1. Introduction
Statistical process control (SPC) is a statistical method application, which implies not only monitoring the status of a process, but also the ability to maintain process stabilization by distinguishing between common causes and assignable causes of variation in a process (Montgomery [
1]). The control chart was first introduced by Walter A. Shewhart in the 1920s and widely used graphical tools in SPC, for monitoring production processes and improvement of process quality. For sophisticated production processes, the Shewhart control chart is insensitive in detecting small process shifts. To cope with this problem, Roberts [
2] introduced the exponentially weighted moving average (EWMA) chart to improve the poor detection ability of the Shewhart control chart when small process shifts are of interest. Subsequently, various and extended EWMA charts are developed to deal shifts in process mean, process variability, and both. Such as:
- (I)
Crowder [
3], Ng and Case [
4], Lucas and Saccucci [
5], Steiner [
6], and Sheu and Lin [
7] developed extended EWMA charts to improve the performance of detecting small process mean shifts.
- (II)
Crowder and Hamilton [
8], Castagliola [
9], Shu and Jiang [
10], Huwang et al. [
11], and Sheu and Lu [
12] designed various EWMA charts to monitor small process variability shifts.
- (III)
Sweet [
13] and Gan [
14] developed two EWMA charts, one to detect mean shifts and the other to detect changes in variability.
The two EWMA charts are applicable when there is insufficient information regarding the effect of assignable causes on the process. However, the use of such charts is time-consuming, and may cause increased costs. A single EWMA chart has therefore been developed for easy use, and reduced time and costs. Domangue and Patch [
15] developed an omnibus EWMA chart to jointly monitor changes in location and spread. Chao and Cheng [
16] proposed the semicircle (SC) chart for detecting changes in the process mean and variability. Chen and Cheng [
17] utilized maximum absolute values of the standardized mean and standard deviation to construct a single Max chart to monitor both the process mean and dispersion. Xie [
18] presented various types of single EWMA charts such as EWMA-Max, MaxEWMA, EWMA semicircle (EWMA-SC), and sum of squares EWMA (SSEWMA) charts. Chen et al. [
19,
20] extended the work of Xie [
18] on the MaxEWMA and EWMA-SC charts, respectively. Teh et al. [
21] proposed a sum of squares double EWMA (SS-DEWMA) chart to improve the performance of the SSEWMA chart in terms of the average run length (
) and standard deviation of the run length (
) criteria.
In recent years, there is a wave of interest in developing auxiliary information-based (AIB) charts in the SPC field. AIB charts are mainly based on accurate estimators, which are estimated through auxiliary information (auxiliary variables) along with the quality characteristic (study variable), and attempts to enhance the detection ability of those not using the auxiliary information. Riaz [
22], Ahmad et al. [
23], and Riaz [
24] developed AIB-Shewhart mean charts using regression estimators, ratio-type, and location estimators, respectively, for monitoring process mean shifts. Similarly, Riaz and Does [
25] developed the AIB-Shewhart dispersion chart by using a ratio-type variance estimator. Like as the AIB-Shewhart charts have increased the performance of the original Shewhart chart. As for EWMA-type charts for monitoring small process shifts, Abbas et al. [
26] proposed the AIB-EWMA chart, and showed that the proposed chart is more sensitive than both, the classical EWMA chart and the AIB-Shewhart chart in detecting small process mean shifts. Haq [
27] utilized regression estimators to develop an AIB-EWMA chart to monitor increases and/or decreases in the process dispersion. For more work on the AIB charts, we refer to Haq and Khoo [
28], Abbasi and Riaz [
29], Riaz et al. [
30], Haq and Abidin [
31], Haq et al. [
32], and Chen and Lu [
33]. Most AIB charts mentioned are designed for separately monitoring the process mean or variability shifts. Haq [
34] first introduced the AIB-MaxEWMA chart for simultaneously detecting both increases and decreases in the mean and/or dispersion of a process. Simulations revealed that the AIB-MaxEWMA chart performs uniformly better than the MaxEWMA chart in detecting all kinds of shifts in both, the process mean and dispersion.
In this paper, a single correlated auxiliary variable is used to develop a novel SSEWMA chart, namely the AIB-SSEWMA chart, for simultaneously monitoring the process mean and/or variability of a normally distributed process. Numerical simulations are evaluated by run length profiles in terms of and . As expected, the AIB-SSWMA chart not only improves the detection ability of the SSEWMA chart, but also outperforms the competitive AIB-MaxEWMA chart.
The rest of the paper is organized as follows: The SSEWMA chart is reviewed in
Section 2. The AIB-SSEWMA chart is proposed in
Section 3.
Section 4 evaluates the performance of the AIB-SSEWMA charts in terms of their initial-state
and
values.
Section 5 compares the proposed chart with the competitive AIB-MaxEWMA chart. A numerical simulation example is presented in
Section 6. Finally, significant conclusions are presented in
Section 7.
