Some Control Charts for the Process Mean and Variance based on Downton's Estimator

Michael B.C. Khoo School of Mathematical Sciences, University Science of Malaysia, 11800 Minden, Penang, Malaysia

Abstract-The EWMA (Exponentially Weighted Moving Average) control chart is a good alternative to the Shewhart chart in the detection of small shifts. The EWMA charts for the sample means and sample ranges are constructed based on the assumption that the data used in the computation of the limits are outlier free. In most real life situations, outliers will occur in the data used in computing the limits. Since the EWMA chart is a weighted average of all past and current observations, it is insensitive to outliers. The detection of outliers using EWMA charts for the process mean and variance becomes even more difficult if the average sample range, E , is used in the computation of the limits bffause the sample range, R, is easily influenced by outliers so that the limits will be stretched. In this paper, the use of Downtoo's eslimator in setting up the limits of the EWMA charts for the process mean (EWMAMD) and the pmcess variance (EWMAVD) will he proposed. Monte Carla simulations using SAS (Statistical Analysis System) version 8 are conducted to show that the new EWMA charts have higher pmbhililies in detecting outliers while maintaining the same rates of Type-1 errors compared lo the standard EWMA chart.

Keywords-Downton's estimator, EWMA, EWMAMD, EWMAVD, in-control, aut-of-control, outliers, Type-I error

1. I N T R O D U ~ I O N

Since the introduction of the EWMA chart by Roberts [l], numerous extensions have been proposed such as the following, to mention only a few: Studies have been made to provide ARL (average run length) tables and graphs for the selection of the values of parameters in the design of optimum EWMA charts [Z] & [3]. Various types of FIR (fast initial response) features have been suggested to make the EWMA charts more sensitive to start-up problems [4] & [ 5 ] . Extensive research have been done to construct attribute charts using EWMA statistics to enhance the sensitivity of the conventional c and U charts (61. There are also research made on deriving EWMA charts based on the sample means, sample ranges, moving ranges and individual measurements [7j.

The EWMA charts for the sample mean (EWMASM) and sample range (EWMASR) are often used in the monitoring of the process mean and variance respectively. The limits of these charts are computed under the assumption that the data representing the underlying process are outlier free. This assumption is usually not true in real situations. Outlien may be single unusual values due to a sporadic special cause [SI. The presence of outliers reduce the sensitivity of a control chart because

the control limits are stretched so that the detection of the outliers themselves become more difficult. The EWMAMD and EWMAVD charts for the process mean and variance respectively, proposed in this paper solve the above problem.

11. AN OVERVIEW OE THE EWMASM AND EWMASR CHARTS

Let x, and R, represent the mean and range of sample f. The EWMASM chart is constructed by plotting

" - [71

Z , =aX,+(l-a)Z,-,, t=1,2 ,.... (1)

X + F , R , (2) - based on limits

where

(3)

The EWMASR chart is constructed by plotting [7]

with upper and lower control limits

and

(4)

UCL, = F,R (5)

?r =aR,+(l-a)Ft,, f=1 ,2 ,...,

LCL, = F,R (6)

and -

- - In the above equations, Z, = x and io = R . x and denote the average sample mean and sample range respectively, where both are estimated from an in-control preliminary data set while a (0 S a < 1) represents the weighting factor. The chart constants, d , and d, which depend on the sample size, n are given in most quality control text books.

111. PROPOSED EWMAMD AND EWMAVD CHARTS

Denote the mean and standard deviation of sample f as x, and S, respectively. The EWMAMD chart for the

0-7803-8519-5/04/$20.00 @ 2004 IEEE 1071

process mean is constructed by plotting the statistic, 2, in equation (l), at sample t based on limits given by - . -

CLEWM,, = z , = x . (9) - UCL,,-, = i , + 3 8 . 2 s = X + K , I * (10)

- and -

LCLEWw,=Z -38. = X - K , E * . (11)

In the construction of the EWMAVD chart for the process variance, the statistic, ?, is plotted at sample i, where

The limits of the EWMAVD chart are

0 1,

Y, =as, + ( l - c ~ ) K . ~ , f = 1,2, ... . (12)

CL,,,,, =Po = C,O' , (13)

(14)

(15)

UCLEWM,, =Yo 138. y, = K , E *

LCLE," - Y - 0 -38- 6 = K , B * . and

Note that

(16)

and

K , =c4 + 3 / ( l - c i [ L ) 2-a . (18)

represent the factors of the EWMAMD and EWMAVD charts while

TABLE 1

FACTORS OF C O m o L LIMITS FOR THE EWMAMD. K,

TABLE 2 FACTORS OFCONTROL LIMITS FOR THE EWMAVD). K ,

is the Downton's estimator [Y]. 5: is estimated from the subgroup 0:'s to reduce the influence of outliers on the control limits. The values of K , , K, and K , for the various sample sizes, n are given in Tables 1, 2 and 3 respectively while the values of c4 can be obtained in most quality control text books.

