LotbyLot Acceptance Sampling for Attributes

1
Chapter 14

• Lot-by-Lot Acceptance Sampling for Attributes

2
14-1. The Acceptance-Sampling Problem

• Acceptance sampling is concerned with inspection
  and decision making regarding products.

3
Three aspects of sampling

• The purpose of acceptance sampling is to sentence
  lots, not to estimate the lot quality
• Although, some plans do this
• Acceptance sampling is not quality control
• Reject or accept lots only
• Even if lots are of the same quality, sampling
  will accept some lots and reject others

4
Three aspects of sampling

• Quality cannot be inspected into the product
• Acceptance sampling is an audit tool that insures
  that the output of a process conforms to
  requirements

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14-1. The Acceptance-Sampling Problem

• Three approaches to lot sentencing
• Accept with no inspection
• 100 inspection
• Acceptance sampling

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14-1. The Acceptance-Sampling Problem

• Why Acceptance Sampling and Not 100 Inspection?
• Testing can be destructive
• Cost of 100 inspection is high
• 100 inspection is not feasible
• Requires too much time
• Can be inaccurate
• If vendor has excellent quality history

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14-1. The Acceptance-Sampling Problem

• 14-1.1 Advantages and Disadvantages of Sampling
• Advantages
• Less expensive
• Reduced damage
• Reduces the amount of inspection error
• Disadvantages
• Risk of accepting bad lots, rejecting good
  lots
• Less information generated
• Requires planning and documentation

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14-1. The Acceptance-Sampling Problem

• 14-1.2 Types of Sampling Plans
• There are variables sampling plans and attribute
  sampling plans (this chapter is about attributes)
• Single sampling plan
• Double-sampling plan
• Multiple-sampling plan
• Sequential-sampling

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14-1. The Acceptance-Sampling Problem

• 14-1.3 Lot Formation
• Considerations before inspection
• Lots should be homogeneous
• Produced by the same machine, same operators,
  common raw materials, approximately the same time

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14-1. The Acceptance-Sampling Problem

• 14-1.3 Lot Formation
• Considerations before inspection
• Larger lots more preferable than smaller lots
• More economical
• Lots should be conformable to the
  materials-handling systems used in both the
  vendor and consumer facilities.

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14-1. The Acceptance-Sampling Problem

• 14-1.4 Random Sampling
• The units selected for inspection should be
  chosen at random.
• If random samples are not used, bias can be
  introduced.
• If any judgment methods are used to select the
  sample, the statistical basis of the
  acceptance-sampling procedure is lost.

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Non-randomization

• Pick a unit from the top layer of each box
• Uncle Charlie

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Randomization

• Example Assign a number to each unit in the lot
• 1, 2, , N
• Select n unique random numbers from 1, 2, , N
• The selected numbers constitute the sample

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14-2. Single-Sampling Plans For
Attributes

• 14-2.1 Definition of a Single-Sampling Plan
• A single sampling plan is defined by sample size,
  n, and the acceptance number c. Say there are N
  total items in a lot. Choose n of the items at
  random. If more than c of the items are
  unacceptable, reject the lot.
• N lot size
• n sample size
• c acceptance number
• d observed number of defectives
• The acceptance or rejection of the lot is based
  on the results from a single sample - thus a
  single-sampling plan.

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Example

• N 10000, n 89, c 2
• From a lot of 10,000, take a sample of size 89
• Observe the number of defectives, d
• If d lt 2, accept
• Otherwise, reject

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14-2. Single-Sampling Plans For
Attributes

• 14-2.2 The OC Curve
• The operating-characteristic (OC) curve measures
  the performance of an acceptance-sampling plan.
• The OC curve plots the probability of accepting
  the lot versus the lot fraction defective.
• The OC curve shows the probability that a lot
  submitted with a certain fraction defective will
  be either accepted or rejected.

