Optimal Fuel:Engine ratio

[Iranon]

No, because you're attempting to treat an entire squadron as a single craft.  While it's clever, the problem is that you are in an area where two of the underlying assumptions (free tradeoff and infinite scaling) hold very poorly.

The "infinite scaling assumption" holds much better if we use the squadron as an independent entity, so that's a reason for doing this rather than one against it.
Whether the "free trade-off assumption" is a problem depends on how we approach design - "Define requirements. Meet them."  vs "Design something inherently efficient that hits most of the sweet spots. Check if it fits our requirements". It's less of a problem for the latter approach, which I find preferable because decisions are made on incomplete information. Easier, too.
Using your example:

Let's say that during design, you end up with a .5 f:e ratio, instead of the .4 you would want.  Normally, you would increase the amount of engine, and decrease the amount of fuel, but in this case, you can't increase the engine on the fighters because you don't have space, and putting the engine on the tankers won't help the fighters at all.  That, plus limited engine granularity makes working with theoretical optima here somewhat academic.

In descending order of laziness: don't change a thing about the designs, field less tankers until we hit the sweet spot. We lose range for general capability, i.e. more craft that actually do something - like shooting or snooping.

If range isn't negotiable but speed is, we can decrease the engine power multiplier.

If there's no wiggle room for either speed or range, we may have other options depending on our current fighter design. Like playing with the fuel carried by non-tankers, or the engines carried by everyone... both affecting and limited by size. Those won't be quite as straightforward, and there may be considerations more important than reaching a theoretical local optimum.

So I agree with you there: Maybe we can't hit the majority of the sweet spots (optimal fuel:engine ratio, concentration of armament to avoid wasteful use of fire control, matching MSP to possible failures if we use engineering bays, maybe even desired compromise between cost-efficient components and redundancy, matching maintenance life to deployment time where applicable) and get something that fits our requirements.
However, I'd prefer to try first. One advantage of "sweet" over "specced" design is that incremental component upgrades don't throw off the assumptions behind our design compromises.

[bean]

The "infinite scaling assumption" holds much better if we use the squadron as an independent entity, so that's a reason for doing this rather than one against it.

Infinite scaling might have been a poor choice of words.  The problem is that we lose a lot of granularity when dealing with fighters, which can prevent us from reaching theoretical optima.

Whether the "free trade-off assumption" is a problem depends on how we approach design - "Define requirements. Meet them."  vs "Design something inherently efficient that hits most of the sweet spots. Check if it fits our requirements". It's less of a problem for the latter approach, which I find preferable because decisions are made on incomplete information. Easier, too.

This may be the problem.  I tend to set moderately firm requirements, and then try to meet them.  Information is always incomplete, but experience and thought can generally result in good requirements.

If there's no wiggle room for either speed or range, we may have other options depending on our current fighter design. Like playing with the fuel carried by non-tankers, or the engines carried by everyone... both affecting and limited by size. Those won't be quite as straightforward, and there may be considerations more important than reaching a theoretical local optimum.

I think that this is going to happen most of the time.  It's an interesting consideration, but definitely secondary to making the rest of the design work.  

So I agree with you there: Maybe we can't hit the majority of the sweet spots (optimal fuel:engine ratio, concentration of armament to avoid wasteful use of fire control, matching MSP to possible failures if we use engineering bays, maybe even desired compromise between cost-efficient components and redundancy, matching maintenance life to deployment time where applicable) and get something that fits our requirements.

We usually can't hit very many.  Part of my issue here is that you're focusing on one part of the design (and a fairly minor one, all things considered) to the point at which I think it will probably hinder the design.

However, I'd prefer to try first. One advantage of "sweet" over "specced" design is that incremental component upgrades don't throw off the assumptions behind our design compromises.

This doesn't make sense.  Particularly with fighters, you can redesign the whole thing every time you get a tech upgrade without penalty.  With larger ships, I just haven't seen this as a problem.  You do sometimes lose a bit of theoretical performance over a clean-sheet design, but not enough to be a problem.  And you still have a ship that's probably better than a "sweet" design.

WRT the specific problem of fighter squadrons and fuel, there's another assumption that I haven't brought up, but which actually pushes you to a worse result.  Assume we have a given strike squadron, which takes two tanker-loads to refuel, and which must hit a target at its maximum range.  The obvious solution is to give it two tankers.  The better solution is to have one tanker go halfway with the squadron, then turn around and return home.  It refuels, and heads back out to meet the squadron on the return leg.  We've just doubled the effectiveness of our tankers at no additional cost.

