Assuming it would be possible (but not required) to share resources between ships during the journey, but doing so takes minimal (but non-zero) resources.
Is it more fuel/space efficient to have a fleet of generation ships (either several self-contained ships, or specialised ships), or to have one large ship for all functions (self-contained)?
Running 10 small sewage processing plants uses more power than 1 big one. Same with maintaining 1 big nuclear reactor vs maintaining 10 small ones. Same with Co2 filters, Gyms, Schools, Creche, etc. Are you going to build 10 playgrounds for children? Do you have 10 autopilots, as well as 10 navigation computers, 10 gyroscopes, 10 airlocks? As well as mandatory redundancies for all these systems. Do you have 10 kitchens and 10 dining rooms?
10 small ships also burn through spare parts faster than 1 larger one. Spare parts (storage or manufacturing) will be one of your biggest logistical challenges. 10 ships with 10 x 2-core nuclear reactors (you need a spare when one is being maintained) will burn through parts faster than 1 x 4-core reactor.
For 8 hour shifts, 40 hour weeks, with sick leave and 2 weeks holiday a year, a position that needs 24/7 coverage will need 5 fold staffing to ensure that a post is always manned. So you'll need 5 x 10 = 50 people to ensure that each ship has someone on duty. So 50 people capable of piloting the a ship. 50 trained nuclear techs. 50 police officers (minimum). 1 big ship may need 2 or 3 concurrent police officers on duty at any one time - but you'll only need a staff of 10-15 total officers on the roster in order to get 2-3 on duty at any time. That simplifies your training regime considerably.
If you have shipbuilding facilities that can only build a ship up to a maximum size it may be tempting to build small generation ships for a fleet - to which I'd suggest build the ship in sections and join it together in a vacuum.
I note in the comments (now moved to chat) there's a discussion about "It's not very efficient if your only ship blows up". So as an aside this can be settled too:
Redundancy vs efficiency is a tricky problem. However there is a middle ground for this problem.
(For those of you who spend your nights clicking through random Wikipedia pages - this is a DHT)
As already pointed out in other answers/comment, a single big ship is more efficient BUT will increase the chances that a single, catastrophic failure leads the entire mission to failure.
A reasonable compromise seems then to use a few large ships instead of a single, humongous one. The individual ship will still be big enough to cash on the scale efficiencies, and the redundancy implicit in having two or more of them increase the chances that at least one of them will make it to the destination.
More or less what NASA did when sending out the planetary probes in the 70's, with multiple Pioneer, Voyager and Mariner probes being sent out in very close intervals.
One of the major challenges for crewed interplanetary spaceflight is the health risk presented by cosmic radiation. This is a big problem even for a 180-day journey to Mars, let alone a multi-generation trip between stars.
The linked Wiki article mentions an indicative figure of about 4 tons shielding per square metre to bring radiation levels on a space station down to roughly earthlike levels. There's a fair bit of uncertainty on that, and requirements for interstellar travel would be different again, but it gives us a ballpark: if you don't have some other way to protect against cosmic radiation, physical shielding is going to be a big part of the weight of your ship.
It's possible to have shielding that's also useful for other purposes, e.g. store the ship's water supply or hydrogen fuel in its skin. But for a generation ship, you probably want near-100% recycling of resources, meaning that the amount of water you'd otherwise need to carry is likely far less than the amount you'd need for shielding. Even if you're using consumables for shielding, you still need enough left over at the end of the trip that you're adequately shielded in the last years, which still means a large increase in the amount you have to carry.
If the weight of shielding is a major design constraint, then your ships are going to look like big balls, because that's the most efficient shape in terms of surface area per volume contained. (Edit: As mentioned in comments, some components can be put outside the shielding, so more like "big balls with stuff hanging off them"). Thanks to the square-cube law, one 200-metre-diameter ship will hold as much payload as eight 100-metre ships of similar shape, but with only half the surface area and hence half the weight of shielding.
(In fact, the big ship requires a bit less than half the shielding of eight small ships, because thickness of shielding is likely to be non-negligible, but let's not worry too much about that.)
So, if you're dependent on physical shielding for radiation protection, and you don't have some super-light unobtainium shielding, you probably want to go with a small number of big ships.
Another option might be magnetic shielding. This is a bit more speculative, but the requirements for that will probably still scale roughly with the surface area to be protected, which again makes bigger more efficient.
If you're launching off of Earth or some planetary object in these ships (like they show at the beginning of Wall-E), I'd recommend smaller ships. But I wouldn't recommend this strategy; either way, launching ships as big as these out of a planet's gravity well is extremely energy expensive. Instead, they should be built in space and then people shuttled up to them.
