R&D Investment Returns

[alimentary]

ORIGINAL: n01487477

Would you mind if I borrowed/adapted this formula for later iterations of Tracker - not 1.9 as it is about due but 2.0(as an estimate of repairs to R&D factories) ?

Please feel free.

[Hanzberger]

ORIGINAL: n01487477

Would you mind if I borrowed/adapted this formula for later iterations of Tracker - not 1.9 as it is about due but 2.0(as an estimate of repairs to R&D factories) ?

OH GOOD, this is why I LOVE Tracker~!!!! Let it do the work. [:D]

[kmitahj]

ORIGINAL: Kitakami

If it is so, then it makes MUCH sense to set a few factories to research the late fighters... they will come online slowly, so the economic impact should not be felt as much, but 37% of many months is a LOT of research time... a couple of airframes could be accelerated quite a bit. I guess the Frank is a given, but as I see things, there are enough R&D factories to research another late (maybe IJN?) fighter.

Just my 2 centavos.

Yes, I agree. That is what I used to aim at in old WitP: make a plan, setup research factories early and then... be patient. Very patient. At the beginning tempo of repair for late-war planes will be agonizingly slow which - as you said - has a good side effect of beeing light on supply usage.
This slowlness is btw clearly seen in the probability distribution table generated in alimentary script. Every entry $sizedist[D][S] in that table tells us precisely what is a probability that our factory will have size S after first D days. So nice... If applied mathematics can be cute then I dare to say that this script is really cute! [:D]

Just practical example from classic WitP age: if we setup Frank research early enough we have about 28 months till availability. Using above estimation: 28*0.37, rounded down to be on the safe side, leave us some 10 months for actual r/d pts generation. If we decide to invest in 5 Frank r/d factories of 30+ size we can expect 300 r/d pts every two months, that is we expect to get 3 months of advance every two months of work. Apply this twice and we use up whole 10 months: 4 months of r/d points generation resulting in 6 months of advance time - not bad I think. This assumes near ideal conditions of course. One thing which may make conditions less then ideal is supply stress in HI area (this may result in less then 10k supply in some research sites which negatively impact repair chances).
And of course there is another serious danger when researching very late-war airframes: allied player may get his 4E hordes in range of research facilities before they will be able to deliver. In classic WitP aggressive allied player using Iwojima or Sakhalin Gambits could start strategic bombing of HI somewhere in late 43...

ORIGINAL: alimentary

The approach I took on the latter challenge was to model the possibility of an
advance in availability date (and the resulting increase in factory repair rate)
by taking the expected value of the research to date _from all of the "other"
factories_. When that exceeds 100 points, availability moves up and repair
rates increase to match.

If you are estimating for 1 factory, this correction has no effect.
If you are estimating for 30 factories, I _think_ that this results
in an unbiased estimator.

Thanks for taking the challenge. To be honest I can't say anything (at least anything consistent) pro or contra regarding correctnies of your approximation. I see it reduces nicely when $facilities=1 but that's it - beyond that my intuition simply refuses to cooperate with me [;)]
I also briefly comtemplated strict solution (that is one in which $days variable would have its own distribution table calculated synchronously with factory repairs dist-table [or rather tables]) but quickly came to conclusion that complexity of such approach is far beyond me.
Still I don't thing it is that importatnt as we are trying to catch here secondary if not tertiary effect relevant - if at all - only for larger number of factories. So it's mostly academic though interesting (imho) problem. From practical point of view the distribution generated for single factory is all we need I think. Beatuy of your script is that for single factory it gave us on the golden plate mathematicaly correct and complete solution which may be used as base for whatever prediction one would be interested in. For example one can easily (and precisely!) answer the question about expected supply cost of r/d factory repair after any given number of days. So, who knows, some day we may even see in Tracker a plot predicting day-after-day supply usage of r/d repairs for given initial setup of r/d factories. [&o]

Btw, before I forget: I think there is a typo in the script in place marked below:

# First, figure the number of research points per factory per turn
# complete repairs.
$p = ($maxsize+1)/30;
if ( $p > 30 ) { <--here: $p > 1
$p = 1;
};

[alimentary]

ORIGINAL: kmitahj

ORIGINAL: Kitakami

Btw, before I forget: I think there is a typo in the script in place marked below:

# First, figure the number of research points per factory per turn
# complete repairs.
$p = ($maxsize+1)/30;
if ( $p > 30 ) { <--here: $p > 1
$p = 1;
};

Quite right. Corrected.

[Hanzberger]

ORIGINAL: Mike Solli

ORIGINAL: alimentary

ORIGINAL: kmitahj



Interesting result! Did you reach that DOA*(1-exp(-1)) in analytical way? If so you seem to be one of that few human beeings left which are adept in an ancient art of mathematics [&o]. When I was working on it I limited myself to running series of simulations on large number of input data (DOA & fab sizes) and was happy to see that no matter what was the starting point all results seem to converge to single factor which I estimated at about 0.63 of DOA. (or in other words 37% of DOA time left for r/d pts generation when starting from fully damaged factory). Your results seem to confirm my rough estimations which made me happy [:)]

Let me see if I can reconstruct the logic...

The formula for R&D factory repair is that the probability of repair is
such that you get a fraction 1/(days_to_nominal_arrival) of the
factory's total fully-repaired size repaired each day.

100 days to arrival, factory size 1 = 1% chance of 1 repair
100 days to arrival, factory size 30 = 30% chance of 1 repair
1000 days to arrival, factory size 30 = 3% chance of 1 repair

In general, if you go 10% of the way to the scheduled arrival
date, you get 10% of the capacity repaired. Or if you go
1% of the way to the schedule arrival date, you get 1% of
the capacity repaired.

So the first 10% of the repairs will take the time remaining
down to 90% of the original time remaining. [As a crude
estimate]

Or the first 1% of the repairs will take the time remaining
down to 99% of the original time remaining. [As a better
estimate]

And the next 1% of the repairs will take the time remaining
down to 99% of _that_.

If you count it out in 10% increments then the formula is

(1 - 1/10) ** 10

If you count it out in 1% increments then the formula is

(1 - 1/100) ** 100

As the number of days to nominal availability increases
without bound, you get

lim x->oo (1 - 1/x) ** x

Now for x very large there is an approximation that applies
here:

1 - 1/x ~= 1 / (1 + 1/x)

[e.g. .99 ~= 1/1.01]

so the above limit is equal to:

lim x->oo 1/(1 + 1/x) ** x

Which is equal to

1 / lim x->oo (1 + 1/x) ** x

Which is clearly equal to 1/e

If that's the fraction of time remaining left at the moment of
full repairs then the time to full repair is 1 - 1/e of the way
through the interval.

Q.E.D.

I am with you Mike, and just when I thought I started to understand a few things....