problem regarding total expected value and probability density and cumulative distribution function

[Phamthe30071997]

Hello everyone,

I read the literature about estimating the optimal quantity order (to satisfy the demand X) from 2 suppliers with different probability of disruption may occur (p) to get the maximum profit.

they have a benchmark model as folowing:


In order to estimate the maximum quantity order from each supplier to maximize the total expected value, they derive respective first order partial derivatives



set the first order partial of each Q1 and Q2 equal to 0, we can get the equation to calculate the optimal Q1 and Q2



I have question that, in the example below:


I do not know the progress that they calculate the optimal Q1 and Q2. I stuck at the cumulative funtion of F(X) in the intergration of g(t)

f(x) is the density function of market demand X
F(x) is the cumulative function of market demand X
Ti is disruption time of supplier i ( i=1,2)
L is selling period ( T will occur within L)
g(t) is density function of disruption time T
G(t) is cumulative function of disruption time T

Thank you in advance for your time