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

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

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.

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.

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