Design of Improved EWMA Control Chart for Monitoring the Process Mean Using New Median Quartile Double Ranked Set Sampling

Department of Statistics, National College of Business Administration and Economics, Lahore, Pakistan

E-mail: wasif.yasin988@gmail.com; m_tayyab82@hotmail.com; drhanif@ncbae.edu.pk

*Corresponding Author

Received 31 July 2021; Accepted 28 October 2021; Publication 06 December 2021

## Abstract

It is essential to monitor the mean of a process regarding quality characteristics for the ongoing production. For enhancement of mean monitoring power of the exponentially weighted moving average (EWMA) chart, a new median quartile double ranked set sampling (MQDRSS) based EWMA control chart is proposed and named as EWMA-MQDRSS chart. In order to study the performance of the developed EWMA-MQDRSS chart, performance measures; average run length, and the standard deviation of run length are used. The shift detection ability of the proposed chart has been compared with counterparts, under the simple random sampling and ranking based sampling techniques. The extensive simulation-based results indicate that the EWMA-MQDRSS chart performs better to trace all kinds of shifts than the existing charts. An illustrative application concerning monitoring the diameter of the piston ring of a machine is also provided to demonstrate the practical utilization of the suggested chart.

Keywords: Average run length, control chart, EWMA, ranked set sampling, simulation.

## 1 Introduction

Statistical process control (SPC) comprises tools that can be beneficial for checking the shift in process parameters. In competitive markets at present, there prevails a great tussle between the various competitive service providers, and sustainability of the value of the goods or amenities is vital for the contentment of the clients to carry on and for capturing the industry. The buyers have various choices that they can select from, thus, product quality plays a key role in attracting the consumers. Control charts are the most refined and frequently adopted SPC tool in industries like medicine, ecology and technology used to monitor study variable during the production and to focus the situation where the process gets out of control (OOC). A process is considered in control (IC) if production is controlled in a way that measurement of process production variation remains between given lower and upper control limits. Effective monitoring of mean during the production process by an economical sampling scheme has gained reasonable attention of the quality statistician to retain the quality of products (Montgomery, 2009).

The use of an efficient sampling technique has a key role in enhancing the shift detection ability of charts to improve the monitoring of process parameters. In this research study, a new and efficient MQDRSS scheme is introduced to design the EWMA-MQDRSS chart for mean monitoring with the aim of enhancing the ability of the chart to identify the small to moderate shifts. The proposed chart is also valuable in such circumstances when some supplementary information about the quality characteristic of interest is already available without any extra cost. The additional information can be useful for ranking mechanism and helpful to draw more representative sample to get an actual measurement of quality characteristic under study. We expect that the suggested chart will be more efficient for diagnosing the unnatural variation in the process mean than its counterparts. The upcoming formation of the manuscript is followed by Section 2 which is being delivered the design structure of efficient EWMA-MQDRSS chart. The performance evaluation and comparative study of the suggested chart is presented in Section 3, followed by the application of the EWMA-MQDRSS chart in Section 4. Finally, this study ends with conclusions and recommendations.

## 2 Design of the Proposed EWMA-MQDRSS Control Chart

In this section, an efficient EWMA charting structure has been designed by exploiting a new ranked based sampling scheme named as MQDRSS. The proposed chart is named as EWMA-MQDRSS chart and it is more beneficial when the ranking mechanism of quality characteristic of interest occurs without any cost. The proposed ranked based MQDRSS procedure is provided in the following steps.

Step 1. Select $m3$ units from population and distribute these units between $m$ sets at random. Each set contains $m$ subsets of $m$ units.

Step 2. Without identifying actual measurements, the $m$ units of each subset of $m$ sets are ranked visually or with auxiliary information or by any cost-free method, and then apply MRSS design on the $m$ sets. This step contains $m$ median ranked set samples of size $m$ each.

Step 3. Again rank each median ranked set sample, obtained during Step 2, and then apply QRSS to select improved DRSS (MQDRSS) of size $m$ for the actual measurement.

Step 4. Repeat the Steps 1–3 independently $r$ times, if essential to select a final sample of size $n=m⁢r$.

The following example illustrates the MQDRSS procedure for selecting the sample with size 6.

Example: To select a MQDRSS of size $n=6$ for $r=1(n=m)$, identify $m3=216$ (6 sets of size 36 each) sampling units. Let $Yi⁢(j)⁢k$ be the $jt⁢h$ smallest ranked unit from $it⁢h$ subset of the $kt⁢h$ set, where $i,j,k=1,2,…,6$. Rank the units of each subset of all 6 sets, according to the variable under study.

