You can model this as a graph problem. You have a bipartite graph $G$, with $2n$ vertices $\{a_1,a_2,\ldots, a_n,b_1,b_2,\ldots b_n\}$. You connect every vertex $a_i$ with every vertex $b_j$. Moreover, you give every edge $(a_i,b_j)$ the weight $\rho_{ij}$. A permutation $\sigma$ corresponds now to a perfect matching in $G$.
Let me first discuss how to solve the decision problem for your question, that is you ask if $M:=\inf_{\sigma\in \Sigma} \sup_{1\le i\le n} \rho_{i,\sigma(i)}\le k.$ You first delete all edges that have too large weights, i.e., all edges $(a_i,b_j)$ with $\rho_{ij}>k$. If there exists a perfect matching in the remaining graph, then $M$ is less then $k$, otherwise not. You can decide if a perfect matching exists by the algorithm of Hopcroft and Kahn (other algorithms are also applicable).
Since you have now an alogorithm for the decision problem you can simply run a binary search over all possible solutions, which are the weights $\rho_{ij}$.
PS I think than cs.SE is a good place for these type of questions.