Shewhart‐type Phase II control charts for monitoring times to an event with a guaranteed in‐control and good out‐of‐control performance

Abstract

Monitoring time to event (failure) data is important in many applications. Proper monitoring and control can make the production process more efficient and provide economic advantages. In this paper, we consider the efficacy of a class of Shewhart‐type control charts for monitoring time to event data following an exponential distribution with an unknown mean, which is estimated from a class of estimators. An estimator is chosen within this class, so that the in‐control performance is maximized with respect to a number of popular criteria in the recent literature, and the proposed optimal charts are compared on the basis of their in‐control and out‐of‐control performance. The comparisons include the traditional Phase II exponential Shewhart chart using the maximum likelihood estimator. Improved in‐control and out‐of‐control performances of these charts can enhance the quality and productivity of manufacturing processes. Since no chart is best under all the criteria, a ranking system is used to choose a chart to use in practice with a good overall performance. Two illustrative examples using real data are given; summary and conclusions are offered.

A parameter‐free exponential control chart based on Kullback‐Leibler information for time‐between‐events monitoring

“…Depending on the application, there are three scenarios about how we concern the change of 𝜆 1 : (1) increasing of the occurrence rate only, (2) decreasing of the occurrence rate only, and (3) both increasing and decreasing of the occurrence , 26 Kumar and Chakraborti, 7 and Kumar et al 11 all suggested the estimation error of parameter influencing the run length distribution of TBE charts. Thus, we investigate three different estimators of 𝜆 1 in parallel and evaluate their impacts on the run length in each of the three scenarios above.…”

“…9 Ali 10 adopted a predictive Bayesian approach to set up dynamic control limit Shewhart charts. Kumar et al 11 designed a Shewhart-type chart with the occurrence rate estimated by a class of estimators and studied the effect of parameter estimation. Charts designed with sequential sampling schemes can be found in Zhang et al, 12 and Qu et al 13 Because Shewhart-type charts are not sensitive for small shift size, several run rules were developed for t charts; see Cheng and Chen, 14 Santiago and Smith, 15 Kumar et al, 16 and Rizzo et al 17 Alternatively, synthetic charts aiming to solve the ineffectiveness related to small shifts were discussed in Scariano and Calzada, 18 Fang et al, 19 Fang et al, 20 and Sun et al 21 Vardeman and Ray 22 first proposed a CUSUM chart based on the exponential distribution and Lucas 23 proposed a CUSUM chart based on the Poisson distribution.…”

Optimal Design of 𝒕𝒓 -chart under Two Perspectives with Estimated Parameter

“…A recent and extensive literature review on the development of estimation effect on the performance of the control charts has been given by Jensen et al (2006) and Psarakis et al (2014). For recent researches on TBE charts with estimated parameter, the readers are referred to Alevizakos et al (2019), Ali (2020), Hu et al (2021), Kumar & Baranwal (2020). Institute of Science, BHU Varanasi, India Traditionally, when the parameter is unknown (case U), the performance of control chart is evaluated in terms of unconditional run length (URL) distribution and its associated characteristics (specially its mean and standard deviation).…”

“…Recently, Kumar et al (2020) has proposed three designs of exponential chart, named as optimal design OD j (𝑗 = 1,2,3), in terms of expected 𝐶𝐴𝑅𝐿 𝑖𝑛 , expected false alarm rate and standard deviation of 𝐶𝐴𝑅𝐿 𝑖𝑛 , i.e., 𝐴𝐴𝑅𝐿 𝑖𝑛 , AFAR and 𝑆𝐷 𝐶𝐴𝑅𝐿:𝑖𝑛 (for more details see Appendix C) by considering the class of sufficient estimators. They concluded that all the three optimal design exponential charts have better IC and OOC performance than the existing exponential chart (exponential chart based on maximum likelihood estimator (MLE)) and require significantly less Phase I observations than the existing exponential chart.…”

Economic-statistical design of synthetic np chart with estimated process parameter

“…Therefore, considerable attention was devoted to investigate the performance of control charts when the process parameters are estimated from an in-control Phase-I dataset. For example, see [1][2][3][4][5][6][7][8][9][10].…”