## 5 Discussion
Assessment of the mixed population at each month of calendar time resulted in an increasing Weibull shape parameter estimate at each assessment. When the two populations were separated and estimated properly, a better fit with more accurate estimates of Weibull shape parameters resulted. [PERSON] and [PERSON] [36] proposed control charts for monitoring changes in the Weibull shape parameter \(\beta\). These charts were based on the range of a random sample from the smallest extreme value distribution. The control chart limits depended only on the sample size, the desired stable average run length (ARL), and the stable value of \(\beta\). They derived control limits for both one- and two-sided control charts. They were unbiased with respect to the ARL. [PERSON] and [PERSON] discuss sample size requirements when estimating the stable value of \(\beta\) from past data. The proposed method was applied to data on the breaking strengths of carbon fibers. For this case, the authors recommended one-sided charts for detecting specific changes in \(\beta\) because they were expected to signal out-of-control sooner than the two-sided charts. [PERSON], [PERSON], and [PERSON] [37] proposed a methodology to assess the reliability factor in the management of industrial equipment designing. The complexity of several industrial procedures and the equipment required establishes that not all of an asset's failure patterns may be easily handled through maintenance service activities done after its manufacture and use. To avoid all kinds of high-impact failures when using the product, there must be a stage of elimination by removing some maintenance needs, and it should be done considering their own foundations from the very first moment when the asset is designed and produced. These failures might emerge in production, quality, safety, the environment, and costs, among others, which are hard to identify and control. All this requires different concepts and tools in industrial maintenance and service and the engineering of reliability during the design phase. [PERSON] [38] proposed conditional Weibull control charts using multiple linear regression. Because, in Weibull analysis, the key variable to be monitored is the lower reliability index (\(R(t)\)), and because the \(R(t)\) index is completely determined by both the lower scale parameter (\(\eta\)) and the lower shape parameter (\(\beta\)), a pair of control charts to monitor a Weibull process were proposed based on the direct relationships between \(\eta\) and \(\beta\) with the log-mean (\(\mu x\)) and the log-standard deviation (\(\sigma x\)) of the analyzed lifetime data. Moreover, because of the fact that, in Weibull analysis, right-censored data are common, and because they introduce uncertainty into the estimated Weibull parameters, in the proposed charts, \(\mu x\) and \(\sigma x\) are estimated based on the conditional expected times of the related Weibull family. After that, both \(\mu x\) and \(\sigma x\) are used to monitor the Weibull process. In particular, \(\mu x\) was set as the lower control limit to monitor \(\eta\), and \(\sigma x\) was set as the upper control limit to monitor \(\beta\). Numerical applications were used to show how the charts work. [PERSON] and [PERSON] [39] analyzed the parameter estimation for the Weibull distribution with right-censored data using the EM algorithm. Maximum-likelihood estimation (MLE) is a method for estimating the parameters of a statistical model for given data set. This method allowed the authors to estimate the unknown parameters of a statistical model. These parameters were obtained by maximizing the likelihood function of the model in question. The estimation of the parameters of the Weibull distribution by the maximum-likelihood method based on information from a historical record with right-censored data showed this difficulty. The solution used the Expectation Maximization (EM) algorithm.