basic principles of quality acceptance

8/11/2019 Basic Principles of Quality Acceptance

1/22

Module 20:

Basic principles of Acceptance sampling

(Most of the material in these notes is based on the material given in Introduction to Statistical

Quality Control by Douglas C. Montogomery)Acceptance sampling is a statistical method that enables us to base the accept-

reject decision on the inspection of a sample of items from the lot. The items of interest

can be incoming shipments of raw materials or purchased parts as well as finished goods

from final assembly. Typical application of acceptance sampling is for lot disposition,

sometimes referred to as lot sentencing, for receiving inspection activities. Accepted lots

are put into production. Rejected lots may be returned to supplier or subjected to other

lot-disposition action. Sampling methods may also be used during various stages ofproduction.

Acceptance sampling is a form of quality testing used for incoming materials or

finished goods. e.g., purchased material & components. The procedure of acceptance

sampling consists of the following steps.

Take one or more samples at random from a lot (shipment) of items

Inspect each of the items in the sample

Decide whether to reject the whole lot based on the inspection results

Acceptance plan: It consists of procedures for inspecting incoming materials or finished

goods, identifies

Type of sample

Sample size ( n)

Criteria ( c) used to reject or accept a lot

An acceptance plan is negotiated and decided by Producer (supplier) & consumer (buyer)

must negotiate.Example 1:

An inspector takes a sample of 20 items from a lot. His policy is to accept a ot if no more

than 2 defectives are found in the sample. Assuming that a lot is 5 % defective, what is

the probability that he will accept the lot or reject the lot?

In this example, we have


2/22

n=20, c =2 and p=0.05

P(acceptance of lot)=f(0) + f(1) + f(2)

3585.0)05.01(05.0)!020(!0

!20)0( )020(0 =

=

f

3774.0)05.01(05.0)!120(!1

!20)1( )120(1 =

=

f

1887.0)05.01(05.0)!220(!2

!20)2( )220(2 =

=

f

P( acceptance of lot)=0.3585 + 0.3774 = 0.1887 = 0.9246

Acceptance sampling has following advantages over 100% inspection including:

Usually less expensive

Less product damage due to less handling

Fewer inspectors required

Provides only approach possible if destructive testing must be used

The lot is disposed off in shorter time so that scheduling and delivery are improved

The problem of monotony and inspector error induced by 100 percent inspection isminimized

Rejection (rather than sorting) of nonconforming lots tends to dramatize the qualitydeficiencies and to urge the organization to look for preventive measures

Proper design of the sampling plan commonly requires study of the actual level of qualityrequired by the user. The resulting knowledge is a useful input to the overall qualityplanning.

The acceptance sampling procedure can be represented as follows.


3/22

The three important aspects of acceptance sampling are:

It is the purpose of acceptance sampling to sentence the lots, not to estimate the lotquality

Acceptance sampling plans do not provide any direct form of quality control. They onlyaccept or reject lots. Process controls are used to control and improve quality

Acceptance sampling is an audit tool to ensure that the output of a process conforms torequirements

Type I and Type II errors:

If the null hypothesis H 0 is that the lot quality is good, then the following situations andthe corresponding types of error can be considered.

Type I & Type IIErrors

State of the Lot

H0 True

Good Quality Lot

Ho False

Poor Quality LotDecision

Accept H 0

Accept the lot

Correct Decision Type II Error

Accepting a PoorQuality Lot

Reject H 0 Reject the lot

Type I ErrorRejecting a GoodQuality Lot

Correct Decision


4/22

Economics of inspection: Let

N =number of items in lot, n=number of items in sample, p=proportion defective in lot

A=damage cost incurred if a defective slips through inspection, I =inspection cost per item

P a=probability that lot will be accepted by sampling plan

Assuming

1) No inspection errors

2) Cost to replace a defective found in inspection is borne by the producer or is smallcompared to the damage or inconvenience caused by the defective

Economic comparison of inspection alternatives :

100% inspection is justified in case of equipment where the cost of finding and correctingdefective increases rapidly for each major stage that the product moves to from production to thecustomer.

