Water is delivered throughout New York City using eight main waterlines that are connected at six pumping stations are shown in the image below. The numbers on each of the arcs indicate the maximum allowable flow of water through each waterline $($in $1000$s of gallons per minute$).$

$1$ to $2(80),2$ to $4(90),4$ to $6(70),1$ to $3(50),3$ to $5(70),5$ to $6(60),2$ to $5(60),$ and $3$ to $4(50).$

Because the city's waterlines are aging, breaks have been occurring more frequently and are related to the increasing demands being placed on the system. Civil engineers have estimated the probability of waterline break occurring as follows:

Probability of failure on the line from station i to station j $=1-$ EXP$(-$ aijFij$/1000)$ where Fij is the flow $($in $1000$s of gallons per minute$)$ on the line from station i to station j and the values for the parameters aij are given as follows:

Parameters from Station to Station

From Station To Station aij

$120\mathrm{.}10$

$130\mathrm{.}17$

$240\mathrm{.}19$

$250\mathrm{.}15$

$340\mathrm{.}14$

$350\mathrm{.}16$

$460\mathrm{.}11$

$560\mathrm{.}09$

Engineers can use control valves to limit how much water flows through each waterline. During peak demand times, a total of $110,000$ gallons of water per minute needs to flow through this system.

Formulate an NLP model for this problem. $($Hint: review Module $4$ Practice Case II for the maximum flow problem.$)$