I've got a question that's a variation of this one:
How to weight three different variables to create a ranking
I'm starting a new question because of the age of the other question. The difference is that in my situation, I want to allow the weights to be negative. In the other question, an answer very helpfully describes how to handle the case where the scores can be negative, but when the weights are negative, the math does not work out.
I expect that having a negative score in a category with a negative weight contributes to having a higher overall score, and having a higher positive score in a category with a negative weight would be contribute to having a lower overall score… Perhaps even a negative score.
One of the things with which I'm struggling, is the case where two weights are given, for example, 1 and -1. When dividing by the sum of the scores, you end up dividing by 0. What is the proper way to deal with this?
EDIT:
Here's what worked: Transfer the sign of the weight to the value, and then treat the weight as if it were positive. So, if value is positive, and then weight is negative, the scored value is negative. If the value is negative, and the weight is negative, the scored value is positive (which makes sense if you have a low value in something with a negative weight). When dividing by the sum of all the weights, you have to divide by the sum of the absolute value of the weights, so that in the case of the weights being +1 and -1 and dividing by zero, instead, you divide by +2.
Best Answer
If you deal with the absolute values for the weights then it's equivalent to having the score negative. Say you have scores $s_{+,-}$ with weights $w_{+,-}$, with $w_-<0<w_+$. Then $$\frac{w_+s_++w_-s_-}{|w_+|+|w_-|}=\frac{|w_+|s_++|w_-|(-s_-)}{|w_+|+|w_-|}$$ Not sure if negative weight make sense.