NMAK15004U Advanced Operations Research: Stochastic Programming

Volume 2021/2022

MSc Programme in Mathematics-Economics


This course introduces the students to optimization under uncertainty by means of stochastic programming. In many real-life situations, decisions have to be made while relevant data is uncertain, noisy, imprecise. Examples are investments in assets with uncertain returns or production of goods with uncertain demand. For these problems, the course presents different mathematical formulations, illustrates the corresponding mathematical properties, shows how to exploit these properties in various solution methods, and discusses how uncertain parameters can be transfortmed into input data (scenarios). 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. Decision making under uncertainty.
  • A2. Formulations of stochastic programming problems.


B. Approximations and scenario generation:

  • B1. Monte Carlo techniques.
  • B2. Property matching.
  • B3. Assessing the quality of a solution.


C. Properties of stochastic programming problems:

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


D. Solution methods:

  • D1. Decomposition techniques for two-stage stochastic programs (e.g., L-shaped decomposition).
  • D3. Decomposition techniques for multistage stochastic programs (e.g., Dual decomposition).


E. Practical aspects and applications:

  • E1. Solution of case studies from e.g., Energy planning, Finance, Transportation, using optimization software such as GAMS, Cplex or Gurobi.
Learning Outcome


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



  • Formulate different types of stochastic programming problems
  • Recognize and prove properties of stochastic programs
  • Represent/approximate the uncertain data by means of scenarios
  • Evaluate the benefits of using stochastic programming
  • Apply the solution methods presented in the course to solve stochastic programs
  • Implement a (simplified version of a) solution method using optimization software



  • Recognize and structure a decision problem affected by uncertainty and propose a suitable mathematical formulation
  • Design a solution method for a stochastic program based on an analysis of its properties and justify the choice
  • Identify a suitable way of representing the uncertain data of the problem, and its effect on the solutions obtained
  • Quantify the benefit of using stochastic programming in a particular decision making problem

Lecture notes provided by the teacher (see Absalon).

Operations Research 1 (OR1) or similar is required.
Recommended but not required: Applied Operations Research.

Academic qualifications equivalent to a BSc degree is recommended.
2x2 hours of lectures per week, 2 hours of classroom exercises or project work supervision. Individual or group-based project work throughout the course.
  • Category
  • Hours
  • Lectures
  • 28
  • Preparation
  • 28
  • Theory exercises
  • 14
  • Project work
  • 55
  • Exam Preparation
  • 80
  • Exam
  • 1
  • Total
  • 206
Feedback by final exam (In addition to the grade)

Lecturer's oral or written feedback (collective and/or individual) 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 one project report 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.

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

Same as ordinary exam, conditional on the approval of the project work. If the required project report was not approved before the ordinary exam it must be (re)submitted no later than three weeks before the beginning of the re-exam week (contact the teacher for further details).

Criteria for exam assesment

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