We are given a $n\times n$ matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Questions: How can we find an upper bound for the maximum value $z^*$ of $z$ over all matrices $M$? Can we prove that $z^*$ is upper bounded by $\alpha n$ for some positive absolute constant $\alpha$?