Linear Programming: Chapter 13
Network Flows: Theory
Robert J. Vanderbei
October 17, 2007
Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544
http://www.princeton.edu/∼rvdb
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Linear Programming Network Flows, Lecture Notes - Mathematics, Study notes of Linear Programming
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Linear Programming: Chapter 13
Network Flows: Theory
Robert J. Vanderbei
October 17, 2007
Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/∼rvdb
Networks
Basic elements:
- N Nodes (let m denote number of them).
- A Directed Arcs
- subset of all possible arcs: {(i, j) : i, j ∈ N , i 6 = j}.
- arcs are directed: (i, j) 6 = (j, i).
Network Flow Problem
Decision Variables:
- xij , (i, j) ∈ A, quantity to ship along arc (i, j).
Objective:
minimize
(i,j)∈A
cij xij
Network Flow Problem–Cont.
Constraints:
- Mass conservation (aka flow balance):
inflow(k) − outflow(k) = demand(k) = −supply(k), k ∈ N m ∑
i : (i, k) ∈ A
xik −
j : (k, j) ∈ A
xkj = −bk, k ∈ N
- Nonnegativity: xij ≥ 0 , (i, j) ∈ A
Dual Problem
maximize −bT^ y subject to AT^ y + z = c z ≥ 0
In network notation:
maximize −
i∈N biyi subject to yj − yi + zij = cij (i, j) ∈ A zij ≥ 0 (i, j) ∈ A
Complementarity Relations
- The primal variables must be nonnegative.
- Therefore the associated dual constraints are inequalities.
- The dual slack variables are complementary to the primal variables:
xij zij = 0, (i, j) ∈ A
- The primal constraints are equalities.
- Therefore they have no slack variables.
- The corresponding dual variables, the yi’s, are free variables.
- No complementarity conditions apply to them.
Connected vs. Disconnected
Connected Disconnected
Cyclic vs. Acyclic
Cyclic Acyclic
Spanning Trees
Spanning Tree–A tree touching every node
Tree Solution
xij = 0 for (i, j) 6 ∈ Tree Arcs
Note: Tree solutions are easy to compute—start at the leaves and work inward...
Online Pivot Tool–Notations
Data:
- Costs on arcs shown above arcs.
- Supplies at nodes shown above nodes.
Variables:
- Primal flows shown on tree arcs.
- Dual slacks shown on non- tree arcs.
- Dual variables shown below nodes.