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Example

• See Fig. 14-2 on pg. 683 and discussion following
• If p .01, Pa .9397
• See computation at bottom of pg. 683
• See Table 14-2
• If p .02, Pa .7366 means that 73.66 of lots
  will be accepted and 26.34 will be rejected

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Effect of n and c on OC curves

• Fig. 14-3, ideal OC curve
• Pa 1.0 until a level of quality that is
  considered bad is reached

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Effect of n and c on OC curves

• Fig. 14-4, OC curve for different values of n
• By increasing the sample size, we get closer to
  the ideal OC curve

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Effect of n and c on OC curves

• Fig. 14-5, OC curve for different values of c
• As c is decreased, the OC curve shifts to the
  left
• When c 0, it is very hard on the vendor

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14-2. Single-Sampling Plans For
Attributes

• 14-2.3 Designing a Single-Sampling Plan with a
  Specified OC Curve
• Let the probability of acceptance be 1 - ? for
  lots with fraction defective p1.
• Let the probability of acceptance be ? for lots
  with fraction defective p2.
• Assume binomial sampling is appropriate.
• For type B OC curves (from large lots)

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14-2. Single-Sampling Plans For
Attributes

• 14-2.3 Designing a Single-Sampling Plan with a
  Specified OC Curve
• The sample size n and acceptance number c are the
  solution to

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14-2. Single-Sampling Plans For
Attributes

• Example
• Consider constructing a sampling plan for which
• p1 0.01
• ? 0.05
• p2 0.06
• ? 0.10
• N 1000
• Using computer software or a graphical approach
  (using an appropriate binomial nomograph) it can
  be shown that the necessary values of n and c are
  85 and 2, respectively.

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Using the nomograph

• Figure 14-9
• Draw a line from p1 .01 on the left side to
  (1a ) .95 on the right side
• Draw a second line from p2 .06 on the left to b
  .10 on the right
• The intersection of the two lines comes close to
  defining the plan n 89, c 2

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Using the nomograph

• Figure 14-9
• Since n and c must be integers, this procedure
  will actually produce several plans that have OC
  curves that pass close to the desired plans
• Holding the first line constant, and holding p2,
  two plans are observed, with values of b
  different than desired, one lower and the other
  higher

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Using the nomograph

• Figure 14-9
• Holding the second line constant, and holding p1,
  two plans are observed, with values of a
  different than desired, one lower and the other
  higher

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Rectifying inspection

• Require corrective action when lots are rejected
• 100 screening of rejected lots
• Defective items are removed
• Affects the outgoing quality

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Rectifying inspection
Rejected lots
Fraction defective 0
Inspection activity
Incoming lots
Outgoing lots
Fraction defective p0
Fraction defective p1ltp0
Fraction defective p0
Accepted lots
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Average outgoing quality

• AOQ is the result of applying rectifying
  inspection
• In a lot of size N, there will be
• n items in the sample that, after inspection,
  contain no defectives (all of the defectives were
  replaced)
• N-n items that, if the lot is rejected, also
  contain no defectives (the balance of the lot was
  inspected 100)
• N-n items that, if the lot is accepted, contain
  p(N-n) defectives

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Average outgoing quality

• AOQ Pa p (N-n)/N
• Example
• N 10000, n 89, c 2, p .01
• Previously determined that Pa .9397
• AOQ (.9397)(.01)(10000-89)/10000
• AOQ .0093
• Since (N-n)/N 1, AOQ Pap

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AOQ curve for rectifying inspection

• See Fig. 14-11 for n 89, c 2
• When incoming quality is very good, average
  fraction defective of outgoing lots is low
• When incoming quality is very poor, average
  fraction defective of outgoing lots is low

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Average outgoing quality limit

• See Fig. 14-11
• AOQL .0155
• No matter how bad the incoming lots are, the
  outgoing quality level will never be worse than
  1.55 fraction defective

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Average total inspection

• ATI n (1- Pa)(N - n)
• N 10000, n 89, c 2, p .01
• Pa .9397
• ATI 89 (1-.9397)(10000-89) 687
• See Fig. 14-12
• ATI curves for n89, c2, N1000, 5000, 10000

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Double sampling plans

• Defined by four parameters
• n1 sample size for the first sample
• c1 acceptance number of the first sample
• n2 sample size for the second sample
• c2 acceptance number of the second sample

