1. Network Optimization Models: Maximum Flow Problems
In this handout:
The problem statement
Solving by linear programming

2. Maximum Flow Problem Given: Directed graph G=(V, E),
Supply (source) node O, demand (sink) node T
Capacity function u: E  R .
Goal: Given the arc capacities,
send as much flow as possible
from supply node O to demand node T
through the network.
Example: O A D B C T 4 5 6

3. Characteristics of a feasible flow
Let xij denote the flow through arc i  j .
Capacity kij of arc i  j is the upper bound on the flow shipped through arc i  j .
Thus, we have the following constraints:
0  xij  kij , for any arc i  j
Every node i, except the source and the sink,
should satisfy the conservation-of-flow constraint, i.e.,
flow into node k = flow out of node k
In terms of xij the constraint is
For any flow that satisfies the conservation-of-flow constraints,
flow out of the source = flow into the sink
This is the amount we want to maximize.

4. Linear Program for the Maximum Flow Problem
Summarizing, we have the following linear program:
0  xij  kij , for any arc i  j
This linear program can be solved by a Simplex method or Excel Solver add-in.

5. x12 4 5 6 1 2 3
x13
x14
x35
x34
x56
x46
x25
Objective function: maximize xmax
Capacity constraints:
x12 ≤ 4, x13 ≤ 5, x14 ≤ 4, x25 ≤ 4, x34 ≤ 4, x35 ≤ 6,
x45 ≤ 5, x56 ≤ 5
Conservation-of-flow constraint:
x12 = x25, x13 = x34+x35, x14+x34=x46, x24+x35=x56
Constraint for the sourse and sink node:
x12+x13+x14=x46+x56=xmax
Non-negativity constraints: xij≥0