Chapter 8 Alternatives to Shewhart Charts

1
Chapter 8Alternatives to Shewhart Charts
2
Introduction

• The Shewhart charts are the most commonly used
  control charts.
• Charts with superior properties have been
  developed.
• In many cases the processes to which SPC is now
  applied differ drastically from those which
  motivated Shewharts methods.

3
8.1 Introduction with Example

•  

4
8.2 Cumulative Sum ProceduresPrinciples and
Historical Development

•  

5
 

•  

6
Cusum Example
Sample Mean
1 1.54 -0.09 1.75 -1.58 0.41
2 0.86 0.57 1.17 1.82 1.11
3 -0.89 0.21 -1.23 1.77 -0.04
4 -1.88 -0.43 -0.42 -1.45 -1.05
5 -1.85 2.03 -0.64 0.31 -0.04
6 -2.53 -0.59 0.60 -0.22 -0.69
7 -0.74 -1.25 -0.40 -1.01 -0.85
8 2.10 1.48 0.86 -1.19 0.81
9 0.56 1.78 -0.81 0.97 0.63
10 -1.53 0.99 -2.38 1.41 -0.38
11 0.53 -0.52 1.71 0.43 0.54
12 -0.81 0.67 0.42 0.46 0.19
13 0.84 -0.71 0.27 0.93 0.33
14 0.22 1.27 0.64 -0.83 0.33
15 2.30 -0.33 0.19 -0.38 0.45
16 2.14 0.51 -1.65 -0.14 0.22
17 1.03 0.30 0.55 1.65 0.88
18 -0.90 1.71 -1.08 0.93 0.17
19 1.56 -0.70 2.06 0.88 0.95
20 1.28 0.98 1.29 0.81 1.09
N(0,1)
N(0.5,1)
7
Cusum Example
8
Runs Criteria and their Impacts

• Runs Criteria
• 2 out of 3 beyond the warning limits (2-sigma
  limits)
• 4 out of 5 beyond the 1-sigma limits
• 8 consecutive on one side
• 8 consecutive points on one side of the center
  line.
• 8 consecutive points up or down across zones.
• 14 points alternating up or down.
• Somewhat impractical
• Very short in-control ARL (91.75 with all run
  rules)

9
Cusum Procedures

•  

(8.1)
(8.3)
10
Cusum Example(Table 8.2)
i x-bar Z S(H) S(L)
1 1.54 -0.09 1.75 -1.58 0.41 0.81 0.31 0.00
2 0.86 0.57 1.17 1.82 1.11 2.21 2.02 0.00
3 -0.89 0.21 -1.23 1.77 -0.04 -0.07 1.45 0.00
4 -1.88 -0.43 -0.42 -1.45 -1.05 -2.09 0.00 -1.59
5 -1.85 2.03 -0.64 0.31 -0.04 -0.08 0.00 -1.17
6 -2.53 -0.59 0.60 -0.22 -0.69 -1.37 0.00 -2.04
7 -0.74 -1.25 -0.40 -1.01 -0.85 -1.70 0.00 -3.24
8 2.10 1.48 0.86 -1.19 0.81 1.63 1.13 -1.11
9 0.56 1.78 -0.81 0.97 0.63 1.25 1.88 0.00
10 -1.53 0.99 -2.38 1.41 -0.38 -0.76 0.62 -0.26
11 0.53 -0.52 1.71 0.43 0.54 1.08 1.20 0.00
12 -0.81 0.67 0.42 0.46 0.19 0.37 1.07 0.00
13 0.84 -0.71 0.27 0.93 0.33 0.67 1.23 0.00
14 0.22 1.27 0.64 -0.83 0.33 0.65 1.38 0.00
15 2.30 -0.33 0.19 -0.38 0.45 0.89 1.77 0.00
16 2.14 0.51 -1.65 -0.14 0.22 0.43 1.70 0.00
17 1.03 0.30 0.55 1.65 0.88 1.77 2.97 0.00
18 -0.90 1.71 -1.08 0.93 0.17 0.33 2.80 0.00
19 1.56 -0.70 2.06 0.88 0.95 1.90 4.20 0.00
20 1.28 0.98 1.29 0.81 1.09 2.18 5.88 0.00
11
Cusum Example
12
ARL for Cusum Procedure(Table 8.3)
Mean Shift h4 h5
0 168.00 465.00 370.40
0.25 74.20 139.00 281.15
0.50 26.60 38.00 155.22
0.75 13.30 17.00 81.22
1.00 8.38 10.40 43.89
1.50 4.75 5.75 14.97
2.00 3.34 4.01 6.30
2.50 2.62 3.11 3.24
3.00 2.19 2.57 2.00
4.00 1.71 2.01 1.19
5.00 1.31 1.69 1.02
13
8.2.2 Fast Initial Response Cusum

