NMAK15004U Advanced Operations Research: Stochastic Programming

Volume 2024/2025
Education

MSc Programme in Mathematics-Economics

Content

In countless real-life situations, decision makers are required to make decisions under uncertainty, that is while relevant information is uncertain, noisy, imprecise. Examples are investments in assets or projects with uncertain returns, scheduling of taks with uncertain duration, or production of goods with uncertain demand. Decision problems with these features are central in the modern finance, energy, and logistics sector, to name a few.

This course introduces the students to optimization in conditions of uncertainty by means of stochastic programming. 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 sound input data (scenarios). The students of this course will independently handle practical problems in project work and exercises, hereby gaining the practical experience necessary to work on complex decision problems under uncertainty. 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 mathematical properties of stochastic programs.
• C2. The value of stochastic programming and the value of information.

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 or Stochastic Dual Dynamic Programming).

E. Practical applications:

• E1. Solution of case studies from e.g., the energy, finance, or logistics sector, using optimization software such as GAMS, Cplex or Gurobi.
• E2. Solution of several practical exercises.
Learning Outcome

Knowledge:

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

Skills:

• 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

Compentences:

• 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).

Required competencies: Linear Programming (Operations Research 1 or similar) in addition to basic Probability Theory.

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
Oral

Lecturer's oral or written feedback (collective and/or individual) on the project work.

Credit
7,5 ECTS
Type of assessment
Oral examination, 30 minutes (30-minute preparation time)
Exam registration requirements

Approval of one project report is a prerequisite for enrolling for examination.

Aid
Only certain aids allowed
• All aid can be used during the preparation time.
• No aid can be used during the oral examination.
Marking scale