ORIGINAL ARTICLE

Optimal designing of an SkSP-V skip-lot sampling planwith double-sampling plan as the reference plan

Muhammad Aslam & Saminathan Balamurali &Chi-Hyuck Jun & Munir Ahmad & Mujahid Rasool

Received: 26 October 2010 /Accepted: 9 September 2011 /Published online: 30 September 2011# Springer-Verlag London Limited 2011

Abstract This paper deals with the optimal designing ofa skip-lot sampling plan of type SkSP-V by consideringthe double-sampling plan as the reference plan. Thedesign parameters are determined so as to minimize theaverage sample number while the specified producer’srisk and the consumer’s risks are satisfied. The tables areconstructed by considering the various combinations ofacceptable and limiting quality levels, and an example isgiven for illustration purpose. The advantages of theproposed plan over the conventional double-samplingplan are also discussed.

Keywords Acceptable quality level . Average samplenumber . Consumer’s risk . Double sampling . Producer’srisk . Skip-lot sampling

1 Introduction

Statistical quality control is the combination of qualitycontrol charts and acceptance sampling. The former isused to maintain the quality of products during themanufacturing process and the latter is used to inspectthe final manufactured products before they can bereleased for consumer's use. During the process ofinspection, it may not be possible to inspect 100% ofproducts, especially in case of electronic components, as itrequires high cost, time, and damage of the products. Onthe other hand, if the product is believed to be good, thenit requires no inspection. Acceptance sampling works as abridge between 0% and 100% inspections. For theinspection of the products, only a small amount from theproducts called the sample size is selected from the bigamount of the products, which is also called a lot. The fateof the entire lot whether to accept or to reject depends onthe information obtained from this sample size. The basicacceptance sampling plan called the single-sampling planis widely used in industry to inspect items due to itseasiness of implementation. Further, in single-samplingplans, the decision of the submitted lot is taken on thebasis of a single sample drawn from the lot. There aremany other types of acceptance sampling plans availablein the literature including double-sampling plan, multiplesampling plan, sequential sampling plan, repetitive groupsampling plan etc. Double-sampling plan is an extensionof single-sampling plan to make a decision on the lot based onthe inspection of two samples. More generally, multiplesampling plans can be considered but repetitive groupsampling plans are usually adopted due to their simplicity.

M. Aslam (*) :M. RasoolDepartment of Statistics, Forman Christian College University,Lahore, Pakistane-mail: aslam_ravian@hotmail.com

S. BalamuraliDepartments of Mathematics, Kalasalingam University,Krishnankoil 626190( Tamil Nadu, Indiae-mail: sbmurali@rediffmail.com

C.-H. JunDepartment of Industrial and Management Engineering,POSTECH,Pohang 790-784, Republic of Koreae-mail: chjun@postech.ac.kr

M. AhmadNational College of Business Administration & Economics,Lahore, Pakistane-mail: drmunir@ncbae.edu.pk

Int J Adv Manuf Technol (2012) 60:733–740DOI 10.1007/s00170-011-3635-5

It should be noted that whatever may be the type ofacceptance sampling schemes, two risks are alwaysassociated with them. As decision is made on the basis ofthe sample from a lot, there is a chance of committingerrors. If a good lot of products is rejected, this leads type-1error. Type-1 error is defined as the probability of rejectinga good lot and it is denoted by α. On the other hand, if abad lot is accepted, this may lead to type-2 error. Type-2error is defined as the probability of accepting a bad lot andit is denoted by β. So, one purpose of an acceptancesampling plan is to minimize the sample size whilesatisfying the producer's risk as well as the consumer's riskat the specified quality levels. The minimum sample size iscalled the optimal sample size.

In a life test, the cost of the inspection is directly relatedto the sample size selected to put on the test. The larger thesample size, the larger the cost of the experiment. So,producers and consumers prefer the plan which providesthem protection and save their cost of the experiment toreach on the final decision.