 $[Y1⁢(1)⁢1 Y1⁢(2)⁢1 Y1⁢(3)⁢1 Y1⁢(4)⁢1 Y1⁢(5)⁢1 Y1⁢(6)⁢1Y2⁢(1)⁢1 Y2⁢(2)⁢1 Y2⁢(3)⁢1 Y2⁢(4)⁢1 Y2⁢(5)⁢1 Y2⁢(6)⁢1Y3⁢(1)⁢1 Y3⁢(2)⁢1 Y3⁢(3)⁢1 Y3⁢(4)⁢1 Y3⁢(5)⁢1 Y3⁢(6)⁢1Y4⁢(1)⁢1 Y4⁢(2)⁢1 Y4⁢(3)⁢1 Y4⁢(4)⁢1 Y4⁢(5)⁢1 Y4⁢(6)⁢1Y5⁢(1)⁢1 Y5⁢(2)⁢1 Y5⁢(3)⁢1 Y5⁢(4)⁢1 Y5⁢(5)⁢1 Y5⁢(6)⁢1Y6⁢(1)⁢1 Y6⁢(2)⁢1 Y6⁢(3)⁢1 Y6⁢(4)⁢1 Y6⁢(5)⁢1 Y6⁢(6)⁢1]$ $[Y1⁢(1)⁢2 Y1⁢(2)⁢2 Y1⁢(3)⁢2 Y1⁢(4)⁢2 Y1⁢(5)⁢2 Y1⁢(6)⁢2Y2⁢(1)⁢2 Y2⁢(2)⁢2 Y2⁢(3)⁢2 Y2⁢(4)⁢2 Y2⁢(5)⁢2 Y2⁢(6)⁢2Y3⁢(1)⁢2 Y3⁢(2)⁢2 Y3⁢(3)⁢2 Y3⁢(4)⁢2 Y3⁢(5)⁢2 Y3⁢(6)⁢2Y4⁢(1)⁢2 Y4⁢(2)⁢2 Y4⁢(3)⁢2 Y4⁢(4)⁢2 Y4⁢(5)⁢2 Y4⁢(6)⁢2Y5⁢(1)⁢2 Y5⁢(2)⁢2 Y5⁢(3)⁢2 Y5⁢(4)⁢2 Y5⁢(5)⁢2 Y5⁢(6)⁢2Y6⁢(1)⁢2 Y6⁢(2)⁢2 Y6⁢(3)⁢2 Y6⁢(4)⁢2 Y6⁢(5)⁢2 Y6⁢(6)⁢2]$ $⋮$ $[Y1⁢(1)⁢6 Y1⁢(2)⁢6 Y1⁢(3)⁢6 Y1⁢(4)⁢6 Y1⁢(5)⁢6 Y1⁢(6)⁢6Y2⁢(1)⁢6 Y2⁢(2)⁢6 Y2⁢(3)⁢6 Y2⁢(4)⁢6 Y2⁢(5)⁢6 Y2⁢(6)⁢6Y3⁢(1)⁢6 Y3⁢(2)⁢6 Y3⁢(3)⁢6 Y3⁢(4)⁢6 Y3⁢(5)⁢6 Y3⁢(6)⁢6Y4⁢(1)⁢6 Y4⁢(2)⁢6 Y4⁢(3)⁢6 Y4⁢(4)⁢6 Y4⁢(5)⁢6 Y4⁢(6)⁢6Y5⁢(1)⁢6 Y5⁢(2)⁢6 Y5⁢(3)⁢6 Y5⁢(4)⁢6 Y5⁢(5)⁢6 Y5⁢(6)⁢6Y6⁢(1)⁢6 Y6⁢(2)⁢6 Y6⁢(3)⁢6 Y6⁢(4)⁢6 Y6⁢(5)⁢6 Y6⁢(6)⁢6]$ (1)

Then select the middle units in boxes from each set and the sampling units in each set are given in rows, as follows:

 $[Y1⁢(3)⁢1 Y2⁢(3)⁢1 Y3⁢(3)⁢1 Y4⁢(4)⁢1 Y5⁢(4)⁢1 Y6⁢(4)⁢1Y1⁢(3)⁢2 Y2⁢(3)⁢2 Y3⁢(3)⁢2 Y4⁢(4)⁢2 Y5⁢(4)⁢2 Y6⁢(4)⁢2Y1⁢(3)⁢3 Y2⁢(3)⁢3 Y3⁢(3)⁢3 Y4⁢(4)⁢3 Y5⁢(4)⁢3 Y6⁢(4)⁢3Y1⁢(3)⁢4 Y2⁢(3)⁢4 Y3⁢(3)⁢4 Y4⁢(4)⁢4 Y5⁢(4)⁢4 Y6⁢(4)⁢4Y1⁢(3)⁢5 Y2⁢(3)⁢5 Y3⁢(3)⁢5 Y4⁢(4)⁢5 Y5⁢(4)⁢5 Y6⁢(4)⁢5Y1⁢(3)⁢6 Y2⁢(3)⁢6 Y3⁢(3)⁢6 Y4⁢(4)⁢6 Y5⁢(4)⁢6 Y6⁢(4)⁢6]$ (2)