2. A Review of the Sum of Squares EWMA Chart
Xie [
18] proposed a sum of squares EWMA chart, called the SSEWMA chart, which not only combines two EWMA statistics into a single variable chart to jointly monitor process mean and/or variability shifts, but also identifies the sources and direction of the shift. Herein, the SSEWMA chart is applied, as described by Xie [
18].
Let be a quality characteristic of interest. It is assumed that has a normal distribution with mean and standard deviation , that is, , where and indicate that the process is in control; otherwise, the process has changed or drifted.
Suppose
,
, and
is the measurements of the variable
arranged in groups of size
at time
. Let
and
denote the sample mean and sample variance, respectively. Then
,
are independent normal random variables with mean
and variance
;
and
, are independent chi-square random variables with
degrees of freedom, and
and
are mutually independent random variables for an in-control process. Defining the following two transformed statistics:
and
where
and
represent the inverse standard normal distribution function and the chi-square distribution function with
degrees of freedom, respectively.
Both
and
are independent standard normal random variables when the process is in control, and the distributions of
and
are both independent of the sample size
. Two EWMA statistics each for the mean and variance can be defined as:
and
where
are the starting values of
and
, respectively. It is known that
and
are independent because
and
are independent, and when the process is in control and
,
and
, where
.
The statistic of the SSEWMA chart, say
, is defined as:
Since
is nonnegative, the initial state of the SSEWMA chart requires only an upper control limit
, which is given by:
where
and
are the mean and variance of
, respectively. To achieve the desired in control
, the corresponding control limit constant
and the smoothing parameters
are determined. The process is considered to be out of control whenever
exceeds
and some action should be taken to identify and eliminate the assignable cause of the process. Xie [
18] provides a simple and quick computation formula for
, given as follows:
The process is considered to be out of control when the statistic
exceeds
, i.e., when the pair
plots outside the circular control region centered at (0, 0) with a radius of
. To avoid the problem of concentric circles, Xie [
18] plots
on a circular control region centered at (0, 0) with a radius
of
, where
,
, and
.
3. The Proposed AIB-SSEWMA Chart
Assume that there exists a corresponding auxiliary variable that accompanies quality characteristic of interest. Let follow a bivariate normally distributed process with means and variances . Let be the correlation between and , that is, , where and are magnitudes of process mean and variability shifts, respectively. Suppose , , is a random sample of size taken from the process at time , for . It is assumed that the underlying process parameters are known for phase II monitoring. The sample means and variances based on and , respectively, are:
, and ,
, and
.
Assume that the underlying process is in-control (
and
); then, following Haq and Khoo [
28], the regression estimator of
is:
with its mean and variance given by
We defined the following statistic for estimator
:
where
is a standard normal random variable, that is,
when the process is in-control.
Moreover, it is known that
and
follow a chi-squared distribution with
degrees of freedom when the process is in-control. Then, for an in-control process,
and
have a standard normal distribution, where
is the distribution function of the standard normal distribution. Following Haq [
34], the difference estimator of
is defined by:
where
is the correlation coefficient between
and
. Hence, the mean and variance of
are:
We defined the following statistic for estimator
:
where
is also a standard normal random variable, that is,
, when the process is in-control.
Both
and
are independent standard normal random variables when the process is in control. Two EWMA statistics,
and
, can be defined from
and
, as follows:
and
where
and
are the respective starting values, and
is a smoothing constant. Note that
and
are also independent because of the independence of
and
. When the process is in control and
, we had both
and
, where
.
The AIB-SSEWMA statistic based on
and
is defined as:
Since
is a nonnegative quantity, the initial state of the AIB-SSEWMA chart only requires an upper control limit
, which is given by:
where
and
are the in-control mean and variance of
, respectively. The control limit constant
and the smoothing parameter
are determined by the desired in control
. In order to speed up the computation of the control limit
for the initial state AIB-SSEWMA chart, the derivation of
and
are shown in the
Appendix A, and represented as follows:
The computation of
for the AIB-SSEWMA chart is similar to that of the SSEWMA chart by Xie [
18]. It is noted that when correlation coefficient
, there does not exist a single correlated auxiliary variable
, that is, the estimators of the mean and variance are only estimated by the study variable
. Then, the AIB-SSEWMA chart reduces to the SSEWMA chart, that is, they are identical when
.
The AIB-SSEWMA chart initiates an out-of-control signal when the statistic
is above the
as if when the pair
plots outside the circular control region centered at (0, 0) with a radius of
. One advantage of using the consecutive pairs of
plotted on a chart instead of the corresponding statistic
is that it can clearly indicate the source and direction of a process shift. To avoid drawing concentric circles, let
Each sample point is plotted on coordinates with circular control region centered at (0, 0) and radius . When a sample point deviates sufficiently from the -axis, it indicates that the shift is due to the process mean. Similarly, a sample point that deviates sufficiently from the -axis shows a shift in process variance. A sample point that is close to either the line or , indicates that the change is likely caused by a shift in both, the process mean and variance.
