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MOVING AVERAGE AND DOUBLE MOVING AVERAGE CONTROL CHARTS FOR PROCESS VARIABILITY USING AUXILIARY INFORMATION
Vikas Ghute1 and Sarika Pawar2
^Department of Statistics Punyashlok Ahilyadevi Holkar Solapur University Solapur (MS)-413255, India [email protected], ssgulame@ gmail.com
1Corresponding Author
Abstract
The memory type control charts based on auxiliary information have been introduced in the literature for improved monitoring of the process parameters for normally distributed process. In this paper, we design moving average and double moving average control charts based on auxiliary information for efficient monitoring the shifts in the process variability. Regression estimator of process variance in the form of auxiliary and study variables is considered to construct charting statistics for the proposed charts. The average run length (ARL) and standard deviation of run length (SDRL) performance of the proposed charts is investigated using simulation study and is compared with the originally proposed Shewhart control charts based on auxiliary information and without auxiliary information. The proposed auxiliary information based moving average and double moving average charts are found to be efficient for monitoring the process variance of normally distributed process. An illustrative example based on simulated data set is provided to show the implementation of the proposed charts in detecting shifts in the process standard deviation.
Keywords: Control chart, average run length, auxiliary information, moving average, double moving average.
1. Introduction
Statistical process control (SPC) is a powerful statistical technique used to determine the performance of the process accurately. It has been widely used in manufacturing and service industries. A control chart is one of the most widely known tools in SPC which is extensively used to monitor process quality. It is designed to identify and detect timely assignable causes in the process. In general, control chart is used to detect changes in the process parameters. Two types of control charts are generally used to monitor production processes namely the location chart and the variability chart. The location chart is used to monitor process mean and the variability chart is used to monitor process variability. It is a standard practice to use Shewhart X chart for monitoring the process mean and R or S charts for monitoring the process variability. Some practioners recommend a control chart based directly on the sample variance S2 control chart for monitoring process variability. A major disadvantage of Shewhart type control charts is that they use only
Vikas Ghute and Sarika Pawar RT&A, No 3 (74)
MA AND DMA CHARTS FOR PROCESS VARIABILITY USING AIB_Volume 18, September 2023
information of last sample observation and ignores the past information of the process which makes it insensitive to small shifts in process parameters.
Memory type control charts are most commonly used in process monitoring to detect small to moderate shifts in the process parameters. They are constructed using past information regarding the production process and are more sensitive to monitor the small and moderate shifts in the process parameters. These control charts includes cumulative sum (CUSUM), exponentially weighted moving average (EWMA) and moving average (MA). Relative to CUSUM chart, the EWMA and MA charts are quite basic. The EWMA chart uses a weighted average as the chart statistic while the time weighted MA chart is based on simple moving average. The moving average statistics of width w is simply the average of the w most recent observations and are more sensitive to monitor the small and moderate shifts in the process parameters. Wong et al. [1] developed simple procedures for the design of an individual MA chart and a combined MA-Shewhart scheme. Khoo and Yap [2] proposed the use of single MA chart for joint monitoring of the process mean and variance by combining X and S charts into a single chart. Adeoti and Olaomi [3] proposed a moving average control chart based on sample standard deviation for detecting small shifts in process variability. Ghute and Rajmanya [4] developed moving average control chart based on Downton's D statistic and Gini's mean difference G statistic for detecting small shifts in the process standard deviation. The proposed moving average control charts are found to be more efficient for monitoring process variability. In multivariate setup, Ghute and Shirke [5] developed a multivariate moving average T2 control chart for monitoring mean vector of multivariate process. It was shown that the proposed chart performs better than the Hotelling's T2 chart in the detection of small to moderate shifts in the process mean vector. Ghute and Shirke [6] also developed a multivariate moving average |5|control chart for monitoring covariance matrix of multivariate normally distributed process. It was shown that proposed chart performs better than Shewhart-type |S| chart in the detection of small to moderate changes in the process covariance matrix.
The double moving average (DMA) control charts have been proposed in the literature to further improve the performance of moving average (MA) control charts. Khoo and Wong [7] introduced the DMA chart by computing moving averages twice for early detection of small to moderate shifts in the process mean. It was shown that the DMA control chart performs better than the MA chart for the detection of small to moderate shifts in the mean. Adeoti et al. [8] proposed a DMA control chart based on sample standard deviation for detecting shifts in the process variability. Sukparungsee et al. [9] developed a mixed Tukey-Double moving average control chart for monitoring process mean of symmetric and asymmetric processes. They compared ARL performance of the proposed chart with the existing charts. It was shown that the proposed chart is an effective competitor to the existing counterparts. Taboran and Sukparungsee [10] designed a new EWMA-DMA control chart for detecting change of mean of the process with normal, Laplace, exponential and gamma distributions. They compared ARL performance of the proposed chart with other existing charts. It was shown that the proposed chart has best detection ability for certain level of shifts in process mean.
In order to increase the sensitivity of the traditional control charts many new modifications and improvements in the control charting procedure have been suggested in the SPC literature. One of such modifications is the development of auxiliary-information-based (AIB) control charts which have an excellent speed in detecting shift in the process parameters than those based without it. Such control charts are based on a statistic that utilizes information from both the study and auxiliary variables. The information on auxiliary variable is generally known prior to the sampling procedure and it assists in estimating the study variable with increased accuracy. There are many AIB control charts available in literature with Shewhart type and memory type charting structures for monitoring process mean and process variability. Riaz [11] first suggested AIB-
Vikas Ghute and Sarika Pawar RT&A, No 3 (74) MA AND DMA CHARTS FOR PROCESS VARIABILITY USING AIB_Volume 18, September 2023
Shewhart that is based on regression-type mean estimator for monitoring shifts in process mean. It
was shown that the AIB-Shewhart chart using the regression estimator performs better than the
Shewhart chart for detecting shifts in the process mean. Riaz and Does [12] developed a Shewhart-
type variability chart using ratio-type variance estimator for the Phase I quality control. It was
shown that AIB Shewhart variability chart is more powerful than the existing Shewhart variance
chart. Riaz [13] proposed a Shewhart-type control chart for an improved monitoring of mean level
of quality characteristic of interest Y using the information on a single auxiliary characteristic X on
product difference pattern. Riaz et al. [14] suggested new AIB-Shewhart chart based on regression
estimator for monitoring the process variability. They have shown that the Shewhart chart using
regression estimator outperforms the other Shewhart charts when detecting increase in the process
variability. Abbas et al. [15] introduced the EWMA chart with the auxiliary information, using
regression estimator for monitoring location of the process. It was shown that proposed chart
performs better than its existing control charts. Abbasi and Riaz [16] made dual use of auxiliary
information to propose new chart for process location. Riaz et al. [17] made dual use of auxiliary
information to propose a chart for process variability. Sanusi et al. [18] studied CUSUM charts
using different estimators based on auxiliary information. Haq [19] proposed new EWMA charts
using auxiliary information for efficiently monitoring the process dispersion. Recently, Amir et al.