•  

14
FIR Cusum vs Cusum(Table 8.4) N(0.5,1)
i z With FIR With FIR w/o FIR w/o FIR
i z SH SL SH SL
(Reset) -  - 2.50 -2.50 0 0
21 -0.08 -0.16 1.84 -2.16 0 0
22 0.57 1.14 2.48 -0.52 0.64 0
23 0.80 1.60 3.58 0 1.74 0
24 0.23 0.46 3.54 0 1.70 0
25 0.08 0.16 3.20 0 1.36 0
26 1.33 2.66 5.36 0 3.52 0
27 1.23 2.46 5.48 0
15
FIR Cusum vs Cusum(Table 8.5) N(0,1)
i z With FIR With FIR w/o FIR w/o FIR
i z SH SL SH SL
(Reset) -  - 2.50 -2.50 0 0
21 -0.28 -0.56 1.44 -2.56 0 -0.06
22 0.07 0.14 1.08 -1.92 0 0
23 0.21 0.42 1.00 -1.00 0 0
24 0.46 0.92 1.42 0 0.42 0
25 0.55 1.10 2.02 0 1.02 0
26 0.77 1.54 3.06 0 2.06 0
27 -0.3 -0.60 1.96 -0.10 0.96 -0.10
28 0.09 0.18 1.64 0 0.64 0
29 0.69 1.38 2.52 0 1.52 0
30 0.44 0.88 2.90 0 1.90 0
31 -0.26 -0.52 1.88 -0.02 0.88 -0.02
32 -0.34 -0.68 0.70 -0.20 0 -0.20
33 -0.28 -0.56 0.00 -0.26 0 -0.26
16
Table 8.6 ARL for Various Cusum Schemes (h5,
k.5)
Mean Shift Basic Cusum Shewhart-Cusum (z3.5) FIR Cusum Shewhart-FIR Cusum (z3.5)
0 465.00 391.00 430.00 359.70
0.25 139.00 130.90 122.00 113.90
0.50 38.00 37.15 28.70 28.09
0.75 17.00 16.80 11.20 11.15
1.00 10.40 10.21 6.35 6.32
1.50 5.75 5.58 3.37 3.37
2.00 4.01 3.77 2.36 2.36
2.50 3.11 2.77 1.86 1.86
3.00 2.57 2.10 1.54 1.54
4.00 2.01 1.34 1.16 1.16
5.00 1.69 1.07 1.02 1.02
17
8.2.3 Combined Shewhart-Cusum Scheme

•  

18
8.2.4 Cusum with Estimated Parameters

• Parameter estimates based on a small amount of
  data can have a very large effect on the Cusum
  procedures.

