ACCEPTANCE SAMPLING FOR ATTRIBUTES

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ACCEPTANCE SAMPLING FOR ATTRIBUTES

• Types Of Sampling Plans
• Mistakes To Avoid, And Their Statistical
  Equivalents
• AQL LTPD
• Single-sampling Plans
•  Average Outgoing Quality
• ISO 2859 Dodge-Romig Plans

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IS THIS SHIPMENT ANY GOOD?

• Can you trust your vendor's quality?
• If so, great!
• If not, inspect each shipment that arrives
• Sorting out good from bad shipments

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OPTIONS FOR VENDOR QUALITY

• Objective ensure vendor delivers quality
  supplies
• Two ways to reach objective
• Inspect vendor's shipments
• Acceptance" sampling
• Have vendor performing QM
• Vendor supplies customer with relevant control
  charts
• Customer certifies vendor in QM
• The first approach is traditional, but the second
  is preferable in general and necessary for lean
  operations

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"ACCEPTANCE" SAMPLING

• Basic idea
• Inspect a random sample of each lot
• Classify each item as ok/not ok
• Conclude entire lot is either
• Ok -- accept it
• Not ok -- reject it and/or sort it

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ATTRIBUTE VS. VARIABLE SAMPLING PLANS

• Simplest sampling plans are attribute
• Based on binomial (or hypergeometric)
  distribution
• May lose variable data on several QC'S
• Requires large sample size
• Plans based on variable data
• Based on normal distribution
• Provide more information on source of quality
  problems
• Require smaller samples for same a, b

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SINGLE, DOUBLE, MULTIPLE SAMPLING PLANS

• Single sampling plans
• Make accept/reject decision based on one sample
• Double sampling plans
• Make accept/reject/take-another- sample decision
  based on first sample
• Make accept/reject decision based on second
  sample (if taken)
• Can have "triple", "quadruple", or any other
  multiple sampling plan
• Multiple sampling plans require more but smaller
  samples for same a, b

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SINGLE (ATTRIBUTE-BASED) SAMPLING PLANS

• Define
• N -- number of parts in shipment
• n -- number of parts in a sample from shipment
• c -- acceptance number
• Acceptance sampling has 3 easy steps
• For each shipment of N parts, a sample of size n
  is taken
• Inspect each of the n parts
• Reject the shipment if the number of defects
  exceeds c units. Otherwise, accept the shipment

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MISTAKES TO AVOID, THEIR STATISTICAL
EQUIVALENTS

• As with product quality control, there are two
  types of mistakes to avoid
• Type I -- Conclude the shipment is bad when in
  fact it is good (false alarm)
• Type II -- Conclude the shipment is good when in
  fact it is bad (overlooked problem)
• Probability of each type of mistake
• Type I -- a Type II -- b
• This is standard hypothesis testing with the
  following null hypothesis
• H0 the shipment is good

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DOUBLE SAMPLING PLANS

• Define
• n1 -- sample size on first sample
• c1 -- acceptance number for first sample
• d1 -- defectives in first sample
• n2 -- sample size on second sample
• c2 -- acceptance number for both samples
• d2 -- defectives in second sample
• Take sample of size n1
• Accept if d1 c1 reject if d1 gt c2
• Take second sample of size n2 if c1 lt d1 c2
• Accept if d1d2 c2 reject if d1d2 gt c2

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DEFINING GOOD AND BAD SHIPMENTS AQL VERSUS LTPD

• Instead of simply "good" versus "bad", we will
  define "really good", "really bad", and "ok, but
  not great" shipments
• p -- True (unknown) percent defective in shipment
• AQL -- Acceptable quality level
• LTPD -- lot tolerance percent defective
• Then
• A really good shipment has p lt AQL
• A really bad shipment has p gt LTPD
• Anything in between (AQL lt p lt LTPD) is ok, but
  not great

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THE OPERATING-CHARACTERISTIC (OC) CURVE

• For a given a sampling plan and a specified true
  fraction defective p, we can calculate
• Pa -- Probability of accepting lot
• If lot is truly good, 1 - Pa a
• If lot is truly bad, Pa b
• A plot of Pa as a function of p is called the OC
  curve for a given sampling plan

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THE OPERATING-CHARACTERISTIC (OC) CURVE