The steps involved in constructing the AIB-SSEWMA chart for simultaneous monitoring of the process mean and variability are summarized as follows:
- (1)
Choose a suitable combination based on the values of sample size and desired in-control .
- (2)
Use Equation (18) to compute the control limit, .
- (3)
Compute the values of , , , , and for each sample by using as the starting values.
- (4)
The consecutive pairs of are computed by Equations (19) and (20), and plotted with coordinates . The circular control region is centered at (0, 0) and the radius is .
- (5)
Check if any point
falls outside the circular control region given in Step (4). If an out-of-control point is detected, identify the source and direction of the shift corresponding to the position on the chart. Simple criteria and symbols are described in
Table 1, and also depicted in
Figure 1 for easy identification of the source and direction of an out-of-control signal. The symbols in
Table 1 are defined as follows: “
” and “
” indicate an increase and a decrease in the process mean, respectively; “
” and “
” indicate an increase and a decrease in the process variance, respectively; “
” indicates a simultaneous increase in the process mean and variance. Similar interpretations apply to the other three cases: “
”, “
”, and “
”.
- (6)
Examine each of the out-of-control points.
4. Evaluation and Performance Comparison
The performance of a control chart is generally measured in terms of its and . An in-control is expected to be sufficiently large to avoid frequent false alarms. However, when a present assignable cause displaces the process parameters, it results in an out-of-control process. Meanwhile, the out-of-control and need to be sufficiently small to rapidly detect the process mean and/or variability shifts. A control chart with a smaller out-of-control and for a given shift is generally indicated to have superior performance.
The and values of the initial state SSEWMA and AIB-SSEWMA charts are computed through Monte Carlo simulations. In this study, the use of exact limits, instead of asymptotic limits, enables the initial state charts to quickly detect initial out-of-control signals effectively. An algorithm in R is developed to calculate and values, which are an average of 50,000 run lengths. Without the loss of generality, the random samples , , for , are drawn from a bivariate normal distribution, that is, , where and are magnitudes of process mean and variability shifts, respectively.
Suppose
and
, when
and
indicate that the underlying process is in-control. Subsequently the charting multiplier
and charting parameter
are adjusted to achieve the desired in-control
.
Table 2 presents the
values for the initial state AIB-SSEWMA chart. The specific charting parameter of
= {0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, correlation coefficient values of
= {0.00, 0.25, 0.50, 0.75, 0.95}, and sample size
= {5, 10} are considered in
Table 2 to achieve an in-control
of approximately 370. Note that
is the correlation coefficient value between
and
in Equation (11), which is determined by
n and
Haq [
27] computes the values of
for different choices of
and
through Monte Carlo simulations. The partial pair values of
from Haq [
27] are (0.25,0.05639), (0.50,0.22933), (0.75,0.53136), and (0.95,0.88808) for
; they are (0.25,0.05980), (0.50,0.24084), (0.75,0.54947), and (0.95,0.89736) for
.
From
Table 2, we observed that a small sample size
needs a large charting multiplier
to achieve the desired in-control
, which was obviously larger for
. For a fixed value of
, the values of
increased as the smoothing parameter
increased. For a fixed value of
, the values of
increased when
at
; however, they decreased when
at
. Moreover, the difference between
was large for a larger
value. For example, when
, the value of
at
for
= 0.05 and 1.0 were 3.533 and 4.909, respectively; however, those at
are 3.544 and 5.503, respectively. Similar results were found regarding the large sample size
in
Table 2.
When and/or , the underlying process containing assignable causes results in an out-of-control process. The initial state AIB-SSEWMA charts are maintained at the desired in-control , and the out-of-control , and were evaluated for specific process shifts. In this study, we considered the process mean shifts = {0.00, 0.25, 0.50, 1.00, 1.50, 3.00}, and the process standard deviation shifts = {0.50, 1.00, 1.25, 1.50}. For each shift combination of under different scenarios , the out-of-control and of the initial state AIB-SSEWMA chart with sample size were calculated under the corresponding parameter combination of , where the smoothing constant = {0.05, 0.10, 0.25} and correlation coefficient = {0.00, 0.25, 0.50, 0.75, 0.95} were investigated to match the desired in-control value by adjusting the control limit constant .
Table 3 and
Table 4 show the
combinations and the corresponding out-of-control
and
of the AIB-SSEWMA charts with in-control
at approximately 185 and 370, respectively. In particular, the AIB-SSEWMA chart with
was identical to the SSEWMA chart proposed by Xie [
18]. That is, the SSEWMA chart is a special case of the AIB-SSEWMA chart when
.
