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# Oc Curves[1]

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• ### Oc Curves[1]

1. 1. Operating Characteristic (OC) Curves Ben M. Coppolo Penn State University
2. 2. Presentation Overview <ul><li>Operation Characteristic (OC) curve Defined </li></ul><ul><li>Explanation of OC curves </li></ul><ul><li>How to construct an OC curve </li></ul><ul><li>An example of an OC curve </li></ul><ul><li>Problem solving exercise </li></ul>
3. 3. OC Curve Defined <ul><li>What is an Operations Characteristics Curve? </li></ul><ul><ul><li>the probability of accepting incoming lots. </li></ul></ul>
4. 4. OC Curves Uses <ul><li>Selection of sampling plans </li></ul><ul><li>Aids in selection of plans that are effective in reducing risk </li></ul><ul><li>Help keep the high cost of inspection down </li></ul>
5. 5. OC Curves <ul><li>What can OC curves be used for in an organization? </li></ul>
6. 6. Types of OC Curves <ul><li>Type A </li></ul><ul><ul><li>Gives the probability of acceptance for an individual lot coming from finite production </li></ul></ul><ul><li>Type B </li></ul><ul><ul><li>Give the probability of acceptance for lots coming from a continuous process </li></ul></ul><ul><li>Type C </li></ul><ul><ul><li>Give the long-run percentage of product accepted during the sampling phase </li></ul></ul>
7. 7. OC Graphs Explained <ul><li>Y axis </li></ul><ul><ul><li>Gives the probability that the lot will be accepted </li></ul></ul><ul><li>X axis =p </li></ul><ul><ul><li>Fraction Defective </li></ul></ul><ul><li>P f is the probability of rejection, found by 1-P A </li></ul>
8. 8. OC Curve
9. 9. Definition of Variables <ul><li>P A = The probability of acceptance </li></ul><ul><li>p = The fraction or percent defective </li></ul><ul><li>P F or alpha = The probability of rejection </li></ul><ul><li>N = Lot size </li></ul><ul><li>n = The sample size </li></ul><ul><li>A = The maximum number of defects </li></ul>
10. 10. OC Curve Calculation <ul><li>Two Ways of Calculating OC Curves </li></ul><ul><ul><li>Binomial Distribution </li></ul></ul><ul><ul><li>Poisson formula </li></ul></ul><ul><ul><ul><li>P(A) = ( (np)^A * e^-np)/A ! </li></ul></ul></ul>
11. 11. OC Curve Calculation <ul><li>Binomial Distribution </li></ul><ul><ul><li>Cannot use because: </li></ul></ul><ul><ul><ul><li>Binomials are based on constant probabilities. </li></ul></ul></ul><ul><ul><ul><li>N is not infinite </li></ul></ul></ul><ul><ul><ul><li>p changes </li></ul></ul></ul><ul><ul><li>But we can use something else. </li></ul></ul>
12. 12. OC Curve Calculation <ul><li>A Poisson formula can be used </li></ul><ul><ul><li>P(A) = ((np)^A * e^-np) /A ! </li></ul></ul><ul><li>Poisson is a limit </li></ul><ul><ul><li>Limitations of using Poisson </li></ul></ul><ul><ul><ul><li>n<= 1/10 total batch N </li></ul></ul></ul><ul><ul><ul><li>Little faith in probability calculation when n is quite small and p quite large. </li></ul></ul></ul><ul><li>We will use Poisson charts to make this easier. </li></ul>
13. 13. Calculation of OC Curve <ul><li>Find your sample size, n </li></ul><ul><li>Find your fraction defect p </li></ul><ul><li>Multiply n*p </li></ul><ul><li>A = d </li></ul><ul><li>From a Poisson table find your P A </li></ul>
14. 14. Calculation of an OC Curve <ul><li>N = 1000 </li></ul><ul><li>n = 60 </li></ul><ul><li>p = .01 </li></ul><ul><li>A = 3 </li></ul><ul><li>Find P A for p = .01, .02, .05, .07, .1, and .12? </li></ul>d= 3 Np 072 7.2 151 6 39.5 4.2 64.7 3 87.9 1.2 99.8 .6
15. 15. Properties of OC Curves <ul><li>Ideal curve would be perfectly perpendicular from 0 to 100% for a given fraction defective. </li></ul>
16. 16. Properties of OC Curves <ul><li>The acceptance number and sample size are most important factors. </li></ul><ul><li>Decreasing the acceptance number is preferred over increasing sample size. </li></ul><ul><li>The larger the sample size the steeper the curve. </li></ul>
17. 17. Properties of OC Curves
18. 18. Properties of OC Curves <ul><li>By changing the acceptance level, the shape of the curve will change. All curves permit the same fraction of sample to be nonconforming. </li></ul>
19. 19. Example Uses <ul><li>A company that produces push rods for engines in cars. </li></ul><ul><li>A powdered metal company that need to test the strength of the product when the product comes out of the kiln. </li></ul><ul><li>The accuracy of the size of bushings. </li></ul>
20. 20. Problem <ul><li>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: </li></ul>
21. 21. Problem <ul><li>A lot size of 300 of which 5 are defective. </li></ul><ul><li>A lot size of 1000 of which 4 are defective. </li></ul><ul><li>A lot size of 2500 of which 9 are defective. </li></ul><ul><li>Use Poisson formula to find the answer to number 2. </li></ul>
22. 22. Summary <ul><li>Types of OC curves </li></ul><ul><ul><li>Type A, Type B, Type C </li></ul></ul><ul><li>Constructing OC curves </li></ul><ul><li>Properties of OC Curves </li></ul><ul><li>OC Curve Uses </li></ul><ul><li>Calculation of an OC Curve </li></ul>
23. 23. Poisson Table
24. 24. Poisson Table
25. 25. Poisson Table
26. 26. 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 Technician’s 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.