Unit-3 Acceptance Sampling - Aligarh Muslim University

Unit-3 Acceptance Sampling Lecture Notes by Sheema Sadia

Department of Statistics & Operations Research, AMU, Aligarh Page 1

Product Control: It is another method of statistical quality control in which

the quality of a product is controlled while the product is ready to dispatch or

sell to the customers. Product control used the technique of acceptance

sampling to detect defects and control the quality of a product ( whether

these are fit to be used or dispatched). It concerns with the classification of

new material, semi finished goods or finished goods into acceptable or reject

able items. This is concerned with the inspection of goods already produced to

judge whether these are fit to be used or dispatched. Acceptance sampling is

normally used for product control which indicates whether the lot should be

accepted without further inspection or another sampling should be taken for

deciding about the quality of the lot.

What is Acceptance Sampling?

Acceptance sampling is an important field of statistical quality control that was

popularized by Dodge and Romig and originally applied by the U.S. military to

the testing of bullets during World War II. If every bullet was tested in advance,

no bullets would be left to ship. If, on the other hand, none were tested,



malfunctions might occur in the field of battle, with potentially disastrous

results.

Dodge reasoned that a sample should be picked at random from the lot, and

on the basis of information that was yielded by the sample, a decision should

be made regarding the disposition of the lot. In general, the decision is either

to accept or reject the lot. This process is called Lot Acceptance Sampling or

just Acceptance Sampling.

Means fate of the lot depends on the sample, if sample is accepted, lot will be

accepted, if sample is rejected, lot will be rejected.

A sample from the inspection lot is inspected, and if the number of defective

items is more than the stated number (the number is decided using statistics

after a decision is taken about confidence level depending upon the place of

application of the product and its criticality) known as Acceptance Number, the

whole lot is rejected.

The purpose of Acceptance Sampling is, therefore, to decide whether to accept

or reject the lot. It does not control the quality during the process of

manufacturing.

So, in simple words “Acceptance sampling is "the middle of the road"

approach between no inspection and 100% inspection.”

Means No inspection can lead to disaster at the time of main operation and

100% inspection can sometimes becomes costly or completely destructive as in

case of bullets or crackers.



Acceptance Sampling Problem

A typical application of acceptance sampling is as follows: A company receives

a shipment of product from a supplier. This product is often a component or

raw material used in the company’s manufacturing process. A sample is taken

from the lot, and some quality characteristic of the units in the sample is

inspected. On the basis of the information in this sample, a decision is made

regarding lot disposition. Usually, this decision is either to accept or to reject

the lot. Sometimes we refer to this decision as lot sentencing. Accepted lots

are put into production; rejected lots may be returned to the supplier or may

be subjected to some other lot disposition action.

Generally, there are three approaches to lot sentencing: (1) accept with no

inspection; (2) 100% inspection—that is, inspect every item in the lot,

removing all defective1 units found (defectives may be returned to the

supplier, reworked, replaced with known good items, or discarded); and (3)

acceptance sampling. The no-inspection alternative is useful in situations

where either the supplier’s process is so good that defective units are almost

never encountered. We generally use 100% inspection in situations where the

component is extremely critical and passing any defectives would result in an

unacceptably high failure cost at subsequent stages, or where the supplier’s

process capability is inadequate to meet specifications.

Acceptance sampling is most likely to be useful in the following situations:

1. When testing is destructive. Eg testing of electrical fuses, crackers etc.

2. When the cost of 100% inspection is extremely high.

3. When 100% inspection is not technologically feasible or would require so

much time that production scheduling can be seriously impacted.



4. When there are many items to be inspected and the inspection error rate is

sufficiently high that 100% inspection might cause a higher percentage of

defective units to be passed than would occur with the use of a sampling plan.

5. When the supplier has an excellent quality history.

6. When there are potentially serious product liability risks and handling of the

product may result in its deterioration.

Advantages of Sampling Inspection

When acceptance sampling is contrasted with 100% inspection, it has the

following advantages:

1. It is usually less expensive because there is less inspection.

2. There is less handling of the product, hence reduced damage.

3. It is applicable to destructive testing.

4. Fewer personnel are involved in inspection activities.

5. It often greatly reduces the amount of inspection error.

6. The rejection of entire lots as opposed to the simple return of defectives

often provides a stronger motivation to the supplier for quality improvements.

