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 Example N(0,1) N(0.5,1) 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 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 -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 0.67 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 15 2.30 -0.33 0.45 16 2.14 0.51 -1.65 -0.14 17 1.03 0.30 0.55 1.65 0.88 18 -0.90 -1.08 0.17 19 1.56 -0.70 2.06 0.95 20 1.28 0.98 1.29 1.09 N(0,1) N(0.5,1)

Cusum Example

Runs Criteria and their Impacts 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 0.00 2 0.86 0.57 1.17 1.82 1.11 2.21 2.02 3 -0.89 0.21 -1.23 1.77 -0.04 -0.07 1.45 4 -1.88 -0.43 -0.42 -1.45 -1.05 -2.09 -1.59 5 -1.85 2.03 -0.64 -0.08 -1.17 6 -2.53 -0.59 0.60 -0.22 -0.69 -1.37 -2.04 7 -0.74 -1.25 -0.40 -1.01 -0.85 -1.70 -3.24 8 2.10 1.48 -1.19 1.63 1.13 -1.11 9 0.56 1.78 -0.81 0.97 0.63 1.25 1.88 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 12 0.67 0.42 0.46 0.19 0.37 1.07 13 0.84 -0.71 0.27 0.93 0.33 1.23 14 0.22 1.27 0.64 -0.83 0.65 1.38 15 2.30 -0.33 0.45 0.89 16 2.14 0.51 -1.65 -0.14 1.70 17 1.03 0.30 0.55 1.65 0.88 2.97 18 -0.90 -1.08 0.17 2.80 19 1.56 -0.70 2.06 0.95 1.90 4.20 20 1.28 0.98 1.29 1.09 2.18 5.88

Cusum Example

ARL for Cusum Procedure (Table 8.3) Mean Shift h=4 h=5 168.00 465.00 370.40 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 4.00 1.71 2.01 1.19 5.00 1.31 1.69 1.02

8.2.2 Fast Initial Response Cusum  

FIR Cusum vs Cusum (Table 8.4) N(0.5,1) z With FIR w/o FIR SH SL (Reset) -  - 2.50 -2.50 21 -0.08 -0.16 1.84 -2.16 22 0.57 1.14 2.48 -0.52 0.64 23 0.80 1.60 3.58 1.74 24 0.23 0.46 3.54 1.70 25 0.08 0.16 3.20 1.36 26 1.33 2.66 5.36 3.52 27 1.23 2.46 5.48

FIR Cusum vs Cusum (Table 8.5) N(0,1) z With FIR w/o FIR SH SL (Reset) -  - 2.50 -2.50 21 -0.28 -0.56 1.44 -2.56 -0.06 22 0.07 0.14 1.08 -1.92 23 0.21 0.42 1.00 -1.00 24 0.46 0.92 1.42 25 0.55 1.10 2.02 1.02 26 0.77 1.54 3.06 2.06 27 -0.3 -0.60 1.96 -0.10 0.96 28 0.09 0.18 1.64 0.64 29 0.69 1.38 2.52 1.52 30 0.44 0.88 2.90 1.90 31 -0.26 -0.52 1.88 -0.02 32 -0.34 -0.68 0.70 -0.20 33 0.00

Table 8.6 ARL for Various Cusum Schemes (h=5, k=.5) Mean Shift Basic Cusum Shewhart-Cusum (z=3.5) FIR Cusum Shewhart-FIR Cusum (z=3.5) 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 2.00 4.01 3.77 2.36 2.50 3.11 2.77 1.86 3.00 2.57 2.10 1.54 4.00 2.01 1.34 1.16 5.00 1.69 1.07 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 2 330.0 310.7 167.8 3 363.4   Basic Cusum FIR Cusum Sheahart-Cusum r ARL 2 330.0 310.7 167.8 3 363.4 341.0 199.0 4 383.6 359.4 222.0 6 406.9 380.5 254.4 8 419.9 392.2 276.3 10 428.2 400.0 292.3 25 450.0 419.5 344.7 50 457.8 426.5 368.9 100 462.2 430.4 383.1 500 466.0 434.7 395.6  

  Lower Upper r ARL 4 2963.5 440.3 6 2298.2 493.9 8 1995.2 531.2 10 1818.8 559.4 25 1390.7 664.1 50 1227.4 728.8 100 1127.8 780.4 500 1011.8 858.6  

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 Sample i x Arcsine Transformation Normal Approximation z(a) SH SL z(na) 1 47 1.169 0.669 1.167 0.667 2 38 -0.286 -0.333 3 39 -0.117 -0.167 4 46 1.014 0.514 1.000 0.500 5 42 0.378 0.392 0.333 6 36 -0.629 -0.129 -0.667 7 8 37 -0.456 -0.500 9 40 0.050 10 35 -0.804 -0.304 -0.833  

8.6 Cusum Procedures for Non-conforming Units: Example Sample i x Arcsine Transformation Normal Approximation z(a) SH SL z(na) 11 34 -0.981 -0.784 -1.000 -0.833 12 31 -1.526 -1.811 -1.500 -1.833 13 33 -1.160 -2.471 -1.167 -2.500 14 29 -1.904 -3.874 -3.833 15 -4.534 -4.500 16 39 -0.117 -4.151 -0.167 -4.167 17 -5.555 -5.500 18 19  

8.7 Cusum Procedures for Non-conformity Data  

8.7 Cusum Procedures for Non-conformity Data: Example Sample i c Transformation Normal Approximation z(T) SH SL z(NA) 1 9 0.573 0.073 0.524 0.024 2 15 2.284 1.857 2.706 2.230 3 11 1.191 2.548 1.251 2.981 4 8 0.239 2.287 0.160 2.641 5 17 2.776 4.564 3.433 5.574 6 5.255 6.325 7 -0.904 3.852 -0.404 -0.931 4.894 -0.431 4.543 5.645 13 1.758 5.801 1.979 7.124 10 -0.115 5.186 -0.204 6.420 0.890 5.575 0.887 6.807 12 1.480 6.556 1.615 7.922  

8.7 Cusum Procedures for Non-conformity Data: Example Sample i c Transformation Normal Approximation z(T) SH SL z(NA) 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 -5.126 -4.337 18 -6.483 -5.496 19 6 -0.494 -6.477 -0.567 -5.563 20 -8.420 -7.085 21 -8.035 -6.789 22 9 0.573 0.073 -6.962 0.524 0.024 -5.765 23 1 -3.175 -9.637 -2.386 -7.651 24 5 -0.904 -10.041 -0.931 -8.082 25 8 0.239 -9.302 0.160 -7.422

8.7 Cusum Procedures for Non-conformity Data The z-values differ considerably at the two extremes: c15 and c2

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 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 -0.0617 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 0.2195 0.5669 15 0.45 0.2759 16 0.22 0.2607 17 0.88 0.4161 18 0.17 0.3533 19 0.95 0.5025 20 1.09 0.6494

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.