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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
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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.
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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.
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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”.
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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.
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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).
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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
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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.
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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”.
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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.
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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)
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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
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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.
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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
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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.
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)( 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 =
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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
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(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.
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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)
Skip-lot Sampling plan of type 2 with
Chain Sampling Plan - SkSPCHS-2
Skip-lot Sampling plan of type 2 with
Multiple Repetitive Group Sampling plan - SkSPMRGS-2
Skip-lot Sampling plan of type 2 with
Two Stage Conditional Repetitive Group sampling plan - SkSPTSCRGS-2
Skip-lot Sampling plan of type 2 with
Repetitive Deferred Sampling Plan - SkSPRDS-2
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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.
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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.
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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.
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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).
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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
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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.
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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
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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
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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.
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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)(
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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
)]()][()([)()(
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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
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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
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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.
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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.
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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
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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.
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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
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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 + ………
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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].
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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
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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.
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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
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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
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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.
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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
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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.
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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
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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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