1. 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

2. OC Curve Defined What is an Operations Characteristics Curve?
the probability of accepting incoming lots.
Vaughn (11)
A graph used to determine the probability of accepting lots as a function of the lots or processes’ quality level when using various sampling plans.
Summers(526)

3. 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
OC curves are not used often by inspectors however here are some advantages.
Griffith(405)

4. OC Curves What can OC curves be used for in an organization?
Accepting a batch of steel screws used in the building of towers bridges or other structures.
The probability of accepting a batch of light bulbs coming out of a furnace.
The probability of accepting a batch of prefabricated floor trusses made from solid wood or from MDF particle board.

5. 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
Summers(526)
Type A and Type B curves may be considered identical for most practical purposes.
Grant(439)
We will not be talking about Type C curves

6. OC Graphs Explained Y axis X axis =p
Gives the probability that the lot will be accepted
X axis =p
Fraction Defective
Pf is the probability of rejection, found by 1-PA

7. OC Curve
Sample OC curve with the sample size n=82 and the number of defects in the sample size A = 2.
Doty(289)

8. 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
PA = 1 - PF

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

10. 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.
If n was infinite and if p was replaced after being inspected we could use the binomial calculation, however this is not true.
Changes in N and because p not replaced the proportion of a defect remaining changes, making binomial distribution very difficult to use.
Vaughn(113)

11. OC Curve Calculation A Poisson formula can be used Poisson is a limit
P(A) = ((np)^A * e^-np) /A !
Poisson is a limit
Limitations of using Poisson
n<= 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.
As n is larger and p is smaller for small sample sizes n>20 and p <= 0.05 Poisson can be used.
This would make calculation fairly easy however a summation of defects from A=0 to the number of defects in the sample size is needed to get the probability of acceptance.
Using Poisson equation makes calculating OC curves very difficult and repetitive. If one uses a Poisson table we can make these curves much easier.
Vaughn(113)

12. Calculation of OC Curve
Find your sample size, n
Find your fraction defect p
Multiply n*p
A = d
From a Poisson table find your PA

13. Calculation of an OC Curve
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
n * p = 60 *.01 = .6 n * p =60 * .02 = 1.2
A = d = 3 A = d = 3
PA = 99.8% PA =87.9
n * p = 60 * .05 = 3 n * p = 60 *.07 = 4.2
PA = PA = 39.5
n * p =60 * .1 = 6
A = d = 3
PA = 15.1
n * p =60 * .12 =7.2
PA = 7.2

14. Properties of OC Curves
Ideal curve would be perfectly perpendicular from 0 to 100% for a given fraction defective.
Doty(292)

15. 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.
When sample sizes are increased the curve becomes steeper and provides better protection for both consumer and producer.
When acceptance number is decreased the curve becomes steeper and the plan provides better protection.
Decreasing the acceptance size is preferred because increasing the sample size increases cost.
Doty( )

16. Properties of OC Curves
The first graph shows the comparison of four sampling plans with 10% samples
The second graph shows a comparison of 4 sampling plans with constant sample sizes
This emphasizes that the absolute size not the relative size of the samples determines the protection given by the sampling plans.
Grant(434,437)

17. 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.
This shows that the larger the sample size the steeper the curve.
The ability of sampling plan to discriminate between lots of different qualities.
The larger the sample size the better the consumer is protected from accepting bad lots and the producer is protected by rejecting good lots.
Grant(439)

18. 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.

19. 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:

20. Problem A lot size of 300 of which 5 are defective.
Use Poisson formula to find the answer to number 2.
1) 2)
n = 300 * .10 = 30 n = 1000 * .10 = 100
P = p = .03
np = np = 3
A = 5 A = 4
P(A) = 100% P(A) = 81.5% 3)
n= 2500 * .10 = 250
p = .03
A = 9
P(A) = 77.3%

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

22. Poisson Table
Doty( )

23. Poisson Table
Doty( )

24. Poisson Table
Doty( )

25. 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.