One commenter questions whether generals and armies were capable of the advanced math implicated by the harmonic transfer method.
But they don't have to do any advanced math. They just take food for the fleet from a single donkey and turn that donkey around as soon as his food supply equals the food necessary for the return trip. Rinse and repeat. Right?
The reality does get a bit more complicated because at least one guide is required for each donkey, so you really don't want to lose a man for every donkey that turns back, and instead do the maneuver in groups. You also need to feed the return guide.
Optimizing under those conditions gets a lot more complicated I think.
You could also use this logic for sherpa-assisted mountain climbing.
This analysis also quite reasonably assumes zero transfer time between donkeys, perfect knowledge to run donkeys to exactly their limit (which are identical between all donkeys), and zero overhead in general. That's all fine, it's an upper limit analysis. But the overhead will be biting you in the exponential regime, unfortunately.
Another thing to remember about "the old days" is that they may not have computers and they may not have the Internet, but they have LOTS of time to think about more efficiency compared to an Internet commenter for whom this is merely a momentary side diversion for a few moments, and a lot more motivation. I feel satisfied with some of the answers to things like "How was Stonehenge built?" that have been found in the last couple of decades (or, if you prefer, "Can you show at least one method that could have been used to build Stonehenge?"), but I think one of the other lessons that was learned (at least by me) is that "take a modern person, give them one try in their busy lives to try the first thing that comes to their mind, and declare the task impossible when that doesn't work" isn't a very good way to understand the ancient world. They had time.
Would an ancient have described the "harmonic transfer technique" in this way? Heck no. Could they have worked their way to it through trial, error, and much simpler thinking? Absolutely. Finding that solution doesn't require calculus. Calculus just supplies a very nice analysis framework and a fantastic communication tool between the post author and us readers.
If you're sending them back one by one you need a guide for each.
If you send them in groups then you're not as optimal (first donkey is unladen in 3 days singly, but two donkeys are unladen in 5 days, say, meaning you need to keep feeding the first until the second is done, or equivalent via load balancing).
You could have trained donkeys that might self-navigate but that's moderately unlikely. Probably better to eat them or give them away free to the countryside.
The British were able to fly bombers from tremendously far away, with a chain re-fuel strategy where the re-fuelling tanker planes are themselves re-fuelled by other tankers which then turn back. Later iterations were optimised by thinking more carefully about who should transfer fuel, to who, and when, as in this example.
Arguably Black Buck was pointless, it was certainly not pivotal in the outcome of the war, but the actual process was fascinating.
But they don't have to do any advanced math. They just take food for the fleet from a single donkey and turn that donkey around as soon as his food supply equals the food necessary for the return trip. Rinse and repeat. Right?
The reality does get a bit more complicated because at least one guide is required for each donkey, so you really don't want to lose a man for every donkey that turns back, and instead do the maneuver in groups. You also need to feed the return guide.
Optimizing under those conditions gets a lot more complicated I think.
You could also use this logic for sherpa-assisted mountain climbing.