How to give an upper bound on this bin packing problem?
Your relaxation is same as original problem.

$$\sum_{i\in S}w_i\leq f_j,\forall j\in S$$

is same condition as 

$$\sum_{i\in S}w_i\leq\min\limits_{j\in S}f_j$$.

because 

$$\min\limits_{i\in S}f_i \leq f_j,\forall j\in S$$.