Cheapest way?
Weigh each component and input the results in an Excel table, linking cost and competitor’s products with known weights.
Then piddle about with the spreadsheet model and work out the best cost/performance ratio.
You can save a significant amount of weight by switching from plain gauge to butted spokes in a wheelset for example, at a cost of £0.50 per spoke. A heavy saddle and post combination can add upwards of 250g to the weight of a bike – I switched to an SDG I-Beam post and made a saving.
However, before you get carried away there are a few things to bear in mind:
1) A spreadsheet won’t tell you whether your new Specialized S-Works tyres that save you 150g at each end will tear a sidewall within a hundred yards (mine did), or that the weak sidewalls combined with being tubeless made the rear of the bike roll and wallow in the corners.
2) A spreadsheet won’t tell you whether you’ve turned a substantial AM bike into a “spastic Bambi” at huge cost (phrase borrowed from the late Steve Worland) that compromises your enjoyment of a ride.
3) Going 1×9 is a great idea, until you try to climb a steep, technical trail with a 32t cassette and a 34t front chainring. You really need a wide cassette range to make it work if you live anywhere with hills.
All of the above are things I learned to my own cost when I tried to pare the weight of a fully built AM bike.
Oh and a fitter rider on a burlier bike is always faster than a less fit rider on a light bike. I found that one out to my cost, too.