THE WAY OF UPGRADING LUMBERMILLS
This guide is about giving info of maybe the most important thing of Fortress Survival, the lumber income. This guide doesn't tell when and how much you're supposed to pool your waller, thats allways a tradeoff which will be noticed in total lumber production later on. This guide focuses mainly to the importance of upgrading the lumbermills and doing it in right order, offering some reasoning and simulation behind it. I presume that anyone reading this guide, knows the basics of lumbermills, lumbering setups, transmuting and the related stuff. The forums guide-section offers some really nice guides which will cover all of the previous and touch upon a huge part of this guide and from where I've also absorbed a lot of info 
Ok, lets get started then 
Let's make couple of presumptions which will apply through the whole guide
- Constant lumbering, it's the most important thing about lumbering.
- If lumbering without Tinker (as second builder), I recommend using the 2 mill setup, which reduces the preparation time before the actual lumbering and regardless of that can still prove to be faster with some micromanagement.
- Big enough base, which won't run out of space it's important for the later comparison
- Not having unoccupied lumber if there is mills to upgrade
After that, there is the big question whether to
- Make 1 x lumbermill lvl 7 as fast as possible and upgrade rest of the mills afterwards starting from the lowest
OR
- Upgrade the lowest lvl mills always as highest priority
Lets consider some of the facts that for example Erdam has pointed out in hes great lumbering guide.
But first we gotta bring some values to the table (got the joke? ;-) no? (╯°□°)╯︵ ┻━┻/(.□ . \) ¡uɐɯ ʇno llᴉɥɔ 'ʞoʞo) eh..
Table 1. Basic lumbermill characteristics.
MillLvl: Lumbermill level
WpsReal: Real Wood per second, given by the current level lumbermill e.g. 1 wood every 1/WpsReal seconds
UpgCost/l: Amount of lumber it costs to upgrade lumbermill to the current level
UpgTime/s: Time it takes to upgrade lumbermill to the next level
Just like it says, this table above holds the ground values that gives lumbermill its characteristics. The simulation will base everything in this table, and this table only, what it comes to upgrading individual mills. We will use this table to examine things little deeper, lets continue to the next table.
Table 2. Advanced lumbermill characteristics.
MillLvl: Lumbermill level
WpsStep: The amount Wps increased from previous level
CumUpgCost/l: Cumulative sum of Upgrade costs till the current level
CumLumberLostByUpg/l: Cumulative sum of lumber left unproduced while upgrading lumbermill towards the current level
1WpsRealCost/l: The amount of lumber it costs to achieve income of 1 Wps with current level lumbermills only
WpsStepCost/l: The amount of lumber it costs to upgrade lumbermill to the next level, including upgrade loss. (lvl5 = 4 x lvl4 upg loss, lvl7=8xlvl5 upg loss only)
1WpsCumStepTotalCost/l: The amount of lumber it costs to achieve income of 1 Wps with current level lumbermills only, taking ONLY in account Wps achieved by upgrades but cumulatively, in the other words, reducing lvl1 lumbermill WpsReal value from all following level "WpsReal"s, before calculating similar way than "1WpsRealCost/l".
This table has a lot of variables, don't get scared here. We're actually gonna use just a few of em and consider if we find meaning for the rest of them. There is always people , who likes extra data though ;-). WpsReal being already explained (below table 1), lets move to the 1WpsRealCost/l (check explanation above). I call it "Real" because that's the real amount of lumber it takes to upgrade enough lumbermills to current level so they achieve the real income of 1 Wood Per Second. Where does this real amount come from? First we look at the Table 1 column UpgCost/l, name is pretty self-explaining. Actually we need that, but as stacking level by level, a cumulative sum, which is presented in Table 2, CumUpgCost/l. That's it? Well, thats what I though at first too. After a chat with Ethaw I realized that the lumber left unproduced during the upgrades, should be taken in to account here too. That introduces us a new column called CumLumberLostByUpg/l. Now we have everything to make the mentioned formula out of it:
1WpsRealCost/l = (CumUpgCost/l + CumLumberLostByUpg/l) / WpsReal.
