$\max\{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha \}$  \\

$\textrm{s.t.}\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$ \\

We should find the best $W >0$ when $F( \cdot )$ is probability distribution function, and $L_0$ and $L_1$ are positive random variables. $\xi$, $\pi_0$, $\pi_1$ are constant. Also, $\pi_0, \pi_1>0$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume, for example, $F_{L_0}$ and $F_{L_1}$ such as below ($\lambda$ and $\mu$ are constant:

  $F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0.$
  $F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0. $