• The ideal sampling plan discriminates perfectly
  between good and bad shipments
• Both a and b are zero in this example!
• This requires a sample size equal to the
  population -- not feasible

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CONSTRUCTING AN (OC) CURVE

• For a specified single sampling plan, the OC
  curve may be constructed using a binomial
  distribution if n is small relative to the lot
  size
• p -- true fraction nonconforming
• n -- sample size
• c -- acceptance number
• We know that

Excel
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CONSTRUCTING AN (OC) CURVE

• Suppose we have a sampling plan defined by the
  following parameters
• n 100
• c 2
• What is the probability of accepting a lot with
  0.5 defectives?

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CONSTRUCTING AN (OC) CURVE
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USING AN (OC) CURVE

• How do we find a and b using an OC curve?
• AQL 0.01
• LTPD 0.05
• Then a 1 Pa(p0.01) 1 - 0.9206 0.0794
• And b Pa(p0.05) 0.1183

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AVERAGE OUTGOING QUALITY

• Consider a part with a long-term fraction
  nonconforming of p
• Samples of size n are taken from a lot of size N
  and inspected
• Any defectives in the sample of size n are
  replaced, accept or reject
• When a lot of is accepted, we expect p(N-n)
  defectives in the remainder of the lot
• When a lot is rejected, it will be sorted and
  defective units replaced, leaving N-n good units
  in the remainder
• This is referred to as "rectifying" inspection

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AVERAGE OUTGOING QUALITY

• If Pa is the probability of accepting a lot, then
  the average outgoing quality is

• The worst possible AOQ is the AOQ Limit or AOQL

Excel
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AVERAGE TOTAL INSPECTION

• Rectifying plans have greater inspection
  requirements
• The Average Total Inspections

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ISO 2859 (ANSI/ASQC Z1.4)

• One of oldest sampling systems
• Covers single, double, multiple sampling
• AQL-based Type I error ranges 9-1 as sample
  size increases
• Minimal control over Type II error
• Type II error decreases as general inspection
  level (I, II, III) increases
• Special inspection levels when small samples
  needed (and high Type II error probability
  tolerated)
• Mechanism for reduced or tightened inspection
  depending on recent vendor performance
• Tightened -- more inspection
• Reduced -- less inspection

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ISO 2859 (ANSI/ASQC Z1.4)

• A vendor begins at a "normal" inspection level
• Normal to tightened 2/5 lots rejected
• Normal to reduced
• Previous 10 lots accepted (NOT ISO 2859)
• Total defectives from 10 lots ok (NOT ISO 2859)
• If a vendor is at a tightened level
• Tightened to normal 5 previous lots accepted
• If a vendor is at a reduced level
• Reduced to normal a lot is rejected

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ISO 2859

• A vendor begins at a "normal" inspection level
• Normal to reduced
• Switching score set to zero
• If acceptance number is 0 or 1
• Add 3 to the score if the lot would still have
  been accepted with an AQL one step tighter else
  reset score to 0
• If acceptance number is 2 or more
• Add 3 to the score if the lot is accepted else
  reset score to 0
• If score hits 30, switch to reduced inspection

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USING ISO 2859

• Choose the AQL
• Choose the general inspection level
• Determine lot size
• Find sample size code
• Choose type of sampling plan
• Select appropriate plan from table
• Switch to reduced/tightened inspection as required

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USING ISO 2859
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USING ISO 2859
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USING ISO 2859
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USING ISO 2859
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DODGE-ROMIG PLANS

• Developed in the 1920's
• Rectifying plans
• Requires knowledge of vendor's long-term process
  average (fraction non-conforming)
• Choice of LTPD or AOQL orientation
• Both minimize ATI for specified process average
• Type II error 10,

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DODGE-ROMIG PLANS

• AOQL plans
• 1) Determine N, p, and AOQL
• 2) Use table to find n and c
• Finds plan with specified AOQL which minimizes
  ATI
• Calculate resulting LTPD with Type II error 10
• LTPD plans
• 1) Determine N, p, and LTPD
• 2) Use table to find n and c
• Finds plan with specified LTPD which minimizes
  ATI
• Calculate resulting AOQL

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DODGE-ROMIG PLANS
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DODGE-ROMIG PLANS