# Operations Research 2: Advanced Operations Research (OR2)

### Course content

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 in GAMS.
• D4. Implementation of a solution method for a given problem in GAMS.
Education

MSc Programme in Mathematic-Economics

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 in GAMS
• Apply the solutions methods presented in the course
• Implement a solution method for a given problem in GAMS (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

2 x 2 hours of lectures and 2 x 2 hours exercises/project work per week for 7 weeks

Previous years, the textbook L. A. Wolsey: Integer Programming, 1998, John Wiley & Sons, Inc. was used.

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

Academic qualifications equivalent to a BSc degree is recommended.

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.

ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
30 minutes oral examination with 30 minutes preparation time.
Aid
Written aids allowed
Marking scale
Censorship form
No external censorship
Several internal examiners
##### Criteria for exam assessment

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

Single subject courses (day)

• Category
• Hours
• Lectures
• 28
• Preparation
• 70
• Theory exercises
• 28
• Project work
• 30
• Exam
• 50
• English
• 206