Control charts are widely used

tools of statistical quality control in industrial environments since its

inception by Shewart in 1920’s. The major function of control charting is to

detect the occurrences of assignable causes so that the necessary corrective

action may be taken before large quantity of non conforming product is

manufactured. A survey conducted by Saniga and Shirland (1977) shows that on

continuous measurement scale the control chart for averages dominates the use

of any other control chart technique. All control charts have a common

structure. A plot of the result of repeated sampling is made on a vertical

scale against the number of samples plotted horizontally. The center line of

the chart represents a long term average of the process statistic or its

standard value. The upper control limit (UCL) and Lower control limit (LCL)

represent the boundaries of typical statistic variation. The process call for

adjustment if the points fall outside the control limits. Departures from

expected process behavior within the limits (non random patterns on the chart)

can be detected by using different run tests for pattern recognition (Nelson

(1985)). On using control charts two kinds of errors may occur: over adjustment

and under adjustment. Uncertainty of inferences based on sampling statistic is

the major cause for these errors. The magnitude of the errors depends on the

decision-making method. It is beneficial that a control chart detect process

change quickly so that the causes of any undesirable changes can be identified

and removed. It is also beneficial that the rate of false alarms generated by

the control chart be low in order to maintain the confidence of process

operations in the chart. Sampling cost will be an issue for most of the

applications, thus it is important that a control chart be able to provide fast

detection of process change and a low false alarm rate with a reasonable rate

of sampling. So the statistical performance of a control chart is often

evaluated by considering, for a given false alarm rate and sampling rate, the

expected time required by the chart to detect various process changes. It has

been found in recent years that the statistical performance of control charts

can be improved considerably by changing the rate of sampling as a function of

the data coming from the process. The basic idea is that whenever there is an

indication of a problem with the process the sampling should be more intensive

and less intensive when there is no indication of a problem. There are many

ways in which the sampling rate can be varied as a function of process data.

One of the ways is to vary the sampling interval: a short sampling interval is

used when there is a indication of a problem and a long sampling interval is

used when there is no indication of a problem. The resulting variable Sampling

interval (VSI) control charts have been studied broadly (see, e.g., Reynolds et

al (1988) ; Zee (1990), Runger and Pignatiello (1991); Baxley (1996); and

Reynolds (1996a, 1996b). when there is an indication of a problem a large

sample size is used and a small sample size is used when there is no indication

of a problem. Variable sampling size (VSS) have been examined in various papers

(see ,e.g., Prabhu et al. (1993); Costa (1994), Park and Reynolds (1994),

Reynolds (1996b), Rendtel (1990), Arnold et al. (1993), Prabhu et al. (1994,

1997), and Arnold and Reynold (1994). Variable sampling Rate (VSS) is a control

chart that varies the sampling interval and the sample size. Tagaras (1998)

gave a survey on VSR chart. Most studies of VSR charts have compared VSR and

FSR charts by fixing the false alarm rate and the average sampling rate and

then comparing the expected time required to detect various shifts in the

process parameter of interest. The conclusion of this comparative study shows

that the VSR chart will detect small and moderate shifts much quicker than the

FSR chart. For very large shifts the VSR feature is not helpful to lessen the

detection time and almost appropriate and handy. However, they do not attempt

to specify the benefits of VSR feature in an economic sense. Even though a VSR

chart will have more design parameters than an FSR chart, some of the issues

associated with chart design are the same for the two charts. In particular,

there is the question of how to choose the design parameters of the chart to

achieve a reasonable balance between the cost of sampling, the cost due to

false alarms, and the cost of not detecting process changes. If an FSR chart is

already in use in a particular application, then the VSR chart for this

application could be setup to have the false alarm rate and the same average

sampling rate as the FSR chart. This approach mostly used to compare VSR and

FSR charts would result in faster detection of most process shifts.

