Inspection of Goods

1
Inspection of Goods

• Why inspect?
• If each part guaranteed to be as good as
  advertised, no need for inspection!
• Tradeoffs
• inspection gt cost of inspection
• no inspection gt risk/cost of defectives
• Breakeven?

2
Inspection of purchased parts from a vendor -
Example

• Cost of a defective part 10
• Inspection cost per part 0.30
• Suppose, on average, 4 percent of the parts are
  defective.
• For a lot-size of N1000
• If we inspect every part, cost
  (0.30)(1000)300.
• If we do not inspect, cost (0.04)(1000)(10)
  400.
• So we should inspect every part!
• What if the inspection cost is higher?
• What if the defective percentage is higher?

3
100 Inspection

• If (cost of part inspection) lt
• (defective percentage)(cost of defective part)
• expected cost due to defective part
• then inspect every part,
• otherwise, do not inspect.
• What if we do not know the defective percentage?
• Multi-vendor
• New vendor, not enough historical data

4
The Purpose of Acceptance Sampling

• Acceptance sampling is a process of inferring the
  quality of a large number of items based on the
  quality of a small sample of the items.
• Purposes
• Determine quality level
• Ensure quality is within predetermined level

5
Acceptance Sampling

• Advantages
• Economy
• Less handling damage
• Fewer inspectors
• Upgrading of the inspection job
• Applicability to destructive testing
• Entire lot rejection (motivation for improvement)
• Disadvantages
• Risks of accepting bad lots and rejecting
  good lots
• Added planning and documentation
• Sample provides less information than 100-percent
  inspection

6
Statistical Sampling--Data

• Sampling to accept or reject the immediate lot of
  product at hand.
• Go / no-go classification
• each part either classified as defective or good
• classification can be based on a set of
  attributes
• Attributes
• Defectives--refers to the acceptability of
  product across a range of characteristics.
• Defects--refers to the number of defects per
  lot--may be higher than the number of defectives.

7
Single Sampling Plan

• Procedure
• 1. Take a random sample of n items from a
• lot and inspect each item .
• 2. Reject the entire lot if more than c items
• defective otherwise, accept entire lot.
• A simple goal
• Determine
• (1) how many units, n, to sample from a lot, and
• (2) the maximum number of defective items, c,
  that can be found in the sample without rejecting
  the lot.

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Risk

• Acceptable Quality Level (AQL)
• Max. acceptable percentage of defectives, defined
  by consumer/producer.
• Producers risk (a )
• The probability of rejecting a good lot.
• Lot Tolerance Percent Defective (LTPD)
• Percentage of defectives that defines consumers
  rejection point.
• Consumers risk (b)
• The probability of accepting a bad lot.

9
Operating Curves

• p actual proportion defective items
• (probability of a part being defective)
• x number of defective items in the batch
• n sample size

Binomial approx.
Poisson approximation
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Example Acceptance probability
Suppose p 0.02, and c3.
Note that probability of acceptance depend on
p, which we do not know!
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Operating Characteristic Curve

• For each value of p, we can compute the
  probability of acceptance

OC(p) Prob(Accepting the lot true proportion
of defective is p) Prob(x lt c true
proportion of defective in lot is p)

• How does the operating characteristic curve
  change when
• c increases?
• n increases (for fixed c/n)?
• lot size N increases?

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Operating Characteristic Curve
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Operating Characteristic Curve
1
0.9
0.8
0.7
0.6
0.5
Probability of acceptance
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
11
12
Percent defective
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Comments

• 1. For any fixed ratio c/n, the large n is the
  better differentiation btw good and bad
• 2. For a fixed n, the larger c is, the larger
  acceptance rate, ??, ??
• 3. For a fixed n, the smaller c is, ??, ??.

15
The Ideal OC Curve

• Suppose AQL 0.02, we want a sampling procedure
  that has an operating characteristic curve like

1
0.9
0.8
0.7
Probability of acceptance
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
11
12
AQL
Percent defective
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Balancing the producer and consumer risks

• The true value of p unknown
• Even if p known, sampling involves randomness,
  and we could still
• reject a lot even if p lt c/n
• accept a lot even if p gt c/n
• The OC curve gives an indication of the values of
  a and b.

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Example Sampling

• Suppose we receive a shipment of 3000 items, AQL
  0.02, LTPD 0.06, n 60, c 3.
• ? probability of rejecting a batch with
  defective rate p0.02.
• probability that four or more defective item
• prob(x ? 4 p 0.02)
• 1- prob(x ? 3 p 0.02) 1-0.9662
  0.0338
• where (np) 1.2

? probability of acceptance a batch with
pLTPD0.06 prob(x ? 3 p0.06) 0.5153
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Example Sampling

• n 120, c 6
• ? prob(x ? 7 p0.02) 1 - prob(x ? 6)
• 1 - 0.9884
• 0.0116
• ? prob(x ? 6 p 0.06) 0.420

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Single Sampling Plan Design

• Determine appropriate values for a and b.
• Choose n and c so that
• Prob(Reject lot) lt a if true p lt AQL
• Prob(Accept lot) lt b if true p gt LTPD
• How?
• (Idea With Poisson approx., can compute n(AQL)
  given c and a, can compute n(LTPD) given c and b.
  For fixed a and b, can make a table of (c,
  LTPD/AQL).)

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Example Acceptance Sampling
Zypercom, a manufacturer of video interfaces,
purchases printed wiring boards from an outside
vender, Procard. Procard has set an acceptable
quality level of 1 and accepts a 5 risk of
rejecting lots at or below this level. Zypercom
considers lots with 3 defectives to be
unacceptable and will assume a 10 risk of
accepting a defective lot. Develop a sampling
plan for Zypercom and determine a rule to be
followed by the receiving inspection personnel.
AQL? a? LTPD? b?
21
Example Continued
n (AQL) 3.286
How can we determine the value of n? What is our
sampling procedure?
For given a and b
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Example Continued
c 6, from Table n (AQL) 3.286, from Table AQL
.01, given in problem n(AQL/AQL) 3.286/.01
328.6, or 329 (always round up)
Sampling Procedure Take a random sample of 329
units from a lot. Reject the lot if more than 6
units are defective.
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Sampling Plans

• 1. Single sampling plan.
• 1. Draw a single random sample n items
• 2. If the number of defective items is more than
  c, we reject the batch.
• 2. Double-sampling plans
• (n1, c1, c2 n2, c3)
• 1. Examine an initial sample of size n1.
• 2. If number defective lt c1 , accept the lot.
• 3. If number defective gt c2 , reject the lot.
• 4. Otherwise, inspect another sample of size n2
  ,
• if the combined number of defectives lt c3 ,
  accept the lot
• otherwise, reject the lot.

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Sequential Sampling Plan

• 1. Items sampled one at a time.
• 2. After inspecting each item, record
• 1. Total number of items sampled,
• 2. Cumulative number of defectives.
• 3. Based on these numbers, either
• accept lot,
• reject lot, or
• continue sampling.
• For any values of AQL, LTPD, a and b, a
  sequential plan can always be devised.

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Average Outgoing Quality

• Given a sampling plan, we can compute the Average
  Outgoing Quality (AOQ), the long-run ratio of
  expected no. of defectives to expected number of
  items successfully passing through the inspection
  plan.
• Assuming the defective items sampled and
  defective lots are replaced,
• AOQ p(N-n)Pr(Accept lot)/N
• The maximum value of AOQ over all possible values
  of p is called Average Outgoing Quality Limit
  (AOQL).
• If this is too high, then the sampling plan
  should be revised.