The cheap 12V Chinese LED PSUs generally go up to at least 14V. You can safely feed up to 15V to the Ardiuno/RAMPS combo if you are not running an LCD or anything else from the 5V rail. But you said you were using a relay, and if the relay coil is powered from 5V then you may already be overloading the Arduino voltage regulator with 12V input.
You should preferably put three IN4001 or similar diodes in series with the fan(s) to drop the fan voltage back to 12V.
You should preferably put three IN4001 or similar diodes in series with the fan(s) to drop the fan voltage back to 12V.