design of x-bar control chart for resampling under ...fs.unm.edu/neut/designofxbarcontrol.pdf ·...

Received April 17, 2019, accepted May 1, 2019, date of publication May 6, 2019, date of current version May 21, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2915007

Design of X-Bar Control Chart for ResamplingUnder Uncertainty EnvironmentMUHAMMAD ASLAMDepartment of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia

e-mail: [email protected]

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant 130-174-D1440.

ABSTRACT This paper presents the designing of the X-bar control chart using resampling scheme underthe uncertainty environment. The necessary measures of the present chart are derived under the neutrosophicsystem. The neutrosophic average run length (NARL) when the process is in-control and out-of-controlare derived under the neutrosophic statistical interval method (NSIM). The neutrosophic control chartcoefficients are determined through the algorithm developed under NSIM. The comparative study showsthat the proposed chart is better than the existing chart in NARL.

INDEX TERMS Monitoring, Shewhart chart, neutrosophic average run length, statistical interval method,shift.

I. INTRODUCTIONThe monitoring of the manufacturing process is very impor-tant to improve the quality of the product. Therefore, to meetspecifications limits or the target, a quick indication is neededwhen the process moves away from the given target. Thecontrol chart is effective tool for the monitoring of themanufacturing process in the industries and service compa-nies. The Shewhart control charts have many applications inthe industries. The Shewhart control charts are easy in theimplementation and save time in deciding the state of theprocess. The attribute and variable Shewhart control chartsare used to reduce non-conforming items. In earlier chart isapplied when data coming from the process is discrete andlater one is used for the monitoring the process when datais measurable. The X-bar control chart is contained moreinformation about the process data than the attribute controlchart such as the np-control chart. [1] pointed out that thecontrol charts are applied to monitor the variation in theprocess. [2] discussed the applications of the control charts inthe industry [3] proposed the variable X-bar control chart. [4]proposed the variable control chart using efficient samplingschemes. [5] designed the control chart for monitoring thevariability in the process. More details on the applications ofthe control charts can be read in [6], [7] and [8].

Although, the Shewhart control charts are designed usingsingle sampling is easy to apply but needs a larger sample for

The associate editor coordinating the review of this manuscript andapproving it for publication was Choon Ki Ahn.

the decision-making. Larger the sample size means the highcost for the monitoring of the process. Further, the controlcharts using the single sample cannot be applied when theexperimenters are in-decision at the information of the firstsample. The repetitive sampling scheme introduced by [9]provides a decision on less sample than single sampling.The repetitive sampling guides the experiments to resampleif in-decision at the first sample information. Because ofmany applications of the repetitive sampling in the controlcharts, several authors proposed the control chart using itand showed the efficiency over the control charts using thesingle sampling. [10] proposed the X-bar control using repet-itive sampling. [11] worked on double EWMA chart usingthis sampling. [12] worked on the process capability indexunder repetitive sampling. More information can be readin [13]–[18] and [19].

Usually, the control charts are designed under the assump-tions that all observations or parameters are known and deter-mined. As mentioned by [1],‘‘observations include humanjudgments, and evaluations and decisions, a continuous ran-dom variable of a production process should include the vari-ability caused by human subjectivity ormeasurement devices,or environmental conditions. These variability causes createvagueness in the measurement system’’. To deal with thesesituations, the control charts using the fuzzy logic are appliedfor the monitoring of the process. [20] proposed the fuzzychart for more than two processes. [21] proposed the fuzzyattribute chart. The fuzzy based attribute and variable control

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M. Aslam: Design of X-Bar Control Chart for Resampling Under Uncertainty Environment

TABLE 1. The NARL when nN ε [2], [5].

TABLE 2. The NARL when nN ε [5], [7].

charts are studied by [22]. [23] proposed the X-bar and rangechart using this approach. More details about such controlcharts can be seen in [24]–[30] and [31].

In practice, it may not possible that all observations aredetermined and precise. To deal with the situation, neutro-sophic statistics (NS) which is based on the neutrosophic

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TABLE 3. The values of NARL when nN ε [8], [10].