[Jorgen_CAB]

This whole tanker/fighter thing is kind of moot in the end. It will all depend on doctrine and the use of your resources as a whole.

You might want to have a long range fighter, let's say you choose to sacrifice some speed with a slightly lower multiplier and slightly more space for fuel. You might thin that it will be an inefficient design, but that depends on what type of tankers you have. You might not include any fast moving fighter/tanker but rely on slower tankers to escort fighter wings or be stationed in space for fighters to refule at during striking missions. There is also the total amount of fuel consumption as a whole to figure. You perhaps can't afford to have fast tankers follow the fighters all they way and like to build long range fighters with better fuel efficiency for that reason etc..

It's not all about one thing the consider.

In general my supply ships actually carry a few 750-1000t fast tankers. These generally have very good fuel efficient engines and is pretty fast (in comparison with normal ships) and can carry enough fuel to act as jumping point for fighter wings or bring fuel to task-groups in need without endanger the supply ship itself. This and I also would deploy more regular fighter/tankers with the carriers. But, building a long range fighter with more fuel efficient engines can be a viable option just to save the amount of fuel you will consume on such missions. Even more important if you don't deploy fighter/tankers with high speed engines.

In essence, the sweet spot is completely depending on the underlying assumptions.

[bean]

This whole tanker/fighter thing is kind of moot in the end. It will all depend on doctrine and the use of your resources as a whole.

Yes and no.  Iranon's analysis was oversimplifed, but he wasn't entirely wrong in looking at the fighter/tanker ratio mathematically.

You might want to have a long range fighter, let's say you choose to sacrifice some speed with a slightly lower multiplier and slightly more space for fuel. You might thin that it will be an inefficient design, but that depends on what type of tankers you have. You might not include any fast moving fighter/tanker but rely on slower tankers to escort fighter wings or be stationed in space for fighters to refule at during striking missions. There is also the total amount of fuel consumption as a whole to figure. You perhaps can't afford to have fast tankers follow the fighters all they way and like to build long range fighters with better fuel efficiency for that reason etc..

This was stated earlier.  He was assuming no offboard refueling, and no fuel supply considerations.  The first is entirely a matter of doctrine, while the second depends on what the fighter is to do.

It's not all about one thing the consider.

...

In essence, the sweet spot is completely depending on the underlying assumptions.

I'm in complete agreement, with the caveat that so long as we are clear about our assumptions we can arrive at optima that we would have struggled to find otherwise.  In some cases (missile engines being the prime example) the assumptions are few and obvious, the analysis is simple, and the conclusion useful.  In other cases, such as optimizing a wing of fighters, the assumptions made are far less obvious.  (We're into a field that goes by names like Operations Research and Systems Analysis.  It's something I'm interested in, and Aurora is actually a decent place to practice it.) 
That said, I'm going to take a look at some of the assumptions involved here, stating them as best I can and looking at how they would impact the outcome.  This can be summed up as two questions:
1. What does the wing have to do?
2. What else am I trying to optimize?
Question 1 could be answered any number of ways.  The most likely are: fixed speed and/or range, a certain amount of hangar space, or a certain number of missile tubes.  These could be fixed by the carriers you have available, your doctrine, or experience with past battles.  Usually, you'll get two out of the three as fixed, although sometimes you will only have speed or range as your variable.
Question 2 is probably getting the one that wasn't fixed in question 1 as high as possible while keeping cost as low as possible.
It seems simple, but the devil is in the details.  Often, the way your fleet is already set up will drive what you do with new pieces to it.
For the wings we've been talking about here, fuel has a massive influence on how you set them up.  If you're assuming that all fuel will be carried by the wing at launch, you'll probably go for maximum theoretical range even if it costs fuel efficiency.  Offboard refueling raises the question of how big the tanker is, which brings that into design considerations.  Are the benefits of offboard refueling (not having to carry fuel at fighter speeds) worth the added complexity and risk?  (Keep in mind that there's no tooling for fighters.) 
And then there's the logistical implications off all this.  Fuel economy is far less important to the system defense wing that flies a mission once every few years from a planet with plenty of fuel than it is to a carrier's wing that flies every few months.