What can we do with this 1WpsRealCost/l then? Can we compare different level mills with this? Lets have a simple example, CumUpgCost/l for lvl7 mill is 1760 (11wps), with that amount of lumber we could on the other hand upgrade 352 lvl2 mills (17,6wps), which would seem much better value. But hold on, the cost is same but are we still missing something? Waffle(est) brought this pretty important fact to think about. When you upgrade mill from lvl1 to lvl2, you do double the wps with 5 lumber. On other words you bough half of it with the upgrade and the lvl1 already gave you the second half with zero cost. lvl7 consists of 32 basic mills and we were comparing it against 352 lvl2 mills. To be fair, we should also add 320 lvl1 mills for lvl7s side (32+320=352) and since they were free, the CumUpgCost/l of 1760 stays same. That will turn out an another 320x0,025 = 8wps for the lvl7s good . So its now 19wps (lvl7+lvl1s) vs 17,6wps (lvl2s) and Lvl7 actually does go ahead. So, comparing pure 1WpsRealCost/l does also compare different amounth of lumbermills against each other (=different amount of free wps). It really does make the lower level 1WpsRealCost/l shine more than it should.
Since this free wps does mess up with our comparison, lets find a way of getting rid of it. WpsStep shows how much each upgrade-step increases wps. Cumulative sum of WpsStep is actually same as WpsReal. Now, lets present 1WpsCumStepTotalCost/l and we calculate it similar than 1WpsRealCost/l but we just ignore the first step from WpsStep:
1WpsCumStepTotalCost/l = (CumUpgCost/l + CumLumberLostByUpg/l) / (WpsReal - a x 0,025),
where a is amout of mills involved (e.g. for lvl5, a = 4) (Again, first step ignored, no dividing with 0).
How to decode 1WpsCumStepTotalCost/l value in the table? Well, since it tells us how much it costs to achieve 1 Wps based on the upgrade lvl we want to achieve it with, so lower is better. Put other way around, with same amount of lumber we can get higher Wps if we prefer the cheaper options. Cheapest would be lvl7 (246,8) after all and ironically 2nd and 3th place goes to lvl2 (255) and lvl3 (275). We gotta remember though, that upgrading proceeds in order, step by step. There is no shortcuts upgrading to lvl 7, so we also gotta get through lvl4 (318) and lvl5 (301), while they drag us with the most expensive wps and rapidly increasing upgrading times (lvl4s). Gereric guideline would be to upgrade mills so, that they would spend as little time as lvl4 or lvl5 as possible, untill lvl7 cap is reached, then the lvl5 is the only way because of income surplus, decent wpsdensity/-cost and transmuting.
Of course all the mills has to be upgraded eventually all the way to lvl 5 and 7, but the name of the game is to upgrade mills to the most profitable state and continue from there level by level, keeping the fresh ones upgraded too. But that was just math and reasoning leaving the actual acts behind. Maybe there is something we haven't taken into consideration. Maybe the actual results would speak for themselfes?
To approach the question from this direction, I have made an excel form which simulates a test-Dummy as a lumberer. The Dummy has only one job which is making mills and upgrading them predefined way. Might get boring after couple of games, eh? Well, they haven't complined this far 
Let's define the environment and rules for our puppet to follow
-
Dummy makes the preparations and starts lumbering so that the first mill is ready and transmuted in its place at 47:30 min mark
- Dummy has infinite mana (you can think it as having alchemist hero + unit + high builder lvl)
- Dummy doen't have to use lumber in anywhere else except upgrading lumbermills (after a certain time, there will be surplus though)
- Dummy uses 2 mill system, where he gets 1 mill out every 5 seconds and he keeps it up till Trojan spawns
- 5 second isn't lowest possible but it feels more like an average that human could keep up with for 47:30, lol
- That 5 seconds also defines a cycle
- When the cycle starts, Dummy upgrades mills, right after that Wps for the current cycle is calculated
- Dummy gets the lumber calculated from Wps at the end of every cycle
- Dummy can't upgrade anything he can't afford
- Dummy can have unoccupied lumber only for max 5 seconds (unoccupied = anything that Dummy could use for a planned upgrade)
- Lvl 5 mill is always started to upgrade when 4 x lvl 4 mills has been completed, etc. lvl 7 from 8 x lvl 5
- Mill upgrade costs, wps, upgrade times as presented at the table above
- We will not take lvl 6 mill in account, because it doesn't support transmuting, which is essential for achieving highest wps
And lets prepare 2 testruns for the simulation, with slightly different upgradinggoals.