Now that we've got that out of the way, here are our options:
So, what matters more, being cautious about the risks of ship malfunction or saving resources? Also, which makes the better story? There's definitely a lot more tension when the sole ship malfunctions than when one ship of a fleet has problems. On the other hand, if conflicts break out between ships that could also be a story in the making.
So I'd say it comes down to which one you can better build a story out of. There are definitely arguments for each.
Is it more fuel/space efficient to have a fleet of generation ships (either several self-contained ships, or specialised ships), or to have one large ship for all functions (self-contained)?
Nothing's less efficient than a single giant sphere which gets destroyed by a passing asteroid.
That's why fuel and space efficiency are never the sole driving factors. Especially in space travel!!
Triple redundancy, over-construction, lots of machine shops and "blank" parts (generic units of material which get machined down to usable parts) will drive the designs. (Plural, since one design might have a fatal, unforeseen flaw which destroys all/most of the ships.)
That is why multiple self-contained ships launched at different times (for example, three different ships of three different designs -- total of nine ships -- launched at three different times) are the most efficient way to ensure that some people get to the destination.
In space, you have exactly one method available to get rid of heat, and that is to radiate it away. All big spacecraft like the space shuttle and the ISS must include radiators to avoid overheating.
Generation ships have a power consumption that's roughly linear to the amount of people on board, and people on board grow with the volume of the ship ($\approx x^3$). The available surface area for waste heat radiation, however, only grows with the surface of the ship ($\approx x^2$). So, as you scale your generation ships up, you will run out of surface area for heat radiation eventually.
Now, you could say: Oh, well, then I just build huge radiators that reach out far into space. This will work for a while, but eventually you get the problem that you need to transport the waste heat from the core of the ship all the way to the tips of the heat radiators. The longer this trip becomes the less efficient the cooling will be (more energy is expended on pumping, and it becomes harder to isolate the return pipes sufficiently from the environment as they go down into the heart of the ship).
Your ship is basically designed as a gigantic space station. It's assembled from modules and hubs that are connected via some standard connector system. Each module is basically a long tube that has its own heat radiators attached, and is connected to one hub at each end. The hubs are designed so that they mate to six modules in a single plane, and to three modules diagonally upwards. Thus one half of the nodes form a single layer of triangles with 2/5 of the modules, the other half of hubs form a second layer with another 2/5 of the modules. These two layers are connected by the last 1/5 of the modules, using the diagonal upwards connections of the hub. This forms many more triangles which are not parallel to the two planes, and thus provide lateral stiffness to the ship.
As you may know, triangular constructions are extremely stiff and never produce any bending forces on the individual beams. That's why you see such triangular construction at each and every construction crane.
You grow this ship simply by adding modules to the edge of the double plane. As such, the effective surface of the ship grows linearly with its usable volume. The mating mechanism between modules and hubs has air locks, valves on all pipes, and electrical switches on all power lines that pass through it. This allows defective modules/hubs to be separated from the ship in any manner that might be necessary. An open connection, however, allows free exchange of whatever the different modules want/need to exchange, allowing the ship to act like a big city.
One big ship.
The answers on this so far are all much broader than the question calls for. If the question was as broad as people seem to think (i.e. which option is better), then it would be unanswerable without more info.
However, it is very specific and addresses only two things: fuel efficiency and space efficiency. It assumes both options are possible with the engineering capabilities of the world in question and doesn't care about redundancy/safety or the psychological impact or whatever other factors are normally looked at when making this type of decision.
For space efficiency, look at duplication of essential facilities (e.g. toilet/WC/loo, recreational facilities, medical facilities, etc.) as well as outside/surface area versus internal volume of the ship (square-cube law) and how this affects efficient use of space as well as available space as function of amount of construction material used. This clearly shows that the large ship is the more space efficient option as it needs less space to house and transport the same number of people.
As for fuel efficiency, assuming no strange, as yet undiscovered effects happen to larger objects in space, then as long as your ship never takes off or lands from/on a planet, the larger ship will be more fuel efficient as well. You need fuel to accelerate to journey speed and decelerate at the end and small amounts now and then to maintain journey speeds amidst various gravitational pulls. The amount of fuel you need is based on the amount of mass you need to shift, and we've already discussed how you need less material (therefore less mass) to construct one large ship than several smaller ones, so again, the one big ship will be more fuel efficient.
Just for the record, though, and completely outside the scope of the question, I'm an engineer and a firm believer in Murphy's law, so safety factor thinking says send at least 3 ships and hope at least one makes it.
A larger spacecraft will be generally more efficient, all else being equal, with diminishing returns in increased efficiency as scale increases. However, a larger spacecraft is a more difficult construction project, will take more time and resources to build, will result in less efficient utilization of shipyards and other construction infrastructure, and will be less able to make use of the latest technological advances. On a colonization program level, a convoy or series of convoys may well be more efficient, even if the individual ships are less so.
