Online process control for attributes Misclassification errors and Repetitive classifications

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On-line process control for attributes
(Misclassification errors and Repetitive
classifications)

• Linda Lee Ho (lindalee_at_usp.br)
• Roberto C. Quinino (roberto_at_est.ufmg.br)
• A.L.G. Trindade (anderson.trindade_at_poli.usp.br)

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On-line process control for attributes
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On-line process control for attributes

• Perfect classification system
• Taguchi et al. (1989)
• Nayebpour Woodall (1993)

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On-line process control for attributesImperfect
classification system
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On-line process control for attributes

• Imperfect classification system
• Borges, Ho Turnes (2001)

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On-line process control for attributes Imperfect
classification system Repetitive classifications
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On-line process control for
attributesImperfect classification
systemRepetitive classification procedure

• Decrease in the impact of the misclassification
  errors.
• Quinino, Ho Trindade (2005)

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Repetitive classification procedure

• Aims
• Determine the optimum frequency of the repetitive
  test on each examined item
• Determination of the optimum sampling interval.

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On-Line Process Control for Attributes(Brief
Review) Taguchis model

• Main assumptions
• Uniform distribution (implicitly)- Changes of
  defective fraction
• Items are produced independently
• A single item is examined at every m items
  produced
• Process is stopped if the examined item is
  non-conforming
• Lag time - L items are produced
• At each stoppage, the process begins in control
  a new production cycle begins.

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On-line process control for attributesNayebpour
Woodalls model (1993)

• Sampling scheme Taguchis model
• Geometric distribution (?)- Changes of defective
  fraction
• Retrospective inspection.

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On-line process control for attributesMisclassifi
cation errors

• e1 conforming item classified as
  non-conforming
• e2 non-conforming item classified as
  conforming

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On-line process control for attributes
Borges, Ho Turnes (2001)

• Geometric distribution (?) - Changes of
  defective fraction
• Misclassification errors
• Discard the signaled item (as non-conforming)
  and L items (lag time)
• No retrospective inspection
• Sampling schemeTaguchis model.

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On-line process control for attributes
Borges, Ho Turnes (2001)

• Costs

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On-line process control for attributes Borges,
Ho Turnes (2001)

• Example

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On-line process control for attributesMisclassif
ication errors-Repetitive classifications

• Non destructive Bernoulli trials
• Examined item classified repetitive and
  independent r times
• At each time item is classified as conforming or
  non-conforming

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On-line process control for attributesMisclassif
ication errors - Repetitive classifications

• Classifications?, diagnostic errors ?
• Unfeasible inspection cost
• Repetitive classifications - decrease in the
  impact of the misclassification errors

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On-line process control for attributesMisclassi
fication errors - Repetitive classifications

• Sampling scheme Taguchis model
• Criteria item is declared conforming if the most
  frequency result is conforming if a gt 0.5r
• If the item is declared non-conforming ? the
  process is stopped.
• Ref.(Quinino, Ho Trindade, 2005a, 2005b)

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On-line process control for attributesMisclassif
ication errors - Repetitive classifications
Inspection procedure (Markov chain)

• Space of states (s,k)

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Repetitive classifications Inspection procedure
Transition matrix P (s,k) ? (s,k)
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Repetitive classifications Inspection procedure
(Markov Chain)

• Q is stationary distribution of Pu u??
• QA-1B - the solution of the system QQP,
  restricted to ?Qi1.

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Renewal Reward Theorem

• T of produced items at each adjustment
• V cost at each adjustment

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T items produced

• T discrete random variable
• ETmQ1(mL)Q2(mL)Q3mQ4

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V Cost of control system

• V discrete random variable
• EVV1Q1V2Q2V3Q3V4Q4

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Cost in control system

• Costs in the proposed model
• cost of inspecting each item
• cost for adjustment the process
• cost of non-discarding non-conforming item
• cost of discarding non-conforming item
• cost of discarding conforming item

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Cost V1 (s1k1) S1 (state I) - the process is
in control K1- item declared conforming - No
adjustment

• Costs
• of inspecting (r times independently) the
  examined item
• of discarding the examined item
• of producing non-conforming items.

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Cost V2 (s1k2) s1 (state I)- The process is
in control k2- (item declared non-conforming) -
process is stopped

• Costs
• of inspecting (r times independently) the
  examined item
• of discarding the examined item
• of producing non-conforming items.
• of adjusting the process
• of discarding L items produced in the lag time
  (state I ? state I or state I? state II)

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Cost V3 (s2k2) s2 - (state II) - the process is
out of controlk2 - (item declared
non-conforming) process is stopped

• Costs
• of inspecting (r times independently) the
  examined item
• of discarding the examined item
• of producing non-conforming items (state I ?state
  II or state II ? state II).
• of adjusting the process
• of discarding L items produced in the lag time
  (state II? state II)

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Cost V4 (s2k1) s2 - (state II) - the process is
out of controlk1 - (item declared conforming)
no adjustment

• Costs
• of inspecting (r times independently) the
  examined item
• of discarding the examined item
• of producing non-conforming items (state I ?state
  II or state II ? state II).

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Objetive Function

• Objetive function
• Optimum parameters

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Numerical example

• CI0.25 CND-NC 20 CA100
• CD-NC2.0 CD-C2.5
• p10.999 p20.001 p0.0001 e1e20.01 L10.

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Cost versus m (perfect classification system)
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Numerical example
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Cost versus m

• (a single inspection - r1)

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Numerical example
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Cost versus m

• r1 and optimum r

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• Optimum parameters

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Numerical example

• Optimum parameters

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All the cases
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Cost versus m

• All the cases

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Conclusions

• In a presence of diagnostic errors Repetitive
  classifications feasible procedure
• A single classification and most frequent result
  of the repetitive classifications not better
  procedures.

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References

• Borges, W, Ho, L. L, Turnes, O. An analysis of
  Taguchi's on-line quality monitoring procedure
  for attributes with diagnosis errors. Applied
  Stochastic Models in Business and Industry. 2001,
  17 261-276.
• Nayepour, M. R. e W. H. Woodall. An Analysis of
  Taguchi's On-line Quality Monitoring Procedure
  for Attributes. Technometrics. 1993. 3553-60.
• Quinino, R.C. Ho, L.L. Trindade, A.L.G.
  Repetitive tests in Taguchis on-line quality
  monitoring procedure for attributes to reduce the
  impact of diagnosis errors (2005a - submitted)
• Quinino, R.C. Ho, L.L. Trindade, A.L.G.
  Controle on-line de atributos com erros de
  classificação uma abordagem econômica com
  classificações repetidas (Sobrapo, 2005b)
• Ross, S. M. Introduction to probability models.
  San Diego London Burlington Academic Press.
  2000. 693 p.
• Taguchi, G., E. A. Elsayed, et al. Quality
  engineering in production systems. New-York
  McGraw-Hill. 1989. 173 p.

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• Thank you for your attention!