operating characteristic (oc) curves ben m. coppolo penn state university

Operating Characteristic Operating Characteristic (OC) Curves(OC) Curves

Ben M. Coppolo Ben M. Coppolo

Penn State UniversityPenn State University

Presentation OverviewPresentation Overview

• Operation Characteristic (OC) curve Operation Characteristic (OC) curve DefinedDefined

• 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

OC Curve DefinedOC Curve Defined

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

lots.lots.

OC Curves UsesOC Curves Uses

• Selection of sampling plansSelection of sampling plans

• Aids in selection of plans that are Aids in selection of plans that are effective in reducing riskeffective in reducing risk

• Help keep the high cost of Help keep the high cost of inspection down inspection down

OC CurvesOC Curves

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

Types of OC CurvesTypes of OC Curves

• Type AType A– Gives the probability of acceptance for an Gives the probability of acceptance for an

individual lot coming from finite productionindividual lot coming from finite production• Type BType B

– Give the probability of acceptance for lots Give the probability of acceptance for lots coming from a continuous processcoming from a continuous process

• Type CType C– Give the long-run percentage of product Give the long-run percentage of product

accepted during the sampling phaseaccepted during the sampling phase

OC Graphs Explained OC Graphs Explained

• Y axisY axis– Gives the probability that the lot will Gives the probability that the lot will

be acceptedbe accepted

• X axis =pX axis =p– Fraction DefectiveFraction Defective

• PPff is the probability of rejection, is the probability of rejection, found by 1-Pfound by 1-PAA

OC CurveOC Curve

Definition of VariablesDefinition of Variables

PPA A = The probability of acceptance= The probability of acceptance

p = The fraction or percent defectivep = The fraction or percent defective

PPF F or alpha = The probability of rejectionor alpha = The probability of rejection

N = Lot sizeN = Lot size

n = The sample sizen = The sample size

A = The maximum number of defectsA = The maximum number of defects

OC Curve CalculationOC Curve Calculation

• Two Ways of Calculating OC CurvesTwo Ways of Calculating OC Curves– Binomial DistributionBinomial Distribution– Poisson formula Poisson formula

• P(A) = ( (np)^A * e^-np)/A !P(A) = ( (np)^A * e^-np)/A !


• Binomial DistributionBinomial Distribution– Cannot use because:Cannot use because:

• Binomials are based on constant Binomials are based on constant probabilities.probabilities.

• N is not infiniteN is not infinite• p changesp changes

– But we can use something else.But we can use something else.


• A Poisson formula can be usedA Poisson formula can be used– P(A) = ((np)^A * e^-np) /A !P(A) = ((np)^A * e^-np) /A !

• Poisson is a limit Poisson is a limit – Limitations of using PoissonLimitations of using Poisson

• n<= 1/10 total batch Nn<= 1/10 total batch N• Little faith in probability calculation when n is Little faith in probability calculation when n is

quite small and p quite large.quite small and p quite large.

• We will use Poisson charts to make this We will use Poisson charts to make this easier.easier.

Calculation of OC CurveCalculation 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 = d A = d

• From a Poisson table find your PFrom a Poisson table find your PAA

Calculation of an OC CurveCalculation of an OC Curve

• N = 1000N = 1000• n = 60n = 60• p = .01p = .01• A = 3A = 3

• Find PFind PA A for p for p = .01, .02, .05, .0= .01, .02, .05, .07, .1, and .12?7, .1, and .12?

NpNp d= 3d= 3

.6.6 99.899.8

1.21.2 87.987.9

33 64.764.7

4.24.2 39.539.5

66 151151

7.27.2 072072

Properties of OC CurvesProperties of OC Curves

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


• The acceptance number and The acceptance number and sample size are most important sample size are most important factors.factors.

• Decreasing the acceptance number Decreasing the acceptance number is preferred over increasing sample is preferred over increasing sample size.size.

• The larger the sample size the The larger the sample size the steeper the curve.steeper the curve.


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

Example UsesExample Uses

• A company that produces push A company that produces push rods for engines in cars.rods for engines in cars.

