Linear Programming: Chapter 13
Network Flows: Algorithms
Robert J. Vanderbei
October 17, 2007
Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544
http://www.princeton.edu/∼rvdb
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Linear Programming Network Flows Algorithm, Lecture Notes - Mathematics, Study notes of Linear Programming
Linear Programming Network Flows Algorithm, Lecture Notes - Mathematics - Prof. J Vanderbei.pdf, Prof. J Vanderbei, Mathematics, Linear Programming, Network Flows Algorithm, Network Simplex Method, Planar Networks, Integrality Theorem
Typology: Study notes
2010/2011
1 / 22
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Linear Programming Network Flows Algorithm, Lecture Notes - Mathematics and more Study notes Linear Programming in PDF only on Docsity!
Linear Programming: Chapter 13
Network Flows: Algorithms
Robert J. Vanderbei
October 17, 2007
Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/∼rvdb
Agenda
- Primal Network Simplex Method
- Dual Network Simplex Method
- Two-Phase Network Simplex Method
- One-Phase Primal-Dual Network Simplex Method
- Planar Graphs
- Integrality Theorem
Primal Method—Second Pivot
Entering arc: (d,e)
Leaving arc: (d,a)
Explanation of leaving arc rule:
- Increase flow on (d,e).
- Each unit increase produces a unit increase on arcs pointing in the same direction.
- Each unit increase produces a unit decrease on arcs pointing in the opposite direction.
- The first to reach zero will be the one pointing in the oppo- site direction and having the smallest flow among all such arcs.
Primal Method—Third Pivot
Entering arc: (c,g) Leaving arc: (c,e)
Optimal!
Dual Network Simplex Method
Second Pivot
Leaving arc: (d,a) Entering arc: (b,c)
Optimal!
Explanation of Entering Arc Rule
Recall initial tree solution:
Leaving arc: (g,a) Entering arc: (d,e)
- Remove leaving arc. Need to find a recon- necting arc.
- Consider some reconnecting arc. Add flow to it. - If it reconnects in the same direction as leaving arc, such as (f,d), then flow on leaving arc decreases. - Therefore, leaving arc’s flow can’t be raised to zero. - Therefore, leaving arc can’t leave. No good.
- Consider a potential arc reconnecting in the opposite direction, say (b,c).
- Its dual slack will drop to zero.
- All other reconnecting arcs pointing in the same direction will drop by the same amount.
- To maintain nonnegativity of all the others, must pick the one that drops the least.
Two-Phase Method–First Pivot
Use dual network simplex method. Leaving arc: (d,e) Entering arc: (e,f)
Dual Feasible!
Two-Phase Method–Phase II
- Turn off display of ar- tificial dual slacks.
- Turn on display of dual slacks.
Two-Phase Method–Third Pivot
Entering arc: (f,c) Leaving arc: (a,f)
Optimal!
Online Network Simplex Pivot Tool
Click here (or on any displayed network) to try out the online network simplex pivot tool.
Second Iteration
- Range of μ values: 2 ≤ μ ≤ 9.
- Entering arc: (a,c)
- Leaving arc: (b,c)
New tree:
Third Iteration
- Range of μ values:
- 5 ≤ μ ≤ 2.
- Leaving arc: (a,g)
- Entering arc: (g,e)
New tree:
Fifth Iteration
- Range of μ values: 1 ≤ μ ≤ 1.
- Leaving arc: (f,b)
- Nothing to Enter.
Primal Infeasible!
Online Network Simplex Pivot Tool
Click here (or on any displayed network) to try out the online network simplex pivot tool.