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

Wasif Yasin, Muhammad Tayyab* and Muhammad Hanif

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 idea of control chart used for monitoring the process mean was given by Shewhart (1924) and this chart is very beneficial to determine the major change in process parameters. The EWMA charting structure is developed by Roberts (1959) for monitoring unnatural changes in the process and showed this as a better alternative of the mean chart to trace small or modest changes in the process mean. The sample measurements of the quality characteristic for constructing control charting structures involve cost and time constraints and ranked set sampling (RSS) is an inexpensive and efficient alternate of simple random sampling (SRS) mostly in conditions where measurement on the selected matters are challenging or costly to obtain, but, on the other hand, the ranking of the elements permitting to the study variable is comparatively easy and economical. The RSS along with its modified forms has been effectively used to improve the efficiency of charts for monitoring of process mean. McIntyre (1952) firstly presented the idea of RSS for estimation of the average production of the pasture. The author established it as a valued sampling technique that provides more effectual estimates than the SRS technique. Median RSS (MRSS) and quartile RSS (QRSS) are the efficient modified RSS schemes suggested by Muttlak (1997, 2003) for parametric estimation. In order to pursue a more efficient sampling scheme, Al-Saleh and Al-Kadiri (2000) and Al-Omari and Al-Saleh (2009) suggested extensions of RSS, named double RSS (DRSS) and quartile double RSS (QDRSS) schemes respectively, for estimating the population mean more effectively than usual RSS.

Abujiya and Muttlak (2004, 2007) investigated that Median double RSS (MDRSS) and Extreme double RSS (EDRSS) schemes based charts perform better in detecting shift in the process mean than the conventional RSS scheme. Al-Omari and Haq (2012) used Shewhart chart for betterment in the process monitoring by using DRSS schemes like double quartile RSS (DQRSS), QDRSS and double extreme RSS (DERSS). Abujiya and Lee (2013) extended the work on EWMA, CUSUM and Shewhart charts by applying RSS techniques, instead of SRS technique and they showed that the charts based on RSS techniques to be more sensitive when judge against its competitors in SRS. Haq et al. (2015) produced EWMA charts under perfect mixed RSS (MxRSS) and imperfect mixed RSS (IMxRSS) and concluded based on average run length (ARL), mean deviation of run length (MDRL) and standard deviation of run length (SDRL) values that the new charts are performing more efficiently when compared with classical EWMA chart in tracing a shift in the process location. Tayyab et al. (2019) presented paired RSS based EWMA type charts and explored that these developed charts are cost-effective and more sensible for detection the shift than that of EWMA-SRS chart. Some efficient charts were also proposed by Noor-ul-Amin et al. (2019) by using Paired DRSS (PDRSS), extreme PDRSS (EPDRSS) and quartile PDRSS (QPDRSS) strategies. ARL and SDRL were used to compare the enactment of proposed and existing charts via simulation study. Noor-ul-Amin and Tayyab (2020) enhanced the efficiency of the EWMA chart by using economical and well-organized RSS type techniques. Ali et al. (2020) used the RSS scheme to design a new non-parametric EWMA sign chart and studied the performance of the suggested chart when data is not normal.

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=mr.

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 jth smallest ranked unit from ith subset of the kth 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)1Y1(2)1Y1(3)1Y1(4)1Y1(5)1Y1(6)1Y2(1)1Y2(2)1Y2(3)1Y2(4)1Y2(5)1Y2(6)1Y3(1)1Y3(2)1Y3(3)1Y3(4)1Y3(5)1Y3(6)1Y4(1)1Y4(2)1Y4(3)1Y4(4)1Y4(5)1Y4(6)1Y5(1)1Y5(2)1Y5(3)1Y5(4)1Y5(5)1Y5(6)1Y6(1)1Y6(2)1Y6(3)1Y6(4)1Y6(5)1Y6(6)1]
[Y1(1)2Y1(2)2Y1(3)2Y1(4)2Y1(5)2Y1(6)2Y2(1)2Y2(2)2Y2(3)2Y2(4)2Y2(5)2Y2(6)2Y3(1)2Y3(2)2Y3(3)2Y3(4)2Y3(5)2Y3(6)2Y4(1)2Y4(2)2Y4(3)2Y4(4)2Y4(5)2Y4(6)2Y5(1)2Y5(2)2Y5(3)2Y5(4)2Y5(5)2Y5(6)2Y6(1)2Y6(2)2Y6(3)2Y6(4)2Y6(5)2Y6(6)2]
[Y1(1)6Y1(2)6Y1(3)6Y1(4)6Y1(5)6Y1(6)6Y2(1)6Y2(2)6Y2(3)6Y2(4)6Y2(5)6Y2(6)6Y3(1)6Y3(2)6Y3(3)6Y3(4)6Y3(5)6Y3(6)6Y4(1)6Y4(2)6Y4(3)6Y4(4)6Y4(5)6Y4(6)6Y5(1)6Y5(2)6Y5(3)6Y5(4)6Y5(5)6Y5(6)6Y6(1)6Y6(2)6Y6(3)6Y6(4)6Y6(5)6Y6(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)1Y2(3)1Y3(3)1Y4(4)1Y5(4)1Y6(4)1Y1(3)2Y2(3)2Y3(3)2Y4(4)2Y5(4)2Y6(4)2Y1(3)3Y2(3)3Y3(3)3Y4(4)3Y5(4)3Y6(4)3Y1(3)4Y2(3)4Y3(3)4Y4(4)4Y5(4)4Y6(4)4Y1(3)5Y2(3)5Y3(3)5Y4(4)5Y5(4)5Y6(4)5Y1(3)6Y2(3)6Y3(3)6Y4(4)6Y5(4)6Y6(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)th ranked unit (in boxes), i.e. Yi(2)* from ith subset (i=1,2,3) and choose (3(m+1)4)th ranked unit (in boxes), i.e. Yi(5)* from ith 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 ith subset (i=1,2,,m2) and Yi(q3)* shows third quartile unit from ith subset (i=m+22,,m) for even sample size. Let Yi(q1)* shows first quartile unit from ith subset (i=1,2,,m-12), Yi(q3)* shows third quartile unit from ith subset (i=m+12,,(m-1)) and Ym(q2)* be the median of mth 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=ξY¯(MQDRSS)t*+(1-ξ)Et-1,0<ξ1, (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)2t)2-ξ]. (7)

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

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

where

Var(Y¯(MQDRSS)et*)=1m2[i=1m/2Var(Yi(q1)*)+i=m+22mVar(Yi(q3)*)] (9)

and

Var(Y¯(MQDRSS)ot*)
  =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+δσn,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 RL 2.5199 2.7190 2.9180 2.9950
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 RL 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 RL 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 RL 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 ARL0=370

CHART

EWMA- EWMA- EWMA- EWMA- EWMA- EWMA-
δ RL 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 performance of the proposed EWMA-MQDRSS chart is better in detecting a shift in the process mean than considered EWMA-DRSS, EWMA-EDRSS, EWMA-MDRSS, EWMA-QDRSS, and EWMA-SRS charts.

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

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Figure 1 The shift detection ability comparison between (A) EWMA-SRS, (B) EWMA-EDRSS, (C) EWMA-DRSS, (D) EWMA-QDRSS, (E) EWMA-MDRSS and (F) EWMA-MQDRSS charts.

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

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Biographies

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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.

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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.

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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.

Abstract

1 Introduction

2 Design of the Proposed EWMA-MQDRSS Control Chart

3 Performance Evaluation and Comparative Study

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4 An Application

5 Conclusion

References

Biographies