I am trying to derive a formula for my collaborative algorithm problem to calculate popularity rating of an item.
I am considering three factors to calculate rating for an item based on three different ratings $a$, $b$ and $c$, each with weight $w_1$, $w_2$ and $w_3$. The cumulative rating is:
$$w_1 a + w_2 b + w_3 c$$
In this equation, I want to give $b$ less weight than $a$, i.e. $w_1 > w_2 > w_3$.
But the problem is that how should I decide/calculate the weights such that they maintain this relative ordering when normalized (I am not looking for linear normalization) i.e.
$$w_1 a > w_2 b > w_3 c$$
Is there a way to calculate $w_1$, $w_2$, $w_3$ dynamically and what normalization process I should use or is recommended?
It would be really helpful to get comments on the approach as well.