The Dyson Vacuum cleaner may well have been an example of analogical reasoning.
James Dyson is well-known for having “re-designed” the vacuum cleaner. He noted that conventional vacuums lost suction when the bags got clogged with dirt and dust. He developed a new way of removing dirt and dust from the air inside the vacuum cleaner by developing cyclonic air currents that literally spin the dirt and dust out of the air due to the difference in density between the air and the dirt and dust. Some people have tried to explain Dyson's innovation (e.g. Art Markman), but the best description is developed by combining a few of the creativity methods described in this website.
The story goes that Dyson drew on his experience in sawmills - where industrial cyclones are regularly used to clean the air of sawdust. Here are the steps that Dyson went through along with the methods that he (probably) used.
In the case of the vacuum, it’s that they lose suction. This could have been apparent from user feedback about existent vacuums, product tests, personal observations, etc.
However, this isn’t quite as easy as it seems, because the only way you can notice is something is bad, is if you have something good to compare it to. After all, fish don’t know they’re wet. (Because by the time they get dry, they're dead.) Anytime you see something bad, it must be because you can measure its badness against something else that you consider better. If you know there's something better, then you know how to fix the bad design.
By looking at the alternative vacuum designs that all suffer the problem identified in step 1, one looks for the common features. For the vacuum, the bag is the common feature. Because it is common to all conventional vacuums, their designers must have all assumed that a bag is necessary. One then simply asks Why is the bag necessary? This is simply an instance of the creativity method “challenge assumptions.”
One may do this step without first identifying a shortcoming. The danger, however, is that one will find and redesign common features of products that in fact do not contribute to improved product balance. This could lead to an even worse product, because it will cost more for all the redevelopment work put into it for only marginal, if any, gains.
The bag serves certain functions in the vacuum cleaner: to clean the air of dirt and dust. This function is impeded when the bag gets clogged. Any technology that can provide the same function as the bag is a candidate replacement for it. One can then evaluate those technologies to look for ones that are innovative, less expensive, of higher quality, or in some other way better.
This is basically an instance of concept design; in particular, looking for functional alternatives is identical to seeking embodiments of functional requirements. So any technique of embodiment design can potentially be used here to seek functional alternatives.
That’s all there is to it. Dyson didn’t look very far to find something. Indeed, as with so many innovative designs, the answer lay in the designer’s own experience1). It might be difficult to imagine that Dyson saw a similarity between domestic vacuums and sawmills, but it becomes simple if you think about function instead of structure. Vacuums suck dust up using air; the vacuum must therefore internally separate the dust from the air. That’s exactly what the function of the vacuum bag is – to trap dirt and let air through. In the sawmill, an industrial cyclone is used to separate sawdust from air. Notice that the functions served by the bag on the one hand and the cyclone on the other are exactly the same. It’s much easier to think of new designs when you think about function.
And that’s exactly what one needs to do to find an innovative solution to this problem: look for ways of providing the same function as the bag provides.
Indeed, this case is also an example of another technique of developing potentially innovative designs: change the state of matter used to provide a function. The bag uses mass (the bag itself) to separate dust from air. One could use liquid to do this – that’s the principle of a “shampooing vacuum.” One could also use gas – which is how the Dyson vacuum works. Finally (since matter and energy are related thanks to Einstein), one can use energy – a static electric charge (like in a Swiffer) that attracts and holds dust.