Disadvantages of Sampling Inspection

Acceptance sampling also has several disadvantages, however. These include

the following:

1. There are risks of accepting “bad” lots and rejecting “good” lots.

2. Less information is usually generated about the product or about the

process that manufactured the product.



3. Acceptance sampling requires planning and documentation of the

acceptance-sampling procedure whereas 100% inspection does not.

Types of Acceptance Sampling Plans

There are a number of different ways to classify acceptance-sampling plans.

One major classification is by variables and attributes.

Acceptance Sampling by Variables:- Variables, of course, are quality

characteristics that are measured on a numerical scale.

Acceptance Sampling by Attributes:- Attributes are quality characteristics that

are expressed on a “go, no-go” basis. Therefore, actual measurements are not

done. For example- the pipes without cracks are accepted and the pipes with

cracks are rejected without measuring the size and shape of the cracks.

The sampling plans for attributes are further classified into three categories.

1. Single-sampling plan

2. Double-sampling plan

3. Multiple-sampling plan

A single-sampling plan is a lot-sentencing procedure in which one sample of n

units is selected at random from the lot of size N and the disposition of the lot

is determined based on the information contained in that sample. For

example, a single-sampling plan for attributes would consist of a sample size n

and an acceptance number c. (defined above) The procedure would operate as

follows: Select n items at random from the lot. If there are c or fewer

defectives in the sample, accept the lot, and if there are more than c defective

items in the sample, reject the lot.



Double-sampling plans are somewhat more complicated. Following an initial

sample, a decision based on the information in that sample is made either to

(1) accept the lot, (2) reject the lot, or (3) take a second sample. If the second

sample is taken, the information from both the first and second sample is

combined in order to reach a decision whether to accept or reject the lot.

Or, A sampling plan in which a decision about the acceptance or rejection of a

lot is based on two samples that have been inspected is known as a double

sampling plan. The double sampling plan is used when a clear decision about

acceptance or rejection of a lot cannot be taken on the basis of a

single sample.

A multiple-sampling plan is an extension of the double-sampling concept, in

that more than two samples may be required in order to reach a decision

regarding the disposition of the lot. Sample sizes in multiple sampling are

usually smaller than they are in either single or double sampling. The ultimate

extension of multiple sampling is sequential sampling, in which units are

selected from the lot one at a time, and following inspection of each unit, a

decision is made either to accept the lot, reject the lot, or select another unit.



For Understanding only-The procedure commences with taking a random

sample of size n1 from a large lot of size N and counting the number of

defectives, d1.

If d1≤a1, the lot is accepted.

If d1≥r1, the lot is rejected.

If a1<d1<r1, another sample is taken.

If subsequent samples are required, the first sample procedure is repeated

sample by sample. For each sample, the total number of defectives found at

any stage, say stage i, is

𝐷𝑖 = 𝑑𝑗

𝑖

𝑗 =1

This is compared with the acceptance number ai and the rejection number ri for

that stage until a decision is made. Sometimes acceptance is not allowed at the

early stages of multiple sampling; however, rejection can occur at any stage.

Lot Formation

The formulation of lot can influence the effectiveness of the acceptance-

sampling plan. There are a number of important considerations in forming lots

for inspection. Some of these are as follows:

1. Lots should be homogeneous. The units in the lot should be produced

by the same machines, the same operators, and from common raw

materials, at approximately the same time. When lots are non-

homogeneous, such as when the output of two different production

lines is mixed, the acceptance-sampling scheme may not function as



effectively as it could and also make it more difficult to take corrective

action.

2. Larger lots are preferred over smaller ones. It is usually more

economically efficient to inspect large lots than small ones.

3. Lots should be conformable to the materials-handling systems used in

both the supplier and consumer facilities. In addition, the items in the

lots should be packaged so as to minimize shipping and handling risks,

and so as to make selection of the units in the sample relatively easy.

Properties of OC curve

1. Ideal curve would be perfectly perpendicular 0 to 100% for a given

fraction defective.

2. The acceptance number c and the sample size n are most important

factors in defining the OC curve.

3. Decreasing the acceptance number (c) is preferred over increasing

sample size (n).

4. The larger the sample size (n), the steeper is the OC curve (i.e. it

becomes more discriminating between good and bad lots)



OC (Operating Characteristic) Curve

An important measure (graphical) of the performance of an acceptance-

sampling plan is the operating characteristic (OC) curve. This curve plots the

probability of accepting the lot versus the lot fraction defective. It shows

the probability that a lot submitted with a certain fraction defective will be

either accepted or rejected. An operating characteristic (OC) curve is a chart

that displays the probability of acceptance versus percentage of defective

items (or lots).

