Worst case using the first fit $2$-approximation for the bin packing problem I am with the phone so I would be more verbose when I will have a pc on hand, if you desire.

The first fit algorithm for approximating the bin packing problem (NP-hard) is a $2$-approximation for the optimum. Can you show me a concrete worst case showing that $2$ is a good (or bad) extimate for the bound?
The approximation ratio of first fit for bin packing is actually 1.7 rather than 2. See Dósa and Sgall, First fit bin packing: a tight analysis. The paper contains references which prove a matching lower bound.