[20] developed auxiliary information based moving average control chart (denoted as AB-MA
chart) for effective monitoring of shifts in the process location parameter. They compared the
performance of proposed chart with existing control chart and shown that chart outperforms in
detecting small and medium shifts in the process location parameter. Amir et al. [21] designed
auxiliary information based double moving average control chart (denoted as ADMA chart) for
effective monitoring of the process location parameter. They compared the performance of the
proposed ADMA chart with its memory-type counterparts and shown that the proposed ADMA
chart performs uniformly better than the EWMA and CUSUM charts when correlation between
auxiliary variable and study variable is high.
Most of the studies on AIB memory-type charts have been concentrated on monitoring process mean. Often, monitoring shifts in the variance of related study variable is also important. The purpose of this paper is to develop auxiliary information based MA and DMA control charts for efficient monitoring of process variability of normally distributed process in Phase II. By getting motivation of improved performance of auxiliary information based MA and DMA charts for monitoring process mean recently proposed by Amir et al. [20] and Amir et al. [21] respectively, in the present paper, we develop new MA and DMA charts using auxiliary information for efficiently monitoring the process variability in phase II. We expect that the proposed MA and DMA control charts will be more sensitive for efficiently monitoring process variability. The regression estimator of the process variance in the form of auxiliary and study variables is considered to construct charting statistics for the proposed MA and DMA charts. The performance of the proposed charts is evaluated in terms of the average run length (ARL) and standard deviation of run length (SDRL) criteria. Monte Carlo simulations are used to study the run length profiles of the proposed AIB-MA-V and AIB-DMA-V charts. The performance of the proposed charts is compared with its Shewhart-type counterparts.
Rest of the paper is organized as follows: In Section 2, traditional S2 chart for monitoring process variability is discussed. The auxiliary information based Shewhart-type control chart for monitoring process variability is presented in Section 3. The design procedure of proposed auxiliary information based moving average and double moving average control charts are presented in Section 4 and Section 5 respectively. In Section 6, a simulation study is conducted to evaluate the performance of proposed control charts in comparison to that of Shewhart-type charts. An illustrative example is presented in Section 7 to demonstrate the implementation of the proposed charts. Finally, Section 8 concludes the findings of the paper.
2. Shewhart control chart for process variability.
In this Section, we discuss the traditional S2 control chart for monitoring process variability that has been constructed without using the auxiliary information. Let Yi = (Yi1, Yi2,... Yin) i — 1,2,.... be independent random samples of size n (n > 2) from a normally distributed process with mean and variance a2. Here our objective is to monitor the changes in the process variance. We assume that ny = ny,0 and ay = ay0 are known, even if, in practice, these parameters have to be estimated from an in-control population. It is assumed that the process is in-control with variance a2,0. When shift in process variance a2 0 occurs, we have change from in-control value a20 to the out-of-control valuea^,!. Let A = ay1/ay 0(0 < A < 1) denotes the amount of shift in the in-control process standard deviationay ^ WhenA = 1, the process is considered to be in-control, otherwise the process is considered to be out-of-control. Let Yi = 1zn=iYy the sample mean at stage i. The S2 chart can be constructed by plotting sample variances
n
n-i-i = 1(Ii)
Si = n-TEn=i(Yy — Y) ,¿ = 1,2..........(1)
With lower and upper control limits set as
LCL = ^¡~^Xn-1,1-a/2, UCL = ^—^Xn-l.a/2 (2)
Where X«/2, Xi-a/2, denotes the upper and lower a/2percentage points of the Chi-square distribution with n — 1 degrees of freedom. The S2 chart for monitoring process variability gives a signal if Si2exceeds the control limits.
3. AIB Shewhart-type V Chart for Process variability
In this Section, we discuss AIB Shewhart-type control chart using regression estimator of the variance for monitoring shifts in process variability. Assume that a process has a quality characteristic of interest Y which is correlated with an auxiliary variable X. The pairs (Yi, Xi)are assumed to follow bivariate normal distribution with N2(ny,^x,a2,a2,p). Here p represents the correlation coefficient between study variable Y and auxiliary variable X. The observations of Y and X are obtained in the paired form for each sample and the population parameters are assumed to be known.
Let (Y1,X1), (Y2,X2),.... (Yn,Xn) represents a sample of size n from the bivariate normal distribution. The auxiliary information based unbiased regression estimator of population variance ay2 of study variable Y using a single auxiliary variable X for a bivariate random sample of size n is given by (Haq,(2017))
7 = Sy2+p2j|(ax2— Sx2) (3)
Where Sy = ^^-^n=1(Yi — Y)2, S| = ;^£n=1(Xi — XX)2 represent sample variances of Y and X — 1 — 1
respectively and Y = ~Zn=1 Yi, X = -^n=1Xi represent sample means of Y andX respectively. The mean and variance of V are as follows:
£(7) = ay2 and Var(V) = a2 = ^ (1 — p4)
Riaz et al. (2014) suggested AIB-Shewhart chart based on regression estimator V of process variance ay2 for monitoring process variability. We denote the chart as AIB-Shewhart-V chart. In the construction of AIB-Shewhart-V control chart for monitoring the process variability, the
Vikas Ghute and Sarika Pawar RT&A, No 3 (74) MA AND DMA CHARTS FOR PROCESS VARIABILITY USING AIB_Volume 18, September 2023
statistic V is plotted on the chart against the sample number. The control limits of the AIB-
Shewhart-V chart are
LCL = a- - Lay and UCL = a2 + LtfJ2^4 (4)
Where the value of L determines the in-control ARL of the AIB-Shewhart-V chart.
