DATA ANALYSIS AND SOFTWARE

Transcription

1 DATA ANALYSIS AND SOFTWARE 3 cr, pass/fail Session (Keijo Ruohonen): QUALITY ASSURANCE WITH MATLAB 1

2 QUALITY ASSURANCE WHAT IS IT? Quality Design (actually part of the larger area Design and Analysis of Experiments ) Quality Control or Statistical Process Control (SPC) Quality Management Acceptance Sampling 2

3 QUALITY CONTROL Goals Using regularly taken small samples to check that a chosen parameter of the process is within given limits (the in-control situation). Allow for noise that from time to time will move sample statistics outside the control limits (cf. testing hypotheses). Alert when there is evidence that the process is out of control. 3

4 Methods Shewhart s control charts Control charts for slow change (CUSUM, EWMA, etc.) Multivariate control charts Capability indices 4

5 CONTROL CHARTS A chart is a graph of values of the used sample statistic against sample number, with a center line (CL), and lower and/or upper control limits (LCL and/or UCL), and possibly other limits, too. Using the limits in one of several possible ways criteria for the out-of-control situation are prescribed, and checked for each new sample. Type I and Type II error probabilities (α and β) are computed. 5

6 Shewhart s X-bar Chart CL is the process mean µ and LCL and UCL are µ ± k σ n where σ is the process standard deviation and n is the (fixed) sample size. The sample statistic used is the sample mean X whose expectation is µ and standard deviation σ/ n. k and n are available to set α and β to prescribed values. 6

7 It will be assumed that the process (population) distribution is N(µ, σ 2 ) even though this may not be the case exactly. Now α = 2 ( 1 Φ(k) ) is the probability of false positive or false alarm, i.e., alerting even when the process is in control. A typical choice is k = 3 whence α = Choice of the sample size n has an effect on β (the probability of false negative or not alerting on an out of control process). When computing β a typical shift of process mean from µ (in control) to µ + σ (out of control) is assumed. Then β = Φ(k n) Φ( k n). 7

8 X-bar chart can be designed by first choosing k = Φ 1 (1 α/2) and then checking through small values of n to force β to be small enough. Typically n is 4 or 5. The formulas are easily extended to the case where only LCL or UCL is in use. A typical X-bar chart (in control) looks like the following (MAT- LAB, k = 3, n = 5): 8

9 7 6 XBAR control chart Data Violation Center LCL/UCL 5 XBAR And one out of control (at sample no. 43, same k and n): 9

10 7 6 XBAR control chart Data Violation Center LCL/UCL 5 XBAR Here the red circle indicates violation of control limits. 10

11 Shewhart s S-chart S-chart controls process standard deviation σ. statistic is the sample standard deviation S = 1 n (X i X) 2. n 1 i=1 The sample Assuming the process distribution is N(µ, σ 2 ), the distribution of (n 1)S 2 /σ 2 is χ 2 with n 1 degrees of freedom, which can be used to compute the expectation and variance of S in the form 11

12 E(S) = c 4 σ and var(s) = c 2 5 σ2 where c 5 = 1 c 2 4 and c 4 = Γ(n 2 ) 2 Γ( n 1 2 ) n 1. Thus the CL is c 4 σ, and the lower and upper control limits are (c 4 ± kc 5 )σ, usually denoted by B 5 σ and B 6 σ. Using the χ 2 -distribution α and β can be computed for the S-chart, and fixed taking proper values of k and n. 12

13 Now α and β both depend on both k and n (but α only very weakly on n). So fixing them is more difficult than for the X-bar chart. Letting F be the cdf of the χ 2 -distribution with n 1 degrees of freedom we have α = 1 F ( (n 1)B6 2 ) ( + F (n 1)B 2) 5 and ( B 2 ) β = F (n 1) 6 (1 + ) 2 ( B 2 ) F (n 1) 5 (1 + ) 2. Here getting out of control means process standard deviation moving from σ to (1 + )σ. 13

14 A usual choice, however, is to take k = 3, and since S-chart is often used in connection with an X-bar chart using the same samples, sample size n is already fixed for the X-bar chart. (Recall that for a normal process distribution X and S are independent.) The purpose of using an S-chart is of course to control process variability. But often the X-bar chart is the important one, and the S-chart is used mainly to keep an eye on the assumptions for the X-bar chart (i.e., the constancy of σ). Often only the upper control limit is used. 14

