I'm currently trying to think of a metric to measure a performance of a product that gives higher points to longer contracts.
For example, let's say that one contract A is a year long contract and the customer broke the contract after six months.
There's another contract B, which is a two year long contract and and the customer broke the contract after a year.
If I take the ratio between the duration of the contract and number of contracted years, both would be 0.5 and would be deemed equal in this case.
However, I would like to find a formula that gives a higher evaluation score on the longer contracted one even though the ratios between the duration and the contracted years are the same.
Thus, I would like something like
$f(\frac{a}{b}) < f(\frac{c}{d})$
when $\frac{a}{b} = \frac{c}{d}$ and $b<d$.
The one I thought was using logarithms like
$\frac{\log{a}}{\log{b}}$
This does get the kind of result that I want, but the difference gets too small as the contract-year increases.