Permutations $r_j$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized

We consider the problem of finding the set of permutations $r_j$ of $\{1,\cdots , n\}$ such that $\sum_{i=1}^n \prod_{j=1}^k r_j(i)$ is maximized or minimized. While the set of permutations maximizing this value are easily determined, finding the set of permutations minimizing this value appears to be an open problem. We show values of $k$ and $n$ for which an explicit solution exists and comment on computational issues in determining the general problem. We also look at the dual problem of finding the permutations such that $\prod_{i=1}^n \sum_{j=1}^k r_j(i)$ is maximized or minimized. As part of this study we also look at a variant of a rearrangement inequality.
Comment: 16 pages. Added Theorem 10