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n150, c11 n2100, c23
Inspect a random sample of n1 50 from the
lot d1 number of observed defectives
Accept the lot
Reject the lot
d1ltc11
d1gtc23
1ltd1lt3
Inspect a random sample of n2 100 from the
lot d2 number of observed defectives
Reject the lot
Accept the lot
d1d2ltc23
d1d2gtc23
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Double sampling plans

• Advantages
• Can reduce the total amount of inspection
• Allows the vendor a second chance
• Disadvantages
• Unless curtailment is used, can lose the
  economical advantage
• More record keeping is needed

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OC curve

• Pgs. 696-698
• Pa probability of acceptance on the combined
  samples
• PaI Probability of acceptance on the first
  sample
• PaII Probability of acceptance on the second
  sample
• Pa PaI PaII

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OC curve

• n150, c11
• n2100, c23
• For p .05
• Compute PaI .279 as shown on pg. 697
• Then, compute PaII .010 as shown on pg. 697
• So, Pa .279 .010 .289

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Average sample number

• ASN n1P1 (n1 n2)(1 P1)
• n1 n2(1 P1)
• where P1 PaI PrI
• See Fig. 14-15
• Compares the ASN for n160, c12, n2120, c23
  and the ASN for n89, c2
• The OC curves of the two plans are nearly
  identical

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Designing a sampling plan

• Grubbs tables are commonly used
• n2 n1 or n2 2n1 and a .05, b .10
• Portion of table for n2 2n1 shown on next slide

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Portion of Grubbs tables
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Example

• Want a double sampling plan with a .05, b
  .10, p1 .02, p2 .12 and n2 2n1
• R p2/p1 .12/.02 6
• Plan 3 comes closest where c11 and c23

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Example, cont.

• Determine n1
• Hold a
• pn1 .60
• n1 pn1/p160/.02 30
• So, n2 60
• Summary n1 30, c1 1, n2 60, c2 3

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Example, cont.

• Another way, hold b constant for plan 3
• n1 pn1/p2 3.89/.12 32.42
• Rounding up, n1 33, n2 66, c1 1, c2 3

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Multiple sampling plans

• See pg. 701 for an example
• After first sample of n1 20, if d1 0 accept,
  if d1 3, reject
• If a decision is made, curtailment can be applied
• If d1 1 or 2, take a second sample of n2 20,
  and if the cumulative number of defectives is 1,
  then accept, or, if the cumulative number of
  defectives is 4, reject
• This continues until the 5th sample at which time
  a decision is made

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Multiple sampling plans

• Advantage is that the average sample number may
  be lower than single- or double-sampling
• Disadvantage is increased complexity

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Sequential sampling plans

• Take samples of size one and continue until a
  decision is made
• This could continue indefinitely
• In practice, truncation is used

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Sequential sampling plans

• Sequential probability ratio test (SPRT)
• See Fig. 14-16
• See equations on pg. 702

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Example

• We want a sequential-sampling plan for which p1
  .01, a .05, p2 .06, b .10
• Limit lines are
• XA -1.22 .028n
• XR 1.57 .028n

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Example, cont.

• Can also use a table to make decisions
• See Table 14-3 on pg. 704
• Calculations for n 45
• XA -1.22 .028(45) .04
• XR 1.57 .028(45) 2.83
• Acceptance and rejection numbers must be integer
• Round down XA to 0 and round up XR to 3

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Example, cont.

• When is the first opportunity to accept?
• -1.22 .028n gt 0
• n gt 43.57 44
• When is the first opportunity to reject?
• For n 1, 1.57 .028 1.598 gt 1
• For n 2, 1.57 .056 1.626 lt 2 2

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.1 Description of the Standard
• Developed during World War II
• MIL STD 105E is the most widely used
  acceptance-sampling system for attributes
• Gone through four revisions since 1950.
• MIL STD 105E is a collection of sampling schemes
  making it an acceptance-sampling system

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.1 Description of the Standard
• Three types of sampling are provided for
• Single
• Double
• Multiple
• Provisions for each type of sampling plan include
• Normal inspection
• Tightened inspection
• Reduced inspection

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.1 Description of the Standard
• The acceptable quality level (AQL) is a primary
  focal point of the standard
• The AQL is generally specified in the contract or
  by the authority responsible for sampling.
• Different AQLs may be designated for different
  types of defects.
• Defects include critical defects, major defects,
  and minor defects.
• Tables for the standard provided are used to
  determine the appropriate sampling scheme.

