Reference no: EM13342663
Sending Water through the Pipe Network
Suppose that you are the most famous fire fighter in Rolla, MO. Right now, there is a fire in one of the districts of Rolla. As the famous fire fighter, you want to stop this fire as fast as possible. To do so, you want to transfer as much water as possible from the water depot in Rolla to the district with fire using the pipe network in Rolla. Particularly, the pipe network is given below.
The water can be transferred from the water depot to city valves (there are 5 valves shown in orange octagons) and from some of the valves to the fire district. The table below shows how much water can be transferred in the pipes connecting the valves, water depot, and fire (the water can only flow in the direction of the pipes as given in the above network).
a) Mathematically formulate a network optimization problem for transferring as much water as possible from the water depot to the fire district. Assume that the water depot is sufficiently large and it has unlimited water in it. That is, mathematically formulate the above network optimization problem as a min-cost flow or max-flow problem (recall that shortest path problems can be formulated as min-cost flow problems). Define your decision variables clearly and the notation you use for your decision variables, write the objective and objective function, and constraints. Combine everything to get the final model.
b) Now suppose that the water depot has a capacity of 150 m3/minute. How would you modify the above network and your model in part a so that you still only have flow balance and arc capacity constraints. Formulate the additional constraints (you do not need to formulate the whole problem from start).
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: Formulate a network optimization problem for transferring as much water as possible from the water depot to the fire district and how would you modify the above network and your model in part a so that you still only have flow balance and arc capa..
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