With no defects, we'll surely have 100% acceptance! But, take a look at 0.05

(5% defective). At this point, there is still 90% acceptance. Then, as the

proportion of defectives increases the probability of acceptance will

decrease.

Or we can say OC curves quantifies manufacturer’s (producer’s) risk and

consumer’s (purchaser’s) risk. This is a graph of the percentage defective in

a lot versus the probability that the sampling plan will accept a lot.

Types of OC curve

There are two types of OC- Curve



1. Type-A OC Curve-The type-A OC curve is used to calculate probabilities

of acceptance for an isolated lot of finite size. The exact sampling

distribution of the number of defective items in the sample is the

hypergeometric distribution. However, binomial and poisson distribution

also provide good approximation.

2. Type-B OC Curve - It gives the probability of acceptance of large lots or

lots drawn from a continuous process. It assumes an infinite lot size. In

this situation, the binomial distribution is the exact probability

distribution for calculating the probability of lot acceptance. However,

poisson distribution also provides good approximation.

Parameters of OC- Curve

When a sampling plan is used for the acceptance of lots, some lots will be

accepted and some lots will be rejected. Since the lot acceptance or

rejection is based on the number of defectives found in the sample it cans

so happen that a good lot may be rejected or a bad lot may be accepted,

these two events may happen due to chance. When good lots are rejected,

it is a loss to the producer and when bad lots are accepted , it is a loss to the

customers. These risks are defined as:-

1. Producer’s risk: The probability of rejecting good lots is called as

producer’s risk. It is denoted by α and 1-α is the probability of

acceptance (usually, α =5% ). The producer risk is the probability that a

consumer refuses the good lot. i.e. Pr (Consumer refuses the lot/lot is

good).

Therefore, Pr (rejecting a good lot)=0.05 and Pr (accepting a good

lot)=0.95



2. Consumer’s Risk: the probability of accepting the bad lot is called as

consumer’s risk and is denoted by β (Usually, β =10%). The consumers

risk is the probability that a consumer accepts the bad lot. i.e. Pr

(Consumer buys a lot/lot is bad),

Risk α and β are the two key parameters of the OC Curve. Each wants

their own risk to be small as possible.

Acceptance Quality Level (AQL)

This is the maximum proportion of defectives that will make the lot acceptable,

the AQL is the fraction defective corresponding to a certain level of probability

of acceptance(1-α). This is considered that amount of worst quality which is

still considered satisfactory for the purpose of acceptance sampling. Or in

other words, it can be considered as a compromise between producer’s

capability and the consumer’s requirements. The sampling procedure is

designed in such a way that the OC Curve gives a high probability of

acceptance at the AQL.

Suppose a lot with relatively small fraction defective say P1 that we do not wish

to reject is regarded as good lot. Generally,

Pr (rejecting a lot of quality P1) =0.05

Pr (accepting of a lot of quality P1) =0.95

P1 is known as the acceptance quality level and a lot of this quality is

considered as satisfactory by the consumer.

For Understanding purpose only Examples of AQL: European Standards state

that medical examination gloves shall have an AQL of 1.5(here,P1=1.5) for

pinholes. This means that it is acceptable for up to 1.5% of all gloves that are



made to contain a hole. US and International standards for medical

examination gloves only require an AQL of 2.5 (here,P1=2.5). This means that it

is acceptable for 2.5% of all gloves that are made to contain a hole. As you can

see, a lower AQL number means that there is less risk of encountering a defect.

However, it means that higher levels of inspection are required, leading to

increased prices. It is one of the responsibilities of European standards makers

to decide what percentage of defects is permissible in any market. For pinholes

in medical gloves the Standards Committee felt that the internationally

accepted AQL of 2.5 did not provide sufficient protection for European

Healthcare Workers and so set a stricter level of 1.5. Of course, an AQL defines

the worst a manufacturer can allow his process to become before action is

taken. Good manufacturers regularly produce product that is significantly

better than the AQL level suggests.

Lot Tolerance Percent Defective (LTPD)

The LTPD is the worst level of quality that a consumer is willing to accept in an

individual lot. The LTPD is the unsatisfactory quality level at which consumer

accepts a lot with very low probability. It is also called Rejectable Quality level

(RQL) or Limiting Quality Level (LQL).

