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

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)

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)

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.

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

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

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

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

OC Curve Calculation Two Ways of Calculating OC Curves Binomial Distribution Poisson formula P(A) = ( (np)^A * e^-np)/A ! Vaughn(112-113)

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)

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)

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

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 = 64.7 PA = 39.5 n * p =60 * .1 = 6 A = d = 3 PA = 15.1 n * p =60 * .12 =7.2 PA = 7.2

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

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(290-291)

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)

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)

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.

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:

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 = .03 p = .03 np = 0.9 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%

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

Poisson Table Doty(364-366)

Poisson Table Doty(364-366)

Poisson Table Doty(364-366)

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.