Acceptance Sampling and Operating Characteristic Curves

© 2006 Prentice Hall, Inc. S6 – 1
Acceptance Sampling
 Form of quality testing used for
incoming materials or finished goods
 Take 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
 Only screens lots; does not drive
quality improvement efforts
N.R. Abeynayake, Ph.D

• One sample of items is selected at
random from a lot and the
disposition of the lot is determined
from the resulting information. These
plans are usually denoted as (n,c)
plans for a sample size n, where the
lot is rejected if there are more than c
defecs. These are the most common
(and easiest) plans to use
Single sampling plans

Operating Characteristic
Curve
 Shows how well a sampling plan
discriminates between good and
bad lots (shipments)
 Shows the relationship between
the probability of accepting a lot
and its quality level

Properties of OC curves
• Ideal curve would be perfectly
perpendicular from 0 to 100% for a given
fraction of defectives.
• Ideally, if up to 4% non-conforming is
acceptable, the probability of acceptance
containing less than 4% non-confirming
should be one and probability of
acceptance a batch containing more than
4% on-confirming should be zero.
Typical OC Curve

Naturally, we would prefer a highly discriminating
sampling plan and OC curve. If the entire
shipment of parts has an unacceptably high level
of defects, we hope the sample will reflect that
with a very high probability (preferably 100%) of
rejecting the shipment.
OPERATING CHARACTERISTIC
(OC) CURVES

• If such situation the shape of OC
curve is:
% Def_1
P(Acceptance)_1
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1.0
0.8
0.6
0.4
0.2
0.0

Return whole
shipment
The “Perfect” OC Curve
% Defective in Lot
P(Accept
Whole
Shipment)
100 –
75 –
50 –
25 –
0 –
| | | | | | | | | | |
0 10 20 30 40 50 60 70 80 90 100
Cut-Off
Keep whole
shipment

• We assume, N= lot size. It is also assumed
that N is very large, as compared to the
sample size n,
• Then the number of defectives, d, in a
random sample of n items is
approximately binomial, B(n,p) where p is
the fraction of defectives per lot. Then,
Probability of Acceptance
d
n
d
p
p
d
n
d
n
d
Y
P 



 )
1
(
)!
(
!
!
)
(

• Thus, the probability of acceptance,
)
(
)
1
(
)!
(
!
!
)
(
0
say
p
p
p
d
n
d
n
c
Y
P C
d
n
d
c
d




 


Probability of Acceptance

Example: N=1000, n=52, c=3
0.01 0.02 0.03 0.04 0.05 0.06 0.07
0.998153 0.979765 0.929537 0.845989 0.738317 0.619594 0.501847
0.08 0.09 0.10 0.11 0.12 0.13
0.393763 0.300280 0.223187 0.162066 0.115198 0.0802873

% Def
P(Acceptance)
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1.0
0.8
0.6
0.4
0.2
0.0
OC Curver for the given sampling plan

(a) Perfect Discrimination for Inspection Plan. (b) OC Curves for
Two Different Acceptable Levels of Defects (c = 1, c = 4) for the
Same Sample Size (n = 100). (c) OC Curves for Two Different
Sample Sizes (n = 25, n = 100) but Same Acceptance Percentages
(4%). Larger sample size shows better discrimination.

• you will be able to see that the OC curve
for n = 25, c = 1 rejects more good lots and
accepts more bad lots than the second
plan.
OC CURVES

In other words, the probability of
accepting a more than satisfactory lot
(one with only 1% defects) is 99% for n =
100, but only 97% for n = 25. Likewise, the
chance of accepting a “bad” lot (one with
5% defects) is only 44% for n = 100,
whereas it is 64% using the smaller
sample size.

• Draw OC curve for n=50, r=2. Show on
your graph the ideal shape of operating
characteristic if up to 5% non confirming
items are acceptable.
Activity (practical)

AQL and LTPD
 Acceptable Quality Level (AQL)
 Maximum proportion of defectives
which the consumer finds definatly
acceptable.
 Lot Tolerance Percent Defective
(LTPD)
 Quality level we consider bad
 Consumer (buyer) does not want to
accept lots with more defects than
LTPD

Producer’s and Consumer’s
Risks
 Producer's risk ()
 Probability of rejecting a good lot
 Probability of rejecting a lot when the
fraction defective is at or above the
AQL
 Consumer's risk (b)
 Probability of accepting a bad lot
 Probability of accepting a lot when
fraction defective is below the LTPD