Without quantifying the actual measurement of the units of these subsets, rank units of each subset of above set again and then choose $(m+14)t⁢h$ ranked unit (in boxes), i.e. $Yi⁢(2)*$ from $it⁢h$ subset $(i=1,2,3)$ and choose $(3⁢(m+1)4)t⁢h$ ranked unit (in boxes), i.e. $Yi⁢(5)*$ from $it⁢h$ subset $(i=4,5,6)$ for actual quantification shown as below:

 $[Y1⁢(1)* Y1⁢(2)* Y1⁢(3)* Y1⁢(4)* Y1⁢(5)* Y1⁢(6)*Y2⁢(1)* Y2⁢(2)* Y2⁢(3)* Y2⁢(4)* Y2⁢(5)* Y2⁢(6)*Y3⁢(1)* Y3⁢(2)* Y3⁢(3)* Y3⁢(4)* Y3⁢(5)* Y3⁢(6)*Y4⁢(1)* Y4⁢(2)* Y4⁢(3)* Y4⁢(4)* Y4⁢(5)* Y4⁢(6)*Y5⁢(1)* Y5⁢(2)* Y5⁢(3)* Y5⁢(4)* Y5⁢(5)* Y5⁢(6)*Y6⁢(1)* Y6⁢(2)* Y6⁢(3)* Y6⁢(4)* Y6⁢(5)* Y6⁢(6)*]$ (3)

The units ${Y1⁢(2)*,Y2⁢(2)*,Y3⁢(2)*,Y4⁢(5)*,Y5⁢(5)*,Y6⁢(5)*}$ in boxes represent MQDRSS of size $n=6$.

Assume that $Y1,Y2,…,Yn$ be a random sample of size $n$ drawn from a distribution with density function $fY$, distribution function $FY$, mean $μ$ and variance $σ2$. The SRS mean is $Y¯SRS=∑i=1nYi/n$ and $E⁢(Y¯SRS)=μ$ along with $Var⁢(Y¯SRS)=σ2/n$. The cycle in this study is replicated once i.e. $r=1$. Let $Yi⁢(q1)*$ shows first quartile unit from $it⁢h$ subset $(i=1,2,…,m2)$ and $Yi⁢(q3)*$ shows third quartile unit from $it⁢h$ subset $(i=m+22,…,m)$ for even sample size. Let $Yi⁢(q1)*$ shows first quartile unit from $it⁢h$ subset $(i=1,2,…,m-12)$, $Yi⁢(q3)*$ shows third quartile unit from $it⁢h$ subset $(i=m+12,…,(m-1))$ and $Ym⁢(q2)*$ be the median of $mt⁢h$ set for odd sample size. Note that $q1=(m+1)/4$, $q2=2⁢(m+1)/4$ and $q3=3⁢(m+1)/4$. It is also assumed that $Y¯(MQDRSS)⁢e*$ and $Y¯(MQDRSS)⁢o*$ be the MQDRSS mean estimators for the even and odd sample sizes, respectively. The respective $Y¯(MQDRSS)⁢e*$ and $Y¯(MQDRSS)⁢o*$ estimators are defined as:

 $Y¯(MQDRSS)⁢e*=1m⁢[∑i=1m/2Yi⁢(q1)*+∑i=m+22mYi⁢(q3)*]$ (4)

and

 $Y¯(MQDRSS)⁢o*=1m⁢[∑i=1(m-1)/2Yi⁢(q1)*+∑i=m+12(m-1)Yi⁢(q3)*+Ym⁢(q2)*]$ (5)

Following the structure of the Classical EWMA Chart (Montgomery, 2009), a sample of size $n$ based on MQDRSS design at each time point $t$ is considered. We also assume that $Y(MQDRSS)⁢t*$ be the sequence of identically, independently and normally distributed random variables, for $t=1,2,…$ and $Y¯(MQDRSS)⁢t*$ denotes the mean of sample at $t$. By using the $Y¯(MQDRSS)⁢t*$, the EWMA-MQDRSS statistic $Et$ (plotting statistic) is defined as

 $Et=ξ⁢Y¯(MQDRSS)⁢t*+(1-ξ)⁢Et-1,0<ξ≤1,$ (6)

where $ξ$ is the smoothing-constant, $E0=μ$ and $E0=Y¯(MQDRSS)*$ when $μ$ is not known. The variance of $Et$ is derived by

 $Var⁢(Et)=Var⁢(Y¯(MQDRSS)⁢t*)⁢[ξ⁢(1-(1-ξ)2⁢t)2-ξ].$ (7)

