optimal scheme for process quality and cost control by...

Research ArticleOptimal Scheme for Process Quality andCost Control by Integrating a Continuous Sampling Plan andthe Process Yield Index

Chunzhi Li Shurong Tong and KeqinWang

School of Management Northwestern Polytechnical University No 1 Dongxiang Road Changrsquoan District Xirsquoan 710129 China

Correspondence should be addressed to Shurong Tong stongnwpueducn

Received 28 April 2018 Revised 2 July 2018 Accepted 30 October 2018 Published 19 November 2018

Academic Editor Lu Zhen

Copyright copy 2018 Chunzhi Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The single level continuous sampling plan (CSP-1) is an in-line process control tool that has been commonly adopted in variousmanufacturing industries However CSP-1 is designed for only satisfying the quality constraint At the same time CSP-1 hasdisadvantages with the high probabilities of both Type I and Type II errors due to its inherent deficiency coming from the operatingprocedure In this work an optimal scheme for process quality and cost control is proposed tomonitor the process cost and improvethe process quality The CSP-1 and the process yield index (119878119901119896) are integrated in the present scheme which work independentlyand complementarily The four parameters (clearance number inspecting fraction sample size and critical value) are designed inthe proposed scheme under simultaneously considering the quality and cost constraints The sole feasible inspection scheme inCSP-1 under the two constraints is found and used for controlling the process quality The probabilities of Type I and Type II errorsare concurrently controlled at the stipulated level with the risk control scheme which is constructedwith two nonlinear inequationbased on the accurate distribution of the index 119878119901119896 A case study is illustrated to validate the effectiveness and practicality of theproposed scheme

1 Introduction

With the advances in manufacturing and control technologyprocess quality can be constant at required level for the in-control process Multiple constraints such as quality costrisk and environmental adaptability are imposed on the pro-cess control The constraints are originated from the inten-sifying market competition and the demand of sustainabledevelopment Many kinds of process control tools are devel-oped to monitor and improve the process quality under vari-ous constraints for the in-control process As far as we knowthe process control tool is designed for satisfying only onecontrol constraint For example the continuous samplingplan (CSP) is commonly adopted in process control variousCSPs are presented for meeting only the quality constraintThe process yield index (119878119901119896) is proposed to only express theprocess capability No process control tool is developed forsolving the optimal problem of multiple constraints

Various CSPs are developed for satisfying the qualityrequirement and used to improve and control the process

quality The current literature on CSP can be generallyclassified into three categories Various CSPs with differentinspecting procedures were presented in the first kind ofliterature The single level CSP was designated as CSP-1 andwas widely adopted for process quality control in manufac-turing [1] To meet different demands in process controlsome reduced CSPs were presented with the objective ofdecreasing the number of units inspected when the probabil-ity of nonconformity for the processwas very low [2ndash5] Sometightened CSPs have been designed to guarantee that theoutgoing quality meets stringent quality requirements stip-ulated by the customer [6ndash8] Nevertheless CSPs designedunder quality constraint have not taken other constraint suchas cost and risk constraints into account The category IIliterature has investigated the influence of the change of theinspection scheme in CSPs on the inspection cost From acost perspective it was demonstrated that implementing aCSP for a stable production process was inappropriate [9]Considering the economic objective the inspection schemein CSP-1 could work most effectively when the probability

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 1917252 14 pageshttpsdoiorg10115520181917252

2 Discrete Dynamics in Nature and Society

of nonconformity was roughly two-thirds of the value of theaverage outgoing quality limit (AOQL) [10] Moreover thedifferent methods have been proposed for optimizing theparameters of inspection schemes in CSP-1 in a view of cost[11ndash14] However there is no agreed conclusion about how toidentify the inspection scheme in CSPs whose inspection costis the minimum The third kind of literature has proposedthe integrated control scheme betweenCSP and other processcontrol tools such as preventive maintenance and specifi-cation limit [15ndash21] The integrated schemes were designedwith the objective of minimizing the cost function Never-theless no integrating process control scheme is presentedfor simultaneously meeting the quality and cost constraintsby combining the CSP and other process control tools

The in-control process is commonly regarded as stochas-tic process There are two kinds of risk when process con-trol tools are adopted in process control One risk is theprobability of making type I error that an in-control processwith good quality is rejected The other is the probabilityof making type II error that an in-control process with badquality is accepted CSPs have an inherent deficiency that thetwo risks are high [22ndash25] The process yield index 119878119901119896 is aneffective performance measure for reflecting the influence ofthe machining centre drift and process deviation fluctuationon the probability of nonconformity The index 119878119901119896 had aone-to-one relationship with the probability of conformityand nonconformity [26] It has been demonstrated that thenatural estimator of 119878119901119896 was asymptotically normally dis-tributed [27] The accuracy of the natural estimator has beeninvestigated with a simulation technique [28] The processyields for some specified cases such as the imprecise sampledata circular profiles autocorrelation between linear profilesand multiple stream processes have been analyzed in someliterature [29ndash32] In the above researches the index 119878119901119896 wasused to only reflect the process capability In recent yearssome variable sampling plans based on the process capabilityindices have been proposed with the objective of buildingthe determination rules for the acceptance or rejection ofproduct lots [33ndash36] Two risks are simultaneously taken intoconsideration in the proposed variable sampling plans byutilizing the inference property of the natural estimator 119878119901119896Nevertheless no literature has been devoted to constructingthe risk control strategy with the index 119878119901119896 under quality andcost constraints for in-line process control

Distinguished from the objective of minimizing cost inthe sampling inspection for lot acceptance decision [34] thein-line process control aims to reach the control goal ofminimizing total cost The total cost constraint restricts thein-constant process to keep constant at right capability levelThus the reasonable inspection cost is permitted in processcontrol which is called cost constraintThe integrated processcontrol scheme needs to be presented with simultaneouslyconsidering the quality constraint and the inspection costconstraint The inspection cost constraint can be translatedinto the maximum affordable inspected fraction for the in-line process control In this work an optimal scheme forprocess quality and cost control by combining CSP-1 and theindex 119878119901119896 is presented CSP-1 is adopted in the integratedscheme due to its simplicity and practicability in operation

In the proposed plan there are four parameters (the clearancenumber the sampling fraction the sampling size and thecritical value of the index 119878119901119896) under the quality and costconstraints The two parameters (the clearance number andthe sampling fraction) in CSP-1 are utilized to guarantee thatthe average outgoing quality is conforming for the in-controlprocess The index 119878119901119896 is used to construct the risk controlscheme The two risks are concurrently controlled at the sti-pulated level by constructing two nonlinear inequations withthe accurate distribution of the estimator 119878119901119896

The rest of this paper is organized as follows In Section 2the concept and estimator of the index 119878119901119896 are introducedIn Section 3 the optimal scheme for process quality and costcontrol and the operating procedure are presentedThemeth-od of identifying the plan parameters is provided In Sec-tion 4 the values of the parameters of the optimal scheme forthree different quality constraints are tabulated for practicalpurposes Comparisons of the operating characteristic (OC)curves between CSP-1 and the optimal scheme are given topresent the advantages of the integrated scheme In Section 5an example of application is provided to validate the effective-ness and practicality of the integrated control plan FinallySection 6 concludes the paper

2 Process Fraction Nonconforming andProcess Yield Index

The process fraction nonconforming 119901 is a crucial per-formance measure for in-line process quality control For aprocess that is well controlled the value of 119901 can be taken as aconstant However the index of 119901 is an unknown variable andneeds to be estimated Let 119865(sdot) be the cumulative distributionfunction (CDF) of the quality characteristic interested so119901 = 1 minus [119865(119880119878119871) minus 119865(119871119878119871)] for the in-control process with atwo-sided specification limits where 119880119878119871 is the upper spec-ification limit and 119871119878119871 is the lower specification limit If thequality characteristic follows a normal distribution we get

119901 = 1 minus Φ(119880119878119871 minus 120583120590 ) minus Φ(119871119878119871 minus 120583120590 ) (1)

where 120583 and 120590 are the process mean and the processstandard deviation respectively and Φ(sdot) is the CDF of thestandard normal distribution119873(0 1) Unfortunately there isno literature devoted to a study of the distribution propertiesof the index of 119901

