DM559: Linear and Integer Programming (7.5 ECTS)

STADS: 15018001

Level
Bachelor course

Teaching period
The course is offered in the spring semester.

Teacher responsible
Marco Chiarandini, Lektor,
Tlf.: 6550 4031 Email: marco@imada.sdu.dk

Timetable
Group Type Day Time Classroom Weeks Comment
Common I Monday 10-12 U91 8,10
Common I Monday 10-12 U20 11
Common I Monday 08-10 U55 12
Common I Monday 10-12 U140 17,19
Common I Wednesday 14-16 U166 7
Common I Wednesday 14-16 U55 14
Common I Wednesday 10-12 U23 15-20
Common I Wednesday 14-16 U23 22 DM559
Common I Thursday 10-12 U23 5,8-12,14-16,18,20
Common I Thursday 08-10 U23 6-7
Common I Thursday 14-16 U140 21
Common I Friday 10-12 U173 5
Common I Friday 10-12 U43 6
H1 TE Monday 08-10 U143 7-11,16
H1 TE Wednesday 14-16 U154 18-21
H1 TE Thursday 08-10 U23 12 TH1H2 DM559
H1 TL Friday 10-12 IMADA ComputerLab 14,16
H1 TE Friday 08-10 U143 14
H1 TE Friday 10-12 U143 21
H2 TE Tuesday 14-16 U141 21
H2 TL Wednesday 16-18 IMADA ComputerLab 14
H2 TL Wednesday 14-16 IMADA ComputerLab 16
H2 TE Thursday 12-14 U31 7
H2 TE Thursday 12-14 U17 8,10
H2 TE Thursday 12-14 U154 9
H2 TE Thursday 12-14 U64 11
H2 TE Thursday 14-16 U154 14
H2 TE Thursday 12-14 U143 16,18,20
H2 TE Friday 10-12 U144 19
H2 TE Friday 12-14 U145 21
M1 TE Tuesday 12-14 U64 12,15,17-21
M1 TE Wednesday 10-12 U8 21
M1 TL Friday 08-10 IMADA ComputerLab 14,16
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Comment:
Ubegrænset deltagerantal. Fælles undervisning med DM545, Lineær- og heltalsprogrammering fra uge 12.

Prerequisites:
None

Academic preconditions:
Students taking the course are expected to:
  • Have knowledge of the content of the course: DM507, "Algorithms and Data Structures" 
  • Be able to program
 


Course introduction
Linear and Integer programming is a field at the intersection between mathematics and computer science that has seen a huge development in the last 60 years. It provides the tools that are at the core of operations research, the discipline that provides analytical methods to help making better decisions. The main focus of linear and integer programming is on resource constrained optimization problems that can be described by means of linear inequalities and a linear objective function. These problems may arise in all contexts of decision making, such as manufacturing, logistics, health care, education, finance, energy supply and many others. The subject of the course therefore has an enormous practical relevance. 

The aim of the course is to enable the student to work first with the elements of linear algebra that are relevant for computer science and especially for linear and integer programming, then with the basics of linear programming and duality theory, and finally, with the main solution techniques, such as the simplex method, branch and bound and cutting planes. Besides giving skills in mathematical modeling, the course has also the goal of  giving to the students experience with practical applications and with a mathematical software system for finding numerical solutions to the applications proposed.

The course builds on the knowledge acquired in the course DM507, "Algorithms and Data Structures" and gives an academic basis for doing a bachelor thesis project and other theoretical or practical oriented study-activities as well as for studying elective courses with elements of optimization that can be chosen as part of the degree. 

In relation to the competence profile of the degree it is the explicit focus of the course to:

  • Give the competence to handle complex and development-oriented situations in academic and work settings 
  • Give skills to describe, analyze and solve computer science problems with the application of methods and modeling formalisms from the core areas of computer science and its mathematical support disciplines
  • Give skills to take and justify computer science decisions 
  • Give skills to describe, formulate and communicate problems and results to either peers and non-specialists or partners and users
  • Give knowledge and understanding of how certain optimization problems can be solved by means of linear and integer programming
  • Give knowledge of how to understand and reflect on theories, methods and practices in the computer science field
 


Expected learning outcome
The learning objective of the course is that the student demonstrates the ability to:
  • make calculations in matrix algebra.
  • solve systems of linear equations both manually and with the computer.
  • apply linear algebra to solve practical problems that resemble those treated in the course.
  • formulate a mathematical (linear) model from a given problem description in words.
  • derive the dual program of a given linear program.
  • apply the simplex method to simple linear programs.
  • apply the branch and bound technique to small example problems.
  • derive Gomory cuts and apply the cutting plane algorithm in small example problems.
  • apply the theory from the course to practical optimization problems such as flows in networks, matching problems, packing problems, simple scheduling problems etc.
  • use computer software for solving linear and integer programs.
  • think innovative by seeing possibilities for applying theoretical knowledge in the industry.
 
 


Subject overview
The following main topics are contained in the course:
  • Matrix algebra
  • Systems of linear equations and Gauss elimination
  • Rank of matrices, matrix inversion
  • Vector spaces, linear transformations
  • Linear programming and the simplex method
  • Duality theory
  • Integer programming and branch and bound and cutting plane algorithms 
  • Min cost flow problem and its applications
  • Software for solving linear and integer programming
 


Literature
    Meddeles ved kursets start.


Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
  1. Mandatory assignments. Is evaluated internal on a pass/fail basis by the teacher.
Assessment and marking:
  1. 4 hours written exam. Evaluated by external censorship by the Danish 7-mark scale.


Expected working hours
The teaching method is based on three phase model.
Intro phase: 46 hours
Skills training phase: 50 hours, hereof:
 - Tutorials: 42 hours
 - Laboratory exercises: 8 hours

Educational activities
  • Reading from text books
  • Solving homeworks
  • Applying acquired knowledge to practical projects
 
Educational form
In the intro phase, concepts, theories and models are introduced and put into perspective. In the training phase, students train their skills through exercises and dig deeper into the subject matter. 

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Remarks
The course is co-read with: DM545 in the second half of the semester.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.