1. Operating Characteristic (OC) Curves Ben M. Coppolo Penn State University

2. Presentation Overview Operation Characteristic (OC) curve DefinedOperation Characteristic (OC) curve Defined Explanation of OC curvesExplanation of OC curves How to construct an OC curveHow to construct an OC curve An example of an OC curveAn example of an OC curve Problem solving exerciseProblem solving exercise

3. OC Curve Defined What is an Operations Characteristics Curve?What is an Operations Characteristics Curve? –the probability of accepting incoming lots.

4. OC Curves Uses Selection of sampling plansSelection of sampling plans Aids in selection of plans that are effective in reducing riskAids in selection of plans that are effective in reducing risk Help keep the high cost of inspection downHelp keep the high cost of inspection down

5. OC Curves What can OC curves be used for in an organization?What can OC curves be used for in an organization?

6. Types of OC Curves Type AType A –Gives the probability of acceptance for an individual lot coming from finite production Type BType B –Give the probability of acceptance for lots coming from a continuous process Type CType C –Give the long-run percentage of product accepted during the sampling phase

7. OC Graphs Explained Y axisY axis –Gives the probability that the lot will be accepted X axis =pX axis =p –Fraction Defective P f is the probability of rejection, found by 1-P AP f is the probability of rejection, found by 1-P A

8. OC Curve

9. Definition of Variables P A = The probability of acceptance p = The fraction or percent defective P F or alpha = The probability of rejection N = Lot size n = The sample size A = The maximum number of defects

10. OC Curve Calculation Two Ways of Calculating OC CurvesTwo Ways of Calculating OC Curves –Binomial Distribution –Poisson formula P(A) = ( (np)^A * e^-np)/A !P(A) = ( (np)^A * e^-np)/A !

11. OC Curve Calculation Binomial DistributionBinomial Distribution –Cannot use because: Binomials are based on constant probabilities.Binomials are based on constant probabilities. N is not infiniteN is not infinite p changesp changes –But we can use something else.

12. OC Curve Calculation A Poisson formula can be usedA Poisson formula can be used –P(A) = ((np)^A * e^-np) /A ! Poisson is a limitPoisson is a limit –Limitations of using Poisson n<= 1/10 total batch Nn<= 1/10 total batch N Little faith in probability calculation when n is quite small and p quite large.Little faith in probability calculation when n is quite small and p quite large. We will use Poisson charts to make this easier.We will use Poisson charts to make this easier.

13. Calculation of OC Curve Find your sample size, nFind your sample size, n Find your fraction defect pFind your fraction defect p Multiply n*pMultiply n*p A = dA = d From a Poisson table find your P AFrom a Poisson table find your P A

14. Calculation of an OC Curve N = 1000N = 1000 n = 60n = 60 p =.01p =.01 A = 3A = 3 Find P A for p =.01,.02,.05,.07,.1, and.12?Find P A for p =.01,.02,.05,.07,.1, and.12? Np d= 3.699.8 1.287.9 364.7 4.239.5 6151 7.2072

15. Properties of OC Curves Ideal curve would be perfectly perpendicular from 0 to 100% for a given fraction defective.Ideal curve would be perfectly perpendicular from 0 to 100% for a given fraction defective.

16. Properties of OC Curves The acceptance number and sample size are most important factors.The acceptance number and sample size are most important factors. Decreasing the acceptance number is preferred over increasing sample size.Decreasing the acceptance number is preferred over increasing sample size. The larger the sample size the steeper the curve.The larger the sample size the steeper the curve.

17. Properties of OC Curves

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

19. Example Uses A company that produces push rods for engines in cars.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.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.The accuracy of the size of bushings.

20. 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: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:

21. 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.

22. Summary Types of OC curvesTypes of OC curves –Type A, Type B, Type C Constructing OC curvesConstructing OC curves Properties of OC CurvesProperties of OC Curves OC Curve UsesOC Curve Uses Calculation of an OC CurveCalculation of an OC Curve

23. Poisson Table

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.