[Iranon]

It can become difficult to keep track of our assumptions were and what we were optimising for.

1) My wing of fighters + tanker-fighters was judged as an alternative to an original design of uniform long-range-fighters. In that context, I wanted to minimise the disadvantages as far as possible (nothing I can do about losing redundancy at range).
2) Fuel efficiency is relevant only as far as it limits range, I'm cramming maximum independent range and capability into limited tonnage. Why? Carting fuel with fighters is inherently wasteful. Right tool for the job: Tanker-fighters extend tactical range, supply vessels perform supply jobs.

The first assumption means our optimisation concerns are the same as for a single craft, because concerns for a wing of identical craft are the same as concerns for a single craft. Because of how I'd define a long-range fighter, it implies that the balance of speed and range is negotiable.
The second assumption means that, if speed and range are indeed negotiable, the optimal f:e ratio is straightforward to calculate. If we're already fitting the biggest engines we can and have room to play with our power multiplier, the sweet spot is at 0.4.
In practice, that's the upper bound rather than the ideal because assumption 2) is too extreme:  fuel efficiency will be a very minor concern instead of no concern at all.

*

Byron gave an example how we can refine our doctrine for a common missile profile if we ditch assumption 1).
We require less fuel for the same range than uniform long-range-fighters would, and can go lower on tankers than a straight equivalent.
Does the same hold true for most of our needs? If yes: does this affect just the ratio of tanker-fighters we want to field, or does it call for different design?
If we keep them away from the melee, we may relax the fighter-sized requirement. If we want to field a decent number and soften their role as a pure range extender, fuel efficiency may enter the picture. The resulting "tactical tanker" may be very different from our mainline fighter stripped of offensive payload for fuel tanks.

Other criticisms are valid if I drop either assumption.
If a mix of fuel efficiency and range is desired instead of pure range, the optimal fuel:engine ratio will be lower than 0.4, we can calculate the exact value if we can define the relative importances.
If speed isn't negotiable and I'm operating at the highest power multiplier, I can freely trade range and capability by adjusting the ratio of tankers. There is no obviously ideal fuel:engine ratio.

*

Formal methods of coming up with the optimal solution for multiple simultaneous requirements would be nice to have, but the maths behind them is rather daunting. Possibly beyond my ability (economics background), definitely beyond my willingness to try.
Simple optimisation methods finding the sweet spots for fewer variables don't have that ability. Fairly easy analysis is sufficient for revealing questionable parts of our designs though. And I mean questionable, it's not an euphemism for wrong: "Is this a design mistake, or the least bad compromise for our tech constraints, assumptions, and resulting doctrines?"

[bean]

It can become difficult to keep track of our assumptions were and what we were optimising for.

1) My wing of fighters + tanker-fighters was judged as an alternative to an original design of uniform long-range-fighters. In that context, I wanted to minimise the disadvantages as far as possible (nothing I can do about losing redundancy at range).

And as such, it has merit.  I'm just not sure that 40% fuel is more than mathematically ideal in that situation. 

2) Fuel efficiency is relevant only as far as it limits range, I'm cramming maximum independent range and capability into limited tonnage. Why? Carting fuel with fighters is inherently wasteful. Right tool for the job: Tanker-fighters extend tactical range, supply vessels perform supply jobs.

The problem is that your carrier is also serving as a tanker here, and it has to carry the fuel for multiple strikes.  I've run into trouble with this before (in slightly different circumstances) so I'm probably more cautious than you are about it.

The first assumption means our optimisation concerns are the same as for a single craft, because concerns for a wing of identical craft are the same as concerns for a single craft. Because of how I'd define a long-range fighter, it implies that the balance of speed and range is negotiable.

This is theoretically true, but I'm not sure how well it will work in practice. 
That said, I took a look at how this might work in my current game, and I was surprised by how well it did.  My existing standard fighter happened (totally accidentally) to have a .4 f:e ratio, so I just copied the design, removed 75% of the fuel, and modified my standard tanker design to have the same speed.  (For reasons I can't remember, it's slower.)  The standard squadron ended up as 9 craft, either 8 fighters and a tanker or 9 fighters.  The 9-fighter squadron had 126 missile tubes and cost 1427.4 BP.  The 8-fighter squadron had 128 missile tubes and cost 1351.6 BP.  However, because the tanker is carrying 75% of the fuel, I can't do the separate tanker range extension thing with a single squadron.  Of course, I'm slightly hemmed in by the fact that each fighter only has a single size 1 engine.

