Multiple Dependent State Sampling-Based Chart Using Belief ...

Research ArticleMultiple Dependent State Sampling-Based Chart Using BeliefStatistic under Neutrosophic Statistics

Ahmed Ibrahim Shawky Muhammad Aslam and Khushnoor Khan

Department of Statistics Faculty of Science King Abdulaziz University Jeddah 21551 Saudi Arabia

Correspondence should be addressed to Muhammad Aslam aslam_ravianhotmailcom

Received 31 May 2020 Revised 26 July 2020 Accepted 1 August 2020 Published 27 August 2020

Academic Editor Francisco Balibrea

Copyright copy 2020 Ahmed Ibrahim Shawky et al +is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

In this paper a control chart scheme has been introduced for the mean monitoring using gamma distribution for belief statisticsusing multiple dependent (deferred) state sampling under the neutrosophic statistics +e coefficients of the control chart and theneutrosophic average run lengths have been estimated for specific false alarm probabilities under various process conditions +eoffered chart has been compared with the existing classical chart through simulation and the real data From the comparison it isconcluded that the performance of the proposed chart is better than that of the existing chart in terms of average run length underuncertain environment+e proposed chart has the ability to detect a shift quickly than the existing chart It has been observed thatthe proposed chart is efficient in quick monitoring of the out-of-control process and a cherished addition in the toolkit of thequality control personnel

1 Introduction

+e control chart is a key technique to statistical processcontrol to ensure the quality of the production process +etechnique of control chart requires the construction of acentral line and two control limits are known as the lowercontrol limit (LCL) and upper control limit (UCL) +equality characteristic of interest is then plotted on this chartfor the quick monitoring of an observation falling outsidethese two limits +e idea of control chart was floated byShewhart A Walter during the 1920s and plenty of controlchart techniques has been developed by researchers butremained unsuccessful to develop a robust control charttechnique Gamma distribution is the commonly employedprobability distribution for estimating time between eventsin physical sciences +e gamma distribution is a well-fitteddistribution to the failure between events and an excellentalternative to normal log normal and nonparametric ap-proaches and routinely used in the control chart literature[1] It has been concluded that the gamma distributionapproaches to the normal distribution when the shape pa-rameter is very large Aslam et al [2] proposed an MDS

sampling chart using the transformation of the gamma tonormal distribution Aslam et al [3] developed a controlchart for the belief estimator using the interested qualitycharacteristic which follows the gamma distribution Zhanget al [4] investigated the gamma chart for monitoring rthevent in the time occurring events Bhaumik and Gibbons[1] developed one-sided prediction intervals for environ-mental-based quality characteristics for gamma distributionAl-Oraini and Rahim [5] developed an economic controlchart using constraints to the optimization problem forgamma-distributed quality characteristics Several researchstudies including [6ndash8] have used gamma distribution indeveloping control chart

+e technique of multiple dependent state (MDS)sampling was announced by Wortham and Baker [9] inwhich the decision for in-control process or out-of-controlprocess is established upon not only the current informationof the process but also considers the previous record of theprocess Soundararajan and Vijayaraghavan [10] developedthe acceptance sampling plan for the fixed number of unitsfor fixed acceptance and limiting quality levels which involvethe least sum of risks Soundararajan and Vijayaraghavan

HindawiJournal of MathematicsVolume 2020 Article ID 7680286 14 pageshttpsdoiorg10115520207680286

[11] proposed a search procedure using conventionalsampling plans for smaller sample sizes Kuralmani andGovindaraju [12] presented the techniques and tables ofMDS sampling plan for the selection of minimum samplesize +e MDS sampling plans for the least sampling sizewere proposed by Govindaraju and Subramani [13] forindicated acceptable quality and limiting quality levelsBalamurali and Jun [14] suggested an MDS sampling planfor the normal distribution for variables Aslam et al [15]designed an MDS sampling control chart for averagemonitoring with two control limits Aslam et al [16] de-veloped an MDS sampling plan for variable quality char-acteristics under process loss consideration Aslam et al [17]suggested an exponential distribution for the MDS samplingchart for a normal approximation using transformationAslam et al [18] developed an attribute control chart forMDS sampling as compared to the traditional control chartsBalamurali et al [19] designed anMDS sampling plan for thegamma-Poisson distribution using the Bayesian statisticsYan et al [20] proposed an MDS sampling plan based uponthe coefficient of variation for the normally distributedquality characteristics Aldosari et al [21] developed anMDSsampling control chart for attribute quality characteristicsusing the repetitive group sampling scheme Zhou et al [22]adopted the MDS sampling scheme using a joint adaptive npchart for improved checking of the manufacturing processAfshari and Sadeghpour Gildeh [23] described the MDSsampling plan for an attribute using the fuzzy environmentAslam et al developed an MDS sampling plan for themeasurable quality characteristics by joining the features ofthe repetitive group sampling and MDS sampling for theprocess capability index

+ere are many situations in the real world when theinformationmay be determinate or indeterminate [24] Suchdata are dealt with the theory of neutrosophic statisticsintroduced by Smarandache [25]+e neutrosophic statisticsis defined as the generalization of conventional statistics (see[26]) During the last few years the application of neu-trosophic statistics has attracted the attention of variousresearchers due to its nice properties as the conventionalstatistics cannot be used when our data consist of vagueincomplete uncertain soggy or unclear observations Yeet al [27] investigated different properties and measure theeffect of indeterminate joint roughness coefficient valuesusing neutrosophic numbers Aslam [28] proposed a neu-trosophic sampling plan for the loss consideration processesAslam and Al-Marshadi [29] developed a neutrosophicsampling plan for the regression estimator Aslam and Arif[30] suggested a neutrosophic sampling plan for examiningelectrical devices Aslam et al [31] proposed a chart forreliability data under neutrosophic statistics Aslam et al[32] developed a control chart scheme for checking theinconsistency under the neutrosophic interval techniqueAslam et al [33] developed an attribute control chart for theneutrosophic statistics Jansi et al [34] developed a

correlation measure for the Pythagorean neutrosophic setsusing the dependent neutrosophic components Mur-alikrishna and Kumar [35] developed a neutrosophictechnique for data relating to linear space+e application ofneutrosophic statistics in the quality control literature can beseen in works including [34ndash43]

Aslam et al [44] proposed the control chart for the gammadistribution using the belief statistic for the single samplingplan By exploring the literature and best of our knowledgethere is no work on the control chart using the belief statisticfor MDS +e process monitoring using the belief statistic forneutrosophic statistics under the gamma distribution usingMDS sampling has not been studied by any researcher yet Inthis paper we will present the MDS control chart using thebelief statistic under neutrosophic statisticsWe expect that theproposed chart will perform better in neutrosophic averagerun length as compared to the previous chart in an uncertainenvironment It is expected that the proposed chart will able todetect shifts earlier than the existing chart In addition it isexpected that the proposed chart will ensure the quality of theproduct according to ISO 9001 2015 Quality ManagementSystems (httpswwwisoorg) +e rest of the article is or-ganized as follows the scheme of the planned chart is given inSection 2 We described the simulation study of the plannedchart in Section 3 Section 4 describes the advantages of theplanned chart A comparison of the planned chart with anexisting control chart has been given in Section 5 Conclusionsare described in the last section

2 Scheme of the Planned Chart

Let TN be the neutrosophic time between events of theinterested quality characteristic from the gamma distribu-tion having cumulative distribution function as

P TN le tN( 1113857 1 minus 1113944

aNminus1

j0

eminus tNb

tNbN( 1113857j

j (1)

where aN and bN are the shape and scale parameters re-spectively Under the particular condition when aN 1then this distribution tends to form the exponential dis-tribution Wilson and Hilferty [45] described that if T isdistributed from the gamma distribution then the trans-formation TlowastN T13

N follows the normal distribution+e random variable TlowastN has the mean and variance as

follows

μTlowastN

E TlowastN( 1113857

b13N Γ aN + 13( 1113857

Γ aN( 1113857 (2)

σTlowastN

b13N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2

11139741113972

(3)

2 Journal of Mathematics

+en the approximately normal distribution of TlowastN canbe described as

TlowastN sim N

b13N Γ aN + 13( 1113857

Γ aN( 1113857 b

13N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦⎛⎝ ⎞⎠

(4)

Data are collected for belief statistics with the assump-tion of a single observation (n 1) of the targeted quality ofinterest We assume that the kth observation be Tk and Ok