19
8.2.5 Computation of Cusum ARLs

•  

20
8.2.6 Robustness of Cusum Procedures

•  

(8.4)
21
 
Basic Cusum Basic Cusum FIR Cusum FIR Cusum Sheahart-Cusum Sheahart-Cusum
r ARL r ARL r ARL
2 330.0 2 310.7 2 167.8
3 363.4 3 341.0 3 199.0
4 383.6 4 359.4 4 222.0
6 406.9 6 380.5 6 254.4
8 419.9 8 392.2 8 276.3
10 428.2 10 400.0 10 292.3
25 450.0 25 419.5 25 344.7
50 457.8 50 426.5 50 368.9
100 462.2 100 430.4 100 383.1
500 466.0 500 434.7 500 395.6
 
22
 
Lower Lower Upper Upper
r ARL r ARL
4 2963.5 4 440.3
6 2298.2 6 493.9
8 1995.2 8 531.2
10 1818.8 10 559.4
25 1390.7 25 664.1
50 1227.4 50 728.8
100 1127.8 100 780.4
500 1011.8 500 858.6
 
23
8.2.7 Cusum Procedures for Individual Observations

•  

24
8.3 Cusum Procedures for Controlling Process
Variability

•  

25
 

•  

(8.5)
26
8.4 Applications of Cusum Procedures

• Cusum charts can be used in the same range of
  applications as Shewhart charts can be used in a
  wide variety of manufacturing and
  non-manufacturing applications.

27
8.6 Cusum Procedures for Non-conforming Units

•  

(8.6)
(8.7)
28
8.6 Cusum Procedures for Non-conforming Units
Example
Sample i x Arcsine Transformation Arcsine Transformation Arcsine Transformation Normal Approximation Normal Approximation Normal Approximation
Sample i x z(a) SH SL z(na) SH SL
1 47 1.169 0.669 0 1.167 0.667 0
2 38 -0.286 0 0 -0.333 0 0
3 39 -0.117 0 0 -0.167 0 0
4 46 1.014 0.514 0 1.000 0.500 0
5 42 0.378 0.392 0 0.333 0.333 0
6 36 -0.629 0 -0.129 -0.667 0 -0.167
7 46 1.014 0.514 0 1.000 0.500 0
8 37 -0.456 0 0 -0.500 0 0
9 40 0.050 0 0 0 0 0
10 35 -0.804 0 -0.304 -0.833 0 -0.333
 
29
8.6 Cusum Procedures for Non-conforming Units
Example
Sample i x Arcsine Transformation Arcsine Transformation Arcsine Transformation Normal Approximation Normal Approximation Normal Approximation
Sample i x z(a) SH SL z(na) SH SL
11 34 -0.981 0 -0.784 -1.000 0 -0.833
12 31 -1.526 0 -1.811 -1.500 0 -1.833
13 33 -1.160 0 -2.471 -1.167 0 -2.500
14 29 -1.904 0 -3.874 -1.833 0 -3.833
15 33 -1.160 0 -4.534 -1.167 0 -4.500
16 39 -0.117 0 -4.151 -0.167 0 -4.167
17 29 -1.904 0 -5.555 -1.833 0 -5.500
18 39 -0.117
19 34 -0.981
 
30
8.7 Cusum Procedures for Non-conformity Data

•  

31
8.7 Cusum Procedures for Non-conformity Data
Example
Sample i c Transformation Transformation Transformation Normal Approximation Normal Approximation Normal Approximation
Sample i c z(T) SH SL z(NA) SH SL
1 9 0.573 0.073 0 0.524 0.024 0
2 15 2.284 1.857 0 2.706 2.230 0
3 11 1.191 2.548 0 1.251 2.981 0
4 8 0.239 2.287 0 0.160 2.641 0
5 17 2.776 4.564 0 3.433 5.574 0
6 11 1.191 5.255 0 1.251 6.325 0
7 5 -0.904 3.852 -0.404 -0.931 4.894 -0.431
8 11 1.191 4.543 0 1.251 5.645 0
9 13 1.758 5.801 0 1.979 7.124 0
10 7 -0.115 5.186 0 -0.204 6.420 0
11 10 0.890 5.575 0 0.887 6.807 0
12 12 1.480 6.556 0 1.615 7.922 0
 
