You can approximate the problem by minimizing the range via integer linear programming.  Let binary decision variable $x_{i,j}$ indicate whether $i\in S_j$.  The problem is to minimize $u-\ell$ subject to:
\begin{align}
\sum_j x_{i,j} &=1 &&\text{for all $i$}\\
\sum_i x_{i,j} &= m &&\text{for all $j$}\\
\ell \le \sum_i a_i x_{i,j} &\le u &&\text{for all $j$}
\end{align}
To obtain a formulation for the min-max or max-min problem, omit the parts involving $\ell$ or $u$, respectively.