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Chapter I

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1

CHAPTER – I

Dodge (1969) points out that The “acceptance quality control system that was developed encompassed the concept

of protecting the consumer from getting unacceptably defective material, and encouraging

the procedure in the use of process quality control by varying the quality and severity of

acceptance inspections in direct relation the importance of the characteristics inspected, and

in inverse relation to the goodness of the quality level as indicated by those inspection”.

This chapter consists of nine sections:

Section 1.1 Introduction

Section 1.2 Deals with terms, notations and terminology in connection with materials

presented in the entire thesis.

Section 1.3 Review on Skip lot sampling plan of type 2 (SkSP-2)

Section 1.4 Review on Multiple Deferred State sampling plan (MDS(r,b))

Section 1.5 Review on Chain sampling plan (ChSP-1)

Section 1.6 Review on Multiple Repetitive Group sampling plan (MRGS)

Section 1.7 Review on Two Stage Conditional Repetitive Group sampling plan (TSCRGS)

Section 1.8 Review on Repetitive Deferred State Sampling Plan (RDS)

Section 1.9 Glossary and symbols


2

Section 1.1

Introduction:

Acceptance Sampling is the technique which deals with procedures in which decision

either to accept or reject lots or processes which are based on the results of the inspection of

samples. According to Dodge (1969), the major areas of acceptance sampling are,

Lot -by-Lot sampling by the method of attributes, in which each unit in a sample is

inspected on a go-no-go basis for one or more characteristics;

Lot-by-Lot sampling by the method of variables, in which each unit in a sample is

measured for a single characteristics, such as weight or strength;

Continuous sampling of a flow of units by the method of attributes; and

Special purpose plans including chain sampling, skip-lot sampling and small sample

plans etc.,

Sampling Plan Design Methodologies

Methodologies Risk Based Economically Based

Non Bayesian 1 2

Bayesian 3 4

In this dissertation, sampling plan design of category 1 (that is risk based non-

bayesian approach) is alone considered. According to Case and Keats (1982), only the

traditional category 1 design is applied by the vast majority of quality control practitioners

due to their wider availability and ease for applications. An attempt is made to study the

result if category 3 is implemented.


3

This study involves the designing of skip lot sampling plan of type SkSP-2 for given set

of conditions with various reference plans. The following are the reference plans considered

for this study. The chapters are designed in such a way that each chapter clearly explains the

construction and evaluation of performance measures for SkSP-2 with all the reference plans

mentioned

1. Multiple Deferred State sampling plan of type MDS (0,1) and

Multiple Deferred State sampling plan of type MDS (0, 2)

2. Chain Sampling Plan of type ChSP – 1

3. Multiple Repetitive Group Sampling plan (MRGS)

4. Two stage Conditional Repetitive Group Sampling plan (TCRGS)

5. Repetitive Deferred State sampling plan (RDS)

The performance measures include Acceptable Quality Level (AQL), Limiting

Quality Level (LQL), Indifference Quality Level (IQL), Average outgoing Quality Limit

(AOQL), and relative slopes (h1, h0, h2) are considered for the selection of parameters. The

operating characteristic curve and average outgoing quality curve for all the plans are clearly

studied and illustrated.


4

Section 1.2

In this Section concepts, terminology and symbols of acceptance sampling in

connection with this dissertation are explained.

American National Standards Institute / American Society for Quality Control

Standard A2 (1987) defines acceptance sampling as the methodology that deals with

procedures by which decisions on the acceptance or non-acceptance are based on the results

of the inspection of samples. According to Dodge (1969), the major areas of acceptance

sampling are,

Lot -by-Lot sampling by the method of attributes, in which each unit in a sample is

inspected on a go-no-go basis for one or more characteristics;

Lot-by-Lot sampling by the method of variables, in which each unit in a sample is

measured for a single characteristics, such as weight or strength;

Continuous sampling of a flow of units by the method of attributes; and

Special purpose plans including chain sampling, skip-lot sampling and small sample

plans etc.,

Sampling Plan, Sampling Scheme and Sampling System

ANSI / ASQC Standard A2 (1987) defines an acceptance sampling plans as “a

specific plan that states the sample size or sizes to be used and the associated acceptance and

non-acceptance criteria”. In acceptance sampling plan the operating characteristics directly

follow from the parameters specified which are uniquely determined.

ANSI/ASQC Standard A2 (1987) defines an acceptance sampling scheme as

“ a specific set of procedures which usually consists of acceptance sampling plans in which

lot sizes, sample sizes and acceptance criteria or the amount of 100 percent inspection and

sampling are related”.


5

Hill (1962) has described the difference between sampling plan and sampling scheme.

According to Hill (1962) a sampling scheme “is a whole set of sampling plans and operations

included in the standard describes the overall strategy specifying the way in which the

sampling plans are to be used”. Stephens and Larson (1967) defines a sampling system

“as an assigned grouping of two or three sampling plans and the rules for using (that is

switching between) these plans for sentencing lots or batches of articles to achieve blending

of the advantageous feature of the sampling plan”.

Cumulative and Non-Cumulative Sampling plans

Stephens (1966) defines non-cumulative sampling plan as one which uses the current

sample information from the process or current product entity for making a decision about

process or product quality. Single and Double sampling plans are examples for non-

cumulative sampling plan. Cumulative results sampling inspection is one which uses the

current and past information from the process towards making a decision about the process.

Chain sampling plan of Dodge (1955) is an example for cumulative results sampling plan.

Operating Characteristic (OC) Curve Associated with each sampling plan there is an OC curve which portrays the

performance of the sampling plan against good and poor quality. The probability that a lot

will be accepted under a given sampling plan which is denoted by Pa(p) and a plot of Pa(p)

against given value of lot or process quality p will yield the OC curve. OC curves are

generally classified under Type A and Type B. The definitions of Type A and Type B OC

curves according to ANSI / ASQC Standard A2 (1987) are as follows:

Type A OC curve

Sampling from an individual (or isolated) lot, with a curve showing probability of

lots, which will be accepted when plotted against lot proportion, defective.


6

Type B OC curve

Sampling from a process, with a curve showing proportion of lots, which will be

accepted when plotted against process proportion defective.

For special purpose plans the OC curve is “a curve showing, for a given sampling

plan, the probability of continuing to permit the process to continue without adjustment as a

function of the process quality

The conditions under which each of Poisson, binomial and hyper geometric models

can be used (Schilling (1982)) are given below:

Binomial Model

This model is exact for the case of nonconforming units under type B situations. This

can also be used for type A situations for the case of nonconforming units whenever

n/N ≤ 0.10, where n and N are respectively the sample and lot sizes.

Poisson Model

This model is exact for nonconformities under both Type A and Type B situations.

Under Type A situation, for the case of nonconforming units, poisson model can be used

whenever n/N ≤ 0.10, n is large and p is small such that np < 5. Under Type B situation, for

the case of nonconforming units, poisson model can be used whenever n is large and p is

small such that np < 5.

Hypergeometric Model

This is exact model for the case of nonconforming units under type A situations and

is used for isolated lots.

To evaluate Pa(p), hyper geometric model is exact for Type A situation (when

sampling with an attribute characteristic from a finite lot without replacement).


7

Under type B situation (when sampling from the conceptually infinite output of units that

the process would turn out under the same essential conditions) Binomial model will be exact

for the case of nonconforming units to calculate Pa(p). Binomial model is also exact in the

case of sampling from a finite lot with replacement.

Poisson model is exact in calculating Pa(p) which specifies a given number of

nonconformities per unit (nonconformities per hundred units). Variable sampling plans use

normal distribution for calculating Pa(p). Hypergeometric, binomial, Poisson and normal

distributions are the most commonly used distributions in acceptance sampling.