The primary difference is AQL is the average quality level over a series of lots

where LTPD is lot-to-lot based.

As we have specified four parameters α, β, AQL and LTPD of OC curve, we can

now construct OC Curves for the various values of the probability of lots being

accepted verses various values of proportion (percent) defective. The curve is

given as



The OC Curve also helps to illustrate the consumer and producer risk

associated with a particular sampling plan, along with AQL and LTPD

Interpretation of the above OC- Curve

Recall that the producers Risk is tied to AQL, and is traditionally set at 5% (95%

probability of acceptance + 5% probability of rejection), while the consumer

risk is tied to LTPD and is set at 10% (10% probability of acceptance, 90%

probability of rejection).

We can see from the graph if the fraction defective is upto 2%, the probability

of acceptance is 95% (and more than 95% for less than 2% of fraction

defective). Similarly for the fraction defective of 8%, it shows that lots at or

worse than the LTPD are accepted at most 10% of the time. In other words,

they are rejected at least 90% of the time.



For understanding only- LTPD is generally defined as that level of quality

(percent defective, defects per hundred units, etc.) that the sampling plan will

accept only 10% of the time.

Rectifying inspection Plan :- Acceptance-sampling programs usually require

corrective action when lots are rejected. This generally takes the form of 100%

inspection or screening of rejected lots, with all discovered defective items

either removed for subsequent rework or return to the supplier, or replaced

from a stock of known good items. Such sampling programs are called

rectifying inspection programs or plan , because the inspection activity affects

the final quality of the outgoing product.

Suppose that incoming lots to the inspection activity have fraction defective op .

Some of these lots will be accepted, and others will be rejected. The rejected

lots will be screened, and their final fraction defective will be zero. However,

accepted lots have fraction defective op . Consequently, the outgoing lots from

the inspection activity are a mixture of lots with fraction defective op and

fraction defective zero, so the average fraction defective in the stream of

outgoing lots is 1p , which is less than op . Thus, a rectifying inspection program

serves to “correct” lot quality.

NOTE:- The two important points related to rectifying inspection plans are:



1. The average quality of the product after sampling and 100% inspection of

rejected lots, called Average Outgoing Quality (AOQ).

2. The average amount of inspection required for the rectifying inspection

plan, called Average Total Inspection (ATI).

Average Outgoing Quality (AOQ)

The average outgoing quality is the quality in the lot that results from the

application of rectifying inspection. It is the average value of lot quality that

would be obtained over a long sequence of lots from a process with fraction

defective p . It is simple to develop a formula for average outgoing quality

(AOQ). Assume that the lot size is N and that all discovered defectives are

replaced with good units. Then in lots of size N ,

1. n items in the sample that, after inspection, contain no defectives,

because all discovered defectives are replaced.

2. nN items that, if the lot is rejected, also contain no defectives.

3. nN items that, if the lot is accepted, contain )(* nNp defectives.

Thus, lots in the outgoing stage of inspection have an expected number

of defective units equal to )(** nNpPa , which we may express as an

average fraction defective, called the average outgoing quality or

N

nNpPAOQ a )(**

(1)

If the lot size N becomes large relative to the sample size n, then

pPN

nNpPAOQ a

a *)(**

(2)

For understanding :-Suppose that N = 10,000, n = 89, and c = 2, and that

the incoming lots are of quality p = 0.01. Now at p = 0.01, we have Pa =

0.9397, and the AOQ is



0093.010000

)8910000(*)01.0(*)9397.0()(**

N

nNpPAOQ a

That is, the average outgoing quality is 0.93% defective.

Average outgoing quality will vary as the fraction defective of the

incoming lots varies. The curve that plots average outgoing quality

against incoming lot quality is called an AOQ curve.

