chapter 8 alternatives to shewhart charts
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Chapter 8 Alternatives to Shewhart Charts. Introduction. The Shewhart charts are the most commonly used control charts. Charts with superior properties have been developed. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 8
Alternatives to Shewhart Charts
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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 Shewhart’s methods.”
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8.1 Introduction with Example
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8.2 Cumulative Sum Procedures:Principles and Historical Development
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Cusum ExampleSample Mean
1 1.54 -0.09 1.75 -1.58 0.412 0.86 0.57 1.17 1.82 1.113 -0.89 0.21 -1.23 1.77 -0.044 -1.88 -0.43 -0.42 -1.45 -1.055 -1.85 2.03 -0.64 0.31 -0.046 -2.53 -0.59 0.60 -0.22 -0.697 -0.74 -1.25 -0.40 -1.01 -0.858 2.10 1.48 0.86 -1.19 0.819 0.56 1.78 -0.81 0.97 0.63
10 -1.53 0.99 -2.38 1.41 -0.3811 0.53 -0.52 1.71 0.43 0.5412 -0.81 0.67 0.42 0.46 0.1913 0.84 -0.71 0.27 0.93 0.3314 0.22 1.27 0.64 -0.83 0.3315 2.30 -0.33 0.19 -0.38 0.4516 2.14 0.51 -1.65 -0.14 0.2217 1.03 0.30 0.55 1.65 0.8818 -0.90 1.71 -1.08 0.93 0.1719 1.56 -0.70 2.06 0.88 0.9520 1.28 0.98 1.29 0.81 1.09
N(0,1)
N(0.5,1)
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Cusum Example
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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)
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Cusum Procedures
(8.1)
(8.3)
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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 0.002 0.86 0.57 1.17 1.82 1.11 2.21 2.02 0.003 -0.89 0.21 -1.23 1.77 -0.04 -0.07 1.45 0.004 -1.88 -0.43 -0.42 -1.45 -1.05 -2.09 0.00 -1.595 -1.85 2.03 -0.64 0.31 -0.04 -0.08 0.00 -1.176 -2.53 -0.59 0.60 -0.22 -0.69 -1.37 0.00 -2.047 -0.74 -1.25 -0.40 -1.01 -0.85 -1.70 0.00 -3.248 2.10 1.48 0.86 -1.19 0.81 1.63 1.13 -1.119 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 -0.2611 0.53 -0.52 1.71 0.43 0.54 1.08 1.20 0.0012 -0.81 0.67 0.42 0.46 0.19 0.37 1.07 0.0013 0.84 -0.71 0.27 0.93 0.33 0.67 1.23 0.0014 0.22 1.27 0.64 -0.83 0.33 0.65 1.38 0.0015 2.30 -0.33 0.19 -0.38 0.45 0.89 1.77 0.0016 2.14 0.51 -1.65 -0.14 0.22 0.43 1.70 0.0017 1.03 0.30 0.55 1.65 0.88 1.77 2.97 0.0018 -0.90 1.71 -1.08 0.93 0.17 0.33 2.80 0.0019 1.56 -0.70 2.06 0.88 0.95 1.90 4.20 0.0020 1.28 0.98 1.29 0.81 1.09 2.18 5.88 0.00
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Cusum Example
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ARL for Cusum Procedure(Table 8.3)
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8.2.2 Fast Initial Response Cusum
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FIR Cusum vs Cusum(Table 8.4) N(0.5,1)
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FIR Cusum vs Cusum(Table 8.5) N(0,1)
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Table 8.6 ARL for Various Cusum Schemes (h=5, k=.5)
Mean ShiftBasic
Cusum
Shewhart-Cusum (z=3.5)
FIR CusumShewhart-FIR Cusum
(z=3.5)
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
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8.2.3 Combined Shewhart-Cusum Scheme
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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.
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8.2.5 Computation of Cusum ARLs
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8.2.6 Robustness of Cusum Procedures
(8.4)
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Basic Cusum FIR Cusum Sheahart-Cusum
r ARL r ARL r ARL2 330.0 2 310.7 2 167.83 363.4 3 341.0 3 199.04 383.6 4 359.4 4 222.06 406.9 6 380.5 6 254.48 419.9 8 392.2 8 276.3
10 428.2 10 400.0 10 292.325 450.0 25 419.5 25 344.750 457.8 50 426.5 50 368.9
100 462.2 100 430.4 100 383.1500 466.0 500 434.7 500 395.6
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Lower Upper
r ARL r ARL4 2963.5 4 440.36 2298.2 6 493.98 1995.2 8 531.2
10 1818.8 10 559.425 1390.7 25 664.150 1227.4 50 728.8
100 1127.8 100 780.4500 1011.8 500 858.6
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8.2.7 Cusum Procedures for Individual Observations
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8.3 Cusum Procedures for Controlling Process Variability
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(8.5)
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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.