Boyles [13] proposed the use of the process yield index119878119901119896 to obtain an exact measure of the process yield for aprocesswith a normal distributionThere is a one-to-one rela-tionship between 119878119901119896 and the process yield The proposedindex 119878119901119896 is defined as

119878119901119896 = 13Φminus1 12Φ(119880119878119871 minus 120583120590 ) + 12Φ(120583 minus 119871119878119871120590 ) (2)

where Φminus1(sdot) is the inverse function of the CDF Φ(sdot) of thestandard normal distribution Let 119862119901 = (119880119878119871 minus 119871119878119871)(6120590)and 119862119886 = 1 minus |120583 minus 119872|119889 where119872 is the middle point of thewhole tolerance range119872 = (119880119878119871 + 119871119878119871)2 and 119889 is half the

Discrete Dynamics in Nature and Society 3

tolerance range 119889 = (119880119878119871 minus 119871119878119871)2 Equation (2) can also beexpressed as

119878119901119896 = 13Φminus1 12Φ (3119862119901119862119886) + 12Φ (3119862119901 (2 minus 119862119886)) (3)

The formula for the relationship between 119901 and 119878119901119896 canbe obtained from (1) and (2)

119901 = 2 minus 2Φ (3119878119901119896) 119878119901119896 gt 0 (4)

Table 1 shows the one-to-one correspondence betweenprocess yield process fraction nonconforming and processyield index

The process mean 120583 and the process standard deviation120590 are usually unknown variables and need to be estimatedusing the sample mean 119909 and the sample standard deviation 119904where 119909 = (1119899)sum119899119894=1 119909119894 and 119904 = radic(1(119899 minus 1))sum119899119894=1(119909119894 minus 119909)2Thus the estimator of 119878119901119896 can be written as follows

119878119901119896 = 13Φminus1 [12Φ(119880119878119871 minus 119909119904 ) + 12Φ(119909 minus 119871119878119871119904 )]= 13Φminus1 12Φ (3119862119901119862119886) + 12Φ (3119862119901 (2 minus 119862119886))

(5)

The estimator 119878119901119896 is such a complicated function that it isimpossible to obtain its exact cumulative distribution func-tion and probability density function Lee et al [14] furnisheda useful approximation to the distribution of 119878119901119896 under anormal distribution as

119878119901119896 asymp 119878119901119896 + 16radic119899 119882120601 (3119878119901119896) (6)

where 120601 is the probability density function of the standardnormal distribution119873(0 1) and119882

=

radic1198992 [119886 (1199042 minus 1205902)120590 ] minus radic119899119887 (119909 minus 120583)

120590 for 120583 lt 119872radic1198992 [119886 (1199042 minus 1205902)

120590 ] + radic119899119887 (119909 minus 120583)120590 for 120583 gt 119872

(7)

where 119886 and 119887 are defined as functions of 120583 and 120590 (or 119862119901 and119862119886)119886 = 1radic2

119880119878119871 minus 120583120590 120601 (119880119878119871 minus 120583120590 )+ 120583 minus 119871119878119871120590 120601 (120583 minus 119871119878119871120590 )= 1radic2 3119862119901 (2 minus 119862119886) 120601 (3119862119901 (2 minus 119862119886))+ 3119862119901119862119886120601 (3119862119901119862119886)

Table 1 The process yield process fraction nonconforming andcorresponding value of the process yield index 119878119901119896Yield () fraction nonconforming () 119878119901119896993066052 06933948 090995628077 04371923 095997300204 02699796 100998367295 01632705 105999033152 00966848 110

119887 = 120601 (119880119878119871 minus 120583120590 ) minus 120601 (120583 minus 119871119878119871120590 ) = 120601 [3119862119901 (2 minus 119862119886)]minus 120601 (3119862119901119862119886)

(8)

The estimator 119878119901119896 approximately follows a normal distri-bution119873(119878119901119896 (1198862 + 1198872)36119899(120601(3119878119901119896))2) Thus the probabilitydensity function of 119878119901119896 can be obtained as

119891119878119901119896 (119909) = radic18119899120587sdot 120601 (3119878119901119896)radic1198862 + 1198872 exp[[

minus18119899 (120601 (3119878119901119896))2

1198862 + 1198872 (119909 minus 119878119901119896)2]]

minusinfin lt 119909 lt +infin

(9)

3 The Optimal Scheme forProcess Quality and Cost Control

31 Quality and Cost Constraints Two constraints the aver-age outgoing quality limit (AOQL) and the inspection capa-bility limit (119860119865119868119871) are predetermined by the practitioner119860119865119868 is the average fraction inspected 119860119865119868119871 is the costconstraint and represents themaximumaffordable inspectionworkload The value of 119860119865119868119871 is converted from the plannedinspection cost The value of AOQL is generally stipulatedby the product designer or the customer For the in-controlprocess the proportion of nonconformance 119901 can be takenas constant Obviously there exists a specified in-controlprocess named limit quality process with quality limit 119901119868119876119871for which conforming outgoing quality can be achieved onlyunder the inspection capability limit 119860119865119868119871 This means that119860119865119868119871 = 1 minus AOQL119901119868119876119871 for the limit quality process withthe quality limit 119901119868119876119871 It needs to be noted that generally119901119868119876119871 gt AOQL

32 Determination of Parameters for the Quality ControlScheme in CSP-1 Thequality control scheme in CSP-1 shouldguarantee that the two constraints AOQL and 119860119865119868119871 aresimultaneously satisfied when 119901 ranges from 0 to 119901119868119876119871According to the performance formulas in CSP-1 proved by


0 04 08 12 16 20

1

2

3

p

AOQ

AOQL

Ｒ 10minus3

pIQL

AOQ3

AOQ2

AOQ1

Ｒ 10minus4

(iL fL)

(i3 f3)

(i2 f2)

(a)

0 04 08 12 16 20

025

05

075

1

p

AFI

Ｒ 10minus3

pIQL

(iL fL)

(i3 f3)

(i2 f2)

AFIL

AFI3

AFI2

AFI1

(b)

Figure 1 Comparisons of the curves of (a) 119860119874119876 and (b) 119860119865119868 between (119894119871 119891119871) and other two types of the AOQL contour schemes

Yang [37] the set of inequalities can be constructed as followsfor the quality and cost constraints respectively

(1 minus 119891) 119901119902119894(119891 + (1 minus 119891) 119902119894) le AOQL (10)

119891(119891 + (1 minus 119891) 119902119894) le 119860119865119868119871 (11)

where 119901 le 119901119868119876119871 119902 = 1 minus 119901 0 lt 119891 lt 1 119894 is the clear-ance number and only takes positive integer and 119891 is theinspection fraction

Obviously all AOQL contour schemes can meet inequal-ity (10) when 119901 ranges from 0 to 1 Figure 1 shows thecurves of the performance measures 119860119874119876 and 119860119865119868 for allAOQL contour schemes under the given constraints AOQLand 119860119865119868119871 and 119860119874119876 is the average outgoing quality Theinspection schemes (1198942 1198912) represent the type of schemeswith119901119871 gt 119901119868119876119871 (1198943 1198913) with 119901119871 gt 119901119868119876119871 (119894119871 119891119871) with 119901119871 = 119901119868119876119871119901119871 is the point where max(119860119874119876) = AOQL occurs Thereare infinite schemes respectively included in the two typesof the inspection schemes (1198942 1198912) and (1198943 1198913) The inspectionscheme (119894119871 119891119871) with 119901119871 = 119901119868119876119871 has only one included

It can be seen from Figure 1(b) that only the inspectionscheme (119894119871 119891119871) can meet the inequality (11) when 119901 le 119901119868119876119871Thus the inspection scheme (119894119871 119891119871) is the sole scheme whichcan simultaneously satisfy the two inequalities (10) and (11)named the optimal scheme

