NETWORK FLOWS AND COMBINATORIAL OPTIMIZATION

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ESI 6417 Linear Programming and Network
Optimization Fall 2006 Ravindra K. Ahuja 370
Weil Hall, Dept. of ISE ahuja_at_ufl.edu Office
[phone], ext 2004 Cell [phone]
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Course Objectives

• Engineers and managers are constantly attempting
  to optimize, particularly in the design,
  analysis, and operation of complex systems. The
  course seeks to
• to present a range of applications of linear
  programming and network optimization problem in
  many scientific domains and industrial setting
• provide an in-depth understanding of the
  underlying theory of linear programming and
  network flows
• to present a range of algorithms available to
  solve such problems
• to give exposure to the diversity of applications
  of these problems in engineering and management
• to help each student develop his or her intuition
  about algorithm design, development and analysis.

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Course Topics

• Linear Programming
• Formulating linear programs
• Applications of linear programming
• Linear algebra, convex analysis, polyhedral sets
• Simplex algorithm
• Revised simplex algorithm
• Duality theory
• Sensitivity analysis
• Integer programming Applications and algorithms
• CPLEX and CONCERT Technology
• Network Optimization
• Shortest path problem
• Minimum spanning tree problem
• Maximum flow problem
• Minimum cost flow problem

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Course Details

• Lectures
• Tuesday Periods 6 and 7 (1235 PM to 245
  PM), and
• Thursday Periods 6 and 7 (1235 PM to 245 PM)
• Place CISE Building
• Office Hours Tuesday, Period 5, 1130 AM to
  1230 PM
• Text Books
• M.S. Bazaraa, J. J. Jarvis, and H.D. Sherali,
  Linear Programming and Network Flows Second
  Edition," John Wiley, ISBN 0-471-63681-9.
• R. K. Ahuja, T. L. Magnanti, and J. B. Orlin,
  1993,
• Network Flows Theory, Algorithms, and
  Applications, Prentice Hall, NJ. ISBN
  0-13-617549-X.
• Recommended website to buy the books
  www.addall.com, www.amazon.com

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Course Details (contd.)

• One practice problem set will be distributed
  every week. Some problems may be specially meant
  for Ph.D. students.
• There will be a 2-hour tutorial every other week
  to clarify students difficulties. There will be
  a 15-minute test in every tutorial. The test will
  be either from the practice problem set or
  something similar.
• Solutions of the problem set to be submitted will
  be provided after the test.
• Some programming assignments may be given during
  the course.

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Grading

• There will be two midterm examination, each of
  two hour duration.
• First midterm will be taken at the end of the
  linear programming part. The second midterm will
  take place at the end of the network optimization
  part on the last day of classes.
• The course grade will be based on two midterm
  exams and weekly tests. The weights for these
  components will be as follows
• First Midterm Exam (Linear Programming) 40
• Second Midterm Exam (Network Optimization) 40
• Weekly tests 20
• M.S. students will be graded separately from
  Ph.D. students.

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Linear Programming Problem

• Features of Linear programming problem
• Decision Variables
• We maximize (or minimize) a linear function of
  decision variables, called objective function.
• The decision variables must satisfy a set of
  constraints.
• Decision variables have sign restrictions.
• Example
• Maximize z 3x1 2x2
• subject to
• 2x1 x2 ? 100
• x1 x2 ? 80
• x1 ? 40
• x1, x2 ? 0

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Syllabus on Linear Programming

• Introduction to Linear Programming
• Applications of Linear Programming
• Linear Algebra, Convex Analysis, and Polyhedral
  Sets
• Simplex Algorithm
• Special Simplex Implementations
• Duality Theory and Sensitivity Analysis
• Integer Programming
• AMPL/CPLEX

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Directed and Undirected Networks

• DIRECTED GRAPH

UNDIRECTED GRAPH
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Syllabus on Graph Preliminaries

• Introduction to Network Flows
• Network Notation
• Network Representations
• Complexity Analysis
• Search Algorithms
• Topological Sorting
• Flow Decomposition

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Shortest Path Problem

• Identify a shortest path from a given source node
  to a given sink node.

• Finding a path of minimum length
• Finding a path taking minimum time
• Finding a path of maximum reliability

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Syllabus on Shortest Path Problem

• Introduction to Shortest Paths
• Applications of Shortest Paths
• Optimality Conditions
• Generic Label-Correcting Algorithm
• Specific Implementations
• Detecting Negative Cycles
• Shortest Paths in Acyclic Networks
• Dijkstras Algorithm and Its Efficient
  Implementations

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Minimum Spanning Tree Problem

• Find a spanning tree of an undirected network of
  minimum cost (or, length).

• Constructing highways or railroads spanning
  several cities
• Designing local access network
• Making electric wire connections on a control
  panel
• Laying pipelines connecting offshore drilling
  sites, refineries, and consumer markets

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Syllabus on Minimum Spanning Tree Problem

• Introduction to Minimum Spanning Trees
• Applications of Minimum Spanning Trees
• Optimality Conditions
• Kruskal's Algorithm
• Prim's Algorithm
• Sollin's Algorithm

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Maximum Flow Problem

• Determine the maximum flow that can be sent from
  a given source node to a sink node in a
  capacitated network.

• Determining maximum steady-state flow of
• petroleum products in a pipeline network
• cars in a road network
• messages in a telecommunication network
• electricity in an electrical network

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Syllabus on Maximum Flow Problem

• Introduction to Maximum Flows
• Introduction to Minimum Cuts
• Applications of Maximum Flows
• Flows and Cuts
• Generic Augmenting Path Algorithm
• Max-Flow Min-Cut Theorem
• Capacity Scaling Algorithm
• Generic Preflow-Push Algorithm
• Specific Preflow-Push Algorithms

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Minimum Cost Flow Problem

• Determine a least cost shipment of a commodity
  through a network in order to satisfy demands at
  certain nodes from available supplies at other
  nodes. Arcs have capacities and cost associated
  with them.

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• Distribution of products
• Flow of items in a production line
• Routing of cars through street networks
• Routing of telephone calls

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Syllabus on Minimum Cost Flow Problem

• Introduction to Minimum Cost Flows
• Applications of Minimum Cost Flows
• Structure of the Basis
• Optimality Conditions
• Obtaining Primal and Dual Solutions
• Network Simplex Algorithms
• Strongly Feasible Basis