• A powdered metal company that A powdered metal company that need to test the strength of the need to test the strength of the product when the product comes product when the product comes out of the kiln.out of the kiln.

• The accuracy of the size of The accuracy of the size of bushings.bushings.

ProblemProblem

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

ProblemProblem

1.1. A lot size of 300 of which 5 are A lot size of 300 of which 5 are defective.defective.



4.4. Use Poisson formula to find the Use Poisson formula to find the answer to number 2.answer to number 2.

SummarySummary

• Types of OC curvesTypes of OC curves– Type A, Type B, Type CType 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

Poisson Table Poisson Table d

np 0 1 2 3 4 5 6 7 8 9 100.02 980 10000.04 961 999 10000.06 942 998 10000.08 923 997 10000.1 905 995 10000.15 861 990 999 10000.2 819 982 999 10000.25 779 974 998 10000.3 741 963 996 10000.35 705 951 994 10000.4 670 938 992 999 10000.45 638 925 989 999 10000.5 607 910 986 998 10000.55 577 894 982 998 10000.6 549 878 977 997 10000.65 522 861 972 996 999 10000.7 497 844 966 994 999 10000.75 472 827 959 993 999 10000.8 449 809 953 991 999 10000.85 427 791 945 989 998 10000.9 407 772 937 987 998 10000.95 387 754 929 984 997 10001 368 736 920 981 996 999 1000

Poisson Table Poisson Table d

np 0 1 2 3 4 5 6 7 8 9 101.1 333 699 900 974 995 999 10001.2 301 663 879 966 992 998 10001.3 273 627 857 957 989 998 10001.4 247 592 833 946 986 997 999 10001.5 223 558 809 937 981 996 999 10001.6 202 525 783 921 976 994 999 10001.7 183 493 757 907 970 992 998 10001.8 165 463 731 891 964 990 997 999 10001.9 150 434 704 875 956 987 997 999 10002 135 406 677 857 947 983 995 999 1000

2.2 111 335 623 819 928 975 993 998 10002.4 91 308 570 779 904 964 988 997 999 10002.6 74 267 518 736 877 951 983 995 999 10002.8 61 231 469 692 848 935 976 992 998 999 10003 50 199 423 647 815 916 966 988 996 999 1000

3.2 41 171 380 603 781 895 955 983 994 998 10003.4 33 147 340 558 744 871 942 977 992 997 9993.6 27 126 303 515 706 844 927 969 988 996 9993.8 22 107 269 473 668 816 909 960 984 994 9984 18 92 238 433 629 785 889 949 979 992 997

4.2 15 78 210 395 590 753 867 936 972 989 9964.4 12 66 185 359 551 720 844 921 964 985 9944.6 10 56 163 326 513 686 818 905 955 980 9924.8 8 48 143 294 476 651 791 887 944 975 990

Poisson TablePoisson Tabled

np 0 1 2 3 4 5 6 7 8 9 105 7 40 125 265 440 616 762 867 932 968 986

5.2 6 34 109 238 406 581 732 845 918 960 9825.4 5 29 95 213 373 546 702 822 903 951 9775.6 4 24 82 191 342 512 670 797 886 941 9725.8 3 21 72 170 313 478 638 771 867 929 9656 2 17 62 151 285 446 606 744 847 916 957

6.2 2 15 54 134 259 414 574 716 826 902 9490.4 2 12 46 119 235 384 542 687 803 886 9396.6 1 10 40 105 213 355 511 658 780 869 9276.8 1 9 34 93 192 327 480 628 755 850 9157 1 7 30 82 173 301 450 599 729 830 901

7.2 1 6 25 72 156 276 420 569 703 810 8877.4 1 5 22 63 140 253 392 539 676 788 8717.6 1 4 19 55 125 231 365 510 648 765 8547.8 0 4 16 48 112 210 338 481 620 741 8358 0 3 14 42 100 191 313 453 593 717 816

BibliographyBibliography

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.

operating characteristic (oc) curves ben m. coppolo penn state university

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