Detecting improvement using Shewhart attribute control charts when the lower control limit is zero.(Report)

1. Introduction

Perhaps the primary use of a control chart lies in the monitoring of the quality of a stable process. Using the terminology of Deming (1991) a stable process can be described as one in which special causes of variability have been eliminated. The control chart is used to signal their recurrence. Another commonly used way to improve the quality of a stable process is to reduce common-cause variability. This task, called continuous improvement, is performed using a range of techniques varying from designed experiments to unstructured observational studies. One convenient method to detect an actual improvement is to use observations below the lower control limit on certain types of control charts, such as the np chart or the c chart, as an indication of an improved process. These charts, however, do not provide a method to signal improvement when p or c is small, or more precisely when the lower control limit is less than or equal to zero.

Acosta-Mejia (1999) reviews several methods that sometimes provide lower limits in these cases. Both Ryan (1989) and Schwertman and Ryan (1997) recommend the modification of the usual Shewhart limits. These modified limits are designed to ensure that the tail probabilities are approximately equal, a property not shared by Shewhart limits when p is small because of the skewness of the binomial and poisson distributions. A third method is to use normalizing transforms as proposed by Eisenhart et al. (1947). A fourth method is to use a runs rule based upon a modified centerline similar to methods suggested by Nelson (1997). Acosta-Mejia (1999) compares the properties of some of these methods using examples with large sample sizes and/or relatively high probabilities of a defective item. Therefore, there is only a small probability of obtaining a zero count in a particular sample. One problem with the methods mentioned above is that they have a high false alarm frequency when P(0), the probability of a zero count in a particular sample, is relatively high.

Another approach to this problem was recommended by Rice (1947) and Duncan (1986). Their solution is to simply increase the sample size, which provides a lower limit. However, our proposed method will allow low-side detection without the need to increase the sample size.

Reynolds and Stoumbos (1999, 2000) also address the problem of monitoring proportions including the situation in which there is a decrease in the proportion defective. They propose the use of Bernoulli and binomial CUSUM charts to detect shifts in the process parameter.

In this paper we approach the problem of monitoring increases and decreases in the count data in a somewhat different way. Our objective is to provide Shewhart control chart users with a simple, flexible add-on to monitor decreases in the process parameter when there is no lower control limit. The method we recommend consists in counting the number of samples with zero counts of defectives or zero defects per unit and signaling if k-in-a-row samples or 2-in-t samples have zero counts. We also show how to choose the appropriate k or t value.

We note that when the Shewhart user encounters a lower control limit that is negative or zero there is insufficient information in a single low observation to signal that improvement has occurred. Our counting rules are a simple way of combining information from recent samples.

These schemes have good properties. In particular, the k-in-a-row method is a likelihood ratio CUSUM that is able to detect large shifts, as is the Shewhart chart, and thus it is optimal in terms of ARL (Average Run Length) in this situation. Later, we show that both methods have a good performance for other shifts. Another attribute of the low side add-on schemes we propose is that they are as robust as traditional CUSUMs and are easy to understand, design and implement. They are also flexible enough to be easily tailored to a meet the user's desires in terms of ARL requirements. While the attributes of our methods could make them a possible alternative to traditional CUSUMs, we feel that their primary purpose is to provide a simple method of detecting improvement for Shewhart chart users who are currently using only one-sided charts.

In the next section, we discuss the proposed methods to monitor decreases in the process parameter. We also compare the performance of our methods with a traditional CUSUM chart. Then in subsequent sections we show how our method can be applied over a wide range of parameter values, provide an example problem, and give our conclusions.

2. Design of the proposed schemes

Our schemes to detect process improvements are based upon counting the number of samples with zero counts when a fraction defective (p) or defect per unit (c) Shewhart control chart is being used. They are add-ons to the Shewhart chart that enable a user to monitor decreases in the process parameter. The two counting schemes we investigate are a k-in-a-row procedure and a 2-in-t procedure. Page (1955) studied these two runs rules for variables and showed they were CUSUM procedures. He also showed that other runs rules would have a penalty anomaly in that the CUSUM of a rejected run will be smaller than the CUSUM of an accepted run for some sequences of data. Thus, other runs rules are not considered in this paper.

In our use of these schemes we count the number of samples in which zero counts of defectives or defects per unit occur. We then signal if k-in-a-row samples have zero counts if we are using the k-in-a-row scheme. If we are using the 2-in-t scheme we signal if two samples within a set of t samples contain zero counts. In this paper we call schemes such as the k-in-a-row or 2-in-t schemes low-side schemes because in this situation they are used to detect improvements in quality. Duncan (1986) also discusses the use of a lower limit on attribute charts to detect inspection error: our schemes can also be used for this application.

Before we design a low-side scheme, we check that the Shewhart Upper Control Limit (UCL) yields an in-control high-side ARL value of at least 250. Because of the skewness of the distributions with low counts the usual three-sigma limits will yield small in-control ARL values when count levels are small; this point is not generally mentioned in popular textbooks in the quality area. Thus, in all but one of the examples in this paper our chosen UCL is one higher than the usual Shewhart three-sigma UCL. While our examples are illustrated using a Shewhart chart with a UCL chosen such that the in-control ARL value is at least 250, we emphasize that our low-side scheme can be easily used with smaller ARL values and in conjunction with other types of high-side schemes such as schemes designed around statistical, economic, or economic-statistical criteria. These have been discussed by Saniga (1989) and Saniga et al. (1995).

An example of our method is given in Table 1, which contains the ARL values for various values of the fraction defective for a situation where the in-control fraction defective is 0.025 and a Shewhart np chart with a sample of size 50 is used. The second column of Table 1 contains the ARL value for the Shewhart chart with UCL = 5. The remaining pairs of columns consider k values increasing from four to nine. The first column in each pair gives the ARL value of the k-in-a-row low-side scheme, and the second column gives the combined ARL value for a Shewhart high-side scheme coupled with the k-in-a-row low-side scheme. A three-sigma chart with n = 50 for an in control p = 0.025 yields UCL = 4 (UCL = 50(0.025) + 3(50(0.025)(0.975))[.sup.0.5] = 4.56). With UCL = 4 the in-control ARL value on the high side, hereafter ARLICH, has a value of 123. Since we propose that the ARL value be at least 250, we use a UCL = 5, which …