Hindawi Publishing CorporationInternational Journal of Manufacturing EngineeringVolume 2013 Article ID 751807 12 pageshttpdxdoiorg1011552013751807

Research ArticleOptimal Designing of Variables Chain Sampling Plan byMinimizing the Average Sample Number

S Balamurali and M Usha

Department of Mathematics Kalasalingam University Krishnankoil Tamil Nadu 626 126 India

Correspondence should be addressed to S Balamurali sbmuralirediffmailcom

Received 13 March 2013 Accepted 9 October 2013

Academic Editors GTS Ho T R Kurfess and I Zeid

Copyright copy 2013 S Balamurali and M Usha This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We investigate the optimal designing of chain sampling plan for the application of normally distributed quality characteristicsThe chain sampling plan is one of the conditional sampling procedures and this plan under variables inspection will be usefulwhen testing is costly and destructive The advantages of this proposed variables plan over variables single sampling plan andvariables double sampling plan are discussed Tables are also constructed for the selection of optimal parameters of known andunknown standard deviation variables chain sampling plan for specified two points on the operating characteristic curve namelythe acceptable quality level and the limiting quality level along with the producerrsquos and consumerrsquos risksThe optimization problemis formulated as a nonlinear programming where the objective function to be minimized is the average sample number and theconstraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operatingcharacteristic curve

1 Introduction

Although in recent years process control techniques and off-line quality control methods have taken important rolesacceptance sampling procedures remain as a major tool formany practical quality control systems In general acceptancesampling is a statistical tool used to help make decisions con-cerning whether or not a batch or lot of products should bereleased for consumer use That is sampling plans are widelyused in industries for estimating lot or process characteristicsand for determining the disposition of lots Variable samplingplans constitute one of the major areas of the theory andpractice of acceptance sampling The principle prerequisitefor variables sampling is that the quality characteristic ofinterest is measured on a continuous scale The primaryadvantage of the variables sampling plan is that the sameoperating characteristic (OC) curve can be obtained with asmaller sample size than would be required by an attributessampling plan Thus a variables acceptance sampling planwould require less samplingThe variables sampling providesmore information than the attributes sampling and thereforethe same protection is attained with partly considerable

smaller sample size When destructive testing is employedvariables sampling is particularly useful in reducing the costsof inspection

However it is to be pointed out that the variablessampling plans have some demerits compared to attributessampling plans The variables sampling plans are of courselittle bit more tedious to apply than the attributes samplingplans and the assumptions on which they are based may notbe met or may not be known to be met It is also obvious thatthe cost of inspecting each item by attributes is less than byvariables The lower cost may also be a result of the simplerprocedure of recording only the number of units failing atest rather than recording variables measurements Howeverin modern quality control systems these reasons are nolonger valid because measurements are made by automatedtest equipment Even the simplest automated test equipmentusually has the capability of the simple calculations and dataaccumulation needed for variables sampling In one of hispapers Collani [1] criticized the variables sampling plans andargued that the acceptance sampling by variables is inappro-priate if one is interested in the fraction nonconforming inincoming batches But at the same time Seidel [2] has proved

2 International Journal of Manufacturing Engineering

in one of his subsequent papers that the sampling by variablesis always optimal

The conventional variables sampling plans have beeninvestigated by many authors and are available in the lit-erature of acceptance sampling Lieberman and Resnikoff[3] developed tables for the selection of plan parameters forvarious acceptable quality levels (AQL) for MIL-STD 414scheme Owen [4] developed variables sampling plans basedon normal distribution when the standard deviation of theprocess is unknown Bender Jr [5] considered variables sam-pling plans for assuring the product quality when the qualitycharacteristic of the product follows normal distributionwithunknown standard deviation and provided a procedure forcalculating the noncentral t-distribution Hamaker [6] hasgiven a procedure for finding the parameters of the unknownsigma variables sampling plans from the known sigma vari-ables sampling plans Govindaraju and Soundararajan [7]developed tables for selecting the parameters of variablessingle sampling plans that match with the OC curves ofMIL-STD 105D schemes Bravo and Wetherill [8] developeda method for designing double sampling plans (DSP) byvariables with OC curves matching with the OC curves ofthe equivalent single sampling plans Schilling [9] has writtenan exclusive book on acceptance sampling which also dealswith the conventional variables sampling plans Baillie [10]developed tables for variables double sampling plans whenthe process standard deviation is unknown Govindaraju andBalamurali [11] extended the chain sampling for variablesinspection However they have not provided any tables foreasy implementation of the plan Also they have restrictedtheir discussion only with known standard deviation Hencethis paper attempts to develop tables for easy application ofchain sampling to variables inspection for a normally dis-tributed quality characteristic for both known and unknownstandard deviation cases

2 Chain Sampling Plan forVariables Inspection

The concept of chain sampling was first introduced by Dodge[12] for the application of attribute quality characteristicsThe characteristics and the properties of the chain samplingplan were investigated by many authors (see for exampleSoundararajan [13]) The chain sampling plan belongs tothe group of conditional sampling procedures like multipledependent (deferred) sampling plans (see Vaerst [14] andWortham and Baker [15]) In these procedures acceptance orrejection of a lot is based not only on the sample from that lotbut also on sample results from past lots The chain samplingplan is applicable in the case of Type B situations (iesampling from a continuous process) where lots expected tobe of the same quality are submitted for inspection seriallyin the order of production The operating procedure andcharacteristics of the attributes chain sampling plan can beseen in Dodge [12]

The performance of an acceptance sampling plan canbe measured by its OC curve which quantifies the risks ofproducer and consumers It would be of interest to examine

whether or not the shape of the variables plan is concavefor good quality levels If the variables plan possesses anunsatisfactoryOC curve similar to that of the zero acceptancenumber attributes plan then it is desirable to improve theshape of the curve A detailed discussion on the shape of theOC curve of a single sampling variables plan with knownstandard deviation can be seen in Govindaraju and Kural-mani [16] It is always necessary to increase the discriminatingpower of theOC curveThis can be achieved by increasing thesample size but this may not be always possible particularlyin costly and destructive testing An alternative solution isto develop a conditional sampling procedure Based on thisidea Govindaraju and Balamurali [11] extended the conceptof chain sampling to variables inspection As pointed outearlier they have not provided any tables for the selectionof parameters and also they have dealt only with knownstandard deviation caseThis paper attempts to provide tablesfor the easy industrial application of this plan for bothknown and unknown standard deviation cases It is also to bepointed out that Govindaraju and Balamurali [11] have givenan approximate solution for finding the plans of unknownstandard deviation case But in this paper we provide adifferent procedure for finding the unknown sigma chainsampling plan The major advantage of this plan is to achievebetter protection to the producer with minimum averagesample number (ASN) and these aspects will be discussed ina separate sectionThe following assumptions should be validfor the application of the variables chain sampling plan

(i) Lots are submitted for inspection serially in the orderof production from a process that turns out a constantproportion nonconforming items

(ii) The consumer has confidence in the supplier andthere should be no reason to believe that a particularlot is poorer than the preceding lots

In addition the usual conditions for the application of singlesampling variables plans with known or unknown standarddeviation should also be valid The operating proceduresof the variables chain sampling plan are given in the nextsection

3 Operating Procedures of Variables ChainSampling Plan

31 Known Sigma Case Suppose that the quality characteris-tic of interest has the upper specification limit U and followsa normal distribution with unknown mean 120583 and knownstandard deviation 120590 Then the following two procedures ofthe variables chain sampling plan are proposed

Step 1 Fromeach submitted lot take a random sample of size119899120590 say (119883

1 1198832sdot sdot sdot 119883119899120590

) and compute119881 = (119880minus119883)120590where 119883 = (1119899

120590) sum119899120590

119894=1119883119894

Step 2 Accept the lot if 119881 ge 119896120590and reject the lot if 119881 lt 119896

1015840

120590 If

1198961015840

120590le 119881 lt 119896

120590 then accept the current lot provided that

the preceding 119894120590lots were accepted on the condition

that 119881 ge 119896120590but reject otherwise (note 119896

120590gt 1198961015840

120590)

International Journal of Manufacturing Engineering 3

In the case of lower specification limit L the operatingprocedure is as follows

Step 1 From each lot take a random sample of size 119899120590 say

(1198831 1198832sdot sdot sdot 119883119899120590

) and compute 119881 = (119883 minus 119871)120590 where119883 = (1119899

120590) sum119899120590

119894=1119883119894

Step 2 Accept the lot if 119881 ge 119896120590and reject the lot if 119881 lt 119896

1015840

120590 If

1198961015840

120590le 119881 lt 119896

120590 then accept the current lot provided that

the preceding 119894120590lots were accepted on the condition

that 119881 ge 119896120590but reject otherwise

Thus the proposed variables chain sampling plan is charac-terized by four parameters namely 119899

120590 119894120590 119896120590 and 119896

1015840

120590 If 119896120590=

1198961015840

120590 then the proposed plan will reduce to the variables SSP

Also when 119894120590tends to infinity the proposed plan becomes

variables SSP with parameters 119899120590and 119896

120590 It is to be pointed

out that the chain sampling plan can be applied for inspectionof lots which are submitted serially in the order of productionor in the order of being submitted The decision of currentlot depends on the results of preceding lots So when 119894

120590ge

2 we need to keep the records of results of previous lotsOf course maintaining records of preceding lots may be adrawback of the chain sampling plan over the single samplingplan however this can be compensated by minimizing theinspection efforts in terms of minimum ASN with desiredprotection

32 Unknown Sigma Case Whenever the standard deviationis unknown we may use the sample standard deviation Sinstead of 120590 In this case the plan operates as follows

Step 1 From each submitted lot take a random sampleof size 119899

119878 say (119883

1 1198832sdot sdot sdot 119883119899119878

) and compute 119881 =

(119880 minus 119883)119878 where 119883 = (1119899119878) sum119899119878

119894=1119883119894and 119878 =

radicsum(119883119894minus 119883)2

(119899119878minus 1)

Step 2 Accept the lot if 119881 ge 119896119878and reject the lot if 119881 lt 119896

1015840

119878 If

1198961015840

119878le 119881 lt 119896

119878 then accept the current lot provided that

the preceding 119894119878lots were accepted on the condition

that 119881 ge 119896119878

Thus the proposed unknown sigma variables chain samplingplan is characterized by four parameters namely 119899

119878 119894119878 119896119878

and 1198961015840

119878 If 119896119878

= 1198961015840

119878 then the proposed plan will reduce to

the variables single sampling plan with unknown standarddeviation

4 Designing Methodology of Variables ChainSampling Plan

Generally variables sampling plans are designed based ontwo points on the OC curve namely (119901

1 1 minus 120572) and (119901

2 120573)

where 1199011is called the acceptable quality level (AQL) 119901

2is

the limiting quality level (LQL) 120572 is the producerrsquos risk and120573 is the consumerrsquos risk Any well-designed sampling planmust provide at least (1 minus 120572) probability of acceptance ofa lot when the process fraction nonconforming is at AQLlevel and the sampling plan must also provide not more than

120573 probability of acceptance if the process fraction noncon-forming is at the LQL level Thus the acceptance samplingplan must have its OC curve passing through two designatedpoints (AQL 1 minus 120572) and (LQL 120573) Some other strategiesare also followed to design the sampling plans besides thestatistical based paradigm which include Bayesian approachand economic based approach For further details readersmay refer Chen and Lam [17] Ferrell and Chhoker [18]Chen [19] Chen et al [20] Balamurali and Subramani [21]Vijayaraghavan and Sakthivel [22] Balamurali et al [23]and Fallahnezhad and Aslam [24] In this paper we havefollowed the designing methodology based on two points onthe OC curve approach The variables chain sampling planis designed based on the two points on the OC curve in thefollowing manner

The fraction nonconforming in a lot is given as

119901 = 1 minus Φ(119880 minus 120583

120590) (1)

where Φ(119884) is the cumulative distribution function of stan-dard normal distribution and is given by

Φ (119884) = int

119884

minusinfin

1

radic2120587exp(

minus1198852

2)119889119885 (2)

Here the quality characteristic of interest is normally dis-tributed with mean 120583 and standard deviation 120590 and theunit is classified as nonconforming if it exceeds the upperspecification limit U So the unknown mean 120583 can be deter-mined if p is specified Let us define the standardized qualitycharacteristic corresponding to the fraction conforming as

119885119901= Φminus1

(1 minus 119901) (3)

Then the OC function of the variables chain samplingplan which gives the proportion of lots that are expectedto be accepted for given product quality p is given by (seeGovindaraju and Balamurali [11])

119875119886(119901) = Pr (119881 ge 119896)

+ [Pr (119881 ge 1198961015840

) minus Pr (119881 ge 119896)] [Pr (119881 ge 119896)]119894

(4)

where Pr(119881 ge 119896) is the probability of accepting a lot basedon a single sample with parameters (119899 119896) and Pr(119881 lt 119896

1015840

) isthe probability of rejecting a lot based on a single sample withparameters (119899 1198961015840) Under Type B situation (ie a series of lotsof the same quality) forming lots of N items from a processand then drawing random sample of size 119899 from these lotsis equivalent to drawing random samples of size 119899 directlyfrom the process Hence the derivation of the OC functionis straightforward

The probability of acceptance of the chain sampling plancan also be written as

119875119886(119901) = Φ (119908

2) + [Φ (119908

1) minus Φ (119908

2)] [Φ (119908

2)]119894

(5)

where 1199081= (119885119901minus 1198961015840

)radic119899 and 1199082= (119885119901minus 119896)radic119899

It is to be pointed out that the parameters of theknown sigma variables chain sampling plan are denoted by

4 International Journal of Manufacturing Engineering

(119899120590 1198961015840120590 and 119896

120590) and for unknown sigma plan they are denoted

by (119899119878 1198961015840119878 and 119896

119878) The determination of parameters (119899

119878 1198961015840119878

and 119896119878) of unknown sigma variables chain sampling plan

is explained as follows It is known that for large samples119883 plusmn 119896

119878119878 is approximately normally distributed with mean

120583 plusmn 119896119878119864(119878) and variance (120590

2

119899)+119896119878Var(119878) (see Duncan [25]

and Balamurali and Jun [26]) That is

119883 + 119896119878119878 sim 119873(120583 + 119896

119878120590

1205902

119899+ 1198962

119878

1205902

2119899) (6)