An electronic component may cost $1 at incoming inspection, $10 at the PCB stage, $100 at thesystem level and $1000 in the field. In such a case it is worthwhile to spend more on inspectionat initial stage rather than spending much more at a later stage.

It also needs to be noted that Neither sampling nor 100% inspection can guarantee that everydefective item in a lot will be found.

Sampling involves a risk that the sample will not adequately reflect the conditions of the lot. TheType I error discussed earlier is also called Producers risk or also as risk. The Type II error isalso known as Consumers risk or risk.

Alternative Total cost

No inspection NpA

Sampling nI + (N-n)pAP a + (N-n)( 1-P a)I

100% inspection NI


5/22

Types of Sampling Plans:

Single sampling plan

Sample size n (randomly selected) Acceptable number c

Double-sampling plans

One of the following decisions, based on the information in the initial sample, ismade

1. Accept the lot2. Reject the lot3. Take a second sample

If a second sample is taken, information in both the samples is used todecide acceptance or rejection

Multiple sampling-plans

Extension of Double-sampling plan, more than two samples may be required Ultimate extension of sampling is Sequential sampling


6/22

Module 21

Single sampling plans for attributes

A lot of size N has been submitted for inspectionSample size n Acceptance number of defective c Lot sentencing is based on one sample of size n

Operating characteristic (OC) curveCurve plots the probability of accepting the lot ( P a) versus the lot fraction defective ( p)

Example 2: N=10000 , n = 89 and c=2The probability of acceptance for different values of p, fraction defective are shown in thefollowing table.

The OC curve is shown in the figure below.


7/22

The OC curve shows discriminatory power of the sampling plan.In the plan n=89, c=2, if the lots are 2% defective, P a is approximately 0.74. This means in 100lots 74 are expected to be accepted and 26 to be rejectedThe ideal OC curve which has perfect discriminatory power is as shown below.

Comparison of two different OC curves with regard to their discriminatory power is done in thefollowing figure.


8/22

The and risks referred to earlier are shown on the OC curve as below.

The following figure shows the effect of sample size on the OC curves. It is noted that thediscriminatory power of the sampling plan increases with sample size.


9/22

The following important observations can be made from the above.

The ideal OC curve cannot be obtained in practice The ideal OC curve can be approached by increasing the sample size. This is seen from the previous slide in which acceptance number c is kept proportional to

n. The greater the slope of the OC curve, the better is its discriminatory power

Effect of c on the OC curve: This is shown in the following figure

Changing c does not dramatically change the slope of the OC curveAs c is decreased the OC curve shifts to the left.Plans with lower values of c provide discrimination at lower levels of fraction defective

Quality Indices for Acceptance Sampling Plans :Acceptable Quality Level (AQL) represents the poorest level of quality for the vendorsprocess that the consumer would consider to be acceptable as a process average.Lot Tolerance Percent Defective (LTPD) is the poorest level of quality that the consumer

is willing to accept in an individual lot. LTPD is also called Rejectable quality Level(RQL) or Limiting Quality Level (LQL)AQL and LTPD are not characteristics of sampling plan. AQL is a property of thevendors manufacturing process. LTPD is a level of the lot quality specified by theconsumer.


10/22

Indifference Quality Level (IQL) is a quality level somewhere between the AQL andLQL. It is frequently defined as the quality level having a probability of acceptance of 0.5for a given sampling planAverage Outgoing Quality (AOQ) is the average quality of rejected and accepted lots

Type-A and Type-B OC Curves:

Type-B OC curve is constructed with the assumption that the sample is taken from alarge lot or from a stream of lots selected at random from a process.Binomial distribution is the exact probability distribution for calculating the probabilityof acceptance.Type-A OC curve is used to calculate probability of acceptance for an isolated lot offinite size.The exact sampling distribution of the number of defective items in the sample is the

hypergeometric distribution.As the size of the lot increases, the lot size has a decreasing impact on the OC curve.If the lot size is at least 10 times the sample size, the two types of curves are almost same.Type-A OC curve always lies below the Type-B OC curve.