Average Sample Number

ANSI / ASQC A2 standard (1987) defines ASN as “the average number of sample

units per lot used for making decisions (acceptance or non-acceptance)”. A plot of ASN

against p is called the ASN curve.

Average Outgoing Quality (AOQ)

A common procedure, when sampling and testing is non-destructive, is to 100%

inspect rejected lots and replace all defectives with good units. In this case, all rejected lots

are made perfect and the only defects left are those in lots that were accepted. AOQ's refer to

the long term defect level for this combined LASP (lot acceptance sampling plan) and 100%

inspection of rejected lots process. If all lots come in with a defect level of exactly p, and the

OC curve for the chosen (n, c) LASP indicates a probability Pa of accepting such a lot, over

the long run the AOQ can easily be shown to be:

N

nNpPAOQ a )()(

Where N is the lot size. Beainy and Case (1981) have given expressions for AOQ

to different policies adopted for single and double sampling attribute plans. In this

dissertation, AOQ is approximated as p * Pa(p). The assumption underlying this expression

is that for all accepted lots the average fraction nonconforming is assumed to be p and for all


8

the rejected lots the entire units are being screened and nonconforming units are replaced.

It is further assumed that the sampling fraction n/N is very small and can be ignored. A plot

of AOQ against p is called the AOQ curve.

Average Outgoing Quality Level (AOQL)

A plot of the AOQ (Y-axis) versus the incoming lot p (X-axis) will start at 0 for

p = 0, and return to 0 for p = 1 (where every lot is 100% inspected and rectified). In between,

it will rise to a maximum. This maximum, which is the worst possible long term AOQ, is

called the AOQL.

Average Total Inspection (ATI)

When rejected lots are 100% inspected, it is easy to calculate the ATI if lots come

consistently with a defect level of p. For a LASP (n, c) with a probability Pa of accepting a lot

with defect level p, one can have

ATI = n + (1 - Pa) (N - n) where N is the lot size.

Acceptable Quality Level (AQL)

ANSI / ASQC A2 standard (1987) defines AQL as “the maximum percentage or

proportion of variant units in a lot or batch that, for the purpose of acceptance sampling, can

be considered satisfactory as a process average.”

The AQL is a percent defective that is the base line requirement for the quality of the

producer's product. The producer would like to design a sampling plan such that there is a

high probability of accepting a lot that has a defect level less than or equal to the AQL.

Inspection level

In addition to an initial decision on an AQL it is also necessary to decide on an

"inspection level". This determines the relationship between the lot size and the sample size.

The standard offers three general and four special levels.


9

Producers Risk (α) (Type I Error)

This is the probability, for a given (n, c) sampling plan, of rejecting a lot that has a

defect level equal to the AQL. The producer suffers when this occurs, because a lot with

acceptable quality was rejected. The symbol α is commonly used for the Type I error and

typical values for usual range, a level which it is generally desired to accept.

Limiting Quality Level (LQL)

ANSI / ASQC A2 standard (1987) defines LQL as “the percentage or proportion of

variant units in a batch or a lot for which, for the purpose of acceptance sampling, the

consumer wishes the probability of acceptance to be restricted to a specified low value”.

The LQL is a designated high defect level that would be unacceptable to the

consumer. The consumer would like the sampling plan to have a low probability of accepting

a lot with a defect level as high as the LQL.

Consumers Risk (β) (Type II Error)

This is the probability, for a given (n, c) sampling plan, of accepting a lot with a

defect level equal to the LTPD. The consumer suffers when this occurs, because a lot with

unacceptable quality was accepted. The symbol β is commonly used for the Type II error and

typical values for usual range, a level which it is seldom desired to accept.

Indifference Quality Level (IQL)

The percentage of variant units in a batch or lot for which, for purpose of acceptance

sampling, the probability of acceptance to be restricted to a specific value namely 0.50. The

point (IQL, 0.50) on the OC curve is also called as “Point of control”.


10

Maximum Allowable Percent Defective (MAPD)

The point on the OC curve at which the descent is steepest is called point of

inflection. The proportion nonconforming corresponding to the point of inflection of OC

curve is interpreted as the maximum allowable percent defective.

Maximum Allowable Average Outgoing Quality

The MAAOQ of a sampling plan is designated as the Average Outgoing Quality

(AOQ) at the MAPD.

AOQ = p Pa (p)

Then MAAOQ = AOQ at p = p* which can be rewritten as,

MAAOQ = p* * Pa (p*)

One of the desirable properties of an OC curve is that the decrease of Pa (p) should be

lower for smaller values of p and steeper for higher values of p, which provides better overall

discrimination. Since p corresponds to the inflection point of an OC curve, it implies that

*2

2

*2

2

*2

2

0))((

0))((

0))((

ppfordp

pPd

ppfordp

pPd

ppfordp

pPd

a

a

a

Soundararajan (1975) has proposed a procedure for designing Single sampling plan

with quality standards p* and K = pt / p*. Suresh and Ramkumar (1996) have designed the

Single sampling plan indexed with MAAOQ.


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Advantages of using MAAOQ and AOQL are as follows

If p* is considered as a quality level to identify a sampling plan with the AOQL, then

it cannot always derive a unique plan. When considering the MAPD and MAAOQ,

with R1 = MAAOQ / MAPD, a monotonic decreasing function in c is obtained, which

will provide a unique plan.

The AOQL is only a physical upper bound for the AOQ of any sampling plan, which

does not satisfy the engineer’s need of quality, Anscombe (1958) remarked that the

AOQL is only a statistician’s guarantee of outgoing quality but not a guarantee of the

consumer’s requirement.

The AOQL attains meaning using the mathematical logic of maximization; hence the

quality pm on which it is defined has no practical significance in acceptance sampling.

However, the MAAOQ is defined on p* which is a favored quality index for

engineer’s and it protects the interest of the consumer.

MAAOQ / p* is simply pa(p*) which is the ordinate of the inflection point.

The calculation of the MAAOQ is comparatively easy compared with that of the

AOQL, because the AOQL is a solution of a complicated expression.

MAAOQ = p* Pa(p*) ≤ AOQL = pm Pa (pm) is implies that higher consumer protection

is guaranteed on an MAAOQ plan.

The AOQL is the maximum AOQ overall incoming lots but, in practice, an upper

bound nearer to the AOQL is only usually attained.

Operating Characteristic Curve

The OC curve is a graph of PA, the probability of accepting the process as a function

of fraction defective p.

Specification Limit

It is the specified Maximum or Minimum acceptable value of the characteristics.


12

Lower Specification Limit (L)

It is the specified Minimum acceptable value of the characteristics.

Upper Specification Limit (U)

It is the specified Maximum acceptable value of the characteristics.

Designing sampling plans

In designing a sampling plan, one has to accomplish a number of different purposes.

According to Hamaker (1960) the most important of which are

To strike a proper balance between the consumers requirement, the producers

capabilities and the inspectors capability.

To separate bad lot from good.

Simplicity of procedures and administration.

Economy in number of observations towards sampling inspection.

To reduce the risk of wrong decisions with increasing lot size.

To use accumulated sample data as a valuable source of information.

To exert pressure on the producer or supplier when the quality of the lot

received is unreliable up to standard.

To reduce sampling when the quality is reliable and satisfactory.