An OC Curve
Probability
of
Acceptance
Percent
defective
| | | | | | | | |
0 1 2 3 4 5 6 7 8
100 –
95 –
75 –
50 –
25 –
10 –
0 –
 = 0.05 producer’s risk for AQL
b = 0.10
Consumer’s
risk for LTPD
LTPD
AQL
Bad lots
Indifference
zone
Good
lots

OC Curves for Different
Sampling Plans
n = 50, c = 1
n = 100, c = 2

• A shipment of 2,000 portable battery units for
microcomputers is about to be inspected by a
Malaysian importer. The Korean manufacturer
and the importer have set up a sampling plan
in which the risk is limited to 5% at an
acceptable quality level (AQL) of 2% defective,
and the risk is set to 10% at Lot Tolerance
Percent Defective (LTPD) = 7% defective. We
want to construct the OC curve for the plan of
n = 120 sample size and an acceptance level of
c ≤ 3 defectives.

• Both firms want to know if this plan will
satisfy their quality and risk requirements.
By varying the percent defectives (p) from
.01 (1%) to .08 (8%) and holding the
sample size at n = 120, we can compute
the probability of acceptance of the lot at
each chosen level. The values for P
(acceptance) calculated in what follows
are then plotted to produce the OC curve
shown in below

• Now back to the issue of whether this OC
curve satisfies the quality and risk needs
of the consumer and producer of the
batteries. For the AQL of p = .02 = 2%
defects, the P (acceptance) of the lot =
.779. This yields an risk of 1 - .779 = .221,
or 22.1%, which exceeds the 5% level
desired by the producer. The risk of .032,
or 3.2%, is well under the 10% sought by
the consumer. It appears that new
calculations are necessary with a larger
sample size if the level is to be lowered.

Type A
Gives the probability of acceptance for an
individual lot coming from finite production. That
is based on lot size (N) using hypergeometric
distributions.
Type B
Give the probability of acceptance for lots coming
from a continuous process. They are closely
approximated by assuming an infinite lot size
using binomial distribution or Poisson
distribution.
Types of OC Curves

Type A and Type B OC Curves for n=13, c=0 (N=50, 200)

• A common procedure, when sampling and
testing is non-destructive, is to 100%
inspect rejected lots and replace all
defectives with good units.
• AOQ's refer to the long term defect level
for this combined lot acceptance sampling
procedure (LASP) and 100% inspection of
rejected lots process.
Average Outgoing Quality

• 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, then
where N is the lot size.
N
n
N
P
p
AOQ a )
( 


• Let
• Probability of acceptance a lot =
• Probability of rejection = ( )
• Proportion of lots accepted = (k- number
of lots of size N)
• Proportion of lots rejected = =
• The number of defects in a lot= Np
• Number of defects originally presents in k lots =
kNp
a
p
a
p

1
a
p
k *
a
p
k
k *
 )
1
(
* a
p
k 
N
N
n
n

• Number of items remaining unchecked
= (N-n)
• No. of defects in the outgoing lots
= p k(N-n) … (2)
Thus =
=
When n <<<<< N then tend to zero and AOQ
tends to
kN
n
N
k
pp
AOQ a )
( 

)
(
N
n
N
ppa

a
pp

where
P = true percent defective of the lot
Pa = probability of accepting the lot
N = number of items in the lot
n = number of items in the sample
AOQ =
(P)(Pa)(N - n)
N

• Example:
• Let N = 10000, n = 52, c = 3, and p is the
quality of incoming lots = 0.03. Find AOQ.
= 0.9295 (From Binomial)
=0.9300 (From HG – N=10000, M=300,
p=0.03 c=3)
• AOQ = = (0.03)(0.930)(10000-52) / 10000 =
0.02775

Setting p = .01, .02, ..., .12, the
following table can be derived.
P Pa AOQ
Binomial HG Binomial HG
0.01 0.0010
0.02 0.0196
0.03 0.0278
0.04 0.0338
0.05 0.0369
0.06 0.0372
0.07 0.0351
0.08 0.0315
0.09 0.0270
0.10 0.0223
0.11 0.0178
0.12 0.0138

p= Icoming quality level
AOQ
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.04
0.03
0.02
0.01
0.00
AOQ curve

0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.2
0.6
1
1.4
1.8
2.1
2.3
2.5
3.5
4.5
Incoming quality (100p)
AOQ
(%)
AOQ - if rejected lots
are screened
AOQ - without
inspection