Alternatively, the VSR chart could be set up to provide approximately the same

detection ability as the FSR chart but with reduced sampling cost. Baxley

(1996) and Reynolds (1996a) examine the use of VSI charts to reduce sampling

costs. Reduction of the problem rank for univariate cases by plotting a

statistic depending on both process mean and variance have been considered by

many authors. Reynolds and Glosh (1981)

proposed plotting a statistic representing the squared standardized deviations

of the observations from the target value. Monitoring the value of this

function is also discussed by Derman and Ross (1994). Surely, these charts are

not studied for evaluation of process study but its uniformity that can be

characterized by a quality less occurring when the process deviates from its

desired value and generates non-uniform products. Unfortunately, these charts

are incapable of distinguishing a shift in the mean from a increase in

variance. There is a close connection between control charts and hypothesis

testing. In a sense, a control chart is a test of the hypothesis that the

process is in a statistical control. A point plotting within the control limits

is equivalent to failing to reject the hypothesis of statistical control, and a

point plotting outside the control limits is equivalent to accepting the

hypothesis of statistical control (Montgomery, 1997). In other words the aim of

quality monitoring is to test the null hypothesis

:

? = 0 (out-of-control state of the

process) against the alternative hypothesis

😕 ? 0 (out-of-state of the process) (Pacella

and Sameraro (2007)), where ? represents the mean shift. This hypothesis

testing framework is useful in many ways, but there are some differences in

viewpoint between control charts and hypothesis testing. There are at least two

remarkable differences between the SPC techniques and hypothesis testing.

First, the rejection of null hypothesis must be supported by a follow-up

investigation to identify assignable causes, where as the common procedure of

hypothesis testing permits rejection on the basis of comparison of the test statistic

with the test critical value only. Secondly, a null hypothesis in SPC can be

rejected not only for salient points on a control chart but also due to non

random nature of the received data, where as hypothesis testing does not imply

any analysis of data patterns. Despite, many specialists admits that “a Shewart

control chart is equivalent to applying a sequence of hypothesis tests (

Chengular et al. (1989)). The centre line represents hypothesized mean value of

the process parameter, the control limits represents the critical values of the

two-sided test for the null hypothesis acceptance region, and each point

represents the critical values of the two -sided test for the null hypothesis

acceptance region, and each point represents a test value for the given sample.

The quality of the product depends upon the combined effect of various quality

characteristics. These characteristics may be correlated among themselves for

example, quality of linen thread, depends upon diameter and breaking strength

of the thread, which are related to each other. Hence, the use of some

available process capability measures separately for each of the quality characteristics

may not yield an adequate idea about the overall process capability. SPC is a

well known and widely publicized tool which is employed in perseverance of

products specification compliance through the regulation of the production

process. SPC is generally affected by external intervention, unlike, automatic

process control (APC) which is generally a closed loop and online sub-system of

the process. Control charts play a prominent role in SPC applications and

although they originated in applied engineering sector, their use has since

spread other areas of production and processing, such as, chemical industry.

