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How Long Should You Burn
In a System? (Part II)
In a previous
HotWire, the burn-in time for a non-repairable
system was obtained purely based on the system failure rate. [1]
These methods did not take into account the warranty period and
the costs associated with burning in the system, nor the
costs of a failure in the field. Such factors are considered in another
HotWire article where the analysis is done at
the component level. [2] In this article we will expand the
method presented there to a system with multiple components.
Alternatively, this method could be used for a component with multiple
failure modes.
Example
A disposable medical device made of three
components in series is experiencing a high number of failures
within the warranty period of 1 year. One of the components is
known to exhibit infant mortality; however, it cannot be
burned in unless it is assembled in the system. The manufacturer
would like to determine whether they could benefit from
subjecting the entire device to burn-in.
Individual component distributions have been determined as
follows:
| Component 1 |
Weibull, Beta = 0.5, Eta = 100 days |
| Component 2 |
Exponential, Mean = 1000 days |
| Component 3 |
Weibull, Beta = 3, Eta = 1000 days |
Before costs are considered, the first step is to determine
whether the system as a whole exhibits a decreasing failure
rate. If that is not the case, then any burn-in time will be
detrimental. In this particular example, because of the
simplicity of the system, this step is trivial. However, as the
complexity of the system increases, it is considered good
practice to examine the system failure rate behavior.
The diagram that follows shows how
reliability block diagrams can be used in
Weibull++ 7 to
determine the system failure rate (for more information, see [3]):
The system failure rate can then be calculated and
illustrated using the Failure Rate vs. Time plot, as shown next.
The overall system failure rate exhibits an initial region with a
decreasing failure rate. It is therefore concluded that the
system could benefit from a burn-in period.
The cost of
burning in the system is expected to be $5 per day, per unit. If a
failure is observed during the burn-in period, the cost is $20.
In comparison, the cost is determined to be $100 if a warranty claim has to be
processed.
Now we can determine the expected cost
per unit at a given burn-in time as follows:
 |
(1) |
where:
CUT_B is the cost per unit time.
CPF_B is the cost per failure during burn-in.
CPF_F is the cost per failure in the field.
Tb is the burn-in time.
T is the operational time (or warranty time).
R(t) is the reliability at time t.
For more information on the derivation of this equation, see
[2]. Note that in our example the reliability
used in Eqn. (1) is the system reliability, rather than the
component reliability.
When you click the ...
button
in the control panel, a window appears with the system reliability equation.
In order to obtain the optimum burn-in period,
Eqn. (1) can be minimized analytically. Alternatively, we could modify
the report template used in [2] to take into account the system
equation.
  Note that two columns have been added: Rel(T) contains the system reliability at the burn-in time and Rel(T+WT) contains the sum of the burn-in time and the warranty time. The cost column can be modified to use these values as shown next.
 The optimum burn-in time is then found to be 0.5 days.
Conclusion
In this article, we have discussed a method for determining the optimum burn-in time for a non-repairable system with multiple components or a component with multiple failure modes. This method took into consideration both the costs and the warranty time, along with the failure distributions for the individual components (or modes) and the reliability-wise configuration.
References
[1] ReliaSoft Corporation. "How Long Should You Burn In a System?"
ReliaSoft Corporation. 2006.
http://www.weibull.com/hotwire/issue69/relbasics69.htm.
[2]
ReliaSoft Corporation. "Quantifying Optimum Burn-in Period."
ReliaSoft Corporation. 2005.
http://www.weibull.com/hotwire/issue58/hottopics58.htm.
[3] ReliaSoft Corporation,
Life Data Analysis Reference, Tucson: ReliaSoft
Publishing, 2007.
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