The second assumption means that, if speed and range are indeed negotiable, the optimal f:e ratio is straightforward to calculate. If we're already fitting the biggest engines we can and have room to play with our power multiplier, the sweet spot is at 0.4.
In practice, that's the upper bound rather than the ideal because assumption 2) is too extreme:  fuel efficiency will be a very minor concern instead of no concern at all.

At this point, I decided to do a sensitivity analysis.  I wanted to check and see how big the sweet spot was.  I re-did the original optimum equations for both standard engines (no size scaling) and missiles, for both constant speed and constant range.  Then I looked at the ranges where the dependent variable would be at least 90% or 95% of optimum.  The results were rather surprising (all numbers expressed as % fuel:engine):
Fixed Speed/Variable Range:

	Standard	Missile
95%	27.1-57.9	21.5-44.9
90%	22.6-67.7	18.1-52.1

Fixed Range/Variable Speed:

	Standard	Missile
95%	21.3-71.3	17.0-54.8
90%	15.8-90.8	12.7-68.8

A few caveats.  First, these are assuming there are no limits on the power multipliers that can be used, and that fuel and engine space can be freely traded with no granularity.  Second, the total propulsion space is assumed to be fixed, as is the first value.  Third, the question is not if you can increase one variable at cost to the other.  It's if you can make one better while keeping the other constant (or improving it slightly.) 
I'm rather surprised at how wide the ranges are.  This tells me that strict adherence to optima in ships is mostly unnecessary, unless you're way, way off optimum.  For missiles, research is cheap and you can usually get closer to the optimum, so it's probably a lot more worthwhile to worry about those.

[bean]

We require less fuel for the same range than uniform long-range-fighters would, and can go lower on tankers than a straight equivalent.

This isn't true.  You don't need less fuel than long-range fighters (except from reduction in the number of vessels involved) and you can replace redundant tankers with more fighters on a 1 to 1 basis, so you need exactly the same amount of fuel, just split up differently.

If we keep them away from the melee, we may relax the fighter-sized requirement.

I actually wouldn't do that, unless you're talking about going from 250 to 500 tons.  I actually tried that with my designs, and I got about 4% more fuel per ton.  It's not really worth it.  Going with FAC-size ships is going to add a bunch of logistical headaches that will probably be more trouble than their worth, at least if you plan on integrating them into the same wing.

Formal methods of coming up with the optimal solution for multiple simultaneous requirements would be nice to have, but the maths behind them is rather daunting. Possibly beyond my ability (economics background), definitely beyond my willingness to try.
Simple optimisation methods finding the sweet spots for fewer variables don't have that ability. Fairly easy analysis is sufficient for revealing questionable parts of our designs though. And I mean questionable, it's not an euphemism for wrong: "Is this a design mistake, or the least bad compromise for our tech constraints, assumptions, and resulting doctrines?"

This I will agree with.  I'm not willing to break out linear programming and the other forms of math involved here, although I could if I really wanted to.  Honestly, I've found the best way to answer these sorts of questions is to do paper studies as far as possible.  Build your hypothetical ships in the ship design window, and see how they work.  (You could either set up a parallel simulation game to your main game, or just SM a copy of the parts you have upcoming with a special name so you know not to use it on a ship you're actually going to build.)  Although in a lot of cases, that may not be necessary because you'll already have most of the parts you need.
I do a lot of this.  I tend to run a big, high-tech 'flagship' game, and I've rebuilt my planned fleet for 6.4 twice over the past few months.

[Iranon]

It looks like we see eye to eye on the major points that don't require digging deeply into assumptions, doctrine and other waffly things. Hooray, progress!

That sensitivity analysis looks useful, there seems to be a fair bit of wiggle room in how far you can deviate from theoretical ideals. The low end is especially interesting, as we save fuel (mostly relevant for ships) and require less engine power tech there. Looks like my previous aiming spot of 0.3-0.33 was reasonable. The numbers seem to imply I can drop considerably lower without wasting much performance, and save fuel in the process.
However: we could achieve the same speed and range on less weight. Important for craft struggling to stay within fighter/FAC limits, also worth looking at in high-performance craft where the last 10% in speed were expensive (in terms of weight).

