In this paper we address the issue of estimating the Shewhart control chart limits when the values of the process parameters are not known. Recently it has been shown that in order for the estimated control limits to perform similarly to the true, but unknown, limits, they should be based on data from at least 400/(n - 1) subgroups, where n denotes the subgroup size. In this paper, we propose an approach for constructing control limits that attempt to match any specific percentile point of run length distribution of the true limits, even when the limits are estimated using data from only a few subgroups. This approach would enable the user to start monitoring the process with an control chart at an earlier stage than would be possible with the standard approach. We compare the performance of the proposed approach with that of the standard approach through Monte Carlo simulation experiments. The simulation results show that the control limits constructed using our proposed method perform similarly to the true limits even when estimated from a small number of subgroups
Keywords: Average Run Length, Control Charts, Control Limits, Monte Carlo Simulation, Multivariate t Distribution, Run Length Distributions, Statistical Process Control
by Gunabushanam Nedumaran, SAS Institute, Inc., Cary, NC 27513 and Joseph J. Pignatiello, JR., Florida State University, Florida A & M University, Tallahassee, FL 32310-6046
The Shewhart control chart is widely used to monitor process means. When the in-control values for the process mean and variance are known ( and , respectively), these known parameter values are used to set up the center line and control limits of the chart. However, in most cases these parameters are unknown, and hence are estimated from some m initial subgroups of size n taken when the process is believed to be stable.
Many writers have recommended that 20 to 30 subgroups of size 4 or 5 be used for estimating the process parameters (see Montgomery (1996, p. 181) and Ryan (1989, p. 74)). Quesenberry (1993), however, shows that this recommendation would result in higher false alarm rates initially for the in-control process and longer average run lengths (ARLs) for both in-control and out-of-control processes when compared to the chart with true control limits. Quesenberry (1993) further showed that m should be at least 100 when n = 5 in order for the estimated limits to perform like true limits with respect to the in-control run length distribution and the ARL.
In most cases, however, it may not be feasible to wait for the accumulation of Quesenberry's (1993) recommended number of subgroups, since the user might wish to monitor the process at an earlier stage. In this paper, we propose a new approach for constructing prospective control limits that attempt to match any specific percentile point of the run length distribution of the chart with true limits for any m. Thus, our proposed method can be used even during the start-up stages of the process. Further, we assess the performance of the proposed approach through Monte Carlo computer simulation experiments.
This paper is organized as follows. In the next section we delineate the motivation for this research and review some pertinent literature. In the following section we describe our proposed approach for constructing control limits for use during on-line process monitoring. An illustrative example is discussed next. The results of the simulation experiments are presented in the subsequent section followed by a summary.
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