Another less conventional control chart is that of missionary condition

monitoring published by Raadnui (2000). The control chart is conventionally

demarcated by upper control limits (UCL) and lower control limits (LCL),

Kreysig (1972), which are often symmetrically placed about a central value

line. If a system is under control it means that its output is statistically

uniform and the product is all from the same universe, Davis (1958). This does

not mean that the product mostly compiles with its quality specifications. The

quality will be in terms of a critical parameter target value (with a tolerance

range), typically, a dimension, strength, weight or electrical resistance, for

example, the observed average of the output samples may not coincide with the

target value or even if it does, the scatter in the values may be too high. The

level of process control must match the quality specifications. If the former

type of fault is noted, it may be corrected by process adjustment under SPC or

APC but if latter type of fault occurs it may entail more fundamental action

with offline intervention, perhaps up to system redesign. An alternative is to

amend the level of control to match specifications. In any application of SPC

the hypothesis being tested is that any detected variation in the data is only

due to stochastic process. Two types of variations may be observed in sample

data: 1) Chance variations, these are due to assignable events within the

process or small variations in environmental conditions, in inputs or in

operator actions. 2) Assignable variations, these are due to accountable causes

and usually produce identifiable patterns on a control chart. For example, they

are produced by deterioration, mechanical faults, a change in source of raw

material or process operator fatigue. The effect of both types of variation are

additive. In the assignable chance cause category, part of this may be due to

random effects that can be termed ‘noise’. It is well known that integration

reduces the effect of signal noise and, similarly, the effect of averaging

sample value of, say, 5-10 production units will assist in mitigating the

effect of random or periodic variations. Clearly, the larger the sample size,

the greater the reduction, but subject to control system and economic

efficiency. SPC cannot be applied to chance causes. In the case of non random

variations in the system output, the identification and classification of

causes are important. In the application of control charts, Davies (1958) has

emphasized that each case needs to be considered in detail and treated on its

own merits to determine the appropriate form of control chart. This is also

true in wider context of SPC–APC applications. For any control systems of any

type, the time scale of control response must be of smaller order of magnitude

than that of production process for it to be effective. For SPC intervention,

the process may need to be stopped, but APC intervention is normally online. The

SPC concepts outlined above are mainly concerned with the observation of output

variable data, where individual measurements are made on specimen samples. The

back theory is based upon a normal probability distribution. If the data are

from number of defective items per output batch, the theory is based upon

Binomial distribution, and for number of defects per unit (as in metal, plastic

sheet), the background theory is based upon poison distribution, davies (1985).

In the first case, it is known as analysis by variables, whereas in the latter

case it is termed as analysis by attributes, Riaz (1997).

The

power of any test of statistical importance is defined as the probability that

it will reject a false null hypothesis. Statistical power is affected mainly by

the size of the effect and the size of the sample used to identify it. It is

easy to detect the larger effects as compared to smaller effects, while large

samples offer greater test sensitivity than small samples. In order to analyze

the minimum sample size required, power analysis can be used so that one can be

practically liable to detect an effect of a given sample. In order to compare

different statistical testing procedures, the concept of power can be used: for

example, between a parametric and a nonparametric test of the hypothesis. Correlation

is a statistical technique that can show whether and how strongly pairs of

variables are related. Correlation is bivariate analysis that measures the

strength of relationship between two variables and the direction of the

relationship. Usually, for Pearson’s correlation it is assumed that the

variables under study should be normally distributed. Other assumptions may

also include linearity and homosedasticity which expects linear relationship

between each of the variables and the data is normally distributed about the

regression line. A basic assumption made in most traditional applications of

control charts is that the observations from the process are independent. When

the mean of observations is being monitored, the mean is assumed to be constant

at the target value until a special cause occurs and produces a change in the

mean. However, for many processes, e.g., in the chemical and process

industries, there may be correlation between observations that are closely

spaced in time. For FSI chart correlation is not a serious problem but it may

become problematic for a VSI chart because some of the observations will be

taken using a relatively short sampling interval. The effects of correlated

observations on the performance of FSI control charts have been studied by

several authors. Goldsmith and Whitfield (1961); Bagshaw and Johnson (1975),

Harris and Ross ( 1991); Yaschin (1993), and Van Brackle and Reynolds (1994)

investigated the effect of correlation on EWMA charts. Vasilopoulous and

Stamboulis (1978); Maragah and Woodall (1992) and Pedgett et al. (1992)

investigated

charts with correlated observations, where the

control limits are estimated from data. Alwan (1992) investigated the

capability of standard control charts for individual observations to identify

special causes which are reflected as isolated extreme points in the presence

of correlation. Chou et al. (2001) studied economic design of

charts for non-normally correlated data. Liu

et al. (2002, 2003) examined the minimum- loss design of

chart for correlated data; they also

attempted to study the effect of correlation on the economic design of warning

limit

charts.

