Decision Risk

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• Decision Risk
• Producers Risk conforming lots are rejected
• Consumers Risk nonconforming lots are
  accepted.

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Lot-by-Lot Acceptance Sampling by Attributes
Single Sampling
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Single Sampling N lot size the number of
items in the lot from which the sample is to be
drawn. n sample size the number of items
drawn at random from the lot c the maximum
allowable number of defective items in the
sample. More than c defectives in the sample will
cause rejection of the lot.
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• Type A OC Function for Single Sampling Plan
• Sampling Plan Specification
• N Lot Size
• n Sample Size
• c Acceptance number
• D true number of defectives in the lot
• X number of defective items in the random
  sample.

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• Type A OC Function for Single Sampling Plan
• Probability Distribution of X
• X H(N, D, n)
• Probability Mass Function of X

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• Type A OC Function for Single Sampling Plan
• OC Function

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• Example Single Sampling Plan A
• Determine and plot the OC Function for a single
  sampling plan specified by

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Example Solution Single Sampling Plan
OC(D)
D
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Concept Suppose that the lot size N is large
(theoretically infinite). Under this condition,
the distribution of the number of defectives d in
a random sample of n items is binomial with
parameters n and p, where p is the fraction of
defective items in the lot. An equivalent way to
conceptualize this is to draw lots of N items at
random from a theoretically infinite process, and
then to draw random samples of n from these lots.
Sampling from the lot in this manner is the
equivalent of sampling directly from the process.
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• Type B OC Function for Single Sampling Plan
• Sampling Plan Specification
• N Lot Size
• n Sample Size
• c Acceptance number
• N is infinite, or at least much larger than n
• D true number of defective items in the lot
• p proportion of the population that is
  defective, i.e.,
• X number of defective items in the random
  sample.

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• Type B OC Function for Single Sampling Plan
• Probability Distribution of X
• X B(n, p)
• Probability Mass Function of X

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Acceptance Sampling The probability of observing
exactly x defective items is for x
0, 1, . . ., n
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OC Function The probability of acceptance is the
probability that d is less than or equal to c,
or
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• Example Single Sampling Plan B
• Determine and plot the OC Function for a single
  sampling plan specified by
• Compare the OC curve for N500, n98, c2.

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Example Solution Single Sampling Plan
OC(p)
p
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Example The OC curve is developed by evaluating
PA(p) for various values of p. The following
table 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 98, 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, in the long run, to accept
74 of the lots and reject 26 of them.
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Example Probabilities of Acceptance for the
Single-Sampling plan n 89, c 2 Fraction
Probability of Defective, p Acceptance, Pa
(p) 0.005 0.9897 0.010 0.9397 0.020 0.736
6 0.030 0.4985 0.040 0.3042 0.050 0.1721
0.060 0.0919 0.070 0.0468 0.080 0.0230 0
.090 0.0109
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Example
-p
10
20
0
30
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-D
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• Single Sample Test Plan Design
• Probability Distribution of X
• P0 specified fraction defective
• P1 minimum acceptable fraction defective
• a producers risk
• b consumers risk

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• Test Procedure
• To test
• H0 p p0
• vs H1 p p1
• at the ? ? 100 level of significance,
• Obtain a random sample of size n
• Inspect the n items and determine the number, X,
  that are defective
• Reject H0 if x gt c, otherwise accept H0