4. AIB Moving Average Control Chart for Process Variability
In this Section, we develop moving average control chart for detecting changes produced in the process variability. The proposed moving average chart is based on auxiliary information based V statistic and chart is denoted as AIB-MA-V chart. The construction of the chart is based on computing the moving averages of V statistics given in Eq. (3). The moving average statistic of span w at time i for a sequence of V statistics are computed as
MAi = Vi+Vi-1+^+Vi-w+1,for i>w (5)
For periods i < w we compute the average of available charting statistic. In other words, average of all V observations up to period i defines moving average. Fori > w, mean and variance of MAt statistic for in-control process are given as
E(MAd = a- and Var(MAd = ^-¡^(1 - P4) The control limits of MA-V chart are as follows:
UCLILCL =
for
<Jy ± L(Jy
(6)
Where L is a constant chosen to specify in-control ARL for the AIB-MA-V chart. The AIB-MA-V chart is constructed by plotting the MAi statistics on the chart against the sample number i. An out-of-control signal is issued when MAtis smaller than LCL or larger than the UCL.
5. AIB Double Moving Average Control Chart for Process Variability
In this Section, we present the design of double moving average control chart for detecting changes produced in the process variability. The proposed chart is denoted as AIB-DMA-V chart. The double moving average (DMA) statistic is based on the twice the subgroup average of the MA statistic. The moving average statistic for sequence of subgroup variances with time i and width w is given in Eq. (5). For interval i < w the DMA statistic can be calculated as the mean of all subgroup variances up to interval i while for interval i > w, the plotting statistic of AIB-DMA-V chart is given by
DMAi=MAi+MAi-i+J+MAi-w+i,for i>w (7)
where w represents the span at time i for computing DMAt statistic. The in-control mean of DMAt statistic calculated for i>w is given by,
E(DMAi)=1E( ^ MAj\=1(wa2) = ay2
ij = i-W+l
The in-control variance of DMAt is given by
7ar(DM.) =
Sky ,*<w
gj2(1-p4) w2(n-1) S7='-
gy2(1_P4)
i2(n-1)
11 w+i - + (i — w + 1) — ,w<i<2w-1
w + 1 ] w
(8)
gy2(1_P4) w2(n-1) '
i > 2w — 1
For the w = 2, the variance of the DMA is calculated using the 1st and 3rd lines of the above Eq. (8). The control limits of the proposed AIB-DMA-V chart are given as
UCL/LCL =
±-
2(1-p4)yi 1 i V n—1 S'=17-
< w
2 ± .Lfz. -w+1±+(i —w + 1)1 , w < i < 2w — 1
w \ n—1 iJ./=i-w+1 ; ^ ' w
(9)
2(1-p4)
± —J21^ ,i > 2w — 1
The control limits for w = 2 are calculated based on the using the 1st and 3rd lines of the above Eq. (9). L is the constant that is set according to the desired in-control ARL for the AIB-DMA-V control chart.
v.
a
y
V.
6. Performance Evaluation and Comparison
Performance of a control chart is typically measured in terms of average run length (ARL) and standard deviation of run length (SDRL). The ARL is the average number of sample points that is plotted on a chart before the first out-of-control signal is detected, whereas, the SDRL measures the spread of the run length distribution. When the process is out-of-control, it is desirable to have small values of ARL and SDRL. The performance of the control chart is measured in terms of in-control ARL (denoted as.flL0) and out-of-control ARL (denoted as.4fiL1). To compare the efficiency of two control charts for detecting the shift in process parameter, the general practice is to adjust their control limits so that their .AL0values become same and then compare .fiL1values at various shifts in process parameter. A chart with smaller .fiL1 is considered to be more efficient to detect pre-assigned shift in the process parameter.
In this Section, we evaluate the performance of the proposed AIB-MA-V and AIB-DMA-V charts for detecting shifts in Phase II monitoring. Monte Carlo simulation approach is used to evaluate the run length performance of the proposed charts in comparison with that of the AIB-Shewhart-V and S2charts. Simulation study based on sample of size n = 10,15,20 with p = 0.3,0.6,0.9 is carried out to assess the performance of the proposed AIB-MA-V, AIB-DMA-V and AIB-Shewhart-V charts. It is assumed that the in-control process is a bivariate normal distribution. Without loss of generality a standard bivariate normal distribution (i.e.N2(0,0,1,1,p) is considered as in-control process distribution. The out-of-control process is a bivariate normal with the same means of both auxiliary and study variables with changed variance of study variable 7. That is, we consider out-of-control procedure asN2(0,0, A, 1,p). Using the simulation, the control limit constant L of the AIB-MA-V and AIB-DMA-V charts are obtained for w = 2,3 and 4 so that.flL0 of the chart is approximately 200. The ARL and SDRL values of the chart are found with 50000 simulations for each shift of magnitude A in the process variance of the study variable 7. The magnitude of shift in the standard deviation of the study variable is considered asA = 1.0(0.1)2.0,2.5,3.0. To compare the performance of the proposed AIB-MA-V and AIB-DMA-V charts with AIB-Shewhart-V chart and Shewhart S2chart, each chart is designed so that .AL0 is approximately 200. The control limits, ARL and SDRL values of the AIB-Shewhart-V chart are obtained using simulation. The exact ARL values of the traditional S2 chart are computed using Excel.
Tables 1 to 3 represent the ARL and SDRL (shown in parenthesis) values of the proposed control chart with n = 10,15,20 according to the variance shift when the correlation coefficient is
0.3, 0.6 and 0.9 respectively. Here control limit constant L is chosen so that the value of ARL0is close to 200.
