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Monitoring of time between events (TBE) instead of the number of events is used in high‐quality processes where the events occur rarely. This article presents a double generally weighted moving average control chart with a lower time‐varying control limit to monitor the TBE (regarded as DGWMA‐TBE chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the DGWMA‐TBE chart is more effective than the DEWMA and GWMA charts in detecting moderate to large shifts. Furthermore, the DGWMA‐TBE chart is very robust for the same range of shifts when the TBE observations follow a Weibull or a lognormal distribution. Finally, examples are also presented to enhance the performance of the proposed chart.

Control charts are widely applied in many manufacturing processes to monitor the quality characteristic of interest. Recently, a homogeneously weighted moving average (HWMA) control chart was proposed as an improvement of the exponentially weighted moving average (EWMA) chart for efficiently monitoring of small shifts in the process mean. In the present article, we extend the HWMA chart by imitating exactly the double EWMA (DEWMA) technique. The proposed scheme is regarded as double HWMA (DHWMA) control chart. The run‐length characteristics of the proposed chart are evaluated by performing Monte Carlo simulations. A comparison study versus the EWMA, DEWMA, HWMA, mixed EWMA cumulative sum (CUSUM), CUSUM, and GWMA charts indicates that the DHWMA chart is more effective in detecting small to moderate shifts, while it performs similarly with its competitors for large shifts. We also study the robustness of the proposed chart under several nonnormal distributions, and it is shown that the DHWMA chart is in‐control robust for small values of the smoothing parameters. Finally, two examples are given to demonstrate the implementation of the proposed chart.

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Control charts based on time between events (TBE) are widely used in highquality manufacturing processes where the events occur rarely. For monitoring such data, a new two-sided TBE control chart called progressive mean control chart (regarded as PM-TBE chart) is proposed in the present study. Through a simulation study and using the average run-length (ARL) measure, it is shown that the proposed chart outperforms the t r , ARL-unbiased gamma, GWMA-TBE, and DEWMA-TBE charts in detecting very small to moderate downward shifts, as well as in some cases small upward shifts. Moreover, the PM-TBE chart is very robust for moderate to large shifts when the true distribution of the TBE observations is a Weibull or a lognormal. Finally, examples are given to display the application of the proposed chart. KEYWORDSaverage run length, Erlang distribution, progressive mean, robustness, time between events | INTRODUCTIONControl charts based on time between events (TBE) are frequently used in high-quality processes where the rate of occurrences is low. The traditional Shewhart control charts for attributes, such as c chart, u chart, p chart, and np chart, are used to monitor the number or the proportion of nonconformities or defective products, 1 whereas TBE control charts are used to monitor the inter-arrival times between the occurrences of events, which are assumed to be exponentially distributed. 2 Many researchers have focused on the study of TBE control charts. Lucas 3 studied the cumulative sum (CUSUM) chart for exponentially distributed observations, while Vardeman and Ray 4 used integral equations to calculate the average run-length (ARL) of the exponential CUSUM chart. Gan 5 investigated the optimal design of the exponential CUSUM chart, and Gan and Choi 6 presented a program for calculating the ARL of the above control chart. Gan 2 proposed the exponentially weighted moving average (EWMA) control chart for exponential data computing its ARL using differential equations, and he found it slightly less sensitive than the exponential CUSUM chart. Chan et al 7 proposed a Shewhart-type chart, called cumulative quantity control chart (CQC chart) for monitoring exponential TBE data. Jones and Champ 8 studied the phase I of Shewhart-type TBE control charts. Xie et al 9 proposed a Shewhart-type control chart, namely, t r chart, based on the Erlang distribution, to monitor the time between r (r ≥ 1) failures. Borror et al 10 studied the robustness of the exponential CUSUM chart assuming that the TBE observations follow a Weibull or a lognormal distribution, and they showed that it is very robust for both small and large shifts of parameters. Liu et al 11 compared various one-and two-sided exponential TBE control charts. Liu et al 12,13 proposed CUSUM and EWMA schemes, respectively, for monitoring exponential data after transforming it to an approximate normal distribution using the double square root transformation. Zhang et al 14 proposed a TBE control chart based on gamma distribution using the Let Y t ∼ iid ExpðθÞ, t=1,2,… r...