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Chiang Mai J. Sci. 2013; 40(3) : 485-498 http://it.science.cmu.ac.th/ejournal/ Contributed Paper Statistical Quality Control Based on Ranked Set Sampling for Multiple Characteristics Adisak Pongpullponsak* and Peerawut Sontisamran Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand. *Author for correspondence; e-mail: [email protected] Received: 11 September 2012 Accepted: 18 July 2013 ABSTRACT Many quality control charts for mean have been developed from ranked set sampling (RSS), one of them, so-called ranked set sampling for multiple characteristics (RSSMC). This study is to compare a new chart based on RSSMC data with the control chart established from simple random sampling (SRS). The RSSMC control chart has better average run length (ARL) than the classical chart when a sustained shift in the process mean for error ranking characteristic by using other characteristics is added to help ranking data. To compare the RSSMC method with median ranked set sampling (MRSS), RSS, SRS, in term of out–of–control ARL performance, two characteristics of simulated data is used. It is found that the first data has error ranking, while the second data shows no error ranking when it is compared with the measuring data that is simulated. These data are subsequently used to build the corresponding control chart. All over the study, we use the simulated normal distribution data at the given mean and variance. Keywords: rank set sampling (RSS), quality control, multiple characteristics, average run length (ARL) 1. INTRODUCTION Ranked set sampling (RSS) was introduced by McIntyre [1], who used the technique in estimation of the land for raising domesticated species without the requirement of transportation and harvesting. Later, Takahasi and Wakimoto [2] developed the theory and method of RSS for various problems in agricultural and forestry fields. In 1972, Dell and Clutter [3] studied the theory under the hypothesis of perfect and imperfect RSS. Patil et al. [4] investigated the theory with the sample units of unlimited population size using different sampling approaches based on RSS. Subsequently, Bohn [5] modified RSS for nonparametric statistics before further developed for collecting of environmental samples which contained high variation within populations. For example, in sampling for forest condition survey, a wide range of tree sizes would be experienced extensively affecting the

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Page 1: Statistical Quality Control Based on Ranked Set Sampling ... · Statistical Quality Control Based on Ranked Set Sampling for Multiple Characteristics Adisak Pongpullponsak* and Peerawut

Chiang Mai J. Sci. 2013; 40(3) 485

Chiang Mai J. Sci. 2013; 40(3) : 485-498http://it.science.cmu.ac.th/ejournal/Contributed Paper

Statistical Quality Control Based on Ranked SetSampling for Multiple CharacteristicsAdisak Pongpullponsak* and Peerawut SontisamranDepartment of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,Bangkok 10140, Thailand.*Author for correspondence; e-mail: [email protected]

Received: 11 September 2012Accepted: 18 July 2013

ABSTRACTMany quality control charts for mean have been developed from ranked set

sampling (RSS), one of them, so-called ranked set sampling for multiple characteristics(RSSMC). This study is to compare a new chart based on RSSMC data with the controlchart established from simple random sampling (SRS). The RSSMC control chart hasbetter average run length (ARL) than the classical chart when a sustained shift in theprocess mean for error ranking characteristic by using other characteristics is added tohelp ranking data. To compare the RSSMC method with median ranked set sampling(MRSS), RSS, SRS, in term of out–of–control ARL performance, two characteristics ofsimulated data is used. It is found that the first data has error ranking, while the seconddata shows no error ranking when it is compared with the measuring data that issimulated. These data are subsequently used to build the corresponding control chart.All over the study, we use the simulated normal distribution data at the given mean andvariance.

Keywords: rank set sampling (RSS), quality control, multiple characteristics, averagerun length (ARL)

1. INTRODUCTIONRanked set sampling (RSS) was

introduced by McIntyre [1], who used thetechnique in estimation of the land forraising domesticated species withoutthe requirement of transportation andharvesting. Later, Takahasi and Wakimoto[2] developed the theory and method ofRSS for various problems in agriculturaland forestry fields. In 1972, Dell andClutter [3] studied the theory under thehypothesis of perfect and imperfect RSS.

Patil et al. [4] investigated the theory withthe sample units of unlimited populationsize using different sampling approachesbased on RSS. Subsequently, Bohn [5]modified RSS for nonparametric statisticsbefore further developed for collecting ofenvironmental samples which containedhigh variation within populations. Forexample, in sampling for forest conditionsurvey, a wide range of tree sizes would beexperienced extensively affecting the

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measurement of parameters such as heightand width of the samples. Utilization ofsimple random sampling (SRS) orsystematic sampling would consume highcost in recording the size or height of eachsample. For this reason, RSS seems to bethe best choice for solving differences ofvariables between populations.