IV. EVALUATION OF PERFORMANCES

The performances of the E W M M D vs. EWMASM and EWMAVD vs. EWMASR charts are compared based on Monte-Carlo simulations using SAS version 8. The following two situations for samDle sizes of n = 5 and 10 - are considered: (i) The '%-control" situation where the data are all ~,

standard normal random variables. (ii)The "Outliers" situation where the data are a mixture

of observations which are grouped into the following four categories: 95% N(O,1) and 5% N(0,9), 90% N(0,l) and 10% N(O,Y), 95% N(0,l) and 5% N(0,25), and 90% N(0, l ) and 10% N(0,25). These four

TABLE 3 FACIORS OF CONTROL LIMITS FOR THE EWMAVD, K ,

S u n l k Ydur O f 0

Si," U 0.1 " 2 0,s 0.4 0 5 0.6 0.7 0 8 0.9 1

2 080 1.2, 1.40 l i b 3.70 l.w 1 9 8 2.12 2.21 2.4, 2.61 3 089 1.21 1.35 1.47 1.58 1.69 1.80 1.91 2.02 i.ll 1 2 8 4 092 1.19 1.31 1.41 1.10 1.59 1.68 1.78 1.81 1.98 2 0 9 I 0 % , . I , 1.28 1.37 i.45 1.53 1.61 169 1.78 1.81 I P L

6 0.95 1.16 1.26 1.24 I 4 I 118 1.56 1.63 1.70 1.7'1 1.81 7 0 % 1.16 1.24 1,) 1 3 8 LIS l i l ,.sa 1.65 1.7, ,.SI 8 0.97 1.15 1.23 1.30 1.36 1.42 148 3.54 1.61 168 1.71 9 0.97 1.14 122 1.28 1.34 1 0 1.45 1.51 1.57 1.M 1.71 10 0.97 113 120 1.27 1.12 1.37 1.43 1.48 154 1.60 1.67

I I 0.98 1.13 1.20 1.21 1.31 1.36 1.41 1.46 1.12 $ 5 7 1.64 I2 0.98 1.12 1.19 1 . Y 1.29 1.M 139 1.44 149 l.55 1.61 I3 098 l . l i 1,s 1.23 1.28 1.1, 1.38 1.42 1.47 1.53 1.59 I4 098 111 1.18 *.U 1.2, 1.32 1.36 I.41 1.46 !*I 1.56 IS 098 1 1 1 1.17 *.U 1.26 111 1.31 1.39 l.44 1.49 1 . m

16 0.98 1.11 I 1 6 1.21 1.25 1.34 1 . 3 1.38 1.41 1.47 133 I? 0.38 1.11 1.16 1.21 1.B 129 1.31 1.37 111 1.46 1.53 I8 0.99 1.10 1.16 L.W 1.24 1.w 1.32 1.36 l.40 1.45 1.50 19 0.w 1.10 I I i ,.I9 1.u 121 1.3, 1.31 1.39 144 1.48 20 099 1.10 1.15 1.11 1.23 1.27 1.24 1M 1.38 1.41 1.41

1072 International Engineering Management Conference 2004

categories of contaminated normal distributions are represented by C05N3, ClON3, C05N5 and C10N5 respectively.

Repeatedly, m = 10 subgroups of size either n = 5 or 10 observations each are generated for situations (i) and (ii), control limits computed and the number of subgroups that fall outside the limits are calculated. This procedure is repeated 10000 times for a total of 10000 x m subgroups. The proportions of out-of-control subgroups (based on 10000 x m subgroups) are computed for the two different conditions of EWMAMD vs. EWMASM and EWMAVD vs. EWMASR charts and their results are displayed in Tables 4 and 5 respectively.

