Linear programming on graph

For a fixed path with weights $w_1, w_2, \dots, w_k$, the sum $\frac{w_1}{t_1} + \frac{w_2}{t_2} + \dots + \frac{w_k}{t_k}$ subject to $t_1 + t_2 + \dots + t_k \le T$ is minimized when $t_i = \frac{T \sqrt{w_i}}{\sqrt{w_1} + \sqrt{w_2} + \dots + \sqrt{w_k}}$: essentially, when $t_i \propto \sqrt{w_i}$, with the scaling factor being whatever we need for the sum to equal $T$. Then the sum is equal to $(\sqrt{w_1} + \sqrt{w_2} + \dots + \sqrt{w_k})^2/T$. This is a direct consequence of Titu's lemma.

Therefore to find the best path, you should find the shortest path with respect to the weights $\sqrt{w_i}$.