 Open Access
 Total Downloads : 457
 Authors : Rajmanya S V, Ghute V B
 Paper ID : IJERTV2IS4847
 Volume & Issue : Volume 02, Issue 04 (April 2013)
 Published (First Online): 25042013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
The Synthetic D Chart Under NonNormality
Rajmanya S V a and Ghute V B b*
aDepartment of Statistics, Sangameshwar College, Solapur, Maharashtra, India.
bDepartment of Statistics, Solapur University, Solapur413255, Maharashtra, India.
*Correspondance to: V. B. Ghute, Department of Statistics, Solapur University, Solapur, India.
Abstract
The synthetic control chart based on Downtons estimator (Synthetic D chart) for monitoring the process variability developed by Rajmanya and Ghute (2013) is based on the assumption that the underlying process distribution is normal. In this paper, the performance of the synthetic D chart under nonnormality is studied and is compared with the synthetic S chart. The comparison result shows that the synthetic D chart under nonnormal distributions is more efficient than the synthetic S chart in detecting shifts in process variability.
Keywords Downtons Estimator, Process variability, Nonnormality, Average run length, Conforming run length, Control chart.

INTRODUCTION
Control charts are statistical process control tools that are widely used for controlling and monitoring a process. Shewharts S chart is most widely used control chart to monitor process variability of a quality characteristic of interest. The S chart is based on the fundamental assumption that the underlying distribution of the quality characteristic is normal. It is easy to implement and is effective in the detection of large shifts in process standard deviation but become less effective for small shifts because it is based on only the most recent observation. Recently, Abbasi and Miller (2011) proposed the control chart based on Downtons estimate (D chart) of process standard deviation as an efficient alternative to S chart for monitoring process variability. It was shown that for normally distributed process, the D chart is equally efficient to the S chart for detecting shifts in the process variability.
In the literature synthetic charting concept is used by many researchers for improving the performance of the control charts. Huang and Chen (2005) developed synthetic S chart for process dispersion by combining the sample standard deviation, S and conforming run length (CRL) charts. The CRL chart is an attribute control chart proposed by Bourke (1991) for monitoring fraction nonconforming. With the objective of improving the performance of D chart, Rajmanya and Ghute (2013) developed synthetic D chart as a combination of D chart and CRL chart. It was shown that synthetic D chart is close competitor of synthetic S chart for detecting shifts in process standard deviation. The synthetic D chart was designed and evaluated under the assumption that the underlying process distribution is normal.
The behaviors of the commonly used control charts have been widely studied when some aspects of the assumption do not hold. One violation of the assumption is nonnormality, which has been discussed by many authors. Borror et al. (1999) examined the performance of the EWMA chart for the mean in nonnormal cases. Stoumbos and Reynolds (2000) studied the effect of nonnormality and autocorrelation on the performance of various individual control charts for monitoring the process
mean and/or variance. Calzada and Scariano (2001) have studied the robustness of the synthetic X chart when the normality assumption is violated and show that for sample size n = 6, the synthetic chart is robust to violations of the normality assumption. Kao and Ho (2007) examined the performance of R chart and found that R chart is robust to nonnormality. Schoonhoven and Does
(2010) studied the design schemes for the X control chart under nonnormality.
The objective of this paper is to investigate the effect of nonnormality on the statistical performance of the synthetic D control chart for monitoring process variability of a continuous process. Nonnormal distribution can be a symmetric distribution or a skewed distribution. In this study, we selected a t distribution as symmetric distribution and Weibull distribution as the skewed distribution to examine and compare the performance of a synthetic D chart under nonnormality with synthetic S chart.