Therefore the probability of accepting a lot at each repetitionis given by

119875 (119881 ge 119896119878) = 119875 119883 le 119880 minus 119896

119878119878 | 119901

= 119875 119883 + 119896119878119878 le 119880 | 119901

= Φ(119880 minus 119896

119878120590 minus 120583

(120590radic119899119878)radic1 + (1198962

1198782)

)

= Φ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(7)

If we let

1199082119878

= ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

then Pr (119881 ge 119896119878) = Φ (119908

2119878)

(8)

Similarly if we let 1199081119878

= ((119885119901minus 1198961015840

119878)radic119899119878(1 + (11989610158402

1198782))) then

we have

Pr (119881 lt 1198961015840

119878) = 1 minus Φ (119908

1119878) (9)

Hence the lot acceptance probability for sigmaunknown caseis given by

119875119886(119901) = Φ (119908

2119878) + [Φ (119908

1119878) minus Φ (119908

2119878)] [Φ (119908

2119878)]119894119878

(10)

where 1199081119878

= (119885119901minus 1198961015840

119878)radic119899119878and 119908

2119878= (119885119901minus 119896119878)radic119899119878

If (AQL 1 minus 120572) and (LQL 120573) are prescribed then the OCfunction can be written as

Φ(11990821119878

) + [Φ (11990811119878

) minus Φ (11990821119878

)] [Φ (11990821119878

)]119894119878

= 1 minus 120572

Φ (11990822119878

) + [Φ (11990812119878

) minus Φ (11990822119878

)] [Φ (11990822119878

)]119894119878

= 120573

(11)

Here 11990811119878

is the value of 1199081119878at 119901 = 119901

1 11990821119878

is the value of1199082119878at 119901 = 119901

1 11990812119878

is the value of 1199081119878at 119901 = 119901

2 and 119908

22119878is

the value of 1199082119878at 119901 = 119901

2

We obtain 11990811119878

11990821119878

11990812119878

and 11990822119878

respectively by

11990811119878

= ((1198851199011

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990821119878

= ((1198851199011

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

11990812119878

= ((1198851199012

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990822119878

= ((1198851199012

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(12)

where 1198851199011

is the value of 119885119901at AQL and 119885

1199012

is the value of119885119901at LQL For given AQL and LQL the parametric values of

the unknown sigma variables chain sampling plan namely119899119878 119894119878 119896119878 and 119896

1015840

119878 are determined by satisfying the required

producer and consumer conditions Alternatively we candetermine the above parameters of the variables chain sam-pling plan to minimize the ASN at AQL which is analogousto minimizing the average sample number in the variablesrepetitive group sampling plans andmultiple dependent statesampling plan (see Balamurali et al [27] and Balamuraliand Jun [26]) Some of the authors have investigated thedesigning of sampling plans by using some other optimiza-tion techniques which are available in the literature (see forexample Feldmann and Krumbholz [28] Krumbholz andRohr [29 30] Krumbholz et al [31] and Duarte and Sariava[32 33] The ASN for the chain sampling plan is the samplesize only Therefore the following optimization problem isconsidered to determine those parameters

Minimize ASN (1199011) = 119899119878

119899119878 119894119878 119896119878 1198961015840

119878

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899119878ge 2 119894119878ge 1 119896

119878+ 120575 ge 119896

1015840

119878+ 120575 ge 0

120575 ge 0 119899119878isin 119873 119894

119878isin 119873

(13)

In the case of known sigma variable chain sampling plan theOC function under the specified AQL and LQL conditionscan be written as

Φ(11990821120590

) + [Φ (11990811120590

) minus Φ (11990821120590

)] [Φ (11990821120590

)]119894120590

= 1 minus 120572

Φ (11990822120590

) + [Φ (11990812120590

) minus Φ (11990822120590

)] [Φ (11990822120590

)]119894120590

= 120573

(14)

Here 11990811120590

is the value of 1199081120590

at 119901 = 1199011 11990821120590

is the value of1199082120590

at 119901 = 1199011 11990812120590

is the value of 1199081120590

at 119901 = 1199012 and 119908

22120590is

the value of 1199082120590

at 119901 = 1199012 That is

11990811120590

= (1198851199011

minus 1198961015840

120590)radic119899120590 119908

21120590= (1198851199011

minus 119896120590)radic119899120590

11990812120590

= (1198851199012

minus 1198961015840

120590)radic119899120590 119908

22120590= (1198851199012

minus 119896120590)radic119899120590

(15)

International Journal of Manufacturing Engineering 5

where1198851199011

is the value of119885119901at AQL and119885

1199012

is the value of119885119901

at LQL The following optimization problem is considered todetermine the optimal parameters of known sigma variablessampling plan such as 119899

120590 119894120590 119896120590 and 119896

1015840

120590

Minimize ASN (1199011) = 119899120590

119899120590 119894120590 119896120590 1198961015840

120590

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899120590ge 2 119894120590ge 1 119896

120590+ 120575 ge 119896

1015840

120590+ 120575 ge 0

120575 ge 0 119899120590isin 119873 119894

120590isin 119873

(16)

We may determine the parameters of the unknown sigmachain sampling plan and the known sigma chain samplingplan by solving the nonlinear equation given in (13) and (16)respectively There may exist multiple solutions since thereare four unknownswith only two equationsGenerally a sam-pling would be desirable if the required number of samples issmall So in this paper we consider the ASN as the objectivefunction to be minimized with the probability of acceptancealongwith the corresponding producerrsquos and consumerrsquos risksas constraints To solve the above nonlinear optimizationproblems given in (13) and (16) the sequential quadraticprogramming (SQP) proposed by Nocedal and Wright [34]can be used Here it is to be pointed out that the optimizationproblems given in (13) and (16) include continuous valuesof 1198961015840120590(or 1198961015840

119878) and 119896

120590(or 119896119878) and integer values of 119899

120590(or 119899119878)

and 119894120590(or 119894119878) This kind of formalization leads to a problem of

mixed integer programming (MINLP) typeThe SQPmethoddoes not allow directly solvingMINLP problems but only thenonlinear programming (NLP) structures The strategy wehave followed is to relax the discrete variables first and solvethe NLP problem and then fix them to the upper integers andresolve the problem again now just to find 119896

1015840

120590(or 1198961015840

119878) and

119896120590(or 119896119878) In such a way we have solved the NLP problem

given above The SQP is implemented in Matlab softwareusing the routine ldquofminconrdquo By solving the NLP based onthe procedure described above the optimal parameters (119899

120590

119894120590 119896120590 and 119896

1015840

120590) for known sigma plan and the parameters (119899

119878

119894119878 119896119878 and 119896

1015840

119878) for unknown sigma plan are determined and

these values are tabulated in Table 1

5 Designing Examples

51 Selection of Known Sigma Variables Chain Sampling Planfor Specified AQL and LQL Table 1 is used to determine theparameters of the known 120590 variables chain sampling plan forspecified values of AQL and LQL when 120572 = 5 and 120573 = 10For example if 119901

1= 2 119901

2= 7 120572 = 5 and 120573 = 10 Table 1

gives the parameters as 119899120590= 18 119894

120590= 3 1198961015840

120590= 1544 and 119896

120590=

1779For the above example the plan is operated as followsFrom each submitted lot take a random sample of size 18

and compute 119881 = (119880 minus 119883)120590 where 119883 = (118)sum18

119894=1119883119894

Accept the lot if119881 ge 1779 and reject the lot if119881 lt 1544If 1544 le 119881 lt 1779 then accept the current lot providedthat the preceding 3 lots were accepted on the condition that119881 ge 1779 with the sample size of 18

52 Selection of Unknown Sigma Variables Chain SamplingPlan for Specified AQL and LQL As mentioned earlier theunknown sigma variables chain sampling plan is operatedas a known sigma variables chain sampling plan but theparameters 119899

119878 119894119878 119896119878 and 119896

1015840

119878are used in the place of 119899

120590 119894120590 119896120590

and 1198961015840

120590 respectively Table 1 can also be used for the selection

of the parameters of the unknown 120590 variables chain samplingplan for given values of AQL and LQL Suppose that AQL =00075 LQL = 0035 120572 = 5 and 120573 = 10 From Table 1the parameters of the variables chain sampling plan can bedetermined as 119899

119878= 47 119894

119878= 1 1198961015840119878= 1925 and 119896

119878= 2190

53 Illustrative Example To illustrate the implementationof the proposed sampling plan for the example given inSection 52 we consider case study data on STN-LCD man-ufacturing process given in Wu and Pearn [35] The upperspecification limit is given as 077mmHere we consider only47 values randomly taken from the original data given inWuand Pearn [35] The data are shown below

0717 0698 0726 0684 0727 0688 0708 0703 0694 0713

0730 0699 0710 0688 0665 0704 0725 0729 0716 0685

0712 0716 0712 0733 0709 0703 0730 0716 0688 0688

0712 0702 0726 0669 0718 0714 0726 0683 0713 0737

0740 0706 0726 0688 0715 0704 0724

(17)

The implementation of the plan is shown below

Step 1 Take a random sample of size 47 The data are givenabove

Step 2 For this data we calculate 119883 = (1119899)sum119899119878

119894=1119883119894

=

0708915 and 119878 = radicsum(119883119894minus 119883)2

(119899 minus 1) = 0017583

Step 3 Calculate 119881 = (119880 minus 119883)119878 = (077 minus

0708915)0017583 = 34741

Step 4 Since 119881 = 34741 gt 119896119878= 2190 we accept the current

lot without considering the result of past lots

Just for the sake of discussion let us assume that the 119881

value (for different data sets) is calculated as 215 In this case

6 International Journal of Manufacturing Engineering

Table 1 Variables chain sampling plans indexed by AQL and LQL for 120572 = 5 and 120573 = 10

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0001

00020 119 1 2883 3013 650 1 2888 301300025 68 2 2810 2965 361 1 2834 29840003 46 1 2776 2961 242 1 2774 29640004 31 2 2740 2885 147 1 2716 29210005 21 2 2610 2860 103 1 2637 28970006 19 4 2597 2807 84 2 2599 28290007 15 3 2534 2789 72 3 2532 27920008 13 3 2495 2765 62 3 2518 27680009 12 3 2521 2736 51 2 2470 27650010 12 6 2517 2697 45 2 2425 27500012 9 4 2331 2686 40 3 2414 26940015 7 3 2237 2657 34 4 2387 26370020 7 6 2414 2539 27 5 2335 25650025 6 6 2370 2485 20 3 2197 25520030 5 6 2264 2454 19 5 2231 2476

00025

0004 230 2 2658 2738 1131 2 2676 27360005 105 2 2598 2703 474 2 2585 27050006 64 2 2530 2675 282 1 2533 269800075 39 2 2441 2641 171 1 2464 26690010 26 3 2379 2579 106 1 2430 26100012 19 2 2335 2555 76 1 2321 26010015 16 4 2296 2491 55 1 2237 25720020 12 4 2260 2425 38 1 2139 25290025 10 6 2206 2366 32 2 2163 24130030 8 5 2114 2334 29 5 2107 23420035 7 6 2307 2297 25 5 2092 23020040 6 6 1884 2274 21 5 1952 2282

0005

00075 259 1 2434 2524 1113 2 2442 25120010 86 1 2326 2486 347 1 2348 24830012 53 1 2277 2457 211 1 2299 24540015 33 1 2211 2421 128 1 2233 24180020 24 4 2161 2316 75 2 2100 23450025 16 3 2046 2281 55 2 2083 22930030 14 6 1989 2224 43 3 1974 22490035 12 6 2002 2182 36 3 1979 22090040 10 4 1976 2156 31 3 1964 21740050 8 5 1893 2098 25 4 1925 2105

00075

0010 505 2 2339 2384 1817 1 2329 23940012 173 1 2259 2369 690 2 2278 23530015 78 1 2185 2335 286 1 2182 23370020 43 2 2146 2251 133 1 2083 22930025 29 5 2008 2198 83 1 2003 22580030 19 2 1968 2178 61 1 1984 22140035 17 5 1888 2123 47 1 1925 21900040 13 3 1837 2107 41 3 1851 21110050 10 3 1792 2052 28 2 1756 20810060 9 5 1792 1982 26 4 1815 1990

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

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International Journal of

2 International Journal of Manufacturing Engineering

in one of his subsequent papers that the sampling by variablesis always optimal

The conventional variables sampling plans have beeninvestigated by many authors and are available in the lit-erature of acceptance sampling Lieberman and Resnikoff[3] developed tables for the selection of plan parameters forvarious acceptable quality levels (AQL) for MIL-STD 414scheme Owen [4] developed variables sampling plans basedon normal distribution when the standard deviation of theprocess is unknown Bender Jr [5] considered variables sam-pling plans for assuring the product quality when the qualitycharacteristic of the product follows normal distributionwithunknown standard deviation and provided a procedure forcalculating the noncentral t-distribution Hamaker [6] hasgiven a procedure for finding the parameters of the unknownsigma variables sampling plans from the known sigma vari-ables sampling plans Govindaraju and Soundararajan [7]developed tables for selecting the parameters of variablessingle sampling plans that match with the OC curves ofMIL-STD 105D schemes Bravo and Wetherill [8] developeda method for designing double sampling plans (DSP) byvariables with OC curves matching with the OC curves ofthe equivalent single sampling plans Schilling [9] has writtenan exclusive book on acceptance sampling which also dealswith the conventional variables sampling plans Baillie [10]developed tables for variables double sampling plans whenthe process standard deviation is unknown Govindaraju andBalamurali [11] extended the chain sampling for variablesinspection However they have not provided any tables foreasy implementation of the plan Also they have restrictedtheir discussion only with known standard deviation Hencethis paper attempts to develop tables for easy application ofchain sampling to variables inspection for a normally dis-tributed quality characteristic for both known and unknownstandard deviation cases

