The combination of PCA and FCM has been shown to be a more effective classifier than FCM alone. Although there are more advanced applications of PCA, in this study PCA is used to enhance the efficiency of training the FCM-BPN. PCA consists of the four following steps: Raw data standardization: to eliminate the difference between the dimensions and the impact of large numerical difference in the original variables where Establishment of the correlation matrix where Determination of the number of principal components: the variance contribution rate is calculated as:
and the accumulated variance contribution rate is
Choose the smallest Formation of the following matrixes:
After PCA, examples are then classified using FCM.
In the proposed methodology, jobs are classified into
FCM classifies jobs by minimizing the following objective function:
Produce a preliminary clustering result. where Remeasure the distance from each job to the centroid of each category, and then recalculate the corresponding membership. Stop if the following condition is met. Otherwise, return to step (2):
where
Finally, the separate distance test (
subject to
After clustering, a portion of the jobs in each category is input as the “training examples” to the three-layer BPN to determine the parameter values. The configuration of the three-layer BPN is set up as follows. First, inputs are the six parameters associated with the
The procedure for determining the parameter values is now described. Two phases are involved at the training stage. At first, in the forward phase, inputs are multiplied with weights, summated, and transferred to the hidden layer. Then activated signals are outputted from the hidden layer as
The network parameters are placed in vector
The Levenberg-Marquardt algorithm is an iterative procedure. In the beginning, the user should specify the initial values of the network parameters
Finally, the BPN can be applied to estimate the cycle time of a job, and then the remaining cycle time of the job can be derived as
In traditional fluctuation smoothing (FS) rules there are two different formulation methods, depending on the scheduling purpose [
Chen [
The new rule is composed of two rules. The first rule is derived by diversifying the slack in the 2f-TNFSVCT rule, aimed at minimizing the variation of cycle time [
The second rule is derived by diversifying the slack in the 2f-TNFSMCT rule, aimed at minimizing the mean cycle time:
To generate a biobjective rule, the two rules need to be combined into a single one, for which the following nonlinear programming model is to be optimized:
s.t.
To evaluate the effectiveness of the proposed methodology, simulated data were used to avoid disturbing the regular operations of the wafer fabrication factory. Simulation is a widely used technology to assess the effectiveness of a scheduling policy, especially when the proposed policy and the current practice are very different. This investigation is not possible to implement in the actual production environment. The real-time scheduling systems will input information very rapidly into the production management information systems (PROMIS). To this end, a real wafer fabrication factory located in Taichung Scientific Park of Taiwan with a monthly capacity of about 25,000 wafers was simulated. The simulation program has been validated and verified by comparing the actual cycle times with the simulated values and by analyzing the trace report, respectively. The wafer fabrication factory is producing more than 10 types of memory products and has more than 500 workstations for performing single-wafer or batch operations using 58 nm~110 nm technologies. Jobs released into the fabrication factory are assigned three types of priorities, that is, “normal,” “hot,” and “super hot.” Jobs with the highest priorities will be processed first. Such a large scale accompanied with reentrant process flows make job dispatching in the wafer fabrication factory a very tough task. Currently, the longest average cycle time exceeds three months with a variation of more than 300 hours. The wafer fabrication factory is therefore seeking better dispatching rules to replace first-in first-out (FIFO) and EDD, in order to shorten the average cycle times and ensure the on-time delivery to its customers. One hundred replications of the simulation are successively run. The time required for each simulation replication is about 30 minute using a PC with Intel Dual CPU E2200 2.2 GHz and 1.99G RAM. A horizon of twenty-four months is simulated.
To assess the effectiveness of the proposed methodology and to make comparison with some existing approaches–FIFO, EDD, SRPT, CR, FSVCT, FSMCT, Justice [
To determine the due date of a job, the PCA-FCM-BPN approach was applied to estimate the cycle time, for which the Levenberg-Marquardt algorithm rather than the gradient descent algorithm was applied to speed up the network convergence. Then, we added a constant allowance of three days to the estimated cycle time, that is,
Jobs with the highest priorities are usually processed first. In FIFO, jobs were sequenced on each machine first by their priorities, then by their arrival times at the machine. In EDD, jobs were sequenced first by their priorities, then by their due dates. In CR, jobs were sequenced first by their priorities, then by their critical ratios. In the proposed methodology, the nonlinear model with
The job speed matrix.
Machine’s bottleneck status | ||||
---|---|---|---|---|
Work progress status | Behind | Rapid | Rapid | Normal |
Just in time | Rapid | Normal | Suspended | |
Advanced | Normal | Normal | Suspended |
Subsequently, the average cycle time and cycle time standard deviation of all cases were calculated to assess the scheduling performance. With respect to the average cycle time, the FIFO policy was used as the basis for comparison, while FSVCT was compared in evaluating cycle time standard deviation. The results are summarized in Tables
The performances of various approaches in the average cycle time.
