Operating Characteristic (OC) Curves

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Operating Characteristic (OC) Curves

• Ben M. Coppolo
• Penn State University

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Presentation Overview

• Operation Characteristic (OC) curve Defined
• Explanation of OC curves
• How to construct an OC curve
• An example of an OC curve
• Problem solving exercise

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OC Curve Defined

• What is an Operations Characteristics Curve?
• the probability of accepting incoming lots.

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OC Curves Uses

• Selection of sampling plans
• Aids in selection of plans that are effective in
  reducing risk
• Help keep the high cost of inspection down

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OC Curves

• What can OC curves be used for in an
  organization?

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Types of OC Curves

• Type A
• Gives the probability of acceptance for an
  individual lot coming from finite production
• Type B
• Give the probability of acceptance for lots
  coming from a continuous process
• Type C
• Give the long-run percentage of product accepted
  during the sampling phase

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OC Graphs Explained

• Y axis
• Gives the probability that the lot will be
  accepted
• X axis p
• Fraction Defective
• Pf is the probability of rejection, found by 1-PA

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OC Curve
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Definition of Variables

• PA The probability of acceptance
• p The fraction or percent defective
• PF or alpha The probability of rejection
• N Lot size
• n The sample size
• A The maximum number of defects

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OC Curve Calculation

• Two Ways of Calculating OC Curves
• Binomial Distribution
• Poisson formula
• P(A) ( (np)A e-np)/A !

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OC Curve Calculation

• Binomial Distribution
• Cannot use because
• Binomials are based on constant probabilities.
• N is not infinite
• p changes
• But we can use something else.

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OC Curve Calculation

• A Poisson formula can be used
• P(A) ((np)A e-np) /A !
• Poisson is a limit
• Limitations of using Poisson
• nlt 1/10 total batch N
• Little faith in probability calculation when n is
  quite small and p quite large.
• We will use Poisson charts to make this easier.

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Calculation of OC Curve

• Find your sample size, n
• Find your fraction defect p
• Multiply np
• A d
• From a Poisson table find your PA

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Calculation of an OC Curve

• N 1000
• n 60
• p .01
• A 3
• Find PA for p .01, .02, .05, .07, .1, and .12?

Np d 3
.6 99.8
1.2 87.9
3 64.7
4.2 39.5
6 151
7.2 072
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Properties of OC Curves

• Ideal curve would be perfectly perpendicular from
  0 to 100 for a given fraction defective.

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Properties of OC Curves

• The acceptance number and sample size are most
  important factors.
• Decreasing the acceptance number is preferred
  over increasing sample size.
• The larger the sample size the steeper the curve.

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Properties of OC Curves
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Properties of OC Curves

• By changing the acceptance level, the shape of
  the curve will change. All curves permit the
  same fraction of sample to be nonconforming.

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Example Uses

• A company that produces push rods for engines in
  cars.
• A powdered metal company that need to test the
  strength of the product when the product comes
  out of the kiln.
• The accuracy of the size of bushings.

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Problem

• MRC is an engine company that builds the engines
  for GCF cars. They are use a control policy of
  inspecting 15 of incoming lots and rejects lots
  with a fraction defect greater than 3. Find the
  probability of accepting the following lots

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Problem

1. A lot size of 300 of which 5 are defective.
2. A lot size of 1000 of which 4 are defective.
3. A lot size of 2500 of which 9 are defective.
4. Use Poisson formula to find the answer to number
   2.

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Summary

• Types of OC curves
• Type A, Type B, Type C
• Constructing OC curves
• Properties of OC Curves
• OC Curve Uses
• Calculation of an OC Curve

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Poisson Table
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Poisson Table
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Poisson Table
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Bibliography
  Doty, Leonard A. Statistical Process
Control. New York, NY Industrial Press INC,
1996. Grant, Eugene L. and Richard S.
Leavenworth. Statistical Quality Control. New
York, NY The McGraw-Hill Companies INC,
1996. Griffith, Gary K. The Quality Technicians
Handbook. Engle Cliffs, NJ Prentice Hall,
1996. Summers, Donna C. S. Quality. Upper
Saddle River, NJ Prentice Hall, 1997. Vaughn,
Richard C. Quality Control. Ames, IA The Iowa
State University, 1974.