Interactive Approach for Generating Shewhart Control Limits

Transcription

1 Interactive Approach for Generating Shewhart Control Limits Authors: Susana Vegas Valeria Quevedo Geoff Vining

2 Introduction

3 Introduction In Phase I, control charts are more of an exploratory data analytical tool whose purpose is to generate the control limits for Phase II. Phase II is the true control procedure, which may be viewed as a series of hypothesis tests. The null hypothesis is that the process is in-control, and the alternative is that it is out-of-control. The typical presentation of Phase I is a preset number of rational subgroups. Such an approach fails to address the tension that exists between having enough information to generate reliable control limits and being able to start active control of the process.

4 Phase I and Phase II Control Charts Phase I control chart cannot provide active (real time) control of the process because we don t know the control limits. Phase I control charts truly are an exploratory data analytic tool. Phase II control charts provide a basis for real time, active control of the process. Assumes that the number of rational subgroups used in the Phase I study is sufficiently large to assume that the resulting estimate of the variance is known. Jensen et al. (2006) For readers interested in more details about Phase I control charts.

5 "Theoretical Basis" for the Adaptive Control Limit Process Phase II control limits assume that the target value and the variance for the process being monitored are known. If the process is in-control, then under the Central Limit Theorem y i μ σ n follows a standard normal distribution. UCL = μ σ LCL = μ 0 3 σ n n The problem is that σ is never truly known, which is the fundamental problem with the transition from Phase I control charts to Phase II.

6 "Theoretical Basis" for the Adaptive Control Limit Process Consider now a Phase I control chart where μ 0 is known but σ 2 is not. s 2 p the pooled sample variance(m rational subgroups, size n). Suppose we can assume that at least the process variance was in control over the entire base period. The resulting control statistic is t = y i μ 0 s p n which follows a t-statistic with m(n 1) degrees of freedom if the process is in-control. UCL = y i + t m n 1, LCL = y i t m n 1, s p s p n n

7 "Theoretical Basis" for the Adaptive Control Limit Process Vining (1998) outlines a procedure for producing adaptive control limits. Jensen et al. As more data become available and the process is still determined to be stable, the control limits should be updated.

8 BASIC ISSUES Have enough rational subgroups available to estimate the control chart parameters well. Ensure that these limits are calculated from the in-control process. The historic approach uses a single base period of size m, choosing m large enough. However, the larger m is, the more likely that the process contains rational subgroups that in fact are out-of-control, thus distorting the estimated control limits. The idea for an adaptive procedure that updates the estimated control chart parameters using only rational subgroups thought to be in-control.

9 The Basic Adaptive Approach Step1: Take an initial base period of m 0 = 20 to 30 rational subgroups. Estimate the control limits. Investigate any rational subgroups that appear to be out-of-control. Drop those rational subgroups for which there is an assignable cause. Recalculate the estimated control chart limits. Continue until no more rational subgroups can be dropped. Step 2: Use the resulting estimated control limits from Step 1 actively to control the process for the next m 1 rational subgroups, where m 1 is Investigate any rational subgroup that appears to be out-of-control. Use only those rational subgroups thought to be in-control from both Steps 1 and 2 to recalculate the estimated control limits.

10 The Basic Adaptive Approach Step 3: Repeat Step 2. Each time, use all the good rational subgroups to recalculate the estimated control limits. Continue to repeat Step 2 until there is at least 100 in-control rational subgroups upon which to base the estimates for the control limits. This procedure seeks to balance using the control chart for active control of the process as soon as possible with ensuring that there are enough rational subgroups to estimate the control limits well. The process described before will be called Iterative generation of control limits and is presented in the following flow chart:

11 Iterative generation of control limits

12 Iterative generation of control limits

13 Monitoring yarn production Parameter to monitor: Cotton tenacity in cn/tex, (breaking strength point of the fibers) Factors that explain variations in the yarn quality: fiber uniformity, failures or defects in the machinery, and the humidity and temperature of the facility. Sample Every week, the factory takes samples -of size ten- from the same production line in the overnight shift.

14 Monitoring yarn production

15 Monitoring yarn production

16 Monitoring yarn production

17 Monitoring yarn production

18 Monitoring yarn production

19 Monitoring yarn production

20 Conclusions The classical approach of calculating the limits with 20 subgroups give as result relaxed control limits that are not really useful to detect variability other than the one inherent to the process. On the other hand, the iterative approach give us more adjusted limits that will allow us to detect the presence of assignable causes that disturbs the process. Also, leaves out subgroups that really don t belong to it. The classical approach doesn t allow us to obtain a good estimation of the process variability. It can be seen, for example in the study case, that using only a moderate number of rational subgroups to calculate the control limits, could mislead us, and give as an out of control process when it is really in control, as we evidence when more subgroups were added. With the traditional calculation of the control limits, we could take a premature decision and discard subgroups as not being part of a control process when they really belong to it, loosing valuable information as a consequence. For a novice practitioner, the iterative approach gives a better overview of the process and helps to understand its behavior over the time.

21 References Jensen, W.A., Jones-Farmer, L.A., Champ, C.W., and Woodall, W.H. (2006) Effects of Parameter Estimation on Control Chart Properties: A Literature Review, Journal of Quality Technology, 38, pp Montgomery, D.C. (2015) Introduction to Statistical Quality Control, 7 th ed. Hoboken, N.J.: John Wiley and Sons. Vining, G.G. (1998). Statistical Methods for Engineers. Belmont, Ca.: Duxbury Press. Vining, G. (2009). Technical Advice: Phase I and Phase II Control Charts, Quality Engineering, 21, pp

22 Gracias! Authors: Susana Vegas Valeria Quevedo Geoff Vining