Guaranteed in‐control performance of the EWMA chart for monitoring the mean

Abstract

Research on the performance evaluation and the design of the Phase II EWMA control chart for monitoring the mean, when parameters are estimated, have mainly focused on the marginal in‐control average run‐length (ARLIN). Recent research has highlighted the high variability in the in‐control performance of these charts. This has led to the recommendation of studying of the conditional in‐control average run‐length (CARLIN) distribution. We study the performance and the design of the Phase II EWMA chart for the mean, using the CARLIN distribution and the exceedance probability criterion (EPC). The CARLIN distribution is approximated by the Markov Chain method and Monte Carlo simulations. Our results show that in‐order to design charts that guarantee a specified EPC, more Phase I data are needed than previously recommended in the literature. A method for adjusting the Phase II EWMA control chart limits, to achieve a specified EPC, for the available amount of data at hand, is presented. This method does not involve bootstrapping and produces results that are about the same as some existing results. Tables and graphs of the adjusted constants are provided. An in‐control and out‐of‐control performance evaluation of the adjusted limits EWMA chart is presented. Results show that, for moderate to large shifts, the performance of the adjusted limits EWMA chart is quite satisfactory. For small shifts, an in‐control and out‐of‐control performance tradeoff can be made to improve performance.

A head-to-head comparison of the out-of-control performance of control charts adjusted for parameter estimation

“…For the Shewhart control chart, all properties of the conditional run length distribution can be determined from a geometric distribution with parameter CAR according to [6]. For the CUSUM control chart we have calculated the CARL values using a modified Siegmund formula approximation as described in Diko, Chakraborti, and Does (2019b), and for the EWMA we use a Markov-Chain approximation such as in Diko, Chakraborti, and Does (2019a). We have also evaluated other properties of the conditional run length distributions such as the standard deviation and several percentiles obtained by simulations.…”

“…For each of the discussed control charts, CARL IN depends on the Phase I parameters estimates as well as the chosen control limit constants C, h, and L. For a given Phase I sample (size), these constants can be adjusted to match the EPC. We consider adjustments of C as provided in Goedhart, Schoonhoven, and Does (2018), for h as provided in Diko, Chakraborti, and Does (2019b), and for L according to Diko, Chakraborti, and Does (2019a), respectively. In Table 1 we provide an overview of the required values for the Shewhart control chart, and for the EWMA and CUSUM control charts designed to detect shifts of d equal to 0.5 (small), 1 (medium) and 1.5 (large).…”

Conditional median run length performance of the synthetic X¯ chart with unknown process parameters

“…For example, in Table 8, when m = 100, the AMRL and SDMRL values of the conditional in-control MRL values of the adjusted synthetic and the adjusted ShewhartX charts are 396. 45,192.09 and 338.70, 132.00, respectively. When the process is out of control, the AMRL and SDMRL values of the conditional MRL values of the adjusted syntheticX chart is generally smaller than the ones of the adjusted ShewhartX chart.…”

“…of the gamma distribution with parameters and (see Zhang et al and Khoo et al). Moreover, because of the relation between the gamma distribution and the chi‐square distribution, is also a scaled chi density function with degrees of freedom (see Capizzi and Masarotto, Saleh et al, and Diko et al). In this paper, we use the gamma distribution for discussion.…”

On the design of control charts with guaranteed conditional performance under estimated parameters

“…In Diko et al 42 , a different approach was used to guarantee a minimum performance of the conditional in‐control ARL. Their method to adjust the control chart limits according to the exceedance probability criterion is a better alternative to parametric bootstrapping.…”