Optimal design of VSI

control charts for monitoring correlated

samples were discussed by Chen and Chiou (2005). Chen et al. (2007) studied the

economic design of VSSI

control charts for correlated data. An

economic design of double sampling

charts for correlated data were studied by

Torng et al. (2009). On examining these studies it may be concluded that if the

control charts are applied under correlated observations the results can be

misleading. In particular, there may be more frequent false alarms when the

process is in control, and the detection of out-of control situations may be

much slower than expected. In many situations were there is correlation between

the observations, assuming the process mean to be constant is not practical

until a special cause occurs. It may be more practical to assume that the

process mean is continuously wandering even though no special cause can be

identified. For some situations of this kind, the purpose of process monitoring

may be the application of some kind of engineering feedback control method in

addition to the detection of special causes (Macgregor (1990)). In order to

detect the special causes in correlated observations, several authors have

studied the alternative to a control chart which plots the original

observations. Two general monitoring approaches are recommended in order to

deal with this problem. First approach is to fit time series model to the data,

and then apply traditional control charts such as Shewart, EWMA

(exponentially-weighted moving average) and CUSUM (cumulative sum) control

charts to the residuals from the time series model. Another approach is to use

traditional control charts to monitor correlated observations with modified

control limits. Alwan and Roberts (1988) through their study shows that using

residuals from the time series model (ARIMA) may be appropriate if the correct

time series model is known and then apply the traditional control charts since

the residuals of time series model of correlated process are independently and

identically distributed with mean zero and variance ?2. Harris and

Ross (1991) fit a time series model to the univariate observations, and then

investigate the correlation effect on the performance of CUSUM and EWMA charts

by using residuals. Montgomery and Mastrangelo (1991) shows that the

Exponentially Weighted Moving Average (EWMA) may be useful for correlated data

by applying control charts to the residuals of time series model; Wardett et

al. (1994) reflects the ability of EWMA charts to detect the shift more quickly

than individual Shewart chart when the correlation is based on an ARMA (1,1)

model. They also suggest that residual charts are not sensitive to small

process shifts. EWMA control charts were studied by Lu and Reynolds (1995) to

monitor the mean of correlated process. They suggested that for the low and

moderate level of correlation, a Shewart control chart of observations will be

better at detecting a shift in the process mean than Shewart. They found that

when there is high correlation in process, control charts based on estimated

parameters mused be used. To monitor multivariate process in the presence of

correlation Pan and Jerret (2004) propose using vector autoregressive model

(VAR) by using residuals of the model. Pan and Jarret (2007) extended Alwan and

Robert’s approach to multivariate cases using residuals from a VAR on

Hotellings T-square control charts to monitor the multivariate process in

presence of correlation. Brain Hwang and Yu Wang (2010) proposes a Neutral

Network Identifier (NNI) for multivariate correlated processes.

In this chapter an attempt has been

made to examine the power of

chart when the assumption of independence of

data is violated. Expressions for the power of

chart are derived for different values of correlation

coefficient and for sample size. It is assumed that the process has a normal

distribution with mean µ and variance ?2 (known). We further assume

that at the time of determining the control limits the process is in a

statistical control.

2.2

Power of

Chart in presence of data correlation

In this development it is assumed that the

process has a normal distribution with mean µ and variance ?2. It is

further assumed that at the time of determining the control limits the process

is in a statistical control, and the same device is used as will be employed

for latter measurements. We further assume that the observations came from the

normal population and the observations are correlated or we can say that the

assumption of independence is violated in this case. Thus the data used for

establishing the limits on the control chart comes from a process that is N(µ,

?2). When the process shifts, the data is assumed to come from a N(

, ?2T2) population. If samples

of size n are taken from the population N(

, ?2T2) and the value of

is plotted with control limits of µ ± 3

,

the power of detecting the change of process is given by the following formula:

. (2.1)

Converting to a standard normal distribution,

we have

. (2.2)

Where

,as we assume that the data comes from the normal

population and the observations are dependent. Thus the variance of

under the correlated data is given by

,

=

,

(2.3)

where

,where s2

is

assumed to be known, r is the correlation coefficient and n is

the sample size.

Now assuming the new variable, equation

(2.1) can be written as:

, (2.4)

=

,

=

,

=

,

=

,

= F

+

F

, (2.5) where

, is the change of process average

and

F(z) =

).

3.

Numerical Illustrations and Results:

For the purpose of illustrating the effect of

correlation on the power of

chart, we have determined the values of

power function

for independent observations (r

=

0) and for different values of correlation coefficient r.

The values of power function have been calculated and the results are presented

in Table -2.1, Table- 2.2 for n = 5 and for n= 7 respectively. In order to give

visual comparison of the power functions for different values of correlation

coefficients r and process average d, a curves have been

drawn and shown in Fig. 2.3 and Fig. 2.4 for n = 5 and n = 7, which illustrates

the relationship between the change of process average d and the power of

detecting this change

in presence of correlation. The power depends

upon the magnitude of process change.