(T1 T2 Tk) for the kth iteration of the vector of ob-servation We further assume that Ok (Tk Okminus1) andB(Ok) are the posterior belief and the prior belief is B(Okminus1)Here our purpose is to select a new observation by updatingB(Ok) using B(Okminus1) We required the updated posteriorbelief for the gamma distribution under the transformedvariable TlowastN T13

N For this purpose the following equationis developed

B OkN1113872 1113873 B TkN

OkNminus11113872 1113873

B OkNminus11113872 1113873e

TlowastN

minus μTlowastNσTlowast

N

B OkNminus11113872 1113873eTlowast

NminusμTlowast

NσTlowast

N + 1 minus B OkNminus11113872 11138731113872 1113873

(5)

It is to be noted that the variable TlowastN is without thesubscript k

Let the statistic proposed by Fallah Nezhad and AkhavanNiaki [46] as

ZkN

B OkN1113872 1113873

1 minus B OkN1113872 1113873

(6)

+e above expression can be written as follows

ZkN ZkNminus1

eTlowast

Nminus μTlowast

NσTlowast

N (7)

Let the specified starting value ofZ0 1 and B(O0) 05are being used then the statistic given below follows nor-mally distributed quality characteristic having zero meanand k variance [46]+en the lower and upper control limitsof the proposed control chart are given by

UCL1N L1N

kN

1113969

(8)

LCL1N L1N

kN

1113969

(9)

UCL2N L2N

kN

1113969

(10)

LCL2N L2N

kN

1113969

(11)

where L denotes the chart coefficient to be estimated usingthe type I error and in-control average run lengths of theprocess

+e technique of the planned chart is elaborated in thesubsequent steps as follows

Step 1 choose an item randomly at the kth subgroupand calculate its quality characteristic TkN

ConvertTlowastkN

T13kN

and then compute

ln ZkN1113872 1113873 ln ZkN

minus 11113872 1113873 +TlowastkN

minus μTlowastkN

σTlowastkN

(12)

Step 2 if LCLN le ln(ZkN)leUCLN announce the

process as the in-control process and ifln(ZkN

)gtUCLN or ln(ZkN)gt LCLN then the process is

confirmed as out of control

+e measures of the planned chart for the in-control andthe out-of-control process are computed with the assumptionthat we change the scale parameter of underlying distributionwhile the shape parameter remains fixed during the entireprocess Let b0N and b1N denote the scale parameter of the in-control and out-of-control process +e probability that the in-control process is stated as the out-of-control process whenactually process is in-control may be expressed as

Pin0N P LCL2N le In ZkN1113872 1113873leUCL2N1113872 1113873 (13)

After some simplification equation (13) can be written asfollows

Pin0N 2 Φ L2N( 1113857( 1113857 minus 1 (14)

Similarly the probability for in-decision can be writtenas

P LCL1N le ln ZkN1113872 1113873le LCL2N1113872 1113873

+ P UCL2N le In ZkN1113872 1113873leUCL1N1113872 1113873

2 ΦN L1N( 1113857 minusΦN L2N( 11138571113858 1113859

(15)

Here it is worth noting that Pin0N does not involve kN+e probability of in-control for the proposed chart is givenby

PinN Pin0N + P LCL1N le ln ZkN1113872 1113873le LCL2N1113872 11138731113960

+ P UCL2N le ln ZkN1113872 1113873leUCL1N1113872 11138731113961lowastP

min0N

(16)


PinN 2ΦN L2N( 1113857 minus 1( 1113857 + 2 ΦN L1N( 1113857 minusΦN L2N( 11138571113858 11138591113858 1113859

times 2ΦN L2N( 1113857 minus 1( 1113857m

(17)

Journal of Mathematics 3

+e evaluation of the functioning of the developedcontrol chart is judged by calculating the average run length(ARL) that is very commonly suggested by the qualitycontrol researchers [47] It is defined as the average of all thesamples until the process shows the deterioration +e ARLof the in-control manner may be determined as follows

ARL0 1

1 minus PinN

(18)

It is very common that no process operates smoothly fora long time without any alteration in the process So theevaluation of the changed process provides us the effec-tiveness of the proposed scheme in haste and prompt in-dication of the out-of-control process Let the scale factor of

the gamma distribution has been moved from b0 to b1 sb0where s is the amount of shift +en the average and thevariance of the changing process are calculated as follows

E TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b130N

Γ aN + 13( 1113857

Γ aN( 1113857 (19)

Var TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b230N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦

(20)

+us the mean and variance of ln(ZkN) at b1N which

follows an approximately normal distribution are given asfollows

E ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113876 1113875 kN

Γ aN + 13( 1113857Γ(a) s13

minus 11113872 1113873

Γ aN + 23( 1113857Γ aN( 1113857 minus Γ aN + 13( 1113857Γ aN( 1113857( 11138572

1113969 (21)

Var ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113874 1113875 kNS23

(22)

So the probability of the out-of-control process for theshifted process at kth sample is calculated as follows

Pin1N P LCL2N le ln ZkN1113872 1113873leUCL2N1113872 1113873 (23)

Pin1N P ln ZkN1113872 1113873ltUCL2N

1113868111386811138681113868 bN b1N1113966 1113967 minus P ln ZkN1113872 1113873ltLCL2N

1113868111386811138681113868 bN b1N1113966 1113967

Pin1N P ln ZkN1113872 1113873ltL2N

kN

1113969 111386811138681113868111386811138681113868 bN b1N1113882 1113883 minus P ln ZkN1113872 1113873ltminusL2N

kN

1113969 111386811138681113868111386811138681113868 bN b1N1113882 1113883(24)

Pin1N ΦL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusΦminusL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

+e probability of in-decision for the shifted process isgiven by

P LCL1N le ln ZkN1113872 1113873le LCL2N1113872 1113873 + P UCL2N le ln ZkN

1113872 1113873leUCL1N1113872 1113873 (26)


+e simplified form of equation (26) is given by

ΦminusL2N

kN

1113969

minus kN middot Γ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠+

+ΦL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

(27)

+e probability of in-control for the shifted process isgiven by

P1inN

1113868111386811138681113868m1N ΦL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13 minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝

minusΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠

+

ΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

+ΦL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times ΦL2N

kN

1113968minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13 minus 1( 1113857


1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎛⎜⎜⎝

minusΦminusL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎞⎟⎟⎠

m

(28)


Similarly the expression for the (k + j)th sample of theshifted process as the change occurs at k minus th sample is givenas

P RL j1113864 1113865 1 minus P1out kN+11113872 1113873 1 minus P

1out kN+21113872 1113873

1 minus P1out kN+jminus11113872 1113873P

1out kN+j

(29)

Here RL is a run-length random variable indicating theout-of-control process

So the ARL of the shifted process is specified as follows

ARL1N P1out kN+1 + 2 1 minus P

1out kN+11113872 1113873P

1out kN+2

+ 3 1 minus P1out kN+11113872 1113873 1 minus P

1out kN+21113872 1113873P

1out kN+3 + middot middot middot

(30)

Suppose r0 be the in-control specified ARL +en theprocedure to compute the control chart coefficient and ARLof the shifted process may be followed as

Step 1 select a series of control chart coefficient LN

Step 2 compute LN such that ARL0N ge r0

Step 3 for a constant value of kN and many values ofshifts s we calculate P1

out kNusing equation (18)

Step 4 compute ARL1N for fixed kN and various shiftedvalues of s

+e values of NARL for various parameters are pre-sented in Tables 1ndash4 From Tables 1ndash4 it can be witnessedthat NARL decreases as k increases from k ϵ[3 5] to kϵ[8 10] For example from Table 1 the values of NARL are[12105 10093] [6251 4035] and [3311 179] for theshifts s 110 120 and 130 and when kϵ[3 5] and whenkϵ[8 10] from Table 2 then the values of NARL are [90677619] [3195 2157] and [1329 813] for r0 200 We alsoobserve that as a decreases from aϵ[195 205] toaϵ[095 105 ] the values of NARL increase

3 Comparison of the Planned Chart with theExisting Chart

In this section the benefits of the neutrosophic control chartunder MDS sampling for the belief statistics will be dis-cussed Aslam et al [44] proposed the control chart for beliefstatistics using the single sampling when the observationscollected are vague incomplete unclear imprecise or un-certain and showed that it was an effective and efficient chartas compared to chart under classical statistics According to[48 49] a control chart having the smaller values of ARL issaid to be a more efficient chart In this section we willcompare the proposed chart with Aslam et al [44]