2 Chain Sampling Plan forVariables Inspection

The concept of chain sampling was first introduced by Dodge[12] for the application of attribute quality characteristicsThe characteristics and the properties of the chain samplingplan were investigated by many authors (see for exampleSoundararajan [13]) The chain sampling plan belongs tothe group of conditional sampling procedures like multipledependent (deferred) sampling plans (see Vaerst [14] andWortham and Baker [15]) In these procedures acceptance orrejection of a lot is based not only on the sample from that lotbut also on sample results from past lots The chain samplingplan is applicable in the case of Type B situations (iesampling from a continuous process) where lots expected tobe of the same quality are submitted for inspection seriallyin the order of production The operating procedure andcharacteristics of the attributes chain sampling plan can beseen in Dodge [12]

The performance of an acceptance sampling plan canbe measured by its OC curve which quantifies the risks ofproducer and consumers It would be of interest to examine

whether or not the shape of the variables plan is concavefor good quality levels If the variables plan possesses anunsatisfactoryOC curve similar to that of the zero acceptancenumber attributes plan then it is desirable to improve theshape of the curve A detailed discussion on the shape of theOC curve of a single sampling variables plan with knownstandard deviation can be seen in Govindaraju and Kural-mani [16] It is always necessary to increase the discriminatingpower of theOC curveThis can be achieved by increasing thesample size but this may not be always possible particularlyin costly and destructive testing An alternative solution isto develop a conditional sampling procedure Based on thisidea Govindaraju and Balamurali [11] extended the conceptof chain sampling to variables inspection As pointed outearlier they have not provided any tables for the selectionof parameters and also they have dealt only with knownstandard deviation caseThis paper attempts to provide tablesfor the easy industrial application of this plan for bothknown and unknown standard deviation cases It is also to bepointed out that Govindaraju and Balamurali [11] have givenan approximate solution for finding the plans of unknownstandard deviation case But in this paper we provide adifferent procedure for finding the unknown sigma chainsampling plan The major advantage of this plan is to achievebetter protection to the producer with minimum averagesample number (ASN) and these aspects will be discussed ina separate sectionThe following assumptions should be validfor the application of the variables chain sampling plan

(i) Lots are submitted for inspection serially in the orderof production from a process that turns out a constantproportion nonconforming items

(ii) The consumer has confidence in the supplier andthere should be no reason to believe that a particularlot is poorer than the preceding lots

In addition the usual conditions for the application of singlesampling variables plans with known or unknown standarddeviation should also be valid The operating proceduresof the variables chain sampling plan are given in the nextsection

3 Operating Procedures of Variables ChainSampling Plan

31 Known Sigma Case Suppose that the quality characteris-tic of interest has the upper specification limit U and followsa normal distribution with unknown mean 120583 and knownstandard deviation 120590 Then the following two procedures ofthe variables chain sampling plan are proposed

Step 1 Fromeach submitted lot take a random sample of size119899120590 say (119883

1 1198832sdot sdot sdot 119883119899120590

) and compute119881 = (119880minus119883)120590where 119883 = (1119899

120590) sum119899120590

119894=1119883119894

Step 2 Accept the lot if 119881 ge 119896120590and reject the lot if 119881 lt 119896

1015840

120590 If

1198961015840

120590le 119881 lt 119896

120590 then accept the current lot provided that

the preceding 119894120590lots were accepted on the condition

that 119881 ge 119896120590but reject otherwise (note 119896

120590gt 1198961015840

120590)

International Journal of Manufacturing Engineering 3

In the case of lower specification limit L the operatingprocedure is as follows

Step 1 From each lot take a random sample of size 119899120590 say

(1198831 1198832sdot sdot sdot 119883119899120590

) and compute 119881 = (119883 minus 119871)120590 where119883 = (1119899

120590) sum119899120590

119894=1119883119894

Step 2 Accept the lot if 119881 ge 119896120590and reject the lot if 119881 lt 119896

1015840

120590 If

1198961015840

120590le 119881 lt 119896

120590 then accept the current lot provided that

the preceding 119894120590lots were accepted on the condition

that 119881 ge 119896120590but reject otherwise

Thus the proposed variables chain sampling plan is charac-terized by four parameters namely 119899

120590 119894120590 119896120590 and 119896

1015840

120590 If 119896120590=

1198961015840

120590 then the proposed plan will reduce to the variables SSP

Also when 119894120590tends to infinity the proposed plan becomes

variables SSP with parameters 119899120590and 119896

120590 It is to be pointed

out that the chain sampling plan can be applied for inspectionof lots which are submitted serially in the order of productionor in the order of being submitted The decision of currentlot depends on the results of preceding lots So when 119894

120590ge

2 we need to keep the records of results of previous lotsOf course maintaining records of preceding lots may be adrawback of the chain sampling plan over the single samplingplan however this can be compensated by minimizing theinspection efforts in terms of minimum ASN with desiredprotection

32 Unknown Sigma Case Whenever the standard deviationis unknown we may use the sample standard deviation Sinstead of 120590 In this case the plan operates as follows

Step 1 From each submitted lot take a random sampleof size 119899

119878 say (119883

1 1198832sdot sdot sdot 119883119899119878

) and compute 119881 =

(119880 minus 119883)119878 where 119883 = (1119899119878) sum119899119878

119894=1119883119894and 119878 =

radicsum(119883119894minus 119883)2

(119899119878minus 1)

Step 2 Accept the lot if 119881 ge 119896119878and reject the lot if 119881 lt 119896

1015840

119878 If

1198961015840

119878le 119881 lt 119896

119878 then accept the current lot provided that

the preceding 119894119878lots were accepted on the condition

that 119881 ge 119896119878

Thus the proposed unknown sigma variables chain samplingplan is characterized by four parameters namely 119899

119878 119894119878 119896119878

and 1198961015840

119878 If 119896119878

= 1198961015840

119878 then the proposed plan will reduce to

the variables single sampling plan with unknown standarddeviation

4 Designing Methodology of Variables ChainSampling Plan

Generally variables sampling plans are designed based ontwo points on the OC curve namely (119901

1 1 minus 120572) and (119901

2 120573)

where 1199011is called the acceptable quality level (AQL) 119901

2is

the limiting quality level (LQL) 120572 is the producerrsquos risk and120573 is the consumerrsquos risk Any well-designed sampling planmust provide at least (1 minus 120572) probability of acceptance ofa lot when the process fraction nonconforming is at AQLlevel and the sampling plan must also provide not more than

120573 probability of acceptance if the process fraction noncon-forming is at the LQL level Thus the acceptance samplingplan must have its OC curve passing through two designatedpoints (AQL 1 minus 120572) and (LQL 120573) Some other strategiesare also followed to design the sampling plans besides thestatistical based paradigm which include Bayesian approachand economic based approach For further details readersmay refer Chen and Lam [17] Ferrell and Chhoker [18]Chen [19] Chen et al [20] Balamurali and Subramani [21]Vijayaraghavan and Sakthivel [22] Balamurali et al [23]and Fallahnezhad and Aslam [24] In this paper we havefollowed the designing methodology based on two points onthe OC curve approach The variables chain sampling planis designed based on the two points on the OC curve in thefollowing manner

The fraction nonconforming in a lot is given as

119901 = 1 minus Φ(119880 minus 120583

120590) (1)

where Φ(119884) is the cumulative distribution function of stan-dard normal distribution and is given by

Φ (119884) = int

119884

minusinfin

1

radic2120587exp(

minus1198852

2)119889119885 (2)

Here the quality characteristic of interest is normally dis-tributed with mean 120583 and standard deviation 120590 and theunit is classified as nonconforming if it exceeds the upperspecification limit U So the unknown mean 120583 can be deter-mined if p is specified Let us define the standardized qualitycharacteristic corresponding to the fraction conforming as

119885119901= Φminus1

(1 minus 119901) (3)

Then the OC function of the variables chain samplingplan which gives the proportion of lots that are expectedto be accepted for given product quality p is given by (seeGovindaraju and Balamurali [11])

119875119886(119901) = Pr (119881 ge 119896)

+ [Pr (119881 ge 1198961015840

) minus Pr (119881 ge 119896)] [Pr (119881 ge 119896)]119894

(4)

where Pr(119881 ge 119896) is the probability of accepting a lot basedon a single sample with parameters (119899 119896) and Pr(119881 lt 119896

1015840

) isthe probability of rejecting a lot based on a single sample withparameters (119899 1198961015840) Under Type B situation (ie a series of lotsof the same quality) forming lots of N items from a processand then drawing random sample of size 119899 from these lotsis equivalent to drawing random samples of size 119899 directlyfrom the process Hence the derivation of the OC functionis straightforward

The probability of acceptance of the chain sampling plancan also be written as

119875119886(119901) = Φ (119908

2) + [Φ (119908

1) minus Φ (119908

2)] [Φ (119908

2)]119894

(5)

where 1199081= (119885119901minus 1198961015840

)radic119899 and 1199082= (119885119901minus 119896)radic119899

It is to be pointed out that the parameters of theknown sigma variables chain sampling plan are denoted by

4 International Journal of Manufacturing Engineering

(119899120590 1198961015840120590 and 119896

120590) and for unknown sigma plan they are denoted

by (119899119878 1198961015840119878 and 119896

119878) The determination of parameters (119899

119878 1198961015840119878

and 119896119878) of unknown sigma variables chain sampling plan

is explained as follows It is known that for large samples119883 plusmn 119896

119878119878 is approximately normally distributed with mean

120583 plusmn 119896119878119864(119878) and variance (120590

2

119899)+119896119878Var(119878) (see Duncan [25]

and Balamurali and Jun [26]) That is

119883 + 119896119878119878 sim 119873(120583 + 119896

119878120590

1205902

119899+ 1198962

119878

1205902

2119899) (6)

Therefore the probability of accepting a lot at each repetitionis given by

119875 (119881 ge 119896119878) = 119875 119883 le 119880 minus 119896

119878119878 | 119901

= 119875 119883 + 119896119878119878 le 119880 | 119901

= Φ(119880 minus 119896

119878120590 minus 120583

(120590radic119899119878)radic1 + (1198962

1198782)

)

= Φ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(7)

If we let

1199082119878

= ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

then Pr (119881 ge 119896119878) = Φ (119908

2119878)

(8)

Similarly if we let 1199081119878

= ((119885119901minus 1198961015840

119878)radic119899119878(1 + (11989610158402

1198782))) then

we have

Pr (119881 lt 1198961015840

119878) = 1 minus Φ (119908

1119878) (9)

Hence the lot acceptance probability for sigmaunknown caseis given by

119875119886(119901) = Φ (119908

2119878) + [Φ (119908

1119878) minus Φ (119908

2119878)] [Φ (119908

2119878)]119894119878

(10)

where 1199081119878

= (119885119901minus 1198961015840

119878)radic119899119878and 119908

2119878= (119885119901minus 119896119878)radic119899119878

If (AQL 1 minus 120572) and (LQL 120573) are prescribed then the OCfunction can be written as

Φ(11990821119878

) + [Φ (11990811119878

) minus Φ (11990821119878

)] [Φ (11990821119878

)]119894119878

= 1 minus 120572

Φ (11990822119878

) + [Φ (11990812119878

) minus Φ (11990822119878

)] [Φ (11990822119878

)]119894119878

= 120573

(11)

Here 11990811119878

is the value of 1199081119878at 119901 = 119901

1 11990821119878

is the value of1199082119878at 119901 = 119901

1 11990812119878

is the value of 1199081119878at 119901 = 119901

2 and 119908

22119878is

the value of 1199082119878at 119901 = 119901

2

We obtain 11990811119878

11990821119878

11990812119878

and 11990822119878

respectively by

11990811119878

= ((1198851199011

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990821119878

= ((1198851199011

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

11990812119878

= ((1198851199012

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990822119878

= ((1198851199012

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(12)

where 1198851199011

is the value of 119885119901at AQL and 119885

1199012

is the value of119885119901at LQL For given AQL and LQL the parametric values of

the unknown sigma variables chain sampling plan namely119899119878 119894119878 119896119878 and 119896

1015840

119878 are determined by satisfying the required

producer and consumer conditions Alternatively we candetermine the above parameters of the variables chain sam-pling plan to minimize the ASN at AQL which is analogousto minimizing the average sample number in the variablesrepetitive group sampling plans andmultiple dependent statesampling plan (see Balamurali et al [27] and Balamuraliand Jun [26]) Some of the authors have investigated thedesigning of sampling plans by using some other optimiza-tion techniques which are available in the literature (see forexample Feldmann and Krumbholz [28] Krumbholz andRohr [29 30] Krumbholz et al [31] and Duarte and Sariava[32 33] The ASN for the chain sampling plan is the samplesize only Therefore the following optimization problem isconsidered to determine those parameters

Minimize ASN (1199011) = 119899119878

119899119878 119894119878 119896119878 1198961015840

119878

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899119878ge 2 119894119878ge 1 119896

119878+ 120575 ge 119896

1015840

119878+ 120575 ge 0

120575 ge 0 119899119878isin 119873 119894

119878isin 119873

(13)

In the case of known sigma variable chain sampling plan theOC function under the specified AQL and LQL conditionscan be written as

Φ(11990821120590

) + [Φ (11990811120590

) minus Φ (11990821120590

)] [Φ (11990821120590

)]119894120590

= 1 minus 120572

Φ (11990822120590

) + [Φ (11990812120590

) minus Φ (11990822120590

)] [Φ (11990822120590

)]119894120590

= 120573

(14)

Here 11990811120590

is the value of 1199081120590

at 119901 = 1199011 11990821120590

is the value of1199082120590

at 119901 = 1199011 11990812120590

is the value of 1199081120590

at 119901 = 1199012 and 119908

22120590is

the value of 1199082120590

at 119901 = 1199012 That is

11990811120590

= (1198851199011

minus 1198961015840

120590)radic119899120590 119908

21120590= (1198851199011

minus 119896120590)radic119899120590

11990812120590

= (1198851199012

minus 1198961015840

120590)radic119899120590 119908

22120590= (1198851199012

minus 119896120590)radic119899120590

(15)

International Journal of Manufacturing Engineering 5

where1198851199011

is the value of119885119901at AQL and119885

1199012

is the value of119885119901

at LQL The following optimization problem is considered todetermine the optimal parameters of known sigma variablessampling plan such as 119899