Hamaker (1960) also points out that these aims are partly conflicting and all of cannot

be simultaneously realized. According to Peach (1947) the following are some of the major

types of designing the plans, based on the OC curves, classified according to types of

protection. The plan is specified by requiring the OC curve to pass through (or nearly

through) two fixed points. In some cases it may be possible to impose certain additional

conditions. The two points generally selected are (p1, 1-α) and (p2, β)


13

p1 or p1-α = the quality level that is considered to be good so that the

producer expects lots of p1 quality to be accepted most of

the time

p2 or pβ = the quality level that is considered to be poor so that the

consumer expects lots of p2 quality to be rejected most of the

time

α = the producers risk of rejecting p1 quality; and

β = the consumers risk of accepting p2 quality.

Tables of Cameron (1952) are an example for this type of designing. Schilling and

Sommers (1981) term p1, as the Producers Quality Level (PQL) and p2 as the Consumers

Quality Level (CQL). Earlier literature calls p1 as the Acceptable Quality Level (AQL) and

p2 as the Limiting Quality Level (LQL or simply LQ) or Rejectable Quality Level (RQL) or

Lot Tolerance Percent Defective (LTPD). Peach and Littauer (1946) have define the ratio

p2/p1, associated with given values of α and β as the “Operating Ratio” (OR). Traditionally

the values of α and β are assumed to take 0.05 and 0.10 respectively.

The plan is specified by fixing one point only through which the OC curve is

required to pass and setting up one or more conditions, not explicitly in terms

of the OC curve. Dodge and Romig (1959) LTPD table is an example for this

type of designing.

The plan is specified by imposing upon the OC curve two or more

independent conditions none of which explicitly involves the OC curves.

Dodge and Romig (1959) AOQL table is an example for this type of designing.


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Designing Procedure adopted in this study

The various sections adopted in this study are

Designing of sampling plans using quality levels

Designing of sampling plans using relative slopes

Designing of sampling plans using Average Outgoing Quality

Review of literature

It is the usual practice to select any sampling plan such that its OC curve passes

through two points namely, producer and consumer quality levels with specified risks

α = 0.05 and β = 0.10. Using the operating ratio (OR = p2 / p1). Cameron (1952) has designed

Single Sampling Plans for attributes and constructed tables for ready-made selection of plan

parameters using unity value approach. Hamaker (1950) has studied the selection of Single

Sampling Plan assuming that the quality characteristics follow Poisson model such that the

OC curve at that quality level. Hamaker (1959) has also studied the selection through

adjustments of parameters to finite lot size. Mandelson (1962) has explained the desirability

for developing a system plans indexed through Maximum Allowable Percent Defective

(MAPD) and shown that p* = c/n for Single Sampling Plan. Mayer (1967) has explained that

the quality standard p can be considered as a quality level along with certain other conditions

to specify an OC curve. Soundarajan (1975) has constructed tables for the selection of Single

Sampling Plan indexed through MAPD and K = pt / p*. Suresh and Srivenkataramana (1996)

have designed procedure for the selection of Single Sampling Plan using producer and

consumer quality levels. Suresh (1993) has also studied the quality levels along with their

relative slopes. Suresh and Deepa (1999) have studied the selection of Special Type Double

Sampling Plan indexed with the point of control p0 and the measure for sharpness of

inspection, K0.

According to Golub (1953) there are many reasons in practice for economic,

administrative or practical reasons n is fixed and small. Golub (1953) has developed a

method of designing a Single Sampling Plan when the sample size is fixed and has given an


15

expression for c such that the sum of two risks namely producer’s risk (α) and consumer risk

(β) is minimum. Minimizing (α + β) is the same as maximizing (1- α) + (1- β).

Soundararajan (1978b) has given an expression for i, one of the parameters of Chain

Sampling Plan (ChSP-1), which minimizes (α + β) for specified sample size, AQL and LTPD

under both binomial and Poisson models. Vijayathilakan (1981) has considered the problem

of assigning different weights for the producer and consumer risks in order to protect any one

of them from sharing a large proportion of total risks such that w1 + w2 = 1. Then the required

plan may be obtained by minimizing w1α + w2β which is equivalent to α + wβ where w = w2

/ w1. If w is greater than 1 the obtained plan will be more favorable to the producer. Raju

(1984) has given a set of tables for finding i values indexed through AQL and LQL for fixed

sample size minimizing α + β with and without weight for ChSP -1 plan. Raju (1984) follows

Golub’s approach for designing Multiple Deferred State Sampling Plan of type

MDS -1 (c1, c2). Soundarajan and Arumainayagam (1989) have considered the selection of

tables for Quick switching system following Golub’s approach. Sathaiya (1985) has applied

Golub’s approach for designing Double Sampling Plans when some of the parameters are

fixed and also sum of weighted risk. Soundararajan and Vijayaraghavan (1989) has applied

Golub’s approach for designing Multiple Deferred State Sampling Plan MDS-1 (0,2) when

sample is fixed for given values of AQL and LQL such that α and β are small.

Average Outgoing Quality Limit is the worst average quality that a consumer will

receive in the long run when defectives are replaced with good ones. According to Suresh

and Ramkumar (1996) the Maximum Allowable Average Outgoing Quality is the outgoing

quality defined with p which is a favored quality index for engineers and it protects the

interests of the consumer. Considering the simplicity, practicability and consumer protection

offered, the MAAOQ has major practical advantages in acceptance sampling compared with

AOQL, which can be considered as a measure for selection of plan parameters. Dodge and

Romig (1959) have proposed procedure for the selection of Single Sampling Plan indexed

through AOQL by minimizing the Average Total Inspection. Soundararajan (1981) has

suggested procedure for the selection of Single Sampling Plan in terms of AQL and AOQL.


16

Gloub’s Approach Soundararajam (1981) has extended the Golub’s approach to single sampling plan

under the conditions of Poisson model to the OC curve. Subramani (1991) has worked on

sampling plans involving minimum sum of producer and consumer risks. Norman Bush et.al

(1953) have suggested certain methods, which describe the direction of the OC curve. Which

Goulb’s ( 1953) approach involves minimization of the sum of risks, which leads to ideal OC

condition, the method of Norman Bush et.al (1953) involves the comparison of some portion

of the ideal OC curve.

The chord AB coincides with that of B’B and the operating characteristic curve

approaches to the ieal OC curve. That is the ideal OC curve passes through (p1, 1- α ) and

( p2 , β ). Singaravelu (1993) has designed plans involving minimum angle for Single

Sampling Plan and Double Sampling Plan.

In certain circumstances, a lot is inspected again using the same or different attribute

sampling plans. For example, the first sampling inspection may be done by the quality

engineers of a company. If a lot is rejected, the production department may again inspect the

lot using a sampling plan before screening the batch. Because of the random nature of the

sampling and sometimes intentional and unintentional differences in inspection, the

difference in the result does cause problems in relationship. The inspection results differ not

only due to sampling variability but also due to several reasons, generally jnown as (non-

sampling) inspection errors causes. Govindaraju and Ganesalingnam (1997) have studied

sampling inspection scheme for resubmitted lots. Govindaraju, Lai and Xie ( 2000) have

studied about the contradicting results under Single Sampling Plan in case of binomial

model.

Unity Value Approach

This approach can be used only under conditions for application of Poisson Model for

OC curve. As noted in Duncan (1986) and in Schilling (1982) the assumption of Poisson

model permits one to consider the OC function of all attribute sampling plans simply as a


17

function of the product np ( in place of the sample size n and submitted quality p separately)

for given acceptance and rejection numbers. That is, the OC function remains the same for

various combinations of n and p provided their product is the same forgiven acceptance and

rejection numbers. As a result one is able to develop compact tables for the selection of

sampling plans as only one parameter np is considered in place of two parameters viz. n and

p. The primary advantage of the unity value approach is that plans can easily obtained one

necessary tables have been constructed. Tables are constructed by unity value approach

which is only widely available in text books. (See for examples Schilling (1982))

Search Procedure

In this approach the parameters of a sampling plan are chosen by trial and error, by

varying the parameters in a uniform fashion depending upon the properties of OC function.