Efficiency of the spacecraft can't be your primary design criteria anyway. There's always going to be a choice between launching and doing something to make the ship slightly more efficient. If you always choose to make the ship more efficient, the result is a generation ship that never leaves, but just sits in orbit being upgraded and added to forever...or more likely, gets canceled before completion.
Redundancy
Redundancy aka strength in numbers, does only hold up against external threads. Meaning a asteroid puncturing your ships can be protected against with lots of little turtle eggs marching to the sea. A engine-design flaw, can not be protected against with systems that all have the same flaw.To small a resource can not be protected against that way. The stranded predecessors can be scavenged though, and thus build "a road in the sky".
Redudancy is also possible within a ship, by modular design and self-recover ability. (The traders in the A-Deepness-in-the-sky universe did this). So bulkheads and decentralized command& controlposts.
For a generationship, the most important thing though, would be the capability to self-repair and bootstrap the environment up again in case of catastrophic failure. Like send everyone into storage, while it grazes for raw-material and repairs its eco-system.
One big ship would be most efficient in operation, as other answers have pointed out. That's only half the equation though.
This massive ship has to be built. It's massive facilities have to be able to handle huge volumes. Giant reactors for power, giant water recycling systems, giant engines to move it, giant everything.
Building large ships is pretty difficult. Parts get so large that they don't fit into existing manufacturing facilities. You can't just bolt smaller ones together; as the whole thing gets larger any joint becomes more and more of a potential weakness as the stresses involved grow too.
So before you cast that giant section of hull you have to build a giant metalworking facility that can produce vast amounts of liquid hot alloy and pour it into a giant mould. You need a kilometre long factory just to contain that extrusion.
This also increases the difficulty of maintaining the ships. Parts are now so big that they require special giant size machines to handle.
You can trade off these issues by having ships that accelerate more gently so they can be made of smaller parts connected together. Ships on Earth do that, the hull is made in sections. But then your journey time is extended, you need more supplies and so on.
Overall smaller, mass produced ships may well be more efficient over the entire lifecycle.
In general, I think there are a lot of efficiencies that occur with scale, but I question the assertion that it's necessarily true that bigger will always be more efficient. Net efficiencies of scale don't always increase without limit for any given engineering constraint as scale goes up.
The classic example is the square-cubed relationship with surface area vs volume. As you scale one axis, volume increases faster than surface area. This is beneficial when you want to maximize how much stuff you can fit inside the ship, for example, but it's problematic for issues like heat dissipation, which relies on surface area to radiate the heat away. You can mitigate this using radiator arrays to stretch out the surface area, but if you also consider that the total amount of heat you'll need to dissipate will increase with the size of the ship, the total radiating area required might grow rather quickly. If you optimize for just the problem of heat dissipation and 'cost' of radiators, there is probably a sweet spot for this factor of efficiency.
Another example is power generation and power transmission. A large centralized power generator may be more efficient to produce, but now you also require longer distance transmission, which has associated inefficiencies. Here is another place where this a trade-off of considerations and the optimum is likely not toward infinity.
There's also the harder-to-quantify fact that scaling systems up will tend to make them more complicated, which may mean greater odds that they can fail (requiring additional redundancy if you want a safety margin) and likely requiring a larger amount of support equipment and support staff. Larger groups of people introduces a whole new slew of risks and challenges, as managing a large group of people over a long period of time to maintain a complex system creates its own challenges in terms of the required long-term social engineering. Smaller ships means smaller societies, which might be more stable in the long run than a large society sharing a single ship and having to coordinate at large scales over a long period of time to be successful.
In short, it's important to consider efficiencies of scale, but it's equally important to recognize the counterbalancing challenges/inefficiencies that occur as scale increases and weigh these factors when engineering a solution.
Breaking with tradition, I will answer a question which I have downvoted.
The question cannot be answered, except by an experienced generation ship designer who has access to data which the question chose not to include.
The point is that there are always limits in engineering. Designs work between a minimum size and a maximum size; outside the boundaries, different designs are needed: and the question does not say how to build a generation ship, how big is big, what are the design goals, how is the ship powered, and so on and so on.
Let's take a much more mundane example, and ask ourselves whether it's more efficient to carry containerized cargo from China to Europe using one big ship or a fleet of smaller ships.
A modern cargo ship at maximum fuel efficiency takes about 35 days to travel from China to western Europe, assuming it fits through the Suez canal. If it is too large for the Suez canal, travel time increases to about 50 days. If it can make use of the Russian North-East Passage, travel time decreases to about 25 days.
Current Suezmax limits are length 400 meters, beam 50 meters, draft 20 meters, height above water 70 meters. Guess what? The current largest container ships are designed to these limits.
But what if you spoke to the Egyptians and convince them to increase the limits? Well, you will soon run into another limit: the ship needs to be able to load and unload cargo in a port, and ports have limits too.
But what if you spoke to the port authorities in Antwerp, Rotterdam, Shenzhen and Shanghai to do the needful to increase the limits of their ports? You will run into other limits: for example, you would much prefer to deliveries from China every 10 days or so, and do not want to have to wait up to 70 days for your merchandize to arrive from China.
Overall, the current large container ships, capable of carrying about 24,000 TEU, are at the limit of what is actually possible and economically reasonable. We could design a container ship capable of carrying 50,000 TEU, but it would be of no use to anybody. We do not know how to design a container ship capable of carrying 500,000 TEU.
Coming back to the generation ship: let's say the current designs top out at ships about 10 km long with a diameter of about 2 km, giving an internal volume of about 31.5 km³. With great increase in cost, designing and building a ship with an internal volume of 100 km³ might be possible. But what if the colonisation effort needs 1,000 km³? Such a ship would exceed design limits, and sending multiple ships would be the only option.
In conclusion, we cannot say that one large ship is more or less efficient than multiple smaller ones unless we know a lot more about those ships than what the question says.
Instead of answering a very open ende question, I ask why not both?
Take a huge block of ice, or an asteroid or comet. Embedded into the mass are several independent habitats and drive units. The asteroid serves as radioation shielding and reaction mass, so upon arrival it will be rather hollow. Two approaches to build such a thing come to mind immediately:
rockets in the back: Build rocket shaped ships, with drives at the bottom and habitat/produiction areas at the top. These are stuck into the backside of the comet. Upon arrival they fill their internal fuel tank, detach and brake individually to not waste reaction mass on brakeing the rest of the asteroid (resteroid?)
Drive modules and habitat modules are separated, the drive modules sit on the skin and slowly eat their way in to asteroid, tha habitats are more to the inside but distributed as much as possible
The idea is that the individual modules can help each other, but that each can be fairly self sufficient if the need arises. They should be as far from each other as the space allows so a catastrophic event is unlikely to affect all of them. The close procimity would mean one could essentially take a train from one module to another, so ne need to spend reaction masse.
An interstellar ship will be one part payload, a thousand parts or more reaction mass so you might as well use that mass for shielding and structural purposes.
As others have pointed out bigger is more efficient, but many ships are more redundant.
A couple of important things with regard to redundancy:
You want redundant design not just copies. What if you design a redundant fleet of ships but all have a design flaw in say all of the main sewerage valves.
There is also the problem (which can be used to create drama to drive the story) of independent evolution on each of the redundant ships. Both of culture of the crews and also of bacteria and viruses. Imagine Covid2099 slowly developing on one ship. Slowly enough that the crew develop immunity, but when they either arrive at the destination system or during a ship to ship transfer there is the risk of spreading an infection. We all know how problematic that can be...
Tl;dr: Huge ships are structurally less stable and have a worse mass ratio.
At the core of the matter lies the square-cube law. You cannot simply scale up a small model because the strength of its support structures grows only quadratic with size (it essentially depends on the area of the cross-sections of struts and walls) while the mass it is supporting grows with the cube of the size (it only depends on the volume). A ship that's twice as large as another ship will have, ceteris paribus, eight times the mass, but the strength of its struts will only be four times the smaller one. On large ships relatively more mass is needed for structural stability. At some point this will begin to outweigh the advantage of large ships regarding the lacking usability of an outer shell area, if it exists at all (the area would likely hold supplies anyway).
This assumes that there is no (anti) gravitation mechanism, and that the generation ship will accelerate at the beginning and the end of the journey, which seems unavoidable. It is also likely that it will need to spin for gravity which will create structural stress beyond many potential drives.
One big ship is more efficient.
The reason for this is that one ship requires less fuel, less crew, less power, and only one shipyard dock to make.
Basically, the concentration of resources is more efficient than making and sending, a bunch of smaller ships.
But
If that one big is drastically bigger than the smaller ships, then it's less efficient.
An example of this is the death star. The death star is so expensive and fuel hogging, that it would be much more efficient to send 2 superstar destroyers than one death star. The reason for this is because the death star is so much bigger than the superstar destroyer, that it requires at least 10 times the amount of power, fuel, and crew (and everything required to keep the crew alive and going).
Another But
Sending several smaller ships (like a couple of sizes under, like 5 cruisers instead of one capital ship) provides a greater tactical advantage.
Basically, one ship can't do flanking maneuvers, all of its weapons don't point at one target so it can't bring as much firepower to bear, and it's less maneuverable, and if it is taken out, that's it, you've lost the battle.