The concept of skip-lot sampling was developed byDodge [10] based on the principle of continuous samplingplan applicable to a continuous series of lots or batches ofmaterials. The initial skip-lot sampling plan was designatedas SkSP-1 and has been used for the inspection for bulkmaterials or products produced in successive lots. As thisscheme requires the smaller number of sample units for theinspection as compared to the single-sampling plan, themain advantage of this type of sampling scheme is toreduce the inspection cost. It is to be pointed out that inSkSP-1 scheme, no reference plan concept is used.Dodge and Perry [11] and Perry [19] developed a systemof skip-lot sampling plans by using the concept ofreference plan, which is designated as SkSP-2. In SkSP-2plan, provision is made for skipping inspection of someof the lots when the quality of the submitted lot isgood. Skip-lot sampling is used for sampling chemicaland physical processes in order to bring about substantialsavings on inspection of products, which normallyconform to specification. This particular sampling planis applicable when the lots are small or where inspection islow and costly. For more details about the SkSP-2 plan,the readers can refer to Burnett [7], Schilling [20], Carr[8], Liebesman and Saperstein [16], Liebesman [15],Vijayaraghavan [22], ISO 2859–3 [14], and Balamuraliet al. [6]. Parker and Kessler [18] modified the procedureSkSP-2 plan and developed modified skip-lot samplingplans. Hsu [13] presented a model involving stochasticand cost components to obtain an optimal skip-lotsampling plan. Recently, Aslam et al. [2] developed tablesfor determining the design parameters of an SkSP-2plan using single-sampling plan as the reference plan.They determined the design parameters including the

parameters of the reference plan such that the specifiedproducer's and consumer's risks are satisfied simulta-neously. Aslam et al. [3] also developed a two-plansampling system for Weibull distribution based on asimilar idea.

MIL-STD 1235C [17] consists of five differentcontinuous sampling plans (CSP) including the CSP-V.All these plans are used to reduce the inspection cost andtime, so they are considered as more economical. CSP-Vis one of the single level continuous sampling plans inwhich the reduced inspection can be achieved by using asmaller clearance interval when reducing the samplingfrequency has no advantage upon demonstration of goodquality. The CSP-V plan provides for alternating sequen-ces of 100% inspection and sampling inspection. For moredetails about the continuous acceptance sampling plans,reader can refer to Stephens [21]. Since the skip-lotconcept is sound and useful, it is economicallyadvantageous to the skip-lot approach in the design ofsampling plans. Recently, Balamurali and Jun [5]developed a new system of skip-lot sampling plandesignated as SkSP-V based on the principles of CSP-Vplan. This plan requires the return to the normalinspection whenever the lot is rejected, but it has aprovision for a reduced normal inspection upon demon-stration of superior product quality. For more detailsabout this plan and its properties, reader can refer toBalamurali and Jun [5]. Some other variants of the skip-lot sampling plans have been considered by Cho andChoi [9], Yang and Cho [24] and so on. Balamurali andJun [5] investigated the properties of SkSP-V plan withsingle-sampling plan as the reference plan. But, it iswell-known that a double-sampling plan reduces thesample size required for inspection and provides anopportunity to the producer for the acceptance of theproduct when the product under study is questionable.The operation of a double-sampling plan with fourparameters is as follows (see [12]):

1. From a lot, select a random sample of size n1 andobserve the number of nonconforming units d1.

2. If d1≤c1 (c1 is acceptance number for the firstsample), then the lot is accepted. If d1>c2 (c2 is theacceptance number for the combined sample) the lot isrejected. If c1<d1≤c2, then go for second sampling.During second sampling, select a random sample ofsize n2 and observe the number of nonconformingunits d2.

3. If d1+d2≤c2, then the lot is accepted. Otherwise, the lotis rejected.

Vijayaraghavan and Soundararajan [23] developed tablesfor the selection an SkSP-2 plan with double-sampling planas the reference plan under Poisson probability model and

734 Int J Adv Manuf Technol (2012) 60:733–740

we designate this plan as SkDSP-2. In the literature, noattempt has been made to find the design parameters ofthe SkSP-V sampling plans with double-sampling planas the reference plan by considering the producer's riskand the consumer's risk simultaneously. So this paperattempts to develop tables for the selection of parametersof an SkSP-V plan with double-sampling plan as thereference plan and the proposed plan is designated asSkDSP-V plan. The rest of the paper is set as follows: theoperation of SkDSP-V plan is given in Section 2. InSection 3, comparison of the proposed plan is made withthe SkSP-2 plan and double-sampling plan. In the lastsection, some concluding remarks are given.