An example for this approach is the one followed by Guenther (1969, 1970) while

determining the parameters of single and double sampling plans under the conditions for

application of binomial, poisson and hypergeometric models for OC curve. The advantage

of search procedure is that the sample size need not be rounded. The disadvantage of this

procedure is that obtaining of plans need elaborate computing facilities.

Designing plans for given IQL

Hamaker (1950a) considered two important features of the OC curves, namely the

place where the curve shows its steeper descent and the degree of its steepness, as the basis

for two indices namely, the Indifference Quality Level p0 and the relative slopes of the OC

curve at (p0, 0.5) denoted as h0, which may be used to design any sampling plan. Hamaker

(1950b) has given simple empirical relations existing between the sample size and the

acceptance number and between the parameters p0 and h0, under the condition of application

of poisson, binomial and hyper geometric distributions for single sampling attribute plan. A

number of papers have been published on selection of acceptance sampling plans for given

p0 and h0. For example, Soundarajan and Muthuraj (1985) have given procedures and tables

for designing single sampling attribute plans towards given p0 and h0.


18

)( 2025 2 1 1 2

p p p p

Designing plans for given MAPD

The proportion nonconforming corresponding to the inflection point of the OC curve,

denoted by p* and interpreted ass the maximum allowable percent defective (MAPD) by

Mayer (1967) is also used as the quality standard along with some other conditions for the

selection of the sampling plans. The relative slope of the OC curve at this point, denoted as

h*, is also used to fix the discrimination of the OC curve of any sampling plan. The

desirability of developing a set of sampling plans indexed by p* has been explained by

Mandelson (1962) and Soundarajan (1971). While choosing a plan for given p*, one is also

to specify the (inverse) measure of discrimination K = pT/p*, where pT is the point at which

the tangent line at the inflection point of the OC curve cuts the p-axis or h*, the relative slope

of the OC curve at p*. Many papers also have been published towards the selection of plans

for given p* and h*. For example Soundarajan and Muthuraj (1985) have given procedures

and tables for designing single sampling attribute plans for given p* and h*.

Designing sampling plan using Minimum angle method

Norman Bush et.al. (1954) has used different techniques to describe direction of the

OC curve to be evaluated with the corresponding portion of the ideal OC curve. They have

taken chord length, that is the line joining the AQL and Pa of 0.5 as

CL = 2212025 pp

The smaller the chord length, the more nearly the curve approaches ideal one. But

in this method of approximation to chord length is poor, so another method is suggested

which considers the cosine of chord length

Cosine =


19

Here the small value of cosine Ө implies the curve approaches to the ideal OC curve.

Further they have considered two point on the OC curve as (AQL, 1- α) and (IQL, 0.05) for

minimizing the consumer’s risk. But Peach and Littauer (1946), have taken two points on

the OC curve as (p1, 1- α) and (p2, β) for ideal condition to minimize the consumer’s risk.

Here another approach of minimization of angle the lines joining the points (AQL, β), (AQL,

1- α), (LQL, β) is giving due to Singaravelu (1993). Applying this method one can get a

better plan which has an OC curve approaching to the ideal OC curve.

The formula for tan ө is given as

tan ө = Opposite side / Adjacent side

= (p2 – p1) / ( 1- α - β) (p2 – p1)

= (p2 – p1) / [pα(p1) - p α (p2)]

Hence for given two points on the OC curve the values of minimum tan ө are calculated.

Designing Procedure Involved

One of the most important methods of specifying the requirements for the selection of

sampling plans in practice is choosing two quality levels namely (AQL(p1) and LQL (p2)

with p1< p2, and two corresponding risks namely α and β of making wrong decisions. The

quality level p1 represents a satisfactory quality, also called the producer’s risk point,

whereas p2 represents unsatisfactory quality also called the consumer’s risk point, whereas p2

represents unsatisfactory quality also called the consumer’s risk point. The probability of

rejecting a lot at p1 is called producer’s risk(α), and consumer’s risk(β).Mathematically, these

can be written as pa(p1) < 1- α and pa(p2) > β

Designing of Minimum angle Method:

The practical performance of any sampling plan is generally revealed through its

operating characteristic curve. When producer and consumer are negotiating for quality

limits and designing sampling plans, it is important to minimize the consumer risk. In order

to reduce the consumer’s risk, the ideal OC curve could be made to pass as closely through


20

(AQL,1-α) and (LQL, β) or (p1, 1-α) and (p2 , β). Norman Bush (1953) have considered two

points on the OC curve as (AQL, 1-α) and (IQL, 0.50) for minimizing the consumer risk.

But Peach and Littauer (1946) have taken two points on the OC curve as (p1, 1-α) and (p2, β)

for ideal condition to minimize the consumers risks. In this paper another approach with

minimization of angle between the lines joining the points (AQL,1-α) and (LQL, β ) has been

done. Applying this method one can get a better plan which has an OC curve approaching to

the ideal OC curve. The procedures and the necessary tables are provided for the ready-made

selection of the sampling scheme through minimum angle criteria with suitable illustration.

The formula for tanθ is given as

sideadjacentsideopposite

tan

Tangent of angle made by the two lines (AQL,1-α) and (LQL, β ) is

)()()(tan

12

12

pPpPpp

aa

Where p1=AQL and p2=LQL. This may also be expressed as

)1()(tan 12

pnnpn

The smaller value of this tanθ closer to the angle θ approaching zero, and the chord AB

approaching AC, the ideal condition through ((AQL, 1-α).

Now θ=tan-1{(n tanθ / n)}

Using this formula the minimum angle θ is obtained, for the given np1 and np2 values.


21

Abbreviations used in this study

Single Sampling Plan - SSP

Chain Sampling Plan - ChSP-1

Skip-lot Sampling plan of type 2 - SkSP-2

Multiple Deferred State Sampling Plan - MDS(r,b)

Multiple Deferred State Sampling Plan with r=0 and b=1 - MDS(0,1)

Multiple Deferred State Sampling Plan with r=0 and b=2 - MDS(0,2)

Conditional Repetitive Group Sampling Plan - CRGS

Multiple Repetitive Group Sampling plan - MRGS

Two Stage Conditional Repetitive Group sampling plan - TSCRGS

Repetitive Deferred Sampling Plan - RDS

Skip-lot Sampling plan of type 2 with

Multiple Deferred State Sampling Plan - SkSPMDS-2(r,b)


Chain Sampling Plan - SkSPCHS-2


Multiple Repetitive Group Sampling plan - SkSPMRGS-2


Two Stage Conditional Repetitive Group sampling plan - SkSPTSCRGS-2


Repetitive Deferred Sampling Plan - SkSPRDS-2


22

Section 1.3

This section deals with the review on Skip lot sampling plan of type 2 (SkSP-2)

Skip-Lot sampling Plan

The continuous sampling plans are applied only on individual units that are produced

in a sequence from a continuing source of supply. The principles of continuous sampling are

even applicable to individual lots received in a steady stream from a supplier. During the

sampling phase, few lots are skippsed from being inspected. The skipped lots automatically

accepted. Such a procedure is passed under analogous skip-lot plan, which was proposed by

Dodge (1955).

The SkSP-1 originally designed by Dodge (1955), is based on the same principles as

followed in Continuous Sampling Plan of type 1 (CSP-1). The CSP- 1 type deals with series

of lots. It is proposed that quality is good or rather accepted, and then only a fraction of the

submitted lot require to be inspected. On the other hand, when a defective unit is found

during sampling phase, then it becomes necessary to revert to 100% inspection once again.

