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There are many types of problems that concern a network of conductors along which some sort of flow is observed. Examples
of these include an irrigation network and a network of streets or freeways. There are often points in the system at which a net
flow either enters or leaves the system. The basic principle behind the analysis of such systems is that the
total flow into the system must equal the total flow out. In fact, we apply this principle at every junction in the
system.
Junction Rule At each of the junctions in the network, the total flow into that junction must equal the total flow out.
This requirement gives a linear equation relating the flows in conductors emanating from the junction.
A network of one-way streets is shown in the accompanying diagram. The rate of flow of cars into intersection \(A\) is 500 cars per
hour, and 400 and 100 cars per hour emerge from \(B\) and \(C\), respectively. Find the possible flows along each
street.
Suppose the flows along the streets are \( f_1\), \(f_2\), \(f_3\), \(f_4\), \(f_5\), and \(f_6\) cars per hour in the directions shown.
Then, equating the flow in with the flow out at each intersection, we get
This gives all
solutions to the system of equations and hence all the possible flows.
Of course, not all these solutions may be acceptable in the real situation. For example, the flows \( f_1, f_2, \dots , f_6 \) are all positive in the present
context (if one came out negative, it would mean traffic flowed in the opposite direction). This imposes constraints on the
flows: \(f_1 \geq 0\) and \(f_3 \geq 0\) become
A proposed network of irrigation canals is described in the accompanying diagram. At peak demand, the flows at
interchanges \(A\), \(B\), \(C\), and \(D\) are as shown.
1.
Find the possible flows.
2.
If canal \(BC\) is closed, what range of flow on \(AD\) must be maintained so that no canal carries a flow of more than 30?
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