Optimal build to maximize output of districts + support districts

[unmerged(87171)]

Apologizes if someone has already posted an analysis of this, but as a fun math problem, I decided to figure out the optimum ratio of production districts to support districts myself. Feel free to call out any mistakes or incorrect assumptions on my part.

My assumption is that the output of a given resource is given by:

(JD ND + JS SD + FJ) SD SM PJ

Constraint
ND + SD = TD

Where:
PJ = Production Per Job
JD = Jobs Per District (City District for specialist resource, Resource District for worker resources)
JS = Jobs Per Support District (meaning either a City District for worker resources, or a Resource District for specialist resources, assuming you've built the appropriate district specialization)
ND = Districts
FJ = Fixed Jobs (i.e. not from districts but other sources like buildings)
SD = Support Districts
TD = Total Districts
SM = Support District Multiplier

So
(JD (TD - SD) + JS SD + FJ) SD SM PJ

Expanding
(JD TD SD - JD SD^2 + JS SD^2 + FJ SD) SM PJ
((JD TD + FJ) SD - (JD - JS) SD^2) SM PJ

Now to find the max (or min) of a function you take the derivative and see where it's zero:

(JD TD + FJ - 2 (JD - JS) ND) SM PJ = 0

Which gives us

ND = (JD TD - FJ) / (2 (JD - JS))

We also need to check the second derivative to make sure it's actual a max and not a min or inflection point.

-2 (JD - JS)

Which will be negative as long as JD is greater than JS (should always be the case, otherwise there would never be any reason to build a regular district) verifying that this is a maximum value.

In the special case of no fixed jobs and the support district not providing any jobs, it just reduces to the solution familiar to Balatro players playing on the Plasma desk, a 1:1 ratio of districts to support districts.

ND = TD / 2 = SD

Also in the case that FJ > JD TD you should only build support districts, but presumably that would be very rare, like trying to maximize resource output on a planet that's not suited for production of that resource.

Of course this assumes you have the population to support the optimal number of districts, if not obviously you build as many as you can fill then make the rest support districts. And this only assumes you trying to maximize one type of output.

 

[HFY]

Is this for supported Specialists (e.g. Researcher subtypes), or for supported Workers (which get +% from eating Trade), or both?

My question for Researchers would be if diverting pops to work those Worker jobs would be better than just sending them to do more Research ... and if so, then maybe there's a strong benefit to Kilostructures which enable jobless acquisition of the resources consumed by Resdearch-boosting support districts without costing jobs.

 

[unmerged(87171)]

Yeah all those externalities aren't taken into account. That's what makes the game so fun, trying to manage all the other tradeoffs, like trade, energy, population, automation, etc. This is just the calculation for getting the maximum output of one resource on one planet, not the tradeoffs for having local deficits, etc.

 

[HFY]

Yeah all those externalities aren't taken into account. That's what makes the game so fun, trying to manage all the other tradeoffs, like trade, energy, population, automation, etc. This is just the calculation for getting the maximum output of one resource on one planet, not the tradeoffs for having local deficits, etc.

If you feel like doing the externality math then my question would be:

- What is the +%x bonus above which +n (food / energy / minerals) and +m Trade is NOT worth the +10% for that type of research?

- Do the +flat research buildings change the math at all?

 

[Drachasor]

You accidentally changed the +FJ to a -FJ at the last step there