Dodge (1955) has extended the concept of CSP-1 to individual lots, under the

conditions where a single determination on analysis is made for each of the specified quality

characteristic subject to the inspection. Single determination on analysis means the

ascertainment of acceptability or non- acceptability of lots. The next procedure is to examine

the case where each lot to be inspected is sampled according to some given lot inspection

plan.


23

Conditions:

The conditions assumed for the application of the plan are

The product comprises of a series of successive lots from same source, normally

expected to be essentially of the same quality.

The specified requirements are expressed as maximum and / or minimum limits for

one or more characteristics.

For a given characteristics, the normal acceptance procedure for each lot is to obtain a

suitable sample of the material and make a lab analysis or teat of it. If the test results

meet the applicable specification requirements then the lot is found to be conforming,

otherwise non-conforming.

Operating Procedure for SkSP-1 plan:

The plan is to be applied separately for each of the characteristics under certain

conditions.

For a given characteristic, one of the two procedures A1 or A2. Is chosen depending

on what is normally to be done when a lot is found to be non-conforming for those

characteristics.

Procedure A1 – applicable when each non-conforming lot is to be either corrected

or replaced by a non-conforming one:

At the outset, test every lot consecutively as purchased and continue such

test until i lots in succession are found to be conforming.

When i lots in succession are found to be conforming discontinue testing

the lot and test only fraction of the lots which are chosen randomly.

If a tested lot is found to be non-conforming revert to testing every

succeeding lot until again i lots in succession are found conforming.


24

Accept each non-conforming lot after it has been made conforming by

reprocessing and correct or replace the non-conforming lot by a

conforming lot.

Procedure A2 - applicable when each non-conforming lot is purchased and continue

such test until i lots in succession are found to be conforming.

At the outset, test every lot consecutively as purchased and continue such test

until i lots in succession are found to be conforming.

When i lots in succession are found to be conforming discontinue testing the

lot and test only a fraction and of the lots which are chosen randomly.

If a tested lot is found to be non-conforming, revert to testing every

succeeding lot until again (i+1) lots in succession are found conforming.

Reject and remove each non-conforming lot.

Skip-lot Sampling Plan of type - SkSP-2

Perry (1970) has developed a system of sampling inspection plan known as SkSP-2.

This plan involves inspection of only some fraction of the submitted lots when quality of the

submitted product is good as demonstrated by the quality of the product. These plans are

applicable to products produced or furnished in successive lots or batches. The SkSP-1 was

primarily intended to be utilized in circumstances leading to a simple and absolute to-no-go

decision on each lot whereas the continuous sampling approach to skipping lots can be

utilized when a standard sampling plan is applied to each lot. A lot, after inspection is either

accepted or reject ed along with an associated producer and consumer risks. These risks have

been factored into the skip-lot procedure by Dodge and Perry (1971) in their development of

SkSP-2. which was intended to a series of lots of discrete item with which a sampling plan

can be considered as standard reference sampling plan.


25

Operating Procedure

A SkSP-2 plan is one that uses a given lot inspection plan by the method of attributes

(single, double sampling, multiple sampling, chain sampling, etc.) called the ‘reference plan’

together with a procedure that calls for normally inspecting every lot , but for inspecting only

a fraction of the lots when the quality is good. The plan includes specific rules based on the

record of lot acceptance and rejections, for switching back and forth between “normal

inspections” (inspecting every lot) and ‘skipping inspection’ (inspecting only a fraction of the

lots).

The OC function associated with SkSP-2 plan is of type B, based on probabilities of

sampling from an infinite universe or process. The conditions associated with sampling from

an infinite universe are based on the notion of a process producing a theoretically continuous

infinite product flow. The OC function for SkSP-2 plan with Multiple deferred state sampling

plan as reference plan is obtained by two approaches namely (1). Power Series Approach

and (2). Markov Chain Approach.

The operating procedure is given below.

1. Start with normal inspection, using the reference plan.

2. When ‘i’ consecutive lots are accepted on normal inspection, switch to skipping

inspection of inspecting a fraction ‘f’ of the lots.

3. When a lot is rejected on skipping inspection, switch to normal inspection.

4. Screen each rejected lot by replacing the non-conforming units by conforming one.

The SkSP-2 plan is specified by the reference sampling plan applied to each lot, i the

clearing interval, f the sampling frequency. Here 0 < f < 1 and i is a positive integer.

Let P denote the operating characteristics curve, whose expression can be approach is due to

Dodge and Perry(1971) whereas the Markov chain approach is due to Perry (1973).


26

The probability of acceptance for SKSP-2 plan is as follows

i

i

a PffPfPfdccifP

)1()1(),,,,( 21

It is noted that Pa ( f ,i) is a function of ‘ i ’ clearing interval; ‘ f ’ sampling fraction.

The mathematical relationship between f, i and P are:

The SKSP-2 having a given reference plan and f = 1 becomes identically the reference

plan itself.

For P and i being fixed

iia PfP

PP

1/11

Where P is the OC function of reference plan, i is the clearing interval

and f is the sampling fraction.

For f and p being fixed

fPf

fPfP ia

1

1 is a decreasing function of i.

For f and i being fixed

fPf

PfP ia

1

11

According to Perry (1973)

For f1 < f2 , i fixed and given reference plan

Pa (f1,i) < Pa (f2,i)

For integers i <j , f fixed and given reference plan

Pa (f,j) < Pa (f,i)

Pa (f,j) > P


27

Suresh and Ramachandran (1990) have suggested a selection procedure for

SkSP-2 with DSP (0,1) as a reference plan. Perry (1970) has developed an SkSP procedure

with two skipping levels f1 and f2. The expression for Pa(f1,f2,i) is derived using Markov

chain Approach.

Parker and Kessler (1981) have modified the existing SkSP-2 plan under which

atleast one unit is always sampled from a lot. The expression for the probability of

acceptance using this plan are derived and compared with the standard skip-lot plans.

Review of Literature

Perry (1970) has tabulated P0.95 / P0.10 for a variety of SkSP-2 plans with various

combinations of i and f values of 2/3 , ½ , 1/3, ¼ and 1/5, and values of i considered for the

designing of plan are 2,4,6,8,10. Vijayaraghavan (1990) has developed the tables for

designing SkSP-2 for specified values of AQL, AOQL. Suresh (1993) has constructed

tables for designing SkSP-2 plan based on the relative slopes at the points (p1, 1-α) and

(p2 , β) considering the filter and incentive effects for selection of plans.


28

The skip-lot procedure is represented diagrammatically as shown below:

Inspect i successive lots

Lots rejected

Lot accepted

Randomly inspect a fraction f of the lots

Lot accepted

Lot rejected


29

Section 1.4

This section deal with the review on Multiple Deferred State sampling plan

MDS (r, b)

Multiple Deferred State sampling plan (MDS (r, b))

The concept of multiple dependent (or deferred) state sampling (MDS) was

introduced by Wortham and Baker (1976). The MDS sampling plan belongs to the group of

conditional sampling procedures. In these procedures, acceptance or rejection of a lot is

based not only on the sample from that lot, but also on sample results from past lots (in the

case of dependent state sampling) or from future lots (in the case of deferred state sampling).

The MDS plan is applicable in the case of Type B situations (i.e., sampling from a

continuous process) where lots are submitted for inspection serially in the order of

production. The operating procedure and characteristics of the attributes MDS sampling plan

can be found in Wortham and Baker (1976) and this plan was studied further by Vaerst

(1982), Soundararajan and Vijayaraghavan (1990).

