One. (20 points) A factory needs to produce two types of products, A and B. According to the process specifications, each unit of product A requires 2 hours, 3 hours, and 4 hours of processing on three different machines A, B, and C, respectively. Each unit of product B requires 4 hours, 4 hours, and 3 hours of processing on machines B, C, and D, respectively. The maximum working hours per day for machines A, B, C, and D are 12 hours, 15 hours, 16 hours, and 24 hours, respectively. If producing one unit of product A yields a profit of 700 yuan, and producing one unit of product B yields a profit of 800 yuan, how should production be arranged daily to maximize the profit?
Let the daily production of product A be $x$ units, and the production of product B be $y$ units. Then we have:
$$
\left\{\begin{array}{l}
2 x \leqslant 12, \\
3 x+4 y \leqslant 15, \\
4 x+4 y \leqslant 16, \\
3 y \leqslant 24 .
\end{array}\right.
$$

The profit obtained is
$$
W=700 x+800 y=100(7 x+8 y) \text {. }
$$

Thus, the problem is to find the values of $x$ and $y$ that maximize $W$.
(2) + (3) gives $7 x+8 y \leqslant 31$. Therefore, $W \leqslant 3100$.

Since $x$ and $y$ are non-negative integers, and from
$$
\left\{\begin{array}{l}
3 x+4 y=15, \\
4 x+4 y=16
\end{array} \text { we solve to get } x=1, y=3\right. \text {. }
$$

Clearly, this solution also satisfies (1) and (4).
Therefore, producing 1 unit of product A and 3 units of product B daily yields the maximum profit.