What, and why, is the right choice in these cases?
(*) my supervisor's suggestion, I changed my schedule.
Should (*) be: in, on, at, under,....?
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At my supervisor's suggestion.
The easy answer is that it's the dictionary-defined convention. In more detail; you have heard your supervisor's suggestion and have aligned yourself with, to, or "at" it.
You might also use "to my supervisor's suggestion" although this is far less common to the extent that it might be considered dialectial.
"In my supervisor's suggestion" would mean you are talking about the suggestion itself, and some detail of it.
"On my supervisor's suggestion" isn't quite right.
"At my supervisor's suggestion" means "I did this because my supervisor suggested it", which i think is the desired meaning.
"Under my supervisor's suggestion" isn't correct, although "Under my supervisor's supervision..." would be a common usage.
Since your referent ('suggestion') is abstract, the preposition has no spacial meaning. The most basic preposition is 'at' (coinciding), so that is a good choice. 'Per' would also work.
On the other hand, "on" may be short for "upon". Would upon his suggestion be OK?
You can also use per:
I changed my schedule as per my supervisor's suggestion.