Conclusion

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After seeing the main definitions of reliability and maintenance, let's finally see how we can use reliability knowledge also to carry out an economic optimization of replacement activities.

Consider a process that follows the power law with β>1. As time goes by, faults begin to take place more frequently and, at some point, it will be convenient to replace the system.

Let us define with τ the time when the replacement (here assumed instantaneous) takes place. We can build a cost model to determine the optimal preventive maintenance time τ* which optimizes reliability costs.

Let’s denote by C_f the cost of a failure and with C_r the cost of replacing the repairable system.

If the repairable system is replaced every τ time units, in that time we will have the replacement costs C_r and so many costs of failure C_f as how many are the expected number of faults in the time range(0;τ]. The latter quantity coincides with the expected value of the number of faults E[N(τ)].

The average cost per unit of time c(τ), in the long term, can then be calculated using the following relationship:

Then follows:

Differentiating c(τ) with respect to τ and placing the differential equal to zero, we can find the relative minimum of costs, that is, the optimal time τ* of preventive maintenance. Manipulating algebraically we obtain the following final result:

Consider, for example, a system that fails according to a Weibull distribution with β=2.2 and α=1500 hours. Knowing that the system is subject to replacement instantaneous and that the cost of a fault C_f=2500 € and the cost of replacing C_r=18000 €, we want to evaluate the optimal interval of replacement.

The application of Table 4.8 provides the answer to the question: