偵測識別批量生產系統中的低良率機器對於最大限度地減少缺陷輸出至關重要。 然而，解決這個複雜問題的最新方法需要大量的專業知識、昂貴的資源或兩者的結合。 為了解決這個挑戰，我們提出了一種使用「Maximum Likelihood Estimation」和「Bootstrap Confidence Intervals」的簡單且經濟高效成本效益高的方法。此方法可以有效估計每台機器的良率，從而能夠精確定位低良率機器並產生優先順序清單以進行進一步調查。 擁有 50-500 台機器的製造商可以透過建立包含約 6-20 倍生產機器批次的資料集來實施此方法。 當滿足此條件時，系統可有效偵測識別最多 5 台低良率機器。 此外，估計的每台機器良率可用於預測各個生產批次的良率，為生產計劃和優化提供有價值的見解。

Identifying low-yield machines in batch production systems is crucial to minimize defective outputs. However, recent methods of addressing this complex issue require considerable expertise, expensive resources, or a combination of both. To solve this challenge, we propose a straightforward and cost-effective method using maximum likelihood estimation and bootstrap confidence intervals. This method efficiently estimates per-machine yield, enabling the pinpointing of low-yield machines and the generation of a prioritized list for further investigation. Manufacturers with 50–500 machines can implement this method by building a dataset containing approximately 6-20 times as many batches as production machines. When this condition is met, the system effectively identifies up to 5 low-yield machines. Additionally, the estimated per-machine yield can be used to predict the yield of individual production batches, providing valuable insights for production planning and optimization.

[1] S. L. Lee et al., “Modernizing Pharmaceutical Manufacturing: from Batch to Continuous Production,” J Pharm Innov, vol. 10, no. 3, pp. 191–199, Sep. 2015, doi: 10.1007/s12247-015-9215-8.
[2] K. P. Cole and M. D. Johnson, “Continuous flow technology vs. the batch-by-batch approach to produce pharmaceutical compounds,” Expert Rev Clin Pharmacol, vol. 11, no. 1, pp. 5–13, Jan. 2018, doi: 10.1080/17512433.2018.1413936.
[3] A. I. Malik and B. Sarkar, “Disruption management in a constrained multi-product imperfect production system,” J Manuf Syst, vol. 56, pp. 227–240, Jul. 2020, doi: 10.1016/j.jmsy.2020.05.015.
[4] P. F. E. Adipraja, C.-C. Chang, W.-J. Wang, and D. Liang, “Prediction of per-batch yield rates in production based on maximum likelihood estimation of per-machine yield rates,” J Manuf Syst, vol. 62, pp. 249–262, Jan. 2022, doi: 10.1016/j.jmsy.2021.11.015.
[5] Jianxin Jiao, M. M. Tseng, Qinhai Ma, and Yi Zou, “Generic Bill-of-Materials-and-Operations for High-Variety Production Management,” Concurrent Engineering, vol. 8, no. 4, pp. 297–321, Dec. 2000, doi: 10.1177/1063293X0000800404.
[6] J. E. See, “Visual Inspection Reliability for Precision Manufactured Parts,” Human Factors: The Journal of the Human Factors and Ergonomics Society, vol. 57, no. 8, pp. 1427–1442, Dec. 2015, doi: 10.1177/0018720815602389.
[7] S. F. Beckert and W. S. Paim, “Critical analysis of the acceptance criteria used in measurement systems evaluation,” International Journal of Metrology and Quality Engineering, vol. 8, p. 23, Oct. 2017, doi: 10.1051/ijmqe/2017016.
[8] A. S. Nookabadi and J. E. Middle, “An integrated quality assurance information system for the design‐to‐order manufacturing environment,” The TQM Magazine, vol. 18, no. 2, pp. 174–189, Mar. 2006, doi: 10.1108/09544780610647883.
[9] D. Powell, M. C. Magnanini, M. Colledani, and O. Myklebust, “Advancing zero defect manufacturing: A state-of-the-art perspective and future research directions,” Comput Ind, vol. 136, p. 103596, Apr. 2022, doi: 10.1016/j.compind.2021.103596.
[10] S. Patel, B. G. Dale, and P. Shaw, “Set‐up time reduction and mistake proofing methods: an examination in precision component manufacturing,” The TQM Magazine, vol. 13, no. 3, pp. 175–179, Jun. 2001, doi: 10.1108/09544780110385528.
[11] F. Psarommatis, J. Sousa, J. P. Mendonça, and D. Kiritsis, “Zero-defect manufacturing the approach for higher manufacturing sustainability in the era of industry 4.0: a position paper,” Int J Prod Res, vol. 60, no. 1, pp. 73–91, Jan. 2022, doi: 10.1080/00207543.2021.1987551.