31 Compression in NARL We will compare the plannedneutrosophic chart using MDS sampling for the beliefstatistics with the existing neutrosophic chart for beliefstatistics provided by Aslam et al [44] As described earlier

NARL is used for examining the efficiency of any chart forthe speedy and rapid indication of the out-of-control situ-ation From Table 5 by matching the ARLs of the plannedchart with the existing Aslam et al [44] chart it can bewitnessed that ARL measures of the planned control chartare lesser than the NARL values of the existing chart for allprocess shifts s For example for a shift of 075 the existingchart is expected to detect the shift from 116th to 218thsample On the other hand the proposed chart will detect theshift from 78th to 192nd sample From this study it isconcluded that the proposed chart has the ability to detect ashift earlier than the existing chart +erefore the use of theproposed control chart in the industry will help to improvethe quality of the product +is finding is the same asmentioned in ISO 9001 2015 Quality Management Systems(httpswwwisoorg)

32 Comparisonby Simulation Study +e comparison of theplanned chart with the existing chart has also been presentedby simulation data given in Tables 6 and 7 +us the first 20observations are created from the in-control process havingkN [3 5] aN [195 205] and bN [2 22] and the next

Table 1 +e values of NARL when a 195 and 205 andk 3 and 5

a ϵ[195 205] k ϵ[3 5]

k1N [32104 32996] [34042 34512] [31128 32105]K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [111 102] [112 102] [116 103]300 [134 110] [138 111] [147 113]280 [145 114] [150 116] [162 119]250 [171 125] [180 128] [200 133]225 [213 142] [229 147] [260 155]200 [299 176] [331 185] [385 201]190 [360 201] [405 213] [475 234]180 [449 237] [514 255] [610 284]170 [586 294] [686 323] [820 366]160 [809 391] [969 441] [1168 507]150 [1193 57] [1471 663] [1781 778]140 [1903 937] [2426 1134] [2938 1351]130 [3311 179] [4388 2269] [5300 2734]120 [6251 4035] [8659 5410] [1043 6543]110 [12105 10093] [17552 14423] [213 17521]100 [20056 20033] [30111 30100] [37194 37138]080 [1374 7263] [20193 1034] [26441 13497]075 [9692 4199] [13973 582] [19224 7817]070 [6490 2381] [9162 3207] [1334 4387]060 [2644 787] [3566 991] [5736 1344]050 [1007 305] [1288 353] [2157 447]040 [386 157] [462 169] [736 192]030 [171 110] [188 113] [257 118]025 [129 102] [136 103] [166 105]015 [101 100] [102 100] [104 100]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


Table 2 +e values of NARL when a 195 and 205 and k 8 and 10

a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]


20 observations are created from the deteriorated processhaving slowastbN [2 22] where s 140 Figure 1 shows theproposed chart Figure 2 shows the Aslam et al [44] chartand Figure 3 shows the Shewhart chart under the classicalstatistics By comparing these figures it can be seen that theproposed chart detects shift near to 33rd sample On theother hand Figures 2 and 3 do not show any shift in theprocess Figures 1ndash3 have been given for simulated datawhich show better detecting ability of the indeterminacydata of the proposed chart From the simulation study it is

concluded that the proposed chart is better in detecting theshift in the process while the existing chart does not detectthe shift +erefore the use of the proposed chart willminimize nonconforming items

4 Application of the Proposed Chart in theHealthcare Department

In this section we will discuss the application of the pro-posed chart using the Urinary Tract Infection (UTI) data


a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]


Table 6 +e simulated data for the proposed control chart

Sr no B (k) z (k) ln (ZkN)

1 [0007 0569] [0007 1323] [minus5016 028]2 [0374 0246] [0598 0326] [minus0515 minus1122]3 [0308 0888] [0445 7959] [minus0811 2074]4 [0702 0991] [2359 105995] [0858 4663]5 [0454 0006] [0832 0006] [minus0184 minus5084]6 [0744 0447] [2906 0808] [1067 minus0214]7 [064 0269] [1781 0368] [0577 minus1001]8 [0035 0972] [0036 34967] [minus3327 3554]9 [0896 0565] [8659 13] [2159 0263]10 [0155 0493] [0183 0973] [minus1698 minus0027]11 [0273 0882] [0375 7501] [minus0981 2015]12 [0917 0727] [11026 2656] [24 0977]13 [0499 0964] [0997 26938] [minus0003 3294]14 [0732 0818] [2735 4492] [1006 1502]15 [0023 0261] [0023 0353] [minus3754 minus1041]16 [0465 0988] [0868 7971] [minus0141 4378]17 [0146 0841] [0171 5303] [minus1768 1668]18 [0873 0972] [6844 344] [1923 3538]19 [095 0021] [19136 0021] [2952 minus385]20 [0605 0723] [1531 2606] [0426 0958]21 [0277 002] [0383 002] [minus096 minus389]22 [0954 0853] [20717 5789] [3031 1756]

Table 5 +e NARL values of the proposed and the existing charts when aϵ[195 205] and k 3 5

a ϵ[195 205] k ϵ[3 5]

Existing Proposed Existing Proposed Existing ProposedkN[28071 28141] [29354 29416] [30003 30012]k1N [32104 32996] [34042 34512] [31128 32105]

K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued


23 [0875 097] [7015 3267] [1948 3486]24 [0577 0876] [1364 706] [0311 1954]25 [0808 0952] [4216 19804] [1439 2986]26 [0902 0995] [9252 206617] [2225 5331]27 [0535 0645] [115 1816] [014 0597]28 [044 0221] [0786 0283] [minus024 minus1262]29 [0965 0999] [27261 933686] [3305 6839]30 [0532 0992] [1135 131102] [0126 4876]31 [0425 0999] [074 1913741] [minus0301 7557]32 [0592 0819] [1450 4517] [0371 1508]33 [0995 0977] [19348 43177] [5265 3765]34 [0456 0479] [0838 0919] [minus0177 minus0085]35 [0672 1000] [2053 2577057] [0719 7854]36 [0683 0968] [2159 30231] [0770 3409]37 [0635 0288] [1738 0404] [0553 minus0907]38 [0358 0991] [0559 111696] [minus0582 4716]39 [0927 0930] [12662 13193] [2539 2580]40 [0634 0847] [1733 5526] [0550 1710]

Table 7 +e simulated data for the existing chart


1 [0352 0515] [0543 1063] [minus061 0061]2 [0237 0825] [031 4718] [minus1171 1551]3 [025 0424] [0334 0735] [minus1096 minus0308]4 [0767 087] [3291 6721] [1191 1905]5 [0569 0042] [1322 0044] [0279 minus3125]6 [0877 0019] [7107 0019] [1961 minus3939]7 [0359 0834] [056 5039] [minus058 1617]8 [0806 0968] [4153 29856] [1424 3396]9 [002 0881] [0021 7412] [minus388 2003]10 [0662 0071] [1958 0077] [0672 minus257]11 [0909 0963] [9933 26211] [2296 3266]12 [0959 0142] [23368 0166] [3151 minus1797]13 [0232 099] [0302 95866] [minus1199 4563]14 [0063 0965] [0067 27253] [minus2701 3305]15 [0918 0146] [11154 0172] [2412 minus1763]16 [0086 0109] [0094 0122] [minus2364 minus2105]17 [0936 0944] [1455 16968] [2678 2831]18 [0198 0265] [0248 036] [minus1396 minus102]19 [0214 0605] [0271 1533] [minus1304 0427]20 [0507 0411] [1028 0699] [0027 minus0359]21 [098 0998] [49554 414687] [3903 6028]22 [0984 0278] [60027 0385] [4095 minus0954]23 [0752 0979] [3034 46245] [111 3834]24 [0857 0713] [5976 2482] [1788 0909]25 [0267 0946] [0364 17361] [minus101 2854]26 [0744 0166] [2904 0199] [1066 minus1614]27 [0577 0974] [1363 36773] [031 3605]28 [079 0589] [3761 1432] [1325 0359]29 [0841 0657] [5275 1914] [1663 0649]30 [0969 0808] [31497 4206] [345 1436]31 [0557 0959] [1257 23411] [0229 3153]32 [0763 0998] [3228 572344] [1172 635]33 [094 0998] [15578 631372] [2746 6448]34 [0165 0848] [0197 5575] [minus1625 1718]


from a big hospital According to Santiago and Smith [50]ldquothe hospital would like to track the frequency of patientsbeing discharged who had acquired a UTI while in thehospital as a way to quickly identify an increase in in-fection rate or conversely monitor whether forthcomingprocess or material changes result in fewer infectionsBecause the root cause often differs based on gender maleand female patients are charted separately and this ex-ample focuses on malesrdquo Aslam et al [44] presented theneutrosophic form of UTI data which is shown in Table 8Suppose that kN [3 5] aN [195 205] andbN [2 22] +e four control limits for the UTI data aregiven as follows

UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969

[minus53915 minus71789]

UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519

100 3020 40Sample number

ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U

Figure 2 +e existing chart proposed by Aslam et al [44]

LCL = ndash519

UCL = 519

20 30 400 10Sample number

ndash10

ndash5

0

5

10

15

ln (Z

t)

Figure 3 +e existing Shewhart chart under classical statistics

UCL1L = 53915

LCL2L = 39823LCL2U = ndash51412

LCL1U = ndash71789LCL1L = 53915

UCL1U = 71789

UCL2U = 51411UCL2L = 39823

402010 300Sample number

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U

Figure 1 +e proposed control chart using simulated data

Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]


Using the given information the value ofZkNwhenZ0 1

and B(O0) 05 is calculated as follows

ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)

+e neutrosophic statistic ln (ZkN) is plotted on four

control limits in Figure 4 +e planned control chartshows the out-of-control process after the 6th sample +emeasures of ln (ZkN

) are also plotted in Figure 5 using thechart of neutrosophic statistics proposed by Aslam et al[44] which shows the out-of-control process at the 32ndsample +e same data were also plotted in Figure 6 using

the Shewhart chart which is incapable to identify the out-of-control situation +erefore the planned chart is effi-cient in identifying the out-of-control situation moreswiftly +e use of the proposed control chart in thehospital will help to identify the UTI patient quickly ascompared to the existing charts

ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915

Figure 4 +e proposed chart for UTI data

Table 8 +e UTI data


1 [0712 0448] [2471 0813] [0905 minus0207]2 [0201 089] [0251 8101] [minus1382 2092]3 [0099 0897] [011 8668] [minus2211 216]4 [0245 0087] [0325 0095] [minus1123 minus2354]5 [0417 0655] [0715 1896] [minus0335 064]6 [0083 0033] [009 0034] [minus2405 minus3373]7 [0555 0397] [1245 0658] [0219 minus0419]8 [0719 0114] [2555 0129] [0938 minus2051]9 [0893 0915] [8361 10723] [2124 2372]10 [0697 0097] [2296 0108] [0831 minus223]11 [0573 0193] [1341 0239] [0294 minus1432]12 [0234 0701] [0305 2346] [minus1187 0853]13 [0932 0072] [13773 0078] [2623 minus255]14 [0533 0165] [1143 0197] [0133 minus1622]15 [0134 0015] [0155 0016] [minus1867 minus4157]16 [0674 0955] [2071 21464] [0728 3066]17 [0008 0563] [0008 1286] [minus4799 0251]18 [0241 0939] [0317 1527] [minus1149 2726]19 [0481 0123] [0928 014] [minus0074 minus1967]20 [0044 0433] [0046 0764] [minus3071 minus0269]21 [0109 0928] [0123 12969] [minus2098 2563]22 [056 0236] [1275 0308] [0243 minus1176]23 [0025 0844] [0026 5411] [minus3666 1688]24 [0027 0153] [0028 0181] [minus3574 minus171]25 [0826 059] [4739 1438] [1556 0363]26 [0125 0872] [0143 6795] [minus1942 1916]27 [0011 0361] [0011 0565] [minus4488 minus0571]28 [0025 0198] [0026 0247] [minus3667 minus1399]29 [0567 069] [1311 223] [0271 0802]30 [0909 0175] [9999 0211] [2303 minus1554]31 [0325 08] [0482 4006] [minus0731 1388]32 [0126 0998] [0144 519999] [minus1941 6254]33 [0474 0728] [0901 2671] [minus0104 0982]34 [035 0022] [0539 0022] [minus0619 minus3809]35 [0425 0049] [074 0051] [minus0301 minus2971]36 [0876 0723] [7063 2614] [1955 0961]37 [0177 0068] [0215 0072] [minus1539 minus2625]38 [0371 099] [0591 10078] [minus0526 4613]39 [0043 0173] [0045 0209] [minus3108 minus1565]40 [0117 0191] [0132 0235] [minus2022 minus1446]

LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U

Figure 5 Aslam et al [44] chart for UTI data


5 Conclusions

In this article the planning of a control chart for gamma-distributed belief statistic using MDS sampling under theneutrosophic statistic has been offered +e parameters of theplanned chart have been estimated for vague data using codeprogramming for R language Neutrosophic average runlengths for indeterminacy intervals under different processsettings for various shift levels have been calculated +ecomparison of the planned scheme with the existing chart hasbeen made which shows the better identifying skill of the out-of-control process It has been perceived that the proposedscheme is a valuable accumulation in the toolkit of the qualitycontrol professionals for the monitoring of neutrosophic dataA real-world example has been added for the practical ap-plication of the planned scheme by the quality control workers+e proposed chart ensures the producercustomer that theproduct manufactured using the proposed control chart will beaccording to the given specification limits and good qualityproduct as mentioned in ISO 9001 2015 Quality ManagementSystems (httpswwwisoorg) +e proposed chart can beapplied in the industry to minimize the nonconformingproduct +e proposed chart has the limitation that it can beapplied when the quality of interest follows the normal dis-tribution +e proposed control chart for nonnormal distri-butions can be considered as future research+e planned chartcan further be extended for the multivariate probability dis-tributions+e proposed control chart using the costmodel canbe studied as future research

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under grant no(G-1407-130-1440) +e authors therefore gratefully ac-knowledge the DSR technical and financial support

References

[1] D K Bhaumik and R D Gibbons ldquoOne-sided approximateprediction intervals for at leastpofm observations from agamma population at Each Of relocationsrdquo Technometricsvol 48 no 1 pp 112ndash119 2006

[2] M Aslam O-H Arif and C-H Jun ldquoA control chart forgamma distribution using multiple dependent state sam-plingrdquo Industrial Engineering and Management Systemsvol 16 no 1 pp 109ndash117 2017

[3] M Aslam N Khan and C-H Jun ldquoA control chart usingbelief information for a gamma distributionrdquo OperationsResearch and Decisions vol 26 no 4 pp 5ndash19 2016

[4] CW Zhang M Xie J Y Liu and T N Goh ldquoA control chartfor the gamma distribution as a model of time betweeneventsrdquo International Journal of Production Research vol 45no 23 pp 5649ndash5666 2007

[5] H A Al-Oraini and M A Rahim ldquoEconomic statisticaldesign of control charts for systems with gamma (λ 2) in-control timesrdquo Computers amp Industrial Engineering vol 43no 3 pp 645ndash654 2002

[6] H Aksoy ldquoUse of gamma distribution in hydrologicalanalysisrdquo Turkish Journal of Engineering and EnvironmentalSciences vol 24 no 6 pp 419ndash428 2000

[7] B Bobee and F Ashkar 3e Gamma Family and DerivedDistributions Applied Inhydrology Water Resources Publi-cations Fort Collins CO USA 1991

[8] I M Gonzalez and E Viles ldquoDesign of r control chart as-suming a gamma distributionrdquo Economic Quality Controlvol 16 no 2 pp 199ndash204 2001

[9] A W Wortham and R C Baker ldquoMultiple deferred statesampling inspectionrdquo International Journal of ProductionResearch vol 14 no 6 pp 719ndash731 1976

[10] V Soundararajan and R Vijayaraghavan ldquoOn designingmultiple deferred state sampling (MDS-1 (0 2) plans in-volving minimum risksrdquo Journal of Applied Statistics vol 16no 1 pp 87ndash94 1989

[11] V Soundararajan and R Vijayaraghavan ldquoConstruction andselection of multiple dependent (deferred) state samplingplanrdquo Journal of Applied Statistics vol 17 no 3 pp 397ndash4091990

[12] V Kuralmani and K Govindaraju ldquoSelection of multipledeferred (dependent) state sampling plansrdquo Communicationsin Statisticsmdash3eory and Methods vol 21 no 5 pp 1339ndash1366 1992

[13] K Govindaraju and K Subramani ldquoSelection of multipledeferred (dependent) state sampling plans for given accept-able quality level and limiting quality levelrdquo Journal of AppliedStatistics vol 20 no 3 pp 423ndash428 1993

[14] S Balamurali and C-H Jun ldquoMultiple dependent statesampling plans for lot acceptance based on measurementdatardquo European Journal of Operational Research vol 180no 3 pp 1221ndash1230 2007