Control limits of the proposed EWMA-MQDRSS chart is given by

 $LCLt=μ-L⁢Var⁢(Y¯(MQDRSS)⁢t*)⁢[ξ⁢(1-(1-ξ)2⁢t)2-ξ]UCLt=μ+L⁢Var⁢(Y¯(MQDRSS)⁢t*)⁢[ξ⁢(1-(1-ξ)2⁢t)2-ξ]},$ (8)

where

 $Var⁢(Y¯(MQDRSS)⁢e⁢t*)=1m2⁢[∑i=1m/2Var⁢(Yi⁢(q1)*)+∑i=m+22mVar⁢(Yi⁢(q3)*)]$ (9)

and

 $Var⁢(Y¯(MQDRSS)⁢o⁢t*)$ $=1m2⁢[∑i=1(m-1)/2Var⁢(Yi⁢(q1)*)+∑i=(m+1)/2(m-1)Var⁢(Yi⁢(q3)*)+Var⁢(Ym⁢(q2)*)]$ (10)

Here, $L$ is a control-coefficient of EWMA-MQDRSS chart, $LCLt$ and $UCLt$ signify the lower and the upper control limits. The value of $L$ is determined, according to the pre-specified value of $ARL0$. Furthermore, a process under EWMA-MQDRSS chart is called OOC, if the plotting-statistic $Et$ lies beyond the $LCLt$ and $UCLt$.

## 3 Performance Evaluation and Comparative Study

To assess the efficiency of EWMA-MQDRSS chart for IC and OOC situations, we used ARL and SDRL as performance measures. The values of run-length (RL) for proposed EWMA-type chart and considered modified DRSS based charts (EWMA-DRSS, EWMA-EDRSS, EWMA-MDRSS and EWMA-QDRSS) are calculated by using Monte Carlo simulation study in R-Language. On the basis of IC process, standard normal distribution is used and 100,000 samples of size $m$ are randomly chosen then estimated lower and upper limits of EWMA-DRSS, EWMA-EDRSS, EWMA-MDRSS, EWMA-QDRSS, EWMA-SRS and EWMA-MQDRSS charts with exact-ranking. With the setting of $ARL0$, the simulated 10,000 Phase-II samples of size $m$ in each, are drawn from $N⁢(μ+δ⁢σn,σ)$ and then ARL with SDRL estimated results of considered and proposed charts are determined. For evaluating the performance of the proposed EWMA-MQDRSS chart, we reported the values of $L$ with various choices of $ξ$ for which $ARL0$ is fixed at 370 and results are given in Table 1. For comparative study, we have taken $m=6,7$, $r=1(n=m)$, $ξ=0.05$ and $ARL0=200,370$ for justifiable reasoning of suggested chart and RL results are placed in Tables 25.

Table 1 Performance evaluation of the proposed EWMA-MQDRSS chart for various choices of $ξ$ when $ARL0=370$ and $m=5$

 $ξ$ $0.05$ $0.10$ $0.25$ $0.50$ $L$ $δ$ $R⁢L$ $2.5199$ $2.7190$ $2.9180$ $2.99$$50$ 0.00 ARL 370.7207 370.9989 371.8835 370.7247 SDRL 379.1414 371.4939 368.3937 365.4463 0.05 ARL 115.7943 150.2544 211.6349 262.4377 SDRL 112.3641 144.5767 210.2592 260.1857 0.10 ARL 41.0214 52.0252 84.3254 136.5405 SDRL 34.7925 46.1766 82.7830 136.8407 0.25 ARL 8.8857 10.1607 13.5716 22.7962 SDRL 6.0978 6.9956 10.5585 20.8639 0.50 ARL 2.9398 3.2663 3.6870 4.5188 SDRL 1.6914 1.8133 2.0755 3.0729 0.75 ARL 1.6529 1.7937 1.9516 2.1144 SDRL 0.7845 0.8549 0.9293 1.0806 1.00 ARL 1.2026 1.2713 1.3373 1.3978 SDRL 0.4296 0.4855 0.5340 0.5868 1.25 ARL 1.0408 1.0638 1.0960 1.1111 SDRL 0.1983 0.2472 0.2959 0.3215 1.50 ARL 1.0073 1.0092 1.0168 1.0180 SDRL 0.0851 0.0954 0.1285 0.1344 1.75 ARL 1.0006 1.0011 1.0015 1.0024 SDRL 0.0244 0.0331 0.0387 0.0489 2.00 ARL 1.0000 1.0000 1.0001 1.0001 SDRL 0.0000 0.0000 0.0100 0.0100 3.00 ARL 1.0000 1.0000 1.0000 1.0000 SDRL 0.0000 0.0000 0.0000 0.0000