Malfunction of machines is one majorproblem faced in many industries, whereinspection on the machines has high costand it is difficult to perform as this mayretard the production process. Accordingly,several statistical methods have beenemployed to enhance the productionefficiency and reduce defective productnumbers, thereby decreasing the cost. Inquality control using quality control chart,SRS has been noted to be not suitable sinceit yields population with highly skeweddistribution causing wide control margins,subsequently low efficiency of control. Forthis reason, RSS has been further developedto solve such a problem. This methodshows higher sampling efficiencycompared with the actual investigation withthe same sample size as it gives narrowercontrol margin which improves theprocess control, consumes low cost andshort period in process. Halls and Dell [6]found that the efficiency of RSS is higherthan SRS efficiency when the same samplesize was examined. In 1997, Salazar andSinha [7] proposed RSS and median rankedset sampling (MRSS), the newly modifiedsampling methods, for investigation ofprocess. Few years later, Muttlak andAl-Sabah [8] developed new quality controlcharts for population mean using RSS andtwo of its modifications before comparedthe average run length (ARL) with that ofthe conventional control chart based onSRS method. Practically, in quality controlby sample are required to be considered.

One example is evaluation of average driedweight of leaves by Ridout and Cobby [9].Since it was very difficult to measure theprecise dry weight of leaves, the spraydeposits of water on both sides, the upperand lower surfaces of the leaves, were usedin estimation. Consideration of spraydeposits on one side may give high rankingerror. Ridout and Cobby [9] employed thecharacteristic of interest in RSS from theproducts containing multiple characteristicsin order to reduce error from ranking andincrease efficiency of sampling. It wasfound that RSS is not appropriate forcontrolling production of goods withmultiple characteristics. The aim of thisstudy is to establish the quality controlchart based on RSS for multiplecharacteristics and compare the efficiencywith the previously mentioned qualitycontrol charts through ARL values.

2. SAMPLING METHODS2.1 Ranked Set Sampling (RSS)

RSS is the method facilitating sampleselection for ranking by considering avariable which is related to a variable ofinterest or a variable that desires to measure.The step of RSS can be described as below.

Step 1. Randomly select n sample unitsper set from the total n sets.

Step 2. Allocate the selected samplesinto the sets by considering a variablerelated to a variable of interest.

Step 3. Choose a sample unit for actualanalysis starting from the smallest rank inthe first set, then the smallest rank in thesecond set, continuing the procedure untilthe last set the largest rank is chosen.

Step 4. Repeat steps 1 through 3 for rcycles until the desired sample size isobtained.

As seen in Figure 1. Samples arerandomly selected for 3 sets, where each

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Figure 1. Sample units for RSS.

set contains 3 sample units, then repeat thisprocedure for 4 times.

Illustrating in Figure 1, each rowcontains ranked sample units, and only thesample units marked as @ is chosen foractual measurement. Therefore, from thetotal 36 sample units which are randomlyselected from 4 cycles, only 12 units willbe used for measuring a variable of interest.Furthermore, for some situations theselected samples could not be ranked dueto the problem of making decision,another characteristic that is related to thecharacteristic of interest or consuminglower costs in process may be used insample ranking. Given X(i:n)j is the ith orderstatistic from the set of size n in the jthcycle. Takahasi and Wakimoto [2] proposedan estimator of population mean for RSSas below;

(1)

which is an unbiased estimator of μ.The variance of rss,j is defined by

var (2)

2.2 Median Ranked Set Sampling (MRSS)MRSS was presented by Muttlak [10].

Using this method, the sample at themedian of the sets is selected, if the set sizen is odd. In case of the set size with evennumber, sample selection is from the (n/2)th order in the first half and the ((n+2)/2)th order in the second half of the set.The method of MRSS can be concluded asfollows.

Step 1. Select n sample units per setfrom the total n sample sets.

Step 2. Allocate sample units into theset by using a variable related to a variableof interest in ranking.

Steps 3. Choose the sample units foractual measurement by selecting the smallestrank in the ((n+2)/2)th order from thesample sizes with an odd number. For thesample sets with an even number, thesmallest rank in the (n/2)th order of thefirst half and the smallest rank in the((n+2)/2)th order in the second half arechosen.

Step 4. Repeat steps 1 through 3 for rcycles until the desired sample size isobtained.