[15] M Aslam N Khan and C-H Jun ldquoA multiple dependentstate control chart based on double control limitsrdquo ResearchJournal of Applied Sciences Engineering and Technology vol 7no 21 pp 4490ndash4493 2014

UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)

Figure 6 Shewhart chart for UTI data


[16] M Aslam C-H Yen C-H Chang and C-H Jun ldquoMultipledependent state variable sampling plans with process lossconsiderationrdquo 3e International Journal of AdvancedManufacturing Technology vol 71 no 5ndash8 pp 1337ndash13432014

[17] M Aslam M Azam N Khan and C-H Jun ldquoA control chartfor an exponential distribution using multiple dependent statesamplingrdquo Quality amp Quantity vol 49 no 2 pp 455ndash4622015

[18] M Aslam A Nazir and C-H Jun ldquoA new attribute controlchart using multiple dependent state samplingrdquo Transactionsof the Institute of Measurement and Control vol 37 no 4pp 569ndash576 2015

[19] S Balamurali P Jeyadurga and M Usha ldquoDesigning ofbayesian multiple deferred state sampling plan based ongamma-poisson distributionrdquo American Journal of Mathe-matical and Management Sciences vol 35 no 1 pp 77ndash902016

[20] A Yan S Liu and X Dong ldquoDesigning a multiple dependentstate sampling plan based on the coefficient of variationrdquoSpringerPlus vol 5 no 1 p 1447 2016

[21] M S Aldosari M Aslam and C-H Jun ldquoA new attributecontrol chart using multiple dependent state repetitivesamplingrdquo IEEE Access vol 5 2017

[22] W Zhou Q Wan Y Zheng and Y-W Zhou ldquoA joint-adaptive np control chart with multiple dependent statesampling schemerdquo Communications in Statisticsmdash3eory andMethods vol 46 no 14 pp 6967ndash6979 2017

[23] R Afshari and B Sadeghpour Gildeh ldquoDesigning a multipledeferred state attribute sampling plan in a fuzzy environ-mentrdquo American Journal of Mathematical and ManagementSciences vol 36 no 4 pp 328ndash345 2017

[24] J Chen J Ye and S Du ldquoScale effect and anisotropy analyzedfor neutrosophic numbers of rock joint roughness coefficientbased on neutrosophic statisticsrdquo Symmetry vol 9 no 10p 208 2017

[25] F Smarandache Neutrosophy Neutrosophic Probability Setand Logic Latern Press Seattle WA USA 1998

[26] F Smarandache ldquoIntroduction to neutrosophic statisticsinfinite studyrdquo 2014 httpsarxivorgftparxivpapers140614062000pdf

[27] J Ye J Chen R Yong and S Du ldquoExpression and analysis ofjoint roughness coefficient using neutrosophic numberfunctionsrdquo Information vol 8 no 2 p 69 2017

[28] M Aslam ldquoA new sampling plan using neutrosophic processloss considerationrdquo Symmetry vol 10 no 5 p 132 2018

[29] M Aslam and A Al-Marshadi ldquoDesign of sampling planusing regression estimator under indeterminacyrdquo Symmetryvol 10 no 12 p 754 2018

[30] M Aslam and O Arif ldquoTesting of grouped product for theweibull distribution using neutrosophic statisticsrdquo Symmetryvol 10 no 9 p 403 2018

[31] M Aslam N Khan and M Albassam ldquoControl chart forfailure-censored reliability tests under uncertainty environ-mentrdquo Symmetry vol 10 no 12 p 690 2018

[32] M Aslam N Khan and M Khan ldquoMonitoring the variabilityin the process using neutrosophic statistical interval methodrdquoSymmetry vol 10 no 11 p 562 2018

[33] M Aslam R A R Bantan and N Khan ldquoDesign of a newattribute control chart under neutrosophic statisticsrdquo Inter-national Journal of Fuzzy Systems vol 21 no 2 pp 433ndash4402019

[34] R Jansi K Mohana and F Smarandache ldquoCorrelationmeasure for pythagorean neutrosophic sets with t and f as

dependent neutrosophic componentsrdquo Neutrosophic Sets ampSystems vol 30 2019

[35] P Muralikrishna and D S Kumar ldquoNeutrosophic Approachon Normed Linear Spacerdquo Neutrosophic Sets amp Systemsvol 30 2019

[36] F Amin A Fahmi and M Aslam ldquoApproaches to multipleattribute group decision making based on triangular cubiclinguistic uncertain fuzzy aggregation operatorsrdquo Soft Com-puting vol 24 no 15 2019

[37] S Dobbah M Aslam and K Khan ldquoDesign of a new syn-thetic acceptance sampling planrdquo Symmetry vol 10 no 11p 653 2018

[38] S Islam and S C Deb ldquoNeutrosophic goal programmingapproach to a green supplier selection model with quantitydiscountrdquo Neutrosophic Sets amp Systems vol 30 2019

[39] M Kashif H Nida M I Khan and M Aslam ldquoDecom-position of matrix under neutrosophic environmentrdquo Neu-trosophic Sets amp Systems vol 30 2019

[40] M Mashuri and M Ahsan ldquoPerfomance fuzzy multinomialcontrol chartrdquo Journal of Physics Conference Series vol 10282018

[41] X Peng and J Dai ldquoApproaches to single-valued neu-trosophic madm based on mabac topsis and new similaritymeasure with score functionrdquo Neural Computing and Ap-plications vol 29 no 10 pp 939ndash954 2018

[42] M-H Shu D-C Dang T-L Nguyen B-M Hsu andN-S Phan ldquoFuzzy and control charts a data-adaptability andhuman-acceptance approachrdquo Complexity vol 2017 ArticleID 4376809 17 pages 2017

[43] C M Villamar J Suarez L De Lucas Coloma C Vera andM Leyva ldquoAnalysis of technological innovation contributionto gross domestic product based on neutrosophic cognitivemaps and neutrosophic numbersrdquo Neutrosophic Sets amp Sys-tems vol 30 2019

[44] M Aslam R A R Bantan and N Khan ldquoMonitoring theprocess based on belief statistic for neutrosophic gammadistributed productrdquo Processes vol 7 no 4 p 209 2019

[45] E B Wilson and M M Hilferty ldquo+e distribution of chi-squarerdquo Proceedings of the National Academy of Sciencesvol 17 no 12 pp 684ndash688 1931

[46] M S Fallah Nezhad and S T Akhavan Niaki ldquoA newmonitoring design for uni-variate statistical quality controlchartsrdquo Information Sciences vol 180 no 6 pp 1051ndash10592010

[47] C D Montgomery Introduction to Statistical Quality ControlJohn Wiley amp Sons New York City NY USA 2009

[48] L AhmadM Aslam and C-H Jun ldquoCoal quality monitoringwith improved control chartsrdquo European Journal of ScientificResearch vol 125 no 2 pp 427ndash434 2014

[49] L Ahmad M Aslam and C-H Jun ldquoDesigning of x-barcontrol charts based on process capability index using re-petitive samplingrdquo Transactions of the Institute of Measure-ment and Control vol 36 no 3 pp 367ndash374 2014

[50] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the t chartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013


[11] proposed a search procedure using conventionalsampling plans for smaller sample sizes Kuralmani andGovindaraju [12] presented the techniques and tables ofMDS sampling plan for the selection of minimum samplesize +e MDS sampling plans for the least sampling sizewere proposed by Govindaraju and Subramani [13] forindicated acceptable quality and limiting quality levelsBalamurali and Jun [14] suggested an MDS sampling planfor the normal distribution for variables Aslam et al [15]designed an MDS sampling control chart for averagemonitoring with two control limits Aslam et al [16] de-veloped an MDS sampling plan for variable quality char-acteristics under process loss consideration Aslam et al [17]suggested an exponential distribution for the MDS samplingchart for a normal approximation using transformationAslam et al [18] developed an attribute control chart forMDS sampling as compared to the traditional control chartsBalamurali et al [19] designed anMDS sampling plan for thegamma-Poisson distribution using the Bayesian statisticsYan et al [20] proposed an MDS sampling plan based uponthe coefficient of variation for the normally distributedquality characteristics Aldosari et al [21] developed anMDSsampling control chart for attribute quality characteristicsusing the repetitive group sampling scheme Zhou et al [22]adopted the MDS sampling scheme using a joint adaptive npchart for improved checking of the manufacturing processAfshari and Sadeghpour Gildeh [23] described the MDSsampling plan for an attribute using the fuzzy environmentAslam et al developed an MDS sampling plan for themeasurable quality characteristics by joining the features ofthe repetitive group sampling and MDS sampling for theprocess capability index

