Whilst many problems will definitely benefit from a more creative approach the use of a logical and analytical process should not be discarded.
There are many issues that lend themselves to this method.
In particular, any problem that is tangible and easy to visualise could probably benefit.
If a piece of equipment is not working it is quite straightforward to consider the make up of all of its parts.
In stripping down the item you can identify each component part and decide on its suitability and replace or repair it.
In this way, it is a simple matter to correct the workings a piece of machinery, for example.
The above assumes that the solution to the failing parts is simple and obvious but this may not be the case.
Once a problem has been broken down into its constituent parts you then have the task of assessing each of them to make sure that they are at fault.
You may then need to use other techniques to get to the bottom of why these items are at fault and then assess them for repair or replacement.
The problem with this method is in its straightforward simplicity.
People will be happy to just go along with this technique and perhaps ignore other methods or find them a little awkward to implement.
Hence, the logical approach may provide narrower solutions than you would like, thus, missing out on perhaps more innovative ideas.
Breaking any problem down into smaller parts will always make the job appear easier and increase motivation to complete it.
This approach is very similar to that used in time or project management in general; where breaking down the task will help with motivation, management and its completion.