[12] L. Biggio and I. Kastanis, “Prognostics and Health Management of Industrial Assets: Current Progress and Road Ahead,” Front Artif Intell, vol. 3, Nov. 2020, doi: 10.3389/frai.2020.578613.
[13] Y. J. Qu, X. G. Ming, Z. W. Liu, X. Y. Zhang, and Z. T. Hou, “Smart manufacturing systems: state of the art and future trends,” The International Journal of Advanced Manufacturing Technology, vol. 103, no. 9–12, pp. 3751–3768, Aug. 2019, doi: 10.1007/s00170-019-03754-7.
[14] S. Cofre-Martel, E. Lopez Droguett, and M. Modarres, “Big Machinery Data Preprocessing Methodology for Data-Driven Models in Prognostics and Health Management,” Sensors, vol. 21, no. 20, p. 6841, Oct. 2021, doi: 10.3390/s21206841.
[15] Y. A. Yucesan, A. Dourado, and F. A. C. Viana, “A survey of modeling for prognosis and health management of industrial equipment,” Advanced Engineering Informatics, vol. 50, p. 101404, Oct. 2021, doi: 10.1016/j.aei.2021.101404.
[16] S. Sundaram and A. Zeid, “Smart Prognostics and Health Management (SPHM) in Smart Manufacturing: An Interoperable Framework,” Sensors, vol. 21, no. 18, p. 5994, Sep. 2021, doi: 10.3390/s21185994.
[17] Z. M. Çınar, A. Abdussalam Nuhu, Q. Zeeshan, O. Korhan, M. Asmael, and B. Safaei, “Machine Learning in Predictive Maintenance towards Sustainable Smart Manufacturing in Industry 4.0,” Sustainability, vol. 12, no. 19, p. 8211, Oct. 2020, doi: 10.3390/su12198211.
[18] F. Tao, Q. Qi, A. Liu, and A. Kusiak, “Data-driven smart manufacturing,” J Manuf Syst, vol. 48, pp. 157–169, Jul. 2018, doi: 10.1016/j.jmsy.2018.01.006.
[19] T. Li et al., “WaveletKernelNet: An Interpretable Deep Neural Network for Industrial Intelligent Diagnosis,” IEEE Trans Syst Man Cybern Syst, vol. 52, no. 4, pp. 2302–2312, Apr. 2022, doi: 10.1109/TSMC.2020.3048950.
[20] A. Lokrantz, E. Gustavsson, and M. Jirstrand, “Root cause analysis of failures and quality deviations in manufacturing using machine learning,” Procedia CIRP, vol. 72, pp. 1057–1062, 2018, doi: 10.1016/j.procir.2018.03.229.
[21] H. Fei, W. Chaojun, and F. Shu-Kai S, “Fault Detection and Root Cause Analysis of a Batch Process via Novel Nonlinear Dissimilarity and Comparative Granger Causality Analysis,” Ind Eng Chem Res, vol. 58, no. 47, pp. 21842–21854, Nov. 2019, doi: 10.1021/acs.iecr.9b04471.
[22] N. Yu, Q. Xu, and H. Wang, “Wafer Defect Pattern Recognition and Analysis Based on Convolutional Neural Network,” IEEE Transactions on Semiconductor Manufacturing, vol. 32, no. 4, pp. 566–573, Nov. 2019, doi: 10.1109/TSM.2019.2937793.
[23] J. O’Leary, K. Sawlani, and A. Mesbah, “Deep Learning for Classification of the Chemical Composition of Particle Defects on Semiconductor Wafers,” IEEE Transactions on Semiconductor Manufacturing, vol. 33, no. 1, pp. 72–85, Feb. 2020, doi: 10.1109/TSM.2019.2963656.
[24] B. Steenwinckel et al., “FLAGS: A methodology for adaptive anomaly detection and root cause analysis on sensor data streams by fusing expert knowledge with machine learning,” Future Generation Computer Systems, vol. 116, pp. 30–48, Mar. 2021, doi: 10.1016/j.future.2020.10.015.
[25] Y. S. Cho and S. B. Kim, “Quality-Discriminative Localization of Multisensor Signals for Root Cause Analysis,” IEEE Trans Syst Man Cybern Syst, vol. 52, no. 7, pp. 4374–4387, Jul. 2022, doi: 10.1109/TSMC.2021.3096529.
[26] P. F. E. Adipraja, C.-C. Chang, H.-S. Yang, W.-J. Wang, and D. Liang, “Detecting Low-Yield Machines in Batch Production Systems Based on Observed Defective Pieces,” IEEE Trans Syst Man Cybern Syst, pp. 1–12, 2024, doi: 10.1109/TSMC.2024.3374393.
