**2. Cost Functionals**

Whereas the weight of an edge gives its flow rate, we now want to estimate the cost of pumping fluid through the associated pipe.

(More precisely this would be called the cost rate or cost per unit time.)

The simplest model would say that the cost rate is equal to the flow rate times the length of the pipe - as the cost of friction with the walls of the pipe is proportional to the length of the pipe.

But there is an economy of scale for wider pipes, which is expressed by a parameter \\(\alpha\\) between 0 and 1.

For a pipe with flow rate \\(w\\) and length \\(L\\), in this model the cost is \\(w^\alpha \cdot L\\).

Whereas the weight of an edge gives its flow rate, we now want to estimate the cost of pumping fluid through the associated pipe.

(More precisely this would be called the cost rate or cost per unit time.)

The simplest model would say that the cost rate is equal to the flow rate times the length of the pipe - as the cost of friction with the walls of the pipe is proportional to the length of the pipe.

But there is an economy of scale for wider pipes, which is expressed by a parameter \\(\alpha\\) between 0 and 1.

For a pipe with flow rate \\(w\\) and length \\(L\\), in this model the cost is \\(w^\alpha \cdot L\\).