2 SkSP-V plan with double-sampling planas the reference plan

As mentioned already, SkSP-V plan is a generalization ofSkSP-2 plan. In SkSP-2 plan, reversion of normal inspec-tion takes place immediately after a lot is rejected onskipping inspection. But the SkSP-V plan requires a returnto normal inspection whenever a lot is rejected duringsampling inspection, but has a provision for a reducednormal inspection upon demonstration of superior productquality. The operation of SkSP-V plan with double-sampling plan as the reference plan is shown in Fig. 1.

The SkSP-V plan is characterized by four parameterswhich are f (0<f<1), i k, x (≤i) along with the parameters ofthe reference plan. The Skip-lot sampling plan of type SkSP-

2 is a special case of SkSP-V sampling plan with k=x=i. Byassuming x=k in SkSP-V plan, we need to determine onlythree parameters of the proposed plan, which are f, i and k.Therefore, the SkDSP-V plan is characterized by sevenparameters n1, n2, c1, c2, i, f, and k.

3 Designing of an SkDSP-V plan

In general, any sampling plan or any sampling system canbe designed for specified two points on the operatingcharacteristic (OC) curve namely, acceptable quality level(AQL) and limiting quality level (LQL), along with thecorresponding producer's risk (α) and the consumer's risks(β). According to the American National Standards Insti-tute/American Society for Quality [1] (ANSI/ASQ Z1.4-2008), the AQL is defined as “the maximum proportion orpercent defective (or the maximum number of defects perhundred units) that, for purposes of sampling inspection,can be considered satisfactory as a process average”. AQLis usually defined as the worst-case quality level, inpercentage or ratio, which is still considered acceptable.As an AQL is an acceptable level, the probability ofacceptance of a lot at the AQL should be high. AmericanNational Standards Institute/American Society for Quality[1] (ANSI/ASQ Z1.4-2008) defines the LQL as “thepercentage or proportion of variant units in a batch or lotfor which, for the purposes of sampling inspection, theconsumer wishes the probability of acceptance to berestricted to a specified low value”. LQL is used as an

Start

NormalInspection

Are i consecutivelots acceptable ?

SkippingInspection

Is a sampled lotacceptable ?

Have last k lotsbeen acceptable ?

Normal withreduced clearance

Are x consecutivelots acceptable ?

Replace all nonconforming units with conforming ones

yes

no

yes no

yes

no

no

yes

Fig. 1 Operation of SkSP-V plan

Int J Adv Manuf Technol (2012) 60:733–740 735

index for consumer protection for designing an acceptancesampling plan. AQL is denoted by p1 and the LQL isdenoted by p2. Based on the principle of two points on theOC curve, the designing methodology of the SkDSP-V planis explained below.

According to Balamurali and Jun [5], the OC function ofan SkSP-V plan is given by,

PaðpÞ ¼fP þ 1� fð ÞPiÞ þ fPkþ1 Pi � Pk

� �

f 1þ Piþk � P2kð Þ þ 1� fð ÞPi

where Pa (p) is the probability of acceptance of a lot underSkSP-V sampling scheme and P is the acceptance proba-bility under the double-sampling scheme, which is given by