For situations involving costly or destructive testing by attributes, it is the usual

practice to use Single Sampling Plan with smaller sample size and an acceptance number

zero to have the decision either to accept or reject the lot. The smaller sample size is dictated

with the cost of test and zero acceptance number arises out of desire to maintain a steeper OC

curve. A single sampling plan has the following undesirable characteristics.

A single defect in the sample calls for the rejection of the lot.

The OC curves of all such sampling plan have a uniquely poorer shape,

in that the probability of acceptance starts to drop rapidly for the

smallest values of percent defective.

Hence Wortham and Baker have (1976) developed the multiple deferred and multiple

dependent state sampling plans. These plans are designated as MDS (r, b). The operation of

this is restricted to those situations in which


30

Production is steady so that results of past, current and future lots are broadly

indicative of a continuing processes.

Lots are submitted substantially in the order of their production.

A fixed sample size, n from each lot is assumed and

Inspection is carried for attributes with quality defined as fraction non-

conforming.

Operating procedure for MDS (r, b)

The operating procedure for this plan is stated as:

From each lot. Select a random sample of n units and observe the number of non-

conforming units d.

If d ≤ r, accept the lot.

If d ≥ r + b reject the lot.

If r+1 ≤ d ≤ r + b, accept the lot if the forthcoming m lots in succession are all

accepted (previous m lots in case of multiple dependent state sampling plan and

forthcoming m lots in case of multiple deferred state sampling plan)

The OC function of MDS (r, b, m) plan is provided as

marabraraa pPpPpPpPpP )]()][()([)()( ,,,

Govindaraju (1984) has constructed tables for the selection of MDS (0,1) plan using

operating ratios, Soundararajan and Vijayaraghavan (1990) have given tables for the

selection of MDS (r, b, m) plan. matching of MDS plan with single and double sampling

plan are also carried. A search procedure was also carried out for the selection of the plan.

Subramani (1991) has studied MDS plan involving minimum sum of risks. Rambert Vaerst

(1980) has developed MDS-1 (c1, c2) sampling plan in which the acceptance or rejection of a

lot is based not only on the results from the current lot but also on the sample results of the

past or future lots.


31

Operating Procedure for MDS-1 (c1, c2)

For each lot, select a sample of n units and test each unit for conformance to the

specified requirements.

Accept the lot if d (the observed number of observation of defectives) is less than or

equal to c1; reject the lot if d is greater than c2.

If c1 < d < c2 , accept the lot preceding or succeeding i lots are accepted with

d < c1 , otherwise reject the lot.

Operating Procedure for MDS (0, 1) Plan

A multiple deferred state sampling plan of Wortham and Baker (1979) with r = 0 and

b = 1 is operated as follows:

From each lot, take a random sample of n units and observe the non-conforming

units, d.

If d = 0, accept the lot; if d > 1, reject the lot. If d = 1, accept the lot, provided the

forthcoming m lots in succession are all accepted (previous m lots in case of multiple

dependent state sampling).

The probability of acceptance based on poisson model is

npmnpnp

a eenpepP 1)(


32

Operating Procedure for MDS (0, 1) Plan

A multiple deferred state sampling plan of Wortham and Baker (1979) with

r = 0 and b = 2 is operated as follows:

From each lot, take a random sample of n units and observe the non-

conforming units, d.

If d = 0, accept the lot; if d > 2, reject the lot.

If d = 2, accept the lot, provided the forthcoming m lots in succession are all

accepted (previous m lots in case of multiple dependent state sampling).

The probability of acceptance based on Poisson model is

2/))()( )1(21

mnpnpmnpnpa enpeenpepP

The Operating characteristic function for MDS (r,b ) is as follows

npmnpnpa

marabraraa

eenpepPbewillceaccepofprobabiltyThe

bandrparmeterstheWhen

pPpPpPpPpPP

1

,,,

)(tan

10

)]()][()([)()(


33

Operating Procedure for MDS (0, 2)

A multiple deferred state sampling plan with r = 0 and b = 2 is operated as follows:

From each lot, take a random sample of n units and observe the non-conforming

units, d.

If d = 0, accept the lot; if d > 2, reject the lot. If d = 2, accept the lot, provided the

forthcoming m lots in succession are all accepted (previous m lots in case of multiple

dependent state sampling).

The probability of acceptance based on Poisson model is

It is noted that Pa ( f ,i) is a function of ‘ i ’ clearing interval; ‘ f ’ sampling fraction.

The Operating characteristic function for MDS (r,b ) is as follows

)1(2

,, ,

2) (

tan

20

)]( )][ ( )( [ )( )(

mnpnpmnp np a

m a r a b r a r a a

e npe e np e pP

bewill ceaccepof probabilty The

band r parmetersthe When

pP pP p P p P p P P

)1 ( 2

2 ) ( m npnpmnpnp

a e np e e np e p P


34

Flow Chart for the operating procedure of Multiple deferred state sampling plan

MDS – (r, b)

Sample n from current lot and observe d

d ≤ r

r < d ≤ b

One lot rejected in the preceding (succeeding) m

lots

No lot rejected in the preceding (succeeding) m

lots

Accept

Reject

d > b

Start


35

Section 1.5 This section deals with the review on Chain sampling plan (ChSP-1)

Chain sampling Plan (ChSP-1) In this section, a brief review on chain sampling plans of Dodge (1955), Dodge and

Stephens (1966) is made.

Sampling inspection in which the criteria for acceptance and non-acceptance of the

lot depends on part of the results about the inspection of immediately preceding lots is

adopted in Chain Sampling Plan. Chain Sampling Plan (ChSP-1) was proposed by

Dodge(1955) making use of cumulative results of several samples which help to overcome

the short comings of the Single Sampling Plan.

When a manufacturing concern produces materials which involve destructive or

costly tests for attributes, it is the usual practice to use a small sample plan so that the cost

of inspection is minimum. Often a single sampling plan with zero acceptance number is

taken for economic consideration but this has the following disadvantages:

A single occasional non-conforming unit may call for rejection of the lot.

The power of discrimination of the plan between good and bad lots, as revealed by

the OC curve, is uniquely poor. That is probability of

Acceptance drops rapidly even for small values of percent non-conforming p.

In contrast, a single sampling plan having c=1 or more, as well as double and multiple

sampling plans lack this undesirable property, but requires larger sample size. For such

situations, Dodge (1955) developed chain sampling plan of type ChSP-1 which is an answer

to the question whether anything can be done to improve the discriminating power of the

acceptance number c for a single sampling plan without appreciably increasing the sample

size.

soundararajan (1978a) has constructed tables for the selection of ChSP-1 plans

under the conditions for application of poisson model for given p1, p2, α and β.


36

Soundararajan and Govindaraju (1982) have presented a new table using which, one can fix

the parameters for the ChSP-1 plan with given acceptable quality level and limiting quality

level involving minimum sum of risks. Govindaraju (1990) has given procedures and tables

for the selection of chain sampling plan of type ChSP-1 for given p1, p2, α and β.

Conditions for application of ChSP-1

1. The cost of destructiveness of testing is such that a relatively small sample sizes are

necessary, although other factors makes larger sample desirable.

2. The product to be inspected comprises a series of successive lots produced by a

continuing process.

3. Normally lots are expected to be of essentially the same quality.

4. The consumer has faith in the integrity of the producer.

Operating procedure for ChSP-1

The operating procedure for ChSP-1 plan is implemented in the following way. For each lot, Select a random sample of n units and test each unit for conformance to

the specified requirements.

Accept the lot if d ( the observed number of nonconformities ie., defectives) is zero

in the sample of n units, and reject the lot if d>1.

Accept the lot if d=1 and if no conformities i.e., defectives are found in the

immediately preceding i samples of size .

