# DM871: Linear and Integer Programming

## Comment

The course is optional for the following curricula: Computer Science, Applied Mathematics, Mathematics and Data Science.

## Entry requirements

## Academic preconditions

Students taking the course are expected to:

- Have knowledge of Linear Algebra, eg having attended DM561, MM505, MM538 or DS827
- Be able to program, eg having attended DM536, DM550, DM574, DS800 or DS831

## 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 aid decision making. 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.

- Give the competence to handle complex and development-oriented situations in academic and work settings
- Give skills to describe, analyze and solve mathematical problems with the application of methods and modeling formalisms from the areas of mathematics and computer science
- Give skills to take and justify decisions on a mathematical basis
- 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 a specific area of mathematics

## Expected learning outcome

The learning objective of the course is that the student demonstrates the ability to:

- 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.

## Content

- Linear programming and the simplex method
- Duality theory
- Integer programming: branch and bound and cutting plane algorithms
- Min cost flow problem and its applications
- Software for solving linear and integer programming problems

## Literature

## Examination regulations

## Exam element a)

## Timing

## Tests

## Compulsory assignments

## EKA

## Assessment

## Grading

## Identification

## Language

## Examination aids

## ECTS value

## Additional information

All tests during the classes must be attended. Tests during the course may be in the form of 24-hour take-home assignments or exam hours in the classroom. The schedules will be agreed with the participants.

Reexam will be a single short-deadline assignment in the form of a short-answer test, corresponding to the mandatory assignments from the ordinary exam.

## Indicative number of lessons

## Teaching Method

At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.

Teaching activities are reflected in an estimated allocation of the workload of an average student as follows:

- Intro phase (lectures) - 32 hours
- Training phase: 18 hours

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.

Educational activities

- Reading from text books
- Solving homeworks
- Applying acquired knowledge to practical projects