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.1 Description of the Standard
• Switching Rules
• Normal to tightened
• Tightened to normal
• Normal to reduced
• Reduced to normal
• Discontinuance of inspection

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.2 Procedure
• Choose the AQL
• Choose the inspection level
• Determine the lot size
• Find the appropriate sample size code letter from
  Table 14-4
• Determine the appropriate type of sampling plan
  to use (single, double, multiple)
• Enter the appropriate table to find the type of
  plan to be used.
• Determine the corresponding normal and reduced
  inspection plans to be used when required.

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• Example
• Suppose a product is submitted in lots of size
  N 2000. The AQL is 0.65. Say we wanted to
  generate normal single-sampling plans.
• For lots of size 2000, (and general inspection
  level II) Table 14-4 indicates that the
  appropriate sample size code letter is K.
• From Table 14-5 for single-sampling plans under
  normal inspection, the normal inspection plan is
  n 125, c 2.

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.3 Discussion
• There are several points about the standard that
  should be emphasized
• MIL STD 105E is AQL-oriented
• The sample sizes selected for use in MIL STD 105E
  are limited
• The sample sizes are related to the lot sizes.
• Switching rules from normal to tightened and from
  tightened to normal are subject to some
  criticism.
• A common abuse of the standard is failure to use
  the switching rules at all.

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Switching rules
Start
And conditions
O Production steady O 10 consecutive lots
accepted O Approved by responsible authority
2 out of 5 consecutive lots rejected
Tightened
Normal
Reduced
Or conditions
O Lot rejected O Irregular production O Lot
meets neither accept nor reject criteria O
Other conditions warrant return to
normal inspection
5 consecutive lots accepted
10 consecutive lots remain on tightened inspection
Discontinue
inspection
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OC curves

• See pg. 713 for OC curves for sample size code
  letter K

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Double sampling

• These are included in the full text of the
  standard

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14-4. Military Standard 105E (ANSI/ASQC
Z1.4 ISO 2859)

• 14-4.3 Discussion
• ANSI/ASQC Z1.4 or ISO 2859 is the civilian
  standard counterpart of MIL STD 105E.
• Differences include
• Terminology nonconformity, nonconformance,
  and percent nonconforming is used.
• Switching rules were changed slightly to provide
  an option for reduced inspection without the use
  of limit numbers
• Several tables that show measures of scheme
  performance were introduced
• A section was added describing proper use of
  individual sampling plans when extracted from the
  system.
• A figure illustrating switching rules was added.

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Dodge-Romig Plans

• For rectifying inspection
• See Table 14-8 on pg. 717 for an example for AOQL
  3
• Indexed by lot size (N) and process average (p)

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Example

• N 5000, p .01
• Want a single sampling plan (w/rectifying
  inspection) with AOQL 3
• Read n 65, c 3 from the table
• These plans minimize ATI
• Pa .9957 at p .01 (determined as previously)
• ATI 65 (1 - .9957)(5000 65) 86.22

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Example, cont.

• Also, note that LTPD 10.3
• This is the point on the OC curve for which Pa
  .10
• That is, this plan provides that 90 of incoming
  lots that are as bad as 10.3 defective will be
  rejected

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LTPD plans

• Can also develop a plan for a specified LTPD
• Table 14-9 is for LTPD 1

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Example

• N 5000, p .25
• We want a single sampling plan (w/rectifying
  inspection) with LTPD of 1
• Find n 770, c 4

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Assignment

• Work odd numbered exercises through 14-15
• Understand MIL-STD 105E and Dodge-Romig tables

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End