TABLE 4. The NARL for various nN .

logic that is the generalization of fuzzy logic, see [32]and [33]. The NS, which is the extension of classical statisticsand can be applied efficiently than classical statistics under

the uncertainty environment. [34] and [35] used the neutro-sophic numbers in rock measuring problems. [36] introducedthe idea of NS in the area of quality control. [37] proposed

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TABLE 5. The comparison of both charts when nN ε [5], [7].

the attribute control chart using the NS. [38] worked onthe variance chart using the NS. [39] designed the chartto monitor reliability under the NS. [40] worked on thegamma chart using the NS and [41] proposed the NS basedattribute chart using the repetitive sampling. [42] proposedthe X-bar control chart using the single sampling. In thisexisting control chart, a decision about the state of the processis taken on the basis of information obtained from the singlesample.

The presence of uncertain observations in the productiondata may mislead the industrial engineers for the monitor-ing of the manufacturing process. For example, the controlchart under classical statistics when the data have uncertainvalues or neutrosophic numbers can show that the processis in control when actually some plotting statistic are withinthe indeterminacy limits. Therefore, in this case, the processmonitoring should be done using the control chart under NS.By exploring the literature and best of our knowledge, thereis no work on the design of X-bar control using repetitivesampling under the neutrosophic statistical interval method(NSIM). This manuscript presents the designing of the X-barcontrol chart using repetitive sampling under the uncertaintyenvironment. We will discuss the advantages of the presentchart over available in the literature. We expect that theproposed control chart will perform better in detecting the

shift in the process than the existing charts under the uncer-tainty environment.

II. DESIGN OF THE PROPOSED CHARTLet XL and XU are the lower and upper values of anindeterminacy interval. Suppose that XNiε {XL ,XU } =i = 1, 2, 3, . . . , nN be a neutrosophic random variableexpressed in indeterminacy interval follows the neutrosophicnormal distribution with neutrosophic mean µN ε {µL , µU }and neutrosophic standard deviation σN ε {σL , σU }. LetXN ε

{XL , XU

}and sN = {sL , sU } be the neutrosophic sample

mean and standard, respectively and considered as the bestlinear unbiased estimator ofµN ε {µL , µU } andσN ε {σL , σU }.We definemN ε {mL ,mU } be the target mean for the in-controlprocess. The neutrosophic upper and lower control limits aregiven as follows

LCL1N = mN − k1NσN√nN; µN ε {µL , µU } ,

σN ε {σL , σU } , mN ε {mL ,mU } (1)

UCL1N = mN + k1NσN√nN; µN ε {µL , µU } ,

σN ε {σL , σU } , mN ε {mL ,mU } (2)

LCL2N = mN − k2NσN√nN; µN ε {µL , µU } ,

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FIGURE 1. The proposed chart for the simulated data.

σN ε {σL , σU } , mN ε {mL ,mU } (3)

UCL2N = mN + k2NσN√nN; µN ε {µL , µU } ,

σN ε {σL , σU } , mN ε {mL ,mU } (4)

Note here that k1N ε {k1L , k1U } and k2N ε {k2L , k2U } are theneutrosophic control chart coefficients associated with neu-trosophic lower control limit (NLCL) and neutrosophic uppercontrol limit (NUCL), respectively.

Based on this information, we propose the following X-barcontrol chart for using the repetitive sampling under the NS.

Step-1:Compute XN ε

{XL =

nLXL∑i=1

/nL , XU =nUXU∑i=1

/nU

}based on XNiε {XL ,XU } = i = 1, 2, 3, . . . , nN .Step-2: Declare the in-control state if LCLN2 ≤ XN ≤

UCLN2.Step-3:Declare the out-of-control state if XN ≥ UCLN1 or

XN ≤ UCLN1.The proposed control chart is the generalized form of

several control charts. For example, it reduces to [42] chartwhen k1N = k2N ; k1N ε {k1L , k1U } , k2N ε {k2L , k2U }. Theproposed chart becomes the traditional repetitive chart pro-posed by [43] under classical statistics when k1L = k1U andk2L = k2U . The proposed chart reduces to the traditional

Shewhart control chart under classical statistics when k1N =k2N and k1L = k1U and k2L = k2U .

