NMAK15004U  Advanced Operations Research: Stochastic Programming

Volume 2018/2019

MSc Programme in Mathematics-Economics


This course is about optimization under uncertainty by means of stochastic programming. Special emphasis is placed on different problem formulations and selected scenario generation methods as well as to understand specific properties of stochastic programming problems and how to exploit these properties in various solution methods. Furthermore, the students of this course will independently handle more practical problems by stochastic programming.


A. Stochastic programming problems:

  • A1. Formulation of two-stage and multi-stage recourse problems, simple recourse, linear and integer problems, chance constrained problems.
  • A2. Examples.
  • A3. Implementation and solution of mathematical programming problems using state-of-the-art optimization software (e.g., GAMS, AMPL or Cplex).
  • 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: Continuity and convexity.


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 Energy planning, Finance, Transportation.
Learning Outcome


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



  • Formulate two-stage and multi-stage recourse 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 SP problems
  • Explain how to exploit the properties of a given class of SP problems in the design of a solution method
  • Adapt a solution method to a given class of SP problems, and make small changes to and extensions of the method
  • Evaluate the quality of scenario trees
  • Discuss the challenges of modeling and solving practical problems
  • Formulate, implement and solve a practical problem and justify the choice of model formulation, scenario generation method and solution method
Operations Research 1 (OR1) or similar is required.
Recommended but not required: One or more of the following courses: Modelling and Implementation and/or Operations Research 2 (OR2)
2 x 2 hours of lectures and 1 x 2 hours exercises/project work per week for 7 weeks
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 (failed project reports can be resubmitted)

All aids allowed

All aids are allowed during the preperation time. During the examination, the student is allowed to bring a piece of paper with keywords.

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 resubmittet no later than two 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.

  • Category
  • Hours
  • Lectures
  • 28
  • Theory exercises
  • 14
  • Project work
  • 44
  • Exam
  • 50
  • Preparation
  • 70
  • Total
  • 206