120590 119894120590 119896120590 and 119896

1015840

120590

Minimize ASN (1199011) = 119899120590

119899120590 119894120590 119896120590 1198961015840

120590

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899120590ge 2 119894120590ge 1 119896

120590+ 120575 ge 119896

1015840

120590+ 120575 ge 0

120575 ge 0 119899120590isin 119873 119894

120590isin 119873

(16)

We may determine the parameters of the unknown sigmachain sampling plan and the known sigma chain samplingplan by solving the nonlinear equation given in (13) and (16)respectively There may exist multiple solutions since thereare four unknownswith only two equationsGenerally a sam-pling would be desirable if the required number of samples issmall So in this paper we consider the ASN as the objectivefunction to be minimized with the probability of acceptancealongwith the corresponding producerrsquos and consumerrsquos risksas constraints To solve the above nonlinear optimizationproblems given in (13) and (16) the sequential quadraticprogramming (SQP) proposed by Nocedal and Wright [34]can be used Here it is to be pointed out that the optimizationproblems given in (13) and (16) include continuous valuesof 1198961015840120590(or 1198961015840

119878) and 119896

120590(or 119896119878) and integer values of 119899

120590(or 119899119878)

and 119894120590(or 119894119878) This kind of formalization leads to a problem of

mixed integer programming (MINLP) typeThe SQPmethoddoes not allow directly solvingMINLP problems but only thenonlinear programming (NLP) structures The strategy wehave followed is to relax the discrete variables first and solvethe NLP problem and then fix them to the upper integers andresolve the problem again now just to find 119896

1015840

120590(or 1198961015840

119878) and

119896120590(or 119896119878) In such a way we have solved the NLP problem

given above The SQP is implemented in Matlab softwareusing the routine ldquofminconrdquo By solving the NLP based onthe procedure described above the optimal parameters (119899

120590

119894120590 119896120590 and 119896

1015840

120590) for known sigma plan and the parameters (119899

119878

119894119878 119896119878 and 119896

1015840

119878) for unknown sigma plan are determined and

these values are tabulated in Table 1

5 Designing Examples

51 Selection of Known Sigma Variables Chain Sampling Planfor Specified AQL and LQL Table 1 is used to determine theparameters of the known 120590 variables chain sampling plan forspecified values of AQL and LQL when 120572 = 5 and 120573 = 10For example if 119901

1= 2 119901

2= 7 120572 = 5 and 120573 = 10 Table 1

gives the parameters as 119899120590= 18 119894

120590= 3 1198961015840

120590= 1544 and 119896

120590=

1779For the above example the plan is operated as followsFrom each submitted lot take a random sample of size 18

and compute 119881 = (119880 minus 119883)120590 where 119883 = (118)sum18

119894=1119883119894

Accept the lot if119881 ge 1779 and reject the lot if119881 lt 1544If 1544 le 119881 lt 1779 then accept the current lot providedthat the preceding 3 lots were accepted on the condition that119881 ge 1779 with the sample size of 18

52 Selection of Unknown Sigma Variables Chain SamplingPlan for Specified AQL and LQL As mentioned earlier theunknown sigma variables chain sampling plan is operatedas a known sigma variables chain sampling plan but theparameters 119899

119878 119894119878 119896119878 and 119896

1015840

119878are used in the place of 119899

120590 119894120590 119896120590

and 1198961015840

120590 respectively Table 1 can also be used for the selection

of the parameters of the unknown 120590 variables chain samplingplan for given values of AQL and LQL Suppose that AQL =00075 LQL = 0035 120572 = 5 and 120573 = 10 From Table 1the parameters of the variables chain sampling plan can bedetermined as 119899

119878= 47 119894

119878= 1 1198961015840119878= 1925 and 119896

119878= 2190

53 Illustrative Example To illustrate the implementationof the proposed sampling plan for the example given inSection 52 we consider case study data on STN-LCD man-ufacturing process given in Wu and Pearn [35] The upperspecification limit is given as 077mmHere we consider only47 values randomly taken from the original data given inWuand Pearn [35] The data are shown below

0717 0698 0726 0684 0727 0688 0708 0703 0694 0713

0730 0699 0710 0688 0665 0704 0725 0729 0716 0685

0712 0716 0712 0733 0709 0703 0730 0716 0688 0688

0712 0702 0726 0669 0718 0714 0726 0683 0713 0737

0740 0706 0726 0688 0715 0704 0724

(17)

The implementation of the plan is shown below

Step 1 Take a random sample of size 47 The data are givenabove

Step 2 For this data we calculate 119883 = (1119899)sum119899119878

119894=1119883119894

=

0708915 and 119878 = radicsum(119883119894minus 119883)2

(119899 minus 1) = 0017583

Step 3 Calculate 119881 = (119880 minus 119883)119878 = (077 minus

0708915)0017583 = 34741

Step 4 Since 119881 = 34741 gt 119896119878= 2190 we accept the current

lot without considering the result of past lots

Just for the sake of discussion let us assume that the 119881

value (for different data sets) is calculated as 215 In this case

6 International Journal of Manufacturing Engineering

Table 1 Variables chain sampling plans indexed by AQL and LQL for 120572 = 5 and 120573 = 10

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0001

00020 119 1 2883 3013 650 1 2888 301300025 68 2 2810 2965 361 1 2834 29840003 46 1 2776 2961 242 1 2774 29640004 31 2 2740 2885 147 1 2716 29210005 21 2 2610 2860 103 1 2637 28970006 19 4 2597 2807 84 2 2599 28290007 15 3 2534 2789 72 3 2532 27920008 13 3 2495 2765 62 3 2518 27680009 12 3 2521 2736 51 2 2470 27650010 12 6 2517 2697 45 2 2425 27500012 9 4 2331 2686 40 3 2414 26940015 7 3 2237 2657 34 4 2387 26370020 7 6 2414 2539 27 5 2335 25650025 6 6 2370 2485 20 3 2197 25520030 5 6 2264 2454 19 5 2231 2476

00025

0004 230 2 2658 2738 1131 2 2676 27360005 105 2 2598 2703 474 2 2585 27050006 64 2 2530 2675 282 1 2533 269800075 39 2 2441 2641 171 1 2464 26690010 26 3 2379 2579 106 1 2430 26100012 19 2 2335 2555 76 1 2321 26010015 16 4 2296 2491 55 1 2237 25720020 12 4 2260 2425 38 1 2139 25290025 10 6 2206 2366 32 2 2163 24130030 8 5 2114 2334 29 5 2107 23420035 7 6 2307 2297 25 5 2092 23020040 6 6 1884 2274 21 5 1952 2282

0005

00075 259 1 2434 2524 1113 2 2442 25120010 86 1 2326 2486 347 1 2348 24830012 53 1 2277 2457 211 1 2299 24540015 33 1 2211 2421 128 1 2233 24180020 24 4 2161 2316 75 2 2100 23450025 16 3 2046 2281 55 2 2083 22930030 14 6 1989 2224 43 3 1974 22490035 12 6 2002 2182 36 3 1979 22090040 10 4 1976 2156 31 3 1964 21740050 8 5 1893 2098 25 4 1925 2105

00075

0010 505 2 2339 2384 1817 1 2329 23940012 173 1 2259 2369 690 2 2278 23530015 78 1 2185 2335 286 1 2182 23370020 43 2 2146 2251 133 1 2083 22930025 29 5 2008 2198 83 1 2003 22580030 19 2 1968 2178 61 1 1984 22140035 17 5 1888 2123 47 1 1925 21900040 13 3 1837 2107 41 3 1851 21110050 10 3 1792 2052 28 2 1756 20810060 9 5 1792 1982 26 4 1815 1990

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

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DistributedSensor Networks

International Journal of

International Journal of Manufacturing Engineering 3

In the case of lower specification limit L the operatingprocedure is as follows

Step 1 From each lot take a random sample of size 119899120590 say

(1198831 1198832sdot sdot sdot 119883119899120590

) and compute 119881 = (119883 minus 119871)120590 where119883 = (1119899

120590) sum119899120590

119894=1119883119894

Step 2 Accept the lot if 119881 ge 119896120590and reject the lot if 119881 lt 119896

1015840

120590 If

1198961015840

120590le 119881 lt 119896

120590 then accept the current lot provided that

the preceding 119894120590lots were accepted on the condition

that 119881 ge 119896120590but reject otherwise

Thus the proposed variables chain sampling plan is charac-terized by four parameters namely 119899

120590 119894120590 119896120590 and 119896

1015840

120590 If 119896120590=

1198961015840

120590 then the proposed plan will reduce to the variables SSP

Also when 119894120590tends to infinity the proposed plan becomes

variables SSP with parameters 119899120590and 119896

120590 It is to be pointed

out that the chain sampling plan can be applied for inspectionof lots which are submitted serially in the order of productionor in the order of being submitted The decision of currentlot depends on the results of preceding lots So when 119894

120590ge

2 we need to keep the records of results of previous lotsOf course maintaining records of preceding lots may be adrawback of the chain sampling plan over the single samplingplan however this can be compensated by minimizing theinspection efforts in terms of minimum ASN with desiredprotection

32 Unknown Sigma Case Whenever the standard deviationis unknown we may use the sample standard deviation Sinstead of 120590 In this case the plan operates as follows

Step 1 From each submitted lot take a random sampleof size 119899

119878 say (119883

1 1198832sdot sdot sdot 119883119899119878

) and compute 119881 =

(119880 minus 119883)119878 where 119883 = (1119899119878) sum119899119878

119894=1119883119894and 119878 =

radicsum(119883119894minus 119883)2

(119899119878minus 1)

Step 2 Accept the lot if 119881 ge 119896119878and reject the lot if 119881 lt 119896

1015840

119878 If

1198961015840

119878le 119881 lt 119896

119878 then accept the current lot provided that

the preceding 119894119878lots were accepted on the condition

that 119881 ge 119896119878

Thus the proposed unknown sigma variables chain samplingplan is characterized by four parameters namely 119899

119878 119894119878 119896119878

and 1198961015840

119878 If 119896119878

= 1198961015840

119878 then the proposed plan will reduce to

the variables single sampling plan with unknown standarddeviation

4 Designing Methodology of Variables ChainSampling Plan

Generally variables sampling plans are designed based ontwo points on the OC curve namely (119901

1 1 minus 120572) and (119901

2 120573)

where 1199011is called the acceptable quality level (AQL) 119901

2is

the limiting quality level (LQL) 120572 is the producerrsquos risk and120573 is the consumerrsquos risk Any well-designed sampling planmust provide at least (1 minus 120572) probability of acceptance ofa lot when the process fraction nonconforming is at AQLlevel and the sampling plan must also provide not more than

120573 probability of acceptance if the process fraction noncon-forming is at the LQL level Thus the acceptance samplingplan must have its OC curve passing through two designatedpoints (AQL 1 minus 120572) and (LQL 120573) Some other strategiesare also followed to design the sampling plans besides thestatistical based paradigm which include Bayesian approachand economic based approach For further details readersmay refer Chen and Lam [17] Ferrell and Chhoker [18]Chen [19] Chen et al [20] Balamurali and Subramani [21]Vijayaraghavan and Sakthivel [22] Balamurali et al [23]and Fallahnezhad and Aslam [24] In this paper we havefollowed the designing methodology based on two points onthe OC curve approach The variables chain sampling planis designed based on the two points on the OC curve in thefollowing manner

The fraction nonconforming in a lot is given as

119901 = 1 minus Φ(119880 minus 120583

120590) (1)

where Φ(119884) is the cumulative distribution function of stan-dard normal distribution and is given by

Φ (119884) = int

119884

minusinfin

1

radic2120587exp(

minus1198852

2)119889119885 (2)

Here the quality characteristic of interest is normally dis-tributed with mean 120583 and standard deviation 120590 and theunit is classified as nonconforming if it exceeds the upperspecification limit U So the unknown mean 120583 can be deter-mined if p is specified Let us define the standardized qualitycharacteristic corresponding to the fraction conforming as

119885119901= Φminus1

(1 minus 119901) (3)

Then the OC function of the variables chain samplingplan which gives the proportion of lots that are expectedto be accepted for given product quality p is given by (seeGovindaraju and Balamurali [11])

119875119886(119901) = Pr (119881 ge 119896)

+ [Pr (119881 ge 1198961015840

) minus Pr (119881 ge 119896)] [Pr (119881 ge 119896)]119894

(4)

where Pr(119881 ge 119896) is the probability of accepting a lot basedon a single sample with parameters (119899 119896) and Pr(119881 lt 119896

1015840

) isthe probability of rejecting a lot based on a single sample withparameters (119899 1198961015840) Under Type B situation (ie a series of lotsof the same quality) forming lots of N items from a processand then drawing random sample of size 119899 from these lotsis equivalent to drawing random samples of size 119899 directlyfrom the process Hence the derivation of the OC functionis straightforward

The probability of acceptance of the chain sampling plancan also be written as

119875119886(119901) = Φ (119908

2) + [Φ (119908

1) minus Φ (119908

2)] [Φ (119908

2)]119894

(5)

where 1199081= (119885119901minus 1198961015840

)radic119899 and 1199082= (119885119901minus 119896)radic119899

It is to be pointed out that the parameters of theknown sigma variables chain sampling plan are denoted by

4 International Journal of Manufacturing Engineering

(119899120590 1198961015840120590 and 119896

120590) and for unknown sigma plan they are denoted

by (119899119878 1198961015840119878 and 119896

119878) The determination of parameters (119899

119878 1198961015840119878

and 119896119878) of unknown sigma variables chain sampling plan

is explained as follows It is known that for large samples119883 plusmn 119896

119878119878 is approximately normally distributed with mean

120583 plusmn 119896119878119864(119878) and variance (120590

2

119899)+119896119878Var(119878) (see Duncan [25]

and Balamurali and Jun [26]) That is

119883 + 119896119878119878 sim 119873(120583 + 119896

119878120590

1205902

119899+ 1198962

119878

1205902

2119899) (6)