Table 2 Performance comparison of EWMA-MQDRSS chart when $m=6$ and $ARL0=200$

 CHART EWMA- EWMA- EWMA- EWMA- EWMA- EWMA- $δ$ $R⁢L$ SRS DRSS EDRSS QDRSS MDRSS MQDRSS 0.00 ARL 200.7897 200.1237 200.8548 200.1214 200.559 200.6491 SDRL 216.5813 217.5797 211.9462 214.9439 213.325 218.1511 0.05 ARL 177.6349 113.1382 135.2745 100.0508 88.1721 73.4607 SDRL 190.0858 116.6684 143.7699 101.8467 88.6285 72.1830 0.10 ARL 129.6552 53.2011 74.5652 43.0920 36.7537 27.8757 SDRL 136.3758 50.8719 73.5582 40.4018 33.0570 24.2009 0.25 ARL 47.9933 13.4452 19.5807 10.8511 8.8225 6.5662 SDRL 45.6784 10.6625 16.1643 8.4339 6.6426 4.6751 0.50 ARL 17.1182 4.4524 6.6145 3.5938 2.9796 2.2705 SDRL 13.9193 3.0037 4.7619 2.3726 1.8480 1.2845 0.75 ARL 9.0689 2.4498 3.5120 2.0071 1.6908 1.3788 SDRL 6.8293 1.4363 2.2523 1.0842 0.8466 0.5982 1.00 ARL 5.6700 1.6498 2.3006 1.4112 1.2530 1.0883 SDRL 3.9387 0.8216 1.3090 0.6316 0.4823 0.2868 1.25 ARL 4.0353 1.2868 1.6928 1.1504 1.0630 1.0110 SDRL 2.6824 0.5195 0.8451 0.3773 0.2478 0.1043 1.50 ARL 3.0331 1.1163 1.3910 1.0421 1.0113 1.0008 SDRL 1.8725 0.3301 0.6108 0.2013 0.1057 0.0282 1.75 ARL 2.4162 1.0370 1.1870 1.0075 1.0014 1.0001 SDRL 1.4216 0.1914 0.4229 0.0862 0.0373 0.0100 2.00 ARL 2.0176 1.0088 1.0904 1.0010 1.0000 1.0000 SDRL 1.1141 0.0933 0.2919 0.0316 0.0000 0.0000 3.00 ARL 1.2566 1.0000 1.0008 1.0000 1.0000 1.0000 SDRL 0.4918 0.0000 0.0282 0.0000 0.0000 0.0000

Table 3 Performance comparison of EWMA-MQDRSS chart when $m=7$ and $ARL0=200$

 CHART EWMA- EWMA- EWMA- EWMA- EWMA- EWMA- $δ$ $R⁢L$ SRS DRSS EDRSS QDRSS MDRSS MQDRSS 0.00 ARL 200.986 201.7473 200.0152 201.3467 200.6361 200.555 SDRL 216.9549 215.5833 218.9451 216.4543 214.4325 220.0323 0.05 ARL 173.1492 107.0433 129.0895 94.9536 78.8984 58.7670 SDRL 184.8944 110.5526 137.8684 97.6608 79.0397 56.5090 0.10 ARL 129.1749 48.3775 67.0559 39.7094 30.6945 21.2337 SDRL 133.7282 45.9790 66.5801 36.5590 26.7145 17.9833 0.25 ARL 48.3291 11.8660 17.2783 9.5450 7.2080 4.9118 SDRL 46.0134 9.2268 14.1144 7.3246 5.3592 3.3502 0.50 ARL 17.1921 3.9060 5.9112 3.2105 2.4942 1.8078 SDRL 13.9757 2.5468 4.2227 2.0291 1.4724 0.9450 0.75 ARL 8.9453 2.1744 3.0994 1.8253 1.4774 1.1714 SDRL 6.7397 1.2100 1.9160 0.9486 0.6811 0.3987 1.00 ARL 5.6581 1.4991 2.0679 1.2953 1.1283 1.0182 SDRL 3.9923 0.7091 1.1283 0.5277 0.3447 0.1344 1.25 ARL 4.0113 1.2013 1.5502 1.0925 1.0218 1.0011 SDRL 2.6510 0.4326 0.7472 0.2975 0.1460 0.0331 1.50 ARL 3.0399 1.0600 1.2656 1.0200 1.0027 1.0001 SDRL 1.8907 0.2416 0.4984 0.1414 0.0518 0.0100 1.75 ARL 2.4131 1.0173 1.1274 1.0027 1.0000 1.0000 SDRL 1.3942 0.1319 0.3458 0.0518 0.0000 0.0000 2.00 ARL 2.0153 1.0024 1.0524 1.0005 1.0000 1.0000 SDRL 1.1071 0.0489 0.2246 0.0223 0.0000 0.0000 3.00 ARL 1.2558 1.0000 1.0003 1.0000 1.0000 1.0000 SDRL 0.4902 0.0000 0.0173 0.0000 0.0000 0.0000