P ¼Xc1

i¼0

n1i

0

@

1

Api 1� pð Þn1�i

þXc2

x¼c1þ1

n1x

0

@

1

Apx 1� pð Þn1�xXc2�x

i¼0

n2i

0

@

1

Api 1� pð Þn2�i

2

4

3

5

ð1ÞUnder AQL (=p1) and LQL(=p2) conditions respectively

Eq. 1 can be written as

P1 ¼Xc1

i¼0

n1i

0

@

1

Ap1i 1� p1ð Þn1�i

þXc2

x¼c1þ1

n1x

0

@

1

Ap1x 1� p1ð Þn1�x

Xc2�x

i¼0

n2i

0

@

1

Ap1i 1� p1ð Þn2�i

2

4

3

5

ð2Þ

P2 ¼Xc1

i¼0

n1i

0

@

1

Ap2i 1� p2ð Þn1�i

þXc2

x¼c1þ1

n1x

0

@

1

Ap2x 1� p2ð Þn1�x

Xc2�x

i¼0

n2i

0

@

1

Ap2i 1� p2ð Þn2�i

2

4

3

5

ð3ÞThe design parameters of the double-sampling plan can

be determined such that the following inequalities shouldbe satisfied

Xc1

i¼0

n1i

0

@

1

Ap1i 1� p1ð Þn1�i

þXc2

x¼c1þ1

n1x

0

@

1

Ap1x 1� p1ð Þn1�x

Xc2�x

i¼0

n2i

0

@

1

Ap1i 1� p1ð Þn2�i

2

4

3

5 � 1� a

ð4ÞXc1

i¼0

n1i

0

@

1

Ap2i 1� p2ð Þn1�i

þXc2

x¼c1þ1

n1x

0

@

1

Ap2x 1� p2ð Þn1�x

Xc2�x

i¼0

n2i

0

@

1

Ap2i 1� p2ð Þn2�i

2

4

3

5 � b

ð5Þ

Similarly, under the conditions of AQL and LQL, theparameters of an SkDSP-V plan namely i, f, k, n1, n2, c1,and c2 will be determined such that the followinginequalities are satisfied.

fP1 þ 1� fð ÞP1i þ fP1

kþ1 P1i � P1

k� �

f 1þ P1iþk � P1

2k� �þ 1� fð ÞP1

i � 1� a ð6Þ

fP2 þ 1� fð ÞP2i þ fP2

kþ1 P2i � P2

k� �

f 1þ P2iþk � P2

2k� �þ 1� fð ÞP2

i � b ð7Þ

where P1 and P2 are obtained by using (2) and (3)respectively.

There may exist many combinations of the designparameters of the SkDSP-V plan for specified requirements.Therefore, we will use the average sample number (ASN)as the criterion to select the suitable combination of thedesign parameters. According to the American NationalStandards Institute/American Society for Quality [1](ANSI/ASQ Z1.4-2008), the ASN is defined as “theaverage number of sample units per lot used for makingdecision (acceptance or non-acceptance)”. Hence, thesuitable combination of the design parameters can bedetermined such that it provides minimum ASN than theother combination of sampling plans. According to manyauthors including Balamurali and Jun [4], we can minimizethe ASN at the AQL as well as at the LQL. But theselection of the design parameters by minimizing the ASNat the LQL is preferred because sample sizes at first andsecond stages are larger at LQL than AQL. The ASN of theSkDSP-V plan is given by

ASNðpÞ ¼ ASNdoubleðpÞð Þ f þ f Piþk � P2k� �

f 1þ Piþk � P2kð Þ þ 1� fð ÞPi

ð8Þ

where ASN double (p) is the ASN of a double-sampling plan,which is given by

ASNdoubleðpÞ ¼ n1 þ n2Xc2

i¼c1þ1

n1i

0

@

1

Api 1� pð Þn1�i ð9Þ

To determine the design parameters satisfying inequal-ities (6) and (7) we will consider the various combinationsof AQL and LQL values. The design parameters of theproposed plan are placed in Table 1. The ASN and theminimum producer's and the consumer's risks are alsogiven in Table 1. For constructing tables, we haveconsidered the parameters in the following intervals. n1=n2=2(1)5,000, c1=0(1)100, c2=c1+1(1)100, i=2(1)10, k=1(1)10, f=0.1(0.05)0.95.