MATERIALS AND METHODS
The D Control Chart
Let X1, X2 ,…, Xn
represents a random sample of size n from a normally distributed process with
process mean and standard deviation and X(1) X(2) … X(n) denote the corresponding
order statistics. Downton (1966) proposed the following estimator of process standard deviation
(i)
(i)
2 n 1
D i (n 1) X
n (n 1) 2
(1)
i1
which is unbiased estimator of for normally distributed quality characteristics. Let 0 and 0 be
the incontrol values of and respectively. When a shift in process standard deviation occurs, we
have change from the incontrol value 0
to the outofcontrol value 1 0
( 0 1) . When
0
0
1, the process is considered to be incontrol . For considered to be outofcontrol and an upper control limit
1 an increase in occurs, process is
0
0
k of D chart is required, and a signal is
issued if
D k . For 1 decrease in occurs, process is considered to be outofcontrol and
0
0
0
0
lower limit k of D chart is required, and a signal is issued if D k . In this study we consider
the case of increase in the process standard deviation.
The average run length (ARL) which denotes the average number of D samples required to detect a change in of the D chart can be calculated as
ARLD
1
0
0
Pr (D k 
0 )
1
k
Pr Z
1
(2)
k
1 F
where, Z D
and F(.) its cumulative distribution function.
Synthetic D Chart
0
0
The synthetic D control chart developed by Rajmanya and Ghute (2013) is a combination of the D chart and the CRL chart. The synthetic D chart consists of D subchart and CRL subchart. For detecting increase in standard deviation, the D subchart has upper control limit UCL k . The CRL subchart has a lower control limit L. These limits are called as design parameters of the synthetic D chart. According to the synthetic procedure, the signal is based on the CRL. The CRL is the number of conforming samples between two consecutive nonconforming samples including the end nonconforming sample.
Let ARLS ( ) denote the average number of D samples required for a synthetic D chart to signal a shift of magnitude in process standard deviation . Then the ARL values of the synthetic D chart for a given shift of magnitude is given by the formula,
ARLS ( )
1
k
1
k L1
(3)
1 F
1 F
When 1, incontrol ARL of the synthetic D chart is
ARLS
(1)
1
1 F(k )
1
1 F (k )L1
(4)
To design synthetic D control chart means to obtain two design parameters L and k+ suitably that
guarantee a minimum ARL for predetermined shift of magnitude
* . The design shift *
is the
magnitude considered larger enough to seriously impair the quality of the products; thus
corresponding
ARLS *)
should be as small as possible.
ARLS (1)
is decided by the requirement
on the false alarm rate. The synthetic D control chart is properly designed by solving an optimization
problem. The objective function
ARLS ( *) minimum ,
(5)
subject to the equality constraint in Eq. (4).
The optimal design procedure for the synthetic D chart uses the approach described in the following steps.

Specify n, * and incontrol ARL.

Initialize L as 2.

Obtain k by solving Eq.(4).

Calculate ARL for * from current value of L and k using Eq.(3) (use *) .

If L is not equal to 2, go to the next step; otherwise, L is increased by one and go back to step (3).

If the current ARL for *
is greater than the preceding one, go to the next step; otherwise, L is
increased by one and go back to step (3).

Take the preceding L and k as the optimal design parameters for the synthetic chart.
To illustrate the design of the synthetic D chart, an example with n =10, * 1.4 and incontrol
S
S
ARL (1) 200 . Table 1 shows that each set of (L, k+) results in different
ARLS ( *) . The ARL
first declines and then increases. The ARLS ( *)
reaches its minimum at 3.0926 when L = 7 and k =
1.5180. So in this case the design parameters of the synthetic D chart are L = 7 and k = 1.5180.
Table1. Sets of (L, k ) and corresponding ARL for * 1.4 .
(L, k )
ARL for *
(2, 1.4098)
4.5127
(3, 1.4530)
3.5726
(4, 1.4780)
3.2783
(5, 1.495)
3.1542
(6, 1.5073)
3.1005
(7, 1.5180)
3.0926
(8, 1.5264)
3.0999
(9, 1.5340)
3.1252
(10, 1.5400)
3.1473
The optimal values of L and k for sample size n = 5, 8 and 10 provided incontrol ARL = 200 and 370 under * 1.2, 1.4 and 1.6 are presented in Table 2.
Table2. Optimal design parameters of the synthetic D chart.
n
*
ARLS
(1) 200
ARLS
(1) 370
L
k
L
k
5
1.2
17
1.843
24
1.927
1.4
10
1.786
11
1.856
1.6
7
1.750
8
1.825
8
1.2
12
1.595
18
1.658
1.4
7
1.550
9
1.612
1.6
5
1.522
5
1.567
10
1.2
12
1.5192
15
1.5638
1.4
6
1.469
7
1.518
1.6
4
1.438
5
1.495
Effect of Nonnormality on Synthetic D chart
The synthetic D control chart developed for monitoring the process variability is based on the assumption that the underlying process distribution is normal. In this section, we examine the effects of nonnormality on the statistical performance of the synthetic D control chart. In order to study the effect of nonnormality, we considered heavytailed symmetric and skewed distributions. Specifically, we simulated observations in the heavy tailed symmetric case from t distribution and in the skewed case from the Weibull distribution.
The probability density function of t distribution with degrees of freedom is
1
x
x
1/ 2
f (x)
2 2
1
, x ; 0.
(6)
/ 2
The probability density function of Weibull distribution with shape parameter and scale parameter is
x
1
x
f (x)
e
, x 0 ; 0, 0
(7)