Now, we derive the necessary measures of the proposedcontrol to evaluate its performance. The neutrosophic prob-ability for the in-control process for the single sampling isgiven by

P0inN1=P(LCL2N ≤ XN ≤ UCL2N

)=P

(LCL2N − mNσN /√nN

≤XN − mNσN /√nN≤UCL2N − mNσN /√nN

);

XN ε{XL , XU

},mN ε {mL ,mU } (5)

Let LCL2N−mNσN/√

nN= ZN be the neutrosophic standard normal

variable. After, some simplification, we have,

P0inN1 = 1−8N (k2N )+8N (k1N ) ;

k1N ε {k1L , k1U } , k2N ε {k2L , k2U } (6)

where 8N (q) shows the neutrosophic cumulative distribu-tion function. The neutrosophic repetitive probability for thein-control process is given by

P0PrepN = P(LCL1N ≤ XN ≤ LCL2N

)+P

(UCL2N ≤ XN ≤ UCL1N

)(7)

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FIGURE 2. The existing chart for the simulated data.

The Eq. (7) can be written as

P0PrepN

= P(LCL1N − mNσN /√nN

≤XN − mNσN /√nN≤LCL2N − mNσN /√nN

)+P

(UCL2N − mNσN /√nN

≤XN − mNσN /√nN≤UCL1N − mNσN /√nN

);

XN ε{XL , XU

},mN ε {mL ,mU } (8)

The simplified form of Eq. (8) is given by

P0PrepN = (8N (−k2N )−8N (−k1N ))+ (8N (k1N )

−8N (k2N )); k1N ε {k1L , k1U } , k2N ε {k2L , k2U }

(9)

Finally, the resampling probability for an in-control processunder NSIM is given by (10), as shown at the bottom of thispage. The neutrosophic average run length (NARL) indicateswhen on the average the process will be out-of-control underthe indeterminacy. The NARL for the in-control process isgiven as

NARL0N =1

1− P0inN;ARL0Nε {ARL0L,ARL0U} (11)

Now, we assume that the process has shifted with a shiftconstantc, to a new neutrosophic meanm1N = mN + cσN ;m1N ε {m1L ,m1U } , σN ε {σL , σU }. The neutrosophic probabil-ity for the in-control process when the process has shifted atm1N = mN + cσN ; m1N ε {m1L ,m1U } , σN ε {σL , σU } for the

P0inN =1−8N (k2N )+8N (k1N )

1− [(8N (−k2N )−8N (−k1N ))+ (8N (k1N )−8N (k2N ))]; P0inN ε

{P0inL ,P

0inU

}(10)

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FIGURE 3. The proposed chart for the real data.

single sampling is given by

P1inN1 = 1−8N(k2N + c

√nN)+8N

(−k2N + c

√nN);

k1N ε {k1L , k1U } , k2N ε {k2L , k2U } (12)

The neutrosophic repetitive probability for the in-control atthe process atm1N = mN+cσN ;m1N ε {m1L ,m1U } , σN ε{σL ,

σU } is given by

P1PrepN |m1N = P(LCL1N ≤ X1N ≤ LCL2N

)+P

(UCL2N ≤ X2N ≤ UCL1N

);

X1N ε{X1L , X1U

}, X2N ε

{X2L , X2U

}(13)

or

P1PrepN = 8N(−k1N + c

√nN)−8N (−k2N

+ c√nN)+8N

(k2N + c

√nN)

−8N(k1N + c

√nN);

k1N ε {k1L , k1U } , k2N ε {k2L , k2U } (14)

Finally, the neutrosophic probability that the process isin-control at m1N = mN + cσN ; m1N ε {m1L ,m1U } , σN ε

