robotoid factories vs mining facils

[LintMan]

I'm wondering what the optimal number of robotoid factories are if you are building a mining world, so I decided to do some investigation and found some interesting things. The same calculations should apply to research centers/computer labs

My biggest assuption is that you can build multiple robotoid factories and get a cumulative bonus from them. I'm pretty sure it works that way.

I'm not 100% sure how all the arrorted other bonuses (such as population and culture) are applied so I might be off in my calculations.

Anyway, here's what I've found:

M = max total # of mining and factory facils you can build on the planet ==> (x + y) = M
x = # of mining facils
y = # of robotoid factories ==> y = (M - x)
b = % bonus that a robotoid factory gives
k = amount of minerals produced by one mining facility (Factor all the other bonuses and planetary value % into this number)

As I understand it: the total resources you get on a planet is (number of mining facils times the amount per facil) plus (number of
robotoid factories times the bonus per factory times the total production of all the facilities) which can be expressed as:

kx + yb(kx)

This simplifies to:

kx(1 + yb)

We can substitute in for y = M-x, and rearrange things a bit to get:

kx((1+bM) - bx)

With some further algebra we get...

k(1+bM)x - bkx^2 (that's x-squared)

If you graphed this, it looks like an upside-down U. We can now find the value of x (number of mining facils) that produces the maximum total resources.

We do that by taking the derivative and then solving for when it is zero...

k(1+bM) - 2bkx = 0

Solving this and cancelling out the k's gives us:

1 + bM
======= = x
2b

This is the optimal number of mining facilities to build, with the rest being robotoid factories. If this number isn't an integer, just round to the nearest int.

Note that there is no k variable remaining. That means that the optimum number of mining facils depends only on the factory bonus % and the total # of facils you can build.

So, some examples:

For a level 3 factory (30% bonus I think) on a planet with 20 open facility slots (so M = 20, b = 0.3):
(1+(0.3)*(20))/(2*(0.3)) = 11.66
So you'd optimally make 12 mining facilities and 8 robotoid factories

For a level 3 factory (30% bonus I think) on a planet with 10 open facility slots (so M = 10, b = 0.3):
(1+(0.3)*(10))/(2*(0.3)) = 6.66
So you'd optimally make 7 mining facilities and 3 robotoid factories

For a level 1 factory (10% bonus I think) on a planet with 20 open facility slots (so M = 20, b = 0.1):
(1+(0.1)*(20))/(2*(0.1)) = 15
So you'd optimally make 15 mining facilities and 5 robotoid factories

For a level 1 factory (10% bonus I think) on a planet with 10 open facility slots (so M = 10, b = 0.1):
(1+(0.1)*(10))/(2*(0.1)) = 10
So you'd optimally make 10 mining facilities and 0 robotoid factories

I haven't verified this yet experimentally, but I'm pretty sure the math is correct at least.

Doess anyone have more information about exactly how/when the assorted bonues are applied in relation to the factories?