DOSSIER DE PRÉSENTATION
POUR TRADUCTION D'UN COURS EXISTANT
JOINDRE LE PLAN DE COURS EN FRANÇAIS
PROGRAM: PhD
TITLE (IN FRENCH): Optimisation de réseaux d’entreprises
TITRE (in French, short) : Optimisation de réseaux
TITLE: Network Optimization in Business
TITLE (short) : Network Optimization
SIGLE [Série 100, 200, etc.] : 80-632-09
Code disciplinaire CLARDER : 1703
Libellé CLARDER : Applied mathematics
PROFESSORS IN CHARGE OF THE PROJECT: Jacques Desrosiers
Department: Méthodes quantitatives de gestion
DURATION : 45 hours
PREREQUISITE : None
REQUEST DATE: Fall 2011
PLAGIARISM AND FRAUD: Please consult article 12 of the Règlement régissant l'activité
étudiante à HEC Montréal (regulation governing student activities at HEC Montréal), to find
out the acts and gestures that are considered plagiarism or another academic violation
(12.1), along with the applicable procedure (12.2) and sanctions, which range up to
suspension and even expulsion from HEC (12.3). Violations are analyzed based on the
facts and circumstances, and sanctions are applied accordingly.
OBJECTIVE: The aim of the course is to present the most important aspects of
mathematical optimization applied to network flow problems. On the one hand, we study
the specialized algorithms, and on the other hand, we take a look at the numerous
applications in this field.
Section réservée à la Direction des programmes
Date d'approbation au Comité des programmes : ________________________________
2016-02-09
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PEDAGOGICAL APPROACH: Classroom lectures
INSERTION IN PROGRAM STRUCTURE: optional
EVALUATION:
Mid-term exam:
Final exam:
4 assignments:
30%
30%
40%
DETAILED DESCRIPTION :
Session 1 – Introduction
 Basic notions; definitions; network flow problems.
Session 2 – Network representations and transformations
 Directed network representation; incidence matrix; adjacency matrix; adjacency lists.
Transformations: undirected arcs; removing nonzero lower bounds and arc capacity;
node splitting. Introduction to reduced costs.
Session 3 – Shortest path problems : label-setting algorithms
 Formulations and applications; tree of shortest path; shortest path in acyclic networks;
Dijkstra’s algorithm; Dial’s implementation. Telephone operator scheduling.
Session 4 – Shortest path problems : label-correcting algorithms
 Optimality conditions and generic algorithms; special implementations; detecting
negative cycles. All-pairs shortest path problem; optimality conditions and FloydWarshall algorithm.
Session 5 – Maximun flows
 Formulation and applications; feasible flow; flows and cuts; generic augmenting path
algorithm.
Session 6 – Minimum cost flows
 Applications. Elementary notions : residual network; flow decomposition. Optimality
conditions based on negative cycles, on reduced costs, and on complementary
slackness. Generic algorithm.
Session 7 – Duality in network flows
 Weak and strong duality theorems; optimal arc flows and node potentials.
Identification of network problems : linear programs with consecutive ones.
Applications in project management and scheduling.
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Session 8 – Network simplex algorithm
 Cycle free and spanning tree solutions; computing node potentials and arc flows;
simplex algorithm for network problems. Unimodularity property.
Session 9 – Lagrangian relaxation
 Definitions and properties. Integrality property. Sub-gradient method. Applications:
shortest path problem with duration constraint; vehicle routing problem with time
windows; cutting stock problem.
Session 10 – Dantzig-Wolfe decomposition
 Definitions and properties. Integrality property. Applications: shortest path problem
with duration constraint; vehicle routing problem with time windows; cutting stock
problem : various formulations (Kantorovich, Gilmore-Gomory, Valerio de Carvalho).
Sessions 11-12 – Column generation and industrial applications
 Selected topics : acceleration strategies; stabilization of the dual variables; constraint
aggregation for set partitioning problems. Improved simplex method.
References
Book : Ahuja, Magnanti & Orlin : Network Flows, Prentice Hall (1993)
Additionnal papers:
M. Lübbecke and J. Desrosiers, Selected Topics in Column Generation, Operations Research,
53(6), 1007–1023, 2005.
J. Desrosiers et M. Lübbecke, A Primer in Column Generation, in : Desaulniers, G., Desrosiers, J.
and Solomon, MM. (eds.) Column Generation, Springer, 1–32, 2005.
Date d'implantation :
2016-02-09
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