Table 4 Performance comparison of EWMA-MQDRSS chart when $m=6$ and $ARL0=370$

 CHART EWMA- EWMA- EWMA- EWMA- EWMA- EWMA- $δ$ $R⁢L$ SRS DRSS EDRSS QDRSS MDRSS MQDRSS 0.00 ARL 370.0326 370.9499 370.1715 372.7233 376.9636 370.7616 SDRL 390.7855 388.4784 384.993 385.2872 390.0622 377.9665 0.05 ARL 313.4847 187.2319 229.7951 163.532 135.0883 107.8952 SDRL 325.4136 184.0379 228.0746 158.6047 132.0583 107.6180 0.10 ARL 212.6695 75.7897 108.5092 61.4427 49.8817 35.7901 SDRL 216.6743 70.9902 103.3676 55.0476 43.4161 29.6895 0.25 ARL 67.2010 16.5372 25.5354 13.1974 10.7230 7.6545 SDRL 60.2927 12.4413 20.3130 9.5039 7.5885 5.1434 0.50 ARL 21.5543 5.2649 8.0259 4.2788 3.4836 2.5779 SDRL 16.5424 3.3811 2.5336 2.6290 2.0544 1.4469 0.75 ARL 10.8764 2.8238 4.1419 2.2533 1.8825 1.4938 SDRL 7.6838 1.5885 2.5336 1.2169 0.9440 0.6660 1.00 ARL 6.7223 1.8559 2.6353 1.5605 1.3482 1.1326 SDRL 4.4488 0.9141 1.4544 0.7139 0.5560 0.3524 1.25 ARL 4.7155 1.4008 1.9305 1.2233 1.1063 1.0195 SDRL 2.9341 0.6033 0.9843 0.4452 0.3165 0.1382 1.50 ARL 3.5408 1.1788 1.5309 1.0697 1.0227 1.0014 SDRL 2.1185 0.4003 0.6927 0.2573 0.1509 0.0373 1.75 ARL 2.7752 1.0639 1.2845 1.0411 1.0027 1.0001 SDRL 1.5760 0.2470 0.5079 0.1179 0.0518 0.0100 2.00 ARL 2.2915 1.0175 1.1401 1.0021 1.002 1.0000 SDRL 1.2349 0.1311 0.3595 0.0457 0.0141 0.0000 3.00 ARL 1.3594 1.0001 1.0021 1.0000 1.0000 1.0000 SDRL 0.5646 0.0100 0.0457 0.0000 0.0000 0.0000

Table 5 Performance comparison of EWMA-MQDRSS chart when $m=7$ and $A⁢R⁢L0=370$

 CHART EWMA- EWMA- EWMA- EWMA- EWMA- EWMA- $δ$ $R⁢L$ SRS DRSS EDRSS QDRSS MDRSS MQDRSS 0.00 ARL 370.0875 370.9484 370.7777 370.8612 370.725 371.5046 SDRL 385.2726 383.7889 380.3405 387.8859 384.3543 393.1296 0.05 ARL 310.4283 166.3566 222.0646 148.7629 119.1026 83.6310 SDRL 321.4991 167.8902 232.1651 148.1591 118.3827 78.0557 0.10 ARL 213.3268 67.1165 98.7937 53.5403 39.9294 27.0221 SDRL 217.2951 62.4886 95.1638 47.4724 34.0929 21.9266 0.25 ARL 67.5817 14.6152 22.0425 11.6448 8.7304 5.9358 SDRL 62.0721 10.7303 17.0660 8.2402 5.9435 3.8783 0.50 ARL 21.5311 4.6158 6.9785 3.7713 2.8505 2.0148 SDRL 16.4577 2.8935 4.6216 2.3092 1.6084 1.0444 0.75 ARL 10.8690 2.4713 3.6330 2.0373 1.6127 1.2470 SDRL 7.6183 1.3560 2.1461 1.0589 0.7637 0.4728 1.00 ARL 6.7976 1.6569 2.3372 1.4206 1.1854 1.0381 SDRL 4.5554 0.7842 1.2372 0.6162 0.4106 0.1924 1.25 ARL 4.7242 1.2750 1.7506 1.1397 1.0375 1.0016 SDRL 2.9632 0.4945 0.8480 0.3511 0.1899 0.0399 1.50 ARL 3.5357 1.1025 1.3878 1.0356 1.0039 1.0000 SDRL 2.1033 0.3111 0.5889 0.1858 0.0623 0.0000 1.75 ARL 2.7870 1.0291 1.1847 1.0049 1.0003 1.0000 SDRL 1.5820 0.1698 0.4144 0.0698 0.0173 0.0000 2.00 ARL 2.2983 1.0069 1.0782 1.0005 1.0000 1.0000 SDRL 1.2267 0.0827 0.2714 0.0223 0.0000 0.0000 3.00 ARL 1.3625 1.0000 1.0006 1.0000 1.0000 1.0000 SDRL 0.5686 0.0000 0.0244 0.0000 0.0000 0.0000

On the basis of ARL with SDRL results presented in Tables 15, a descriptive list of key findings is given as:

• In Table 1, when value of $ξ$ decreases, the shift finding ability of the proposed EWMA-MQDRSS chart escalates for all shifts. For example, for $ξ=0.5,0.25,0.1$ and $0.05$ at $δ=0.10$, the OOC ARL $(ARL1)$ values are 136.5405, 84.3254, 52.0252 and 41.0214 respectively. It indicates that designed chart performs well for several shifts with $ξ=0.05$. Therefore, the value of $ξ=0.05$ is choose for all comparison cases.

