I want to write an optimisation problem then solve it, to get optimised weights to compute a final score using a weighted arithmetic mean.
The problem is as follows. I have an entity (an input vector $V$) that I need to match with none/some/all others entities bases on their scores. Assuming there are $N$ candidate matches for $V$, if the score from $V$ to $C_i$ $i \in \{ 1, N \}$ noted $S_i$, is higher than a certain threshold, it is considered match, otherwise it is not. Consequently, we may end-up that none/some/all candidates are matches.
$S_i$ is a final weighted score that needs to be computed based on three scores, with different weights. The three scores $S_i^1$, $S_i^2$ and $S_i^3$ are outputed by a machine learning platform, each of which represents a certain attribute:
$$S_i = W_i^1 S_i^1 + W_i^2 S_i^2 + W_i^3 S_i^3$$
So I want to model this as an optimisation problem whose decision variables are the weights. I am thinking to set the objective function as $\sum_i S_i$. Nevertheless, I am not convinced by my approach.
My question is are there any state-of-art/classical problems that I could refer to refine my modeling? How could classify this problem? It is not a 100% matching problem as I have only one entry in the left part of the bipartite graph.
Thanks in advance for any feedback.