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Let $n$ be a positive integer and $W$ be the set $W=\{w_1,\ldots,w_n\}$ for positive $w_i$.

I am looking for a function $f:W\mapsto V$ (may be a bijection?) where $V=\{v_1,\ldots,v_n\}$ that satisfies the following two conditions simultaneously:

$$\sum_{i=1}^nw_i\leqslant\max\limits_{i=1,\ldots,n}\{f(w_i)\},$$ and $$\sum_{i=1}^nf(w_i)\leqslant\max\limits_{i=1,\ldots,n}\{w_i\}.$$

Can we find such a function $f$?

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    In general, no. Take $n=2$, the $w_i$ to be $1,2$, and the $v_i$ to be $3,4$. It is easy to see that there is no function $f:W\to V$ satisfying your requirements. – 2017-01-11

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