combinatorics – Partitioning a set with minimum weight

Let $X$ be a finite collection of natural numbers.($X$ is a multiset.)
Consider $R$ as the set of all $(A,B)$ such that $X$ is disjoint union of $A$ and $B$.
We now assume that $S$ be the set of elements $(A,B)$ of $R$ in such a way that $Sigma_{ain A} a$ is minimum among all elements of $R$.

What is the lower bound for $(A,B)in S$ in terms of the minimum of $X$?