This Average outgoing quality curve is plotted for n=89 and c= 2. From

examining this curve we note that when the incoming quality is very

good (i.e. fraction defective is low), the average outgoing quality(i.e.

fraction defective for outgoing lot is also less and it increases with

increase in fraction defective of incoming lot which is on x-axis) is also

very good. In contrast, when the incoming lot quality is very bad (i.e.

fraction defective is high), most of the lots are rejected and screened,

which leads to a very good level of quality in the outgoing lots. In

between these extremes, the AOQ curve rises, passes through a

maximum, and descends. The maximum ordinate on the AOQ curve

represents the worst possible average quality that would result from the

rectifying inspection program, and this point is called the average



outgoing quality limit (AOQL). (The AOQL is seen to be approximately

0.0155. That is, no matter how bad the fraction defective is in the

incoming lots, the outgoing lots will never have a worse quality level on

the average than 1.55% defective. Let us emphasize that this AOQL is an

average level of quality, across a large stream of lots. It does not give

assurance that an isolated lot will have quality no worse than 1.55%

defective. )

Average Sample Number (ASN):

The average sample number is the expected value of the sample size

required for coming to a decision about the acceptance or rejection of

the lot in an acceptance-rejection sampling plan. For example, if a single

sampling acceptance-rejection plan is used, ASN=n. However, in double-

sampling, the size of the sample selected depends on whether or not the

second sample is necessary.

Average Total Inspection (ATI):The expected number of items inspected

per lot to arrive at a decision in an acceptance rectification sampling

inspection plan calling for 100% inspection of the rejected lots is called

the average amount of total inspection. Obviously, it is also a function of

the lot quality p as ASN.

If the lots contain no defective items, no lots will be rejected, and the

amount of inspection per lot will be the sample size n. If the items are all

defective, every lot will be submitted to 100% inspection, and the

amount of inspection per lot will be the lot size N. If the lot quality is 0 <

p < 1, the average amount of inspection per lot will vary between the

sample size n and the lot size N. If the lot is of quality p and the

probability of lot acceptance is Pa, then the average total inspection per

lot will be ))(1( nNPnATI a (3)



Here,

ATI = ASN + (Average size of inspection of the remainder in the

rejected lots) (4)

Therefore from equation (4), we can say that if the lot is accepted on

the basis of the sampling inspection plan then, ATI = ASN otherwise

ATI > ASN. (For more details see page.645 of Montogomery, (2009):

Introduction to Statistical Quality Control, 6th Edition)

Single Acceptance Sampling Plan

Suppose that a lot of size N has been submitted for inspection. A

single-sampling plan is defined by the sample size n and the

acceptance number c. Thus, if the lot size is N = 10,000, then the

sampling plan means that from a lot of size 10,000 a random sample

of n = 89 units is inspected and the number of nonconforming or

defective items d observed. If the number of observed defectives d is

less than or equal to c = 2, the lot will be accepted. If the number of

observed defectives d is greater than 2, the lot will be rejected. Since

the quality characteristic inspected is an attribute, each unit in the

sample is judged to be either conforming or nonconforming. One or

several attributes can be inspected in the same sample.

A unit that is nonconforming to specifications on one or more

attributes is said to be a defective unit. This procedure is called a

single-sampling plan because the lot is sentenced based on the

information contained in one sample of size n.



Flowchart

Probability of Acceptance (Pa):

Suppose that the lot size N is large. Under this condition, the

distribution of the number of defectives d in a random sample of n

items is binomial with parameter n and p, where p is the fraction

defective items in the lot. Therefore, the probability of observing

exactly d defectives is

)( defectivesdP = dnd ppdnd

n

)1()!(!

!

The probability of acceptance is simply the probability that d is less

than or equal to c ( cd ) , or

)( cdPPa =

c

d

dnd ppdnd

n

0

)1()!(!

! (5)

For example, if the lot fraction defective is p = 0.01, n = 89, and c = 2,

then



The OC curve is developed by evaluating equation (5) for different values of P . Table 1

displays the calculated value of several points on the curve. The OC curve shows the

discriminatory power of the sampling plan. For example, in the sampling plan n = 89, c =

2, if the lots are 2% defective, the probability of acceptance is approximately 0.74. This

means that if 100 lots from a process that manufactures 2% defective product are

submitted to this sampling plan, we will expect to accept 74 of the lots and reject 26 of

them.

Table-1

The OC Curve

It is an essential measure of the performance of an acceptance sampling plan.

The OC curve plots the probability of accepting the lot versus the lot fraction

defective. Thus, the OC curve displays the discriminating power of the

sampling plan. That is, it shows the probability that a lot submitted with a

certain fraction defective will be either accepted or rejected. We have

already seen the this form of OC curve earlier in this unit.



Single Sampling with type-B curve

Let us suppose that we want to construct a sampling plan such that the

probability of acceptance 1aP is )1( for lots with fraction defective 1p , and

the probability of acceptance 2aP is for lots with fraction defective 2p . By

using binomial distribution we have

11P

c

d

dndpp

dnd

n

011 )1(

)!(!