Table 1: Run length profiles of the charts with n = 10 and ARL0 = 200
Shift
A
S'2 chart
AIB-V AIB-MA-V chart AIB
chart w = 2 w = 3 w = 4 w = 2
DMA-V chart w = 3 w = 4
p = 0.3
1.0 200.65 199.95 199.03 200.54 200.93 199.90 200.73 200.38
(197.87) (197.86) (200.71) (200.13) (200.79) (199.54) (200.53) (198.33)
1.1 72.98 46.54 36.11 31.68 29.23 34.65 30.36 29.01
(72.48) (45.94) (35.64) (30.92) (28.56) (33.80) (29.01) (26.72)
1.2 25.23 16.68 12.11 10.38 9.46 11.69 10.40 10.25
(24.18) (16.25) (11.42) (9.56) (8.65) (10.68) (8.76) (8.01)
1.3 11.00 8.03 5.89 5.18 4.83 5.89 5.55 5.80
(10.40) (7.48) (5.22) (4.46) (4.04) (4.89) (4.04) (3.76)
1.4 6.09 4.66 3.62 3.29 3.12 3.75 3.76 4.03
(5.52) (4.11) (2.99) (2.57) (2.42) (2.76) (2.43) (2.37)
1.5 3.92 3.17 2.58 2.40 2.28 2.76 2.87 3.12
(3.34) (2.61) (1.93) (1.72) (1.61) (1.84) (1.72) (1.82)
1.6 2.83 2.38 2.02 1.91 1.85 2.22 2.34 2.52
(2.26) (1.82) (1.37) (1.23) (1.19) (1.36) (1.35) (1.48)
1.7 2.20 1.92 1.70 1.62 1.58 1.87 1.99 2.12
(1.66) (1.33) (1.05) (0.95) (0.91) (1.07) (1.12) (1.23)
1.8 1.81 1.63 1.49 1.44 1.41 1.64 1.73 1.83
(1.22) (1.02) (0.82) (0.75) (0.72) (0.88) (0.95) (1.04)
1.9 1.57 1.46 1.36 1.32 1.29 1.48 1.55 1.62
(0.96) (0.82) (0.67) (0.62) (0.59) (0.74) (0.81) (0.88)
2.0 1.42 1.33 1.26 1.23 1.21 1.35 1.40 1.46
(0.77) (0.67) (0.55) (0.51) (0.49) (0.63) (0.69) (0.74)
2.5 1.10 1.06 1.06 1.05 1.05 1.09 1.11 1.13
(0.33) (0.24) (0.26) (0.24) (0.23) (0.31) (0.34) (0.37)
3.0 1.03 1.33 1.02 1.02 1.01 1.03 1.03 1.04
(0.17) (0.66) (0.13) (0.13) (0.12) (0.17) (0.18) (0.20)
L --- 3.431 3.090 2.909 2.789 3.742 4.045 4.342
p = 0.6
1.0 200.65 200.97 200.80 199.06 200.21 200.76 199.56 200.79
(197.87) (201.34) (201.53) (200.18) (201.40) (200.75) (198.82) (198.20)
1.1 72.98 42.69 33.12 28.76 26.55 31.50 27.92 26.79
(72.48) (42.50) (32.96) (28.04) (25.97) (30.78) (26.32) (24.47)
1.2 25.23 14.79 10.66 9.29 8.55 10.37 9.35 9.21
(24.18) (14.34) (9.95) (8.54) (7.63) (9.36) (7.75) (6.93)
1.3 11.00 6.94 5.22 4.59 4.36 5.26 5.08 5.33
(10.40) (6.43) (4.52) (3.84) (3.62) (4.23) (3.60) (3.34)
1.4 6.09 4.14 3.26 2.98 2.83 3.40 3.45 3.77
(5.52) (3.59) (2.61) (2.30) (2.12) (2.42) (2.15) (2.18)
Table 1 continued
Shift
X
52 chart
AIB-V AIB-MA-V chart
chart w = 2 w = 3 w = 4
AIB-DMA-V chart w=2 w=3 w=4
p = 0.6
1.5 3.92 2.83 2.34 2.19 2.11 2.52 2.68 2.92
(3.34) (2.29) (1.68) (1.52) (1.42) (1.62) (1.55) (1.69)
1.6 2.83 2.14 1.87 1.78 1.72 2.05 2.19 2.36
(2.26) (1.56) (1.22) (1.11) (1.06) (1.22) (1.25) (1.38)
1.7 2.20 1.76 1.58 1.52 1.49 1.74 1.86 1.97
(1.66) (1.16) (0.92) (0.84) (0.81) (0.96) (1.03) (1.13)
1.8 1.81 1.52 1.40 1.37 1.34 1.54 1.63 1.72
(1.22) (0.89) (0.72) (0.67) (0.64) (0.80) (0.88) (0.96)
1.9 1.57 1.37 1.29 1.26 1.24 1.39 1.47 1.53
(0.96) (0.72) (0.59) (0.56) (0.53) (0.66) (0.74) (0.80)
2.0 1.42 1.27 1.21 1.19 1.18 1.29 1.35 1.40
(0.77) (0.58) (0.49) (0.46) (0.45) (0.57) (0.63) (0.68)
2.5 1.10 1.05 1.05 1.04 1.04 1.07 1.09 1.10
(0.33) (0.22) (0.23) (0.21) (0.20) (0.28) (0.30) (0.33)
3.0 1.03 1.26 1.01 1.01 1.01 1.02 1.02 1.03
(0.17) (0.57) (0.12) (0.11) (0.10) (0.14) (0.16) (0.17)
L --- 3.360 3.045 2.877 2.774 3.680 4.015 4.332
p = 0.9
1.0 200.65 200.46 200.25 200.41 199.09 199.40 199.88 199.83
(197.87) (199.21) (198.55) (199.40) (200.01) (198.33) (199.47) (198.62)
1.1 72.98 24.31 17.98 15.48 14.28 17.01 14.96 14.38
(72.48) (23.72) (17.24) (14.62) (13.42) (16.04) (13.29) (11.85)
1.2 25.23 6.72 5.02 4.51 4.20 5.00 4.86 5.16
(24.18) (6.19) (4.30) (3.68) (3.33) (3.91) (3.26) (3.08)
1.3 11.00 3.17 2.60 2.42 2.33 2.76 2.93 3.21
(10.40) (2.62) (1.92) (1.69) (1.59) (1.78) (1.66) (1.75)