It can be concluded from Figure 1 that the value of119860119874119876 increases gradually to the quality constraint AOQLand the inspection workload also increases gradually to theinspection capability limit 119860119865119868119871 when the process quality isclose to the quality limit 119901119868119876119871 for the optimal scheme (119894119871 119891119871)

The scheme (119894119871 119891119871) can achieve the conforming outgoingquality 119860119874119876 lt AOQL with a smaller inspection workloadthan 119860119865119868119871 for an in-control process with 119901 lt 119901119868119876119871 and119860119874119876 = AOQL under the inspection capability limit 119860119865119868119871for an in-control process with 119901 = 119901119868119876119871

The values of the parameters 119894119871 and 119891119871 cannot be easilyachieved with inequalities (10) and (11) Li et al [38] proposedthe formulas to solve the parameters of the specified AOQLcontour scheme for the specified 119901 Thus the parameters 119894119871and 119891119871 can be obtained as follows

119894119871 = 119902119868119876119871119901119868119876119871 minus AOQL (12)

119891119871 = 119902119894119868119876119871119902119894119868119876119871 + 119901119868119876119871119894119902minus1119868119876119871 minus 1 (13)

where 119902119868119876119871 = 1 minus 119901119868119876119871Table 2 shows various inspection schemes (119894119871 119891119871) under

three quality constraints and various cost constraints Thevalues of 119894119871 decrease and the values of 119891119871 increase when thevalues of the quality limit 119901119868119876119871 become greater under thegiven quality requirement AOQL

In addition it can be observed from Figure 1(b) that forthe process with 119901 gt 119901119868119876119871 the minimum inspection work-load demanded formeeting the quality constraint exceeds theinspection capability limit 119860119865119868119871 the production should bestopped Nevertheless both the probabilities of making thetype I and II errors are high when the process is controlledwith the AOQL contour scheme in CSP-1 The risk controlscheme for controlling concurrently the two risks in theoptimal scheme should be redesigned under the quality andcost constraints


Table2Th

equalitycontrolschem

e(119894 119871119891 119871)

intheo

ptim

alschemeu

nder

specified

AOQLand119860119865119868119871(119901 119868119876119871)

AOQL=0

00018

AOQL=0

00143

AOQL=0

0122

119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 11987105000

000

036

5554

0119

205000

000286

697

0119

505000

00244

8001218

066

67000

054

2776

03086

066

67000

429

348

03092

066

6700366

3903145

07500

000

072

1851

044

1707500

000

572

232

044

2507500

004

8826

04498

08000

000

090

1388

05342

08000

000

715

174

05351

08000

00610

1905437

08333

000

108

1110

060

1108333

000

858

139

06022

08333

00732

1506117

08571

000

126

925

06515

08571

00100

1115

06527

08571

00854

1206629

08750

000

144

793

06908

08750

001144

9906920

08750

00976

1107028

08889

000

162

693

07222

08889

001287

8607235

08889

01098

907347


33 Determination of Parameters for the Risk Control SchemeThe in-control process with 119901 le AOQL should be acceptedwith a higher probability than 1 minus 120572 120572 is the probability ofmaking the type I error that the process with high qualitylevel is rejected The in-control process with 119901 gt 119901119868119876119871 shouldbe rejected with a higher probability than 1 minus 120573 120573 is theprobability of making the type II error that the process withlow quality level is accepted However the two types of riskscannot be simultaneously controlled with the AOQL contourschemes Thus the risk control scheme based on the index119878119901119896 is designed under the quality and cost constraints

There are two key points (AOQL 120572) and (119901119868119876119871 120573) thatneed to be considered at the same time in the risk controlscheme The OC curve should pass through the two desig-nated points tomeet the two constraints AOQL and119860119865119868119871 Let119878AOQL be the value of the process yield index correspondingto the quality level 119901 = AOQL and let 119878119868119876119871 be the valueof the process yield index corresponding to the quality level119901 = 119901119868119876119871 Therefore the two key points (AOQL 120572) and(119901119868119876119871 120573) for the OC function can be designated as (119878AOQL 120572)and (119878119868119876119871 120573) It means that if the estimator of 119878119901119896 for the in-control process is greater than the given value of 119878AOQL theprobability of accepting the in-control process will be greaterthan 1 minus 120572 and if the estimator of 119878119901119896 for the in-controlprocess is lower than the fixed value of 119878119868119876119871 the probabilityof accepting the in-control process will be less than the givenvalue of 120573

For the quality characteristic following a normal distri-bution and having a lower specification limit 119871119878119871 and upperspecification limit119880119878119871 the OC function with 119878119901119896 = 1198781015840 can begiven as

119875 (119878119901119896 ge 1199040 | 119878119901119896 = 1198781015840) = intinfin1199040

radic18119899120587

sdot 120601 (31198781015840)radic1198862 + 1198872 exp[[

minus18119899 (120601 (31198781015840))2

1198862 + 1198872

times (119909 minus 1198781015840)2]]119889119909 minusinfin lt 119909 lt +infin

(14)

Thus the two nonlinear inequalities specified by the tworisks can be obtained as follows

1198751 (119878119901119896 ge 1199040 | 119878119901119896 = 119878AOQL) = intinfin1199040

radic18119899120587sdot 120601 (3119878AOQL)radic1198862AOQL + 1198872AOQL

exp[minus18119899 (120601 (3119878AOQL))21198862AOQL + 1198872AOQL

times (119909 minus 119878AOQL)2]119889119909 ge 1 minus 120572

(15)

1198752 (119878119901119896 ge 1199040 10038161003816100381610038161003816119878119901119896 = 119878119868119876119871 ) = intinfin1199040

radic18119899120587sdot 120601 (3119878119868119876119871)radic1198862119868119876119871 + 1198872119868119876119871 exp[minus

18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871times (119909 minus 119878119868119876119871)2]119889119909 le 120573

(16)

The two parameters the critical value 1199040 and the samplesize 119899 must simultaneously satisfy the two nonlinear inequal-ities (15) and (16)There are infinite values of the combination(119899 1199040)which can meet inequalities (15) and (16)The specifiedvalue of (119899 1199040) which can simultaneously satisfy (17) and (18)is the boundary of all the feasible combinations (119899 1199040) Itmeans that the combination of (119899 1199040) meeting (17) and (18)can satisfy inequalities (15) and (16)

intinfin1199040



times (119909 minus 119878AOQL)2]119889119909 = 1 minus 120572

(17)

intinfin1199040

radic18119899120587120601 (3119878119868119876119871)

radic1198862119868119876119871 + 1198872119868119876119871 exp[minus18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871

times (119909 minus 119878119868119876119871)2]119889119909 = 120573(18)

The solution (119899 1199040) for (17) and (18) is sole For exampleunder two constraints AOQL = 000018(119878119860119874119876119871 = 12485)and 119901119868119876119871 = 00009(119878119868119876119871 = 11067) and two given risks120572 = 01 and 120573 = 01 the sole solution (119899 1199040) = (225 11730)can be obtained from (17) and (18) It implies that the valueof the estimator 119878119901119896 is calculated with the 225 sample data If119878119901119896 ge 11730 the process is judged to be controllable with thecurrent inspection scheme (119894119871 119891119871) under the quality and costconstraints and the current inspection scheme (119894119871 119891119871) willcontinue If 119878119901119896 lt 11730 the inspection with the inspectionscheme (119894119871 119891119871) will become uneconomic under the givenconstraints AOQL and 119860119865119868119871 and the production should bestopped or the maintenance will be triggered

34 Operation Procedure The roles of CSP-1 and the index119878119901119896 in the optimal scheme are independent and complemen-tary when considering simultaneously the two constraintsAOQL and 119860119865119868119871 for the process quality and cost controlThe objective of employing an inspection scheme (119894119871 119891119871) isto reduce the proportion of nonconformance and achieve theconforming outgoing quality The combination of (119899 1199040) isused to control simultaneously the two types of risks and can


be regarded as a new stopping rule for CSP-1 The operatingprocedure for the optimal scheme is as follows