Tables 4 and 5 show that the out-of-control proportions for the in-control case of the EWMAMD and EWMAVD charfs where 0.1 5 a 5 1 are either similar or slightly lower than the corresponding values of their EWMASM and EWMASR counterparts respectively. This indicates that the Type-I error of the two proposed charts are either similar or lower than their conventional counterparts. All of the charts generally show an increase in the false alarm rate as a increases. The EWMAMD and EWMAVD charts are also superior to the EWMASM and EWMASR charts in the detection of outliers for 0.1 L a S 1 because the former two charts have higher out-of-control proportions for all four categories of contaminated normal distributions than the corresponding values of the latter two charts. For example, when a = 0.5 and n = 10, the out-of-control proportions for EWMAMD are (0,0013, 0.0016, 0.0047, 0.0058) while that of EWMASM are {0.0009, 0.0010, 0.0021, 0.0020) for the four categories of C05N3, ClON3, COW5 and CION5 respectively.

V. DISCUSSION

Outliers usually act only on occassional observations from a subgroup and not on the subgroup as a whole. The occurrence of outliers must be detected, investigated and the special cause removed if possible [SI. The EWMAMD and EWMAVD charts produce better results in the detection of outliers and having Type-I errors of not greater than that of the EWMASM and EWMASR charts respectively. The limits of the EWMAMD and EWMAVD charts computed from the Downton's estimator are less influenced by outliers compared to that of the EWMASM and EWMASR c h a m whose limits are computed based on the average sample range. Wider limits make the detection of outliers difficult using the EWMASM and EWMASR charts.

VI. CONCLUSION

This paper proposes two new EWMA control charts for the process mean and variance which are referred to as the EWMAMD and EWMAVD charts. It is shown by

simulation that these new charts allow easier detection of outliers in the subgroups and are superior alternatives to the standard EWMASM and EWMASR charts for quality control practitioners.

ACKNOWLEDGMENT

This research is supported by the University Science of Malaysia "Fundamental Research Grant Scheme (FRGS)" no. 304ipmathsi670039.

REFERENCES

S. W. Rabens, "Control chart based on geometric moving averages", Technomehics, vol. 1, pp. 239-250, 1959. S. V. Crowder, 'A simple method for studying run length distributions of exponentially weighted moving average charts", Technometricr, vol. 29, pp. 4 0 1 4 0 7 , 1987. S. V. Crowder, "Design of exponentially weighted moving average schemes", Journal of Quolity Technology, vol. 21, pp. 155-162,1989. I. M. Lucss and M. S. Saccucci, "Exponentially weighted moving average control schemes: prop~t ies and enhancemenu", Technomerrics, vol. 32, pp. 1-12,1990, T. R. Rhoadr, D. C. Montgomery and C. M. Mastrangelo, "A fast initial response scheme far the exponentially weighted moving average ~ontrol chart", Quoliry Engineering, vol. 9, pp. 317-327,1996. C. M. Borror, C. W. Champ and S . E. Rigdon, ~'Poisson E W A control charts", Journal of Quolily Technology, vol. 30, pp. 352-361, 1998. C. H. Ng and K. E. Care, "Development and evaluation of control chane using exponentially weighted moving averages", Joournol of Quoliry Technology, vol. 21, pp. 242-250,1989. D. M. Rocke, '"gQ and Ru charts: robust control cham",

TheSmtisticion, vol. 41, pp. 97-104, 1992. M. 0. Abu-Shawierh and M. B. Abdullah, "Estimating the process standard deviation bared on Downton's estimator", Quolily Engineering, vol. 12, pp. 357-363.20w).

International Engineering Management Conference 2004 1073

PROPC I 3 w

ple She, n ne Dirfribulioi

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4

a 0.5 0.6 0.7 0.8 0.9

I

0

-~

PROPOF I

1074

ple Sire, n ?g Distribution

0 1 0.2 0.3 0.4 0.5 0.6 a

0.7 0.8 0.9

0.1 0.2

0.4 a 0.5

0.6 0.7 0.8 0.9 1

0.3

-~

I".( __ 5

0.0000 o.ooo1 0.0002 0.0005 0.0009 0.0013 0.0017 0.0021 0.0024 0.0028

0.Mw

0.0003 0.ow5 0.0009 0.0013 0.0017 0.0022 0.0025 0.0028

0.0001

__

NS OF 0 I"-<

5

0.0000 0.0000 0.0001 0.0004 0.wO8 0.0014 0.0019 0.0023 0.0027 0.0029

0 . W 0.0000 0.0002 0.0004

0.0013 0.0020 0.0023 0,0028 0.0032

__

0.0008

10

0 . m 0.m0 0.0001 0.0004 0.0007 0.0011 0.0014 0.0017 0.0019 0.002l

0.Mw 0.0w0 0.0001 0.0004 0.0008 0.0011 0.0015 0.0018 0.0021 0.0025 __

)F-CONT ml

&gL 10

0 . m 0.0000 0.0001

0.0006 0.0009 0.W11 0.0014 0.0017 0.0018

0.0wO 0.0000 0.0002 0.0004 0.0007 0.0012 0.0017 0.0022 0.0025 0.0028

__

0.0003

~

TABLE 4 L FOK THE EWMAMD AND EWMASM CHART

n

5 C05N3 Cl0M CO5N5 CION5 0.00011 0.0000 0.0000 0.0000 0.0001 0.0004 0.0010 0.0022 0.0030 0.0037 0.0046 0.0056 0.0064

n.ooou 0.0001 0.0004 0.wO9 0.0018 0.W26 0.0034 0.0044 0.0050 0.0056 __.

