Operating Characteristic (OC) curves are powerful tools in the
field of quality control, as they display the discriminatory
power of a sampling plan. In this article, we explain the
applications of OC curves in reliability engineering.
In quality control, the OC curve plots the probability of
accepting the lot on the Yaxis versus the lot fraction or
percent defectives (p) on the Xaxis [1], as
illustrated in Figure 1.
Figure 1: OC Curves in Quality Control
Based on the number of defectives in a sample, the quality
engineer can decide to accept the lot, to reject the lot or
even, for multiple or sequential sampling schemes, to take
another sample and then repeat the decision process.
In reliability engineering, the OC curve shows the
probability of acceptance (i.e. the probability of
passing the test) versus a chosen test parameter. This parameter
can be the true or designedin mean life (MTTF) or the
reliability (R), as shown in Figure 2.
Figure 2: Probability of Acceptance Versus MTTF or
Reliability
The probability of acceptance, P(A), can be represented by
the cumulative binomial distribution [2]:
which gives the probability that the number of failures
observed during the test, f, is less than or equal to the
acceptance number, c, which is the number of allowable
failures in n trials. Each trial has a probability of
succeeding of R, where R
is the reliability of each unit under test (an analog to the
probability of success in each Bernoulli trial). The reliability
OC curve is developed by evaluating the above equation for
various values of R.
In Table 1, we calculate the cumulative binomial probability
for different levels of reliability and c = 2 allowable
failures in n = 10 test samples.
Table 1: Probability of Acceptance for
Various Values of Reliability when (n, c) = (10,
2)
Reliability 
P(A) 
0.00 
0.0000 
0.05 
0.0000 
0.10 
0.0000 
0.15 
0.0000 
0.20 
0.0001 
0.25 
0.0004 
0.30 
0.0016 
0.35 
0.0048 
0.40 
0.0123 
0.45 
0.0274 
0.50 
0.0547 
0.55 
0.0996 
0.60 
0.1673 
0.65 
0.2616 
0.70 
0.3828 
0.75 
0.5256 
0.80 
0.6778 
0.85 
0.8202 
0.90 
0.9298 
0.95 
0.9885 
1.00 
1.0000 
A similar method can be used to develop an OC curve for a
reliability demonstration test. We can use
Weibull++'s
Design of Reliability Tests (DRT) tool to simplify the
calculation of the probability of acceptance values. The DRT can
be accessed by either choosing Design of Reliability Tests
from the
Tools menu or clicking the DRT icon on the General
toolbar.
On the NonParametric Binomial page of the DRT, we can
calculate the confidence level for each value of reliability.
The probability of acceptance used to construct the OC curve
will equal 1 CL, where CL is the confidence level, since the
binomial lower onesided confidence limit is equal to the beta
risk and equal to 1 CL [2 , 3
and also see
Type I and Type II Errors and Their Applications in the
June, 2008 issue of the Reliability Hotwire].
For example, if reliability R = 90%, number of units
n = 10, and number of allowable failures c = 2,
the confidence level is CL = 0.0702, as shown in the calculation
in Figure 3. So, in this case, P(A) = 1 CL = 1 0.0702 = 0.9298.
Figure 3: Weibulls++ Design of Reliability Tests Tool, Used
to Calculate Confidence Level
By plotting the probability of acceptance versus reliability,
we get the OC curve for this test, as shown in Figure 4.
Figure 4: OC Curve Corresponding to Table 1 Values when (n,
c) = (10, 2)
The selection of a reliability demonstration test plan
depends upon the degree of risk that is acceptable and upon the
cost of testing [4]. Comparison of different OC
curves can provide an indication of relative risk.
In Figure 5 we examine the Type I (alpha or producers risk)
and Type II (beta or consumers risk) errors for two different OC
curves with a constant ratio of allowable defects to sample size
c/n = 0.2. (For a discussion of Type I and Type II
errors, see
Type I and Type II Errors and Their Applications in the
June, 2008 issue of the Reliability Hotwire.)
Figure 5: Effect of Sample Size on Alpha and Beta Risks
For a low reliability, e.g. R = 0.72:
For (n, c) = (10, 2) the
probability of acceptance P(A) is 0.43.
For (n, c) = (30, 6) the
probability of acceptance P(A) is 0.22.
So in this region, by increasing the sample size we are
lowering the beta risk (i.e.
the risk of accepting a test when in fact the reliability is
lower than the acceptable minimum).
For a higher reliability, we observe a similar behavior: an
increased sample size reduces the alpha error of rejecting a
test when in fact we have reliability higher than required.
The steeper the OC curve, the smaller the alpha and beta
errors, in other words the greater the discriminatory power. If
the alpha and beta errors were zero, the "ideal Reliability OC
curve" would look like a step function, as shown in Figure 6. In
practice, this can never be obtained, unless the whole
population is tested and there are no errors in identifying
failed versus passed units.
Figure 6: The Ideal OC Curve
The selection of the appropriate OC curve for a test depends
on the relative risk factors and the associated costs that the
organization is willing to afford at each phase of a project.
For example, during product development, an OC curve with high
alpha and beta risks might be selected, and during
manufacturing, a more conservative curve with lower alpha and
beta risks might be chosen.
We can design OC curves for specified levels of alpha and
beta risk. Again assuming binomial sampling, we can derive the
acceptance number, c, and sample size, n, by
specifying the required alpha and beta values in the system of
nonlinear equations below:

(1) 

(2)