Dodge (1955) has given the operating characteristic function for ChSP-1 plan as

Pa(p) = P0 + P1 (P0)i,

where Pi = probability of finding i nonconforming units in a sample of n units for i

The Chain sampling Plan is characterized by the parameters n and i.

When i= ∞, the OC function of a ChSP -1 plan reduces to the OC function of the

Single Sampling Plan with acceptance number zero and when i = 0, the OC function of

ChSP-1 plan reduces to the OC function of the Single Sampling Plan with acceptance

number 1.


37

Flow chart for the operating procedure of Chain Sampling Plan(ChSP-1)

Sample n from current lot

No defective

found

2 or more

defectives found

One defective

found

One defective in previous i lots

No defectives in previous i lots

Accept

Reject

Start


38

Section 1.6

This section deals with the review on Multiple Repetitive Group Sampling Plan (MRGS)

Multiple Repetitive Group Sampling Plans

The concept of repetitive group sampling (RGS) plans was introduced by Sherman in

which acceptance or rejection of a lot is based on the repeated sample results of the same lot

is based on the repeated sample result of the same lot. Recently, Shankar and Mohapatra

and Joseph (1993) have proposed a new repetitive group sampling plan as an extension of

conditional repetitive group sampling plan in which acceptance or rejection of a lot on the

basis of repeated sample is dependent on the outcome of the single sampling inspection under

RGS inspection system of the immediately preceding lots.

MRGS is an extension of CRGS plan in which acceptance or rejection of a lot on the

basis of repeated sample results is dependent on the outcome of inspection under a RGS

inspection system of the preceding lots.

For convenience, the proposed plan will be designated as Multiple Repetitive group

sampling plan. Second, an attempt has also been made to model and analyse the dynamics of

the proposed inspection through GERT (Graphical Evaluation and Review Technique)

approach which has been successfully used by several authors for studying a few types of

quality control plans. A brief account of researchers in quality control through GERT

methods have been given by Shankar [1988]

The advantage of GERT analysis is twofold. Firstly, this procedure gives a visual

picture of the inspection system and secondly, it offers through characterization of the plan.

The formula for the OC and ASN function of the plan are derived and illustrated numerically.

Finally, Poisson unity values have been tabulated for the construction and selection of the

plan.


39

For the sake of simplicity and convenience, we reproduce the operating procedure of

conditional RGS plans due to Shankar and Mohapatra[1993] subsequently; discuss the

operation of the proposed plan.

Operating Procedure for Multiple Repetitive Group Sampling Plans:

1. Draw a random sample of size n and determined the number of defectives (d) found

therein.

2. Accept the lot, if d ≤ c1

Reject the lot if d > c2

3. If c1 < d ≤ c2 , repeat the step (1) and (2) provided i successive previous lots

are accepted under RGS inspection system, otherwise reject the lot.

Thus MRGS plans are characterized by four parameter, namely, n, c1, c2 and

acceptance criterion i. Here, it may be noted that when c1 = c2 , the resulting plan is simple

single sampling. Also, for i=0, one can have the RGS plan of Sherman (1965). It may

further be noted that the conditions of the application of the proposed plan is same as

Sherman RGS plan.

The operating characteristics function Pa(p) of Multiple Repetitive Group sampling

plan is derive by Shankar and Joseph(1993) using poisson model as

Pa(p) =

iac

ic

ica

PPPPP

11

Pa(p) = acc

ca

PPPPP

11 where i=1

Where Pa = !/exp1

0Xnpnp

c

X

X

and Pc =

2

0 !)(c

x

xnp

xnpe

1

0 !)(c

x

xnp

xnpe

and h0 = -p/PA(p) . dPA(p)/dp


40

Section 1.7

This section deals with the review on Two Stage Conditional Repetitive Group

Sampling Plan (TSCRGS)

Repetitive Group Sampling Plan

Sherman (1965) has introduced a new acceptance sampling plan called Repetitive

Group Sampling (RGS) plan in which a sample is drawn the number of defectives counted ,

then according to a fixed criterion, the lot is either accepted , rejected or the sample is

completely disregarded and we begin over again with a sample. This is continued until the

fixed criterion tells us to either accept or reject the lot. Further details of RGS plan are given

below.

Operating Characteristic function

The operating characteristics function Pa(p) Repetitive Group sampling plan is

Pa(p’) = p[d < c1/p =p’]

be the probability of acceptance in a particular group sample, when p = p’.

Let Pr(p’) = p[d > c2/p =p’]

be the probability of rejection in a particular group sample, when p = p’

Let PA(p’) and PR(p’) be the probabilities for eventually accepting and rejecting

the lot respectively, when p = p’.

The lot may be accepted on the basis of the first sample or second sample , thus we get

PA = Pa + ( 1 - Pa - Pr ) Pa + ( 1 - Pa - Pr )2 Pa + ………


41

Which is a geometric series summing to

PA(p) = Pa(p) / [ Pa(p) + Pr(p) ]

Which is the OC function.

Similarly we can get

PR = Pr / ( Pa + Pr )

Such that PA + PR = 1

Conditional Repetitive Group Sampling Plan (CRGS)

The concept of repetitive group sampling (RGS) plan was introduced by Sherman

(1965) in which acceptance or rejection of a lot is based on the repeated sample results of the

same lot. Recently, Shankar Mohapatra (1993) have developed a new repetitive group

sampling plan designated as Conditional RGS plan in which disposal of a lot on the basis of

repeated sample results with dependent on the outcome of the inspection of the immediately

preceding i lots. They derived the OC and ASN functions through Graphical Evaluation and

Review Technique (GERT) approach. The detailed procedures and tables for construction

and selection of RGS plans have been given by Soundararajan and Ramasway (1982).

Recently, Singh et al (1989) modeled RGS plans through the GERT (graphical evaluation

and review technique) approach. First we propose a new RGS inspection system in which

disposal of a lot on the basis of repeated sample results is dependent on the outcome of the

inspection of the immediately preceding lots. For convenience the proposed plan is

designated as conditional repetitive group sampling plan. Second, as attempt is also made to

model the dynamics of the conditional RGS plans through the GERT approach which has

been used by several authors for studying some quality control plans. Some reference may

be made to Kase and Otha [1977], Otha [1978] and Shankar [1993].


42

The advantage of the GERT analysis is two-fold. First, this procedure gives the

visual picture of the inspection system and second, it offers through characterization of the

plan. The formulae for the OC and ASN function of the plan have been derived and

illustrated numerically. Finally, Poisson unity values have been tabulated for construction

and selection of the plan.

The operating procedure for CRGS plan

1. Draw a random sample of size n and test each unit for conformance to the

specified requirements.

2. Accept the lot if d (the observed number of defectives in the sample) is less

than or equal to c1.

3. Reject the lot, if d is greater than c2.

4. If c1 ≤ c2, repeat the steps (1), (2), (3) and (4) provided previous i lots are

accepted, otherwise reject the lot.

For CRGS plan the probability of eventually accepting the lot is given by

rac

c

x

xr

c

x

xa

iacaa

PPP

xnpnpP

xnpnpP

wherePPPpP

1

/))(exp(1

/))(exp(

]1/[)(

2

1

0

0


43

Two stage Conditional Repetitive Group Sampling Plan

The concept of repetitive group sampling (RGS) plan was introduced by Sherman1 in

which acceptance and rejection of the lot was based on the repeated sample results of the

same lot. The detailed procedures and tables for construction and selection of RGS plan have

been given by Soundararajan(1982) and Ramaswamy (1982), and Singh, et. al (1989). Later

on, Shankar and Mohapatra (1993) developed conditional RGS plan as an extension of the

classical RGS plan. Mohapatra (1993) compared Conditional RGS plan with the RGS plan

and observed that the conditional RGS plan is better in sample size efficiency than the RGS

plan. The purpose of present investigation is two-fold. Firstly, following Stephens and Dodge

(1964), the proposed plan uses different sample sizes in the normal and the tightened phases

of inspection . Secondly, the dynamic characteristics of the proposed plan have been

modelled and analysed through graphical evaluation and review technique (GERT), which

has been used by several authors to study quality control systems. A brief account of such

studies has been given by Shankar (1993). The formula for operating characteristic and

average sample number (ASN) functions of the plan has been derived by applying Mason’s

rule on the GERT network representation inspection system. Lastly, Poisson unity values

have been tabulated to facilitate the operation and construction of the plan.