To ensure the accuracy of the simulation algorithm, the following example where
was used to compare our numerical results of the AIB-SSEWMA chart with those of the SSEWMA chart by Xie [
18]. For
, the control limit constants
of both charts at in-control
of 370 were 3.533. This parameter combination gives the out-of-control
and
of the initial state AIB-SSEWMA chart at
and
of 9.28 and 7.49, respectively (cf.
Table 4). Meanwhile, the out-of-control
and
for the same parameter combination from Xie [
18] were 9.28 and 7.49, respectively, which were similar to our simulated results.
From
Table 3 and
Table 4, it was observed that the performance of both the SSEWMA chart and the AIB-SSEWMA chart was better when detecting different shifts with smaller
. Moreover, the AIB-SSEWMA chart performed uniformly better than the SSEWMA chart, which was obvious in detecting the small process mean and/or variability shifts. However, they performed almost equally in detecting large process shifts. As the value of
increased, the effectiveness of the AIB-SSEWMA chart in detecting different kinds of shifts increased. For example, considering the small process shifts
and
maintained at in-control
; then, the out-of-control
values at
for
= 0.00, 0.25, 0.50, 0.75, and 0.95 were 9.28, 9.08, 8.21, 5.97, and 2.19, respectively.
This section investigated the diagnostic abilities of the proposed AIB-SSEWMA chart against the existing SSEWMA chart. For each shift combination of
= {0.00, 0.25, 0.50, 1.00, 1.50, 2.00, 3.00} and
= {0.50, 0.75, 1.00, 1.25, 1.50, 3.00}, 1000 out-of-control signals were simulated for both charts. The guidelines to count the number of out-of-control signals triggered by both charts have been reported by Haq [
34]. According to
Table 4, we considered the sample size
at in-control
; the parameter combinations of
were (0.05, 3.533) and (0.05, 3.534) used for the SSEWMA chart when
and AIB-SSEWMA chart when
, respectively.
Table 5 presents the number of out-of-control signals, which shows that the AIB-SSEWMA chart had better diagnostic abilities than that the SSEWMA chart. For example, in the case of an upward shift, there existed both process mean shift
and variability shift
. The SSEWMA and AIB-SSEWMA charts triggered 323 and 421 out-of-control “
” signals out of 1000 signals, respectively.
6. Illustrated Example
A simulated dataset was used to illustrate the implementation of the existing SSEWMA and proposed AIB-SSEWMA charts. The monitored quality characteristics and the auxiliary variable of the underlying process,
,
, and
, followed a bivariate normal distribution with
,
, and specific
. In this study,
was designed for the SSEWMA chart, and
was considered for the AIB-SSEWMA chart. Then, we generated 30 bivariate samples of five observations each, from a bivariate normal distribution. The first 10 samples were from
and
, referred to as in-control for the SSEWMA chart and AIB-SSEWMA chart, respectively. Assume that the underlying process mean shifts from
to
, and the process standard deviation changes from
to
due to some assignable causes. Shifts of
in the process mean and
in the process variability were considered for the last 20 samples.
Table 7 lists the 30 successive pairs of
representing the SSEWMA statistics when
; similarly, the pairs of
represent the AIB-SSEWMA statistics when
.
To compare the performance of the sum of squares EWMA chart with and without the use of an auxiliary variable, the in-control
of the two charts were maintained as 370. According to
Table 4, the parameter combination
of the SSEWMA chart with time-varying control limits was selected as (0.05, 3.533) when the specific shift combination
was (0.25, 1.25). When the auxiliary variable was available, the parameter combination
of the AIB-SSEWMA chart with time-varying control limits was selected at (0.05, 3.534) using the same shift combination for a fair comparison. The corresponding radius
of the SSEWMA chart and
of the AIB-SSEWMA chart are also listed in
Table 7 and illustrated in
Figure 2.
Figure 2a,b plots the SSEWMA and AIB-SSEWMA charts, respectively, with the 30 samples from
Table 7. The two charts show that the process remained in control for the first 10 samples. However, when an assignable cause produced small shifts in both the process mean and variability, the first out-of-control signal was detected with the SSEWMA chart on sample 24, while 20 samples were only required to enable the AIB-SSEWMA chart to signal the out-of-control. The remaining out-of-control samples of the AIB-SSEWMA chart in
Figure 2b deviated far from the upper part of the
-axis and the right side of the
-axis, as they were shifted due to a joint increase in the process mean and variability. Similar results also occurred in the SSEWMA chart. Moreover, if the deviation of an out-of-control sample from the
or
axes was not obvious,
Figure 1 provides a guide to identify the direction and source of a shift in the AIB-SSEWMA chart. For shifts in the process mean
and variability
, the illustration results indicate that the AIB-SSEWMA chart was more sensitive than the SSEWMA chart in detecting small process shifts.