# NMAA09044U Operations Research 2: Advanced Operations Research (OR2)

Volume 2023/2024
Education

MSc Programme in Mathematic-Economics

Content

This course gives an introduction to Integer Programming, which is a widely used methodology for discretely constrained optimization and decision-making.

As a decision support tool, Integer Programming is of vital importance to the industry, with applications in a variety of fields, including finance, logistics, production planning and emerging areas such as the green transition.

The course provides a theoretical foundation to Integer Programming and an overview of well-known solution methods. Moreover, it involves hands-on experience with modeling and implementation, many examples and real-world applications (from industry and/or academic research).

The contents of the course are as follows:

A. Problem formulation and modeling:

• A1. Formulate mathematical optimization models for classical OR problems.
• A2. Linearization of non-linear constraints.
• A3. Quality of different model formulations.
• A4. Modeling practical OR problems.

B. Integer Programming:

• B1. Integer Programs (IP), Binary Integer Programs (BIP), and Mixed-Integer Programs (MIP).
• B2. Properties of Integer Programs.
• B3. Examples of Integer and Mixed-Integer Programs.

C. Solution methods for Integer Programming Problems:

• C1. Relaxation and duality.
• C2. Decomposition.
• C3. Branch and bound.
• C4. Dynamic programming.
• C5. Cutting planes.
• C6. Column generation.

D. Practical aspects:

• D1. External talks: Relation between academia and practice.
• D2. Case studies: Energy planning/Vehicle routing/Travelling salesman.
• D3. Implementation of a given problem using an appropriate software package.
• D4. Implementation of a solution method for a given problem.
Learning Outcome

Knowledge:

• Mathematical optimization problems, including LP, IP, BIP and MIP; classical problems such as Travelling Salesman, Knapsack and Network Flow problems.
• Properties of Integer Programming problems
• Solution methods for Integer Programming Problems

Skills:

• Characterize different classes of mathematical optimization problems, including LP, IP, BIP and MIP problems
• Formulate models for LP, IP, BIP and MIP problems
• Implement a given problem using appropriate software
• Apply the solutions methods presented in the course
• Implement a solution method for a given problem (in a simplified fashion)
• Understand and reproduce the proofs presented in the course

Competences:

• Evaluate the quality of different model formulations
• Discuss the challenges of solving IP problems
• Explain how to exploit the properties of a given class of IP problems in the design of a solution method
• Adapt a solution method to a given class of IP problems
• Describe similarities and differences between solution methods
• Discuss the challenges of modeling and solving practical problems
• Formulate, implement and solve a practical problem and justify the choice of model formulation and solution method

Previous years, the textbook L. A. Wolsey: Integer Programming, 1998, John Wiley & Sons, Inc. was used. Do not buy course material before consulting the course responsible.

Operations Research 1 (OR1) or similar is required.

Academic qualifications equivalent to a BSc degree is recommended.
2 x 2 hours of lectures and 2 x 2 hours exercises/project work per week for 7 weeks.
• Category
• Hours
• Lectures
• 28
• Preparation
• 70
• Theory exercises
• 28
• Project work
• 30
• Exam
• 50
• Total
• 206
Written
Oral
Individual
Collective

Individual written feedback will be given on mandatory assignments in order for students to improve subsequent submissions and resubmissions of assignments.

Collective oral feedback will be given on students’ presentations in class.

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

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

Aid
Written aids allowed

The use of Large Language Models (LLM)/Large Multimodal Models (LMM) – such as ChatGPT and GPT-4 – is permitted.

Marking scale