I think you need about 50-60K ballistae for the main attack, lose about 15-20K. They should kill 1.1M warriors without ever getting touched by them.

So that's the equivalent of training 400-500K archers and upkeeping 300-350K.
If 20K ballys is all that you'd lose in the attack, that's WAY cheaper than an estimated 300-500K archers.

But then again, it could all go wrong and u trained 60K ballys for nothing lol.