RESULTS AND DISCUSSIONS Performance comparisons for Nonnormal process
The synthetic D chart is designed by using the values of design parameters (L, k) with
* 1.2 from Table 2, which are, technically, only appropriate for normally distributed process data.
The design parameter k of the charts is then adjusted so that both synthetic charts have approximately the same incontrol ARL value 200. The ARL values of the synthetic S and synthetic D charts are computed using 10000 simulations when underlying process distribution is normal, heavy tailed and skewed. To study the effect of nonnormality on the performance of the synthetic D control chart, we have considered the process data from t distribution with 5 and Weibull distribution with shape
parameter = 2 and scale parameter = 1.
Table 3 presents the ARL profiles of the synthetic S and synthetic D charts for increase in the process standard deviation when underlying process data follows standard normal distribution with in control ARL = 200, * 1.2 and sample sizes n = 5, 8 and 10.
Table3. ARL comparison of synthetic charts under normal distribution.
Shift 
n = 5 
n = 8 
n = 10 

Syn S k =1.722 L = 18 
Syn D k =1.843 L = 17 
Syn S k =1.5263 L = 12 
Syn D k =1.595 L = 12 
Syn S k =1.4663 L = 12 
Syn D k =1.5192 L = 12 

1.0 
200 
200 
199 
200 
200 
201 
1.1 
42.62 
43.92 
33.29 
32.80 
27.96 
28.61 
1.2 
15.54 
15.89 
10.49 
10.75 
8.58 
8.66 
1.3 
8.09 
8.34 
5.07 
5.27 
4.22 
4.41 
1.4 
5.25 
5.38 
3.31 
2.49 
2.77 
2.89 
1.5 
3.86 
3.92 
2.50 
2.56 
2.09 
2.16 
2.0 
1.77 
1.79 
1.28 
1.31 
1.18 
1.18 
From Table 3, when underlying process distribution is normal, we observe that for any range of shifts, the synthetic S control chart produces slightly smaller outofcontrol ARL than that of the synthetic D chart. The synthetic S chart performs slightly better than the synthetic D chart.
Table 4 and Table 5 present the ARL profiles of the synthetic S and synthetic D control charts for increase in the process standard deviation when underlying process data follows t distribution and Weibull distribution respectively each with inontrol ARL = 200, * 1.2 and sample sizes n = 5, 8
and 10.
Table4. ARL comparison of synthetic charts under t distribution.
Shift 
n = 5 
n = 8 
n = 10 

Syn S k =2.0535 L = 18 
Syn D k =2.10 L = 17 
Syn S k =1.78138 L = 12 
Syn D k =1.771 L = 12 
Syn S k =1.746 L = 12 
Syn D k =1.6706 L = 12 

1.0 
200 
200 
200 
200 
200 
200 
1.1 
83.25 
74.89 
73.64 
63.31 
70.28 
56.08 
1.2 
39.25 
34.84 
31.79 
24.73 
28.47 
21.18 
1.3 
21.71 
18.91 
16.11 
12.64 
14.09 
10.16 
1.4 
13.51 
11.88 
9.42 
7.50 
8.04 
5.99 
1.5 
9.29 
7.96 
6.30 
4.97 
5.32 
4.08 
2.0 
3.18 
2.93 
2.12 
1.91 
1.85 
1.63 
Table5. ARL comparison of synthetic charts under Weibull distribution.
Shift 
n = 5 
n = 8 
n = 10 

Syn S k =1.7678 L = 18 
Syn D k =1.8475 L = 17 
Syn S k =1.5648 L = 12 
Syn D k =1.5935 L = 12 
Syn S k =1.5014 L = 12 
Syn D k =1.5152 L = 12 

1.0 
200 
200 
200 
200 
200 
200 
1.1 
47.51 
44.32 
36.61 
33.76 
32.03 
29.07 
1.2 
18.05 
16.44 
12.19 
11.23 
9.97 
9.14 
1.3 
9.28 
8.63 
5.91 
5.68 
5.00 
4.54 
1.4 
5.98 
5.61 
3.84 
3.52 
3.14 
3.01 
1.5 
4.35 
4.07 
2.76 
2.65 
2.34 
2.25 
2.0 
1.88 
1.85 
1.36 
1.33 
1.22 
1.19 
From Table 4 and Table 5, when underlying process distribution is heavy tailed and skewed we observe that, the synthetic D chart consistently produces smaller outofcontrol ARL values than that of the synthetic S chart for entire range of shifts in the process standard deviation.
Conclusions
In this paper, a synthetic D chart based on Downtons estimator is studied under nonnormality of process data. By comparing synthetic S and synthetic D charts when process distribution is heavy tailed or skewed, we found that the synthetic D chart detects the shifts in process standard deviation
quicker than the synthetic S chart. Hence for nonnormal processes the synthetic D chart is uniformly better than the synthetic S chart.
References

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