16. Let $x_{1}, x_{2}, x_{3}, x_{4}$ be non-negative real numbers satisfying
$$
x_{1}+x_{2}+x_{3}+x_{4}=1
$$

Find the maximum value of $\sum_{1 \leqslant i<j \leqslant 4}\left(x_{i}+x_{j}\right) \sqrt{x_{i} x_{j}}$, and determine the values of $x_{1}, x_{2}, x_{3}, x_{4}$ when the maximum value is achieved.
16. The maximum value sought is $\frac{3}{4}$, if and only if
$$
x_{1}=x_{2}=x_{3}=x_{4}=\frac{1}{4} \text {. }
$$

From $4\left(x_{i}+x_{j}\right) \sqrt{x_{i} x_{j}}$
$$
\begin{aligned}
= & x_{i}^{2}+6 x_{i} x_{j}+x_{j}^{2}-\left(\sqrt{x_{i}}-\sqrt{x_{j}}\right)^{4} \\
\Rightarrow & 4 \sum_{1 \leqslant i<j \leqslant 4}\left(x_{i}+x_{j}\right) \sqrt{x_{i} x_{j}} \\
& =3\left(\sum_{i=1}^{4} x_{i}\right)^{2}-\sum_{1 \leqslant i<j \leqslant 4}\left(\sqrt{x_{i}}-\sqrt{x_{j}}\right)^{4} \\
& \leqslant 3\left(x_{1}+x_{2}+x_{3}+x_{4}\right)^{2}=3,
\end{aligned}
$$

The equality holds if and only if $x_{1}=x_{2}=x_{3}=x_{4}$.
Given $x_{1}+x_{2}+x_{3}+x_{4}=1$, the condition for equality is $x_{1}=x_{2}=x_{3}=x_{4}=\frac{1}{4}$.