NMAK15004U Advanced Operations Research: Stochastic Programming

Volume 2019/2020

MSc Programme in Mathematics-Economics


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.
Learning Outcome


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



  • 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



  • 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

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.
2x2 hours of lectures per week, 2 hours of exercise sessions per week, and project work for 7 weeks.
  • Category
  • Hours
  • Exam
  • 1
  • Lectures
  • 28
  • Preparation
  • 123
  • Project work
  • 40
  • Theory exercises
  • 14
  • Total
  • 206

Collective and/or individual feedback on the project work.

7,5 ECTS
Type of assessment
Oral examination, 30 minutes
30 minutes oral examination with 30 minutes preparation time.
Exam registration requirements

Approval of two project reports is a prerequisite for enrolling for examination.

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
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Same as ordinary exam. If the required project reports were not approved before the ordinary exam they must be (re)submitted and approved no later than three weeks before the beginning of the re-exam week.

Criteria for exam assesment

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