Bayesian Method for Quality Control with Weibull Distribution
Article Sidebar
Main Article Content
Abstract
Weibull distribution is one of the continous probability density function. This distribution is known as a flexible distribution. One of it’s flexibility is can be transformed into other distributions such as exponential distribution depends on parameter selected. Similar to other distribution, Weibull distribution also characterized by cumulative distribution function, mean, variance and moment generating function. One of the well-known application of this distribution is in the field of quality control utilizing reliability data and the well-known tool in quality control is control chart. Therefore, reliability data is not follow normal distribution then Shewhart control chart cannot be applied. To solve this problem, control chart can be formed with the control limit is obtained by Bayesian method. In applying Bayesian method, prior distribution of shape parameter is assumed to be uniform distribution and variables for reliability is assumed to be invers of Weibull distribution. The prior distribution is combine with likelihood function then posterior distribution is obtained which in then used as the control limits by find it’s mean.
Article Details
RANGE: Jurnal Pendidikan Matematika
References
Abdel-Motaleb, H. (2022). Statistical Process Control. Cutting Tool Engineering, 74(5), 32–35.
Ahmad, Z., & Hussain, Z. (2017). Flexible Weibull Extended Distribution. MAYFEB Journal of Materials Science , 2(June 2017), 5–18.
Aichouni, M., & Bachioua, L. (2014). Control Charts for Non-Normal Data : Illustrative Example from the Construction Industry Business 2 Johnson ` s Distributions in Quality Improvement. The 16th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, 2, 71–76.
Al-Bossly, A. (2020). Bayesian statistics application on reliability prediction and analysis. Journal of Statistics Applications and Probability, 9(1), 19–34. https://doi.org/10.18576/jsap/090103
Al-Zahrani, B., Marinho, P. R. D., Fattah, A. A., Ahmed, A. H. N., & Cordeiro, G. M. (2016). The (P-A-L) extended weibull distribution: A new generalization of the weibull distribution. Hacettepe Journal of Mathematics and Statistics, 45(3), 851–868. https://doi.org/10.15672/HJMS.20156110040
Basheer, G. T., & Algamal, Z. Y. (2021). Reliability Estimation of Three Parameters Weibull Distribution based on Particle Swarm Optimization. Pakistan Journal of Statistics and Operation Research, 17(1), 35–42. https://doi.org/10.18187/pjsor.v17i1.2354
Benkov, M., Bedn, D., & Bogdanovsk, G. (2024). Process Capability Evaluation Using Capability Indices as a Part of Statistical Process Control.
Black, G., Smith, J., & Wells, S. (2011). The impact of Weibull data and autocorrelation on the performance of the Shewhart and exponentially weighted moving average control charts. International Journal of Industrial Engineering Computations, 2(3), 575–582. https://doi.org/10.5267/j.ijiec.2011.03.002
Erto, P., & Giorgio, M. (2013). A note on using Bayes priors for Weibull distribution. http://arxiv.org/abs/1310.7056
Guo, J. (2019). An Introduction of Bayesian Method for Reliability Data Analysis Zhaojun ( Steven ) Li , Ph . D . & Jian Guo. January 2018.
Hsu, Y. C., Pearn, W. L., & Lu, C. S. (2011). Capability measures for Weibull processes with mean shift based on Erto’s-Weibull control chart. International Journal of Physical Sciences, 6(19), 4533–4547. https://doi.org/10.5897/IJPS11.757
Jeang, A. (2015). Robust product design and process planning in using process capability analysis. Journal of Intelligent Manufacturing, 26(3), 459–470. https://doi.org/10.1007/s10845-013-0802-6
Kumar, A. R., & V, K. (2017). A Study on System Reliability in Weibull Distribution. Ijireeice, 5(3), 38–41. https://doi.org/10.17148/ijireeice.2017.5308
Lai, C. (2006). Springer Handbook of Engineering Statistics. Springer Handbook of Engineering Statistics, May. https://doi.org/10.1007/978-1-84628-288-1
Limpert, E., & Stahel, W. A. (2011). Problems with using the normal distribution - and ways to improve quality and efficiency of data analysis. PLoS ONE, 6(7). https://doi.org/10.1371/journal.pone.0021403
Lu, D., Wang, L., Sun, Y., & Wang, X. (2018). Statistical Bayesian method for reliability evaluation based on ADT data. IOP Conference Series: Materials Science and Engineering, 351(1). https://doi.org/10.1088/1757-899X/351/1/012008
Luko, S. N. (1999). A review of the weibull distribution and selected engineering applications. SAE Technical Papers, September 1999. https://doi.org/10.4271/1999-01-2859
Madiana, A. A. (2022). Process Capability Analysis for Quality Control at the Mixing Stage of Animal Feed Products at PT. Gold Coin Indonesia. Mecomare, 10(4), 170–179. www.trigin.pelnus.ac.id
Montgomery, D.C. (2013). Introduction to Statistical Quality Control. Hoboken: John and Wiley
Nabilah, A. N., Soelistyo, A., & Kusuma, H. (2014). 済無No Title No Title No Title. Paper Knowledge . Toward a Media History of Documents, 5(2), 40–51.
Petropoulos, F., Apiletti, D., Assimakopoulos, V., Babai, M. Z., Barrow, D. K., Ben Taieb, S., Bergmeir, C., Bessa, R. J., Bijak, J., Boylan, J. E., Browell, J., Carnevale, C., Castle, J. L., Cirillo, P., Clements, M. P., Cordeiro, C., Cyrino Oliveira, F. L., De Baets, S., Dokumentov, A., … Ziel, F. (2022). Forecasting: theory and practice. International Journal of Forecasting, 38(3), 705–871. https://doi.org/10.1016/j.ijforecast.2021.11.001
ROOHI. (2003). Some Problems in Realibility Theory.Ph.D. 1–101.
Şengöz, N. G. (2018). Control Charts to Enhance Quality. Quality Management Systems - a Selective Presentation of Case-Studies Showcasing Its Evolution. https://doi.org/10.5772/intechopen.73237
Sheldon, N., Bain, L. J., & Engelhart, M. (1991). Introduction to Probability and Mathematical Statistics. The Statistician, 40(4), 465. https://doi.org/10.2307/2348752
Wu, S., Castagliola, P., & Celano, G. (2021). A distribution-free EWMA control chart for monitoring time-between-events-and-amplitude data. Journal of Applied Statistics, 48(3), 434–454. https://doi.org/10.1080/02664763.2020.1729347
Zhang, L., Jin, G., & You, Y. (2019). Reliability assessment for very few failure data and weibull distribution. Mathematical Problems in Engineering, 2019. https://doi.org/10.1155/2019/8947905