Response to the letter by Professors Woodall and Knoth

Abstract

Dear Professor Edwards: I am thankful to Professors Woodall and Knoth for their comments on my article “On Efficient Change Point Detection using a Step Cumulative Sum Control Chart” and wouldmake an effort to address each of their concerns. Referring to some statements from Page (1961), Duncan (1986) and Montgomery (2013), Professors Woodall and Knoth argued that smaller shifts in the process parameters are not desired to be detected. Based on the said arguments, they concluded that classical cumulative sum (CUSUM) performs better (overall) as compared to the step-CUSUM, if designed for shifts of size 2.5 standard error or larger. In my opinion, shifts of sizes more than 2.5 standard error are quite large and memory-type control charts are not usually designed for such shifts. In example 1.9.1 for monitoring the diameter of 5mm bolts, Hawkins and Olwell (1998, 28) used the design parameter k 1⁄4 0:25 for demonstration of CUSUM chart which is corresponding to optimizing the chart for shift size dTarget 1⁄4 0:5: Further, Hawkins and Olwell (1998, 32) recommended choosing a value of k that targets “a shift large enough to have a meaningful impact on the process operation but small enough not to be obvious to the naked eye.” Writing about the situations where detecting small shifts is important, they further added “If however a shift of one standard deviation is large enough to affect the process, then not being able to detect it reliably could be a problem, and then tuning to d 1⁄4 3 could be a bad choice.” In another well-known book on statistical process control, while recommending the design parameter of CUSUM chart in Table 9.3, Montgomery (2013, 422) gave results at k 1⁄4 0:5 which corresponds to optimizing the chart for dTarget 1⁄4 1: Likewise, on choosing the design parameter of CUSUM chart, Ryan (2011, 266) stated “We have seen that an X-chart is effective in detecting a largemean shift such as a 3or 4-sigma shift. Therefore, there would be no point in setting k 1⁄4 1:5 or k 1⁄4 2:0: Theusual choice is k 1⁄4 0:5, which is the appropriate choice for detecting a 1-sigma shift.” Considering these recommendations about the design parameter of CUSUM chart, it seemsunreasonable to design/compare the performance of twomemory-type control charts for larger shifts of sizesmore than 2.5 standard error. Professors Woodall and Knoth also commented on the usefulness of combined Shewhart-CUSUM chart for situations where the step-CUSUM is performing better than classical CUSUM. I should emphasize that in the original paper Abbas (2023, 12), the performance comparison (in terms of zeroand steadystate average run lengths) of step-CUSUM with the combined Shewhart-CUSUM is already given in Tables 17 and 18. The overall comparison using the extra quadratic loss (EQL) metric showed that the step-CUSUM is slightly better than the combined Shewhart-CUSUM chart. In addition, the design structure and implementation procedure of a combined Shewhart-step-CUSUM is also proposed. This combination works even better than the step-CUSUM in terms of EQL. In their second last paragraph, Professors Woodall and Knoth mentioned structural details and some drawbacks of progressive mean (PM), double progressive mean (DPM) and mixed EWMA-CUSUM charts. As can be observed in Abbas (2023, Section 4), I tend to agree with their assessment of the steady-state performance of PM and DPM charts. Although these control charts offer excellent zero-state ARL performance, the performance deteriorates for delayed shifts. Nevertheless, these structures can be useful for some shortrun processes. Regarding mixed EWMA-CUSUM chart, I believe more research needs to be done on identifying its optimal design parameters for particular values of dTarget: Only then would it be possible to compare the performances fairly and pinpoint areas of superiority. Finally, I appreciate Professors Woodall and Knoth’s input and welcome this positive debate. I want to express my gratitude to the editor of Quality Engineering for extending the invitation to write this letter.