The interaction between differentials on a graded ring and chain-homotopy equivalences

Question

I am wondering about the following question:

Given a differential graded algebra $A$, how many other differentials can we put on the underlying graded ring of $A$, which are also chain-homotopy equivalent to $A$ ?

I'd like some references, perhaps, that may address the above question. Or if anyone here knows about results related to this question, please cite them here.

Thanks!