## 4 Case Study  
One of the objectives of this case study is to apply the EM algorithm to find the solution of the function obtained by the maximum likelihood method in the parameter estimation of the Weibull distribution using right censored data. The case study analyzes the failure history of five centrifugal pumps used by a petrochemical company for pumping similar density oil, specifically emphasizing bearing failures. Over seven years, these pumps experienced recurrent failures, resulting in significant downtime and a decrease in the overall reliability of the pumping system. Centrifugal pumps play a critical role in various industries by transporting fluids. Their design allows them to handle various fluid viscosities and temperatures, making them versatile for different applications. Reliability is essential for ensuring the efficient and uninterrupted operation of these pumps. The bearing system within centrifugal pumps is particularly crucial, as its performance directly impacts the overall reliability of the pump. The collected data revealed that bearing failures accounted for significant failures, representing approximately 38%. This highlights the importance of investigating and addressing bearing failures to improve the reliability and performance of the pumping system. Bearing failures can have severe consequences, including costly downtime, repairs, safety hazards, and environmental risks. Estimating the parameters of the Weibull distribution based on limited and censored data enables plant operators to proactively monitor and maintain critical components such as bearings to ensure continuous and safe operations. The failure data collected were analyzed to determine the frequency of each failure mode over the seven years. This analysis provides insights into the relative importance of each failure mode in contributing to the overall failure rate. Failures between regular inspections conducted by the maintenance staff, which occur at least every 8 h, were considered complete data. Hence, it was assumed that the exact moment of failure was well known, as the time between inspections was relatively short compared to the total observed time. The last recorded data point for each pump represents the end of the study rather than an actual failure. This type of situation, as mentioned earlier, is known as right censored data, where the observed data are incomplete due to the study ending without observing all potential failures. Treating the last recorded time for each pump as right censored data is a good approach, since it indicates that the failure times for those pumps are unknown, as they were still operating when the study concluded. The expectation maximization algorithm was employed to estimate the parameters of the Weibull distribution using the maximum likelihood method with right censored data. The iterative process was implemented using the R statistical program. The least squares method [32] was utilized to determine the initial solution \(\theta^{(0)}\). The iterative method terminated when the difference between iteration \(k+1\) and \(k\) was smaller than 0.1. Table 1 presents the expected values for \(\beta\) and \(\eta\) for the bearings of the five analyzed centrifugal pumps obtained through the EM algorithm. The confidence intervals for each parameter are also provided. Given the small data sample size, the bootstrap-t method was employed to determine the confidence intervals, with a confidence level of 95% [33]. For all bearings, the shape parameter \(\beta\) was found to be greater than 1. The scale parameter \(\eta\) exhibited values ranging between 509.95 and 656.96 operation days. In the bootstrap method, 1000 resampling iterations were performed to obtain reliable estimates. Figure 2 visually presents the confidence interval of the estimated parameters of the Weibull distribution specifically for bearing 1. These intervals were derived from the information obtained through the bootstrap method.