Water is delivered throughout New York City using eight main water lines that are connected at six pumping stations as shown in Figure. The numbers on each of the arcs indicates the maximum allowable flow of water through each water line (in 1000s of gallons per minute).

Because the city’s water lines are aging, breaks have been occurring more frequently, related to the increasing demands being placed on the system. Civil engineers have estimated the probability of a water line 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 1000s of gallons per minute) on the line from station i to station j and the values for the parameters aij are given as follows:

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

a. Create a spreadsheet model to determine the flow pattern that meets the required demand for water in the most reliable way.
b. How much water should flow through each water line?
c. What is the probability that no water line will fail while operating in thisway?

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