11/22

OC curve for single-sampling with c=0: The curves are shown in the figure below

Sampling plans with c=0 have OC curves that are complex through out their range. Probability of acceptance drops very rapidly even for small values of lot fraction

defective. This is hard on the vendor and some times uneconomical for the consumer.

From slide 14, for n=89, AQL=1%, P a=0.78 for c=1 and P a=0.41 for c=0

Designing a Single-Sampling Plan with a Specified OC Curve

Suppose we wish to construct a sampling plan such thatP a=1- for lots with fraction defective p1

and P a= for lots with fraction defective p2 Then ,n and c are solution to the equations shown in below

The equations are non-linear and not easy to solve.The nomograph shown below can be used for this purpose.


12/22

The illustration is for the case p1=0.01, =0.05, p2=0.06 and =0.1

OC curve can also be drawn using Poisson Approximation.

The Poisson distribution yields a good approximation when n is at least 16, N at least 10times n and p is less than 0.1.

The Poisson distribution as applied to acceptance sampling is

where, P(r) is the probability of exactly r defectives in sample of n.!

)()(r

r npnper P

=


13/22

Module 22

Rectifying inspection

Corrective action required after rejection of lots.Generally, rejected lots are 100% inspected, with defective items removed for rework orreplaced by good items.Such sampling programs are called Rectifying Inspection Programs

Average Outgoing Quality (AOQ) is the quality in the lot that results from the applicationof rectifying inspection.

If the lots are from a process with p fraction defective, then In lots of size N , ,1. n items contain no defectives as defectives are replaced2. N-n items contain no defectives if the lot is rejected3. N-n items contain p(N-n) defectives if the lot is accepted

Expected number of defectives is P a(N-n) Therefore, AOQ = P a p(N-n)/N

If N is large relative to n, Approximately, AOQ= P a p The curve that plots AOQ against incoming quality, p is called AOQ curveFrom the AOQ curve, we note that average outgoing quality is good both for good and

very bad incoming qualities.


14/22

The maximum ordinate on the curve represents the worst possible average quality thatresults from the rectifying plan.This point is called the Average Outgoing Quality Limit (AOQL)AOQL is an average level of quality, across a large stream of lots and it does not giveassurance that an isolated lot will have quality no worse than AOQL.If the lot has no defectives, the amount of inspection per lot = n

If all are defectives, the amount of inspection per lot = N If the lot quality is 0< p


15/22

Double Sampling PlanDecision of acceptance or rejection based on two samplesA lot may be accepted if the sample is good enough or may be rejected if the sample isbad enoughIf the first sample is neither good enough nor bad enough, decision is based on evidence

of first and second sample combined.This is shown in the following .


16/22

OC Curve of a double sampling plan:

Example 3:

Draw the OC curve of the single-sampling given with sample size 300 and acceptance number 5.We first fix tentative probabilities of acceptance (Pa)Consider the table shown below

n np P P a P ap

300 0.98

300 0.95

300 0.70

300 0.50

300 0.20

300 0.05

300 0.02

Search the cumulative Poisson distribution table under c=5 column for a value close to therequired P a value and note the corresponding np value . For this np calculate the correspondingp. Complete the table and draw the required curve.


17/22

n np P P a P ap=AOQ

300 2.0 0.0067 0.983 0.0065

300 2.6 0.0087 0.951 0.00827

300 4.4 0.0147 0.70 0.0106

300 5.6 0.0187 0.512 0.00957

300 7.8 0.025 0.210 0.00526

300 10.5 0.035 0.05 0.00175

300 12.0 0.04 0.02 0.0008

From the above table the OC curve can be constructed which is given in the following


18/22

Example 4:In a double sampling plan, N=5000, n1=100, c1=0, n2=100, c2=1

(a) Use Poissons table to compute the probability of acceptance of a 1% defective lot.(b) assume that a lot rejected by this sampling plan will be 100% inspected. What will be the

AOQ if the submitted product is 1% defective? Considering the inspection of bothrejected lots, what will be the avarage number of articles inspected per lot if thesubmitted product is 1% defective?

