Let $n\geq 4;\ $ let $x_1,\ldots,x_n,\ y_1,\ldots,y_n\in \mathbb{R}:\ x_i,\ i=1,\ldots,n $ are not all equal.
Proposition: To find a permutation $\sigma(x_1),\ldots \sigma(x_n)\ $ of $x_1,\ldots,x_n\ $ which gives the least value of $\ y_1\sigma(x_1) + \ldots + y_n\sigma(x_n)$ greater than $x_1 y_1 + \ldots + x_n y_n$, we only need to swap the position of two values $x_i, x_j$ in $x_1, \ldots, x_n.$
It seems to be true based on experimentation, but I've no idea how to prove it. We may assume WLOG that $y_1<\ldots<y_n,$ but this isn't very impressive. What next?
Here is some Python code I wrote:
import itertools
num1 = [1,2,3,4]
num2 = [1,2,3,4]
print(sum([i*j for (i, j) in zip(num1, num2)]))
permutations = list(itertools.permutations(num1))
permutations_new=[]
for i in permutations:
permutations_new.append(list(i))
print(permutations)
print(len(permutations_new), permutations_new)
for k in range(len(permutations_new)):
print(sum([i*j for (i, j) in zip(permutations_new[k], num2)]))