Oh, math! I love math! I got sniped a little bit, but I’mma post this anyway.
Most expensive Heavy Cannon or Great Bombard is 822.25, and least expensive Heavy Cannon 316.25 and Great Bombard is 379.5. Most expensive Rocket is 686.4, and least expensive Rocket is 377.3. Considering the cost of a Heavy Cannon or Great Bombard is listed as 200 Wood, 500 Coin and 100 Wood, 600 Coin respectively, and a Rocket is 100 Wood, 500 Coin, this seems pretty reasonable.
Oh, man, that was fun!
More of this please! <3
Conclusion: If you don’t upgrade your factories to produce resources you’ll always “save” resources by producing cannons. If you do upgrade your factories, which pays itself off in a few minutes, you’ll “save” resources as long as you get the upgrade, but not a lot. If you combine cards with the upgrade you’ll start seeing rather big savings. It works best for Swedes or Chinese(!).
There’s the positive that Heavy Cannons are better than the craftable cannons in that they are very tanky and have very few damage modifiers. That they’re continuously produced also helps you with your macro, not having to remember to go back and produce more every once in a while. Sort of like Lakota with houses or Ottomans with Settlers.
There’s the negative of not being able to reinforce quickly by batch-training, the speed, and that Heavy Cannons use up 7 pop instead of 6.
Math and reasoning ahead: Heavy Cannon (and Great Bombard) has a training time of 115 seconds (according to AoE fandom wiki). All civs can reduce this by 25% using the Factory upgrade, 86.25.
Ottomans, French and Swedes can reduce this with Engineering School to either 86.25 (according to the wiki. Maybe it’s based on before Definitive edition? Or Engineering School’s been nerfed?) or 103.5 (my math, 115 with 10% reduction).
Chinese (can gain 1 factory through consulate, allying with Russians) or Swedes can reduce this with TEAM Engineering School to 109.25 (my math, wiki doesn’t provide this number). Sweden (or Ottomans or French with a helpful Chinese or Swedish ally) can reduce this with both cards to 69 (Wiki) or 98.32 (my math, multiplicative).
Upgrade+Engineering School+TEAM Engineering School according to my math would be 73.74. If it’s multiplicative. If it’s additive it goes down to 69 (like the wiki says). I know resource reduction cards are multiplicative (try reducing the cost of estates with multiple cards, the order you do it in matters, always do biggest first for best result), but training time? Who even knows.
77.6 with upgrade and just Engineering School multiplicative. For completion’s sake.
Does the Chinese card Acupuncture reduce the training time of Heavy Cannon’s by 20% as it’s said to reduce “Unit” by 20%? It doesn’t seem to have the Flying Crow negative added to Heavy Cannon. If so, this would be 57.5 with upgrade and TEAM Artillery, making Chinese Heavy Cannon’s the cheapest in the game.
Edit: Actually, given that you can train things instantly with other cards, it’s probably additive. I’ll use the additive numbers. The Wiki numbers probably just included the 25% faster training time upgrade without saying it. Leaving my mistakes up there, but working with the (probably) correct numbers going forward.
Notably British Rockets take 98 (or 90, wiki claims both numbers, in different places) seconds to produce instead, but they don’t have access to Engineering School. 73.5 with upgrade. 68.6 additive or 69.8 multiplicative with TEAM Engineering School from an ally.
Factories produce 5.5 resource per second without upgrade, or 7.15 upgraded. So now we just multiply.
Source: https://ageofempires.fandom.com/wiki/Factory and the pages for Heavy Cannon, Great Bombard, and Rocket