{σL , σU } using the repetitive sampling is given by (15), asshown at the bottom of this page. The NARL atm1N = mN +cσN ; m1N ε {m1L ,m1U } , σN ε {σL , σU } is given by

NARL1N =1

1− P1inN;ARL1Nε {ARL1L,ARL1U} (16)

Tables 1-3 show the values of NARL for nN ε [2], [5],nN ε [5], [7], nN ε [8], [10], various values of c and variousvalues of specifiedNARL0N, say r0N . Table 4 is presentedthe values of NARL for various values of nN ε [nL , nU ]. FromTables 1-4, we note the following trends in the control chartparameters under the NS.

1) For a fixed value of nN ε [nL , nU ], the indeterminacyinterval of NARL decreases as c increases.

2) For other fixed neutrosophic parameters, the indeter-minacy interval of NARL decreases as nN ε [nL , nU ]incerses.

3) From Table 4, we note that as nN ε [nL , nU ] increases,indeterminacy interval of NARL increases.

A. ALGORITHMThe following algorithm under the NSIM is used to eval-uate the values of k1N ε {k1L , k1U } , k2N ε {k2L , k2U } andARL1Nε {ARL1L,ARL1U}.

1) Specify the suitable values of ARL0Nε{ARL0L,

ARL0U}, c and nN ε [nL , nU ].2) Determine P0inN ε

{P0inL ,P

0inU

}and NARL0N using

Eq. (10) and Eq. (11), respectively.3) Determine the values of k1N ε {k1L , k1U } , k2N ε{k2L , k2U } such that NARL0N ≥ r0N .

4) Choose those combinations of k1N ε {k1L , k1U } , k2N ε{k2L , k2U } where NARL0N is very close to r0N .

P1inN =1−8N

(k2N + c

√nN)+8N

(−k2N + c

√nN)

1−[8N

(−k1N + c

√nN)−8N

(−k2N + c

√nN)+8N

(k2N + c

√nN)−8N

(k1N + c

√nN)] ; P1inN ε {P1inL ,P1inU}

(15)

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FIGURE 4. The existing chart for the real data.

5) Use k1N ε {k1L , k1U } , k2N ε {k2L , k2U } to find ARL1Nε

{ARL1L,ARL1U} for various c.

III. COMPARATIVE STUDIESThe efficiency of the proposed control chart with the availablecompetitor control chart under the NSIM proposed by [42] isdiscussed in this section. [42] showed that X-bar under NSperforms better than the X-bar chart under classical statisticsin ARL. We will discuss the advantages of the proposedchart with [42] in terms of NARL and using the simulateddata at the same values of specified parameters of bothcontrol charts. For this comparison, let ARL0Nε {370, 370}and nN ε [5, 7]. The values of the proposed chart andAslam and Khan (2019) are shown in Table 5.

From Table 5, it can be noted that the proposed controlchart has the smaller values of NARL than the existingcontrol chart at all values of c. Note that when c = 0.05,the indeterminacy interval of NARL values for the pro-posed control chart is ARL1Nε [262.337533, 249.664424]while it is ARL1Nε [349.323906, 345.119982] from the

Aslam and Khan (2019) control chart. From this comparison,it is concluded that the proposed control will detect the shiftin the process between the 249th sample and the 262nd samplewhile the existing chart will indicate the shift in the pro-cess between 345th sample and the 349th sample. Therefore,the proposed control chart is more sensitive to detect a shiftin the process as compared to the competitor chart under theuncertainty.

Now, the efficiency of the proposed chart with the exist-ing chart is discussed using the simulated data drawn fromthe neutrosophic normal distribution with neutrosophic meanµN ε {0, 0} and σN ε {1, 1}. The first 20 observations are gen-erated assuming the process is in-control and next sameobservations are generated for the shifted process whenc = 0.07, nN ε [5, 5] and ARL0Nε {370, 370}. The statisticXN ε

{XL , XU

}is computed and plotted on the control chart

given in Figure 1 for the proposed chart and Figure 2 forthe existing control chart. From Figures 1-2, we note thatthe proposed control chart detects the shift at around the36th sample while the competitor’s chart shows that all values

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TABLE 6. The data of diameter.

of statistic within the control limits. Therefore, we concludethat the proposed chart has the ability to detect the shift in theprocess earlier than the existing control chart.