Military standard plans are commonly used in defense establishments by the military,

as a consumer. In application of the MIL-STD-105D system, it is intended that a switch to

tightened inspection, in case of poor quality, provides a psychological and economic

incentive for the producer to improve the level of the product quality submitted. The

proposed two-stage plan based on normal and tightened inspection may be useful as an

addition to MIL-STD-105D system for quick discrimination of good and bad quality lots.

The present study may also be used when from a continuous process is inspected and/or

sampled continuously to determine the quality of goods. Thus, the proposed procedure could

be used for both the process control and goods acceptance or rejection in various stages of

production in defence organizations.


44

Following the notations and concepts similar to those of Sherman (1965), and

Shankar and Mohapatra (1993), the proposed two-stage conditional RGS plan is carried out

through the following steps:

(a) Draw a random sample of size n, from the lot for normal inspection and Determine

the number of defectives (d) found therein.

(b) Accept the lot, if d < c1

Reject the lot, if d > c2

(c) If c1 < d < c2 then repeat the above steps provided previous i lot are

accepted under normal inspection. Otherwise reject the lot.

Here, it may be noted that a lot with number of defectives < c1 is accepted at both the

normal and tightened inspection states. Furthermore, the process is automatically switched to

normal inspection after acceptance/rejection of the current lot.

The proposed plan is characterized by five parameters, namely n1, n2, c1, c2 and i.

For i = 0, the resulting plan is two-phase inspection RGS plan due to Shankar. Moreover,

for i = 0 and k = n2 / n1 = 1 (i.e. n1 = n2), one has RGS plan due to Sherman (1965) .

The operating characteristic function for two stage CRGS is given below

Where

Where


45

Flow chart for the operating procedure of Conditional Repetitive Group Sampling Plan

Sample n from current lot and observe d

If d ≤ c1

If d > c2

If c1 < d ≤ c2

Previous i lots are accepted

Accept

Reject

True

False

Start


46

Section 1.8

This section deals with the review on Repetitive Deferred Sampling Plan

Repetitive Deferred Sampling Plan

In this plan the acceptance or rejection of lot in deferred state is dependent on the

inspection results of the preceding or succeeding lots under Repetitive Group Sampling

(RGS) inspection. RGS is the particular case of RDS Plan.

For situation involving costly or destructive testing and the lots are submitted a

continuous production, it is advantageous to use SkSP-2 Plan with RDS Plan as reference

plan. Since both the plans ensure a relatively smaller sample size.

Conditions for application of RDS plan

1. Production is steady so that result of past, current and future lots are briefly indicative of

a continuing process.

2. Lots are submitted substantially in the order of their production.

3. A fixed sample fixed, n from each lot is assumed.

4. Inspection by attributes with quality defined as fraction non-conforming.

The condition for RDS plan is given below

a. When ‘i’ consecutive lots are accepted on normal inspection, switch to skipping

inspection of inspecting a fraction ‘ f’ of the lots.

b. When a lot is rejected on skipping inspection, switch to normal inspection.

c. Screen each rejected lot and correct or replace all defective units found.

d. The OC function associated with SkSP-2 plan is of type B, based on Probabilities of

sampling from an infinite universe or process.


47

The operating procedure for RDS plan:

1. Draw a random sample of size n from the lot and determine the number of defectives (d)

found there in.

2. Accept the lot if d < c1 , Reject the lot if d > c2 .

3. If c1 < d < c2 , accept the lot provided ‘i’ preceding or succeeding lots are accepted under

RGS inspection plan, otherwise reject the lot.

Here c1 and c2 are acceptance numbers such that c1 < c2 when i = 1 this plan reduces to

RGS plan.

The operating characteristic function Pa (p) for RDS plan is derived by

Sankar and Mahopatra ( 1991) using Poisson model as

Pa ( 1- Pc )i + Pc Pai

Pa (p) = ( 1- Pc )i c1 e – x x r r=0

Where Pa = P [ d < c1 ] = r ! c2 c1

e – x x r e – x x r r=0 r=0

Where Pc = P [ c1 < d < c2 ] = — r ! r !

where x = np

Thus the RDS plan is characterized with parameters namely n, c1 , c2 and the

acceptance criterion i.

The conditions associated with sampling from an infinite universe are based on the

notion of a process producing a theoretically continuous infinite product flow. The OC

function for SkSP-2 plan with RDS as reference plan is


48

,

)1()1(

)1()1(),,,,( 21 i

c

iac

ica

i

i

a PPPPP

PWherePffPfPfdccifP

where P is the OC function for the reference sampling RDS plan

Therefore the operating characteristic function for SkSPRDS-2 plan is designated as follows

i

ic

iac

ica

i

ic

iac

ica

ic

iac

ica

a

PPPPP

ff

PPPPP

fP

PPPPf

dcciifP

1

11

1

11

1

11

)1()1(

)1(

)1()1(

)1()1(

)1(

),,,,( 211

It is noted that dcciifPa ,,,,( 211 ) is a function of ‘ i ’ clearing interval; ‘ f ’ sampling

fraction. c1, c2 are the acceptance numbers, d represents rejection number or number of

defectives, i1 is the number of consecutive lots to be considered for RDS plan.


49

Section 1.9 Glossary of Symbols

N - Lot size

P - Lot or process quality

Pa(p) or PA - Probability of acceptance of single lot for given p

p1 - Acceptable Quality Level (AQL)

p2 - Limiting Quality Level (LQL)

p0 - Indifference Quality Level (IQL) such that

Pa(p0) = 0.50

α - Producers Risk

β - Consumers Risk

h0 - Relative slope of the OC Curve at IQL

(Absolute value)

ASN - Average sample number

n - Sample size

c - Acceptance number in single sampling plan

c1 and c2 - Acceptance numbers in SkSPRDS-2 plan

i - clearance number

r - Maximum number of defectives for unconditional acceptance

in Multiple Deferred state sampling plan

b - Maximum number of additional defectives for conditional

acceptance in Multiple Deferred state sampling plans

m - Number of future lots in which conditiona acceptance is based

for MDS-(r, b, m) sampling plans

d - Non-conforming units in MDS(r,b)

ich - Number of lots that are to be consecutively accepted in SkSP-2

plan, number of previous samples for Chain Sampling

Plans or clearance number

f - Fraction of the lots sampled in the skipping Phase of SKSP-2

plan


50

n1 - First stage sample size

n2 - Second stage sample size

k - The ratio between second stage and first stage sample size in

TSCRGS plan

σ - Population Standard Deviation

x - Sample mean

im - number of lots concecutively accepted in SkSP-2 plan, number

of previous samples for Multiple deferred state Sampling plan

or clearance number

imrgs - number of lots concecutively accepted in SkSP-2 plan, number of

previous samples for Multiple repetitive group Sampling plan

or clearance number

itscrgs - number of lots concecutively accepted in SkSP-2 plan, number of

previous samples for two stage conditional repetitive group

Sampling plan or clearance number


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