0.0001 0 . W 6 0.0014 0.0026 0.0036 0.0049 0.0057 0.0070 0.0073

nmno 0.0001 0.0005 0.0011 0.0021 0.0031 0.0042 0.0054 0.0059 0.0067 ~-

0.0002 0.0011 0.0033 0.0061 0.0088 0.0113 0.0135 0.0149 0.0158

0.0000 0.00Ol 0.0009 0.0025 0.0045 0.0068 0.0086 O.Oll2 0.0124 0.0136 ~.

0.0003 0.0015 0.0040

0.0105 0.0133 0.0154 0.0178

0.0068

0.0189

0.0000 0.0002 0.0013 0.0031 0.0056 0.0084 0.0102 0.0122 0.0142 0.0151

~

TABLE 5 -FOR M E EWMAVD AND EWMASR CHMR

0,

COSM CION3 C05N5 CION5 0.wO3 0.0004 0.0060 0.0074 0.0030 0.0047 0.0235 0.0310 0.0078 o.om 0.0392 0.0515 n n i w n m ~ 4 nn41(2 nnM6

~ ~~. .- ~ ~. . 0.0176 0.0248 0.0544 0.0746 0.0208 0.0238 0.0258 0.0272 0.0276

0.0002

0.0054 0.0103 0.0149 0.0186 0.0216 0.0235 0.0251 0.0266

0.0016

~-

0.0300 0.0337 0.0366 0.0382 0.0395

0.0003

0.0078 0.0142 0.0206 0.0256 0.0294 0,0328 0.0339 0.0366

0.0025

~-

0.0582 0.0607 0.0629 0.0653 0.0655

0.0020 0.0124 0.0260 0.0373 0.0442 0.0498 0.0532 0.0571 0.0589 0.0598 --

0.0804 0.0854 0.0880

0.0922

0.0026 0.01M 0.0332 0.0469 0.0569 0.0639 0.0694 0.0746

0.0777

0.0900

0.0766 -

SED ON m = 10 SUBGROUPS

10 C05N3 CION3 CU5NS CION5 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0007 0.0013 0.0022

0.0036 0.0041 0.w45

0.0027

0.m1 0.0003 0.0009 0.0016 0.0025 0.0036 0.0043 0.0049 0.0053

0.0002 0.0010 0,0025 0.0047 0.0070 0.0090 0.0108 0.0124 0.0130

0.0000 0.w0 0.0000 0.0000 0.0000 0 . m 0.0001 0.0002 0.04 0.0005 0.0005 0.0010

0.0015 0.0017 0.0032

0.0024 0.0025 0.0055 0.0030 0.0030 0.0065 0.0031 0.0034 0.0068

0.0009 0.0010 0.0021

0 . ~ 2 0 0.0021 0.0045

~~~

0.0002 0.0012 0.0032 0.0058 0.0082 0.0107 0.0127 0.0146 0.0151

0.0000 0.0001 0.0003 n.0010 n.wzo 0.0032 0.0045 0.0054 0.0065 0.0069 -.

SED ON m = 10 SUBGROUPS rr

0.0344 0.0379 0.0402 0.0426 0.0431

0.0007 0.0047 0.0128 0.0209 0.0277 0.0323 0.0365 0.0401 0.0419 0.0439 __-

0.0495 0.0534 0.0568 0.0586 0.0602

0.M8 0.0070 0.0172 0.0275 0.0356 0.0418 0.0466

0.0535 0.0543

0.0506

__-

International Engineering Management Conference 2004

0.1065 0.1079 0.1083 0.1100 0.1100

0.0097 0.0349 0.0587 0.0747 0.0820 0.0868 0.0908 0.0936 0.0948 0.0966 ~.

0.1458

0.1478 0.1469 0.1470

0.0112 0.0406 0.0686 0.0877 0.0977 0.1036 0.1067 0.1079 0.1085 0.1098

0.1473

__