As seen in Figure 2. Samples are

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488 Chiang Mai J. Sci. 2013; 40(3)

In Figure 2, each row contains rankedsample units, and only the sample unitsmarked as @ is chosen for actualmeasurement. Therefore, from the total 36sample units which are randomly selectedfor 4 cycles, only 12 units will be used for

Figure 2. Sample units for MRSS - case I.

measuring a variable of interest.As seen in Figure 3, samples are

randomly selected for 4 sets, where eachset contains 4 sample units, then repeat thisprocedure for 4 times.

In the illustration above, each rowcontains ranked sample units, and only thesample units marked as @ is chosen foractual measurement. Therefore, from thetotal 36 sample units which are randomlyselected for 4 cycles, only 12 units will be

Figure 3. Sample units for MRSS - case II.

used for measuring a variable of interest.For the sample sizes with an odd

number, given X(i:n)j is the (n/2) orderstatistic of the ith order from sample sizen in the jth cycle.

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In case of an even number, X(i:n)j is the(n/2)th order statistic of the ith orderfrom sample size n(i = 1,2,...,L where L =n/2), and the ((n+2)/2)th order statisticof the th order from sample size, whereMuttlak [10] proposed an estimator ofpopulation mean for MRSS as below;

(3)

The variance of mrss, j is defined by

(4)

2.3 Ranked Set Sampling for MultipleCharacteristics (RSSMC)

Generally, in RSS only one charac-teristic is used in ranking samples, whereranking errors may easily occur. Thus itis recommended to consider multiplecharacteristics in ranking to reduce errors.Ridout and Cobby [9] described theprocedure for selection of characteristicsused in ranking below:

Step 1. Select characteristics of interest,then designate as i = 1, i = 2 and so on.

Step 2. Let nij denote the number ofsamples ranked in the jth order for theith characteristic, and Vi is the samplevariance of the ith characteristic in thesample number nij. For a balanced

sampling, variance of the ith characteristicequals zero (Vi = 0). When no ranking erroroccurs, a new set of size n is establishedby calculating the Vi values from eachcharacteristic in the sample set to estimatethe values of V1 and V2 .

Step 3. Select the sample thatminimizes V =V1 +V2 from each sampleset; in case of absence of ranking errors,select sample in the same way as MRSS; ifthere is more than one such sample, selectone of them at random.

Step 4. Repeat steps 1 through 3 for rcycles until the desired sample size isobtained.

From the steps described above, theprocedure of RSSMC can be performed asfollows, giving the weight as the variableof interest or the variable to be actualmeasurement, and the height and width asvariables that may relate to the weight. Forthis reason, we use the height and width asvariables for ranking as shown below.

Considering RSS for multiple charac-teristics, a selected sample is the one withthe smallest variance or the one that is themedian of the samples ranked by multiplecharacteristics. Muttlak and Al-Sabah [8]developed a control chart using the samplesthat are the median of MRSS data.

Characteristic Sample Sample SampleSample Sample Sample Sample Sample Sample Sample#1 #2 #3 #4 #5 #1 #2 #3 #4 #5

Height (A1) 63.17 72.61 65.99 69.28 79.62 V1 48.54 6.13 17.14 0.73 90.03Width (A2) 19.96 16.99 12.12 14.82 16.99 V2 14.33 0.66 16.46 1.83 0.66Weight 56.22 50.93 44.17 49.71b 51.43 V 62.87 6.79 33.60 2.57c 90.70

Table 1. A sample set from using the RSSMC method when the sample size n = 5a.

Note: a In each set, the data are obtained from the same method until we have n samplesfrom n sets repeated for r rounds; b is the sample selected by calculating fromVi = (aij − i )

2; c is a minimum value of V1 +V2 when A is the variable of characteristicused in ranking.

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490 Chiang Mai J. Sci. 2013; 40(3)

Given X(i:mc)j is the order statistic thathas the smallest sum variance value forsample size n in the jth cycle, where theRSS estimator for multiple characteristicscould be calculated by

(5)

The variance of rssmc, j is defined by

(6)

3. QUALITY CONTROL CHARTSStatistical process control (SPC) is an

effective method used to improve qualityof an industry and production. Theobjective of SPC is to rapidly determinefor any changes or problems in the process.SPC composes of two major toolsinvolving in control of the productionprocess, and sampling for acceptable andcontrol chart. The control chart is a processcontrol technique that has been widelyemployed. It requires only few samples ininvestigation of the process, making SPCbecomes a high efficient approach ininspection of changes in the process thatmay affect to the product quality.

