Stochastic Programming - Index of

NETWORK PROBLEMS 291
Figure 10
Section 6.4.
Transportation network for sewage, used for the example in
All three cities produce sewage, and they have local treatment plants to take
care of some of it. Both the amount of sewage from a city and its treatment
capacity vary, and the net variation from a city is given next to the node
representing the city. For example, City 1 always produces more than it can
treat, and the surplus varies between 10 and 20 units per unit time. City 2,
on the other hand, sometimes can treat up to 5 units of sewage from other
cities, but at other times has as much as 15 units it cannot itself treat. City
3 always has extra capacity, and that varies between 5 and 15 units per unit
time.
The solid lines in Figure 10 represent pipes through which sewage can be
pumped (at a cost). Assume all pipes have a capacity of up to 5 units per unit
time. Node 4 is a common treatment site for the whole area, and its capacity
is so large that for practical purposes we can view it as being infinite. Until
now, whenever a city had sewage that it could not treat itself, it first tried to
send it to other cities, or site 4, but if that was not possible, the sewage was
simply dumped in the ocean. (It is easy to see that that can happen. When
City 1 has more than 10 units of untreated sewage, it must dump some of it.)
New rules are being introduced, and within a short period of time dumping
sewage will not be allowed. Four projects have been suggested.
• Increase the capacity of the pipe from City 1 (via City 2) to site 4 with x 1
units (per unit time).
• Increase the capacity of the pipe from City 2 to City 3 with x 2 units (per
unit time).
• Increase the capacity of the pipe from City 1 (via City 3) to site 4 with x 3
units (per unit time).
• Build a new treatment plant in City 1 with a capacity of x 4 units (per unit
time).