1.4 6.09 2.04 1.78 1.72 1.68 1.97 2.14 2.32
(5.52) (1.47) (1.11) (1.02) (0.97) (1.11) (1.15) (1.28)
1.5 3.92 1.56 1.44 1.41 1.39 1.59 1.71 1.83
(3.34) (0.94) (0.75) (0.71) (0.68) (0.82) (0.89) (0.98)
1.6 2.83 1.33 1.26 1.24 1.23 1.37 1.46 1.53
(2.26) (0.66) (0.55) (0.52) (0.51) (0.63) (0.70) (0.76)
1.7 2.20 1.20 1.16 1.15 1.14 1.24 1.29 1.35
(1.66) (0.48) (0.42) (0.40) (0.39) (0.50) (0.56) (0.61)
1.8 1.81 1.13 1.11 1.09 1.09 1.16 1.20 1.23
(1.22) (0.38) (0.33) (0.31) (0.31) (0.40) (0.46) (0.49)
1.9 1.57 1.08 1.07 1.06 1.06 1.10 1.13 1.16
(0.96) (0.30) (0.26) (0.25) (0.25) (0.33) (0.37) (0.41)
2.0 1.42 1.05 1.04 1.04 1.04 1.07 1.09 1.11
(0.77) (0.24) (0.21) (0.21) (0.20) (0.27) (0.30) (0.33)
2.5 1.10 1.01 1.01 1.01 1.01 1.01 1.02 1.02
(0.33) (0.10) (0.08) (0.08) (0.07) (0.11) (0.12) (0.14)
3.0 1.03 1.05 1.00 1.00 1.00 1.00 1.00 1.00
(0.17) (0.24) (0.04) (0.04) (0.04) (0.05) (0.06) (0.06)
L --- 3.176 2.952 2.842 2.763 3.579 3.984 4.333
Table 2: Run length profiles of the charts with n = 15 and ARL0 = 200
Shift 52 chart A AIB-V chart AIB-MA-V chart w = 2 w = 3 w = 4 AIB-DMA-V chart w = 2 w = 3 w = 4
p = 0.3
1.0 200.49 199.97 199.21 199.29 199.17 200.77 199.40 200.93
(201.15) (200.70) (200.38) (198.87) (199.22) (199.73) (198.08) (199.18)
1.1 59.44 37.14 27.73 23.65 21.79 25.84 22.80 21.72
(58.56) (36.65) (27.04) (22.94) (20.94) (24.94) (21.19) (19.33)
1.2 17.34 11.87 8.41 7.20 6.64 8.03 7.33 7.48
(16.74) (11.39) (7.75) (6.38) (5.71) (6.92) (5.71) (5.20)
1.3 7.29 5.39 4.02 3.60 3.41 4.10 4.06 4.38
(6.76) (4.82) (3.32) (2.84) (2.63) (3.08) (2.56) (2.51)
1.4 3.95 3.16 2.52 2.34 2.24 2.69 2.83 3.10
(3.38) (2.63) (1.85) (1.64) (1.52) (1.71) (1.60) (1.72)
1.5 2.59 2.19 1.87 1.77 1.73 2.04 2.20 2.40
(2.02) (1.62) (1.19) (1.07) (1.02) (1.17) (1.21) (1.33)
1.6 1.93 1.69 1.52 1.46 1.44 1.68 1.80 1.94
(1.34) (1.08) (0.84) (0.77) (0.74) (0.88) (0.96) (1.05)
1.7 1.56 1.43 1.33 1.29 1.28 1.45 1.54 1.63
(0.93) (0.78) (0.63) (0.58) (0.56) (0.70) (0.78) (0.84)
1.8 1.35 1.27 1.21 1.19 1.18 1.30 1.37 1.43
(0.70) (0.58) (0.48) (0.45) (0.44) (0.57) (0.63) (0.68)
1.9 1.23 1.17 1.14 1.12 1.11 1.20 1.25 1.29
(0.52) (0.45) (0.38) (0.36) (0.34) (0.47) (0.51) (0.55)
2.0 1.15 1.11 1.09 1.08 1.07 1.14 1.17 1.20
(0.41) (0.35) (0.30) (0.28) (0.27) (0.38) (0.42) (0.46)
2.5 1.02 1.01 1.01 1.01 1.01 1.02 1.03 1.03
(0.14) (0.12) (0.10) (0.10) (0.09) (0.13) (0.16) (0.18)
3.0 1.00 1.11 1.00 1.00 1.00 1.00 1.00 1.00
(0.05) (0.35) (0.04) (0.04) (0.03) (0.06) (0.06) (0.07)
L --- 3.268 2.991 2.837 2.750 3.608 3.964 4.299
p = 0.6
1.0 200.49 200.57 200.93 200.57 200.71 199.40 199.46 199.77
(201.15) (198.84) (199.00) (198.47) (199.68) (198.47) (198.53) (198.65)
1.1 59.44 33.38 24.94 21.55 19.84 23.27 20.92 19.95
(58.56) (33.15) (24.29) (20.67) (18.89) (22.06) (19.16) (17.47)
1.2 17.34 10.23 7.41 6.44 5.97 7.13 6.66 6.84
(16.74) (9.67) (6.74) (5.60) (5.08) (6.03) (5.03) (4.57)
1.3 7.29 4.69 3.61 3.25 3.10 3.68 3.74 4.05
(6.76) (4.16) (2.95) (2.49) (2.32) (2.64) (2.28) (2.25)
1.4 3.95 2.77 2.28 2.13 2.08 2.46 2.65 2.90
(3.38) (2.22) (1.61) (1.43) (1.36) (1.52) (1.46) (1.59)
1.5 2.59 1.96 1.71 1.64 1.61 1.89 2.06 2.23
(2.02) (1.37) (1.03) (0.95) (0.92) (1.05) (1.10) (1.23)
1.6 1.93 1.55 1.42 1.39 1.36 1.57 1.70 1.81
(1.34) (0.92) (0.73) (0.68) (0.65) (0.80) (0.88) (0.97)
1.7 1.56 1.34 1.26 1.24 1.23 1.37 1.46 1.53
(0.93) (0.67) (0.54) (0.52) (0.50) (0.63) (0.70) (0.76)
Table 2 continued
Shift 52 chart AIB-V AIB-MA-V chart AIB-DMA-V chart
X chart w = 2 w = 3 w = 4 w = 2 w = 3 w = 4
p = 0.6
1.8 1.35 1.21 1.16 1.15 1.14 1.25 1.30 1.36
(0.70) (0.50) (0.42) (0.40) (0.39) (0.51) (0.57) (0.62)
