| Re: Resource integral hold for IS? |
Mon, 07 August 2017 04:42   |
|
 magic9mushroom |
 | Commander |
Messages: 1361
Registered: May 2008 | |
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Update: I solved for the optimal holds as a function of planet habitability, pop efficiency, factory settings, population growth rate, and pop available (yes, all of them are relevant). I got working formulae for the case where the optimal hold is between 0 and 25% of capacity, and for the case where the optimal hold is between 100% and 300%.
The case where the hold is between 25% and 100% is much, much harder; it's a casus irreducibilis cubic over most of the practical cases, with three real solutions of which only one is the desired hold (the others, TTBOMK, represent a local worst-case and a local best-case that's worse than the actual best-case). As such, I do not believe a general algebraic formula is very useful in this case; your best bet if you really want to wring out the absolute max is to graph (growth)/(resources lost) vs. hold for specific values of all those parameters and find the maximum numerically. Assuming, of course, that I didn't bugger up the arithmetic somewhere, which is unfortunately a distinct possibility when trying to wrestle with cubics.
So much for my vaunted aptitude with algebra and calculus. 
The one exact solution I've found that's practically useful (i.e. doesn't require adjusting holds every turn to use) is the first case, between 0 and 25%. It turns out that it's always better to have either a "zero hold" (100 colonists kept on the planet as a flagpost + setting the overflow) or a 25% hold (setting the overflow such that the colonists on planet + overflow = 25%) than anywhere in-between (of course, these holds converge when the overflow becomes large). The changeover point is when
pop = (1 - hab * (2 - PGR)) * (2 * PE + FE)/(PE + FE)
where
pop = the amount of population you have at the world (in freighters and on the ground), expressed as a multiple of the world's maxpop (so 550,000 pop on/around a 50% planet with OBRM = 1)
hab = the planet's habitability, expressed as a fraction (so 25% hab = 0.25)
PGR = your growth rate, expressed as a frac
...
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