Step 1 (process quality control) Calculate the value of theparameters (119894119871 119891119871) under the given values of AOQL and119860119865119868119871 Implement the inspection scheme (119894119871 119891119871) accordingto the procedure in CSP-1 for the process with the qualitycharacteristics interested

Step 2 (process risk control) Calculate the value of theparameters (119899 1199040) under the given values of AOQL and119860119865119868119871Keep the latest number 119899 of inspection data consecutively inthe order of production when the inspection scheme (119894119871 119891119871)in CSP-1 is performed Calculate the value of 119878119901119896 with thelatest number 119899 of inspection data The determination iscarried out as follows

(i) continue Step 1 if 119878119901119896 ge 1199040(ii) stop production if 119878119901119896 lt 1199040

where 1199040 is the threshold used to guarantee that the two risksare controlled at the stipulated level 119899 is the number of the lastinspection data recorded consecutively during inspectionincluding the screening inspection stage and the fractioninspection stage in CSP-1 The number 119899 of inspection datais used to calculate the value of 1198781199011198964 Analyses and Comparisons

In the proposed optimal scheme for process quality and costcontrol process quality control can be achieved by perform-ing the inspection scheme (119894119871 119891119871) and process risk controlcan be attained with the combination of (119899 1199040)The combina-tion of (119899 1199040) also plays the role of stopping rule in theinspection scheme (119894119871 119891119871)The values of 119899 and 1199040 are differentfor various values of 120572 and 120573 for specified constraints AOQLand 119860119865119868119871 Tables 3ndash5 show the values of the combinationof (119899 1199040) for 120572 = 001 005 and 01 and 120573 = 001 005and 01 under three quality requirements AOQL =000018 000143 and 00122 and three inspection capabil-ity limits 119860119865119868119871 = 06667 08 and 08571 According to theone-to-one correspondence between 119860119865119868119871 119901119868119876119871 and 119878119868119876119871three different process yield limits 119878119868119876119871 = 11534 11067and 10750 (119860119865119868119871 = 06667 08 and 08571) under 119878AOQL =12485 (AOQL = 000018) 119878119868119876119871 = 09520 08966 and 08585(119860119865119868119871 = 06667 08 and 08571) under 119878AOQL = 10628(AOQL = 000143) and 119878119868119876119871 = 06967 06245 and 05734(119860119865119868119871 = 06667 08 and 08571) under 119878AOQL = 08354(AOQL = 00122) respectively are considered to examinethe behavior of (119899 1199040) It can be observed from Tables 3ndash5that the value of the sample size 119899 becomes smaller as the 120572or the 120573 becomes largerThis phenomenon can be interpretedtomean that if the practitioner reduces the expected values atwhich high quality processes are rejected andor low qualityprocesses are accepted the sample size for the judgement onthe quality and capability of the processes will reduce For agiven 119878AOQL 120572 and 120573 the sample size becomes smaller asthe process yield limit decreases (the value of 119878119868119876119871 becomessmaller) For a fixed 120572 120573 and 119878119868119876119871 the value of 119899 becomes

smaller when the value of AOQL becomes larger (the valueof 119878AOQL becomes smaller) We can explain these phenomenaby saying that the determination of the quality and capabilityof the processes can be done easily using a smaller numberof inspection data when the difference between the qualitylimit 119901119868119876119871 and the quality constraint AOQL becomes largerFor example for 119878AOQL = 12485 120572 = 001 and 120573 = 001 therequired number of inspection data is 1688 for 119878119868119876119871 = 11534and only 485 for 119878119868119876119871 = 10750

Figures 2ndash4 show theOCcurves to depict a comparison ofthe optimal schemewith other twoAOQL contour inspectionschemes in CSP-1 It can be seen from Figures 2ndash4 that forvarious given values of the quality constraint AOQL andvarious values of the process quality limit 119901119868119876119871(119878119868119876119871) (whichrepresents the cost constraint) the OC curves for the pro-posed integrated schemes are more ideal than the OC curvesfor the other two schemes in CSP-1When 120572 and 120573 take largervalues for example when 120572 = 01 and 120573 = 01 the OC curvesstill have a more ideal shape than the curves for the othertwo schemes in CSP-1 at a higher yield level of 119878119868119876119871 When120572 and 120573 are both given smaller values for example 120572 = 001and 120573 = 001 the OC curves are more ideal than the curvesfor CSP-1 at various quality limit levels of 119901119868119876119871(119878119868119876119871) TheOCcurves for the optimal schemes move towards the right whenthe values of 119901119868119876119871 become bigger (the values of 119878119868119876119871 becomesmaller) which shows that the risk control scheme can supplythemore rational probabilities of the acceptance and rejectionfor the quality control scheme in the optimal scheme than theAOQL contour scheme in CSP-1

It can be noted that the inspection workload has notincreased in the proposed integrated scheme because thenumber of inspection data 119899 used to calculate the estimatorof 119878119901119896 is recorded during the CSP-1

5 Example Application

In order to present the way in which the proposed optimalscheme can be applied in practice the following exampletaken from a compressor manufacturing enterprise is con-sidered A cylinder is the key functioning part of an air con-ditioning compressor Cylinder thickness is an important di-mension for ensuring compressor performance Based onthe quality requirements given by the designer the tolerancerange of the cylinder thickness is set to 27784 plusmn 0002 andthe quality requirement AOQL is set to 000018 The currentinspection scheme (119894 119891) = (1540 05) is adopted in CSP-1 tocontrol the outgoing quality For using the optimal schemethe values of the four constraints are specified as AOQL =000018 (119878AOQL = 12485) 119860119865119868119871 = 08571 (119878119868119876119871 = 10750)120572 = 005 and 120573 = 005 From Table 3 the optimal schemecan be found as (119894119871 119891119871 119899 1199040) = (925 06515 242 11553)Using the proposed control scheme the probability of accept-ing an in-control processwith high quality level (eg the non-conforming fraction of the in-control process is lower thanthe value of AOQL = 000018) is greater than 1minus120572 = 095Theprobability of accepting an in-control process with low qual-ity level (eg the value of the index 119878119901119896 is lower than the valueof 119878119868119876119871 = 10750) is less than the value of 120573 = 005 Per-forming the quality control scheme (119894119871 119891119871) = (925 06515)


Table 3 The values of (119899 1199040) at 119878AOQL = 12485 (AOQL = 000018) and three levels of 119878119868119876119871(119860119865119868119871) for 120572 = 001 005 and 01 and 120573 =001 005 and 01120572 120573 119878119868119876119871 = 11534 119878119868119876119871 = 11067 119878119868119876119871 = 10750

(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3

(iL fL n s0)=(2776 03086 512 11985)(iL fL n s0)=(2776 03086 1688 11985)(i f)=(17420 0005)(i f)=(1540 05)

(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)

Figure 2 Comparisons of OC curves for the optimal schemes and inspection schemes in CSP-1 under AOQL = 000018 (a) at119860119865119868119871 = 06667(b) at 119860119865119868119871 = 08 and (c) at 119860119865119868119871 = 08571

242 sample items are recorded consecutively as shown inTable 6 The normal probability plot of the 242 sample itemsis displayed in Figure 5 Figure 6 shows the histogram of the242 observed data with the lower and upper specificationlimits Obviously based on the normality test in Figures 5and 6 the in-control process is shown to be close to a normaldistribution

Carrying out the calculation with the 242 inspection datawe get 119909 = 277842 119904 = 0000517 119878119901119896 = 12144 Obviouslythe estimator of 119878119901119896 = 12144 is greater than the value of1199040 = 11553 The optimal scheme which can simultaneouslymeet the two constraints AOQL = 000018 and 119860119865119868119871 =08571 will be continued Table 7 shows a comparison of the


Table6Th

e242ob

servations

collected

consecutively

from

theinspectionprocessinCS

P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)

(iL fL n s0)=(348 03092 265 10035)(iL fL n s0)=(348 03092 874 10035)(i f)=(2178 0005)(i f)=(19405)

(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)

(iL fL n s0)=(115 0652773 09498)(iL fL n s0)=(115 06527 239 09498)(i f)=(2178 0005)(i f)=(19405)