!

(1)

2P

c

d

dndpp

dnd

n

022 )1(

)!(!

!

(2)

From equation (1) and (2), we obtained the two points on the OC curve. The

two lines are drawn on the OC curve, one connecting 1p and 1-α , and the

other connecting 2p and β. The intersection of these two lines gives the

region of the OC curve in which desired sampling plan is located. Although any

two points on the OC curve could be used to define the sampling plan, it is

customary to use the AQL and LTPD points for this purpose. Where the level of

lot quality specified are 1p = AQL and 2p = LTPD, the corresponding points on



the OC curve are usually referred to as the producer’s risk point and

consumer’s risk point respectively. Thus, α would be called the producer’s risk

and β would be called the consumer’s risk. Producer’s risk

pP

c

d

dndpp

dnd

n

011 )1(

)!(!

!1

If the process average fraction defective is 𝑝1 as claimed by the producer, then

the average amount of total inspection per lot is given by

ATI=𝑛 + (𝑁 − 𝑛)𝑃𝑝

Since 𝑛 items have to be inspected in each case and the remaining (𝑁 – 𝑛)

items will be inspected only if 𝑑 > 𝑐,i.e., if the lot is rejected when the lot

quality is 𝑝1 and the probability for this is 𝑃𝑝 .

But in most of the situation, 𝑝1 is likely to be less than 0.10 and 𝑛 is likely to be

sufficiently larger then Poisson approximation to binomial distribution can be

used. Therefore, equation (1) and (2) can be written as

𝑃𝛼1 = 1 − 𝛼 = (𝑛𝑝1)𝑑𝑒 (−𝑛𝑝1)

𝑑 !

𝑐𝑑=0 (3)

𝛽 = (𝑛𝑝2)𝑑𝑒 (−𝑛𝑝2)

𝑑 !

𝑐𝑑=0 (4)

𝑃𝑝 = 1 − (𝑛𝑝1)𝑑𝑒 (−𝑛𝑝1)

𝑑 !

𝑐𝑑=0

Then, ATI=𝑛 + (𝑁 − 𝑛) 1 − (𝑛𝑝1)𝑑𝑒 (−𝑛𝑝1)

𝑑 !

𝑐𝑑=0

Single Sampling Plan with type – A OC Curve:

This type of OC curve is used to calculate probabilities of acceptance for an

isolated lot of finite size. Suppose that the lot size is 𝑵, the sample size is 𝒏,

and the acceptance number of defective is 𝒄. The exact sampling distribution

of the number of defectives items in the sample is the hyper –geometric.

Let suppose, a lot of incoming quality 𝒑, the number of defective pieces is 𝑵𝑷

and non defective pieces is 𝑵 – 𝑵𝑷 = 𝑵 (𝟏 – 𝑷). The probability of getting



exactly 𝒅 defectives in a sample of size n from this lot is given by the hyper –

geometric distribution

𝑃 𝑑 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑠 = 𝑁𝑃𝑑

𝑁−𝑁𝑃𝑛−𝑑

𝑁𝑛

Remark:

The computation of hyper-geometric probabilities is too difficult and, in

practice, the binomial

approximation to hyper-geometric is used.

𝑃 𝑋 = 𝑥 = 𝑚𝑥

𝑁 − 𝑚𝑛 − 𝑥

𝑁𝑛

If 𝑵 and 𝒏 are large but 𝒎 or (𝑵 – 𝒎) is relatively small, then hyper-

geometric distribution can be approximated by binomial distribution with

parameters 𝒎 and 𝒑 =𝒏

𝑵. Under these conditions, we have

𝑃 𝑋 = 𝑥 = 𝑚𝑥

𝑁 − 𝑚𝑛 − 𝑥

𝑁𝑛

→ 𝑚𝑥

𝑛

𝑁 𝑥

1 −𝑛

𝑁 𝑚−𝑥

Double Sampling Plan

Another sampling scheme propounded by Dodge and Romig is the ‘second

sampling method’. In this method, a second sample is required before the lot

can be sentenced because the first sample is not conclusive on either side

(about accepting or rejecting the lot). Some notations for double sampling plan

are as follows:-

N→denotes lot size from which samples are taken.

𝑛1 →denotes the sample size of the first sample.

𝑛2 →denotes the sample size of the second sample.



𝐶1 → Acceptance number of the first sample, i.e., the maximum allowable

number of defectives in the first sample if the lot is to be accepted without

taking another sample.