No of defective articles = 5000x0.01=50 No of non-defective articles= 5000-50=4950(a) n1p=100x0.01=1

From Poissons table, P I(0)=0.368Accept the lot if

I) Zero defective in the first sampleII) One defective in the first sample and zero defective in the second sample

PI(1)= 0.736-0.368=0.368For the second sample, N=5000-100=4900,No of defectives=49P=49/4900=0.01, n 2p=100x0.01=1PII(0)=0.368Therefore, probability of acceptance,Pa=PI(0) + P I(1)xP II(0)

= 0.368+0.368x0.368=0.5034(b) AOQ=P axp=0.5034x0.01=0.005034 or 0.5034 %

Average no of articles inspected =PI(0)n1+PI(1)(n1+n2) + (1-Pa)N

i.e. 0.368 x100+0.368x0.368x200+ (1-0.5034)5000=2547

Designing a single sampling plan with stated value of P 0.5The quality P 0.5 is the lot or process quality that has a probability of acceptance of 0.5 and iscalled as the indifference quality.The single acceptance plan is designed by using the following approximate formula

n = (c + 0.67) / P 0.5A vendor selects P 0.5 = 0.02 and acceptance number 1. How many lots are likely to accepted if hehas 1000 lots each of quality 2.5% and 1.5%?n=(1+0.67)/0.02 = 84.

1) np = 84(0.015)=1.26. From cumulative Poissons table ( c=1), P a=0.6412) np = 84(0.025) = 2.1, From table, P a = 0.38.

No. of lots likely to be accepted = 0.641(1000) + 0.38(1000)= 1021


19/22

Module 23

Multiple Sampling Plans A multiple sampling plan is an extension of double sampling plan more than two

samples are required If at any stage the number of defective items is less than or equal to the corresponding

acceptance number, the lot is accepted. If at any stage the number of defective items is greater than or equal to the corresponding

rejection number, the lot is rejected.These plans generally involve less total inspection compared to single or double sampling plansguaranteeing the same protection, but usually require higher administrative costs and highercaliber inspection personnel

Sample Samplesize

Combined samples

Size Acc. No. Rej. No.

First n 1 n 1 c1 r 1

Second n 2 n 1+n 2 c2 r 2

Third n 3 n 1+n 2+n 3 c3 r 3

Fourth n 4 n 1+n 2+n 3+n 4 c4 r 4

Fifth n 5 n 1+n 2+n 3+n 4+n 5 c5 c5+1


20/22


21/22

The equations for the two limit lines for specified values of p1, 1- , p2, and are

(Acceptance line) (Rejection line)where

And

Numerical example: Design a sequential sampling plan for the following specifications:

= 0.05, p1 = 0.1 = 0.2, p2 = 0.3

X A = -1.154 + 0.186 n and

X B = 2.054 + 0.186 n Instead of using a graph to determine the lot disposition , the sequential sampling plancan be displayed in the form of a table.The entries in the table are found by substituting values of n into the above equations.Acceptance and rejection numbers must be integers. Acceptance no. is the next integerless than or equal to X A. Rejection no. is the next integer greater than or equal to X B.

Characteristics of a good Acceptance Plan he indices such as AQL, AOQL, etc should reflect the needs of the consumer and

producer. The sampling risks (OC curve) should be known. Both producer and consumer should

have adequate protection. Minimum total cost Flexible enough to take into account changes in N,p, etc. Should be useful in estimating individual lot quality and long run quality


22/22

basic principles of quality acceptance

Documents