# Construct an Operating Characteristic Curve

### Construct an Operating Characteristic Curve

The specific method used to construct an OC Curve depends on whether the measure (or requirement statement) of reliability is made in terms of probability of success versus failure or in terms of duration.  Probabilistic reliability OC curve construction draws on the Binomial distribution, whereas duration reliability utilizes the Exponential and Poisson distributions.

##### Binomial OC Curves for Success/Failure based Reliability

A traditional method for constructing a demonstration test via acceptance sampling would solve the following equations simultaneously for n, the number of samples required, and c, the allowable number of failures to meet the requirement:

$$1-\alpha = \Sigma^c_{i=0} (^n_i)p^i_{max}(1-p_{max})^{n-i}$$

$$\beta = \Sigma^c_{i=0}(^n_i)p^i_{min}(1-p_{min})^{n-i}$$

where α is the producer’s risk or the probability of failing the test if they deliver a product that has a high reliability $$(p_{max})$$ , and β is the consumer’s risk or the probability of passing a system if the product actually has low reliability $$(p_{min})$$.

In defense applications, evaluators are typically worried most about ensuring that the requirement has been met (minimizing consumer risk) and less about controlling producer risk.  Therefore, evaluators construct these curves by first solving the consumer’s risk equation for allowable failures ranging from 0 to some fixed number (often no more than 5 -10 because of limited test resources).  In Excel this is easy to do using the Goal Seek Function with the binomial distribution for a fixed number of failures and allowing the total test size to change to match the desired consumer risk.  A full curve can then be generated by changing the true probability of failure and calculating the probability of passing the test for a fixed sample size and acceptance criteria (c allowable failures).  Several tools are available to help you construct Binomial OC Curves.

##### Exponential OC Curves for Duration based reliability

To construct an OC Curve for a duration based requirement, evaluators use a similar process.  First the total test duration for a fixed number of failures must be set using the Chi-squared lower confidence bound:

$$Test \ Duration = \frac{\chi^2_{\alpha/2,2c+2^\theta}}{2}$$

Where c is the fixed allowable number of failures and $$\theta$$ is the required mean time until failures.  In Excel this can be generated using: CHIINV(α/2, 2c+2).

For example, for a howitzer with a 62.5 hour mean time between failures requirement, the minimum test duration for a 3 failure test would be 485 hours, or approximately 27 missions.  The full curve can then be generated by varying the true mean failure parameter and calculating the probability of passing using the Poisson distribution.  In other words, we need to calculate the probability of seeing c or fewer failures given a known failure rate.  This is given by the cumulative distribution of the Poisson distributions.  In Excel the probability of passing a mission is calculated using: POISSON(c, TestDuration/θ, TRUE).

Several tools are available here to help you construct Exponential OC Curves.

Exponential Operating Characteristic Curves