Minimize: $F=56 x+24 y$

Subject to:
\[
\begin{array}{l}
x+2 y \geq 14 \\
x+y \geq 12 \\
2 x+y \geq 16 \\
x, y \geq 0
\end{array}
\]

Step 1 ：We are given the objective function to minimize: \(F=56x+24y\).

Step 2 ：The constraints are: \(x+2y \geq 14\), \(x+y \geq 12\), \(2x+y \geq 16\), and \(x, y \geq 0\).

Step 3 ：We can solve this problem using linear programming.

Step 4 ：The optimal solution is obtained when the value of the objective function is minimum.

Step 5 ：The minimum value of the objective function is 384 and the values of \(x\) and \(y\) that achieve this minimum are \(x=0\) and \(y=16\).

Step 6 ：Final Answer: The minimum value of the function \(F=56x+24y\) subject to the given constraints is 384, achieved at \(x=0\) and \(y=16\). So, the final answer is \(\boxed{384}\).