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ESI 6417

Linear Programmingand

Network Optimization

Fall 2003

Ravindra K. Ahuja370 Weil Hall, Dept. of ISE

ahuja@ufl.edu352-392-3615

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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 8 and 9 (3 PM to 4:55 PM), and Thursday: Period 8 (3 PM to 3:50 PM)

Place: Weil 273 Office Hours: Tuesday, Period 7, 2 PM to 3 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 15 minutes test every week where one question from the practice problem set will be given to solve.

Some programming assignments may be given during the course.

Solutions of the problem set to be submitted will be provided after the test. Occasionally tutorial sessions will be held to to clarify student’s difficulties.

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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: 35%Second Midterm Exam: 35%Weekly tests: 30%

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 to2x1 + x2 100x1 + 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

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2

3

4

5

6

7DIRECTED GRAPH:

UNDIRECTED GRAPH: 1

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3

4

5

6

7

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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.

1

2 4

3 5

6

10

25

35

20

15

35

40

30

20

s t

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

Dijkstra’s 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).

35

40

25

10

20

15

301

2

3

4

5

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.

1

2 4

3 5

6

10

25

35

20

15

35

40

30

20

s t

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.

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

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2

3

4

5

6

710 10

-5

-15

5

3 6

4

3 12

2

4

6

50

10

5

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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