Therefore the probability of accepting a lot at each repetitionis given by

119875 (119881 ge 119896119878) = 119875 119883 le 119880 minus 119896

119878119878 | 119901

= 119875 119883 + 119896119878119878 le 119880 | 119901

= Φ(119880 minus 119896

119878120590 minus 120583

(120590radic119899119878)radic1 + (1198962

1198782)

)

= Φ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(7)

If we let

1199082119878

= ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

then Pr (119881 ge 119896119878) = Φ (119908

2119878)

(8)

Similarly if we let 1199081119878

= ((119885119901minus 1198961015840

119878)radic119899119878(1 + (11989610158402

1198782))) then

we have

Pr (119881 lt 1198961015840

119878) = 1 minus Φ (119908

1119878) (9)

Hence the lot acceptance probability for sigmaunknown caseis given by

119875119886(119901) = Φ (119908

2119878) + [Φ (119908

1119878) minus Φ (119908

2119878)] [Φ (119908

2119878)]119894119878

(10)

where 1199081119878

= (119885119901minus 1198961015840

119878)radic119899119878and 119908

2119878= (119885119901minus 119896119878)radic119899119878

If (AQL 1 minus 120572) and (LQL 120573) are prescribed then the OCfunction can be written as

Φ(11990821119878

) + [Φ (11990811119878

) minus Φ (11990821119878

)] [Φ (11990821119878

)]119894119878

= 1 minus 120572

Φ (11990822119878

) + [Φ (11990812119878

) minus Φ (11990822119878

)] [Φ (11990822119878

)]119894119878

= 120573

(11)

Here 11990811119878

is the value of 1199081119878at 119901 = 119901

1 11990821119878

is the value of1199082119878at 119901 = 119901

1 11990812119878

is the value of 1199081119878at 119901 = 119901

2 and 119908

22119878is

the value of 1199082119878at 119901 = 119901

2

We obtain 11990811119878

11990821119878

11990812119878

and 11990822119878

respectively by

11990811119878

= ((1198851199011

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990821119878

= ((1198851199011

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

11990812119878

= ((1198851199012

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990822119878

= ((1198851199012

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(12)

where 1198851199011

is the value of 119885119901at AQL and 119885

1199012

is the value of119885119901at LQL For given AQL and LQL the parametric values of

the unknown sigma variables chain sampling plan namely119899119878 119894119878 119896119878 and 119896

1015840

119878 are determined by satisfying the required

producer and consumer conditions Alternatively we candetermine the above parameters of the variables chain sam-pling plan to minimize the ASN at AQL which is analogousto minimizing the average sample number in the variablesrepetitive group sampling plans andmultiple dependent statesampling plan (see Balamurali et al [27] and Balamuraliand Jun [26]) Some of the authors have investigated thedesigning of sampling plans by using some other optimiza-tion techniques which are available in the literature (see forexample Feldmann and Krumbholz [28] Krumbholz andRohr [29 30] Krumbholz et al [31] and Duarte and Sariava[32 33] The ASN for the chain sampling plan is the samplesize only Therefore the following optimization problem isconsidered to determine those parameters

Minimize ASN (1199011) = 119899119878

119899119878 119894119878 119896119878 1198961015840

119878

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899119878ge 2 119894119878ge 1 119896

119878+ 120575 ge 119896

1015840

119878+ 120575 ge 0

120575 ge 0 119899119878isin 119873 119894

119878isin 119873

(13)

In the case of known sigma variable chain sampling plan theOC function under the specified AQL and LQL conditionscan be written as

Φ(11990821120590

) + [Φ (11990811120590

) minus Φ (11990821120590

)] [Φ (11990821120590

)]119894120590

= 1 minus 120572

Φ (11990822120590

) + [Φ (11990812120590

) minus Φ (11990822120590

)] [Φ (11990822120590

)]119894120590

= 120573

(14)

Here 11990811120590

is the value of 1199081120590

at 119901 = 1199011 11990821120590

is the value of1199082120590

at 119901 = 1199011 11990812120590

is the value of 1199081120590

at 119901 = 1199012 and 119908

22120590is

the value of 1199082120590

at 119901 = 1199012 That is

11990811120590

= (1198851199011

minus 1198961015840

120590)radic119899120590 119908

21120590= (1198851199011

minus 119896120590)radic119899120590

11990812120590

= (1198851199012

minus 1198961015840

120590)radic119899120590 119908

22120590= (1198851199012

minus 119896120590)radic119899120590

(15)

International Journal of Manufacturing Engineering 5

where1198851199011

is the value of119885119901at AQL and119885

1199012

is the value of119885119901

at LQL The following optimization problem is considered todetermine the optimal parameters of known sigma variablessampling plan such as 119899

120590 119894120590 119896120590 and 119896

1015840

120590

Minimize ASN (1199011) = 119899120590

119899120590 119894120590 119896120590 1198961015840

120590

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899120590ge 2 119894120590ge 1 119896

120590+ 120575 ge 119896

1015840

120590+ 120575 ge 0

120575 ge 0 119899120590isin 119873 119894

120590isin 119873

(16)

We may determine the parameters of the unknown sigmachain sampling plan and the known sigma chain samplingplan by solving the nonlinear equation given in (13) and (16)respectively There may exist multiple solutions since thereare four unknownswith only two equationsGenerally a sam-pling would be desirable if the required number of samples issmall So in this paper we consider the ASN as the objectivefunction to be minimized with the probability of acceptancealongwith the corresponding producerrsquos and consumerrsquos risksas constraints To solve the above nonlinear optimizationproblems given in (13) and (16) the sequential quadraticprogramming (SQP) proposed by Nocedal and Wright [34]can be used Here it is to be pointed out that the optimizationproblems given in (13) and (16) include continuous valuesof 1198961015840120590(or 1198961015840

119878) and 119896

120590(or 119896119878) and integer values of 119899

120590(or 119899119878)

and 119894120590(or 119894119878) This kind of formalization leads to a problem of

mixed integer programming (MINLP) typeThe SQPmethoddoes not allow directly solvingMINLP problems but only thenonlinear programming (NLP) structures The strategy wehave followed is to relax the discrete variables first and solvethe NLP problem and then fix them to the upper integers andresolve the problem again now just to find 119896

1015840

120590(or 1198961015840

119878) and

119896120590(or 119896119878) In such a way we have solved the NLP problem

given above The SQP is implemented in Matlab softwareusing the routine ldquofminconrdquo By solving the NLP based onthe procedure described above the optimal parameters (119899

120590

119894120590 119896120590 and 119896

1015840

120590) for known sigma plan and the parameters (119899

119878

119894119878 119896119878 and 119896

1015840

119878) for unknown sigma plan are determined and

these values are tabulated in Table 1

5 Designing Examples

51 Selection of Known Sigma Variables Chain Sampling Planfor Specified AQL and LQL Table 1 is used to determine theparameters of the known 120590 variables chain sampling plan forspecified values of AQL and LQL when 120572 = 5 and 120573 = 10For example if 119901

1= 2 119901

2= 7 120572 = 5 and 120573 = 10 Table 1

gives the parameters as 119899120590= 18 119894

120590= 3 1198961015840

120590= 1544 and 119896

120590=

1779For the above example the plan is operated as followsFrom each submitted lot take a random sample of size 18

and compute 119881 = (119880 minus 119883)120590 where 119883 = (118)sum18

119894=1119883119894

Accept the lot if119881 ge 1779 and reject the lot if119881 lt 1544If 1544 le 119881 lt 1779 then accept the current lot providedthat the preceding 3 lots were accepted on the condition that119881 ge 1779 with the sample size of 18

52 Selection of Unknown Sigma Variables Chain SamplingPlan for Specified AQL and LQL As mentioned earlier theunknown sigma variables chain sampling plan is operatedas a known sigma variables chain sampling plan but theparameters 119899

119878 119894119878 119896119878 and 119896

1015840

119878are used in the place of 119899

120590 119894120590 119896120590

and 1198961015840

120590 respectively Table 1 can also be used for the selection

of the parameters of the unknown 120590 variables chain samplingplan for given values of AQL and LQL Suppose that AQL =00075 LQL = 0035 120572 = 5 and 120573 = 10 From Table 1the parameters of the variables chain sampling plan can bedetermined as 119899

119878= 47 119894

119878= 1 1198961015840119878= 1925 and 119896

119878= 2190

53 Illustrative Example To illustrate the implementationof the proposed sampling plan for the example given inSection 52 we consider case study data on STN-LCD man-ufacturing process given in Wu and Pearn [35] The upperspecification limit is given as 077mmHere we consider only47 values randomly taken from the original data given inWuand Pearn [35] The data are shown below

0717 0698 0726 0684 0727 0688 0708 0703 0694 0713

0730 0699 0710 0688 0665 0704 0725 0729 0716 0685

0712 0716 0712 0733 0709 0703 0730 0716 0688 0688

0712 0702 0726 0669 0718 0714 0726 0683 0713 0737

0740 0706 0726 0688 0715 0704 0724

(17)

The implementation of the plan is shown below

Step 1 Take a random sample of size 47 The data are givenabove

Step 2 For this data we calculate 119883 = (1119899)sum119899119878

119894=1119883119894

=

0708915 and 119878 = radicsum(119883119894minus 119883)2

(119899 minus 1) = 0017583

Step 3 Calculate 119881 = (119880 minus 119883)119878 = (077 minus

0708915)0017583 = 34741

Step 4 Since 119881 = 34741 gt 119896119878= 2190 we accept the current

lot without considering the result of past lots

Just for the sake of discussion let us assume that the 119881

value (for different data sets) is calculated as 215 In this case

6 International Journal of Manufacturing Engineering

Table 1 Variables chain sampling plans indexed by AQL and LQL for 120572 = 5 and 120573 = 10

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0001

00020 119 1 2883 3013 650 1 2888 301300025 68 2 2810 2965 361 1 2834 29840003 46 1 2776 2961 242 1 2774 29640004 31 2 2740 2885 147 1 2716 29210005 21 2 2610 2860 103 1 2637 28970006 19 4 2597 2807 84 2 2599 28290007 15 3 2534 2789 72 3 2532 27920008 13 3 2495 2765 62 3 2518 27680009 12 3 2521 2736 51 2 2470 27650010 12 6 2517 2697 45 2 2425 27500012 9 4 2331 2686 40 3 2414 26940015 7 3 2237 2657 34 4 2387 26370020 7 6 2414 2539 27 5 2335 25650025 6 6 2370 2485 20 3 2197 25520030 5 6 2264 2454 19 5 2231 2476

00025

0004 230 2 2658 2738 1131 2 2676 27360005 105 2 2598 2703 474 2 2585 27050006 64 2 2530 2675 282 1 2533 269800075 39 2 2441 2641 171 1 2464 26690010 26 3 2379 2579 106 1 2430 26100012 19 2 2335 2555 76 1 2321 26010015 16 4 2296 2491 55 1 2237 25720020 12 4 2260 2425 38 1 2139 25290025 10 6 2206 2366 32 2 2163 24130030 8 5 2114 2334 29 5 2107 23420035 7 6 2307 2297 25 5 2092 23020040 6 6 1884 2274 21 5 1952 2282

0005

00075 259 1 2434 2524 1113 2 2442 25120010 86 1 2326 2486 347 1 2348 24830012 53 1 2277 2457 211 1 2299 24540015 33 1 2211 2421 128 1 2233 24180020 24 4 2161 2316 75 2 2100 23450025 16 3 2046 2281 55 2 2083 22930030 14 6 1989 2224 43 3 1974 22490035 12 6 2002 2182 36 3 1979 22090040 10 4 1976 2156 31 3 1964 21740050 8 5 1893 2098 25 4 1925 2105

00075

0010 505 2 2339 2384 1817 1 2329 23940012 173 1 2259 2369 690 2 2278 23530015 78 1 2185 2335 286 1 2182 23370020 43 2 2146 2251 133 1 2083 22930025 29 5 2008 2198 83 1 2003 22580030 19 2 1968 2178 61 1 1984 22140035 17 5 1888 2123 47 1 1925 21900040 13 3 1837 2107 41 3 1851 21110050 10 3 1792 2052 28 2 1756 20810060 9 5 1792 1982 26 4 1815 1990

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

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DistributedSensor Networks

International Journal of

4 International Journal of Manufacturing Engineering

(119899120590 1198961015840120590 and 119896

120590) and for unknown sigma plan they are denoted

by (119899119878 1198961015840119878 and 119896

119878) The determination of parameters (119899

119878 1198961015840119878

and 119896119878) of unknown sigma variables chain sampling plan

is explained as follows It is known that for large samples119883 plusmn 119896

119878119878 is approximately normally distributed with mean

120583 plusmn 119896119878119864(119878) and variance (120590

2

119899)+119896119878Var(119878) (see Duncan [25]

and Balamurali and Jun [26]) That is

119883 + 119896119878119878 sim 119873(120583 + 119896

119878120590

1205902

119899+ 1198962

119878

1205902

2119899) (6)

Therefore the probability of accepting a lot at each repetitionis given by

119875 (119881 ge 119896119878) = 119875 119883 le 119880 minus 119896

119878119878 | 119901

= 119875 119883 + 119896119878119878 le 119880 | 119901

= Φ(119880 minus 119896

119878120590 minus 120583

(120590radic119899119878)radic1 + (1198962

1198782)

)

= Φ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(7)

If we let

1199082119878

= ((119885119901minus 119896119878)radic

119899119878

1 + (11989621198782)

)

then Pr (119881 ge 119896119878) = Φ (119908

2119878)

(8)

Similarly if we let 1199081119878

= ((119885119901minus 1198961015840

119878)radic119899119878(1 + (11989610158402

1198782))) then

we have

Pr (119881 lt 1198961015840

119878) = 1 minus Φ (119908

1119878) (9)

Hence the lot acceptance probability for sigmaunknown caseis given by

119875119886(119901) = Φ (119908

2119878) + [Φ (119908

1119878) minus Φ (119908

2119878)] [Φ (119908

2119878)]119894119878

(10)

where 1199081119878

= (119885119901minus 1198961015840

119878)radic119899119878and 119908

2119878= (119885119901minus 119896119878)radic119899119878

If (AQL 1 minus 120572) and (LQL 120573) are prescribed then the OCfunction can be written as