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8.6 Cusum Procedures for Non-conforming Units
(8.6)
(8.7)
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8.6 Cusum Procedures for Non-conforming Units: Example
Samplei x
Arcsine Transformation Normal Approximation
z(a) SH SL z(na) SH SL
1 47 1.169 0.669 0 1.167 0.667 02 38 -0.286 0 0 -0.333 0 03 39 -0.117 0 0 -0.167 0 04 46 1.014 0.514 0 1.000 0.500 05 42 0.378 0.392 0 0.333 0.333 06 36 -0.629 0 -0.129 -0.667 0 -0.1677 46 1.014 0.514 0 1.000 0.500 08 37 -0.456 0 0 -0.500 0 09 40 0.050 0 0 0 0 0
10 35 -0.804 0 -0.304 -0.833 0 -0.333
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8.6 Cusum Procedures for Non-conforming Units: Example
Samplei x
Arcsine Transformation Normal Approximation
z(a) SH SL z(na) SH SL
11 34 -0.981 0 -0.784 -1.000 0 -0.83312 31 -1.526 0 -1.811 -1.500 0 -1.83313 33 -1.160 0 -2.471 -1.167 0 -2.50014 29 -1.904 0 -3.874 -1.833 0 -3.83315 33 -1.160 0 -4.534 -1.167 0 -4.50016 39 -0.117 0 -4.151 -0.167 0 -4.16717 29 -1.904 0 -5.555 -1.833 0 -5.50018 39 -0.11719 34 -0.981
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8.7 Cusum Procedures for Non-conformity Data
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8.7 Cusum Procedures for Non-conformity Data: Example
Samplei c
Transformation Normal Approximation
z(T) SH SL z(NA) SH SL
1 9 0.573 0.073 0 0.524 0.024 02 15 2.284 1.857 0 2.706 2.230 03 11 1.191 2.548 0 1.251 2.981 04 8 0.239 2.287 0 0.160 2.641 05 17 2.776 4.564 0 3.433 5.574 06 11 1.191 5.255 0 1.251 6.325 07 5 -0.904 3.852 -0.404 -0.931 4.894 -0.4318 11 1.191 4.543 0 1.251 5.645 09 13 1.758 5.801 0 1.979 7.124 0
10 7 -0.115 5.186 0 -0.204 6.420 011 10 0.890 5.575 0 0.887 6.807 012 12 1.480 6.556 0 1.615 7.922 0
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8.7 Cusum Procedures for Non-conformity Data: Example
Samplei c
Transformation Normal Approximation
z(T) SH SL z(NA) SH SL
13 4 -1.353 4.703 -0.853 -1.295 6.128 -0.79514 3 -1.857 2.345 -2.210 -1.658 3.969 -1.95315 7 -0.115 1.730 -1.826 -0.204 3.265 -1.65716 2 -2.443 0.000 -3.769 -2.022 0.743 -3.17917 3 -1.857 0.000 -5.126 -1.658 0 -4.33718 3 -1.857 0.000 -6.483 -1.658 0 -5.49619 6 -0.494 0.000 -6.477 -0.567 0 -5.56320 2 -2.443 0.000 -8.420 -2.022 0 -7.08521 7 -0.115 0.000 -8.035 -0.204 0 -6.78922 9 0.573 0.073 -6.962 0.524 0.024 -5.76523 1 -3.175 0.000 -9.637 -2.386 0 -7.65124 5 -0.904 0.000 -10.041 -0.931 0 -8.08225 8 0.239 0.000 -9.302 0.160 0 -7.422
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8.7 Cusum Procedures for Non-conformity Data
• The z-values differ considerably at the two extremes: c15 and c2
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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.
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8.8.1 EWMA Chart for Subgroup Averages
(8.9)
(8.10)
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8.8.1 EWMA Chart for Subgroup Averages
(8.11)
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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
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Table 8.12 EWMA Chart for Subgroup Averages: Example
i x-bar wt CL1 0.41 0.1013 0.37502 1.11 0.3522 0.46883 -0.04 0.2554 0.51404 -1.05 -0.0697 0.53785 -0.04 -0.0617 0.55086 -0.69 -0.2175 0.55797 -0.85 -0.3756 0.56198 0.81 -0.0786 0.56419 0.63 0.0973 0.5653
10 -0.38 -0.0214 0.566011 0.54 0.1183 0.566412 0.19 0.1350 0.5667
i x-bar wt CL13 0.33 0.1844 0.566814 0.33 0.2195 0.566915 0.45 0.2759 0.566916 0.22 0.2607 0.566917 0.88 0.4161 0.566918 0.17 0.3533 0.566919 0.95 0.5025 0.566920 1.09 0.6494 0.5669
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8.8.2 EWMA Misconceptions
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8.8.3 EWMA Chart for Individual Observations
(8.9)’
(8.10)’
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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.
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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.