Question $1(20$ marks$)$

The arterial watermain network of a city is shown below. Each node except the intake station

represents a major water consumption unit. Associated with each segment of the watermain

between two nodes are two parameters: the distribution capacity $Q_i($unit: million liters per hour$)$

and reliability $r_i,i=1,$dots,$12.$ Here the reliability measures the probability that the watermain

segment is operational. The probability that segment $i$ is not available $($due to watermain break,

for example$)$ is thus $1-r_i.$ Suppose the availabilities of the segments are statistically independent

of each other.

The network must be designed to ensure that each consumption unit can have the water they

need with high reliability.

Let us consider only the need of unit F$.$ Formulate a mathematical model and develop a solution

strategy to analyze the probability distribution of the flow capacity from the intake station to the

destination $($unit F$).$

Hint: $(1)$ Note that watermain breaks randomly with a probability of $1-r_i.$ If one segment is

broken, the network topology is changed.