Φ(11990821119878

) + [Φ (11990811119878

) minus Φ (11990821119878

)] [Φ (11990821119878

)]119894119878

= 1 minus 120572

Φ (11990822119878

) + [Φ (11990812119878

) minus Φ (11990822119878

)] [Φ (11990822119878

)]119894119878

= 120573

(11)

Here 11990811119878

is the value of 1199081119878at 119901 = 119901

1 11990821119878

is the value of1199082119878at 119901 = 119901

1 11990812119878

is the value of 1199081119878at 119901 = 119901

2 and 119908

22119878is

the value of 1199082119878at 119901 = 119901

2

We obtain 11990811119878

11990821119878

11990812119878

and 11990822119878

respectively by

11990811119878

= ((1198851199011

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990821119878

= ((1198851199011

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

11990812119878

= ((1198851199012

minus 1198961015840

119878)radic

119899119878

1 + (119896101584021198782)

)

11990822119878

= ((1198851199012

minus 119896119878)radic

119899119878

1 + (11989621198782)

)

(12)

where 1198851199011

is the value of 119885119901at AQL and 119885

1199012

is the value of119885119901at LQL For given AQL and LQL the parametric values of

the unknown sigma variables chain sampling plan namely119899119878 119894119878 119896119878 and 119896

1015840

119878 are determined by satisfying the required

producer and consumer conditions Alternatively we candetermine the above parameters of the variables chain sam-pling plan to minimize the ASN at AQL which is analogousto minimizing the average sample number in the variablesrepetitive group sampling plans andmultiple dependent statesampling plan (see Balamurali et al [27] and Balamuraliand Jun [26]) Some of the authors have investigated thedesigning of sampling plans by using some other optimiza-tion techniques which are available in the literature (see forexample Feldmann and Krumbholz [28] Krumbholz andRohr [29 30] Krumbholz et al [31] and Duarte and Sariava[32 33] The ASN for the chain sampling plan is the samplesize only Therefore the following optimization problem isconsidered to determine those parameters

Minimize ASN (1199011) = 119899119878

119899119878 119894119878 119896119878 1198961015840

119878

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899119878ge 2 119894119878ge 1 119896

119878+ 120575 ge 119896

1015840

119878+ 120575 ge 0

120575 ge 0 119899119878isin 119873 119894

119878isin 119873

(13)

In the case of known sigma variable chain sampling plan theOC function under the specified AQL and LQL conditionscan be written as

Φ(11990821120590

) + [Φ (11990811120590

) minus Φ (11990821120590

)] [Φ (11990821120590

)]119894120590

= 1 minus 120572

Φ (11990822120590

) + [Φ (11990812120590

) minus Φ (11990822120590

)] [Φ (11990822120590

)]119894120590

= 120573

(14)

Here 11990811120590

is the value of 1199081120590

at 119901 = 1199011 11990821120590

is the value of1199082120590

at 119901 = 1199011 11990812120590

is the value of 1199081120590

at 119901 = 1199012 and 119908

22120590is

the value of 1199082120590

at 119901 = 1199012 That is

11990811120590

= (1198851199011

minus 1198961015840

120590)radic119899120590 119908

21120590= (1198851199011

minus 119896120590)radic119899120590

11990812120590

= (1198851199012

minus 1198961015840

120590)radic119899120590 119908

22120590= (1198851199012

minus 119896120590)radic119899120590

(15)

International Journal of Manufacturing Engineering 5

where1198851199011

is the value of119885119901at AQL and119885

1199012

is the value of119885119901

at LQL The following optimization problem is considered todetermine the optimal parameters of known sigma variablessampling plan such as 119899

120590 119894120590 119896120590 and 119896

1015840

120590

Minimize ASN (1199011) = 119899120590

119899120590 119894120590 119896120590 1198961015840

120590

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899120590ge 2 119894120590ge 1 119896

120590+ 120575 ge 119896

1015840

120590+ 120575 ge 0

120575 ge 0 119899120590isin 119873 119894

120590isin 119873

(16)

We may determine the parameters of the unknown sigmachain sampling plan and the known sigma chain samplingplan by solving the nonlinear equation given in (13) and (16)respectively There may exist multiple solutions since thereare four unknownswith only two equationsGenerally a sam-pling would be desirable if the required number of samples issmall So in this paper we consider the ASN as the objectivefunction to be minimized with the probability of acceptancealongwith the corresponding producerrsquos and consumerrsquos risksas constraints To solve the above nonlinear optimizationproblems given in (13) and (16) the sequential quadraticprogramming (SQP) proposed by Nocedal and Wright [34]can be used Here it is to be pointed out that the optimizationproblems given in (13) and (16) include continuous valuesof 1198961015840120590(or 1198961015840

119878) and 119896

120590(or 119896119878) and integer values of 119899

120590(or 119899119878)

and 119894120590(or 119894119878) This kind of formalization leads to a problem of

mixed integer programming (MINLP) typeThe SQPmethoddoes not allow directly solvingMINLP problems but only thenonlinear programming (NLP) structures The strategy wehave followed is to relax the discrete variables first and solvethe NLP problem and then fix them to the upper integers andresolve the problem again now just to find 119896

1015840

120590(or 1198961015840

119878) and

119896120590(or 119896119878) In such a way we have solved the NLP problem

given above The SQP is implemented in Matlab softwareusing the routine ldquofminconrdquo By solving the NLP based onthe procedure described above the optimal parameters (119899

120590

119894120590 119896120590 and 119896

1015840

120590) for known sigma plan and the parameters (119899

119878

119894119878 119896119878 and 119896

1015840

119878) for unknown sigma plan are determined and

these values are tabulated in Table 1

5 Designing Examples

51 Selection of Known Sigma Variables Chain Sampling Planfor Specified AQL and LQL Table 1 is used to determine theparameters of the known 120590 variables chain sampling plan forspecified values of AQL and LQL when 120572 = 5 and 120573 = 10For example if 119901

1= 2 119901

2= 7 120572 = 5 and 120573 = 10 Table 1

gives the parameters as 119899120590= 18 119894

120590= 3 1198961015840

120590= 1544 and 119896

120590=

1779For the above example the plan is operated as followsFrom each submitted lot take a random sample of size 18

and compute 119881 = (119880 minus 119883)120590 where 119883 = (118)sum18

119894=1119883119894

Accept the lot if119881 ge 1779 and reject the lot if119881 lt 1544If 1544 le 119881 lt 1779 then accept the current lot providedthat the preceding 3 lots were accepted on the condition that119881 ge 1779 with the sample size of 18

52 Selection of Unknown Sigma Variables Chain SamplingPlan for Specified AQL and LQL As mentioned earlier theunknown sigma variables chain sampling plan is operatedas a known sigma variables chain sampling plan but theparameters 119899

119878 119894119878 119896119878 and 119896

1015840

119878are used in the place of 119899

120590 119894120590 119896120590

and 1198961015840

120590 respectively Table 1 can also be used for the selection

of the parameters of the unknown 120590 variables chain samplingplan for given values of AQL and LQL Suppose that AQL =00075 LQL = 0035 120572 = 5 and 120573 = 10 From Table 1the parameters of the variables chain sampling plan can bedetermined as 119899

119878= 47 119894

119878= 1 1198961015840119878= 1925 and 119896

119878= 2190

53 Illustrative Example To illustrate the implementationof the proposed sampling plan for the example given inSection 52 we consider case study data on STN-LCD man-ufacturing process given in Wu and Pearn [35] The upperspecification limit is given as 077mmHere we consider only47 values randomly taken from the original data given inWuand Pearn [35] The data are shown below

0717 0698 0726 0684 0727 0688 0708 0703 0694 0713

0730 0699 0710 0688 0665 0704 0725 0729 0716 0685

0712 0716 0712 0733 0709 0703 0730 0716 0688 0688

0712 0702 0726 0669 0718 0714 0726 0683 0713 0737

0740 0706 0726 0688 0715 0704 0724

(17)

The implementation of the plan is shown below

Step 1 Take a random sample of size 47 The data are givenabove

Step 2 For this data we calculate 119883 = (1119899)sum119899119878

119894=1119883119894

=

0708915 and 119878 = radicsum(119883119894minus 119883)2

(119899 minus 1) = 0017583

Step 3 Calculate 119881 = (119880 minus 119883)119878 = (077 minus

0708915)0017583 = 34741

Step 4 Since 119881 = 34741 gt 119896119878= 2190 we accept the current

lot without considering the result of past lots

Just for the sake of discussion let us assume that the 119881

value (for different data sets) is calculated as 215 In this case

6 International Journal of Manufacturing Engineering

Table 1 Variables chain sampling plans indexed by AQL and LQL for 120572 = 5 and 120573 = 10

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0001

00020 119 1 2883 3013 650 1 2888 301300025 68 2 2810 2965 361 1 2834 29840003 46 1 2776 2961 242 1 2774 29640004 31 2 2740 2885 147 1 2716 29210005 21 2 2610 2860 103 1 2637 28970006 19 4 2597 2807 84 2 2599 28290007 15 3 2534 2789 72 3 2532 27920008 13 3 2495 2765 62 3 2518 27680009 12 3 2521 2736 51 2 2470 27650010 12 6 2517 2697 45 2 2425 27500012 9 4 2331 2686 40 3 2414 26940015 7 3 2237 2657 34 4 2387 26370020 7 6 2414 2539 27 5 2335 25650025 6 6 2370 2485 20 3 2197 25520030 5 6 2264 2454 19 5 2231 2476

00025

0004 230 2 2658 2738 1131 2 2676 27360005 105 2 2598 2703 474 2 2585 27050006 64 2 2530 2675 282 1 2533 269800075 39 2 2441 2641 171 1 2464 26690010 26 3 2379 2579 106 1 2430 26100012 19 2 2335 2555 76 1 2321 26010015 16 4 2296 2491 55 1 2237 25720020 12 4 2260 2425 38 1 2139 25290025 10 6 2206 2366 32 2 2163 24130030 8 5 2114 2334 29 5 2107 23420035 7 6 2307 2297 25 5 2092 23020040 6 6 1884 2274 21 5 1952 2282

0005

00075 259 1 2434 2524 1113 2 2442 25120010 86 1 2326 2486 347 1 2348 24830012 53 1 2277 2457 211 1 2299 24540015 33 1 2211 2421 128 1 2233 24180020 24 4 2161 2316 75 2 2100 23450025 16 3 2046 2281 55 2 2083 22930030 14 6 1989 2224 43 3 1974 22490035 12 6 2002 2182 36 3 1979 22090040 10 4 1976 2156 31 3 1964 21740050 8 5 1893 2098 25 4 1925 2105

00075

0010 505 2 2339 2384 1817 1 2329 23940012 173 1 2259 2369 690 2 2278 23530015 78 1 2185 2335 286 1 2182 23370020 43 2 2146 2251 133 1 2083 22930025 29 5 2008 2198 83 1 2003 22580030 19 2 1968 2178 61 1 1984 22140035 17 5 1888 2123 47 1 1925 21900040 13 3 1837 2107 41 3 1851 21110050 10 3 1792 2052 28 2 1756 20810060 9 5 1792 1982 26 4 1815 1990

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

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International Journal of

International Journal of Manufacturing Engineering 5

where1198851199011

is the value of119885119901at AQL and119885

1199012

is the value of119885119901

at LQL The following optimization problem is considered todetermine the optimal parameters of known sigma variablessampling plan such as 119899

120590 119894120590 119896120590 and 119896

1015840

120590

Minimize ASN (1199011) = 119899120590

119899120590 119894120590 119896120590 1198961015840

120590

Subject to 119875119886(1199011) ge 1 minus 120572

119875119886(1199012) le 120573

119899120590ge 2 119894120590ge 1 119896

120590+ 120575 ge 119896

1015840

120590+ 120575 ge 0

120575 ge 0 119899120590isin 119873 119894

120590isin 119873

(16)

We may determine the parameters of the unknown sigmachain sampling plan and the known sigma chain samplingplan by solving the nonlinear equation given in (13) and (16)respectively There may exist multiple solutions since thereare four unknownswith only two equationsGenerally a sam-pling would be desirable if the required number of samples issmall So in this paper we consider the ASN as the objectivefunction to be minimized with the probability of acceptancealongwith the corresponding producerrsquos and consumerrsquos risksas constraints To solve the above nonlinear optimizationproblems given in (13) and (16) the sequential quadraticprogramming (SQP) proposed by Nocedal and Wright [34]can be used Here it is to be pointed out that the optimizationproblems given in (13) and (16) include continuous valuesof 1198961015840120590(or 1198961015840

119878) and 119896

120590(or 119896119878) and integer values of 119899

120590(or 119899119878)

and 119894120590(or 119894119878) This kind of formalization leads to a problem of

mixed integer programming (MINLP) typeThe SQPmethoddoes not allow directly solvingMINLP problems but only thenonlinear programming (NLP) structures The strategy wehave followed is to relax the discrete variables first and solvethe NLP problem and then fix them to the upper integers andresolve the problem again now just to find 119896

1015840

120590(or 1198961015840

119878) and

119896120590(or 119896119878) In such a way we have solved the NLP problem

given above The SQP is implemented in Matlab softwareusing the routine ldquofminconrdquo By solving the NLP based onthe procedure described above the optimal parameters (119899

120590

119894120590 119896120590 and 119896

1015840

120590) for known sigma plan and the parameters (119899

119878

119894119878 119896119878 and 119896

1015840

119878) for unknown sigma plan are determined and

these values are tabulated in Table 1

5 Designing Examples

51 Selection of Known Sigma Variables Chain Sampling Planfor Specified AQL and LQL Table 1 is used to determine theparameters of the known 120590 variables chain sampling plan forspecified values of AQL and LQL when 120572 = 5 and 120573 = 10For example if 119901