Avg. cycle time (hrs) | A (normal) | A (hot) | A (super hot) | B (normal) | B (hot) |
---|---|---|---|---|---|
FIFO | 1254 | 400 | 317 | 1278 | 426 |
EDD | 1094 | 345 | 305 | 1433 | 438 |
SRPT | 948 | 350 | 308 | 1737 | 457 |
CR | 1148 | 355 | 300 | 1497 | 440 |
FSMCT | 1313 | 347 | 293 | 1851 | 470 |
FSVCT | 1014 | 382 | 315 | 1672 | 475 |
NFS | 1456 | 407 | 321 | 1452 | 421 |
Justice | 1126 | 378 | 322 | 1576 | 489 |
2f-TNFSMCT | 1369 | 379 | 306 | 1361 | 399 |
2f-TNFSVCT | 1465 | 416 | 318 | 1551 | 500 |
The proposed methodology | 1076 | 289 | 269 | 1132 | 388 |
The performances of various approaches in cycle time standard deviation.
Cycle time standard deviation (hrs) | A (normal) | A (hot) | A (super hot) | B (normal) | B (hot) |
---|---|---|---|---|---|
FIFO | 55 | 24 | 25 | 87 | 51 |
EDD | 129 | 25 | 22 | 50 | 63 |
SRPT | 248 | 31 | 22 | 106 | 53 |
CR | 69 | 29 | 18 | 58 | 53 |
FSMCT | 419 | 33 | 16 | 129 | 104 |
FSVCT | 280 | 37 | 27 | 201 | 77 |
NFS | 87 | 49 | 19 | 44 | 47 |
Justice | 120 | 26 | 20 | 69 | 32 |
2f-TNFSMCT | 75 | 37 | 17 | 47 | 19 |
2f-TNFSVCT | 38 | 38 | 29 | 33 | 24 |
The proposed methodology | 86 | 26 | 15 | 54 | 21 |
According to the experimental results, the following points can be made: For the average cycle time, the proposed methodology outperformed the baseline approach, the FIFO policy. The average advantage was about 16%. In addition, the proposed methodology surpassed the FSVCT policy in reducing cycle time standard deviation. The most obvious advantage was 59%. As expected, SRPT performed well in reducing the average cycle times, especially for product types with short cycle times (e.g., product A), but might give an exceedingly bad performance with respect to cycle time standard deviation. If the cycle time is long, the remaining cycle time will be much longer than the remaining processing time, which leads to the ineffectiveness of SRPT. SRPT is similar to FSMCT. Both try to make all jobs equally early or late. The performance of EDD was also satisfactory for product types with short cycle time. If the cycle time is long, it is more likely to deviate from the prescribed internal due date, which leads to the ineffectiveness of EDD. That becomes more serious if the percentage of the product type is high in the product mix (e.g., product type A). CR has similar problems. The proposed rule was also compared with the traditional one without slack diversification. Taking product type A with normal priority as an example, the comparison results are shown in Figure
Comparing the slack-diversifying rule with traditional rules without slack diversification.
For capital-intensive industries like wafer fabrication, efficient use of expensive equipment is very important. To this end, job dispatching is a challenging but important task. However, for such a complex production system, to optimize the scheduling performance is a tough task. As an innovative attempt, this study presents a nonlinear programming and artificial neural network approach to optimize the performance of a slack-diversifying dispatching rule in a wafer fabrication factory, to optimize the average cycle time, and to optimize cycle time standard deviation.
The proposed methodology merges two existing rules—2f-TNFSMCT and 2f-TNFSVCT, and constructs a nonlinear programming model to choose the best values of parameters in the two rules. A more effective approach is also applied to estimate the remaining cycle time of a job, which is empirically shown to be conducive to the scheduling performance. To further enhance the accuracy of the remaining cycle time estimation, other dynamic parameters must be considered. In addition, some advanced methods for the cycle time estimation, such as data mining methods [
After a simulation study, we observed the following phenomena. Through improving the accuracy of estimating the remaining cycle time, the performance of a scheduling rule can indeed be strengthened. Optimizing the adjustable factors in the two rules appears as an appropriate tool to enhance the scheduling performance of the rule. Slack diversification is indeed conducive to the performance of a fluctuation smoothing rule.
However, to further assess the effectiveness and efficiency of the proposed methodology, the only way is to apply it to an actual wafer fabrication factory. In addition, other rules can be optimized in the same way in future studies.
This work was supported by the National Science Council of Taiwan.