32
8.7 Cusum Procedures for Non-conformity Data
Example
Sample i c Transformation Transformation Transformation Normal Approximation Normal Approximation Normal Approximation
Sample i c z(T) SH SL z(NA) SH SL
13 4 -1.353 4.703 -0.853 -1.295 6.128 -0.795
14 3 -1.857 2.345 -2.210 -1.658 3.969 -1.953
15 7 -0.115 1.730 -1.826 -0.204 3.265 -1.657
16 2 -2.443 0.000 -3.769 -2.022 0.743 -3.179
17 3 -1.857 0.000 -5.126 -1.658 0 -4.337
18 3 -1.857 0.000 -6.483 -1.658 0 -5.496
19 6 -0.494 0.000 -6.477 -0.567 0 -5.563
20 2 -2.443 0.000 -8.420 -2.022 0 -7.085
21 7 -0.115 0.000 -8.035 -0.204 0 -6.789
22 9 0.573 0.073 -6.962 0.524 0.024 -5.765
23 1 -3.175 0.000 -9.637 -2.386 0 -7.651
24 5 -0.904 0.000 -10.041 -0.931 0 -8.082
25 8 0.239 0.000 -9.302 0.160 0 -7.422
33
8.7 Cusum Procedures for Non-conformity Data

• The z-values differ considerably at the two
  extremes c?15 and c?2

34
8.8 Exponentially Weighted Moving Average Charts

• Exponentially Weighted Moving Average (EWMA)
  chart is similar to a Cusum procedure in
  detecting small shifts in the process mean.

35
8.8.1 EWMA Chart for Subgroup Averages

•  

(8.9)
(8.10)
36
8.8.1 EWMA Chart for Subgroup Averages

•  

(8.11)
37
8.8.1 EWMA Chart for Subgroup Averages

• Selection of L (L-sigma limits), ?, and n
• For detecting a 1-sigma shift, L 3.00, ? 0.25
• Comparison with Cusum charts
• Computation requirement About the same
• EWMA are scale dependent, SH and SL are scale
  independent
• If the EWMA has a small (large) value and there
  is an increase (decrease) in the mean, the EWMA
  can be slow in detecting the change.
• Recommendation of using EWMA charts with Shewhart
  limits

38
Table 8.12 EWMA Chart for Subgroup Averages
Example
i x-bar wt CL
1 0.41 0.1013 0.3750
2 1.11 0.3522 0.4688
3 -0.04 0.2554 0.5140
4 -1.05 -0.0697 0.5378
5 -0.04 -0.0617 0.5508
6 -0.69 -0.2175 0.5579
7 -0.85 -0.3756 0.5619
8 0.81 -0.0786 0.5641
9 0.63 0.0973 0.5653
10 -0.38 -0.0214 0.5660
11 0.54 0.1183 0.5664
12 0.19 0.1350 0.5667
i x-bar wt CL
13 0.33 0.1844 0.5668
14 0.33 0.2195 0.5669
15 0.45 0.2759 0.5669
16 0.22 0.2607 0.5669
17 0.88 0.4161 0.5669
18 0.17 0.3533 0.5669
19 0.95 0.5025 0.5669
20 1.09 0.6494 0.5669
39
8.8.2 EWMA Misconceptions

•  

40
8.8.3 EWMA Chart for Individual Observations

•  

(8.9)
(8.10)
41
8.8.4 Shewhart-EWMA Chart

• EWMA chart is good for detecting small shifts,
  but is inferior to a Shewhart chart for detecting
  large shifts.
• It is desirable to combine the two. The general
  idea is to use Shewhart limits that are larger
  than 3-sigma limits.

42
8.8.6 Designing EWMA Charts with Estimated
Parameters

• The minimum sample size that will result in
  desirable chart properties should be identified
  for each type of EWMA control chart.
• As many as 400 in-control subgroups may be needed
  if ? 0.1.