Assuming the cost funciton on the following format:
$ \color{blue}{ C(\tau) = C_\mbox{PM} / \tau + \lambda_{\mathrm{E}}(\tau) [C_\mbox{CM} + C_\mbox{EP} + C_\mbox{ES} + C_\mbox{EM}]} $
where the effective failure rate could be expressed as :
$ \color{blue}{\lambda_{\mathrm{E}}(\tau) = \left ( \frac{\Gamma(1+1/\alpha)}{\mathrm{MTTF}} \right )^\alpha \tau^{\alpha-1}} $
we may find an analytical solution by taking the derivative, setting the drivative equal to 0, and solve with respect to τ:
$ \color{blue}{\tau = \frac{{\mathrm{MTTF}}}{ \mathrm{\Gamma} (1+1/\alpha )} \sqrt[\alpha]{ \frac{C_{\mathrm{PM}} }{ (\alpha-1)(C_{\mathrm{CM}} + C_{\mathrm{EP}}+ C_{\mathrm{ES}}+C_{\mathrm{EM}}) } }} $
This standard solution is often appropriate for replacement or overhaul based on time. The approximation of the effective failure rate is reasonable if τ ≪ MTTF. For larger value of τ better approximations for the effective failure rate exist.