# A Selectively Fuzzified Back Propagation Network Approach for Precisely Estimating the Cycle Time Range in Wafer Fabrication

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## Abstract

**:**

## 1. Introduction

- (1)
- Some existing methods establish the confidence interval of the cycle time [12,13,14,15]. However, even with advanced computing technologies such as deep learning and big data analysis, the accuracy of predicting the cycle time is still not satisfactory. As far as the impact is reached, the established confidence interval of the cycle time has no reference value.
- (2)
- In some studies, the parameters of a cycle time forecasting method were fuzzified to generate a fuzzy forecast, which represents the range of the cycle time. Most of these methods only fuzzified a single parameter to simplify calculations [4,16]. However, fuzzifying more parameters can further shorten the range of the cycle time.

## 2. Literature Review

## 3. Methodology

- (1)
- Preprocess the collected data: two major tasks in this step are feature selection and data normalization.
- (2)
- Construct a SFBPN to forecast the cycle time of a job.
- (3)
- Train the SFBPN using an existing algorithm to derive the cores of the network parameters.
- (4)
- Apply the random search and local optimization algorithm to derive the lower and upper bounds of the thresholds.
- (5)
- Estimate the cycle time ranges of all of the jobs.
- (6)
- Evaluate the forecasting precision.

#### 3.1. Data Preprocessing

#### 3.2. Forecasting the Cycle Time of a Job Using a SFBPN

**Theorem**

**1.**

**Proof.**

**Property**

**1.**

#### 3.3. Determining the Values of Network Parameters

**NLP Problem**)

**Theorem**

**2.**

**Proof.**

#### 3.4. Fuzzifying Thresholds on Hidden-Layer Nodes

**Theorem**

**3.**

**Proof.**

Algorithm 1: Random search and local optimization |

Step 1. Set t (time index) to 1. Step 2. Set $A{R}_{\mathrm{min}}$ (the minimum of AR so far) to a large positive value. Step 3. Set ${\theta}_{l3}^{h}={\theta}_{l2}^{h*}+{\xi}_{l}$ and ${\theta}_{l1}^{h}={\theta}_{l2}^{h*}-{\zeta}_{l}$ for all l; ${\xi}_{l}$ and ${\zeta}_{l}$ are random numbers within [0, v]. Step 4. Derive the values of ${\theta}_{1}^{o*}$ and ${\theta}_{3}^{o*}$ according to Theorem 3. Step 5. Calculate ${o}_{j1}$ and ${o}_{j3}$ based on the updated network parameters. Step 6. Evaluate AR. Step 7. If $A{R}_{\mathrm{min}}>AR$, set $A{R}_{\mathrm{min}}$ to AR and record the values of ${\theta}_{l3}^{h}$, ${\theta}_{l1}^{h}$, ${\theta}_{1}^{o*}$ and ${\theta}_{3}^{o*}$. Step 8. t = t + 1. Step 9. If t > T (the number of iterations), go to Step 10; otherwise, return to Step 3. Step 10. Stop. |

## 4. Case Study

#### 4.1. Background

#### 4.2. Application of the Proposed Methodology

#### 4.3. Comparison with Existing Methods

**QP**)

- (1)
- All of the compared methods maximized the hit rate for the training data. However, the average ranges achieved using these methods differed significantly, as illustrated by Figure 7. In this regard, the proposed methodology outperformed the existing methods by establishing the narrowest range for the cycle time.

- (2)
- It is questionable whether the advantage of the SFBPN approach over the existing methods is significant. To investigate this, the following hypotheses were tested:

**Hypothesis**

**1**

**(H1).**

**Hypothesis**

**2**

**(H2).**

- (3)
- For the test data, none of these methods optimized the hit rate and the average range simultaneously. Hit rate was usually enhanced at the expense of wide ranges of fuzzy cycle time forecasts. For this sake, CFI might be a better measure for forecasting precision. In this regard, the proposed methodology surpassed the existing methods through reducing the CFI by up to 65%, as shown in Figure 8.

- (4)
- In the random search and local optimization algorithm, it is interesting to know whether the ranges of random numbers affected the forecasting performance of the proposed methodology. To investigate this issue, various ranges of ${\xi}_{l}$ and ${\zeta}_{l}$ were tried so as to observe changes in the forecasting precision. The results are summarized in Figure 9. Obviously, with a wider range, it became more difficult to find the optimal solution, which led to a poorer forecasting precision.