Statistical quality control was establishedin 1924 after Shewhart [11] introduceda control chart for fractional nonconformingunits. Subsequently, in 1950 Aroian andLevene [12] proposed the first example ofestablishing 3 parameters for consideration,which are sample size, control limit, andtime between sampling. After that, Weiler[13] employed sample sizes in constructionof a model used in estimation of theminimum mean before changes in processoccur. By using his established model, timebetween sampling and probability ofinspecting the impact when the process isout of control, so-called the average runlength (ARL), could be estimated. In 1997,

Salazar and Sinha [7] developed an averagecontrol chart based on RSS when thepopulation has normal distribution and theprocess has various shift values. They usedthe Monte Carlo model in calculating ARLwhen changes in process occur for averagecontrol charts based on both RSS andMRSS in cases of perfect and imperfectranking. Later, Muttlak and Al-Sabah [8]modified RSS into more efficient methods;extreme ranked set sampling (ERSS), pairedranked set sampling (PRSS), and selectedranked set sampling (SRSS), when thepopulation has normal distribution andthe process has various shift values. Theycalculated various ARL values usingcomputer simulations and concluded thatevery average control chart based on theirestablished methods is more useful thanthat of Shewhart. The control chart usedin this study is as follows.

3.1 Quality Control Chart Using SRSLet Xij when i = 1, 2,...,n and j = 1,

2,...,r, is the ith unit in the jth order ofsample size n, and Xij ~ N (μ, σ 2). Whenthe population mean and variance are μand σ 2, respectively. The Shewhart controlchart for

(7)

is defined by

(8)

when UCL, CL, and LCL are uppercontrol limit, centerline, and lower controllimit, respectively. After obtained thechart, the population mean j , j = 1, 2,...,rcan be plotted into the upper control chart[14]. However, in actual measurement,

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Chiang Mai J. Sci. 2013; 40(3) 491

the mean μ and variance σ 2 are unknownso both μ and σ are estimated from thecollected data, where the unbiasedestimator for μ is

(9)

but

(10)

where

(11)

is biased estimate for σ . We can use

/c4 as unbiased estimate for where c4 iscalculated from

(12)

and the control chart for sample mean canbe expressed as

(13)

3.2 Quality Control Chart Using RSS

The RSS mean rss,j at the jth cyclecan be plotted in the control chart basedon RSS proposed by Salazar and Sinha [7]as follows,

(14)

when

(15)

where the unbiased estimate for RSS [2] is

(16)

The estimate for suggested by Muttlakand Al-Sabah [8] will be as

(17)

when

(18)

and

(19)

is the estimate for population mean of theith order statistic. Subsequently, thecontrol chart can be constructed by using

rss and as the equation below,

(20)

3.3 Quality Control Chart Using MRSS

The MRSS mean mrss, j of the jth cyclecan be plotted in the control chart basedon MRSS proposed by Muttlak [10] asfollows

(21)

where

(22)

In practical, the values of μ and σmrss

are unknown so an unbiased estimator μis calculated from MRSS data with normaldistribution as shown below,

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492 Chiang Mai J. Sci. 2013; 40(3)

(23)

And the estimate for σmrss

suggested byMuttlak and Al-Sabah [8] is given by

(24)

where X(i:m)j is the estimator for population

mean of the ith order statistic. Thus rssand σ̂

mrss can be used to construct the

control chart from MRSS as follows

(25)

4. RESULTS4.1 Quality Control Chart Using RSSMC

The control chart based on RSSMCand the RSSMC estimator was firstproposed by Pongpullponsak andSontisamran [15]. However, performanceof the proposed estimator was unsatisfactory.For this reason, the aim of this study is toimprove the estimator as could beconcluded following.

The RSSMC mean rssmc, j of the jthcycle which can be plotted on the controlchart based on RSSMC is calculated by

(26)

where

(27)

In actual measurement, the μ andσ

rssmc are unknown so the estimator for

μ based on RSSMC data, when thedistribution is normal, is given by

and the estimator for

where X(i:mc) j is the estimate for populationmean of the ith order statistic. The control

chart can be constructed using the rssmcand σ̂

rssmc as follows

(28)

4.2 Control Limits Using SRS, RSS, MRSSand RSSMC Methods

The study of limits of control chartby using SRS, RSS, MRSS and RSSMCshows how the constructing RSSMC canreduce the variation due to samplingmethod. In this study, we use the simulationdata which has the normal distribution withmean μ and variance σ 2 by the statisticalpackage R. After simulating data, we selectthe samples using SRS, RSS, MRSS andRSSMC and use them to construct thequality control chart. From using differentsampling methods with the sample sizen = 3, 4, the results show that(1) When the sample size is increased, the

control limits from using SRS,RSS,MRSS, and RSSMCare narrowedwhich satisfy the principal ofconstructing control chart for mean.