+ere are many situations in the real world when theinformationmay be determinate or indeterminate [24] Suchdata are dealt with the theory of neutrosophic statisticsintroduced by Smarandache [25]+e neutrosophic statisticsis defined as the generalization of conventional statistics (see[26]) During the last few years the application of neu-trosophic statistics has attracted the attention of variousresearchers due to its nice properties as the conventionalstatistics cannot be used when our data consist of vagueincomplete uncertain soggy or unclear observations Yeet al [27] investigated different properties and measure theeffect of indeterminate joint roughness coefficient valuesusing neutrosophic numbers Aslam [28] proposed a neu-trosophic sampling plan for the loss consideration processesAslam and Al-Marshadi [29] developed a neutrosophicsampling plan for the regression estimator Aslam and Arif[30] suggested a neutrosophic sampling plan for examiningelectrical devices Aslam et al [31] proposed a chart forreliability data under neutrosophic statistics Aslam et al[32] developed a control chart scheme for checking theinconsistency under the neutrosophic interval techniqueAslam et al [33] developed an attribute control chart for theneutrosophic statistics Jansi et al [34] developed a

correlation measure for the Pythagorean neutrosophic setsusing the dependent neutrosophic components Mur-alikrishna and Kumar [35] developed a neutrosophictechnique for data relating to linear space+e application ofneutrosophic statistics in the quality control literature can beseen in works including [34ndash43]

Aslam et al [44] proposed the control chart for the gammadistribution using the belief statistic for the single samplingplan By exploring the literature and best of our knowledgethere is no work on the control chart using the belief statisticfor MDS +e process monitoring using the belief statistic forneutrosophic statistics under the gamma distribution usingMDS sampling has not been studied by any researcher yet Inthis paper we will present the MDS control chart using thebelief statistic under neutrosophic statisticsWe expect that theproposed chart will perform better in neutrosophic averagerun length as compared to the previous chart in an uncertainenvironment It is expected that the proposed chart will able todetect shifts earlier than the existing chart In addition it isexpected that the proposed chart will ensure the quality of theproduct according to ISO 9001 2015 Quality ManagementSystems (httpswwwisoorg) +e rest of the article is or-ganized as follows the scheme of the planned chart is given inSection 2 We described the simulation study of the plannedchart in Section 3 Section 4 describes the advantages of theplanned chart A comparison of the planned chart with anexisting control chart has been given in Section 5 Conclusionsare described in the last section

2 Scheme of the Planned Chart

Let TN be the neutrosophic time between events of theinterested quality characteristic from the gamma distribu-tion having cumulative distribution function as

P TN le tN( 1113857 1 minus 1113944

aNminus1

j0

eminus tNb

tNbN( 1113857j

j (1)

where aN and bN are the shape and scale parameters re-spectively Under the particular condition when aN 1then this distribution tends to form the exponential dis-tribution Wilson and Hilferty [45] described that if T isdistributed from the gamma distribution then the trans-formation TlowastN T13

N follows the normal distribution+e random variable TlowastN has the mean and variance as

follows

μTlowastN

E TlowastN( 1113857

b13N Γ aN + 13( 1113857

Γ aN( 1113857 (2)

σTlowastN

b13N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2

11139741113972

(3)



TlowastN sim N

b13N Γ aN + 13( 1113857

Γ aN( 1113857 b

13N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦⎛⎝ ⎞⎠

(4)




B OkN1113872 1113873 B TkN

OkNminus11113872 1113873

B OkNminus11113872 1113873e

TlowastN


N


NminusμTlowast

NσTlowast


(5)



ZkN

B OkN1113872 1113873

1 minus B OkN1113872 1113873

(6)


ZkN ZkNminus1

eTlowast

Nminus μTlowast

NσTlowast

N (7)


UCL1N L1N

kN

1113969

(8)

LCL1N L1N

kN

1113969

(9)

UCL2N L2N

kN

1113969

(10)

LCL2N L2N

kN

1113969

(11)




ConvertTlowastkN

T13kN

and then compute

ln ZkN1113872 1113873 ln ZkN


minus μTlowastkN

σTlowastkN

(12)








Pin0N 2 Φ L2N( 1113857( 1113857 minus 1 (14)




2 ΦN L1N( 1113857 minusΦN L2N( 11138571113858 1113859

(15)




min0N

(16)




(17)



ARL0 1

1 minus PinN

(18)



E TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b130N

Γ aN + 13( 1113857

Γ aN( 1113857 (19)

Var TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b230N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦

(20)



E ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113876 1113875 kN

Γ aN + 13( 1113857Γ(a) s13

minus 11113872 1113873


1113969 (21)

Var ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113874 1113875 kNS23

(22)





1113868111386811138681113868 bN b1N1113966 1113967


kN


kN

1113969 111386811138681113868111386811138681113868 bN b1N1113882 1113883(24)

Pin1N ΦL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusΦminusL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)



1113872 1113873leUCL1N1113872 1113873 (26)



ΦminusL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠+

+ΦL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

(27)


P1inN

1113868111386811138681113868m1N ΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝

minusΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠

+

ΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

+ΦL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times ΦL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎛⎜⎜⎝

minusΦminusL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎞⎟⎟⎠

m

(28)




1out kN+21113872 1113873


1out kN+j

(29)




1out kN+11113872 1113873P

1out kN+2


1out kN+21113872 1113873P


(30)














a ϵ[195 205] k ϵ[3 5]

k1N [32104 32996] [34042 34512] [31128 32105]K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [111 102] [112 102] [116 103]300 [134 110] [138 111] [147 113]280 [145 114] [150 116] [162 119]250 [171 125] [180 128] [200 133]225 [213 142] [229 147] [260 155]200 [299 176] [331 185] [385 201]190 [360 201] [405 213] [475 234]180 [449 237] [514 255] [610 284]170 [586 294] [686 323] [820 366]160 [809 391] [969 441] [1168 507]150 [1193 57] [1471 663] [1781 778]140 [1903 937] [2426 1134] [2938 1351]130 [3311 179] [4388 2269] [5300 2734]120 [6251 4035] [8659 5410] [1043 6543]110 [12105 10093] [17552 14423] [213 17521]100 [20056 20033] [30111 30100] [37194 37138]080 [1374 7263] [20193 1034] [26441 13497]075 [9692 4199] [13973 582] [19224 7817]070 [6490 2381] [9162 3207] [1334 4387]060 [2644 787] [3566 991] [5736 1344]050 [1007 305] [1288 353] [2157 447]040 [386 157] [462 169] [736 192]030 [171 110] [188 113] [257 118]025 [129 102] [136 103] [166 105]015 [101 100] [102 100] [104 100]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]



a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]







a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)









































TlowastN sim N

b13N Γ aN + 13( 1113857

Γ aN( 1113857 b

13N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦⎛⎝ ⎞⎠

(4)




B OkN1113872 1113873 B TkN

OkNminus11113872 1113873

B OkNminus11113872 1113873e

TlowastN


N


NminusμTlowast

NσTlowast


(5)



ZkN

B OkN1113872 1113873

1 minus B OkN1113872 1113873

(6)


ZkN ZkNminus1

eTlowast

Nminus μTlowast

NσTlowast

N (7)


UCL1N L1N

kN

1113969

(8)

LCL1N L1N

kN

1113969

(9)

UCL2N L2N

kN

1113969

(10)

LCL2N L2N

kN

1113969

(11)




ConvertTlowastkN

T13kN

and then compute

ln ZkN1113872 1113873 ln ZkN


minus μTlowastkN

σTlowastkN

(12)








Pin0N 2 Φ L2N( 1113857( 1113857 minus 1 (14)




2 ΦN L1N( 1113857 minusΦN L2N( 11138571113858 1113859

(15)




min0N

(16)




(17)



ARL0 1

1 minus PinN

(18)



E TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b130N

Γ aN + 13( 1113857

Γ aN( 1113857 (19)

Var TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b230N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦

(20)



E ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113876 1113875 kN

Γ aN + 13( 1113857Γ(a) s13

minus 11113872 1113873


1113969 (21)

Var ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113874 1113875 kNS23

(22)





1113868111386811138681113868 bN b1N1113966 1113967


kN


kN

1113969 111386811138681113868111386811138681113868 bN b1N1113882 1113883(24)