15 Shewhart s R-chart The sample statistic used in the R-chart is the sample range R = max(x 1,..., X n ) min(x 1,..., X n ). The purpose of the chart is exactly the same as for the S-chart: to control process variability. The S-chart is nowadays probably the more popular of the two. Even for a normally distributed process the distribution of R is quite difficult, as is the computation of α and β. 15

16 Anyway, as for the S-chart, the expectation and standard deviation of R can then be given in the form E(R) = d 2 σ and var(r) = d 2 3 σ2, where d 2 and d 3 depend only on n. Thus the CL is d 2 σ, the lower and upper control limits are (d 2 ± kd 3 )σ, denoted by D 1 σ and D 2 σ. Again a usual choice is k = 3. n d 2 d

17 Here are S- and R-charts of the same sample input data: S control chart Data Violation Center LCL/UCL R control chart Data Violation Center LCL/UCL 3 8 S R Note how the lower control limits are set to zero when B 5 and D 1 are negative, as is of course in order. 17

18 Shewhart s P-chart The control statistic is the number D of defective products in a size n batch (sample), taken from time to time. (Often the statistic is P = D/n, whence the name.) The in-control distribution of D is the binomial distribution with (known) parameters p (proportion of defectives) and n (sample size). Then E(D) = np and var(d) = np(1 p), CL is at np and the control limits are np ± k np(1 p). Usually k = 3, and n is larger than, say, for the X-bar chart. 18

19 If getting out of control is interpreted as p moving to p = (1 + )p then the error probabilities are α = 1 F (UCL) + F (LCL) and β = F (UCL) F (LCL), where F is the cdf of the binomial distribution with parameters p and n, and F with parameters p and n. 19

20 Shewhart s C-chart For the C-chart a single specimen (unit) is investigated and the number C of defects (nonconformities) in the specimen is observed. The in-control distribution of C is Poisson s distribution with (known) parameter λ. Thus E(C) = var(c) = λ. CL is at λ and the control limits are λ ± k λ where again often k = 3. 20

21 Getting out of control is now interpreted as λ moving to λ = (1 + )λ, and the error probabilities are α = 1 F (UCL) + F (LCL) and β = F (UCL) F (LCL), where F is the cdf of Poisson s distribution with parameter λ, and F with parameter λ. There is a variant of the C-chart, the U-chart, where a sample of n units are inspected, instead of just one. This behaves as a C-chart for the parameter nλ. 21

22 Starting Shewhart s Charts Charts can be started immediately when estimates for their parameters are available. One (and recommended) way is to first make sure the process is in control, take a sufficiently large sample of values of the variable to be controlled, and then estimate the parameters using the well-known methods. 22

23 Another popular way is to again make sure the process is in control, take a number of small samples (at least some samples with sample size n = 4 or n = 5, say), compute the usual control statistics used in the chart (sample means for the X-bar chart, sample standard deviations for the S-chart, etc.), and then take their mean as the CL. Note that to get the control limits using the latter method the X-bar chart and the S-chart must be started together. In MATLAB both methods are available. The latter method has the advantage that after the estimation, it may be checked by applying the obtained control limits to the samples.

24 Multiple Control Limits in Shewhart s Charts The error probabilities α and β can be set to prescribed values choosing k and n properly (when possible). Since however often n is fixed and traditionally k will be 3, other methods for changing the behavior of the chart are needed. One possibility is to define several control limits, e.g. taking k = 1, 2, 3, and specify rules for the out-of-control situation. There are two well-known sets of such rules, the Western Electric rules and the Nelson rules. Both sets are supported by MATLAB. First the ten WE-rules: 23

25 we point above cl + 3*se we of 3 above cl + 2*se we of 5 above cl + se we of 8 above cl we below cl - 3*se we of 3 below cl - 2*se we of 5 below cl - se we of 8 below cl we of 15 between cl - se and cl + se α = 1% we of 8 below cl - se or above cl + se 24

26 And then the eight Nelson rules: n point below cl - 3*se or above cl + 3*se n of 9 on the same side of cl n of 6 increasing or decreasing n alternating up/down n of 3 below cl - 2*se or above cl + 2*se, same side n of 5 below cl - se or above cl + se, same side n of 15 between cl - se and cl + se n of 8 below cl - se or above cl + se, either side 25