IV. INDUSTRIAL APPLICATIONThe application of the proposed control chart is given forthe automobile data taken from the automobile industry.

Similar data is used by Aslam and Khan (2019). Accord-ing to Aslam and Khan (2019) ‘‘In this industry, the insidediameter measurement of automobile engine piston rings is acontinuous random variable is obtained by the measurementprocess’’. In addition, as mentioned by [44] ‘‘all observationsand measurements of continuous variables are not precisenumbers but more or less non-precise. This imprecision is

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different from variability and errors. Therefore, also lifetimedata are not precise numbers but more or less fuzzy. Thebest up-to-date mathematical model for this imprecision isso-called non-precise numbers’’. Therefore, it may possiblethat some observations of the diameter obtained from themeasurement have imprecise observations. The data is shownin Table 6 is taken from Aslam and Khan (2019) for easy ref-erence. For this data, let nN ε [5, 5] and ARL0Nε {370, 370}.The neutrosophic control limits and statistics for this datagiven as

σN ε [0.008896, 0.009399] , mN ε [74.001, 74.001]

LCLL1 = (73.98971) , LCLL2 = (73.99219)

LCLU1 = (73.99142) , LCLU2 = (73.99168)

UCLL1 = (74.01229) , UCLL2 = (74.00981)

UCLU1 = (74.01058) , UCLU2 = (74.01032)

The statistic XN ε{XL , XU

}is computed and reported in the

last column of Table 6. The values of XN ε{XL , XU

}is plotted

on the control chart in Figure 3 for the proposed control chartand in Figure 4 for the existing control chart. We note thatsome values of the statistic fall in indeterminacy interval andone point is atUCLL1. FromFigure 4, we note that the processis an in-control state. By comparing Figure 3 and Figure 4,it is concluded that the proposed chart indicates that there issome issue to be fixed in the process as one point is at thecontrol limit while the existing control chart indicates that noaction is needed as the process is an in-control state.

V. CONCLUDING REMARKSWe proposed the X-bar control chart using the repetitivesampling under the NS. The necessary measureless werederived to evaluate the performance of the proposed control.From the comparison study, we conclude that the proposedcontrol chart has edge over the existing control chart. Theproposed chart has the ability to perform well under theuncertainty environment. Therefore, the use of the proposedcontrol chart when observations are imprecise, incompleteand uncertain is more effective to be applied than the existingcontrol chart under classical statistics and the NS. The useof the proposed control chart in the industry will be helpfulin reducing the non-conforming items. The proposed controlchart using other sampling schemes can be considered asfuture research. The proposed control chart for monitoringmedian ormonitoring location and scale for some non-normaldistributions can be considered as future research.

AUTHOR CONTRIBUTIONSConceived and designed the experiments: MA. Performedthe experiments: MA. Analyzed the data: MA. Contributedreagents/materials/analysis tools: MA. Wrote the paper: MA

ACKNOWLEDGMENTThe authors are thankful to editor and reviewers for theirvaluable suggestions to improve the quality of manuscript.

They also gratefully acknowledge the DSR technical andfinancial support.

DATA AVAILABILITYThe data is given in the paper.

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MUHAMMAD ASLAM received the M.Sc.degree in statistics from GC University Lahore,in 2004, with the Chief Minister of the PunjabMerit Scholarship, the M.Phil. degree in statisticsfrom GC University Lahore, in 2006, with theGovernor of the Punjab Merit Scholarship, and thePh.D. degree in statistics from the National Col-lege of Business Administration and EconomicsLahore, in 2010, under the kind supervision ofProf. Dr. M. Ahmad.

