Take $$f=(x_{1}y_{1})+(x_{2}y_{2})\dots+(x_{n}y_{n})$$
Suppose I assign $x_i=i$ at every $i\in\{1,\dots,n\}$.
Then suppose I need to assign $y_i$ from $\{1,\dots,n\}$ so that $y_i\neq y_j\iff i\neq j$ then is it clear that assigning $y_i=n-i+1$ gives minimum value?
That is is $f=n+2(n-1)+\dots+(n-1)2+n$ minimum?
If this is minimum is this unique among all assignments?