In short: Most shortlists are decided by whichever criterion was thought about last. This makes the weights explicit first, then applies them evenly to every option.
Weights before scores
Deciding what matters before looking at the options is the only part of this that resists motivated reasoning. Set the weights first and leave them alone.
One scale, applied evenly
Each option is scored on the same criteria and the same scale, and the weighted total is divided by the total weight so the numbers stay comparable however you weighted them.
The number is not the decision
A weighted score makes your reasoning visible; it does not make it right. If the result feels wrong, that reaction is information - usually that a weight is not what you said it was.
Common questions
What scale should I score on?
Any consistent one. Nought to ten is easy to think in. What matters is using the same scale for every option on every criterion.
What if two options tie?
Then the criteria you listed do not separate them, and the tie-breaker is something you have not written down yet. Add it as a fifth consideration and decide by hand.
Can I use fewer than four criteria?
Leave the unused weights at zero. Four is the ceiling here, not the requirement.