Chapter 8

Alternatives to Shewhart Charts

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.”

8.1 Introduction with Example

8.2 Cumulative Sum Procedures:Principles and Historical Development

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)

Cusum Example

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)

Cusum Procedures

(8.1)

(8.3)

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

Cusum Example

ARL for Cusum Procedure(Table 8.3)

8.2.2 Fast Initial Response Cusum

FIR Cusum vs Cusum(Table 8.4) N(0.5,1)

FIR Cusum vs Cusum(Table 8.5) N(0,1)

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

8.2.3 Combined Shewhart-Cusum Scheme

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.

8.2.5 Computation of Cusum ARLs

8.2.6 Robustness of Cusum Procedures

(8.4)

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

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

8.2.7 Cusum Procedures for Individual Observations

8.3 Cusum Procedures for Controlling Process Variability

(8.5)

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.

8.6 Cusum Procedures for Non-conforming Units

(8.6)

(8.7)

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

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

8.7 Cusum Procedures for Non-conformity Data

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

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

8.7 Cusum Procedures for Non-conformity Data

• The z-values differ considerably at the two extremes: c15 and c2

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.

8.8.1 EWMA Chart for Subgroup Averages

(8.9)

(8.10)

8.8.1 EWMA Chart for Subgroup Averages

(8.11)

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

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

8.8.2 EWMA Misconceptions

8.8.3 EWMA Chart for Individual Observations

(8.9)’

(8.10)’

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.

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.