1= 2 119901

2= 7 120572 = 5 and 120573 = 10 Table 1

gives the parameters as 119899120590= 18 119894

120590= 3 1198961015840

120590= 1544 and 119896

120590=

1779For the above example the plan is operated as followsFrom each submitted lot take a random sample of size 18

and compute 119881 = (119880 minus 119883)120590 where 119883 = (118)sum18

119894=1119883119894

Accept the lot if119881 ge 1779 and reject the lot if119881 lt 1544If 1544 le 119881 lt 1779 then accept the current lot providedthat the preceding 3 lots were accepted on the condition that119881 ge 1779 with the sample size of 18

52 Selection of Unknown Sigma Variables Chain SamplingPlan for Specified AQL and LQL As mentioned earlier theunknown sigma variables chain sampling plan is operatedas a known sigma variables chain sampling plan but theparameters 119899

119878 119894119878 119896119878 and 119896

1015840

119878are used in the place of 119899

120590 119894120590 119896120590

and 1198961015840

120590 respectively Table 1 can also be used for the selection

of the parameters of the unknown 120590 variables chain samplingplan for given values of AQL and LQL Suppose that AQL =00075 LQL = 0035 120572 = 5 and 120573 = 10 From Table 1the parameters of the variables chain sampling plan can bedetermined as 119899

119878= 47 119894

119878= 1 1198961015840119878= 1925 and 119896

119878= 2190

53 Illustrative Example To illustrate the implementationof the proposed sampling plan for the example given inSection 52 we consider case study data on STN-LCD man-ufacturing process given in Wu and Pearn [35] The upperspecification limit is given as 077mmHere we consider only47 values randomly taken from the original data given inWuand Pearn [35] The data are shown below

0717 0698 0726 0684 0727 0688 0708 0703 0694 0713

0730 0699 0710 0688 0665 0704 0725 0729 0716 0685

0712 0716 0712 0733 0709 0703 0730 0716 0688 0688

0712 0702 0726 0669 0718 0714 0726 0683 0713 0737

0740 0706 0726 0688 0715 0704 0724

(17)

The implementation of the plan is shown below

Step 1 Take a random sample of size 47 The data are givenabove

Step 2 For this data we calculate 119883 = (1119899)sum119899119878

119894=1119883119894

=

0708915 and 119878 = radicsum(119883119894minus 119883)2

(119899 minus 1) = 0017583

Step 3 Calculate 119881 = (119880 minus 119883)119878 = (077 minus

0708915)0017583 = 34741

Step 4 Since 119881 = 34741 gt 119896119878= 2190 we accept the current

lot without considering the result of past lots

Just for the sake of discussion let us assume that the 119881

value (for different data sets) is calculated as 215 In this case

6 International Journal of Manufacturing Engineering

Table 1 Variables chain sampling plans indexed by AQL and LQL for 120572 = 5 and 120573 = 10

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0001

00020 119 1 2883 3013 650 1 2888 301300025 68 2 2810 2965 361 1 2834 29840003 46 1 2776 2961 242 1 2774 29640004 31 2 2740 2885 147 1 2716 29210005 21 2 2610 2860 103 1 2637 28970006 19 4 2597 2807 84 2 2599 28290007 15 3 2534 2789 72 3 2532 27920008 13 3 2495 2765 62 3 2518 27680009 12 3 2521 2736 51 2 2470 27650010 12 6 2517 2697 45 2 2425 27500012 9 4 2331 2686 40 3 2414 26940015 7 3 2237 2657 34 4 2387 26370020 7 6 2414 2539 27 5 2335 25650025 6 6 2370 2485 20 3 2197 25520030 5 6 2264 2454 19 5 2231 2476

00025

0004 230 2 2658 2738 1131 2 2676 27360005 105 2 2598 2703 474 2 2585 27050006 64 2 2530 2675 282 1 2533 269800075 39 2 2441 2641 171 1 2464 26690010 26 3 2379 2579 106 1 2430 26100012 19 2 2335 2555 76 1 2321 26010015 16 4 2296 2491 55 1 2237 25720020 12 4 2260 2425 38 1 2139 25290025 10 6 2206 2366 32 2 2163 24130030 8 5 2114 2334 29 5 2107 23420035 7 6 2307 2297 25 5 2092 23020040 6 6 1884 2274 21 5 1952 2282

0005

00075 259 1 2434 2524 1113 2 2442 25120010 86 1 2326 2486 347 1 2348 24830012 53 1 2277 2457 211 1 2299 24540015 33 1 2211 2421 128 1 2233 24180020 24 4 2161 2316 75 2 2100 23450025 16 3 2046 2281 55 2 2083 22930030 14 6 1989 2224 43 3 1974 22490035 12 6 2002 2182 36 3 1979 22090040 10 4 1976 2156 31 3 1964 21740050 8 5 1893 2098 25 4 1925 2105

00075

0010 505 2 2339 2384 1817 1 2329 23940012 173 1 2259 2369 690 2 2278 23530015 78 1 2185 2335 286 1 2182 23370020 43 2 2146 2251 133 1 2083 22930025 29 5 2008 2198 83 1 2003 22580030 19 2 1968 2178 61 1 1984 22140035 17 5 1888 2123 47 1 1925 21900040 13 3 1837 2107 41 3 1851 21110050 10 3 1792 2052 28 2 1756 20810060 9 5 1792 1982 26 4 1815 1990

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Propagation

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DistributedSensor Networks

International Journal of

6 International Journal of Manufacturing Engineering

Table 1 Variables chain sampling plans indexed by AQL and LQL for 120572 = 5 and 120573 = 10

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0001

00020 119 1 2883 3013 650 1 2888 301300025 68 2 2810 2965 361 1 2834 29840003 46 1 2776 2961 242 1 2774 29640004 31 2 2740 2885 147 1 2716 29210005 21 2 2610 2860 103 1 2637 28970006 19 4 2597 2807 84 2 2599 28290007 15 3 2534 2789 72 3 2532 27920008 13 3 2495 2765 62 3 2518 27680009 12 3 2521 2736 51 2 2470 27650010 12 6 2517 2697 45 2 2425 27500012 9 4 2331 2686 40 3 2414 26940015 7 3 2237 2657 34 4 2387 26370020 7 6 2414 2539 27 5 2335 25650025 6 6 2370 2485 20 3 2197 25520030 5 6 2264 2454 19 5 2231 2476

00025

0004 230 2 2658 2738 1131 2 2676 27360005 105 2 2598 2703 474 2 2585 27050006 64 2 2530 2675 282 1 2533 269800075 39 2 2441 2641 171 1 2464 26690010 26 3 2379 2579 106 1 2430 26100012 19 2 2335 2555 76 1 2321 26010015 16 4 2296 2491 55 1 2237 25720020 12 4 2260 2425 38 1 2139 25290025 10 6 2206 2366 32 2 2163 24130030 8 5 2114 2334 29 5 2107 23420035 7 6 2307 2297 25 5 2092 23020040 6 6 1884 2274 21 5 1952 2282

0005

00075 259 1 2434 2524 1113 2 2442 25120010 86 1 2326 2486 347 1 2348 24830012 53 1 2277 2457 211 1 2299 24540015 33 1 2211 2421 128 1 2233 24180020 24 4 2161 2316 75 2 2100 23450025 16 3 2046 2281 55 2 2083 22930030 14 6 1989 2224 43 3 1974 22490035 12 6 2002 2182 36 3 1979 22090040 10 4 1976 2156 31 3 1964 21740050 8 5 1893 2098 25 4 1925 2105

00075

0010 505 2 2339 2384 1817 1 2329 23940012 173 1 2259 2369 690 2 2278 23530015 78 1 2185 2335 286 1 2182 23370020 43 2 2146 2251 133 1 2083 22930025 29 5 2008 2198 83 1 2003 22580030 19 2 1968 2178 61 1 1984 22140035 17 5 1888 2123 47 1 1925 21900040 13 3 1837 2107 41 3 1851 21110050 10 3 1792 2052 28 2 1756 20810060 9 5 1792 1982 26 4 1815 1990

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Manufacturing Engineering 7

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0010

0015 248 1 2212 2257 786 1 2187 22670020 74 1 2090 2220 247 1 2071 22260025 42 1 2027 2177 135 1 2011 21860030 27 1 1906 2161 93 2 1954 21240035 21 2 1846 2096 66 1 1900 21250040 18 3 1804 2054 53 2 1846 20660045 15 3 1762 2027 46 3 1811 20260050 13 3 1731 2001 39 2 1824 20040060 11 5 1662 2942 30 3 1730 19550070 9 5 1573 1903 26 4 1728 18980080 7 3 1486 1891 21 5 1579 1869

002

0030 187 2 1896 1976 528 1 1886 19910035 96 2 1835 1945 272 2 1831 19460040 59 1 1779 1939 169 2 1770 19200045 45 2 1749 1889 117 1 1734 19190050 34 2 1693 1868 88 1 1683 19030060 22 2 1563 1833 63 2 1670 18200070 18 3 1544 1779 44 2 1577 17920080 14 3 1434 1749 36 3 1525 17450090 13 4 1502 1697 27 2 1423 1737010 10 2 1436 1691 22 1 1386 1766011 10 4 1457 1632 23 4 1476 1636012 7 1 1235 1725 17 1 1379 1699013 8 5 1360 1580 17 4 1356 1596015 6 4 1110 1560 14 5 1279 1544

0030

0040 341 2 1771 1821 1027 5 1771 18160045 157 1 1711 1811 416 2 1704 17990050 96 1 1650 1795 253 1 1679 17890060 52 2 1566 1736 126 1 1585 17600070 33 1 1508 1728 81 1 1539 17240080 26 3 1427 1657 62 2 1514 16590090 23 5 1449 1609 44 2 1401 16410100 17 3 1378 1593 36 2 1384 16090110 13 2 1268 1588 29 2 1305 15900120 11 1 1270 1605 26 3 1274 15490130 11 4 1204 1514 23 3 1289 15190150 10 6 1273 1443 18 3 1230 14750200 6 5 0950 1365 12 5 1134 1354

004

0060 145 2 1568 1663 339 2 1564 16640070 73 1 1513 1643 166 1 1507 16470080 50 2 1478 1588 102 1 1425 16250090 34 2 1394 1564 71 1 1374 16000100 27 3 1324 1529 53 1 1315 15800110 24 5 1309 1489 42 1 1294 15540120 18 3 1233 1478 34 1 1234 15390130 14 2 1130 1475 31 2 1271 14610140 14 4 1158 1423 27 3 1193 14330150 11 2 1108 1428 23 3 1130 14150170 10 4 1070 1360 17 2 1061 13960200 8 5 0990 1295 15 4 1120 1290

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Manufacturing Engineering

Table 1 Continued

1199011

1199012

Known sigma Unknown sigma119899120590

119894120590

1198961015840

120590119896120590

119899119878

119894119878

1198961015840

119878119896119878

0050

0070 188 1 1487 1582 427 2 1485 15700080 93 1 1408 1558 203 1 1422 15570090 60 2 1359 1509 127 2 1365 15100100 46 3 1331 1471 86 1 1331 15060110 33 2 1288 1453 63 1 1261 14910120 26 1 1270 1450 49 1 1213 14730130 20 1 1157 1452 45 3 1235 13950140 19 3 1160 1375 34 2 1148 13930150 17 4 1108 1348 30 2 1165 13650160 13 2 1011 1356 25 1 1134 13840170 12 2 1045 1330 24 3 1112 13120200 11 6 1059 1229 17 3 1023 1258

Table 2 Variables chain sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables chain sampling plan (known sigma)119899120590

119894120590

1198961015840

120590119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 79 1 282801 283801 099007 000991 7900025 001 91 1 256203 257203 099024 000956 910005 002 77 1 231004 232004 099012 000974 7700075 003 69 1 215205 216205 099003 000976 69001 004 64 1 203504 204504 099007 000929 64002 005 124 1 184403 185403 099018 000997 124003 006 193 1 171300 173200 099007 000977 193004 008 172 1 157299 158299 09906 000986 172005 010 156 1 145798 146798 099017 000999 156

we can accept the current lot provided previous lot must havebeen accepted with the condition that 119881 gt 119896

1015840

119878= 1925

Otherwise the current lot is rejectedFurther it is to be pointed out that the proposed variables

chain sampling plan is more efficient in terms of minimumASN than the variables SSP for low values of producerrsquosrisk (120572) and consumerrsquos risk (120573) In order to prove this weprovide two tables Table 2 gives the optimal parameters ofvariables chain sampling plan for some selected combinationsof AQL and LQL and for 120572 = 1 and 120573 = 1 and Table 3gives the parameters of variables SSP under the same set ofconditions By comparing these two tables one can easilyobserve that variable chain sampling plan involves minimumASN compared to the variables SSP

6 Advantages of the Variables ChainSampling Plan

This section describes the advantages of the variables chainsampling plan over the conventional variables single sam-pling plan Two acceptance sampling plans will be calledequivalent when they possess nearly identical OC curves Acustomary procedure for achieving such equivalency consistsof constructing the sampling plans so that their OC curvescoincide in two suitably chosen points namely (119901

1 1 minus 120572)

and (1199012 120573) Suppose that for given values of 119901

1= 05 120572 =

5 1199012= 15 and 120573 = 10 one can find the parameters of

the known sigma variables chain sampling plan from Table 1as

(i) 119899120590= 33 119894

120590= 1 1198961015840120590= 2211 and 119896

120590= 2421 ASN = 33

For the same values of the AQL and LQL we can determinethe parameters of the single and double sampling variablesplan (from Sommers [36]) as

(ii) 119899120590= 53 and 119896

120590= 235 ASN = 53

(iii) 119899120590= 39 119896

119886120590= 241 and 119896

119903120590= 231 ASN = 43

It can be observed that variables chain sampling planachieves a reduction of over 38 in sample size than thevariables single sampling plan and a reduction of 23over double sampling plan with the same AQL and LQLconditions This indicates that the variables chain samplingplan achieves the same OC curve with minimum sample sizecompared to the variables single and double sampling plansFurther Figure 1 shows the OC curves of the variables chainsampling plans with parameters 119899

120590= 18 119896

1015840

120590= 1544 and

119896120590= 1779 for different values of 119894

120590 This figure apparently

shows that the variables chain sampling plan increases theprobability of acceptance in the region of principal interestthat is for good quality levels and maintains the consumerrsquos