(c)


performances between the original inspection scheme andthe optimal scheme where 119871(119901) represents the probability ofacceptance It can be seen from Table 7 that the performance119860119865119868 increases by 0103172 119860119874119876 decreases by 0000028and the value of the probability of acceptance increases by0062855

6 Conclusions

An optimal scheme for the process quality and cost control isproposed to monitor the process capability and improve pro-cess quality The CSP-1 and 119878119901119896 which play independent andcomplimentary roles are integrated in the optimal scheme


0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)

Figure 4 Comparisons of OC curves for the optimal schemes and inspection schemes in CSP-1 under AOQL = 00122 (a) at119860119865119868119871 = 06667(b) at119860119865119868119871 = 08 and (c) at 119860119865119868119871 = 08571

The sole feasible inspection scheme in CSP-1 for meetingconcurrently the quality and cost constraints is one of theAOQL contour schemes in CSP-1 which occurs at the pointof 119901119871 = 119901119868119876119871 Two types of risk under quality and costconstraints are simultaneously controlled at the stipulatedlevel with the risk control scheme The risk control schemeis constructed with two nonlinear inequalities based on

the accurate distribution of the natural estimator 119878119901119896 Thecombination of the two risk control parameters (119899 1199040) playsthe role of the stopping rule in the inspection scheme (119894119871 119891119871)There is a one-to-one correspondence between the valuesof the four parameters (119894119871 119891119871 119899 1199040) and the given values ofthe four constraints (quality cost and the two risks) Theproposed optimal scheme shows the advantages over the


277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent

Probability Plot of icknessMean 277842StDev 0000517N 242AD 03675Pminusvalue 06473

Figure 5 The normal probability plot of the 242 observations

27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y

Histogram of icknessLSL USL

Figure 6 The histogram of the 242 observations with double specification limits

Table 7 Comparison of the performances of 119860119865119868 119860119874119876 and 119871(119901)(119894 119891) = (1540 05) (119894119871 119891119871 119899 1199040) =(925 06515 242 11553)

119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837

original process control tools in the measures of 119860119865119868 119860119874119876and the probability of acceptance

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H F Dodge ldquoA sampling inspection plan for continuous pro-ductionrdquoAnnals of Mathematical Statistics vol 14 pp 264ndash2791943

[2] KGovindaraju andCKandasamy ldquoDesign of generalizedCSP-C continuous sampling planrdquo Journal of Applied Statistics vol27 no 7 pp 829ndash841 2000

[3] G J Lieberman and H Solomon ldquoMulti-level continuous sam-pling plansrdquo Annals of Mathematical Statistics vol 26 pp 686ndash704 1955

[4] P Guayjarernpanishk and T Mayureesawan ldquoThe design oftwo-level continuous sampling planMCSP-2-CrdquoAppliedMath-ematical Sciences vol 6 no 89-92 pp 4483ndash4495 2012


[5] S Balamurali and K Subramani ldquoModified CSP-C continuoussampling plan for consumer protectionrdquo Journal of AppliedStatistics vol 31 no 4 pp 481ndash494 2004

[6] K Govindaraju and S Balamurali ldquoTightened single-level con-tinuous sampling planrdquo Journal of Applied Statistics vol 25 no4 pp 451ndash461 1998

[7] C Kandasamy andK Govindaraju ldquoSelection of tightened two-level continuous sampling plansrdquo Journal of Applied Statisticsvol 20 no 2 pp 271ndash284 1993

[8] V S S Kumar ldquoA tightened M-level continuous sampling planfor Markov-dependent production processesrdquo IIE Transactionsvol 16 no 3 pp 257ndash261 1984

[9] C R Cassady L M Maillart I J Rehmert and J A NachlasldquoDemonstrating demingrsquos kp rule using an economic model ofthe CSP-1rdquoQuality Engineering vol 12 no 3 pp 327ndash334 2000

[10] A J Duncan Quality control and industrial statistics Reviseded Richard D Irwin Inc Homewood Ill 1959

[11] M Eleftheriou and N Farmakis ldquoExpected cost for continuoussampling plansrdquo Communications in StatisticsmdashTheory andMethods vol 40 no 16 pp 2969ndash2984 2011

[12] H F YuW C Yu andW PWu ldquoAmixed inspection policy forCSP-1 and precise inspection under inspection errors and re-turn costrdquo Computers amp Industrial Engineering vol 57 no 3pp 652ndash659 2009

[13] A Haji and R Haji ldquoThe optimal policy for a sampling plan incontinuous production in terms of the clearance numberrdquoCom-puters amp Industrial Engineering vol 47 no 2-3 pp 141ndash1472004

[14] C H Chen and C Y Chou ldquoEconomic design of continuoussampling plan under linear inspection costrdquo Journal of AppliedStatistics vol 29 no 7 pp 1003ndash1009 2002

[15] W Xu Y Y Yu andQ S Zhang ldquoAn evaluationmethod of com-prehensive product quality for customer satisfaction based onintuitionistic fuzzy numberrdquo Discrete Dynamics in Nature Soci-ety vol 2018 no 1 p 12 2018

[16] B Bouslah A Gharbi and R Pellerin ldquoJoint economic designof production continuous sampling inspection and preventivemaintenance of a deteriorating production systemrdquo Interna-tional Journal of Production Economics vol 173 pp 184ndash1982016

[17] J Zhang ldquoOptimal control problem of converter steelmakingproduction process based on operation optimization methodrdquoDiscrete Dynamics in Nature and Society Art ID 483674 13pages 2015

[18] R Mehdi R Nidhal and C Anis ldquoIntegratedmaintenance andcontrol policy based on quality controlrdquoComputers amp IndustrialEngineering vol 58 no 3 pp 443ndash451 2010

[19] CH Chen C Y Chou andT S Cheng ldquoJoint design of contin-uous sampling plans and specification limitsrdquoThe InternationalJournal of AdvancedManufacturing Technology vol 21 pp 235ndash237 2003

[20] CH Chen andC-Y Chou ldquoJoint design of economicmanufac-turing quantity sampling plan and specification limitsrdquo Eco-nomic Quality Control vol 17 no 2 pp 145ndash153 2002

[21] CH Chen andC Y Chou ldquoDesign a continuous sampling planbased on quadratic quality loss functionrdquo Asia Pacific Manage-ment Review vol 6 no 4 pp 485ndash489 2001

[22] S Muthulakshmi ldquoStopping rules to limit inspection effort inCSP-C continuous sampling planrdquo International Journal ofMathematical Archive vol 3 pp 656ndash662 2012

[23] Y L Fan ldquoTwo more excellent stopping rules for continuoussampling planrdquo Journal of Applied Statistics vol 19 no 6 pp24ndash30 2000

[24] Y L Fan ldquoContinuous sampling plan CSP-1 with stopping in-spection rule (n-1)rdquo Acta Mathematicae Applicatae Sinica vol19 no 1 pp 135ndash143 1995

[25] R B Murphy ldquoStopping rules with CSP-1 sampling plansrdquo In-dustrial Control vol 16 no 5 pp 10ndash16 1959

[26] R A Boyles ldquoProcess capability with asymmetric tolerancesrdquoCommunications in StatisticsmdashSimulation and Computationvol 23 no 3 pp 615ndash643 1994

[27] J C Lee H N Hung W L Pearn and T L Kueng ldquoOn thedistribution of the estimated process yield index Spkrdquo Qualityand Reliability Engineering International vol 18 pp 111ndash1162002

[28] C W Wu and M-Y Liao ldquoEstimating and testing process yieldwith imprecise datardquo Expert Systems with Applications vol 36no 8 pp 11006ndash11012 2009

[29] W L Pearn and C C Chuang ldquoAccuracy analysis of the esti-mated process yield based on Spkrdquo Quality and ReliabilityEngineering International vol 20 no 4 pp 305ndash316 2004

[30] F K Wang ldquoProcess Yield for Multiple Stream Processes withIndividual Observations and SubsamplesrdquoQuality and Reliabil-ity Engineering International vol 32 no 2 pp 335ndash344 2016