• Tables 25 demonstrate that the values of $ARL1$ for proposed chart decline quickly at a fixed $ARL0$ when the process turns into OOC. For $ARL0=200$ with $m=6$, the $ARL1$ value of proposed chart is decreased speedily at 27.8757 for $δ=0.1$.

• The SDRL values of EWMA-MQDRSS chart also decrease speedily when the value of $δ$ increases.

• For a given value of $δ$, the value of $ARL1$ of EWMA-MQDRSS chart declines rapidly as $ARL0$ decreases, e.g. for $δ=0.05$, $ARL0$ values are 370, 200 and the corresponding $ARL1$ values are 107.8952, 73.4607 respectively when $m=6$.

• When the value of $m$ increases, the value of $ARL1$ of offered EWMA-type chart decreases. For instance, for $m=6,7$ at $δ=0.25$ with $ARL0=370$, $ARL1$ values are 7.6545, 5.9358 respectively. This indicates that a greater value of $m$ increases the shift finding ability of EWMA-MQDRSS chart. The decrease in the values of $ARL1$ of the proposed chart is quicker than all existing charting structures.

• The recommended EWMA-MQDRSS chart capably detects small, moderate, and up to a certain degree large shifts for process mean.

## 4 An Application

In this section, a practical example to illustrate the application of the proposed EWMA-MQDRSS chart is provided using a real data set. Suppose that we need to determine statistical control the diameter of piston ring of a machine, produced through a forgoing process (Montgomery, 2009). The 40 samples with size five each, are selected for this process. The measurement of diameters is considered in mm. All samples are combined such that to have a population of 200 measurements and this dataset is utilized for selecting SRS, DRSS, EDRSS, MDRSS, QDRSS, and MQDRSS based samples. We applied the Shapiro-Wilk normality test $(p-value=0.2958>0.05)$ and found that the dataset is normally distributed. We set $ARL0=370$ with $ξ=0.05$ for EWMA-MQDRSS and considered competitor charts.

Thirty samples of size 6 in each are generated using MQDRSS, SRS, QDRSS, EDRSS, MDRSS and DRSS methods, from the population, and these thirty samples are described as Phase-I samples. These sample measurements are exploited for computing the control limits and plotting statistics of EWMA-DRSS, EWMA-EDRSS, EWMA-MDRSS, EWMA-QDRSS, EWMA-SRS and EWMA-MQDRSS charts. Later on, twenty new samples of size 6 in each are again generated from above mentioned charting structures, by adding $0.002$, in the piston rings data, for monitoring of phase-II. The graphical display of the proposed and considered EWMA-type charts, using data, is presented in Figure 1.

The results given in Figure 1 show that the proposed EWMA-MQDRSS and considered EWMA structures are capturing shifts after sample number 30 which indicates the process is IC for the first 30 samples. It is also observed that EWMA-SRS, EWMA-EDRSS, EWMA-DRSS, EWMA-QDRSS, EWMA-MDRSS and EWMA-MQDRSS charts trigger the OOC signal at sample number 48, 40, 38, 36, 35 and 32, respectively. This demonstrates the better shift capturing ability of the designed EWMA-MQDRSS chart as judged against its EWMA-type counterparts. This superiority of proposed chart for real life example validates the run-length results of Section 3 and conclude that suggested EWMA-MQDRSS chart is the best among all its EWMA counterparts.

## 5 Conclusion

In this study, a new memory type chart has been designed to monitor the process mean, by introducing a more efficient MQDRSS scheme and named as EWMA-MQDRSS chart. This chart, for various controlled and out-of-controlled situations, has been evaluated and comparison has been made with conventional EWMA-SRS and DRSS based EWMA type charts (EWMA-EDRSS, EWMA-MDRSS, EWMA-QDRSS and EWMA-DRSS). The performance measures ARL and SDRL are used for this purpose. It is discovered from the simulation results of OOC RL performance that the proposed EWMA-MQDRSS chart outperforms the considered charts by detecting shift first in mean of the process. An illustrative application concerning the monitoring the diameter of the piston ring of a machine determines the superiority of the proposed chart for identification of small and moderate shifts while justifiably sustaining its worth for large shifts. These findings of current study demonstrate that the proposed EWMA-MQDRSS chart is used effectively for mean monitoring than its counterparts and also recommended most beneficial when ranking the units is economical and easier before selection of units for actual measurement. The proposed study can be extended with auxiliary information and a new HEWMA chart could also be designed.