Pin1N ΦL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusΦminusL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)



1113872 1113873leUCL1N1113872 1113873 (26)



ΦminusL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠+

+ΦL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

(27)


P1inN

1113868111386811138681113868m1N ΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝

minusΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠

+

ΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

+ΦL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times ΦL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎛⎜⎜⎝

minusΦminusL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎞⎟⎟⎠

m

(28)




1out kN+21113872 1113873


1out kN+j

(29)




1out kN+11113872 1113873P

1out kN+2


1out kN+21113872 1113873P


(30)














a ϵ[195 205] k ϵ[3 5]

k1N [32104 32996] [34042 34512] [31128 32105]K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [111 102] [112 102] [116 103]300 [134 110] [138 111] [147 113]280 [145 114] [150 116] [162 119]250 [171 125] [180 128] [200 133]225 [213 142] [229 147] [260 155]200 [299 176] [331 185] [385 201]190 [360 201] [405 213] [475 234]180 [449 237] [514 255] [610 284]170 [586 294] [686 323] [820 366]160 [809 391] [969 441] [1168 507]150 [1193 57] [1471 663] [1781 778]140 [1903 937] [2426 1134] [2938 1351]130 [3311 179] [4388 2269] [5300 2734]120 [6251 4035] [8659 5410] [1043 6543]110 [12105 10093] [17552 14423] [213 17521]100 [20056 20033] [30111 30100] [37194 37138]080 [1374 7263] [20193 1034] [26441 13497]075 [9692 4199] [13973 582] [19224 7817]070 [6490 2381] [9162 3207] [1334 4387]060 [2644 787] [3566 991] [5736 1344]050 [1007 305] [1288 353] [2157 447]040 [386 157] [462 169] [736 192]030 [171 110] [188 113] [257 118]025 [129 102] [136 103] [166 105]015 [101 100] [102 100] [104 100]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]



a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]







a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)









































ARL0 1

1 minus PinN

(18)



E TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b130N

Γ aN + 13( 1113857

Γ aN( 1113857 (19)

Var TlowastN

1113868111386811138681113868 b1N1113872 1113873 s13

b230N

Γ aN + 23( 1113857

Γ aN( 1113857minusΓ aN + 13( 1113857

Γ aN( 11138571113888 1113889

2⎡⎣ ⎤⎦

(20)



E ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113876 1113875 kN

Γ aN + 13( 1113857Γ(a) s13

minus 11113872 1113873


1113969 (21)

Var ln ZkN1113872 1113873

11138681113868111386811138681113868 b1n1113874 1113875 kNS23

(22)





1113868111386811138681113868 bN b1N1113966 1113967


kN


kN

1113969 111386811138681113868111386811138681113868 bN b1N1113882 1113883(24)

Pin1N ΦL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusΦminusL2N

kN

1113969

minus kNΓ aN + 13( 1113857Γ aN( 1113857 s13

minus 11113872 1113873


1113969

1113874 1113875

kNS23

1113969⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)



1113872 1113873leUCL1N1113872 1113873 (26)



ΦminusL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠+

+ΦL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

(27)


P1inN

1113868111386811138681113868m1N ΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝

minusΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠

+

ΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

+ΦL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times ΦL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎛⎜⎜⎝

minusΦminusL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎞⎟⎟⎠

m

(28)




1out kN+21113872 1113873


1out kN+j

(29)




1out kN+11113872 1113873P

1out kN+2


1out kN+21113872 1113873P


(30)














a ϵ[195 205] k ϵ[3 5]

k1N [32104 32996] [34042 34512] [31128 32105]K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [111 102] [112 102] [116 103]300 [134 110] [138 111] [147 113]280 [145 114] [150 116] [162 119]250 [171 125] [180 128] [200 133]225 [213 142] [229 147] [260 155]200 [299 176] [331 185] [385 201]190 [360 201] [405 213] [475 234]180 [449 237] [514 255] [610 284]170 [586 294] [686 323] [820 366]160 [809 391] [969 441] [1168 507]150 [1193 57] [1471 663] [1781 778]140 [1903 937] [2426 1134] [2938 1351]130 [3311 179] [4388 2269] [5300 2734]120 [6251 4035] [8659 5410] [1043 6543]110 [12105 10093] [17552 14423] [213 17521]100 [20056 20033] [30111 30100] [37194 37138]080 [1374 7263] [20193 1034] [26441 13497]075 [9692 4199] [13973 582] [19224 7817]070 [6490 2381] [9162 3207] [1334 4387]060 [2644 787] [3566 991] [5736 1344]050 [1007 305] [1288 353] [2157 447]040 [386 157] [462 169] [736 192]030 [171 110] [188 113] [257 118]025 [129 102] [136 103] [166 105]015 [101 100] [102 100] [104 100]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]



a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]







a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)









































ΦminusL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠+

+ΦL1N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969


minus 11113872 1113873


1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

(27)


P1inN

1113868111386811138681113868m1N ΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝

minusΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠

+

ΦminusL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦminusL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

+ΦL1N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

minusΦL2N

kN

1113969



1113969

kNS23

1113969⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times ΦL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎛⎜⎜⎝

minusΦminusL2N

kN



1113969

kNS23

1113968⎛⎜⎜⎝ ⎞⎟⎟⎠⎞⎟⎟⎠

m

(28)




1out kN+21113872 1113873


1out kN+j

(29)




1out kN+11113872 1113873P

1out kN+2


1out kN+21113872 1113873P


(30)














a ϵ[195 205] k ϵ[3 5]

k1N [32104 32996] [34042 34512] [31128 32105]K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [111 102] [112 102] [116 103]300 [134 110] [138 111] [147 113]280 [145 114] [150 116] [162 119]250 [171 125] [180 128] [200 133]225 [213 142] [229 147] [260 155]200 [299 176] [331 185] [385 201]190 [360 201] [405 213] [475 234]180 [449 237] [514 255] [610 284]170 [586 294] [686 323] [820 366]160 [809 391] [969 441] [1168 507]150 [1193 57] [1471 663] [1781 778]140 [1903 937] [2426 1134] [2938 1351]130 [3311 179] [4388 2269] [5300 2734]120 [6251 4035] [8659 5410] [1043 6543]110 [12105 10093] [17552 14423] [213 17521]100 [20056 20033] [30111 30100] [37194 37138]080 [1374 7263] [20193 1034] [26441 13497]075 [9692 4199] [13973 582] [19224 7817]070 [6490 2381] [9162 3207] [1334 4387]060 [2644 787] [3566 991] [5736 1344]050 [1007 305] [1288 353] [2157 447]040 [386 157] [462 169] [736 192]030 [171 110] [188 113] [257 118]025 [129 102] [136 103] [166 105]015 [101 100] [102 100] [104 100]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]



a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]







a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)










































1out kN+21113872 1113873


1out kN+j

(29)




1out kN+11113872 1113873P

1out kN+2


1out kN+21113872 1113873P


(30)














a ϵ[195 205] k ϵ[3 5]

k1N [32104 32996] [34042 34512] [31128 32105]K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [111 102] [112 102] [116 103]300 [134 110] [138 111] [147 113]280 [145 114] [150 116] [162 119]250 [171 125] [180 128] [200 133]225 [213 142] [229 147] [260 155]200 [299 176] [331 185] [385 201]190 [360 201] [405 213] [475 234]180 [449 237] [514 255] [610 284]170 [586 294] [686 323] [820 366]160 [809 391] [969 441] [1168 507]150 [1193 57] [1471 663] [1781 778]140 [1903 937] [2426 1134] [2938 1351]130 [3311 179] [4388 2269] [5300 2734]120 [6251 4035] [8659 5410] [1043 6543]110 [12105 10093] [17552 14423] [213 17521]100 [20056 20033] [30111 30100] [37194 37138]080 [1374 7263] [20193 1034] [26441 13497]075 [9692 4199] [13973 582] [19224 7817]070 [6490 2381] [9162 3207] [1334 4387]060 [2644 787] [3566 991] [5736 1344]050 [1007 305] [1288 353] [2157 447]040 [386 157] [462 169] [736 192]030 [171 110] [188 113] [257 118]025 [129 102] [136 103] [166 105]015 [101 100] [102 100] [104 100]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]



a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]







a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)









































a ϵ[195 205] k ϵ[8 10]