736 Int J Adv Manuf Technol (2012) 60:733–740

Table 1 Parameters ofSkDSP-V plan for specifiedAQL and LQL

Optimal parameters

p1 p2 i k f n1 n2 c1 c2 ASN at p2 (1−α)% β%

0.001 0.005 2 1 0.10 575 575 0 1 643.44 0.9625 0.0995

0.010 2 1 0.65 252 252 0 1 301.47 0.9529 0.0999

0.015 3 1 0.80 165 165 0 1 199.20 0.9730 0.0999

0.020 3 1 0.50 124 124 0 1 149.48 0.9898 0.0994

0.030 2 1 0.85 94 94 0 1 109.51 0.9899 0.0673

0.005 0.010 2 1 0.10 631 631 0 7 1032.34 0.9517 0.0999

0.025 2 1 0.10 114 114 0 1 127.73 0.9631 0.0992

0.050 2 1 0.55 50 50 0 1 59.703 0.9604 0.0989

0.100 2 1 0.75 24 24 0 1 29.014 0.9860 0.0996

0.150 2 1 0.85 19 19 0 1 21.894 0.9898 0.0530

0.01 0.020 2 1 0.10 315 315 0 7 516.13 0.9522 0.0994

0.050 2 1 0.10 57 57 0 1 63.94 0.9631 0.0942

0.100 2 1 0.75 24 24 0 1 29.014 0.9506 0.0996

0.200 2 1 0.55 12 12 0 1 14.392 0.9898 0.0881

0.300 2 1 0.80 10 10 0 1 11.2078 0.9896 0.032

0.05 0.100 2 1 0.10 62 62 0 7 102.908 0.9565 0.0970

0.250 2 1 0.15 10 10 0 1 11.5823 0.9579 0.0900

0.500 2 1 0.25 4 4 0 1 4.910 0.9881 0.0948

0.100 0.200 2 1 0.10 31 31 0 7 52.2517 0.9573 0.0773

0.500 2 1 0.25 4 4 0 1 4.91 0.9554 0.0948

Table 2 Parameters of double-sampling plan for specifiedAQL and LQL

Optimal parameters

p1 p2 n1 n2 c1 c2 ASN at p2 (1−α)% β%

0.001 0.005 692 692 0 3 1047.6 95.09 9.967

0.010 291 291 0 2 404.17 98.11 9.946

0.015 165 165 0 1 199.245 96.65 9.974

0.020 124 124 0 1 149.626 98.02 9.854

0.030 87 87 0 1 103.54 98.98 8.409

0.005 0.010 1237 1237 0 18 2415.9 95.26 9.941

0.025 138 138 0 3 209.186 95.16 9.822

0.050 58 58 0 2 80.595 98.15 9.522

0.100 24 24 0 1 29.11 98.15 9.673

0.150 18 18 0 1 21.07 98.92 6.279

0.01 0.020 618 618 0 18 1207.73 95.34 9.893

0.050 69 69 0 3 104.6 95.19 9.493

0.100 29 29 0 2 40.25 98.17 8.84

0.200 12 12 0 1 14.47 98.16 8.289

0.300 9 9 0 1 10.4 98.94 4.663

0.05 0.100 117 117 0 17 228.96 95.23 9.531

0.250 13 13 0 3 20.29 96.29 9.006

0.500 4 4 0 1 5.0 95.42 0.781

0.100 0.200 55 55 0 16 107.98 95.35 9.148

0.500 6 6 0 3 9.844 97.50 7.837

Int J Adv Manuf Technol (2012) 60:733–740 737

4 Examples

Table 1 can be used to select the optimal parameters of anSkDSP-V plan for specified AQL(=p1) and LQL (=p2) withα=5% and β=10%. Suppose one wants to determine

parameters of an SkDSP-V plan according to the conditionsgiven that p1=0.01, p2=0.05, α=0.05 and β=0.10. FromTable 1, one can find the optimal parameters as i=2, k=x=1, f=0.1, n1=n2=57, c1=0 and c2=1 corresponding to theabove mentioned AQL and LQL conditions. ASN of thisplan is 63.94 which is minimum. Based on these parame-ters, the SkDSP-V plan is operated as follows.

(a) At the outset, start with the normal inspection withdouble-sampling plan (n1=n2=57, c1=0 and c2=1) asthe reference plan. During the normal inspection, lotsare inspected one by one in the order of beingsubmitted to inspection.

(b) When two consecutive lots are accepted on normalinspection, then switch to skipping inspection.

(c) During skipping inspection, inspect only one lot forevery ten lots selected at random. Skipping inspectionis continued until the sampled lot is rejected.

(d) When a lot is rejected on skipping inspection beforeone sampled lot is accepted, revert to normal inspec-tion as per (a) above.

(e) When a lot is rejected after one lot has been acceptedthen revert to normal inspection with reduced clear-ance number 1 as per (f) below.

(f) During normal inspection with clearance number 1,lots are inspected one by one in the order of beingsubmitted and continue the inspection until a lot isrejected or one lot is accepted whichever occurs earlier.

(g) When a lot is rejected, immediately revert to normalinspection with clearance number 2 as per (a) given above.

(h) When one lot is accepted, discontinue normal inspectionand switch to skipping inspection as per (c) above.