Hewas a Lecturer of statistics with the Edge College System International,from 2003 to 2006. He was a Research Assistant with the Departmentof Statistics, GC University Lahore, from 2006 to 2008. Then, he joinedthe Forman Christian College University as a Lecturer, in 2009, where hewas an Assistant Professor (2010–2012), and also an Associate Professor(2012–2014). He was an Associate Professor of statistics with the Depart-ment of Statistics, Faculty of Science, King Abdulaziz University, Jeddah,Saudi Arabia, from 2014 to 2017. He taught summer course as a VisitingFaculty of statistics with Beijing Jiaotong University, China, in 2016. He iscurrently a Full Professor of statistics with the Department of Statistics,King Abdul-Aziz University. He has published over 320 research papersin national and international well-reputed journals, including, for example,the IEEE ACCESS, the Journal of Applied Statistics, the European Journalof Operation Research, the Information Sciences, the Journal of ProcessControl, the Journal of the Operational Research Society, the Applied Math-ematical Modeling, the International Journal of Fuzzy Systems, Symmetry,the International Journal of Advanced Manufacturer Technology, Communi-cations in Statistics, the Journal of Testing and Evaluation, and the PakistanJournal of Statistics. His papers have been cited more than 2800 times withan h-index 30 and i-10 index 84 (Google Scholar). His papers have been citedmore than 1300 times with h-index 21 (Web of Science Citations). He is theauthor of a book published in Germany. He has been the HEC approved Ph.D.supervisor, since 2011. He has supervised five Ph.D. theses, over 25 M.Phil.theses, and three M.Sc. theses. He is currently supervising a Ph.D. thesisand over five M.Phil. theses in statistics. He is a Reviewer of over 50 wellreputed international journals. He has reviewed over 140 research papers forvarious well reputed international journals. His areas of interests includereliability, decision trees, Industrial Statistics, acceptance sampling, rankset sampling, neutrosophic statistics, and applied statistics. He introducedthe area of Neutrosophic Statistical Quality Control (NSQC) for the firsttime. His is the Founder of neutrosophic inferential statistics (NIS) andNSQC. His contribution is the development of neutrosophic statistics theoryfor the inspection and process control. He originally developed theory inthese areas under the Neutrosophic Statistics. He extended the ClassicalStatistics theory to Neutrosophic Statistics originally, in 2018. Dr. Aslam is amember of the editorial board of the Electronic Journal of Applied StatisticalAnalysis, the Asian Journal of Applied Science and Technology, and thePakistan Journal of Commence and Social sciences. He is also a memberof the Islamic Countries Society of Statistical Sciences. He has appointedas an External Examiner for the 2016/2017–2018/2019 triennium at TheUniversity of Dodoma, Tanzania. He has received Meritorious ServicesAward in research from the National College of Business Administration andEconomics Lahore, in 2011. He received the Research Productivity Awardfor the year 2012 by the Pakistan Council for Science and Technology. Hisname is listed at the 2nd Position among Statistician in the Directory ofProductivity Scientists of Pakistan 2013, and is listed at the 1st Positionamong Statistician in the Directory of Productivity Scientists of Pakistan2014. He got 371 positions in the list of top 2210 profiles of Scientist ofSaudi Institutions 2016. He is selected for the Innovative Academic Researchand Dedicated Faculty Award 2017 by SPE, Malaysia. He has received theKing Abdulaziz University Excellence Awards in Scientific Research for thepaper entitled ‘‘Aslam,M., Azam,M., Khan, N. and Jun, C.-H. (2015).ANewMixed Control Chart to Monitor the Process; and the International Journalof Production Research, 53 (15), 4684-4693. He received the KingAbdulazizUniversity Citation Award for the paper entitled ‘‘Azam, M., Aslam, M. andJun, C.-H. (2015). Designing of a Hybrid Exponentially Weighted MovingAverage Control Chart Using Repetitive Sampling; and the InternationalJournal of Advanced Manufacturing Technology, 77:1927-1933, in 2018.

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