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Single Sampling Plan
x
reject
number of defective items
c . . .
accept
2
1
0
n
0
n sample size x number of defective items c
maximum number of defective items for acceptance
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Operating Characteristic (OC) Function No
te that and
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OC Curve
1
1 - ?
?
p1
1
0
p
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Lot-by-Lot Acceptance Sampling by Attributes
Double Sampling
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Double Sampling A double-sampling plan is a
procedure in which, under certain circumstances,
a second sample is required before the lot can be
sentenced. A double- sampling plan is defined by
four parameters n1 sample size on the first
sample c1 acceptance number of the first
sample n2 sample size on the second sample c2
acceptance number for both samples
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Double Sampling - Advantages The principal
advantage of a double-sampling plan with respect
to single sampling is that it may reduce the
total amount of required inspection. Suppose
that the first sample taken under a
double-sampling plan is smaller than the sample
that would be required using a single-sampling
plan that offers the consumer the same
protection. In all cases, then, in which a lot is
accepted or rejected on the first sample, the
cost of inspection will be lower for double
sampling than it would be for single sampling.
It is also possible to reject a lot without
complete inspection of the second sample (called
curtailment of the second sample).
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Double Sampling - Disadvantages Double sampling
has two potential disadvantages 1. Unless
curtailment is used on the second sample, under
some circumstances double sampling may require
more total inspection than would be required in a
single-sampling plan that offers the same
protection. 2. Double-sampling is
administratively more complex, which may increase
the opportunity for the occurrence of inspection
errors. Furthermore, there may be problems in
storing and handling raw materials or component
parts for which one sample has been taken, but
that are awaiting a second sample before a final
lot dispositioning decision can be made.
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Double Sampling - The OC Curve The performance
of a double-sampling plan can be conveniently
summarized by means of its operating- characterist
ic (OC) curve. A double-sampling plan has a
primary OC curve that gives the probability
of acceptance as a function of lot of process
quality. It also has supplementary OC curves
that show the probability of acceptance as a
function of lot acceptance and rejection on the
first sample.
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Double Sampling Operation of the
double-sampling plan with n1 50, c1 1, n2
100, c2 3
Inspect a random sample of n1 50 from the lot
d1 number of observed defectives
Accept the lot
Reject the lot
d1 gt c1 3
d1 ? c1 1
Inspect a random sample of n2 100 from the lot
d2 number of observed defectives
Accept the lot
Reject the lot
d1 d2 ? c2 3
d1 d2 gt c2 3
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Example If denotes the probability of
acceptance on the combined samples, and
and denote the probability of acceptance on the
first and second samples, respectively, then
is just the probability that we will
observe d1 ? c1 1 defectives out of a random
sample of n1 50 items. Thus
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Example If p 0.05 is the fraction defective in
the incoming lot, then To obtain the
probability of acceptance on the second sample,
we must list the number of ways the second sample
can be obtained. A second sample is drawn only if
there are two or three defectives on the
first sample - that is, if c1 lt d1 ? c2.
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Example 1. d1 2 and d2 0 or 1 that is, we
find two defectives on the first sample and one
or less defectives on the second sample. The
probability of this is P(d1 2, d2 ? 1) P(d1
2) x P(d2 ? 1) (0.261)(0.037) 0.009
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Example 2. d1 3 and d2 0 that is, we find
three defectives on the first sample and no
defectives on the second sample. The probability
of this is P(d1 3, d2 ? 0) P(d1 3) x P(d2
0) (0.220)(0.0059) 0.001
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Example Thus, the probability of acceptance on
the second sample is The probability of
acceptance of a lot that has fraction defective p
0.05 is therefore
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Double Sampling - The OC Curve OC
Curves for the double-sampling plan with n1
50, c1 1, n2 100, c2 3
Probability of acceptance on combined sample
Probability, P
Probability of rejection on first sample
Probability of acceptance on first sample
Lot fraction defective, p
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Rectifying Inspection Programs
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• Rectifying Inspection Programs
• Acceptance-sampling programs require corrective
  action when lots are rejected.
• 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 vendor, or replaced from
  a stock of known good items. Such sampling
  programs are called rectifying inspection
  programs.

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• Rectifying Inspection Programs (continued)
• The inspection activity affects the final quality
  of the outgoing product. Suppose that the
  incoming lots to the inspection activity have
  fraction defective, po. 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 p0.
  Consequently, the outgoing lots from the
  inspection activity are a mixture of lots with
  fraction defective p0 and fraction defective
  zero, so the average fraction defective in the
  stream of outgoing lots is p1, which is less that
  p0, and serves to correct lot quality.

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Rectifying Inspection
Rejected lots
Fraction defective0
Incoming lotsFraction defectivep0
Outgoing lotsFraction defectivep1
Fraction defectivep0
Accepted lots
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• Rectifying Inspection Programs (continued)
• Used in situations where the manufacturer wishes
  to know the average level of quality that is
  likely to result at a given stage of the
  manufacturing operations.
• Used either at receiving inspection, in-process
  inspection of semi-finished products, or at final
  inspection of finished goods
• The objective of in-plant usage is to give
  assurance regarding the average quality of
  material used in the next stage of the
  manufacturing operations.

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• Handling of Rejected Lots
• The best approach is to return rejected lots to
  the vendor, and require it to perform the
  screening and rework activities.
• Has the psychological effect of making the vendor
  responsible for poor quality
• May exert pressure on the vendor to improve its
  manufacturing processes or to install better
  process controls.
• Screening and rework take place at the consumer
  level because the components or raw materials are
  required in order to meet production schedules.

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• Average Outgoing Quality
• Widely used for the evaluation of a rectifying
  sampling plan.
• Is the quality in the lot that results from the
  application of rectifying inspection
• Is the average value of lot quality that would be
  obtained over a long sequence of lots from a
  process with fraction defective p.

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Average Outgoing Quality (AOQ) The average
fraction defective, called average outgoing
quality is Where the lot size is N and that
all defectives are replaces with good units.
Then in lots of size N, we have

1. N items in the lot that, after inspection,
   contain no defectives, because all discovered
   defectives are replaced
2. N n items that, if the lots is rejected,
   contain no defectives
3. N n items that, if accepted, contain (N-n)p
   defectives

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Example 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 That is, the average outgoing
quality is at 0.93 defective.
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Example 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.
AOQ
fraction defective, p
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Average Total Inspection (ATI) Another
important measure relative to rectifying
inspection is the total amount of inspection
required by the sampling program. 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. Of the lot quality is 0
lt p lt 1, the average amount of inspection per lot
will vary between the sample size n and the lot
size N.
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Average Total Inspection (ATI) Consider our
previous example with N 10,000, n 89, and c
2, and p 0.01. Then since Pa 0.9397, we
have Remember this is an average number of
units inspected over many lots with fraction
defective p 0.01.
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Determination of optimum quality level, p
cost toachieve p
totalcost
cost ofinspectionper lot
cost
0
p
1
incoming quality level p