[27] R. R. Mohan, K. Thiruppath, R. Venkatrama, and S. Raghuraman, “Quality Improvement through First Pass Yield using Statistical Process Control Approach,” Journal of Applied Sciences, vol. 12, no. 10, pp. 985–991, May 2012, doi: 10.3923/jas.2012.985.991.
[28] Q. Su, L. Liu, and D. E. Whitney, “A Systematic Study of the Prediction Model for Operator-Induced Assembly Defects Based on Assembly Complexity Factors,” IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, vol. 40, no. 1, pp. 107–120, Jan. 2010, doi: 10.1109/TSMCA.2009.2033030.
[29] J. C. Soares, S. Sousa, and A. Tereso, “Industry practices on the rework of defective products: survey results,” The TQM Journal, vol. 32, no. 6, pp. 1177–1196, Apr. 2020, doi: 10.1108/TQM-06-2019-0162.
[30] H. Zhou and C. R. Weinberg, “Modeling Conception as an Aggregated Bernoulli Outcome with Latent Variables via the EM Algorithm,” Biometrics, vol. 52, no. 3, p. 945, Sep. 1996, doi: 10.2307/2533055.
[31] K. Lange, “A Gradient Algorithm Locally Equivalent to the Em Algorithm,” Journal of the Royal Statistical Society: Series B (Methodological), vol. 57, no. 2, pp. 425–437, Jul. 1995, doi: 10.1111/j.2517-6161.1995.tb02037.x.
[32] R. Dwivedi, N. Ho, K. Khamaru, M. J. Wainwright, M. I. Jordan, and B. Yu, “Singularity, misspecification and the convergence rate of EM,” The Annals of Statistics, vol. 48, no. 6, Dec. 2020, doi: 10.1214/19-AOS1924.
[33] C. Biernacki, G. Celeux, and G. Govaert, “Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models,” Comput Stat Data Anal, vol. 41, no. 3–4, pp. 561–575, Jan. 2003, doi: 10.1016/S0167-9473(02)00163-9.
[34] W. Xiang, A. Karfoul, C. Yang, H. Shu, and R. Le Bouquin Jeannès, “An exact line search scheme to accelerate the EM algorithm: Application to Gaussian mixture models identification,” J Comput Sci, vol. 41, p. 101073, Mar. 2020, doi: 10.1016/j.jocs.2019.101073.
[35] E. Shireman, D. Steinley, and M. J. Brusco, “Examining the effect of initialization strategies on the performance of Gaussian mixture modeling,” Behav Res Methods, vol. 49, no. 1, pp. 282–293, Feb. 2017, doi: 10.3758/s13428-015-0697-6.
[36] M. Ouzineb, F. Z. Mhada, R. Pellerin, and I. El Hallaoui, “Optimal planning of buffer sizes and inspection station positions,” Prod Manuf Res, vol. 6, no. 1, pp. 90–112, Jan. 2018, doi: 10.1080/21693277.2017.1422812.
[37] M. Chincholkar and J. W. Herrmann, “Estimating manufacturing cycle time and throughput in flow shops with process drift and inspection,” Int J Prod Res, vol. 46, no. 24, pp. 7057–7072, Dec. 2008, doi: 10.1080/00207540701513893.
[38] H. Cho and C. Kirch, “Bootstrap confidence intervals for multiple change points based on moving sum procedures,” Comput Stat Data Anal, vol. 175, p. 107552, Nov. 2022, doi: 10.1016/j.csda.2022.107552.
[39] P. Hall, “On the Number of Bootstrap Simulations Required to Construct a Confidence Interval,” The Annals of Statistics, vol. 14, no. 4, Dec. 1986, doi: 10.1214/aos/1176350169.
[40] P. Hall, The Bootstrap and Edgeworth Expansion. in Springer Series in Statistics. New York, NY: Springer New York, 1992. doi: 10.1007/978-1-4612-4384-7.
[41] M. R. Chernick, Bootstrap Methods: A Guide for Practitioners and Researchers, Second Edi. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2007. doi: 10.1002/9780470192573.
[42] C. Karras, A. Karras, M. Avlonitis, I. Giannoukou, and S. Sioutas, “Maximum Likelihood Estimators on MCMC Sampling Algorithms for Decision Making,” in Artificial Intelligence Applications and Innovations. AIAI 2022 IFIP WG 12.5 International Workshops, vol. 652, I. Maglogiannis, L. Iliadis, J. Macintyre, and P. Cortez, Eds., Springer International Publishing, 2022, pp. 345–356. doi: 10.1007/978-3-031-08341-9_28.