mth221 r2 network flows case study Essay

7726 Words Dec 12th, 2014 31 Pages
Network Flows

Author: versity. Arthur M. Hobbs, Department of Mathematics, Texas A&M Uni-

Prerequisites: The prerequisites for this chapter are graphs and trees. See
Sections 9.1 and 10.1 of Discrete Mathematics and Its Applications.

In this chapter we solve three very different problems.

Example 1
Joe the plumber has made an interesting offer. He says he has lots of short pieces of varying gauges of copper pipe; they are nearly worthless to him, but for only 1/5 of the usual cost of installing a plumbing connection under your house, he will use a bunch of T- and Y-joints he picked up at a distress sale and these small pipes to build the network shown in Figure 1. He claims that it will deliver three gallons per minute
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For the second reason, we need only realize that problems of this sort are likely to be programmed on a computer, and computers are much better at examining local situations than global ones.
However, using the first view, we can detect rules that must be satisfied by a flow described as in the second view. To state these rules easily, we need to define several terms.


Applications of Discrete Mathematics

Let A be a subset of V in directed graph G = (V, E), and let B = V − A.
Let c(A, B) be the sum of the capacities of the edges directed in G from vertices in A to vertices in B, and let c(B, A) be the sum of the capacities of the edges directed in G from vertices in B to vertices in A. Similarly, let f (A, B) be the amount of flow from A to B, i.e., the sum of the flows in the edges directed from vertices in A to vertices in B. Let f (B, A) be the amount of flow from B to A. Then the net flow F (A) from A is defined by
F (A) = f (A, B) − f (B, A).
For example, in Figure 4, if A = {s, b}, then f (A, B) = 2 and f (B, A) = 0.
Hence F (A) = 2. Similarly, F ({b}) = 2 − 2 = 0 and F ({s, t}) = 2 − 2 = 0, while F ({b, t}) = 2 − 2 − 2 = −2.
Note that F ({s}) is the total number of units of flow moving from the source to the sink in the graph. Our objective is to find a flow for which F ({s}) is maximum.
Since every unit of flow

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