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I have been working on an application where we have tried two alternative methods. Both methods are used for the same purpose, but the first one had several problems. Despite getting results we had to look for another method that solved those problems to some extent.
Can I mention the rejected method along with the accepted method in a research paper?
I cannot see any reason not to mention what could be labelled "negative results". If you have tried a method and it failed there can be two reasons, either the choice of method was bad (which is likely a trivial unpublishable result) or the method is in some way not appropriate due to reasons not before understood. In the latter case, there are clear advantages to convey this result to others. This may spawn new research or simply help others avoid using the method. Hopefully you will be able to provide deeper insights into why things did not work out as you first thought. In essence, anything that progresses our understanding is worth bringing forth in a publication.
Yes, especially if most people would expect the method to work, but you can clearly describe why it does not work. It has to be done carefully though, because the focus should be on actual results, not on failed methods (this is not a negative result, it's really a failed attempt to get any result).
In my opinion, the best place to put this is the introduction. After you state your problem, say which method one would try in a first step, then argue why it doesn't work, and then present you approach to the problem. This approach works especially well if someone suggested the method as a feasible approach to your problem, and you cite that in your introduction.
But keep it short: I would think that two to three sentences should be sufficient to describe it. If the failed method is so little known in your field that it requires careful explanation, it's probably not worth being mentioned at all.
Firstly I will qualify this by declaring that I am not strictly an academic.
I have, however, overseen research for a number of years. This has been government-private partnership industrial research that has been done under full scale production processes of construction - perhaps the largest contiguous experimental research work ever undertaken in hot and tropical climates. This was a 20+ year project involving a number of programmes - some 120,000 data elements and a team of analysts to PhD level.
Needless to say, it is important that this information be published, presented and disseminated - otherwise it is of no value. What is paramount in this process is the benefit that this information and message can give to the audience.
Above all each and every statement must be supported by logic and the data. Try to let the data speak for itself. Try not to wander into speculative territory - some things cannot be explained. And the reader must have enough information to make their own informed decision. We take the view that the researcher has a lot of power - the power to persuade and change an audience and industry to new and better ways. This is research in action.
This brings us to the question posed. For the audience to make an informed decision, it is essential to report on aspects that did not turn out as expected. Often we learn more from things that did not work than from those that did. For example, we established a procedure to carry out full scale temperature monitoring in one experiment, but found, to our surprise, that we still encountered significant ambient (environmental) effects, which prompted us to redesign the experiment until this phenomenon was effectively eliminated. This type of information, to your audience, is quite valuable. It also will build more confidence in your work.
However, it should not be something that one dwells on. Address it, and move on to more important things. Our work tends to go to practitioners, whereas yours may go to other academics. In this situation speculation, although I personally stay away it, may be warranted because it may lead you to new research avenues. These are matters your supervisor or associates will better understand.
Ultimately, research is about objectively seeking the truth - truth, like perfection, being something we may approach but never reach - with successes and failures along the way. Research is amongst the highest of human endeavours, in my view. Try to let your work and your writing convey these principles.