𝐶2 → Acceptance number for both sample, i.e., the maximum allowable

number of defectives in combining samples if the lot is to be accepted without

taking another sample.

𝑑1 → Number of defective in the first sample.

𝑑2 → Number of defective in the second sample.

Procedure:-

Take a random sample of size 𝑛1 , from the lot size 𝑁, and the number of

defectives in the sample 𝑑1 is observed.

If 𝑑1 ≤ 𝐶1, the lot is accepted on the first sample.

If 𝑑1 > 𝐶1 the lot is rejected on the first sample.

If 𝐶1 < 𝑑1 ≤ 𝐶2, a second random sample of size 𝑛2 is drawn from the

lot, and the number of defectives in this second sample 𝑑2, is observed.

Now the combined number of observed defectives from both the first

and second sample, 𝑑1 + 𝑑2, is used to determine the lot sentence.

If 𝑑1 + 𝑑2 ≤ 𝐶2 , the lot is accepted and if 𝑑1 + 𝑑2 > 𝐶2 , the lot is

rejected.



The OC curve for Double Sampling Plan :-

We now describe the construction of Type –B OC curve for double sampling.

Noted that, in double sampling, the OC curve consists of three curves. The

lowest of the three curves gives the probability of acceptance on the first

sample. The middle curve gives the probability of accepting the lot on the

combined sample. The upper curve gives the probability of rejection on the

first sample. The middle curve is considered as the actual OC curve of the

double sampling plan. Sometimes, we also plot the probability of rejection on

the first sample using the right-hand side scale.



if 𝑃𝑎 denotes the probability of acceptance on the combined sample, 𝑃𝑎(𝐼)

and

𝑃𝑎(𝐼𝐼)

denote the probability of acceptance on the first and second sample,

respectively, then

𝑃𝑎 = 𝑃𝑎(𝐼)

+ 𝑃𝑎(𝐼𝐼)

where, 𝑃𝑎(𝐼)

is the probability that we observed 𝑑1 ≤ 𝐶1 defectives out of a

random sample of 𝑛1 . Thus

)( I

aP

1

1

111

0111

1 )1()!(!

!C

d

dndpp

dnd

n

(A)

and to obtain the probability of acceptance on the second sample, we must list

the number of ways that the second sample can e obtained. A second sample

is drawn only if this condition holds i.e 𝐶1 < 𝑑1 ≤ 𝐶2.

The Average Sample Number Curve:- In single sampling plan, the size of the

sample inspected from the lot is always constant, whereas, in double sampling,

the size of the sample selected depends on whether or not the second sample

is necessary. In a double sampling plan, the number of sample items

inspected depends on the probability of acceptance of the lost on the first

sample and the probability of drawing the second sample. The probability of

drawing a second sample varies with the fraction defective in the incoming lot.

Therefore, the general formula for the average sample number in double

sampling, if we assume complete inspection of the second sample, is given by

𝐴𝑆𝑁 = 𝑛1𝑃𝑑 + 𝑛1 + 𝑛2 (1 − 𝑃𝑑) (B)



𝐴𝑆𝑁 = 𝑛1 + 𝑛2(1 − 𝑃𝑑) (C)

Where, 𝑃𝑑 is the probability of making a lot of dispositioning decision on the

first sample. That is

𝑃𝑑 = 𝑃(lot is accepted on the first sample) +𝑃(lot is rejected on the first

sample)

𝑷𝒅 = 𝑷𝒂(𝑰)

+ 𝑷𝒂(𝑰𝑰)

Using Eq (C), we can compute one point on the ASN curves for a given lot

fraction defective ( p) . For different values of p, the ASN value can be

computed and an ASN curve can be drawn.

AOQ for Double Sampling Plan:- Usually, under rectifying inspection, the

defectives found are replaced by good items in the samples as well as the

rejected lots, which are 100% inspected. The AOQ curve can be drawn for

double sample sampling by computing AOQ for a given fraction nonconforming

from

𝐴𝑂𝑄 =𝑃𝑎

𝐼 𝑁 − 𝑛1 𝑝 + 𝑃𝑎 𝐼𝐼 𝑁 − 𝑛1 − 𝑛2 𝑝

𝑁

𝑃𝑎 𝐼 𝑁 − 𝑛1 𝑝:- The Number of defectives found in lots accepted based on the

first sample.