1.9 1.23 1.13 1.11 1.09 1.09 1.16 1.20 1.24
(0.52) (0.38) (0.33) (0.31) (0.31) (0.41) (0.46) (0.50)
2.0 1.15 1.08 1.06 1.06 1.06 1.10 1.13 1.16
(0.41) (0.30) (0.25) (0.25) (0.24) (0.33) (0.37) (0.41)
2.5 1.02 1.01 1.01 1.01 1.01 1.01 1.02 1.02
(0.14) (0.10) (0.09) (0.08) (0.08) (0.11) (0.13) (0.15)
3.0 1.00 1.08 1.00 1.00 1.00 1.00 1.00 1.00
(0.05) (0.30) (0.03) (0.03) (0.03) (0.04) (0.05) (0.06)
L --- 3.206 2.958 2.826 2.746 3.563 3.950 4.289
p = 0.9
1.0 200.49 199.36 199.00 200.65 199.10 199.60 200.81 200.19
(201.15) (199.21) (198.65) (200.28) (198.94) (200.21) (199.91) (198.89)
1.1 59.44 17.49 12.61 10.81 9.83 11.87 10.41 10.13
(58.56) (16.93) (11.79) (9.81) (8.69) (10.73) (8.60) (7.65)
1.2 17.34 4.44 3.42 3.10 2.94 3.49 3.58 3.92
(16.74) (3.93) (2.70) (2.27) (2.11) (2.40) (2.04) (2.05)
1.3 7.29 2.15 1.85 1.78 1.74 2.05 2.25 2.47
(6.76) (1.58) (1.15) (1.05) (1.00) (1.13) (1.15) (1.27)
1.4 3.95 1.48 1.37 1.34 1.33 1.53 1.66 1.78
(3.38) (0.84) (0.67) (0.62) (0.61) (0.74) (0.82) (0.900
1.5 2.59 1.23 1.18 1.17 1.16 1.28 1.35 1.43
(2.02) (0.53) (0.43) (0.41) (0.40) (0.53) (0.59) (0.65)
1.6 1.93 1.11 1.09 1.08 1.08 1.15 1.19 1.24
(1.34) (0.34) (0.30) (0.29) (0.28) (0.38) (0.44) (0.49)
1.7 1.56 1.05 1.05 1.04 1.04 1.08 1.10 1.13
(0.93) (0.23) (0.22) (0.20) (0.19) (0.28) (0.32) (0.36)
1.8 1.35 1.03 1.02 1.02 1.02 1.04 1.06 1.07
(0.70) (0.16) (0.15) (0.14) (0.14) (0.20) (0.24) (0.27)
1.9 1.23 1.01 1.01 1.01 1.01 1.02 1.03 1.04
(0.52) (0.12) (0.11) (0.10) (0.10) (0.15) (0.18) (0.20)
2.0 1.15 1.01 1.01 1.01 1.00 1.01 1.02 1.02
(0.41) (0.09) (0.08) (0.07) (0.07) (0.11) (0.13) (0.15)
2.5 1.02 1.00 1.00 1.00 1.00 1.00 1.00 1.00
(0.14) (0.02) (0.02) (0.02) (0.02) (0.03) (0.03) (0.04)
3.0 1.00 1.01 1.00 1.00 1.00 1.00 1.00 1.00
(0.05) (0.09) (0.01) (0.00) (0.01) (0.01) (0.01) (0.01)
L --- 3.061 2.895 2.809 2.741 3.505 3.934 4.299
Table 3: Run length profiles of the charts with n = 20 and ARL0 = 200
Shift A 52 chart AIB-V chart AIB-MA-V chart w = 2 w = 3 w = 4 AIB-DMA-V chart w = 2 w = 3 w = 4
p = 0.3
1.0 199.83 200.49 199.47 199.24 200.78 200.14 200.10 200.53
(200.31) (200.63) (200.17) (200.61) (200.42) (199.60) (198.72) (198.39)
1.1 49.93 30.87 22.43 19.05 17.72 20.97 18.30 17.63
(49.41) (30.30) (21.83) (18.29) (16.74) (19.91) (16.66) (15.13)
1.2 13.04 9.05 6.38 5.56 5.17 6.21 5.80 6.07
(12.55) (8.54) (5.67) (4.68) (4.24) (5.08) (4.19) (3.79)
1.3 5.26 4.03 3.09 2.80 2.70 3.21 3.32 3.64
(4.72) (3.50) (2.38) (2.06) (1.92) (2.20) (1.90) (1.96)
1.4 2.91 2.40 2.00 1.89 1.85 2.18 2.36 2.60
(2.41) (1.83) (1.33) (1.18) (1.12) (1.25) (1.25) (1.39)
1.5 1.95 1.72 1.53 1.48 1.46 1.70 1.84 1.99
(1.38) (1.12) (0.84) (0.76) (0.75) (0.88) (0.95) (1.04)
1.6 1.53 1.39 1.30 1.27 1.25 1.42 1.52 1.62
(0.89) (0.73) (0.59) (0.55) (0.53) (0.66) (0.74) (0.80)
1.7 1.29 1.21 1.17 1.15 1.14 1.25 1.32 1.39
(0.62) (0.51) (0.42) (0.40) (0.39) (0.51) (0.57) (0.63)
1.8 1.17 1.12 1.09 1.08 1.08 1.15 1.20 1.24
(0.44) (0.36) (0.31) (0.29) (0.29) (0.39) (0.45) (0.49)
1.9 1.10 1.07 1.05 1.05 1.05 1.09 1.12 1.15
(0.33) (0.27) (0.23) (0.22) (0.21) (0.30) (0.35) (0.39)
2.0 1.05 1.04 1.03 1.03 1.03 1.05 1.07 1.09
(0.24) (0.20) (0.18) (0.17) (0.16) (0.23) (0.27) (0.30)
2.5 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.01
(0.06) (0.05) (0.04) (0.04) (0.01) (0.06) (0.07) (0.08)
3.0 1.00 1.04 1.00 1.00 1.00 1.00 1.00 1.00
(0.02) (0.20) (0.01) (0.01) (0.01) (0.02) (0.02) (0.03)
L --- 3.173 2.934 2.808 2.735 3.545 3.926 4.279
p = 0.6
1.0 199.83 199.36 199.39 199.37 199.88 199.44 199.19 200.34
(200.31) (199.22) (197.10) (199.03) (198.13) (198.89) (196.75) (198.49)