- Dummy 1: 32 x -> 32 x -> 32 x -> 8 x -> 1 x -> upgrades rest of the mills priorizing the lowest possible
- Dummy 2: Simply priorizes the lowest possible mill always
Off we go then.
First chart shows how the lumber production builds up till the timer hits zero mark. Not too huge difference is noticed, but the scale is huge and there is some though, seems to be for the advantage for "Lowest mill first". Well, since its all data from excel, we can do another graph which tells the difference of the lumber production between Dummy1 and Dummy2. That way we can see from which point the productions starts to differ from each others.
There we got a chart of "Lumber production difference". Positive value is advantage for the Dummy2, "lowest first". It starts to pull quite a lead after 37:30, dropping some and pulling it to the top all the way with a huge ~1000 lumber lead, at the time around 24:30 left on timer. From here it's still quite a hard trying to figure out whats really happening, but the Upgrading styles of Dummy1 and Dummy2 are supposed to walk their own paths after both of them have started upgrading the 32th lvl 2 mill. And we can see that the difference starts around 42:30 and stops around 24min mark. Why does't production difference keep raising till the end of the game? Is neather of the ways better than the other? 
Wait.. Could we get any more accurate results out of this?
We can zoom between the timeline we studied from the previous chart. And lets add Wood Per Second Difference (Wpsd) on top of that. It will tell, If Dummy1 or Dummy2 has Wood Per Second (Wps) advantage over each other and how much at any moment. Huge spikes showing other getting short but huge wps advantage tells about the upgrading of lvl7 mill, which disables 8 lvl5 mills temporary (8,8wps for 12s, pretty notable for a reason). And the smaller spikes to show lvl 5 mills upgrading the same way.
If we check the lumber production, also zoomed this time, the difference is now notable on the charts. The first chart just had the the scaling all the way to the 200 000 and this one only up to the 30 000. We can see again that Dummy2 (Lower first) gets so huge wps difference lead at beginning that EVEN Dummy1 (Lvl7 first) does get hes Lvl7 first, Dummy2 actually gets hes first right after and the second, third and fourth before Dummy1 gets hes. Difference at 33:30 mark is around 500+ lumber and gap grows to ~1000 lumber at the time of 24min left.
As well as the Wpsd, also the fact that both Wps figures shows that right after 25min mark Wps of Dummy1 and Dummy2 becomes the same. Yes, thats the point where the lumberlead stopped growing, doh. It's because lumber income is based on Wood per second and since the figures becomes as one and will stay like that to the end of game. And thats because both Dummys got their mills finally upgraded to the point where you only gotta upg the last "fresh"ones to the max right away. And since they both have exactly as many mills, their millsetup looks alike now, but Dummy2 (lower first) just got hes "maxed" earlier.
Different building times does affect flat amount. The faster you are, the more you benefit by upgrading lowest first.
Finall conclusions:
Upgradingtimes do play a HUGE role when comparing different upgrading methods, since even hundreds of mills go through the same 265-277 seconds of upgrades and produce nothing at that period, tnx Ethaw.
In older version upgrading times were 125 seconds longer total and it increased the flat lumber production advantage from ~1000 to ~2230
The point where production difference flattens, can barely affect but the flat amount gets bigger with higher mills production rate
Removing the free wps (lvl 1 mills) was right way aproaching the 1wps-cost, tnx Waffle(est)
Lower level mills were more effective to upgrade first because that way mills spend major part of time in cheap-wps state and offer overall faster income to do upgrades as nonstop as possible till lvl 7, like an upgraderush over the most expensive wps-states.
There might be still something you'd like to ask.
So is the other ways of increasing the flat amount of production difference?
Basically no, if we keep comparing 2 Dummies with environment and rules which applies on them and are same for both
Different amount of early lumber-boost could make difference, in addition to those random easter eggs, lvl1-4 mill as starting item could be worth testing.
Also, 5s / mill isn't close to optimum, preparations before starting to lumber can be made faster, cog of times or
tinker as second builder can be quite a game changer,lumber&gold bounty.
To-Do-list
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If you guys find any major or minor mistace or falsity, let me know and I'll try to fix it.
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Reminder to my brains, NEVER start polishing outfit (for example with
html) before all the content is there and the basic formatting done with editor/bbCode, since there is no coming back fromhtmlwithout breaking everything, and the editor is million times faster with basic formation, lol
Attached Files
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lumbersimulator0.1.zip 742.01KB
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