𝑃𝑎 𝐼𝐼 𝑁 − 𝑛1 − 𝑛2 𝑝:- The number of defectives found in lots accepted based

on the second sample.

Average Total Inspection:- For Double Sampling Plan, ATI is computed as

𝐴𝑇𝐼 = 𝑛1 + 𝑛2 1 − 𝑃𝑎 𝐼

+ 𝑁 − 𝑛1 − 𝑛2 (1 − 𝑃𝑎)

where 𝑃𝑎 = 𝑃𝑎(𝐼)

+ 𝑃𝑎(𝐼𝐼)



Procedure of single sampling plan is outlined as follows:

(1) Pick up randomly number of items from the lot of N and inspect them.

(2) If the number of defectives found in the sample size is ≤ A1, accept the lot.

(3) If the number of defectives in sample of n items > A1, inspect the remaining

(N – n) items.

(4) Correct or replace all the defective products found.

Example 1:

Let the values of N, n and A1, be as follows:

N = 400

n = 20

A1= 2

A Sample of 20 products shall be taken from a batch of 400 pieces randomly.

The 20 pieces shall be inspected and if the number of defectives found is ≤ 2,

the lot of 400 pieces will be accepted without further inspection.

If the number of defectives in the sample of 20 is more than 2 then all the

remaining products. (400 – 20) = 380 should be inspected and all the defectives

should either be corrected or replaced by good ones before the whole lot of

400 is accepted.

where, N = Number of items/products in the given lot.

n = Number of units of the product randomly selected from the batch of size N.

A = Acceptance number. It is the number of maximum defectives allowed in a

sample of size n.

So A1, A2, A3, A4, A5 are the acceptance numbers to be used in double and

multiple sampling plans.

Sampling Plan # 2. Double Sampling Plan:



A sample consisting of n units of products is inspected, if the number of

defective is below A1 the lot is accepted, if it is above the second acceptance

number A2 (where A2< A1) the lot is rejected. If the number of defectives falls

between A1 and A2 the result is inconclusive and a second sample is drawn.

The rule again is similar to that of a single sampling plan. If the total number of

defectives of the two samples is below the pre-determinded acceptance

number A2, the lot is accepted otherwise rejected.

Example 2:

Let N = 600 A1 = 2, n1 = 30, n2 = 50, A2 = 4 then the interpretation of the above

data is given below:

(i) The lot consists of 600 products.

(ii) Take a sample of 30 randomly from 600 and inspect them.

(iii) If the number of defectives is < 2 accept the lot of 600 without further

inspection

(iv) If the number in 30 is more than 2 but < 4 take the second sample of 50

products from

From the remainder of N i.e., from (600-30) and inspect 50.

(v) If the total of defective in 50 and 30 together < 4 then accept the batch of

600 otherwise reject the lot of 600.




sample of size n.



Sampling Plan # 3. Multiple or Sequential Sampling Plans:



This is similar to the double sampling plan except that with the second sample

we have again an inconclusive range between A3 & A4. Below A3 the lot is

accepted above A4 it is rejected and if the total number of defectives is

between A3 and A4, a third sample is taken and so on.

Eventually after a number of samples the inspector must come to a final

decision and one critical limit is set (as in single sampling plan) which

determines whether the lot is accepted or rejected.




sample of size n.



To explain the procedure of this plan let us assume the following data:

Procedure:

(1) Take the sample of 60 items from the lot of N pieces and inspect them.

(2) If there is no defective accept the lot of N without any further inspection.

(3) If it contains > 3 defectives reject the lot of N.

(4) If it contains < 2 defectives take another sample of 30 pieces at random and

inspect them.

http://cdn.yourarticlelibrary.com/wp-content/uploads/2016/08/clip_image011_thumb2.png



(5) If the total number of defectives from the first and second sample

combined comes out to be 1 accept the lot of N.

(6) If the total number of defectives from the first and second samples

combined comes out to be > 4 reject the lot of N.

(7) If the total number of defectives < 4 and > 1 take third sample of 30 and

inspect the combined lot of 120.

(8) If the total number of defectives in the combined lot of 60+30+30 = 120

items is < 2, accept the lot of N.

(9) If the total number of defectives > 5, reject the lot of N pieces.

(10) If the total number of defectives is more than 2 and < 5, take another

sample of 30 and inspect. In this way sampling is continued.

Documents

Unit-3 Acceptance Sampling - Aligarh Muslim University