27 EWMA-chart The EWMA-chart (exponentially weighted moving average chart) is an example of charts controlling slow shifts to out of control. Another famous chart of this type is the CUSUMchart (cumulative sum chart). Suppose an alert of slow shift of the sample statistic Y is of interest. To set an EWMA-chart we need to know E(Y ) = m and var(y ) = v 2. The i th control statistic Z i is then obtained from the i th sample statistic Y i recursively as Z i = λy i + (1 λ)z i 1 and Z 0 = m. 26

28 Here λ is a parameter of the method. Note that for λ = 1 the usual Y-chart is obtained, and λ = 0 results in a constant chart m (useless, of course). Therefore 0 < λ 1. Summing we get Z i = λ the sum of weights is i 1 (1 λ) j Y i j + (1 λ) i m, and j=0 λ i 1 (1 λ) j + (1 λ) i = 1. j=0 It follows that E(Z i ) = m and var(z i ) = λ( 1 (1 λ) 2i) v 2. 2 λ 27

29 CL is then m and control limits LCL i and UCL i depend on i: λ ( 1 (1 λ) 2i) m ± kv 2 λ Therefore the chart is self-starting. Note that for large values of i the limits are approximately m ± kv λ/(2 λ). Usually k is 3 or somewhere near 3. The smaller the value of λ the better the chart alerts for slow shifts of Y. Here are examples of an EWMA chart with varying λ: 28

30 11 10 EWMA control chart Data Violation Center LCL/UCL EWMA control chart Data Violation Center LCL/UCL EWMA 7 6 EWMA λ = λ = 0.4 In these EWMA-charts m and v are estimated from the data and k = 3. (The sample statistic Y here is X with n = 5.) 29

31 EWMA control chart Data Violation Center LCL/UCL EWMA control chart Data Violation Center LCL/UCL EWMA EWMA λ = λ = 0.8 Note how the chart is smoother for smaller values of λ making it possible to discern a trend among the noise. 30

32 CAPABILITY INDICES As all indices, capability indices attempt to characterize a property using a single number, in this case the capability of the process to behave according to given specifications. For this purpose first lower and/or upper specification limits (LSL and/or USL) are fixed. There are many capability indices. basic indices (Kane s indices). MATLAB supports the 31

33 Kane s Capability Indices If process expectation µ and standard deviation σ are known then Kane s indices are C P = USL LSL, 6σ C PL = µ LSL, 3σ C PU = USL µ 3σ and C PK = min(c PL, C PU ). 32

34 Another form for C PK is C PK = C P 1 µ 3σ which shows its relation to C P. USL + LSL 2 If (as is usually the case) µ and σ are not known, they must be replaced by the estimates ˆµ = X and ˆσ = S b = 1 n n (X i X) 2. i=1 (Note the traditionally used biased variance estimator Sb 2, the unbiased S 2 could be used as well.) 33

35 In this way the estimated capability indices Ĉ P, Ĉ PL, Ĉ PU and Ĉ PK are obtained. These are then used to characterize the behavior of the process. The percentage of products within the specification limits is called the process yield. If only lower or upper specification limit is in use and the process distribution is normal, then the yield is simply ( ( LSL µ )) Y L = Φ % or σ ( USL µ ) Y U = 100Φ %. σ 34

36 These are directly connected to the indices C PL and C PU (or their estimates): Y L = 100Φ(3C PL )% and Y U = 100Φ(3C PU )%. If both specification limits are used then the yield is ( ( USL µ ) ( LSL µ )) Y = 100 Φ Φ % σ σ but this is not connected directly to C PK. There is only the inequality 100 ( 2Φ(3C PK ) 1 ) % Y < 100Φ(3C PK )%. where the lower bound is reached when µ = (USL + LSL)/2. 35

37 For all capability indices larger values are better than smaller. For C P, C PL and C PU the following acceptable minimum values are often cited: Status of process C P C PL or C PU In use New In use and critical New and critical

38 LITERATURE 1. BESTERFIELD, D.H.: Quality Control. Pearson (2008) 2. MONTGOMERY, D.C.: Statistical Quality Control. A Modern Introduction. Wiley (2008) 3. RUOHONEN, K.: Laadunvalvonta ja tarkastusotanta (Finnish lecture notes, 2003) (Available online) 4. WALPOLE, R.E. & MYERS, R.H. & MYERS, S.L. & YE, K.: Probability & Statistics for Engineers & Scientists. Prentice Hall (2007) 37