# Advanced Operations Research: Stochastic Programming

### Course content

This course introduces the students to optimization under uncertainty by means of stochastic programming. In many real-life problems input data is often uncertain, noisy, imprecise. For these problems, the course illustrates different problem formulations, discusses how uncertain parameters can be transfortmed into "scenarios", discusses specific properties of stochastic programs, and shows how to exploit these properties in various solution methods. Furthermore, the students of this course will independently handle practical problems in project work. The content can be summarized as follows.

A. Stochastic programming problems:

• A1. Formulations of stochastic programming problems.
• A2. Examples.
• A3. Implementation and solution of mathematical programming problems using state-of-the-art optimization software (e.g., GAMS, AMPL, Cplex or the like).
• A4. Analysis of the solution.

B. Scenario generation:

• B1. Moment matching.
• B2. Sampling.
• B3. Scenario tree construction.
• B4. The quality of scenario generation methods.

C. Properties of stochastic programming problems:

• C1. The value of stochastic programming: EVPI and EEV.
• C2. Structural properties of stochastic programs.

D. Solution methods:

• D1. L-shaped decomposition.
• D2. Integer L-shaped decomposition.
• D3. Dual decomposition.

E. Practical aspects and applications:

• E1. Implementation of a real-life problem using optimization software.
• E2. Implementation of a solution method using optimization software.
• E3. Case studies from e.g., Energy planning, Finance, Transportation.
Education

MSc Programme in Mathematics-Economics

Learning outcome

Knowledge:

• Formulations of stochastic programming problems
• Scenario generation methods
• Properties of stochastic programming problems
• Solution methods

Skills:

• Formulate different types of stochastic programming problems
• Implement and solve a stochastic programming problem using suitable software
• Apply selected methods to describe the uncertainty of the problem (so-called "scenario generation" methods)
• Apply the solution methods presented in the course
• Implement a (simplified version of a) solution method using optimization software
• Understand and reproduce the proofs presented in the course

Compentences:

• Work out simple proofs using the same techniques as in the course
• Discuss the challenges of solving stochastic programming problems
• Explain how to exploit the properties of a given class of stochastic programs in the design of a solution method
• Adapt a solution method to a given class of stochastic programming problems, and make small changes to and extensions of the method
• Evaluate the quality of scenario trees
• Formulate, implement and solve a practical problem and justify the choice of model formulation, scenario generation method and solution method

2x2 hours of lectures per week, 2 hours of exercise sessions per week, and project work for 7 weeks.

See Absalon for a list of course litterature.

Operations Research 1 (OR1) or similar is required.
Recommended but not required: One or more between Applied Operations Research and Operations Research 2 (OR2)

Academic qualifications equivalent to a BSc degree is recommended.

Collective and/or individual feedback on the project work.

ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
30 minutes oral examination with 30 minutes preparation time.
Aid
Only certain aids allowed

All aid can be used during the preparation time.

No aid can be used during the exam, except for a small outline of the presentation if applicable.

Marking scale
Censorship form
No external censorship
Several internal examiners
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 28
• Theory exercises
• 14
• Project work
• 40
• Exam
• 1
• Preparation
• 123
• English
• 206