I always hit the supply cap every day even with my productions going full tilt. That's 5/5/5 + troop training. I used to simply let it be but then I figured, if it's going to help me do more in tourney, why not?
What's your city look like overall? I've got 15 fully-upgraded workshops (level 31), which I feel is a lot based on other advanced cities I look at, and even that's barely enough to break even. 6/5/4 on manufactories, 6 armories, and of course constantly training troops puts my daily supply costs at anywhere from 650k to 1840k (depending on what I'm training), while I produce around 1480k on average with a minimum 125% culture bonus (this is including AWs and event buildings).
T1 production is ~26k per day
T2 production is ~84k per day
T3 production is ~251k per day
Armory production is ~288k per day
Training Grounds can run 8.13 training slots per day, costing up to ~618k (but could be as little as 0 if I'm training Orcs)
Mercenary Camp can run 11.5 slots per day, costing ~876k
Barracks can run 15.65 slots per day, costing ~1190k
It really depends on what I'm training. I tend to favor my Barracks though because of its speed, and the units perform well enough compared to their specialized counterparts in all tournaments except for Marble, Planks, and Magic Dust, where Blossom Mages, Orc Strategists, and Cerberus respectively all perform much better than Priests, Mortars, and Barbarians that it's worth the slower training speed to recruit them instead.
It could be argued that I could increase the levels of my Endless Excavation and/or Prosperity Towers, but I've also got plenty of supply instants to cover me for a good while as I continue increasing my Flying Academy, Victory Springs, and Needles of the Tempest instead (level 30 in about a week!). Maybe my priorities will change if they ever introduce instants for troops!