• The maximum cordinate on the AOQ curve
represents the worst possible quality that
results from the rectifying inspection
program.
• This maximum value (or percentage of
defective which the consumer finds
definitely acceptable) is called average
outgoing quality limit (AOQL).
• From the above curve AOQL = 0.0372 at p
= 0.06
Limit (AOQL )

Example 2: Find AOQ for the sampling plan n=75,
c=1 & N is large
% of defectives (p’) Pa AOQ =p1
0.2 0.99 0.198
0.4 0.963 0.3852
0.6 0.925 0.555
0.8 0.878 0.7024
1 0.827 0.827
1.2 0.772 0.9264
1.4 0.718 1.0052
1.6 0.663 1.0608
1.8 0.61 1.098
2 0.558 1.116
2.1 0.533 1.1193
2.2 0.509 1.1198
2.3 0.486 1.1178
2.4 0.463 1.1112
2.5 0.441 1.1025
3 0.343 1.029
3.5 0.262 0.917
4 0.199 0.796
4.5 0.15 0.675
5 0.112 0.56

1. If a sampling plan replaces all defectives
2. If we know the incoming percent
defective for the lot
We can compute the average outgoing
quality (AOQ) in percent defective
The maximum AOQ is the highest percent
defective or the lowest average quality
and is called the average outgoing quality
level (AOQL)

• What is the total amount of inspection when
rejected lots are screened?
• If all lots contain zero defectives, no lot will be
rejected.
• If all items are defective, all lots will be
inspected, and the amount to be inspected is N.
• Finally, if the lot quality is p (0 < p < 1), the
average amount of inspection per lot will vary
between the sample size n, and the lot size N.
Choosing a sampling plan to minimize
Average Total Inspection (ATI)

• Let pa= probability of lot acceptance,
p=probability of defectives in the lot.
Then
d
n
d
c
d
p
p
d
n
n
N
n
ATI 













  )
1
(
1
)(
(
0
)
1
)(
( a
p
n
N
n
ATI 



Thus the average fraction
inspected (AFI) is given by N
ATI
AFI 

• let N = 10000, n = 52, c = 3, and p = .03. We know
from the OC curve that = 0.930. Then ATI = 52
+ (1-.930) (10000 - 52) = 753.
Example,
a
P
P ATI
0.01 70
0.02 253
0.03 753
0.04 1584
0.05 2655
0.06 3836
0.07 5007
0.08 6083
0.09 7012
0.10 7779
0.11 8388
0.12 8854
0.13 9201
0.14 9453

P
ATI
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
10000
8000
6000
4000
2000
0
Scatterplot of ATI vs P

• Suppose a vendor supplies pens with your
company logo that you give away at trade
shows. You receive the pens in lots of 5000
and have been frustrated by the fact that many
of them don't work properly. You decide to
implement a sampling plan so you can either
accept the entire lot or reject the entire lot. You
are hoping to send a message to your supplier
that poor quality pens will not be accepted.
You and the supplier agree that the AQL is
1.5% and the RQL is 10%.
Example,

• Interpreting the results
• For each lot of 5000 pens, you need to randomly
select and inspect 52 of them. If you find greater
than 2 defectives among these 52 pens, you
should reject the entire lot. For 2 or less defective
pens, accept the entire lot.
• In this case, the probability of acceptance at the
AQL (1.5%) is 0.957 and the probability of
rejecting is 0.043. When the sampling plan was
set up, the consumer and supplier agreed that
lots of 1.5% defective would be accepted
approximately 95% of the time to protect the
producer. The probability of accepting at the RQL
(10%) is 0.097 and the probability of rejecting is
0.903. The consumer and supplier agreed that
lots of 10% defective would be rejected most of
the time to protect the consumer.

• The AOQ level is 1.4% at the AQL and 0.95% at
the RQL. This is because when lots are very good
or very bad the outgoing quality will be good
because of the rework and reinspection for poor
lots.

• Average Outgoing Quality Limit (AOQL) = 2.603 at
4.300 percent defective represents the worse
case outgoing quality level.
• The ATI per lot represents the average number of
pens inspected at a particular quality level. For
the quality level of 1.5% defective, the average
total number of pens inspected per lot is 266.2.
For the quality level of 10% defective, the average
total number of pens inspected per lot is 4521.9.
•

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