[31] F K Wang ldquoMeasuring the process yield for circular profilesrdquoQuality and Reliability Engineering International vol 31 no 4pp 579ndash588 2015

[32] F KWang andY Tamirat ldquoProcess yield analysis for autocorre-lation between linear profilesrdquo Computers amp Industrial Engi-neering vol 71 no 1 pp 50ndash56 2014

[33] A Lepore B Palumbo and P Castagliola ldquoA note on decisionmaking method for product acceptance based on process capa-bility indices C119901119896 and C119901119898119896rdquo European Journal of OperationalResearch vol 267 no 1 pp 393ndash398 2018

[34] M S Fallah Nezhad and S Seifi ldquoRepetitive group samplingplan based on the process capability index for the lot acceptanceproblemrdquo Journal of Statistical Computation and Simulation vol87 no 1 pp 29ndash41 2017

[35] Y Tamirat and F K Wang ldquoSampling Plan based on the Ex-ponentiallyWeightedMoving Average Yield Index for Autocor-relationwithin Linear ProfilesrdquoQuality andReliability Engineer-ing International vol 32 no 5 pp 1757ndash1768 2016

[36] I Negrin Y Parmet and E Schechtman ldquoDeveloping a sam-pling plan based on Cpk-unknown variancerdquo Quality andReliability Engineering International vol 27 no 1 pp 3ndash14 2011

[37] G L Yang andG L Yang ldquoA renewal-process approach to con-tinuous sampling plansrdquo Technometrics vol 25 no 1 pp 59ndash671983

[38] C Z Li S R Tong and K QWang ldquoOptimal CSP-1 boundaryscheme based on the estimator of the proportion of confor-mance for specified in-control processrdquo Quality Technology ampQuantitative Management 2018

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of nonconformity was roughly two-thirds of the value of theaverage outgoing quality limit (AOQL) [10] Moreover thedifferent methods have been proposed for optimizing theparameters of inspection schemes in CSP-1 in a view of cost[11ndash14] However there is no agreed conclusion about how toidentify the inspection scheme in CSPs whose inspection costis the minimum The third kind of literature has proposedthe integrated control scheme betweenCSP and other processcontrol tools such as preventive maintenance and specifi-cation limit [15ndash21] The integrated schemes were designedwith the objective of minimizing the cost function Never-theless no integrating process control scheme is presentedfor simultaneously meeting the quality and cost constraintsby combining the CSP and other process control tools

The in-control process is commonly regarded as stochas-tic process There are two kinds of risk when process con-trol tools are adopted in process control One risk is theprobability of making type I error that an in-control processwith good quality is rejected The other is the probabilityof making type II error that an in-control process with badquality is accepted CSPs have an inherent deficiency that thetwo risks are high [22ndash25] The process yield index 119878119901119896 is aneffective performance measure for reflecting the influence ofthe machining centre drift and process deviation fluctuationon the probability of nonconformity The index 119878119901119896 had aone-to-one relationship with the probability of conformityand nonconformity [26] It has been demonstrated that thenatural estimator of 119878119901119896 was asymptotically normally dis-tributed [27] The accuracy of the natural estimator has beeninvestigated with a simulation technique [28] The processyields for some specified cases such as the imprecise sampledata circular profiles autocorrelation between linear profilesand multiple stream processes have been analyzed in someliterature [29ndash32] In the above researches the index 119878119901119896 wasused to only reflect the process capability In recent yearssome variable sampling plans based on the process capabilityindices have been proposed with the objective of buildingthe determination rules for the acceptance or rejection ofproduct lots [33ndash36] Two risks are simultaneously taken intoconsideration in the proposed variable sampling plans byutilizing the inference property of the natural estimator 119878119901119896Nevertheless no literature has been devoted to constructingthe risk control strategy with the index 119878119901119896 under quality andcost constraints for in-line process control

Distinguished from the objective of minimizing cost inthe sampling inspection for lot acceptance decision [34] thein-line process control aims to reach the control goal ofminimizing total cost The total cost constraint restricts thein-constant process to keep constant at right capability levelThus the reasonable inspection cost is permitted in processcontrol which is called cost constraintThe integrated processcontrol scheme needs to be presented with simultaneouslyconsidering the quality constraint and the inspection costconstraint The inspection cost constraint can be translatedinto the maximum affordable inspected fraction for the in-line process control In this work an optimal scheme forprocess quality and cost control by combining CSP-1 and theindex 119878119901119896 is presented CSP-1 is adopted in the integratedscheme due to its simplicity and practicability in operation

In the proposed plan there are four parameters (the clearancenumber the sampling fraction the sampling size and thecritical value of the index 119878119901119896) under the quality and costconstraints The two parameters (the clearance number andthe sampling fraction) in CSP-1 are utilized to guarantee thatthe average outgoing quality is conforming for the in-controlprocess The index 119878119901119896 is used to construct the risk controlscheme The two risks are concurrently controlled at the sti-pulated level by constructing two nonlinear inequations withthe accurate distribution of the estimator 119878119901119896

The rest of this paper is organized as follows In Section 2the concept and estimator of the index 119878119901119896 are introducedIn Section 3 the optimal scheme for process quality and costcontrol and the operating procedure are presentedThemeth-od of identifying the plan parameters is provided In Sec-tion 4 the values of the parameters of the optimal scheme forthree different quality constraints are tabulated for practicalpurposes Comparisons of the operating characteristic (OC)curves between CSP-1 and the optimal scheme are given topresent the advantages of the integrated scheme In Section 5an example of application is provided to validate the effective-ness and practicality of the integrated control plan FinallySection 6 concludes the paper

2 Process Fraction Nonconforming andProcess Yield Index

The process fraction nonconforming 119901 is a crucial per-formance measure for in-line process quality control For aprocess that is well controlled the value of 119901 can be taken as aconstant However the index of 119901 is an unknown variable andneeds to be estimated Let 119865(sdot) be the cumulative distributionfunction (CDF) of the quality characteristic interested so119901 = 1 minus [119865(119880119878119871) minus 119865(119871119878119871)] for the in-control process with atwo-sided specification limits where 119880119878119871 is the upper spec-ification limit and 119871119878119871 is the lower specification limit If thequality characteristic follows a normal distribution we get

119901 = 1 minus Φ(119880119878119871 minus 120583120590 ) minus Φ(119871119878119871 minus 120583120590 ) (1)

where 120583 and 120590 are the process mean and the processstandard deviation respectively and Φ(sdot) is the CDF of thestandard normal distribution119873(0 1) Unfortunately there isno literature devoted to a study of the distribution propertiesof the index of 119901

Boyles [13] proposed the use of the process yield index119878119901119896 to obtain an exact measure of the process yield for aprocesswith a normal distributionThere is a one-to-one rela-tionship between 119878119901119896 and the process yield The proposedindex 119878119901119896 is defined as

119878119901119896 = 13Φminus1 12Φ(119880119878119871 minus 120583120590 ) + 12Φ(120583 minus 119871119878119871120590 ) (2)

where Φminus1(sdot) is the inverse function of the CDF Φ(sdot) of thestandard normal distribution Let 119862119901 = (119880119878119871 minus 119871119878119871)(6120590)and 119862119886 = 1 minus |120583 minus 119872|119889 where119872 is the middle point of thewhole tolerance range119872 = (119880119878119871 + 119871119878119871)2 and 119889 is half the



119878119901119896 = 13Φminus1 12Φ (3119862119901119862119886) + 12Φ (3119862119901 (2 minus 119862119886)) (3)


119901 = 2 minus 2Φ (3119878119901119896) 119878119901119896 gt 0 (4)




(5)


119878119901119896 asymp 119878119901119896 + 16radic119899 119882120601 (3119878119901119896) (6)


=




(7)





(8)



minus18119899 (120601 (3119878119901119896))2

1198862 + 1198872 (119909 minus 119878119901119896)2]]


(9)





0 04 08 12 16 20

1

2

3

p

AOQ

AOQL

Ｒ 10minus3

pIQL

AOQ3

AOQ2

AOQ1

Ｒ 10minus4

(iL fL)

