Reliability HotWire

Issue 103, September 2009

Hot Topics

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.

Copyright 2009 ReliaSoft Corporation, ALL RIGHTS RESERVED