**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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- Spring '16
- Poisson Distribution, Binomial distribution, 175, ) Curves, OC Curve, OC Curves