9.6 Counted Data Cusum Control Charts

Transcription

1 9.6 Counted Data Cusum Control Charts The following information is supplemental to the text. For moderate or low count events (such as nonconformities or defects), it is common to assume the distribution of the counts follows a Poisson distribution. In this case, we are assuming that the probability that a defect occurs in a very small volume of product is proportional to the volume of the product, and defects occur independently of each other. This assumption is usually reasonable for production processes when the process is in a state of statistical control. When the process goes out of control, the counts will increase and their distribution may depart from being Poisson. For example, this would occur if defects cluster spatially when the process is out of control. Properties of counted data from a production process can be monitored by a counted data cusum. Counted data cusums should be used in moderate and low count in-control situations (although it can often be practical to use in high count situations). When defects or other undesirable events are being monitored with a counted data cusum, the ideal case would be a count of zero. The average count, of course, will always be greater than zero. Thus, there is no direct analogy to the aim value or the variance in a counted data cusum. Although counted data cusums can, in practice, be designed to detect either increases or decreases in the number of defects in a sampling interval, the primary application is to detect increases. Therefore, the following discussion will be restricted to one-sided schemes to detect an increase in the counts per sampling interval. Design of a Counted Data Cusum Scheme Let Y i be the observed defect count in the i th sample. Thus, Y i is a non-negative integer. The mean acceptable (sample) count µ a and a high (sample) count µ d are set by the process expert and will be used to design the counted data cusum. The high count level is a level which the cusum is designed to detect quickly. The cusum should be designed so that the ARL will (i) be large if the current mean count per sample remains at the mean acceptable count and (ii) be small if the current mean count per sample increases to an unacceptable level. The cusum formula applied to a counted data property (e.g., defects) is S i = max(0, Y i K + S i 1 ) The parameter K is the boundary count level below which the counted data cusum is not designed to react. Note that the cusum only increases if Y i > K. The value of K should be chosen to be between the mean acceptable count and the high count level. It is recommended that the value of K be selected as the integer closest to (High count level) (Acceptable count level) ln(high count level) ln(acceptable count level) = µ d µ a lnµ d lnµ a 194

2 For a given ARL at the acceptable count level, this value of K will give the shortest ARL at the high count level. The value of K need not be an integer although ARL Tables and do not show fractional values of K. Like the continuous data cusums, we need to select a decision interval H. That is, if the cusum reaches or exceeds H, then an out-of-control signal occurs. The value of H can be selected from Table (no FIR) or Table (FIR) to give an appropriately large ARL when the process is running at or below the acceptable count level. It should also be chosen to give an appropriately small ARL value when the process is running at or greater than the high level that is to be detected quickly. The FIR feature should be used to give quicker detection in case of potential process start-up problems. For a FIR, we will use S 0 = H/2. Example In the following example suppose { µa = the acceptable count level = 4 µ d = the high count level = 7 The counted data cusum is designed with a K value close to µ d µ a lnµ d lnµ a = After rounding to the nearest integer, we use K = ln(7) - ln(4) = In the Lucas article Counted Data CUSUMs, the notation k b is the reciprocal of this ratio. For this example, k b = 1/ This would be the value of k to enter in SAS. To use ARL Table or 28.02, the acceptable count level µ a and the high count level µ d must first be expressed in normalized units by dividing by K. Thus, the normalize values are µ a = µ a K = 4 5 =.8 and µ d = µ d K = 7 5 = 1.4 These values correspond to the values in the Mean, as a multiple of K columns. Next, go to the K = 5 rows and select H that has a large ARL in the µ a = 8 column and a small value in the µ d = 1.4 column. For example: in Table 28.01, when K = 5 and H = 10, the ARLs are 422 and 5.59 for normalized values of µ a = 0.8 and µ d =

3 196

4 197

5 198

6 199

7 200

8 Upper One-Sided Cusum for Count Data The CUSUM Procedure Cumulative Sum Chart Summary for accident Individual Decision Individual Decision date Value Cusum Interval date Value Cusum Interval

9 Lower One-Sided Cusum for Count Data The CUSUM Procedure Cumulative Sum Chart Summary for accident Individual Decision Individual Decision date Value Cusum Interval date Value Cusum Interval Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower Lower 202

10 DM LOG; CLEAR; OUT; CLEAR; ; * ODS PRINTER PDF file= C:\COURSES\ST528\SAS\cusum_ct.pdf ; ODS LISTING; OPTIONS NODATE NONUMBER PS=500 LS=76; DATA lucas; DO date = 1 to 120; INPUT OUTPUT; END; LINES; ; SYMBOL1 V=dot WIDTH=2; PROC CUSUM DATA=lucas; XCHART accident*date= 1 / MU0=2 SIGMA0=1.412 NPANELPOS=121 h=5.6 k=0.41 DATAUNITS DELTA=1 HAXIS=0 to 120 by 1 NOMASK; INSET ARL0 ARLDELTA H K SHIFT / POS = sw; LABEL accident = Accidents Cusum date = Month (beginning January 1970) ; TITLE CUSUM for Number of Accidents per Month ; TITLE2 (Data taken from Lucas 1985) ; PROC CUSUM DATA=lucas; XCHART accident*date= 1 / MU0=2 SIGMA0=1.412 NPANELPOS=121 h=5.6 k=0.41 DATAUNITS DELTA=1 HAXIS=0 to 120 by 1 SCHEME=onesided TABLESUMMARY TABLEOUT; INSET ARL0 ARLDELTA H K SHIFT / POS = ne; LABEL accident = Accidents Cusum date = Month (beginning January 1970) ; TITLE Upper One-Sided Cusum for Count Data ; PROC CUSUM DATA=lucas; XCHART accident*date= 1 / MU0=2 SIGMA0=1.412 NPANELPOS=121 h=5.6 k=0.41 DATAUNITS DELTA=-1 HAXIS=0 to 120 by 1 SCHEME=onesided TABLESUMMARY TABLEOUT; LABEL accident = Accidents Cusum date = Month (beginning January 1970) ; TITLE Lower One-Sided Cusum for Count Data ; RUN; 203

11 204

12 205

13 206

14 207

15 208

16 209

17 210

18 211

19 212

20 213

21 214

22 215

23 216