## 5. Conclusions and Future Research Directions

- (1)
- For training data (i.e., learned examples), all of the compared methods were able to include the actual values in the corresponding fuzzy cycle time forecasts or cycle time confidence intervals. However, only the proposed methodology could minimize the average ranges of the fuzzy cycle time forecasts.
- (2)
- For the test data (i.e., unlearned examples), CFI was a better measure for the forecasting precision. In this regard, the advantage of the proposed methodology over existing methods was up to 65%.

- (1)
- Although the random search and local optimization algorithm is likely to find a promising solution within a short time, it cannot guarantee the global optimality of the solution.
- (2)
- The case used to illustrate the proposed methodology is relatively small. A larger case needs to be analyzed to further elaborate the effectiveness of the proposed methodology.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Method | Type of Cycle Time Forecasts | Fuzzified Parameters | Optimization Method | Precision |
---|---|---|---|---|

Wang and Zhang [14] | Crisp | No | Optimized feature selection | Low |

Chen and Wu [16] | Fuzzy | Threshold on the node of the output layer | Equations based on the linear function of the output | High |

Wang et al. [21] | Crisp | No | Deep learning | Low |

Wang et al. [24] | Crisp | No | Optimized feature selection | Low−moderate |

The proposed methodology | Fuzzy | Thresholds on nodes of hidden and output layers | Random search and local optimization algorithm | Very high |

Variable | Definition |
---|---|

${x}_{j1}$ | job size (pieces) |

${x}_{j2}$ | fab work-in-process (WIP; jobs) |

${x}_{j3}$ | queue length before the bottleneck (jobs) |

${x}_{j4}$ | queue length on the processing route (jobs) |

${x}_{j5}$ | average waiting time of recently completed jobs (h) |

${x}_{j6}$ | fab utilization |

Data Part | AR (h) | HR | CFI (h) |
---|---|---|---|

Training | 260 | 100% | 260 |

Test | 261 | 43% | 613 |

Data Part | AR (h) | HR | CFI (h) |
---|---|---|---|

Training | 1736 | 100% | 1736 |

Test | 1736 | 100% | 1736 |

Data Part | AR (h) | HR | CFI (h) |
---|---|---|---|

Training | 617 | 100% | 617 |

Test | 616 | 100% | 616 |

**Table 6.**The forecasting precision using the FBPN method proposed by Chen and Lin [17].

Data Part | AR (h) | HR | CFI (h) |
---|---|---|---|

Training | 301 | 100% | 301 |

Test | 301 | 43% | 709 |

6σ Confidence Interval | FLR-QP | FBPN | SFBPN | |
---|---|---|---|---|

Mean | 1736.5 | 616.8 | 301.1 | 260.0 |

Variation | 6.4 × 10^{−24} | 506.6 | 77.6 | 4856.5 |

Observations | 80 | 80 | 80 | 80 |

Pearson correlation coefficient | 0.005 | −0.091 | −0.168 | |

Degree of freedom | 79 | 79 | 79 | |

t statistic | 189.5 | 42.46 | 5.12 | |

P(T ≤ t) one-tail | 4.4 × 10^{−107} | 1.87 × 10^{−56} | 1.07 × 10^{−6} | |

t Critical one-tail | 1.66 | 1.66 | 1.66 | |

P(T ≤ t) two-tail | 8.7 × 10^{−107} | 3.74 × 10^{−56} | 2.14 × 10^{−6} | |

t Critical two-tail | 1.99 | 1.99 | 1.99 |

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**MDPI and ACS Style**

Wang, Y.-C.; Tsai, H.-R.; Chen, T.
A Selectively Fuzzified Back Propagation Network Approach for Precisely Estimating the Cycle Time Range in Wafer Fabrication. *Mathematics* **2021**, *9*, 1430.
https://doi.org/10.3390/math9121430

**AMA Style**

Wang Y-C, Tsai H-R, Chen T.
A Selectively Fuzzified Back Propagation Network Approach for Precisely Estimating the Cycle Time Range in Wafer Fabrication. *Mathematics*. 2021; 9(12):1430.
https://doi.org/10.3390/math9121430

**Chicago/Turabian Style**

Wang, Yu-Cheng, Horng-Ren Tsai, and Toly Chen.
2021. "A Selectively Fuzzified Back Propagation Network Approach for Precisely Estimating the Cycle Time Range in Wafer Fabrication" *Mathematics* 9, no. 12: 1430.
https://doi.org/10.3390/math9121430