The high end is costly in addition to sacrificing performance. The ship using 90.8% of its engine weight on fuel may be 90% as fast as the performance-optimal version, which doesn't sound too bad. However,  it also uses considerably more fuel, and several times as much as it needs to for its performance (3.5 times as much as the other 90% speed example). This is bad.
It also needs a higher power multiplier, if we researched that at the expense of an engine tech things get downright terrible. And a fuel-to-engine ratio of 0.908 would not seem unreasonably large to people who didn't consider the mechanics, there is probably far worse out there.

This isn't true.  You don't need less fuel than long-range fighters (except from reduction in the number of vessels involved) and you can replace redundant tankers with more fighters on a 1 to 1 basis, so you need exactly the same amount of fuel, just split up differently.

My phrasing was a little off: yes, you don't save fuel in the particular you outlined. Bit of an edge case, often enough it will. You can take just enough tankers to get the job done, while long-range fighters will carry dead weight if their tanks are larger than they need to be. You can also go halfway, top up the fighters, park the tankers, top up the fighters on the way back - this does save fuel.

You also get even more range if you deploy only part of your fighters but all the tankers - some types may be useless for the job, or you may want to keep some fighters around to protect your carrier.

Usual disclaimer: details differ in practice because granularity; this usually works in favour of the approach that leaves the designer more freedom (in this case the short-range + tankers option).

[bean]

However: we could achieve the same speed and range on less weight. Important for craft struggling to stay within fighter/FAC limits, also worth looking at in high-performance craft where the last 10% in speed were expensive (in terms of weight).

I decided to have a look at the math on this one.  I looked at sizes of 105% and 110% of optimum, based upon an optimum size at 40% f:e for a given speed and range.  The results were very surprising.  The 105% size band went from 19.2 to 77.7%, while the 110% band went from 14.0 to 100.2%.  I don't think I'm going to do missile engines, as that would require more math (which I've already done enough of today) and the results would be fairly similar.

[Iranon]

Just to clarify, you refer to allocating 5% or 10% more propulsion tonnage while keeping the whole ship at the same size?

In other words, 5% more propulsion tonnage than the performance-optimum would give us the same range/speed at 59.2% of the former fuel consumption?
1.05*(19.2/119.2)/(40/140)=0.592

Edit: There used to be something very wrong here. Sorry for the inconvenience.

[bean]

Just to clarify, you refer to allocating 5% or 10% more propulsion tonnage while keeping the whole ship at the same size?

In other words, 5% more propulsion tonnage than the performance-optimum would give us the same range/speed at 59.2% of the former fuel consumption?
1.05*(19.2/119.2)/(40/140)=0.592

Edit: There used to be something very wrong here. Sorry for the inconvenience.

That's exactly right.  I held displacement constant for two reasons.  First, it's usually how I design ships.  Second, the math for variable displacement would be a lot messier.

[Iranon]

Oh, definitely the right way to do this... otherwise we'd have to know which percentage of the ship total we allocated to propulsion, which would  change every time we fiddled with it,  and it'd be all-around terrible to work with.
Does the maths lend itself to easy generalisation so we could plug in f:e ratios and get associated weight-efficiency and fuel-efficiency?

Standardising for the most weight-efficient setup, we'd have...

0.4, 1.0, 1.0
0.192, 0.952, 1.689
0.14, 0.909, 2.115

[bean]

Does the maths lend itself to easy generalisation so we could plug in f:e ratios and get associated weight-efficiency and fuel-efficiency?

You can.  The equation I've been using so far is S=E+.4(1/E)^2.5, where S is the total size and E is the engine size.  I then did an intersect on my graphing calculator at S=1.47 and S=1.54, for the 105% and 110% figures.  F:E was calculated from (S/E)-1.  
A bit of algebra gives me the equation E=(2.5*FE)^(-2/7).  Plugging this back into the first equation gives us S=(2.5*FE)^(-2/7)+.4(2.5*FE)^(5/7).  I checked this against my starting equation, and it works.  Pulling out the coefficients and normalizing the whole thing gives us Weight Efficiency = .549762*(FE^(-2/7)+FE^(5/7)).  This is actually a rather neat equation.  I may see if I can do likewise for constant speed and constant range.