OCCurves[1] - Operating Characteristic(OC Curves Ben M...

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Unformatted text preview: Operating Characteristic (OC) Curves Ben M. Coppolo Penn State University 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. 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 • What can OC curves be used for in an organization? 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 OC Graphs Explained • Y axis – 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 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 OC Curve Calculation • Two Ways of Calculating OC Curves – Binomial Distribution – Poisson formula • P(A) = ( (np)^A * e^-np)/A ! 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. OC Curve Calculation • A Poisson formula can be used – 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. 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 • • • • • N = 1000 n = 60 p = .01 A=3 Find PA for p = . 01, .02, .05, .07, . 1, and .12? Np .6 d= 3 99.8 1.2 87.9 3 64.7 4.2 39.5 6 151 7.2 072 Properties of OC Curves • Ideal curve would be perfectly perpendicular from 0 to 100% for a given fraction defective. 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. Properties of OC Curves 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. 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 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. Summary • Types of 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 d np 0.02 0.04 0.06 0.08 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 980 961 942 923 905 861 819 779 741 705 670 638 607 577 549 522 497 472 449 427 407 387 368 1 1000 999 998 997 995 990 982 974 963 951 938 925 910 894 878 861 844 827 809 791 772 754 736 2 3 4 5 6 1000 1000 1000 1000 999 999 998 996 994 992 989 986 982 977 972 966 959 953 945 937 929 920 1000 1000 1000 1000 1000 999 999 998 998 997 996 994 993 991 989 987 984 981 1000 1000 1000 1000 1000 999 999 999 999 998 998 997 996 1000 1000 1000 1000 1000 1000 1000 999 1000 7 8 9 10 Poisson Table d np 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 0 333 301 273 247 223 202 183 165 150 135 111 91 74 61 50 41 33 27 22 18 15 12 10 8 1 699 663 627 592 558 525 493 463 434 406 335 308 267 231 199 171 147 126 107 92 78 66 56 48 2 900 879 857 833 809 783 757 731 704 677 623 570 518 469 423 380 340 303 269 238 210 185 163 143 3 974 966 957 946 937 921 907 891 875 857 819 779 736 692 647 603 558 515 473 433 395 359 326 294 4 995 992 989 986 981 976 970 964 956 947 928 904 877 848 815 781 744 706 668 629 590 551 513 476 5 999 998 998 997 996 994 992 990 987 983 975 964 951 935 916 895 871 844 816 785 753 720 686 651 6 1000 1000 1000 999 999 999 998 997 997 995 993 988 983 976 966 955 942 927 909 889 867 844 818 791 7 8 9 10 1000 1000 1000 1000 999 999 999 998 997 995 992 988 983 977 969 960 949 936 921 905 887 1000 1000 1000 1000 999 999 998 996 994 992 988 984 979 972 964 955 944 1000 1000 999 999 998 997 996 994 992 989 985 980 975 1000 1000 1000 999 999 998 997 996 994 992 990 Poisson Table d np 5 5.2 5.4 5.6 5.8 6 6.2 0.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 0 7 6 5 4 3 2 2 2 1 1 1 1 1 1 0 0 1 40 34 29 24 21 17 15 12 10 9 7 6 5 4 4 3 2 125 109 95 82 72 62 54 46 40 34 30 25 22 19 16 14 3 265 238 213 191 170 151 134 119 105 93 82 72 63 55 48 42 4 440 406 373 342 313 285 259 235 213 192 173 156 140 125 112 100 5 616 581 546 512 478 446 414 384 355 327 301 276 253 231 210 191 6 762 732 702 670 638 606 574 542 511 480 450 420 392 365 338 313 7 867 845 822 797 771 744 716 687 658 628 599 569 539 510 481 453 8 932 918 903 886 867 847 826 803 780 755 729 703 676 648 620 593 9 968 960 951 941 929 916 902 886 869 850 830 810 788 765 741 717 10 986 982 977 972 965 957 949 939 927 915 901 887 871 854 835 816 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. ...
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