I am trying to combine a few different metrics into a single metric, then attempt to compare that single metric when I change the input metrics values. These input values must also have an adjustable weight (or 'importance').
These metrics might have different ranges and distributions.
For example, I could have the metrics:
$A$ [Range: 0, 1] Grows Linearly
$B$ [Range: 0, 1] Grows Exponentially
$C$ [Range: 0, 1000] Grows Linearly
With weights $Wa, Wb, Wc$
Combining them linearly without any normalization is clearly flawed, eg. $AW_a + BW_b + CW_c$, as the weights are meaningless due to the different magnitudes, distributions etc. There may be a way or normalizing these values between $0...1$, but I do not know if that will fix the problem of comparing distributions.
I've also attempted $(A^{W_a})(B^{W_b})(C^{W_c})$ but I think this also has problems when one of the terms is 0, and also fails to account for different distributions sensibly. I figure just adding 1 would fix the first problem, however the second problem still stands.
How would I combine these scores? Even some direction as to further reading would be appreciated.