## References

[1] Abujiya, M. R. and Lee, M. H. (2013). The three statistical control charts using ranked set sampling. In: 5th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO). Hammamet Tunisia, 1–6.

[2] Abujiya, M. R. and Muttlak, H. A. (2004). Quality control chart for the mean using double ranked set sampling, Journal of Applied Statistics, 31(10), pp. 1185–1201.

[3] Abujiya, M. R. and Muttlak, H. A. (2007). Monitoring the process mean and variance using ranked set sampling and its modifications, J. Stat. Theory Appl., 6(4), pp. 408–422.

[4] Ali, S., Abbas, Z., Nazir, H. Z., Riaz, M., Zhang, X. and Li, Y. (2020). On designing non-parametric EWMA sign chart under ranked set sampling scheme with application to industrial process, Mathematics, 8, 1497. doi.org/10.3390/math8091497.

[5] Al-Omari, A. I. and Al-Saleh, M. F. (2009). Quartile double ranked set sampling for estimating the population mean, Econ. Qual. Control, 24(2), pp. 243–253.

[6] Al-Omari, A. I. and Haq, A. (2012). Improved quality control charts for monitoring the process mean using double-ranked set sampling methods, Journal of Applied Statistics, 39(4), pp. 745–763.

[7] Al-Saleh, M. F. and Al-Kadiri, M. A. (2000). Double ranked set sampling, Statistics and Probability Letters, 48(2), pp. 205–212.

[8] Haq, A., Brown, J. and Moltchanova, E. (2015). A new exponentially weighted moving average control chart for monitoring the process mean, Quality and Reliability Engineering International, 31(8), pp. 1623–1640.

[9] McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets, Australian Journal of Agricultural Research, 3, pp. 385–390.

[10] Montgomery, D. C. (2009). Introduction to Statistical Quality Control, John Wiley and Sons, New York.

[11] Muttlak, H. A. (1997). Median ranked set sampling, Journal of Applied Statistical Sciences, 6(4), pp. 245–255.

[12] Muttlak, H. A. (2003). Investigating the use of quartile ranked set samples for estimating the population mean, Applied Mathematics and Computation, 146, pp. 437–443.

[13] Noor-ul-Amin, M. and Tayyab, M (2020). Enhancing the performance of exponential weighted moving average control chart using paired double ranked set sampling, Journal of Statistical Computation and Simulation, 90(6), pp. 1118–1130.

[14] Noor-ul-Amin, M., Arif, F. and Hanif, M. (2019). Joint monitoring of mean and variance using likelihood ratio test statistic under pair ranked set sampling scheme, Iranian Journal of Science and Technology, Transections A: Science, 43(5), pp. 2449–2460.

[15] Roberts, S. W. (1959). Control chart tests based on geometric moving averages, Technometrics, 1(3), pp. 239–250.

[16] Shewhart, W. A. (1924). Some applications of statistical schemes to the analysis of physical and engineering data, Bell Technical Journal, 3, pp. 43–87.

[17] Tayyab, M., Noor-ul-Amin, M. and Hanif, M. (2019). Exponential weighted moving average control charts for monitoring the process mean using pair Ranked Set Sampling schemes, Iranian Journal of Science and Technology, Transactions A: Science, 43(4), pp. 1941–1950.

## Biographies

Wasif Yasin is a Ph.D. student at the National College of Business Administration & Economics (NCBA&E), Lahore, Pakistan. He did his M.Phil and M.Sc. degree from the University of the Punjab, Lahore. He is currently working as Secretary Regional Transport Authority, Sahiwal. His research interests include sampling techniques and Statistical Process Control.

Muhammad Tayyab received the master’s degree in Statistics from College of Statistical and Actuarial Sciences, University of the Punjab, Lahore, M. Phil Statistics from Allama Iqbal Open University, Islamabad and philosophy of doctorate degree in Statistics from National College of Business Administration & Economics (NCBA&E), Lahore, Pakistan. He is currently working as Senior Subject Specialist (Statistics) in department of Education. His research areas include mathematical statistics, sampling techniques and control charting structures. He has been serving as a reviewer for many highly-respected journals.

Muhammad Hanif completed his Master’s degree from New South Wales University, Australia in Multistage Cluster Sampling. He completed his Ph.D. in Statistics from the University of Punjab, Lahore, Pakistan. He has more than 40 years of research experience. He is an author of more than 200 research papers and 10 books. He has served as a Professor in various parts of the world i.e. Australia, Libya, Saudi Arabia, and Pakistan. He is presently a Professor of Statistics and Vice-Rector (Research) at NCBA & E, Lahore, Pakistan.