1.1 49.93 27.68 19.98 17.16 15.97 18.90 16.74 16.07
(49.41) (27.23) (19.26) (16.30) (15.05) (17.97) (14.94) (13.58)
1.2 13.04 7.80 5.61 4.94 4.63 5.49 5.28 5.57
(12.55) (7.26) (4.86) (4.10) (3.73) (4.38) (3.64) (3.36)
1.3 5.26 3.53 2.76 2.57 2.45 2.90 3.07 3.38
(4.72) (2.99) (2.07) (1.81) (1.67) (1.89) (1.70) (1.78)
1.4 2.91 2.13 1.84 1.75 1.71 2.03 2.21 2.41
(2.41) (1.56) (1.14) (1.04) (0.99) (1.13) (1.16) (1.27)
1.5 1.95 1.57 1.43 1.39 1.37 1.59 1.73 1.86
(1.38) (0.94) (0.72) (0.68) (0.66) (0.80) (0.87) (0.96)
1.6 1.53 1.30 1.23 1.21 1.20 1.35 1.43 1.52
(0.89) (0.62) (0.50) (0.48) (0.46) (0.60) (0.67) (0.73)
1.7 1.29 1.16 1.13 1.12 1.11 1.20 1.26 1.32
(0.62) (0.43) (0.36) (0.35) (0.33) (0.46) (0.51) (0.57)
1.8 1.17 1.09 1.07 1.06 1.06 1.12 1.15 1.19
(0.44) (0.31) (0.26) (0.25) (0.25) (0.34) (0.39) (0.44)
Table 3 continued
Shift A S2 chart AIB-V chart AIB-MA-V chart w=2 w=3 w=4 AIB-DMA-V chart w=2 w=3 w=4
p = 0.6
1.9 1.10 1.05 1.04 1.03 1.03 1.07 1.09 1.12
(0.33) (0.22) (0.20) (0.18) (0.18) (0.26) (0.30) (0.34)
2.0 1.05 1.03 1.02 1.02 1.02 1.04 1.05 1.07
(0.24) (0.16) (0.15) (0.14) (0.14) (0.20) (0.23) (0.26)
2.5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
(0.06) (0.04) (0.04) (0.03) (0.03) (0.05) (0.06) (0.07)
3.0 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00
(0.02) (0.16) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02)
L --- 3.118 2.906 2.799 2.731 3.510 3.919 4.277
p = 0.9
1.0 199.83 200.67 199.75 199.31 200.17 199.85 200.14 199.57
(200.31) (200.68) (197.77) (198.84) (199.80) (200.61) (197.25) (196.52)
1.1 49.93 13.74 9.66 8.19 7.45 9.04 8.07 7.99
(49.41) (13.20) (8.88) (7.18) (6.32) (7.85) (6.29) (5.47)
1.2 13.04 3.33 2.62 2.43 2.36 2.78 2.97 3.27
(12.55) (2.77) (1.90) (1.62) (1.53) (1.70) (1.54) (1.61)
1.3 5.26 1.69 1.52 1.48 1.46 1.71 1.89 2.05
(4.72) (1.07) (0.82) (0.75) (0.72) (0.85) (0.92) (1.01)
1.4 2.91 1.25 1.20 1.19 1.17 1.31 1.41 1.51
(2.41) (0.56) (0.46) (0.44) (0.42) (0.55) (0.63) (0.69)
1.5 1.95 1.09 1.08 1.07 1.07 1.14 1.19 1.24
(1.38) (0.32) (0.28) (0.27) (0.26) (0.37) (0.43) (0.48)
1.6 1.53 1.04 1.03 1.03 1.03 1.06 1.08 1.11
(0.89) (0.20) (0.17) (0.16) (0.16) (0.24) (0.28) (0.32)
1.7 1.29 1.01 1.01 1.01 1.01 1.02 1.04 1.05
(0.62) (0.12) (0.11) (0.10) (0.10) (0.15) (0.19) (0.22)
1.8 1.17 1.01 1.00 1.00 1.00 1.01 1.01 1.02
(0.44) (0.08) (0.07) (0.06) (0.06) (0.10) (0.12) (0.15)
1.9 1.10 1.00 1.00 1.00 1.00 1.00 1.01 1.01
(0.33) (0.05) (0.05) (0.04) (0.04) (0.06) (0.08) (0.10)
2.0 1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.01
(0.24) (0.04) (0.02) (0.03) (0.02) (0.04) (0.06) (0.07)
2.5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
(0.06) (0.01) (0.01) (0.00) (0.01) (0.00) (0.01) (0.01)
3.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
(0.02) (0.03) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
L --- 3.002 2.868 2.790 2.733 3.471 3.917 4.280
Observations from Tables 1-3:
• For any range of shifts in process standard deviation, the proposed AIB-DMA-V and AIB-MA-V charts consistently produces smaller out-of-control ARL values than the AIB-Shewhart-V chart and Shewhart S2 chart. For example, in Table 1, with A = 1.3, the traditional S2chart requires on an average 11 samples to signal, the Shewhart AIB-V chart requires 8.03 samples to signal, the ARL reduces to 5.89 with w = 2, 5.18 with w = 3 and 4.83 with w = 4 for proposed AIB-MA-V chart and ARL reduces to 5.89 with w = 2, 5.55 with w = 3 and 5.80 with w = 4 for proposed AIB-DMA-V chart. That means, proposed
AIB-DMA-V and AIB-MA-V charts early detect shifts in process standard deviation earlier than the other two charts.