Table 3 Comparison of Average Sample Number

p1 p2 Average sample number at LQL

SkDSP-V plan DSP

0.001 0.005 643.44 1047.6

0.010 301.47 404.17

0.015 199.20 199.245

0.020 149.48 149.626

0.030 109.51 103.54

0.005 0.010 1032.34 2415.9

0.025 127.73 209.186

0.050 59.703 80.595

0.100 29.014 29.11

0.150 21.894 21.07

0.01 0.020 516.13 1207.73

0.050 63.94 104.6

0.100 29.014 40.25

0.200 14.392 14.47

0.300 11.2078 10.4

0.05 0.100 102.908 228.96

0.250 11.5823 20.29

0.500 4.910 5.0

0.100 0.200 52.2517 107.98

0.500 4.91 9.844

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Fraction Non-conforming, p

Pro

bab

ility

of

Acc

epta

nce

, Pa(

p)

SkDSP-V Plan

SkDSP-2 Plan

DSP

Fig. 2 OC curves of SkDSP-V,SkDSP-2, and double-samplingplan

738 Int J Adv Manuf Technol (2012) 60:733–740

(i) Replace or correct all the nonconforming units foundwith conforming items in the rejected lots.

5 Comparative study

We would like to compare the proposed SkDSP-V planwith double-sampling plan (DSP). Table 2 can be usedto select the optimal parameters of a double-samplingplan. For example, if p1=0.01, p2=0.05, α=0.05 and β=0.10, then the optimal parameters of the double-samplingplan are determined from Table 2 as n1=n2=69, c1=0 andc2=3 which satisfies the above mentioned AQL and LQLconditions. ASN of this plan is 104.6.

For the purpose of comparison, we have providedTable 3 which gives the ASN of SkDSP-V and double-sampling plans for several combinations of AQL and LQL.

From this table, it can be observed that the SkDSP-V plan provided in this paper will have minimumASN when compared to the other sampling plan suchas the double-sampling plans. Similar reduction inASN can be achieved for any combination of AQLand LQL values. This implies that the SkDSP-V plandeveloped in this paper will give desired protectionwith minimum inspection so that the cost of inspectionwill greatly be reduced. Thus the SkDSP-V planprovides better protection than the reference plancompared here.

In order to show the better efficiency of the SkDSP-V plan, we consider three OC curves including the OCcurve of SkDSP-2 plan. We consider the OC curve ofSkDSP-2 plan just to show the efficiency of theproposed plan. Figure 2 shows the OC curves of theSkDSP-V plan with parameters i=2, k=x=1 and f=0.1,SkDSP-2 plan with parameters i=2, f=0.1 with double-sampling plan with parameters n1=n2=57, c1=0 and c2=1as the reference plan along with the OC curve of thereference plan. The SkDSP-V plan is selected in such away that it satisfies the two points on the OC curvecondition (p1=0.01, 1-α=0.95) and (p2=0.05, β=0.10)but at the same time, the same values of the parametersare used for other plans even though the other twoplans do not pass through the specified two points onthe OC curve. That is, we want to show the betterperformance of the proposed plan with the sameparametric values. From this figure, it can be easilyobserved that for good quality, i.e., for smaller values offraction nonconforming, the OC curve of the SkDSP-Vplan has more probability of acceptance than the othertwo plans. When quality deteriorates, the OC curve ofthe SkDSP-V plan coincide with other plans. Hence,SkDSP-V plan gives better protection to the producerwhile safe-guarding the consumers

6 Concluding remarks

In this paper, we have developed tables and designingmethodology for selecting the parameters of a new systemof skip-lot sampling plan of type SkSP-V plan with double-sampling plan as the reference plan for specified values ofAQL and LQL. The two points on the OC curve approachis adopted to find the design parameters of the proposedplan. It is found that the proposed plan requires the lessnumber of sample units for the inspection purpose than theconventional double-sampling plan. So, the proposed planis useful in reducing the cost and the time of the inspectionof the material or the product where skip-lot sampling isused. An example has also been presented for the industrialapplication of the proposed plan. The extension of theproposed plan to sequential acceptance sampling plan maybe a fruitful area for the future research. We will considerthe present approach in time-truncated experiment for life-testing purpose in terms of mean or median ratios in future.

Acknowledgments The authors are thankful to the editor and all theanonymous reviewers for their valuable comments.

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