(i3 f3)

(i2 f2)

(a)

0 04 08 12 16 20

025

05

075

1

p

AFI

Ｒ 10minus3

pIQL

(iL fL)

(i3 f3)

(i2 f2)

AFIL

AFI3

AFI2

AFI1

(b)




119891(119891 + (1 minus 119891) 119902119894) le 119860119865119868119871 (11)







119894119871 = 119902119868119876119871119901119868119876119871 minus AOQL (12)

119891119871 = 119902119894119868119876119871119902119894119868119876119871 + 119901119868119876119871119894119902minus1119868119876119871 minus 1 (13)





Table2Th


e(119894 119871119891 119871)

intheo

ptim

alschemeu

nder

specified

AOQLand119860119865119868119871(119901 119868119876119871)

AOQL=0

00018

AOQL=0

00143

AOQL=0

0122

119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 11987105000

000

036

5554

0119

205000

000286

697

0119

505000

00244

8001218

066

67000

054

2776

03086

066

67000

429

348

03092

066

6700366

3903145

07500

000

072

1851

044

1707500

000

572

232

044

2507500

004

8826

04498

08000

000

090

1388

05342

08000

000

715

174

05351

08000

00610

1905437

08333

000

108

1110

060

1108333

000

858

139

06022

08333

00732

1506117

08571

000

126

925

06515

08571

00100

1115

06527

08571

00854

1206629

08750

000

144

793

06908

08750

001144

9906920

08750

00976

1107028

08889

000

162

693

07222

08889

001287

8607235

08889

01098

907347





119875 (119878119901119896 ge 1199040 | 119878119901119896 = 1198781015840) = intinfin1199040

radic18119899120587

sdot 120601 (31198781015840)radic1198862 + 1198872 exp[[

minus18119899 (120601 (31198781015840))2

1198862 + 1198872


(14)


1198751 (119878119901119896 ge 1199040 | 119878119901119896 = 119878AOQL) = intinfin1199040




(15)

1198752 (119878119901119896 ge 1199040 10038161003816100381610038161003816119878119901119896 = 119878119868119876119871 ) = intinfin1199040


18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871times (119909 minus 119878119868119876119871)2]119889119909 le 120573

(16)


intinfin1199040




(17)

intinfin1199040

radic18119899120587120601 (3119878119868119876119871)

radic1198862119868119876119871 + 1198872119868119876119871 exp[minus18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871

times (119909 minus 119878119868119876119871)2]119889119909 = 120573(18)

















(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018










119878119901119896 = 13Φminus1 12Φ (3119862119901119862119886) + 12Φ (3119862119901 (2 minus 119862119886)) (3)


119901 = 2 minus 2Φ (3119878119901119896) 119878119901119896 gt 0 (4)




(5)


119878119901119896 asymp 119878119901119896 + 16radic119899 119882120601 (3119878119901119896) (6)


=




(7)





(8)



minus18119899 (120601 (3119878119901119896))2

1198862 + 1198872 (119909 minus 119878119901119896)2]]


(9)





0 04 08 12 16 20

1

2

3

p

AOQ

AOQL

Ｒ 10minus3

pIQL

AOQ3

AOQ2

AOQ1

Ｒ 10minus4

(iL fL)

(i3 f3)

(i2 f2)

(a)

0 04 08 12 16 20

025

05

075

1

p

AFI

Ｒ 10minus3

pIQL

(iL fL)

(i3 f3)

(i2 f2)

AFIL

AFI3

AFI2

AFI1

(b)




119891(119891 + (1 minus 119891) 119902119894) le 119860119865119868119871 (11)







119894119871 = 119902119868119876119871119901119868119876119871 minus AOQL (12)

119891119871 = 119902119894119868119876119871119902119894119868119876119871 + 119901119868119876119871119894119902minus1119868119876119871 minus 1 (13)





Table2Th


e(119894 119871119891 119871)

intheo

ptim

alschemeu

nder

specified

AOQLand119860119865119868119871(119901 119868119876119871)

AOQL=0

00018

AOQL=0

00143

AOQL=0

0122

119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 11987105000

000

036

5554

0119

205000

000286

697

0119

505000

00244

8001218

066

67000

054

2776

03086

066

67000

429

348

03092

066

6700366

3903145

07500

000

072

1851

044

1707500

000

572

232

044

2507500

004

8826

04498

08000

000

090

1388

05342

08000

000

715

174

05351

08000

00610

1905437

08333

000

108

1110

060

1108333

000

858

139

06022

08333

00732

1506117

08571

000

126

925

06515

08571

00100

1115

06527

08571

00854

1206629

08750

000

144

793

06908

08750

001144

9906920

08750

00976

1107028

08889

000

162

693

07222

08889

001287

8607235

08889

01098

907347





119875 (119878119901119896 ge 1199040 | 119878119901119896 = 1198781015840) = intinfin1199040

radic18119899120587

sdot 120601 (31198781015840)radic1198862 + 1198872 exp[[

minus18119899 (120601 (31198781015840))2

1198862 + 1198872


(14)


1198751 (119878119901119896 ge 1199040 | 119878119901119896 = 119878AOQL) = intinfin1199040




(15)

1198752 (119878119901119896 ge 1199040 10038161003816100381610038161003816119878119901119896 = 119878119868119876119871 ) = intinfin1199040


18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871times (119909 minus 119878119868119876119871)2]119889119909 le 120573

(16)


intinfin1199040




(17)

intinfin1199040

radic18119899120587120601 (3119878119868119876119871)

radic1198862119868119876119871 + 1198872119868119876119871 exp[minus18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871

times (119909 minus 119878119868119876119871)2]119889119909 = 120573(18)

















(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









0 04 08 12 16 20

1

2

3

p

AOQ

AOQL

Ｒ 10minus3

pIQL

AOQ3

AOQ2

AOQ1

Ｒ 10minus4

(iL fL)

(i3 f3)

(i2 f2)

(a)

0 04 08 12 16 20

025

05

075

1

p

AFI

Ｒ 10minus3

pIQL

(iL fL)

(i3 f3)

(i2 f2)

AFIL

AFI3

AFI2

AFI1

(b)




119891(119891 + (1 minus 119891) 119902119894) le 119860119865119868119871 (11)







119894119871 = 119902119868119876119871119901119868119876119871 minus AOQL (12)

119891119871 = 119902119894119868119876119871119902119894119868119876119871 + 119901119868119876119871119894119902minus1119868119876119871 minus 1 (13)





Table2Th


e(119894 119871119891 119871)

intheo

ptim

alschemeu

nder

specified

AOQLand119860119865119868119871(119901 119868119876119871)

AOQL=0

00018

AOQL=0

00143

AOQL=0

0122

119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 11987105000

000

036

5554

0119

205000

000286

697

0119

505000

00244

8001218

066

67000

054

2776

03086

066

67000

429

348

03092

066

6700366

3903145

07500

000

072

1851

044

1707500

000

572

232

044

2507500

004

8826

04498

08000

000

090

1388

05342

08000

000

715

174

05351

08000

00610

1905437

08333

000

108

1110

060

1108333

000

858

139

06022

08333

00732

1506117

08571

000

126

925

06515

08571

00100

1115

06527

08571

00854

1206629

08750

000

144

793

06908

08750

001144

9906920

08750

00976

1107028

08889

000

162

693

07222

08889

001287

8607235

08889

01098

907347





119875 (119878119901119896 ge 1199040 | 119878119901119896 = 1198781015840) = intinfin1199040

radic18119899120587

sdot 120601 (31198781015840)radic1198862 + 1198872 exp[[

minus18119899 (120601 (31198781015840))2

1198862 + 1198872


(14)


1198751 (119878119901119896 ge 1199040 | 119878119901119896 = 119878AOQL) = intinfin1199040




(15)

1198752 (119878119901119896 ge 1199040 10038161003816100381610038161003816119878119901119896 = 119878119868119876119871 ) = intinfin1199040


18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871times (119909 minus 119878119868119876119871)2]119889119909 le 120573