(2) From comparisons of all limits ofcontrol chart obtained from SRS,RSS,MRSS and RSSMC, both controlcharts based on SRS and MRSS have themaximum control chart width whereasthe minimum width is seen only in theRSSMC control chart. Thus we canconclude that the centerline which is theother tool should be included inconstruction of a control chart. It is

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found that if the width of the controlchart is large, the variation will be largetoo. We also observe that the controlchart for RSSMC constructing in this

Table2. Control limits of SRS, RSS, MRSS and RSSMC for n = 3, 4.

study has the minimum width whichresults in the minimum variation. Thedetails are described as following inTable 2 and Figures 4-7.

Figure 4. Quality control chart using SRS when sample size n=3 (a) and n=4 (b).

a b

Figure 5. Quality control chart using RSS when sample size n=3 (a) and n=4 (b).

a b

Figure 6. Quality control chart using MRSS when sample size n=3 (a) and n=4 (b).

a b

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Figure 7. Quality control chart using RSSMC when sample size n=3 (a) and n=4 (b).

a b

4.3 Comparisons of Efficiency of theControl Charts Constructed UsingSRS, RSS, MRSS and RSSMC

In constructing control chart, the rangeof average run length (ARL) is a criterionneeded to be critically concerned. For thisreason, to evaluate performance of theconstructed control charts the ARL, wherethe process is under control with mean μ0

and standard deviation σ0, and the processmay occasionally begin to be out of controle.g. shifting of the mean from μ0 to μ0 +δσ0 /√

−n = μ , is used in consideration. Inthe study, we assume that the process isfollowing a normal distribution with meanμ0 and variance μ2

0 if the process is undercontrol, and the shift on the process meanis δ = (√ −n / σ0)|μ – μ0|. If δ = 0 the processis under control and in this case when thepoint is outside the control limits, then itis a false alarm.

In sample collection, sample sizes ofeach cycle for SRS, RSS, MRSS andRSSMC are n = 3, 4, 5, 6, 7, as describedin sections 2.1 – 2.3. After constructingthe control charts according to sections3.1 - 3.3, and 4.1, the ARL for control limitsof each chart is estimated by averaging thecycle times until the first out-of-controlgroup is obtained.

Normally, imperfect ranking will

occur if the variable of interest containingerrors in ranking the units is ranked. Toevaluate the constructed control chartperformance, the data used in the studyconsist of 3 related variables with normaldistribution. For each value of ARL wesimulate 10,000 replications and thecomputer simulations are run for n=3, 4,5, 6, 7 and δ = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5,3.0.

From the results shown in Table 3 andFigure 8, conclusion can be as follows.(1) In case that the process is under control,

i.e. δ = 0, using MPRSS the number offalse alarms will not be increased whencompared with SRS, RSS and MRSS.In fact, there is small decrease in theARL; for example, for n=3, ARL=341.993 as compared to 346.090,345.470 and 347.886 for SRS, RSS andMRSS respectively.

(2) If the sample size increases, the ARL willdecrease if δ > 0; for example if thesample size is 4 and δ = 1.0 the ARL is15.944 as compared with 22.995 in thecase of n=3.

(3) The ARL value for the RSS will decreasemuch faster than the case of SRS if δincreases. This increase in ARL valuewill depend on the correlation betweenthe variable of interest and the

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concomitant variable that we use toestimate the rank of the variable ofinterest.

(4) For the data with error ranking, the

ARL for RSSMC is less than that of SRSbecause some variables used in rankingare related to a variable of interestwithout actual measurement.

Table 3. ARL value of each sampling method for n = 3, 4, 5, 6, 7 when δ is 0, 0.5, 1.0,1.5, 2.0, 2.5, 3.0.

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Table 3. (Continue).

Shift on the process mean

AR

L f

or

n =

7

Shift on the process mean

AR

L f

or

n =

5

Shift on the process mean

AR

L f

or

n =

3

Shift on the process mean

AR

L f

or

n =

4

Shift on the process mean

AR

L f

or

n =

6

Figure 8. Comparisons of ARL value ofcontrol chart using SRS, RSS, MRSS andRSSMC when n = 3, 4, 5, 6, 7.