k1N [3022 37678] [36019 42229] [33592 36508]K2N [20893 21015] [20616 21804] [21514 22392]Shift (s) ARLN400 [1 1] [1 1] [1 1]300 [102 101] [102 101] [103 101]280 [104 101] [104 101] [104 102]250 [108 103] [108 104] [109 104]225 [117 108] [117 109] [119 11]200 [137 119] [138 121] [142 123]190 [152 127] [154 131] [16 133]180 [176 140] [18 145] [188 149]170 [214 160] [222 168] [235 174]160 [282 196] [298 21] [319 218]150 [408 264] [447 291] [484 306]140 [677 414] [774 476] [85 509]130 [1329 813] [1617 989] [1804 108]120 [3195 2157] [4216 2811] [4794 3151]110 [9067 7619] [13059 10763] [15349 12565]100 [20115 20212] [3019 30382] [3731 37448]080 [4794 2673] [612 3594] [7535 4264]075 [2569 1279] [3064 1647] [3774 1925]070 [1377 655] [1552 806] [1897 922]060 [434 234] [453 262] [531 284]050 [181 131] [182 137] [199 143]040 [114 104] [113 105] [117 106]030 [101 1] [101 1] [101 1]025 [1 1] [1 1] [1 1]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]


a ϵ[095 105] k ϵ[3 5]

k1N [34983 36534] [3449 38141] [36313 40783]K2N [19746 21033] [20774 2185] [21067 22228]Shift (s) ARLN400 [155 118] [162 12] [165 121]300 [227 147] [244 152] [254 154]280 [259 159] [281 166] [295 169]250 [336 189] [370 199] [394 205]225 [450 234] [506 252] [546 263]200 [669 325] [773 36] [849 382]190 [816 388] [954 437] [1057 468]180 [1022 482] [1213 553] [1356 598]170 [1321 626] [1595 733] [1802 802]160 [1771 859] [2181 1032] [2493 1145]150 [2475 1262] [3116 1558] [3608 1756]140 [3614 2002] [4675 2556] [5485 2932]130 [5512 346] [736 459] [876 5365]120 [8683 6446] [12038 8928] [1453 10653]110 [13674 12213] [1976 17702] [24152 21544]100 [20078 20053] [30132 30136] [37179 37222]080 [2352 15447] [36215 22988] [44756 28183]075 [20685 11191] [31834 16428] [39137 19976]







a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)













































a ϵ[095 105] k ϵ[8 10]

k1N [31112 3779] [37071 40566] [3522 37427]K2N [20442 21009] [20576 21815] [21181 22313]Shift (s) ARLN400 [107 102] [107 103] [108 103]300 [124 111] [125 112] [127 113]280 [132 115] [134 117] [136 118]250 [152 127] [157 13] [161 132]225 [185 144] [195 149] [201 153]200 [253 18] [275 19] [286 196]190 [302 205] [334 22] [349 228]180 [373 244] [423 265] [445 276]170 [485 304] [564 338] [597 355]160 [668 407] [802 465] [856 493]150 [99 599] [1232 705] [1329 757]140 [16 991] [208 1214] [2271 1323]130 [2858 1901] [3904 2438] [4333 2709]120 [5637 4268] [8105 5777] [9208 6584]110 [11595 10455] [17378 1502] [20415 17731]100 [20013 20181] [30558 30331] [37116 37054]080 [11709 7562] [16859 10795] [20429 12977]075 [7876 4372] [10853 6082] [13147 7262]070 [5062 2477] [6661 3353] [8064 3968]060 [1934 819] [2342 1038] [2809 1195]050 [713 315] [807 369] [944 408]040 [280 160] [300 174] [336 183]030 [140 111] [143 114] [151 116]025 [114 103] [115 104] [118 105]015 [1 1] [1 1] [1 1]010 [1 1] [1 1] [1 1]005 [1 1] [1 1] [1 1]

Table 3 Continued

a ϵ[095 105] k ϵ[3 5]

070 [17125 7655] [26351 11072] [3216 13347]060 [10135 3233] [15615 4521] [18725 5349]050 [5153 1272] [7921 17] [9329 1966]040 [2291 498] [3467 623] [401 698]030 [884 217] [128 249] [145 267]025 [521 157] [726 172] [81 18]015 [178 106] [215 108] [229 109]010 [118 1] [128 1] [132 101]005 [101 1] [101 1] [101 1]






a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)












































a ϵ[195 205] k ϵ[3 5]


K2N [20148 21321] [20843 22099] [22992 2337]Shift (s) ARLN400 [128 106] [111 102] [132 107] [112 102] [134 107] [116 103]300 [176 124] [134 110] [188 128] [138 111] [195 131] [147 113]280 [198 134] [145 114] [213 14] [150 116] [222 143] [162 119]250 [25 158] [171 125] [275 168] [180 128] [289 173] [200 133]225 [327 197] [213 142] [367 213] [229 147] [391 222] [260 155]200 [475 274] [299 176] [548 305] [331 185] [592 322] [385 201]190 [574 328] [360 201] [671 37] [405 213] [729 393] [475 234]180 [713 405] [449 237] [847 466] [514 255] [928 498] [610 284]170 [918 524] [586 294] [111 612] [686 323] [1226 661] [820 366]160 [1232 712] [809 391] [1519 851] [969 441] [1696 928] [1168 507]150 [174 1034] [1193 57] [2193 1267] [1471 663] [2476 1398] [1781 778]140 [2608 1627] [1903 937] [3374 2051] [2426 1134] [3861 2294] [2938 1351]130 [4189 2825] [3311 179] [5590 3686] [4388 2269] [6499 4190] [5300 2734]120 [7212 5492] [6251 4035] [9988 7468] [8659 5410] [11835 8658] [1043 6543]110 [12796 11571] [12105 10093] [18500 1656] [17552 14423] [22415 19669] [213 17521]100 [20002 20441] [20056 20033] [30017 30626] [30111 30100] [37082 37183] [37194 37138]080 [15258 9867] [1374 7263] [22795 14372] [20193 1034] [28115 17231] [26441 13497]075 [11934 6759] [9692 4199] [17729 9732] [13973 582] [21811 11609] [19224 7817]070 [9073 4544] [6490 2381] [13407 6461] [9162 3207] [16456 7664] [1334 4387]060 [4926 1971] [2644 787] [7188 2717] [3566 991] [8773 3178] [5736 1344]050 [2463 822] [1007 305] [3526 1086] [1288 353] [4265 1245] [2157 447]040 [1121 344] [386 157] [1556 426] [462 169] [1855 475] [736 192]030 [464 16] [171 110] [61 182] [188 113] [709 195] [257 118]025 [291 122] [129 102] [368 132] [136 103] [419 137] [166 105]015 [127 1] [101 100] [14 101] [102 100] [149 101] [104 100]010 [103 1] [1 1] [105 1] [1 1] [107 1] [1 1]005 [1 1] [1 1] [1 1] [1 1] [1 1] [1 1]


Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)








































Table 6 Continued








UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)









































UCL1N L1N

kN

1113969

[53915 71789]

LCL1N minusL1N

kN

1113969


UCL2N L2N

kN

1113969

[39823 51412]

LCL2N minusL2N

kN

1113969


(31)

UCLU = 671

UCLL = 519

LCLU = ndash671

LCLL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


LCL = ndash519

UCL = 519


ndash10

ndash5

0

5

10

15

ln (Z

t)


UCL1L = 53915



UCL1U = 71789

UCL2U = 51411UCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U


Table 7 Continued


35 [0873 0451] [6898 0822] [1931 minus0196]36 [0836 0688] [5091 2207] [1628 0792]37 [0983 0928] [5662 12949] [4036 2561]38 [0783 0956] [361 21757] [1284 308]39 [0631 0883] [1712 7528] [0538 2019]40 [0361 0998] [0565 488171] [minus0571 6191]




ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)










































ZkN ZkNminus1e

TlowastN


N [0985 54568] (32)





ln (Zt)Lln (Zt)U

UCL1U = 71789UCL2U = 51411

LCL1U = ndash71789

LCL2U = ndash51411

UCL1L = 53915UCL2L = 39823

LCL2L = 39823


ndash5

0

5

10

ln (Z

t) N

LCL1L = 53915


Table 8 +e UTI data



LCLU = ndash671

LCLL = ndash519

UCLU = 671

UCLL = 519


ndash10

ndash5

0

5

10

ln (Z

t) N

ln (Zt)Lln (Zt)U



5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)








































5 Conclusions


Data Availability




Acknowledgments


References
















UCL = 519

LCL = ndash519


ndash10

ndash5

0

5

10

ln (Z

t)








































Multiple Dependent State Sampling-Based Chart Using Belief ...

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