Packing $n$ objects into $m$ bins whose size is variable
One simple "bad" input that needs to be considered for worst-case analysis of this problem is as follows.

Let $c=(\sqrt{17}-1)/2 \approx 1.56$.
There are three objects of size $c$, $1$, and $1$.
There are two bins of size $2$ and $c$.
Initially $a=1$.
Some heuristics will place the largest object into the big bin, necessitating $a$ increasing from 1 to at least $(c+1)/2 = 2/c &gt; 1.2807$.
An optimal packing leaves $a=1$, placing the two small objects into the big bin and the big object into the smaller bin, with no waste.
Hence such heuristics will have a worst-case actual-to-optimal ratio of more than 1.2807.

Notwithstanding this worst-case analysis, a heuristic like the one described in the edited question is likely to work well for practical examples, even if its worst-case performance is at least 28% worse than optimal.
Similar heuristics are often used in practice, sometimes with adjustments like "instead of the general criterion, prefer a bin that is only a tiny bit larger than the object to be placed, if one such exists".
Most implemented heuristics I'm familiar with are a big mess of such case analysis, but work extremely well in practice.
However, it is often quite hard to come up with badly behaved inputs for such complex algorithms, and it is even harder to prove that they are the worst possible.