(16)


intinfin1199040




(17)

intinfin1199040

radic18119899120587120601 (3119878119868119876119871)

radic1198862119868119876119871 + 1198872119868119876119871 exp[minus18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871

times (119909 minus 119878119868119876119871)2]119889119909 = 120573(18)

















(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









Table2Th


e(119894 119871119891 119871)

intheo

ptim

alschemeu

nder

specified

AOQLand119860119865119868119871(119901 119868119876119871)

AOQL=0

00018

AOQL=0

00143

AOQL=0

0122

119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 119871119860119865119868119871

119901 119868119876119871119894 119871

119891 11987105000

000

036

5554

0119

205000

000286

697

0119

505000

00244

8001218

066

67000

054

2776

03086

066

67000

429

348

03092

066

6700366

3903145

07500

000

072

1851

044

1707500

000

572

232

044

2507500

004

8826

04498

08000

000

090

1388

05342

08000

000

715

174

05351

08000

00610

1905437

08333

000

108

1110

060

1108333

000

858

139

06022

08333

00732

1506117

08571

000

126

925

06515

08571

00100

1115

06527

08571

00854

1206629

08750

000

144

793

06908

08750

001144

9906920

08750

00976

1107028

08889

000

162

693

07222

08889

001287

8607235

08889

01098

907347





119875 (119878119901119896 ge 1199040 | 119878119901119896 = 1198781015840) = intinfin1199040

radic18119899120587

sdot 120601 (31198781015840)radic1198862 + 1198872 exp[[

minus18119899 (120601 (31198781015840))2

1198862 + 1198872


(14)


1198751 (119878119901119896 ge 1199040 | 119878119901119896 = 119878AOQL) = intinfin1199040




(15)

1198752 (119878119901119896 ge 1199040 10038161003816100381610038161003816119878119901119896 = 119878119868119876119871 ) = intinfin1199040


18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871times (119909 minus 119878119868119876119871)2]119889119909 le 120573

(16)


intinfin1199040




(17)

intinfin1199040

radic18119899120587120601 (3119878119868119876119871)

radic1198862119868119876119871 + 1198872119868119876119871 exp[minus18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871

times (119909 minus 119878119868119876119871)2]119889119909 = 120573(18)

















(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018












119875 (119878119901119896 ge 1199040 | 119878119901119896 = 1198781015840) = intinfin1199040

radic18119899120587

sdot 120601 (31198781015840)radic1198862 + 1198872 exp[[

minus18119899 (120601 (31198781015840))2

1198862 + 1198872


(14)


1198751 (119878119901119896 ge 1199040 | 119878119901119896 = 119878AOQL) = intinfin1199040




(15)

1198752 (119878119901119896 ge 1199040 10038161003816100381610038161003816119878119901119896 = 119878119868119876119871 ) = intinfin1199040


18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871times (119909 minus 119878119868119876119871)2]119889119909 le 120573

(16)


intinfin1199040




(17)

intinfin1199040

radic18119899120587120601 (3119878119868119876119871)

radic1198862119868119876119871 + 1198872119868119876119871 exp[minus18119899 (120601 (3119878119868119876119871))21198862119868119876119871 + 1198872119868119876119871

times (119909 minus 119878119868119876119871)2]119889119909 = 120573(18)

















(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018






















(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (2776 03086) (119894119871 119891119871) = (1388 05342) (119894119871 119891119871) = (925 06515)

119899 1199040 119899 1199040 119899 1199040001 001 1688 11985 740 11730 485 11553

005 1271 11895 551 11610 362 11406010 1046 11850 459 11520 305 11308

005 001 1229 12060 523 11850 345 11702005 843 11985 370 11730 242 11553010 673 11925 296 11640 195 11446

01 001 996 12120 431 11940 279 11808005 661 12045 290 11820 188 11661010 512 11985 225 11730 147 11553


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (348 03092) (119894119871 119891119871) = (174 05351) (119894119871 119891119871) = (115 06527)119899 1199040 119899 1199040 119899 1199040

001 001 874 10035 371 09720 239 09498005 655 09945 281 09585 181 09327010 554 09885 236 09480 153 09214

005 001 639 10125 266 09870 168 09674005 437 10035 186 09720 120 09498010 358 09975 150 09615 97 09373

0 1 001 508 10200 214 09960 135 09799005 346 10110 144 09825 92 09624010 265 10035 113 09720 73 09498


(119860119865119868119871 = 06667) (119860119865119868119871 = 08) (119860119865119868119871 = 08571)(119894119871 119891119871) = (39 03145) (119894119871 119891119871) = (19 05437) (119894119871 119891119871) = (12 06629)119899 1199040 119899 1199040 119899 1199040

001 001 324 07590 129 07140 78 06800005 246 07470 99 06960 61 06590010 206 07395 85 06855 52 06452

005 001 229 07710 89 07320 53 07026005 163 07590 65 07140 39 06800010 133 07500 53 07012 32 06647

0 1 001 187 07800 71 07455 42 07188005 127 07680 49 07267 29 06961010 100 07590 39 07140 24 06800


0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









0 06 12 180

025

05

075

1

p

L(p)

Ｒ 10minus3


(a)

0 06 12 18x 10minus3

0

025

05

075

1

p

L(p)


(b)

06 12 180p

0

025

05

075

1

L(p)

x 10minus3


(c)





Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









Table6Th

e242ob

servations

collected

consecutively

from


P-1

277834

277844

277839

277834

277844

277838

277851

277845

277836

27784

277842

277834

277842

277836

277832

277846

27785

277844

277850

277837

277844

277840

277844

277844

277835

277834

277839

277852

277845

277848

277836

277841

277848

277844

277849

277835

277836

277838

277846

27785

277846

277841

277845

277854

277849

277842

277837

277834

277846

277847

277844

277852

277844

277841

277845

277844

277841

277838

277830

277840

277843

277844

277843

277839

277844

277843

277847

277844

277844

27783

277842

277845

277840

277852

277837

277843

277849

277849

277835

277846

277834

277839

277849

277839

277848

277844

277835

277845

277849

277836

277834

277844

277842

277846

277837

277844

27784

277842

277846

277849

277836

277830

277842

277845

277844

277847

277846

277844

277843

277848

277847

277846

277831

277841

277844

277851

277836

27785

277844

277842

277849

277846

277836

277832

277842

277843

277846

277836

277850

277842

277834

277843

277849

277836

277834

277838

277845

277845

277838

277851

277842

277834

277840

277844

277839

277832

277841

277847

277847

277841

277846

277842

277834

277840

277846

277841

277829

277842

277844

277849

277844

277845

277845

277834

277837

277842

277837

277835

277844

277846

277852

277846

277842

277843

277835

277844

277849

277839

277832

277844

277849

277849

277842

277844

277844

277835

277841

277844

277838

277834

277841

277844

277848

277848

277851

277844

277847

277843

277844

277834

277833

277844

277844

277847

277841

277847

277843

27784

277844

277842

277841

277843

277844

277844

277837

277842

277846

277839

277844

277851

277836

277836

277836

277842

277842

277845

277845

277842

277844

277844

277843

277846

277840

277845

277842

277838

277836

277836

277836

277836

277836

277837


0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









0 0005 001 00150

025

05

075

1

p

L(p)


(a)

0 0005 001 00150

025

05

075

1

p

L(p)


(b)

0 0005 001 00150

025

05

075

1

p

L(p)


(c)



6 Conclusions



0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









0 004 008 0120

025

05

075

1

p

L(p)


(a)

0 004 008 0120

025

05

075

1

p

L(p)


(b)

0 004 008 0120

025

05

075

1

p

L(p)


(c)





277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018









277825 277835 277845 277855 27786

0005001

00501

025

05

075

09095

0990995

ickness

Perc

ent



27782 27783 27784 27785 277860

5

10

15

20

25

30

35

40

ickness

Freq

uenc

y




119860119865119868 0602509 0705681119860119874119876 0000107 0000079119871(119901) 0794982 0857837


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018


















































Journal of












Journal of







Volume 2018






Volume 2018








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