• The sensitivity of the proposed control charts increases as the span of moving average increases. For example, in Table 3, with A = 1.2 and w = 2, the proposed AIB-DMA-V chart requires on an average 6.21 samples to signal, the ARL reduces to 5.80 and 6.07 with w = 3 and w = 4 respectively. The similar performance is observed for AIB-MA-V chart.
• The out-of-control ARL values of proposed AIB-DMA-V chart decreases when the correlation between study and auxiliary variables increases. For example, from Table 1, with fixed A = 1.3and w = 2, the proposed AIB-DMA-V chart requires on an average 5.89 samples to signal when p = 0.3, the ARL reduces to 5.26 and 2.76 when p = 0.6 and p = 0.9 respectively. The similar performance is observed for AIB-MA-V and Shewhart-AIB-V charts.
• The performance of the proposed AIB-DMA-V and AIB-MA-V charts keeps improving with an increase in sample size n. For example, from Tables 1-3, with fixed A = 1.2, w = 3 and p = 0.6, the proposed AIB-DMA-V chart requires on an average 9.35 samples to signal when n = 10, the ARL reduces to 6.66 and 5.28 when n = 15 and n = 20 respectively.
In this Section, we provide an illustrative example in order to demonstrate the practical application of AIB-MA-V, AIB-DMA-V charts for monitoring process variability using auxiliary information. Here we consider a simulated dataset to present implementation of AIB-Shewhart-V, AIB-MA-V and AIB-DMA-V control charts. To identify the performance of these control charts, the in-control ARL value is set as^fiL0 = 200. We have considered the paired information on (Y,X) where X is used as auxiliary variable and Y as the study variable of interest. The bivariate data set in the form of 15 sub-groups each of size n = 10 are simulated fromN2(ny,nx,ay,a2,p) distribution. For in-control state, the first 7 samples are generated fromW2(0,0,1,1,0.6). Thus the process is stable with respect to process variability for first 7 samples. We add 8 samples to simulate an out-of-control process. Starting from sample 8, new samples are generated from a process by introducing a shift A = 1.5 in ay. Based on 15 subgroups, the values of the control chart statistic for AIB-MA-V and AIB-DMA-V control charts for span w = 3 and AIB-Shewhart-V chart are displayed in Table 4. The control limits of AIB-Shewhart-V chart are computed using Eq. (4) while those of AIB-MA-V and AIB-DMA-V charts are computed using Eq. (6) and Eq. (9) respectively. Implementation of the said charts is presented in Figure 1.
From Figure 1, it can be seen that the process remains in-control at the first seven samples. For detecting shift of size A = 1.5 in process standard deviation, the AIB-Shewhart-V chart does not produce any out-of-control signal for detecting the shift. So AIB-Shewhart-V chart fail to detect a shift in process standard deviation when the shift occur. The AIB-MA-V chart shows first out-of-control signal at point 13. AIB-DMA-V chart shows first out-of-control signal at point 11 which is earlier than that of AIB-MA-V chart. Hence the proposed AIB-DMA-V chart is effective in detecting shifts in process standard deviation than AIB-MA-V and AIB-Shewhart-V charts.
7. An Example
Table 4: Chart statistics based on simulated data
Sample Number
AIB-V
AIB-MA-V
AIB-DMA-V
1 2
3
4
5
0.91 0.63 1.56 1.65 0.74
0.91 0.77 1.03 1.28 1.31
0.91 0.84 0.91 1.03 1.21
Table 4 continued
6 0.95 1.11 1.23
7 0.56 0.75 1.06
8 1.88 1.13 1.00
9 2.30 1.58 1.15
10 0.80 1.66 1.46
11 2.07 1.72 1.65
12 1.71 1.53 1.64
13 2.02 1.93 1.73
14 1.54 1.76 1.74
15 1.03 1.53 1.74
sample number
Figure 1: AIB-Shewhart, AIB-MA and AIB-DMA Charts for process variability
In this paper, we have proposed the AIB-MA and AIB-DMA control charts to efficiently monitoring the variability of normally distributed process. These charts are based on regression estimator of the process variance that utilizes information on a study variable as well as any related auxiliary variable. The construction, performance assessment and an illustrative example of proposed charts are presented in this paper. Using extensive Monte Carlo simulations, ARL and SDRL of the proposed AIB-MA-V and AIB-DMA-V chart has been computed with various choices of correlation coefficient p and span w. From the simulation results it is observed that with an increase in the value of w, the performance of the AIB-MA-V and AIB-DMA-V charts is improved. The performance of the proposed charts keep improving with an increase in the values of sample size n, level of correlation between study variable and auxiliary variable p and size of shift A in process standard deviation at a fixed^fiL0. The SDRL values of the charts are approximately the same as ARL values. The performance of the proposed charts is also compared to its existing counterparts incorporated in this study. It has been found that AIB-DMA-V and AIB-MA-V charts perform uniformly better than the AIB-Shewhart-V andS2 charts for different kind of shifts in the process variability. Among AIB-DMA-V perform better than the AIB-MA-V chart for a span ofw = 2,3; while for span ofw = 4, performance of both proposed charts is similar.
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Vikas Ghute and Sarika Pawar RT&A, No 3 (74) MA AND DMA CHARTS FOR PROCESS VARIABILITY USING AIB_Volume 18, September 2023
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