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Manufacturing Engineering 9

Table 3 Variables single sampling plans indexed by AQL and LQL for 120572 = 1 and 120573 = 1

1199011

1199012

Variables single sampling plan (known sigma)119899120590

119896120590

119875119886(1199011) 119875

119886(1199012) ASN

0001 0005 82 283305 099006 000992 8200025 001 94 256703 099001 000980 940005 002 80 231505 099016 000970 8000075 003 72 215806 099004 000932 72001 004 66 203907 099015 000957 66002 005 130 184907 099018 000996 130003 006 205 171806 099009 000971 205004 008 182 157805 099007 000982 182005 010 165 146305 099025 000984 165

Table 4 ASN values of the variables SSP DSP and variables chain sampling plans

1199011

1199012

Average sample numberKnown sigma Unknown sigma

SSP DSP ChSP SSP DSP ChSP0001 0003 74 594 46 381 3024 24200025 00075 62 501 39 267 2142 1710005 0015 53 430 33 196 1576 12800075 0025 39 313 29 129 1021 83001 005 19 149 13 54 415 39002 008 21 169 14 50 396 36003 009 30 240 23 66 525 44004 010 39 313 27 82 651 53005 012 39 320 26 76 609 49

0010203040506070809

1

001 002 003 004 005 006 007 008 009 01Fraction nonconforming p

i120590 = 2i120590 = 3i120590 = 4i120590 = infin

i120590 = 1

Prob

abili

ty o

f acc

epta

ncePa(p

)

Figure 1 OC curves of a variables chain sampling plan for different119894120590values

risk at poor quality levels This is also an important feature ofthe variables chain sampling plan

Further it is also to be noted that the variables chain sam-pling plan is economically superior to the double samplingplans in terms of average sample number (ASN) Obviouslya sampling plan having smallerASNwould bemore desirableThe variables double or multiple sampling plans are notpractically very useful Variables sampling standards avoid

presenting such plans due to increased complexity involvedin operating them but the variables chain sampling plan hasno such complexity Table 4 shows the ASNs for variablessingle sampling plan variables double sampling plan alongwith variables chain sampling plan for some arbitrarilyselected combinations of AQL and LQL These ASN valuesare calculated at the producerrsquos quality level for the knownsigma plans The ASN of the variables single and doublesampling plans can be found in Sommers [36]

From this table one can easily understand that thevariables chain sampling plan will have minimumASNwhencompared to the single and double sampling plans Similarreduction in the ASN can be achieved for any combinationof AQL and LQL values This implies that variables chainsampling plan will give desired protection with minimuminspection so that the cost of inspection will greatly bereduced Thus the variables chain sampling plan providesbetter protection than the variables single and double sam-pling plans

7 Average Run Length of Variables ChainSampling Plan

Schilling [37] has pointed out that average run length (ARL)is a missing and meaningful measure for characterizing andevaluating the sampling plans under Type B situations as inthe process control procedures The ARL gives an indication

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Manufacturing Engineering

Table 5 ARLs of variables chain sampling plan for different 119894120590values

119901Average run length

119894120590= 1 119894

120590= 2 119894

120590= 3 119894

120590= 4 119894

120590= infin

0 infin infin infin infin infin

0005 8876834 716975 601334 517815 60780001 86888 55466 41018 32719 38290015 8189 5025 3721 3013 1064002 2007 1266 977 825 5060025 812 542 441 392 315003 441 314 270 250 2290035 288 219 198 189 183004 214 173 162 158 1560045 173 147 141 139 139005 148 132 128 128 1280055 133 122 120 120 120006 123 115 115 114 1140065 116 111 111 111 111007 111 108 108 108 1080075 108 106 106 106 106008 106 104 104 104 1040085 104 103 103 103 103009 103 102 102 102 102010 102 101 101 101 101

of the expected number of samples until a decision is madeThe ARL can be easily calculated once the probability ofacceptance (119875

119886(119901)) of the plan is known for any process

fraction nonconforming p It is clear that the distributionof the run length L follows the geometric distribution withprobability mass function

119891119866(119871) = [119875

119886(119901)]119871minus1

[1 minus 119875119886(119901)] (18)

Its mean and variance are respectively given by

ARL = 119864 (119871) =1

(1 minus 119875119886(119901))

Var (119871) =119875119886(119901)

(1 minus 119875119886(119901))2

(19)

where Pa(p) is given in (5)Table 5 gives the values of ARL of the chain sampling

plan with 119899120590= 16 1198961015840

120590= 1501 and 119896

120590= 1841 for different

values of 119894120590This table apparently shows that when the process

fraction nonconforming is small the ARL is high and for theincreased values of fraction nonconforming the ARL is lowBy comparing the ARL values for different 119894

120590values when 119894

120590

increases the ARL values reduce even for the lower fractionnonconforming Also it is clear from the table that 95 ofthe lots will be accepted at fraction nonconforming 2 by thevariables chain sampling plan (119894

120590= 1) at an average rate of 20

inspections whereas with the single sampling plan (119894120590= infin)

at 2 nonconforming only 80 of lots will be accepted at

an average rate of 5 inspections Also 90 of the lots will berejected at the fraction nonconforming 7 by the variableschain sampling plan at an average rate of 111 inspections andat the fraction nonconforming 93of the lots will be rejectedby the single sampling plan at the rate of 108 inspections

8 Conclusions

The purpose of this paper is to develop conditional samplingprocedures for the inspection of normally distributed qualitycharacteristics Variables sampling plans generally requirea smaller sample size than do attributes plans If the OCcurve of the variables plan is unsatisfactory then its shapecan be improved by chaining the past lot results The pro-posed variables chain sampling plan is one of the variablesconditional sampling plans which also ensure the protectionagainst the consumer point of view This plan is also simpleto apply rather than double and multiple sampling variablesplans Also this plan provides better protection than theconventional single and double sampling variables plans withminimum sample size Such a variables chain sampling planwill be effective and useful for compliance testing Howeverit is also to be pointed out that the variables chain samplingplan developed in this paper is based on the assumptionthat the quality characteristic of interest follows a normaldistribution Whenever the normality assumption is not trueor invalid using this variables chain sampling plan can bequite misleading Hence this study can be further extendedfor nonnormal distributions as future research

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Manufacturing Engineering 11

Nomenclature

119875 Submitted lot quality q = 1minusp1199011 Acceptable quality level (AQL)

1199012 Limiting quality level (LQL)

120572 Producerrsquos risk120573 Consumerrsquos risk119875119886(119901) The probability of lot acceptance when the

fraction nonconforming is p119899120590 Sample size for known sigma plan

119896120590 Acceptance criterion for known sigma

plan1198961015840

120590 Rejection criterion for known sigma plan

119899119878 Sample size for unknown sigma plan

119896119878 Acceptance criterion for unknown sigma

plan1198961015840

119878 Rejection criterion for unknown sigma

plan119894120590 Number of preceding lots considered for

accepting current lot for known sigmaplan

119894119878 Number of preceding lots considered for

accepting current lot for unknown sigmaplan

V The value which is to be compared withacceptance criterion for making decision

119883 The sample meanS The sample standard deviation120590 The population standard deviationΦ(sdot) The cumulative distribution function of

standard normal distributionASN The average sample numberARL The average run lengthSSP Single sampling planDSP Double sampling plan

Acknowledgment

The authors would like to thank the anonymous reviewersand the editor for their valuable comments and suggestionswhich significantly improved the presentation of this paper

References

[1] E V Collani ldquoA note on acceptance sampling for variablesrdquoMetrika vol 38 pp 19ndash36 1990

[2] W Seidel ldquoIs sampling by variables worse than sampling byattributes A decision theoretic analysis and a new mixedstrategy for inspecting individual lotsrdquo Sankhya B vol 59 pp96ndash107 1997

[3] G J Lieberman and G J Resnikoff ldquoSampling plans forinspection by variablesrdquo Journal of the American StatisticalAssociation vol 50 pp 72ndash75 1955

[4] D B Owen ldquoVariables sampling plans based on the normaldistributionrdquo Technometrics vol 9 no 3 pp 417ndash423 1967

[5] A Bender Jr ldquoSampling by variables to control the fractiondefective part IIrdquo Journal of Quality Technology vol 7 no 3pp 139ndash143 1975

[6] H C Hamaker ldquoAcceptance sampling for percent defective byvariables and by attributesrdquo Journal of Quality Technology vol11 no 3 pp 139ndash148 1979

[7] K Govindaraju and V Soundararajan ldquoSelection of single sam-pling plans for variables matching the MIL-STD 105 schemerdquoJournal of Quality Technology vol 18 pp 234ndash238 1986

[8] P C Bravo andG BWetherill ldquoThematching of sampling plansand the design of double sampling plansrdquo Journal of the RoyalStatistical Society A vol 143 pp 49ndash67 1980

[9] E G Schilling Acceptance Sampling in Quality Control MarcelDekker New York NY USA 1982

[10] DH Baillie ldquoDouble sampling plans for inspection by variableswhen the process standard deviation is unknownrdquo InternationalJournal of Quality and ReliabilityManagement vol 9 pp 59ndash701992

[11] K Govindaraju and S Balamurali ldquoChain sampling plan forvariables inspectionrdquo Journal of Applied Statistics vol 25 no1 pp 103ndash109 1998

[12] H Dodge ldquoChain sampling inspection planrdquo Industrial QualityControl vol 11 pp 10ndash13 1955

[13] V Soundararajan ldquoProcedures and tables for construction andselection of chain sampling plans (ChSP-1)mdashpart Irdquo Journal ofQuality Technology vol 10 no 2 pp 56ndash60 1978

[14] R Vaerst ldquoA procedure to construct multiple deferred statesampling planrdquoMethods of Operations Research vol 37 pp 477ndash485 1982

[15] A W Wortham and R C Baker ldquoMultiple deferred state sam-pling inspectionrdquo International Journal of Production Researchvol 14 no 6 pp 719ndash731 1976

[16] K Govindaraju and V Kuralmani ldquoA note on the operatingcharacteristic curve of the known sigma single sampling vari-ables planrdquo Communications in Statistics Theory and Methodsvol 21 pp 2339ndash2347 1992

[17] J Chen and Y Lam ldquoBayesian variable sampling plan for theWeibull distribution with type I censoringrdquo Acta MathematicaeApplicatae Sinica vol 15 no 3 pp 269ndash280 1999

[18] W G Ferrell Jr and A Chhoker ldquoDesign of economicallyoptimal acceptance sampling plans with inspection errorrdquoComputers and Operations Research vol 29 no 10 pp 1283ndash1300 2002

[19] C H Chen ldquoEconomic design of Dodge-Romig AOQL singlesampling plans by variables with the quadratic loss functionrdquoTamkang Journal of Science and Engineering vol 8 no 4 pp313ndash318 2005

[20] J Chen K Li and Y Lam ldquoBayesian single and double variablesampling plans for the Weibull distribution with censoringrdquoEuropean Journal of Operational Research vol 177 no 2 pp1062ndash1073 2007

[21] S Balamurali and J Subramani ldquoEconomic design of SkSP-3skip-lot sampling plansrdquo International Journal of Mathematicsand Statistics vol 6 no 10 pp 23ndash39 2010

[22] RVijayaraghavan andKM Sakthivel ldquoChain sampling inspec-tion plans based on Bayesian methodologyrdquo Economic QualityControl vol 26 no 2 pp 173ndash187 2011

[23] S Balamurali M Aslam C H Jun and M Ahmad ldquoBayesiandouble sampling plan under gamma-poisson distributionrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 8 pp 949ndash956 2012

[24] M S Fallahnezhad and M Aslam ldquoA new economical designof acceptance sampling models using Bayesian inferencerdquoAccreditation and Quality Assurance vol 18 no 3 pp 187ndash1952013

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 International Journal of Manufacturing Engineering

[25] A DuncanQuality Control and Industrial Statistics Richard DIrwin Homewood Ill USA 5th edition 1986

[26] S Balamurali and C Jun ldquoMultiple dependent state samplingplans for lot acceptance based on measurement datardquo EuropeanJournal of Operational Research vol 180 no 3 pp 1221ndash12302007

[27] S Balamurali H Park C H Jun K J Kim and J LeeldquoDesigning of variables repetitive group sampling plan involv-ing minimum average sample numberrdquo Communications inStatistics Simulation and Computation vol 34 no 3 pp 799ndash809 2005

[28] B Feldmann and W Krumbholz ldquoASN-minimax double sam-pling plans for variablesrdquo Statistical Papers vol 43 no 3 pp361ndash377 2002

[29] W Krumbholz and A Rohr ldquoThe operating characteristic ofdouble sampling plans by variables when the standard deviationis unknownrdquo Allgemeines Statistisches Archiv vol 90 no 2 pp233ndash251 2006

[30] W Krumbholz and A Rohr ldquoDouble ASN Minimax samplingplans by variables when the standard deviation is unknownrdquoAStA Advances in Statistical Analysis vol 93 no 3 pp 281ndash2942009

[31] W Krumbholz A Rohr and E Vangjeli ldquoMinimax versions ofthe two-stage t testrdquo Statistical Papers vol 53 no 2 pp 311ndash3212012

[32] B P M Duarte and P M Saraiva ldquoAn optimization-basedframework for designing acceptance sampling plans by vari-ables for non-conforming proportionsrdquo International Journal ofQuality and Reliability Management vol 27 no 7 pp 794ndash8142010

[33] B Duarte and P Saraiva ldquoOptimal design of acceptancesampling plans by variables for non-conforming proportionswhen the standard deviation is unknownrdquo Communications inStatistics Simulation and Computation vol 42 pp 1318ndash13422013

[34] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

[35] C Wu and W L Pearn ldquoA variables sampling plan based onCpmk for product acceptance determinationrdquoEuropean Journalof Operational Research vol 184 no 2 pp 549ndash560 2008

[36] D J Sommers ldquoTwo-point double variables sampling plansrdquoJournal of Quality Technology vol 13 pp 25ndash30 1981

[37] E G Schilling ldquoAverage run length and the OC curve ofsampling plansrdquoQuality Engineering vol 17 no 3 pp 399ndash4042005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

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