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5. CONCLUSIONS AND DISCUSSION5.1 Conclusions

Generally, for two groups of statisticalprocess control (SPC), univariate statisticalprocess control (USPC) and multivariatestatistical process control (MSPC), RSS isproved to be more efficient when units aredifficult and costly to measure, but will beeasy and cheap to rank with respect to avariable of interest without actualmeasurement. In this study, we use theRSSMC in developing control charts forsample mean. When compared thesecontrol charts with the control chartsmentioned above, we find that they havemore efficiency and satisfy Muttlak andAl-Sabah’s [8] statistical quality controlbased on RSS. The following are somespecific conclusions.(1) Quality control chart using RSSMC

dominates the classical charts. If theprocess starts to get out of control, thischart reduces the number of ARLssubstantially.

(2) To overcome the problem of errors inranking and/or to increase the efficiencyof estimating the population mean, wesuggest that using RSSMC instead ofRSS and MRSS can reduce the errorsin ranking. The RSSMC dominates allother methods in terms of reducing theARL if the process starts to get out ofcontrol.

Finally, we recommend using theRSSMC for construction of both USPCand MSPC, in case that the data have errorranking or unknown some characteristicsrelated to a variable of interest withoutactual measurement, as it can reduce theARL compared to the previouslymentioned control charts.

5.2 DiscussionComparisons of the control chart for

population mean developed in this studywith the RSS, SRS, MRSS and ERSScontrol charts reveal that our constructedcontrol chart is more efficient and satisfyMuttlak and Al-Sabah’s [8] statisticalquality control based on RSS. In theirreport, they documented that MRSS andERSS have more efficiency as the ARLvalues are less than those of the othermethods when the shift on the processmean occurs because some data have errorranking.

6. SuggestionThe results in the present study show

that the ARL obtained from using RSSMCmethod is less than the ARL of othermethods. However, when the variable usedfor ranking is highly related to a variableof interest without actual measurement, theARL obtained from MRSS and RSS arenot different from that of RSSMC. Forthe future research, we will optimize thecost and sample unit for RSS with multiplecharacteristics.

REFERENCES

[1] McIntyre G.A., A method for unbiasedselective sampling, using, ranked sets,Aust. J. Agric. Res., 1952; 3: 385-390.

[2] Takahasi K. and Wakimoto K., On theunbiased estimates of the populationmean based on the sample stratified bymeans of ordering, Ann. Inst. Stat.Math., 1968; 20: 1-31.

[3] Dell T.R. and Clutter J.L., Ranked setsampling theory with order statisticsbackground, Biom., 1972; 28: 545-553.

[4] Patil G.P., Sinha A.K. and Taillie C.,Ranked set sampling, A Handb. Stat.,1993; 1: 51-65.

[5] Bohn L.L., A review of non-parametricranked set sampling methodology,

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Comm. Stat. - Theory and Methods,1996; 25: 2675-2685.

[6] Halls L.K. and Dell T.R., Trial ofranked set sampling for forage yields,For. Sci., 1966; 12: 22-26.

[7] Salazar R.D. and Sinha A.K., Controlchart x-bar based on ranked setsampling, Comun. Tecica, 1997, No.1-97-09 (PE/CIMAT).

[8] Muttlak H.A. and Al-Sabah W.S.,Statistical quality control based onranked set sampling, J. Appl. Stat.,2003; 30: 1055-1078.

[9] Ridout M.S. and Cobby J.M., Rankedset sampling with non-randomselection of sets and errors in ranking,Appl. Stat., 1987; 36: 145-152.

[10] Muttlak H.A., Median ranked setsampling, J. Appl. Stat. Sci., 1997; 6:245-255.

[11] Shewhart W.A., Some applications ofstatistical methods to the analysis ofphysical and engineering data, BellTechnol. J., 1924; 3: 43-87.

[12] Aroian L.A. and Levene H., Theeffectiveness of quality control charts.J. Am. Stat. Assoc., 1950; 45: 520-529.

[13] Weiler H., On the most economicalsample size for controlling the meanof a population, Ann. Math. Stat.,1952; 23: 247-254.

[14] Montgomery D.C., Introduction toStatistical Quality Control, 5thedn,New York, John Wiley & Sons, 2009.

[15] Pongpullponsak, A. and SontisamranP., Statistical quality control based onranked set sampling for multiplecharacteristics, Proc. Int. Conf.Sustainable Greater Mekong Subregion, 2010; 26-27, August, Bangkok,Thailand.